NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
The Kluwer International Series on
ASIAN STUDIES IN COMPUTER AND INFORMATION SCIENCE Series Editor
Kai-Yuan Cai Being University ofAeronautics andAstronautics, Being, CHINA
Editorial Advisory Board war
vii
Han-Fu Chen, Institute of System Science, Chinese Academy of Sciences Jun-Liang Chen, Beijing University of Post and Telecommunication Lin Huang, Peking University Wei Li, Beijing University of Aeronautics and Astronautics Hui-Min Lin, Institute of Software Technology, Chinese Academy of Sciences Zhi-Yong Liu, Institute of Computing Technology, Chinese Academy of Sciences Ru-Qian Lu, Institute of Mathematics, Chinese Academy of Sciences Shi-Tuan Shen, Beijing University of Aeronautics and Astronautics Qing-Yun Shi, Peking University You-Xian Sun, Zhejiang University Lian-Hua Xiao, National Natural Science Foundation of China Xiao-Hu You, Southeast University Bo Zhang, Tsinghua University Da-Zhong Zheng, Tsinghua University Bing-Kun Zhou, Tsinghua University Xing-Ming Zhou, Changsha University of Technology >>.
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NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
by
Qiang Lu
Yuanzhang Sun Shengwei Mei
Tsinghua University, Beijing, China
0 KLUWER ACADEMIC PUBLISHERS
vii
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IT Electronic Services Library of Congress Cataloging-in-Publication Data
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Lu, Qiang, 1936Nonlinear control systems and power system dynamics / Qiang Lu, Yuangzhang Sun, Shengwei Mei. p. cm.-- (Kluwer international series on Asian studies in computer and information science ; 10) Includes bibliographical references and index. ISBN 0-7923-7312-X (alk. paper) 1. Automatic control. 2. Nonlinear control theory. 3. Electric power system stability. I. Sun, Yuanzhang, 1954- II. Mei, Shengwei, 1964- III. Title. IV. Series.
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vii
TJ213.L72 2001 629.89-dc2l
2001016020
Copyright © 2001 by Kluwer Academic Publishers
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, Massachusetts 02061
Printed on acid free paper. Printed in the United States of America The Publisher offers discounts on this book for course use and bulk purchases. For further information, send email to <[email protected]>.
To Our Alma Mater Tsinghua University
SERIES EDITOR'S ACKNOWLEDGMENTS
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I am pleased to acknowledge the assistance to the editorial work by Beijing University of Aeronautics and Astronautics and the National Natural Science Foundation of China
Kai-Yuan Cai Series Editor Department ofAutomatic Control Beijing University of Aeronautics and Astronautics Beijing 100083 China
Contents
Preface ...............................................................................................mod
Chapter 1
Introduction .........................................................................................1 --'
1.1 Overview ................................................................................................ 1 1.2 Outline of the Development of Control Theory ..................................... 3 1.3 Linear and Nonlinear Control Systems ................................................ 12 1.0
.-' ...
1.4 Modeling Method of Approximate Linearization ................................ 16 1.5 Stable and Unstable Equilibrium Points .............................................. 19
1.6 References ........................................................................................... 22
Chapter 2 Basic Concepts of Nonlinear Control Theory ................................25 k/'1
2.1 Introduction .......................................................................................... 25 2.2 Coordinate Transformation of Nonlinear Systems ............................... 26 2.2.1 General Concepts of Coordinate Transformation ..................... 26 2.2.2 Coordinate Transformation of Linear Systems ......................... 28 2.2.3 Nonlinear Coordinate Transformation and Diffeomorphism .... 29 2.2.4 Mapping .................................................................................... 30 2.2.5 Local Diffeomorphism .............................................................. 30
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2.2.6 Coordinate Transformation of Nonlinear Control Systems....... 32
2.3 Affine Nonlinear Control Systems ....................................................... 33 2.4 Vector Fields ........................................................................................ 34 2.5 Derived Mapping of Vector Fields ....................................................... 36 2.6 Lie Derivative and Lie Bracket ............................................................ 38
2.6.1 Lie Derivative ........................................................................... 38
viii
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
2.6.2 Lie Bracket................................................................................ 41 2.7 Involutivity of Vector Field Sets .......................................................... 45
2.8 Relative Degree of a Control System ................................................... 47 2.9 Linearized Normal Form ...................................................................... 50 2.10 Summary ............................................................................................ 56 2.11 References ......................................................................................... 58
Chapter 3 Design Principles of Single-Input Single-Output Nonlinear Control Systems ..............................................................59 3.1 Introduction .......................................................................................... 59
3.2 Design Principles of Exact Linearization via Feedback....................... 60 'p.
3.2.1 Linearizing Design Principle as Relative Degree r Equals n for an nth-order System .............................................. 61 3.2.2 General Linearization Design Principle .................................... 70 3.2.3 Conditions for Exact Linearization ........................................... 72 3.2.4 Algorithm of Exact Linearization ............................................. 80 3.3 Zero Dynamics Design Principle ......................................................... 90 3.3.1 First Type of Zero Dynamic Design Method ............................ 91 3.3.2 Second Type of Zero Dynamic Design Method ........................ 97 3.3.3 Discussion of Some Problems................................................. 101 3.4 Zero Dynamics Design Method for Linear Systems .......................... 104 3.5 Design of Disturbance Decoupling .................................................... 109 I'D
3.6 References ......................................................................................... 120
Chapter 4 Design Principles of Multi-Input Multi-Output Nonlinear Control Systems ............................................................121 4.1 Introduction........................................................................................ 121 4.2 Relative Degrees and Linearization Normal Forms ........................... 122 4.2.1 Relative Degree ........................................................................ 122 4.2.2 Linearization Normal Form ...................................................... 125
4.3 Zero Dynamics Design Principle ....................................................... 136 4.4 Design Principles of Exact Linearization via State Feedback............ 147
4.4.1 Conditions for Exact Linearization via State Feedback.......... 148
Contents
ix
4.4.2 Algorithm of Exact Linearization via State Feedback ............ 151 4.5 References ......................................................................................... 164
Chapter 5 Basic Mathematical Descriptions for Electric Power Systems ...................................................................165 5.1 Introduction ........................................................................................ 165 5.2 Rotor Dynamics and Swing Equation ................................................ 166 5.3 Output Power Equations for a Synchronous Generator ..................... 170 5.4 Output Power Equations for Synchronous Generators in a Multi-Machine System ................................................................ 179 5.4.1 Output Power Equations for a Generator in a One-machine Infinite-bus System ....................................... 179 5.4.2 Practical Output Power Equations for Synchronous Generators in a Multi-machine System .................................. 181 5.5 Electromagnetic Dynamic Equation for Field Winding ..................... 185 5.6 Mathematical Description of a Steam Valving Control System......... 186 5.7 Mathematical Description of a DC Transmission System ................. 191 5.7.1 Dynamic Equations of a DC Transmission Line ..................... 191
5.7.2 Mathematical Model of a DC Control System ........................ 196 5.8 References ......................................................................................... 198
Chapter 6 Nonlinear Excitation Control of Large Synchronous Generators .......................................................................................199 6.1 Introduction ........................................................................................ 199 6.2 Development of Excitation Control ................................................... 200 6.3 Nonlinear Excitation Control Design for Single-Machine Systems .................................................................... 208 6.3.1 Exact Linearization Design Approach .................................... 209
6.3.2 Discussions on the Implementation of Nonlinear Excitation Control ................................................. 217 6.3.3 Effects of Nonlinear Excitation Control .................................. 219 6.4 Nonlinear Excitation Control Design for Multi-Machine Systems ..................................................................... 223
x
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
6.4.1 Dynamic Equations of Multi-Machine Systems ..................... 223 6.4.2 Exact Linearization Design Method for Excitation Control ............................................................ 225 6.4.3 Practical Nonlinear Excitation Control Law ........................... 234
6.4.4 Discussion on the Nonlinear Excitation Control Law ............. 235 6.4.5 Effects of the Nonlinear Excitation Control ............................ 237 6.5 References ......................................................................................... 244
(!1
Chapter 7 Nonlinear Steam Valving Control .................................................245
7.1 Introduction ........................................................................................ 245 7.2 Nonlinear Steam Valving Control in a One-Machine Infinite-Bus System .................................................... 246 7.2.1 Mathematical Model ............................................................... 246 7.2.2 Exact Linearization Method .................................................... 249 7.2.3 Physical Simulation Results of Nonlinear Valving Control in a One-Machine Infinite-Bus System ..................... 256
7.2.4 Digital Simulation Results of Nonlinear Steam Valving Control in a One-Machine Infinite-Bus System ....... 259
7.3 Nonlinear Steam Valving Control in a Multi-Machine System ......... 261 7.3.1 Mathematical Model ............................................................... 261 7.3.2 Exact Linearization Method .................................................... 263 .-..
7.3.3 Effects of Nonlinear Steam Valving Control in a Multi-Machine System ........................................................... 271
7.4 Discussion on Some Issues ................................................................ 273
7.5 References ......................................................................................... 276
Chapter 8 Nonlinear Control of HVDC Systems ...........................................277 8.1 Introduction ........................................................................................ 277 8.2 Characteristics and Conventional Control of Converter Stations ...... 278 8.2.1 Voltage-Current Characteristics on Rectifier Side .................. 278 8.2.2 Voltage-Current Characteristics on Inverter Side .................... 279
8.2.3 Conventional Control with Constant DC Current at Rectifier and Constant Extinction Angle at Inverter .............. 280
Contents
xi
8.2.4 Conventional Control with Constant DC Current at Rectifier and Constant DC Voltage at Inverter....................... 282 8.2.5 Power Modulation in DC Transmission Systems.................... 284 8.3 Nonlinear Control of Converter Stations ........................................... 285 8.3.1 Nonlinear Control with Constant Current and Constant Extinction Angle .............................................. 285 8.3.2 Nonlinear Control with Constant Current at Rectifier and Constant DC Voltage at Inverter....................... 297 8.4 Nonlinear Control of DC Systems and Stability of AC/DC Systems ............................................................................ 302 8.4.1 Modeling for Nonlinear Stabilizing Control Design of AC/DC Systems ................................................................. 302 8.4.2 Nonlinear Control Design for Stabilizing
AC/DC Systems ..................................................................... 304
8.4.3 Effects of Nonlinear Control for Stabilizing AC/DC Systems ..................................................................... 306 8.5 References ......................................................................................... 308
Chapter 9 Nonlinear Control of Static Var Systems ......................................309 9.1 Introduction ........................................................................................ 309 9.2 Fundamentals of Reactive Power Compensation ............................... 310 9.2.1 Reactive Power Flow in a Transmission System .................... 310 9.2.2 Two Basic Types of Reactive Power Compensators ............... 312 9.2.3 Effects of the Midpoint Compensator on the Stability Limits ............................................................ 314 9.3 Configuration of Static Reactive Compensators ................................ 319 9.3.1 Thyristor-Controlled Reactor (TCR) ....................................... 319 9.3.2 Thyristor-Switched Capacitor (TSC) ...................................... 325
9.4 Conventional Control Strategies of SVS ........................................... 330 9.5 Nonlinear Controller Design for SVS ................................................ 333 9.5.1 Modeling of SVS Control Systems ......................................... 333 9.5.2 Exact Linearization Design Approach .................................... 335 9.5.3 Effects of the Nonlinear Control of SVS ................................ 339 9.6 References ......................................................................................... 342
xi
i
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Chapter 10 Nonlinear Robust Control of Power Systems ...............................343 10.1 Introduction ...................................................................................... 343
10.2 Basic Concepts ................................................................................. 344 10.2.1 L2-space ................................................................................ 344 10.2.2 L2 -gain .................................................................................. 345
10.2.3 Penalty Vector Function ........................................................ 348 10.2.4 Dissipative Systems .............................................................. 348
10.3 Nonlinear Robust Control ................................................................ 350 10.3.1 Description of Nonlinear Robust Control ............................. 350 10.3.2 General Form of the Nonlinear Robust Control Law............ 351 10.3.3 Hamilton-Jacobi-Isaacs Inequality ........................................ 355
10.4 HJI Inequality of Linear Control System - Riccati Inequality....... 358 10.5 Nonlinear Robust Excitation Control (NREC) ................................. 359 10.5.1 Introduction ........................................................................... 359 10.5.2 Regulation Output Linearization ........................................... 361 10.5.3 Analysis of Robustness of the Closed Loop System ............. 363 'v,
10.5.4 Nonlinear Robust Excitation Control .................................... 365
10.5.5 Simulation Results ................................................................ 368 10.5.6 Summary ............................................................................... 369 10.6 References ....................................................................................... 371
Index .................................................................................................373
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List of Figures
Figure 1.1 Figure 1.2 'ti
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Figure 1.3 Figure 2.1 boo
Figure 2.2
Structural diagram of linear optimal control system ..................... 8
R-L--C Circuit ............................................................................. 12 A one-machine infinite-bus system ............................................ 14 a, b, c coordinates and d, q, 0 coordinates of a synchronous generator ........................................................ 27 Mappings between X coordinate system and
Z coordinate system ................................................................... 30 Figure 3.1
Diagram illustrating the design principle of exact linearization via feedback .................................................. 68
Figure 3.2
Relations among the coordinate transformations of
Figure 3.3
Structural diagram of a system with
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the spaces X, W and Z ............................................................... 85 outputs decoupled from disturbances .........................................112 Relationship between different reference axes
Figure 5.2
used to measure the motion of a generator's rotor .................... 167 Coordinate axes a, b and c fixed on the stator and
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Figure 5,1
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coordinate axes d, q and 0 fixed on the rotor ............................ 171
Figure 5.3 Figure 5.4
A one-machine, infinite-bus power system ............................... 180
Figure 5.5
Electric potential vector diagram of
Figure 5.6
Positive directions of the current and voltage of
Figure 5.7
a field winding .......................................................................... 185 Physical configuration of the steam valving control system
A 6-generator power system and its equivalent circuit ............. 182 an n-generator power system .................................................... 183
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for a large generator set with reheater ....................................... 187
Figure 5.8
Transfer function block diagram of the control system
for a steam turbine with reheater .............................................. 188
xiv
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Figure 5.9
Transfer function block diagram of a steam valving control system with reheater ..................................................... 189
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Figure 5.10 Transfer function block diagram of a steam valving control system with reheater ........................................ 190 Figure 5.11 Transfer function diagram for a steam valving control system. 190 Figure 5.12 Basic configuration of a DC transmission system ..................... 191 Figure 5.13 Equivalent circuit of a DC transmission line ............................. 192 Figure 5.14 Converting loop of the rectifier side ......................................... 193 Figure 5.15 Converting process of a rectifier ............................................... 193 Figure 5.16 Equivalent circuit for a rectifier ................................................ 193 Figure 5.17 Converting loop of an inverter .................................................. 194 Figure 5.18 Converting process of an inverter ............................................. 194 Figure 5.19 Equivalent circuit for an inverter when Vd, is expressed by the inverter firing angle /3.................................... 194
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Figure 5.20 Equivalent circuit for an inverter when Vd; is expressed by the extinction angle y........................................... 195 Figure 5. 21 Schematic diagram of a rectifier's a -regulator ....................... 196 Figure 5.22 Schematic diagram of a inverter's (3-regulator .......................... 197 Figure 6.1 The structure of generator self-shunt excitation ........................ 200 Figure 6.2 The block diagram of transfer function of single variable excitation control .............................................. 201 Figure 6.3 A single-input single-output closed-loop system ...................... 202 Figure 6.4 The excitation regulator transfer function dividing the amplification into static and dynamic amplifications ................ 204 Figure 6.5 The transfer function block diagram of PSS ............................. 205 Figure 6.6 Schematic diagram of a generator LOEC (analogous) in a one-machine, infinite-bus system ...................................... 207 Figure 6.7 The schematic diagram of microcomputer nonlinear excitation controller ......................... 219 Figure 6.8 The one-machine, infinite-bus system diagram and its parameters ...................................................................... 219 Figure 6.9 The generator power-angle curve under nonlinear excitation control....................................................... 221 Figure 6 10 The structure diagram of the 6-machine system ....................... 238 C1.
.,.
List of Figures
xv
Figure 6.12 Figure 6.13 Figure 6.14 Figure 6.15 Figure 6.16 Figure 6.17 Figure 7.1
The system's dynamic response curves with PSS ..................... 241
Figure 7.2 Figure 7.3
A one-machine, infinite-bus system ......................................... 256
Figure 7.4
Physical simulation results for improving the transient stability by
Figure 7.5
using steam valving nonlinear control under permanent faults. 258 Computer simulation results of
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The comparison of generator small disturbance response curves under various excitation controls strategies ................... 239 iU.
Figure 6.11
.-.
The system's dynamic response curves with LOEC ................ 241 The system's dynamic response curves with NOEC................. 242.
The system's dynamic response curve with PSS ...................... 242 The system's dynamic response curves with LOEC ................ 243 The system's dynamic response curves with NOEC................. 243 The structure diagram of transfer function for nonlinear control of steam valves .............................................. 255 Physical simulation results for improving transient stability by
using nonlinear steam valving control under temporary faults. 257
nonlinear steam valving control ................................................ 259
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Figure 7.6 Figure 7.7 Figure 7.8 Figure 7.9 Figure 7.10
Computer simulation results with permanent fault .................... 260
Dynamic responses under three-phase fault .............................. 272 Dynamic response curves under three-phase fault ................... 273 Dynamic responses under three-phase fault .............................. 274 Physical testing results of re-synchronizing by using the nonlinear steam valve control .................................... 275
Figure 8.1 Figure 8.2 Figure 8.3 Figure 8.4
Vd,-Id, characteristic of rectifier with various firing angles....... 278 Vd, Id; characteristic of rectifier with various firing angles ....... 279
Vd,. Id, characteristic of inverter with various extinction angles 280
Converter controller characteristic with constant
current and constant extinction angle ...................................... 281
Figure 8.5
Converter controllers with constant current and constant extinction angle .................................................... 281
Figure 8.6
Converter controller characteristic with constant current and constant voltage ....................................... 283
Figure 8.7 Figure 8.8
Inverter controllers with constant DC voltage ........................... 283
The transfer function block diagram of the power modulator in a converter ........................................... 284
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
Figure 8.9 Figure 8.10
Figure 8.12 Figure 8.13
Figure 8.14 Figure 8.15
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Figure 8.11
A 6-machine AC/DC power system .......................................... 294 System dynamic response under conventional controllers with constant current rectifier control and constant extinction angle inverter control ................................ 295 System dynamic response under nonlinear controllers with constant current at rectifier and constant extinction angle at inverter .......................................... 296 System dynamic response under conventional controllers with constant current at rectifier and constant voltage at inverter..... 300 System dynamic response under nonlinear controllers with constant current at rectifier and constant voltage angle at inverter .............................................. 301 An AC/DC power system .......................................................... 302 ,..
xvi
System dynamic response under nonlinear controllers for AC/DC system stability ....................................................... 307
Figure 9.1
Reactor and capacitor treated as load ........................................ 310
Figure 9.2 Figure 9.3 Figure 9.4 Figure 9.5
Reactor and capacitor modeled as power supply ........................311
Figure 9.6 Figure 9.7
The equivalent diagram of transmission system ....................... 314
Figure 9.8 Figure 9.9 Figure 9.10 Figure 9.11 Figure 9.12
Single phase diagram of TCR and its waveforms ..................... 320
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A node with three branches ....................................................... 312 Simple circuit and its vector diagram ........................................ 312 '-n
The circuit with the capacitive compensator ............................. 313 Effects of reactive power compensation on the L/?
transmitted power of symmetrical lossless line ........................ 319 Current and voltage waveforms at different gating angle ......... 321 The control characteristic of BTCR ............................................. 322 The configuration of TCR-FC type compensator ..................... 323
Voltage/current characteristics of Thyristor-Controlled
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Reactor-Fixed Capacitor (TCR-FC) type SVS ........................ 324 Bsvs characteristics of Thyristor-Controlled Reactor-Fixed Capacitor (TCR-FC) type SVS ........................ 325 Figure 9.14 TSC type .................................................................................... 326 Figure 9.15 TSC applied structure ................................................................ 326 Figure 9.16 Effects of n2/n2-1 on natural frequency ................................... 327 Figure 9.17 The configuration of TCR-TSC type SVS compensator.......... 328 Figure 9.18 Equivalent circuit of SVS .......................................................... 331 Figure 9.19 The fundamental control system of TCR-FC type .................... 332 Figure 9.13
List of Figures
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Figure 9.20 Figure 9.21 Figure 9.22 Figure 9.23 Figure 9.24
xvi i The conventional PID control structure of TCR-FC type SVS. 332 Single-machine, infinite-bus system with SVS ......................... 334 The block diagram of the nonlinear controller of SVS ............ 339 The block diagram of SVS conventional control system .......... 339 Effects of different types of controllers on system performance ........................................... 341
Figure 10.1
Dynamic response of the system with PSS ............................... 370 Figure 10.2 Dynamic response of the system with LOEC............................ 370 Figure 10.3 Dynamic response of the system with NOEC ........................... 370 Figure 10.4 Dynamic response of the system with NREC ........................... 370 Figure 10.5 Dynamic response of the system with NREC under the values of parameters having 50% errors .................. 370
List of Tables
Table 6.1
Comparison of Transient Stability Limit ....................................... 222
Table 6.2
Generator parameters of the 6-machine system ............................ 237 The critical clearing time under
Table 6.3
different excitation control mode .................................................. 240 Table 6.4
The critical transmission power under various excitation controls ............................................................ 241
Table 7.2
Dynamic simulation results under temporary short circuit fault... 256 Dynamic simulation result under permanent faults ....................... 258 .R.
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Table 7.1
Table 7.3
Critical clearing time for different control strategies .................... 271 Table 8.1 Operating parameters of the DC transmission system............... 293 Appendant Table 9.1 Errors with approximate linearization of SVS............ 330 Table 9.1
Characteristic values of SVS with three types compensators........ 340
Preface
We may agree with the opinion that in operating an electric power
6"'
system the top priority should be given to the dynamic security and stability before pursuing other targets, such as economical operation, optimal load flow and fair deregulation (power market), etc. However, in the past decade, quite a number of power systems, both in China and abroad still suffered from too many collapses, most of them affected a large area and lasted even over 10 hours, which caused grievous damages not only to the economies of the nations, but also to the residents' comforts and the public order. These fro
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facts show clearly that it is of paramount importance for us to greatly improve the operation security, more precisely, the dynamic stability and -10
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especially the transient stability of power systems.
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The development of science and technology calls for belief that the stability of power systems can be decisively enhanced by adopting new and
advanced control theories, approaches and the corresponding control schemes. (CD
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Now the question is: what should we do to achieve that goal? The first step is to build a bridge over the big gap between the mathematicians, the theorists and the engineers, the technicians; to build a bridge over the big gap between the profound theorems, the abstruse mathematical notations and the industrial designs, the engineering implementation. We have tried, in our small way, to do just this and come to write this book. We wish that this volume could become a small impulse to the purpose mentioned -above and set a small example to show that the endeavor which the scientists and engineers of the world will make along this way could lead to something new and better for modem electric power systems, as well as for other
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vii
nonlinear control systems. This book presents a comprehensive description of nonlinear control of electric power systems using nonlinear control theory which is developed by the approach of the differential geometry and the notions of the dissipativity and the differential game. It is intended as a text for undergraduate seniors and graduate students, as well as a reference to engineers and researchers, who are interested in the (OD
application of modem nonlinear control theory to practical engineering
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NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
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control designs. While readers in the area of power systems may feel a great interest in the chapters which deal with power system control problems; readers in other engineering disciplines may use some of these chapters as examples of application to facilitate their own control designs. For readers' convenience, this book is organized in a self-contained way to introduce, in sufficient detail, the essences of modem nonlinear control theory as well as the corresponding developed algorithms to the readers in L1.
practical engineering control areas, and to explore how these theory and algorithms work in practice. Mathematical preliminaries in ordinary differential equations and modern linear control principle are required for reading the book.
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The book consists of five parts, and the organization has been coo
influenced by the objective of this book.
Part 1 (Chapter 1 and Chapter 2) starts with the characteristics and
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special problems associated with nonlinear systems by comparing nonlinear systems with linear ones. In this part some elementary notions of nonlinear control theory are introduced, emphasizing on differential geometric approach. These notions provide the necessary background for subsequent discussions. Part 2 (Chapter 3 and Chapter 4) presents a comprehensive discussion of the design principles and methods for single-input single-output and for
multi-input multi-output nonlinear systems. The discussion involves the 's.
design principles and approaches associated with exact linearization via state
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feedback, zero dynamics, and output-disturbance decoupling. The design principles and methods presented in this part constitute the tool kit, which will be used to solve practical engineering control problems. In this book they are used in power system control designs. Part 3 (Chapter 5) serves as the connection between nonlinear control theory and power system control designs. This part addresses the modeling ,of power systems. In particular, nonlinear mathematical models of power systems are given in this part.
Part 4 (Chapter 6 through Chapter 9) discusses the application of ..1
'.3
nonlinear control theory to various power system control designs, which include nonlinear excitation control of large generators, nonlinear steam valving control of turbine-generator units, nonlinear control of HVDC transmissions systems, and nonlinear SVS control. All the mathematical models, design methods, control strategies, and effects of nonlinear
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controllers are given in this part. The last part (Chapter 10) deals with the nonlinear robust control theory,
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which is mainly based on the notions and principles of dissipativity and differential game, and its application to multi-machine systems. This part presents a rich collection of new results on nonlinear control of power systems. Each chapter in Part 4 and the last part is virtually a research
Preface
monograph.
At the moment of completing the book, I cherish the memory of Jingde Gao more than ever before. It was he who first propounded the subject of power systems nonlinear control, and benefited me significantly for my research work and in my academic career.
road
I wish to express my sincere gratitude to Professor T. J. Tarn, to Professor J. Zaborszky, to Professor Y. N. Yu, to Professor T. Mochizuki, to
Professor Felix Wu and to Professors Daizhan Cheng, Huashu Qin from whom I learnt many of the methods and methodologies which have been
applied in the book. I wish to thank Professor Kaiyuan Cai for his vii
,ti
encouragement. I am indebted to Dr. Zheng Xu, Dr. Xin Jiang, Professors Gengshen Hu, and Dayu He who reviewed the manuscript and made many valuable suggestions. Finally, my thanks go also to my graduate students Wei Hu, Yusong Sun, Wencong Wang, Feng Liu, Hua Xie, Baoping Mao, Jin
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Ma, Juming Chen, Tianqi Guan, Zhitao Wang et al., the manuscript preparation would not have been possible without their assistance. On behalf of the authors I wish to express our gratitude to KLUWER
ACADEMIC PUBLISHERS for affording the opportunity to publish this book.
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The work presented in this book -mostly grew out of the projects supported by Chinese National Key Basic Research Fund under grant
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G1998020300, by NSFC under grant 59837270, and by Chinese National Key Science and Technology Tackling Project under grant No. 97-312-0111-1a. The work is also supported partly by NEDO International Joint Research, Japan under grant 99EA I.
QiangLU Being, May 2000
Chapter 1
Introduction
OVERVIEW
1.1
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Out of many problems to be resolved and improved in modem power systems, the economy and reliability of power systems are the two main categories. Reliability consists of two different aspects. One is how to choose the ways of connection of power plants, substations, and power networks to minimize the probability of occurrence of such accidents that lead to power cut. Problems pertaining to this aspect can be called the static or structural reliability of power systems. The other is the stability of power systems, that is, the ability of power systems to keep in synchronization among the generators under small or large disturbances. Problems in this aspect can be called the dynamic reliability or dynamic security of power
in'
systems [25, 54]. Stability is the most important issue of power systems, because once the stability of large power systems is devastated, it can result in power collapse
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in one or several large areas, which may cause great loss in economy and
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great damage to the residents. Many countries have had disastrous lessons on this respect. Therefore, to improve the stability of power systems was, is and will continue to be a pressing and important task. [n'
One of the major ways to improve the stability of power systems is taking highly sophisticated technology of control. A great deal of research
works has been done to improve and develop the control techniques of power systems [1, 20, 26, 31, 32], the excitation control for example, during the past 50 years. In the excitation control field, single variable feedback was
the main control technique before 70's, i.e. taking the deviation of the `+'
terminal voltage of the generator as the feedback variable. Controllers of this
2
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
kind have two forms: proportional control and proportional-integraldifferential (PID) control. With the development of power systems, the limitations of this single
input control technique have become increasingly obvious. In the 1970s, deMello and Concordia [10] proposed a control technique which, besides the deviation of the terminal voltage, took a supplementary feedback variable as '_'
another input, which could be the speed deviation Aw, the frequency deviation if, or the deviation of active power APe . Thus the excitation control of generators evolved from single-input control to dual-input control, which is the control usually called PSS (power system stabilizer). Meanwhile, a control technique called forcing excitation was developed by r0-
('Q
researchers of the former Soviet Union using their own design method, called "multi-variable partition in D domain". This control technique, besides the deviation of terminal voltage, added the generator current, the p,'
vii
v0,
cad
deviations of frequency, of active power and their first order derivative as its feedback variables. However, it has not achieved its expected control effects due to the limitations of the theory and method it applied.
't7
In 1970 and 1971 Yu and Anderson proposed the approach of linear optimal control to power system excitation control [20, 55]. In the 1980s, while introducing the PSS excitation control and trying to make some improvements, the linear optimal excitation control (LOEC) technology was developed in Chinese power systems, and EDF, which took the deviations of terminal voltage, the speed (or frequency), and the active power as the feedback variables. The gain for each feedback variable is obtained from the solution of the Riccati equation, concerned with a quadratic performance index. As a result this issue is called the LQR (Linear Quadratic Riccati) problem. This LOEC excitation control has been applied successfully in some large power plants in China and France, which have improved the stability under small disturbance and low-frequency oscillation of power systems. ,..'
vii
In fact, the PSS and LOEC control techniques could also be used to steam valving control and water gate control, in one word, the governor Ga.
control. But, due to various reasons, this multi-input variable control has not been duly considered and used in governors. Productivity and technology, however, will never stay at a certain stage
E00°
of their development. They will unceasingly go from development to
vii
C7O
'C3
ate)
a...
development. As we know, a power system is a nonlinear dynamic system. Both PSS and LOEC have a common characteristic that their models are established by linearizing the nonlinear equations of power systems at a certain operating point (a fixed equilibrium point X e ). It is obvious that these approximately linearized models are relatively accurate only when the actual state X(t) is rather close to X. When such a condition does not hold, it will inevitably lose its validity, and so does any type of linear controllers.
Introduction
3
Consequently various types of linear control schemes will not be able to
"co
meet the requirements of improving the transient stability of power systems. To overcome this fundamental problem,- we should try to design controllers of power systems using nonlinear control theory and approach.
Caw°
We have so far, taking power systems as examples, discussed the r.,
coo
s0.,
problems brought about by the design method of approximate linearization. In fact, up to now, almost all of the controllers of nonlinear systems were designed by using this method. It is not true that people did not know the disadvantages of this method, but because the nonlinear control approach come into being and be applied in practice only in recent years and it really is a new engineering branch. It seems that it is the time now to improve the control techniques of power systems and other nonlinear systems with the nonlinear control theory.
t7'
-ay
'(y
(DD
fro
s0.
0
.T.
In summary, the general trend of development and improvement of control theory and its applications is to evolve from single-variable control to multi-variable control, from linear control to nonlinear control. It is for this reason that we present readers with this volume which embraces the relevant recent results both in theory and practice. We hope that the nonlinear control theory and techniques can be rapidly improved to a new level. We also hope that the "seeds" of nonlinear control theory and techniques can sprout, grow and finally give out plentiful and substantial fruits in the soil of engineering control areas.
OUTLINE OF THE DEVELOPMENT OF CONTROL THEORY
1.2
.a'
'ti
development.
coos
In order to get a general picture of this book we intend to first make a brief introduction to the features of control theory at its different stages of (IQ
Control is a general term for the theory and techniques to change the dynamic performance of a system by imposing certain inputs on the systems,
so as to satisfy certain requirements to their best. The improvement of chi
control techniques always depends on the improvement of control theory, (IQ
which, up to now, has experienced the following developing stages: The first stage could be called the control theory in complex variable s
coo
or frequency jw domain, which is also called classical control theory in some relevant literature [8]. This control theory in fact is not so "classical" as its name suggests. It is only in the late 1950s that its theoretical system became matured and has made some progress even today. The reason why it is usually called "classical" is that it is rather classical comparing with the modem control theory developed rapidly in the beginning of the 1960s. This
4
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
control theory in s domain generally consists of four parts, which are modeling theory, response analysis, stability analysis, and synthesis correction respectively. The most significant feature of this theory lies in its own modeling method, which is based on such a fundamental fact that if x(t) is a function of time t as its independent variable and the integral jm o
dt k
dkx(t)e_,,dt
a4'
(where k is a natural number, s is a complex variable) is absolutely convergent, that is _00 <jo,ddtkxk
:1.
then under the condition of zero initial state, the following equation can be obtained according to the definition of Laplace transformation L(dkxkt))= jo
dkxk(t)e_S'dt=skX(s)
dt
dt
(1.1)
where symbol denotes the Laplace transformation, X(s) is the Laplace transform function of x(t). In the light of formula (1.1), we can see more clearly that for a singleinput single-output linear time-invariant system, its dynamic behavior can be described by the following constant-coefficient ordinary differential equation d "x(t)
d 1""'1x(t)
dt "
dt ("-"
b,
+...+a, dx(t) +a°x(t) dt
(1.2)
d'u(t)+...+b du(t)+b u(t) dt' dt ° `
where, u(t) is the control input variable, x(t) the output variable. Perform the Laplace transformation on both sides of Eq.(1.2), with the zero initial state, we obtain +...+a,s+ao)X(s) =(b,s' +b,_,Sr-'
(1.3)
Defining the system's transfer function G(s) as the ratio between the CAD
Laplace transformations of the output and the input of the system when the initial state is zero, we can get the transfer function of system (1.2) from Eq.(1.3) as G(s)=X(s)_
U(s)
(1.4) a,,s" +a,,_,s"-'
This equation, as a fraction of polynomials of s, is the general form of
Introduction
5
transfer functions, which is also the basic form of mathematical model in the
classical control theory. In modeling, the transfer function of the whole closed loop system is of course not worked out in one shot. Instead we firstly a;-
decompose the system into several typical blocks described by first or second-order differential equations, working out the transfer functions of these blocks, forming the transfer function block diagram of the whole
per''
closed-loop system by assembling these transfer functions according to the
Vin'
ii,
COQ,
r.'
structure of the actual system, and then from which we get the transfer function of the closed loop system. Since transfer function G(s) is a function of complex variable s, this control theory is therefore named as control theory in s domain. We also know that jw(-oo < w < +oo) can be taken as a complex variable s with its real part being zero. If we substitute s = jco for s into the transfer function G(s) , we can have the system's frequency characteristic G(jw). Therefore the analysis method of this control theory is also called frequency domain analysis.
(]_
.+S
Ft'
The special modeling method, using transfer function to describe a linear control system, governs the following properties and the application scope of the classical control theory. Firstly, the most significant feature of this theory or method is that it transforms high-order differential equation of time t into polynomials of complex variable s, so the mathematical tools used are fairly simple, which are mainly Laplace transform method and algebraic polynomials. Secondly, since the transfer function is obtained from the Laplace transformation of the linear time-invariant ordinary differential equation, they are equivalent to each other. Therefore the systems that the rail
'-t
transfer functions can model are only linear constant control systems. Thirdly, since the transfer function is the ratio between the Laplace transformation of output and that of input, this theory or method is only applicable to single-input single-output systems. Fourthly, the modeling '17
-fl
.y"
"'ti
cad
method of transfer functions or frequency characteristics can only be used to
investigate and analyze the system from the viewpoint of the relation CD,
0CD
between the output and input, so it conceals the internal dynamic behavior of the system. All the advantages, properties, and limitations of the classical control theory discussed above are governed by its special modeling method.
R°'
mss'
The second stage is generally called modern control theory, or more precisely, it should be called multi-variable linear control theory, which was developed very rapidly from the beginning of the 1960s and nowadays has been extensively applied in the world. Since the 1960s, as the scale and complexity of the engineering control
systems became greater and greater, and the requirements of control CAD
oe;
Cep
precision and dynamic performances of the systems became more stringent, more and more limitations of the classical control theory mentioned above
were discovered. The further advance in productivity and technology appealed urgently for a new control theory and synthesis method which
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
6
U
°
o
could be well applicable to multi-input multi-output dynamic systems. On the other hand, computer technology has made a rapid progress since the 1960s, which provided the technological means for large-scale numerical analysis. All of these facts above formed the backgrounds and conditions for the development of control theory at its second stage, namely the multiO
o
°
o o
.U
O
U
moo'
o
U
variable linear control theory. The monograph "Introduction to Matrix Analysis" [39] by R. Bellman
in 1960 and the treatise "Mathematical description of Linear Dynamic O
0
Systems" [40] by R.E. Kalman in 1963 have made great contributions to the foundation of this control theory. The most significant feature of this theory, )>~
O
°
U
O
if allow us to summarize in one sentence, is an integration of modeling o
theory in state space and mathematical method with linear algebra. The modeling theory and method in state space can be summarized as {x, (t), x2 (t), , x,, (t)}
or a
O
5:`
U
0
OO
follows. Firstly, we define a set of variables
o
CD-.CD
0
vector X(t) = [x, (t) x2 (t) ... X" (t)]T as the system's state variable set or state vector which can uniquely determine the dynamic behavior of the system by
O
these minimum number of variables. A space R" formed with all of the state variable axes is called the state space. One state of the system can be represented by a certain point in the state space, and the dynamic process o
O
N
starting from an initial state X,, will be characterized as a trajectory in the space starting from X.. If the dimension of the state vector of the dynamic system is n, we call it an n`h-order system. It is well known that, an n'h-order linear dynamic system can be modeled as an n`h-order constant-coefficient ordinary differential equation. But if we model a dynamic system as an n'order one, what appear in the model are only the input and output of the system. In order to investigate multi-variable systems, modern control theory requires using n first-order differential equations to describe an nth-order linear dynamic system. The standard form of the mathematical model is 0
O
o
°o.
OO
O
0
O
'O'f
o
O
(t) = a, x, (t) + a12 X2 (t) + ... + al ^ xn (t) + hi l u l (t) + ... + b, u, (t) +
dX1
dt
+
+
dX2(t) = a21x1 (t) + a22X2 (t) +... + a2n X. (t) + b21u1 (t) +... + b2ru,(1) dt
(1.5)
dt
The output equations are +
+
II
y1(t)= CII X1 (t) + C12 X2 (t) +... + C1n X (1)
+
+
+
II
ym(t) = CmIXI (t)+
Cm2X2(t)+... +
where, x; (t), i = 1, 2, , n, represent the state variables; y, (t), l =1, 2, , m, the output variables; uk (t), k =1, 2, , r, the control variables; a,,, b,,, and
Introduction
7
c,,, j =1, 2, , n, I =1, 2, , m, the coefficients. If all of the coefficients are
constant, the system is linear and time-invariant. Otherwise, the system is linear time-varying. Eq.(1.5) can be rewritten in the following matrix form X(t) = AX(t) + BU(t) Y(t) = CX(t)
(1.6)
[17
t!'
where, X(t) is an n-dimensional state vector; Y(t) an m-dimensional output vector; U(t) a r-dimensional control vector; A, B, and C are coefficient matrixes. The standard form given in Eq.(1.5) or (1.6) is the state space equation of a linear dynamic system, or state equation for short. With the initial state X(to) = X0, the solution of the state equation is represented by a trajectory starting from point Xo in the state space, which is the state locus. It is a description of the dynamic process of the system, which is commonly called the transient process in the literature of power systems. N..
'-,
...
O-.
°}3
.-.
Since all the linear dynamic systems can be modeled as the state equation in matrix form as shown in Eq.(1.6), almost all the methods in matrix algebra (linear algebra) can be used to investigate and analyze the various problems of linear dynamic systems, such as the problems of c°)
controllability, observability, dynamic performance, stability, identification
CD0°4A
of parameters and compensation techniques, etc. So, as mentioned previously, state space description method, combined with linear algebra fro
vii
+U+
theory, formed the complete multi-variable linear control theoretical system, and the control theory has stepped onto its second stage of development. It is worth noting that there exists a matured and widely-used branch in .+,
coo
this theoretical system, i.e. the linear optimal control [16]. Problems of optimal control can be defined as the following. To a given linear system or nonlinear system linearized at one of its equilibrium points as follows X(t) = AX(t) + BU(t) Y(t) = CX(t)
(1 7)
4-+^
('1
For the LOC (Linear Optimal Control) design of the system, a quadratic form performance index is usually chosen as
(1.8) DC'
J = to (X' (t)QX(t)+ UT(t)RU(t))dt
mow
where, Q is a positive definite or positive semi-definite weighting matrix and R a positive definite weighting matrix, and the superscript T denotes the transpose of the vector. The goal is to find the control law U'(X(t)) such that the performance index J given in formula (1.8) achieves its extremum. Whether the extremum is maximum or minimum depends on the physical meanings of the problem itself and the index. The control U'(X(t)) that
8
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
makes the index J achieve its extremum is called the optimal control. As we know, the quadratic index J shown in expression (1.8) is a functional of state vector X(t) and control vector U(t). So from the viewpoint of mathematics, to solve the optimal control problem is to find the conditional extremum of functional J[X(t),U(t)] subject to the constraint of state equations. This is a --4
3?'
9r,
typical problem of variation with constraints and the Euler-Lagrange equation is the foundation for solving that problem. In 1957, R. Bellman also proposed the dynamic programming approach to the linear optimal control
"'O
930
p.,
problem [38]. Pontryagin (A. C. H o H T p x r u H) generally solved this problem by discovering the so-called Pontryagin's maximum principle in 1958. No matter which method was used, the same final result was reached that the solution to this linear optimal problem was found to be a certain linear combination of state variables X(t), i.e.
U' =-K'X(t)
(1.9)
where, K' is the optimal gain matrix which is expressed as K' = R-'BTP*
(1.10)
This is a constant matrix as the system under consideration is linear and time-invariant and P' is the positive definite solution of the Riccati matrix equation ATP+PA-PBR-'BTP+Q = 0
(1.11)
Therefore, the linear optimal control system can be graphically represented by the structural block diagram given in Fig. 1.1. The linear optimal control problem with quadratic performance index is called the LQR problem, that is the linear-quadratic-Riccatian problem. The multi-variable linear optimal
To solve: PA + ATP - PBR-'BTP + Q = 0 Figure 1.1
Structural diagram of linear optimal control system
9
Introduction +-+
control theory made such a rapid progress that it could be said that all the
...
--,
coo
theoretical problems were solved during the period from the beginning of the 1960s to the mid of the 1970s. In summary, the modem control theory as mentioned above has some significant features as follows: (1) It models the system under consideration
as a set of basic linear differential equations in time domain; (2) The mathematical tools used are mainly the theory of linear time-invariant ordinary first order differential equations and linear algebra, rather than
COD
o")
Laplace transformation and algebra of polynomials used by the classical control theory; (3) The modeling theory and mathematical methods make it possible to handle the linear multi-input multi-output system ; (4) A systematical theory and design methods of optimal control have been established which enables the control laws so acquired to guarantee the performance index functional achieving its extremum; (5) To systems whose
parameters possibly vary in large ranges, optimal control design method combined with parameter identification could achieve the so-called adaptive 0
800`O
control systems. These are the main characteristics and development backgrounds of the second stage of control theory, i.e. the linear multiin'
variable control theory. The third stage of the development is nonlinear control theory [3, 17, 28, 46]. As we know, most engineering control systems are nonlinear in nature. Admittedly many systems can be linearized at one of their equilibrium points, still satisfying the basic engineering requirements. These systems can certainly be analyzed and synthesized using theories and methods of linear ;-.
t3.
coo
+N+
L.,
control system. But some system models such as the models of power
WHO
systems are not fitting and proper when established by adapting the approach
v:,
"C7
of approximate linearization as their stability and dynamic responses are analyzed under large disturbance [24]. Other systems such as robot arm systems, automatic pilot systems, and some chemical processes control systems could not get the ideal control effects when the approximately -40)
linearized mathematical models are used in design. In one word, the development of engineering and science calls for a new systematical nonlinear control theory urgently. coo
During the past decade, differential geometry has manifested to be an effective means of analyzing and designing nonlinear control systems, and formed a new scientific field [5, 7-9, 11, 14, 15, 19, 21-23, 27, 29, 30, 41 ..cd
-45, 49, 51-53]. In that field, a branch-exact linearization via feedback has been rapidly developed and has seen success in some challenging boo
engineering applications. Its essential points are stated as:
An affine nonlinear) system is described by the following state equations and output equations ') The concept of an affine nonlinear system will be discussed in Chapter 2.
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
10
dxl (t) dt
=f(xl,...,x
)+g
+...+
dx2 (t)
dt
n
(xl,...,x 11
n
+g (x l ,...,x n )u 2
)u l
21
9.1(x1,..., xn)um
(xl,...,xn
--(xl,...,x)+g12
+...+
)u +g22(x l 1
,...,x n )u2
9'2(x1,...,xn)um
=Jn(xl,...,xn)+g1n(xl,...,xn)ul
dt
+g2n(xl,...,xn)u2
+...+gmn(xl,...,xn)um
y1 (t) = hl (xl , ... , xn ) y2(t)=h2(xl,...,xn)
ym(t)=h.(xl,...,xn)
where, u1 , u2 ,
and
are the state variables; y, , , ym the output variables; , um the control variables; f,, i =1, 2, 3, , n;; h j =1, 2, 3, , m; x1,
, x,,
the scalar functions of the state variables. Rewrite the above equation into the compact form as X(t) = f (X(t)) +
g 1(X)u
(1.12)
Y(t) = h(X(t))
where X = [x, x2
xn IT is a state vector
y = [y, Y2
Y. IT is an output vector
f(X)=[fi(X) J2 (X) ... fnMIT gi(X)=[gri(X) 9!2(X) ... g;,,(X)]T h(X) = [h1 (X) h2 (X) ... hm (X )]T
The target is to find a nonlinear feedback U = a(X) + 13(X)V
(1.13)
such that the nonlinear system (1.12) can be transformed into a completely controllable linear system via a proper coordinate transformation Z(t) = cb(X(t)), that is
Z(t) = AZ(t) + By Y(t) = CZ(t)
(1.14)
where, a(X) is an m -dimensional vector function, R(X) an m x m matrix
Introduction
11
function, V an m-dimensional vector, Z an n -dimensional vector, A an n x n state coefficient matrix, B an n x m control coefficient matrix, Y(t) an m -dimensional output vector, C an m x n output coefficient matrix. It is obvious that this linearization approach is entirely different from
that obtained by discarding high-order terms of the Taylor expansion of nonlinear function f(X) at a given equilibrium point, that is, to replace the increment of a nonlinear function f (X) at a given equilibrium point Xa
Af(X)=.f(X)-f(X0) with the total differential of the function at the same point df(X) = af(xo)Ax, +...+ of(Xo)Ax,, CX1
ax',
.°u
--J
fro
;-H
As a result, the approach which will be emphatically discussed is called not approximate but exact linearization. R. W. Brockett was a representative founder of exact linearization [44]. B. Jakubczyk, et al. gave the sufficient and necessary conditions of local exact linearization of nonlinear multi-variable control system [6], which were predigested by R. Su and L. R. Hunt [43]. A. Isidori and T. J. Tam and D. Cheng gave the even briefer sufficient and necessary conditions of the global exact linearization of affine nonlinear systems [8], which hinted the corresponding algorithm. In the application area, G. Meyer used the theory of exact linearization in the helicopter automatic flight control [12, 13]; T. J. Tarn successfully used it in the control systems of the robot arms [5, 47]; The authors of this book cultivated its new applications in nonlinear control of power systems [33, 36, 37, 56]. In recent years a new academic branch, the nonlinear robust control or say nonlinear disturbance attenuation control, is developing [3, 4, 34, 35, 48, 50]. Drawing on the concept of H. control of linear systems, some authors named that branch as nonlinear H. control. The statement of this problem is as follows. Consider an affine nonlinear system modeled by equations of the form ,...
X = f(X)+g,(X)W +g2(X)U Z = h(X) + K(X)U
(1.15)
The Eq.(1.15) describes a dynamic system with state X E R", control input
U e R' and disturbance input W e R' ; and Z E R' denotes the penalty vector.
The statement of the nonlinear robust control problem is that to find the control strategy U' for the system in Eq.(1.15) such that under the action of control input U' the Lz-gain G2 of the closed loop system from disturbance
12
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
input W to penalty output Z is less than or equal to a prescribed positive number y, which can be expressed as follows
VT>0
foDIZ(t)IIZdt _ y2foIIW(t)II2dt
(1.16)
In fact, formula (1.16) can be considered as a performance index for the control system (1.15). In this sense the nonlinear robust control is a kind of nonlinear optimal control. As the approach to linear robust control problem
.Go
is deduced to getting the solution of an algebraic Riccati equation, the
.gon
o.0
CDi
nonlinear robust control one is deduced to solving the so-called HamiltonJacobi-Isaacs inequality, a partial differential one. However, up to date there is no general algorithm for obtaining its solution. In Chapter 10 of this book, we will see how the problem is solved for the nonlinear robust control of a multi-machine system, and how this control law works in practice.
The foregoing paragraphs have presented an overview of the three stages of development of control theory. We hope that this brief introduction will be helpful to readers in understanding the main developing trends of control theory from a high position and in generally mastering the contents of this book.
1.3
LINEAR AND NONLINEAR CONTROL SYSTEMS ."a
'a'
Control systems, according to their dynamic characteristics, or mathematical models, can be classified into two categories: linear and nonlinear. >a.
It is of course correct if we define linear control systems as those that can be modeled by linear differential equations, and nonlinear systems as those that can be modeled by nonlinear differential equations. This definition, te".
.`3
however, does not bring more information about the problem. In order to deepen the understanding of this classification issue, let us first take a look at
the R - L - C circuit (Fig. 1.2). Take the outer voltage source V(t) as the
input, the voltage imposed on the capacitor R
VV (t)
as the output, the
L
i(t)
V'(t) -C
V(t)
T Figure 1.2
R-L-C Circuit
Introduction
13
differential equation of this system is LCd2V(t)+RC dt
dt2
t)+V (t)=V(t)
(1.17)
`
where R, L and C denote the resistance, inductance and capacitance, respectively. If we set x,(t)=V,(t) X2 (t) =
dxi(t) di
dt
then, Eq.(1.17) can be rewritten as z, (t) = x2(t)
HIV
-IV
iR
x2(t)=-rI X,(t)- x2(t)+ I V(t)
(1.18)
or as
r
Lx2 (t)]
0
1 LC
0 I
1
- RL
[x2 (t)] +
V(t)
(1.19)
LC
The previous equation can be written as its compact form (1.20)
i(t) = AX(t) + BV(t)
This is the general form of state equations of linear control systems. In Eq.(1.20), if A and B are both constant matrices, the system modeled by this equation is a time-invariant linear system, which is also called the constant linear system. Suppose that we want to consider the aging effects of the components,
that is to consider the effects of parameter variations on the circuit as time vii
goes by, then, R, L and C should be considered as functions of time t, which are represented as R(t), L(t), and C(t) respectively. In this case, A and B in Eq.(1.19) are no longer constant matrices, instead they are function matrices of time t, and the R-L-C circuit under discussion should be modeled as j ((t) = A(t)X(t) +B(t)V(t)
(1.21)
This equation gives the general form of state equations of time-varying linear systems.
From the discussion above, we know that linear systems have two varieties, namely, time-varying and time-invariant systems. Now we turn to discuss the mathematical models of nonlinear systems.
To make the discussion more illustrative, we still take the R-L-C `C3
circuit as an example. Now consider a special capacitor whose capacitance C is an inverse function of the voltage V. That is
14
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
C=.f(VV)=k, 1 fr
=k 1
(1.22)
x,
where k, is a constant. Substitute Eq.(1.22) into Eq.(1.18), and let a, _ -1/(k,L) , a2 = -R/L, a, =1/(k,L) , which are obviously constants, we have
x,(t)=x2(t)
(1.23)
x2 (t) = a, x, (t) + a2 x2 (t) + a, x, (t)V (t)
Apparently, the state equation of this circuit can not be written in the form of X = AX+BU, instead, it can only be written as
x,(t)1=[
I x2(t)J
X2 (t)
1 +[
2
La,x, (t)+a2x2(t)J
ax,(t).1
(1.24)
V(t)
In the previous equation, let f(X(t))
g(X(t)) = [a3
= [aixi () +ta2x2 (t)]
(t)]
u(t) = V (t)
Eq.(1.24) can be expressed as
X(t) = f(X(t)) + g(X(t))u(t)
(1.25)
,02
.'3
This is the general form of state equations of nonlinear systems, which are often encountered in engineering. Let us take another example of a simple power system to illustrate this nonlinear model. Suppose a simple power system where a large steam-turbine generator .=.
set connects to an infinite bus, as shown in Fig. 1.3. According to the 5
N governor F*X 6 1
2
infinite system
4
Figure 1.3 A one-machine infinite-bus system
I- generator; 2- transformer; 3- transmission line; 4- steam turbine; 5- steam valve; 6- regulated valve; 7- governor
Newton's second law, the motion equations of the machine's rotor can be written as
Introduction
15
HddtZt)
M,(t)-Me(t)
d8(t) =w(t)
(1.26)
(1.27)
dt
where, 8(t) is the generator's rotor angle; Mm the mechanical input torque; M, the electric output torque; H the moment of inertia of the shaft of the machine; co the speed. From the electric machinery, we know that the electric torque can be expressed as Me (t) = XVS sin S(t)
(1.28)
(CD
chi
where E9 is the transient potential of q-axis of generator; V. the voltage of the infinite bus, which is a constant; XL the sum of the transient inductance of the generator, the inductance of the transformer and the inductance of the transmission lines. To be simple, assume the transient potential Eq is constant. So the ..N
system's control input variable is the mechanic torque of the steam turbine M.. Now, Egs.(1.2,7) and (1.26) can be written as S(t) = w(t) tiw(t) = -a, sin S(t) + a2M, (t)
(1.29)
where _ a'
E9VS
HX'
aZ
H
Let x, (t) = S(t)
xz (t) = w(t)
u(t) = M. (t)
then Eq.(1.29) can be rewritten as x,(t) _ z2 (t)]
x2(t)
0 u(t) - [- a, sin x, (t)] + [a2l
(1.30)
Therefore, the state equation of the power transmission system can be written in the affine nonlinear equation form X=f(X)+g(X)u
(1.31)
The only difference is that g(X) in Eq. (1.30) is a constant vector. It
should be noted that Eq.(1.31) is just the general form of
mathematical models of nonlinear systems of one particular class. This form
is featured with its nonlinearity relative to the state vector X(t) and its
16
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
linearity relative to the control vector u(t). The most general form of state equations of nonlinear control systems is dx,(t) =J1(x1(t),...
xn(t),u1(t),...
dt dxdt2
=
f2(x1(t),...,x
um(t))
(t),u1 (t), ... U. (t))
4(x1(t),...
(1.32)
xn(1),u1(t),...,um(t))
dt Y1 (t) = h,(Xi(t)r ...>xn(t))
(1.33) y. (t) =
L3.
where Eq.(1.32) is the state equations,,and Eq.(1.33) the output equations, in which x, (t), , x,,(t) are the state variables, u, (t), , um (t) the control variables, y, (t), , ym (t) the output variables, f, , f2 ..... f could be considered as the generalized velocity functions, and h, h2, ., hm the output functions. Eqs.(1.32) and (1.33) can be written in the matrix form as X(t) = f(X(t), U(t)) Y(t) = h(X(t))
(1.34)
where X(t) is an n -dimensional state vector, U(t) an m -dimensional control variable, f an n -dimensional vector function of X.
MODELING METHOD OF APPROXIMATE LINEARIZATION E"1
1.4
.+'
Strictly speaking, almost all the practical dynamic systems are nonlinear.
In the design of a control system, the traditional method is to linearize the
original system at one of its equilibrium points, say X, to get its approximately linearized mathematical model, which is a transfer function or '+'
a linear state equation set; then the system can be analyzed and designed cad
according to the methods of linear control theory. Generally speaking, a nonlinear function is feasible to be linearized if it is monodromic and differentiable in the neighborhood of the point Xr. For example, we suppose there is a nonlinear function of two variables 100
y_,
x, and x2 Y=f(x1,x2) Rewrite it using Taylor expansion
Introduction
17
Y =! (xl, x2) = J (xa +&l,xe2 +A2) of of =f(xel,xe2)+ Al + axe A2 ax, IX=x,
+1
+...
(7f Axl +&2)(2)
2! ax,
7x2
In the previous equation,
x=x,
is the selected equilibrium point, and
(al Ox + of Ox2)(2) = a2 f Ox; + aZ f Ox2 +2 7x2
ax,
(1.35)
x=x
7x2
ax,
"f
OxOx2
ax ax2
4.+
Those first-order and high-order partial derivatives as given above are evaluated at the equilibrium point If the second-order and highorder derivative terms are ignored, we have
Ox,+O
Y=f(xel,xe2)+L
2 X.
X,
from which we immediately get the linear expression of Ay in the form as f(xl,x2)-f(Xel,xe2)
AY =
Ox, +
of
0x2 =k, Ox, +k2i\x2
(1.36)
2 x=x,
1 x=x
where k, =
2f__1
ox,
= const
k2 =
of = const 7z2
x=x,
x=x,
This is the principle and method of approximate linearization. From formula 1'?
(1.36) we know that the right side of the expression Ay is the total of function y = at point Xe . So the
differential
essence of approximate linearization is to replace the nonlinear function's increment Of (X) near the equilibrium point Xe for the state equation of the system by the total differential df(X) near the same point. For example, in a one-machine, infinite-bus power system modeled by Eq.(1.30), we can write the increment of nonlinear function Me = a, sins at 8o as a(al sin 8)
?C'
AM,,
L'.
08 = a, cos50A8 = ks08
(1.37)
Thus the system modeled by Eq.(1.30) can be expressed by the corresponding deviation form as follows
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
18
AS(t) = AC(t)
Am(t) = k,A8+a,AM,,,
or expressed with the form of X = AX + BU as
]m IAwi _ [ka
(1.38)
0] [Am] + [a2
In this way, we get the approximately linearized state equation of the onemachine power system. The approximate linearization has its serious limitations which can not ?.'
be ignored. As the online operating point of the system is far from the selected equilibrium point, the state equations obtained by approximately linearized method will lose its qualification for describing the original nonlinear system. The inevitable result is that the controllers designed
C15
..,
according to the mathematical model obtained by this means can not exert its expected effects on the practical systems when they are operating at a point
far from the equilibrium state used in design. They even create negative vii
...,
actions in some situations. Let us recall the power system just discussed as an example. From Eq.(1.37) we know that in matrix A the element a2,(= k8) equals a, cos80 . If we choose 80=60' as the equilibrium for forming the
model, now suppose the power angle swings to
130°
tossed by large
'c'
disturbance, then the ratio of the designed value of a21(a, cos60°) to its postdisturbance value (a, cos130°) is about 230%. It would not surprise that the
a.,
G:.
Ll.
well-designed controllers according to the mathematical model obtained through approximate linearization can only be able to improve the systems' stability under small disturbance, but reveals their inability under large CAD
disturbance. Why, however, the method of approximate linearization is still widely used in engineering design even today in spite of so many limitations? There
19-
?'_
are several contributing factors: firstly, indeed some of nonlinear control systems usually will not slip very far away from its specified equilibrium point used in design, and its mathematical model obtained via approximate linearization satisfies the engineering requirements; secondly, to some systems, although it is obvious that the state point of them could run very far away from the selected equilibrium point, the linearized model is easy to handle for design, while the nonlinear control theory and method have not been commonly mastered by the technicians and engineers since it only appears in recent years and the relevant literature about it is written in coo
ors
.>7
abstruse mathematical forms.
It is time for researchers, engineers and technicians to master and use nonlinear control approach.
Introduction
1.5
19
STABLE AND UNSTABLE EQUILIBRIUM POINTS The most general description of nonlinear control systems is (1.39)
X(t) = f(X(t),U(t))
where, X E R" is the state vector, U E R' the control vector. If state feedback is used, the control variables u, , u 2 , , um are functions of the state variables X, expressed as U = U(X). Substituting this vector function into
Eq.(1.39), we obtain the general mathematical description of nonlinear control systems in the form as X(t) = f(X(t))
(1.40)
In the above equation, let
.-.
f(X)=0 The zeros of i(X) are precisely the solutions of the above algebraic equation. Each solution Xe determines one equilibrium state of the nonlinear system. Since each certain state corresponds to a certain point in the state space, the equilibrium state X. is also called an equilibrium point.
From Eq.(1.40), we know that the physical meaning of f(X) is the generalized velocity set. Because Xe is one of the solutions of the equation
f(X) = 0 , that implies, the "velocity" of every state variable x; of the system at point Xe along its own axis is zero, namely, =
dt = dt
l(xel,xe2,...,x")0
f2(xelIX.21
=4
`-'
dx,{ dt 72
(Xel,xe2,...,xen)
=0
,.,
This means that absolutely without any disturbances, the state point of the system will stay at the point X. for ever. This is why the Xe is called an equilibrium point of the system. A hyperbolic linear system has only one equilibrium point, i.e. the coordinate origin. Since the state equation is i(t) = AX and its equilibrium equation is AX = 0 , therefore X,=O, because hyperbolic means A has only non-zero real part eigenvalues and hence A is non-singular. As a result, if a 'C3
a°.
20
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
linear system is stable at Xe= 0, then the system is stable. According to the Lyapunov's stability principle, the necessary and sufficient condition for a CAD
linear system to be asymptotically stable is that the real part of every eigenvalue of matrix A is negative, i.e.
R{2;}<0
Q;0
A nonlinear system may have multiple equilibrium points because
Coo
coy
equation set f(X) = 0 usually has multiple solutions. Therefore the question for stability of a nonlinear system can only be stated as whether the system is stable at a certain equilibrium point Xe,, or whether the equilibrium point X., is stable. Only in the case that the equilibrium point is unique the global stability may be considered. This is an important difference between linear and nonlinear systems ' concerning stability.
Then how to assess the stability of a nonlinear system at one of its coo
equilibrium points Xe;? Two approaches exist.
The first one is to linearize a nonlinear system X = f (X) at one of its equilibrium points XG,, as described in Section 1.3, and the matrix A of
0
.nom
the corresponding state equation X = AX is obtained. If the real parts of all the eigenvalues of A are negative, then the nonlinear system is stable in a sufficiently small neighborhood of this equilibrium point Xe; . Otherwise it is unstable. This approach is called the Lyapunov's first method. The second one can be stated as follows: suppose X. is an equilibrium
point of system X = f(X). Perform a coordinates shift like this Z(t) = X(t) - Xe
Thus if the state of the system stays at an equilibrium point X, -then it must have a shift to set Z=0. The principle to assess the stability of point Xe is
(1) Consider the state equation
dZ(t) &
= f(X, + Z(t))
(1.41)
If a differentiable positive scalar function V(Z) can be found such that the derivative of V(Z) with respect to t V(Z) = a>(Z) dZt = 5V(Z) f(Z)
where f(Z) = f(X8 + Z)
(1.42)
Introduction
21
satisfies
for Z=0 for Z#0
V(Z)=0 V(Z)<0
then the nonlinear system under consideration is asymptotically stable at point Xr. (2) If a positive scalar function V(Z) can be found to satisfy
V(Z) =a
)f(Z)=0
V(Z) > 0
for Z=0 Z
for
0
then point Xe(Z = 0) is an unstable equilibrium point. `.<
f-+
This approach is called Lyapunov's second method, or called Lyapunov's direct method, which is considered as the very important method to assess the stability of nonlinear systems.
mil'
It should be pointed out that from the viewpoint of the security operation of power systems, there is still a problem related to the stability of power systems, that is the complex nonlinear characteristics such as some aperiodic and seemingly random electromechanical oscillations that occur suddenly or fitfully in power systems. There are some practical examples in the operating records of power systems in China and in other countries. As reported in the reference [54], the American northwest and southwest power systems had worked normally :n-
for a short period after they were interconnected, but before long they
'C7
'^t
O..
CD,
oscillated 6 times in one minute, which was not the sorts of low-frequency oscillation observed by the operators and finally resulted in separating the system into two parts. Such kind of oscillation was also observed in power systems in Canada and the South Scotland. A sort of random oscillation has also been observed on the interconnecting lines of some interconnected power systems in China. At first, the operators and researchers thought this (IQ
tea.
a)=
vii
was the result of lacking damping torque, but when the power system stabilizers were equipped to strengthen the damping effects of the power system, the oscillation still occurred occasionally. Nowadays, with certain knowledge of nonlinear systems, it is realized that besides the low-frequency
oscillation caused by negative damping ratio, there exists another crisis of (DD
aperiodic oscillation in power systems.
It is easy to understand that since the complex nonlinear phenomena determined by the nonlinear
such as the aperiodic oscillation is
characteristics of power systems, it can hardly counteract this sort of
0-0
P',
oscillation by using the techniques of linear control, such as linear feedback excitation controller using single supplementary feedback variable (PSS) or multi-variable linear controller (LOEC). Instead, some nonlinear control may be used to counteract or compensate the nonlinear characteristics of the
22
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS i..
power system, which could possibly eliminate the aperiodic oscillation in power systems. What will be investigated in the following chapters and sections in this book is the design method of such nonlinear controllers, with emphasis on how to find and apply the proper nonlinear state feedback control such as to exactly linearize the power system under consideration. In order to eliminate the random oscillation in power systems, the role to be played by nonlinear control can not be replaced by that of any linear control.
t]»
"CJ
t].
The importance of using nonlinear control has many facets, of course, avoidance of the aperiodic oscillation in power systems discussed here is only one of them.
REFERENCES
1.6
2. 3.
International Journal of Robust and Nonlinear Control, Vol.4, pp. 553-574, 1994. A. Isidori, Nonlinear Control Systems: An Introduction (31 Edition), Springer-Verlag, New York, 1995. A. Isidori and A. Astolfi, "Disturbance Attenuation and H. Control via Measurement Feedback in Nonlinear Systems", IEEE Trans. AC, Vol. 37, No. 10, pp.1283-1293, COD
4.
A. A. Fouad and Vijay Vittal, Power System Transient Stability Analysis Using the Transient Energy Function Method. Prentice -Hall, 1992 A. Isidori, "H Control via Measurement Feedback for Affine Nonlinear Systems",
o<.
1.
1992.
.y. p(7
o°°°°
"..
0.,
:-l
9. 10.
0.1
8.
y.,
7.
'ate
6.
A. K. Bejczy, T. J. Tam and Y. L. Chen, "Robot Arm Dynamic Control by Computer", IEEE Int. Conf. Robotics and Ant., St. Louis,1985. B. Jakubczyk and W. Respondek, "On Linearization of Control Systems", Bull. Acad. Polon. Sci., Math., 1980. C. A. King, J. W. Chapman and M. D. Ilic, "Feedback Linearizing Excitation Controller on a Full-scale Power System Model", IEEE Trans. PWRS, Vol. 9, pp. 1102-1109, 1994 D. Cheng, T. J. Tam and A. Isidori, "Global Linearization of Nonlinear Systems Via Feedback", IEEE Trans. AC, Vol. 30, No. 8, pp. 808-811, 1985. D. P. Atherton, Stability of Nonlinear Systems, John Wiley, NewYork, 1981.
Off.
5.
F. P. Demello and C. Concordia, "Concepts of Synchronous Machine Stability as Affected by Excitation Control", IEEE Trans. Power Appar. Syst., pp. 316-329, April, 1969.
11.
F. W. Warner, Foundations of Differential Manifolds and Its Lie Groups, Scott, Foresman, Glenview, 1970.
12.
-.,
G. Meyer, L. R. Hunt and R. Su, "Design of a Helicopter Autopilot by Means of Linearizing Transformations", Guidance and Control Panel 35' Symposium, Lisbon, Portugel (AGARD Conf. Proc. No. 321), 1983.
13.
G. Meyer, R. Su and L. R. Hunt, "Application of Nonlinear Transformation to
14.
Automatic Flight Control", Automatic, Vol. 20, No.1, 1984. H. D. Chiang, M. W. Hirsch and F. F. Wu, "Stability Regions of Nonlinear Autonomous Dynamical Systems", IEEE Trans. AC, Vol. 33, No. 1, pp.16-27, 1988.
15.
H. J. Sussmann, "Lie bracket, real analyticity and geometric control", in Geometric
16.
theory of nonlinear control systems, Birkhauser, Boston, pp. 1-116, 1983. H. Kwakernak and R. Sivan, Linear Optimal Control Systems, Wiley, New York, 1972.
23
Introduction
18.
Qtr
17.
H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamic Control Systems, SpringerVerlag, New York, 1990. H. S. Tsien, Engineering Cybernetics, New York, McGraw-Hill Book Company, Inc., 1958.
I. M. Mareels and D. J. Hill, "Monotone Stability of Nonlinear Feedback Systems", Journal of Mathematical Systems, Estimation, and Control,. Vol. 2, No. 3, pp. 275-291,
20.
J. H. Anderson, "The Control of A Synchronous Machine Using Optimal Control
.w+
19.
1992.
"R.
coo
000
"."
25. 26.
P0.
afro
1F.
24.
'No
23.
Theory", Proc. IEEE 90, pp. 25-35, 1971 J. Hammer, "Nonlinear Systems, Stabilization, and Coprimeness", Int. J. Contr. Vol.42, pp. 1-20, 1985. J. M. Coron, "Linearized Control Systems and Applications to Smooth Stabilization", SIAM J. Control Optim. Vol, 32, pp. 358-386,1994. J. Wang, W. J. Rugh, "Feedback Linearization Families for Nonlinear Systems", IEEE Trans. AC, Vol. 32, pp.935-940, 1987. J. Zaborszky, G. Huang, B. Zheng and T. C. Leung, "On the Phase Portrait of a Class of Large Nonlinear Dynamic Systems Such As the Power System", IEEE Trans. AC, Vol. 33, No. 1, pp.4-15, 1988. M. A. Pai, Power System Stability, New York: North Holland, 1981. M. Klein, G. J. Rogers, S. Moorty, and P. Kundur. "Analytical Investigation of Factors
mix-
22.
.;H.; a-:
21.
'.0
coo
Influencing Power System Stabilizer Performance", IEEE EC, Vol. 7, pp.382-388, 1992.
28. 29.
M. Spivk, A Comprehensive Introduction to Differential Geometry, Vol. I, Publish or Peremish, Boston, 1970. M. Vidyasagar, Nonlinear Systems Analysis, Englewood Cliffs, NJ: Prentice-Hall,1978.
M. Zribi, J. Chiasson, "Exact Linearization Control of a PM Stepper Motor", Proc. coo
27.
U°°
eon
oho °'"
.-.
N. Kalouptsidis and J. Tsinias, "Stability Improvement of Nonlinear Systems by Feedback", IEEE Trans. AC, Vol. 29, pp. 364-367, 1984. P. Kunder, Power System Stability and Control, McGraw-Hill, Inc. 1994.
0.0
31. 32.
epU
American Control Conference, Pittsburgh, 1989. 30.
P. M Anderson. and A. A. Fouad, Power System Control and Stability, IEEE Press,
34.
Q. Lu, S. Mei, T. Shen and W. Hu, "Recursive Design of Nonlinear H. Excitation
II!
Controller", Science in China(series E), Vol. 43, No, I, pp23-31, 2000. Q. Lu, S. Mei, W. Hu and Y. H. Song, "Decentralized Nonlinear H. Excitation Control Based on Regulation Linearization", IEE Proc-Gener. Transm. Distrib., Vol 147, No. 4, pp245-251, 2000. 'r-
35.
u<+3
New York, 1994. Q. Lu, and Y. Sun, "Nonlinear Stabilizing Control of Multimachine Systems", IEEE Trans. PES, Vol. 4, No.1, pp. 236-241, 1989.
aim
33.
'c3
37.
0.U
Application, Brussels, Sept., 1988. Q. Lu, Y. Sun, Z. Xu and Y. Mochizuki, "Decentralized Nonlinear Optimal Excitation Control". IEEE Trans. PWRS, Vol. 11, No. 4, pp. 1957-1962, 1996. R. Bellman, Dynamic Programming, Princeton University Press, Princeton, 1957. R. Bellman, Introduction to Matrix Analysis, McGraw-Hill Book Company, New York, 0.0
1960.
40.
E^.
38. 39.
Q. Lu, Y. Sun and Gordon K.F. Lee, "Nonlinear Optimal Excitation Control For Multimachine Systems", IFAC Symposium on Power System Modeling and Control
4J=
36.
R. E. Kalman, "Mathematical Description of Linear Dynamical Systems", J. SIAM 'L3
Control. Ser. A, Vol. 1, pp. 152-192, 1963.
42.
Boa
R. Marino, "On the Largest Feedback Linearizable Subsystem", System & Control Letters, Vol. 6, pp.345-351, 1986. R. Marino, W. M. Boothby, D. L. Elliott, "Geometric Properties of Linearizable Control Systems", Math. Systems Theory, Vol. 18, pp. 97-123, 1985. a°,
41.
43. 44. 45.
'yam
24
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS R. Su, "On the Linear Equivalence of Nonlinear Systems", System & Control Letters, Vol. 2, pp.48-52, 1982. R. W. Brockett, "Nonlinear Systems and Differential Geometry", Proc. IEEE, Vol. 64, No.1, 1976. S. P. Banks, Mathematical Theories of Nonlinear Systems, Prentice Hall, Hertfordshire, 1988.
50.
51.
Control", IEEE Trans. AC, Vol. 33, No. 6, pp.770-784, 1992. W. A. Boothby, An Introduction to Differential Manifold and Riemannian Geometry, Academic, New York, 1975. W. Kang, "Nonlinear H. Control and Its Applications to Rigid Spacecraft", IEEE Trans. AC, Vol. 40, pp. 1281-1285, 1998.
0.i
49.
[i7
48.
`-r
47.
T. Basar, "Disturbance Attenuation in LTI Plants with Finite Horizon: Optimality of Nonlinear Controllers", Systems & Control Letters, Vol. 13, pp.183-191, 1989. T. J. Tam, A K Bejczy, A Isidori and Y L Chen, "Nonlinear Feedback in Robot Arm Control", Proc. 23' IEEE Conf. Dec. Contr., Las Vegas, 1984. Van Schaft, "L2 -Gain Analysis of Nonlinear Systems and Nonlinear State Feedback H.
Nib
46.
W. Mielczarski and A. Zajaczkowski, "Nonlinear Field Voltage Control of a a0.
Synchronous Generator Using Feedback Linearization", Automatica Vol. 30, pp.16251630, 1994.
!.1U1
v0.
