Power System Dynamics and Stability

Power System Dynamics and Stability PETER W. SAUER M. A. PAl Department of Electrical and Computer Engineering Univers...

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Power System Dynamics and Stability

PETER W. SAUER M. A. PAl Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign

Prentice Hall Upper Saddle River, New Jersey 07458

Library of Congress Catalogi ng-i n-Pu blication Daw Sauer, Peler W. Power system dynamics and stabi lity / PeTer W. Sauer and M. A. Pai. p. em . Incl udes bibliographical references and index.

ISBN 0-1J-<;78830-0 I. Electric power system s tability. 2. Electric machinery, Synchronous-Mathematical models. 3. Electric power stystemsControl. L Pai, M. A. 11. Title.

TK1010.s38

1998 97- 17360

621.31 '01 ' l ---dc2 1

C IP

Acquisitions editor: ERlC SVENDSEN Editor-in·chief: MARCIA HORTON Managing editor: BAYAN/ MENDOZA DE LEON Director of productio n and manufact uring: DIl V/D W RICCARDI Production editor: KATflARITA LAMOZA Cover director: JAYNE CONTE Manufacturing buyt r: JULIA MEEHA N EdhoriaJ assistant : ANDREA AU © 1998 hy Prentice-HalL Inc. Simon & Schuster I A Viaco m Company Upper Saddle Ri ver, New Jersey 07458 All righT S reserved . No p3I1 of this book lIlay be reproduced. in any fonn or by any means. without permission in writi ng from the publisher. The all1ilor and publisher of this book have used their beSI efforts in prepanng Ihis book. These efforts include the development. research, and testing of the theories and programs to determine their effect iveness. T he author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or The documentation contained in {his book. The nuthor nod publis her shaJl not be liable in any even! for incidental or consequential damages in connection with. or arisiog OOl of, the furnI shing. perfonnancc. or use of these programs. This boo k waS TEX I:;

~

p n~pared

using LATEX and TEX. It was publi shed from TEX fi les p repared by the authors.

lrademark of the Ameri can Mathematical Society.

Printed in the Uflited States o f America

10 9 8 7 6 5 4 3 2

ISBN

0-13-678830-0

Prentice-Hall Imcrnational ( UK) Limited, London Pre ntice-Hall o f Austr.llia Ply. Limited , Sydney Prentice-Hal l Canada Inc., Toront o Prentice· Hall Hispanoamericana, S_A .. Mexico Prentice-Hall of India Pm'ate Limited, New Delhi Prentice-Hall of Japan. Inc. , Tokyo Simon & Schuster Asia Pte. Ltd., Singapore EdilOra. Pn::ntice·l-laH do Brasil, Ltda., Rio de Janeiro

To Sylvia and N andini

Contents Preface

Xl

1 INTRODUCTION 1.1 Background .. . 1.2 Physical Structures .. 1.3 Time-Scale Structures 1.4 Political Structures .. 1.5 The Phenomena of Interest

1

2

7

3

4

1

3 4 5 6

ELECTROMAGNETIC TRANSIENTS 2.1 The Fastest Transients .. 2.2 Transmission Line Models 2.3 Solution Methods 2.4 Problems . . . . . . .. .

7 8

13 20

SYNCHRONOUS MACHINE MODELING 3.1 Conventions and Notation . . . 3.2 Three-Damper-Winding Model" 3.3 Transformations and Scaling . . 3.4 The Linear Magnetic Circuit 3.5 The Nonlinear Magnetic Circuit. 3.6 Single-Machine Steady State . . . 3.7 Operational Impedances and Test Data 3.8 Problems . . . . . . . . . . . . . , .. .

23

SYNCHRONOUS MACHINE CONTROL MODELS 4.1 Voltage and Speed Control Overview 4.2 Exciter Models . . . . . . 4.3 Voltage Regulator Models

65 65 66 70

v

23 24 26 35 43 50 54

61

vi

CONTENTS 4.4 4.5 4.6

5

6

77 83

87

SINGLE-MACHINE DYNAMIC MODELS

89

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

89

Terminal Constraints . . . . . . . . . .. . The Multi-Time-Scale Modcl . . . . . . . Elimination of Stator/ Network Transients The Two-Axis Model . . . . . . . The One-Axis (Flux-Decay) Model The Classical Model .. . .. Damping Torques. .. . . . . . Synchronous Machine Saturation Problems ........... .

MULTIMACHINE DYNAMIC MODELS 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

7

Turbine Models . . . . Speed Governor Models Problems ... ... .

The Synchronously Rotating Reference Frame. Network and R-L Load Constraints . . . . Elimination of Stator/Network Transients Multimachine Two-Axis Model Multimachine Flux- Decay Model Multimachine Classical Model . Multimachine Damping Torques Multimachine Models with Satu.ration Frequency During Transients . . . . . Angle References and an Infinite Bus.

93 96 101 104 106 107 113 120 123 123 126 128 139 142 146 149 150 156 157

MULTIMACHINE SIMULATION

161

7.1 7.2

161 165 165 165 166 167 167 169 178 183

7.3

7.4 7.5 7.6

Differential-Algebraic Model . Stator Algebraic Equations 7.2.1 Polar form . . . . . 7.2.2 Rectangular form .. . 7.2.3 Alternate form of stator algebraic equations Network Equations . . . . . . 7.3.1 Power-balance form 7.3.2 Current-balance form Industry Model . . . . . . . . Simplification of the Two-Axis Model Initial Conditions (Full Model) . . .

188

CONTENTS 7.7

8

9

Vll

Numerical Solution: Power-Bala.nce Form 7.7 .1 SI Method . . . . . 7.7.2 PE method .. 7.8 Numerica.l Solution: Current-Balance Form 7.9 Reduced-Order Multimachine Models . 7.9.1 Flux-decay model . . . . . 7.9.2 Structure-preserving classical model 7.9.3 Internal-node model 7.10 Initial Conditions 7.11 Conclusion .. 7.12 Problems

198 199 202 203 206 207 209

SMALL-SIGNAL STABILITY 8.1 Background .. . .. . 8.2 Basic Linearization Technique . 8.2.1 Linearization of Model A 8.2.2 Linearization of Model B 8.3 Participation Factors . 8,4 Studies on Parametric Effects 8 .4.1 Effect of loading 8.4.2 Effect of KA 8.4.3 Effect of type of load . 8.4.4 Hopf bifurcation 8.5 Electromechanical Oscillat ory Modes 8.6 Power System Stabilizers. .. 8.6.1 Basic approach .. 8.6 .2 Derivation of K1-K6 constants [78 ,88] 8.6.3 Synchronizing and damping torques 8.6.4 P ower system stabilizer design 8.7 Conclusion .. . . . . 8.8 Problems

221 221 222 223 234 235 241

ENERGY FUNCTION METHODS 9.1 Background 9.2 Physical and Mathematical Aspects 9.3 Lyapunov's Method 9.4 Modeling Issues . .. 9.5 Energy Function Formulation .. 9.6 Potential Energy Boundary Surface (PEBS)

283 283 283 287 288 291

214 216 218 218

242 244 244 247 249

254 254 255 262 270 277 277

294

CONTENTS

VIIl

9.6. 1 Single-machine infinite-bus system . . . . . . . . . . . 294 9.6. 2 Energy function for a single-machine infinite-bus system298 9.6.3 Equal-area criterion and the energy function .. .. 302 9.6.4 Multimachine PEBS . . . . . . . . . . . . . . . . .. 306 9.6.5 Initialization of VPE(O) and its use in PEBS method 309 9.7 The Boundary Controlling u.e.p (B CU ) Method. 313 9.8 Structure-Preserving Energy Functions . 318 9.9 Conclusion 319 9.10 Problems . . . . . . . . . . 320

A Integral Manifolds for Model A. l Manifolds and Integral Manifolds . . . . . A.2 Integral Manifolds for Linear Systems .. A .3 Integral Manifolds for Nonlinear Systems

323 323 324 335

Bibliography

341

Index

353

PREFACE The need for power system dynamic analysis has grown significantly in recent years. This is due largely to the desire to utilize transmission networks for more flexible interchange transactions. While dynamics and stability have been studied for years in a long-term planning and design environment, there is a recognized need to perform this analysis ill a weekly or even daily operation environment. This book is devoted to dynamic modeling and simulation as it relates to such a need, combining theoretical as well as practical information for use as a tex:t for formal instruction or for reference by working engineers. As a text for formal instruction, this book assumes a background in electromechanics, machines, and power system analysis . As such, the text would normally be used in a graduate course in electrical engineering. It has been designed for use in a one-semester (fifteen-week), three-hour course. The notation follows that of most traditional machine and power system analysis books and attempts to follow the industry standards so that a transition to more detail and practical application is easy. The tex:t is divided into two basic parts. Chapters 1 to 6 give an introduction to electromagnetic transient analysis and a systematic derivation of synchronous machine dyuamic models together with speed and voltage control subsystems. They include a rigorous explanation of model origins, development, and simplification. Particular emphasis is given to the concept of reduced-order modeling using integral manifolds as a firm basis for understanding the derivations and limitations of lower-order dynamic models. An appendix: on integral manifolds gives a mathematical introduction to this technique of model reduction . Chapters 6 to 9 utilize these dynamic models for simulation and stability analysis. Particular care is given to the calculation of initial conditions and the alternative computational methods for simulation. Small-signal stability analysis is presented in a sequential IX

x

PREFACE

manner, concluding with the design of power system stabilizers. Transient stability analysis is formulated using energy function methods with an emphasis on the essentials of the potential energy boundary surface and the controlling unstable equilibrium point approaches . The book does not claim t o be a complete collection of all models and simulation techniques, but seeks to provide a basic understanding of power system dynamics. While many more detailed and accurate models exist in the literature, a major goal of this book is to explain how individual component models are interfaced for a system study. Our objective is to provide a firm theoretical foundation for power system dynamic analysis to serve as a starting point for deeper explorat ion of complex phenomena and applications in electric power engineering. We have so many people t o acknowledge for their assistance in our Careers and lives that we will limit our list to six people who have had a dhect impact on the Un.iversit y of illinois power program and the preparation of this book : Stan Helm, for his devotion to the power area of electrical engineering for over sixty years ; George Swenson , for his leadership in strengthen.ing the power area in the department ; Mac VanValkenburg, for his fatherly wisdom and guidance; David Grainger, for his financial supp ort of the power program; Petar Kokotovic, for his inspiration and energetic discussions; and Karen Ch.itwood, for preparing the manuscript . Throughout our many years of collaboration at the University of Ulinois, we have strived to m aintain a healthy balance between education and research . We th ank t he Universit y administration and the funding support of the National Science Foundation and the Grainger Foundation for making this possible.