"'^
Y. N. Yu, K. Vongsuriya and L. N. Wedman, "Application of an optimal Control vii
W.¢
Theory to a Power System", IEEE Trans. Power Appar. Syst., pp. 55-62, Jan., 1970. Y. Sun, Q. Lu and J. Gao, "A New Nonlinear Modulation Control for HVDC Power
0.o
56.
arm
54. 55.
fir.
53.
W. Respondek, "Linearization, feedback and Lie Brackets", in Geometric theory of nonlinear control systems, Technical University of Wroclaw, Poland, pp.131-166, 1985. W. Respondek, Geometric Methods in Linearization Nonlinear Systems in Mathematical Control Theory, Banach Center Publications, Polish Scientific Publishers, Warsaw, pp. 453-467, 1985. Y. N. Yu, Electric Power System Dynamics, Academic Press, 1983.
Transmission Systems", CSEE/IEEE International Conference on Power System U7°
52.
'°'
Technology, Beijing, Sept., 1991.
Chapter 2
Basic Concepts of Nonlinear Control Theory
INTRODUCTION
2.1
Each scientific branch has its own specific basic concepts, which show extraordinary significance because they are the basic elements and components of this theory and the essential part of the theoretical framework.
'off
D'°,
°r,'
::r
.fl
In modem linear control system theory, the basic concepts include dynamic control systems, inputs and outputs, feedback, state variables and state vectors, state space and state equations, dynamic responses and state trajectory, stability, reachability, controllability and observability, performance index, optimal control, and the basic concepts of linear algebra, etc. All these basic concepts that readers may have profoundly grasped form the foundation not only for the modem control system theory but also for the nonlinear control theory studied in this book. Besides the above, there are some specific concepts and definitions of nonlinear control theory. It is the primary objective to clarify them in this ---
chapter.
ova
-d""
First, the concepts of nonlinear coordinate transformation, nonlinear mapping in state space and diffeomorphism are illustrated in comparison with those in linear systems so as to get a deeper understanding. The next concept is the affine nonlinear control system, which is the most common and important type of nonlinear control system in applications and is the primary focus of our research. Following that, vector fields in state space and derivative calculation of
vector fields, i.e. Lie derivative and Lie bracket, will be further discussed. The concepts and definitions of the vector fields, Lie derivative and Lie bracket are equally important to nonlinear control system theory as those of
26
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
(DD
the functions and derivative functions to calculus. With the vector field and Lie bracket it is possible to discuss the concept of involutivity which is a very important property of vector field sets and will be used in the condition of exact linearization of nonlinear control systems. To discuss the issue of
exact linearization we will illustrate the concept of relative degree of a control system first and then discuss the normal form of linearized nonlinear control systems. It should be noted that some, notations introduced in this chapter are both the carriers and presentations of those basic concepts and definitions, some of them may be unfamiliar to readers. As an important step to master r.3
the nonlinear control theory, it is necessary to understand and become CAD
04)
familiar with these notations in the process of grasping those basic concepts and definitions [1]. The internal relations of these concepts will be revealed at the end of this chapter.
2.2
COORDINATE TRANSFORMATION OF NONLINEAR SYSTEMS
2.2.1
General Concepts of Coordinate Transformation o-3,
t".
The method of coordinate transformation is often used and is not unfamiliar to us. For example, both Cartesian coordinates and polar coordinates can be used to describe a particle's movement in a space. There
o5'
CAD
exists a deterministic relation between them, and this relation is the CS'
O+3
..r SFr
coordinate transformation. Another example is the mathematical description of the dynamics of a synchronous generator. Either a, b, c coordinates fixed on the generator's stator or d, q, 0 coordinates fixed on the rotor can be used, the deterministic relation of which is the coordinate transformation between them. It needs to be emphasized that the transformation relation between two coordinate systems has to be invertible and deterministic to be useful. Let us exemplify this as follows. From Fig. 2.1 one can see that the variables of a generator presented in a, b, c coordinates and d, q, 0 coordinates have the following relation. sty
..O
Iago =TIQk and
(2.1)
Basic Concepts of Nonlinear Control Theory
Figure 2.1
27
a, b, c coordinates and d, q, 0 coordinates of a synchronous generator
labs = T-'Idqo
(2.2)
where T is the transfer matrix, i.e. cosy
2
cosy - z)
cos(y +
3
3
- sin y - sin(y -
2 jr)
2
- sin(y + 7r)
3
1
2
2
NIA
,__,
1
2
(2.3)
3
.-.
_ 2
,r)
3
'vi
T
2
1
"l7
where y is the included angle between positive directions of a-axis and daxis (also shown in Fig. 2.1). The inverse matrix of T is
-sing
cosy 2
T-' _ cosy - 3 ,r)
cosy +
2
- sin(y -
ir) - sin(y +
3
1
2
7r)
1
v,r)
1
3
2 3
From Eqs. (2.1) to (2.4) we know that a specific current value in a, b, c coordinates at a certain time corresponds to a specific current value in d, q, 0 coordinates, and vice versa. That is id = W 1 \ia, ib,
'I
is
(V2(ta, 1b, ic)
l0 = W3(1a, tb, lc)
(2.5a)
28
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
and ', = ql (td, jq. 10 )
(2.5b)
lb = 91(id, l9, i0) is ` g1(id, lq, t0 )
where yrl, w21 yr, and vi, cG2, ) are continuous, differentiable and singlevalued function. In other words, the points in these two discussed coordinate systems are "bi-directional one-to-one correspondence". This transformation
will fail to hold, if matrix T is not invertible. Hence the transformational invertibility and bi-directional one-to-one correspondence between the two coordinate transformations are the preconditions of discussion, without which the transformation will be meaningless.
2.2.2
Coordinate Transformation of Linear Systems
In linear control system theory considerations are generally given to linear coordinate transformation only. Given a linear system
X=AX+BU
(2.6)
Y = CX
where X is an n-dimensional state vector (X a R"), U a r-dimensional control vector (U a R'), Y an m-dimensional output vector (Y E Rw) ; A, 40-
B and C are constant matrixes with the corresponding dimensions. Assume that coordinate transformation is needed for some purpose. Set
Z = TX where Z e R", and T is an n x n constant matrix.
(2.7)
In order to replace X in Eq. (2.6) with the new coordinates Z in Eq. (2.7), the following steps are taken. First, from Eq. (2.7) we obtain
+U,
X = T-'Z (2.8) then substituting Eq. (2.8) into the original state equations (2.6), we obtain state equations with the new state variables Z
Z=AZ+BU Y=CZ
(29)
where
A=TAT ' B = TB
C =CT-'
(2.10)
Basic Concepts of Nonlinear Control Theory
29
Obviously, Eq. (2.8) holds only when T is a nonsingular matrix, i.e. it is invertible.
2.2.3
Nonlinear Coordinate Transformation and Diffeomorphism
As we have seen in the preceding section, in the case of a linear system, only linear coordinate transformation is usually considered. If the system is nonlinear, it is more meaningful to consider nonlinear coordinate transformation. Nonlinear coordinate transformation can be described in the form as
Z = fi(X)
(2.11)
where Z and X are vectors of equal dimensions, cD is a nonlinear vector function which can be expanded as ql1(xl,x2,...,xn)
Z1 =
Z2 = 02(xl a x2,..., xn)
Z. =
Pn(x1,x2,...,x
(2.12)
)
The first condition we assume for the nonlinear coordinate transformation in Eq. (2.12) is that its inverse transformation exists and is single-valued, i.e. (2.13)
X=fi-'(Z)
The second condition is that both c(X) and (D-1 (Z) are smooth vector functions, i.e. the function of each component of both cD and c-' has continuous partial derivatives of any order'). In short, the first condition is invertible and the second is differentiable. Of course, it is unnecessary to mention the second condition (Cx) especially for a linear transformation ..-.
since this condition is met naturally.
If these two conditions are satisfied the expression Z = c(X) should be 0
:ti
Lbw
a valid coordinate transformation and this expression c(X) is called a diffeomorphism between two coordinate spaces. For example, consider the following transformation Z1
Z2 Z3
x3
-(D(xl,x2,x3)=
x2
I+x1 -ex'
') A function fulfilling this conditions is called a smooth or C° function.
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
30
This function is single-valued and smooth for all points in space X E R' and its inverse transformation exists in the form as -1+z3 +e`' x2
z2
x3
z,
X,
is a which is single-valued and smooth as well. Accordingly diffeomorphism between X space and Z space and Z = (D(X) is a valid transformation.
2.2.4
Mapping
From a geometric viewpoint the coordinate transformations Z = o(X) and x = (D-' (Z) can be regarded as a mapping between two spaces with same dimension X and Z (see also Fig. 2.2). Therefore coordinate transform-
Figure 2.2 Mappings between X coordinate system and Z coordinate system
ation is referred to as a mapping as well. If this transformation is a diffeomorphism, then the mapping is also called a diffeomorphism mapping, or diffeomorphism for short.
2.2.5
Local Diffeomorphism 'LS
From previous description one can see that a valid coordinate transformation (or mapping) is surely a diffeomorphism. However, the necessary and sufficient conditions mentioned above that a diffeomorphism must be satisfied may hold only for a neighborhood of a specific point X° rather than all points in the space (any X E R" ). The phrase neighborhood here
Basic Concepts of Nonlinear Control Theory
31
does not only refer to a very small part (subspace). It may be sufficiently large. But we can just call it a local diffeomorphism so long as it is not defined on the whole space no matter how large it is. In engineering field what required are often local diffeomorphisms to solve problems. Let us take the following transformation for instance x, 1
[z'](D(x,,x2)
x2
Z2
- x, +1
The function cD(x,, x2) is not defined on the line x, = -1. So we define
it on the domain U = I x > -1) where the function c(X) is singlevalued and smooth, and the existing inverse function is single-valued, smooth as well, which can be denoted as z,
[:']=-'(ZZ2)=[Z
2
+ z,
Thus we know that the above expression Z = cD(X) is a local coordinate transformation or a local diffeomorphism defined on a certain domain. How to test whether a nonlinear mapping d>(X) is a local
diffeomorphism at X°? The answer is that (D(X) is a local diffeomorphism in an appropriate neighborhood-of X° including X° if the vector function c(X) has a nonsingular Jacobian matrix at X°. That is to say, it is a valid coordinate transformation or mapping. Let us present the property of the above nonlinear function by a proposition.
Proposition 2.1 Suppose c1(X) is a smooth function defined on a certain subset U of space R". If the Jacobian matrix at X= X° a0p,
ar,
ax2
awe
a
ax
F ap,
ax
awl
...
ate,
ax awl
&2
X=X°
L ax,
ax2
axn
X=X°
is nonsingular, c1(X) is then a local diffeomorphism in an open subset including X°. Now let us illustrate it with an example. Example 2.1 Consider the following function
U°
32
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
[z2]-(D(x"x2)-[sinx221
which is defined for all points in space R2 . Its Jacobian matrix _ rl
ax
o
cosx2
is nonsingular at X° =[0 0]T and has rank 2. Hence this function cp(X) is a
local diffeomorphism in a subset including x° = [0 O]''. What can be confirmed further is that this open subset including X° is
U°_{(x,,x2)IIx2I <2} since the inverse function of cD(X) is single-valued (and smooth) on this subset.
2.2.6
Coordinate Transformation of Nonlinear Control Systems
Given a single-input single-output system as xi
x2 =A(XII...Ix")+92(x1
(2.14)
x"
and the output equation
y=h(xl,...,x") which can be written in the compact form as
X = f(X)+g(X)u
(2.15)
y = h(X)
where X E R" and u e R are state variables and control variable respectively, y e R denotes the output variable, f and g are nonlinear function vectors and h is a nonlinear function of X. Choose a coordinate transformation as (2.16)
Z = cb(X)
Its derivative with respect to variable t will be
dc(X)=ab(X)dX dt
aX
dt
(2.17)
33
Basic Concepts of Nonlinear Control Theory
Substituting Eq. (2.15) into (2.17) for X , we have
Z=
a(D (X)
ax y=h(X)
(f(X)+ g(X)u)
(2.18)
Assuming CD(X) is a diffeomorphism, then we can get from Eq. (2.16) that
(2.19)
X=(D-'(Z)
According to Eq. (2.19), a Z-coordinate system model can be obtained by substituting X with Z in Eq. (2.18), i.e.
Z = f(Z) + g(Z)u y = h(Z)
(2.20)
where
f(Z)
a a(X) f(X)
Ix-m (Z]
g(Z) = a a(X) g(X)
(2.21)
x=m (z)
and
h(Z) = h(X) Ix=,-'(Z)
Here Eqs. (2.20) and (2.21) represent the common form of coordinate transformation of a nonlinear control system as Eq. (2.15).
2.3
AFFINE NONLINEAR CONTROL SYSTEMS vii
In the engineering world, many nonlinear systems such as power systems, robot control systems, helicopter control systems and chemical 4)'
control systems etc., have the following form of state equations fl(xl,...,xn)+911(xl,...,xn)ul +...+gm1(xi,...,xn)um xl = x2 =
f2(x1,...,xn)+g12(x1,...,xn)u1 +...+gm2(xl,...,xn)um
xn =
fn(x],...,xn)+gln(xl,...,xn)ul +...+gmn(xl,...,xn)um
(2.22a)
and output equations y1
=h,(x,,... x»)
(2.22b)
ym =hm(x...... xn)
which can be written in the compact form as
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
34
X(t) = f(X(t)) + Y_ M g; (X(t))u; (t)
(2.23)
Y(t) = h(X(t))
`f'
where X E R" is state vector, u ; (i =1, , m) control variables, h(X) mdimensional output function vector; f(X) and g; (X) (i =1, , m) are ndimensional function vectors. A nonlinear control system, just like Eq. (2.22) or (2.23), possessing the feature that it is nonlinear to state vector X(t) but linear to control variables u, (i =1, , m) , is called an affine nonlinear system.
Cad
Example 2.2 Consider a one-machine infinite bus system as shown in Fig. 1.3. The system equations are given by Eq.(1.30). Comparing Eq. (1.30) with Eq. (2.23)'we know that
X =co['] (t)
(2.24)
f(X) = - a, sin S(t)] g(X) =
]
1
(2.25a) (2.25b)
2
L
(2.26)
U = M.
Now we realize that the power system modeled by Eq. (1.30) is an affine nonlinear system.
2.4
VECTOR FIELDS
r'-
In this section the concept of vector fields will be discussed. In Eq. (2.23) f(X) is an n-dimensional smooth vector function, i.e.
f(X)=
.f2(x,>...,x,)
(2.27)
Each component of f(X) is a smooth function of variables X = [x, x I' . Then we know that each specific point in the state space corresponds to a certain smooth vector at this point
35
Basic Concepts of Nonlinear Control Theory
f(XO)=[fi(Xl) ... f,,(X°))T
'-h
f(X) is called a vector field of the state space. Likewise g; (X) = [g, (X) g;" (X)]T, i =1, , m in Eq. (2.23) is a vector field as well. A vector field of variables X of a space is a rule f that assigns to each point in the space a fixed vector of f(X). For instance,
x3(1+x2) CD (X) =
(2.28)
x1
x2(1+x,) and 0
(2.29)
'Y(X) = 1 +x2 - x3
are two vector fields defined on X e R3 . According to the concepts 'C3
illustrated in Section 2.2, a vector field can also be taken as a coordinate transformation or a mapping from R" to R". To explain it more clearly, we change Eqs. (2.28) and (2.29) as x3 (1 + x2 )
Z=cb(X)=
(2.30)
x,
1x2(1+x,) an d 0
=`Y(X)= 1+x2
(2.31)
- x3
From the above it is immediately evident that cb(X) and T(X) here indicate coordinate transformations or mappings from space X ER 3 to Z ER 3 and to 8 E R3, respectively. Therefore, in light of the concept of coordinate transformation discussed in Section 2.2, the vector fields that we 'C3
discussed should be at least a local diffeomorphism. It should be noticed that just as a constant can be taken as the particular
example of a function, a constant vector can also be taken as a vector field that does not vary with points of the space. We are, however, not unfamiliar with the concept of vector field. It can
be thought as the geometric view of what commonly referred to as the smooth vector function in a state space.
O0)
36
2.5
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
DERIVED MAPPING OF VECTOR FIELDS
Derived mapping of a vector field plays an important role in exactlylinearized algorithms. Its definition and concept will be discussed in this v°4
section. Suppose CD is a same-dimensional diffeomorphism mapping from an
n-dimensional space in X coordinates to that in Z coordinates such that (2.32)
Z = CD(X)
CD:
i.e. ZI =q
SD:
V 2 (xl
, x2 , ...,
(2.33)
Z And a vector field in X space is given as fi (XI, X2 1 ... , X")
f(X)=
(2.34)
f, (X, x2,..., CDR
so_
Under the mapping CD expressed by Eq. (2.33), the mapping of the vector field f(X) from X space to Z space is called the derived mapping ,.s
of f(X) under the mapping CD and denoted as C,(f). To put it more , the derived mapping of vector field f(X), is a type of
plainly, cD, (f)
::-
transformation and operation that moves the vector field f(X) from X space to Z space based on the transformation Z = (D(X) .
With the above explanation we can give the following rigorous definition of derived mapping of vector fields now.
Definition 2.1
Given a diffeomorphic coordinate mapping CD :
Z=C(X)
ZI =(01(xI,x2,' xn) Z2 = V2(x1, x2,...,xh) Zn =
and a vector field in X space
-Pn(x1,x2,...,xn)
Basic Concepts of Nonlinear Control Theory
37
fl(XI, x2,...,x.)
f(X)= fn (XI, X2,..., Xn)
Let J1 be the Jacobian matrix of f(X) , i.e. a4p,
ax ...
Jm = acD(X)iaX =
&I
ax.
E"'
Then 4).(f), the derived mapping of f(X) under the mapping (D, is defined as
(2.35)
(D.(f)=J ,(X)f(X)IX=m-'(z)
From Definition 2.1 we know that the derived mapping of f(X), w-'
(D. (f) is a new vector field of Z coordinate.
Example 2.3
Given a space mapping z = cD(X)
z, = x, +x2 -1
(1):
(2.36)
z2 = x2 - 1
23 =x1+x2+x3-2 cad
and a vector field -I
f(X) =
(2.37)
1
2x,x3 + 2x2x3
How to calculate the derived mapping of f(X), 4).(f)? According to the Definition 2.1, we first seek for J,,
' - ax
_
1
1
0
0
1
0
1
1
2x3'
(2 . 38) ono
J
ac(X)
.7i
Hence 1
1
0
0
1
0
1
1
0
Jmf(X)=
-x3 2
1
2x3 (x, +x2)
X, +x2j
(2.39)
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
38
Then the inverse mapping of 0, V' is x, = z, - z 2
(2.40)
x2 = z2 + 1
x3 = (Z3 - Z, + 1)2
As a result, the desired derived mapping is 10.(f) = J0(X)f(X) 1x=4,-1(Z) 0 1
Z+
LIE DERIVATIVE AND LIE BRACKET
2.6
arm
In the study of nonlinear control systems, we can not go without the concepts and operations of Lie derivative and Lie bracket, which are one of the essential geometric methods of nonlinear systems. They will be clarified as follows.
2.6.1
Lie Derivative
Given a differentiable scalar function of X A(X) = A,(x x...... X")
and a vector field f1(xl,x2,...,xn) .f2(xl,x2,...,xn)
f(X) _
fn (xl , x2 > ... , xnv)
How to define the derivative of this scalar function 2(X) along the vector field f(X) ? The answer is that the derivative is defined as the scalar product (point product or inner product) between VA(X) and f(X), where V2(X) is the gradient of the function 2(X) VA(X) = I
That is
ap (x)
as (X)
ax
axe
...
a 2(X)
ax
Basic Concepts of Nonlinear Control Theory
39
or
< aA(X),f(X) >
ax
< V2(X),f(X) >
(2.41)
This formula defines a new scalar function which is called the Lie derivative of 2(X) along f(X) and denoted by L f2(X) . Now we give the precise definition of Lie derivative.
Given a differentiable scalar function 2(X) of and a vector field f(X)=[f, the new scalar
Definition 2.2
X=[x, ... function, denoted by L12(X) j s obtained by the following operation
Lf2(X)= a2(X)f(X)-Ya2,(X) f; (X) ax
ax;
(2.42)
and called the Lie derivative of function A(X) along the vector field f(X).
Example 2.4 the vector fields
Find the Lie derivative of the function h(X) = x3 along e'!
0
and
g(x) = e.'
f (X) =
,
..y
0
x+ x, -x2 y..,
According to the definition of Lie derivative we first obtain the gradient vector of the scalar function h(X) = x, as Oh(X) = a(x3) = [0
0
ax
ax
1]
The desired Lie derivatives are er2
Lgh(X)
aaX)g(X)=[0
0
1]I ex,
I=0
Lfh(X)_[0
r-,
0
0
1]
0
x, +x2 =x,-x2 El XI -x2
From Definition 2.2, we know that Lie derivative is a scalar function, so it is possible to repeat use of this operation to obtain the Lie derivative L f2(X) along another vector field g(x), i.e.
LgL f
(X ) =
a(L,2(x)) ax
g(x)
(2.43)
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
40
Certainly, we can also derive the k`' order Lie derivative of 2(X) along f(X) recursively as
L f(L fi(x)) = L fi(x) 2
=
a(L fA (x)) ax f(x)
(2.44) L
f (x)
The k'"-order Lie derivative of 2(X) along f(X), L2(X) is still a scalar function and thus can be used to get Lie derivative along another vector field g(X), i.e. k
LgLf2(X) =
a(Lk2(X))
ax
(2.45)
g(X)
Let us use an example to show the method of obtaining high order Lie derivative.
Example 2.5
Given a scalar function
h(X)=x3 and vector fields 0
f(X) = x, +x2 x, -x2
ex,
g(X) = ex'
and
e X3
find high order Lie derivative of h(X) along f(X) and g(X)
.
According to Definition 2.2, we obtain
Lfh(x)= a(h(X)) f(x)=[o ax
0
1]
x, I x1
= x, - x2
From formula (2.44) we know that LZfh(X)=
a(Lfh(x)) f(X) = a(x, - x2)f(x) ax ax 0
=[I -1 0] x, + x2 Lxi - x2 J
-(x, +X2 )
With the above second-order Lie derivative of h(X) along f(X), we can further obtain
Basic Concepts of Nonlinear Control Theory
a(
LJh(X)
ax
41
x2) X + X 2 X, -X ' 0
=[-I
0] x, +x2
-2x2
x, - x2
=-2x2(x, +x2) According to formula (2.45) we obtain 2
L9Lfh(X) =
a(- x, - x2 )
ax
= -er' -
2.6.2
g(X) = [-1
- 2x2
2x2exi
Lie Bracket
Assume two vector fields of the same dimension are given as
fi(x...... x,,)l .f2(xl,...,x»)
g(X) =
and
I
III
f(X) =
frr(xl,...,x,r)
mss'
fit
How to define the derivative of the vector field g(X) along the other vector field f(X) ? What should be clarified first is that the derivative of one vector field along another is still a vector field rather than a scalar function. And it is defined as follows. Let [f(X),g(X)] denote the derivative of g(X) along f(X), then ''b
ag,
L'9 2
ag2
x
cps
If, g1(X) = ax,
,r
ax
of
of
cps
ag,
rf,
fi
a
f2
al2
,r
...
- ax
fig, l g2
ax,r
(2.46)
...
Oxag'
g In
...
ax,,
af, a,
afn axn
...
gn
which can be abbreviated as Vet
If, g1(X) = ax
f ax
(2.47) g
where ag/aX and of/aX are the corresponding Jacobian matrixes.
42
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Expression (2.47) shows a new vector field, namely Lie bracket, for its accustomed denotation [f, g], which can also be denoted by adfg . The precise definition of Lie bracket is given below.
Definition 2.3 Suppose two vector fields f(X) = [f f2 - f ]T and g ]T. The following operation denoted by [f(X), g(X)] or g(X) _ [g, g2 adfg
agf-a ax
[f,g]=adfg=
g
obtains a new vector field which defines the Lie bracket of g(X) along f(X). `C3
E-'
The following two examples are presented to explain the computing
method.
Example 2.6
...
Given vector fields 0
I X3(1+X2)
+X1 )- x3
f(X)=
g(X)= 1+x2
and
x, 1 x2(1
compute `".
[f(X), g(X)] = ad fg(X) C"'
According to the definition of Lie bracket, we first figure out the ,ti
Jacobian matrixes of f(X) and g(X) I
f(X) ax
f
J s= ag(X) ax
O2
0
0
x2
1+X,
0
0
0
1
0
0
0
-1
0
0
From Definition 2.3
[f(X), g(X)] = ad fg(X) = 0
=0 0
a gm f(X) - af(X) g(X) ax ax
0
0
x3(1+x2)
0
x3
1+x2
0
1
0
x,
1
0
0
1+x2
x2
1+x,
0
-x3
-1 X2(1+x1 )
0 0
X,
-x2(1+x1 )
0
-
0
(1+x,)(1+x2)
43
Basic Concepts of Nonlinear Control Theory 0 X1
-(1+x,)(1+2x2)
Since the Lie bracket of g(X) along f(X) is a new vector field, it can be used to calculate the Lie bracket along f(X) once more. Similarly we can find k"` order Lie bracket inductively as ad f g(X) = [f,[f,g]](X) (2.48)
ad jg(X) _ If, ad j'g](X) Suppose two vector fields f(X) and g(X) are as in Example 2.6, calculate ad 2g . From Eq. (2.48) we know that nPO
Example 2.7
ad fg(X) = [f(X),ad,-g(X)] y!'
f(X) - af(X) ax (ad fg(X )) 0^Q
ax
AIM
= a(ad fg(X ))
From Example 2.6, we have 0
ad f9 =
x,
-(1+x,)(1+2x2) Therefore 0
0
1
0
0
L- (1 + 2x2)
- 2(1 + x,)
0
7C)
adfg=
0
1 + x211
0
x3
1
0
0
x2
1 + x,
0
0
x3(1 + x2)
-2x,(l+x,)-x3(1+x2)(1+2X2) [x1x3 -(1+x1)(1+x2)(1+2x2)1 0
x1(1+x 1) - x1x3 + (1 + x1)(1 + x2)(1 + 2x2) x3 (1 + x2 )
-x3(1+x2)(1+2x2)-3x1(1+x,)
44
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
The following gives the main operational rules of Lie bracket. Rule 1 Lie bracket is skew symmetric, i.e. (2.49)
If, g] = -[g, f ] This can be shown from the definition clearly.
Rule 2
If c, and c2 are two real numbers, then
[f,c,g, +c2g2](X)=c,[f,g1](X)+c2If,g2](X) Rule 3
(2.50)
If p(X) is another vector field, then
If,[g,P]I+[P,If,g]I+[g,[P,f]]=0
(2.51)
,+y
Formulae (2.50) and (2.51) can be deduced from the definition directly. Rule 4 If f and g are vector fields, A(X) is a scalar function, then the Lie derivative of ,,(X) along vector field If (X), g(X)] is L1J.g]A(X) = LJLs'%(X) - LgL f..%(X)
(2.52)
O.=
We give the proof for Rule 4 below while leaving the proof for the other three rules to the reader as an exercise.
Proof for rule 4: From the definition of Lie derivative we have
ax ax
V'1
a.. (X) ag _ of f g) ox (aX ox - a2(X) ag f _ a,,(X) of
LtJ,gl't(X) =
(2.53)
ax ax g
According to the definition we know L9X
a f=X (ax g).f
...
a(L9A)
V'1
e(,
LJLgA(X)
(2.54)
f) - g
(2.55)
and
a(L 1 A) U.&
g=
aax ax (aa,
V'1
L g Lf A(X) =
The above two expressions can be further evolved as
a (aa,) + aA ag )f ax ax ax ax gTax(
L JL9A(x) = (gT
(2.56)
ax )f+axaxf
L 9 L fA(X) = f T
From these two formulae we get
a (aa. )g + aA of g ax ax
ax ax
(2.57)
can
45
Basic Concepts of Nonlinear Control Theory
L fLg2(X) - LgL fZ(X) =
aa. ag f - aa, of
ax ax
g
ax ax
(2.58)
Comparing formula (2.53) with formula (2.58) we have L(f g] A(X) = La,,gA(X) = LfL9A.(X)- L 9 L f %(X) Thus Rule 4 above has been proved.
These rules about Lie derivative and Lie bracket yield a kind of algebraic operation, i.e. Lie algebra. According to Rule 1,
If M, g(X)] # [g(X), f(X)] Thus we know that Lie algebra does not satisfy commutative law and is a type of non-commutative algebra.
2.7
INVOLUTIVITY OF VECTOR FIELD SETS
c0)
Topics in this section will be: first, what is involutivity; second, how to comprehend its geometrical meaning. Let k n-dimensional vector fields be of the following forms 9k1(x11...1x11)
g11
gk(X)=
g1(X)= to form the matrix g11 g21 ... gkl `.Y
G = g12 922 ... 9k2
= [g1(X) g2(X) ... gk(X)]
(2.59)
gL, g2n ... gb,
If the matrix has rank k at X = X° and the augmented matrix [g1
g2
...
gk
1909,1 1
(2.60)
has the same rank k at X = X° to the arbitrary number pair i and j where I < i, j :- k , then the vector field set (2.61)
is called an involutive one or we say it has the property of involutivity. Sp {g 1, g 21 , g k } , the spanned space of the above vector field set is also called the distribution in geometry. Hence the distribution satisfying the above conditions is called an involutive distribution.
46
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
The definition of involutivity has been clarified so far, but how to understand it geometrically? Since the matrices in expressions (2.59) and (2.60) have an equal rank value, the new vector field obtained by Lie bracket operation of two arbitrary vector fields from the original k vector fields (i.e. evaluating the derivative of one vector field along the other) is not linearly independent but linearly dependent with the previous k vector fields. In other words, the new vector field [g,, gJ ] is still in the original spanned space of .-J
the k vector fields and does not form a new direction. This is just the .r..'
geometrical meaning of involutivity. S3.
It is very important to confirm the involutivity of vector field sets in nonlinear control systems. A simple example is used as follows to explain how to check the involutivity of a vector field set.
Example 2.8 Let the vector fields f(X) and g(X) be given as in Example 2.6. We need to check the involutivity of vector field set L."
{g(X),ad fg(X)} near X° =0. From Example 2.6 we know that 0
0
g(X)= I+x2
ad jg(X)=
x,
-(I+x,)(1+2x2)
-x3 At X = 0 , the matrix
0
0 .-.
[g(X) ad rg(X )] =
1
0
0
-1
has rank 2 (i.e. r=2).
To check the involutivity of {g(X), ad fg(X)} at X = 0 , we first calculate the Lie bracket of vector field ad fg along g at X = 0 [g, ad fg]=
0
0
0
0
I
0
0
1+x2
0
-x3
-(1+2x2) -2(1+x,) 0
0
0
0
-0
1
0
X,
0
0
At X = 0 , the augmented matrix
-1 -(1+x, )(1+2x2)
47
Basic Concepts of Nonlinear Control Theory
[g
ad fg
[g, ad fg] ] =
0
0
0
1
0 -1
-3
0
0
has the rank that still equals 2. Therefore the set {g, ad fg} is involutive.
2.8
RELATIVE DEGREE OF A CONTROL SYSTEM
Another important concept, relative degree of a control system denoted by r, is introduced in this section. Suppose a single-input single-output nonlinear control system
X = f(X)+g(X)u
(2.62)
y=h(X) where X E R", U E R, y r= R, f(X) and g(X) are vector fields. If
(i) The Lie derivative of the function Lfh(X) along g equals zero in a
neighborhood 52 of X = X ° , i.e. LgLfh(X)=0,
k
`dxErl
(ii) The Lie derivative of the function L't'h(X) along vector field g(X) is not equal to zero in 12 , i.e. LgL'f'h(X) :t 0 then this system is said to have relative degree r in 12
Example 2.9
Given a nonlinear system with state equations X
x2
-
[c,(1 - x? )x2 - c2x,
r0 + LIJu
where c, and c2 are constants and the output equation is
y=h(X)=x, Calculate the relative degree of the system. 5,P
According to the definition of relative degree, to ascertain the relative degree r, we first calculate the Lie derivative of the function of zero-order
Lie derivative of h(X) along f(X) along g and the result is
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
48
LgL°jh(X) = Lgh(X) = _
[1
0]
ahX) g(X) = [ax ,
ax z J[O]
[?] = 0
Then we compute LgL fh(X). First we calculate
Lfh(X)= Oh(X)f(X) x2
_ [1
0] [c, (1-
= x2
x;)x2 - c2x,
thus the following result is induced LgL fh(X)
a(Lfhh
X ))
g(X) = [0
1]
[?] =1
rte.
Hence the given system has the relative degree r = 2.
a0+
We have introduced the definition and calculating method of relative degree. An interesting question is: what is the characteristic of an affine nonlinear system if it has a relative degree that equals the state vector's dimension n ? Assume a system X = f(X) + g(X)u y = h(X)
(2.63)
where X E R", has relative degree r = n. The following expressions are true according to the definition:
LgL°fh(X)=LgLfh(X)=LgL2h(X)=
=LgL'f2h(X)=0
LgL'f'h(X) ;& 0
(2.64a) (2.64b)
We now construct a mapping from X space to Z space. If we choose z, = h (x, ... , x,,)
(2.65)
then
Substituting Eq. (2.63) into above equation for k we obtain
ah(X) f(X)+ ah(X) g(x)u
ax
ax
This formula can be rewritten according to the definition of Lie derivative as
Basic Concepts of Nonlinear Control Theory
it = L fh(X) + LgL°fh(X )u
49
(2.66)
From the formula (2.64), we know that in (2.66)
LgL°Jh(X) = 0 Therefore,
i, =Lfh(X) If setting
i, = L fh(X) = zz
(2.67)
then we have
iz = L fh(X) + LgL fh(X )u Since r = n, we also know from formula (2.64) that
LgLfh(X)=0 So
iz = LZfh(X ) Once again we set iz = L z fh(X) = z3
2.68)
By analogy, we can certainly obtain
i, = L fh(X) = z;.1
(2.69)
L'f'h(X) = z,,
(2.70)
until
Since r = n, things will change if we follow Eq. (2.70), i.e.
i,, = L'jh(X) + LgL'f'h(X )u
(2.71)
From the formula (2.64b), we know
LgL'f'h(X) # 0 Therefore Eq. (2.71) can also be written as
i = a(X) + b(X)u
2.72)
where
a(X) = L'fh(X)
b(X) = LgL'f'h(X) # 0
(2.73)
Integrating the contents from Eq. (2.63) through Eq. (2.73) yields an X-to-Z coordinate transformation and a Z-coordinates system. The chosen
50
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
coordinates are zl = h(X)
z2 =Lfh(X)
(2.74)
z = L'j'h(X) which can also be written as
Z = cb(X) where 1(X) is required to be a local diffeomorphism. This fact can be seen dL'f' h (X) are linearly independent. Thus the new dynamic system described by Z-coordinates is by showing dh (X), dL 1 h (X ),
,
Zl = z2 ci,
Z2 = z3
(2.75) zn-1 = Z
a =a(X)+b(X)u
X =(D -'(Z)
and the output equation is
y=h(X)=zl
fit
err
coo
The answer is clear now. If an SISO affine nonlinear system has relative degree r = n where n denotes the number of the system's order, then a coordinate mapping shown in the expressions (2.74) can transfer the original nonlinear system into that, as Eq. (2.75), in which the first (n -1) equations are linearized and do not include the control variable u apparently and only the last equation involving u is nonlinear. The above fact is very important
for the exact linearization of affine nonlinear systems and will be further discussed in subsequent sections.
2.9
LINEARIZED NORMAL FORM
The instance that an affine nonlinear system has relative degree r = n (n is the number of the system's order) has been discussed in the above section where the original system can be transformed into the form of Eq. (2.75) by the local change of coordinates shown in formula (2.74). In general, however, the relative degree r may not just equal n but r < n . In this section
we are going to explore that as r < n the original system model can be transformed into what kind of form by appropriate coordinate transformation. For the sake of completeness, let us consider a nonlinear system
Basic Concepts of Nonlinear Control Theory DC'
51
X = f(X)+g(X)u y = h(X)
where X e R" and the relative degree r is less than n. The mapping Z = i(X) will be chosen by the following steps. First we choose the first r components of the transformation as zl = pI (X) = h(X) Z2 = V2(X) = Lfh(X)
( 2 . 76 )
z, = tp,(X) = Lrf'h(X)
Then the remaining n - r components of the transformation are chosen as Zr+I ='Pr+I(X)
......
(2.77)
Z. = qn (X)
such that Lg.p; (X) = 0
r + 1<- i<_ n
(2.78)
holds. After the transformation as shown in formula (2.76) the first r equations of the original nonlinear system are transformed into
i; = L'fh(X)
i
(2.79)
In view of the definition of the relative degree of the system we have
LgL'f'h(X)=0
i
(2.80)
Considering the mapping in formula (2.76) we know that
i
(2.81)
i, = Lrfh(X) + LgLrf'h(X )u
(2.82)
i; = L'fh(X) = zi+I and the rch equation must be
a(Z) = Lrfh(X)
b(Z) = L., Lf'h(X)
(Z)) LgLrf'h((D-'(Z))
0 . As a
(2.83) 1w,1
From the definition of relative degree, we know that LgLf'h(X) result in Eq. (2.82) if we set
then Eq. (2.82) can be written as
i, = a(Z) + b(Z)u
(2.84)
Generally speaking a(Z) and b(Z) above are nonlinear functions of Z . Now let us consider the remaining n - r dynamic equations. It is clear from Eq. (2.77) that
LL')
52
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
z rtl _
aqr+l(X) f(X)+ alr+t(X) g(X)u ax ax
x=
zr+l = Lfp,+, (0-'(Z)) + Lg(0r+l ((D-' (Z))u
zr+i = Lf qr+i ((D-'(Z)) + Lg ', l ((D-' (Z))u
r+i
Z = Lf(o,, (ID -'(Z))+Lglp,, (ID -'(Z))u
(D-l(Z)
(2.85)
(2.86)
(2.87)
Since we have known from Eq. (2.78) that p,+,, , p,, satisfy
L1(X)=O
' r+l
From Eq. (2.85) through Eq. (2.87) we obtain Zr+l = Lf Vr+l ((D '(Z))
(2.88)
-'(Z))
z» =
To get the normalization form, we set in the above formulae that q,,, (Z) = Lfq',+l ((D -'(Z))
(2.89)
q,, (Z) = Lf-Pn((-'(Z))
thus the dynamic equations from the (r+1)' through the n' have the following forms Zr+l = qr+l (Z)
(2.90) Z = q,, (Z)
The above arguments yield the following
Proposition 2.2
Consider the system
X = f(X)+g(X)u y = h(X)
where X E R" and the relative degree r is less than n. If the mapping Z = O(X) is chosen as zl =fo1(X)=h(X) z2 = fp2(X) = Lfh(X )
Basic Concepts of Nonlinear Control Theory
53
Zr = cpr(X) = Lrf'h(X) Zr+I -qVr+1(X)
.'T
zn =q',,(X) in which (pr+,,
, q.
satisfy
Lg(p;(X)=0
r+l<-i<-n
and the Jacobian matrix at X=X°
J'= acD(x) ax
_Xa
o-!
is nonsingular, setting a(Z) = Lrfh((D-' (Z)) b(Z) = LgLr 'h((D-' (Z))
and qr+1 (Z) = L fVr+1 ((D -'(Z))
then the original nonlinear system can be transformed into the following form it = Z2 i2 = Z3
it = a(Z)+b(Z)u
(2.91)
it+l = qr+1 (Z)
i,, = q,, (Z)
The model of the system expressed as Eq. (2.91) is called a normal form.
This type of the normal form is important for designing nonlinear control systems which will be evident in the following chapters. Of course,
Eq. (2.75) in Section 2.7 shows a normal form as well but that is in the special case of r = n. The following example illustrates the whole process of finding the normal form of the transformed nonlinear system.
Example 2.10 Consider a system
y = 2x3
54
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
j
From the above we know
f(X)
x,
2e''
=
2x1x2
g(X)
h(X) = 2x3
1
Now we are going to find the system's normal form. Step 1. Figure out the relative degree r. So we calculate
ah(X) _ [0
ax
0
2]
0
2]
2e X2
ah(X)g-[0
Lgh(X) =
1
=0
0
then - x,
LgL jh(X) = Lg([0 0
2] 2x,x2 ) = Lg (2x2) = 2 # 0 x2
thus we have that r = 2 . Step 2. Choose a coordinate transformation Z = (D(X)
Z, =r'1(X)=h(X)=2x3 z2 = (p2(X) = Llh(X) = 2x2
then we choose the components of the transformation from z,+i through z , which is z,+, = z,, = z3 in this case. To find the normal form, we select z3 = (p3(X) so that Lg-P3 (X) =
-P1
aX
g(X) = 0
Thus a partial differential equation that can determine rp3 is drawn forth as ac'3 ax,
a(P3
aV3
ax2
ax3
r2ex'1
]
1
I=0
0
that is
a(0'2es'+aT' =0 ax,
ax2
It is not too difficult to find that the above equation has a solution
q3(X)=1+x,-2e"
55
Basic Concepts of Nonlinear Control Theory
Now the possible coordinate transformation Z = c(X) has been achieved z, = 2x3 z2 = 2x2
z3 = 1+x, -2e" Step 3. Check whether 0 (X) is a diffeomorphism. What we should do is to check the nonsingularity of the Jacobian matrix
ac(X)
J
ax
2]
0
= 0 1
-2e 2
00
From this formula we can see that the J. is nonsingular for any X in the state space. Therefore the chosen (t(X) is a valid mapping as required.
Step 4. Calculate the inverse transformation X = 4) -'(Z) . That is
x, =-1+z3+2e" 1
x2 = 2 z2
x3=2z, 1
Step 5. Write the new Z-coordinates system. So we first calculate a(Z) = L .h((D-' (Z)) = 2(-1 + z3 + b(Z) = LgL fh(cD -'(Z)) = 2
2e'"2 )z2
and then we obtain
i2 = 2(-l+z3 +2e'z2)z2 + 2u From Eq. (2.88), we have
i3 = Lf(P3l(')-' (Z)) = -x, - 4x,x2e"
Jx=ro-,(z)
that is .IN
-1n
i3 = (1- z3 - 2e''' )(1 + 2z2e'`' ) y"'
To sum up, the obtained normalized form is i, = z2
i2 = 2(-1 + z3 +2e 2-2 )z2 + 2u
i3 = (1- z3 - 2e'"' )(1 + 2z2e'"' ) 'C3
,..'
0p'1
Now, let us consider another aspect of the problem. As r < n, from Eq. (2.78) we know that the components of coordinates from (r+l )h through nt', , q' are chosen in accordance with the following conditions (Pr+, ,
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
56
Lgrp;(X)=0
r+15iSn
(2.92)
which implies that the partial differential equations (2.92) should be solved to obtain the normal form as in Eq. (2.91), and this is not easy. If we avoid only according to the condition solving Eq.(2.92) and choose rpr+,, that 0-0(X°)/8X is nonsingular, then from Eq. (2.85) through Eq. (2.87) we have Zr+I = qr+l (Z) + pr+I (Z)u (2.93)
i,, =
p,, (Z)u
where qr+, , . q, are shown as in Eq. (2.89) and p,+,,, r+l a , p,, are a
pr+I (Z) = Lg P,+I ((D -'(Z))
(2.94) pn (Z) = Lg7n ((D -1 (Z))
Thus, the obtained dynamic system shown in the new coordinates Z is
i,=Z2 Zr-1 = Zr
Zr = a(Z) + b(Z)u
(2.95)
Zr+I = qr+I (Z) + pr+l (Z)u
i. = q,, (Z) + p,, (Z)u
where, a(Z) and b(Z) have been given in Eq. (2.83);
qr+, (Z), , q,, (Z) are as Eq. (2.89) and pr+, (Z), , p,, (Z) are given in Eq. (2.94). We know so far that the partial differential equations (2.92) should be solved, if we want to get the normal form similar to Eq. (2.91). Otherwise the chosen coordinates from rpr+I , , q),, can not assure that pr+, (Z), .. , p,, (Z) equal zero, and only a incomplete normal form as shown in Eq. (2.95) can be obtained.
2.10
SUMMARY
°,0
A series of fundamental concepts of nonlinear control theory is discussed in this chapter. None of them can be separated from others and there exist close logical relations inherent between them. One of the essence and pith of modern nonlinear control theory is that a nonlinear system can be exactly linearized or partially linearized by means of appropriate coordinate
transformation and nonlinear feedback. Hence coordinate transformation
57
L!7
Basic Concepts of Nonlinear Control Theory
,..r
becomes one of the most fundamental concepts, which has been detailed at the beginning of this chapter. From a geometric point of view coordinate transformation is no more than an inter-mapping between the points in two +-'
.),
0.°
real spaces. Therefore it can not be discussed without the concept of <w-
v,'
O.2
0D'
mapping. A mapping between two spaces, not only from a mathematical viewpoint but also an engineering one, is viewed valuable and meaningful, if there exists bi-directional one-to-one correspondence between their points. c=)
Since this property of the chosen mapping belongs to the field of
Hi.
v,'
diffeomorphism in differential geometry, it is necessary to illustrate the concept of diffeomorphism and the method to check whether a transformation (mapping) is a global or local diffeomorphism in discussing `i'
cad
the concept of coordinate transformation.
Having discussed the concept of the coordinate transformation, we should put our immediate focus on the mainly investigated objects, i.e. affine
nonlinear systems, in which both f(x) and g,(x) are function-composed
cad
vectors. From the dimension of geometric structure, each specific point X° in the state space corresponds to specific vectors f(X°) and g,(x°), thus f(x) or g,(x) describes a vector field. What can be seen as well is that the concept of vector field in differential geometry is as important and essential as that of function in calculus. Since we can not advance further only with fundamental concepts, some essential derivation and computation methods and tools should be mastered. Just as we can not study linear systems without linear algebra, we need Lie algebra in studying nonlinear systems. Lie derivative and Lie bracket are two a..
c00
...
0
fundamental concepts of Lie algebra, which have been illustrated in a
coo
coo
.>.
Cep
r-!
00)
detailed way in Section 2.6. Although what should be discussed following the description of vector field is the involutivity of a vector field set, we discuss this content after the section about Lie derivative and Lie bracket because they will be used in defining the involutivity of a vector field set. The concept and definition of derived mapping of vector fields are clarified in a specific section since it plays a significant role in exact linearization °-'
algorithms. The final two sections, i.e. Sections 2.8 and 2.9 direct us substantially to
coo
the orientation of nonlinear system's exact linearization. It is easy to sense the tight relation between relative degree, normal form and exact linearization after reading these two sections. The primary idea of this section is that the eight elementary concepts discussed in this chapter are not isolated from each other. We should be able
'C7
to grasp the essence of them and constitute a "lively" system of them according to their inherent relations.
58
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
2.11 REFERENCES 1.
A. Isidori, Nonlinear Control Systems: An Introduction (3'a Edition), Springer-Verlag, New York, 1995.
Chapter 3
Design Principles of Single-Input Single-Output Nonlinear Control Systems
INTRODUCTION
3.1
o,0
-+'
Coo
aim
C..+'
rya
From this chapter, we can find a notable characteristic of this book, i.e. the book expatiates upon combining the nonlinear control principle with the design method. As viewed from engineering practice, the ultimate purpose of a new control theory's establishment and development is to design and manufacture the more novel and better types of controllers. The classical control theory has this purpose, so does the linear optimal control theory. The new system of nonlinear control theory, which we are presenting to the reader, will not depart this practical purpose. From another point of view, a control theory can only embody its value when the control systems designed following the theory are applied to the engineering practice and put into full play; At the same time, only by the extensive application of a theory, we will be able to find out its weak points and to improve and develop the theory. The outstanding feature of practicality of the control theory distinguishes C°,
cr.
itself from some of other basic theoretical branches (such as pure
coo
coo
mathematics, astronomy, etc.). In this chapter, we shall put forward and classify some design methods for the single-input single-output (SISO) affine nonlinear systems. They are the state feedback exact linearization design method, zero-dynamics design
r°°
method and the disturbance decoupling one. The first two design methods mainly improve the system's stability and dynamic performance; the last one is to gain high capability for the control system to resist disturbances. For the sake of better comprehension, each design method is illustrated with
60
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
examples. .ti
00.,
w'.
p.'
In the state feedback exact linearization design principle, we shall put forward and solve two kinds of problems. The first one is how to confirm
whether a certain nonlinear system can be exactly linearized into a completely controllable linear system, which is the necessary and sufficient
OP'
condition for exact linearization. In this aspect, we will quote a famous theorem - Frobenius theorem, and from this theorem we will obtain the
5.0
qualification of state feedback exact linearization. The other important one is the algorithm problem, namely the problem of how to get the state nonlinear
44i
feedback law. If this problem can not be solved, the application of the .s!
principle will not be realized. This algoristic issue is solved by Ref. [1], and the feedback linearization algorithm discussed in this chapter mainly comes from this paper. The plot of the zero-dynamics method is ingeniously conceived. Though not perfect in theory, the method may give quite good design result. The choice of a design method is usually made after the careful analysis of a specific system. If the designer hesitates to decide which one is better, he could turn to simulation results for help. (IQ
F'+
3.2
DESIGN PRINCIPLES OF EXACT LINEARIZATION VIA FEEDBACK +r"
P',
As have been discussed previously, for a nonlinear system if we use the mathematical model of approximate linearization at a fixed point and make a ^C3
design with the linear system approach, we can obtain a control law.
t~7
C"O
'ti
However it generally can not make the system retain satisfactory stability and dynamic performance at a state that is far from the point at which the linearization is carried out. In order to solve this problem one may use the adaptive technology to remedy a slow-dynamic processing nonlinear system, and the technology is to make the controller's parameters change with the state to realize the online adjustment. But to a power system whose dynamic process is much faster, the application of the above method does not work very well. In the engineering control field people are looking forward to developing a new theory and method, which can make the nonlinear system exactly linearized in its global state space or in a large enough region of state space. Using this theory and method to design control systems we can overcome the disadvantages brought by approximate linearization. In recent years, the research results of nonlinear control system theory shows that: using nonlinear state feedback and suitable coordinate transformation can exactly linearize an affine nonlinear system satisfying certain conditions, and
o,.
"CJ
``d `c3
off'
this state feedback can ensure the control system's stability and good
Design Principles of Single-Input Single-Output Nonlinear Control Systems
61
dynamic performance.
In this section, we will discuss under what conditions a SISO affine nonlinear system can be exactly linearized via state feedback. Meanwhile, we will clarify how to choose this coordinate transformation and how to achieve this state feedback law. In the subsequent chapters, we can see how the principles and the methods clarified in this section can be extended to"the multi-input multi-output affine nonlinear systems.
3.2.1
'Z'
Linearizing Design Principle as Relative Degree r Equals n for an nth-order System
For the sake of completeness, let us consider a nonlinear system described by equations of the form X(t) = f(X(t)) + g(X(t))u(t) y(t) = h(X(t))
(3.1)
where X e R" is the state vector; u e R the control vector; y e R the
...
't7
output vector; f(X) and g(X) are the n-dimensional vector fields in the state space; h(X) is the scalar function of X. We suppose the relative degree of the system r equals the number of the dimension of the system's state vector n, i.e. r = n. The above case has been preliminarily discussed in Section 2.8. From Eq. (2.74) we know that in order to establish the normal form of the dynamic system in a new coordinate system Z, the coordinate mapping should be selected like this
T2(xl,x2,...1Xn)
L.th(X)
L(Pn(xv,x2,...xn)J
L'f'h(X)
...
Z=cb(X)=
h(X)
...
where '(X) is a local diffeomorphism. From the discussion in Section 2.8, under the condition that the relative degree of the system equals n, according to the coordinate transformation in Eq. (3.2), the original system Eq. (3.1) can be transformed into the following normal form Zl = Z2 Z2 = Z3
in-t = Zn
(3.3)
62
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
a =a(X)+/3(X)ulX_@_,(Z)
where a(X) and /3(X) are the nonlinear scalar functions of X. The first n1 equations of Eq. (3.3) have been linearized. The last equation, which involves the control variable u is nonlinear. In order to exactly linearize the system (3.3) completely, we set (3.4)
v = a(X) + /3(X)u
`.T
From the statements in Section 2.8, we know /3(X) * 0. The following two results can be obtained: Firstly, we get a completely controllable and exactly linearized system in the new coordinates Z = [z, z2 which is in the form as Z, =Z2 Z2 = Z3
(3.5)
Z"
a =V The form of Eq. (3.5) is called the Brunovsky normal form. It can be written as
Z=AZ+Bv
(3.6)
where
010...00
0
z,
Z2
A= :
.
.
...
0
Z=
.
.
B=
.
000...01
0
000...00
1
Secondly, according to the definition of the relative degree, one knows that in Eq. (3.3) /3(X) # 0, so that from formula (3.4) the expression of the control u will be
__a(X)+ ,k'
u
/3(X)
1
v
/3(X)
From Section 2.8 we know that a(X) = L"fh(X)
(3.8)
,6(X) = L,L f'h(X) # 0
(3.9)
So far, in the expression of the control law given by expression (3.7),
there is only v which is not specified yet. Subsequently we will go on explaining how to determine the input variable v in (3.7).
Design Principles of Single-Input Single-Output Nonlinear Control Systems
63
From Eq. (3.5) and (3.6), v is the "control" input of the linear system of the Brunovsky normal form, so the most reasonable way is using the linear optimal control design method with the quadratic performance index (LQR method) to produce the v. Now let us briefly review the LQR design principle and method. Consider the following linear control system
(3.10)
Z(t) = AZ(t) + BV(t)
where Z is the n-dimensional state vector, V the m-dimensional control vector, A and B are an n x n matrix and an n x m matrix respectively. Matrices A and B satisfy the following condition, namely the matrix
D=[BIABI A'B ...I A'-'B] has rank n. This condition means that the system (3.10) is controllable. The performance index of the system (3.10) is quadratic, i.e.
J= ! f u (ZTQZ+VTRV)dt
(3.11)
where Q is a semi-positive definite n x n weighting matrix, R is a positive definite m x m weighting matrix. The problem can be granted as obtaining the state feedback vector vl(Z1, Z2,. .,Zn)
V = V(Z(t)) -
(3.12)
which is able to make the performance index J reach its extremum (maximum for the system in Eq. (3.10) or minimum) for the system in Eq. (3.1). This is called the LQR problem. The performance index J of formula (3.11) is the function of Z and V, Yet Z = [Zt Z2 .. and V = [v, VZ _ . V. ]T are the functions of time '17
t, so that J is the functional of Z(t) and V(t) .-From the viewpoint of mathematics, the LQR problem is a constrained variational problem, that is, to find the extremum conditions of the functional J shown in (3.11), subject to the constraints - Z(t) + AZ(t) + BV(t) = 0
(3.13)
Using Lagrange approach of variational method, our general approach to solve the problem will involve following main steps:
Step 1.
In connection with the performance index functional in
expression (3.11) and the constraint equation (3.13), we make an auxiliary functional
64
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
J= Ja [(ZTQZ+VTRV)+AT(t)(AZ+BV-Z)ldt
(3.14)
where A is the n-dimensional Lagrange multiplier vector function, which is also called co-state vector. Step 2. Define a scalar function - Hamiltonian function.
'-IN
I
H(Z,A,V)= (ZTQZ+VTRV+)STAZ+ATBV)
(3.15)
Then the auxiliary functional j in expression (3.14) can be written as
f
u (H(Z,A,V)-ATZ)dt
(3.16)
Let F denote the integrand in Eq. (3.16), i.e. ILL
F = H(Z,A,V)-ATZ
(3.17)
Step 3. In view of Euler-Lagrange Equation in variational principle, the necessary conditions for obtaining extreme functions of functional J are
aFd aF0 dt az OF _d aF = 0 aZ
a-dtaV
(3.18)
aFd aF0
aA dt aA Substituting (3.17) into equation (3.18), we can get the extremum conditions of the performance index functional J
aH(Z,A,V)+A-o aZ aH(Z,A,V)
0
av
(3.19)
Z=AZ+BV Step 4. Substitute the Hamiltonian function (3.15) into (3.19), which express the extremum condition of the performance index functional J, and take a linear transformation A(t) = PZ(t)
(3.20)
where P is a non-singular n x n parameter matrix. So we can obtain the control vector V which makes the performance index functional J reach its extremum
V' = -R-'BTP'Z(t) = -K'Z(t)
(3.21)
65
Design Principles of Single-Input Single-Output Nonlinear Control Systems
where V' denotes the optimal control vector, K' is the optimal feedback gain matrix, i.e.
(3.22)
K' =R-'B Tp*
P' is the solution of the Riccati algebraic equation (3.23)
ATP + PA - PBR-'BTP + Q = 0
We have, so far, reviewed the LQR principle in the simplest way. The more detailed discussion can be found in Ref. [2]. From the above description we can come to an important conclusion: optimal control solution V in the sense of quadratic performance index of a linear system is still a linear feedback of state variables, i.e.
k k12 ... ki, 1 ... k2,,
z,
Z2
k2
...
k22
v2
...
V. =
V.
km, km2 ... k
Z,
(3.24)
So there is
v, =-k;,z,-k,zz2-...-k,,,z,,
(3.25)
Now let us return to the design principle of the SISO affine nonlinear control system. One of our main purposes is to seek for a state feedback +U+
expression u = u(X(t)) of the nonlinear control system (3.1). For this purpose we need to substitute formula (3.25) into (3.7) for v. In equation (3.7) we set
v = v; = -k z, - k' z2 -
- k;, z
(3.26)
According to the coordinate transformation (2.74) we know
z, =h(X) (3.27)
z2 = Lf h(X)
z,, =L'J'h(X)
Substituting the coordinate transformation (3.27) into Eq. (3.26), yields v = -k; h(X) - k;L fh(X) -
- k,,L'f'h(X)
(3.28)
Now substituting (3.28) into (3.7), and considering the formulae (3.8) and (3.9), we can get the expression of nonlinear state feedback u in the system (3.1) as follows u-- L"fh(X) -k,h(X)+k;Lfh(X)+ +k;L"f'h(X) (3.29) LgL"f'h(X)
LgL"f'h(X)
66
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
or
U=- L"" h(X)+k; h(X)+k;L
(3.30)
LgLf'h(X)
where k,*, kz, , k,, could be obtained from formula (3.23) and Eq. (3.24). Now let us briefly summarize the above content.
Consider a SISO nt-order nonlinear control system described by equations of the form X = f(X) + g(X)u y = h(X)
(3.31)
If the relative degree of the system r = n, then in terms of the nonlinear state feedback u=-a(X)+ /3(X)
1
/(X)
v
(3.32)
where a(X) = LJh(X) /3(X) = LgL"f'h(X)
and the diffeomorphism coordinate transformation h(X)
Z=cb(X)_
Lh(X) L'r'h(X)
the original nonlinear system could be transformed into a controllable linear system Z = AZ + By
(3.33)
o..
CAD
where the matrices A and B are Brunovsky normal form. We can see that the control variable u of the original system and the input variable v of the exactly linearized system (3.33) have the relation expressed by (3.32). As long as v in Eq. (3.32) is determined, the control vector u will be determined accordingly. In order to keep the satisfactory dynamic performance of the nonlinear control system, we should make v the optimal control of the linear system (3.33), i.e.
= v' =-B Tp*Z(t) = -K'Z(t) v=v* where P' is the solution of the Riccati matrix equation. Considering the coordinate transformation Z =
Design Principles of Single-Input Single-Output Nonlinear Control Systems
67
(3.34) LgL°f 'h(X)
r-'
This will naturally bring about a question: whether the "control" v' which make the exactly linearized system (3.33) reach its optimal performance index can make the control law u in Eq. (3.34) optimal in some sense to the original nonlinear system? The answer is positive. From the above statement we have known that the v` is an optimal control to the system described by Eq. (3.5) or (3.10) in the sense of the performance index expressed by formula (3.11). And the formula of coordinate mapping Z = c(X) has been defined by (3.2). In view of (3.5) and (3.2) we realize that
"--
v= i ax
dt
Substituting the above expression and (3.2) into (3.11), we have the performance index as follows
J=
dip (X) r dip (X) R( dt ) dt ))dt
1
2 ((r (X
(CD
°~n
Let's set, as usual, the weighting matrix Q = diag(q, , q2, , and the weighting coefficient R=1, in the above formula. Thus the performance
'Oo
index for optimization could be written as
J = 2l0 ((D r(X)QD(X)+(dOn(X))2)dt That is the formula of performance index expressed by state variable 'r''
coordinates X(t). Now, let us answer the question mentioned above: the control law u acquired according to the approach proposed by the above section is also an
optimal control to the original nonlinear system in the sense of the performance index as
-IN
f o (r
J=
dt
2 )dr
2
or
1 r. J =-Jo (Zq;(P?(X)+(Z
2
x;) )dt
In order to get a deeper understanding of the above design method, let us illustrate it by the diagram in Fig. 3.1. Let us concretely explain the calculating process of the state feedback exact linearization and the obtaining of the state feedback law in terms of an example.
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
68
(X)
a(X) = LIh(X)
h(X)
/3(X)
4-2
/3(X) =
I
X(1) a(X)
,B(X)
+-v' Q(X)
U
I
X=f(X)+g(X)u
y(t)
X f(X)+g(X)a(X)
-g(X),6(X)K (X)
i(Z)
X=
IV.
Z = AZ+Bv IA,B
K'(P(X)
=-K'Z Z = O(X) N
Figure 3.1
7
v' =-K'Z
BTP
P'
to solve:
ATP+PA-PBBTP+Q=
Diagram illustrating the design principle of exact linearization via feedback
Example 3.1 Consider the system 0 e'' X = x1 +x2 + er' u
X, -X2
11 0
Y=xs
where 0 e`' f (X) = x1 + x2 , g(X) = ex'
x, -x2
h(X) = x3
0
Calculating and checking the Lie derivatives of the given system, we have
LgL°fh(X) = Lgh(X) = 0
L fh(X) = x, - x2
LgL fh(X) = 0
L fh(X) _ -(x1 + x2 ) ">t
N`,
LgL fh(X) = -(1 + 2x2 )ex'
L3fh(X) = -2x2 (x, + x2 )
From the above we know that the relative degree of the system r = n = 3 for any point of X satisfying I+ 2x2 # 0 . Choosing the following coordinate transformation 0 z1 = h(X) = x3 ..J
z2 = L fh(X) = x, - x2 z3 =L f h(X) _ -(x, + x2 )
Design Principles of Single-Input Single-Output Nonlinear Control Systems
69
and the state feedback
u= -+ LgL fh(X)
^..
L3fh(X)
1
LgL fh(X)
2x2 (x, + x2) + v
v
(1+2x2)eX'
we get the dynamic system expressed in the new coordinates Z as Zt = z2 z2 = Z 3
Z3 =V
This is a controllable linear system. From the linear optimal control principle, the optimal control is achieved as v = -K'Z = -k; z, - k2z2 - k; z3
= -k; x3 - k; (x, -x2)+k; (x, +x2) and
K' = R-'BTP'
R is the weighting coefficients, and choose R = 1.0; P' is the solution of the Riccati matrix equation ATP + PA - PBBTP + Q = 0
where 0
0
0
1
B= 0
0
0
.-.
0
1
moo
0
A= 0
1
If the weighting matrix Q is chosen as a diagonal one, i.e. Q = diag(l, 1, 0)
we have P.
2.29 2 .14
= 2.14 1.0
3 .9
2 .29
1.0
2.29 2.14
Then we get k; = 1.0
k; = 2. 29
k; = 2.14
Finally, the state feedback law could be achieved as +
U=--2x2(x, +x2)+2.14(x, +x2)-2.29(x, -x2)-1.0x3 (1 +2x2 )e" `CS
CAD
From this example, one may get a deep impression of the complexity of the control law in terms of the state variable X.
Ci)
70
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
3.2.2
General Linearization Design Principle
The above discussion tackles the exact linearization design problem when the system's relative degree r equals n. Now we will discuss the exact linearization design problem under general conditions.
v'+
.fl
In some practical engineering problems, the output functions of the system may not be clear; or to an output function y = h(X) which has a clear physical meaning, its relative degree r is less than n. Under the above two conditions, either we can hardly write the output function or 4-.
unable to transform the original nonlinear system into the Brunovsky normal form by using the known output function h(X). Next we will discuss the method to solve the problem. When r
order to distinguish the above r = n condition, this "output function" is noted as (3.35)
y = w(X) c+?
4"'
.-.
QUO
cad
The "output" function w(X) expressed by formula (3.35) seemingly has no clear physical meaning, but due to the same reason mentioned above, as long as this "output" function w(X) satisfies the following two conditions: Condition 1 Lgw(X) = 0
LgLjw(X)=0
(3.36)
LgL"f2w(X)=0
but (3.37)
LgL'f'W(X) # 0
Lgi,w(X)=0 LgL j'w(X) # 0
for
0<_i
OHO
This condition can also be written as (3.38)
Condition 2 The vector field w(X)
L,w(X)
T(X)= LZw(X) [L! w(X)]
(3.39)
Design Principles of Single-Input Single-Output Nonlinear Control Systems
71
satisfies that at the point of X = X° its Jacobian matrix J,v(X°) =
aT(X)
ax
X=X°
0
is nonsingular. Let us emphasize it again, as long as this vector field 'P (X) satisfies the above two conditions, it must be a local diffeomorphism defined
in an open subset U° of X°, and also be the coordinate transformation which is being sought for, namely, zi
Z=`Y(X)=
z2
w(X) L fw(X) L fw(X)
(3.40)
L'f' w(X)
Thus we can transform the original affine system into a Brunovsky normal form, and obtain the corresponding state feedback
u= -
L'fw(X) LgE
'w(X)
k,
(3.41) LgLj'w(X)
Now the problem is clear: in order to get the needed "output" function w(X), we must solve the partial differential equation set (3.38). It can be seen from (3.38) that the last equation in this condition is an inequality, i.e. LgLL'w(X) # 0
So the solution of equation (3.38) is not unique, which means getting any w(X) which satisfies both the condition (3.38) will meet our requirement. We have, so far, two problems to solve: The first one is for a given
affine nonlinear system, how to confirm the existence of the "output" function w(X) which satisfies the above condition 1 (and condition 2); The second one is if we know the existence of this "output" function w(X), whether we can use more practical and simple method to get the function without having to solve the partial differential equation (3.38). The first
problem is the necessary and sufficient condition problem for exact linearization of a nonlinear system; the second one is the algorithm problem of the exact linearization. In the following sections we will discuss and solve these two important problems.
72
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
3.2.3
Conditions for Exact Linearization
When discussing the necessary and sufficient conditions for exact linearization of an affine nonlinear system, we need a theorem known as Frobenius Theorem. So we will first introduce this theorem, but not strictly prove the theorem because it is out of the scope of this book. If readers are interested in the verification of the theorem, please refer to the textbooks of Differential Geometry [3, 4].
Theorem 3.1 Frobenius Theorem Consider the following partial differential equation set ah(X) ax,
ah(X)
Y" (X) +
axe
ah(X) ax
axn
ax2
Yin (X) = 0
axn
ah(X) Ykl (X) +
ax2
Yk2 (X) +
, k. j =1, 2,
in which y; (X), i =1, 2, x2
ah(X)
ah(X) Y22 (X) + ... + ah(X) yen (X) = 0
Y21 (X) +
ah(X) ax,
Yi2 (X) +...+
+
ah(X)
(3.42)
.vkn (X) = 0
axn
, n, is .the scalar function of X = [x,
x ]T ; h(X) is the function to seek for; k < n. If Eq. (3.42) is written in the compact form, we have
aXX)[yl(X) Y2(X) ...yk(X)l=0
Y2 (X),
'h(X) _ [ah(X) ah(X)
ax
&,
axe
ah(X)]
is the gradient vector of h(X); Y, (X),
axn
, Yk Mare the n-dimensional vector fields defined on X space, CD,
where
(3.43)
IY,1(xi,x2,...,x,,)
Y,(X) =
k
Suppose the matrix
''n
[yl (X) Y2 (X) ... Yk Mil =
Y1I (X) Y21(X) ... Yk1 (X) Y12 (X) Y22 (X) ... yk2 (X)
Yln(X) Y2n(X) ... Ykn(X)
has rank k at the point X = X°. If, and only if the augmented matrix [y1
y2
...
Yk
[V j, Y; l l
Design Principles of Single-Input Single-Output Nonlinear Control Systems
where
73
denotes a Lie bracket of any two column vectors in the above
..b
original matrix, still has rank k for all X in a neighborhood of X°, there must exist n-k scalar functions, defined in a neighborhood of V, which are the solutions of the given partial differential equations (3.43), and such that the Jacobian matrix ah, (X)
&,
axn
a&2
ah"_k (X) ah"_k (X) ax,
ax ax
ax"
ah2(X) ... ah2(X)
C -)X,
ah"_k (X)
ah, (X)
axe
ah2(X) J1, =
ah, (X) ...
axe
ah"-k (X) ax"
has rank n-k at X = X°.
Having comprehended the Frobenius Theorem as plainly explained
above, one may easily associate the theorem with the definition of involutivity of a vector field set in Section 2.7. The conditions proposed by Frobenius Theorem is actually the condition of involutivity of a vector field set {Y, , Y21 , Yk } . In terms of the definition of the vector field set's
involutivity, the Frobenius Theorem can be more briefly expressed as follows.
Consider the following partial differential equation set of the function h(X)
ah()[YI(X)
Y2(X) ...Yk(X)]=0
where X E R" ; Y; (X), i =1,2, , k, are linearly independent vector fields. If and only if the vector field set D={Y,(X) Y2(X) ...Yk(x)} is involutive at X = X°, there exist exactly n - k scalar functions , (X) satisfying the given partial differential equation set, h, (X),h2 (X), and the gradient vectors
"ax,
Vh
(x)=[ahm(x)
ah",(X) axe
ahm(x)] az"
m=1,2, .,n - k
are linearly independent. With the Frobenius Theorem, we can further discuss the conditions for exact linearization of an affine nonlinear system. From Section 3.2.2, for an affine nonlinear system X = f (X) + g(X)u
74
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
if we can find an "output" function w(X) such that the relative degree of the system r = n, this is, w(X) satisfies the following conditions at a neighborhood of X° LgLJ Zw(X) = 0
ti=
Lgw(X) = LgL fw(X) = LgLZfw(X)
LgL"f'w(X) # 0
(3.44)
CDR
c-.
and at the point X° the Jacobian matrix aw(x) ax a(L,w(x)) Jw = ax a(L1,'w(x)) ax
is nonsingular, then the original nonlinear system can be transformed into a
Brunovsky normal form. In order to get the "output" function w(X)
.fl
satisfying the above conditions, we must solve the partial differential Eq. (3.44). Now the problem that we are facing is: how to confirm the existence of the solution w(X) of the Eq. (3.44) before solving the partial differential equation. This is the condition problem for state feedback exact linearization -_r
of the affine nonlinear system. To get the necessary and sufficient conditions for state feedback exact :p,
linearization of system (3.1), we will make some change in the conditions (3.44). From equation (2.52) we know LIf g1W(X) = L j,1gw(X) = L fLgw(X) - LgL fw(X)
According to (3.45), there Lgw(X) = 0 in equation (3.44), i.e.
(3.45)
are two equivalent conditions for
I, gw(X) = 0
(3.46)
LgL jw(X) = 0
(3.47)
and
Using Eq. (3.45) once more, we have Ladig w(X) = Lf.ad,g,w(X) = Lf Lad,gw(X) - Lod g L fw(X)
(3.48)
Expanding the above formula we have Lpdfgw(X) = L,Lgw(X) + LgL2fw(X) - 2L fLgL fw(X)
From the above discussion we arrive at
(3.49)
75
Design Principles of Single-Input Single-Output Nonlinear Control Systems
Lgw(X)=LgLfw(X)=0
So there are also two conditions which are equivalent to equation (3.49) and
L,d`gw(X) = 0
LgL fw(X) = 0 .fl
Repeatedly using the relation of (3.45), the conditions stated by (3.44) can be restated as: at the point X = X° the following conditions must hold LgW(X) = L d gw(X) = Lad,rgW(X) =
(3.50)
= Lod,r_,gW(X) = 0
and .^+
(3.51)
`'"
had
Ladf,gw(X) # 0
Equation (3.50) is a set of partial differential equations for w(X). According to the definition of Lie derivative we can write it as C-wax(X)
[g(X) ad fg(X) ad fg(X)
ad; 2g(X)] = 0
(3.52)
Now comparing the above condition with the Frobenius Theorem we immediately know that if the vector fields g(X), ad fg(x), ad fg(X) , , ad 72 g(X) are linearly independent, and the vector field set `^-'
D={g(X),adfg(X),adfg(X), ,ad 2g(X)} .,r
is involutive, there must exist a w(X) satisfying partial differential equations (3.44). That is to say, there must exist a w(X) such that the system's relative degree r = n. So after obtaining the w(X), as long as we acd
choose the coordinate transformation w(X) L fw(X)
zI
Z=
z2
_ ,D(X) = L fw(X)
...
Z,
L'J `w(X)
and the state feedback
u(X)=-
Lj,v(X) LgL! w(X)
+
1
v
LBLJ W(X)
Gig
the original system can be transformed into a completely controllable linear system - a Brunovsky normal form. It can be seen from Eqs. (3.50) and (3.51) that if the "output" function w(X) exists, then the two conditions must hold
ax
[g(X) ad fg(X) ad fg(X) ... adf 2g(X)] = 0
(3.53)
76
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
and at the point of X=X° aw(X)
ax
ad j' g (X)
(3 . 54)
0
From the Frobenius Theorem, we have known that g(X), ad fg(X),adf g(X),
-.r;
adt Zg(X) are linearly independent at X = V. So the necessary condition for equations (3.53) and (3.54) to hold is the n vectors
, ad f zg(X) are linearly independent at X = V. On the whole, the necessary and sufficient conditions for the existence of the "output" function w(X) whose relative degree r = n are the following: Firstly, vector fields g(X), ad fg(X),ad fg(X), , adr 2g(X),ad 7'g(X) are g(X), ad fg(X), ad fg(X),
.
t0.
t+:
Gt.
linearly independent in a neighborhood of V. Secondly, the vector field set D = {g(X), ad fg(X), ad f g(X),
, adi Zg(X))
is involutive in this neighborhood of X°. Up to this point we arrive at the following theorem.
Theorem 3.2
Given the system X = f (X) + g(X)u
where X E R" is the state vector, u e R the control variable, f and g are both n-dimensional vector fields. If and only if (1) The rank of the matrix
[g(X) adfg(X) ad fg(X) ... adf 2g(X) ad!'g(X)] does not change and equals n near X°. (2) The vector field set D = {g(X), ad fg(X), ad fg(X), ..., adf 2g(X)}
is involutive at X = X°, then there must exist a function w(X) whose relative degree r is equal to the system's order n at x = V.. That means t1.
the given system can be exactly transformed into a completely controllable
linear system - a Brunovsky normal form in an open neighborhood of X=X°. 40.
We may expand our understanding of the conditions for exact linearization with an example.
Design Principles of Single-Input Single-Output Nonlinear Control Systems
77
Example 3.2 Consider the system X3(1+X2)
X=
+
x,
u
x2(1+x1)
Step 1. Check whether the system can be transformed into a linear controllable system by means of coordinate transformation and the state feedback. We first compute ad,g(x) and ad,g(x) ad,g(X) = ag(X) f(x) ax
0
=0 0
- of (X) g(X)
0
0
ax x3(1+x2)
1
0
x1
0 -1
0
x3
1 +x2
-1
0 x2 1 +x1
x2(1+x1)
0
0
1+x2
0
- x3
0
=
x,
- (1 + x 1)(1 + 2x2 )
2
adjg(X) =
=
a(adJg(X))
ax
f(X) -
af(X) aX
adfg(X)
0
0
0
x3(1+x2)
1
0
0
x1
-(1+2x2) -2(1+x1) 0 11 x2(1+x1) 0 - 1
x3 0
,..
x2 1+x1
1+x2
0
0
0
x,
-(1+x1)(1+2x2)
(1+x1)(1+x2)(1+2x2)-x1x3 x3(1+x2)
=
:..
-x3(1+x2)(1+2x2)-3x1(1+x1) From the above we know that at X = 0 the following matrix [g(X) ad fg(X) ad jg(X)]X-u =
0
0
1
0
0 -1
1
0 0
has the rank 3 which is equal to the system's order n. So the condition (1) of Theorem 3.2 holds. Now we go on to check the condition (2) of Theorem 3.2 by calculating the Lie bracket
78
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
a(ad fg(X))
[g(X) ad g(X)1 =
g(X)
8X
- 3f(X) ad fg(X) ax
0
0
0
0
1
0
0
(1+x2)
-(1+2x2) -2(1+x,) 0
- 0 0
-x3
0
0
0
0
1
0
x,
0 -1 -(1+x )(1+2x2) 0
- x,
-(1+x,)(3+4x2) We know that the matrix [g(X) ad fg(X) [g ad fg]], 0
0
0
1+x2
x,
-x,
-(1+x,)(1+2x2) -(1 +x,)(3+4x2)
-x3
has the rank which is equal to 2 for all X. Therefore, the set D = {g(X) ad fg(X)}
is involutive for all X. So the condition (2) of Theorem 3.2 holds. Therefore we can say that the system has an "output" function w(X) which makes the system's relative degree equal 3. Step 2. Calculate w(X) which is the solution of the partial differential function 7R) [g(X) ad fg(X)] = 0
Substituting the expressions of g(X) and ad fg(X) into the above partial differential equation ow(X) (1+x2)- 8w(X) &3
ox2
(X)(l+x,)(1+2x2)=0
(X) x,+ 2
x3 = 0
0 -X3
We can find out the following solution to the partial differential equations without much difficulty w(xl,x2,x3)=x1
Step 3. Look for suitable coordinate transformation z=mM. Through calculation we know that as w(X) = x, there are
Design Principles of Single-Input Single-Output Nonlinear Control Systems
x3(1+x2)
Lfw(X)=[1 0 0]
=x3(1+x2)
x, x2(1 + x,)
(1 +x2)
[0 x3 1+x2]
x1
=x1x3+x2(1+x,)(1+x2)
rX'2 (1+x1)
So the coordinate transformation (D should be
z, =P,(X)=w(X)=x, z2 = VAX) = L fw(X) = x3 (1 + x2 ) Z3 =V3 (X) = Lfw(X) = x,x3 + x2 (1 + x1)(1 + x2 )
Step 4.
Obtain the control law. The form of the control law is U=.
-L3fw(X)+v LgLfw(X)
Find out LgLfw(X) = [x3 +x2(1+x2) (1+2x2)(1+x,)
x1]
= (1+x,)(1+x2)(1+2x2)-x1x3 x3(1 + x2)
LgL33W(X)=[x3+x2(1+x2) (1+2x2)(1+x,) x1]
x,
x2(1+x1)
=x;(1+x2)+x2x3(1+x2)2 +x1(1+x,)(1+2x2)+x,x2(1+x,)
so that u=
-(xs (1+x2)+x2x3(1+x2)2 +x,(1+x,)(1+2x2)+x,x2(1+x,))+v (1 +x1)(1 +x2)(1 + 2x2) - x1x3
In the above formula we set
v=-K'Z and
K =BTP* =[0 0 1]P' _[Psi
P32
P331
where P' is the solution of the Ricatti equation
010'
010
0
100
001 P+P001 -P0 [001]P+ 0 1 0 =0 000 000 000 1
79
80
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Solving the equation we have
k; = p;, =1.0
k2 = Paz = 2.29
kg = P33 = 2.14
Finally the control law is acquired as follows
U=- (X" (1+x2)+x2x3 (1+x2)2 +x, (l +x,)(1+2x2)+x,x2(1+x,)) (l+x,)(1+x2)(1+2x2)-x,x3
_ x, +2.29x3(1+x2)+2.14(x,x3 +x2(1+x,)(1+x2))
(I +x,)(1+x2)(1+2x2) -x,x3
We can see that the control law is a complicated nonlinear function of the state variables x,, x2 and x3.
3.2.4
Algorithm of Exact Linearization S..
The above section shows that: after we make sure that the conditions for
existence of "output" function w(X) whose relative degree r equals n are fulfilled, in order to obtain the concrete expression of w(X) we must solve a set of partial differential equations as follows Vw(x)g(x) = 0 Ow(X)adf g(X) = 0
(3.55)
Vw(X)ad72g(X) = 0
and yet (3.56)
Vw(X)ad f'g(X) # 0
'l,
tee'
[17
L:.
where' Ow(X) = aw(X)/aX is the gradient vector of w(X). In a practical engineering problem, solving Eqs. (3.55) and (3.56) is not so easy as Example 3.2 shows. In order to meet the needs of practical control system design, we ought to find an algorithm which does not need to solve the partial differential equations but can get the "output" function w(X) which can transform the nonlinear system (3.1) into a Brunovsky normal form via exact linearization, and which can get the state feedback law. Our general algorithm will involve four main steps: Step 1. According to the system in Eq. (3.1), calculate each order Lie
Bracket of g(X) along f(X): adrg(X), adfg(X), , adf'g(X), and form n vector field sets
Design Principles of Single-Input Single-Output Nonlinear Control Systems D, _ {g(X)} D2 = {g(X), ad jg(X)}
81
(3.57)
D. = {g(X), adf g(X), , adf-'g(X)}
Step 2. such that
Choose n linearly independent vector fields D,, D2,
, D"
E D, D2 E D2
(3.58)
E D"
Formula (3.58) means selecting the scalar function i ? j, i, j =1, 2,
k('>(X)
of X,
, n such that the subsequent formulae hold
D, +k;'>(X)g(X)=0 D2 + 42> (X)g(X) + k22> (X)ad jg(X) = 0
`g(X)=0
D;
(3.59)
(X)ad""'g(X)=0
Formula (3.59) implies that the vector field D, is a linear combination of vector fields g(X), adfg(X), adfg(X), , ad'f'g(X), i=1, G1.
.L"
Step 3. Let us introduce a symbol (D`, (X°) , which represents the integral curve of the vector field f(X)_[f(X) f2(X) f,,(X)]T with the 0
initial value V, which is also the solution to the set of differential equations dx, (t)
.fi(xi,x2,...,x")
dx2 (t)
=J2(x1,x...... x
n)
dtt) =
with the initial value X° = [x,° x2
x°]T
.
The task of Step 3 is to calculate the mapping X = F(W) from the state
space R" of the new coordinates W to the state space R" of the former
n vector fields D D2, , A,,
...
coordinates X. This mapping can be expressed by the integral curve of the i.e.
by the solution
a).6,,
corresponding differential equation, i =1, 2, , n, namely the mapping
of the
82
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
F(w, w2
(3.60)
ocI
If we still feel a little puzzled about the meaning of formula (3.60), the following discussion will make it clear. Formula (3.60) can be calculated by the following steps: first calculate the integral curve (DD- (X0) of the vector field D with the initial value X0
and the independent variable w . That is to compute the solution of the differential equation x,(0)
d
x2 x1
=D
X(0) = x2(0)
x
-X0
(3.61)
x (0)
The solution (DD. (X0) of the differential equation (3.61) is the vector function of the independent variable w , namely x,(wn)
(D D.(XOX\wn)
x2(wn)
Equation (3.60) tells us that the next step is to calculate the integral curve of the vector field with the independent variable and the initial
condition cp°; (X0) which is to find out the solution to the differential equation x,
d
X2
(3.62)
dwn-, X1,
Solving the differential equation (3.62) we can get the integral curve (DD._, (X 0) ° De (X0), which is the vector function of the independent variable
i.e.
(X
q)
(D
D.
We use the above as the initial value and compute (DD:-;
(X0) ° D;=; (Xo) ° D; (X0) = X(wi-2,
w)
83
Design Principles of Single-Input Single-Output Nonlinear Control Systems
Similarly, we can obtain
vector
field DI , which is the solution to the differential equation x' xl(w2,... wn)
d
x2
Dl
( X 0 ) . . _..,OD, (X°)
X(0) = (D
dw,
x2(w2,...,wn)
(3.63)
=
Di
`XJW2,...I WO]
Solving equation (3.63) we obtain the mapping (3.60)
defined by
x1
X= x2 =F(W)= F2(wl,w2, .,wn)
(3.64)
xn
From the above we can calculate the inverse mapping of F wl(x1,x2, ',xn) w2(x1,x2, ,xn)
Wl
W=
IV2
=F-'(X)=
(3.65)
Lwn(xl,x2, ,xn)J
LwnJ
Step 4. Calculate the derived mapping F.-' (f) of the vector field f(X) of the system (3.1) under the mapping F-' in Eq. (3.65). From the definition of the vector field derived mapping in Section 2.5 we know F.-'(f) = J F-' f(X)IX=F(W)
(3.66)
In the above formula, JF_, represents the Jacobian matrix of F-'(X), i. e. awl (X) axl
JF =
'1(X) ax2
awl (X) axn
awl(X) aw2(X) ... 3w2(X) axl vc2 n 11
(3.67)
,,..
UWn(X) n(X) axl
ax2
5w,(X) axn
Let .fl'0) (W)
f(0)(W)= fz°)(W) = F. ' (f) = J FA f(x)l x=F(W) fn(o)
(W)
(3.68)
84
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
In order to compute the coordinate transformation and state feedback for exactly linearizing the nt'-order affine nonlinear system, we also need to define R R2, , one by one. First we define the transformation R, as follows f2(0) (W)
z1(')
z(° =(°' f(W) 2
(3.69)
z(1) = f(o) (W) n
1
Z(1) = Wn n ..o
In the above formula f(') (W), i = 2, , n, has been given in Eq. (3.68). Subsequently we find out f(1)(W)
f(') (W)=(R)).f(0)(W)=JR f(0)(W)
(3.70)
The Jacobian matrix in (3.70) is
(3.71)
JR =
awl
L awl
SWn J
Then we continue to define the transformation Ri as ZI(i) = f{ (i-1) (W)
f
Z(2j)
-1) (W)
Z(i ) = f "U-1) (W)
(3.72)
-1
Zi) = z (i1)(= n
From the above we can calculate the f(i)(W) f (i' (W)
f(i)(W) = f(i )(W)
=j .(W)f(i (W)
(3.73)
fn(i) (W)
JR, in formula (3.73) denotes the Jacobian matrix of the mapping Ri which
has the same form as J. in Eq. (3.71). Following the computing procedure shown by (3.72) we can calculate each transformation we need until the (n -1)1h transformation
Design Principles of Single-Input Single-Output Nonlinear Control Systems z("-1)
Z2,-1)
85
= f,("-" (W) { 3(11-2) (W) =J
f
(»-1)
z("-'
Z"
11
(3.74)
(W) )
is actually the eventual coordinate transformation we need. From (3.74) we know that the transformation is the change of coordinates from W space to Z("-') space. What we need is to calculate the mapping from X space to Z("-') space. Note that calculating the coordinate transformation from X space to Z("-° space is equivalent to calculating the composite transformation T, defined by T = R,,-, F-1 (refer to Fig. 3.2). The
Figure 3.2
Relations among the coordinate transformations of the spaces X, W and Z
transformation T can be calculated in terms of the following formulae
f2(-2) (W)=TA(X)
zin-1)
W=F" (X)
f3(n-2)(W) W=F"i(X) c..
z(2-1)=
=T2 (X)
(3.75) Z(. -I)
n
1
=
f,(1-2)
(W)IW=F-I(X) = Tn-I(X)
N
z("-1) = w n W."-, (X) = Tn (X)
According to the coordinate transformation T, we could transform the vector fields f(X) and g(X) of the original nonlinear system (3.1) to f(X) and g(X) which represent the transformed vector fields, i.e. i(X) = JT(X)f(X) = [fi(X) f2(X) g(X) = JT(X)f(X) =181(X)
0
...
.._
Of
fn-1(X) f (X )]T
(3.76)
(3.77)
86
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
where g(x) m o, and aT,(X)
aT, (X) aT, (X)
8x2
axt
azn
aT2(X) aT2(X)
JT(X) =
ax!
aT2(X)
ax2
axn
aTn(X) aTT(X) ax1
Step 5.
ax2
(3.78)
aT°(X) ...
aXn
The final coordinate transformation Z=F(X) could be
immediately obtained by setting >C.
Z, = w (X) Z2 = f,(( )
(3.79)
Zn-1 = 3 (X)
Z. =f2(X) The inverse transformation is (3.80)
X = F-' (Z)
If we define U
f (X)
1
v)IX-F i(Z)
- ( 81(X) + 81(X) then the original nonlinear system (3.1) is transformed into a Brunovsky normal form ii = Z2 Z2 = Z3
(3.81) Zn
in = V
and the state feedback law of the original nonlinear system is achieved as follows __ _
1,(X) 1 v 81(X) + g1(X) 2C0
U(X)
(3.82)
Formula (3.82) gives the control law which is sought for. Above is the exact linearization algorithm proposed by Ref. [1]. Now
+-'
we can understand that the essence of this algorithm is to transform the problem of solving partial differential equations into the problem of of a series of ordinary obtaining the solution
87
Design Principles of Single-Input Single-Output Nonlinear Control Systems
cep
0
differential equations, and then to carry on the transformation step by step using the solution. It is worthy of noting that before taking the steps of exact linearization transformation, we should check whether the discussed affine nonlinear system satisfies the exact linearization conditions of Theorem 3.2.
After obtaining the affirmative answer, we can carry on the deduction of exact linearization. Let us explain the whole process of this algorithm with an example. Example 3.3
Consider the system
x,+3x; +Sxl
x [x2 ]
1
+[1+X11 ] u
l +X2
0
-x, +x2
(3.83)
x2 (0) = 0
X1 (0) = 0
vii
First we should check whether the system satisfies the conditions for exact linearization. To save space, we suppose that the system satisfies the
conditions of Theorem 3.2. Now let us carry on the deduction of exact linearization via state feedback.
Step 1. Calculate the Lie bracket of g(X) along f(X). Since n = 2, we only need calculate adfg(X) (1+x; )-2x,
ad rg( X )
0
x,+3xl +Sx5 0 0 [
[(1 +0x1 2 )
[
1+x ,
-x +x2
1 2x2
l
J
J
-1
=
1
l_+_X;
A..
-(1+x; )-22x,(x, +2x, /3+x; 15) +(l+x, )-'(1+2x, +x, ). where The two vector field sets composed are D,=(g) and D2 = {g, ad rg} 1
D, _{ 1 +x ;
-1
1
}
D 2 ={l+x,
0
l+x
0
r..
t71
I}
1
z
Step 2. Select the simplest n = 2 linearly independent vector fields ED, and DZ E D2. We determine the functions k;',(X), k;2)(X) and
k(2) (X) based on formula (3.59).
1
L
0
[01
88
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS I
D2 +k;2'(X) l+x2 +k(22) (X) 0
In order to make D, and D2 as simple as possible, we choose
k;' (X)=-(1+x k,2)(X)=-(1+x,)2 k22)(X)=-(1+x;) Then we have
Step 3.
D2 =[0 If
Of
D, =[1
The next task is to compute the mapping using formula (3.60) .(DD, (X,)
X0 =0
So we need to calculate the integral curve (DW, (Xo) .Solving the differential
equations dx,
chv2
dx2
with [°1
[x2 (0)
I
=
[01
dw2
we have
x2]7 =[0 w2]f
[x1
Then we calculate cb o 5' (Xo), i.e. solve the differential equations z
r, l dw,
[o]
dx2
dw,
with the initial condition X(0) = [0
w2 ]
T
Thus we obtain [x21
F(wl+x'z)-[F2(w'],w2)][x'2]
Obviously, the inverse mapping is w2 J
[F (x, xz) =
W1 (X I X2) X. J [w2 (xI , x2 )J = [x2
Design Principles of Single-Input Single-Output Nonlinear Control Systems
Step 4.
89
Calculate the derived mapping F'(f), thereby obtain
0)(W). First we need to compute the Jacobian matrix JF_, of F-' ,(X)
aw1(X) axz _
ax,
JF'
1
ax,
0
01
aw2(X) aw2(X) axe
Thus, from formula (3.68) we have
1f f(0)(W)=
ro)(W) -JF`f(X)IX=F(W)
0]
0
1
W1+3W1+SW
W1+W,3 +SW1 3
1
N_"'
[1
l+w
l+w
_WI +W2
-w,+w2
Since in this example n = 2, it is necessary to compute the R,. From (3.69) we have R, transformation Z;') = A°' (W) = w2 - w, = W2 Zco 2
Accordingly, the transformation T = R,F-' is >
Z,
( z - W, ) = W2 I
W-F_'(X)
(1)
Z2 = W21 W=F-'(X)
2 = X2 -X,
_ - x2
w++
From transformation T we can get the final transformation f = T.(f) = JT(X)f(X) x , +2X 1
2x2
0
+x S
'
1+x
1
- x, + x2
X1+?x; +-x;
t+x
5
2
1
+2x2(-X, +x2)
-x, +x2 = T.(9) = JT(X)f(X)
=r-1
2x21 1
1
-1
l+x; = l+x 0
0
90
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Step 5.
Define z,
=.f2(X) =-x, +x2
z2 =
and take the state feedback as u
f (X)
i (3.84)
(X) + S1 (X) v
=-(x,+3x1 +5x,)+2x2(-x,+x2)(1+x, )-(1+x, )v 2
Thus the system of the example has been exactly transformed into a linear and controllable system in the new coordinates Z Il =z2 Z2 =V'
In order to strengthen the understanding of the problem, we may substitute the control law u(X) (3.84) into the system (3.83), then we can get a closed loop system whose state vector is X X = f(X)+g(X)u(X)
(3.85)
After that, using the inverse coordinate transformation given by the Step 5 of this example X, = z,2 - z2 x2 = z,
we transform the system (3.85) from X space into Z space and have the Brunovsky normal form, i.e. a, = z2
z2=v
3.3
ZERO DYNAMICS DESIGN PRINCIPLE In the foregoing section, we discussed the necessary and sufficient
conditions for the exactly transforming a SISO affine nonlinear system into a
completely controllable linear system (Brunovsky normal form), and introduced how to calculate an appropriate coordinate transformation and the corresponding nonlinear state feedback law, which will be able to greatly improve stability and dynamic performance of the nonlinear system even when the system state changes in a wide range. Actually the state feedback calculated by the above method compensates or "counteracts" the original system's nonlinear characteristics and transforms it into a controllable linear
c^0
Design Principles of Single-Input Single-Output Nonlinear Control Systems
91
any
system with good dynamic performance. In dealing with this problem, getting the proper nonlinear state feedback is essential for obtaining the system's exact linearization, while getting the appropriate coordinate transformation is only the external appearance of the way in which the Q..
linearization problem is solved. In the algorithm provided in above section, seeking for coordinate transformation is closely connected with getting the state feedback. The control law obtained by the algorithm can completely and exactly linearize the original nonlinear system. It is clear that the control law designed by this method is quite complicated, which can be seen from the given examples. In this section we shall discuss and explore another design method, which could be called zero dynamics design method. This method does not need to exactly linearize all system's state equations, but just a part of them. In fact dynamic behavior of a system could be classified as external dynamics and internal dynamics. From the perspective of application, we are mainly concerned with the system's external dynamics, which should have stability and good performance. But as to the internal dynamics, what we need is only stability. According to this idea the designed control law may sometimes be more practical and simple. This is the basic idea of the zero dynamics design method. Next we will discuss this design principle and method. There are two types of the zero dynamics design method, which will be discussed one by one. G1.
pry
o
3.3.1
First Type of Zero Dynamic Design Method
Consider a nonlinear system X = f(X)+g(X)u y(t) = h(X(t))
(3.86)
a,.
The relative degree r from the output equation h(X) to the system is less than the system's order n. Under this condition, from formula (2.91) we know that the original system (3.86) can be transformed into the following normal form
.-.
III
2-1 = Zr Zr = Lrfh((D-' (Z))+LgLrj'h((D-' (Z))u
Zr+i = Lfco, i(I '(Z))
(3.87)
92
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
(3.87)
z = L f(p (10-' (Z)) where Lrfh((D-'(Z)) denotes Lrfh(X)Ix=(D-,(z), and so on. z,
...
h(X) Lfh(X) 0000,
...
O00
Zr
(D(X) _
(3.88)
L.i' (X)
zr+l
(Pr+i(X)
L z. J
The selected functions (pr+ , (p (X) satisfy (3.89)
LgcPr+1 (X) = Lg(Pr+2 (X) _ ... = Lg(p (X) = 0
And the Jacobian matrix
n(X)
Jm
ax
x=x°
is nonsingular. For the sake of convenience, we introduce here the following notations:
denotes the first r state variables in Eq. (3.87),
n the remnant n-r state
variables, ' = [z1 Z2 ... Zr ]!
n=[zr+l zr+2 ...
(3.90) z,,]T
(3.91)
Then Eq. (3.87) can be written as Z, =z2
(3.92)
Zr- = Zr
z, =a(t, n)+b(b, n)u n)
n
where L fh(X)j,=0-,(4'
n) = Lf h((I)
' (r , n))
b(t, Ti) = Lg Lr!' h(X)I x=a -,(;, n) = Lg Lf' h((D '
(3.93)
n))
and
qr+l(r'f n)
Lf(Pr+,(b'(`', n)) (3.94)
n) q,,
n)
Ti))
Design Principles of Single-Input Single-Output Nonlinear Control Systems
93
Generally speaking, we can always choose the output equation h(X(t))
which is equal to zero at the equilibrium point X°, that is h(X)=O. Therefore the output y(t)=h(X) is the actually dynamic deviation of the practical output (dynamic response) from the output at an equilibrium point.
If we use the control means to impose on that dynamic deviation of the output of the system keeps zero at any time, i.e.
y(t)=h(X(t))=0
0
then from external dynamics of the control system we can consider that the system is so stable that under the influence of any disturbance the output of the system does not change. Let us take a power system as an example to explain this point. As to the generators connected to a power system, we hope that they can keep their voltages V, constant at given values V., under the influence of disturbances. The output equation y(t) = h(X(t)) = V, (t) - VxcF can be chosen to this end. In this case if we make
y(t) = h(X(t)) = 0, t >- 0, then the purpose to keep the generator's terminal voltage constant under the influence of disturbances will be fulfilled. Here is another example. To the generators in a power system, we hope that the generators can not only keep their terminal voltages constant, but also hope keep their rotor speeds invariant. Then we may choose the following output equation y(t) = h(X(t)) = (V, (t) - V,EF )2 + (w(t) - w°)2
where w° is the rated rotor speed of a generator. For the time being, we will not discuss whether or not the requirement
o=.
of , y(t) = h(X(t)) = 0 for t >- 0, can be actually fulfilled in practice but clearly it is reasonable and logical that the condition which makes the external dynamics (output) of a system zero at any time is regarded as an 0
objective function, i.e. a performance index of a control system in controller design. As we have mentioned above, the condition of the system output
p.,
w`<
..k
keep at zero at any time has two meanings: the first is that the "external dynamics" of the system is asymptotically stable; the second is that the system's output has the optimal dynamic characteristics. Since zero is less than or equal to the minimum of the quadratic performance functional, the ce,
index of y(t) = h(X(t)) = 0 must be an optimal one. It is easy to know that in such a condition the whole control system will be stable with optimal output if the "internal dynamics" of the system is stable too. Since y(t) = z, (t) has been set to zero at any time, under this condition from Eq. (3.92) we then obtain
94
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS dZl
(t) = 0 dt
z2 (t) = z3 (t) =
dz2 (t) = 0
t>0
dt
Similarly, the first r components of coordinates Z are (3.95)
C(t) = [zl (t) z2 (t) ... Zr (t)l T = 0
for all t >- 0, and there exists
.I.
(3.96)
zr (t) = 0
Under this condition, from Eq. (3.92) we know that control variable u can be determined by the following formula, n) + b(t, rt)u = 0
(3.97)
Obviously, here b(4, n) * 0 , therefore we can solve the Eq. (3.97) for u and obtain the state feedback law U
a(Z)
- Ljh(X)
b(Z) z=m(X)
LRLJ'h(X)
(3.98)
Moreover, the first r equations of system (3.92) will vanish. In this way (3.99)
'i= q(O, Ti)
Expanding the above formula, we have Zr+l =
Zr+I,..., Zu)
(3.100) Z,, = q,,
(0,...,0, Zr+1,...,z,,)
From formula (3.94), dynamics (3.100) can be specified by zr+1 = Lf (pr+1(,D-' (0, n))
(3.101)
z = LJq (,D-' (0, 3l))
coo
cal
'C3
'"'O
.-.
.fl
`-'
CAD
Logical inference tells us that since external dynamics of the system equals zero under the effect of the control strategy as (3.98) shows, the differential equation set (3.101) describes actually the internal dynamics of the system. Those equations which determine internal dynamics of the system are called the zero dynamics equations of the original system (3.86), or are simply called "zero dynamics". If the zero dynamics of system (3.86) is stable, then the whole system must be stable by adopting the control law (3.98), and the output variable y(t) will keep constant under any disturbance. But we know
that for practical control system, the condition of keeping the system's output zero at any time is only the ideal index we hope to reach in the design.
However for any practical control system in operation there exist such factors as failure areas, limited effects and time lag, etc., which are not
Design Principles of Single-Input Single-Output Nonlinear Control Systems
95
considered in design. So the index of ),(t)=0, t can hardly be accurately achieved. Now let us use an example to explain the above first zero dynamics design method. Example 3.4
Perform the following system's zero dynamics design. 3
-x2
X= -x2
0
+
-l
u
The output is y(t) = h(X) = x, (t)
Step 1.
For the given system we calculate 0
Lgh(X) _ [1
0 0] -1 = 0
Lgh(X)=x3-x2 0
LgLfh(X)=[0 -3x22 1] -1 =3x2+1 2
Therefore we know that the relative degree of the given system r = 2. From formula (3.88) we can calculate the first r (r = 2) components of coordinate transformation z, =,p, (X) = h(X) = x, 3
zz = (02 (X) = L fh(X) = x3 - x2
Step 2.
Select the n'h (n = 3) component of coordinate transformation Z3 = q3 (X)
where q'3 (X) should satisfy the equation LfV3 (X) = 0
43
[
ax,
ag)3
a2
aQ3 ax3
So the equation which (o3 (X) should satisfy is
_aP3+43=0 82
The solution is
ax3
96
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS S03 (X) = xz + x3
Then we get the coordinate transformation Z, = x,
Z2 = x3 - x23 z3 = x2 + x3
caw
Step 3. Calculate the zero dynamics of the system. From Eq. (3.101) we know that the zero dynamics of the system should be calculated as follows: first we calculate x3 - xz3
Lfcp3(X)=[0
1
1]
=x; -(x2+x3)
-x2 z X1
- x3
According to the calculated coordinate transformation Z = 1(X) we transform the X in above formula to Z -z3
Lf0'3(X)Ix=m (y)
From (3.101) we know Z3 = Lf,P3(cIi'(0, n)) _ -z3
Therefore the zero dynamics is Z3 = -Z3 ,..'
As a result we know that the zero dynamics of the system is asymptotically stable and so is the whole system. Step 4. Seek for control strategy. According to formula (3.98), in order to obtain state feedback law we firstly calculate x3 -
3
x2z =3x2+x?-x3 X1z - x3
And LgL fh(X) has been calculated in Step 1 LgL fh(X) = 3x22 +1
Finally from (3.98) we obtain the control law u as
u= - x; +3x2 - x3
97
Design Principles of Single-Input Single-Output Nonlinear Control Systems
3.3.2
Second Type of Zero Dynamic Design Method
Using the first design method needs to solve the partial differential system r+1<_i<_n
Lgrp;(X)=0
(3.102)
to get zr+, = rpr+, (X), , z. = op (X). If we want to avoid solving the partial differential equation set (3.102), generally we should get the incomplete normal form as shown in Eq. (2.95). For the sake of completeness, we recall the normal form -1 - -2
(3.103) =r = Lrfh(4)-'(Z)) +LgLrr'h(c-'(Z))u rt)u n=
where
q,, (Z)
Lfpr+1
=
q(
(" '(Z))
qn(Z)
Lfcpo(ID
Pr+] (Z)
LgcGr+i (D -'(Z))
P (Z)
-'(Z))
Lfh(0-' (z)) denotes Lfh(X)lx,. ,(Z) . The other notation should be interpreted in a similar sense.. t,..
In order to get the zero dynamics of the system and the control law which make the output y(t) = h(X(t)) = 0
in Eq. (3.103) we define be obtained as
t >- 0
,(t) = 0 and zr(t) = 0, thus the state feedback can
u=-
Lfh(X)
(3.106)
LgL'I'h(X)
Then the zero dynamics of the system is Lrh((D-'(0, n)) 'e'
4= q(0, n)-P(0. n)
LgLrf'h((D -'(0, q))
As in (3.92) and (3.93), we still use the following symbols
(3.107)
98
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
a(
n) = L
b(
n) = LgL'f'h(0
r1))
(3.108)
then Eq. (3.107) can be written as n= q (0 , 4) - p (0, 4) a(0'
1i)
(3 . 109)
b(0, r1)
The above are the state feedback and zero dynamics of the system obtained
by the second design method. As in the first method, the stability of a specific system's zero dynamics equation set should be tested. Let us take another example to explain the second zero dynamics design method.
Example 3.5
Find the zero dynamics and the state feedback of the
system
i=
U
1: +x2
.s.
y=h(X)=x4 First, to the given system we calculate Lgh(X) = Lgx4 =0
LgL fx4 = 2(1 + x3 )
L fx4 =X 2 + xz
coo
Therefore we see that LgLfx4 #0 is valid for any x except x3 = -1, i.e. we can find a group of local coordinate transformations such that the relative degree of the system r = 2 excluding the points of x, = -i . For the same reason in Example 3.4 we can calculate the first r = 2 components of the coordinate transformation z, =,p,(X)=h(X)=x4 zZ =Vz(X)=Lfh(X)=x,z +x2 1=1
Then we select the last two components of the coordinate transformation =(3(X)=x3 Z3 Z4 = V4 (X) = x, t.-
In order to check whether the selected mapping Z = c3(X) is a local diffeomorphism, we should check whether the Jacobian matrix
ax
-
0
2x,
00 00 1
.-.
a0( X)
1
-0 1000 01 0
Design Principles of Single-Input Single-Output Nonlinear Control Systems
99
n"1
Vi'
is nonsingular. Obviously this matrix is nonsingular to all X, therefore the selected mapping Z = (D(X) is a proper coordinate transformation. Now we calculate the inverse mapping x = (D-' (Z) of Z = c3(X) xl = Z4
x2=22-242 x3 = Z3
x4 = Zl CS'
From Eq. (3.103) we know that in the new coordinates Z the system appears as it = Z2
i2 = L22 h(C)-' (Z)) + Lg L f h((D- (Z)) u
Z3 = Lf93 ((D-' (Z)) + Lg'P3 ((D-' (Z)) u
Z4 = Lf'P4 (0-'(Z)) + LgcP4 ((D-'(Z)) U
For the above equations we calculate the Lie derivative L2fh(X) = Lf (x; + x2) = 2x, (x2 - x2) + xl
to set L2f h( ' (Z)) = 2z; (z2 - 2z;) + z4
Computing L f(P3(X) = Lfx3 = -x3
we have LftP3 (d-' (Z)) = -z3 (IQ
Computing
Lg(P3(X)=Lgx3 =1
we know Lg'P3 (t-' (Z)) = 1
Computing L1co4 (X) = L fxl = xl (x2 - xl )
yields -'(Z)) = z2z4 - 2z4
up to this point we can calculate Lg'P4(X)=Lgxl =0
') where Lgc04 (X) = 0 is only a special case.
100
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS Cad
C'"
After completing the calculation of Lie derivative and coordinates transformation, we can write the system model in the new coordinates Z as i, = z2 z2 =2 Z4 (z2 -2 z4) + z4 + 2(1 + z3 )u
Z3 - -Z3 + U
-22
Z4 = ZZZ4
4
In the above equations we set z, = z2 = 0 and i2 = 0 , thus obtain
2:4(:2 -2:4)+ 4
u(Z)
2(1+:3)
So we have z4 (4z4 -1)
U(0' n) =
2(1 + z3 )
Substituting the expression of u(0, n) into the above equations related to i3 and i4 , and setting z, = z2 = 0, we can get the zero dynamics of the system Z3
= -Z 3 +
z4(4z4 - 1) 2(1 + z3 )
Z4 = -2Z43
Next, we should discuss the zero dynamics stability by means of numerical
analysis method. If the zero dynamics is stable, then the whole control system is stable. The numerical analysis will be omitted here for the sake of space limitation. In this example, what we still need is calculating the state feedback law u(X). u(X) can be obtained by the transforming Z in the above expression
of u(Z) into X, i.e.
--
2z4(z2 -2z4)+z4 2(1 + z3 )
Z=4'(X)
Alternatively we can directly use formula (3.98) U
=-
Lfh(X)
In both cases, we get the same result 11 -
-2x1 (x2 -x1)+x, 2(1+x3)
Design Principles of Single-Input Single-Output Nonlinear Control Systems
3.3.3
101
Discussion of Some Problems
About the control law Comparing (3.98) with (3.106) we can see that the state feedback laws obtained by adopting the first design method and the second design method 1.
0.`c
have the same form of expression
u=-
L'fh(c-'(Z)) LgL7'h((D-' (Z))
z=m(X)
"fit
However, the coordinate transformations selected by the two methods are coo
different, so the state feedback laws are generally not the same. (or
The above state feedback expressions can be directly written in the original coordinates X as L'fh(X)
u=-LgLf'h(X) y.,
Since theoretically this state feedback has the ability to keep the output y(t) = h(X(t)) of the system at zero, namely, if the above state feedback u(X) is used, the X(t) will be forced to satisfy the following constraints y(t) = h(X) = 0 ,y(t) = L fh(X) = 0
Y(r-')(t) = Lf'h(X) = 0
where r is the relative degree of the system. Up to this point we arrive at the following proposition.
Proposition 3.1
Consider the system X(t) = f (X(t)) + g(X (t))u(t)
y(t) = h(X(t))
with relative degree r. The state feedback which forces the output y(t) to equal zero is Lf h(X(t)) u(X(t))
LgLf'h(X(t))
If the above control strategy is utilized, the state X(t) of the system will be restricted in the following subset
S2= {XeR"I h(X)=Lfh(X)= =L'f'h(X)=0) p
102
ca. :0
"
The advantages and disadvantages of the two design methods In the two design methods, because the (r+l)"' through nt' components M.
0
0
to
2.
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
0 0 "
of the coordinate transformation Zr+l = 'Pr+l (X) Zn = (P, (X) En
CD
CD
..q
CD
are chosen differently, the zero dynamics of the system are not the same. CD
r+1<-i<-n CL
CL
IA
Lgrp;(X)=0
IA
CD
:3
0
Using the first method needs to solve the partial differential equation set
OR
CA
COD
0 qt
N
0 CL
g-
0
0
O0.
ca.
o
a
but the zero dynamics is simpler than one induced by the second method and so is the stability analysis of the zero dynamics. That is why we say that the two methods have their own advantages and disadvantages. Here we use an example to explain the difference between the two methods for the different CD
W In
system's zero dynamics. Recalling the system in Example 3.4, we use the El
second design method and choose z3 =cp3(X)=x2
The coordinate mapping is z, = X1
Z2 =x3 -x23 z3 = x2
Its Jacobian matrix 1
0
0
0
1
C.
J= 0 - 3x2 1 0
C/)
is nonsingular. So the selected coordinate transformation is valid. The equation for z3 is =3 = L r c3 (X)I X=0-1 (Z) + L, P3 (X)1x=0-1(Z) u((D-' (X))
Calculating Lf -p3 (X) = Lf x2 = -x2
Lgx2 = -1
we have Z3 = (-x2 - u)Ix=,-I(z)
From the expression of Z = cb(X) obtained above we can find out the inverse mapping
Design Principles of Single-Input Single-Output Nonlinear Control Systems
103
X, = Zl x2 = z3 X3 = Z2 + Z3
The control u expressed by the coordinate Z is U(0- (Z))
L2 h((D-' (Z))
x , + 3x2 -x3
Lg L f h((I)-' (Z))
3x2 +1
X=4 (Z)
zi +3Z3 -Z2 -Z3 1+3z3
From the above Z3 =(-X2 -u)IX=m''(Z)
we know that the zero dynamics of the system is (since z, = z2 =0) Z3 = -Z3 +
3z3 - z3
1+3z3
However from Example 3.4 we know that the zero dynamics of the system obtained by the first design method is very simple, that is Z3 = -Z3
Achieving the simple zero dynamics is at the cost of solving partial differential equation Lg'P3(X) = 0
3.
Remarks CI!
When we use the second method of zero dynamics to design a controller, from (3.107) we know that the zero dynamics of the system is n= q(0, n) - P(0, n) u(cp-' (0, n))
which implies that we should substitute
= 0 and
_ L'Jh(fi-'(0, xi)) U=-LgL,'h(3-1(0,
n))
into the last equation of (3.103), which is
n)u
However, this does not mean that the system's state feedback can be obtained by the inverse transformation as u
Lfh((D-'(0,n)) L5 LJ' h((D-' (0, n))
Z(x)
104
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
The state feedback law should be complete, i.e.
Lfh(O-'(4, n))
_
u
n))
LRLf'h(X)
z=G(x
Precisely due to the above complete control strategy the controller is able to to be zero, i.e. b =O. The former (complete control strategy) is constrain the cause; the later ( = 0) is the effect. We must not substitute = 0 into the control strategy. 4.
'C7
The limitation of the zero dynamics design method The zero dynamics design method does have its limitations. The control
law obtained by the design method cannot assure that the differential N
equation set of zero dynamics is stable. If the zero dynamics of the system is 'C7
unstable, we have to abandon the method and use the exact linearization design method.
3.4
ZERO DYNAMICS DESIGN METHOD FOR LINEAR SYSTEMS
Just like a straight line can be taken as a kind of special curve, we can consider a linear system as a kind of special nonlinear system. Therefore the zero dynamics design method as mentioned above could also be applied to a linear system. As discussed previously, a control system designed by the zero dynamics method is optimal with respect to system's output response. Compared to the LQR method, this zero dynamics algorithm has its own
advantages: it can avoid problems of randomly selecting the weighting matrix and can relieve us from solving Riccati equation. Now let us discuss this design method. Consider a system described by equations of the form cam
X = AX(t) + Bu(t) y(t) = CX(t)
(3.110)
where X(t) is an n-dimensional state vector; A an n x n constant matrix; B an n-dimensional constant vector; C an n-dimensional constant row vector; y(t) a single output variable; u a control variable. Comparing the above state equation and output equation with Eq. (3.1) we can immediately write f(X) = AX
g(X) = B
h(X) = CX
Design Principles ofSingle-Input Single-Output Nonlinear Control Systems
105
Without loss of generality, we assume that the relative degree r < n. .A+
According to the method discussed in Section 3.3, at first we choose the first r elements of the coordinate transformation as follows z1 = h(X) = CX
z2 =Lf(CX)=CAX
(3.111a)
Zr = Lj' (CX) = CA'-' X
Then choose the last n - r elements of the coordinate transformation. The simplest selection is
(3.111 b)
.`3
rte,
Putting the above two parts into (3.111) completes a linear transformation, which is denoted as Z(t) = TX(t)
(3.1 t2)
where [C
CA CA 2
T=
CA'-'
(3.113)
(1, 0, ..., 01
(0,1,0,...,0] [o`-.
,0,...,0]
I,-r
is a nonsingular matrix. The inverse transformation of (3.112) is X(t) = T-'Z(t)
(3.114)
From the second zero dynamics design method described in Section 3.3.2 we know that in order to discover the control law and zero dynamics of the system, we need to calculate the Lie derivative L'jh(X)=L'f(CX)=CA'X sec
L5L'f'h(X) = LR(CA'-'X) = CA'-'B and
(3.115)
.
106
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS L ff Op', (
==
a(-') AX = A;X
ax
LgPr++(X)= X)B=bi
(3.116)
15 i5n-r
where A, is the i' row vector of matrix A ;
b,
is the tt component of
vector B . .ice
Now, substituting the above calculating results (3.115) and (3.116) into dynamic system (3.103) and applying the transformation (3.114), we can get
the state-space description of the system in the new coordinates Z as follows
it
= Z2
(3.117)
Zr- = Zr
ir =CA'XIX_-'z+CA'-'Bu and
Zr+1 =AI XI X=T-'Z +blu
Zr+i = Ai
XIX=T-'Z
(3.118)
+biu
Z=
+bn_ru
Eq. (3.118) can be written in its compact form (3.119)
i1= ACT-'Z+B;u
where A, is the (n-r)xn matrix composed of the first n-r rows of matrix A ; B , the vector composed of the first n-r components of vector B . In order to get the control u and the zero dynamics of the system we J`+
set = [Z' (t) Z2 (t) ... Zr (t)]T = 0
t>-0
This also means that L1.
Zr =0
Thus, from Eqs. (3.117) and (3.112) we obtain the control strategy u in the form (3.120a)
u = -KX(t)
(3.120b)
CA'-'B
.-.
U=- CA' X(t) namely, the state feedback is
where the state feedback gain vector K is an n-dimensional constant row
Design Principles of Single-Input Single-Output Nonlinear Control Systems
107
vector, and is given by K=
CA' CA'-'B
(3.121)
.'3
To get the zero dynamics of the system, we should transform the X in formula (3.120) into Z = [ , ii], namely u,
=-KT-'Z=-KT1141
In the above formula we define
(3.122)
= 0 and K Z = KT-', then we have
-KZ[ rJt
u
C3.
If we divide the KZ into two parts, one part corresponds to to ii such that
KZ =[K,
and the other (3.123)
then we can get the expression of u n
un= Substituting the above formula into Eq. (3.119), we have the zero dynamics of the system as jay
n=
A;T-' the (n-r)x(n-r) and (n-r)x(n-r) constant matrices, respectively. The two terms on the right side of the above equation can be combined so that
M''
n= Q n
(3.124)
CDD
.x.
where Q is the (n-r)x(n-r) constant matrix and is completely determined by the matrix A, vector B, output vector C of the original system (3.110), feedback gain vector K. and the transformation matrix T. The Eq. (3.124) represents the zero dynamics of linear control system (3.110). From the principle discussed in Section 3.3 we know that: if all the real parts of eigenvalues of the matrix Q in (3.124) are negative, then the whole linear control system is asymptotically stable. Now let us explain the above method with an example.
Example 3.6 Consider the system
108
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
X=
-1
0
0
-1
0
0
u
(3.125)
+ 1.10) -0,J
I.,X:,
Y(t) = x2 + x3
(3.126)
00U
Try to adopt the zero dynamics design method to find out the state feedback and check the whole control system's stability. Step 1. Choose the coordinates mapping Z, = y(t) = x2 + X3
Step 2.
Calculate i, ii = Lf(x2 +x3)+ Lg(X2 +x3)u
then we have i, = -x2 + 2u
(3.127)
Step 3. From the above calculation we know that the system's relative degree r =1. Choose the (r+l )`h element through the e one of the mapping Z2 = X,
Z3 =x2
Now we choose the coordinates mapping Z, = X2 + X3 z2 = x,
Z3 = x2
Its inverse is X, = Z2
x2 = z3
x3 = zi - z3 0
Obviously, the chosen mapping is bi-directionally one-to-one corresponded, and smooth (having continuous partial derivatives of any order). So without calculating its Jacobian matrix we know it is a diffeomorphism.
Step 4. Obtain the dynamic system expressed by equations in the coordinates Z. We compute 2 = L f (x,) + Lg (x, )u = -x,
.3 = Lf (x2) + Lg (x2)u=-x2 +u
In the coordinates Z, the system appears as i, = -z3 +2u i2 = -z2 Z3 = -Z3 + u
(3.128)
(3.129)
109
Design Principles of Single-Input Single-Output Nonlinear Control Systems
Using the symbols defined in Section 3.3, here we have
=z,(t)
and
i=[z2(t) Z3(t)]T Step 5. Calculate the state feedback law. In Eq. (3.129) setting i, = 0 , we can get the state feedback law u = 2x2
(3.130)
Therefore uZ is uZ = z3 /2.
Step 6.
Calculate the zero dynamics of the system. Substituting
formula (3.130) into (3.129) we can get the zero dynamics n= Q n, which is 2 =
2
3=-23
Then we know that matrix Q is
-1 Q
0
0 -1 2
Its eigenvalues are A, = -1, .1,2 = -1/2 . Therefore the zero dynamics of the system is asymptotically stable and so is the whole system.
3.5
DESIGN OF DISTURBANCE DECOUPLING
Undoubtedly, we hope the control system designed has the excellent capability to resist exogenous disturbances. Ideally the output response of a
system will not be influenced by disturbances at any time. If a control system indeed has such a characteristic, we say that the system is a disturbance decoupling one. This section studies under what conditions a nonlinear system can have the characteristic that its output is decoupled from
disturbances by means of control, and how we can get the analytic `ti
in'
representation of the control law. In control theory, output decoupling from disturbances is called DDP (Disturbance Decoupling Problem). This problem is mainly discussed in this section. Let us first discuss the sufficient condition for output decoupling from disturbances. Consider a system modeled by equations of the form X(t) = f(X) + g(X)u + D(X)p y(t) = h(X)
(3.131)
110
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
,O.
where p represents the disturbance variable, an undesired input; D(X) the ndimensional vector field directly related to disturbance. The meanings of the other symbols are the same as those discussed in the previous sections. Now the problem is to find out under what conditions there exists a state feedback u = a(X) + b(X)v
(3.132)
L7.
such that output y(t) is decoupled from disturbance p. Without loss of generality, here we suppose the relative degree r of the system is less than
the system's order n. Let the it Lie derivative of h(X) along f(X) and then along D(X) equal zero, namely (3.133)
r+%
LDL'fh(X)=0 for 0<_i<<-r-1 and X near X° (IQ
(-)
Choosing the transformation z, = h(X) as we did for obtaining linearization normal form in Chapter 2, we have iI =
a (X)(f(X)+g(X)u+D(X)p)
The above formula can immediately be written as p,;
i, = Lfh(X)+Lgh(X)u+Loh(X)p If the relative degree r > 1, the assumption of formula (3.133) is considered, then we know that the last two terms on the right side of the above formula are equal to zero. Then we have i, = L fh(X) = z2
(3.134a)
A similar situation will happen for all other subsequent equations, and thus we have i2 = Z3
(3.134b) Zr-I = Zr
Since LgLrf'h(X) # 0,
0, for Zr we still get it = Lrfh(X) + LgLrf'h(X)u .
Then we choose the remaining coordinate mappings as follows Zr+l = Pr+1 (X)
(3.135) Z = (P,, (X)
As described in Section 3.3.1, we make ((pr+ ,rp,,) in (3.135) satisfy Lsp,+;(X)=0
for 1:5 i<_n-r
(3.136)
Summing up the above, we can obtain the system dynamic model in the new
Design Principles of Single-Input Single-Output Nonlinear Control Systems
111
coordinates Z ZI
= Z2
Z2 = Z3
Zr-1 = Zr
Zr = (Lrr h(X)+LgLrf'h(X)u)I x=m' (z) Zr+1
(3.137).
1(X)+LDcOr+I(X)P)(x= -i(z)
.-r
Z Eqs. (3.137) can also be written in its compact form Z1
= Z2
Z2 = Z3
(3.138) Zr-1 = Zr
NC'
Zr
= a(t , rl) + b(b, tl) u 'a.
A =q(C,
rt)P
where Lfh(-D-'(Z))
a(C, r) = Lfh(X)1x=0-,(Z)
b(b, rl) = LgLr-'h(X)I f
(3.139)
v(z) = L g
LfcOr+1 (X)
(3.140)
x= (z) LD'Pr+I (X )
s( ' n\ =
LD(Pr+2 (X )
/
(3.141)
ix=m-.(z)
In (3.138) we define u
_ a(t,
rl)
+
1
(3.142)
Then Eq. (3.138) becomes the form as Zl
= Z2
Z2 = Z3
Zr-1 = Zr
(3.143)
112
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS = v
rt)p
rt
Expanding the above formula, we have Zl =Z 2
Zr-1 = Zr
(3.144)
i, =V
Zr+1 =q1(Z)+s1(Z)p Zn = q,,-r (Z) +
(Z)p '0.1
Col,
From (3.144) we know that disturbance variable p has no connection
it can not influence z, (t) . This is because the z,,]T disturbance p can only affect and influence ri = [zr+, , but the
with
z , (t) ,
namely,
relation between rt and z, (t) has been cut off by the equation it (t) = v in
qty
`O^.
(3.144). We will, of course, not forget that z, (t) is the output y(t). So (3.144) specifies a control system whose output has been completely decoupled from the disturbance p. We can use the structural diagram in Fig. 3.3 to show the control system with its output separated from disturbances. From Fig. 3.3 we can see clearly that the disturbance p can not affect z, (t) , i.e. can not influence output At). '"'
-fl
z1 (t) = y(t) gL f 1h(
L'fh(QV'(Z))
1(Z))
y(t)
I z,(t)
I Z'(t)
I Zr+1 (1)
z,+j ( )
Figure 3.3
Structural diagram of a system with outputs decoupled from disturbances
We can draw the following conclusion from the above discussion Consider the system
Design Principles of Single-Input Single-Output Nonlinear Control Systems
113
X(t) = f (X) + g(X)u + D(X)p y(t) = h(X)
where p is the disturbance variable. If the system satisfies the condition
0<-i
LDL'fh(X)=0 om.
where r is the relative degree of h(X) for the system, then the control strategy L'fh(X)
1 LgLf'h(X)+
(3.145)
.-.
u
LgLf'h(X)v
is able to make the output y(t) independent of the disturbance p.
o00
So far we have obtained the sufficient condition for a SISO affme nonlinear system to have the solution for the problem of output decoupling from the disturbance. Now let us show that this condition is also necessary. Let (3.146)
u = a(X) + b(X)v
tip
C1.
To get the control law which can make the output decoupled from the
disturbance at any time, we substitute (3.146) into the system model (3.131), and arrive at X(t) = f (X) + g(X)a(X) + g(X)b(X)v + D(X)p y(t) = h(X)
(3.147)
Cs.
It has been assumed that the output variable y(t) is independent of the disturbance p. Of course, this assumption also holds under the condition of v = 0. Therefore Eq. (3.147) can be written as X(t) = f (X) + g(X)a(X) + D(X)p y(t) = h(X)
(3.148)
Now we can take the f(X)+g(X)a(X) in (3.148) as the new "f(X)" denoted by f(X) ; Similarly, take D(X) as the new "g(X) " denoted by g(X) ; and take p as the new "u " denoted by u . Thus, Eq. (3.148) could be rewritten as X(t) = f(X) + g(X)u y(t) = h(X)
(3.149)
Eq. (3.149) is still in the form of a normal affine nonlinear system. We know from Section 2.2.6 Y(t) = a
(X)) -
aaX)(f(X)+g(X)u)
= L Jh(X) + Lgh(X)u
From the above formula we can see: if the condition
(3.150)
114
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
(3.151)
Lgh(X) = 0
is fulfilled, the output y(t) must be independent of u (disturbance p). Now suppose the condition has been satisfied, we calculate
y(t)=
=Ljh(X)+LgLjh(X)u
From the above formula we know that if the condition (3.152)
LLL jh(X) = 0
is satisfied, the y(t) must be independent of p. The rest may be deduced similarly, until (3-153)
y('' (t) = L jh(X) + Lg L'' ' h(X )u
Obviously, if y(t) is decoupled from p, then we must have (3.154)
LgL'i'h(X) = 0
Summing up equations (3.151) through (3.154) we can draw a conclusion: if
feedback u = a(X) + b(X)v renders y(t) decoupled from p, the following condition must be satisfied.
0
LgL'jh(X)=0 or
0<_iSr-1
LgL(f+g°)h(X)=0
(3.155)
In (3.155) a(Lf'h(X)) f(X) + a(Lf'h(X)) g( X)a (X) ax ax 0:5 i:5 r -1 = L'fh(X) + LgL'' 'h(X)a(X)
L'(f+R°' h(X )
==
(3.156)
We know from the definition of relative degree that in the above formula LgL'' 'h(X) = 0. In fact formula (3.156) becomes L(f,g°)h(X)=Lfh(X)
05iSr-1
(3.157) r-+
Substituting (3.157) into (3.155), the necessary condition of the output decoupling from the disturbance becomes
(3.158)
0:5 i<- r -1 S3.
LDL'fh(X) = 0
Up to this point we conclude that (3.158) is the necessary condition for the output decoupled from the disturbance. So formula (3.158) is a necessary and sufficient condition for decoupling output from the disturbance. Let us use another way to express the necessary and sufficient condition
Design Principles of Single-Input Single-Output Nonlinear Control Systems
115
given by (3.158) for y(t) being independent of p. We know LDL fh(X) = a(LI
h(X))
A
D(X)
0:5 i< r-1
If we define a(L jh(X))
ax a(L fh(X)) LI(X) =
(3.159)
ax a(Lrj' h(x)) ax
we can immediately realize that the necessary and sufficient condition for output being independent of disturbance shown by (3.158) can be written in the following concise form (3.160)
O(X)D(X) = 0
Now let us explain the geometrical meaning of the above formula.
Formula (3.160) indicates that: matrix rl(X) multiplying vector D(X) equals a zero vector for each point in a neighborhood of X°. This means that every row vector of Q(X) is orthogonal to the vector D(X) in the °a.
neighborhood of X°. We know that if two vectors X and Y are
0.a
orthogonal to another vector Z, then all of vectors on the plane spread by
these two vectors are all orthogonal to the vector Z. Plane is a onedimensional subspace, namely, all vectors in the (2-1)-dimensional subspace 540.
spread by vectors X and Y are orthogonal to Z, and Z is, of course, not included in the (2-1)-dimensional subspace. By analogy, all r row vectors in S2 S2 21 ,'2 r form a (r -1) -dimensional subspace, which is orthogonal to D. Obviously, D is not included in the (r -1) -dimensional subspace, which can be expressed, in geometric terms, more precisely as "vector D is included in the null space of matrix S2 ". Through the above geometric explanation of condition (3.160), we can sum up the necessary and sufficient conditions for decoupling the output from disturbance by the following proposition.
matrix 0 are orthogonal to D, then the r vectors
Proposition 3.2
Consider a system %(t) = f (X) + g(X)u + D(X) p
y(t) = h(X) i«:
where D(X) is a vector field, p a disturbance. Suppose the relative degree
of the system equals r in a neighborhood of X°, then the output of the
116
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
system can be decoupled from the disturbance via the state feedback, u
=-
L.h(X)
1
LgLrf'h(X) + LgLf'h(X)
(3.161)
v
if and only if, for all the X in the neighborhood of X°, vector field D(X) is contained in the null space of the matrix ah(X)
ax a(L fh(X))
ax
S2(X) =
3(L1 h(X ))
0
-ax
The v in the expression of state feedback u can be obtained by using the optimal control design method of a linear system which is in the form it
= Z2
(3.162)
The optimal "control" is
v=-k;z, -k;z2 -...-krzr
(3.163)
= -k, where K4 = [k,
kz
... k.
is the optimal gain vector of the linear system (3.162), which has the form (3.164)
K' = BTP`
for weighing matrix R = I . In the above formula B is the r -dimensional vector 0.1
B=[0 0
... 0
If
P' is the solution of the Riccati matrix equation
ATP+PA-PBBTP+I=0 I is identity matrix, and the matrix A is
Design Principles of Single-Input Single-Output Nonlinear Control Systems
117
00
1
...
010 0 .
A=
.
.
000
... 0 ...
1
000...0 Substituting (3.163) into (3.161), we can obtain the control strategy
+kLfh(X)+k,h(X)
u=-L'fh(X)+kL'f'h(X)+
L5L'f'h(X)
(3.165)
which could render the output response of the system decoupled from the disturbance. The steps taken in this way we follow a principle as formulated
in the following steps, which are not very complicated: firstly set up a mathematical model of a practical dynamic system and transform it into the form shown in (3.1); secondly determine the relative degree of the system; thirdly calculate the every order Lie derivative of h(X) along f(X) until the
(r-1)' order Lie derivative; fourthly calculate the gradient vectors
"C7
8(f fh) / A of the above every Lie derivative to get the matrix Q(X) ; finally check whether the condition of proposition 3.1 is satisfied. If so we can use state feedback (3.165) to get a control system whose output has the strong robustness to the disturbance. Since Proposition 3.1 is a necessary and sufficient condition, failure of the condition to hold implies the given system does not have the solution of disturbance decoupling. For the system of this
kind, we can try to change the method of modeling, or to change the fib= 45.0
expression of output y(t) based on the corresponding physical concepts, and then redesign and recheck.
Next, let us apply the principle mentioned above to solving the PP,
disturbance decoupling problem (DDP) of a linear system. Consider the linear system X(t) = AX(t) + Bu(t) + Dp(t) y(t) = CX
(3.166)
^c7
where B and D are the n-dimensional constant vectors; C is an ndimensional row vector; u(t) e R the control variable;' p(t) e R is the disturbance variable. Comparing (3.116) with (3.1) we know f(X) = AX(t) g(X) = B D(X) = D h(X) = CX(t)
From the system relative degree definition in Chapter 2 we know that: if the
118
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
condition (3.167)
LgLj' (CX) # 0
holds, then the system (3.166) has the relative degree r. Under certain special conditions, if LL(CX) # 0 we know the system's relative degree
r=1 From Proposition 3.2 we know if system (3.166) satisfies the condition (3.168)
S2(X)D = 0
30C
the given system is a system whose output can be decoupled from rig
disturbance via feedback. In order to check whether the condition (3.168) holds, we should calculate matrix S2 (X) . Since f(X) = AX, from the results of calculation we know L'fh(X)=Lf(CX)=CA'X
(3.169)
From (3.159) we know that the matrix S) (X) is in the form ah(X)
a(CX)
ax
ax
a(Lfh(X))
CA
ax
CA2
a(CA'-' X))
CA'-'
(o...
fl(X) _
ax :-:
a(L'j'h(X))
1
(3.170)
ax
L
ax
rc
a(CAX)
Then the condition (3.168) can be written as
rc CA
D=O
(3.171)
---
___
CA2
That is to say if the condition (3.171) holds, the linear system (3.166) can become a system with decoupling of output y(t) from the disturbance p(t) 0.1
via state feedback. The state feedback law has been given by (3.165). According to (3.169) the numerator polynomial on the right side of formula (3.165) can be exactly written as
(CA' +k;CA'''
(3.172)
Then let us calculate the denominator on the right side of (3.165). LgL'j'h(X) = Lg (CA'-'X) = CA'-'B * 0
(3.173)
Design Principles of Single-Input Single-Output Nonlinear Control Systems
119
Therefore we can get the state feedback u
=-(CA' +k;CA''
(3.174)
CAr-'B
which ensures the system (3.166) the property of disturbance decoupling. A point needs to be explained. In expression (3.174), k2 ... kr ] K* _ (3.175)
K' = BTP'
where P' is the solution of the Riccati equation (3.176)
ATP + PA - PBBT P + I = 0
and A in (3.175) and (3.176) are not the matrix A and vector B in the 'Ay
state equation (3.166), they are actually
00
0
0
... I];
rxr
Now let us elaborate it with an example.
Example 3.7 Consider the linear system -1 X(t) = 0 0
y(t) _ [0
1
0
0
x,
-1
0
x2 + 1 u(t) + -1 p(t)
0
0
x3
0
1
(3.177)
]
1
1] X(t)
it,
Find the control law u = u(X) which can make the output y(t) decoupled from disturbance p(t). Step 1. Check whether the given system satisfies the condition for
Lgh(X)
aaX
)B=CB=[0
fl.
.-.
decoupling output from disturbance. (i) Find out the system's relative degree
Then we know the relative degree r =1. (ii) Calculate the n(X) in (3.170) S2(X)
and check the condition
=
a(aX) = [0
1
1]
120
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS 1
n(X)D(X)=[O
I
I11=0
1]
L I J
we can know that the given system satisfies the condition shown by (3.177). Step 2. Seek for the control u = u(X) (i) Calculate CA = [0
-1
0]
(ii) Calculate K' = k, = BTP"
1, so P'=1, k, =1.
In this example A=0, (iii) Calculate
CA°B=CB=2 From (3.174) we know that the state feedback is
-CA CB
2 X3 (t)
-,CX(t)=-2[0
0
.=i
u
0
3.6 REFERENCES I. 2. 3. 4.
D. Cheng, T. J. Tarn and A. Isidori, "Global Linearization of Nonlinear Systems Via Feedback", IEEE Trans. AC, Vol. 30, No. 8, pp. 808-811, 1985. H. Kwakemak and R. Sivan, Linear Optimal Control Systems, Wiley, New York, 1972. M. Spivk, A Comprehensive Introduction to Differential Geometry, Vol. 1, Publish or Peremish, Boston, 1970. W. A Boothby, An Introduction to Differential Manifold and Riemannian Geometry, Academic, New York, 1975.
Chapter 4
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
4.1
INTRODUCTION
In the previous chapter, the control design principle and algorithm for SISO affine nonlinear systems are elaborated. This type of systems has only one input, i.e. control variable u and one output y(t). As we know, however, multi-machine power systems are large nonlinear ones with multiple inputs
and multiple outputs (MIMO). Take a system with m generator sets for example, if merely the excitation control problem is considered, there will be
m control variables - excitation voltages; if the opening control of steam valves or water gates is also involved in addition to excitation control, there will be 2m control variables. The outputs may be the terminal voltage and rotor speed of each generator, thus the output number will also be 2m . If the control of Static Var Compensation devices and the DC transmission lines are also included, there will be more control variables and outputs. Besides
electric power systems, a good many types of control systems such as industrial robot control system and aircraft automatic control system also belong to MIMO affine nonlinear systems. Of course, these systems are much more complicated than SISO systems in both modeling and algorithm. However, many of the basic concepts and design principles and approaches
discussed in the previous chapters can be directly extended to MEMO nonlinear systems. In this chapter, the design principles of exact linearization
and zero dynamics for MIMO affine nonlinear control systems will be discussed and their corresponding algorithms will be introduced.
122
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
RELATIVE DEGREES AND LINEARIZATION NORMAL FORMS
4.2
In this chapter we will see how the concepts of system relative degree and linearization normal form discussed in Chapter 2 are extended to MIMO control systems. To be brief, systems with two inputs and two outputs will mainly be considered below. Readers can easily extend the research results to systems with m control variables and m outputs.
4.2.1
Relative Degree
Consider a nonlinear system with m inputs and m outputs X(t) = f(X) + g, (X)u, + g2 (X)u2 + y, (t) = h, (X)
+ gm (X)u",
Ym(t)=hm(X)
°+,
0C7
where X E R" ; f (X) and g, (X), i =1, 2, , m are n -dimensional smooth vector fields; u; is the i" control variable, y;(t) the i`h output, hi (X) a scalar function of X. The concept of the relative degree of the system will be introduced below.
y0,
Rio
Firstly, it should be made clear that for each output y;(t) = hi(X) there exists a corresponding relative degree r.. Thus, the relative degrees of an MIMO system form a set, i.e. ,rm}
Secondly, similar to what are described in Chapter 2, each sub-relative degree satisfies the following conditions in a neighborhood of X°, which are: (1)
LgLfhi (X)=0 Lg2Lchi(X) = 0
LgpLfhi(X)=0
k
C.]
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
123
Lg L'f-'h; (X)
Lg L'f-'h;(X) L9. L" _'hi (X)
i=1,2, ,m do not all equal zeros. The above two conditions can be easily extended out from the definition FUN
of relative degree in SISO systems. For MIMO systems, the following condition should be appended. (3) The matrix
B(X) =
Lg f 'h,(X)
Lg,L'f'h,(X)
Lg,L'I'hz(X)
Lg,Ll'hz(X)
Lg
Lg
Lj-'hm(X)
Lg.L'j
is nonsingular in the neighborhood of X°. In fact, the above condition (2) is included in condition (3).
Summarizing all the above, we can give the following definition of the relative degree set r for MIMO systems. `i7
Definition 4.1 For MIMO systems as shown in Eq. (4.1), if the following conditions hold in a neighborhood of X°, namely, for k; < r, -1, Lg,L«jh,(X)=0
i=1,2,...,m j=1,2,...,m
and the m x m matrix Lg L'f'h,(X) ... 'hz(X) ...
B(X) = Lg,LI
Lg,L71hm(X)
Lg.Lt 'h,(X) Lg.Lj 'hz(X)
... L5.Lj hm(X)
is nonsingular near X°, then r = {r r2, , rm } is the relative degree set of the system, and each sub-relative degree r; corresponds to output yr(t)=h,(X). Let us illustrate it with an example.
Example 4.1
Consider the system +I
xz z3
x, +x2 +3x2/3
X2z
z,
x,x3/z +xZx3/2
u, +
0
x,t/zxt/3 3
2
x2/3
x xl/2 3 1
+x 2
u2
xl/2 3
124
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
y, =h,(X)=x, y2=h2(X)=x2+x3 Calculate its relative degreesr, and r2 in the neighborhood of point
X°_[0
1
l]'.
According to Definition 4.1, we first calculate xz'/3
LgL°fh,(X)=[1 0 0]
=x'/3
X0
3
,21111 x2
2
x1 +x2+3x2/3 0
Lg,L fh, (X) _ [1
x2/3
0] I
=x1+x2+3x23
x32(x1+x2) x2u3
Lg, Lfh2 (X) _ [0
jV3/2 3
0
1]
1
I2 X3 1/3
2
1/2
X, +x2 +3x23
Lg,L°fh2(X)=[0
1
x22/3
1]
=X23 +42(x1 +x2)
[ 42(x1 +x2) J
From those we can see that at point X0=[0
1]T
1
.-.
Lg,L°fh,(X)=1
Lg
Lg,Lfh2(X)
is nonsingular at point X° = [0
1
Lg
L°fh2(X)
If.
Therefore, from Definition 4.1 we know that the system's relative degree in the appropriate neighborhood of X° = [0
1
If is
r={r1,r2}={1, 1}
In fact, the neighborhood of X° = [0 the X for x2 > 0,x3 > 0 .
1
If is very large and includes all
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
4.2.2
125
Linearization Normal Form
In order to make the discussion brief and without loss of generality, let us deal with systems with two inputs and two outputs. That is to say, the nonlinear system to be discussed is in the form t.0,
X(t) = f(X(t))+g1 (X(t))u, +g2 (X(t))u2 (4.2)
Y, (t) = h1(X(t)) Y2 (t) = h2 (X (t))
v,'
Assume that its relative degrees satisfy r = r + r2 = n, where n is the dimension of the state vector X. Under this condition, the coordinates mapping Z = c(X) should be chosen as z, _ q'1 (X) = h, (X)
z2 =,p2 (X) = Lfh,(X)
Zr =q'.(X)=r}-'k (X) (4.3)
Zr,+, _ 11/1(X) = h2 (X) zr,+2
a
=11/2(X)=Lfh2(X)
z,, =11/,,(X)=LJ'h2(X)
From the above mapping, we can obtain
1=
aX
X
_ ah,(x)
ax
(f ( X) + g1(X)u1 + 92 (X)u2)
(4.4)
= Lf h1 (X) + Lg hl (X)u1 + Lg=h, (X)u2
Furthermore from the definition of relative degree (Definition 4.1), if r, > 1, then Lgh,(X)=Lg2h,(X)=0 So Eq. (4.4) can be written as (4.5)
Wl = L fh(X) = 4/2
We can derive similar results for rp; until (4.6)
-1 = rPr,
As stated by the definition of relative degree, are at least not all equal to zeros. So we have
Lg,
h, (X) and Lg2 Lrj ' h, (X)
01=Lrrh,(X)+Lg Lrf'h,(X)u, +Lg2Lrj'h,(X)u2 Similar to the above we can get
(4.7)
126
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
(4.8)
V/r,
VV, = L)h2(X)+Lg,Lrf'h2(X)u, +Lg, rl h2(X)u2
combining Eqs. (4.5) through (4.8), we know that when the relative degrees satisfy r = r, + r2 = n, under the coordinate transformation shown in Eq. (4.3), the system (4.2) can be transformed into the following normal form 01 =P2 Or,-1 = lP,
(p,. = Lrfh,(X)+Lg Lj'h',(X)u, +Lg2 Lrf'h,(X)u2
(4.9a)
W2
Vri
yrr= = Lrjh2(X)+Lg Lrj'h2(X)u,
and the output Y, (t) _ -P 1(X (t ))
Y2 (t) = V/1 (X(0)
(4.9b)
The above Eq.(4.9) is called the first type normal form of MIMO affine nonlinear system. It corresponds to the condition that the sum of system
+-N
relative degrees r = r, + r2 + + rm = n . Let us discuss the condition that the sum of system relative degrees r,, +'A
i =1, 2, , m is less than the system's order n . If for a system as shown in Eq. (4.2), its relative degrees satisfy r = r, + r2 < n, then after choosing a coordinate transformation as shown in Eq.(4.3), the last (n - r) coordinates
could always be found as i =171(X) ......
(4.10)
17n-r = 17 -r (X)
such that the Jacobian matrix of the vector function O(X)=[,P1(X), ..., p, (X); V, (X), ..., V , ,(X); 171(X),
(4.11)
is nonsingular at X = X°. Thus, we have chosen a set of qualified local coordinates mapping. Under this condition, the following (n - r) equations should be added to the system described in new coordinates besides the equation as shown in Eq. (4.9), i.e.
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
127
r), = Lj77i(X)+Lg r7i(X)u, (4.12)
Lj'ln_r(X)+Lg,rln-r(X)u, +Lg,77n-r(X)u2
Therefore, the system after transformation is 01 =qP2 (Pry
gyp, = Lrjh1(X)+Lg Lrf'h,(X)u, 'VI=W2
(4.13a)
Wr,= Wr, =L'jh2(X)+Lg,L'j-'h2(X)u,+ Lg,Lrr'h2(X)u2 r/, = Ljrl,(X)+Lg rl,(X)u, +Lg 171(X)u2
n-r = L1rln-r(X)+
Lg,rl'-r(X)u2
together with output equations Y1 = Vi Y2 =W1
(4.13b)
.-.
What is expressed by Eq. (4.13) can be called the second type normal form. Let us come to the third type normal form. It can be verified that, if the vector field set {91(X)192(X), (X),.. ., g.(X)} shown in (4.1) is involutive, then n-r (X) can surely be found such that coordinates mappings 71, (X), rl, (X),
Lgrl,(X)=0 L" 77, (X) = 0
(4.14)
Lgarl;(X)=0
Thus, (4.13) can be transformed into '1 =P2 Or,-] = V"
rpr = Lr}h,(X)+Lg Lr}'h,(X)u, +Lg,L' 'h,(X)u2 =W2 Wr,-1 -Wr,
yrr2 =L)h2(X)+LgL'}- h2(X)u,+Lg Lj'h2(X)u2 Tll = L1211(X)
n-r = Liri-r (X )
(4.15a)
128
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
with output equations yI = VI
(4.15b)
YZ = V1
In Eq. (4.15),
r/, (X), 171 (X),
,
(X)
are the solutions of the partial
differential equation set (4.14). What is expressed by Eq. (4.15) can be in'
called the third type normal form. It should be pointed out that the third type normal form in Eq. (4.15) is
simpler than the second one in Eq. (4.13) in the last (n - r) equations. However, this simplicity is at the cost of solving the set of partial differential equations in Eq. (4.14). Based on all the above, the following propositions could be formed.
+O-'
Proposition 4.1 Suppose that there exists a system as shown in Eq. (4.2), with the sum of relative degrees r = r, + r2 = n. Choose a coordinate transformation Z = cD(X) as follows z1
=97'(X)= h (X)
Z2 = P2 (X) = L fh, (X) Zr
-'Pr,(X)=L}-'h,(X)
Z,+I =y/1(X)=h2(X) Z".2 = V2(X) If h2(X)
(4.16)
z, =wr,(X)=Ef-'h2(X)
-e.
a.+
then the system can be transformed into the first type normal form in the new coordinates Z 0I =IP2
rpr = Z. (Z) + b11(Z)u1 + b12 (Z)u2
(4.17)
WI =1V2 1Vr,-I = yfrz
,r, = a2 (Z) + b2, (Z)u1 + b22 (Z)u2
where
a(4.18)
Z=IV, ...
-Pr
WI ... V, IT
A(Z)=[,(Z)]=[LihjX)]
lei
a2
)
Ljh2(X)
x=m-I(Z)
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems g(Z) = (bI1(Z)
b12(Z)
__
C/1
Proposition 4.2
Lg L! h1(X) Lg,L[ ' h2(X)
Lg Lrr Ihl(X)
(4.19)
Lg L'!I "2(X)
Suppose there exists a system as shown in Eq. (4.2) 0-0
rt.
k1(Z) b22(Z) ]
129
with the sum of relative degrees
r = r, -i-r2 < n .
Apart from the chosen
coordinate transformation as shown in Eq. (4.16), the last (n-r) coordinates can always be found as Z,+l = 771 = 771(X)
(4.20)
='in-, (X)
Zn
such that the Jacobian matrix of the vector related to the coordinate mapping (D(X) = [p, (X) ... P, (X) Il/, (X) ... V/r, (X) 771(X) ... 77n-r (X )iT
is nonsingular at X°, thereby the original system can be transformed into the second type normal form, which is formed by replenishing the form shown by Eq. (4.17) with the following (n - r) equations (4.21 a)
n= 4(Z)+P,(Z)u1 +P2(Z)u2
which may also be written as (4.21b)
7
n=9(Z)+[PI(Z)+P2(Z)] [UJ where "_ Inl
_
77n-rlT
772
91 (Z)
Lf771(X)
(4.22)
9(Z) 9n-r(Z) Pu (Z)
Lf77n-,(X) x=m-'(z) Lg.171(X) (4.23)
Lp 1(n-r)(T) T21 (Z)
Lgl7]nr(X ) x=m-'(Z) Lg,>11(X )
(4.24)
i'2 (Z) P2(n-r) (Z)
Proposition 4.3
Lg,77n-r(X) x=o-1(z)
Suppose there exists a system as shown in Eq. (4.2)
with the sum of relative degrees
r = r, + r2 < n .