Peter W. Sauer and M. A. Pai Urbana , Ulinois

Chapter 1

INTRODUCTION 1.1

Background

Power systems have evolved from the original central generating station concept to a modern highly interconnected system with improved technologies affecting each part of the system separately. The techniques for analysis of power systems have been affected most drastically by the maturity of rugital computing. Compared to other rusciplines within electrical engineering, the foundations of the analysis are often hidden in assumptions and methods that h ave resulted from years of experience and cleverness . On the one hand , we have a host of techniques and models mixed with the art of power engineering and , at the other extreme, we have sophisticated control systems requiring rigorous system theory. It is necessary to strike a balance between these two extremes so that theoretically sound engineering solutions can be obtained. The purpose of this book is to seek such a middle ground in the area of dynamic analysis . The challenge of m odeling and simulation lies in the need to capture (with minimal size and complexity) the "p henomena of interest." These phenomena must be understood before effective simulation can be performed. The subject of power system dynamics and stability is clearly an ext remely broad topic with a long history and volumes of published literature. There are many ways to ruvide and categorize this suhject for both education and research. While a substantial amount of information about the dynamic behavior of power systems can he gained through experience working with and testing individual pieces of equipment, the complex problems and operating pr actices of large interconnected systems can be better understood if 1

CHAPTER 1. INTRODUCTION

2

this experience is coupled with a mathematical model. Scaled-model systems such as transient network analyzers have a value in providing a physical feeling for the dynamic response of power systems, but they are limit ed to small sizes and are not flexible enough to accommodate complex issues. While analog simulation techniques have a place in the study of system dynamics , capability and flexibility have made digital simulation the primary method for analysis. There are several main divisions in the study of power system dynamics and stability [IJ. F. P. deMello classified dynamic processes into three categories: 1. Electrical machine and system dynamics

2. System governing and generation control 3. Prime-mover energy supply dynamics and control In the same reference, C. Concordia and R . P. Schulz classify dynamic studies according to four concepts: 1. The time of the syst em condition: past, present, or future

2. The time range of the study: microsecond through hourly response 3. The nature of the system under study: new station, new line, etc. 4. The technical scope of the study: fault analysis , load shedding, subsynchronous resonance , etc.

All of these classifications share a common thread: They emphasize that the system is not in steady state and that many models for various comp onents must be used in varying degrees of detail to allow efficient and practical analysis . The first half of this book is thus devoted to the subject of modeling, and the second half is devoted to the use of interconnected models for common dynamic studies. Neither subject receives an exhaustive treatment ; rather, fundamental concepts are presented as a foundation for probing deeper into t he vast number of important and interesting dynamic phenomena in power systems.

1.2. PHYSICAL STRUCTURES

1.2

3

Physical Struct ures

The major components of a power system can be represented m a blockdiagr am format, as shown in FigUIe 1.1.

Mechanical

Press ure Control

Supply Control

Fuel Source

-.

t

Fuel

Furn.ace &

Boil er

......-'--1.. ~

Speed Control

r---.

t

Steam

Electrical

Voltage Control

Turbine

r.

-...

t

Torque

Generator

.. .. Network Control

Load Control

Network

Loads

t )1.

i

t

P,Q

t Energy Control Center

Figure 1.1: System dynamic structure While this block diagram representation does not show all of the complex dynamic interaction between components and their controls, it serves to broadly describe t he dynamic structures involved . Historically, there has been a major div ision into the mechanical and electrical subsystems as shown. This division is not absolute, h owever , since the electrical side clearly contains components with mechanical dynamics (tap-changing-under-Ioad (TCUL) transformers, motor loads , etc.) and the mechanical side clearly contains components wi t h electrical dynamics (auxiliary moto r drives , process controls, etc .). Furthermore , both sides are coupled through the monitoring and control functions of the energy con tral center.

CHAPTER 1. INTRODUCTION

4

1.3

Time-Scale Structures

Perhaps the most imp ortant classification of dynamic phenomena is their natural time range of response. A typical classification is shown in Figure 1.2 . A similar concept is presented in [6J.

I Lightni ng Prol'sgation

I

I Switching Su rges

Slator Transients and Subsynchronoua Rcsonwl cc

Tran.~ienl

Stability

I Governor and Load Fre quency Con lmi

I Doil er and Lons· Telm Dynamics

I 1()-7

I ()-5

1()-3

0. 1 TIme

10

lQ3

(~ec)

Figure 1.2: Time ranges of dynamic phenomena This time-range classification is important because of its impact on component modeling. It should be intuitively obvious that it is not necessary t o solve the complex transmission line wave equations to investigate the impact of a change in boiler con t rol set points. This brings to mind a statement made earlier that "t he system is not in steady state." Evidently, depending on t he nature of the dynamic dis t urbance, portions of the power system can be considered in "quasi-steady state." This rather ambiguous term will be explained fully in the context of time-scale modeling [2J .

1.4. POLITICAL STRUCTURES

1.4

5

Political Structures

The dynamic structure and time-range classifications of dynamic phenomena illustrate the poten tial complexity of even small or moderate-sized problems. The problems of power system dynamics and stability are compounded immensely by the current size of interconnected systems. A general system structure is shown in Figure 1.3. While this structure is not necessarily common to interconnected systems throughout the world, it represents a typical North American system and serves to illustrate the concept of a ((laTge~sca.le system."

Company 1-20 Generators

• Pool 2-10 Companies

Coordinating Council 3-5 Pools

• Intemational 9 Coordinating Councils Figure 1.3 : System organizational structure

If we speculate about the possible size of a single interconnected system containing nine coordinating councils, four pools per coordinating council , six companies per pool, and ten generators per company, the total possible number of generating stations can exceed 2000. The bulk power transmission network (138- 765 kV) then typically consists of over 10,000 buses . Indeed , the current demand in the nine coordinating councils within the North American Electric Reliability Council (NERC) exceeds 500,000 MW [3]. At an average 250 MW per generator, this roughly confirms the estimate of over

CHAPTER 1. INTRODUCTION

6

2000 generators in the interconnected North American grid. Dynamic studies are r outinely performed on systems ranging in size from the smallest company to the largest coordinating council. These are made at both t he pl anning! design and operating stages. These studies provide information about local capabilities as well as regional power interchange capabilities. In view of the putential size, dynamic studies m ust be capable of sufficiently accurate representation without prohibi tive computational cost. The nature of sys tem engineering problems inherent in such a complex task was emphasized in t wo benchmark reports by the U. S. Department of Energy (DOE) and t he Electric Power Research Institute (EPRI) [4, 5J. These reports resulted in a meeting of international leaders to iden tify dir edions for t he future of this technology. These rep orts set the stage for a whole new era of power system planning and operation. The volume of follow-on r esear ch and industry application has been tremendou s. Perhaps the most significant imp act of these report s was the stimulatjon of new ideas

that grew in to student inter est and eventual manp ower.

1.5

The Phenomena of Interest

The dynamic p erformance of power systems is important to b oth the system organiza.tions, from an econom.ic viewpoint , and so ciety in general, from a reliabili ty viewp oint . The analysis of power system dynamics and stab ility is increasing daily in terms of n umber and freq uency of studies, as well as in com plexity and sl7.e. D yn am ic phenom ena have heen di scu ssed according

to basic function , time-scale properties , and problem size. These three fundamental concepts are very closely related and represent the essence of the ch allenges of effedive simulation of power system dynamics. When properly performed, m odeling and simulation capture the phenomena of interest at minimal cost. The first step in this process is understanding the phenomena of inter est . Only with a solid physical and mathematical underst anding can the m odeling and simulation properly reflect the critical system b ehavior. This means that the origin of mathematical models mu st be understoo d , and t heir purpose must be well defined . Once this is accomplished, the minimal cost is achieved by model reduction and simplification without significant loss in aCClITacy.

Chapter 2

ELECTROMAGNETIC TRANSIENTS 2.1

The Fastest Transients

In the time-scale classification of power system dynamics, the fastest transients are generally considered to be those associated with lightning propagation and S\,l\,ritching surges. Since this t.ext is oriented toward system analysis rather than component design, these transients are discussed in the context of their propagation into other areas of an interconnected system. While quantities such as conductor temperature, motion, and chemical reaction are important aspects of such high-speed trans"ients, we focus rnainly on a circuit view, where voltage and current are of primary importance. \Vhile the theories of insulation breakdown, arcing, and 1ightning propagation rarely lend themselves to lncorporation into standard circuit analysis [7], some simulation software does include a portion of these transients [8, 9]. From a. system viewpoint, the transmission line is the main component Lhat provides the interconnection to form large complex models. While the electromagnetic transients programs (EMTP) described in IS] and i9i are unique for their treatment of switching phenomena of value to designers, they include the capability to study the propagation of transients through transmission lines. Tbis feature makes tbe E'.1TP program a system analyst's tool as well as a designer's tool. The transmission liDe models and basic network solution methods used in these programs are discussed In the following sections.

7

8

2.2

CHAPTER 2. ELECTROMAGNETIC TRANSIENTS

Transmission Line Models

Models for transmission lines for use in network analysis are usually categorized by line lengths (long , medium , short) [10]-[12] , This line length concept is interesting, and presents a major challenge in the systematic formulation of line models for dynamic analysis, For example, most students and engineers have been introduced to t he argument that shunt capacitance need not be included in short-line m odels because it has a negligible effect on "the accuracy," Thus, a short line can be modeled using only series resistance and inductance, resulting in a single (for a single line) differential equation in the current state variable, With capacitance, there would also b e a differ ential equation involving the voltage state variable. Reducing a model from two or three differential equations to only one is a pro cess that has to be justified mathematically as well as physically. As will be shown, t he "long-line" model involves partial differential equations , which in some sense represent an infinite number of ordinary differential equations . The reduction from infinity to one is, indeed , a major reduction and deser ves fur ther attention. Since this text deals with dynamics, it is important to be careful with familiar models and con cept s. Lumped-parameter models are normally valid for transient analysis unless they are the result of a reduction technique such as Thevenin equi valencing. Investigation of the various traditional transmission line models illust rates this point very well. The traditional derivation of the "long-line" model begins with the construction of an infinitesimal segment of length Ll.x in Figure 2.1. T his length is assumed to be small enough that magnetic and electric field effects can be con sidered separately, resulting in per-unit length lille parameters R' , L', G', C', These distributed parameters have the uuits of obms/ mi, henries/ mi, etc., and are calculated from the line configuration . This incremental lumped-parameter model is, in itself, an ap proximation of the exact interaction of the electric and magnetic fields. The hope, of course, is that as the incremental segment app roaches zero lengt h , the r esulting model gives a good approximation of Maxwell 's equations , For certain special cases, it can be shown that such an approach is indeed valid ([13], pp, 393-397) , The line has volt ages and currents at its sending end (k ) and r eceiving end (m), The voltage and current anywhere along the line are simply

v=v(x,t) i=i(x,t)

(2, 1) (2 ,2)

2.2. TRANSMISSION LINE MODELS

9

+

r-..

.

+

...- .r ----i

r - - - - -M - - - - - - i.. ~I

I

x=d

.\=0

Figure 2. 1: Transmission line segment

so that Vm ~ v(o,t)

(2.:! )

ito, t)

(2.1 )

= 7J(d, t)

(2.5)

i k = i(d, t)

(2.6)

im =

71k

where d is the line length. The dynamic equation for the voltage drop across tbe infinitesimal segment is I

8i

I

+ L 6x-8t

(2.7)

+ (71) + C'6x'!.(71 + (71) 8t

(2 .8)

6" = R 6xi

and the clurent t hrough the shunt is

6i - c'6x (71 Substituting (2.7) into (2.8),

,

6 i = G 6xv

ai 1+ C '6xa71 + G,6x[ R ,6xi + L ,.6xat at iJ

2

+ C 6x [ R 6x at· + L 6", a at 2'J .'

I

'/.