If its vector field set
{g1(X),g2(X)} is involutive, the last (n-r)coordinates 171(X), can be chosen to satisfy
772(X),...,?7.-,(X)
130
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS Lg 77; (X) = 0
(4.25)
Lg77;(X)=0 rip
such that the original system can be transformed into the third normal form,
which is formed by replenishing the form shown by Eq. (4.17) with the following (n - r) equations rt= q(Z)
(4.26)
In addition, for the above three propositions, their output equations are the same as Y1 =(P,
(4.27)
Y2 = 4/1
The problem discussed in this section so far is, for an affine nonlinear
system with m inputs and m outputs, if the sum of its relative degrees
...
in'
phi
r = r, + r2 + r,, is equal to or less than system's order n, how to transform the system into the three types of normal forms. What if a system has no relation degree, i.e. if the conditions described in the relative degree definition for MIMO systems (Definition 4.1) are not satisfied, can such a system still be transformed into a type of normal form? The answer is yes. As we can see in the following, the normal form for such a system is called
the fourth type of normal form. As before, we first discuss a system with two inputs and two outputs as shown below X(t) = f(X(t)) + g, (X)u, + g2 (X)u2
Y,(t)=h,(X(t)) y2(t)=h2(X(t)) To begin with, choose the first two coordinates as z, =rp,(X)=h,(X) z2 =P2(X)=h2(X) and we will get dynamic equations i, = Lfh,(X)+Lg h,(X)u, +Lglh,(X)u2 i2 = Ljh2(X)+Lg h2(X)u, +Lg=h2(X)u2
(4.28)
(4.29)
(4.30)
which can also be written as +
r ml [Z2]-[Lf h2(X)]+[Lgh2(X) Lgh2(X)j u2J
Now assume that for the matrix in the above formula
(4.31)
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
B(X)-1Lg, h2 (X)
131
(4.32)
Lg,h2(X)I
.Do
its two row vectors do not equal zero at X° but are linearly dependent. That is to say, for all the X in a neighborhood of X°, matrix B(X) is singular. From the definition of relative degree for MIMO systems (Definition 4.1) we know that, the system in Eq. (4.28) has no relative degree. Therefore, it can not be transformed into the above three types of normal forms. Next, we will discuss how to handle this case.
Since the two row vectors of matrix B(X) are linearly dependent, a scalar function r(x) defined in the neighborhood of X° can be found such that [r(X)
Lg h, (X)
Lg,h, (X) = [0
Lg,h2(X)
Lg,h2(X))
1]
0]
(4.33)
Now let us define a scalar function w(X) w(X) = [r(X) 1) Lfh2(X)I=r(X)Lfhl(X)+Lfh2(X)
(4.34)
k
Suppose that the matrix B, (X) = L
Lg
(X)
L" w(X)
is nonsingular in the neighborhood of X° 1).
i2 can be rewritten by adding and subtracting same terms to and from the right side of the second equation of Eq. (4.30), namely i2 = L f h2 (X) + Lg h2 (X )ul + Lgz h2 (X)u2 + r(X)(L fh, (X)
+Lg h1(X)u, +Lgsh1(X)u2)-r(X)(Lfh,(X)+Lg h1(X)u, +Lg2h1(X)u2)
Manipulating above formula while considering the relations implied in Eqs. (4.33) and (4.34), we have i2 = w(X)-r(X)(Lfhl(X)+Lg hl(X)u, +Lg,h,(X)u2)
(4.35)
We choose the third coordinate transformation as
z3 =P3(X)=w(X)
(4.36)
correspondingly, the dynamic equation will be obtained as i3 =L fw(X) + Lg w(X)u, + Lgsw(X)u2
(4.37)
Our next step is to choose the rest (n - 3) coordinates quite arbitrarily
') If this matrix is still singular, the corresponding processing method will be discussed at the end of this section.
132
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS z4 = -P4 (X)
(4.38)
Z. = P (X)
such that the Jacobian matrix of the vector function CD(X) _ [(Pl (X)
P2 (X)
... Pn (X)]T
co3 (X)
is nonsingular at X°, such that the chosen mapping cD(X) is an eligible coordinate transformation. If a vector n is used to represent the (n - 3) coordinates from z4 through z, , i.e. n=[z4
z5
... Z
(4.39)
then (n - 3) dynamic equations can obviously be written as A=q(X)+p1(X)u1+p2(X)u2
(4.40)
where LfV4(X)
91(X)
(4.41)
q(X)
Lg,tP4(X)
Pu(X)
...
Pl (X) 14,
P,(,,-3) (X)
P21(X)
=
(4.42)
(X)
Lg,
9P4(X)
...
P, (X) _
(4 . 43)
From the first equation in equation set (4.30), and Eqs. (4.35), (4.37) and (4.40), we will immediately get the dynamic system described by the form in
the new coordinates Z as =(Lf4(X)+Lg,hl(X)ul +Lg,hl(X)U2)I,--,,(z)
Zl
tit
z2 =(Z3 -r(X)Lfh1(X)+Lg,h1(X)u1 +Lg,hi(X)u2)IX-m-'(z) Z3=(Lfw(X)+Lg,x(X)ul+Lg2%(X)u2)Ix=1, (Z)
(4.44)
n=(q(X)+P1(X)ul +P2(X)u2)I-'(z)
together with Yl (t) = Zl = hl (X (t)) Y2 (t) = z2 = h2 (X (t))
4.45)
Now we can summarize the above into the following proposition.
Proposition 4.4
Suppose there exists a system as shown in Eq. (4.2).
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
133
If it has no relative degree, namely the row vectors of matrix B(X) - [Lg'hz (X)
Lg, hz (
)]
are linearly dependent, then we can get the fourth normal form in the new coordinates Z = (D(X)
i,
a,
+b12(Z)u2
i2 = Z3 - r (Z)(a1 (Z) + b11 (Z)u1 + b12 (Z)u2) i3 = a3 (Z) + b31 (Z)u, + b32 (Z)u2
(4.46a)
it = q(Z) + p, (Z)u, + Pz (Z)uz
with output equations Yl = z1
(4.46b)
Yz = Z2
where A(Z) = a1(Z) _ a1() Ca3 (Z)J - La3 (ix=-(z)
g(Z) =
__
L f17, ()
Lf
(4.47a) X=m-, (z)
_ b11(X) b12(X)1
b11 (Z)
b12 (Z)
631 (Z)
b3z(Z)] - Lb31(X)
_ [L" h, (X)
x)
b,2(X)J,=.-,(z) (4.47b)
Lg3 h1 (X)
Lg=w(X) ]
Lg w(X)
x-m (z)
L fcp4(X)
(4.48)
4(Z) _ ILf c (X)
x=m-'(z)
N1 ILg,q'4(X) ooh
(Z)
(4.49)
Lg,tV"N10-1g) L92 V34 (X)
(4.50)
Pz(Z 1Lg2t,11 (X) x=o-,(z)
and for w(X) = r(X)L fh,(X)+L fh2(X)
r(X) satisfies the condition in Eq. (4.33). The coordinate mapping is
134
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
Z, =,p1(X)=h,(X) z2 = TP2 (X) = h2 (X) z3 = qP3 (X) = w(X)
A= n(X) Nom,
where n= [z4 ... z ]T . The fourth type normal form as shown above in Eq. (4.46) is obtained with the assumption that for all the X in a neighborhood of X°, the matrix [B, (X)
Lgh1(X)
Lg2hI(X)1
Lg w(X)
L,, w(X)
11.
is nonsingular. If the row vectors of the above matrix are still linearly dependent, a scalar function r, (X) of X can surely be found such that [rl (X)
11
L,,4(X) L"h, (X)1= lL Lg w(X)
Lg,w(X) JJ
[0
0]
(4.51)
Define a function p(X) as p(X) = r, (X)L f h, (X) + If w(X)
(4.52)
Suppose that matrix Lgh,(X) Lg,h,(X)
.tr
B2(X)=
Lg,u(X)
L,,/j(X)
is nonsingular at X°. Now let us make the following choice for coordinate transformation z, =Ta1(X)=hl(X) z2 = (p2 (X) = h2(X) z3 = qq3 (X) = w(X)
(4.53a)
z4 =gP4(X)=u(X)
Besides, choose z5 = 95(X)
......
(4.53b)
z^
such that the Jacobian matrix of the vector function D(X) = NP1 (X)
92 (X)
...
P. (X)] T
is nonsingular at X°. Taking steps similar to those for Eq. (4.46), we can have a normal form i1 = a1 (Z) + b1 I (Z)u1 + b12 (Z)u2
_
22 = Z3 - r(Z)(al (Z) + bll (Z)u1 + b12 (Z)u2 )
(4.54)
CA)
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
135
i3 = Z4 - r, (Z)G', (Z) + b (Z)u, + b12 (Z)u2 )
...
i4 = a4 (Z) + b41(Z)u, + b42 (Z)u2 = 9(Z) + ' P , (Z)u, + P2 (Z)u2
with Y, = Z,
(4.55)
Y2 = Z2
where Z = [Z,
Z2
Z4
Z3
a4 (Z) = a4 (X)
11]T
IX=a-, (z) = L f u(X) IX=,-' (z)
(4.56)
(4.57)
b41(Z) = b4, (X)
Lg,N(X) IX=m-'(z)
(4.58)
b42 (Z) = b42 (X) Ix=m-'(z)
Lg, p(X) IX=d '(Z)
(4.59)
Lfco5(X)
(4.60)
9(Z) L ooh
X=m-' (Z)
Lg,'Ps(X) (4.61)
Lg, Pn(X)
X=m-,(2)
101
Lg, q 5 (X) (4.62)
Pz (Z) _ X=a-' (z)
,u(X) = r, (X)L fh,(X) + L fw(X)
and the scalar function
(4.63)
satisfies Eq. (4.51). The coordinate
r, (X)
transformation is Z, =h1(X) Z2 =h2(X) Z3 = w(X)
Z4=u(X) Tl= n(X)
where TL=[ZS
...
In summary this section presents four types of normal forms for MIMO
136
nonlinear systems
C3.
affine
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
obtained in
terms
of certain coordinate
transformations. These four types of normal forms will be used in zero dynamics design approaches, which is the subject of the next section.
4.3
ZERO DYNAMICS DESIGN PRINCIPLE
For MIMO systems, the fundamental notion of zero dynamics design is to search control laws u, (X), u2 (X), , u,,, (X) , in order to make the control system satisfy such a performance index that its output responses y, (t), y2 (t), , y, (t) keep equaling zero at any time t >_ 0 . Under this ..,
condition, the stability of the zero dynamics equation set (called zero .-r
'r3
dynamics for short) of the system should be tested. What this zero dynamics equation set describes is in fact the internal dynamic characteristics of the
Mao
system. Obviously, if the zero dynamics is stable, the "external" and
-'O
"internal" parts of the closed loop system are both stable, and so is the whole system. Consequently, the control system could be regarded as optimal as
coo
ro.
-CS
0.-
viewed from output responses. Generally outputs of a control system are chosen as deviations of state variables from their references values (being constants or dynamic trajectories), in other words, the tracking errors. As a result, if a controller thus designed can keep the outputs equaling zero at any time, then, at least, the external dynamics of the control system is optimal.
In the foregoing section, we have obtained four different types of transformation normal forms aiming at different conditions of system relative degrees. In the following, zero dynamics design method will be discussed for these different normal forms. (`JD
Let us begin with a system that has the third type of normal form as shown in Proposition 4.3. The starting point of zero dynamics design is to make the outputs satisfy y,(t)= y2(t)=0 for any t>_0 namely, h,(X(t)) = h2(X(t)) = 0
for any t >_ 0.
such that L"h,(X(t)) = 0 h,(X(t)) = Lrh2(X(t)) _ ... = L''h2(X(t)) = 0 h,(X(t)) = L jh,(X(t))
4.64)
From Eq. (4.3) we know Eq. (4.64) means that for any time t >_ 0 tP, MO) = tP2 MO) = VI MO) = V2 (X(t)) =
q', MO) = 0 V", MO) = 0
4.65)
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
137
hold. Furthermore Eq. (4.65) implies that 0
dam'
dt dyr,
(4.66) 0
dt
With Eq. (4.66), we can obtain from the two equations about Eq. (4.17) that a, (Z) + b (Z)ul + b1z (Z)u2 = 0
gyp, and Wrl in
(4.67)
a2 (Z) + b21(Z)u, + b22 (Z)u2 = 0
Cal (Z) ] + bl1(Z) a, (Z)] b2l (Z)
°`I
which can be written in a matrix form [u,
bl2 (Z) bzz
(Z)]
=0
Luz
I=
(4.68)
From Eq. (4.68) we can figure out the control vector U ui 1 _ _ bi' (Z)
uz]
f al (Z)
b_12 (Z)
bzl(Z) bzz(Z)
(4.69)
La2(Z)]
III
which can also be written as U
(4.70)
(Z)A(Z)
Thus, for the third type of normal form we have calculated out the control vector U(Z) corresponding to such a performance index of "zero outputs". In fact, what we really need is the state feedback law U(X). Therefore, we
need to substitute Z = cD(X) for Z in the expression of U given in Eq. _>~
(4.69) or Eq. (4.70). Thus, we obtain bll (X)
U = ul Cut ] -
[b21 (X)
bl2 (X) b22
al (X)11
'
(X)]
(4.71)
[a2 (X)J
i.e.
r-7
U = [u11= -B-1(X)A(X)
(4.72)
z
where, from Eqs. (4.18) and (4.19) we know A(X) = L}b,(X) L;h2(X)]
mss"
B(X) = Lg Lr l(X)
LgLj h2(X) LgA
(4.73)
1h2(X) (X)
(4.74)
138
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
l-,
7C'
In Eq. (4.72) and the state feedback expressions given in Eqs. (4.73) and (4.74), since the task of control law is to force the system outputs y, (t) and y2(t) into zero, which means Yz(r'-i)
Yz (t) = Yz (t) _ ... =
`-'
Y,(t)=Y,(t)_...=
(t) = 0 (IQ
y.'
as the result of control, X(t) should satisfy the following constraints Ys'
h,(X(t)) = LfA(X(t)) _
= Lrj'h,(X(t)) = 0 hz(X(t))=Lfhz(X(t))=...=Ly1hz(X(t))=0
Namely, if the control strategy in Eq. (4.73) is adopted, the system state vector X(t) will be confined within a subset 91 as shown below
i=1,2) .h?.
S2={XER" f p'+
The last step of the design is to examine the stability of the zero dynamics equations. Substituting Eq. (4.65) into Eq. (4.26), we can immediately get the zero dynamics as follows (4.75)
or be concisely written as `-.
(4.76)
n= q(0, rl)
The stability of differential equations in Eq. (4.76) should then be examined. Described above is the design method for systems with the third type of normal form. Now let us come to the problem that a system has the second type of normal form.
Suppose a system has the second type of normal form as shown in .14
Proposition 4.2. Under this condition, in a way similar to the above we have state feedback as shown in Eq. (4.71) or Eq. (4.72). To search the system zero dynamics equations we still adopt the following symbols
Z=[ , i]T
(4.77)
where = [(P,
L171
... fir, wi ...
yir= ]T
... tl"-rIT
(4.78) (4.79)
Thus, Eq. (4.70) can be rewritten as
U=ru ' L
z
=_'
'(
,
n)A( ,
n)
(4 . 80)
JI
According to the zero dynamics design principle, it should be set that r=o, i.e.
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
U(0, n) = -B-'(0, n) A(0, n)
139
(4.81)
r,'
Substituting Eq. (4.81) into Eq. (4.21) while setting =o in Eq. (4.21), we can obtain the zero dynamics equation set as follows n= 4(0, n) - p(0, n)B-' (0, n)A(0, n)
(4.82)
where P11(O, n)
P(O, n) _ [PI (O, TO
P21(O, n)
(4.83)
P2 (O, n)l = I0.
hl(n-r)(0, n)
P2(,,-r)(0, n)
In the following let us discuss another probably more important condition as viewed from practical applications, i.e. the condition that the system has the fourth type of normal form as shown in Proposition 4.4. According to the main idea of design principle of zero dynamics, for Eq. (4.46b) we set that Y1(t) = z1 (t) = 0
Y2 (t) = z2 (t) = 0
for any t ? 0
(4.84)
for any t > 0
(4.85)
Thus i1(t) = i2 (t) = 0
Substituting Eqs. (4.84) and (4.85) into the second equation of Eq. (4.46a), we know that under this condition it holds that for any t >- 0
z3 (t) = 0
(4.86)
Thereby M t) = 0
Substituting the above obtained it = 0 and i, = 0 into Eq. (4.46a), we have al (Z) + b,1(Z)ul + bl2 (Z)u2 = 0 a, (Z) + b31(Z)u1 + b32 (Z)u2 = 0
(4.87)
which may be rewritten as Cal(Z) + 11(Z)
a3(Z)
b31 (Z)
b12(Z)
rui =0
b32(Z)
Lugs
(4.88)
or
A(Z) + B(Z)U = 0
(4.89)
where the meaning of A(Z), B(Z) and U will be clear by comparing with Eq. (4.88). From Eq. (4.89), we can figure out the control vector U(Z)
140
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS b,2(Z)
U=-B-1(Z)A(Z)=_ Cb3,
biz (Z)]
(Z)
'ra,(Z)
(4.90)
L°3 (Z)]
Obviously, what we need is the control law U(X). Thus, we need to replace Z in Eq. (4.90) by function of X, namely, substituting Z=(D(X) (IQ
C)'
r..
for Z into Eq. (4.90). Referring to (4.46) and (4.47), we get the state feedback U=
Lg,h,(X) LA(X)
Lg w(X)
L1b,(X)
Lg,w(X)] [L fw(X)]
(4.91a)
.°o
By adopting the control strategy in the above formula X(t) will be confined within the following subset
S2= {XER"Ih,(X)=h2(X)=w(X)=0}
(4.91b)
where it is known from Eq. (4.34) that w(X) = r(X)L fh,(X)+ L fh2(X)
(4.92)
The r(X) in the above formula satisfies the relation in Eq. (4.33). We have, so far, reached our main goal, that is, having acquired the state feedback law as shown in (4.91) which will make the outputs y1(t) and y2 (t) of the system keep the tendency of zero outputs for all t > 0.Our next task is to check whether the zero dynamics of the system is asymptotically stable. To this end, following steps we are familiar with, we shall divide the coordinate Z into two parts, i.e. Z = [z,
z2
(4.93)
n]T
z3
From Eqs. (4.84) and (4.86) we can see that
Z=[0 0
0
(4.94)
n]T
Substituting Eq. (4.94) into Eq. (4.90) and the last equation set of Eq. (4.46a), we have n = 4(0, 0, 0, n) - p(0, 0, 0, n)B-' (0, 0, 0, n)A(0, 0, 0, n)
(4.95)
where 9'(0,
0, n)
LfcP4(X)
(4.96)
_
4(0, 0, 0, n) _ 4n-3(0, 0, 0, n)
Lf lpn(X)
X-0-' (0, 0.0.',)
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems (0, 0, 0, n)
141
...
p21 (0, 0, 0, n)
(0,0,0,rl)_
(0, 0, 0, n)
(X)
L.,
Lg,,pn(X)
`rte
PI(n-3) (0, 0, 0, n)
Lgz cn(X) x=m(0,0, o, n)
b (0, 0, 0, n)
K. (0, 0, 0, n)
63 (0, 0, 0, tl)
b32 (0, 0, 0, n)J
L91
-[L A(0, 0, 0, n) = a,
[
h,
(X) w(X)
L
(0, 0, 0, n)
3 (0, 0, 0,
(4.98)
zw(X)]
---
9(0, 0, 0, n) =
(4.97)
Ls,-P4(X)
_ L fh, (X)
n)]
L fw(X)
(4.99) x=4'
(o, o, a, n)
-t4
What we just described is the zero dynamics design method for a system that can be transformed into the fourth type of normal form as O..'
C-'
described in Proposition 4.4. If a system has a normal form as shown in Eq. (4.54), in order to obtain its state feedback and zero dynamics equations, the .fl
design methods and procedures described previously in this section can almost be entirely followed. Thus, without giving any unnecessary details, we can obtain
U=
B i(X)A(X)
'[[u2J (4.100)
Ib4l(X)
b42 (X),
...
-
a4(X)J
Substituting Eqs. (4.57) through (4.59) into the above formula we can have the concise expression of U(X)
uzi
_
Lg h,(X)
-[Lg,p(X)
Lg h,(X)l
Lfh,(X)l
(4.101)
>'C
u _-[u,
Lg,,u(X)J [Lfu(X)J
The state feedback expression given above will confine X(t) within the following subset
(={XaRnIh1(X)=h2(X)=w(X)=p(X)=0} V'1
where, p(x) has been given in Eq. (4.52), r, (X) satisfies the relation given by Eq. (4.51), and w(X) has been given in Eq. (4.34). Substituting the relations of z, =z2 =z3 =z4 =0 and U =-B-'(0,0,0,0, n)A(0,0,0,0, n) into the last equation set of Eq. (4.54), we will obtain the zero dynamics equation set of the system as n= q(0, 0, 0, 0, n) - p(0, 0, 0, 0, r0-B-'(0, 0, 0, 0, rt)A(0, 0, 0, 0, n)
(4.102)
142
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
where it (0, 0, 0, 0, Ti)
L fcos (X )
(4.103)
y (o, 0, 0, 0, n) _
L1(X)
(0, 0, 0, 0, n)
X=0- (0,0,0,0,,1)
Al (0, 0, 0, 0, n)
p21(0, 0, 0, 0, Ti)
PI(,,-4) (0, 0, 0, 0, n)
T.(-4)(0101010'r')
p(0, 0, 0, 0, n) = Lg,,Ps(X)
Lg,'PJX)
B(0, 0, 0, 0, Ti) =
(4.104)
Lg2T5(X) x=m-'(o,o,o.o,n)
b i (0, 0, 0, 0, n) ' b12 (0, 0, 0, 0, n) b41(0, 0, 0, 0, n) b42 (0, 0, 0, 0, n) Lg2 h, (X)1
Lg ft(X)
Lg=,u(X)Jx=m-'(o,0,0,0,'0
(4.105)
...
Lg h, (X)
-
A(0, 0, 0, 0, Ti) =
a(OOnan n) [ Q4(0. 0,0,0,)] I
Lh(X) Lf u(X) 1
(4.106) X=m-1(0.0.0,0.n)
With the above illustration of a nonlinear system with two inputs and 'o,
E:0
°o.
two outputs, the zero dynamics design method is discussed under the conditions that relative degree is less than the order of the system or the system has no relative degree, etc. These design methods tell us how to +y+
search state feedback law U and corresponding zero dynamics equations. Readers can easily extend the above algorithms to systems with m inputs ON''
and m outputs. To enhance the understanding and grasp of the zero dynamics design method, next we expand the algorithm of the zero CAt
dynamics design method to a more general form. Assume that there exists an nd' order nonlinear system in the form of m
...
X(t) = f NO) + Y g1 M OX (t)
(4.107a)
i=1
..,
YI (t) = h, (X(0) Yz (t) = h2 (X(t))
(4.107b)
Ym(t) = hm(X(t))
Before discussing the general method of zero dynamics design, let us first give specific definitions for the symbols to be adopted. (1) U is an m -dimensional vector U = [u1
u2
...
um ]T
(4.108)
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
143
(2) g; (X) is an n -dimensional vector g(X) = [g,1(X) g,2 (X) ...
(3) g(X) is an n x m
(4.109)
g;,, (X)]T
matrix composed of g; (X) , i=1,2,.-.,m X00
gl l (X) g2 (X) ... gml (X) g12(X) gn(X) ...
g(X)=[g1(X) g2(X) ... g,,(X)]=
^^C
gln(X) g2, (X)
(4.110)
... gmn(X)
(4) h(X) is an m -dimensional function vector h(X) _ [hl (X)
h2 (X)
...
(4.111)
hm (X )]T
(5) Suppose that W(X) is a k -dimensional vector W(X)=[w,(X) w2(X)
wk(X)f
(4.112)
then L fW(X) is defined as a k-dimensional vector
LfW(X)=[Lfw,(X) Lfw2(X)
..
Lfwk(X)]T
(4.113)
...
Lgawl(X) Lg.W2(X)
(4.114)
...
L..Wk(X)
Besides LgW(X) is defined as a k x m matrix Lg3wl(X)
W2(X)
L92W2(X)
Lg,wk(X)
Lgxwk(X)
91
...
...
LgW(X) =
Lg wl(X)
After making clear the meaning of the above symbols, we will introduce the general approach of zero dynamics design for affine nonlinear
systems with m inputs and m outputs given by Eq. (4.107). The general a¢)
approach involves the following main steps: Step 1. Consider the equation L fh(X) + Lgh(X)U = 0
(4.115)
Suppose that in the above equation the rank of the m x m matrix Lgh(X) is
invariant near X° and equal to r and that adjustments have been done to the component arrangement sequence of vector field h(X) _ [h, (X), , hm(X)]T
such that the first r, row vectors of matrix Lgh(X) are linearly independent. From linear algebra we know that, there must exist an (m - r,) x m matrix rl1(X)
...
ro-,(X)
1
...
1
...
1
[R, (X) 1] _
(4.116)
r(.-,), (X) ...
such that
r(m-r
)r (X)
1
(m-r, )xm
144
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
[R, (X) 1] Lgh(X) = 0
(4.117)
Now HI(X) is used to represent the vector composed of the first r, components of vector field h(X), i.e. H1(X)_[h1(X) ... h,(X)]T
(4.118)
W(X)=[R(X) 1] Lrh(X)
(4.119)
Set
where obviously W, (X) is an (m - r,) -dimensional vector. At the end of this step, set H 2 (X)
=CWH, M T
(X)]
where H2(X) is an m -dimensional vector. Step 2. Consider the equation LfH2(X)+LgH2(X)U=0
4.120)
(4.121)
Suppose that in the above equation the rank of the m x m matrix LgH2 (X) equals r2, which is equal to or greater than r, , namely r2 >- r, (for all the X near X°), and that adjustments have been done to the component arrangement sequence of vector W, (X) such that the first r2 row vectors of m x m matrix LgH2(X) are linearly independent. Similarly, there must exist
an (m-r2)xm matrix [R2(X) 1], where R2(X) is an (m-r2)xr2 matrix, such that
[R2(X) 1]LgH2(X)=0
(4.122)
Let H2I (X) represent the vector composed of the first r2 components of vector H2(X), and set W2(X) as W2(X)=[R2(X) 1] L fH2 (X)
(4.123)
which is an (m - r2 )-dimensional vector.
Finally, set
H3(X)=I
(4.124)
where the dimension of H,(X) is M. Step 3. The following steps are basically the same as the above ones. To take the k`h step for example, consider the equation LJH, (X) + LgHk (X)U = 0
(4.125)
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
145
Assuming the rank of LgHk (X) is rk , construct a vector (4.126)
Hk+1(X) _ [Wk(X)l
where HkI (X) represents the first rk components of Hk .
If the calculation is continued to the I" step, where the rank of is m, namely, where matrix LgH,(X) is nonsingular, the
LgH,(X)
calculation terminates at this step. Besides, since the matrix LgH, (X) is invertible, we can solve L JH, (X) + LgH, (X)U = 0
for the control vector U as u,
U= uz =-(LgH,(X))-'LfH,(X)
(4.127a)
Lum J
Through the adoption of the above control strategy, X will be confined within the subset below 111={XeR"
(4.127b) ,,,
a.'
04; E-°
Obtaining Eq. (4.127a) is the main aim of our design. Finally, we need to find the system zero dynamics equation set in order to examine its stability. To obtain the zero dynamics equation set, we need the following calculation. (i) For the mapping Z = d>(X) h(X) W, (X)
Z=rnl= b(X)=
W, (X)
(4.128)
(X)
V (X)
where X=
p = (I + 1)m - (r, + r2 +
+ r,),
calculate
its
inverse
mapping
-' (Z) = (D-'(;, q).
(ii) Substitute X = - (e, n) for X into the Eq. (4.127a), and set = 0, which result in U = -(Ls H, (X))-' L f H i (X)
x- _m_, (o, n)
(iii) Substitute Eq. (4.129) into the equation set
(4.129)
146
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS .M.
ii= 4(0, n)+p(0, n)U
(4.130)
we obtain (o.n>
where I Lfco 1(X)
.^-N
il= (9(O, II) -P(0, A)(LgH,(X))-'LfH,(X)) IX_m
(4.131)
(4.132)
Lf-p"(X)
X=m-I(Z)
Lgz,pP+i(X)
Lg,-p"(X)
Ls,-p"(X)
...
Lg.gP+i(X)
..
(4.133) X=m-,(Z) t17
Lg,q'p+,(X)
The subscript p in the above formula is the same as that in Eq. (4.128). Eq. (4.131) stands for the system's zero dynamics. If the equation set in Eq. (4.131) is asymptotically stable at X°, so is the whole control system in the
neighborhood of V. The above algorithm is called the zero dynamics algorithm. If for each step k >-1, matrix LgHk (X) has constant rank for all the X near X° , and NCD
If.
so does the Jacobian matrix of H k (X) , then X° is called a point of
regularity for the system zero dynamics algorithm. Now let us summarize the above algorithm by the following proposition.
Proposition 4.5 For a nonlinear system as shown in Eq. (4.107), suppose X° is a point of regularity for the zero dynamics algorithm. Then the algorithm ends at the step 1:5 n, in the sense that matrix LgH, (X) is nonsingular at the point X°. In such a case the equation L fH,(X) + LgH,(X)U = 0 ti,
can be solved for U = U' (X) U' (X) = -[LgH, (X)]-' L jH, (X)
1.:
where U' (X) is the unique control vector that can force the output responses Y(t) = h(X) to be zero for any time t >- 0. Namely, U' (X) can guarantee confining the state vector X(t) within the following subset
0 Substituting the above expression of U"(X) and the constraint of X into the previous dynamic equation set
147
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
X = f (X) + g(X)U'
we will have the zero dynamics of the system. If the solution of this equation set is asymptotically stable, then so is the whole system.
So far we have described in detail the zero dynamics design algorithm for MIMO nonlinear systems. To avoid misunderstanding, one thing needs to
a...
be brought up again. Some people may think that for control systems I.,
designed according to the zero dynamics algorithm, their dynamic responses in actual operation would be really kept to zero. Or reversely, if the dynamic responses of a system designed with this method were not always equal to zero in simulation or actual operation, they would doubt the reasonability and validity of zero dynamics design approach itself. In fact, there are some misunderstandings in both two thoughts. The objective of zero dynamics
design method is to make outputs
y, (t), y2 (t),
, ym (t) equal zero at any
time. This objective is in fact a dynamic performance index, which should be correctly understood as to let the outputs always keep their minimum values under the effect of control. Since many actual factors in real control systems have not been considered in design, as the time delay and tiny dead areas in
control signals or some elements of the system, the amplitude-limiting v0,
poi,
characteristics of control variables, the nonlinear characteristics in some of the system parameters themselves (for example, the saturation characteristics
of generator reactance X4) and errors in measuring components of the controller, the control system designed and manufactured with this method can not really make the dynamic outputs strictly equal zero. However, there is one thing we can be sure, that is a control system correctly designed and manufactured according to this method should have the ability to make its dynamic outputs maintain minimum values.
4.4
DESIGN PRINCIPLES OF EXACT LINEARIZATION VIA STATE FEEDBACK
MIMO affine nonlinear systems, which satisfy certain condition can also be transformed into the following Brunovsky normal form by means of coordinates mapping and appropriate state feedback Zl = Z2
..
Z2 = Z3
(4.134)
Zn-m = Zn-m+l Z'?-m+l
= Vl
148
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
i = vi,, .1:
The approach of exact linearization has a broad application prospect in C)- te,
engineering practice. In this section, the conditions and algorithms of exact linearization will be discussed for MIMO systems.
4.4.1
Conditions for Exact Linearization via State Feedback
First, the actual meaning of "exact linearization via state feedback" should be discussed. Given the nonlinear system (4.135)
X = f (X) + > g; (X)u; i=l
If matrix g(X) is set as g(X)=[g1(X) 92(X) ...
gm (X)1
and control vector U as U=
Iul
u2
... U. lT
we can rewrite Eq. (4.135) as X = f(X) +g(X)U III
(4.136) ^°o
The problem of exact linearization via state feedback is presented as follows.
x^,
For a system as shown in Eq. (4.136), set feedback U = A(X)+B(X)V (where A(X) is an m -dimensional vector field, B(X) an m x m matrix, and V an m -dimensional vector) and coordinates mapping (D: Z = I (X), such that the feedback system X = f(X) + g(X)A(X) + g(X)B(X)V
(4.137)
is converted into a linear and controllable system (4.138)
Z = ADZ+B,V
=°'
,pi
...
The above is the presentation of exact linearization via state feedback. In fact, from the first type of normal form given in Proposition 4.1, we have no difficulty in obtaining a linearized system as shown in Eq. (4.134), as long as in Eq. (4.17) setting that a, (Z) + b (Z)u1 + bu (Z)u2 = VI a2 (Z) + b21 (Z)ul + b22 (Z)u2 = v2
(4.139)
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
149
we can get the following linearized system it = z2
z2 = Z3
Z4-I = zr zr, = V,
(4.140)
zq+I = Zr,+2 zr,+2 = Zr,+3
z.-, = z z = v2
From Eq. (4.139) we have the corresponding "control" u,
(Z)
_
Lu2]
b21(Z)
r-a,(Z)+v, (Z)] L-a2(Z)+v2]
b12(Z) b22
(4.141)
To obtain the state feedback law, we only need to substitute coordinate X for Z according to Z = (D(X) . Namely, u,
(bi,(X) U-Cut]-Lb21(X)
-a, (X)+v,
b12(X)
L-a2(X)+v2]
b22(X)]
(4.142)
From Eq. (4.18) and Eq. (4.19) we know, Eq. (4.142) can be concretely written as
u2
Lg,Lj Ihz(X)
Lg,L 'h,(X)
-Ljh1(X)+v,
Lg=LI'h2(X)
L-Lrjh2(X)+v2
.a3
U=[u11_ Lg,L71h,(X)
(4.143)
Thus, we have the desired state feedback law. However, from Proposition 4.1 we can see, the conditions for the above linearization are (1)
Lg h,(X)= Lg Lfh,(X)=
= Lg Lfh,(X)=0
Lg,h,(X)=Lg Ljh,(X)=
=L,e)h,(X)0 Lk h2(X)=0
Lgh2(X)=Lg Lg. h2 (X) = Lg. L J h2 (X) =
k,
k2
(4.144)
= Lg_ Lkf h2 (X) = 0
r,+r2 =n
(2) The matrix
B(X)=[I Ltl is nonsingular at X°.
(X) h2(X)
Lg,L (X)1
LgzLJ h2(X)
(4.145)
150
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
Obviously, the above conditions for linearization are not convenient for application. This is because that for a system with m inputs and m outputs, to search output functions w, (X), w2 (X), , wm (X) which satisfy the conditions for exact linearization, we need to solve a set of partial differential equation
Ls,Lfw2(X)=Ls,Lfw2(X)...=Lg.Lfw2(X)=0
k1
Ls,Lfw.(X)=Ls,Lfwm(X)=...=Lg.Llwm(X)=0 1=0,1,...,km; k,
and also need to test the singularity of the matrix Ls 14'w1(X) 1w2(X)
Lg Lf
Ls=L!
Ls.L ,wi(X) ... Ls.LIw2(X)
Ls,Lf wm(X)
Lg,Lf 1wm(X)
...
Ls,Lri
'w2(X)
(4.147)
Ls.LI-lwm(X)
at X°.
COD
Similar to the condition of SISO systems in Section 3.2, here we again face two problems. The first is, for a known nonlinear system, whether the solution of the partial differential equations (4.146) that makes the matrix given by (4.147) nonsingular at X° exists, and whether the existence of the solution of above equations can be decided via the properties of the vector fields f (X), g, (X), , gm (X) . This problem is equivalent to the necessary and sufficient conditions for exact linearization of a MIMO system. The second problem is that, if it is known that the solution of exact linearization problem exists, whether we can use a practical method to find corresponding coordinates mapping Z = (D(X) and corresponding state feedback vector U($) = A(X) + B(X)V instead of solving the partial differential equations (4.146). This problem is the algorithm of exact linearization. In this section, we will continue to handle the first problem above. As to the second one, namely algorithm problem, it will be dealt with in a separate section. The necessary and sufficient conditions for the exact linearization of MIMO systems via state feedback is in principle agreeable to what discussed in Chapter 3 for SISO systems, but the form is more complicated. In the following, we will give the necessary condition for the former problem while omitting its proof process, if readers are interested in the proof of corresponding theorems, please refer to Ref. [1]. The following theorem gives the necessary condition for that the system (4.135) can be exactly linearized in the neighborhood of X°. i." 4-.
Theorem 4.1 For MIMO affme nonlinear systems
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
151
AI
X=f(X)+Ig1(X)u,
(4.148)
1=1
choose index number m = n, > n2 >
> nN ,
n, = n , where n is the order
of the system. If the system can be exactly linearized, then (1) the matrix composed of n vector fields IDA15,,...,gA,;adf gl,...,adfgA,;...,adf-'g,,...,adf `gAA]
is nonsingular at the point X°. (2) for the following n sets of vector fields D, = (g,) DA
DA+, _ {D n ;ad fg,} DA,+II, = (DA,;aU f6,,...,ad fgn,) D
D
adN-'Q
ad fN-' 9n,
each one is involutive near X°. Applying Theorem 4.1 to a special condition that m =1, we will have the following corollary.
Corollary 4.1 The necessary conditions for solving exact linearization of a SISO affine nonlinear system in a neighborhood of X° via state feedback is
1) the vector fields g(X), ad fg(X), , adf 'g(X) are linearly independent in the neighborhood of X°.
2) the set of the vector fields
is
involutive at point X°. It turns out that these two conditions in Theorem 4.1 are also sufficient
for exact linearization via state feedback. Obviously, the contents of Corollary 4.1 and Theorem 3.2 in Chapter 3 are completely identical.
4.4.2
Algorithm of Exact Linearization via State Feedback
In the previous section, the necessary and sufficient conditions have been presented for a nonlinear system with m inputs and m outputs can be
152
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
exactly linearized into a linear and controllable system Z = A.Z+B,V in a neighborhood of X° via state feedback U = A(X) + B(X)V and coordinates mapping Z=(D(X). These necessary and sufficient conditions are
characterized by means of the properties of the vector fields f(X) and g, (X), , g,,, (X) . In this section, an algorithm will be shown to obtain the
control law U'(X), mapping
and corresponding linear system -
CD
Brunovsky normal form, without solving partial differential equations as shown in Eq. (4.144). Now let us come to this algorithm.
For systems as shown in (4.135), first choose N index >_ n,,, and in, =n, numbers n n2, , n, which satisfy m = n, >_ n2 >_ n3
Step 1.
where n and m are the dimension numbers of state vector X(t) and control vectors, respectively. Then, calculate various orders of Lie bracket of g, (X) along f(X), a set of n vector field D, ={g,(X)} Dm = {g,
Dm+, = {D., ad f g, (X)) Dm +,,
= {D., ad fg, (X),
ad f g., (X))
(4.149)
Dm.,,,., = {Dm+n,,ad fg' (X)}
D_,_,
={Dm+,,,,adjg,
(X),...,ad'g,,,(X)}
D = {Dm+..+
Step 2. Choose n linearly independent vector fields n,, ...,
such
that D, E D,
(4.150)
DED Eq. (4.150) means to choose scalar function k(')(X) of X such that D, +k;')(X)g,(X) = 0 D2 + k; 2) (X)g, (X) + kit) (X)g2 (X) = 0 Dm
+k,(°,)(X )g,
(X)+...+ k(') (X)gm (X) = 0
(4.151)
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems +kim+n(X)gj(X)+kmm+' (X)adf
Dm+1
153
g, (X) = 0
j=1
j=1
i
"+n:)(X)gj(X)+
Yj=1
k(")(X)gj(X)+n
fgj(X)+...
-6n +Y_ j=1
j=1 nn
+ L km+ .+nn_,+j (X )ad j -ig j (X) = 0 j=1
p..
It should be noted that since the solution D , Nn of (4.151) is not unique, we had better to choose the simplest D; a D; . If vector fields D , i, are not properly chosen, the analytical expression of the integral i..
curves corresponding to these vector fields may hardly be obtained (see Step 3), nor the corresponding nonlinear control laws. Therefore, in the course of processing some skills are often needed to simplify the calculation. For some
VIA
complicated nonlinear systems, perhaps it is especially difficult to find D1 a D, (X), D2 a D2 (X), , Dn a D" (X) in a global sense, where X is an arbitrary point in the global state space. However, it is relatively easy to find the 61, , Dn in a local sense. We can search D, a D, (X°), D2 a D2 (X 0),
a
Step 3.
in a neighborhood of X°.
Utilizing the concepts and notations of integral curve
introduced in the exact linearization algorithm of SISO systems, search the mapping X = F(W) from state space R" described in new coordinate W
to the previous state space R" described with X. This mapping can be expressed by the integral curves of n vector fields D1, , D. as F(w1 w2,...,wn)=cii°'
...cp°-(X°)
(4.152)
0
Similar to the calculation procedure of formula (3.60), formula (4.152) can be calculated in terms of the following steps. First, calculate the integral curve Ow-'.X) with initial value of X° and independent variable of w . Namely, solve the following differential equation set
SIG
d
dwn
='D.
X(0) =
= Xo
(4.153)
The solution (D°. (X°) of the equation set (4.153) is a vector function with
independent variable of w
154
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS xl (w, ) x2(wn) D"(X°)= X(wn)= xn(wn)
/
Eq. (4.152) says that, the next step is to search th e integral curve of vector field ii,,, whose independent variable is and initial condition (DD (X°) . Therefore, the differential equation set xl (wn )
x,
X2 7
D
X(O) = (D . (X ° ) =
= Dn-i
d
dwn-1 X.
X 2 (wn )
(4.154)
xn(wn)
should be solved to obtain the integral curve 4)Dr,, -e-(X°), which is a vector function of independent variable (w,,-,,
i.e. xl (wn-l , wn )
(DD"_,
o (DD" X° = X w
w
x2(wn-1,wn) xn (wn-l , wn )
Then, use this as initial condition to calculate the integral curve of ii., (DD.-,
4)D^_=
W-
o
w^_,
°
(DD^( X° W.
) = X (wn-2 wn-1 > W11)
Analogically, we can obtain the integral curve of vector field D, , whose independent variable is w1 and initial condition (D, -01,
The
integral curve will be the solution to the following differential equation set x1 (w2 ..., wn )
x,
d
...
dw,
X2
= Dl
X(0)=(bD. o ...o0D"(X°)=
x2(w2 ...,Wn)
(4.155)
Xn(w21...Iwn)
X.
Solving Eq. (4.155) for X, we will obtain the mapping defined in formula (4.152), i.e. F.(w1..... w,,)
x1
X=
x2
= F(W) =
F2(w1,...,w11)
(4.156)
X.
From the above formula we can calculate the inverse transformation of F
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
W=
W2 WI
= F-' (X) =
155
(4.157)
4x'11 J
Step 4. Under the mapping F-' as shown in Eq. (4.157), compute the derived mapping F.-'(f) for the system vector field f(X) shown in Eq. (4.135). From the definition of the derived mapping of the vector field in Section 2.5 we know that (4.158)
F.-'(f) = JF_,f(X) Ix=F(w)
where JF_, represents the Jacobian matrix of F-1 (X) in Eq. (4.157), i.e. ow, (X)
JF =
&I
&2
aw2(X)
aW2(X)
aw2(X)
OX,
3X2
axn
awn(X)
aw,,(X)
-
F& F
and set
aw,(X)
OW, (X)
.
..
aw,1(X) 3Xn
&2
1
P01(w) f(0)(W)= f? I (W) = F.-'(f) =JF_,f(X) Ix=F(W)
(4.159)
W) fl(°) (W)-
As engaging exact linearization for a MIMO affine nonlinear system in Eq. (4.135) to find the proper coordinate transformation and state feedback, we need to progressively define transformation Rj, j =1, , N -1, for each
j, where N has been determined in the first step. For the sake of `.y
understanding, let us show the transformation R, m
ill
0+)1 (W)
1,,+11,
= f(°' (W) n
Z°'
(4.160)
(
Znl,+I = wl,-ll'+I 7'11')
= wl,
In the above transformation R1, fl.(°) (W) , i = n, +1,...,n, can be found in Eq. (4.159). Then we should calculate f(')(W)
156
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
f°)(W)=JR f(0)(W) where JR is the Jacobian Matrix in the following form raz;" aw"
az(1)
az(1)
L awl
awn
...
awl
JR =
j
Now the general transformation R J becomes R; :
f( j-0 (W) i =1,2 ... n - rI " +'
; J)
z;J-')=w,
(4.161)
from which we can calculate f (J) (W) I
f(i)(W)=
(W)
f2(J)(W) = JR (W)f(i-,)(")y)
(4.162)
fJ)(W) Following the procedure in Eq. (4.161), we can continue the calculation until the (N- 1) `h transformation RN- which is found out, f (N-2)(W) i = 1,2,... n - n1
roe
z(N-1)
(N-2)
(4.163a)
it can be specifically written as RN-, : ZIN-') = f (N- 2)(W) _(N-1) = 4(M-2)(W) 1-2
n2
(4.163b) (N-I) 1
_(N-2) (W)(= = n-n,+1
l (N-I) = y (N-2) (=
wn-n'+I )
wn )
Mapping RN-, is actually the coordinate transformation we eventually want to acquire. From (4.163) we can see that, this transformation is from space W to Z(N-') while what we need is transformation from space X to ZN-' .
In fact, the transformation from X to Z(N-') is equal to the composite transformation of F-' and RN-1, which is defined as T = RN-,F-' . The
157
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
transformation T can be calculated by means of the following formulae fn,(N+1-2)(W)
Z(N-1) =
W=F-'(X)
(N-2) =
(4.164a)
i = n - n, + 1, ... n
- x'l l
Z;
i =1, 2, ...,n -n,
which can also be written as Z(N-q=
4('N-11
W=F '(X)
=T1(X)
(4.164b) Z(N-1) ni+
_Jn
W=F ' (X) =Tni+ 1, (X)
(N-) _
Zn-n,+1
W=F-' (X) = Wn-n,+1 (X) = Tn-, +1 (X )
ZnN 1) = Wn
W=F-'(X)
=W n(X)=T n( X )
L"'
ton
Up to now, the final coordinate transformation T has been obtained. Under this transformation, the vector fields f(X) and g, (X), , gm (X) of the original nonlinear system (4.135) can be converted into i(X) and g(X), i.e. J (X) n1 = m ones
(4.165)
{f,(X)
'(X) = J T (X)f (X) =
ni+l (X)
ones
n2
f,(X) and ion
g(X) = JT(X)g(X)
1"'(') ax
aT, (X) ] ax n
1
821
()
8.1 (X)
812(X)
822(X)
8m2(X)
g2,, (X)
gmn (X)
b01
ax
8 1,, (X)
0
... ... ... ... ...
0
...
811(X)
821(X)
812 (X)
822 (X) 82m (X)
0
811 (X)
aTn (X)
aTn (X)
81m (X) 0 0
r
0
k. 1 (X) 8_m 2 (X )
8mm (X) 0 0
0
(4.166)
158
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
It can be verified that, if the original system satisfies the conditions for exact linearization, then the rank of the above matrix g(X) is m (m = n,) .
Namely, the m x m matrix in the top part of the last matrix in (4.166) is nonsingular.
Step 5.
Till now, the final coordinate transformation Z = F(X) can
immediately be found by setting that Z, = wn-n,-1(X)
Z. = wn{'(X )
(4.167)
Z,,,+1 = Z (X)
Zn-I = f,2(X) III
Zn = f,,
, w,, (X) have been calculated out by Eq. (4.157), and IX.
where
(X)
(X),
`-=
A., (X), ... ,1 (X) by Eq. (4.164).
The inverse of the coordinate transformation Z = F(X) shown in Eq. (4.167) is (4.168)
X = F-' (Z) ..:
If we denote the function vector composed of the first m components in Eq. (4.165) as a(X), i.e. a1(X) f1(X) a2(X) fz(X) = a(X) _ ...
(4.169)
am(X)
L(X)
Denote the m x m matrix in Eq. (4.166) by b(X) .,O
b(X) = b2, (X) Lbml(X)
... bim(X) ... b2m(X) ...
...
"'Z
b1I(X)
g1I (X) 812 (X)
821 (X)
k,2 (X)
...
bmm(X)
glm(X)
82m(X)
... gmm(X)
8- (X) k.' (X)
(4.170)
and
U = -b-' (X)a(X) + b-' (X)V Jx=F-,(z)
where V =[v,
vm]T
(4.171)
, then the original nonlinear system (4.135) will be
transformed into the Brunovsky normal form
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
159
Z, = z2 Z2 = Z3 (4.172)
and the state feedback law will be obtained for the original nonlinear system as U = -b-' (X)a(X) + b-1(X)V (4.173)
Here we need to point out that if the sequence of coordinate transformation in Eq. (4.167) is changed to Z1 = 7.1 (X) Zn-m = f, (X)
(4.174)
Zn-m+l = Wn-m+1 (X)
Z. = W. (X)
then the original nonlinear system Eq. (4.135) will be transformed to the following normal form. it = VI Zm =Vm
(4.175)
im+l = Z1
Now an algorithm for exact linearization has been introduced, which is
put forward and presented in [1]. We can see that, this algorithm is essentially identical compared with that for exact linearization of SISO systems. In the following let us illustrate the whole procedure of this algorithm with an example.
Example 4.2
Given a system -1
JC,
i2 =
N
x3
1
2x1x3V2 + 2x2x3/2
x,(0)
0
x2(0) =
1
x3 (0)
1
x1 + x2 - 3x2/3
XZ 3
+
0
- 2x332x2 3
ju1 +
3x2 3
Ju2
- 2x1x3V2 -
First, we should examine whether the system satisfies the conditions for
160
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
0
exact linearization. To shorten the length, let us suppose it has been tested that the given system satisfies the conditions in Theorem 4.1. Now let us perform the calculation of exact linearization via state feedback. Step 1. Calculate the index number of the system. Since the system inputs number m equals 2, we choose n, = m = 2. Then we can only choose n2 =1 such that the requirement is satisfied, that is, n, + n2 equals n. Therefore, we know N = 2, nN = n2 =1. According to (4.149), we form 3 vector field sets as follows D, = D, _ {g, (X)} D. =D2 = {g1(X),g2(X)} D = D3 _ {91 (X)1 g2 (X), ad fg1(X)}
Thus, we should calculate the first order Lie bracket of f(X) and g,(X), ad1g, (X) 1
x-2/3
3
2
ad Jg, (X) =
0
- 2X-2/3 x32 -2X23 x32 0
may
Step 2. Choose the simplest three vector fields D, a D,(X), D2 a D2(X), D3 a D3 (X) . According to Eq. (4.151) we calculate functions k;°), 1
D, + k;') (X)
=0
0 - 2x1/2x 1/3 3 2
Choose k;') (X) = x2113 , then
_
- 1
D,=
0
EE DI
J2 2x;2
From
x1 + x2 - 3x2
x2 3
D2+k;2)(X)
+k(22) (X)
0 2x3/2x'2 3
we can choose k;2) (X) =
3
3
1=0
3x23
- 2x,x3/2
- 2x2x3/2
(x, + x2 )x2' , k22) (X) = - 3 x22/3 such that
_ D2=
-1 1
0
eD2
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
161
Finally, based on x, + x2 - 3x 2J3
x23
D3 +k;3)(X)
+k(3)(X)
0
3x23
"'x
-212x23 1
-2x,x3/2 -2x212
-2/3 x2
3
+ k(31 (X) 3
=0
0
-
x22/312 -2x 231/2 3
we may choose ki 31 (X) _ -1x2 /3 - xjl/3
k33) (X) = X-113
k23) (X) = 0
3
such that 1
D3-0eD3 0
Step 3.
According to Eq. (4.152), search the mapping F(w,w2,w3)=(D°' o(I) o0'(X°)
We first calculate the integral curve 4)61 (X°), namely, solving the following differential equation dx, dW3
dx2 dw3 dX3
1
0 x2(0) = 1 x,(0)
]11
= 0 0
x3(0)
1
dw3
which yields
Then we compute (V, -°' (X°), i.e. solve the differential equation
162
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
dx,1 dW2 x,(0)
d 2
dw2
0
[W3]
x2(0) =
1
x3(0)
1
dX3
dw2
which gives - w2 + w3
x,
x2 =
w2 + 1
x3
Finally, we figure out
QDD,
WI
1
o,D°= o D (x°), namely solving the differential W, 3
equation dx,
dw,
x,(0[-W2+W3]
1
2
x2 (0) =
0
dw,
x3(O)
2x3/2
w2 +1 1
dX3
dw,
From the above we have obtained the mapping defined in Eq. (4.152) Fl(w1,w2,w3)
XI
-W,-w2+ W3 w2+1 (w, +1)2
Xz =F(w"w2,w3)= x3
Correspondingly, the inverse mapping of F is ,/2 x3
wl w2
-1
x2+1 ,+X2+x31'2-2 Ix
w3
Step 4. Calculate the derived mapping F.-'(f), and then calculate f(0)(W) . We first seek for the Jacobian matrix JF' for F-', i.e. Caw,
J_,= F
awl
8w,
8w,
lx2
ax3
&2 &2
33
f7X,
aX2
aX3
lax,
(x2
OX3
From Eq. (4.159) we have
3
I(w2+1)' 0
2(w2+1)'
f(°)(W)1 f(0)(W)= fz(°) (W) = JR_AX) Ix=R(w)= 40) (W)
G.)
Design Principles of Multi-Input Multi-Output Nonlinear Control Systems
163
-w,+w,+1 I
-w,+w,+1
Since in this example n, = 2, n2 =1, N = 2, we only need to search the transformation R,,,_, = R, . From Eq. (4.161) we obtain R,
Z10) = f'(0) (W) _ -W, + W, +1 ZZt) = Z(o) = W2
Z3') = Z3°) = W3
Correspondingly, transformation T = R,F-' becomes
T: Zi') = A()) (W) _ (-W1 + w3 + 1) I w=R_'(X) = x, + x2 a) Z2 = W2 IW=F-i(X)= X'2 (1)
Z3
1/2 -x'3IW=R_i(X)-x,+x2+x, -2
The Jacobian matrix of T is 1
1
0
JT= O
1
0
1
2 x31/2
1
From the transformation T we calculate the last group of transformations 0
f(X)=JT(X)f(X)= f{2(X) =
I
x, +x2
73(X)
X /3 z
g1 = T. (g1) = JT(X)g,(X) =
0 0
X, + x2
gz = T' (92) = JT(X)gz(X) =
3x2/3
0
Step 5. From the above calculation, by comparing Eqs. (4.169) and (4.170) we have
164
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
a(X)=R(X)1=r01 1 (X)
b(X)=
x23
xl +x2
0
3x,3
From Eq. (4.173) we know, that the nonlinear control law of this example should be 3 (XI +x2)x'
x-13 2
-3(x,+x2)xz
U = -b-'(X)a(X) + b-'(X)V -13 x-2/3 2
0
1 x-2/3 3
[-v",I
2
From Eq. (4.174) we set Z, = AM = x, + x2 Z2 = w2 (X) = x2 -1 3 = w3 (X) = x, + x2 + x3/2 - 2
and let u = (-b-' (X)a(X) + b -' (X)V)'x_F_,
then the system in this example will be exactly linearized into the following normal form Z, = V1
Z2 = V2
Z3 =Z1
4.5 REFERENCES I.
D. Cheng, T. J. Tam and A. Isidori, "Global Linearization of Nonlinear Systems Via Feedback", IEEE Trans. AC, Vol. 30, No. 8, pp. 808-811, 1985.
Chapter 5
Basic Mathematical Descriptions for Electric Power Systems
INTRODUCTION
5.1
0.2
It is a significant task to integrate the latest research achievements on nonlinear control theory with the electric power system dynamics in order to form a modern disciplinary system of nonlinear control of power systems, f7,
coo
with the goal of improving the dynamic performance and both small and large disturbance stability so as to provide better quality and more security of power supply.
8,0
In the previous chapters, we have systematically presented the basic concepts of modem nonlinear control theory and some principles and
coo
apt
methods for designing SISO and MIMO affine nonlinear control systems. All these concepts, principles and methods, being of universal validity, are applicable to a wide range of nonlinear engineering control systems. The main task of the subsequent chapters is to built up a bridge between the ,.t
theory of power system dynamics and these concepts, principles and methods, so as to open a new discipline and to design various types of
coo
controllers to improve the stability of large-scale power systems. To achieve this goal, we must first interface the design principles of nonlinear control with the mathematical models of power systems. This chapter will present the elementary and applied mathematical models of various elements of a power system such as synchronous generators, relevant controlled subsystems, and DC transmission systems, etc., which will be commonly used in the subsequent chapters.
Electric power systems are huge and complex dynamic systems composed of different types of generator sets (hydro generator sets and
166
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
coo
vii
:D+
'in
ono
.fl
(Dr-
CAD
'+=
STS
ay;
(''D 7,05w
ono
"'1
Wit,
thermal generator sets, etc.) [1, 3, 4], numerous substations, electric loads, power networks with different voltage levels, and transmission lines of different transmission modes (AC and DC transmission lines). It is necessary from the very beginning to have a comprehensive and deep understanding of the objects under consideration, and to describe them with mathematical language as accurately as possible. Thus we can perform our investigation and analysis on a reliable basis. On this basis, we still have to make some further studies to distinguish the major points from all other relevant factors and conditions, and to make some necessary approximations by ignoring the less important factors so as to form the applicable mathematical models accurate enough for engineering purposes. Of course, the more comprehensive and deeper the understanding, the better the approximation will fit the objective reality. As mentioned above, electric power systems are not only composed of many links and components, but they have their own dynamic characteristics, especially for synchronous generators, which not only have electro-magnetic transients [7], but also electro-mechanical dynamics. This determines that the mathematical model of a power system is a high-dimensional nonlinear equation set [9]. In the following sections, we will introduce respectively, in the context of a multi-machine power system, the rotor dynamics equations, ..G
=°,
power equations, the field winding's electro-magnetic equations of a
5.2
CAD
.`3
vii
Qom
synchronous generator, the equations of the steam valve governors, and the equations of a DC transmission line and its control systems which will be used for the components and control system modeling.
ROTOR DYNAMICS AND SWING EQUATION ASR
According to the law of Newton's dynamics, we can immediately obtain the relations among angle and angular acceleration of a generator's rotor and the torques imposed on the shaft of the generator set as
Ja=Mm-M.-MD
(5.1)
where, a is the angular acceleration of the rotor of a generator set; M. the CAD
mechanical torque imposed on the driving shaft supplied by the prime mover; M, the electromagnetic torque of the generator; MD the damping torque in
direct proportion to the variations of angular speed (strictly speaking, the damping torque of a generator is not only related to the variations of angular speed, but also a function of the state variables of the generator. But from the viewpoint of engineering approximation, it is considered to be only related
to the variations of the speed); J the moment of inertia of the rotor of the generator set (including the rotors of generator and prime mover).
Basic Mathematical Description for Electric Power Systems
167
CAD
Fig. 5.1 represents the relationship among the fixed reference F-axis used to measure the rotor's motion of a generator set, the synchronous reference S-axis rotating with the synchronous speed a , and the q-axis of generator.
Figure 5.1
Relationship between different reference axes used to measure the motion of a generator's rotor
As shown in Fig. 5.1, assuming that the axes q, S and F are superposed at t = 0, the relation between 9, y and S at the moment of t can be described by (5.2a)
S(t) = y(t) - 9(t)
.^.
and
r(t) = Jow(t)dt
(5.2b)
9(t) = wot
(5.2c)
where S denotes the rotor angle of the generator. From Fig. 5.1, we can obtain the angular speed of generator as CO =COO +
ds
(5.3)
-.fl
and then, the angular acceleration a in Eq. (5.1) as a=
do) dt
=
d ZS
(5.4)
dt2
..w
Thus the equation governing the motion of generator's rotor, which is called the swing equation, can be rewritten as d'.5
J dtz = M. -Mr -MD
(5.5)
168
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Eq. (5.5) can be applied in any unit system. If all variables in the equation are measured in per unit of value, then Eq. (5.5) can be written as
J,
28 W,
= Mm.
(5.6)
- Mr. - MD.
In the above equation, the base value of torque MB is MB - S-101
joule/mechanical radian
2=/60
where S is the power rating of generator with unit of KVA; n denotes generator's mechanical synchronous rotations per minute; the base value of time TB is 1/(2zfo) seconds. So, in the systems of 50 Hz, TB =1/314 seconds. And the per-unit value of the moment,of inertia in Eq.(5.6) is J. = 2,r' fo
GD 2
(60)2
radian
where GD2 denotes the flywheel torque of rotor with unit of In practical engineering applications, not all variables in Eq. (5.5) are measured in per-unit relations. Usually time is expressed in seconds; angle 8 in electrical radian; the torques such as Mm, Mr and MD are expressed in per unit of value; and the moment of inertia, denoted by H, is expressed in seconds. Thus Eq. (5.5) can be rewritten as kg-M2.
H dt 27rfo d 2
=Mm -Mr -MD
(5.7)
If H and time t are measured in seconds, and Mm, Mr and MD still use per unit
of value, but 8 uses electrical degrees, then Eq. (5.5) will have the following form
H d28 360fo dt2
= M. -M, -MD
(5.8)
In the above two equations, unit of the moment of inertia H is in seconds, so 0
H equals the moment of inertia J. (in per unit of value) multiplied by the base time TB, i.e.
H = J.TB = J.
1
21r fo
2.74GDZn2 (n 2 GD2 10 ' seconds )2 = S 60 S
°-05
In terms of physics: H is denoted by the time period it takes for a rotor rotating from a totally static state to reach its rated speed as a 1.0 per unit
coo
driving torque is imposed on it at the moment t = 0. For the purpose of engineering applications, we usually do not use the torque values for modeling power systems, but transform them into power values, since the following relation exists between power P and torque M :
Basic Mathematical Description for Electric Power Systems
M=
169
-P CO
where w denotes the angular speed, and the synchronous speed wo is equal to 1.0 in per unit of value, so the per-unit power equals per-unit torque as w = (Do . The base value PB of per-unit power is the power rating with unit of KVA. Thus Eqs. (5.7) and (5.8) can be respectively rewritten as one of the following equations, either
H d2S 2rcfo dt2
(5.9)
P. - Pe - P»
where S is in radians, H and t in seconds, the mechanical power Pm, the electromagnetic power Pe , and the damping power P° in per unit of value, or
H d28 =P -P -P
(5.10)
°
e
m
360fo dt2
where S is in electrical degrees, and the units of other variables are the same as in Eq. (5.9). The damping power P° for practical calculations can be approximated as P°
(5.11)
2 fo
dtS
where D is the damping coefficient and is regarded as a constant, using radian as its per unit of value, and radian/second as the unit of dS /,*
.
For the relation between S and w : if S is in radians and w is in radian/second, then from Eq. (5.3) we have .--t
dS = CO - WO
(5.12)
dt
It should be noted that the value of H in the technical literature by the authors from some western countries is half as much as that used in some oriental countries [10]. Due to this reason, the corresponding equations of Eqs. (5.9) and (5.10) in the above mentioned literature must be rewritten as 2H d 2S 1-=P m 2;rfo dt2
-Pe -PD
(5.13)
=P -P -P
(5.14)
and
2H d2S 360fo dt2
m
e
D
This difference should be noted in the modeling and calculating processes.
170
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
OUTPUT POWER EQUATIONS FOR A SYNCHRONOUS GENERATOR
5.3
In this section the equations of active and reactive output power of a generator are given, which are the basic relations in the analysis for the electromechanical dynamics of generators and power systems.
According to its definition, the real value of the instantaneous active output power in NMS units for a three-phase synchronous generator is (5.15)
Pr =Vaia +Vbb +Vcic
where va , vb , and v, respectively denote the instantaneous values of generator's terminal voltages of the phase a, b, and c, and is , ib , and is Q;.
denote the instantaneous values of generator's stator currents of three phases, Vii
respectively. Eq. (5.15) can also be rewritten in the form of vectors' inner product, i.e. (5.16)
P. ==V ec1abc
where the vectors are Vatic
Va
'a
Vb
Iabr = Ib
vc
ic
and the superscript "T" of Vatic in Eq. (5.16) represents the transpose. +L
0
Since the base value of power PB = 3V1= 3(v8 /-,f2-. iB / -n) , where V and I denote the effective values of the rated phase voltages and currents of
the stator's windings, and v8 and iB denote the base values of the instantaneous peak values of phase voltage and current, the per-unit instantaneous output power will be Pe = 3 (vaia +Vblb +Vctc) = 3 V. bclabc
(5.17)
where the variables of voltages and currents are measured in per unit of value.
As we know that the stator windings of a three-phase synchronous generator is a symmetric three-phase system of a, b, and c, and each phase winding is 120° apart from the others in space, while its rotor windings are asymmetric. If the stator's system of a, b, and c is chosen as the coordinate C.'4
system for the mathematical model of a generator, then magnetic -S0
conductivity of the magnetic circuits on each coordinate axis will be a periodic function of time t resulting from the rotor's rotation and the salient pole effect. Therefore the self inductances of the phase windings, the mutual
Basic Mathematical Description for Electric Power Systems
171
bow
"C7
inductances between different phases and that between the field winding and the stator windings will all be the periodic functions of time t. This will bring about lots of inconvenience for the analysis of many problems. If we fix the
coordinate axes d and q on the rotor rotating together with them, and let them superpose with the two rotor's symmetric axes (see Fig. 5.2), then the mathematical model of the generator will be greatly simplified. As a result, for an ideal generator, the magnetic circuit conductivity of the windings of axes d and q will be constants, so every parameter of an ideal generator will be constant and completely independent of time t. Thus the mathematical model of a generator becomes a time-invariant dynamic system.
Figure 5.2 Coordinate axes a, b and c fixed on the stator and coordinate axes d, q and 0 fixed on the rotor
From Fig. 5.2, we can intuitively obtain the transformation equations between the two coordinate systems of a, b, c and of d, q, 0 as a
C4b
(5.18)
and the inverse transformation as
1Sad
(5.19)
b`C1q
o
In the above two formulae, d, Q , 4o and a I
Sb I
7c
respectively
correspond to variables such as voltages, currents and flux linkages etc. in the systems of d, q, 0 and a, b, c. The transformation matrix C in formula (5.18) is
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
172
C
2
2
3
3
2
2
cosy - ir) cos(y + ,r)
cosy
- sin y - sin(y - 3 ir) - siny + 3 ,r)
3
L
1
1
1
2
2
2
(5.20)
j
C-', the inverse matrix of C, is
-siny
cosy
C-1 = cosy - 3 ;r) - sin(y - 3 ;r)
I 1
(5.21)
cosy + 3 ,r) - siny + 3 ,r) I
The meaning of angle y in the above two equations can be referred to Fig. 5.2.
Integrating Eqs. (5.18), (5.19) and (5.17), we can know that in order to
transform the instantaneous active power represented by voltages and currents in the coordinate system of a, b, and c into the expression of the voltages v,,, vq and currents i,,, ig in the coordinate system of q, d, and 0, it is necessary only to perform the following transformation Pr
=2 (C i Vdgo)TC , Idgo
3
which can be rewritten as (5.22)
P, = 3 V0(C-1)TC,-1Idg0
where V0}T
Vdgo =[Vd Vq
IdgO - Ltd iq
i077T
J
From Eqs. (5.20) and (5.21) we know that matrix C has the following property
(C')TC' =
3/2 0 0
0
0
3/2
0
0
3
(5.23)
C/1
Substituting expression (5.23) into (5.22), we can obtain the per-unit output power represented by the instantaneous values of voltages and currents in coordinate system d, q, and 0 as Q':
P, =Vdd + Vgtq +
2vOt0
(5.24)
As we know, the armature windings of a synchronous generator are in three phases. If the windings are Y-connected without neutral line, then in Eq.
Basic Mathematical Description for Electric Power Systems
173 C00
(5.24) i0 =0'), and the per-unit active power of a three-phase synchronous generator can be expressed as (5.25)
Pc = Vdld +Vgiq
"CS
In the subsequent discussion, we will give the expressions of d-axis voltage vd and q-axis voltage v9 on the basis of the transformation equations shown in formula (5.19). According to the basic theory of electromagnetics, when the resistances of the armature windings are ignored, the relationship between the phase voltages and the flux linkages can be expressed as va
J
Vb
vc
d
=-dt
Wa
(5.26)
Wb We
where Wa , Wb , and V. respectively denote the instantaneous values of flux
linkages of the armature windings a, b, and c. Now substituting the coordinate transformation expressions given in (5.19) into (5.26), we have C-'
Vd Vy
d =-(c' dt
O
Wd
Wd
Ww
W9
C-1 dt
W0
dt(c
Wd Wq W0
W0
where yd and Wq respectively denote the flux linkages of, ie fictitious d and q axes armature windings.
As we know from Linear Algebra that the above equations can be rewritten as Vd
Wd l
Vq
Wq
dt
dt (c )
W0
VO
Wd Wy
(5.27)
WO
From the expressions of C and C-' given in formulae (5.20) and (5.21), we
know that since the relation between the speed w and the angle y is co = dy / dt (see Eq. (5.2b)), the matrix (CC') in Eq. (5.27) will be
0 -w 0 CC-' = w
0
0
0
0
0
(5.28)
where t` means d (C-') l dt . Substituting formula (5.28) into (5.27), we have vd _ d
dtWd - wWq
(5.29a)
') In reality, the stator windings of a synchronous generator almost always use a Y-connection.
174
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
(5.29b)
Vq = dtY'q +coyid
'F..
What the Eq. (5.29) gives is an expression of the relation between the voltages and the flux linkages of the armature windings of a generator obtained via a coordinate transformation from the coordinate system of a, b, and c into that of d, q, and 0. For better understanding, we can explain the equations as follows: the first term in the right side of Eq.(5.29a) denotes the
induced electric potential in the fictitious d-axis stator winding by the
C/1
in:
0"Q
V1'
variation of the d-axis flux linkage, so it is usually called the "transformer electric potential"; the second term denotes the electric potential in the d-axis stator winding caused by the incision of the d-axis stator winding by the qaxis flux linkage with a rotating speed of co . Obviously it is the main part of voltage vd. The similar explanation may also be applied to Eq.(5.29b). In Eq. (5.29), assuming the generator rotates constantly with the synchronous speed (i.e. co(t) =c)o =1.0 ), and ignoring the so-called transformer electric potentials d yrd I dt and d yrq I dt , the relations between the armature winding voltages and the corresponding flux linkages of axes d and q can be described as Vd = -Y/q (5.30a) (5.30b)
Vq =Vld
Similarly, if we apply the coordinate transformation from a, b, and c into d, q, and 0 for the expressions indicating the relation between the ib, and ii,, and the excitation current IJ, we can get equations for flux linkages on
original flux linkages in phases a, b, and c, the armature currents
iQ ,
axes dand gas cad
(5.31a)
=XdIJ-Xdid
yrq = -xgiq
(5.31 b) CDR.
Wd
C:1
In the above equations, xd and xq denote the stator windings' self inductive reactances of axes d and q; x,,d is the mutual inductive reactance between the d-axis stator winding and the field winding. It should be noted that the effect
of the damping windings is ignored in Eq. (5.31), whose effect on the electromechanical dynamics is just approximately considered by a damping power " PD in the swing equation (see Eq. (5.11)). '+p
O0-,
In expression (5.31a), it is obvious that xadIJ denotes the electrical potential induced in the stator windings when the flux of the excitation current cuts through the air gap and rotates with speed w, i.e. Eq = zeaI J
Thus expression (5.31) can be rewritten as
Basic Mathematical Description for Electric Power Systems
175
Yld = Eq - Xdld
(5.32a)
11/q = -xglq
(5.32b)
Substituting the formula (5.30) into (5.32), we have Vg = Eq - Xdld
(5.33a)-
vd = xgiq
(5.33b)
0
May
Based on the expression (5.25) and the above two formulae, we can obtain the expression of the generator power as Pe = vdld + vglq = Egiq + (xq - xd )Idlq
(5.34a)
which also can be rewritten as Pe = Egiq +(xq -xd)idig +
(5.34b)
When the generator is connected to a power system and is loaded with active power, the load current will cause an angular difference S between
the terminal voltage of the generator and the idling (no-load) electrical potential Eq. In this case, the expressions for the instantaneous terminal voltages can be expressed as')
r
sin(y - S)
va Vb
= -V
WAN
vi
sin(y-S-3,r) sin(y -6 + 2 )r) 00'
vd
Cv.j
2 cosy 3
`C7
In the above expression, V, denotes the amplitude of the generator terminal voltage. Transforming it into its components on axes d and q, we shall have (see Eq. (5.18) and expression (5.20)) cosy - 3tr) cos(y + 3 )c)
Ve vb
-sing -sin(y-2,r) -sin(y+2ir) 3
vc
(5.35)
3
V, sing [V, cos S]
From expressions (5.33) and (5.35), the d, q components of the armature winding currents can be calculated as Eg - V, cos S id =
Xd
'" Assume that the generator's terminal voltage is sinusoidal.