,

t

(2.9 )

Dividing by 6x,

6v 6x

(2.1 0)

CHAPTER 2. ELECTROMAGNETIC TRANSIENTS

10

L'li

_ a' v + a' [R' L'lxi +C' [R' L'lx

+ L ' L'lx ai] + C' av

at

at

~: + L ' L'lx ~:;]

(2. 11 )

Now, the originaJ assumption t hat t he magnetic and electric fields may be analyzed separately to obtain the distributed parameters has more credibility when the length under consideration is zero. Thus, the final step is to evaluate (2.1 0) and (2.11 ) in the limit as L'lx ap proach es zero, which gives lim L'l v = av = R'i L'lx ax

+ L ' a;

(2. 12)

lim L'li = ai = a'v L'lx ax

+ C ' av

(2.13 )

at

£>%_0

£>%~O

at

Equations (2.12) and (2.13) are the finaJ distributed-parameter models of a lossy transmission line. There are two special cases when these partiaJ differential equations have very nice known solutions . The first is t be special case of no shunt elements (C' G' 0) . From (2. 13), the line current is independent of x so that (2.12) simplifies to

=

=

v(x, ·t ) = v(o, t)

di

+ R x i + L x dt I

I

(2.14 )

which has a simpl e series lumped R-L circuit representation. This special case essentially neglects all electric field effects. The secon d special case is the lossless line (R ' = G' = 0), which has the general solution [1 3, 14J

+ vpt) 2ch(x + vpt )

i(x, t ) = - J, (x - vpt) - h (x

(2.15)

v(x,t) = 2cI1(x -vpt ) -

(2.16)

where I, and h are unknown functions tbat depend on the boundary conditions, and tbe phase velocity and tbe characteristic impedance Vp

=

Jr:-c,

Zc

=

..jL'J c'

( 2.17 )

lf only the terminal response (Vk, ik, Vrn , i m ) is of interest, tbe following metbod, often referred to as Bergeron 's method, has a significant value in practical implementations. The receiving end curren t is

(2 .18)

2.2. TRANSMISSION LINE MODELS

11

Now, j,( -lIpt) can be expressed as a function of v(o, t) and h(lIpt) from (2 .16) to obtain (2. 19 ) (2 .20) To determine h(lIpt), it is necessary to evalua.te the sending end current at time dl lip seconds before t as

i.(t - ~) Using (2 .16) at '" =

v.

= - f,(d-lIpt + d)-/2(d+lIpt-d)

d and

(t - ~)

at time

= zcj, (d -

dl"p IIpt

(2.21 )

before t,

+ d) -

zch( d + IIpt

-

d)

(2 .22)

so that (2.21) can be evaluated as (2.23) This solves for h(lIpt) to obtain the expression for the current im(t):

(2.24) This expression has a circuit model as shown in Figure 2.2, where (2.25) A similar derivation (see Problem 2.1) can be made to determine the sendingend model shown in Figure 2.3, where (2 .26)

12

CHAPTER 2. ELECTROMAGNETIC TRANSIENTS 1m

+

Vm

Zc

Figure 2.2 : Receiving-end terminal model Ik

+

Zc

Figure 2.3: Sending-end terminal model These circuit models are illustrated with other components in the next section. Before leaving this topic, we consider the special case in which the voltages and currents are sinusoidal functions of the form

v(x, t) = V(x) cos(w,t + 8(x) ) i(x, t)

= I(x) cos(w,t + ",(x))

(2 .27) (2.28)

where w, is a constant. Substitution of these functions into the partial differential equations yields the model of Figure 2.4, with phasors [10] _

1

V(x) = yi2V(x)L8v (x) -

1

I(x) = yi2I(x)L8i (x)

(2.29) (2.30)

2.3. SOLUTION METHODS

13

-

Ik

•

+

-

Vk

1m

Z sinhyd

+

yd

-y -2

tanh

(~ )

-

Y tanh 2

(~ )

(if )

(f: )

-

-

Vm

-

Figure 2.4: PI lumped-parameter circuit for sinusoidal voltages/currents where

Z

= R'd + jw,L' d,

y = C'd

+ jw,C'd,

__ /zY -y -

V7.2

(2 .31 )

There is always a great temptation to convert Figure 2.4 into a lumpedparameter time-domain R· L-C circuit for transient analysis. While such a circuit would clearly be without mathematical justification , it would be some approximation of the more exact partial differential equation representation. The accu.racy of such an approximation would depend on the phenomena of interest and on the relative sizes of the line parameters. In later chapters, we will disc uss the concept of "network transients" in the context of fast and slow dyn amics.

2.3

Solution Methods

Since most power system models contain nonlinearities, transient analysis usually involves some form of numerical integration. Such numerical methods are well documented for general networks and for power systems [15J[19J. The trapezoidal rule is a common method used in EMTP and other

CHAPTER 2. ELECTROMAGNETIC T RANSIENTS

14

transient analysis programs. For a dynamic system of the form dy

(2.32)

dt = fly, t)

the trapezoidal rule approximates the change of state y over a change of time bot as

y(ti + bot)

~

bot y(ti) + 2[J(y(ti + bot), ti + bot)

+ f(y(ti), till

(2.33)

+

This is an implicit integration scheme, since y(ti bot) appears on the righthand side of (2.33). To illustrate the method, consider t he pure inductive branch of Figure 2.5. The state representation is I

+ L

v

Figure 2.5: Pure linear inductive branch

di 1 =-v dt L The trapezoidal rule approximate solution is ~

.(

tii+

bo)

.()

t ~tti

6.t [V(ti + 6.t) v(ti)] +2 L +T

(2.34)

(2.35)

This approximation can be written as the circuit constraint of Figure 2.6. The circuit is linear for given values of i(ti)' V(ti), and 6.t. A similar circuit can be constructed for a pllre linear capacitor (see Problem 2.2). Since t.hese are linear elements, there is really no need to employ such an approximation, but recall the lossless-line circuit representation of Figures 2.2 and 2.3. The combination of the Bergeron lossless-line m odel and the trapezoidal rule for lumped parameters is appealing . There are two difficulties. First, it may be necessary to consider t ransmission line losses. Second, the lossless-line terminal constraints require knowledge of voltages and currents at a previous

2.3. SOLUTION METHODS

15

time (t - :: ), which may not coin cide with a multiple of the integration step size tJ.t . The first problem is usually overcome by simply lumpin g a series or shunt resistance at either end of the transmission line terminal model. It is common to break the line into several segments, each using the circuits of Figures 2.2 and 2.3 together with a corresponding fraction of the line losses. The second problem is usually overcome either by rounding off the time d /vp to the nearest multiple of tJ.t, or by linear interpolation over one time step tJ.t (see discussion published with [14]). This is illustrated in the following example.

i (ti

+ 111)

+

2L

v (ti + I1t)

I1t

•

i (ti) + v(ti)

~l

Figure 2.6: Circuit representation of trapezoidal rule (linear L)

Example 2.1 Consider a single-phase lossless transmission line connected to an R-L load , as shown in Figure 2.7: t

= 0.0001 sec

iI

+ v,

+ Vj

L' = 1.5 x 10-3 Hlmi

+

0.25H

C = 0.02 x 10--6 Flmi

d= 100mi

Figure 2.7 : Single lossless line and R-L load diagram

4000

16

CHAPTER 2. ELECTROMAGNETIC TRANSIENTS

11,

=

230,000\1'2

v'3

cos

(2 6 ) 7r

Ot

Find it, i 2 , and ""2 if t he switch is closed at t = 0.0001 sec using the trapezoidal rule with a time step 6.t = 0.0001 sec. Initial conditions are i , (O) = i 2(0) = 111(0) = ""2(0) = o. Solution: For the parameters given, Zc

2L = 5000 ohm, -d = 0.00055 sec, 6.t vp

= 274 ohm, -

.;

vp = 182,574 Illi sec

The circuit connection when the switch is closed is found using t = ti 0.0001 as shown in Figure 2.8.

i , (I; + 0.0001)

+ i2 (t; - 0.00045 ) v, (Ii + 0 .0001 )

+ V I (I; + 0.0001 _

274 'L ~ V2 (Ii - 0.00045 )

274 i2 (Ii + 0.0001 )

i, (I; - 0.00045) V,(li - 0.00045) + -'-'-'-c=_ _ 274

•

+

400Q

274Q

+

YO (ti + 0.0001)

5000 Q

V3

(I; + 0.0001

. L2

V3 (ti) (ti) + - 5000

Figure 2.8: Single line and R -L load circuit at t = t; For t; = 0 (t

= 0.0001 sec)

From the iuitial conditions,

i , ( -0.00045) = 0

+ 0.0001

+

2.3. SOLUTION METHODS VI

17

(-0.00045) = 0

i,( -0.00045 ) = 0 v,( -0 .00045) = 0 i,(O) = 0 V3(0) = a v,(O.OOOl) = 187,561 V The circuit to be solved at t = 0.0001 sec is shown in Figure 2.9. it (0.0001 )

+ 187 ,66 1 V

+

_ I VI (O.OOOl)

OA

274>1

;, (0.0001)

+

OA

v, (0.000 1)

400>1

274 >1 5000n

OA

Figure 2.9: Single line and R-L load circuit at t = 0.0001 sec Solving the circuits gives

i,(O.OOOl) = 685 A vI (O.OOOl ) = 187, 661 V i,(O.OOOl) = 0 v,(O.OOOl) = 0 v3(0 .0001 ) =

a

The sending-end current h as changed instantaneously from zer o to 685 A as the switch is closed .

CHAPTER 2. ELECTROMAGNETIC TRANSIENTS

18 For ti

= 0.0001 (t = 0.0002 sec)

From the initial conditions and the solution at time t

= 0.0001 sec,

i, ( -0.00035) = 0 VI

(- 0.00035)

=0

i2( - 0.00035) = 0 V2( - 0.00035) = 0 i2(0 .0001)

=0

v3(0.0001) = 0 v,(0.0002)

= 187,261 V

The circnit for this time is the same as before, except that the source has changed. Solving the circuit gives i l (0.0002)

= 683 A

vI(0.0002) = 187,261 V i2( 0.0002) = 0 v2(0.0002) = 0 v3(0.0002 )

=0

This will continue until the traveling wave reaches the receiving end at ti = 0.00055 sec. Since we are using a time step of 0.0001 sec, it will nrst appear at ti 0.0006 sec.

=

For ti = 0.0006 (t = 0.0007 sec) Need :

il( 0.00015) , v,(0.000l5), i2(0.00015) , v2(0.00015), i2( 0.0006) , v3(O .0006), v,(0.0007) There is now a problem , since i" i 2 , and V2 are not known at t = 0.00015. The voltage vI(0 .00015 ) can be found exactly, since it is equal to the source v,. The other "sources" in the circuits must be approximated. There are at least two approximations that can be used. The first approximation is t o use

i,( 0.00015) ", i,(O.0002) = 683 A

2.3. SOLUTION METHODS

19

The second approximation uses linear interpolation as

. ( ). ( ) 0.00015 - 0.0001(. ( ) ( )) ,,0 .00015 """ 0.0001 + 0.0002 _ 0.0001 ,,0.0002 -;, 0.0001 = 685

+ 0.5 x (683 -

685)

= 684 A

Using this approximation for i , (0 .00015 ), the circuits to be solved at ti 0.0006 (t = 0.0007 sec) are shown in Figure 2.10 . i I (0.0007 )

181.293 V

V I (0.0007)

OA

2740

i2 (0.0007)

+

684A +187,494 A 274

4000

t

2740 V2

(0.0007)

+ 50000

'--v--'

V)(o.OOO7)(

-

1368 A

~ OA

I

Figure 2.10: Single line and R-L load at t

=

0.0007 sec

Solving these circuits,

i,(0.0007)

= 662 A

v1(0 .0007) = 181 , 293 V

i2(0 .0007) = 66 A v2(0 .0007) = 356, 731 V

The traveling wave has reached the receiving end and has resulted in nearly a doubling of voltage, because the receiving end is initially nearly an open circuit. The analysis continues with linear interpolation as needed . When a study contains only one line (like this example), the interface problem

20

CHAPTER 2. ELECTR.OMAGNETIC TRANSIENTS

between At and d/ vp can be avoided by choosing At to be an integer fraction of d/ vp: 1 d

At =- N lip Typical values of N ra.nge between 5 and 10,000 .

o The 11Ilique feature of the combination of Bergeron circuits wit h trapezoidal rule circuits is the heart of most EMTP programs. This enables transmission line plus load transients to be solved using simple "de" circuits. Most EMTP programs contain many other features, including thIee-phase representations and other devices . Its use is normally limited to small-sized systems in which the very fa.st transients of switching are the phenomena of interes t. The purpose of this chapter was to present the basic concepts for dealing wit h the fastest transients from a systems viewpoint. In most studies, these dynamics are approximated further as attention shifts to electromechanical dynamics and subsystems with slower response times.