(5.36a)
176
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
iq=V` sins
(5.36b)
Xq
ors
Substituting the expressions (5.35) and (5.36) into (5.25), the generator output power equation represented by the idling electrical potential Eq, the terminal voltage V,, and the included angle (power angle or rotor angle) 8 between Eq and the terminal voltage vector can be obtained as V2
Pe =
EXV` d
X 2 sin28
sins+( q
(5.37a)
d
If the generator is a cylindrical rotor machine, as xd = xq , the expression of the active power can be written as Pe =
EV` ' g sing xd
(5.37b)
I-+
Now let us make a summary to see what approximations we have made in order to obtain the practical expression of the output active power shown in Eq. (5.37). Firstly, the voltage losses caused by the armature winding
resistances r were ignored in Eq. (5.26). Secondly, the flux linkages ...
produced by the currents in the damping windings were neglected in Eq. (5.31). Next, we discarded the component of the electric potential in Eq. (5.30) caused by the varying flux linkages in the armature windings, which are usually called electric potentials of transformer effects. Furthermore, we always assume that the waveform of the terminal voltage is absolutely sinusoidal. And finally, the rotor speed co was approximated by co, =1.0 in the deduction of the power equation, that means the generator's active power expressed in Eq. (5.37) is only applicable to the synchronous operation of generators and power systems. Despite all those assumptions mentioned '-'
a°-
above, it has been manifested by engineering practice that the power equation (5.37) can meet the accuracy requirements of the mathematical models using to analyze the stability problems and assess the dynamic performances of power systems. Next we will deduce a generator's reactive power equation. Under the condition that the voltage component of the zero axis vo = 0, according to Eq. (5.35) we have the following expression
V,=w;+v9
(5.38a)
1= io +i,2
(5.38b)
and similarly
where V, and I denote the magnitudes of the terminal voltage and the armature winding current of a generator, respectively. According to the
Basic Mathematical Description for Electric Power Systems
177
definition of reactive power, we have
Q= S2 -p2 = (vd +vq)(ld +tq)-(udid +vglq)2 = V qd -vdiq
(5.39)
In the above equation, S = V1 I is the apparent output power of a generator. Substituting Eqs. (5.35) and (5.36) into Eq. (5.39), we can get the expression of the reactive power, that is the generator supplies to the electric network as 2
2
Q = Eg ' cos S -
+ 1) V' (1 Xd 2
+ (Lxq
Xq
xd
cos 23
2
(5.40a)
7n'
Xd
1) V`
If the generator is a cylindrical rotor machine, namely Xd = xq, then
Q= EgV, cosS- V2 Xd
(5.40b)
"'h
Xd
CAD
}''
Now we will deduce the expression of generator output power using transient electric potential E9 instead of idling electric potential Eq. Ignoring the damping effects of the damping windings and the solid rotor of the generator, the flux linkage equations can be written as yr fd =-xJid +x flf
(5.41)
If
(5.42)
Vd r-+
In Eq. (5.41), w fd is the total flux linkage of the field winding. It consists of two parts, one is the excitation flux linkage x. If produced by the excitation '-'
4r+
current If ; the other is the field winding flux linkage -xid produced by
-'o
Lei
0
the cutting through air gap flux due to the d-axis component of the armature current id, and is also called the flux linkage from the armature reaction. The minus sign before it means the effect of the armature reaction from a positive id is of demagnetization. Eq. (5.42) has emerged before, and there is no need to explain it again. Eliminating the excitation current If in Eqs. (5.41) and (5.42), we have Yd -
x xf
Vfd - (xd -
2 x_
)ld
(5.43)
Xf
In the above equation, let Xd =xd - Xd xf
(5.44)
178
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
E'q =
xad
(5.45)
Y1fd
xf
then Eq. (5.43) can be rewritten as (5.46)
Yd=Eq - xdtd
'C3
where, Eq is the electric potential behind the transient reactance x,,, and also is the q-axis transient electric potential. Since x, < x f , Eq' is less than and directly proportional to the rotor total flux linkage 'd fd . Considering Eq. (5.30b) in Eq. (5.46), we have (5.47)
Vq = E' - x,, id
Comparing Eq. (5.34a) with Eq. (5.47),,we can obtain the following equation (5.48)
Eq = Eq + (xd - xd )id
From Eqs. (5.47) and (5.35), the expression of id corresponding to Eq. (5.36) can be obtained as =
1
Eq'
- V, cos 5
(5.49)
x
d
d
Substituting Eqs. (5.49), (5.36b) and (5.35) into the generator output power equation shown by Eq. (5.25), the following power equation can be obtained P r = x y V / sin ,5 + (s - X) d
q
d
sin 28
Z
2
(5.50a)
00"
Assuming xq = xd approximately, the power equation can be simplified as E9 V, Pe
xd
sin8
(5.50b)
The generator output power equations shown by Eqs. (5.37) and (5.50) ,..
are correct for generators running both in steady and transient state. vii
However, when a sudden change occurs to the armature current (e.g. a short circuit in the electric network), there will be a sudden change of the flux
linkage of armature reaction -xid. Since the flux linkage of the field winding yr fd , the total flux linkage of a closed winding, can not change suddenly, hence a corresponding abrupt change has to occur in the excitation current If according to Eq. (5.41) . Thus the idling electric potential Eq = xodl f
will have a sudden change. But at this moment, the electric
behind transient reactance remains unchanged. For this reason, Eq. (5.50) is used more often for dynamic modeling and analyzing the transients of a generator. potential
Eq = W fd xad /x f
Basic Mathematical Description for Electric Power Systems
179
Correspondingly, the generator reactive power can be expressed as Q=EqV, xd
z - )' cos28
cos8-(1 + 1) V,z +(1 xd
Xq
2
1
zq
xd
(5.51a)
2
Assuming xq = xd approximately, the above equation can be simplified as
QEqV, xd
2
cos8- v
(5.51b)
xd
OUTPUT POWER EQUATIONS FOR SYNCHRONOUS GENERATORS IN A MULTI-MACHINE SYSTEM
5.4
I..-O
Abiding by the principle of "from simple to complex", in this section we shall first give the output power equations for a generator in a one1U,
-fl
machine, infinite-bus system, then present the output power equations of a generator in a power system with n generators. omt,
5.4.1
Output Power Equations for a Generator in a One-machine Infinite-bus System (ii
r-+
The general practical expressions of the output power of a generator 'i+
c''
have been given in Eqs. (5.37) and (5.50). These equations reveal the relations among the output power of a generator and the idling electric 't7
potential (or the electric potential E9 behind the transient reactance xd ), the terminal voltage V, , the included angle S between Eq and the terminal voltage vectors, and the generator's reactances Xd (or xd) and xq. In this
...ate--.c
tag
section we shall give the output power equation in this context that a
ova
d.+
generator connects to an infinite-bus system with a main transformer and a transmission line (as shown in Fig. 5.3). If 8 denotes the included angle between the electric potential vector Eq or E9 of the q-axis and the infinite-bus voltage V, , then according to the same equation (5.35), the voltage of the infinite bus also can be divided into two components of axes d and q (5.52a) v.,, = V. sin 5
180
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
XTL
{
10
Pg+jQg
EKE
Figure 5.3 A one-machine, infinite-bus power system
V,- amplitude of terminal voltage; VS amplitude of infinite-bus voltage xr-- reactance of the transformer; xT.,- reactance of the transmission line
(5.52b)
v,,q = V. toss
When the transformer reactance xT and the transmission reactance xTL are taken into consideration, the armature currents can be expressed as id
-
Eg - VScosS
(5.53a)
XdE
iq =
Vs
sine
(5.53b)
XqE + XTL ,
XqE = Xq + XT +X,. If id is represented by Eq, I;'
where Xd E = Xd +X'. then we have
id =
Eg -V, cos8
4:
(5.54)
where x = xd + xT + xTL .
comet
'fi't
4-i
In Eq. (5.25) the active power injected into the infinite bus is also the active power Pe supplied by the generator if the active power losses caused by the resistances of the transformer and transmission line are ignored, i.e. Pe = void +v,,giq
(5.55)
Now substituting Eqs. (5.52) and (5.53) into Eq. (5.55), we get the 4-+
expression of the active power of the generator to the infinite bus as z
Pe =
E-V S si n 8 + v's 2 ( xd - xg) s i n 28 XdZxgE XdE
(5 . 56a)
For a cylindrical rotor generator, the expression will be Pr = EgVs
sin g
(5 . 56b)
XdE
If Eqs. (5.52), (5.53b) and (5.54) are substituted into Eq. (5.55), the power equation can be rewritten as
Basic Mathematical Description for Electric Power Systems
181
,
Pp = x9Vs
sin8+ 2s (x° dExxg sin 15
dE
(5.57a)
qE
If a further approximation xd = xq is made, Eq. (5.57a) can be reduced to
(5.57b)
Pe = E9Vs sins
xL
.N,
.^s
It is necessary to mention that angle S in Eqs. (5.56) and (5.57) is the angle between the q-axis electric potential vector and the infinite bus voltage vector.
Corresponding to Eqs. (5.56) and (5.57), the reactive power of the generator delivered into the infinite bus can be expressed as xdE
2
xdZxgE
2
(5.58a) (J1
Q=EgVscos6-VS(xd1+xgI)+Vs(xd-x°)cos2S xd£xgE
2
X;':
X XqE
V
-xq )cos2S
2
C/1
Q=
k^k
or
(5.58b)
For a cylindrical rotor machine, Eq. (5.58a) can be reduced to xdE
z
Vs
`.,
Q= gV s cos S -
(5.58c)
xdE
vii
Making further approximation by xa = xq in Eq. (5.58b), the equation for generator reactive power delivered to the infinite bus can be written as
Q - Ei_'VS cosS - Vs xa xA 2
5.4.2
(5.58d)
Practical Output Power Equations for Synchronous Generators in a Multi-machine System
Y$3
+U+
A power system with 6 generators and 22 nodes is shown in Fig. 5.4(a). As we know from the theory of power system dynamics [2, 8], if the load of each bus is modeled with a constant impedance and all the external buses are eliminated in Fig. 5.4(a), we can obtain the equivalent diagram shown in Fig. 5.4(b). The electric potential vector of each generator shown in the figure can be either the idling electric potential Ell, or the electric potential Eq,
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
182
Z66
Z11
EI
Zlfi 1
Zls
E6
Zls Zl6
Z12
Et
8
5 E5
2
3
(a)
(b)
Figure 5.4 A 6-generator power system and its equivalent circuit
behind the transient reactance. Which one should be used is dependent on the nature of the problems under consideration. 1, ,
tai
"
12,
If we express the current vectors') for generators 1, 2, . . ., 6, with 16, from Fig. 5.4(b), these vectors can be expressed as I, =T1 Yn +E2Yi2 +E3Y13 +E4Y14 +E5Y;s +E6Y16
(5.59) 16 = E, Y61 +E2Y62 +E3Y63 + E4 Y64 +EsY6s +E6Y66
where Y,, (i, j =1, , 6) is the corresponding component of the network's admittance matrix. Extending Eq. (5.59) to an n-generator power system, we can obtain the delivery of current of each generator to the power network as I; =E;Y;,+Y_ EjY;;
i=1, 2,
n
(5.60)
j=1 jx
In Eq. (5.60), 1, is the complex current vector that the i'" generator injects
into the power network; Ei and E; denote the complex vectors of the idling electric potentials Eq, and Eq; or the transient electric potentials E,';
and Ev of the i' and j' generators, respectively; Y;i the self-admittance of node i; Y;; the mutual admittance between nodes i and j, and obviously Yy =Y;;.
The apparent power W; of the i' generator sending to the power system can be written in complex form as
W; =P;+jQ; =E;I;
(5.61)
I) Complex vectors can express the currents, since their waveform is assumed to be sinusoidal.
Basic Mathematical Description for Electric Power Systems
183
where P, and Q. denote the active and reactive power of the i' generator, respectively; 1, is the conjugate of 1; . Substituting Eq. (5.60) into Eq. (5.61), we have +
W;=P, +jQ;=Ei(E;Y;+ZEj1')
i=1,2,...,n
(5.62)
j=1
,x,
If we express the included angle between electric potential E; and the reference node voltage VRu. by S, (see Fig. 5.5); the angles of self-
Figure 5.5
Electric potential vector diagram of an n-generator power system
admittance Y; and mutual admittance Y, by b;; and 0,, respectively, then Eq. (5.62) will result in yy = P, + jQ, =
E?Ye-jm,, + E;ej5' (Y
E,e-J"51
Yje-jme)
i =1 2 ... n
(5.63)
j=1 ,m;
Separating the above equation into its real part and imaginary part, we have
W;=P,+jQ;=EY;cos¢;;+E;EjYjcos(8;j-0,j) j=1
(5.64)
j#i
EjYjsin(S;j-Ojj))
+j(-E; 2 Y; sin 0,,+E; j=l
jxi
where S, = S; - S, and therefore o, = -Sj, ; besides, Yy = Yj; . Eq. (5.64) suits both nonsalient-pole machines and salient-pole machines, if E is replaced by E,, then x; should be used for the generator reactance. For a nonsalient-pole generator, the idling electric potential Ey has to be substituted into the equation as the electric potential E, then correspondingly the reactance of the generator should be xd. For a salient-pole generator, if the electric potential E in the equations is regarded as the electric potential EC, of its equivalent nonsalient-pole generator, then x9 should be used for the
184
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
reactance of the corresponding generators for calculating the impedance or admittance matrix of the power network. Expressing YJ in the rectangular form, we have
Yj =Yjeill =Gi+jBi,
(5.65)
`.0
Then the above equations can also be rewritten as P = E?G;; + E; Y_ EI (G;j cos 8, + B, sin 8,,,)
i=1,2,
n
(5.66a)
,n
(5.66b)
j=1
t?,
Jxi
Q, =-E ; B,, + E,
E j (G;J sing, - B,, cos 8;j)
i =1, 2,
J=1
jxi
'0.
+O+
In Eq. (5.66), Gi; = Y; cos O;; and B;; = Yi sin O;; denote the self-conductance and self-susceptance of node i, respectively; and G;j=Yjcoso and
denote the mutual conductance and mutual susceptance between nodes i and j, respectively. Replacing loads with constant B;j = Yj sin 0;,,
t1.
impedance and eliminating external nodes are equivalent to moving loads to internal generator nodes. In other words, in Eqs. (5.66a) and (5.66b) these active and reactive powers are expressed by E,?G;; and - E, B,, , respectively. Therefore, the mutual admittance YJ between different nodes only involves the impedances of the transformers and transmission lines. The proportions
of resistance in these impedances are very small, and their major components are reactances, i.e. the impedance angles are nearly 90°. Therefore the values of the mutual conductance G ;j = Yj cos o are small `i'
enough to be ignored. Assuming G;, = 0 in Eq. (5.66), the expressions for a
generator active and reactive powers in an n-generator system can be obtained as follows P . = E?G;; +E;Y_ E,, B. sing,,
i =1, 2, , n
(5.67a)
i=1
Q,=-E?B;;-E;Y_EjB;jcos8;j
III
jmi
(5.67b)
j=1
jxi
These practical equations (5.67) will be used frequently in the subsequent chapters. According to the similar method as mentioned above, the expressions of the currents Id, and Iq, in a multi-machine system can be obtained from Eq. (5.60) as
Basic Mathematical Description for Electric Power Systems
185
n
Id,=-E;Y,,smO,;+Y_EjY;jsin(8;J-qj)
(5.68a)
J=1
jxi 11
Iq; = E;Y;; cosq,, +Y_EJY,j cos(C,j -0j)
(5.68b)
J=j
j=i or n
Id; =-E,Bi;+Y_ EJYJsin(8;j-cy)
(5.68a)
ti;
l=1
jxi
Iqi =EiGii+EEJYjcos(8J-!/lJ)
(5.68b)
j=I
J:,
ELECTROMAGNETIC DYNAMIC EQUATION FOR FIELD WINDING
5.5
Fig. 5.6 shows the positive directions of the voltage and current of a field winding of a generator.
if
Vf
Figure 5.6
Positive directions of the current and voltage of a field winding
According to Fig. 5.6, the voltage equation of the field winding can be written as Vf=rfIf+dyrfd
(5.69)
In the above equation, V f is the voltage of the field winding, 1 f the field current, winding.
ylfd
the total flux linkage, and rf the resistance of the field
Multiplying each term in Eq. (5.69) by xQd/rf and taking some manipulation finally we can have
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
186
xQd
f x-°a =x,lf +xf
d-Vfa xf
V
rf
rf
(5.70)
dt
The ratio between the reactance x f and the resistance rf of the field winding in the second term of the right side of Eq. (5.70) actually is the time constant in seconds of the field winding, and is denoted by TA, namely Tao =xf/rf
In view of formula (5.45) we realize that Vfa
in Eq. (5.70) is
exactly the Eq. And it is clear from Eq. (5.32) that the first term at the right side of Eq. (5.70) is exactly the idling electric potential Eq. Moreover, if we set V f x°a = E f
rf
we can rewrite Eq. (5.70) as dEq
(5.71 a) = E9 +Tao dt If the rated no-load voltage of the field winding Vfo is chosen as the base value of the per unit of value for the field winding voltage Vf, and Ef
Vfo xQalrf as the base value of Ef which is directly proportional to Vf, then
E1,, the per-unit value of E f in Eq. (5.71a) equals Vf., the per-unit value of V f , i.e.
Ef. =Vf. Expressing in per unit of value, Eq. (5.71 a) can be rewritten as Vf =Eq+Tao
dE9
dt
(5.71b)
Eq. (5.71b) is the electromagnetic dynamic equation for the field winding of a generator.
5.6
MATHEMATICAL DESCRIPTION OF A STEAM VALVING CONTROL SYSTEM
Researches [5, 11] show that advanced steam valving control can be used to improve transient stability of power systems. Although the nonlinear optimal steam valving control will be discussed in Chapter 7, the
mathematical model for a steam valving control system is discussed and
Basic Mathematical Description for Electric Power Systems
187
established in this section. Nowadays, steam turbines with reheater are commonly used for largecapacity steam turbine generator sets. Their physical configuration is shown in Fig. 5.7. a
steam flow
2
r A
high-pressure oil
medium-pressure
servomotor
oil servomotor
+
-I
medium-slide valve
electric-hydraulic transducer
E-I
-'1
fast valving controller feedback
signals
electric-hydraulic
steam valve
transducer
regulator
'c3
,.,
Figure 5.7 Physical configuration of the steam valving control system for a large generator set with reheater 1-main steam valve; 2-high-pressure regulating valve; 3-high-pressure turbine; 4-reheater; 5-interception valve; 6-medium-pressure regulating valve; 7-medium-pressure turbine; 8-low-pressure turbine; 9--condenser;
The mark numbers from
1
to 9 in Fig. 5.7 indicate the relevant
n'.
equipment that the steam flow passes through in sequence. The corresponding block diagram of transfer function is shown in Fig. 5.8, where T,,, TM and TL denote respectively the time constants of the high-pressure,
medium-pressure and low-pressure turbines; C,,, CM and CL denote
O..
coo
respectively the power fractions contributed by the high-pressure, mediumpressure and low-pressure turbines, with CH+CM+CL=1.0, where Cn is approximately 0.3, CM-0.4, and CL-0.3; P11, PM and PL are the output mechanical powers of high-pressure, medium-pressure and low-pressure and TMs stand for respectively the time constants turbines respectively; of the oil-servomotors for regulating the valves of the high-pressure and medium-pressure turbines; pe and uM represent the regulated valve openings of the high-pressure and medium-pressure valves; T is the time constant of the reheater; G, and G2 indicate respectively the controller of °-y
4))
the high-pressure regulated valve (the steam controlled valve) and the medium-pressure regulated valve (the so-called fast valve); u and um are the oil-pressure control signals from the electric-hydraulic transducer;
u,
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
188
PH
PA/
I steam flow
PAr
IR
I
1
TH
Sri 1+Ts
I + TA,
H
L l+Tfs
PA/
1
I
1+T,'
u
T
IL
ux
G,
1
1+TH Rs
HH
--------u
1.2
I.2
I
xIC_ ,
l.2
I I
HH --------03
uH U,
u
4
G,
Transfer function block diagram of the control system
Figure 5.8
for a steam turbine with reheater
CAD
ANC'
and u2 are the electric control signals from G, and G2, respectively. Under normal operating conditions, the output signal of the fast valve controller to the medium-pressure regulated valve is cut off by the normal opened contact r of a relay. In case of a system fault, the relay monitoring the fault will switch on its normal opened contact, so the medium-pressure regulated valve is subjected to control signals from G2 , resulting in the so-called fast
:-j
c).0
valving. For the convenience of analysis, we may use one inertia block with the time constant marked by TML to represent both medium-pressure and lowpressure turbine blocks in the transfer function block diagram, and the power contributing fraction of the equivalent block is denoted by CML, CMI, CM+CL 'ti
being approximately 0.7, and its equivalent output mechanical power is denoted by PML. After that, by merging the limit block of the electric-
f3.
hydraulic transducer with the limit block of the medium-slide valve (see Fig. 5.8), we can obtain a simplified block diagram of transfer function shown in Fig. 5.9.
The following two remarks should be given. 1) during designing the controller for the steam control valve, the effect of the valve opening limit should not be considered, because the primary task for designing a controller is to make it function effectively in the full scope of the steam control valve. Only the control law acquired in this way can be effective in the whole
Basic Mathematical Description for Electric Power Systems
189
Fem...
Figure 5.9 Transfer function block diagram of a steam valving control system with reheater
Fn'
.yam
may
!CD
,-+
region of the generator operation state. Therefore the limit blocks in Fig. 5.9 will not be considered1) in modeling for designing nonlinear steam valve controllers. 2) since the time scale of the electromechanical transient process of a power system is much shorter than the repeater time constant TR (about 10-20 seconds), so the output steam flow of the reheater can be regarded as constant in studying the control problem of the electromechanical transients 0
>1V
`n.
of power systems. With these two points in mind, the block diagram of ,'3
CAD
..+
transfer function of a steam valving control system with the reheater can be further simplified as shown in Fig. 5.10. According to the transfer functions shown in Fig. 5.10, the dynamic equations of the control system composed of the steam control valve and high-pressure turbine can be obtained as
PH =
PH=-1PH+1CHPH TH TH
(5.72a)
1 1
(5.72b)
THg
PH +
TH.-
ul
,
PH (0) =PHI
and the dynamic equations of the control system are composed of the medium-pressure regulated valve and medium-pressure and low-pressure turbines (under normal operating conditions, the relay contact r is not closed, so the medium-pressure valve keeps away from control) II The same principle can also apply to the excitation control system design. But obviously the
effect of limits must be considered in the digital or physical simulation test, otherwise the control effect would be exaggerated beyond allowance.
190
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
PH
1
1
1+T, s
1+THgs
PHO PMO
1
1+TMLs
1+ Ugs
coo
Figure 5.10 Transfer function block diagram of a steam valving control system with reheater
PM/ =
,
PML _ TMg
PM +
1 C.pM
PML +
(5.73a)
TML
TML
TMg
u2
,
PM (0) = PM0
(5.73b)
The total output mechanical power P. supplied by the prime mover should be the summation of the output mechanical power of the highpressure turbine Py and the output mechanical power of the mediumP. = PH + PML
can
III
pressure and low-pressure turbines P,,,,L, i.e. (5.74)
PMLO =CMLP."
Figure 5.11 T11 E =TH +THg ; T,,,:
Transfer function diagram for a steam valving control system +T,; PHO-initial steady value of the mechanical power of
the high-pressure turbine; PMLO-initial steady value of the mechanical power of the mediumand low-pressure turbines; P,0-initial steady value of the total mechanical power
Basic Mathematical Description for Electric Power Systems
191
oho
..,
`-o
In the block diagram of transfer function shown in Fig. 5.10, all the values of the time constants THg, TH, T,,,g and TML are small (about 0.2 seconds). Therefore the mathematical models can be further simplified by replacing Fig. 5.10 by Fig. 5.11 with THE = TH +THg and T,,,g = TM, +TM8
.
From Fig. 5.11, we could write the differential equations for the steam valving control system as follows PH (t) = - 1
PH (t) + CH ul (t)
,
PH (O) = CHPm0
(5.75a)
,
PML(O)=CMLP.
(5.75b)
THE
THE
INn
UPI
PML(t)=- 1 PML(t)+CMLu2(t) THE
THE
5.7
MATHEMATICAL DESCRIPTION OF A DC TRANSMISSION SYSTEM
5.7.1
Dynamic Equations of a DC Transmission Line
oho
A DC transmission system mainly consists of converter transformers, converters (rectifier and inverter) [6], DC transmission lines, smoothing reactors and the AC and DC side filters. Its basic configuration is shown in Fig. 5.12. _ _.smoothing
smoothing wave reac ce ;----
wave DC line
converter transformer
DC filter
DC filter
11 11
-GD AC filter
DC filter
L
AC filter
DC line
power converter
Figure 5.12
DC filter
power converter
Basic configuration of a DC transmission system
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
1 92
If Vd, and Vdi denote respectively the mean voltages at DC terminals of the rectifier and inverter, the DC transmission system can be represented by "'S
an equivalent circuit shown in Fig. 5.13, where Ld, and Ld, represent respectively the inductance of the smoothing reactors on the rectifier and inverter sides; Ldand Rd respectively half of the inductance and resistance of the DC transmission line; Cd, the total line-to-earth capacitance of the DC transmission line; Id, and Id; respectively the DC currents of the rectifier and inverter sides; VV the voltage on the line capacitance. All the above variables are measured in per unit value.
Figure 5.13 Equivalent circuit of a DC transmission line
According to the DC line equivalent circuit shown in Fig. 5.13, from circuit theory, the dynamic equations of a DC transmission line can be written as LdrE ddr = -RdIdr +Vd, -V. LdiEdi
Cfr
uV
= -Rd I d, - Vdr + V,
(5.76)
= 1dr- Idi
+
where LdrE = Ldr + Ld and Ldt = Ld, + Ld
.
Generally a converter is composed of three-phase controllable bridge-
type circuit, as shown in Fig. 5.14. Fig. 5.15 illustrates the converting (O)
process from valve 1 to valve 3 as shown in Fig. 5.14. Angle a in the figure .`3
V)'
is called the rectifier firing angle (or control angle), and p is the commutation overlap angle.
When the AC side voltages e0(t), e,,(t) and e,(t) are symmetric, the rectifier is equivalent to the circuit shown in Fig. 5.16, with the voltage as V,0 =
3
-
Vor(i) cos a
(5.77)
and the interior equivalent resistance as 3 R, =-X,
(5.78)
Basic Mathematical Description for Electric Power Systems
193
v,,
e. (t)
e, (t)
wt
Figure 5.14 Converting loop of the rectifier side
Figure 5.15
Converting process of a rectifier
where x, represents the commutation reactance of a rectifier. R,
id,
Figure 5.16 Equivalent circuit for a rectifier p,+
From Fig. 5.16, the DC output voltage of the rectifier can be written as
Vdr=3V-V,(t)cosa-3xrld, It
(5.79)
It
where Va,.(t) is the line voltage on the secondary side of the rectifier '-h
011
...
transformer (close to the rectifier), i.e. V,(t)=If3Er, where E, is the effective value of the phase voltage on the secondary side of the rectifier transformer.
Oar
Similarly, the converting circuit of the inverter side is shown in Fig. 5.17. The converting process from valve 4 to valve 6 as presented in Fig.
5.18 is shown in Fig. 5.17, where 6 is called the advance angle (or
I'0
inverting angle), y the extinction angle.
Similar to the rectifier, the inverter can also be represented by an equivalent circuit shown in Fig. 5.19, with the equivalent voltage as
194
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
Wt 10
u)t
',t
Figure 5.17 Converting loop of an inverter (At the moment from valve 4 to valve 6)
Vdo =
Figure 5.18 Converting process of an inverter
32 Vim; (t) cos /3 Ir
(5.80)
and the internal equivalent resistance as
Mgt;
R. =
3 x;
(5.81)
where x, denotes the commutation reactance of inverter.
From Fig. 5.19, the DC input voltage Vdi of the inverter can be expressed as Vdi = 3 r2- VQi(t)cos/3+ 3 xildi X n
(5.82)
In the above equation, V, (t) = %F3E,, where E; is the effective value of the Ri
Figure 5.19
Equivalent circuit for an inverter when Vdi is expressed by the inverter firing angle j6
195
Basic Mathematical Description for Electric Power Systems
phase voltage at the side of converter transformer close to the inverter. If the extinction angle is denoted by y , the inverter can be represented by the equivalent circuit shown in Fig. 5.20 with the voltage V;io =
3,F2-
if
V,(t)cosy
(5.83)
and the internal resistance -R,.
From Fig. 5.20, we can obtain another expression for Vd;, the DC - R.
Idi
= Vdio
Vdi
T Figure 5.20
Equivalent circuit for an inverter when Vd, is
expressed by the extinction angle y
voltage on the inverter side
3,2 (5.84) 3 xildi if Vo,(t)cos7- if According to the condition that the voltages expressed in Eqs. (5.82) and (5.84) are equivalent to each other, the inverter's extinction angle y can be expressed as [IQ
Vdi =
y = cos-' (cos /3 +
2x,
(5.85)
1d,) C17
r17
Substituting Eqs. (5.79) and (5.82) into Eq. (5.76), the dynamic equations for a DC transmission line can be given as -Rd Id, +
dld;
LdiE
dt = -Rd 1d, -
C,dVd =1dr-1d,
3 if
n xr1dr -V Vo; (t) cos/1 -
3
if
x,,, + V,
l/1
LdrE ddtr
(5.86)
196
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Mathematical Model of a DC Control System
5.7.2
cry
When the DC system is operating under steady state, the steady DC current can be calculated according to Fig. 13, Eqs. (5.79) and (5.82) to give 3
(Va, (t) cos a - Va; (t) cos Q)
Id=ld,=le;=-
(5.87)
2Rd+3x,+3x; r
,r
The above equation shows that the DC current Id or the power Pd transmitted
over the DC transmission line can be adjusted by regulating the rectifier firing angle a or the inverter firing angle Q . On the rectifier side, in order to regulate the firing angle a, the signal amplifier, phase controller, and firing circuits are the basic elements for the a -regulator as shown in Fig. 5.21.
Figure 5. 21
Schematic diagram of a rectifier's a -regulator
f].
'C7
In Fig. 5.21, Ua represents the control signal of the a -regulator. When the constant DC current control is adopted, we have the following equation ua =1d, - I&, = AI&
(5.88)
.vim
where Id, is the steady value of the DC current. When the proportional control of the deviation of the firing angle a is used, the transfer function of the signal amplifier block can be written as
Ga(s)=
a
--'t
Ua
=
ka
1+Tas
(5.89)
where ka represents the magnification factor of the regulator; Ta the time constant of the regulator; and Aa the deviation of angle a .
Basic Mathematical Description for Electric Power Systems
197
Expressing Eq. (5.89) in form of differential equation, we have dt
(-a+ao +kaua)
T0
(5.90)
where ao represents the given value of the rectifier firing angle in its nominal operation state.
In a DC transmission system, the configuration of the phase control circuit for the inverter side (see Fig. 5.22) is similar to the rectifier side:
phase control and trigger circuit
Figure 5.22
Schematic diagram of a inverter's (3-regulator
In Fig. 5.22, up is the control signal of the /3 -regulator. When the constant extinction angle control is adopted for the inverter side, we will have up =Y-Yo =L\Y
(5.91)
where yo represents the allowable rated extinction angle in the inverter's nominal operation. Then the transfer function of the signal amplifier block is (5.92) ONE
-k,6 OQ = 1+T,6s ua [rte
G. (s) = =O,
where k. is the gain of the /i -regulator; and T,6 the time constant of the regulator. Rewriting Eq. (5.92) in form of differential equation, we have d,8=T (-,6 +,80 -kflu")
(5.93)
T,6 000
where fl, represents the given firing angle of the inverter in its nominal operation state.
198
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
5.8 REFERENCES 1.
A. E. Fitzgerald and C. Kingsley, Electric Machinery, Second Edition, McGraw-Hill,
2. 3. 4.
7.
A. R. Bergen, Power Systems Analysis, Prentice-Hall, New Jersey, 1986. B. Adkins, The General Theory of Electrical Machine, Chapman and Hall, 1964. C. Concordia, Synchronous Machines, John Wiley & Sons, 1951. E. F. Church, Steam Turbines, McGraw Hill, 1950. E. Uhlman, Power Transmission by Direct Current, Wiley-Intersci-ence, 1975. G. R. Slemon, Magnetoelectric Devices, John Wiely & Sons,1966.
S.
G. W. Stagg and A. H. El-Abiad, Computer Methods in Power System Analysis,
1961.
5.
6.
McGraw-Hill, 1968.
10. 11.
J. Machowski, J. Bialek and J. Bunmby, Power System Dynamic, Chichester, U.K, .SC
9.
Wiley, 1997. V. A. Venikov, Transient Phenomena in'Electric Power Systems, Pergamon Press, New York, 1964. Y. N. Yu, Electric Power System Dynamics, Academic Press, 1983.
Chapter 6
Nonlinear Excitation Control of Large Synchronous Generators
6.1
INTRODUCTION .`t
Improving power system stability is of great significance, since if the
cab
vii
stability is lost, power collapse may occur in a large area and serious damages will be brought to national economy and the residents' comforts.
Not to mention earlier examples, only since 1960's, large-area power ..O
collapses took place in many large power networks around the world and led
to disastrous losses. This inevitably resulted in grave concern by each r+,
ors
country on the stability and security of power systems. Over a long period of time unremitting researches have been carried out on the subject of power
5°.y
system stability. ,.t
Early in the 1950's, a number of scholars emphasized the importance of synchronous generator's magnetic field regulation to improve power system stability [1, 2). Since then great attention has been paid to research in this field, which mainly covers two areas: one is focused on melioration of the "ate
SS.
t°.
main excitation system, while the other is aimed at the improvement of excitation control strategy.
C1.
CAD
cad
In recent times, static excitation technology has seen great advance. Since the static exciter with thyristor-controlled self-shunt (self-excited) excitation has the advantages of simple structure, high reliability and low cost, it is widely adopted by many large power systems around the world. The design principle and approach of large generator nonlinear excitation control to be discussed in this chapter will be mainly aimed at this type of excitation. For the static excitation shown in Fig. 6.1, since an intermediate inertial
200
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS Rectifying transformer
Adjuster
Figure 6. / The structure of generator self-shunt excitation coo
O'+
element of exciter is omitted, it is featured by the rapidity of its regulation. This rapidity of excitation has two-opposite influences to the power systems. On the one hand, if the excitation control technology and strategy are not
correspondingly improved and proportional or voltage deviation PID
.....
(proportional integral differential) control are still used, the wide adoption of ,4?
rapid excitation will exacerbate the damping characteristics of power Ll.
systems even to the extent of negative damping, which will cause further low frequency oscillation in power systems. On the other hand, however, it is the rapid response of this excitation control that provides favorable conditions for the full play and good effect of advanced control strategies. Summing up
all the above, we can see that the wide adoption of rapid excitation makes
the new excitation control research practically more important and meaningful.
6.2
DEVELOPMENT OF EXCITATION CONTROL The adoption of nonlinear excitation control is the inevitable
development trend of excitation control. In order to motivate the approach to
nonlinear excitation system design, let us return for a moment to the development history of excitation control. Roughly speaking, since 1940's, the excitation control has gone through three development stages.
Stage 1: single variable control
The control strategy of this stage is regulating according to the proportion of generator terminal voltage deviation AY, or to the proportional, integral, and differential of AV, (PID regulation). The transfer functions are respectively as follows.
Nonlinear Excitation Control of Large Synchronous Generators
201
Proportional regulation u
_ kP
(6 . l a)
AY,
PID regulation ( 6 . 1b) z VU
In the above two equations, the generator terminal voltage deviation is AV, =VR
-V,(t), where VRer is the reference voltage, and V, (t) the average
of three-phase effective values of real-time generator terminal voltage. Corresponding to the formulae (6.1a) and (6.1b) the block diagrams of transfer function for the closed-loop system are shown in Fig. 6.2(a) and (b).
V
Controlled object
V.f 'AY'
V'(t)
Controlled object
I+k,s (b)
(a)
Figure 6.2
kP +kDs
The block diagram of transfer function of single variable excitation control (a)
Proportional regulation
(b)
PID regulation
For the single variable control as shown in formula (6.1). or Fig. 6.2,
applying the Frequency Response Method or Root Locus Method of ''',
classical control theory, we can determine the proper value range of each gain, k,, kD and k, in the transfer functions. Since this method is assumed to be familiar to the readers, no details are given here. In the following, we will make some conceptual explanation for the PID regulation as shown in '-.+
Fig. 6.2(b). a-.
,.0
CS'
From formula (6.Ib) or Fig. 6.2(b) it can be seen that, the transfer function structure of the regulator is the proportional block kP added by differential block kDs and then in series with the inertial block 1/(1 + k,s) . As the time constant k, is large enough, the first term in the denominator
caw
polynomial can be omitted, and the inertial block can be approximated by an integral block l/k,s. Thus, we call this regulation mode as PID regulation of terminal voltage deviation AV, . In the following paragraphs, we will discuss the physical essentials of this regulation mode. First, as is well known, the block diagram of transfer function of a linear variable feedback regulation system can always be changed into the form in Fig. 6.3. In this figure XR(s), ,Ca
Y(s), and E(s) are respectively the Laplace transformation functions of input variable xR(t), output variable y(t) and regulation error e(t);
202
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
0,2
kEG(s) is the forward path transfer function, and H(s) is the feedback path transfer function, which can all be represented in the fraction of polynomials
of s. According to classical control theory, for the closed-loop system in Fig. 6.3, as the gain kp increases, the dominant zeros) of the characteristic E( S)
X R (S)
'Ey
tH(s)Y(s)
Y(S)
G(s)
kp
H(
Figure 6.3 A single-input single-output closed-loop system
polynomial of closed-loop system will move rightward in the complex plane. When the gain kp exceeds its critical value kc,; , a pair of characteristic roots of the closed-loop system will appear in the right half complex plane, which means that the system will turn unstable and its dynamic response will be an amplifying oscillation which may be called the spontaneous oscillation.
Hence, the gain
of single variable feedback proportional regulation system must be confined within a certain interval, i.e. k,
ka,
regulation steady state error (6.2)
s(ao) = lime(t)
is small enough. According to the prescription of relevant criterions in China
for example, the steady state error e(oo) of generator terminal voltage regulation in power systems should be no more than 0.5%. Now let us
n+,
discuss the problem of what requirement for the excitation regulator gain k, was brought forward by the criterion just mentioned. For the system shown in Fig. 6.3, its closed-loop transfer function is Y(s) XR(s)
=
kEG(s)
I+k,H(s)G(s)
(6.3)
From this we know that the transfer function from error e(t) to input xR(t)
') Namely the dominant roots of the characteristic equation of the closed-loop system.
Nonlinear Excitation Control of Large Synchronous Generators
203
will be E(s) = XR (s) - H(s)G(s) X' (s) XR (s)
1
1 + kPH(s)G(s)
(6.4)
From above formula we have E(s ) =
1 + kPH(s)G(s) XR (s)
( 6 . 5)
Suppose the input xR(t) to be a unit step function, with Laplace transformation function of XR (s) =1/s , then the Laplace transformation of the error function to the unit step input of system in Fig. 6.3 is E(s) _
1
1
1+k,H(s)G(s) s
(6.6)
From the Final Value Theorem in the classical control theory, the steady state error e(cc) is known to be s(oo) = lime(t) = limsE(s) s-,0
1-
(6.7)
Substitute formula (6.6) into formula (6.7), we have s(oo) = limsE(s) = lim
1
S-3O I + k,H(s)G(s)
S-*O
(6.8)
Represent the H(s)G(s) in the above formula by normal polynomial fraction of s, then formula (6.8) can be written as 1
6(00)=lim
(6.9)
l+kP Thus we have e(co) =1/(1 + kP)
(6.10)
The above expression tells us that the steady state error of a SISO (singleinput single-output) system to unit step function input is approximately equal to the reciprocal of closed-loop gain k, , i.e. e(ao) u 1/kp
(6.11)
Now we have the answer to the question presented above to let the steady state error of generator terminal voltage to a unit step function input be less than 0.5%, the closed-loop amplification of excitation regulation system should be no less than 200. Generally, this value is far beyond the critical gain k,,i of proportional excitation regulation. Thus for proportional excitation regulation, unsolvable contradiction occurs between the
204
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
wavy
>o0
4-r
requirement of system stability (to avoid the spontaneous oscillation) and 0
that of steady state voltage regulation precision. .O^'
`O,
If we divide the regulator gain into dynamic amplification kD (whose influence to control objects has no additive time lag) and static amplification ks (whose influence to control objects should pass an additive heavy inertial
block). The block diagram of transfer function of excitation regulator corresponding to this is shown in Fig. 6.4(a), which can be converted to Fig. 6.4(b).
(b)
(a)
Figure 6.4 The excitation regulator transfer function dividing the amplification into static and dynamic amplifications
According to Fig. 6.4(b) and Initial Value Theorem, we know that at the
dynamic process, of the beginning of the moment, namely when t = 0+, if e(t) is a unit step function, i.e. E(s) =1/s, then the control u(0+) at this time is u(0+)
= lim u(t) = lim sU(s) = lim(s 1o' _-» = lim
(kD + kS) + kDTs 1
1+Ts
s
(6.12a)
(kn +ks)+kOTs _ kD
r-»
1 + Ts
,CD
Besides, from the Final Value Theorem it is known that, for the input of unit step function input E(s) =1/s, the control u at steady state is u(ao) = lim u(t) = lim sU(s)
r..
S-,o
+I±
= 1im(s
(kD+k5)+kDTs 1 _ 1 +Ts s)
k° +S k
(6.12b)
'C3
"i7
According to Eq. (6.12), if the regulator gain is divided into two parts as static ks and dynamic k°, during the dynamic process the control effect is equivalent to proportional regulation with gain of kD , while at steady state its effect is equivalent to proportional regulation with gain of kD +ks . The transfer functions shown in Fig. 6.4(b) is consistent with that of the PID regulator shown in Fig. 6.2(b). Thus, we may realize that PID regulation is essentially equivalent to proportional regulation which divides the whole
205
Nonlinear Excitation Control of Large Synchronous Generators
regulator amplification into two parts as dynamic and static ones.
Although to a certain extent the PID regulation as stated above mitigates the contradiction above mentioned, it can hardly improve stability of power systems effectively. This is because, as we know from the above, the effect of the PID regulator in the power system transient is no more than a proportional regulator which can, in some sense, automatically change its gain.
Stage 2: linear multi-variable control To further improve power system dynamic performance and small disturbance stability, the multi-variable feedback excitation control get gradually developed.
The Power System Stabilizer (PSS) In 1969, F. D. deMello and C. Concordia [4) put forward the supplementary excitation control called Power System Stabilizer (PSS). In the 00..
,.f
control strategy, besides the normal part of proportional, integral and .-y
differential of generator terminal voltage deviation A V, , a second-order lead "CJ
compensation element with respect to generator angular speed deviation Aw or the frequency deviation Af is used as the supplementary feedback block. The block diagram of transfer function of this excitation controller is shown in Fig. 6.5. We can see from Fig. 6.5, the PSS, in fact, is a linear control with a supplementary single signal input Acv or Af .
0
VAEF
I Af (Aw)
9
1+T,s
II
l+kDs l+kDs Hp Flp 1 +k, s l+k , s
III k
IV
Controlled Object
C
+5%
Ts
Ts
-5%
Figure 6.5 The transfer function block diagram of PSS V, : generator terminal voltage; Af : frequency deviation; VREF : reference voltage; Aw : angular speed deviation; VV : control voltage
In the voltage deviation PID path of the above diagram, to meet the requirement of generator steady state voltage regulation precision, its gain kR should be adjusted to be within 100-200; the TD in transfer function of this path is generally selected between 0.3 and 1.0, while the T, may be chosen between 4.0 and 6.0. In Fig. 6.5, the PSS path is made up of two
206
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
'.0
'C3
'LS
first-order lead compensation elements of I and II, an amplification element III, a reset element so called as "wash out " element IV and limiter of ±5%. For the elements I and II, the relationship between the coefficients kD and c/)
k, should always be kept as k >> k,, and the actual values need to be t].
`:1
carefully designed according to calculation for specific power systems. The selection of gain ks in element III also requires calculation, and its value is
c4:-00
a<<
coo
.^s
3a)
usually within 1520. The so-called wash out element IV is actually a differential element with inertia, whose function is to let the steady state value of frequency or speed deviation have no influence on the control voltage V.. The parameter T in this wash out element also needs to be C"-
designed for a specific power system. In order to keep the phase shift of the r0'
frequency characteristics of this element small enough (no more than 5 degrees) at the selected frequency in design, the value of T should not be too small, and it is usually within 2.83.5. From analysis we will see that, with properly designed parameters k,, k,, ks and T, the existence of PSS path can make the dominant `J'
'L7
0-)
r..
CEO
-C3
eigenvalues of the closed-loop system move leftwards on the complex plane, CAD
t3.
,_,.
,w_,.
+-'
-..r
t].
t3.
-+,
cad
fro
.oh
+-i
't3
t1.
.07
coo
thus improving the power system damping characteristics and the small disturbance stability. It should also be pointed out that even for the goal of improving power system small disturbance stability, the supplementary control with single supplementary signal input still has the following two shortcomings. First, when parameters kD, k,, ks and T in the PSS path are already determined, the controller can take good effect for a certain corresponding narrow band of oscillation frequency. However, the PSS's effect will apparently be diminished, either because of the problem in parameter adjustment or changes in power system operation states and network structure, such that the actual system oscillation frequency is beyond the band in which PSS is able to attenuate the low-frequency oscillation of the power system. For this reason, in the past two decades, there have been researches dedicated to "adaptive PSS", which could automatically ameliorate its parameters. The second shortcoming of PSS t].
--,
controller is that, even under the condition of small disturbances, theoretically, this type of control with single supplementary signal will not a°>
r-'
r.pi
achieve optimal control effect. At most it can only obtain relatively good 1:J
S3.
chi,
result with proper design.
The Linear Optimal Excitation Control (LOEC) :.h
'ate
In the early 1970's some scholars presented the linear optimal excitation control, i.e. LOEC [6, 14]. In this field further research has been made by Chinese scientists [5]. The industrial prototype devices of LOEC first developed by Tsinghua University in the early 1980's are now operating in the hydraulic power plants of Northeast of China.
207
Nonlinear Excitation Control of Large Synchronous Generators
The design principle of linear optimal controller has been briefly
FN-
introduced in Section 1.2. If some readers expect further knowledge of it, the relevant references [10, 13] may be consulted. From the principle of linear optimal control in section 1.2 we see that, the control strategy u' of LOEC is the linear combination of state variables of the power system. For a one-
machine and infinite-bus system, supposing that the generator excitation system is a self-shunt (self-excited) system, and choosing the state variables as X (t) = [AV,
0w OPT ]T
where AV,, ew- and AP, are the deviation of generator terminal voltage, angular speed and active power, respectively, then the optimal excitation control strategy can be represented by
u =OVf =-(k,OV, +k.&w+k,AP,)
where AV f is the variance of excitation voltage, and k,, k. and k, are optimal gain coefficients. The schematic diagram of an analogous LOEC '++
o-.0
corresponding to the above equation is shown in Fig. 6.6. The LOEC compensates the aforementioned shortcomings of PSS and ,4?
:-.
wry
.fl
makes progresses in both design principle and control technologies. However, when applying the linear optimal control principle to multimachine system excitation control design, we can hardly get an optimal decentralized control strategy, but a suboptimal decentralized one, which L1.
.°o
should be considered as a defect. R
R
AV,
V
15V
100V/20V
Comparative amplification R
P ower
0
measurement
99?
0 0
Devi ation
4
calc ulation
R
Ap.
S pee d
measurement
Dev iation calc ulation
i i
I
V Pulse Fast filtering
I R D_ Synthetic
R_
F 0
R
phase shift
=
amplification
AO )
Figure 6.6 Schematic diagram of a generator LOEC (analogous) in a one-machine, infinite-bus system
Vf
Field winding
Va, e.
F
av
208
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Stage 3: nonlinear multi-variable control For excitation control measures in the aforementioned two stages, whether they are PID, PSS or LOEC, there exists a common problem that the V1'
design of these types of controllers is based on the mathematical model coo
which is approximately linearized at a specific operating point of the system. This modeling method of approximately linearizing a nonlinear system at a specific operating point and its limitations have been discussed in detail in
Section 1.3. From those discussion, we can see that various types of excitation controllers based on the approximate linearization model (transfer
function G(s) or linear state equation X = AX + BU) inevitably have a common shortcoming that as large disturbances cause the power system's actual operating point to move far away from the equilibrium point chosen in
the design with subsequence of large magnitude oscillations, the control effects will greatly be weakened. In such cases, these types of controllers, PSS for example, even put negative action on transient stability of the power systems. To avoid this situation, an amplitude limitation of ±5% is added to its PSS path, which means the output of this additive path is automatically
locked out in case of large disturbances. How to solve the problem mentioned above? The nonlinear optimal control and nonlinear robust control developed in modern control theory provide just the right answer. Now let us once more dwell on the reason of the above problems. The reason lies in the fact that a power system is a typical nonlinear system while
designers attempt to adopt an approximately linearized model at a certain point as the basis of controller design. Thus, if we abandon the common approximate linearization models and directly adopt nonlinear power system
models, which are exact for dynamics under both large and small (IQ
disturbances, the above problems will be readily solved. In the following, we
will discuss, in a detailed way the principle, approach and realization of c'3
nonlinear excitation controllers.
6.3
NONLINEAR EXCITATION CONTROL DESIGN FOR SINGLE-MACHINE SYSTEMS
ago
The general principle of nonlinear control for single-input single-output (SISO) nonlinear system has been illuminated in Chapter 3. In this section, we will specifically discuss its application to nonlinear excitation control for a one-machine, infinite-bus system.
209
Nonlinear Excitation Control of Large Synchronous Generators
6.3.1
Exact Linearization Design Approach
According to relevant equations given in Chapter 5, we can write the state equations of a one-machine, infinite-bus system with excitation control as
S=w, - wo
(6.13a)
to = H P, E9
=-
,
H(w-coo )
(6 . 13b)
L
dE
xd
E9 +
Td
-H Sx-si n 8
Tdo
xd V, cos S +
,
1f
(6 . 13c)
V
Tao
x,f
where Td = T, xdE /xd £ is the time constant of the field winding when the C))
C/]
stator circuit is closed; and P. is the mechanical power. Since only 0.E
excitation control is considered here, P. = Pmo is assumed where P,,,o is the Cep
mechanical power at initial steady state. The meanings of other symbols have been explained in Chapter 5 and those explanations are therefore not repeated here. Eq. (6.13) describes a normal affine nonlinear system j" = f(X)+g(X)u
X(0) = Xo
(6.14a)
where
X =[E9 w 8]r 1
X. =[Eyo
8o]'
xdE
'V' sin8+ H Pmo
- H (co - wo) -
(6.14b)
xd V, cos 8
Tdo
BIZ
a.,
f(X) =
xd
E' + 1
Td
wo
co
(6.14c)
H x'
w - coo
g(X) = [1/Tda
0
(6.14d)
0]T
(6.14e)
u = Vf
Hereinafter, we will look for a coordinate transformation Z = (D(X)
and nonlinear state feedback Vf =a(X)+b(X)v
210
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS .r.
such that system (6.14) will be transformed into a linear and controllable system in the form ZI = Z2
az = Z3 Z3 = V
From the above, if the linear "optimal control law" v`(X) is calculated, and then the nonlinear control is obtained as VJ = a(X)+b(X)v*(X)
Let us achieve this goal step by step. First, in light of Theorem 3.2, examine whether system (6.14) satisfies the conditions for exact linearization. Therefore, the Lie brackets adJg and ad fg should be calculated. According to (6.14c) and (6.14d), we have 1
wo Vsin8
adJg=-
-
0
Td/
xd -X,'
Tdo
xds
V, sin ,5
--H
- ooH Esx;V, cos8
1
0
D
H x;
1
1
Too
0 0
0
(6.15)
0
.-.
)f(X)- (X )
adJ2 g=(aX
0 L
0
L
3
X2,
zs
xdZ Tdo
w0V,
HxTdo
((ro-m )cos8+(1 + D) sin8) ° COOV,
Hx;Tdo
Thus we know the matrix [C] is
Td
sin (5
H
1
(6.16)
Nonlinear Excitation Control ofLarge Synchronous Generators
211
[C]=[g adfg adfg] 1
I TIfo 0
xdE
xdI 2
2
xdE Tdo 0V'
xdE Tdo 0V'
HxdETdo
dETdo
0
I
W
sins
+
Td
0
wOV'
D
H
sins
Hi£Tdo
and the determinant of [C] is r..
det(C) 1 (
w0V'
)2 sin2.5
(6.17)
Tdo HxcsTdo
From above formula, we can see that for any point X that satisfies S #nn (n=0,1,2,--), the value of det(C) is not equal to zero. Namely, the rank of matrix [C] is n=3 in the set of £
$40
0 = {Eq, w,,5 I E9 and co are arbitrary while S # 0,;r}
Therefore, in the set of Si the exact linearization condition (1) in Theorem 3.2 is satisfied. To examine the condition (2) of Theorem 3.2, we need to calculate the
Lie bracket of g(X) and adfg(X) 10
0
[g,ad,g] = 0
0
0
0
0 WOV'
HxdsTdo
(6.18)
sin s
0
Since zero vector can be a member of any vector field set, the vector field set D = {g(X), ad fg(X)}
}I,
is involutive, and the condition (2) of Theorem 3.2 is fulfilled. Thus, it is concluded that when generator rotor angle S # 0 or tr, the generator nonlinear excitation control system can be exactly linearized via state feedback. Our approach to exact linearization of a nonlinear excitation control system in a one-machine, infinite-bus power system will involve five main steps. Step 1. Construct the following three vector field sets 1
Tdo
III
DI ={
0 0
}
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
212
I(rte
1
Tdo
D2 = {
I
0 o'-
0
0
xd
xd
1
i
2
r
2
D3 = {
sins
wyV r HxdZTdo
0
3
xA Tdo
X I Tdo Tdo
[HXTdO
((w-COO)cosS+(1 + D)sing) } Td
H
- HxLETdo sing CooV,
0
Step 2.
J
0
I
"L7
Choose three linearly independent vector fields
D, , D2, D3
such that D, E D 62 C D2, D3 C D3 . Namely, choose the scalar functions ^'Y
k,' (X);
k,2) (X), k"I (X); k,') (X), kZ3 (X), k33I (X) , such that
D
0
1/Tdo
D12 11
I+k(')(X)
D13
D21
0
= 0
0
0
Tdo
D22 +k(21(X)
0
+
D23
0
0
and xdz D31 32
1
x- FTdo Tdo
I
I+k,3) (X)
D33
0
0)0v''
k2'' (X)
Hx Tdo
sin 9
0
0
A
xr2 Tdso
+ k3 (X)
0`'0V'
Hx Tdo
((m-a°)cosg+(I +D)sing) Td'
Wo V'
Hx;Tdo
sing
H
000
xa
Nonlinear Excitation Control of Large Synchronous Generators
213
We choose k;') (X) _ -Teo '(21
(X) _ -k22) (X) x`'E xaE Teo
k22)
(X)
HxTTO
1
w0V,
sing
ki(3)(X)=-k(3)(X)xeE
x
k23) (X) _ -k23) (X)
sin S
k3 3)(X) = HxzTTO
1 -k(3)(X) z
x
Tao
2
Tao xeE
((w - w0) cos S + (T, + H )sin 6) Td'
1
sing
w0V,
and thus obtain the following
_
_
1
D,= 0 cD,
D3= [00 eD3
0
1
0
Step 3.
_
0
D2= 1 JED2
Look for the mapping =[E'
F(w1,w2,u3)=(DD' fiDI (XO)
3
w0
9
differential equation d dw3
E9
0
w=0 S
1
and have
E'y = E'yo w=w0 S=w3+(50 Next to calculate q) D:
°
(D D;
(XO )
Then, we solve d dw2
and get
S01T
. For this purpose, we need solve the following .-r
First, calculate
X0
(D 15
Ey0 w0
XO = w3
+801
214
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
E'9 = E 9° Q)=W2+0)o
S=w,+50 Finally, calculate (D D,
°(D D,(X°)
°CD D'
Then, we solve E9
d dw,
Ey0
1
w=0 8
X0 = w2 + co, W3 +5o
0
and obtain X = F(w, , w2, w3) as follows
Ey=w,+Eyo=F1 (6.19)
CO = w2 +w0 =
8=w3+S0 From the above we get the following F-' W, = E9 - Ey0 w2
= w - wo
(6.20)
W3 =GS - So
Step 4. Calculate the derived mapping F.-'(f) of f(X) under F-'. For this purpose, first we calculate the Jacobian Matrix Jr, of F-' 1
0
JF_, = 0
1
0 0
0
0
1
Therefore we have f(0)(W)= .f2(o)(W .f3(°)(W) j
- 11 E' + Td
I
xd
T.
xd
V,, cosS
(6.21)
Xd
-H(w-wo)-HPe+ oP,,,o X=F(N)
where PP = E,'VS sin 8/Xd E .
Next, define transformation R, as
215
Nonlinear Excitation Control of Large Synchronous Generators D (CO -
z;')
H
0)°) - S0 H
H Pmo)IX=F(W)
(n - f3ro) _ z2 _ (CO -Co.) IX=F(W)
(6.22)
(1)
Z3 = W3 = (S -'5.) IX=F(W)
and we can continue to calculate f(')(W) Since
.
f(')(W) = JR f(0)(W) "-y
we should first get the Jacobian matrix J R of R, r.0° V,
Hx J R,
sins - D - w°E,V' cosS H Hx
0
1
0
0
0
1
Therefore, .fi(') (X)
H
ss Egsin8+
H
w -w o
L
P.,
(6.23)
i X=F(W) :A.
H(w-wo)-
f(') (W) =
Now we can define R2 as z12) =fit) (X) _ (-
D
(w - 010) -
H
wo
VS
Hx
Ey sin S +
wo
H
Pmo) I X=F(W )
(6.24)
f(')(X)=(w-wo)Ix=F(W) z(2)=z(')= 3 - 3 (8 -So) LF(W) Z2
=
In the following, we define the composite transformation T as VY
zit)=Ti(X)_-D(w-w°)-(0
H
H
E9 sins+Pmo H
(6.25)
Z22) =T2(X)=w-wo z32)
= T3 (X) = S -'50
The Jacobian Matrix of T is
- w0 JT
-I
Vr
Hx',E
sin 8 - D - w° Vs H
E' cos S HxA '
0
1
0
0
0
1
216
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
From Eqs. (3.76) and (3.77), f(X) and g(X) can be calculated with the following algorithm
-
Hx
sing - D - o)o VS Hx' H fib
f(X)=J1(X)f(X) coo V
E'v cosS
0
1
0
0
0
1
f (X)
12 ( f3(X)
As we know
fi(X)=- i Eq
A(X)=th
Tdo
A(X)=w-Col
Therefore,
w0 V' E sing-DwOVS E'(w-w )cosg q
Td, H x:
q
H
f(X)=
Hx;,
°
(6.26)
w
0)-wo and
w°V - HxdZTd0 sin S
(6.27)
g(X) = JT(X)g(X) =
0 0
Step 5. The final coordinate transformation is Z, =w3 =g -go (6.28)
Z2 = J 3 = w - w° w°Vs
Z3 = f 2 =0)=- D (w - coo) E' sin g + 0 P HxdI H H
The exactly linearized system is i, = Z2 i2 = Z3
(6.29)
Z3 =v
From Example 3.2, the optimal control v* of the linear system is known to be v` _ -z, - 2.29z2 - 2.14z3 _ -Ag - 2.29Aw - 2.14Ad)
_ -f Awdt-2.29Aw-2.14Aa)
(6.30)
217
Nonlinear Excitation Control of Large Synchronous Generators
where
ow=w-a
AS=s-50
Ad) = d
(w-wo).
From Eq. (3.82), the nonlinear excitation control law is in the form of u =V1 _
f, (X)+v'
(6.31)
Si(X)
Substituting Eqs. (6.26), (6.27) and (6.30) into (6.31), making some manipulation and considering the generator active power output P, = (VsEq/x")sins ,
/x , we obtain the
and the reactive power output Q. = (V Eq following nonlinear excitation control law as VJ = Eq -Td
Ev Pr
V2
H
Eq
x,£
wa
Pr
(Q. + - )Ow+-T Q (2.29iw+ J'Awdt+(2.14--)tw) H
(6.32) where Td = Tvo x,E /xdE
If we replace Eq with xo,IJ , where I J is the per
.
P3'
unit excitation current with the no-load rated excitation current as the base value, after rearrangement, the excitation control law can be rewritten as
Vj =x,I -Tdxa, P (Qr+C,)Ow+
Tox, P (J''&wdt+C2ero)
Pr
0
(6.33)
r
where Vs2
2.29H
x:x
WO
C2 = 2.14 -
(6.34a)
D
(6.34b)
Formulae (6.32) and (6.33) are the desired generator nonlinear excitation control law for one-machine, infinite-bus systems.
Discussions on the Implementation of Nonlinear Excitation Control CJ'
6.3.2
6.3.2.1. Feedback variable measurements To implement of the control law shown in expression (6.32) or (6.33),
we need to measure the active power Pr, reactive power Q., the speed
218
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
deviation Aw and the no-load potential Eq or excitation current if. The first two variables P, and Qe are easily measurable; the measurement of the deviation of the shaft angular speed Ow may be replaced by that of
frequency deviation tf,, which is also easy to measure [3]. For the excitation current, the Hall Effect Current Transducer may be used, which can yield a DC voltage output in proportion to the measured currents. We
may also adopt another method to realize the nonlinear control law in expression (6.32).
From the power network theory we can see that the generator internal voltage drop Eq - V, may be approximately represented by OVG = E q - V, =
Per+Qexd V,
where r is the equivalent resistance of the generator's stator winding. If this resistance is neglected, we will have Eq - V, = Qexd/V
Thus,
Eq=Qyd+V,
(6.35)
Substituting formula (6.35) into (6.32), we obtain the following expression of the control strategy 2
Vf = 1 (Qexd +V,)(P, -Td(Q,
+V )Ow x.
PP
Vs
+
°(2.3Aw+fuiwdt+(2.14-H)At))
(636 . a)
for PP#0
0
which may also be written as QeXd ) .:C
Vj = P (V +
Td(Qe +C,)&o
+VTd (faAwdt+C20w))
for Pe # 0
co,
where C, and C2 have been determined by Eqs. (6.34a) and (6.34b), respectively. In Eq. (6.36), all the variables can be measured directly.
6.3.2.2. The structure of microcomputer nonlinear excitation controller Obviously, to implement the nonlinear excitation control law given by Eqs. (6.32) and (6.36), we cannot still adopt the common analogous scheme
and should utilize sub-micro or ultra-micro computers instead. The
219 in'
Nonlinear Excitation Control of Large Synchronous Generators
schematic diagram of microcomputer nonlinear excitation controller is shown in Fig. 6.7.
From Fig. 6.7, we can see that a double-CPU is used to implement nonlinear excitation control. The CPU, is used to calculate the relevant feedback variables and to form the control law, while CPU2 is utilized to form six phase-shifting pulses according to the control law. These magnified output phase-shifting driving pulses are delivered into the thyristor gates in the controllable bridge rectifier. The microcomputer nonlinear excitation controller is formed with these components. rSynchronizing signal
FN
It
RAM
0
tri
RAM
EPRO
ra ro
EPROM
ti 1
Q.,f
N
Phase shift pulse generation
Phase-shift
0
of V,
I,, If, P.,
4'.
H
Conversion
E.
CPU2
Interface
F_'
4
Multiplexer
Sample and holding
CPU, Calculation
O
Figure 6.7 The schematic diagram of microcomputer nonlinear excitation controller
6.3.3
Effects of Nonlinear Excitation Control
Fig. 6.8 gives the
connection diagram and parameters of the
transmission system discussed. In this figure, the generator is a 600MW steam turbine unit, which is connected to the receptive infinite bus through two parallel high voltage long transmission lines. The generator excitation system is a thyristor-controlled self-excited system.
rec, AS Nonlinear excitation Controller
AP,
xd-0.825
700=2.9
xe =0.14 xT 0.10
Q,
xLf x 1.46
If
H=6.0 second
Figure 6.8 The one-machine, infinite-bus system diagram and its parameters
220
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS '3-
MoD
In the following, we will separately discuss the improvement in the power system small and large disturbance stability by nonlinear excitation controllers.
6.3.3.1. Improvement in the power system small-disturbance stability
The small disturbance stability refers to, for a system at a certain ?'-
operating point, the ability of its state to go back to the previous point after an arbitrarily small disturbance. From Eqs. (6.28) and (6.29) we know, under CAD
+-+
the condition that the damping torque -DAw/H is neglected and the 0.v
poi
mechanical power P. keeps a constant value Pmo , which equals the initial steady state value of electromagnetic power Pro . The simple transmission system in Fig. 6.8 controlled by the nonlinear law in the expression (6.31) is equivalent to the following completely controllable linear system III
AS = Aa
Ad, = H APr
(6.37)
APr=-Hv wo
In the above equation, APr = Pr (t) - Pro where Pro is the initial steady state generator active power in per unit, and equals the mechanical power Pmo ; v
is the controlling variable of linear system (6.37). Substitute (6.30) into (6.37), we have As = Aw
Aw= -
(6.38)
HAP,
APr =-H(AS+2.3Aw-2.14 H APr) 0
Eliminating Aw in (6.38), and taking Laplace transformation for both sides of the equation, we have APr (s) =
H 1+2.3s AS(s) 2.14w01+0.47s
(6.39)
According to the Final Value Theorem of Laplace transformation, if the
active power angle makes a unit step change, the deviation of the active power, when achieving a new steady state, will be
H
1 + 2.3s
I
APr (oo) = 1im sAPr (s) = lim s-->o s-,o 2.14wo 1+ 0.47s s
0.47H wo
(6.40)
Nonlinear Excitation Control of Large Synchronous Generators
221
,..+
1-+
'y.
Eq. (6.40) states that, if the active power angle takes a unit step variation, the generator active power steady state value will accordingly have
an increment of 0.47H1wo . That means the P-8 curve is a straight line which passes the origin of P-8 plane with slope of tga = 0.47H/wa (refer
dad)
(per
to Fig. 6.9).
60
8°
120
180
L5 (degree )
Figure 6.9 The generator power-angle curve under nonlinear excitation control
The domain of the power-angle characteristic curve is just the set a which was shown in Section 6.3.1. If the power angle 8 is staying in that
hi,
set, then the condition for exact linearization via nonlinear excitation control of a one-machine, infinite-bus system must be satisfied, more specifically, the domain of the angle 8 is the open interval (0,;r) 1). Now we can come to the conclusion: if the nonlinear excitation control given by expression (6.33) is adopted, then whenever the angle 8 between generator potential vector E. and infinite bus voltage vector V, will be theoretically within 0° - 180°, the power system will be stable under small disturbances. Of
course, the stable region of an actual system will be confined by the (DD
precision, response speed of the excitation controller; it is also limited by allowable generator's terminal voltage deviation in normal operation and the peak value of the excitation current (the ceiling field current). The conclusion stated above has been verified by computer simulations 'y.
=t.
with the adoption of the nonlinear excitation control introduced in this section. The system can remain stable as the rotor angle 8 is less than 180°.
') The boundary points of 8=0 and 8 =,r are not included.
222
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS A'+
6.3.3.2. Enhancement of power system large-disturbance stability In the above, the improvement in small-disturbance stability of power systems by nonlinear excitation control (NEC) was discussed. Since this controller design is based on power system nonlinear mathematical model without approximate linearization, it is suitable for the situation when the system's state varies in a wide range. As a result, it can improve large-
woo
C,w0
disturbance stability of the system, which is the most important advantage of the NEC over any types of linear excitation controllers. Cs'
For the system in Fig. 6.8, physical dynamic simulations have been
-°o
(1a
c00
'.3
a.3
done on the system transient stability under different excitation strategies. In the experiment, the following conditions were adopted for testing various excitation control laws. (1) Three-phase fault lasting 0.5 s.. on generator end high-tension bus; (2) Normal operation mode with terminal voltage of V, = 1.0; (3) Peak value of the forced excitation confined to four times of the noload rating value.
In Tab. 6.1, the transient stability limits of these three types of excitation strategies are listed.
From Tab. 6.1 we can see that comparing with other types of linear excitation controls, the nonlinear excitation control can remarkably improve transient stability of power systems. Table 6.1
Comparison of Transient Stability Limit
Excitation Control Mode
Transient Stability Limit P.,,,ac(p.u.)
Stability Limit Improvement (%)
PID Excitation Control
0.74
0.0
Linear Optimal Excitation Control
0.85
14.8
Nonlinear Excitation Control
1.00
35.0
These results illustrated above confirm that compared with PSS and linear optimal excitation controllers, the nonlinear excitation controllers can remarkably improve small disturbance stability and even more prominently enhance large disturbance stability. The technology of NEC developed in this chapter is the first attempt in
the world to unify excitation control designs for improving both small disturbance (dynamic) stability and large disturbance transient stability of power systems.
223
Nonlinear Excitation Control of Large Synchronous Generators
6.4
NONLINEAR EXCITATION CONTROL DESIGN FOR MULTI-MACHINE SYSTEMS
In Chapter 4, the general principle of MIMO nonlinear system control design was discussed. This section will specifically deal with the NOEC design principle and approach for generators in multi-machine power systems.
6.4.1
Dynamic Equations of Multi-Machine Systems
_:c
In this section, we will establish the mathematical model for power systems with n generators. It is assumed that generators are equipped with the fast (static, self-excited) excitation. Namely, the exciter time constant T, almost equals zero. Representing the angle between the generator potential vector Eq, and the voltage vector V,. by S,, according to Eqs. (5.9), (5.11) and (5.12), we can write the rotor motion equations for the n-machine system as follows S=w; - wo
P. -
w=
.
(6.41)
(w; -Co.) - H P,
(6.42)
H
where the subscript i of variables and parameters is the generator serial number, and i =1,2, , n. In the above equation, from formula (5.65) we have Pr;=G;;Ey;2+EQ;Y_ Y;jjE,2sin(S;, -a.)
(6.43)
jmi
where G,; is the conductance of the t' bus, and Y,j is the admittance between the tt" bus and they"' bus; S. = S, - S, ; a,, is the complement of the impedance angle. From Eq. (5.71), we can write the equation describing the electromagnetic dynamic process of the i' generator's field winding as EQ =
-
1 TdOi
E9; + 1 V f
(6.44)
Tdo
As we know, the relation between the q-axis transient potential E' and the q-axis potential Eq is Eq = E9 + Id (xd - xd)
(6.45)
Substituting Eq. (6.45) into Eq. (6.44) while considering the relation in Eq.
(5.68), we will obtain the electromagnetic dynamic equation of the
i'
224
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
generator's field winding as 1+(xdi-xdi)B,,Eqi 4i
Td0i
(6.46)
+xdT xdijY;jEj cos(8j-aj)+Tl doi j° doi jxi
Vj;
i=1,2,...,n
where B,, is the susceptance of the ith bus.
Combining Eqs. (6.46), (6.42) and (6.41), we will have the following state equations of a multi-machine system with generator excitation control 1 + (xdi - xdi )Bii Eq,
Eq, Tdoi
w0
d)i
H;
(Gii Eyi 2 +Eqi
j=i jai
xdi - xdi
+
n
Tdoi
1
Y,jEgj cos(8 j - a,) +Tdoi VI
jxi
Yj E, si (Sij - aij )) - I (coi - Q)0) + COO p.,, (6.47) H, H;
i=1,2,. ..,n
b =w, -wo
The above equations may be written in the following form of affine nonlinear systems X = f(X) +
(6.48a)
g, (X)u,
where _ 1 +(xd,
xd,)B,t Eq, +xdi xdt do,
dot
YjEJ cos(8,j -a,j) J=2
, _ 1+(x,, -xdn Bnn Eqn + xdn -xdn IYnjE, cos((Snj -anj) Tdon
Td On
j-t
j,,n
Poi
f(X) =
Hn
C=7
(Gnn E2 + E,,, [LYnj Eq, sin((Snj - anj)) - " (con -w0)+
w
+
WO
Hn
j=i
(6.48b)
Hn
Pmn
j:n
w,-moo a),, -coo IT
0
0
...
91(X)=
--2 d
0
Ti,,,
g2(X)=
0
0 0
gn(X)=
1)'Tdoi
0
<- n'" (6.48c) `.S
1
#-1st
...
1
Tdat
o
Nonlinear Excitation Control of Large Synchronous Generators X=1E11,
El.; m cot,
...,
tj
O)n
tj2
...
225
5T
(6.48d)
In the above we have established the mathematical model of rnultimachine power systems for generator excitation control analysis and design. From this model, the nonlinear excitation control law will be derived for generators in multi-machine systems as follows.
6.4.2
Exact Linearization Design Method for Excitation Control
CDR
In this section, the exact linearization design principle for MIMO affine
nonlinear systems discussed in Chapter 4 will be applied to design the nonlinear excitation controllers. According to Theorem 4.1, for the system with dimension of the state
C17
CAD
vector of 3n as determined in Eq. (6.48), we first choose its index number as n, = n2 = n3 = n , and thus it holds that n, + n2 + n3 = 3n , where n is the number of generators in multi-machine systems. If the theoretical process is strictly followed, (refer to Section 4.4.2), we should then, according to the expressions of f(X) and g(X) presented in formulae (6.48b) and (6.48c), calculate the matrix D3 and determine its nonsingularity; next, we should determine the involutivity of the 3n vector field sets from D,,, through D3 to make sure that the system in Eq. (6.48) satisfies the conditions for
exact linearization. However, here we plan to adopt procedures more
gyp
practical, namely, to directly perform exact linearization and calculations for (IQ
the system shown in Eq. (6.48) according to the exact linearization o..
f10
algorithms discussed in Section 4.4.2. If a mapping Z = (D(X) and a state feedback U(X) = a(X) + b(X)V could be found such that the system in Eq. CS'
`CS
(6.48) was linearized, not only it proved that the system in Eq_ (6.48) satisfies the conditions of exact linearization, but also the goal was achieved to find nonlinear state feedback. Step 1. For the system in Eq. (6.48) choose the index number as N=3
n, =n2 = n3 =n,
Y_n; = 3n
where n is the number of generators, and 3n is the dimension of the state
.fl
vector of the whole multi-machine system in Eq. (6.48). First, the first-order Lie bracket of f(X) and g; (X) are calculated.
226
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
adfgi(X)=- gi(X) 1
aEq,
asn
((9n
In
Jn
n
a((w,
la{sn
In
Jn+l
Jn ...
aE9,
9n
J,+,
J n+1
01 g1,
aE,',
...
0
2n
a(w, 2n
J2,,
)2n
J2n
aE9n
aw,
dawn
-00'8,
asn
J 2n+!
J 2n+1
J 2n
...
8Egl 0f2n+1
af3n
3n
aE9,
aE9n
I
aE9;
'4 2n+1
... aE9n
aEq,
as,
awn
as,
fl.
a
a aw,
Jn
gii,
2n+1
2n+l
awn
Jn+1
J 2n
aE9; g'i' E9, a g
0j
3n
asn
gii, 0, .., 0,
aE9;
0
]
L(z)1(X), 0, ... Of
=[Lii)(X), ... Lllil(X),
(6.49)
Then we find out the second-order Lie bracket ad2
a(adfgi)
fgi(X)=
ax
f(X)-
0
0
af(X)
ax
where 0
p
p
a(adlgi)
ax
f (X) =
aL(') (n+1)i
aL(') (n+1)i
aEq,
aE9n
n),
L
o
p
p
...
aE9n p
p
... p
0
0
...
Lni(X),
p
...
p
(6.50a)
a4l
aL;;)
a(S,
asn
ni
ni
as,
asn
aL(') (n+I)i
aL(') (n+I)i
as,
asn
p
aL((2 n)i
aE91
p
0 ...
...
Pin
aL((2
0
(adfgi)
...
aL(( 2n),
p
...
A% 0
0
...
0
aL(') (2n)i 0,51
0
L(n+I)i(X),
...
L(2n);(X),
0,
f
{n .fn+l
.f2n+1
An
...,
Of
(6.50b)
Denote the second term of the second-order Lie bracket in (6.50a) as
227
Nonlinear Excitation Control of Large Synchronous Generators
...
aX (adfg,(X))=[L,(X) L;(X)
(6.50c)
,'k
then the expression of the second-order Lie bracket (6.50a) can be rewritten as
ad, g, (X)
From
the
L(2)
L(n.
previously
calculated
...
Lip+1p
-Lc3nu}T
(6.50d)
L(i,,), }T
brackets
Lie
CAD
=1Lir1
...
ply
=[(L,-Li,) (Nz,-Lz) ... (L(y,,-L(''r)
of various
order
A'+
'ti
ad fg , ad fg ad f2g , ad fg , we can achieve the following 3n vector field sets D, _ {g, (X)} D2 = {g1 (X), 92 (X)}
D" = {g1(X), g2 (X), ..., 9. M) D"+1 = {D" , ad fg, (X)}
Den ={D,,,ad
fg"(X)} 2
D2"+1 = {D2n, ad fgl(X)} Dan
Step 2.
Choose 3n linearly independent vector fields D, a D; where
i =1,2,...,3n . From Eq. (4.151) we know that
D,
should satisfy the
following conditions D, +k;')(X)g,(X) = 0
t71
D2 +k, 2)(X)g, (X)
D +k;")(X)g, (X) + k2")(X)g2(X)+ + k;"'(X)g (X) = 0 +k(n+l)(X)g,
D2,,
+k;+i')(X)adfg, =0 +
D,,+1
+...+k,2" (X)g" +k;2")(X)adfg1 +k22")(X)adfg,,
g,
1
fg
2
1
l
=0 t,
g
+k(2"+)ad2 2.,+1
fg1
,,+1
fg1
=0
g+...+k(3n)g +k(3")ad fg 2" n fg1 +...+k(3n)ad ,,+1
+k23,+iad2g1
(6.51)
+...
+...+k;;")adjg =0
1,
228
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
5e!
According to the algorithm discussed in Section 4.4.2, in order to obtain D, , the simplest way is to solve for each k(') (X) in Eq. (6.51) in the
of
neighborhood
X ° _ {.3j, w E9, 15, - a° = 0; i, j =1, , n}
that
such
D, E D -1-63n r= Dan and D , D,,, are as simple as possible. To this end, let us discuss the D matrix at point X° D=[g, ... gn ad1g,(X°) ... adfgn(X°) adfg,(X°) ... ad2gn(X°)] From Eqs. (6.49) and (6.51) we have r gl' ... 0
0
-)(X°) ... 41))(X°)
42)(X°) ...
42)(X°)
Ln)(X0)
...
L(,2,)(X°)
...
gnn ,......... ........ ........... ........ .... .......... ......... ........... ..................... 42.1 (XI) ............................ ......................................... ...........
°
(2)
L(n.1)1(X)
(2) ° ... An.l)n(X )
...
D=
0
2w°
H GnnE4n
0
L 2,)j(X°)
...
.............. .......... ......... ....... ........... ........... ............... _... ....................5 ........................................................................
11
0
q1
0
0
(6.52)
Thus we know that the matrix D is an upper-triangle one. Substituting formula (6.52) into formula (6.51) and solving for k('), we will have the following results. Set
k") _ -1/g, I then
Di =[1
0
..
0
0
...
0
0
Of E D,(X°)
Set again ki z) = 0
k2t) _ -1922
then D2 =[0
1
0
In the same way, we will obtain
0
0
0
Of ED2(X°)
229
Nonlinear Excitation Control of Large Synchronous Generators D3 =[0
0
1
D. = [0 ...
0 the
0
0
0
...
0
0
0
0
...
Of ED3(Xo) 0]T
E
one
coo
Moreover, as long as we select kin+>(Xo)
L11>(Xo) kn(+i)(Xo) g11
knn+1)(Xe)=-Loll (Xo)kn+Il)(Xo) gnn
and
HI
k(n+l)(Xo) = n+i
2w0G11Eg1
then
=[0 ...
0
1
...
0
Of
...
0
0
the(n+1)'"one
For the same reason, we will get Dn+2 = [0
0
0
0
0
1
the(n+2)'"one
00
D2n =[0
1
Of
0
Of EDn+2(Xo)
0
E D2n (Xe)
T
the (2n)' one
Similarly, if we choose ,-+
0
gn
k (2n+q n+I
k (2n+1) n+2 ...
g22
k(2n+1) 2n
0 gnn
where k(zn+1) n+1
k2n+1) 2n
k(2n+1) 2n+1
__
H ,L(2) (n+I)1 2woG11 Eg1
(2n)1 _ _ H nL(2J
2w0GnnErgn
HI 2w0G1, Eq,
+
230
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
then ...
0
...
0
0
0
...
Of E Den+t m
Den+1 = [0
1 the (2n+1)'" one
In this way, we can have 0
D2n+z = [0
...
0
0
0
... Of E
11
the (2n+2)" one
Dan = [0
...
0
0
...
0
...
0
0
117 E D3
the (3n)'" one
Step 3. Calculate the mapping from new state space W into the original one X. According to the algorithm presented in Section 4.4.2, we need to seek the solution of F( w, w2 ...,w3n)=d)Dw,,
DD: O
D
0 ... o (D Dm, (X0)
II`
'-s
The method to solve the above equation is the same as that for onemachine systems presented previously in this chapter. After calculation we obtain the following transformation F Eq1 = w1 + E1,01 ,II
E9 = w + Eqo 0)1 = W.., +coo
......
(6.53)
CU = w2n + (DO 81 = w2n+1 + 801
8n = wan + SOn
whose inverse mapping F-' is wl = EQt - Eyot
w = Ey - Egon wn+1 = 0)1
00
(6.54) w2n = n - 0)0
,II
w2n+1 = 81 - 801
W3,, = 8 - 80n
'71
`-+
'i7
Step 4. From the inverse mapping F-' of F presented in Eq. (6.54), we will find that its Jacobian Matrix is an identity one. Furthermore, we can calculate
Nonlinear Excitation Control of Large Synchronous Generators
231
(6.55)
f(0) = JF_,f(X) = f(X)
Step 5. From formula (4.161), we can successively define transformations R, and R2 according to formula (6.55) as follows. R,; (1)
(o) = = fn+1 4.
m
{'ro) = ./ 2n
ZI
zit
0)
w0
D1
l = - H' (w, - a)0) - Hl
= .fen = - Dn 001 - wo) -
tf
n
Coo
Pn + Hl P.n01
Coo
CO0
H Pen + Hit Pmon H.
(o)
Zn+1 - f1 = f2-1 = wl - 0)° (1) Z2n =
3
(.
(6.56)
= fan - 0)11 - w0
Zin+1 = wzn+1 = Sl - 1501 01
Zin - w3n = Sn - Son
The Jacobian Matrix of R namely, JR is aEg1
alit)
az(')
azazaz;')1
aEQ
aw,
awn
a(51
az(n
&Z a)
...
... n
moo) n
...
aEQ
aEQ,
.. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
moo)
n
...
it
aw
aw,
n
fin)
...
as,
(O'
&(1)
JR =
Wit n
as,,
.................................................... .................................................
0
1
0
0
0
0
0
1
0
0
...
0
...
(6.57)
0
......................................... ..... .............. .............. ..................... ........................ ........................
0
1
0
0
1
0
Next, let us define transformation R,,,_, = R2 according to (4.161). After calculation, we have R2 = R1
So, the Jacobian Matrix of R2 is JR, = JR
Therefore, it holds that
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
232
f(2) = f(X) = JR f(X) _[a1(X)
an(X)
(6.58) ...
221,4
z3,,
...
Z11+1
Z2ur
...
and
gal ='g(X)=JR.Lg,(X) 91 (X) ...
b,(X)
b.. (X)
0
0
..
0
(6.59)
0
In Eq. (6.58) a, (X)
a(X)
I
chit)
"El
l
&(l)
a "go
&,(I) +...+a +a f ",
C-z (l)
l+...+ J f21,+
ot
&n) f+...+a
z(1)
f,,r1+...+a,
&a) f,,,,+...+a
f21,1+...+
u
I
(6.60)
(l)
&(l)
An
f,+a"
qn
{
3n
i
moo)
l
azi(l)
n
From the state equation (6.47), we have f (X) = Eql _
Tdol
.-.
fn(X)=Eqn-
1
1
ut
u
Td 0n
f,,.l (X) = w1
(6.61) An (X) _ thn f2-1 (X) = S1 (X)=in
A3
Substituting Eq. (6.61) into Eq. (6.60) while noticing the relation z;l)
Z =f2 (X) determined by Eq. (6.56), with the mechanical power Pmo, assumed to be a
Nonlinear Excitation Control of Large Synchronous Generators
233
constant, we will have mo (aPel
Eql + aPel
HI aEgl
a8l +
Sl +...+ aPI Eg + ap aEg
(ape,
Coo
u +...+
HIT40I aEgI
a(X)
_
coo
ap E
H aEgl
+ open qI
MI
+
HI
aPel
aEq
(6.62) H,,
I
u + ... +
H,,Te0 aEq.
ape»
w,
u")
s + "' + open Eq + aPC1 aE as ( ape
wo
`
-D
aS»
O)n
Uj
aEq»
Substituting formulae (6.57) and (6.48c) into formula (6.59), we have ()0
b(X)
b
bl
apes
(00
HITao1 aEEI
ap,I
HITe01 aEq» (6.63)
b»,
... b
coo
ape»
wo
H,,Ta0,, aEq,
L
ape»
H»Tdo» aEq
From the expression (4.173) we know that if we set (6.64)
a(X) + b(X)U = V
where U presents the control vector of the nonlinear system, V is the "control" input vector of the following exactly linearized system ZI =
(-6 65 )
Zz = Z3,, Z2»+, = VI
Z3n = Vn
Substituting expressions (6.62) and (6.63) into (6.64), we can get a very simple relation as HI
pel
HI
w,
VI
V = v2 =
H H2
pet H 2 z
w2
(6.66)
V CO°Pu,
H
»
H »
The next task for us is to calculate the control vector U in accordance with the expression (6.66). From expression (5.34b), the active power can be expressed as
234
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS Pe; = E.;lgi +(xq, -xa;)Id;Iq;
If we accept the approximation that xq; = x', , the above equation will be simplified to P. = E;Iq;
(6.67)
PQ1 = Eq;Iq; +Eq;Iq;
(6.68)
Therefore,
'+S
Substituting expression (6.44) into the above equation and making some manipulation, we obtain
=Iqi(-
1
1
i
Eq;+ 1 Vj)+Eq;Iq;
Tao;
Tao;
10 Eq;'+E;iI q ; +
Tao;
(6.69) Iq;Vf
Tao;
.,.
Substituting the above equation into Eq. (6.66) and approximately letting D; = 0, we have
Iq, Eq,
WO
H;Tao;
- wo E lq H;
I q, Vf, = v,
w0
q,
(6 . 70)
H;Tao;
From the above equation, the control variable V f, can be solved as ui = Vr = Eq; -Tao; iglgi
E ( 1 - Te,1 .
q
where
6.4.3
1 q;
Iq,
,
Eq;
1 - H; Tao; -v;
wo
Iqi
(6 . 71)
H;Tao; 1 v
I
w0
ql
To; = (x;,; /xa; )Taol
Practical Nonlinear Excitation Control Law
In the previous section, the nonlinear excitation control law has been ,r.
obtained for generators in a multi-machine system. To facilitate the engineering implementation of the above excitation control law, we need to transform the variables in expression (6.71) into measurable variables. From expression (6.35) we have Pe; .= Eq; Iq;
Moreover, from expression (6.35) it holds that
(6.72a)
235
Nonlinear Excitation Control of Large Synchronous Generators
Eq = y +
Q-,xd,
(6.72b)
V,i
where V, is the terminal voltage of the
generator, and Qe, is the
i'h
reactive power output of the same generator.
Substituting formulae (6.27a) and (6.27b) into expression (6.7 1), we obtain the nonlinear excitation control law for a multi-machine system.
u,Vf (Vi
+ Q.,xd,
P.
)(PP,
- T' (Vn + Q.,xdi) d ( Vn
VII
at
H,Tdot
P,,
Vn +
v,)
(6.73)
a0
Q,, x&
V,
for Pr1 * 0
In the above equation, v, is the "control" input of the exactly linearized system in Eq. (6.65) which should take as the optimal control v; of the linearized system. Similarly with Eq. (6.30), we have `r'
v, =-JoOa,dt-2.34w,-2.14Adi
(6.74)
Substituting Eqs. (6.74) into (6.73), we finally get the following nonlinear optimal excitation control (NOEL) law /for a multi-machine system
I
V =
1 (V.n
Qe,xd,)(PW
P,
V
-Td, (Vi + Quxd, V,
d
Pe,
) dt (
Qeixd,
V" +
+ HiTda, (Joowidt+2.30wi
+2.14&i,))
Vi
(6.75)
for Pe, # 0
wo
Discussion on the Nonlinear Excitation Control Law CSC
6.4.4
From the expressions (6.71) and (6.74) we can see that the nonlinear excitation control law for generators in a multi-machine system has the following characteristics.
Firstly, the control law is decentralized. That means the control input u; of the iih generator is merely relevant to the outputs of this generator, they are: active power Pet , reactive power QC,, terminal voltage V,, and rotor speed w, , while it has no direct relation with the state variables or outputs of other units [8, 9].
Secondly, the control law is independent of transmission network
236
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYAAAffCS
parameters. That is to say that the control variable of the i'h unit is merely relevant to its own parameters, i.e. moment of inertia H,, field winding time constant Tdo; , d-axis reactance xd; and transient reactance xd, where the transmission network parameters are not visibly involved. This enables the CAD
control strategy to have complete adaptability to changes in network parameters and structures.
The above two advantages of nonlinear control law are what the 'CS
scientists and technologists in the field of power system stability and control have long been expecting and seeking [5, 7, 11, 12, 15, 161. However, how should we conceptually comprehend these characteristics of control laws obtained from nonlinear mathematical model and exact linearization method? As we know, for multi-machine power systems, the variations in operation
variables of any generator (such as the variation in power output), or in r.,
network structure (such as tripping or closing a line) will in varying degrees cause corresponding changes in the power output of each generator and in the power flow of the whole system. This fact tells us that, if each generator in the system is regarded as a subsystem, the state or output variable vector of each subsystem will form a "holographic unit". Namely, in the set of state variables or output variables of each subsystem, the information about the operation states of other subsystems and the power flow distribution of the
"O-
".Y
A..
coo
t'1
3C)0,
whole system is included implicitly. This objectively yields such a possibility that in each subsystem merely the state or output variables of its own are adopted as a feedback while the effect achieved is to ameliorate the stability and dynamic performances of the whole system. The control with the above characteristics is said to be decentralized or completely decoupled between subsystems. The exact linearization design approach adopted in this section has just revealed and realized this intrinsic characteristic of a multimachine system. However, it should be pointed out that, which has been used until now, the conventional approximate linearization modeling method 0
O0,
to replace the increment of a nonlinear function at a point x0, Af = ti.
f (x , x,,) - f (x,o, , by the total differential, df = (af 18x,) Ox, + (af /ax2) Axe + + (af/ax,,)Ax,,, at the same point, obscures that inherent characteristics of power systems. This is one important reason why up to now, in the field of linear optimal decentralized control and robust control for multi-machine systems, breakthrough has not been achieved yet and is -C,
hardly to be achieved. mss'
Thirdly, the equation of excitation control strategy presented in Eq. (6.71) or (6.74) for multi-machine systems is also suitable for one-machine, infinite-bus systems since the latter are the special type of multi-machine systems. 146
237
Nonlinear Excitation Control of Large Synchronous Generators
6.4.5
Effects of the Nonlinear Excitation Control
The multi-machine system to be studied is shown in Fig. 6.10, with its parameters given in Tab. 6.2. In the 6-machine system studied, machine No. 6 is a synchronous compensator, while machine No.1 is the reference unit; the transformer and line parameters are indicated in Fig. 10; the load center is located at bus 16 and 18.
For the 6-machine system presented in Fig. 6.10, its small and large disturbance stability are studied under three different excitation control schemes which are listed below. Table 6.2
Generator parameters of the 6-machine system
Generator number
xd
4
Xq
H
D
Tdo(S)
2
0.015
0.015
0.015
140.82/2
3.0
9.0
3
0.321
0.321
0.0382
30.0/2
3.0
8.375
4
0.138
0.0396
0.0396
79.5/2
3.0
7.24
5
0.77
0.77
0.121
15.62/2
3.0
6.2
6
1.633
1.633
0.197
2.62/2
3.0
6.92
The first one is a PSS (power system stabilizer), namely the control with single supplementary variable feedback [4]. According to the PSS design method, we obtain the transfer functions of the supplementary controllers for machine 2, 3, 4 and 5, respectively, as follows 15.48 3s 1+0.43s 2 = 1+0.01s 1+3s (1+0.045s) _ 63.35 3s 1+0.42s 2 G3 (s) 1+0.0lsl+3s 1+0.155s) 14.3 3s 1+0.34s )2 G4(s)=
G,(s)
(6.76)
1+0.0ls l+3s 1+0.126s
G,(s)
_
25.6
3s
1+0.436S 2
1+0.0ls l+3s 1+0.21s )
The PSS input signal is the deviation of speed Ow, and the output limit is
±5%.
The second one is a linear optimal excitation controller (LOEC) [10, 13].
According to the principle of linear optimal control, the LOEC strategies for No. 2, 3, 4 and 5 are given respectively as
Part A
2.87+j 1.4
Lo
j 0.0 T6
0
8
61'0!+50'0 2 )
IL
°
`
0.72+j0.47
Io
0.01+10.04
0.7+JO.5
.021+j 0.088
.016+10.06
22
.0 t2
9
-10.002
i-
0.
88+10. 88
Lit
.OS7+70.1 0.058+10.22
10.064
A
i °o
14
Figure 6.10 The structure diagram of the 6-machine system (Power base=IOOMVA)
2.26+11.69
L==
10.022
Lo 3. 76+12.2
0.003+10.0! 9
0.015+10.10 17
IS
18
I
Part B
Leo 4.3+10.28
..r 10.OS8
I
r ' I o.oolT j 0.034 j 0.01
8£Z 238 NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
239
turn
Nonlinear Excitation Control of Large Synchronous Generators AVJ2 = -73.8AV,2 -35.6APe2 +5.8Aw2 AVJ3 =-69.2AVt3 -23.3APr3 +6.1Aw3
(6.77)
tip
AVf, = -62.9AV,4 -10.00PC4 +2.3iw, AVJ5 = -58.4AVf5 -8.0APe5 +1.6Aw5
The third one is a nonlinear optimal excitation controller (NOEC).
Machine 2, 3, 4 and 5 are equipped with NOECs. The expression of
control laws has been given by Eq. (6.75). In the following we will demonstrate the digital simulation results of the 6-machine system to show the effectiveness of the NOEC to improve the dynamic performance and stability of the system.
1. The improvement in small disturbance response by NOEC
Relative
Relative
The total sum of the system's loads is 1500MW. The disturbance is set as a sudden load shedding of 70MW at bus 21. The generator's swing curves
- 80
d (b)
40
Relative
0
a
-40 -80 0.0
(c)
1.0
1
I
I
2.0
3.0
4.0
1
5.0
Time (s)
The comparison of generator small disturbance response curves under various excitation controls strategies (a) With LOEC (linear optimal excitation control) (b) With PSS (power system stabilizer) (c) With NOEC (nonlinear optimal excitation control)
Figure 6.11
240
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
under the above three types of excitation control schemes are shown in Fig. 6.11.
From the above diagram, we can see that the NOEC mode can notably improve the system dynamic performance.
2. The amelioration in large disturbance stability by NOEC For the system shown in Fig. 6.10, digital simulations are carried out under the following two large disturbances. 1) Short circuit fault. Three-phase short-circuit on bus 11 occurs at 0 second and is cleared at time t, Under this fault, the critical clearing time t, corresponding to different excitation control schemes is listed in Tab. 6.3. Table 6.3
The critical clearing time under different excitation control mode Critical Clearing Time (seconds)
,3°
Control Mode
Critical Clearing Time Improvement (%)
2, 3, 4, 5# unit normal control
0.074
0.0
2, 3, 4, 5# unit with PSS
0.080
8.1
2, 3, 4, 5# unit with LOEC
0.082
10.8
2, 3, 4, 5# unit with NOEC
0.090
21.6
1
3-i
From Tab. 6.3 we can see that when NOECs are installed on all the °\p
machines 2, 3, 4 and 5, the critical clearing time will be improved by 21.6% as compared to the case when all these machines equipped with proportional excitation controllers, and by 12.5% and 10% respective to the case when PSS and LOEC controllers are installed. Fig. 6.12 through Fig. 6.14 illustrate the dynamic responses of the power system when the fault clearing time t, = 0.09 seconds. We can see from Fig. 6.12 through Fig. 6.14 that under large disturbance of this fault, the power system can remain stable only when NOEC is used. 2) Line tripping. At 0 second, the line between bus 11 and 12 is tripped off. Under this fault, we continually increase the load at bus 16 to get the C/,
critical transmission power from area A to area B (see Fig. 6.10). The tested results corresponding to the above three excitation control schemes are shown in Tab. 6.4. From this table, we can see that when NOEC is
-;.;r
adopted by all the generators, the critical transmission power will be respectively improved by 8% and 7% compared with the case when PSS and LOEC are adopted by all the generators. Fig. 6.15 to Fig. 6.17 respectively v'.
give the line tripping response curves under the above three excitation schemes when the transmission power delivered from area A to area B equals 692MW. From these diagrams, it is demonstrated that in such case the
Nonlinear Excitation Control of Large Synchronous Generators
241
system remains stable only when the nonlinear excitation control is used. Table 6.4
The critical transmission power under various excitation controls
Transmission Power limit improvement 0.0
Transmission Power limit (MW) 637
Excitation Control Mode 2, 3, 4, 5" unit with PSS 2, 3, 4, 5" unit with LOEC
647
1.5
2, 3, 4, 5" unit with NOEC
692
8.1
120 r-
Relative angle (degree)
m
631
40
- 40
- 80 - 120
-160
-200
I
I
1.0
2.0
I
I
I
3.0
_-O
I
0.0
4.0
5.0
Time (second)
Figure 6.12
The system's dynamic response curves with PSS
120
Relative angle (degree)
80
L
I
I
I
I
0.0
1.0
2.0
3.0
4.0
I
5.0
Time (second)
Figure 6.13 The system's dynamic response curves with LOEC
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Relative angle (degree)
120
4V r
d 21
I
I
I
I
I
I
0.0
1.0
2.0
3.0
4.0
5.0
Time (second)
Figure 6.14 The system's dynamic response curves with NOEC
120
Relative angle (degree)
242
80 40
- 40
-80 -120 - 160
-200
L
0
110
2.0
310
4.0
5'0
6.0
Time (second) Figure 6.1 S
The system's dynamic response curve with PSS
243
Nonlinear Excitation Control of Large Synchronous Generators X20
Relative angle (degree)
801
L
0.0
1.0
2.0
f
1
1
3.0
4.0
5.0
t
6.0
Time (second)
Figure 6.16 The system's dynamic response curves with LOEC
Relative angle (degree)
40
d
0
ao
N
- 40
-2001-
0
1.0
2.0
3.0
4.0
5.0
6.0
Time (second)
Figure 6.17 The system's dynamic response curves with NOEC
244
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
REFERENCES
1.
2. 3.
4.
3== a'-
.-.
6.5
C. Concordia, Synchronous Machines, John Wiley & Sons, 1951. E. W. Kimbark, Power System Stability, Wiley, New York, 1956. F. P. de Mello, "Measurement of Synchronous Machine Rotor Angle from Analysis of Zero Sequence Harmonic Components of Machine Terminal Voltage", IEEE PWRD, Vol. 9, pp. 1770-1777, 1994.
F. P. Demello and C. Concordia, "Concepts of Synchronous Machine Stability as Affected by Excitation Control", IEEE Trans. Power Appar. Syst., pp. 316-329, April, 1969.
5.
°'o
H. Jiang, H. Cai, J. F. Dorsey and Z. QU. "Towards a Globally Robust Decentralized Control for Large-Scale Power Systems", IEEE Trans. Control Systems Technology, Vol. 5, pp.3 09-319, 1997.
Theory", Proc. IEEE 90, pp. 25-35,1971., M. C. Han and Y. H. Chen, "Decentralized Control Design: Uncertain Systems with Strong Interconnections", Int. J. Control. Vol. 61, No. 6, pp.1363-1385, 1995. M. G. Singh, Decentralized Control, North-Holland, 1987. Q. Lu, Y. Sun, Z. Xu and Y. Mochizuki, "Decentralized Nonlinear Optimal Excitation Control". IEEE PWRS, Vol. 11, No. 4, pp. 1957-1962, 1996.
9.
10.
Q. Lu, Z. H. Wang, and Y. T. Han, "Integrated Optimal Control of Large 070
8.
P',
...
to)
7.
J. H. Anderson, "The Control of A Synchronous Machine Using Optimal Control
C1°
6.
Turbogenerator and Tests on Micro-Alternator Systems", J. Tsinghua Univ., No.2
'L7
coo
Non
erg
13. 14.
S. Jain and F. Khorrami, "Decentralized Adaptive Control of a Class of Large-Scale Interconnected Nonlinear Systems", IEEE AC, Vol. 42, No.2, pp.136-154, 1997. S. Jain and F. Khorrami, "Robust Decentralized Control of Power Systems Utilizing Only Swing Angle Measurements", Int. J. Control. Vol.66, No. 4, pp. 581-602, 1997. `"°
12.
fir.
1981. 11.
Y. N. Yu, Electric Power System Dynamics, Academic Press, 1983,
Y. N. Yu, K. Vongsuriya and L. N. Wedman, "Application of an optimal Control a..
Theory to a Power System", IEEE Trans. Power Appar. Syst., pp. 55-62, Jan., 1970.
Y. Wang, D. J. Hill and G. Guo, "Robust Decentralized Control for Multimachine
a'.
15.
Power Systems", IEEE Trans. Circuits and Systems, Vol. 45, No. 3, pp. 271-279, 1998. 16.
Y. Wang, G. Guo and D. J. Hill, "Robust Decentralized Nonlinear Design for C/2
Multimachine Power Systems", Automatica, Vol. 33, No. 9, pp.1725-1733, 1997.
Chapter 7
Nonlinear Steam Valving Control
7.1
INTRODUCTION (DD
In the last chapter, the method of the nonlinear excitation control (NEC)
design and its effects have been presented. It has been manifested that
p°..
adopting the NEC scheme brings higher-level stability and better dynamic performance of power systems. However, the improvement of transient stability is limited due to the ceiling excitation voltages and the voltage response time limit. In order to make further improvement on the transient stability, the nonlinear control theory should be applied to the control on valve opening of steam turbines or hydroturbines [1-8]. Because of the water hammer effect, using the water gate opening control to improve the transient stability is not so effective. In this chapter, the nonlinear steam valving control will be studied. It should be particularly pointed out that there exists, more or less, an opinion that the continuous steam valving control has little effect on the
system stability. This viewpoint has lagged behind the development of
`.<
modern technology. Indeed, during the years when the electrical-hydraulic governor has not been implemented widely and the sensors of governors were only flyballs, the speed resolution was very low, correspondingly the accuracy of driving and servo mechanisms was rather poor, and the dead band and time lags for regulation were relatively large. In this case, the control on prime move torque could not improve the stability significantly.
In the recent decade, however, great changes have taken place in the
one
governor system of prime movers. Electrical-hydraulic governors have taken the place of mechanical-hydraulic ones. Important improvement has also been made for the servo systems. The dead band of modem high-pressure
246
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
and high-temperature steam turbine valve is only about 0.1 to 0.2 seconds. The inertia-time-constant of electrical hydraulic converter (oil supply system) is also about 0.15 to 0.3 seconds, and the inertia time constant of steam flow is in the range of 0.1 to 0.3 seconds. Under these conditions, the control on valve opening contributes significantly not only to improving the transient stability, but also to improving the dynamic performance and to suppressing the system oscillation as well. Moreover, under certain conditions the effect of steam valving control may even be better than excitation control. Since
Q..
.a..
CID
the latter only controls the field voltage, while the variation of the flux linkage generated by excitation current would correspondingly cause the change of synchronizing torque of the generator. The excitation current varies with field voltage exponentially; the inertia time constant is at least 2.5 to 4.5 seconds when the stator (armature) winding is short-circuited. The field winding is a large inertia element to the control input. From the above analysis it can be seen that the response time of the governor for a modern
steam turbine is comparable to, at least not slower than, the excitation control system. In a word, studying and implementing the more advanced steam valving control for modern steam turbines will significantly enhance the stability of power systems. The principle and methods for nonlinear steam valving control design ;-f
will be described in this chapter. The effectiveness of the new control strategy will also be presented.
7.2
NONLINEAR STEAM VALVING CONTROL IN A ONE-MACHINE INFINITE-BUS SYSTEM
7.2.1
Mathematical Model
The physical structure of the governor system for steam turbine with reheater (RH) is shown in Fig. 5.5. During the investigation of the problem of steam valving control, let us
assume that the generator has been equipped with advanced excitation controller so that the q-axis transient voltage Eq keeps constant throughout the entire dynamic process. Thus the state equations of the valving control system are composed of only two parts. The first part is the generator rotor motion equations (swing equations). The second part is the valving control system dynamics equations. The swing equations:
Nonlinear Steam Valving Control
247
From Eqs. (5.3), (5.9) and (5.11), the motion equations of generator rotors can be expressed as follows d8(t)
= w( t )
dt
- ao
dw(t) u?o P . ( t ) = dt H
(7 . 1 a) too P. (t ) -D (w(t) - w0) H H
( 7. 1 b)
where, 8(t) is the rotor angle in rad; w(t) the angular speed in rad/s; P. and PF are the mechanical input power from the prime mover and the active
power of the generator, respectively, in per unit; D is the damping coefficient in per unit; H the moment of inertia of rotating system of the
machine in second; wo =2'r fo the rated speed in rad/s. f0 the rated frequency. The expression of active power has been given by Eq. (5.57a). By substituting Eq. (5.57a) into Eq. (7.1b), we have
H
D (0)(t) _0)0)_w E' VS P. (t) - H sin 8(t)
`-'
-w
0.'E
dw(t) dt
H x'
(7.2)
>i3
The dynamic equations for a steam regulating system: First, we consider only the case in which the high-pressure (HP) turbine
is controllable, the intercept valve 6 in Fig. 5.7 does not take part in +,,
controlling. That is, we study the nonlinear steam valving control problem, disregarding the effect of "fast valving". The dynamic equations constituted by high-pressure turbine and control valve (CV) are
boy
dP_(t)
=-T PH(t)+ H
diu1 (t) =
dt
I THg
lUH (t) +
(7.3a)
H'UM(t)
TN
1 THg
PHO (t) +
1 THg
ul
(7.3b)
own
where, PH (t) is the mechanical power generated by the HP turbine in per unit; pH (t) is the high pressure regulated valve opening in per unit; TH and THg are time constants of HP turbine and the oil-servomotor respectively, in
seconds; u, is the control signal from G, (see Fig. 5.8); CH is the power fraction contributed by HP turbine being roughly 0.3. The total output mechanical power supplied by the prime mover is the
summation of the mechanical power generated by the high-pressure (HP), medium-pressure (MP) and low-pressure (LP) turbines. That is, P., = PH + PML = CH P. + PMT
(7.4)
In the case when the medium-pressure valve keeps away from control process, the power PML generated by MP and LP turbines keeps constant as
248
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS i..
PMLO during the dynamic process being under stability investigation.
Combining Eqs. (7.1a), (7.2), (7.3) and (7.4), we can write the state equations describing the steam valving control system of a synchronous machine connected to an infinite bus as PH (t) _ -
1
,
PH (t) +
0.,
TN
1
(7.5a)
P, (t)
PHO +
THg
THg
PH(t)_-
1
THg
H PH(t)+-PH(t) TH
(7.5b)
D
EQVS
[w(t) - wo ] H PH(t)+±0-P'L'H H
w(t)
0
Hx
sin 8(t)
(7.5c) (7.5d)
S(t) = w(t) - w0
s-.
0.-
where, PML° =CMLPmo represents the initial steady state mechanical power generated by MP and LP turbines, Pmo is the mechanical power of the initial steady state.
If the practical equation (5.75) is adopted, then the state equations for steam valving control of a thermo-generator set connected to an infinite bus can be written as
=_ '
PH + CH
THE
h=
C
wo PH
H
(7.6a)
Pm0 + CH u,
THE
THE
+ !0 H CML Pmo
D (w - wo ) H H
E9V'
sins
7 6b
.°.
PH
(7.6c)
8 = co - coo
where THE = TH + THg .
The above equations can be written in the general form of an affine nonlinear state equation
X(t) = f(X(t))+g(X(t))u
(7.7)
where, 0.'
THE
f(X) =
(ti
I PH+CPm0 T,
D(w-wo HOPH +OC H ML PmO - H w-wo
Hx
Vs
s in S
(7 . 8a)
249
Nonlinear Steam Valving Control
C g(x) _ [TH 0 0
11
T
(7.8b)
HE
(7.8c)
X=
[I'H
S]T
0)
(7.8d)
The state equations given by Eq. (7.6) disregard the control of the medium-pressure valve, i.e. disregarding the effect of so called "fast valving".
7.2.2
Exact Linearization Method
-'d
From the Theorem 3.2 in Chapter 3, it can be seen that the conditions for exact linearization of system (7.6) are (i) The matrix
C =[g(X) adg(X) adjg(X)] is non-singular at a neighborhood n of Xo (ii) The vector field set
.
{g(X),ad fg(X)}
is involutive in the neighborhood Q. From the Lie bracket expression (2.47), we can calculate IT
ad fg(X) =
C. T 2.
CH
adg(x) j
[T
E
LOCH
- HT.
(7.9)
0
-(woCH + HT.'
co,CH HTHE
H 2THE
Therefore, the matrix C=[g adfg addg] becomes CH
T. C=
0
0
Cf11
THz E to0CH
0
3T. CH
[)OCH
COODCH
+ woCH HTHE
So the determinant of matrix C will be
T
(7.10)
250
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Co#0
det(C) = -
,
for all X
...
The above determinant is not equal to zero for all X, so the condition for exact linearization (i) as depicted above in this section is satisfied. Since the g(X) and ad fg(X) are both constant vector fields, their Lie-
bracket [g, ad fg] is a zero vector. Of course, the zero vector field can be a
number of any vector field set, so the set {g, ad fg} is involutive. That implies that the above condition (ii) has been fulfilled. According to the above analysis we know that the nonlinear equations
for the steam valving control can be exactly linearized in a large range. According to the algorithm given in Section 3.2.4 we can transform the system (7.6) into a controllable linear system, which will involve five main steps:
Step 1.
For system (7.6), the following three vector fields can be
composed: D, ={g(X)}
D2 ={g(X),ad fg(X)}
D3= (g(X), adfg(X), adf2g(X))
where the adfg(X) and ad fg(x) are given by Eqs. (7.9) and (7.10).
t71
Step 2. Since the g(X) fields, we can choose D, =[I 0 Of a D,
,
ad fg(X) and ad 2g(X) are constant vector Of a D2
1
D3 = [0
0 1]T a D3
171
D2 = [0
where j5, , D2 , D3 are linearly independent vector fields. Step 3. Calculate the mapping F(w1,w2,w3)=(IW,
oI
W2
o D. (X0)
(7.11)
where, X = [Pt5Co S]T, X0 = [CH Pmo we S0 ]T . From Eq. (3.61), we know
that 4V, (X0) is the solution of the following equations dPH (t) dw3
do (t))
=0
-
CH PmO
=0
X(0) =
dw3
co,
80
d.5(t) dw3
By solving them, we get PH =CHPm0
co=0)0
8=w3+50
In order to calculate D °Z o (Dw 3 (X0) , we must solve the following differential equations
Nonlinear Steam Valving Control dPH (t)
251
=0
dw2 dW2
CHPmo
.-.
do(t)
=1
X(O) =
co,
w3 +5.
da(t) = 0 dw2
Then we have 8=W3+80
o = w2 + w0
PH =CHPmo
Furthermore we calculate 0D, o cD, o 0w Xo dPH (t)
,
=1
dw, CH Pm0
'ZS
dw(t) dw, 'ZS
d8(t)
X(0) = w2 +wo
=0
W3 +80 0
dw,
Therefore, we obtain mapping F as PH=w,+CH Pmo
(7.12)
CO = w2 + CO'
S=w3+80 The inverse mapping F-' of F is W, = PH -CHPmo (7.13)
w2 = CO - w°
W3 = 8 -C10
Step 4. To calculate the derived mapping F.-' (f) . From (7.13) the Jacobian matrix of F-', JF_, is an identity matrix. Therefore, f (0) (W) = F.-' (f )IX=F(W) = f(X)IX=F(W)
--W, 1
333
THE
.f,(°) (W) .f2(°) (W)
f'(0) (W)
D
w0
H
wt +
0 Pmo -
H
H
w2 W2
Let us compute the transformation R,
690
-E-V,
H x'
sin(w3 +80)
(7.14)
252
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Zi1) =f(0)(W) = Lo w, +±0 Pmo
H
H
E°V,
-D w2 - W0 H H x,
sin(w3 +8°)
Z2') = f-3(0 (W) = w2
(7.15)
(1)
Z3 = w3
The Jacobian matrix of R, should be calculated in order to derive the transformation Ri_1 = R2 :
ale
H
s , cos(w3 +'50)
H
H
(7.16)
JR = 0
1
0
0
0
1
Therefore the f(')(W) can be determined as fl(l) (W)
f(l)(W)=JR f(0)(W)=
f2(1) (W)
.f3(')(W)
H
f,(°)(W)-H f(0)(W)-
H
xiV! cos(w3 +s°)f3(°)(W) a,
f2(0) (W) f3(0)
(W)
From the above we get the transformation R2 as follows Z(2) =f2(o)(W)
H
w, +
H
P,Oo -Dw2
H
Co. E9 Vs
H
sin(w3 +80)
(7.17)
Z22) = f3(0) (W) = w2
(2) - Z3 0) = w3
Z3
The following composite transformation T can be obtained. We know from Eq. (3.75) Z,(2)
=f (')(W) w=F- (x)
_coOP +9°C P H H
Z22) (2) Z3
H M1.
f3(1)(W)IW=F-'(X)
ie0
D(o)-(00)- 0)0 E-Vssin8 H
-H
-0O-CJ°
`w3(W)W_F_i(X) =8-80
Based on the transformation T, the following transformation can be
Nonlinear Steam Valving Control
253
made with respect to f(X) and g(X)
f(X)=JT(X)f(X)=[fi(X) f2(X) f3(X)]T HY z PH+H
HT
.-Hfz(X)-HP,
(7.19)
f2(X)
=1
Aw T
1(X) = [g, (X) 0 O]T
=LOWOC TH
0
(7.20)
0
In the Eq. (7.19), f2(X)=th=AM.M; Aw=co-r,; F, =
E9vs
cos8 Aw, so Eq.
X'E
(7.19) can be rewritten as w0 PH HT",
f (X) l f (X) _
- HD Aw - H0 Pe + O)OCH HTjjM
f2(X)
D
Aw
f3 (X)
Aw
Step 5.The final transformation Z = T(X) is obtained as (7.21a)
Z2 = f3(X)=w-wo =Am
(7.21b)
°11
z, =w3(X)=8-80 =A8
z3 = f2(X)=Aw (7.21c) The next task for us is to obtain the control law. From expression (3.82), we know the nonlinear control law has the form of f' (X) + v*
(7.22)
p02
u, (X) _
k, (X)
where, v' is the "optimal control" input of the following linear system ZI = Z2
(7.23)
a2 = z3
Z3 =v
According to the linear optimal control theory stated in Section 3.2.1, if the weighting matrix is chosen as a diagonal one, i.e. Q = diag(50, 10, 0)
R =1.0
then the optimal gains for the system (7.23) are k; = 7.07
k2 = 7.36
k3 = 3.84
254
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
As a result, the optimal solution for the linear system (7.23) is (7.24)
v' =-7.07z, -7.36z2 -3.84z3 =-7.0708-7.360co-3.84Ao)
Substituting the above expression into formula (7.22), and rewrite 08 as Jo Acodt, we can get the state feedback 7.36HT w°C"
C" 'THE
J° 't o dt-
0
-7.07
7 25 a) (,
ti. t-.
CSI
T
H
If the damping coefficient in expression (7.25a) is ignored, i.e. let D = 0, then we can obtain the control law for the steam valving control as U, = P. - PmO +
Pc
C"
- 7.36 HT y. AO) w0C"
7 25b
i,>
-7.07 HT"£ J0'Ocodt-3.84 "T' Ow r0°C"
cv°C"
Expression (7.25b) presents the nonlinear steam valving control law of the HP turbine with reheater for a generator set connected to an infinite bus. 's'
If we replace, in expression (7.25b), the equivalent time constant of the valving control system of the HP turbine T E with the equivalent time constant of the regulated valve of MP turbine T,,,E, and coincidently replace
the power fraction contributed by the HP turbine C with the one contributed by MP and LP turbines C,,,, then we can obtain the nonlinear valving control law for the MP turbine, or the so-called fast valving control law as
u2-Pm-PmO+T_ CML
CO.CML
- 3.84 - 7.07 HT- J' &codt 0
(7.26) HTML Ad) wOCML
wOCML
In light of the expressions (7.25) and (7.26), we may realize that the nonlinear steam valving control strategies u, (control for the high-pressure Sam"-"
valve) and u2 (control for the medium-pressure valve) of a generator set connected to an infinite bus possess the following properties: 4-+
(1) The nonlinear valve opening control law u, (u2 as well) in expression (7.25b) involves the terms of the speed deviation Oco, the integral of the Om and the derivative of the Aw, which are of negative 7.+
feedbacks. So when a fault occurs in the power system, the increase of the .-O
speed will cause the valve opening decrease to decelerate, which is beneficial to the stability of the power system.
255
Nonlinear Steam Valving Control
.C3
(2) The control law also involves the derivative of active power P,, which is a positive feedback, so the valve opening will be automatically
2.C
reduced when the active power decreases in case a fault occurs.
f17'
(3) The feedback coefficients are only related to the time constants of oil-servomotors, the time constants of HP, MP and LP turbines, the power fractions contributed by HP, MP and LP turbines, and the moment of inertia H. They are independent of parameters of the power network. The change of topology and parameters of power network will be sensed by the controller via the variation of speed Aw, active power AP, and mechanical power (4i
AP., etc., which can be correctly regulated by the control strategies expressed in Eqs. (7.25) and (7.26). This feature of the control law, which the parameter of the network is not explicitly included in expressions, has made the control completely adaptable to the change of the topology and parameters of the power network. According to the nonlinear steam control law given by the Eq. (7.25), we can draw the transfer function block diagram as shown in Fig. 7.1. iP,(t)=P (t) - P (to)
Tur CH U,
7.4HTHE w0CH
w(t)
HTHE 7.1
woCH S
3.SHT,,s w0CH
CH
Figure 7.1
1
1
1+THs
1+THRs
The structure diagram of transfer function for nonlinear control of steam valves
256
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
7.2.3
Physical Simulation Results of Nonlinear Valving Control in a One-Machine Infinite-Bus System
A one-machine, infinite-bus system under investigation is shown in Fig. 7.2. The parameters have been indicated in the diagram. V,
Vs
xT
Figure 7.2 A one-machine, infinite-bus system xd =0.7989, xd -0.2992, xq x.479, H=11.73s, Tdo=Ss D=1 .0, x7. =0.1134, xL, =xL 2 =0.5682, SB=300KVA
7.2.3.1 The effect on improving transient stability
chi
vii
According to the system shown in Fig. 7.2, a physical dynamic simulation has been performed under the conditions of a temporary three-phase short circuit fault and two-phase short circuit fault respectively (The fault occurred at 0 second, the line was tripped at 0.15 second and re-closed successfully at 0.75 second). Tab. 7.1 lists the test results for transient stability limits under various fault cases by adopting nonlinear steam valving control and conventional control laws. We can see from the table that if the +O+
nonlinear control strategy is adopted, the transient stability limit will +C+
increase by 21% comparing with the conventional control. Table 7.1
Dynamic simulation results under temporary short circuit fault
Fault type
Control scheme
Maximum power
Increased by %
Three-phase temporary fault
Conventional
1.099
0.0
Nonlinear
1.33
21.6
Two-phase temporary fault
Conventional
1.3
0.0
Nonlinear
1.58
21.5
Fig. 7.3 shows the test results of nonlinear controller on improving the transient stability limit under three-phase fault for the system given in Fig. 7.2. As shown in the figure, the system collapses soon after the fault occurs
at the point K when the active power output P., achieves 1.0, if the generator -is equipped with conventional governor. However the system
Nonlinear Steam Valving Control
257
180 .N+
120
vV
60
0
1.04
°z 1.0
0.98
2.0 1.30
0.0 2.0 1.3 ci. 4E
0.0
1
I
0.0
I
1.0
1
2.0'
I
I
1
3.0
4.0
5.0
Time (s)
Figure 7.3 Physical simulation results for improving transient stability by using nonlinear steam valving control under temporary faults Curve 1-with conventional governor Curve 2-with nonlinear controller
((D
C'1
`01
remains stable even when the active power Pro reaches 1.3 under the same fault case as the nonlinear steam valving control is applied to the governor (see curve 2 in Fig. 7.3). Moreover, the dynamic performances, the lasting time of transient process and the number of times of oscillation have also been significantly improved.
7.2.3.2. The effect on improving the transient stability under permanent fault A transmission system shown in Fig. 7.2 is used as an example to test the effect of the nonlinear valving control developed in this chapter for improving the transient stability. The permanent faults (three-phase short circuit and two-phase short circuit) occur at the point K at 0 second, the line will be tripped at 0.15 second and re-closed at 0.75 second (the faults still exist) and tripped again at 0.9 second. The transient stability limits under various fault cases are listed in Tab. 7.2. It can be seen from the table that
258
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
nonlinear steam valve control can effectively improve the stability limit under three-phase and two-phase permanent faults. Table 7.2
Dynamic simulation result under permanent faults
Fault type
Control scheme
Power limit
Increased by %
Conventional
0.655
0.0
Nonlinear
0.929
41.8
Two-phase Permanent fault
Conventional
1.072
0.0
Nonlinear
1.257
17.25
z
Three-phase
Permanent fault
40)
The physical simulation results for the dynamic responses of the system under three-phase fault are shown in Fig. 7.4. As shown in this figure, when the fault occurs, the electric power' output P, drops immediately. In the meantime, the mechanical power input P. is reduced. And the governor with the nonlinear control function can automatically limit the opening of the 00)
a.+
180 120
47 0
1.04
ci
1.0
0.98
2.0
°'
ti
P,,=0.93
0.93
2.0
Pmu 0.93
0.93
I
4
0
1.0
f
I
2.0 3.0 Time (s)
1
I
4.0
5.0
.N-.
Figure 7.4 Physical simulation results for improving the transient stability by using steam valuing nonlinear control under permanent faults Curve 1-with conventional governor Curve 2-with nonlinear controller
259
Nonlinear Steam Valving Control
valve to reduce the post-fault mechanical power lower than its pre-fault C3.
value Pmo , hence preventing the power system from the loss of the dynamic stability of a post-permanent fault. Moreover the dynamic performances of the system are improved radically. `CS
Digital Simulation Results of Nonlinear Steam Valving Control in a One-Machine Infinite-Bus System
7.2.4
Fig. 7.5 shows the computer simulations carried out to compare the effect between the nonlinear valve control and the linear optimal valve control.
150.0
Angle
100.0
50.0
0,0 i 0.0
I
I
1
1.0
2.0
3.0
1
1
4.0
5.0
4.0
5.0
Time (s)
Mechanical Power (p.u.)
2.0
1:5
1.0
0.5
1.0
2.0
3.0
Time (s) Computer simulation results of nonlinear steam valving control Curve 1-with linear optimal steam valving control Curve 2-with nonlinear steam valving control 0
Figure 7.5
260
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
The results are shown in Fig. 7.5 with a temporary three-phase fault occurred on point K (the pose-fault switching operation sequence is the same .ti
as assumed in Section 7.2.3). The swing curves of rotor angle and gyp
mechanical power are exhibited in the figure for comparing the effect of the linear optimal steam valving control with the nonlinear control. As shown in the figure, when the linear optimal governor is used, the regulated valve
opening increase too fast at the moment of fault clearing, such that the system loses the stability. When the nonlinear valving control is adopted, the
regulated valve acts properly such that stability is maintained with less
ivy
:='
COD
oscillation and shorter transient process time, which manifests that the latter is far superior to the former. Fig. 7.6 shows the digital simulation results of the dynamic response of the rotor angle S(t) and mechanical power P, (t) under three-phase permanent fault. It can be seen from the results that when linear optimal steam valving control is used, it can not keep the stability of the system. As the nonlinear steam valving control is applied, it can automatically limit the
Angle
150.0
0.01
0.0
Mechanical Power (p.u.)
'"
I
I
2.0 3.0 Time (s)
4.0
L
2.0
I
1.0
1.5
1.0
0.5
1
1.0
I
1
3.0 2.0 Time (s)
1
4.0
1
5.0
Figure 7.6 Computer simulation results with permanent fault Curve 1-with conventional governor Curve 2-with nonlinear governor
261
Nonlinear Steam Valving Control
valve opening so as to maintain the dynamic stability after a permanent fault,
and to ensure the maximum transmission power (the limit of transmission power). Comparing Fig. 7.4 with Fig. 7.6, it can be seen that the physical simulation results coincide with the computer simulation results.
NONLINEAR STEAM VALVING CONTROL IN A MULTI-MACHINE SYSTEM
7.3
In this section, the theory and design method of decentralized control on the steam valving control in a multi-machine power system will be discussed based on the nonlinear control theory of MIMO systems explored in Chapter
4. An example of multi-machine system simulation results will be used to illustrate the effect of the nonlinear control law to improve power system stability and the post-fault dynamic performances.
7.3.1
Mathematical Model
Consider an n-machine system. Based on the rotor motion equations
and the generator power output equations in a multi-machine system described in Sections 5.2 and 5.4, we can obtain the swing equations of the i"' generator S; = w; - w0
wi =
w0
H;
PH +
(7.27a) w0
CM P.Oi
D, H;
H;
- W, (E,; G;; +E,,EvB, Hi
(CO
0
(7.27b)
j=1 jmi
where, 8, is the power angle between the q-axis electrical potential vector Eqi and a reference bus voltage vector Vu,. in the system, in rad; w; is the speed of the i'h generator, in rad/s; P,,, is the mechanical power of HP turbine, in per unit; E,, and E, are the q-axis internal transient electric potential of the 1" and j'h generator, respectively, in per unit; 8;, = 8; - 8j, in rad; PmO; is the initial mechanical power of the i'h generator, in per unit; Hi, C,,, are moment of inertia in second and the power fraction of HP turbine, respectively; G;; and B j are self-conductance of the i'h bus and the mutual-conductance between the i'h and the j' bus respectively. If only the HP controlled valve is considered without consideration of
262
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
the fast valving control, according to Eq. (5.75) the dynamic equation of
PH, = 1 PH, + CH
'ti
steam valving control system is Pm0i
+L_ THE
THE
C'1
THE,
p (7.28)
PH,
where THE, = THa. +TH, is the equivalent time constant of HP turbine, THg, the
t].
CAD
COD
time constant of oil-servomotor of regulated valve of HP turbine, TH the time constant of HP turbine, uH, the electrical control signal from the controller for the regulated valve.
Combining Eq. (7.27) with Eq.(7.28), we get the state equation of the CAD
steam valving control system for n-machine system in the form as
X=f(X)+g1(X)u1 +.. +gn(X)uH.
(7.29a)
which is an affine nonlinear system, where X =1PH
PH... PH^ ; w1 , w2, ..., w, ; '51,
'521
..., 5]?
(7.29b)
CH
PH, +
THE
!
Pm of
THE,
CH
1
THE
PH. +
m
Pm0n
f(X) =
(7.29c)
...
wn - wo
L
0
CH
THE,
0
0 0 0
0
0
0 0
F01
CH=
THE,
g1(X) _
j
CH. THE,
0 f
g2(X)
0
g n (X) _
0
I
0 0
0
(7.29d)
Nonlinear Steam Valving Control
263
and
= H PN, + -WO
H.
CML, Pmoi
- Hi (wi - wo )
H (E' G;;+E;>EvBjsin(8,-Sj) j=,
jxi
In (7.29c) we assume E9, , , EQ are constants. Eq. (7.29a) can be written in the compact form as
X = f(X)+
(7.30)
g,(X)uH,
Now we have established the mathematical model of a nonlinear steam valving control system in an n-machine system. The principle and design method of exact linearization will be discussed in the next section.
7.3.2
Exact Linearization Method
First of all, let us examine that whether the affine nonlinear system given in Eq. (7.29) satisfies the conditions for exact linearization. As stated in Section 4.4.2, for the system in Eq. (7.29), we choose the N indexes. They are N
3
1=,
;=,
n,=n, n2=n, n3=n, so N=3, and >n;=Y_n;=3n, 1-1
where n is the number of generators in a power system. Based on the 'C3
Theorem 4.1 presented in Chapter 4, the conditions for linearization are (i) The following 3n vector fields
g,(X), ...,
adfg,(X), ...,
adfg,(X), ..., adfgn(X) (7.31)
are linearly independent. (ii) Every one of the following 3n vector fields sets D, = {g, (X)}
D = {g, (X), g 2 (X), ..., g (X)} {D,,, ad fg, (X)}
(7.32)
D2
ad fg,(X),adfg2(X), z
={DZ,,,adfg,(X)} D3n
={D2i,,adZtg,(X),ad
264
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
is involutive near X0. In order to verify whether the above two conditions are satisfied, the Lie bracket for system (7.29) will be calculated. They are r
tea.
adfgi(X)=LTH,,0,...,0;_
1T
TH.,0,0,...,0;0,...,o
T
adlg" (X) = 0, ..., 0, CH,
Tl
;
0, ..., 0, _ OCH ; 0, ..., 0
(7.33)
H,,THE T
dz (X) = alg,
CH' ;
0, ...
adz g"( X)=
p ...
p
0; _ ( w0CH, z
H,T,E,
T,
+
aOD,CH, z
H, T Z
WOCH, ), 01 ...10
0,...,
H
01
T
f
CH.. 0, ..., 0, _ a OCH + (
COOD"CH
T,3
); 0,...,0,
a'OCH-
From Eqs. (7.29d) and (7.33), the determinant of matrix C is
Cs z det(C) = (-1)"r1
2;
0, for all X
+_1 H. THE,
The above formula implies that the 3n vector fields of (7.31) are Q..
linearly independent. So the condition (i) is satisfied. Moreover, the 3n vector fields expressed in Eqs. (7.29d) and (7.33), adjg..... adfg", ad 2g , ad f2 g. are all constant vector fields, the Lie-bracket of any two of them is a zero vector which can be the member of any set. So the set of the
mot`
3n vector fields is involutive. That implies the condition (ii) is satisfied. It is
concluded that the nonlinear valving control system for a multi-machine power system (7.29) can be exactly linearized to a controllable linear system Brunovsky normal form.
Subsequently, let us find the nonlinear valve control strategy for a multi-machine system according to the algorithm given in Section 4.4.2.
Step 1. For the system given in (7.29), N = 3 index numbers have been chosen, n, = nz = n, = n , where n is the number of machines, so is the
number of the dimension of the control vector U = [uH, uH2
uH ]T . The
Lie-brackets of each g,(X),..,g"(X) along f(X) from the first order through
(N-1)"' order have been calculated in (7.33). The 3n sets of the vector fields D, , , D3,, have been built up. Step 2. such that
We choose 3n linearly independent vector fields,
Nonlinear Steam Valving Control
265
D, _ [1, 0, 0; 0 D2 = [0, 1, 0,
0; 0
0] T E D,
..., 0;
0, , Of E D2
Dan = [0, ..., 0; 0,
0; 0, ...,
, 0; 0,
....
(7.34)
0, 1] T E Dan
Step 3. We calculate the mapping F(w1,w2,' ,w3")=(D°'
.(DD,,
o...oCj)D'"(X0)
where, X0 =[CH, Pm01, ..,CH.PmOn; (00,-400; 810, "',9,01T. Similarly to the Step 3 of Section 7.2.2, the mapping F is PH =wI+CHIPm01
+
PH = W. + CH. P.O. ()l = Wn+1 + COO
III
(7.35) co,, = w2n + 0')0 81 = w2n+l +1510
15n = w3n + 15"0
The inverse mapping of F is
F-'
.
wl = PH - CH, Pmo1 0.@
Wn = PH. - CH,, Pmon
Wn+1 =[O1(00
(7.36) W2. = 0)n - 0)0 W2n+l = 151 - 810
w3n = 8n - 8n0
Step 4. Calculate the derived mapping of f(X) under F-', F.-'. As know from Eq. (7.36), the Jacobian matrix JF_, = 7F-' /oX is an identity matrix, thus F.-'(f) = JF-'f(X)IX=F(W) Set
f(X)IX=F(W)
266
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
f (X)
...
fl(0) (W)
4(°)(W) fn( (W)
fn (X) fn+1 (X)
(7.37)
f(0)(W)= f"(.0' (W)
(W)
f2n(X) f2n+1 (X)
A. (X)
X=F(W ) 'fl
, f3 (X) have been given by Eq. (7.29c). Then we look for transformations RI and R2 In the Eq. (7.37),
f,, (X),
RI (1)
ZI
ro)
- fn+1 (W)
=(
(cot
H,
PH, + H, C., Pm01 - H'
- a)o )
N]=
-H (E']Gll+E9,E,B,)sin8,)))Ix=F(w) HI j=2
Z(0)
Hn PH.+H C.. P.M
Nn
(art-Co.) (7.38)
n-1
0 (E Hn Zn+)1
E,Bn)sin 8,y))
x=F(w)
)=1
= f2(n+, (W)=(0)l - W0)Ix=F(W)
Z(n1) =Jan°)(W)=(Con -CVO)Ix=p(w) (1)
_
Z2n+l = w2n+1 = (8 - Slo) x=F(w) (1) _ _ Z3,, - Wan - (sn - Sn0) x=F(W)
cps
where s;; = s; - s; . In light of Eq. (7.38), the f())(W) can be calculated
f°'(W) = JR (W)f(0)(W)
f(1)(W) =
f3(W)
(7.39)
Nonlinear Steam Valving Control
0)° HI coo
Hn
267
+i (X) -
HI
0 E'1 Ev BI; cos 81;./zn+l (X) HI j=2
o°
L(0)(X)- D"
Hn
Hn
%=,
f-) (X) A. M W1 -(0)°
COn - O)0
L
JX=F(W)
From f°)(W), we obtain the transformation R2 42)
H,
Zn2)
H-
P, + NI
CO,
HI
(w, -coo)
(E9, G + Ey, Y-Ev BI; sin SI; ))I x=F(w) J=z
= A (n') (W ) (
Hn P". +
Hn
C. P.. Hn (0n -
- jH° (E"GM + EQn
n-I i=1
n
WO)
Ew B,y sin 8,, ))I x=F(w)
(7.40)
Zn+l = f,(.,+) 1(W = (0) 1 -' o)IX-F(W)
Z2(n) = f ,(n') (W) = (a ,, - (AO )IX=F(W ) (2)
Z2n+I = w2n+1 = (SI - 810) x=F(W)
(2)
Z3n -w3n=(Sn-Sn0)x=F(W)
From R2, the composite transformation T can be derived as
(n Tj(X)=ZI(2) =!'r+l(W)W=F-'(X) =
0)0
H,
PH +
H,
'
0
H,
PMLO
D' (CO, - (00 ) HI
(EgiG,,+E',Y_EvB,;sins,;) j=2
(7.41)
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
268
T (X)
W=F'(X)
Pnno -H
PHq+
n-1
0),
- Hn
j=1
=w, -CVo T,+1(X)=(W)W=F-,(X) f2('+'.
T2n (X) = 3n) (W)
W=F-,(X) = (0n -W0
=8, -8io
T2.+[ (X) =
T3n (X) = wan I W=F-'(X) = 8n -15,10
The Jacobian matrix can be derived from (7.41) as H, 0
o
0
o
...
°L0
D,
p
w° ape,
w° ape,
H, a8,
H, a8n
0 ...
p
p
D.
- w open
Hn
Hn a8,
I
H,
H.
_
Co. aPen
Hn a8n
.............. ........ ........ .............. ................................... ......... ........ ..... .... ................ ...... .......... .......................
0
...
0
00
...
1
1
Jr(X)= Onxn
(7.42)
0.-
............... ......................o.............. .......................................... .............. ......................... ..................... 1
o
...
1
0
0
...
1
Onxn
Onzn
Now, we should take the transformation of f(X) and g(X) , which will be !(X) = Jr (X)f (X)
[.f (X) ... f, (X) f,+(X) ... 7 2,, (X) K
_ [7 i
r ... b
...
0),
[p
wl - (00
(X) ... .fan (X )l ...
r C)n - Cvo l
where w0
H;Tn
PH -
D,
Hi
wi - !0 ( ape, (0), - COO +
0.I
woCH
as,,
H;T,
...+-(w,, -CUO))+ Pei
Hi as,
ape
Ev Bj sin CSj
E"Gjj + Eqi j=1
jxi
PmOi
i=1,2,.-,n
r
(7.43)
Nonlinear Steam Valving Control
269 O)o C"
I
0
0
...
H,T,,E
0
g(X) = JT(X)g(X) =
n lines
(0OCH.
......................................................................
(7.44)
0
0
n lines
.
0 0 ............................................................. 0
0
n lines 0
0
,DC
...
Step 5. The final coordinate transformation is obtained as follows Zl =
(X) = 8l - 8l0 pmt
Zn = wa,, (X) = (5n - (5nO
Zn+l = J 2n+1 (X) = 0)l - 0)0
(7.45) Z2» =f3n(X)=Con -a0 Z2n+1 =L+l (X) =")1
Q"'
Z3n = f2n (X) = d)n
Comparing Eqs. (7.43) and (7.44) with Eqs. (4.169) and (4.170), we know that
AN an(X)j
,V2
f2(X)
a(X) =
J,(X)
(O PH-D'Aw,tr)o
H,T.
PH
wOCHi
(7.46) CH,
H,
Hl 88,
Hn
Hn
as,,
H1TH1
P.
DnA(wn-rHO(----Aal+...+BP-Lmn)+woCH.Pmon
0
...
88,
88n
H,THE
0
H1THE
b(X) =
0
wOCH,
0
H2THi ...
0
0
(OCH.
(7.47)
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
270
The inverse matrix of b(X) is
H,T,, 0
0
wOCH
0
H2THL,
b-' (X) _
0
(7.48)
wOCH2
HHTnz.
wOCH, J
As known from Eq.(4.173), the control vector U(X) = -b-'a + b-'V' is THE
u H, uH2
CH'
_
(Pm2
oOCH' w0CH2
UH,,
v1
oOCH'
-Pmo2)+TRZ' Pa +T 2D2 Awe + CH,
,
Ato1 +
+
U(X)
H1T
THE D1
(Pm1 -Pm01)+=Pe1 +
H2THE2
v2
a 0CH2
(7.49)
THE, Dn
Awn + (Pmn - Pmon) + T HE" Pan + wOC, N CH. wOCH
HnTHE_ vn
aP"
where, PU = aP`; Aw, + aP" A0)2+---+ Awn; Am = w, - wo . as, as2 as,
v, v,,
, v,,
are the quadratic optimal "control" (an input) of the linear
system (7.50)
i=1,2, ,n v, = -(k; z, +k;,+iZn+i +k2n+iZ2n+i)
i
(7.51)
where, k, , k,,+i and k2n+i are the optimal "control" gains. As discussed in Section 7.2.1, for the linear system given in (7.50), if the weighting matrix is chosen as Q = diag(50, 10, 0)
R =1.0
Then the optimal "control" gains are k, = 7.07
7.36
k2n+i = 3.84
Substituting Eq. (7.51) into Eq. (7.49) and considering Eq. (7.45) and rewriting AS, as Jo Awi dt , we acquire the control law for the HP turbine
Nonlinear Steam Valving Control
271
valve control of the i' turbo-generator set in a multi-machine system as
T'
7.36H;T, w0CH
(7.52a)
CH'
-7.07
H,THE'
o.0CH
J'4co'.dt- H1THS' (3.84- D' )erar p Hi 0o0CH
If we let D; = 0 in the above formula, we have THE,
uH, =(Pmi -Pma)+Pe! -
7.36H;T Ow; H'THE'
r
wOCH
(7.52b)
w0CH'
CH
-7.07 H,THE, J'Ow,dt p
3.840w
r
w0CH
ti.
Following the derivation steps for obtaining the expression (7.52), we can write the control law for the fast valving control of a steam turbine with reheat of i"' generator set in a multi-machine system as
T,
7.36H;T"
UMi =(Pmi - PmOi)+CPei AII.
-7.07
H,TMS' wOCML,
w0
C
Qwi ML
H,T &' J'Ew.dtp r
3.840cbi
OOCML,
Effects of Nonlinear Steam Valving Control in a Multi-Machine System
7.3.3
(CD
In order to compare the effects of different control strategies in a multimachine power system, the conventional governor, linear optimal governor and nonlinear governor are adopted respectively by No.2 generator set in a 6-machine system shown in Fig. 6.10, while other generators are equipped with conventional governors. Table 7.3
Critical clearing time for different control strategies Control strategy
64-
Critical clearing time
Improved by %
0.198
0.00
0.206
4.04
0.220
11.1
Conventional governor
for No.2 generator r".
Linear optimal valving control for No.2 enerator
Nonlinear valving control for No. 2 generator
272
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
A permanent three-phase fault is assumed to occur on the bus 11 (see Fig. 6.10) for the large disturbance test. The fault occurs at 0 second lasting for a short duration, the 500KV transmission line between bus 11 and bus 12 +.+
.._.10 p..
is tripped and not reclosed again. The critical clearing time for different control strategies are indicated in Tab. 7.3. It is shown in Fig. 7.7 the
dynamic responses of rotor angles and mechanical powers of all the generators in the 6-machine system, when all the generators are equipped with conventional governors. We can see from the figure, the entire system loses stability in very short period of time.
Fig. 7.8 exhibits the dynamic response curves of the system in which
the No.2 generator set is equipped with linear optimal steam valving -CDs
controller and the conventional governors are applied to the other generators. 160.0
Angle
80.0
0.0
-80.0
Power (p.u.)
-160.0
6.0
4.0
2.0
0.0 i
i
I
I
I
I
0.0
1.0
2.0
3.0
4.0
5.0
F"'
Time (s)
Figure 7.7 Dynamic responses under three-phase fault No.2, 3, 4, 5 generators are equipped with conventional governors. Fault occurres
at 0 second, the line between bus 11 and bus 12 is tripped at 0.20 second, SB=100MVA
Nonlinear Steam Valving Control
273
80.0
0.0
-80.0
-160.0
Mechanical Power
6.0
4.0
2.0
0.0 I
I
I
I
I
I
0.0
1.0
2.0
3.0
4.0
5.0
Time (s)
Figure 7.8 Dynamic response curves under three-phase fault No.2 generator is equipped with linear optimal governor; the others are equipped with conventional governors. Fault occurred at 0 second, the line between bus 11 and bus 12 is tripped at 0.21 second, SB 100MVA
r-:
,..
.N.
Although the linear optimal valving control is adopted, the system is still out of synchronism. The effect of nonlinear steam valving control is shown in Fig. 7.9. The nonlinear control is used on the No.2 generator, and the other generator sets are still equipped with conventional governors. The entire system remains stable with the help of the new control strategy. "C7
7.4
DISCUSSION ON SOME ISSUES p.'
Comparing the Eq. (7.26) with Eq. (7.53), we realize that the expressions of control strategies for one-machine infinite-bus system and for 'C3
fir"
274
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
Angle
80.0
- 160.0
Mechanical Power
6.0
4.0
2.0
f
P.3
P.4
0.0 I
I
I
I
0.0
1.0
2.0
3.0
1
4.0
5.0
Time (s)
`D.
1'+
Figure 7.9 Dynamic responses under three-phase fault No.2 generator is equipped with nonlinear governor; the others are equipped with conventional governors. Fault occurred at 0 second, the line between bus 11 and bus 12 is cut off at 0.21 second, SB I OOMVA
multi-machine system are the same.
Similar to the nonlinear excitation control law, the nonlinear steam valving control law of u, for the i' generator in a multi-machine system relates only to the local state variables and local output variables such as the mechanical power Pm, , the active power P, and the rotor speed w,. The control law is independent of the state or output variables of other generators. So, the requirement of decentralized control has been achieved. The control law is independent of the parameters of power network. In other words, the control variable relates only to local parameters such as inertia constant H;, time constants of oil-servomotor and HP turbine and T,,, , and the fraction power of HP turbine c,,, . It does not involve the parameters of power networks, which makes this type of control strategy automatically
275 "'A
Nonlinear Steam Calving Control
adaptive to the network parameters and change of network configurations.
The results, including both physical simulations and digital one for a one-machine, infinite-bus system and the digital simulation
for a
O's
multimachine system mentioned above, show that the nonlinear steam valving control technology can improve the stability of a power system ISA
-"Al
°p'
remarkably. Moreover, when a permanent fault occurs, the regulated steam
cA-
valve opening can be automatically limited to maintain the post-fault dynamic stability of the power system. ."r
L."
In addition, by applying the nonlinear steam valving control, the generator which loses its stability can be pulled into synchronism r'3
'C7
automatically in certain short time (several seconds). This result is obtained by physical dynamic simulation experiments. The power system for testing is shown in Fig. 7.2, and the physical simulation results are illustrated in Fig. 7.10. From it we can see that as the power system suffers from a severe fault and loses stability, the steam valve closes as fast as possible to reduce the mechanical power P. (see Fig. 7.10), about 3 seconds later the generator is pulled into synchronism and stable operation is resumed.
0.0
cep
0.e'
coo
..4.i
The nonlinear steam valving control technology developed in this
r-,
b tlz
G.-
-0. 92
1.0
-1.0
0
1.0
2.0
3.0
4.0
5.0
Time (s)
Figure 7.10 Physical testing results of re-synchronizing by using the nonlinear steam valve control
276
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
Chapter can not only improve both dynamic and transient stability of power systems, but also can resynchronize the generators. This kind of functions and effects can not be achieved by other types of linear control strategies.
1.
.--.
E. F. Church, Steam Turbines, McGraw Hill, 1950.
2.
H. T. Akers, J. D. Dickinson and J. W. Skooglund,
4.
"Operation and Protection of Large Steam Turbine Generators under Abnormal Conditions", IEEE PAS, Vol. 87, pp. 1180-1199, April, 1968. IEEE Committee Report, "Dynamic Models for Steam and Hydro Turbines in Power System Studies", IEEE PAS, Vol. 92. pp.1904-1915, Nov./Dec. 1973.
IEEE Working Group Report, "Dynaipic Models for Fossil Fueled Steam Units in
5'0°0 Sao
f+1
3.
>0.
REFERENCES
7.5
V'1
(]w
Power System Studies", IEEE PWRS, Vol. 6, No. 2, pp. 753-761, May. 1991. 5.
IEEE Working Group Report, "Hydraulic Turbine and Turbine Control Models for
C"'
System Dynamic Studies", IEEE PWRS, Vol.7, No. 1, pp. 167-179, Feb. 1992.
M. S. Baldwin and D. P. McFadden, "Power Systems Performance as Affected by Turbine-generator Control Response during Frequency Disturbance", IEEE PAS, '-1
6.
.-.
Vol.100. pp. 2468-2494, May, 1981. l-:
7.
oo)
P. Kundur, D. C. Lee and J. P. Bayne, "Impact of Turbine Generator Overspeed Controls on Unit Performance under System Disturbance Condition", IEEE PAS, Vol.104. pp. 1262-1267, June 1981. T. D. Younkins and L. H. Johnson, "Steam Turbine Overspeed Control and Behavior During System Disturbance", IEEE PWRS, Vol. 6, No. 2, pp. 753-761, 1991. BOO
8.
Chapter 8
Nonlinear Control of HVDC Systems
INTRODUCTION
8.1
.::
A DC Electric Power Transmission System, with its inherent advantages, such as fast regulation, flexible operation, high-power transmission, and the ability to rapidly regulate active power flow, has been widely applied to highpower transmission over long distance, regional power systems inter- connection,
.J.""
?'.
(~D
.-.
and undersea electric power transmission. So far, dozens of DC transmission lines have been put into operation in the world. As of 2000 in China, there were two main DC transmission lines in operation, which are the TianshengqiaoGuanzhou and the Gezhouba-Shanghai ±500-KV HVDC transmission lines. It is expected that more AC/DC power systems will be developed along with the s..
°o.
power system expansion in China. [z.
For many years, many researches have been done to improve the mar,
performance and stability of a AC/DC parallel system by taking advantage of the fast regulation capability of a DC transmission system, which has ..t
r."
become the direction of the research on AC/DC systems [1-9, 11-15]. This chapter starts with a brief review of the control principle of the DC transmission system and some conventional control methods. It then discusses how to apply the nonlinear control approach explained in Chapter 3 and Chapter 4 to DC converter station control [16]. Finally the chapter concludes by showing some relevant digital simulation results.
278
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
8.2
CHARACTERISTICS AND CONVENTIONAL CONTROL OF CONVERTER STATIONS
8.2.1
Voltage-Current Characteristics on Rectifier Side
According to the mathematical description in the section 5.7 and formula (5.79), the DC voltage on rectifier side (at the sending end) can be written as Vdr =
N
n
Va,(t)cosa- 3x,1,
(8.1)
yr
where, v is DC voltage of a rectifier, Vim, the AC (line-to-line) voltage of
one
the rectifier, I, the DC current, x, the leakage reactance of the converter transformer in the rectifier station, a the firing angle. When the current in DC line equals zero, the DC voltage equals 3 NF2
Vdro =
V,,(t)cosa
(8.2)
Jr
From Eq. (8.1) we know that as the DC current Idr increases, VV decreases. The magnitude of the decrease is proportional to both Idr and x, . Therefore, the DC voltage Vd, has a linear relation with the DC current I,,
if the firing angle a remains constant. But when a varies, the relation between DC voltage and DC current is shown in Fig. 8.1.