2.4

Problems

2.1 Starting with (2 .15) and (2 .16), derive the circuit representation of Figlue 2.3 for the sending-end terminals of a lossless transmission line. 2 .2 Given the continuous time-domain circuit shown :

--'----1' I

+

v

C

T

use the trapezoidal rule approximation to find an algebraic "dc" circuit representation of the relationship between v(t, + At) and i(t, + At) . 2.3 Given the sinusoidal source and de-energized lossless t ransmission line shown :

2.4, PROBLEMS

i, v,,

21

lOn L'

+

~ 2.18 x 10-3

Hlmi

c ~ 0,01 36 X 10-6 FlIDi

+

VL

300n

d = 225 mi v"

~

188,000 cus (21(60 t) volts

dra.w the "Bergeron" algebraic ((de)) circuit and find vL) iLl is for 0 t < 0,04 sec using a time step of 6t = ls.!L. Plot VL . -"p

<

2.4 Given the sinusoidal source and de-energized lossless transmission line shown: 1=

0.000 1 sec

Ion

---4c================r----+ .r.: x Hlmi O

~

v,,

1.5

10-3

c ~ am x 10-6 F/mi

v

d ~100mi

v,

230.000 V2 = ="",,-'-" cos (2n60 1) volt'

V3

use Bergeron's method with linear interpolation to find 11 and i using a time step of 6t = 0.0001 sec. Solve Jor a total time of 0.02 sec. P lot the results .

CHAPTER 2. ELECTROMAGNETIC TRANSIENTS

22

2.5 Repeat Problem 2.4 using the lumped-parameter model shown: t=

0.000 I sec

lOn Cd

v-'

2

I

L'd ry-,

~ Cd

I I

•+

2 v

using the trapezoidal rule approximation with a time step of 10- 6 sec. 2.6 Euler 's forward integration scheme solves ordinary differential equations (dxfdt = f(x )) using a time step has

X(ti

+ h) =

X(ti)

+ (dx/dt)h

where dx f dt is evaluated at time ti. Use this solution scheme to derive an algebraic "de" circuit to solve for the current through a lurnpedparameter R-L series circuit at each time step for any given applied voltage.

Chapter 3

SYNCHRONOUS MACHINE MODELING 3.1

Conventions and Notation

There is probably more literature on synchronous machines than on any other device in electrical engineering. Unfortunately, this vast amount of material often makes the subject complex and confusing . In addition, most of the work on reduced-order modeling is based primarily on physical intuition, practical experience, and years of experimentation. The evolution of dynamic analysis has caused some problems in notation as it relates to COillman symbols that eventually require data from manufacturers. This text uses the conventions and notations of [20], which essentially follows those of many publications on synchronous machines [21]-[27]. When the notation differs significantly from these and other conventions, notes are given to clarify any possible misunderstanding. The topics of time constants and machine inductances are examples of such notations. While some documents define time constants and inductances in terms of physical experiments, this text uses fixed expressions in terms of model parameters. Since there can be a considerable difference in numerical values, it is important to always verify the meaning of symbols when obtaining data. This is most effectively done by comparing the model in which a parameter appears with the test or calculation that was performed to produce the data. In many cases, the parameter values are provided from design data based on the same expressions given in this text. In some cases, the parameter values are provided from standard tests that may not precisely relate to the expressions given in

23

24

CHAPTER 3. SYNCHRONOUS Mil. CHINE MODELING

trus text. In this case, there is normally a procedure to convert the values into consistent data [20J. The original Park's transformation is used together with the "xad" perunit system [28, 29J. Trus results in a reciprocal transformed per-unit model where l.0 per-unit excitation results in rated open-circuit voltage for a linear magnetic system. Even with this standard choice, there is enough freedom in scaling to produce various model structures that appear different [30J. These issues are discussed further in later sections. In this chapter, the macrune transformation and scaling were separated from the topic of the magnetic circuit representation . This is done so that it is clear wruch equations and parameters are independent of the magnetic circuit representation .

3.2

Three-Damper-Winding Model

This section presents the basic dynamic equations for a balanced, symmetrical , three-phase synchronous macrune with a field winding and three damper windings on the rotor . The simplified schematic of Figure 3.1 shows the coil orientation, assumed polarities , and rotor position reference. The stator windings have axes 120 electrical degrees apart and are assumed to have an equivalent sinusoidal distribution [20]. While a two- pole machine is shown, all equations will be written for a P -pole machine w'th w = ~ wshaft expressed in electrical radians per second. The circles with dots and x's indicate the windings. Current flow is assumed to be into the ((x " and o~t of the "dot ." The volt age polarity of the coils is assumed to be plus to minus from the lex " to the "dots." Tills notation uses "motor" current notation for all the windings at this point. The transformed stator currents will be changed to "generator" current notation at the point of per-unit scaling. The fundamental Kirchhoff 's , Faraday's and Newton 's laws give

·

dAa

·

dAb

Va

= ' aT, + dt

Vb

= 'b T, + dt

'U c

=

· ' cT,

+

dAc dt

(3 .1) (3.2)

(3 .3)

3.2.

THREE·DAMPER· WINDING MODEL

25

b-axis

ir'\r.,.,--.L....-+__ a-axis

•

•

a

d-axis

Figure 3.1: Synchronous machine schematic

· d)' fd Vfd = 'fdTfd+ - dt ·

d)"d

'ldT,d

+ -dt

· V'q = "qT'q

+ dt

· "qT'q

+ dt

Vld

V'q

dB shaft dt

J~ dw Pdt

=

d),'q

d)"q

2

(3.4) (3.5) (3.6) (3 .7)

-w

(3.8)

T= -T, - Tfw

(3 .9)

P

where A. is flux linkage) r is winding resistance) J is the inertia constant , P is the number of magnetic poles per phase, T= is the mechanical torque applied to the shaft, -T, is the torque of electrical origin, and Tfw is a friction windage torque. A major modeling challenge is to obtain the relationship between flux linkage and current . These relationships will be presented in later sections.

26

CHAPTER 3. SYNCHRONO US MACHINE MODELING

3.3

Transformations and Scaling

The sinusoidal steady state of balanced symmetrical machines can be transformed to produce constant states . The general form of the transformation that accomplishes this is Park's t ransformation [20], I::. = TdqoVabcl

tdqo

'Uo.bc 6. [VoVbvcJ\

tabc

'fJdqo

.

I::. T . = dqo 'l.abc,

Adqo = TdooAob,

(3.10)

1::.[ ~aZb'/.c ... ], ,

Aob' ~ lAoAb A,]'

(3.11)

Adoo I::. [AdAoAo]'

(3.12)

I::.

where

Vdqo 6.

.

=

.1::. [ ... ], = 'l.d'/'q'l,o

[vdVqVO] t,

tdqo

1

and .

I::.

Tdqo =

2

'3

Pn

sin( ~Oshaft - 2;)

P

cost ~ 0shaft -

sm 20shaft [

cos 2;shafl

2;)

sin ( ~Oshaft cost ~ OSh~aft

+ 2;) + 2;)

1 (3. 13)

1

2

witb tbe inverse

. Po

- 1 _

T dqo -

[

sm 2 shaft .' P 27r sm( 2 Os haft - 3) . (PO sm 2 shaft + 3h)

cos ~ 0sbaft cost ~ 0shaft - 23") cost ~ 0shaft

(3.14)

+ 2;)

From (3.1)- (3.9), Kirchhoff's and Faraday's laws are

(3. 15) which, when transformed using (3. 13) and (3.14), are

(3 .16) After evaluation, tbe system in dgo coordinates has the forms

(3.17)

(3.18)

3.3: TRA NSFORMATIONS AND SCALING Vo =

.

d)" o dt

Ts'l.o+ -

· Vjd = Tfd'!d

+ -dt-

· Vl d = Tld'id

+T

· Vlq = Tlq'lq

+T

V'q

· T'q"q

d)" jd

d)..ld

d)..lq

d)..'q

+T

27

(3. 19) ( 3.20) (3.21) (3.22) (3.23)

2 dBshaft - - w dt P

(3.24 )

J '!-.. dw = T= -Te-Tjw P dt

(3.25)

To derive an expression for Te, it is necessary to look at the overall energy or power balance for the machine. This is an electromechanical system t hat can be divided into an electrical system, a mechanical system, and a coupling field [31]. In such a syst em, resist ance causes real power losses in the electrical system ) friction ca.uses heat losses in t he mecha.nical system, and hysteresis causes losses in the coupling field. Energy is stored in inductances in the electrical syst em, t he r otating mas s of the mechanical system, and the magnetic field that couples the two. Any energy that is not lost or stored must be transferred. In t his text we make two assumptions about this energy balance. First , ail energy stored in the electr ical syst em inside the machine t erminals is included in the energy st ored in the coupling field. Second, the coupling field is lossless. The first assumption is arbitrary , and the second assumption neglects phenomena such a.s hysteresis (but not sat uration). A cliagram that shows such a power b alance for a single machine using the above notation is given in Figure 3.2 , with input powers for both t he elect rical and mechanical system s.

The elect rical powers are Pin = vaia + vbib elec

+ v ,i, + vidifd + Vl dild + vlqilq + v2qi'q

(3.26)

(3.27)

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

28

Ptrans elec (3.28 )

dW( dl

Electrical

Coupling FicJd

System

J dCiJ (1)2 p '" de

Mechanical System

Figure 3.2: Synchronous machine power balance The summation of the electrical system is simply Kirchhoff's plus Faraday's laws , and the summation of the mechanical system is Newton's second law. In terms of the transformed variables, since (3.29 )

p. In elec

3 2

.

-Vd'/,d

. + -3 v 2 qt q

+ 3vo i o + 'Ufd i Jd + Vldild + vlqi 1q + v 2q i 2q Plos t elec

3 ·2 "2Ts'l.d

(3.30 )

3 ·2 3 -2 + ZT..'I'l. q + Ts'l.o

(3.31) Ptrans elec

(3.32)

3.3. TRANSFORMATIONS AND SCALING

29

The power balance equation in the coupling field in tbese dqo coordinates gives the time derivative of the energy stored in the coupling field as

dAd 3 . dAq . + Te1dBshaft + -32 '. d dt dt- + -, 2 q-dt + 3,

3P . . [22 (Ad' q - Aq'd)

0

dAo - dt (3.33)

For independent states Bshaft, Ad, Aq, Ao, Ajd, AId, Alq, A2q the total derivative of Wj is

dWf dt (3.34) For trus total derivative to be exact [32], the following identities must hold:

8Wj = 3P - ('. /\d~q 8B s haft 2 2

~-'--

-

8Wj 8 Ad

, . ) + Te,

3.

- - = - '/,d

Aqtd

etc.

2

(3.35 )

With appropriate continuity assumptions [33], the coupling field energy can be obtained from (3.33) as a path integral

J

+ [% ~ (Adiq -

Wj = WJ

+

+

J

%iqdA q +

J

ilq dAlq

+

Aqid) +

J

3iodA o +

Tel dBshaft + J%iddAd

J

ifddAjd +

J

i 2qdA 2q

J

ilddAld (3.36)

For this integral to be path independent , tbe partial derivatives of all integrands with resp ect to other states must be equal [34], i.e., etc.

(3.37)

These constraints can also be obtained from those of (3.35) by taking the second partials of Wj with respect to states. The assnmption that the coupling field is conservative is sufficient to guarantee that these constraints are satisfied. Nevertheless, these constraints should always be kept in mind

30

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

when deriving t he magnetic circuit relat ionships b et ween flux linkage and current . Assuming that these constraints ar e satisfied, the following ar bitrar y p ath is chosen from the de-energized (WJ = 0) condition to the energized condition: 1. Integrate rotor p osition t o some arbitrary 0shaft while all sources ar e de-energized. T his adds zero to WI since Ad, A. , and Te must be zero.

2. Integrate each source in sequence while maintaining 0shaft at it s arbitrary position. W ith this chosen path, WI will be the sum of seven integrals for the seven independent sources Ad, Aq, AD, Aid, AId, A1q , A2.' Each integrand is the respective source current that must be given as a fun ctioll of the states. Since this is not done un til later sections, we m ake the following as sump t ion. Assume that the relationships between Ad, A., AD , Aid, AId, AI., A2. and id, i q , i Ol i fdl i 1d, i 1ql i 2q aTe independent of Bshaft ' For this assumption, Wi will be independent of 0shaft so that, from (3.35 ), (338) To complete the dynamic model in the t ransformed variables, it is desirable to define an angle that is constant for constant shaft speed . We define t his angle as follows:

(3.39) where w, is a constant normally called rated synchronous speed in electrical radians per second, giving dE! - =w -w (3.40) dt ' The final unsealed model i n the new variables is

dAd dt dA. dt dAD - dt

- -

-r,i d + WA.

+ "d

(3.41 )

- rsi q - WAd

+ Vq

(3.42 )

- rs-io

+ Vo

(343 )

3.3. TRANSFORMATIONS AND SCALING d).,jd dt dAld dt dA' q dt dA2q dt

31

- rjdijd

+ Vjd

(3.44)

- r'di 'd

+ V,d

(3.45)

-1'lqi 1q

+ Vl q

(3.46)

-T Zq i 2q

+ 'U2q

(3.47)

do

dt =

J~ ~~

(3.48)

WS

W -

G) (~)

+

= Tm

(Adiq - Aqid) - Tjw

(3.49)

It is customary to scale the synchronous machine equations using the traditional concept of per-unit [28,29]. This scaling process is presented here as a change of variables and a change of parameters. We begin by defining new abc variables as t:.

Va

Va =

VBABC

t:.

I

Au

t:.

=

~

..pb t:.

1

11: t:.

Vb , VBABC

h=t:.

- ta

IBABc'

a -

'lj;a

Vb

,

/\ BABC

-'b

c -

.

t:.

I

Vc VBAB C' -'c

IB ABC'

c -

IBAB e'

Ab , /\BABC

..pc t:.

;>; c

(3.50)

/\BABC

where VBABC is a rated RM 5 line to neutral stator voltage and

SB

t:. IBABc

= 3V

BABC

(3.51)

'

with 5B equal to the rated three-phase volt amperes and WB equal to rated speed in electrical radians per second (ws). This scaling also converts the model to "generator" notation. The new dqo variables are defined as t:.

Vd

Vd=

11:qt:.-

,

VBDQ Id ..pd t:.

t:.

=

-td IBDQ

Ad , ilBDQ

Vq

,

VBDQ

,

I

t:. q -

..p t:. q -

- 'q IBDQ' ).,q ilBDQ

11: t:. 0

-

t:.

I 0

-

..po t:.

Vo VBDQ' -to IBDQ'

Ao /\ BDQ

(3 .52)

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

32

where VBDQ is rated peak line to neutral voltage , and t;

IBDQ

25

B = 3'TVBDQ '

t; V

ABDQ

BDQ = - WB ---"

(3.53)

with 5B and WB defined as above, and again the scaling has converted to generator notation . The new rotor variables are defined as t; Vfd =

vfd

V

BF'D

It; fd =

V'dt;

'

' fd

Afd ,

V BIQ

VBlD

,

I

Id

IBF'D ,pfd t;

~, V1q D.~) V2q

,p'd

~

t; 'Id = -IBID -, lt q A,d ,

A BlD

AB F'D

,p'q

~~)

I

IBlQ

~~,

6. ~, VB2Q

t;~

1

2q -

,p2q

ABlQ

B2Q

'

~~

(3.54)

AB2Q

where the rotor circuit base voltages are t; VBF'D =

5B

-1 -- ' BF'D

t; 5 B VBlD = -1- -' BlD

t; V BlQ =

5B

t;

1-- '

VB2Q

BIQ

SB

= -1 -

(3.55)

B2Q

and the rotor circuit base flux linkages are A BF'D

t; VBF'D

=

WB

t; VB1Q

,

A

ABlQ= - - ,

" B2Q

WB

t; VB2Q

= - -

WB

(3.56)

with SB and WB defined as above. The definitions of the rotor circuit base currents will be given later, when the flux linkage/ current relationships are presented . In some models, it is convenient to define a scaled per-unit speed as t; W !I = (3.57) WB

where WB is as defined above. This completes the scaling of the model variables. The model parameters are scaled as follows. Define new resistances R t; T., , - ZBDQ '

(3.58) where Z BDQ

t; VBDQ I ' BDQ

=

Z BFD

t; VBF'D

= I BF'D ' t; VBIQ

ZBIQ = -1- -' BIQ

t; VBID

Z BID = -1- - ' BlD

Z B2Q

t; VB2Q

= -IB2Q

(3.59)

3.3. TRANSFORMATIONS AND SCALING

33

The shaft inertia constant is scaled by defining

(3.60) It is also common to define other inertia constants as

M = 2H,

M' =2H

W,

(3.61)

where the constants Hand M' have the units of seconds, while M has the units of se<:onds squared. The shaft torques are scaled by defining tJ.

Te

TELEC = TE '

(3.62 )

where

(3.63) Using these scaled variables and parameters, the synchronous machine dynamic equations at this stage of development with WB = W, are

(3.64) (365) (3.66) (3.67)

(3.68) (3.69)

w, dt 1 d1j;2q -w, dt

-

do dt

2Hdw

(3.70)

=w-

w,

~dt = TM - (1j;d1q - 1j;q1d) - TFw

,

(3.71) (3.72)

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

34

It is important to pause at this point to consider the scaling of variables. Consider a balanced set of scaled sinusoidal voltages and currents of the form:

Va =

hvs cos (w,t + 0,)

Vt, =

hv, cos

2;)

(3.74)

Vc =

hvs cos (W,t + 0, + 2;)

(3.75)

Ia =

hI, cos (w,t + ¢,)

(3.76)

Ib =

hIs cos (w,t + ¢,

_ 2;)

(3.77)

Ie =

hIs cos (w,t +4>, + 2;)

(3.78)

(3.73)

(w,t +0, _

U sing the transformation (3.13),

(P

)

(3.79)

0,)

(3 .80)

hVs VBABC) . VBDQ sm 2"0shaft - wst - 0,

Vd

=

(

Vq

=

(h~~:BC )

cos (

~ 0shaft -

w,t -

(3.81)

Vo = 0

(PO

. Id = ( hIsIBABC) I Sill -2 shaft - wet BDQ (

(P

'" )

'1',

hI,IBABC) cos -Oshaft - w,t - ¢, IBDQ 2

)

(382) (3.83) (3.84)

By the definitions of VB ABC, VBDQ, IBABC, and I BDQ ,

,l2v,vBABC _ v: VBDQ -"

hI,IBABC IBDQ

= Is

(3.85)

Using the definition of 0 from (3.39),

Vd= Vs sin (0-0,)

(3 .86 )

3.4. THE LINEAR MAGNETIC CIRCUIT

35

Vq = V, cos (6 - 0,)

(3 .87)

I d = I, sin (6 - ¢,)

(3.88)

Iq = I, cos (6-¢,)

(3.89 )

These algebraic equations can be written as complex equations

+ jVq)ej(5-~/2) (Id + jlq)ej(5-~ / 2)

(Vd

= V,e j6 ,

(3.90)

= I, e;¢'

(3.91)

These are recognized as the p er-unit RMS phasors of (3. 73 ) and (3.76). It is also imp ortant, at t his point , to note that the m odel of (3.64)- (3.72) was derived using essentially four general assumptions. These assumptions are summarized as follows. l. Stator has three coils in a balanced symmetrical conflguration centered

120 electrical degrees apart. 2. Rotor has four coils in a balanced symmetrical configuration located in pairs 90 electrical degrees apart. 3. The relationship between the flux linkages and currents m ust reflect a conservative coupling field . 4. The relationships between the flux linkages and curr ents must b e independent of 0shaft when expressed in the dqo coordinat e system. The following section s give t he flux linkage/current relationships t hat satisfy these fOU I assumpt ions and thus complete t he dynamic model.

3.4

The Linear Magnetic Circuit

This section p resents the special case in which the machine flux linkages are assumed to be linear functions of currents:

+ L,,(OshaftJirotor L,,( Oshaft)iabe + LTT(Oshaft)irotor

.Aabe = L.. (Oshaft )i abc .Arotor =

(3.92 ) (3.93)

where

(3 .94)

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

36

If space harmonics are neglected, the entries of these inductance matrices can be written in a form that satisfies assumptions (3) and (4) of the last section. Reference [20] discusses t his formulation and gives the following standard first approximation of the inductances for a P-pole machine. L'.

L,,(Bshaft) = Ll,

+ LA -

LB cos PB shaft

-!LA - LB cos(PBshaft _ ';) [

+ 2;) LB cos(PBshaft + 2;)

- tLA - LB cos(PBshaft _ ';) Ll,

+ LA -

j

-~LA - LB cos PBshaft Ll, + LA - LB cos(PBshaft _ 2;)

L .. (Bshaft) =

+ ';)

-~LA - LB cos PBshaft

-~LA - LB cos(PBshaft

-~LA -

LB cos(PBshaft

(395)

L~,(Oshaft) ~ L,'d sin tBshaft

L'l dsin(~Bshaft - 23")

Lddsin(~Bshaft

+ ';)

L,1q cos tOshaft

°

L dq cos ( ~ shaft - ';) L dq cos( tOshaft + ;.".)

L'.

Lrr(Oshaft) =

(3.96)

Lfdfd

L fdld

0

0

Lfdld 0

L'd'd 0

0

0

L 'qlq

L'q'q

0

0

L 'q2q

L'q'q

(3.97)

The rotor self-inductance matrix Lrr( 0shaft) is independent of Bshaft. Using the transformation of (3.10) , Ad

=

(Ll<

+ Lmd)id + L'fdifd + L"di'd

(3.98)

AId =

~L'fdid + Lfdfdifd + Lfdldil d

(3.99)

A,d =

~L"did + Lfdldifd + L'dldild

(3.100)

3.4. THE LINEAR MAGNETIC CIRCUIT

37

and

+ L~q) iq + L"

Aq = ( Ll'

Alq

3L · = 2 .!lqtq

qi 'q

+ L,2q i 2q

. . + L lqlq1,lq + L lq2q~2q

(3.101) (3.102) (3.103)

and

(3 .104) where

(3.105) This set of flux linkage/ current relationships does reflect a conservative coupling field, since the original matrices of (3.95)-(3.97) are symmetric, and the partial derivatives of (3.37) are satisfied by (3 .98)-(3.104). This can be easily verified using Cramer's rule to find entries of the inverses of the inducta.nce matrices. In terms of the scaled quantities of the last section, 01.