Rectifier voltage
'PA
Inverter current Id,
Figure 8.1
Vd,-Id, characteristic of rectifier with various firing angles
279
Nonlinear Control of HVDC Systems
8.2.2
Voltage-Current Characteristics on Inverter Side
According to formula (5.82), the relation between DC voltage and advance angle 8 on the inverter side (at the receiving end) is Vd, =
3,5 Va,(t)cos,6+ 3 x,ld, 9
(8.3)
)r
v'.
where, Vd, is the DC voltage of the inverter; Id; the DC current; V., the AC voltage (line-to-line) of the inverter. Similarly, when DC current equals zero, the DC voltage of the inverter is = 3,12 VthO
a
Va, (t)cos fl
(8.4)
Eq. (8.3) shows that when DC current Id, increases, the DC voltage Vd, will increase proportionally if advance angle /3 remains constant. The voltage-current characteristics with different advance angles 6 are shown in Fig. 8.2.
!3z
tai
Inverter voltage Vd,
constant
Q3
161
Inverter current Id,
Figure 8.2
Vd,-Id, characteristic of rectifier with various firing angles
From Eq. (5.84), if the DC voltage Vd, is expressed by using the extinction angle y rather than fl, Vd, can also be written as 3NF2 V.(t)cosy-3x,le, n if
The corresponding voltage-current characteristic is shown in Fig. 8.3.
(8.5)
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Inverter voltage
280
Inverter current Id;
Figure 8.3
8.2.3
Vd; Id; characteristic of inverter with various extinction angles
Conventional Control with Constant DC Current at Rectifier and Constant Extinction Angle at Inverter
--h
GHQ
From formulae (8.1) and (8.3), we can obtain various voltage-current characteristics of the converters by modulating the firing angle a of the rectifier and the extinction angle y of the inverter. However, the control modes with constant DC current of the rectifier and constant extinction angle
of the inverter are the most widely adopted in normal operating state. Constant DC current of the rectifier I, is achieved by modulating the firing
.0+,
angle a. Extinction angle of the inverter is maintained no less than the minimal extinction angle ym; by regulating the advance angle /3. The control characteristics of the rectifier and the inverter with such modes of operation are shown in Fig. 8.4. In Fig. 8.4, the point x, is the normal operating point with constant DC
current of the rectifier and constant extinction angle of the inverter. In this operating mode, the DC current reference of the rectifier Id,, is required to be larger than that at the inverter Ids . The difference EI, =Idn -I,, is called the current margin, which should be maintained at (10-15%)1,,,,. At the same time, when operating at the point x, , the characteristic of the constant current of the rectifier determines the operating current of the DC system and the characteristic of the constant extinction angle of the inverter determines the operating voltage of the DC system. As the AC voltage V., on the rectifier side declines to Va, , or the AC voltage V., on the inverter
Nonlinear Control of HVDC Systems
ate'
voltage
281
Figure 8.4
vii
DC current Id
Converter controller characteristic with constant current and constant extinction angle Solid line-Rectifier controller characteristic with constant DC current Dashed line-Inverter controller characteristic with constant extinction angle
side goes up to Vim., the operating point moves from x, to x2 (see Fig. 8.4), consequently, the operating mode of the rectifier has to change from constant
DC current into constant firing angle for minimum GYin, meanwhile the operating mode of the inverter must be changed into constant DC current mode. In such circumstance, the current of the DC system is determined by the constant current characteristic of the inverter, and the voltage of the DC system is determined by the constant firing angle characteristic of the rectifier. The conventional control with constant DC current and constant extinction angle is shown in Fig. 8.5(a), (b).
ka(1+TQ,s)
kfi
1+TT5
1+Tes
(a)
(b)
Figure 8.5 Converter controllers with constant current and constant extinction angle (b) Constant extinction angle at inverter (a) Constant current at rectifier
282
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Conventional Control with Constant DC Current at Rectifier and Constant DC Voltage at Inverter
8.2.4
vac
coo
Because the commutation reactance of the converter is relatively high, a large amount of reactive power is consumed during the converter's operation that can be expressed by the following equations 3,12-
Ir
_ 3vf2 rr
Q'
V
I2p+sin2a-sin2(a+p)
Jr
Vale
( 8 .6)
4(cos a - cos(a + p))
2p+sin2y-sin2Q 1.N
Q, -
(8.7)
4(cos Y - cos ,6)
where Qr is the reactive power consumed by rectifier and Q, by inverter.
in.
P+.
coo
In the two equations above, p is the commutation angle of the converter (see Fig. 5.15 and Fig. 5.18). It can be easily found that the reactive power is injected from the AC side to the DC side at both the rectifier and the inverter. Therefore, adequate var compensation is needed at
both the rectifier and the inverter. The reactive power consumed by the converter varies along with the different operating modes. It decreases
Sao
greatly when the converter stations are operating with light load. Thus, if the poi
reactive power supplied by the compensators can not be automatically adjusted according to the variation of the operating mode, there will be "l7
c°°
CAD
Sao
F+,
redundant reactive power that will be injected into the AC system and drive the voltage on the AC side of the converter stations up. On the contrary, it may cause the voltage on the AC side to fall down dramatically and even threaten the voltage stability of the AC system. According to the formulae (8.6) and (8.7), the stability of the AC voltage at converter stations can be improved by adjusting the firing angle a of the rectifier and the advance angle Q of inverter, because the reactive power consumed by the converter stations can be regulated rapidly and to a large extent by modulating the two angles. Consequently, the contradictory between reactive powercompensation and voltage stability maintenance could be solved. The control
with constant DC current of the rectifier and constant DC voltage of the inverter is such a regulating solution to the problem. The characteristics of these controllers are shown in Fig. 8.6. L'3°
(°i
CAD
The regulating process of the constant voltage adjustment will be introduced briefly in the following. When the AC voltage of the inverter
chi
.fl
?'+
'C7
.`3
rises, the DC voltage will also go up (see formula (8.3)). In order to keep the DC voltage constant, the regulator will automatically increase the advance angle /1 of the inverter. From formula (8.7), the reactive power consumed by the inverter will increase as a consequence. Thus, the AC voltage will
Nonlinear Control of HVDC Systems
voltage
283
DC current Id
Figure 8.6 Converter controller characteristic with constant current and constant voltage Solid line-Rectifier controller characteristic with constant DC current Dashed line-Inverter controller characteristic with constant voltage
drop and the voltage oscillation is damped. It should be noted that if the
00-
I..
control mode of inverter is constant extinction angle rather than constant DC voltage, the AC voltage cannot be maintained constant. Assuming the AC voltage of the inverter goes up, the adjustment to constant extinction angle will diminish the advance angle ,6 of the inverter in order to keep the extinction angle y at its original level. Then, according to formula (8.7), the reactive power consumed by inverters will decrease and the AC voltage will
a-9
increase as a result. Thus a positive feedback is formed and the voltage stability is damaged. From the analysis above, it can be concluded that the control mode with constant DC current of the rectifier and constant DC voltage of the inverter is an effective method to enhance the AC voltage stability of converters. Therefore, it is regarded as another basic regulating mode in DC transmission systems.
When the control mode with constant DC current and DC voltage is adopted, the block diagram of the constant DC current regulator is the same BCD
as that in Fig. 8.5(a) and the block diagram of the constant DC voltage regulator is shown in Fig. 8.7.
Vdis
Vdi
4
'In l+Tfts
Figure 8.7 Inverter controllers with constant DC voltage
284
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
8.2.5
Power Modulation in DC Transmission Systems
CD.
coo
mar.
DC power modulators have been extensively applied to improve the stability of the AC/DC systems and to enhance the stability limits of the AC lines. Power modulation consists of two parts: active power modulation and reactive power modulation, both are generally called supplementary control of DC control system. The supplementary control signal can be obtained from the active power increment AP,, of the AC line in parallel, or from the reactive current increment 4I9 of the converter stations. If there are no AC transmission lines in parallel with DC lines between two converter buses, the supplementary control signal can thus be taken from the frequency increment of the AC bus voltages in the converter stations (4f, or 4f, ), or
from the frequency difference of the AC bus voltages at the two stations ( 4fr - 4f, ). Amended by one or more phase compensation blocks and combining with the DC current deviation signal, the supplementary signal can be applied to control the transmission power in DC lines, to improve the damping characteristic and the dynamic performance of AC/DC transmission systems. The design principle of power modulator is similar to that of the power system stabilizer (PSS) and its block diagram is shown in Fig. 8.8.
inn
Many researches have shown that if the DC power modulator is properly designed, the power transportation limit of the AC transmission line I&
Id
Af, (4f ) 4lri AP,.
4Iq Figure 8.8 The transfer function block diagram of the power modulator in a converter Afr -Frequency increment of the AC bus voltage at rectifier; Af. -Frequency increment of the AC bus voltage at inverter;
Af,=Af,-Af,.; AP_ -Active power increment of the AC line in parallel with the DC line; AIy -Reactive current increment at converter stations (Rectifier and Inverter)
Nonlinear Control of HVDC Systems
285
can be greatly enhanced. As a result, the stability of the AC/DC transmission system will be effectively improved and the low frequency oscillation in the
system can also be restrained. A successful case of power modulation pas
Off''
systems is the one applied in the American Pacific DC tie-line, which has improved the transmission ability of the AC line operating in parallel with the DC line from 2100MW to 2500MW [10].
NONLINEAR CONTROL OF CONVERTER STATIONS
8.3
°-n
0
`C7
From the mathematical model (Eq. (5.86)) of the DC transmission system provided in Chapter 5, both the rectifier and the inverter have complex nonlinear control characteristics. For the rectifier, the variation
.r7
(tip
ivy
can
Imo
range of the firing angle a is 5°-90°. For the inverter, the variation range of the advance angle 8 is 20°-60°. Besides, the DC voltages of the rectifier and the inverter are the functions of the voltages on their AC sides that have complex nonlinear relations with the whole system. Therefore, there will be difficulties for the proportion control or the PID control equipped in the previous sections to improve the stability of the AC/DC transmission system effectively, especially under large disturbances. In order to fully utilize the fast regulation characteristics of DC converters to improve the performance of AC/DC transmission system and enhance the stability of the system under CAD
"21
both small and large disturbances, it is necessary to take the nonlinear characteristics of DC converters and the effects of AC voltage variations at yam,
converters into consideration in the process of modeling and designing .a;
controllers for DC converters. Thus, in this chapter, nonlinear system control a-+
theory described in previous chapters will be applied to design nonlinear controllers for DC converters.
Nonlinear Control with Constant Current and Constant Extinction Angle .*.
8.3.1
As mentioned in the foregoing sections, the control mode with constant
current at rectifier and constant extinction angle at inverter is the basic
operating modes of DC converter stations. In order to keep the DC transmission system operating in this mode, there must be a proper control method. When the distributed capacitance CdC between a DC line and the ground is ignored, the DC current Id, of the rectifier is equal to that of the
286
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
inverter Id;, i.e. I, = Id, = Id . Thus, according to Fig. 5.13, formula (8.1), formula (8.3), Eq. (5.90) and Eq. (5.93), we can obtain a simplified state equation set for the DC transmission system Id = Ll (-Rd_Id +
32
d_
;r
VV(t)cosa - 2
)r
V,,(t)cos/I)
a=T (-a+a,+uQ(t))
(8.8)
T. +'60 + Up (0)
,6 a
where Ld. =La.1+Ld;_; Rd,. = 2Rd +(3/s)x, +(3/ r)x, . In Eq. (8.8), the voltages V, (t) and V,, (t) at the AC sides of DC converters have complex relations with
the state variables of the whole AC/DC transmission system. The variation of the state variables on the AC side affects the behavior of the DC system via the change of the AC voltage (amplitude and phase). For a rectifier, in order to maintain the DC current at the reference-value, it seems reasonable to set up the performance index which makes the difference between the actual DC current Id(t) and the reference current Ids (8.9)
Y1 (1) = Id(t) - IdS
reaches its minimum. Similarly, for the inverter, in order to keep the extinction angle constant, we adopt the following performance index min yr (t) = min (y(t) - yo)
(8.10a)
where, yr (t) is the difference between the actual extinction angle y(t) and the reference value yo. From Eq. (5.85), Eq. (8.10a) can be rewritten as yr (t) = cos-'(cos/I +
2x'
JV.,(t)
Id) -yo
(8.10b)
The target of the control strategies for both converters is to minimize the indexes above. Using Eqs. (8.8) through (8.10), the DC control system model can be written in the following vector form
X= f(X)+g,(X)ual)+gp(X)u() y, = h, (X)
(8.11 a)
y,=h,(X) where X = [Id a 6]T;
ua') and
rectifier and the inverter, respectively.
represent the control variables of the
Nonlinear Control of HVDC Systems Ld_ I
287
(-Rdr 1d + 3
V, cosa -
3NF2
V, cosfj)
(-a+ao)
f(X) =
(8.1 lb)
Ta
(-Q+f 0)
TT
T
ga (X) = L0
(8.11c)
0
Ta
1 T
gp(X)=I0
(8.11 d)
0 To
(8.11e)
h, (X) = Id(t) - Ids
h,(X)=cos-'(cos/3+
Id) ' 12 Va, (t)
(8.11f)
- yu
¢Ora1
Now, according to the design method presented in Chapter 4, the coordiC's
nate transformation for the system in Eq. (8.11) is derived as the following. Choose the first coordinate transformation as
Z, =yr(t)=h,(X)=1d(t)-Ids and the second as Z2
=A,-= ah, X +ah, Var ,,,
dt
aX
aVa,
(8.12a)
+ah,
aVa,
Va;
(8.12b).
=Lfh,(X)+Lgah,(X)uaM
ah, +Lg,h,(X)u() + V + ah, Va; ava, aVa,
Ld_ 1
`I.
Lfh, =
(-Rdr Id+
_-'C2
I4,
According to the definition of Lie derivative, we can obtain Var(t)cosa-
3
Va;(t)cos/3)
Lga h, = 0
Lg,h,=0 cad
Since h, (X) is independent of the voltages Va, and V on the AC side of the converters, we have
ah,=0 L9 V,
ah,=0 ava,
288
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
The second coordinate transformation can thus be rewritten as z2 =
Ld_ I
(-Rd,
Id + 3
Va;(t)cosl) = Id
V.,(t)cosa - 3
(8.12c)
Choose the third coordinate transformation as z3 = y7 (X) = hr (X) = cos-' (cos ,8 +
2x,
Y2Vai(t)
Id) - Yo
(8.12d)
Therefore, the coordinate transformation has been completed as follows Z = [zi
z3]T = c(X)
zi
whose Jacobian matrix is .-.
0
1
0.'a'
Jm =
ax
- 3,F2 rLdu
- Rd Ld
=
- 2xi / Vai
0
V., ar sin a
3L V sin /3
,rLd_
ar
sin ,6
0
1-cost y
1-cost y
And the determinant value of J. is sin a sin /j
det(J m) _ -
Ld 1-cost y Obviously, the voltage amplitude Va, (t) on the AC side of the rectifier is always greater than zero. So, within a proper operating range of the DC system, say, amity <- a <,r, P + Yin <_ /3 < IT , Y ? Ymin > 0 , the determinant values will
never equal zero, where amin is the allowable minimal firing angle of the
rectifier usually 5'-7'5 y,,, is determined by the turn-off delay of the thyristor, and is generally 15°. So the chosen coordinate transformation Z = (D(X) can be expressed as
zI =Id(t)-Ids z2 =
L1 (-Rd_ Id + d_
z3 = cos-' (cos /j +
3
Var (t) cos a - 3Nf2 Va; (t) cos /3) )T
2xi
(8.12e)
)T
Id) -Yo
`. Vai (t)
It qualifies as a coordinate transformation for the system in Eq. (8.lla). With this transformation, the system in Eq. (8.11 a) can be transformed into the following system
Nonlinear Control of HVDC Systems
289
=1 _ -2
a.,
2 = dt Lfh,(X)= L2rh,(X)+LgaLfh,(X)ua') aVa,
.or +aLfhi RC'
+LgpLfh,(X)u("")+aLfh,
aVa,
3 = dr =Lfhr(X)+Lg.hr(X)uQ') ah
ah
+Lgrhr(X)u()+BVr Va,+aV Va; a,
a;
The output equations are Y1 = Zi
(8.13)
Yr = Z3
Let
v =L2 h,(X)+LgaLfh,(X)U') (8.14a)
3Lfh, Var +
V.
am"'
tea`
+LgpLfh,(X)u() + aaVh, ar
ai
r.,
v = Lfhr(X)+Lghr(X)uj + Lg Lfhr (X)ua) + 9
(8.14b)
ahr Var + ahr Vai
aVa;
a Var
CIO
It is clear from Eq. (8.14) that in order to obtain the control strategies u .O+
C77
and uQ') we need to solve Eq. (8.14) for u( ) and uQ' . To complete this task, (D'
what we need to do is to find out the relevant partial derivatives and Lie derivatives. Now we compute them one by one.
Since the expressions of the functions hr and h, have been given by the Eqs. (8.11 e) and (8.11 f), it is easy to get ahr
1
aVa,
aVa;
sing
aL fh, _ 34-2
aLfh,
ahr
=0 cos a
I d
(8.15a)
cos /i
avai
ZLd_
avar
2x;
V;
2'dy
Next step is to compute the relevant Lie derivatives. In Eq. (8.14): L2f
h (X) _ r
Ca(Lfh,) a(Lfh,) a(Lfh,)I f1(X) al d
as
aQ
f2(X)
f(x)
Substituting formulae (8.12b) and (8.11 b) into formula (8.15a), we obtain
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Id
LZ f hi(X) =
-
Rd=-
L,,,
I-
=-
Rd_
3
V sin a
3
f3J V, a! sin
7rLd,
7ZEd_
3J2 V,(a-ao)sina-
Id +
)rTaLds
Ld_
3'N[2
7rTpLdr
Va,(/l-,60)sin fl (8.15c)
Additionally, the other Lie derivatives in formula (8.14) is 7{!
L8, If h, (X) =
3 -
- Rer
)rLd,
Lds
3
Varsina
0
r
ll
1
Va,Sln /3I
Lds
T. 0
3-52
(8.15d)
V,a sin a
2[TaLdr
-
Lg, L f h, (X) = I - Rd'
=
3
3
V0,sina
7r4j:
Ld,.
L
3,F2
Va; sin
7r Lds
(8.15e)
Va; sin
,rTfLds
1d
Lfh,, (X)=[-
2x; r 2 - Va
0
sin!] -a-ao sin y
; si n
Ta
Q-/6o
(8.15f)
T6
-2x
L8eby(X)=r
ld - T9 sin y
-2x;
L NF2Va;
sin y
r- 2x;
L,, hr(X) = L
fl0)
sin ,5
Vajsiny
2 "aismy
tray
290
0
sin Q
=0
sin y
0 sin /3
0
siny] 1/T '6
-
(8.15g)
sin /3
T. siny
(8.15h)
Nonlinear Control of HVDC Systems
291
Then by substituting Eqs. (8.15f) through (8.15h) and (8.15a) into Eq. (8.14b), we have
I_
I4:
v2 _
V° sing °
sin/3
+
-
Tp sin y
0
Solving the above equation for
T16
sin y
u (r) +
x;
1° V°'
V°? sing
we immediately get
u(r) = Tp siny v, + 2x'Tp Id p sin ,8
sin/
Va; sin,Q
Jx;Tpld V,, sin /3
Vai +U3-Qo)
(8.16a)
According to Eq. (8.14a), following the similar calculation, we have 3F2
v) _- RdE Id+ LdE
Var(a-ao)sina
2rTaLdE
3,F2
Var sinaua
1 Ld E
3
+
3
Va; sin fl u (r) +
7fTpLdE
cos a Var -
nLdy
3
cos fl Vai
rrLdE
Similarly, solving the above equation for u"), we obtain the expression of u(1) =
-
O+,
U(I) as follows TCTcxLdE
3VGVar
sins
2aRdE
vl -
3NF2 Var
sina
Id - TaVai sin/3 (6 TfVar sina
-,86)
+TaVaisin/ju(y)+TacosaVar-TaCOSfl Vai +(a-ao) TpVar sina p Var sina Var sina
7aLdE
3/Var sina
;rTa(2Rd + 3 xr)
v) -
7rTa
3y GVarsina +
Id +
L cosa
Var
3Var J2 sina 3,F Vai Vai (13-,6o)sin,6-Tp sin/iu )) (cos,6Va, +Tp V,,,. sin a
(8.16b)
(
3K x; Id)+(a-ao)
Thus, the system in Eq. (8.11) can be transformed into the first normal form as
Zl = Z2 Z2 =V1
(8.17a)
Z3 = V2
The corresponding output equations are
yl = z, yr = Z3
(8.17b)
292
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
From Eq. (8.9) and Eq. (8.10a), we recognize that y, is the difference between the actual DC current Id and the reference current Ids of the rectifier, and y, is the difference between the actual extinction angle y and the reference extinction angle y,. According to LQR method the optimal control law of the system in Eq. (8.17) should be v, _ -k,z, - k2z2
(8.18a)
v2 = -k3 z3
.fl
where, k, , k2 and k3 are optimal feedback gain coefficients. When the weighting matrix Q of the quadratic performance index is chosen as identity matrix and R = I , the optimal feedback coefficients are k,=1
k2=r3-
(8.18b)
k3=1
ray
Substituting Eq. (8.18b) into Eqs. (8.16a) and (8.16b), considering the expression of j,, in Eq. (8.8) and the expression tic
=
(cos / 3 Vp ; + T V a ,
3
.ti
Vd
(/3 - f30) sin fl - T V , sin fi ) +
x; ld
a
TO
7rTL 3hVp, (t) sin a +
Ta cosa
Vp,
2Rd+3x, (8.19a)
Lds
-
iffa
Vd, +(a-a0)
3-hVo,(t)sina ma.
Va(t)sina
(-r3-
.N.
s0.
which is obtained according to Eq. (8.3) we can easily obtain the nonlinear control law with constant current of the rectifier and constant angle y of the inverter in the DC power transmission systems. The nonlinear control law with constant current of the rectifier is
It can be seen from the equation above that in order to implement the E"''
nonlinear control law (8.19a) on the rectifier side, it is required to deliver the signal of the variable Vd; from the inverter side to the rectifier side. This is very difficult to achieve due to the requirement for real time communication. Therefore, a suitable transformation is required for expression (8.19a). We know that Vd, is the function of Vd, and Id, that is G."
.Y^
Vd, = V, - 2Rd Id
where 2Rd is the resistance of the DC line (see Fig. 5.13). Thus, we obtain Vd, = Vd, - 2 Rdld
Nonlinear Control of HVDC Systems
293
or
Vd = Vdr - 2R tV,
Substituting the above equation into Eq. (8.19a) and after some manipulation, we can get the constant current control law of the rectifier as u
p
(t) =
7rTa Ld
(Al, + (,[3,' ' -
3NF2Var (t) sin a
r )A.1d ) 7rLd: (
+ Ta cosa y
;rTa
V, (t) sin a
3hVar (t) sin a
where Da = a - ao
8 19b)
ydr +Da
.
And the constant extinction angle control law of the inverter is 2T x ;
(t) = - T cos f3 Ay sin f3
Aid -
'Va; (t) sin /3
Tf x! Id
V. + A/3 (8.20)
112Vaj (t) sin /3
In the two equations above
AId=ld-Ids,Dy=y-yo, of3=f3-f3o.
'T7
In order to test the control effect of the nonlinear control laws, we take the 6-generator AC/DC power system shown in Fig. 8.9 as an example in digital simulation. All the serial numbers of the buses and parameters of the AC lines and the generators are the same as those in Fig. 6.10. Despite of the similarity of the two systems, some modifications have been made. First, the AC line between Bus 11 and Bus 12 in Fig. 6.10 is substituted by a ±500KV DC line. Secondly, the two AC lines between Bus 22 and Bus 20, Bus 22 and Bus 21 are eliminated, which turns the AC transmission systems shown by Fig. 6.10 into an AC/DC power system with single DC transmission line connecting the two AC subsystems. The parameters of the DC transmission lines are shown in Tab 8.1. Table 8.1
Operating parameters of the DC transmission system
DC line rated voltage
500kV
DC line rated current
900A
DC line resistance (p.u.)
0.02153
DC system reactance (including smoothing reactance)(p.u.)
0.3000
Commutation reactance of rectifier (p.u.)
0.0342
Commutation reactance of inverter (p.u.)
0.03247
Transformer ratio at rectifier
0.86446
Transformer ratio at inverter
0.96450
rn
04
NONLINEAR CONTROL SYSTEMS AND POWER
294
Lx-
2. 26+j 1. 69 W X0,012
20
Lm
0. 72+j0.47
21
0.01+j0.04
Ls,
0. 7+jO. 5
19
L
.017+j0. 0
0.037+j0.18
058+j0.22
I8
w
_..L
Ls
116
j1.993
Region B
5.0+ j2.918
Lie
4. 3+j2. 6
1
j0.038
A 6-machine AC/DC power system
Figure
Nonlinear Control of HVDC Systems
295
In simulation study, the q-axis component of the voltage behind transient reactance Ey and the mechanical power of each generator are supposed to be constant. The block diagram of the conventional control with constant current and constant extinction angle is shown in Fig. 8.5, where ka =30.0, Ta, =0.005s, k,6 = 30.0. The time constant of the constant current regulator Ta and that of the constant extinction angle regulator T,6 are 0.02s and 0.05s, respectively. The introduced disturbance is to trip off a generator set of 100MW in the 5t' power plant at O.Os and the generator is restored and put into normal
Rotor angle (degree)
120.0
20.0
-80.0
-180.0
1_
0.0
1.0
i
2.0
3.0
1
4.0
Time (Sec.) (a)
line (p.u.)
6.0
5.0
Active power on
uo
4.0
3.0
1
0.0
l
I
1.0
2.0
I
'3.0
I
4.0
1
5.0
Time (Sec.) (b)
System dynamic response under conventional controllers with constant current rectifier control and constant extinction angle inverter control (a) Rotor angle dynamic response of No.2 No.6 generators (b) Active power dynamic response of the DC line C/1
Figure 8.10
296
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
operation at 0.5s.
The dynamic responses of the system under conventional control and nonlinear control are shown in Figs. 8.10 and 8.11 respectively.
From the control laws in Eqs. (8.19) and (8.20) and the simulation results, it can be seen that:
(1) Compared with the conventional control laws, the nonlinear one presented in this chapter can maintain the rectifier DC current Id and the inverter extinction angle y constant more accurately.
Rotor angle (degree)
120.0
20.0
-80.0
-180.0
0.0
1
1
I
I
1
1.0
2.0
3.0
4.0
5.0
Time (Sec.) (a)
line
6.0
Active power on
uo
5.0
4.0
..J
3.01
0.0
1
1.0
i
L
2.0
3.0
1
4.0
I
5.0
Time (Sec.) (b)
Figure 8.11
System dynamic response under nonlinear controllers with constant current at rectifier and constant extinction angle at inverter (a) Rotor angle dynamic response of No.2-No.6 generators (b) Active power dynamic response of the DC line
Nonlinear Control of HVDC Systems
297
0-2
(2) The control laws shown in Eqs. (8.19) and (8.20) are complicated nonlinear functions, so a computer or a SCM (Single Chip Micro-computer) is needed to realize these controls. (3) The simulation results show that compared with the conventional control of constant DC current and constant extinction angle, the nonlinear control can not only improve the performance of the DC system, but also enhance the stability of the AC/DC power systems remarkably.
8.3.2
Nonlinear Control with Constant Current at Rectifier and Constant DC Voltage at Inverter
As mentioned previously, the constant voltage control on the inverter side has become a kind of basic control mode since it can improve voltage stability of the AC system on the inverter side. However, because of the complex nonlinear relations among the DC voltage, the DC current, the advance angle, and the voltage on the AC side, the linear feedback method based on the approximately linearized model at a fixed equilibrium point can not effectively exert the inverter's voltage control action for improving the
system stability under large disturbances. In this section, therefore, the nonlinear system control theory discussed in Chapter 4 will be applied to design the nonlinear controller via the feedback with constant current at the rectifier end and constant voltage at the inverter end.
To meet the requirement of keeping the DC voltage of the inverter constant in any circumstances, we expect the deviation (8.21)
YV (t) = Vdi (t) -
to be the minimum where yv. (t) is the deviation of the actual DC voltage Vd; of the inverter from its reference value V,,f . From Eq. (5.82), y, (t) can be rewritten as YV(t) = 3vr2 V.,(t)cos/+
XiI, -Vrcf
(8.22)
000
Combining Eqs.(8.8), (8.9) and (8.22), n we can write the control system model with constant rectifier current and constant inverter voltage in the following nonlinear system form X = f(X) + g.(X)ua`) + g
(8.23a)
O(X)U(V)
Y1 =h,(X)
(8.23b)
YV =hh(X)
where, X = [Id a /3f is the state vector; uQ'° and
U
16
are control variables
298
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
of the rectifier and the inverter, respectively; f (X) , g,,(X), g,6(X) and h,(X) are shown in Eqs. (8.11b) through (8.11e); hv(X) can be expressed as hv(X) = 12C2 Va,(t)cos/3+ 3 x,ld -Vds
(8.24)
n Applying the method discussed in Chapter 4 to the system (8.23), we can obtain the coordinate transformation to transfer the original system into a linear and controllable system. X
First, we choose the two coordinates expressed by Eqs.(8.12a) and (8.12b), and the third transformation is chosen as z3 =Yv =hv(X) (8.25)
= 3 2 Va;(t)Gos/3+ 3 xiId -VVf
Then the mapping 0, composed of Eqs.(8.12a), (8.12b) and (8.25), is
zi =Id(t)-Ids Z2- Ll L. z3 =
(-Rd,
Id+3
V=,(t)cosa-3
Va,(t)cos,8)=1d
(8.26)
39
with its Jacobian matrix 0
1
Jm=
a
=
- Rdc Ldr
0
3VoI
;rLd=
3X x;
0
sin a
3F2 Val .
sin ,8
)rLd=
- 3,
jr
Va sin,6
and the determinant value det(J m) =
18
Va; Va, sin a sin ,l3
X 2Ld= .O+
The determinant value does not equal zero when the DC system is operating within its possible operation area, i.e. aa,;, <- a< it, p + ym;n s /3 < tr . Eq. (8.26) is a qualified coordinate transformation of the
DC control system (8.23). Through this coordinate transformation, the system can be transformed into the first type of normal form as it = Z2
i2 = VI i3 = V2
The output equations are
(8.27a)
299
Nonlinear Control of HVDC Systems Y, = Z,
(8.27b)
Yv = Z3 (CD
In view of formula (8.19a) and considering formulae (8.18a) and (8.12e), the nonlinear constant current control law ua') for the rectifier appears as ua '>()t
_
2rTaLd..
,rTa (2 Rd +3x,/>r)
.
v,
-
(t) sin a + Ta cosa
VV(t)sina
Ad
3r2Va, (t) sin a
2la Vd,+(a-ao) Va,(t)sina
V
(8.28a)
Calculating the derivative of z3 in (8.26), substituting it into a3 = v2 and rewriting the result, we can obtain the constant voltage control law for the inverter as K_
(v)
of (t)=-
3' IVa; (t) sin /3
.Va; (t) sin fl
Va
pVa; (t))sin,B
(8.28b) 0000
TP x,
+
Tfl cos,6
v2+
Id+(Q-fin)
From Eq. (8.18a) and Eq. (8.18b), the control law of the linear system (8.27a) can be written as V1 _ -Z, - -13Z2
(8.29)
vz = -Z3 ONO
where the expressions of z, , z2 and z3 are given by Eq. (8.26)
X36
ono
Substituting Eq. (8.29) into Eq. (8.28) and using the coordinate transformation in Eq. (8.26), we have the nonlinear control law with constant current for the rectifier and constant voltage for the inverter as follows. The control law for the rectifier with constant current is rT.Ld_ tie
u«f (t)=- 3,F2 V_ (t) sin a +
-
(mod + (V S -
Va.
)mod
)rLd
Ta cos a Va,(t)sina
sx
YrTa
3,r2Va,(t)sina
(8.30)
Vd. +(a-ao)
The control law of the inverter with constant voltage is (V)
7Ta
Ta cos /3
V.,(t)sinfl ai
3-F2 Va,.(t)sin/3 Tfi x,
(8.31)
300
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
In order to test the control effect of these nonlinear control laws, we have made the following comparison by still using the case described in Fig. 8.8. Under the same conditions, the computer simulations have been carried out on the system with the conventional control method (as shown in Fig.
.nor,
8.5(a) and Fig. 8.7, where ka 30.0, kai=0.005s, Ta 0.02s, T,0.05s). The simulation results of different control modes are shown in Fig. 8.12 and Fig. 8.13.
120.0 661
angle
821 831
Rotor
20.0
0
0
04
-80.0
-180.0 .
0.0
I
I
I
1..0
2.0
3.0
Time (Sec.) (a)
AC voltage of the converter (p.u.)
1.2
V., 1.1
1.0
0.9
I
0.8 0
I
1.0
t
2.0 Time (Sec.) (b)
I
3.0
System dynamic response under conventional controllers with constant current at rectifier and constant voltage at inverter (a) Rotor angle dynamic response of No.2-No.6 generators (b) The AC voltage dynamic response of inverters
Figure 8.12
Nonlinear Control of HVDC Systems
301
Rotor angle (degree)
120.0
20.0
-80.0
-180.01 0.0
I
I
1.0
2.0
I
3.0
4.0
AC voltage of the converter
Time (Sec.) (a)
V.,
0.81
1
0.0
1.0
1
1
I
2.0
3.0
4.0
1
.5.0
C/]
Time (Sec.) (b)
System dynamic response under nonlinear controllers with constant current at rectifier and constant voltage angle at inverter (a) Rotor angle dynamic response of No.2-No.6 generators (b) The AC voltage dynamic response of inverters
Figure 8.13
w-'
any
From the simulation results, it can be seen that the nonlinear control is superior over the linear conventional one not only in suppressing the inverter AC bus voltage fluctuation, but also in restraining the power oscillation in
AC/DC systems. Nonlinear control technology provides new promising ways to further improve the operation security and dynamic performance of the AC/DC power systems when it is used for DC converters.
302
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
NONLINEAR CONTROL OF DC SYSTEMS AND STABILITY OF AC/DC SYSTEMS
8.4
coo
In the previous section, we have discussed the conventional and the nonlinear control methods mainly from the viewpoint of stable operation of the AC/DC link itself. Although these control methods can damp the power
l<w
oscillations in some extent, they can hardly improve the stability of the
overall system. Thus, in what follows, we will concentrate on the improvement of the performance of the whole AC/DC power system. This can be achieved because the DC regulation system can change the power flow distribution quickly. By using the nonlinear system control theory presented in Chapter 3, the design method of the nonlinear supplementary control for DC converters will be discussed.
8.4.1
Modeling for Nonlinear Stabilizing Control Design of AC/DC Systems
The most typical AC/DC power system is shown in Fig. 8.14. In this diagram, PL, and PL2 are the local loads of the equivalent generators 1 and
2, respectively, which can be seen as two power systems of different capacities.
P A,. Pe, P.
3F
RE
I
P
Pe2
H
II equivalent machine
2' equivalent
machine
PL,
PL2
Figure 8.14
An AC/DC power system
The swing equation of the equivalent generator 1 is S, = w,
(8.32a)
tiw, = H1
(P., - Da w, - P11)
and the equation of the equivalent generator 2 is
Nonlinear Control of HVDC Systems
303
8z =wz &2 =
wo
(Pm2
H2
D2
(8.32b)
C02 - Pet)
coo
The electromagnetic power Pe, and Pee of the equivalent generator 1 and 2 can be respectively represented as Pe, =PL, +P,+PP (8.33a) (8.33b)
Pee = PL2 - Pd, - P.
The meanings of the symbols in the above equations are shown in Fig. 8.14. Subtracting Eq. (8.32b) from Eq. (8.32a), we can obtain the swing equation expressed by relative rotor angle 812 and relative rotation speed 0)12 as
812 = wit QJp
(8.34)
Pn,2)-(DL C, - D2 wz) d),2 =(O Pmi H2 H, H, H2
-(°2 H,
PLI -
O PLZ)+( + H,
H2
" )P -( -9 + H2
H,
w0 H2
)Pdc
When the stability of the AC/DC power system is studied, the dynamic characteristic of the DC line is usually not taken into consideration, and the DC power regulation is regarded as a first-order inertia element. Thus the equation of the DC power can be written as 1
Pd, =
T Td
(-Pde +PdeI. +u&)
(8.35)
where P. is the reference value of the DC power, Td the equivalent time `ate)
constant of the DC system, and u. the control variable of the DC system. As stated previously, the target of the nonlinear control studied here is to keep the whole system stable by regulating the transmission power on the DC line especially when the AC system is disturbed by a large disturbance. For the system shown in Fig. 8.14, we can evaluate the system stability by checking the
absolute value of the relative angle between the two equivalent generators within 180°. This relative angle is used as a stability index of the connected power systems. After a large disturbance, the smaller the relative angle is, the more stable the system will be. Therefore, we can take the relative rotor angle increment between the two equivalent generators as the output equation for the nonlinear controller design. That is
(8.36) y(t) =8,0) -81zo Combining Eqs.(8.34), (8.35), (8.37), we can obtain the state equations and the output equation of a AC/DC nonlinear control system
304
o,.
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS 512 = 6)12
612=( H,
-(0H,PL I 0.b
Pd. =
LoPml-L0Pm2)-(D6)1-D26)2) H2 H, H2 (00 0 -(Wo + 0 )Pd, + )PW - H2 H, H2 H, H2
.
(8.37a)
PL2+(u?0
P& + P&REF + ud,
T Td BCD
The output equation is y(t) = h(X(t)) = 512 (t) -8,0
8.4.2
(8.37b)
Nonlinear Control Design for Stabilizing AC/DC Systems
According to the design method of the SISO nonlinear system control discussed in Chapter 3, for the system in Eq. (8.37), we can calculate its Lie derivatives in every order, namely Lg h(X) = 0 LgL fh(X) = 0
(8.38)
6)o 6)o LgLfh(X)=--(---) Td H, H2 z
1
0
From the above results, the relative degree of the system in Eq. (8.37) equals 3, i.e. r = 3. Therefore, the coordinate transformation (D can be calculated as follows Z1 =
h(X) = 512 -'5120
3
-
= L2f h(X ) =
ro12
(1 PL, -
- ( H,0 P
°
HI
H2
6)0 Pm 2)
ml
H2
PL2)+(T 0 + H,
w° H2
- ( D' HI
6)
D2
1
H2
Ono
=2=Lfh(X)=6)12
(8.39)
6)) 2
'!Lo-)Pd,
)Pac -(Oj0 +
H2
H,
The Jacobian matrix of this coordinate transformation is az,
az, 6)P&
'
0
6)512
aCU12
aZ2
aZ2
aZ2
0
1
x512
x6)12 aZ3
a"d,
aZ3
a512
x6)12
aids
III
aZ3
CO° + cw°
- (H1
aPa.
H2) x512
+
Jm
aZ,
where we have made the following assumptions
0 0
- DH - ( Hl CO
+ 6)o H2
1
P
)( + 6)P&
305
Nonlinear Control of HVDC Systems
D,IH,=D2/Hz=D/H The transmission power in AC line is the function of relative rotor angle 812 and the DC transmission power Pdc , i.e. Pa = P.(812, Pte) .
The determinant value of the Jacobian matrix J, can be easily figured out as det(J
-(
0 +'00 )(1+ apO2 )
H,
H2
aPdc
Obviously, if a? /aPQ is not equal to -1, the relations shown in Eq. (8.39) can be considered as a coordinate transformation of the system in Eq.
(8.37), which can transform the AC/DC nonlinear control system in Eq. (8.37) into a linear and controllable system in the subset S2={812,a12,PP I aPOClaPd t-
-1}.
The resultant linear system is z1 = z2
(8.40)
Z2 = Z3
Z3 = V
and the output equation is Y(t) = zl
Substituting the derivative of z3 in Eq. (8.39) into the Eq. (8.40) and
after some manipulation, we can obtain the nonlinear supplementary stabilizing control law for the converters as follows 3 udc=-Lfh(X)+v=
HI
H2
to°
a`0
H,
H2 Td
tea`
LBLfh(X)
D°12 +(Co +-0)Pn,+v H
+ Pdc-Pdc;,,.
(8.41)
V" _ -z1 - 2.29z2 - 2.1423 = -0812 - 2.29(012 - 2.14w12
_ -J w12 dt -2.29a12 -2.14r
O^0
G00
where the optimal control v` for the linear system (8.40) can be obtained from Example 3.2, which is (8.42)
Substituting the above equation into Eq. (8.41), we obtain the nonlinear stabilizing control law of the DC system as udc =
HI H2 7-d (f 1 a)12dt + 2.29ro12 + (2. l4 H1 + H2 WO °
D)[o12
H
-7 dP + P. - PP
CAD
Eq. (8.43) is the nonlinear control strategy that we are seeking for. Let us make some observation of Eq. (8.43)
(8.43)
306
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
.'301
CAD
CAD
"C7
o:.
(1) The nonlinear control law studied here consists of the proportion, integral and derivative elements of the AC system frequency difference between the two sides of the DC line with the derivative element of the active power P,, on the AC line in parallel with the DC line. The structure is not complicated and easy for implementation. (2) The nonlinear control law has no direct relations with the structure
and parameters of the network, but is related to the equivalent inertia
`CC
constants of the equivalent generators at the two ends (the sending end and receiving end) and the DC system equivalent time constant T,. Therefore, this controller has comparatively strong adaptability to the network structure and parameters. (3) If the two systems linked by the AC/DC parallel transmission lines are of large capacity and complex structure, w,Z in the control law (8.43) can be replaced by the difference between the AC bus voltage frequencies of the rectifier and the inverter. (4) If the AC lines are tripped off in parallel with DC lines, from the design method studied above, the nonlinear control law can be obtained as
8.4.3
H HZ
Td
a.;
ud, =
(f w12dt+2.29W12+(2.14- D)W12)+(Pk -P&REF) (8.44) H
Effects of Nonlinear Control for Stabilizing AC/DC Systems
Nod
In order to demonstrate the ability of the nonlinear stabilizing control of the DC system (in formula (8.44)) to improve the stability of the AC system,
the digital simulation is carried out on the system shown in Fig. 8.9. In (IQ
formula (8.44), the AC system frequencies of the converters are taken from those of the AC bus voltages at the converters. The equations are
(_f
1
dOr
1
dO,
Jr - fo + 2, dt
f =f0+21r
p (8.45)
di
where Br and 0, are respectively the phase angles of the AC bus voltage in the rectifier and the inverter and fo is the steady state frequency. The disturbance is assumed to occur by tripping off two 100MW generator units in the No.5 plant at O.Os and then putting the unit back into normal operation at 0.5s. This means about 23.6% of the total capacity of the AC system at the inverter side is affected directly. The nonlinear control with constant current and constant extinction angle (see Eqs. (8.19b) and (8.20)), or the nonlinear
Nonlinear Control of HVDC Systems
Rotor angle (degree)
307
Power on
line (p.u.)
6.0
5.0
4.0
3.0
1
0.0
1.0
2.0
4.0
3.0
5.0
Time (Sec.) (b)
metro
Figure 8.15 System dynamic response under nonlinear controllers for AC/DC system stability (a) Rotor angle dynamic response of No.2 No.6 generators (b) The power dynamic response of the DC line (y1
'-'0
``i
control with constant current and constant voltage (see Eqs. (8.30) and (8.31)), which have been discussed in Section 8.3, cannot keep the system
stable. The reason is that both controllers can not regulate the DC
0.p
transmission power in sufficient extent, and the performance index, constant current, constant extinction angle and constant voltage operation, can not be
satisfied under large disturbances. However, the nonlinear controller designed for AC/DC system stability improvement in this section can keep the system stable under the same disturbance. The simulation results are
308
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
shown in Fig. 8.15. From Fig. 8.15(b) we can easily observe that this nonlinear controller can regulate the amount of the power flow of the DC line sufficiently, so the stable operation of the AC/DC system is achieved.
8.5 REFERENCES A. E. Hammad, "Analysis of Power System Stability Enhancement by Static Var :7.
I.
°.r
G. W. Stagg and A. H. El-Abiad, Computer Methods in Power System Analysis,
n~y
4.
.,e
3.
Compensators", IEEE PWRS, Vol.1, No.4, pp.222-227, 1986. CIGRE Working Group 14.01, "Use of Static or Synchronous Compensator on HVDC Systems", Electra, Vol. 91, pp.51-82, 1983. E. Uhlman, "Stabilization of an AC Link by a Parallel DC Link", Direct Current, No. 8,
2.
ova
1964.
McGraw-Hill, 1968. IEEE Committee Report, "Dynamic Performance Characteristics of North American HVDC Systems for Transient and Dynamic Stability Evaluations", IEEE PAS, Vol. 100, pp. 3356-3364, 1981. 6. IEEE Committee Report, "HVDC Control for System Dynamic Performance", IEEE PWRS, Vol. 6, No. 2, pp. 743-752, May, 1991. 7. J. F. Clifford, A. H. Schmidt, "Digital Representation of a DC Transmission System and Its Controls", IEEE PAS, Vol. 89, pp. 97-105, 1970. 8. K. R. Padiyar, HVDC Power Transmission System: Technology and System Interactions, John Wiley & Sons, 1990. 9. P. K. Dash, A. C. Liew and A. Routary, "High-Performance Controller for HVDC Transmission Links", IEE Proc.-Generation, Transmission and Distribution, Vol. 141, No. 5, pp.422-428, 1994. 10. R. L. Cresap, D. N. Scott, W. A. Mittelstadt and C. W. Taplor, "Operation Experience with Modulation of the Pacific HVDC Intertie", IEEE PAS, Vol. 97, No.4, 1978. 11. R. T. Byerly, D. T. Poznaniak and E. R. Taylor, "Static Reactive Compensation for Power Transmission System", IEEE Power Engineering Society Winter Meeting Paper 82 WM 179-0, New York, Jan. 31-Feb. 5, 1982. 12. S. Kapriclian, K. Clements and J Turi, "Application of exact linearization techniques for steady-state stability enhancement in a weak AC/DC system". IEEE PWRS, Vol. 7, No. 2, pp. 536-543, May, 1992. 5.
°".
-=W
ti'
0'O
con
t17
0
Sam
ono
F'>-
coo
`y°
EAU
.C7
`v°
J"'
°--
nab
0.U
O>e
T. Machida, "Improving Transient Stability of AC System by Joint Usage of DC
ion
13.
Systems", IEEE PAS, Vol. 85, No. 3, pp.226-232, 1966. T. Smed, G. Anderson, "Utilizing HVDC to Damp Power Oscillations", IEEE PWRS, Vol. 8, No. 2, pp.620-627, 1993. 15. X. Jiang, A. M. Gole, "An Energy Recovery Filter for HVDC Systems", IEEE PWRS, Vol. 9, No.1, pp.119-127, 1994. 16. Y. Z. Sun, Q. Lu and J. Gao, "A New Nonlinear Modulation Control for HVDC Power .-:
14.
°3'
coo
ti'
Transmission Systems", CSEE/IEEE International Conference on Power System 0+0
Technology, Beijing, Sept., 1991.
Chapter 9
Nonlinear Control of Static Var Systems
9.1
INTRODUCTION
Nowadays static reactive power compensator or Static Var System (SVS) has been widely used in power systems to improve the stability of the system
voltages as well as the stability under small and large disturbances. The primary function of SVS is to maintain a substantially constant voltage of its '-$)
installed bus by generating or absorbing the required amount of reactive power, which also benefits to the reactive power balance of the power system.
If a static Var compensator is installed at a midpoint of a long-distance transmission line, it is possible to hold the terminal voltage of its connection point essentially constant by continuous regulation of the reactive power. Then, the "electrical distance" of the transmission line can be regarded as being halved. Thus an increase in the stability limits of the power system can be expected. With its ability to continuously regulate the reactive power, high speed of response and simple configuration, static Var compensator has become more and more widely used [1-6]. This chapter will provide a detailed discussion on the theory and control
of the reactive power compensation. It starts with the basic concept of reactive power compensation and marches on to the configuration of the Static Var system and its modeling. Then, the fundamental principles and methods of nonlinear control system discussed previously will be applied to
analyze the performance of SVS, and to solve the control problem. Computer simulations are performed, and the results show the effectiveness (IQ
of the SVS with the nonlinear control for improving the dynamic performances and stability of the power system.
310
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
9.2
FUNDAMENTALS OF REACTIVE POWER COMPENSATION
9.2.1
Reactive Power Flow in a Transmission System
In this section we define some principle symbols and terms used with the purpose of describing the fundamentals of reactive power compensation. If a reactor or capacitor connected to a node of network is treated as a
load as shown in Fig. 9.1, then the, active and reactive power supplied at voltage V can be defined as (9.1)
P + jQ = VI
where V is the nodal voltage vector, and I the conjugate complex of the load current vector.
(b)
(a)
Figure 9.1
Reactor and capacitor treated as load (a) Reactor (b) Capacitor
From Eq. (9.1), the active and reactive power consumed by a pure reactor and a pure capacitor can be expressed respectively as Reactor:
PL+jQL=i
V2
(9.2a)
then QL=a- L
V2
(9.2b)
Pc +jQc = -jcoCV2
(9.3a)
PL = 0
Capacitor:
Nonlinear Control of Static Var System
311
then
Pc=0
(9.3b)
Qc=-wCV2
where L is the inductane, C the capacitance, w = 2nf , f the frequency of the bus voltage of the power network. We give the following definition. If Q > 0, then the load is said to be of
reactive characteristic, it absorbs reactive power from the system. And, if Q<0, the load is said to be of capacitive characteristic, i.e. it supplies
00.
reactive power to the system. If we treat the reactor or capacitor element connected to the system as source, then the direction of the current vector will be opposite to that shown in Fig. 9.2, and the corresponding current and power are
(a)
(b)
Figure 9.2 Reactor and capacitor modeled as power supply (b) capacitor (a) reactor
Reactor: IL
V
- i wL
QL _ _ V2
wL
Capacitor:
(9.5)
1,=-jwCV Qc =wCV2
CADS'
When Q > 0 , the power source generates reactive power; when Q < 0 , it absorbs reactive power from the system. In order to avoid possible confusion of the concepts discussed above, in this section we will define that the reactor consumes reactive power, and the capacitor generates reactive power.
However, in a equation, the signs (positive or negative) of reactive
0U4
power of a device in the model will depend on whether it is treated as a load or as a source. For a node with three branches as shown in Fig. 9.3, if all the branches
are treated as source, the signs of currents flowing to the node are .fl
positive.And then the balance equation of reactive power flow can be given as
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
ail
312
Figure 9.3 A node with three branches I1+I2 + I3 = 0
P1+P2+P3=0 Q 1 +Q2 + Q3 = 0
9.2.2
Two Basic Types of Reactive Power Compensators
A-.
BCD
A simple compensator circuit and its corresponding vector diagram are given in Fig. 9.4. Obviously, there exist differences both in the magnitudes and in the phase angles between the voltages of sending end and receiving end. The vector diagram shows that the voltage drop along the line is caused by the transmission of reactive power over the line, and the difference of
phase angle between the voltage vectors is caused mainly by the active
jx
I
(b)
Figure 9.4 Simple circuit and its vector diagram AV: Voltage drop caused by active power
313
Nonlinear Control of Static Var System
power transmitted along the line. Fig. 9.5 shows the effects of the shunt capacitive compensator installed at the receiving terminal of the transmission line. The vector diagram in Fig. 9.5(a) shows the case where the magnitude
of the current IC of the compensation capacitor is equal to that of the inductive load current IL, in this case there exists a voltage difference AV between the magnitudes of the terminal voltages V2 and V, , which is caused by the transmitted active power. If the system is over-compensated, i.e. the reactive power generated by the capacitor exceeds that absorbed by the load, as shown in Fig. 9.5(b), the magnitude of the capacitor current I. is given by Il c I = III I + II4a I . The over-compensated capacitive current I.,, CCD
'L"
may offset the voltage drop caused by the active power transmitted along the
line, and make the volatge magnitude of sending end equal that of the ...
receiving end, i.e. I V, I = IV2I . In summary, there are two basic types of capacitive compensators: the
first one is called the load compensator which provides reactive power to jx
I
V2
(b)
Figure 9.5 The circuit with the capacitive compensator (a) Compensator cosrp = I (b) Over-compensated IV I = IVz I AV:
Voltage drop caused by active power
314
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
>C.
`C7
offset the power-factor to about 1.0 although the voltage drop between the sending and receiving ends of the transmission line still exists. The second one is the system compensator which offsets the voltage drop along the line compeletely by making the magnitude of the capacitor current Ic larger than that of the load current IL according to the requirement of the voltage level of the system.
9.2.3
Effects of The Midpoint Compensator on the Stability Limits
9.2.3.1 Fundamental equation of a transmission line
The study of the effects of the reactive power compensator on the power flow limits of a transmission line starts with its fundamental equation. From the equivalent diagram of transmission system given in Fig. 9.6, 1 is the I(0)=I,
I(x)=1,
V(0)=V,
0
V(x)=V,
1,
I.
x
l
Figure 9.6 The equivalent diagram of transmission system
distance of the transmission line; Vx and Ix, represent the voltage and current at point x, respectively. And the voltage and current at sending end are represented as V(x) = VS, 1(x) = I.. Then the fundamental equations governing the propagation of the energy along the trasmission line can be given as
NIA
V(x)= V,cos,6x- jZOI,sinQx I (x) = 1, cos fx - j L'
0
sin ,&
(9.7)
(9.8)
315
Nonlinear Control of Static Var System
;-p
where Z° = 4171-F is the surge impedance in 0, /3 = w L'C' is the wave number, i.e. the number of complete waves per unit of line length in rad/km. L' is the inductance per unit of line length in H/km; C' is the capacitance per unit of line length in F/km. x is the distance of point x along the line _aU
measured from the sending end of the line in km, shown in Fig. 9.6. Eqs. (9.7) and (9.8) describe the variation of the voltage V and the line current I along the line, and indicate that both can be represented as a function of the voltage VS and current I, at the sending end. e+'
9.2.3.2 The flow of active power on a lossless line buy
If the voltages at both ends of a transmission line are known as V, = VsejO
(9.9a)
V, = V,e'(-B) = V, (cos9 - j sin 0)
(9.9b)
(14
C,.
where 0 is the load angle, and 0 > 0 implies the phase of the voltage at the receiving end lags behind that at the sending end. The current at the receiving end is given by Eqs. (9.9b) and (9.7) =V,cos/31-V,(cosB-jsin 0) (9.10) jZ0 sin fit
The active and reactive power at the sending end can be expressed respectively as
P, =
V,V,
sin 0
ZOsin/31
V,V,cosB+V,ZCOs/31
(9.12)
N
Q,
(9.11)
L
Z0 sin /31
And the active and reactive power at the receiving end can be written as V,V1
P,
Q
Z0 sin /11
sin 9
V,V,cos0+V,ZCOS/31
Z0sin6l
(9.13) (9.14)
If the transmission line is a symmetrical one, i.e. the voltages at both ends have the same magnitude, that is V, =V, =V, then Eqs. (9.11) through (9.14) can be reduced to
P=
V
z
Z0 sin /31
sing
(9.15)
316
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
-Vzcos0+V2cos/31
(9 . 16)
Z0sin/31
P=
-V z Zo sin /31
(9.17)
sing
VZcos0+VZcos,61
Q
(9.18)
Zo sin /3l
From above equations, we get P, = -P., Q, = Q, , which means there are no active and reactive power losses in the transmission line. A transmission line under this condition is said to be "a lossless line". The surge-impedance loading (SIL, sometime called as natural power flow) is defined as 2
(9.19)
SIL=Po=VNO 0
where VN,O,, is the nominal or rated voltage of the line. If the rated voltage is
used as the base value, then the per unit of value of VNOM is 1.0. Assuming
the voltages of both ends of the transmission line are from Eq. (9.15) through Eq. (9.18) we have P, = - f,_ = sin 0 P0
P0
Q, Po
Po
V, = V, = VNOM = 1.0,
(9.20)
sin /31
cos B
cos /31
sin/31
sin/31
(9 . 21)
NIA
,may..
If we treat the midpoint of the line as the sending end of the right half line, then from Eq. (9.7), the voltage at the receiving end can be expressed with the voltage V. and current I. of the midpoint of the line, namely
1- jZoIm sin Q1
V, = V. cos
(9.22)
Since the transmission line is a symmetrical one, substituting Q. = 0 and P. = P, -P, = P into above equation we have 1
2
0., Ira
V, = V. cos
- jZo Y sin V.
1
(9.23)
2
where P. is the conjugate of the midpoint voltage vector V.. If Vm = Vm + jO is adopted as the reference vector, then from Eq. (9.23) we have V 2 = V.', cos z A+ Zo 2
z P22
sin 2 Al
(9.24)
Nonlinear Control of Static Var System .-3
317
Taking nominal voltage VNOM as the base value and solving Eq. (9.24) for the midpoint voltage V. in per unit value, we obtain V.
_
1
1
V NOM
2 cos 2
,Bl
+
Ql 4 cos `
-(
P
Z
) tg
P.
(9.25)
2
2
2
2 ,Bl
In order to maintain the midpoint voltage V. = VNOM, we need to determine the amount of the required compensation reactive power Qc at
the midpoint of the transmission lines. The balancing equation of the reactive power at the midpoint of the line is (9.26)
Qc + Q, + Q, = 0
If the transmission line is a symmetrical one (Q, = Q.), then the required compensation reactive power at the midpoint is Qc = -2Q, = -2Q,
(9.27)
Substituting Eq. (9.27) into Eq. (9.21), we obtain 9
Q PO
/31
cos - - cos -
=2 o
2 sin
2
l
for VS=V,= 1.0
(9.28)
181
2
If we treat the midpoint as the receiving end of the left half line, as well as the sending end of the right half line, from Eq. (9.20), we get P P°
sin
B
21 Q sin
for V, = V, =1.0
(9.29)
2
From Eq. (9.25) we know that the maximum voltage along the line appears at the points x 1/4 and x =31/4, which is (as V, = V, =1.0 ) V(-,P=0)_V(4'P=0)_ 1
VNOM
VNOM
cos
(9.30) '61
4 O0,
If a shunt capacitor is installed at the midpoint of a symmetrical line, it
will be effective for controlling the voltage V. at the midpoint of the transmission line as well as the active power flow of the line, which will be discussed hereafter. Applying Eq. (9.12) to the right half of the line, then in per unit system, we get
318
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
_ Qsr =
0) 1 .C . " 2- + (
(V
VNOM
Cos
2
VNOM
0.°
(9.31)
sin161
PO
2
CID
CAD
Likewise, the reactive power requirement of the left half line is identical, thus the total reactive power requirement at the midpoint of the line is Qc = -2Q,,
(9.32)
0
cos20 2
cos 2
V.
+
Ql 2 cos
VNOM
2
4cos2
61
Qc 2PO
SIN
Nam
.@O
Substituting for Q. in Eq. (9.31) from Eq. (9.32), and making use of the relations of terminal voltages V, = V, = VNOM , we can have the per unit value of the midpoint voltage V. as tg Ql 2
(9.33)
2
1
Because the target value of V. to be compensated is known a priori and also the terminal voltages V. = Vr = VNOM , from Eq. (9.11) the active power flow of the line can be given as P, _
sin
Vmz
0 2
VNzOM
PO
Sln
(9.34)
2
Eq. (9.34) describes the effect of the midpoint voltage on the power CEO
flow of the line. This can be best illustrated by the following example. Consider a 600km transmission line operated at 765kv line-line voltage, the parameters of the line are as follows: L'=0.874mH/km, C'=13.2uF/km,
,.t in'
ti.-
duo
'-`
III
ZO =275SZ, PP =2280MW, (3 =0.0733 rad/km. In order to maintain the midpoint voltage at 1.0 pu, the reactive power requirement calculated from Eq. (9.31) to Eq. (9.34) is illustrated in Fig. 9.7. Fig. 9.7 shows that to maintain the voltages of both terminals and the
midpoint at 1.0 pu, if the active power flow on the line is less than the
«..
C=D
vii
natural power flow (i.e. SIL) P, , the compensator must absorb the surplus reactive power generated by the line. And if the active power flow is larger than PO , the compensator should supply reactive power to the system. The following conclusions are drawn from the above analysis: (1) A lightly loaded line has a surplus of reactive power, in order to maintain the system voltage normal, the compensator must consume reactive power, i.e. the compensator should be of inductive characteristic. (2) A heavily loaded line needs to absorb the reactive power from the
Nonlinear Control of Static Var System
319
0.6,0.4
0.2
P/Po 0.0
0.5
1.0
1.5
-0.2
-0.4 -0.6L-
w__,
Figure 9.7 Effects of reactive power compensation on the transmitted power of symmetrical lossless line
compensator to maintain the system voltage normal.
(3) The adjustable reactive power compensator can hold the system >>°
voltage virtually constant under the variation of the power flow by providing
reactive power required by the system, and hence further enhance the transmission power limit (stability limit) of the line.
9.3
CONFIGURATION OF STATIC REACTIVE COMPENSATORS
In practice, there are many different types of static compensators for transmission application, and the main types will be briefly described in this section.
9.3.1
Thyristor-Controlled Reactor (TCR)
r..
The single-phase diagram of a thyristor-controlled reactor (TCR) is shown in Fig. 9.8. The thyristor controller consists of two anti-parallel thyristor valves, which conduct on alternate half-cycles of the supply frequency. And a reactor is connected in series with the thyristor controller. Each valve is further composed of many thyristors, as shown in Fig. 9.8, and connected in series and parallel to withstand high voltage and large current. If the supply voltage vs is
320
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS L ITCR
Figure9.8
Single phase diagram of TCR and its waveforms
VS = V sin wt
(9.35)
From Fig. 9.8, the voltage equation of the circuit is given as
Ldi-VS.=0
(9.36)
NIA
Solving above equation for i we have V i(t) = L JVsdt +c =-cos Cot + C 1
wL
where c
is
(9.37)
the integration constant. And from the initial condition SIC
i(wt = a) = 0 (see Fig. 9.8), we get
---(cos a - cos wt) L
(9.38)
0(Q
where, a is the gating delay angle in rad. The waveforms of the current I,CR, anode-cathode voltage of thyristor V,cR, and the voltage at the reactor at different gating angle a are shown in
Fig. 9.9. The waveform of IrcR indicates that the TCR is operated as a continuously adjustable reactor. The effects of increasing the gating angle a are to reduce the fundamental component of the current I,cR. As the gating angle a approximates to 180°, I,cR approximates to zero. Full conduction is obtained with a gating angle of 90°, where the waveform of current is
Nonlinear Control of Static Var System
321
ITCR
N VTCR
V, 1
U \I/ a near 90°
Figure 9.9
a=120°
011-4
a=150°
1
a near 180°
--
Current and voltage waveforms at different gating angle
0
nearly sinusoidal, so its magnitude of fundamental component (elementary wave) reaches its maximum value as shown in Eq. (9.38). Fig. 9.9 shows that the current waveform involves harmonic components. However, we are mainly concerned with the fundamental component of the current in the design of compensators for a transmission system. According to the Fourier analysis, the fundamental component of the current can be written as (9.39)
i, (t) = a, cos cot + b, sin cot
r-.
From Eq. (9.38), current i(t) is an even function, i(t) = i(-t) , i.e. b, =0; and considering i(t + T /2) = -i(t), we can figure out the actual value of a, as a,
_
2,r-2a+sin2a
(9. 40)
In view of the following formula T12 0 =Tf
f(r ) cos 27rr d r
(9 . 41)
[r]
a,
where a is expressed in radian. Substituting Eq. (9.40) into Eq. (9.39), the magnitude of the fundamental component of current is
N-1
V 2,r - 2a + sin 2a '
wL
(9.42)
rr arc
Obviously, the fundamental component is a function of gating angle a. Dividing both sides of Eq. (9.42) with the magnitude of voltage, the
322
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
fundamental frequency susceptance BR of TCR will be I' V
=BTCR - BL
2,r-2a+sin2a
(9.43)
/T
However, if the reactor is represented as Z = jx, its susceptance is Y=1/Z= jlx = jBL, i.e.
BL =-1 =X
(9.44)
1
wL
C/]
Cep
!3.
where L is the inductance of the reactor. The relation between BrcR and gating angle a is shown in Fig. 9.10. Static compensator is also named as static reactive power compensator; as a continuously adjustable reactor, TCR is the simplest type of SVS.
X
X
a
2
Figure 9.10 The control characteristic of BTCR
SVS can be generally thought of as a device with adjustable parameters (susceptance), the relationship between its terminal voltage v,,, and current IS,,S is expressed as
(9.45)
Isvs = jBsvsV svs C%1
As aforementioned, for the simplest type of SVS, Bv,, = B,cR , if a shunt fixed capacitor bank (FC) is added to TCR, then BSUS =B,R +BB, where Bc = cvC , C is the capacitance. Such a type of static compensation PL,
4-i
system SVS is called TCR-FC type compensator in short. The configuration of TCR-FC type compensator is illustrated in Fig. 9.11. From Fig. 9.11, BSS in Eq. (9.45) will be B SVS
__
B (BC + BTCR
Bo+Bc+BTCR
(9.46)
323
Nonlinear Control of Static Par System
The configuration of TCR-FC type compensator C.7
Figure 9.1 1
Since the gating angle a can vary continuously between 180° and 90°, the value of B,r.R will change continuously between BL and 0. As a=180°, B,cR = 0, and the maximum value of susceptance of SVS becomes (9.47)
BQ Bc
B svs
= B+Bc ICJ
Similarly, as a=90°, B,r.R = BL, and the minimum susceptance of SVS given by
i
B
B (BC + BL)
(9.48)
svs=Bo+Be+BL
t17
Analyzing Eq. (9.46) we know that the susceptance B,, of static Var alb
system is a nonlinear function of B,,,. However, if susceptance B c / BL <<1, and BL /BQ << 1, B,,, can be approximated by a linear function of BrcR (see the Note A at the end of this section), the function is as follows (9.49)
Bsvs =(I - B )BC + (1- 2B B BL )BTCR 0
0
From above equation, the limiting values of the susceptance of SVS can be given as Bc
Bsvs = (1- Bv )Bc
as
BTCR = 0
BC B0
B
= (1-
BL ) (B c
+ BL )
,
as BTCR = B L
(9.50a)
(9 . 50b)
324
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
From Fig. 9.11 the secondary voltage of transformer is 1
Vz = Vsvs = Isvs
(9.51)
j(BC + BTCR )
From Eqs. (9.45) and (9.46), voltage VZ is B°
VZ = VS S = V
(9.52)
BQ + BC + BTCR
And from the limiting values of compensator susceptance, the maximum and VS
= -1.173
BC = 0.857
(a)
a=180*
Bsvs 1 0
generating
\'\
/ (_`0°
I
\
Be-0 31
max. VAR
/
absorbing
max VAN
-1.0 -0.5 Generate reactive power
0
0.3
Absorb reactive power (b)
Figure 9.12
Voltage/current characteristics of Thyristor-Control led Reactor-Fixed Capacitor (TCR-FC) type SVS
Nonlinear Control of Static Var System
325
minimum values of secondary voltage of transformer should be (see Fig. 9.12(a)) BQ
V2mex =V
B
(9.53a) c
when BTCR = 0 and B, << Bo .
B, V2 min = V
Bo + Bc + BL
V (l
_
Bc + BL) Bo
(9.53b)
'!1
when BTCR = BL and BL << Bo. U°.
,CD
A typical TCR-FC type SVS can be described in Fig. 9.12(a). From the parameters given in Fig. 9.12(a) and above equations, the boundary values of the operating range of this type of SVS are shown in Fig. 9.12(b). Fig. 9.13
shows the nonlinear and approximate linearized characteristic relation between Bsix and B,,,.
Figure 9.13
`s7
Bsvs characteristics of Thyristor-Controlled Reactor-Fixed Capacitor (TCR-FC) type SVS C.)
1-Linearized characteristic; 2-Actual nonlinear characteristic
Thyristor-Switched Capacitor (TSC) `-i
9.3.2
The principle of the thyristor-switched capacitor (TSC) is shown in Fig. 9.14.
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
326
C
XL
vi
vC
XC
2
vc
0
1V
Figure 9.14 TSC type
Figure 9.15 TSC applied structure (XL =w0L,Xc =I[%C)
ti.
C"4
't3
(ND
At the instant when the switch is on, switching transients may be of two different types, which are discussed below. (1) When the thyristor is turned on, if the voltage at the capacitor is not equal to that of the supply voltage, there will be an infinitely large quantity of current to charge the capacitor voltage up to the supply voltage in a very short time. The thyristor, which is used as switch, can hardly withstand such a surge current, and be destroyed.
(2) If the thyristor is gated as the capacitor voltage equals that of the supply voltage, the current will have a discontinuous step change to- its
`CD
chi
steady state value, and reach its steady state in a very short time. In this case, even though the magnitude of the current is less than its steady state value, the rate of the current rise di/dt may be indefinitely large, and may exceed the maximum permissible value of di i dt , and cause a damage of thyristor. For the reason above mentioned, a special damping inductor is inserted in series with a capacitor in the main circuit as shown in Fig. 9.15. If the supply voltage is (9.54)
V (t) = V sin coot
At the instance t=0, the thyristor conducts, the instantaneous current is given as
i(t) =1AC cos(wot + a) - IAC cosa
-nBc(Vco - n Vsina)sina,,t n-1 where, Vco is the initial value of the capacitor voltage at t=0.
wn=nooo= LC' n= &c' 1X111
The magnitude of current IAC is
(9.55)
327
Nonlinear Control of Static Var System
_ 1AC
BCBL
VBc+BL
_
n2 -VBcn2-1
(9.56)
VC
_V
III
In steady state, the voltage on the capacitor is n2
(9.57)
n2-1
where n2 /(n2 -1) is a magnification factor, which relates to the inductance of
the damping inductor and accounts for the natural frequency of the TSC circuit. The effect of n2 /(n2 -1) on the natural frequency is plotted in Fig. 9.16. n2
2-1 3.3
2.2
1.1
I
I
1
5
t0
n= w,
wo
Figure 9.16 Effects of n2 In 2 -1 on natural frequency
From Fig. 9.16, if the natural frequency of the LC circuit of TSC is 'C7
too
+U-
selected as 3 times of the power frequency, i.e. w,, = 3wa , the magnification factor will approximate to 1.0. And if w,, < 3w0 the magnification factor will increase rapidly. Hence, the natural frequency n is normally selected to be a value which is greater than 3 in an actual TSC system. In practice, a TCR compensator together with several parallelconnected TSC, as shown in Fig. 9.17, will constitute a hybrid SVS system, or a TCR-TSC system. Assuming that such a SVS consists of one TCR and n TSCs connected in parallel, and the susceptance of each TSC is Ll.
BCBLC
BC1
= Bc + BLC
(9.58)
328
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
vs
Figure 9.17 The configuration of TCR-TSC type SVS compensator
Then the resultant susceptance of n parallel-connected TSCs should be (9.59)
B,,, = nBCi = n BcBLC B(. + BLC
From Fig. 9.17, the resultant susceptance of TCR-TSC type SVS becomes goo
B SVS _ B (BC + BTCR)
(9.60)
Bo + BC + BTCR
As the gating angle a varys continuously between 180° and 90°, the value of B,CR will vary continuously between 0 and BL. Then the
maximum and minimum susceptance of TCR-TSC type SVS can be obtained as Bsvsm
_
Bsvsmin -
B,Bc ,
(9:61)
By + Bc
BBy°
(9.62)
BL
+ BL
If BC / BL <<1 and BL / B, <<1, then the relation between Bs,, and BWWR can be expressed approximately by the following linear equation +(1-2BC,,+BL)BTCR
Bsvs =(1a
a
(9.63)
Nonlinear Control of Static Par System
329
And the ceiling values of the susceptance of TCR-TSC type SVS can be approximated as Ba,
(9 . 64 a)
Bsvs6n =(I - BL )BL
(9.64b)
Comparing Eq. (9.50) with Eq. (9.64) we can see that the adjustable extent of susceptance of TCR-TSC is larger than that of TCR-FC type compensator.
Note From Eq. (9.46), Bsys is a nonlinear function of Bane
,
B, (Bc +BTCR ) BSVS _ B, + Bc +BTCR where, Bs,s
is
(N.1)
the resultant susceptance of SVS, B7rR the susceptance of reactor
controlled by the semiconductor switch (Thyristor), and the meanings of other symbols can be found in Fig. 9.21. The following is the process of linearization of Eq. (N.1), which will involve two steps.
Step 1. The numerator and denominator in the right side of Eq. (N.1) are divided simultaneously by B, , and the equation is rewritten as BSVS
1 + Bc + BTCR B,
(BC + BTCR )
(N.2)
We note that for Bc + BTCR
B,
<<
(N.3)
which is often the case for transmission lines, so the factor in the right side of Eq. (N.2) will be
(N.4)
-< 1.0
+BC+BTCR
B,
Considering the relation expressed by formula (N.3), we also know that
1Bc+BTCR <1
(N.5)
Bo
Thus, we can derive an alternative expression for Eq. (N.2) as follows BSVS = (1-
Bc +BBTCR )(BC
= ((1 - Bc) B,
+ BTCR )
- BTCR\ )(Bc + BTCR )
(N.6)
B,
After some algebraic calculation the Eq. (N.6) can be rewritten as
BSVS = (I - BC )BC + (1 - Bac )BTCR - BBCR BTCR a
a
(N.7)
330
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
Since the last term in the right side of Eq. (N.7) is a function of the square of BrcR , hence B,, is not a linear function of BTCR . Further linearization is needed. Step 2. If in Eq. (N.7) the following relation holds, BTCR
B,
=
BL
(N 8) T
B,
then Eq. (N.7) can be rewritten as
c +(I -
Bsvs = ( I
2B + BL )BTCR
(N.9)
,,
Eq. (N.9) shows the linear relationship between BSv, and BR, which is exactly the same as Eq. (9.49). The error e introduced by substituting //BL / B. for B7cR / B. in Eq. (N.7) is Iel = BTCR
(BTCR B0 - BL)
(N.10)
From above equation, while Brt'R is equal to its maximum value BL or its minimum value 0, Iel will be equal to zero, i.e.
Isl = 0
for B.R = BL or B^R = 0 r.+
An example is adopted to illustrate the calculation error introduced by the approximate linearization, which shows that the calculation precision of the linearization method given in Eq. (9.49) can meet the requirement of engineering practice.
Suppose B. =-6.0, BB =0.857, BL =-1.173, the calculation results are given in appendant Tab. 9.1.