'I'd =

W,(Ll,

+ Lmd)( - IdIBDQ) + --"---"';..:-=-= w,L'fdIfdIBFD =VBDQ VBDQ

W,L,'dltd I BID +-'---7:'-=--=-C='

(3.106)

VBDQ

..p fd =

w,~L'fd( - IdIBDQ)

-'-''---'=--

VBFD

--''-'''-''-'-

w,Lfdfd1fdIBFD + -=--,,-,:~=-::..:....:=VBFD

+w,Lfd'dI'dIBID

(3.107)

VBFD

W, ~ L"d( -IdIBDQ )

w,Lfdld1fdlBFD

VBlD

VElD

..p'd = -=-L-C"7:'---=-==-=-= + -=---"-==:-=-::..:...:=-

+W,L 'd' dhd I BID

(3.108)

VBlD .1.

'I'q

=

W.(Ll,

+ L~q )( -1q1BDQ) + --"'----=,:;:!....O"-.::'-= W,L" qh qIBIQ VBDQ

w,L, 2q I 2q IB2Q

+ -'--~'-=~~

VBDQ

VBDQ

(3.109)

CH APTER 3. SYNCIIRONOUS MACH TNE MODBUNG

38

w, ~ L,'q ( - lo I BDQ ) -

- -

w .•L , q1q I ' qIH1Q

- - -t .

-- VBt Q

11B1Q ... W, L' q2q i 2q I H2Q , VB1 Q

( 3. 110)

"

W'2 L ,2q(- I o IB OQ) ... W,Llq2 qh qIB1Q VB2Q

VB2 Q

.

B 2Q +w, L 2q2q l2q1 - --

(.1. 11 1)

"H2 Q

..po

w, /'1,( -lo J BDQ )

=

(3 1 12)

FB DQ

i\lthougb the viJ,lues of [BFD, JHW , Jnt Q, 1B 2Q have no t yet ueen sp ecifieu) their relationship to theiT respective vult age bases assures t hat the scaled transformed syst em (3.1. 06 )- (3. 112 ) is reciprocal (sYlllmetri c ind1[ctance lllatri ces). The rotor current b;tses arc chosen at this poin t to make as many off-diagonal terms equ al as possible. To do this , define l BFD

~ Lmd -L

I BDQ , I ntO

A Lrnd

sid .6.

.'I'l d

Lmq I

- L.'i l q

(3. 113)

= L - l BD Q

(3.1 14)

B OQ,

a nd the following scaled parameters: X

d Tn

c.

v

~ ~.~L~~ -

ZBJJQ

w .• L Jdld

1 1 ft/, -::: -

v

X

....'fdlri.

-- ,

v

~ -

W,

L

WsL'I7l. q

Z

(3.115)

BDQ

l> w,Lldl d];"J d = -,---

ZlJFD

~\.l q

t: mq -

1

(3.116)

ZBFlJ L '\'l d

'q 'q '

X

6 Ws Llq2 q L slq

1q2. -

Z U1Q

Z B 10 L s2q

(3.1 17)

It is convenient also to define the scaJed leakage reaclances of the rotor windings as

(3.118) X nq

l> X = 2" -

v ... \ mq

(3119 )

3.4. THE LINEAR MAGNETIC CIRCUIT

39

Similarily, we also define (3.120) (3 .121) The resulting scaled

'I/J -

I relationship is

+ X=d1fd + X=dl,d X=d( -1d) + Xfd1fd + CdX=d!,d Xmd( -1d) + cdX=d1fd + X,d!,d

(3.122)

+ Xmq1'q + X=qh q Xmq( -1q) + X'q!,q + cqXmqhq Xmq( -1q) + CqXmq!,q + X 2qh q

(3.125)

'l/Jd = Xd( -ld) 'l/Jfd = 'I/J'd =

(3.123) (3.124)

and

'l/Jq = Xq( -1q)

'I/J'q

=

'l/J2q =

(3.126) (3.127)

and (3.128) While several examples [30] have shown that the terms Cd and cq are important in some simulations, it is customary to make the following simplification [20]: (3.129) This assumption makes all of the off-diagonal entries of the decoupled inductance matrices equal. An alternative way to obtain the same structure

without the above simplification would require a different choice of scaling and different definitions ofleakage reactances [30]: Using the previously defined parameters and the simplification (3.129), it is common to define the following parameters [20] (3.130)

(3.131 )

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

40

x; X"q

e:. e:.

, e:.

T'qu e:.

" Tqo

1

+ I Xm.d

Tdo =

" Tdo

Xl,

e:.

Xl,

+

l 1d

1

x ,mq

(3.133)

+ xu:; ' + xa; '

Xfd

(3.134)

w,Rfd X ,q

(3 .135)

W~Rlq

1 W,R'd

e:.

(3.132)

+~ ' + X'

1 WsR2q

( Xlld

+ _, X.rnd

( Xl 2q

1

_1 )

+X

ljd

+ -, + 1 I ) xtnq

(3.136 )

(3.137)

x llq

and the following variables E'q

Efd

e:.

Xmd ..p - - fd Xfd

(3.138)

e:.

X mdv - - fd Rfd

(3. 139)

, e:. Ed = _ Xmq ..p,q X 'q

(3.140)

Adkins [23] and several earlier refer ences define X: as in (3.131) and T~~ as in (3. 135 ). This practice is based on the convention that single primes refer to the so-called "transient" p eriod, while the double primes refer to the supposedly faster "subtransient" period . Thus , when a single damper winding is modeled on the rotor , this is interpreted as a "subtransient" effect and denoted as such with a double prime. Some published models use the notation of Yo ung [35], which uses an E~ definition that is the same as (3. 140) but with a positive sign . In several publications, the terminology Xfd is used to define leakage reactance r a.t her than self-reactance. The symbols Xad and Xaq are common alternatives for the magnetizing reactances Xmd and X mq. The dynamic model can contain at most only seven of the fourteen flux linkages and currents as independent state variables. The natural form of

3.4. THE LINEAR MAGNETIC CIRCUIT

41

the state equations invites the elimination of currents by solving (3.122)(3.128) . Since the terminal constraints have not yet been specified, it is unwise to eliminate la, I q , or 10 at this time. Since the terminal constraints do not affect I fd , lId, Ilq, Iz q, these currents can be eliminated from the dynamic model now . This is done by rearranging (3.122)-(3.128) using the newly defined variables and parameters to obtain 7/1d =

X"I -

d d

(X; - Xl,) E' (X~ - X;) 01. + (X~ _ Xl,) q + (X~ _ Xl,) 'l'ld

1, Ifd = -[Eq X=d X~ -

hd =

+ (Xd -

X;

(X~ _ Xl,»

, Xd)(Id - hd)]

(3.142)

I

[7/1ld

+ (Xd -

(3.141)

I

XI,)Id - Eq]

(3.143)

and

(3.144) 1, - X [-Ed =q

+ (Xq -

Xq, -Xq" (X' _ X )' [7/1'q q

l,

, Xq)(Iq -Izq)] t

I

+ (Xq -

(3.145)

Xl,)Iq

+ Ed]

(3.146)

and

(3.147) Substitution into (3.64)-(3.72) gives the dynamic model for a linear magnetic circuit with the terminal constraints (relationship between Vd, IdJ Vcp 1q) VOl 10 ) not yet specified. Since these terminal constraints are not specified, it is necessary to keep the three flux linkage/current algebraic equations involving Id, I q, and 10 , In addition, the variables Efd, TM, and TFW are also as yet unspecified. It would be reasonable, if desired at this point, to lllii.ke Eid alld TM constant inputs, and TFW equal to zero. We will continue to carry them along as variables . With these clarifications, the linear magnetic circuit model is shown in the following boxed set.

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

42

(3. 148) (3.149) 1 d,po --d W,

I

t

= R,Io + Vo

dE~

(3.150)

I

d'l/J2q

[

q

+(X~ - X t .)lq II

X; )2(,p2q

X~ Xq) 1q - (X' _ X

+ (Xq -

I

Tqod t = - Ed

t,

+ E~)l

(3. 153 )

I

I

Tq0dt = -,p2q - Ed - (Xq - Xt ,)Iq

-do =w dt 2H dw

(3.154)

w '

-;;;-dt = TM ,

(3.155)

(,pd1q

,pqId )

-

-

TFW

" (X; - Xt,) , ,pd = - Xd 1d + (X~ _ Xl,) Eq

(3.156) (X~ - X;) Xis) ,pld (3.157)

+ (X~ _

(X'. - X") (X'''q -X) " l. , q ,pq = -Xq 1q - (X' _ X ) Ed + (X ' _ X ) ,p2. (3.158) q

,po

= -Xl,Io

b

q

b

(3.159)

Although there are time constants that appear on all of the flux linkage derivatives, the right-hand sides contain flux linkages multiplied by constants. Furthermore, the addition of the terminal constraints could add more terms when 1d, I q, and 10 are eliminated. Thus, the time constants shown are not true time constants in the traditional sense, where the respec-

3.5. THE NONLINEAR MAGNETIC CIRCUIT

43

tive states appear on the right-hand side multiplied only by -1. It is also possible to define a mechanical time constant T, as

T, _J2Hw,

(3. 160)

w, =tl T,(w - w,)

(3.161)

tl

and a scaled transient speed as

to produce the following angle/speed state pair (3 .162) (3.163) While this will prove useful lat er, in the analysis of the time-scale prop erties of synchronous machines, the model normally will be used in the form of (3.148)-(3.159). This concludes the basic dynamic modeling of synchronous machines if saturation of t he magnetic circuit is no t considered. The next section presents a fairly general m ethod for including such nonlinearities in the flux linkage/current relationships.

3.5

The Nonlinear Magnetic Circuit

In t his section, we propose a fairly generalized treatment of nonlineari ties in the magnetic circuit . The generalization is motivated by the multitude of various representations of saturation that have appeared in the literature. Virtually all methods proposed to date involve the addition of one or more nonlinear terms to the model of (3.148)-(3.159). The following treatment retUInS to the original abc variables so that any assumptions or added terms can be traced through the transformation and scaling processes of the last section. It is clear that , as in the last section, the flux linkage/current relationships must satisfy assumptions (3) and (4) at the end of Section 3.3 if the result s here are to be valid for the general model of(3.64)-(3.72 ). Toward this end, we propose a flux linkage/ current relationship of the following form:

Aabc = L,,(Oshaft)iabc

+ L,,(Oshaft)irotor

- Sabc( iabc, Arotor. 0shaft)

(3.164)

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

44

Arotor = L!r (Oshaft )iabe + Lrr (Oshaft )irotor -Srotor( iabe, Arotor, 0shaft)

(3.165)

where all quantities are as previously defined, and Sabe and Srotor satisfy assumptions (3) and (4) at the end of Section 3.3. The choice of stator currents and rotor flux linkages for t he nonlinearity dependence was made to allow comparsion with traditional choices of function s. With these two assumptions, Sabe and Srotor must be such that when (3.164) and (3.165) are transformed using (3.10), the following nonlinear flux linkage/current relationship is obtained Ad

= (Ll, + L=d)id + L,fdifd + L,ldild -

Sd(idqa, Arotor) (3 .166)

3 Afd = 2 L,fdid+ Ljdjdijd + Ljdldild - SjJ(idqa,Arotor)

(3 .167) (3.168)

(3.170) (3.171) and (3.172) This system includes the possibility of coupling between all of the d, g, and 0 subsystems. Saturation functions that satisfy these two assumptions normally have a balanced symmetrical three-phase dependence on shaft position. In terms of the scaled variables of the last two sections , and using (3.129 )