-0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.173
9.4
Bsms (Exact values)
B
(Approximation)
1.0
0.979
0.735
0.761
0.495
0.543
0.269
0.325
0.058
0.107
-0.139 -0.300
-0.111 -0.299 .-+
BTCR
Errors with approximate linearization of SVS .`r
Appendant Table 9.1
CONVENTIONAL CONTROL STRATEGIES OF SVS
In order to investigate the effectiveness of different control strategies used in Static Var Compensator, SVS is represented by an equivalent circuit shown in Fig. 9.18. From above equivalent circuit the voltage applied to SVS can be written as
Nonlinear Control of Static Var System
Figure 9.18
331
Equivalent circuit of SVS Veq
(9.65)
Vsvs
1+
Bsvs Be,
If the equivalent susceptance of system B, the adjustable reactor Vs,,s can be expressed as Vsvs = Veq (1-
<< Beq, then the voltage of
Bsvs
(9.66)
eq
NCO
While SVS is turned off, i.e. B. = 0. From Eqs. (9.65) and (9.66), the terminal voltage Vs, of SVS should be equal to the equivalent voltage of system Vq , i.e. VsS(Bs s =0) = Veq . Thus, the variation of SVS voltage (i.e. the variation of the system voltage) AV is AV = Vsvs (Bsvs) - Vsvs (Bsvs = 0) = Veq (1- B
vs) _ Vq eq
Bsvs
Veq
(9.67)
eq
From above equation, there exists a linear relationship between the compensation susceptance B., and the variation of system voltage AV. If we define the gain of SVS control system as
(9.68)
KN = - Veq Beq mo.
then from Eq. (9.67) the increment of the system voltage can be obtained as
(9.69)
AV = K,,,B,,,s +0+
Note that the gain of the control system KN, is determined by the
1+T,s
Kr+KDs
F--
Bsvgn n
a
Bsvs
I
Bo
2Bc+ BJL
Bs,, - (1- BC
Bc
Brce
Figure 9.20 The conventional PID control structure of TCR-FC type SVS
Bsvymin
Bsvsmax
Figure 9.19 The fundamental control system of TCR-FC type
BQ
°
2Bc+B,
ll
- { I - Bc I Bc
BL
-r-
Ba
a
Bc
2
W W
NONLINEAR CONTROL
VRC.F
KN
Bsvsmax
V,
332 POWER
Nonlinear Control of Static Par System
333 °,,
parameters of the equivalent circuit, Peg and Beg , which are the functions of CAD
a+0
in'
the configuration of the system and the mode of operation. Hence the gain KN, is a variable, and may change within a certain range along with the variation of the operating state of the system. For a TCR-FC type SVS, the block diagram of a fundamental control system is given in Fig. 9.19. CAD
In order to hold the voltage at the connection point substantially constant, the commonly-adopted control strategy is PID control, and its structure is shown in Fig. 9.20.
C)(
It should be pointed out that the conventional control strategy which
CAD.
aims at holding the voltage at the connection point constant will also
v-'
BCD
Q..
(CD
0.y
°.y
contribute to the enhancement of the power flow limit (stability limit) of the transmission line. This type of static compensator, which is adjusted by the integral and differential algorithm according to the deviation of voltage, is in fact a voltage control device of the power system. Along with the rapid development of power systems, the ever-growing capacity of power systems and the increasing complexity of their structure, the problem of stability has become more prominent. Hence, it is necessary to investigate new control strategies, so as to make SVS not only be able to adjust the system voltage, but also play an important role in the improvement
'C3
of the stability of long-distance transmission lines under small and large disturbances. That implies we should introduce other state variables in the control laws, and apply the nonlinear control approach to the design of SVS control system.
9.5 NONLINEAR CONTROLLER DESIGN FOR SVS
9.5.1
Modeling of SVS Control Systems
The simplest case, a TCR-FC type SVS connected to the midpoint of the transmission line of a single-machine, infinite-bus system, is adopted to investigate the performance of SVS under different conditions. The principle of such a system and its corresponding equivalent circuit are shown in Fig. 9.21. From Eqs. (5.9) and (5.10), the motion equation of the generator can be written as
334
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS V
XT XL ,
M
Eq
XL2
X1
VM
X2
a
V,
4
BTCR
Br
(a) One-line diagram of SVS (b) The equivalent circuit figure Figure 9.21 Single-machine, infinite-bus system with SVS W (t) - w°
a(t) =
w(t)= H (P,
D w0
(a- coo) - Pr)
(9.70)
where s is the rotor angle of generator, co the speed of the rotor shaft in rad/s; P., Pp are the mechanical and electromagnetic power of the generator,
respectively; H is the inertia constant and, D the damping coefficient. If we neglect the electromagnetic transient period of the transmission
line and SVS devices, according to the system in Fig. 9.21(b), the electromagnetic power of generator can be expressed as P, =
E,V,
x, +x2 +x,x2B
sins
(9.71)
If we further assume that the transient voltage of the generator E' and the mechanical power P. constant, then the above extended single-machine,
infinite-bus system can be described by the following nonlinear state equations s(t) = w(t) - w0
E,V, D sins) ru(t)= 0 (Pm - (w-w°)H x, +x2 +x,x2Bsvs w°
where s(t), w(t) are state variables, and BS, is the control variable. If the control variable adopted is
(9.72)
Nonlinear Control of Static Par System
335 1
u=
(9.73)
X1 + X2 +xI x2Bsvs
Then Eq. (9.72) can be rewritten in an affine nonlinear form as X = f(X)+ g(X)u
(9.74)
where, x = [w s]T
(P. - D (w-wo))
f(X)=
g(X)
00
[__Ev. H 7sing
(w-wo)
9.5.2
0
Exact Linearization Design Approach (IQ
"i7
From Theorem 3.2 of Chapter 3, the non-singularity of the following matrix C = [g(X) adf g(X)] near X0 is the unique condition for the exact '-r
linearization of system Eq. (9.74). From the calculation formula of Lie bracket Eq. (2.47) we get
- H E9V,(w-wo)cosS- 22-T-EqV, sing
CIE
ad fg(X) _
(9.75)
E'V sins
Thus matrix C can be given as
H E',V,sinS -
E'V(w-wo)cos8- H2 E'V,sin8
C= ±'-E,'V, sing
0
H
Correspondingly, the value of the determinant of matrix C can be obtained as .".
z
det(C) = --TO E92V z sin 2 S 22
H is nonzero for all 8 and co in the Because the value of above determinant neighborhood S2 = {8, co 10
,
the nonlinear control system of SVS
can be exactly linearized for all 0
Step 1. From the system (9.74), we can obtain the following two vector fields
336
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
D, _ {g(X)}
DZ = {g(X), ad fg(X)}
where adfg(X) is obtained in Eq. (9.75). Step 2. Select two linearly independent vector fields D, and D2 and let them satisfy D, a D,, D2 ED,. Then the extended vector fields can be expressed as
) Hw0 v d12
d2,
1W. +k2(X) H
+ k, (X)
=0
0
w0 sin H E'V, v 0
E'V,(a -w°)cos8- H2 E9V,sin8 =0 H EqV, sin ,5
If we define three scalar functions k1 (X), k2 (X) and k3 (X) as follows
k,(X)=
H w°Eq'V, sin 8
H2 k2(X)co2Er2VZ s
0
Coo
Sin2.5
H2 E'V, sing) q (- H E'V,(w-w°)cos8-
H k3(X)=w°E'V, sins
then we have III
D, _ [1 01T E D,
Step 3.
D2 = [0 1]T E D2
Calculate the mapping la"
F(w, w2)=(DD' o (D "I (X,)
By solving the following differential equations dw d,v,,
[O X0
s0
dw2
we have
w=w°
8=w2+8°
Substituting them into 'Z o q)°s (X(,) will lead to solution of mapping
Nonlinear Control of Static Var System
337
.F(w,, w2)
a)=WI+a0 8=W2+80
and its inverse F-' W, = aJ - a)°
W2 =8-s°
Step 4.
Solve the derived mapping F(f)of f(X) under F-'.
Because the Jacobian matrix of F-' is an identity matrix, we have F.-'(f)
f (0) (W) = f'(°) (W )l = lf2'0) (W)
H
WI
W1
Next, we define a transformation R, as `-'
'
zi') = f2(o) (W) = WI (1)
Z2 -w2
Alb
Due to the fact that the system (9.74) is a second order system, its composite transformation T can be directly derived as Z,') =a)-)0 =AO)
ZM =8-8° =As The Jacobian matrix of T is also an identity matrix, then from Eq. (3.76), i(X) and g(X) can be given respectively as (co
=[9Pm H H IA M CO - 0), (X)1II
002
i(X) = J T (X)f (X) _
g(X)=JT(X)g(X)=[g'0X)]_
- a)0 )
- H E'Vssins
Finally, the coordinate transformation adopted is z, = w2
=,5-9.
(9.76)
Z2 =f2(X)=aJ-aJ0
And the control input applied to the nonlinear system controller is given by
1(X) + y loo
u=
(9.77)
S, (X)
Then the system Eq. (9.74) can be exactly linearized into following linear system
338
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS z, = Z2 z2 = V
(9.78)
The optimal control law with quadratic performance index for above linearized system will be
v*=-k,z, -k2z2 =-k,AS-k2Aw
(9.79)
'-n
If the weighting matrix Q of the performance index of quadratic form is adopted as Q = diag(1, 0) , and the control-weighting coefficient R is 1.0, we get k, = 1.0, k2 = . From Eqs. (9.77) through (9.79) the solution of the nonlinear control for system Eq. (9.74) is obtained as -Pm + H
AS+ H (-AS-jAw)
°°
U=
W.
(9.80)
EEVs sin 8
Substituting the above equation into Eq. (9.73), we get the nonlinear expression of SVS susceptance
E'VSsin8-(x, +x2)(Pm + H AS+Hmow) wa
wo Bsvs =
H
x,x2(Pm +HAS+
w,
W.
(9.81)
L000)
The meanings of x,,x2 can be found in Fig. 9.21. Using Eq. (9.70), we can rewrite the above equation as E-VS
Bsvs = - xx2 + x x2
H Pe
sin 8
XI
H
H
)o
wo
(9.82)
rvo
This expression gives the nonlinear control law for the Static Var compensator. And if the parameters in Fig. 9.21 satisfy the following conditions
B` <
,
Ba
then the relationship between approximately as
B, By
B.,
2
B
__
TCR
BC -BaBC
BB -2B, -BL
+
and
B,,,
Bo Bsvs
BQ -2BC -BL
can be expressed
(9.83)
The relation between BTCR and the gating angle of thyristor a is shown graphically in Fig. 9.10. And the corresponding block diagram of the
Nonlinear Control of Static Var System
339
1----------------------
Bye
Bar.
tea.
P,
;
Eq. (9. 82
S r-*
Figure 9.22
The block diagram of the nonlinear controller of SVS
nonlinear controller of SVS is illustrated in Fig. 9.22.
9.5.3
Effects of the Nonlinear Control of SVS
In order to investigate the benefits provided by the nonlinear control of SVS, the dynamic performance of compensated system in Fig. 9.21 under small and large disturbances is studied in detail with computer simulations. Three types of compensators are studied here, they are 1. Fixed shunt capacitor In simulation a fixed shunt capacitor with compensating capacity +100MVA is applied to the bus M of Fig. 9.21. 2. TCR-FC type SVS with conventional controller. A TCR-FC type SVS with reactive power ±I00MVA is connected at the bus M of Fig. 9.21. Conventional control law is expressed by the block diagram shown in Fig. 9.23. Kc
TZs+1 TZs
Figure 9.23
The block diagram of SVS conventional control system
3. TCR-FC type SVS with nonlinear controller
All the operation conditions are the same as those for conventional
340
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
controllers and the nonlinear control law given by Eq. (9.82). The computation results of the characteristic value of above three types of SVS under small disturbances are given in Tab. 9.1. Tab. 9.1 indicates that the application of nonlinear control law to SVS
will enhance the stability of the existing power system under small disturbances. Table 9.l
Characteristic values of SVS with three types compensators Control strategy
Eigenvalue
Fixed Capacitor
-0.075±j4.46
TCR-FC with conventional control
-0.1385±j6.839
TCR-FC with nonlinear control
-1.0255±j0.095
v,'
To investigate the dynamic responses of a power system under large disturbances, we assume that a three-phase short circuit fault occurs at the bus of high-voltage side of the generator at t=0s, and the fault is cleared at
(DO
CD'
t=0.1 s. Fig. 9.24 shows the response curves of the power angle of the generator, voltage at connection point and the reactive power supplied by three types of SVS respectively. The simulation result implies that in comparison with other two types of SVS, the static compensation system with nonlinear control is much more effective on improving the transient stability of a power system and on keeping the voltage at the connection point of SVS virtually constant.
Nonlinear Control of Static Var System
341
Power Angle (degree)
I;n
j\
100 F-/
/
\
\
Voltage at connection point (perunit)
L 1.0
0.5
fl V014tLU 11 Il l
P.-
Reactive power (perunit)
0.0
I
I
V 11
II
r
1
I I
Il
I
11 i_ 1
(c)
L_ 0.0
1.0
I
l
1
I
2.0
3.0
4.0
5.0
".7
Figure 9.24 Effects of different types of controllers on system performance Solid line-TCR-FC with nonlinear controller Dotted line-TCR-FC with conventional controller Dashdotted line-Fixed capacitor
342
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
9.6 REFERENCES A. E. Hammad, "Analysis of Power System Stablity Enhancement by Static Var "0'
1.
Compensators", IEEE PWRS, Vol. 1, No.4, pp. 222-227, 1986. 2.
A. Olwegard, et al., "Improvement of Transmission Capacity by Thyristor Controlled Reactive Power", IEEE PAS, Vol. 101, No. 8, pp. 3930-3939, 1981.
CIGRE Working Group 31-01, "Modeling of Static Shunt Var Systems for System
0.N
3.
Analysis", Electra, No 51, pp.45-74, 1977.
6.
K. R. Padiyar, et al., "Damping Torque Analysis of Static Var System Controllers", IEEE PWRS, Vol. 6, No.2, pp. 458-465, 1991. M. R. Iravani, "Coupling Phenomenon of Torsional Modes", IEEE PWRS, Vol. 4, No.3, pp. 881-888, 1989. S. C. Kapoor, "Dynamic Stability of Long Transmission Systems with Static
-°r
5.
,?,
4.
Compensators and Synchronous Machines", IEEE PAS, Vol. 98, No. 1. pp.124-134, 1979.
Chapter 10
Nonlinear Robust Control of Power Systems
10.1
INTRODUCTION CAD
In the previous nine chapters of this book, we explored the theory, methods, models, control laws and results of simulation for nonlinear E,'
optimal control. The research results in the field of nonlinear optimal control of power systems achieved by the authors of this book are not only in theory, but also in practice. A type of digital nonlinear optimal excitation controllers of generators (NOEC) developed by the National Key Laboratory of Power
vii
Systems in Tsinghua University, China, has been put into operation in a series of Chinese power stations. And the NOEC have played an important role in improving dynamic performance and stability of power systems. Moreover, a new type of digital nonlinear optimal governors (NOG) for hydro turbines has also been developed by the same Laboratory above mentioned. The application of the NOEC and NOG to the Three Gorge coq
Power Station is hopeful. It should be pointed out that in previous discussions of the theory and
approach, we paid little attention to the question of how to reduce the harmful influence of disturbance to a control system. The disturbances, in a broad sense, include both exogenous disturbance and internal disturbance. The internal disturbance includes the impreciseness of a mathematical model (so-called unmodelling dynamics), the measurement errors of a controller
"via
and errors of parameter values in the expression of a control law, etc. Improving the robustness of a system means enhancing the ability for attenuating the influence of disturbances to outputs of a control system. The study of nonlinear robust control of a power system is the subject matter of this chapter.
344
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS "'S
Now, let us further explain the concepts of robustness-and robust control '°')
of a system. The precise mathematical definition will be given in the subsequent sections. Coo
"ice
If the outputs of a stable closed loop system is not sensitive to disturbance inputs of the system, in other words, if the influence of a..
disturbance inputs to the outputs of a system is small enough, then we say that this system has enough robustness to disturbances. If under the action of a control, a stable closed loop system can sufficiently reduce the influence of disturbance inputs to outputs of the system, then we say this control is a robust control. 'C7
According to the discussions above, since the actual meaning of 'CS
"robustness" of a dynamic system is the system's ability to attenuate the influence of the disturbances from inputs to outputs of the system, the concrete forms of disturbances are scarcely considered in modeling and designing a robust control system. coo
From the concepts of robustness, we know that any control system .'2
under normal operation must have such two characteristics: one is stability,
and the other is, more or less, the possession of robustness. In reality, a control system without any robustness can not perform normally because the
outputs will out of their permissible region in view of engineering as an unavoidable disturbance affects the system. It is an important task for us to '.-
improve robustness of various engineering control systems to meet the needs
of continuous growth of quality requirements of control systems. That is a°)
°°+
why robust control especially nonlinear robust control has become one of the key research subjects in the field of modem control theory. In our arrangement of this chapter, we study the robust control problem
of affine nonlinear systems by starting from the linear robust control
10.2
fl.
..2
fit
problem and then turn to resolution of the nonlinear one. In a sense the linear system robust control problem could be regarded as a special nonlinear one. In this way, the discussions will be systematized.
BASIC CONCEPTS
"-3
10.2.1 L2-space To begin with, let us explain and define the concept of LZ -space. For an n-dimensional vector Z(t), if it is defined in a half open interval
Nonlinear Robust Control of Power Systems
345
[O, cc), then it canbe denoted by Z : [0, oo) a R" .
Definition 10.1 A space is called a L2 -space denoted by L 2 [0, co) , if everyvector Z : [0, oo) -a R" in the space satisfies that the improper integral (10.1)
fo IIZ(t)II2dt
is convergent, where IIZ(()II = ('-z? )u2 , and the form of the improper integral
(10.1) is called L2 -norm of Z(t). El
The definition shows an important property that if a vector function R" is a member of L"2[0,ao) that is Z(t) E LZ[0,oo), then the
Z : [0, w)
improper integral J - IJz(t)JJ2 dt must be convergent.
10.2.2 LZ-gain L2 -gain is as important to the robust control problem as quadratic performance index to the optimal control problem. L2 -gain is a quantifiable standard to judge the robustness of a system. Considering a nonlinear system described by equations of the form \.J
X(t) = f(X(t)) + g(X(t))U(t) Y(t) = h(X(t))
(10.2)
where, X E R" is a state vector; U E R"' an input vector; Y e R° an output vector; f(X), g(X) and h(X) are smooth (C") vector functions in the state space, which satisfies f(0) = 0, h(0) = 0. And assume that X(0) = 0 is an asymptotically stable equilibrium point of the system. Now, we can define L2 -gain for system (10.2).
Definition 10.2 The L2 -gain of system (10.2) is defined as T
L2-gain
$ IY(t)II2 dt )2 ( =G2= sup sup rT Te(O,=) (NU1I2d,mo IIU(t)II2 dt 1
(10.3)
Jo
In formula (10.3), III and IIUII denote the Euclid norm of output vector Y(t) and input vector U(t), respectively, "sup" means the supremum. That
346
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
is, if y = [y, Y2 ... y p ]T and U = [u, u2 ... u," ]T , then IIYjI _ (t y2 )V 2 and 1=1
Nil = (u,
)1,2
o
1=1
From (10.3) we know that the L2 -gain of system (10.2) reflects essentially the amplification of the amplitude of the system output vector to the amplitude of the system input vector for the whole dynamic process, and L2 -gain has distinct sense in engineering which can be seen clearly from the following explanation. Concerning formula (10.3), for VT > 0 , and setting T
Y2dt JJJJ
7
1
2
IJUJI
7>0
(10.4)
dt ear
'DD
one knows that formula (10.4) indicates that the influence of input to output is limited in a certain interval expressed by number y. Obviously, the less the value of y is, the less the influence of input U(r) to output Y(t) will be. The inequality (10.4) can be rewritten as
joiIYl,2dtSy2foIIUII2dt
`dT>0
(10.5)
jo(,2IIUjI2-11Y112)dt>-0
vT>0
(10.6)
or
The above two expressions, especially (10.6) will be often mentioned in subsequent sections. The discussion above suggests the following proposition.
Proposition 10.1 Given a nonlinear system as described by Eq. (10.2), VT > 0, if to (y2 11U112 -11YIl2) >- 0 holds, then the L2 -gain G2 of the
nonlinear system from input U to output Y is less than or equal to the prescribed positive number y. O In order to get a deeper understanding of the concept L2-gain, we would better discuss the relation between L2 -gain and H. norm of the transfer function matrix of a linear system [18]. Consider a linear system modeled by equations of the form X(t) = AX(t) + BU(t) Y (t) = CX(t) + DU (t)
(10.7)
where, X E R" is a state vector; U e R" a input vector; Y E R" a output vector; A, B, C and D are constant matrices with proper dimensions.
Nonlinear Robust Control of-Power Systems
347
Suppose X(O) = 0 is the asymptotically stable equilibrium point of the system.
Since the system (10.7) is assumed to be asymptotically stable at the initial state X(0) = 0 , then for every input U(t) e LZ [0,00) the output (the state response) Y(t) from the initial state X(O) = 0 exists and is a member of L2 -space too, namely Y(t) e Lz [0,00) . Thus, in the neighborhood of X(O) = 0 e R", the local input-output mapping G : LZ (0,oo) -+ LP, [O,oo) is well
defined.
Definition 10.3
The L2 -gain of a linear system described by Eq.
f ° Y(t)II' dt
G2 =
sup (° UELi [o,w)
fIIU(t)I12
N1-
(10.7) can be defined as
)z
(10.8)
dt
lo1 1UII2,**o
In the above definition (10.8), the necessity of the condition for f o IIUII2 dt # 0 is obvious; besides the condition U E C, [0,00) means that the improper integral .i IIU(t)II2 dt is convergent.
Comparing Definition 10.3 in Eq. (10.8) with Definition 10.2 in formula (10.3), it is easily seen that the former is a special case of the latter.
After knowing the definition of L2 -gain of a linear system, from the linear H. theory [5, 19], we have, so far, obtained the relation between the L2 -gain and the H. norm of a linear system. Theorem 10.1 For an asymptotically stable linear system (10.7), the H. norm of its transfer function matrix T(s) = C[sl - A]-'B + D (see also the Note below) is equal to L2 -gain, i.e. IITII. = G2 . Note
For a linear system
X(t) = AX (t) + BU (t) Y (t) = CX (t) + DU (t) the definition of the transfer function matrix is
(N-1)
Y(s) = T(s)U(s)
(N-2)
BU (s) = [sI - AIX(s)
(N-3)
X(s) = [sl - A]-'BU (s)
(N-4)
From Eq. (N-1) we have
or
348
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
where I is identity matrix. Besides, from Eq. (N-1), we have
Y(s) = CX (s) +DU (s)
(N-5)
By substituting (N-4) into (N-5) and considering Eq. (N-2), we get the following result
(C[sI - A]-'B + D)U(s) = T(s)U(s)
(N-6)
Comparing both sides of (N-6), the transfer function matrix T(s) becomes
T(s) = C[sI - A]-'B + D
(N-7)
10.2.3 Penalty Vector Function Consider an affine nonlinear system modeled by equations of the form X(t) = f(X(t)) + g, (X(t)) W + g2 (X(t))U Z(t) = h(X(t)) + K(X(t))U
(10.9)
where, X E R" is a state vector, U e R"' a control vector input; W E R' a disturbance vector input; f(X), g, (X) , g2 (X) , h(X) and K(X) are all smooth vector functions with relevant dimensions defined in state space;
=1-
oho
Z(t) E RP is a penalty variable vector of the system. The so-called penalty vector is, in fact, a kind of outputs of the system, which may represent the composition of errors and control cost. In a word, penalty vector is such a kind of outputs whose value of energy weighed usually by L2 -norm is expected to be minimum. We may naturally raise such a question: Whether we can find and how to find a control strategy such that under the disturbance W the energy of penalty output of the nonlinear system can still remain small enough or even can reach its minimum? This is the motivation for the nonlinear robust control problem, and it is the pursuit of the solution of the problem that promotes the development of the nonlinear robust control theory.
10.2.4 Dissipative Systems The concepts of dissipativity, supply rate and storage function of a
CAD
dissipative system are involved in the principle of nonlinear robust control [4, 9, 10, 13], which will be discussed in this section. In order to elucidate the essential concepts of dissipative systems, we will not touch on disturbance and penalty function problems for the moment. First let us consider a nonlinear system described in the form as
349
Nonlinear Robust Control of Power Systems
X(t) = f(X(t)) + g(X(t))U(t) Y(t) = h(X(t))
(10.10)
where, X E R" is a state vector; U E R' an input vector; Y E RP an output vector; f(X) , g(X) and h(X) are defined as smooth vector functions in the
C17
state space.
coo
Definition 10.5 The nonlinear system described by Eq. (10.10) is said to be dissipative with respect to a given supply rate S(U,Y), if there exists a nonnegative scalar function V(X) : R" R satisfying V(0) = 0 , such that the inequality V(X(T ))-V(X(T0)) < f'S(U(t),Y(t))dt
(10.11)
T >To
holds, and the function V(X) is called the storage function of the dissipative system.
0
How- to comprehend the above definition of a dissipative system? If the nonnegative storage function V(X) is taken as an energy function and S(U,Y) the energy supply rate of the system, i.e. the supply power, then the right side of (10.11) can be regarded as the supply energy gained by the system in the period of [TO,7 ] ; and the left side is the increment of storage energy in the same period of time. It is clear that for a period of time [To,71, if the increment of storage energy of the system is less than the supply energy gained by the -system, then there must be some energy dissipated in the period [7, T,] . Therefore, it is reasonable to assign to the system the
i:.
s"7
r'.
vii
'via/
attributes of dissipativity. After defining and explaining the definition of a dissipative system, let us consider a sort of dissipative system with a certain supply rate.
If system (10.10) is an asymptotically stable dissipative system at X(0) = 0, and the supply rate is (10.12)
S(U,Y) =7211U112 -1IYI12
then according to Definition 10.5 and the characteristic of V(X), i.e. V(X(T)) > 0 for T> 0 , and V(X(0)) = V(0) = 0 , the inequality 0 <- V(X(T)) <_ f o (y2IIU(t)112 - IlY(t)f12 )dt, T > 0, y > 0
(10.13)
must be held, and formula (10.13) can be rewritten as 0:- f01'a (y211U(t)112
-IIY(t)112)dt,
y>0
(10.14)
From Proposition 10.1 in Section 10.2.1 we realize what the inequality
350
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
(10.14) really means is just that the L2 -gain G2 of the nonlinear system as shown in Eq. (10.10) is less than or equal to a prescribed number y. From the above an important proposition is therefore formed as follows.
Proposition 10.2 Consider a nonlinear system as shown in Eq.(10.10). If X(0) = 0 is an asymptotically stable equilibrium point of the system, and the system is dissipative with respect to the supply rate 2IIU(t)IIZ
then the L2 -gain G2 of the system from input U to output Y is less than or equal to the S=y
- IfY(t)II2, i.e.
to (y 2IIU(t)II2 - IIY(t)II2 )dt >- 0,
prescribed positive number y.
10.3
NONLINEAR ROBUST CONTROL
10.3.1 Description of Nonlinear Robust Control From the discussion in Section 10.1, we have comprehended the meaning of nonlinear robust control. The strict definition of nonlinear robust control problem will be given in this section. Considering a nonlinear system described by the differential equations of the form
X(t) = f(X(t))+g1(X(t))W(t)+g(X(t))U(t) Z(t) = h(X(t)) + K(X(t))U(t)
(10.15)
Cs.
where, X E R" is a state vector; U E Rm a control vector; W E R' a disturbance vector; f (X) , g, (X) , g2 (X) , h(X) and K(X) are smooth vector functions defined on the state space; and Z E R° is a penalty vector. The aim of the nonlinear robust control is twofold: firstly, to find a control strategy which makes the closed loop system (10.15) be asymptotically stable at X(O) = 0 when W = 0; secondly, the control strategy can make the L2 -gain G2 from disturbance W to penalty vector CS'
Z be less than or equal to a prescribed positive number y, that is (see Proposition 10.1 in 10.2.1) f0(y2IIWII2-IIZII2)dt - 0
VT>0
(10.16)
p.'
The first purpose is the prerequisite for a nonlinear robust control system - or for any control system. The controllers that satisfy the first
Nonlinear Robust Control of Power Systems
351
°,m
(IQ
condition are called admissible controllers. Our discussions in subsequent sections are all based on the admissible controllers. The second condition is the essence of robust control issue, which indicates that the control law sought for must have the ability of disturbance attenuation. The degree of the effect of the ability is characterized by the value of y, the smaller the value of y is, the stronger the effect of disturbance attenuation will be. If y reaches its minimum, then we can get the so-called optimal robust control. From the engineering design point of view, we do not always seek for the "optimal" robust control law, but seek for the appropriate or satisfactory one. Obviously, if the value of y prescribed is too small which is less than the actually existing minimum of itself, then the robust control issue has no solution. Why the problem of nonlinear robust control is sometimes also named by some literature as nonlinear H. control? This needs to be explained. We know that there are no transfer functions for a nonlinear control system, nor, therefore, H. norm of transfer function matrix. However from previous statements we know that a control is said to be a robust one of a nonlinear system, if it is such that the L2 -gain from disturbance W to penalty Z less than or equal to a given positive number y, which could be named in short form as L2 -gain control. Since from Proposition 10.1 in ..1
..o
.fl
Section 10.2.1 we know that in linear case the H. norm IITII. precisely equals
L2 -gain,
namely,
IITIL = G2 ,
some
borrowed
authors
this
4-+
terminology of "H control" when discussing robust control problem of a nonlinear system [2, 4, 16, 23, 25]. However we prefer to use the terminology of L2 -gain control, robust control or disturbance attenuation .fl
0
control for a nonlinear system [8].
.Fr
103.2 General Form of the Nonlinear Robust Control Law For the sake of completeness, let us recall that given a nonlinear system in the form as X(t) = f(X(t))+g1(X(t))W(t)+K(X(t))U(t)
(10.17a)
Z(t) = h(X(t)) + K(X(t))U(t)
(10.17b)
where, X E R" is a state vector; U E R' a control vector; W E R' a disturbance vector, which is a harmful input whose influence should be reduced;
L()
352
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Z E R° is a penalty vector, which may include dynamic tracking errors and
the cost of the control U. So the L2-norm of Z(t) should be as small as o.,
possible. f(X), g,(X), g2(X), h(X) and K(X) are all smooth vector
functions defined on R", satisfying f(0) = 0, h(o)=O. The task of this section is to find a general form of state feedback U = U.(X)
such that as W = 0 the system is asymptotically stable at the equilibrium point X(0) = 0, and L2 -gain G2 of the system from the disturbance W to the penalty Z is less than or equal to a described positive number 7. In other words, according to Proposition 10.1, the system to be designed should satisfy the following condition (10.18)
dT > 0
.(U, W) =1o (IIZII2 - r 211W 112 )dt <_ 0
So far, the problem becomes to find the control strategy U(X) such that the inequality (10.18) holds. Now, the problem of seeking for a robust control U which renders the objective functional (called value functional) J(U, W) = !o (IIZII2 _
yzIIWII2
)dt
nonpositive for each W subject to the differential equations X(t) = f (X(t)) + g, (X(t)) W + g2 (X(t))U
within duration [0, T] and with the fixed initial condition X(0) = 0 , can be viewed as a two players, zero sum, differential game [1, 22, 24]. In the above expression of the value functional the minimizing player governs the control U and the maximizing player controls the disturbance W. From the viewpoint of mathematics the problem stated above belongs to
the field of variational problem with constraints, that is to find out the
u-.
CAD
extreme value of the functional (10.18) and the extremals W* and U * with the constraints of equation (10.17a). Our general approach to the variational problem will involve following main steps: Step 1. Set up the augmented functional i Jo
(IIZII2 -72IIWII2 +AT(f(X)+g1(X)W +g2(X)U -X))dt
where A : R" --* R is a Lagrange multiplier vector, or called a co-state vector.
Step 2. According to the above augmented functional j, write down the system's Hamiltonian function H(X, A, W, U) : R" x R" x R' x R' -> R H(X,A,W,U) =IIZII2
2IIWIi2
(10.19) + A' (t)(f (X) + 9 , (X)W + 9 2 (X)U)
353
Nonlinear Robust Control of Power Systems
By substituting (10.17b) into (10.19), we can write the function H as H(X,A,W,U)=AT(t)(f(X)+g,(X)W+gz(X)U) + Ih(X)+K(X)UII2
(10.20)
-Y2 11W11"
Step 3. Look for the expressions of the extremals W' and U' of the functional J. From variational principle we know that the extremals U' and W' of the functional J necessarily satisfies the conditions aH(X,A,W,U)
au
=o
(10.21)
aH(X,A,W,U) =0
(10.22)
cwt
aw Substituting (10.20) into (10.21) and (10.22) solving the obtained
equations for W and U, we can get the extremals W' (X, A) and U'(X,A), i.e. 12
1W'(X,A) U'(X,A)
2y
gi(X)A (10.23)
-R-'(X)(I gi(X)A+KT(X)h(X)) J
where R(X) = KT (X) K(X), and which is assumed that it is nonsingular for
each X. So R-' (X) in (10.23) exists.
In order to examine the features of the extremals W ' ((A) and U' (X, A) in Eq. (10.23), let us replace W and U in (10.20) by W' and U' in (10.23). The result is denoted by H', that is
H'(X,A)=H(X,A,W`,U')
(10.24)
Comparing the expression of H'(X,A) thus obtained with the formula (10.20), the following relation can be discovered: H(X,A.W,U) = H'(X,A)+IIU - U'If A -y2IIW - W',+2
(10.25)
where IIU-U*11' =[U-U']TR[U-U']
(10.26)
and R = K TK has been assumed to be nonsingular for each X. Obviously, the right side of the equality (10.26) is a positive definite quadratic function. And the third term of the right side of identity (10.25) (-y2IIW-w'Z) is a negative definite one; and the second term IIU -U'II2A is
354
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
positive definite. As a result when W=W*, the Hamiltonian function reaches its maximum, and when U = U` , it reaches its minimum. That is H(X,A,W,U'(X,A)) 5 H(X,A,W'(X,A)),U*(X,A))
(10.27)
5 H(X,A,W'(X,A)),U)
The relation expressed by Eq. (10.27) tells us that the (W`, U') precisely is the saddle point of the Hamiltonian function H(X,A, W, U) of the system.
Supposing there exists a differentiable (C') function V(X) : R" -- R, and replacing the co-state variable vector AT in (10.26) by the gradient vector') of V(X)
SIC
aV(x) VX=,ax
(10.28)
yields the identity (10.29)
H(x,VX,W,u)=H*(x,VX)+IIU-u'11 Suppose the function V(X) > 0, for all X # 0,
V(O) = 0, and such that
H'(X,V,t,W',U')=H'(X,VX)<-0
(10.30)
Thus, from Eq. (10.29) we know that the following inequality must be held -y211W-W*112
.-.
(10.31)
,-,
.-'
From Eq. (10.25), we can see that the Hamiltonian function H of the system reaches its minimal value as U = U*. So U` is the optimal strategy for the player in the game who demands that the value of Hamiltonian
anti
.-y
_+,
function H of the system reaches its minimum; oppositely, W' is the
'-'
f1.
"optimal" strategy for the player who demands that the function H reaches its maximum. It is clear that if U = U' and W = W', then from (10.29) and (10.30) we immediately obtain H(X VX , W,U) s 0 . As a result, the issue to
find out a
function V(X) : R" -* R such that acquiring two "optimal" strategies
H' (X, V,) S 0 for
W' (X, VX) _ « 1(X) = 2y 2 gi (X)VX
(10.32)
U*(X, VX) =«2(X) = -R-'(X)(2 gz (X)VX +KT(X)h(X))
(10.33)
C'
The gradient - v,
=
2V (x) is defined as a row vector,
ax
Nonlinear Robust Control of Power Systems
355
can be properly characterized as a "two players, zero sum, differential game " problem as proposed in above discussions. However W is the disturbance,
so the "optimal" strategy at I (X) for the maximizing player W, in fact, should be interpreted as the worst possible disturbance affecting the system.
So far we have discussed the properties of U' and W*. And the problem tends to be much clearer to find out nonnegative function V(X)
which is at least one-order differentiable (C') and V(O)=O, such that H' (X, V,)< 0. After that, based on the obtained V(X), the control law (state feedback) can be acquired according to Eq. (10.33) , which is the optimal control strategy under the worst possible disturbance W*. Now the problem
becomes: whether the control law U' =a2(X) given by Eq. (10.33) can render the systems L2 -gain less than or equal to a prescribed number y under the worst possible disturbance W' ? It is the essential question of the nonlinear robust control principle which will be answered in next section.
10.3.3 Hamilton-Jacobi-Isaacs Inequality In last section we characterized the issue of finding a control law such that the L2 -gain G2 of the system is less than or equal to a prescribed positive number by finding a nonnegative C' scalar function V(X) satisfying V(0) = 0, such that
H'(X,VX)s0
(10.34)
a1'
Equation above is well known as Hamilton-Jacobi-Issacs inequality abbreviated as HJI inequality in subsequent discussion. The specific expression of the HJI inequality will be introduced below,
which is obtained by replacing W and U in expression (10.20) with expressions about W' and U' in Eqs. (10.32) and (10.33). By some manipulating, we have
H*(X,VX,W',U') =VXf(X)+hT(X)h(X)+y2«; (X)a,(X)-«2(X)R(X)o 2(X) = VXf(X)+hT (X)h(X)+ 4 V" k (X) V.1,
where
(10.35)
356
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
i(X) = f(X) - gz (X)R-' (X)K T (X)h(X) h(X) =[I - K(X)R-' (X)K T (X)]h (X)
R(X) = z gi (X)gT(X) - gz (X)R- (X)gz (X) 7
0
So the HE inequality (10.34) can be rewritten as (X)xI (X)-«z(X)R(X)a2(X)s0 r-.
Vxf(X)+hT(X)h(X)+Y2
(10.36)
where, «1(X)and a2(X) have been defined by formulae (10.32) and (10.33), respectively. The inequality (10.36) can also be expressed in the form as VXf(X)+hT(X)h(X)+4VXR(X)V,r <0
(10.37)
`MM
>,w
where, f , h and k are given by formula (10.35). C5'
So far it has been verified that if the control strategy U* =a 2 (X) shown in Eq. (10.33) is the robust control law, i.e. which can make the system in Eq.
(10.17) have a L2 -gain equaling or being less than a prescribed number r, then the positive definite function V(X) : (R" -). R, V(O) = 0) must satisfy the ...
'C3
p.'
HJI inequality. This implies that the HJI inequality being a necessary condition for U* =« 2 (X) is the robust control strategy.
Now, let us further verify that the HJI inequality is also a sufficient condition for a 2 (X) being the robust control for the system in Eq. (10.17).
In order to prove that we begin by substituting U* =« 2(X) shown in Eq. (10.33) into state equation (10.17). Hence we obtain the closed loop system in the form of X(t)=f(X(t))+g1 (X(t))W(t)+g2(X(t))a2(X(t))
(10.38)
Z(t) = h(X(t)) + k(X(t))a 2(X(t))
(10.39)
Let us set r(X) = VX (f(X)+g2(X)% 2(X))+hT(X)h(X)
(10.40a)
+4V,,( g1(X)g; (X)+g2(X)R-'(X)g2(X))Vz CAD
Substituting expression (10.33) into (10.40a) and after some algebraic manipulation, we have
r(X)= VXf(X)+hT(X)h(X)+I VXR(X)VX From expressions (10.35) and (10.40), we realize that r(X) = H'(X, V,T )
(10.40b)
357
Nonlinear Robust Control of Power Systems
From HJI inequality (10.34), we know that H' (X, VX) <- 0, therefore r(X) <- 0. From the expression (10.40a), we can see that the r (X) includes three terms, the second and third terms are non-negative (R(X) = KT(X)K(X) has been assumed to be nonsingular and so is positive definite). Therefore, it must have
VX(f(X)+g2(X)a2(X)) <0
for each X# O. The above inequality says that aV(X) dx _
as
-V(X)<-0
aX
dt
W=0
Using Lyapunov's direct method, it is proved that the closed loop system
k = f(X)+g2(X)a2(X)
as W = 0
is asymptotically stable at X = 0 , which shows that the control U* =a 2 (X)
is an admissible one. The next step will verify that the control strategy «2 (X) can make the system in Eq. (10.17) have a L2 -gain equaling or being
less than a prescribed number r. For this purpose we write the expression of H(X,V,T , W, U) according to Eq. (10.19) as H(X, VX,W,U*) = IIZII2 -YZIIWIIZ + VI (f(X)+g1 (X)W +g2(X)U*) (10.41)
And it is clear from inequality (10.31) that
H(X,V, ,W,U')5_y2IIW-W*II2
(10.42)
The inequality (10.42) says that H(X, V,T , W, U') <- 0
Comparing the above inequality with Eq. (10.41), we immediately obtain VX
(f(X)+g,(X)W+g2(X)U')<-YZIIWIIZ -IIZII2
The above inequality means exactly
aV(x) dx ax
, dt < Y211WI12 -IIZII2
dV(X) dt
as U(X) = U' = a2(X)
as U(X) = U
(10.43)
}..
To calculate definite integrals for both side of the inequality (10.43) over the interval from t = 0 to t = T, we have
358
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
dT > 0
V(X(T))- V(X(0)) . Jo (y2IIWI12 -IIZ112)dt
(10.44)
Since V(X(T)) > 0 and V(X(0)) = 0, the inequality (10.44) becomes J0(y2JJW1I2 _JJZ112)dt?0,
y>0,U=U`
(10.45)
In view of the discussion on the problem statement in Section 10.3.1 the
above inequality confirms that the L2 -gain G2 from disturbance W to penalty vector Z is less than or equal to the given positive number y,
CAD
c0)
off,
0
in'
which precisely means that the state feedback U(X) =a 2(X) stands for the robust control strategy by which the problem of nonlinear robust control is solved. So far the verification is completed. Now, let us return to make a few notes about the HJI inequality shown by Eq. (10.37) and its solution V(X), We recognize from Eq. (10.37) that it is a nonlinear partial differential inequality with quadratic terms such as VXRV7e . The solution of it has to satisfy the condition V(X) > 0 for each X # 0 , and V(O) = 0 . Unfortunately, up to now, there is no literature that contribute any general algorithm to solve it. That research subject is still open. In subsequent sections we will discuss how to solve the problem of robust control design of an affine nonlinear control system, excitation robust control for example, in an engineering practical way.
c0)
10.4
HJI INEQUALITY OF LINEAR CONTROL SYSTEM RICCATI INEQUALITY
To make our discussion complete, let us stay for a few moment in this section to make a discussion on the linear robust control problem.
Consider a linear system described by the following differential equations of the form X(t) = AX(t) + B, W (t) + B2 (X)U(t)
Z(t) = CX(t) + KU(t)
(10.46)
'CS
where X E R" is a state vector; U E R' a control vector, W E R' a disturbance vector; and Z E R" a penalty vector; A E R"', B, E R"s, B2 E R"", C E R/'"" and K E RP'" are matrices. The problem of linear robust control is to find a control strategy, such
that the L2 -gain from disturbance input W to the penalty output Z is minimized, and therefore, the closed loop system is asymptotically stable at X(0) = 0 when W = 0.
Nonlinear Robust Control of Power Systems
359
From Definition 10.3 we know that the LZ -gain of the linear system in Eq. (10.46) from the input W to the penalty output Z is defined as G1 = NTm L =
Sup
(f
wec;io,-,
Io
11
IjZdt dt Z
fll'
)
(10.47)
IIw
-..
where T., is the transfer function of the closed loop system from W to Z. Since linear system in Eq. (10.46) can be considered as a special case of nonlinear system in Eq. (10.17), it is easy to obtain the result of linear robust control according to the nonlinear robust control approach stated in Section 10.3. The following is the proposition for linear robust control problem.
Proposition 10.3 Consider the system in Eq. (10.46) and suppose R = KTK is nonsingular. If there exists a positive definite and symmetric matrix P such that
PA+AP+C;C, +PRP <0
(10.48)
where
A = A-BZR-'KTC
C, =(I-KR-'KT)C R=
B1B1 -BZR-'B?
y>0
Z
Then, the problem of linear robust control is solved by the state feedback U(X) _ -R-' (BZP + KTC)X
The inequality (10.48) is called the Riccati inequality [15, 181.
This proposition proposes a sufficient condition for the existence of a state feedback which asymptotically stabilize the system when W = 0 and A
renders IITZW L < y .
10.5
NONLINEAR ROBUST EXCITATION CONTROL (NREC)
10.5.1 Introduction We learned from Chapter 6 that the excitation control of large generators is one of the most effective and economical techniques for improving both dynamic performance and transient stability of power
360
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
(T'
.,,
t7'
p..
coo
((DD
C/+
4=c
a.:?
systems. For this reason, researchers have been working in this field for a long time to develop various control strategies. The research results on nonlinear optimal excitation control (NOEC) have been presented in Chapter 6. However, that design approach mentioned above is based on the model with fixed structure and parameters. In power systems, exogenous and internal uncertainties (disturbances) always exist, such as sudden load shedding, generation tripping, small and large disturbances, change of parameters and network structures, etc. In order to attenuate the influence of such disturbances so as to improve dynamic performance and transient stability of power systems, many fruitful results have been achieved in
ca.
(NOD
vow
,,,
robust control of power systems [6, 11, 12, 14, 17, 24, 26-32]. In this section, a novel design approach based on a nonlinear robust model involving disturbances is explored. As stated in Section 10.3.3, the control laws being searched to solve the CS"
nonlinear robust control problem rely upon the solutions of so-called Hamilton-Jacobi-Issacs inequality. However, there is no general approach O..
5c3
and algorithm to obtain the solution of nonlinear partial differential inequality-HJI inequality. In this section, combining the state feedback
`<x
S]'
exact linearization approach with linear robust control theory, a new design approach to robust nonlinear controller design is introduced. The improved design approach is essentially based on feedback linearization principle and the control strategy is obtained for the corresponding linearization system (so-called Brunovsky normal form as discussed in Chapter 3 and Chapter 4). con
Then a significant question arises, that is, whether the control strategy obtained in terms of linear robust control principle for a linearized (via state
A..
feedback) system possesses the property of robustness for the original nonlinear system? The results developed in this section will answer this question by showing that the nonlinear robust control strategy achieved in the light of linear robust approach for the exactly linearized system is equivalent to a solution of disturbance attenuation problem for the original (f4
'L3
Gt.
nonlinear system in the sense of a kind of LZ -gain. Thus, it can be seen that this section provides an appropriate approach to solving the robust control
WM;CD
problem for an affine nonlinear system. As an example, the approach
F-+
(D'
in'
'CS
te-.
proposed and verified in this section is applied to solve the nonlinear robust excitation problem of power systems. The analytical expression of excitation control law is obtained. Moreover, the feedback variables in the expression of the control law are completely local measurements. Thus it is feasible for `CS
=ti
practical applications. The simulation results on a 6-machine system demonstrate the effectiveness of the proposed method.
Nonlinear Robust Control of Power Systems
361
10.5.2 Regulation Output Linearization Consider an affine nonlinear system described by equations of the form X = f (X) + g, MW + 912 (X)U Y = h(X)
(10.49)
where X E R", U e R', W e R", Y E R4 are state, control, disturbance and regulation output vectors, respectively; f, g g2, and h(X) are smooth vector functions with suitable dimensions, satisfying f(0) = 0 and h(0) = 0 . The problem of nonlinear robust control of system (10.49) is to find a
smallest number r' > I, and a control strategy U = U' (X) , such that for V7>1. 2dr
VT >_ 0
holds and the closed-loop system is asymptotically stable provided W=O. Here it should be pointed out that studying the nonlinear robust control problem for the system in Eq. (10.49) corresponds to studying the same problem one for the system in Eq. (10.15) with the penalty output h(X
Z(t)
0`t))1 + CU(t)]
For the sake of convenience, here we study the feedback linearization of the system in Eq. (10.49) with X E R" , W E R2 , u e R , y e R . The
results achieved can be extended without much difficulty to more general affine nonlinear systems.
Denote by g,(X)=[g,i(X) 9,2(X)] and W=fw, w,]". Now we assume that the relative degree p of the system from control u to output y is equal to n, and the relative degree related to disturbance w, and w2 are p, and p2 , respectively, and denote ,u the minimum between p, and p2 , of course p S n . Thus, by means of coordinates transformation and state feedback v = a(X) + ,l3(X)u
(10.50) V'>
Z = T(X)
the nonlinear affine system X = f(X) + g2 (X)u
y = h(X)
can be transformed into a linear system in the form
(10.51)
362
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
Z= AZ+B2v Y. =CZ
(10.52)
where h(X) ...
Lf'h(X) = T(X) _ LJh(X) LJ''h(X)
Lf'h(X) a(X) = L'fh(X) /3(X) = Lg,L;'h(X)
L'Jh(X) denotes the r"' order Lie derivative of h(X) along f(X), 0:5 r:5 n, and A, B2, and C are Brunovsky normal form as discussed in Chapter 3 and Chapter 4 [3, 7, 20, 21]. Then the system in Eq. (10.49) can be written as
Z=AZ+B2v+ a(X)g'(X)W
(10.53)
YZ =CZ Set
Lg h(X)
aX) g, Mw
=
L g LJ%-2 h(X)
L,,, L'-2h(X) J
Lg LJ -1 h(X)
L,,
J
...
W=
Lgsh(X)
Lg Lf h(X)
L,,, L' h(X)
LguL-'h(X) J
L',2 L' j' h(X)
2
I
According to the definition of the relative degree, the above matrix becomes 0 0
W = Lg L''h(X)
0 0
Lg,L"7'h(X)
Lg Lfh(X)
Lg=Lfh(X)
Lg L'J'h(X)
Lg,L'J'h(X)
Then the system in Eq. (10.53) can be written as
(10.54)
363
Nonlinear Robust Control of Power Systems
Z = AZ+B,W +B2v Yz =CZ
where B, _
)x(w)
0
O(n-N+l
(10.55)
0<w l) (, N+n
p-n
'(n-k+i)x(n-N+l)]
(1;°o
Now, we can design a controller for the system in Eq. (10.55) according
to the linear robust control theory presented in Section 10.4. The above robust control problem has a solution if the following Riccati inequality ATP+PA + Y2 PB1B; P-PB2BTp+CTC <0
has a nonnegative solution P*. Then, the optimal control strategy vi is v' _ -BZ P'Z
and the worst possible disturbance W' in (10.55) is
W' = 1 Br p'Z
(10.56)
According to (10.50) and (10.56), the control law u' in coordinate X can be written as
u' =-f-'(X)(a(X)+BZP'T(X))
(10.57)
where u' stands for the nonlinear robust control law for the original system in Eq. (10.49).
10.5.3 Analysis of Robustness of The Closed Loop System
oho
Now, let us answer the question: in what sense the control law (10.57) contributes the robustness for the original nonlinear system in Eq. (10.49)? In view of the two players-differential game principle presented in Section 10.3.2, for the nonlinear system in Eq. (10.49), the robust control problem can be formulated as follows. VT > 0, find U` that minimizes the performance index
J(U,W)=
2 2f (IlYll+IIU11-y2NWll)dt 2
(10.58)
under the worst possible disturbance W' maximizing the index as shown in (10.58), which is denoted by infsupJ(U,W) <_ 0 (10.59) Ti
W
364
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
subject to the constraint of equation (10.49), where "inf' denotes the infimum.
In a similar way to the statement above, for the linearized system mss-
(10.55), the same problem can be expressed as to find V' that minimizes the performance index J(V,W)=fr(Ily.I,,.+lIVII,_Y2II"IIZ)dt
(10.60)
under the worst possible disturbance W' , i.e. inf supJ(V, W) <_ 0 VW
(10.61)
subject to the constraint of equation (10.55). Here the "two players" are the "control" input V and the disturbance W. Then, substituting (10.54) into (10.60) and considering (10.61), we have ! (IICT(X)112 +Ila(X)+Q(X)U'II2)dt 2
ax
(10.62)
g1(X)W11 )dt <_Y2c25 IIWII2 dt
where co =supr(X), r(X) is the maximum eigenvalue of the function XER'
matrix gl'(X) a' T(X) n(X) gI (X). Inequality (10.62) tells us that U" is the
ax
ax
solution of the robust control problem of the following system X = f(X)+g1(X)W +g2(X)U ( CT(X) ) Y= a(X)+ Q(X)U)
(10.63)
Hi.
and the corresponding closed loop system has a L2 -gain (from W to Y) which is less than or equal to yc°. In the light of the above statement, we could see that the control law (10.57) is the best control strategy in the sense of the following problem infsupJ(U,W)=infsup)o(IMI2-Y2IIWII2)dt _ 0
subject to the equations of the original system in Eq. (10.49). In the above equation the Y is the regulation output of the system (10.63). Therefore, it can be concluded that the control law in Eq. (10.57) has robustness for the system in Eq. (10.49).
Nonlinear Robust Control of Power Systems
365
10.5.4 Nonlinear Robust Excitation Control The new design approach based on regulation output linearization introduced above can be applied to design robust excitation control for multi-machine systems.
10.5.4.1 The robust model Consider a multi-machine power system with n generators, each one described by a third-order dynamic model with disturbances S;=rv;-WO
(10.64)
w, = H (Pmi - P'; - P. + P..n) Eq = 1 (-Eq; + Vfi +Vd,r,) Tdo,
where 4,"
BjEysinSj
PP, =G;,Eq; +Ey;
PD, = DJ (w, -wo) wa Eq, = E,, + Id, (xd, - Xd,)
E' sin(S..-O.) j=1, jx,
91
Pds denotes the power disturbance, Vd;s the voltage disturbance of the field winding. The other symbols of variables and parameters in Eq. (10.64) have been defined in Section 6.3. The system in Eq. (10.64) can be written as follows
X = f(X)+g;(X)u; +EEQj(X)wj
(10.65)
where X = [S1 w, E,7, S2 GJ2 E92 ... S, a.
u, = V,,
01
ct -COO
...
(Pm1-41 -PDI) 0
--Eql 1
Tdo'
f(X) =
w-wo
0
g,(X) =
1
0
0
Hn 1 Eqn Tdo,,
Qjj (X) =
Tdo,
0j
...
H
E,',,]'
366
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS w,l = Pds,
w,2 = VO,s,
where 1(Tdo; is the (3i)t component of g;(X), the element gryo in QO(X) is the (3i-2+j)' component, qi10 = w0 /H; and qi20 =1/TT0, ; i=1, 2,..., n; j=1, 2.
10.5.4.2 The robust control law y.,
From Eqs. (10.64) and (10.65), we know that the description of the ia'generator can be expressed by the form as (10.66)
X; =f,(X)+g;(X)uf+q,1(X)w;l+gi2(X)w,2 CCD
where
uj=u,, X,=[S,w,Eq,1T, f,(X)=[w,-wo g,(X)=[0 0
1T
0)0-
(P.,P"-Pa
_
1
qn(X)=[0 Eo of
q,2(X)=(0 0
H,
Teo,
T
Too;E,l T
Tao,
Choose the output
y,=h,(X)=S,-So then the Lie-derivatives can be figured out as follows
Lfh,(X)=
ahaXX)f,(X)=[1,0,0]f,(X)=w;
-wo (10.67) ,.1
Lfh,(X)=aLaXX)f,(X)=10,1101f,(X)=H (Pm,-1,-PD,) and
Lghi (X)=LgLI, hi(X)=0
LgLfhi(X)# 0
where L'f, h; denotes the r`h order Lie derivative of h, along f,.
Thus we know that the relative degree p of the system from ufi to
y;
is equal to 3. Based on the statement above, we can set up the following coordinates transformation Z = T(X) as
Z' =8' -.5. z2 =Lfh,(X)=w,-coo z3 = L2rhi(X)= H (Pm, -P,, - PD,
Then, from Section 10.5.3, W can be written as
(10.68)
367
Nonlinear Robust Control of Power Systems L9 h; (X)
Lq,2h,(X)
OT(X) [q, (X) q,2 (X)]W = LI Ljh;(X) LQ2LlhX) W
L9 Lfh(X) L12Ljh,(X)
i..
where W = [w;, wi2]T . Calculating each element of the above matrix yields Lq hi (X) = Lq,h;(X) =
LgLfhi(X)=1
LgLjhi(X)=-D;roo
0
Lq,, ,hi(X)=IQ;/Tdo;
where 19; is q-axis armature electric current. So, we have
W=
0
0
1
0
W
-D,wo 1q,fTdo;
Then, via coordinates transformation (10.68) and state feedback
v; =a(X)+J3(X)u where a(X) = L ?f h; (X) ,
8(X) = Lg, Lf h; (X). The system in Eq. (10.66) can
be transformed into
Z=AZ+B,W+B2(X)v;
(10.69)
y=CZ where 0
1
0
1
0
0
0
C=[]
0
0
B,= 0 0
0]
0
0
1
0
0
.-.
000
0
A= 0
Z=[z,
0
B2= 0
11
1
Z2
1
Z3]T
For the linear system in Eq. (10.69), based on linear robust control .°'
theory introduced in Section 10.4, the optimal control strategy and the worst possible disturbance are expressed respectively as v, = -B2
P.Z = -(p31z1 + p32Z2 + P33z3 ) I-\
w; = 12B;P'Z
(10.70)
7
where P' is a positive definite symmetrical solution matrix of the following Riccati inequality
ATP+PA+2 PB,B;P-PB2BZP+CTC<0
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
368
Furthermore, from formula (6.67) we know Pi -- Eq;Iq,, so Pei = E'q;Iq, +Egilgi
.
Then from (10.50) the expression of "control" vector v can be
expressed by Eq,lq, - Egtlgt
.-.
Coo V,
wo
wt
H,
H, (Tdo; V2
v=
V, j
1q) VD
H1Tdot
(10.71) H. (Teon
ErIq -Egrti,)
co,
H d)
HfTdon
J
f BnV f
8)o
Thus, approximating D, = 0 in the light of Eq. (10.50) we obtain the nonlinear robust excitation control strategy for each generator as follows Vj=-H;Tdo;y.+E9 _Tdo;Ev [q,
wolq;
(10.72)
Iqi
Substituting (10.70) into (10.72), we obtain Vf = H;Tdoi (Ps,OSi +PszOwi +Pss coo APei)+Eq, wolgi Hi
TdoiEgi
Iv, (10.73)
I9+
`s'
Since in (10.73) the variables Eqj and 'qi are not convenient to be measured directly, the following transformation is required. Using the results of power system modeling discussed in Section 6.4.3, we have (10.74)
Eqi = V + xd,Q., / V,;
By using PC, -- Eq;lq; and substituting Eq. (10.74) into Eq. (10.73), we
finally acquire the nonlinear robust excitation control law (for P, # 0) as follows
+ (V n
J
Vii
0t) dt+PszOwi +Pss -O -AP,,) H,
Qexdi) - x' Tdoi (V j + a Vii
xdi Pei
Qexdi )2 d ( Vu
dt
72
SIC
COOP,;
I0.
Vr = HiTdoi (V, +Qeixdi )(Pat
Puyii
(10.75)
)
+Qixdi
where V,,, P; , Qe; and co, are terminal voltage of generator, active electrical power, reactive power and rotor speed, respectively, which can be measured directly and locally.
10.5.5 Simulation Results The system under study has been shown in Fig. 6.10 and the data of the
Nonlinear Robust Control of Power Systems
369
CS'
system is included in Table 6.2. In order to investigate the effectiveness of the different types of excitation controllers, we will make comparison among the following control configurations. Configuration 1: No.2-No.5 generators are the PSS-equipped machines, the transfer functions of PSSs are given by expression (6.76); Configuration 2: the above-mentioned generators are equipped with linear optimal control designed by LQR approach with the feedback gains of LOEC given by Eq. (6.77); Configuration 3: the generators are equipped with the nonlinear optimal excitation controllers (NOEC), the control law is shown in Eq.
(6.75). Finally, the generators are equipped with the nonlinear robust excitation controllers (NREC) proposed in this section, and the control 0
strategy is given by Eq. (10.75). A three-phase temporary short circuit is assumed to occur on No.11 bus (see Fig. 6.10) for 0.15 seconds and then the faulted line is tripped without
coo
'n.
reclosure. The simulation results are shown by Figs. 10.1 through 10.4, which indicate the rotor angle responses for the four control configurations respectively. Furthermore, Fig. 10.5 shows the simulation results for the control configuration that No.2-No.5 generators are equipped with the nonlinear robust controllers proposed in (10.75), whose design values of the parameters H, xd and Tdo are intentionally given 50% errors. The purpose
of the action taken is to test how the robust control strategy works/in attenuating the disturbances. in'
We can see from Figs.
10.1
and 10.2 that the system lost its
Coy
coo
'17
C].
'=r
synchronism soon after the fault occurs when linear PSS or LOEC is used. The system remains stable by employing the NOEC or NREC (see Figs. 10.3 and 10.4). Comparing Fig. 10.4 with Fig. 10.3, we can see that the dynamic responses are better as the NREC schemes are used. Moreover, from Fig. 10.5 it can be clearly seen that in spite of the 50% parameter errors, the system still has excellent dynamic performance.
10.5.6 Summary In this section, a new design approach to affine nonlinear robust control has been developed, which is mainly based on robust control principle and feedback linearization. As an important example, it has been successfully applied to nonlinear robust excitation control of multi-machine systems. All of the variables appearing in the expression of the control law are local measurements. This fact implies that the robust excitation control is decentralized. Simulation results for a six-machine system verifies the effectiveness of the nonlinear robust excitation control strategy proposed in this section.
370
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS
150 100
50
rotor
LL]
rotor
1(!
-200 -250 0
2
1
3
4
0
5
1
Figure 10.1
2
3
4
5
time(second)
tine(second)
Figure 10.2 Dynamic response of the system with LOEC
Dynamic response of the, system with PSS
150
rotor angle(degree)
rotor angle(degree)
100
v
50 0
-50
S
2
1
4
3
0
1
5
Figure 10.3 Dynamic response of the system with NOEC
Figure 10.4
150 100 I
50
rotor
0
1
I
0
1
2
3
4
2
3
4
5
t ae (second)
tae (second)
5
t ine(second)
Figure 10.5 Dynamic response of the system with NREC under the values of parameters having 50% errors
Dynamic response of the system with NREC
Nonlinear Robust Control of Power Systems
371
10.6 REFERENCES
3. 4.
`ti
A. Friedman, Differential Games, Wily-Interscience, New York, 1971.
A. Isidori, "H Control via Measurement Feedback for Affine Nonlinear Systems",
Caw
1.
2.
International Journal of Robust and Nonlinear Control, Vol.4, pp. 553-574, 1994. A. Isidori, Nonlinear Control Systems: An Introduction (3nd Edition), Springer-Verlag, New York, 1995. A. Isidori and A. Astolfi, "Disturbance Attenuation and H. Control via Measurement Feedback in Nonlinear Systems", IEEE Trans. AC, Vol. 37, No. 10, pp.1283-1293, 1992.
B. A. Francis, A Course in H. Control Theory, New York: Springer-Verlag, 1987.
7.
Dissipativity Framework for Power System Stabilizer Design", IEEE Trans. PS, Vol. 11, No. 4, pp. 1963-1968, November, 1996. D. Cheng, T. J. Tam and A. Isidori, "Global Linearization of Nonlinear Systems Via Feedback", IEEE AC, Vol. 30, No. 8, pp. 808-811, 1985.
yam
dog
Stability", IEEE Trans. AC, Vol. 25, pp. 931-936, 1980. D. J. Hill and P. J. Moylan, "Dissipative Dynamical Systems: Basic Input-Output and State Propertiies", J.Frankin Inst, Vol. 309, pp.327-357, 1980. D. J. Hill and P. J. Moylan, "The Stability of Nonlinear Dissipative Systems", IEEE Trans. AC, Vol. 21, pp. 708-711, 1976. J.
Hill, "Nonlinear Output Stabilization Control for oro
cow
G. Guo, Y. Wang and D.
woo
11.
300
10.
°q'
9.
D. J. Hill and P. J. Moylan, "Connections between Finite-Gain and Asymptotic .c.
8.
C. A. Jcobson, A. M. Stankovic, G. Tadmor and M. A. Stevens, "Towards a
C10
'.0
5. 6.
`"a
N-.
Multimachine Power Systems", IEEE Trans. Circuit and Systems, Part 1, Vol. 47, No. 1, pp. 46-53, 2000. 12. H. Jiang, H. Cai, J. F. Dorsey and Z. QU. "Towards a Globally Robust Decentralized Control for Large-Scale Power Systems", IEEE Trans. Control Systems Technology, Vol. 5, pp.309-319, 1997. 13. J. C. Willems, "Dissipative Dynamical Systems, Part I: General Theory", Arch. Rat. Mech. Anal., Vol. 45, pp. 321-351, 1972. 14. J. C. Willems, "Dissipative Dynamical Systems, Part II: Linear Systems with Quadratic Supply Rates", Arch. Rat. Mech. Anal., Vol. 45, pp.352-393, 1972. 15. J. Doyle, K. Glover, P. Khargonekar, B. A. Francis, "State-Space Solution to Standard HZ and H, Control Problem", IEEE AC, Vol. 34, No. 8, pp.831-842, 1989. 16. J. William Helton, Matthew R. James, Extending H. Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives, SIAM, Philadelphia, GYM
4)y
M-.
..O
1999.
18. 19.
Control System", IEEE Trans. EC, Vol. 7, No. 1, pp. 108-113. 1992. K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control, Upper Saddle River, NJ:Prentice-Hall, 1996. M. James, "Computing the H. Norm for Nonlinear Systems", Proc. 12'h IFAC World Congr, Sydney, Australia, 1993.
Q. Lu, S. Mei, T. Shen and W. Hu, "Recursive Design of Nonlinear H. Excitation Controller", Science in China(series E), Vol. 43, No. 1, pp23-31, 2000. Q. Lu, S. Mei, W. Hu and Y. H. Song, "Decentralized Nonlinear H. Excitation Control Based on Regulation Linearization", IEE Proc-Gener. Transm. Distrib., Vol 147, No. 4, pp245-251,2000. R. Issacs, Differential Games, Wiley, New York, 1965. R. Marino, W. Respondek, A. J. van der Schaft and P. Tomei, "Nonlinear H. Almost Disturbance Decoupling", System & Control Letters. Vol. 23, pp. 159-168, 1994. I17
21.
K. Ohtsuka, T. Taniguchi, T. Sato et. al., "A H. Optimal Theory-Based Generator
.>>
20.
0
17.
22. 23.
s,,
r_2
372
NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS
25.
Van Schaft, "L2 -Gain Analysis of Nonlinear Systems and Nonlinear State Feedback H. Control", IEEE Trans. AC, Vol. 33, No. 6, pp.770-784, 1992. Y. Wang and D. J. Hill. "Robust Nonlinear Coordinated Control of Power Systems", Automatica, Vol. 32, No. 9, pp.611-618, 1996
ate,
...
S. Chen and O. P. Malik, "H., Optimization-Based Power System Stabilizer Design", IEE Proc.-Generation, Transmission and Distribution, Vol. 142, No.2, pp.179-184,
UWa
24,
1995.
5'0
dCab
yo,
O°j
26.
Y. Wang, D. J. Hill and G. Guo, "Robust Decentralized Control for Multimachine
28.
Power Systems", IEEE Trans. Circuits and Systems, Vol. 45, No. 3, pp. 271-279, 1998. Y. Wang, G. Guo and D. Hill, "Robust Decentralized Nonlinear Controller Design for Multimachine Power Systems", Automatica, Vol. 33, No. 9, pp. 1725-1733, 1997.
29.
Y. Wang, L. Xie and C. E. De Souza, "Robust Control of a Class of Uncertain
¢C4
p..,
0.'
^f7
ltd
chi ooh
Nonlinear Systems", Systems & Control Letters, Vol. 19, 139-149, 1992a. Y. Wang, L. Xie, D. J. Hill and R. H. Middleton, "Robust Nonlinear Controller Design for Transient Stability Enhancement of Power Systems", Proc. 31" IEEE Conf. On Decision and Control, Tuson, Arizona, pp. 1117-1122, 1992. Z. Qu, "Robust Control of a Class of Nonlinear Uncertain System", IEEE. Trans. AC, Vol. 37, pp. 1437-1442, 1992. .-;
32.
'.' tit
...j
..r
ran
On-
Ga.
f'1
31.
[7'1
.ti
X23
30.
:co,
27.
Z. Qu, J. F. Dorsey, J. Bond. and J. McCalley, "Robust Transient Control of Power Systems", IEEE Trans. Circuits and Systems, Vol. 39, pp.470-476, 1992.
Index
rotor angle, 167
W
rotor speed, 167
wo
synchronous speed, 167
Hs
control, 351
H
norm,351
L2
-gain, 345
fn-
8
generator, 174 Xq
q-axis reactance of a generator, 174
d x transient inductance of generator, 175
-space 344
AC and DC side filters, 191
d-axis current, 172
if
excitation current, 174
Iq
q-axis current, 172
Admittance matrix, 182 Advanced steam valving, 186 Affine nonlinear control systems, 33 Apparent output power, 177 Armature reaction, 177 Asymptotically stable, 93
Vd
d-axis voltage, 172
Vq
q-axis voltage, 172
Eq
idling electrical potential, 174 damping constant, 168 inertia coefficient of a generator
D
H
I
'L1
L2
Id
Brunovsky normal form, 62
set, 168
VREF
reference node voltage, 183
Tdo
time constant of field winding, 186
V/kl
total flux linkage of excitation winding, 177
Ey
transient electric potential, 175
Me
a°°
MD
mechanical input torque, 167 the electric output torque, 167 damping torque, 167
°_'
PP
active electrical power, 169
Pm
mechanical power, 169
PD
damping power, 169
Qe
reactive power, 177
Vf
voltage of field winding, 185
V,s
voltage of the infinite bus, 179
xd
d-axis synchronous reactance of a
Closed loop system, 201 Composite transformation, 85
I!1
M.
Computer simulations, 259
Conditions of exact linearization via state feedback, 73 Constant DC current control, 196 extinction angle, 194 extinction angle control, 197 impedance, 181 Control strategy or control law, 187 valve (CV), 189 Conventional control, 256 Converter station, 277
Can
000
Nonlinear Control Systems and Power System Dynamics
374
Excitation flux linkage, 178 Exogenous disturbances, 109 External dynamics, 91
Transformation, 25 mapping, 36 Cylindrical rotor machine, 176
Fast valving, 188 controller, 187 Feedback variable measurements, 217 Field winding, 166 Firing circuits, 196 Flux linkage, 171 Flyballs, 245 Frobenius theorem, 60 P.'
0'0
0°0
..+
Implementation of nonlinear excitation .N.
control, 217 cap
Index number, 151 Inductance, 170 a0.
oµ'
cop
Initial steady state generator active power, 220 Instantaneous peak values of phase voltage and current, 170 Integral curves, 153 Intercept valve, 247 Internal dynamics, 91 Inverse mapping or transformation, 38 Inverter, 191 Involutivity, 26 Jacobian matrix, 31 t10
0
transient process, 223 Exact linearization, 26
Ideal generator, 171 Impedance angle, 184
>G.
185
HVDC , 277
.O.
Effective value, 170 Effects of the valve opening limit, 188 Electrical-hydraulic, 245 transducer, 187 Electromagnetic dynamic equation for field winding,
Hamiltonian function, 64 Hamilton-Jacobi-Isaacs (HJI), 356 High-pressure regulated valve or steam controlled valve, 188 High-pressure, medium-pressure and low-pressure turbines , 187
.fl
d, q coordinates of a synchronous generator 27 Damping effects, 177 windings, 174 DC control system, 196 transmission, 191 DDP, 109 Dead band, 245 Demagnetization, 177 Derived mapping, 36 Diffeomorphism, 25 Dissipative systems, 348 Disturbance decoupling, 61 Dynamic characteristic, 303 deviation, 93 equations, 247 performance, 205 response, 202
r=7
transformer, 191 circuit of inverter, 193 process, 192 Coordinate
via state feedback, 60 conditions, 87 design method, 59
Large-capacity steam turbine generator set, 187
Index
375
linearized normal form, 50 linearly independent vector fields, 152 Line-to-earth capacitance of DC transmission line, 192 Load compensator, 313 Local diffeomorphism, 30
c,y
Penalty function, 3428 Performance index, 2 Per-unit instantaneous output power, 170 Phase controller, 196 Physical dynamic simulations, 222 PID (proportional integral differential), 200 Power fraction, 187 modulation, 284 Power System Stabilizer (PSS), 205 .-.,
(LOEC), 222 quadratic regulation (LQR), 63
Oil-pressure control signals, 187 Oil-servomotor, 247 One inertia block, 189 One-machine, infinite-bus system, 17
poi
nee
Large-disturbance stability, 222 Liebracket, 25 derivative, 25 Limit block, 188 Linear multi-variable control, 205 Linear Optimal Excitation Controller
Prime movers, 245 Magnetic circuits, 170
q-axis transient electric potential, 178
conductivity, 170
Mathematical model of multi-machine power systems, 223
Reactive
Mechanical power, 209 Mechanical-hydraulic, 245
power flow, 310 Reactor, 310 Rectifier, 191 transformer, 193 ,O,
co.
MKS units, 170 Multiple inputs and multiple outputs (MIMO), 121
Reheater, 246 Relative degree, 26 Riccati algebraic equation, 65 p».
121
°F'
medium-pressure regulated valve or fast valve, 187 MIMO affine nonlinear control systems,
power compensation, 282
Robust control, 343 Robustness, 343 Rotor windings, 170
Natural power, 316
Salient-pole generator, 183
Nominal operation state, 197 Nonlinear excitation control law, 217 optimal excitation control (NOEL), 223 state feedback, 209 Normal affine nonlinear system, 209 Normal opened contact, 188
Self admittance, 182 Self inductance, 170 inductive reactance, 174
0'!
Mutual inductance, 170
chi
Servo mechanisms, 245 systems, 245
Signal amplifier, 196 SISO (single input single output), 59
C'7
Nonlinear Control Systems and Power System Dynamics
376
Variational method, 63
_a'
£,'
affine nonlinear system, 59 Small disturbance stability, 205 Smoothing reactors, 191
Vector
.lam
:;,
.G+
Speed resolution, 245 State feedback, 59 law, 94 Static reactive power compensator, 309 Static Var system (SVS), 309 Stator current, 170 windings, 170 Steady state, 196 Steam flow, 246 s..,
^-.
G..
.'+
turbine, 187 turbine with reheater, 187 valving control system, 189 Storage function, 348 Surge-impedance loading(SIL), 316 Swing equations, 166 Symmetric three-phase system, 170 Synchronous operation, 176 System compensator, 314 Terminal voltage, 170 Three-phase controllable bridge-type circuit, 192 short circuit fault, 240 Thyristor -switched capacitor (TSC), 325 -controlled reactor (TCR), 319 Time lags, 245 Time scale of electromechanical transient process of a power system, 189 Transformation matrix, 107 Transformer electric potential, 174 Transient .n»
a0.
«0.
.-+
...
C/)
gyp'
fl'
.fl
`;P
E--
state, 178 !-+
,f+
stability of power systems, 186
Two-phase short circuit fault, 256 Valve opening, 187
field, 26 field sets, 26 Voltage losses, 176 stability, 282 -current characteristics, 278
Water hammer, 245 Whole region of generator operation state, 189
Zero dynamics design method, 91