(cd = cq = l ), ..pd = Xd( - Id) + Xmd1jd ,pfd

+ X=dIld -

S~l )(ytl

(3.173)

= X=d( - Id) + Xfd1fd + Xmdhd - S}~(ytl

(3.174)

+ X=d1fd + X1dhd - Sl~)(Yl)

(3.175)

,pld = Xmd( - Id )

3.5. THE NONLINEAR MAGNETIC CIRCUIT

45

and

'1fiq = Xq ( - l q)

+ X~qhq + x=q!,q -

S~')(Y')

(3.176)

'1fi' q = X~q ( - Iq)

+ X,q 1'q + X~ql2q - S;~)(Yl)

(3.177)

'1fi2q = X=q ( -1q)

+ X~q hq + X 2q12q - S~~lt Yl )

(3 .178 )

and

(3.179) where

(3.180)

S~I)

t. Sd / ABDQ,

S}~ ~ Sfd / ABFD , 5;~)

t. 5 1d/ A BlD

t. 5 q/ A BDQ , 5(1) t. 5 1q / A B1Q , 5(1) t. 5 2q / A B2Q 5 q(1) = 1q = 2q =

(3 .181) wit h each 5 evaluated using Adqo, Arotor written as a function of Y , . Using new variables E~ and E~J and rearranging so th at rot or currents can be

eliminated, gives

(3.183) (3.184) and

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

46

~ _

'l/Jq ~

" _ (Xq" -: Xl, ) , Xqlq (X' _ X ) Ed q

l~

( Xq ' - " Xq) _ (2) X ) 'l/J2q Sq (Y2 ) (3.18 5)

+ (X'q _

is

(3. 186)

(3.187) and

(3188) where (3.189) and

5(2)

~

d

-

S(I) _ 5(2 ) _ d

fd

5 (2) b. 5(1) _ 5(2) _

q

q

5( 2) b. 5(1) o

0

lq

(X~ - X~) 5(2)

(X~

_ Xl,)

ld'

(X~ - X~' ) 5 (2) (X'q _ X I, ) 2q'

(3.190)

with each 5(1) evaluated using Y1 , written as a function of Y2 • Elimination of rot or currents from (3.64)- (3.72) gives the final dynamic model with general nonlinearities.

3.5. THE NONLINEAR MAGNE T IC CIR CUIT

47

(3. 191)

1 d.,pq W -W -dt- = R , I q - -.,pd + v:q w, ,

(3.192) (3. 193)

X'd - X,"d dE q' / ,.' Tdo = - Eq - (Xd-Xd )[Id- ( ' _) (.,pl d+( X r dt X d -Ji.1,2 I

I

- E; + S~~)(Y2 ))] - S}~)(1T2) + E Jd " d.,pld Tdod t =

- .,pld

' + Eq'-r(Xd -

XI, )Id -

I

I

I

dEd

'

(2)

SId

(3.194)

(Y2 )

(3.195 )

/I

Xq - }(q

,. 1

XI, )Id

I

Tq0dt = - Ed + (Xq - X q) [I q - (X~ _ Xl,F (.,p2,+ (Xq - XI, )I q

+E~ + S~!) (1T2 ))] + S~!\Y2)

(3 .196) (3 .197) (3 198) (3.199)

with the three a.lgebra.ic equations , 01.

'I'd = -

X "I (X ;-XI')E' ( X~-X; ) ol. d d + ( X~ _ Xl, ) q + ( X~ _ Xl, ) 'l'ld

- S~2)(Y2)

(3.200)

" ( X~' - Xl, ) , ( X~ - X~') 1{Iq = - Xq Iq - ( 'K' _ X ) Ed + (X ' _ X )1{I2 q J

q

t.

- S~2) ( Y2 ) .,po

= -XI, I o - si2 )( y2 )

1'2 = [Id E~ 1{I,d Iq Ed .,p2q Jo]t

q

llf

(3.201)

(3.2 02 ) (3.203 )

48

CIlAPTER 3. SYNCHRONOUS MACHINrJ MODELING

As in the last section , new speeds could be defined so tha t each dynami c sLate includ es a time constant . It is important to note , however, that, as in t he last section , terms on the right-hand side of the dynamic state model imply that these t ime constants do noL necessarily completely identify the speed of res ponse of each variable. This is even more evident with the addition of nonlinearities. One purpose for beginning this sect ion by returning to the abc variables was to trace t he nonlineari!ies through t he transformation and scaling proce~s. This € lLS Ur es that the result ing model with nonlinearit ies is, in some sense, consistent. This was partly motivated by the proliferati on of different met hods to account for saturation in the literature . For example, the lite r ature talks a b out " X~/' sat urating, o r being a function of t.he dynamic states. This could imply that many constants we have defined would change when saturation is considered. With the presentation given above, it is clear that all constants can be left unchanged, while the nonlinearities are included in a set of funct ions to be specified based on some design calculation or test

X=d

procedure.

It is interesting to compare these general nonlinearity functions with

other meth ods that have ap peared in the literature [20,22,23,26,27,35,36] - [50] . Reference [37] discu sses a typical representation that uses: 5(2) d

-

()

and keeps S}~ and

0 1 5(2 ) - 0 5(2) - 0 5 (2) = 0 q I 2q I 0

5 (2) -

)

ld

-

(3204 )

sl!) expressed as (3.205 )

(3.206) where (3.207 ) and .,." '/' d

"

(X; - Xl,) E•' + (X~ - X;) '/'l d X' - X X' _ X

(3208)

(X:-Xt') , (X~-X~') ..p2. X' _ X Ed + X ' _ X

(3209)

.1."" '1'. - -

.1.

d

q

is

d

ill

is

q

l.f

3.5. THE NONLINEAR MAGNETIC CIRCUIT

49

The saturation function SG should be correct under open-circuit conditions. For steady state with

= Iq = 10 = I 'q = Izq = 0 ..p. = - E~ = ..p~ = - Vd = ..p,. = ..p2. = 0 Id

""d 'I' -- E'• -- ""~II 'I'd

V;q -- ""'l'ld -- Efd - S(2) fd

--

(3.210)

the open-circllit terminal voltage is (3.211) and the field current is E'

I jd

=

+ S(2)

• fd Xmd

(3.212)

From th e saturation representation of (3 .205),

S}~ = SG(V,.J XmdIfd = V, ••

(3.213)

+ SG(Vi.J

(3.214)

The function SG can then be obtained from an open-circuit characteristic , as shown in Figure 3.3. While this illustrates the validity of the satur ation function under open-circuit conditions, it does not totally support its use under load . In addition, it has been shown that this representation does not satisfy the assumption of a conservative coupling field [51J. slope = Xmd

v VI OC

I

~

Sc,(VJ Xmd

lfd

Figure 3.3: Synchronous machine open-circuit characteristic

50

3.6

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

Single -Machine Steady State

To introduce steady sta.te, we a.ssume constant states and look at the algebraic equations r esulting from t he dynamic model. We will analyze t he system under the condition of a linear magnetic circuit. Thus, beginning with (3.148)-(3 .159), we observe that, for constant stat es, we must h ave constant speed w and const ant angle 5, thus r equiring w = w, and, therefore,

Vd = - R,ld - ,po

(3215)

Vo = -R,lq + ,pd

(3.216)

Assuming a balanced three-phase operation, all of the "zero" variables a nd damper winding currents are zero. The fact that damper-winding currents are zero can be seen by recalling that the right-hand sides of (3.152)-(3. 154) are actually scaled damp er-wjnding currents. Usjng these t o sjmplify (3. 151 ), (3.153), (3.157), and (3.158), the other algebraic equations to be solved are

o = - E~ - (Xd - X~)ld + Efd o = - ,p'd + E~ - (X~ - Xt.)ld o = -E~ + (Xq - X~)Iq o = - ,p2q - E~ - (X; - X/,)lq o = TM - (,pdIq - ,pq Id ) - Trw

(3.217) (3. 218)

(3.2 19) (3.220 ) (3. 221)

,pd = E; - X~Id

(3.222)

= -E~ -

(3.223)

,po

X~Io

Except for (3.221), these are all linear equations that can easHy b e solved for various steady-state repr esentations . Substituting for ,pd and ,pq jn (3.215) and (3.216) gjves

+ E~ + X~Io

(3224 )

Vq = - R , Iq + E; - X~ld

(3.225)

Vd = - R,Id

These two real algebraic equations can be written as one complex equation of the form

(3. 226)

3.6. SINGLE-MACHINE STEADY STATE

51

where E

-

[( E~ - (Xq - X~)Iq)

-

j[(Xq - Xd)Id

+j (E~ + ( Xq -

X~)Id)]ei(Ii -"/2)

+ E~] ej(Ii-~/ 2)

(3.227)

Clearly, many alternative complex equations can be written from (3.224) and (3 .225), dep ending on what is included in the "internal" voltage E. For b alanced symmetrical sinusoidal steady-state abc voltages and currents, the quantities (Vd + jVq)e i (Ii - ,,/2) and (Id + jIq)ei (Ii-" /2) are the per-unit RMS phasors for a phase voltage and current (see (3.73 )-(3.91 )). This gives considerable physical siguificance to the circuit form of (3.226 ) shown in Figure 3.4. The internal voltage E can be further simplified, using (3.217), as j E = j[(Xq - Xd)Id + Efd]e (Ii-"/2l =

[(Xq - Xd )Id

+ Efd]e''Ii

(3.228)

+

E +

F igure 3.4: Synchronous machin e circuit representation in steady state An important observation is

(3 .229)

0 = angle on E Also from (3 .142) and (3 .143),

(3.230)

lfd = Efd/Xmd

Several other p oints are worth noting. First , tlte open-circuit (or zero stator current ) terminal voltage is (Vd + jVq)e i (Ii- ,, /2 ) 11d =I.=0= Efde,li

( 3.231)

Therefore, for Efd = 1, t he open-circuit t erminal voltage is 1, and field current is 1/ X md. Also, Vd IIp1.=0

= E~ 11 =I. =0= - 1/;q 11 =1. =0= 0

Vq 11d =I.=0 =

d

d

E~ 11.=1,=0= ..pd 11d =1,=0= Efd

(3.232) ( 3.233)

52

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

The electrical torque is (3.234) Tills torque is precisely the "real power" delivered by the controlled source of Figure 3.4. Tha.t is, for 1 = (Id + j1q )ei (6-"/ 2), TELEe

= TM = Real[Er]

(3.235)

We can then conclude that the electrical torque from the shaft is equal to the power delivered by the controlled source. In steady state, the electrical torque from the shaft equals TM when Tpw = O. From the circuit with V = (Vd + jVq )e i (6-" / 2), (3.236) For zero stator resista.nce and round roteIl

(3.237) or (3.238) where the angle OT is called the torque angle, (3.239) with (3.240) Under these conditions, and this definition of the torque angle, OT motor and liT > 0 for a generator.

< 0 for

Example 3.1 Consider a synchronous machine (without saturation) serving a load with V = ILlO° pu

1=

0.5 L - 20 0 pu

a

53

3.6. SINGLE-MACHINE STEADY STATE

It has Xd = 1.2, Xq = 1.0, X~d = 1.1, Xd = 0.232, R, = 0 (all in pu). Find 0, OT, Id, I q , Vd , V., ..pd, ..p., E~, Efd, Ifd (all in pu except angles in degrees) .

Solution: -(1.2 - 0.232)Id + E fd

E'

•

(0 .768Id + E~)LO

1L90 x 0 .5 L - 20° + 1LlO° _

1.323 L29.1°

so 0=29 .1° OT = 29.1° - 10° = 19.1° Id

+ jI. = 0.5 L Id Vd

+ jV. =

20° - 29.1° + 90°

= 0.378

= 0.5L40.9°

= 0.327 29.1° + 90° =

1LlOo -

I.

1nO.9°

Vd = 0.327 V. = 0.945 ..pd

= Vq + 01. = 0.945

..p. = - Vd - Old = - 0.327

To find E~ and E Id , return to I E

I

0.768 x 0.378 E~

+ E~ =

= 1.033

1.033 = -(1.2 - 0.232) Eld

1.323

X

0.378 + Eld

= 1.399

To find Ifd, it is easy to show that 1 fd -- -Eld- _- 1.399 -- 127 . X=d 1.1 These solutions can be checked by noting that, in scaled per unit , equal to Pou,,· TELEC

-

..pdI. - ..pqld = 0.4326

POUT

-

Real (V 7*) = Real (0.5L

+ 30°) =

0_433

TELEC

is

CH APTER J. SYNCHRONOUS MACHINE MODELING

54 Also, QOUT

Imag(V J') = Imag(0 .5 L300) = 0.25

+ jVq)e - j (6- 1C/2)(Id Imag((Vd + jVq)(Id - jIq)) ..pdId + ..pql q = 0.25 Imag((Vd

jIq)e;(6-1C(2))

o The steady-state analysis of a given problem involves certain constraints. For example, depending on what is specified, the sol ution of the ,teadystate equations may be very difficult to solve. T he solution of steady-state in multimachine power systems is usually called load flow, and is discussed in later chapters . The extension of this steady-state analysis to include saturatjon js left as an exercise.

3.7

Operational Impedances and Tes t Data

The synchronous machine mo del derived in this chapter was based on the initial assumption of three stator windings, one field winding, and three damper win dings (ld, lq, 2q) . In addition , the machine reactances and time constants were defined in terms of this machine structure . This is consistent wit h [20J and many other references. It was noted earlier, however, that many of t he machine reactances and time constants have been defined through physical tests or design parameters rather than a presupposed physical structure and model. Regardless of the definition of constants, a given model contains quantities that must be replaced by numbers in a specific simulation . Since designers use considerably more detailed modeling, and physical tests are model independent , there could be at least three different ways to a rrive at a value for a constant denoted by the symbols used in the model of this chapter. For example, a physical test can be used to comput e a value of T~o if T~~ is defined through the outcome of a test. A designer can compute a value of T~o from p hysical pa.rameters such that the value would approximate the test value. The definition of T~o in this chapter was not based on any test and could, therefore, be different from that furnished by a manufacturer. For this reas on, it is important to always verify the definitions of all constants to ensure that the numerical value is a good approximation of t he constant used in the model.

3.7. OPERATIONAL IMPEDANCES AND TEST DATA

55

The concept of operational impedance was introduced as a means for relating test data to model constants. The concept is based on the response of a machine to known test voltages. These test voltages may be either dc or sinusoidal ac of variable frequency. Tbe stator equations in the transformed and scaled variables can be written in the Laplace domain from (3 .64)- (3.72) with constant speed (w = w u ) as -

W,u-

.5-

..pq + -..pd

Vd

= -R.Id -

Vq

= -R.Iq + -..pd + -..pq Ws W!J

VA

= -R.Io + -w. ..po

-

w&

-

W!J!I -

-

s-

(3.241)

Ws

S -

(3.242) (3.243)

where s is the Laplace domain operator, which, in sinusoidal steady state with frequency W o in radians / sec, is S

= JW o

(3.244)

If we make the assumption that the magnetic circuit has a linear flux linkage/current relationship that satisfies assumptions (3) and (4) of Section 3.3, we can propose that we have the Laplace domain relationship for any number of rotor-windings (or equivalent windings that represent solid iron rotor effects). When scaled, these relationships could be solved for ..pd, ..po' and ..po as functions ofl d , I q, la, all rotor winding voltages and the operator s. For balanced , symmetric windings and ali-rotor winding voltages zero except for Vfd , the result would be

,pd = - Xdop(s)Id ,pq

=-

Xqop(s)Iq

..po = -Xooplo

+ Gop(S)Vfd

(3.245) (3.246) (3.247)

To see how this could be done for a specific model, consider the threedamp er-winding model of the previous sections. The scaled Kirchhoff equations are given as (3.67)-(3.70), and the scaled linear magnetic circuit equa· tions are given as (3 .122)-(3.128). From these equations in t he Laplace domain with operator sand V,d = V 'q = V2q = 0,

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

56

(3.249) S

-

- [X~q( - Iq)

w, s

-

- [X~q( -Iq)

W,

+ X'qIq + CqXmq1,q ) = -R'qI'q

+ cqX=qI,q + X 2q-I 2q ] =

-R 2q I 2q

(3.250) (3.251) (3.252)

The two d equations can be solved for Ifd and I'd as functions of s times Id and V fd. The two q axis equations can be solved for I'q and 1,q as functions of s times I q . When substituted into (3.1 22) and (3.125), this would produce the following operational functions for tills given model:

X dopeS)

= Xd

;;;X!d(Rld + ;;;X'd - 2;;;Cd X =d + Rid + ;;;X/d)] (3.253) [ (Rfd + ':, Xfd)(R'd + ':,X'd ) - C,CdX=d )2 _

G (s) = op

[

X~d(R'd + ..L X 'd - ..LCdX=d) ] "" "', (Rfd+ ':,Xfd)(R1d+ ':,X,d) - (':,Cd X =d)2

(3.254)

Xqop(S) = Xq ;;;X!q(R 2q + ~X2q - 2 ~cqXmq + R 1q + ttXlq) (R2q + ':,X2q ){R ,q + ':,X,q ) - (':.CqX=q)2

X ""p{s ) = Xl, Note that for

S

(3.256 )

= 0 in this model Xdop{O ) = Xd Gop{O) = Xmd Rfd Xqop {O) = Xq

and for

S

=

00

(3.255)

(3.257) (3.258) (3.259)

in this model with Cd = cq = 1

= X;

(3 .260)

Xqop {oo) = X;

(3.261)

Xdop{ oo)

For this model, it is also possible to rewrite (3.253)- (3.256) as a ratio of polynomials in s that can b e factored to give a time constant representation.

3.7. OPERATIONAL IMPEDANCES AND TEST DATA

57

The purpose for introducing this concept of operational functions is to show one possible way in which a set of parameters may be obtained from a machine test. At standstill, the Laplace domain equations (3.241)-(3.243) and (3.245)-(3.247) are

Vd = - (R.

+ -':"'Xdop(S)) Id + -':"'Cop(S)VJd Ws

(3.262)

w~

-(R.+:.Xqop(s))Iq

(3.263)

Vo = - (R'+ :,Xoop(S))10

(3.264)

Vq

To see how these can be used with a test, consider the schematic of Figure 3.1 introduced earlier. With all the abc dot ends connected together to form a neutral point, the three abc x ends form the stator terminals. If a scaled voltage Vtest is applied across be, with the a terminal open, the scaled series Ic( -h) current establishes an axis that is 90 0 ahead of the original a-axis, as shown in Figure 3.5.

(neutral)

""'P------'r-L--j--+

(neutral)

a-axIS

(neutral)

Figure 3.5: Standstill test schematic With this symmetry and 0shaft = "; mech. rad (found by observing the

58

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

field voltage as the rotor is turned), the scaled voltage Va will be zero even for nonzero Ie = - Ib and lfd, since its axis is perpendicular to both the band c-axis and the field winding d-axis. Also, the unsealed test voltage is Vtest

and, by symmetry with Os haft =

= Vb

-

(3.265)

llc

't mechanical radians, (3.266)

The unsealed transformed voltages are

v'3

v'3

v'3

Vd = 2Vb - 2ve = 2Vtest 'V q

= Va

=0

(3.267) (3.268)

and the unscaled transformed currents are

.

v'3

v'3

'd = 2'b - 2'e = iq

=

io

v'3'b = v'3'test

=0

(3.269) (3.270)

For a test set of voltage and current,

v'3v'2v,0 cos(wot + 00) = v'2lto cos(wot +
Vtest =

(3.271)

itest

(3.272)

with scaled RM S cosine reference phasors defined as

(3.273) (3 .274) the scaled quantities Vd and Id are

(3.275) (3.276)

3.7. OPERATIONAL IMPEDANCES AND TEST DATA

59

or

Vd

= -V3 IV testl cos(wot + 0 2 pu

(3.277)

Id

= -V3Il'testl cos(wot + ¢o)

(3.278)

0 )

pu

For

5

= jwo, the ratio of test voltage to current is ' )LI Z( o JW o

V test pu

__

-

Itest

2 Vd Id

(3.279)

pu

which can be written as a reaJ plus imaginary function

(3.280) Therefore, from (3.262), with V

fd

= 0 and

5

= jwo,

(3.281) Clearly R, is one-half the ratio of a set of de test voltage and current . After R, is determined , the operationaJ impedance for any frequency Wo is

(3.282)

A typicaJ test result for a salient pole macrune is shown in Figure 3.6 [20, 52J (X d",,(jW o ) is a complex quantity) . The plot typicaJly has three levels with two somewhat distinct breakpoints, as shown . For round rotor macrunes, the plot is more uniform, decreasing with a major breakpoint at 0.01 Hz. A similar test with different rotor positions could be used to compute Xqop(jw o). These tests, as well as others, are described in considerable detail in [20,52 , 53J. Reference [53] also includes a discussion on curve-fitting procedures to compute "best-fit" model parameters from the frequen cy response test data. For example, the operationaJ impedance X dop( 5) for the model

60

CHAPTER 3. SYNCHRONOUS MACHINE MODELING

used earlier is given by (3.253). H a plot similar to Figure 3.6 (together with a plot of the angle on X dop( s)) is available, the model parameters can be computed so that (3.253) is some "best-fit" to Figure 3.6. It is possible to make the fit better for different frequency ranges. Thus, the data to be used for a given model can be adjusted to make t he model more accurate in certain frequency ranges. The only way to make the model more accurate in aJl frequency ranges is to increase the dynamic order of the model by adding more "damper" windings in an attempt to reach a b etter overall fit of X dop( s) to the standstill frequency response. Similarly it is clear that the key to model reduction is to properly eliminate dynamic states while still preserving some phenomena (or frequency range response) of interest. This is discussed extensively in later chapters.

1.0

O. l

000

rad/sec

Figure 3.6: Salient-pole machine operational impedance

In summary, it is important to note several important points about synchronous machine dynamic modeling. First, the literature abounds with various notational conventions and definitions. The notations and conventions used in this text are as standard as possible given the proliferation of models. It basically follows that of [20J- [29]. Second, it is imp ortant to repeat that standard symbols have been defined through the model proposed in this text (and [20]-[29]) with a few noted excep tions. The procedures followed by the industry often define constants through well-defined tests. In some cases, these definitions do not coincide precisely with the model definitions in this text. Thus, when using data obtained from manufacturers, it is important to clarify which definition of symbols is being used . H frequency response data are given, the model parameters can be computed using curve-fitting techniques.

3.B.

3.8

PROBLEMS

61

Problems

3.1 The text uses Tqdo to transform abc variables into dqo variables. Consider the following alternative transformation matrix:

Pdqo

- I'{ _

COS

2

[

(~

~ 0shaft cos ~fOshaft - 2;~ . Po 2" sm "2 shaft -"3

. P 8m 2

~shaft

I

.j2

+ >:j

cos °shaft . Po 2" 8m "2 shaft + "3

1

I

V'2

.j2

Show that PjqO = P;;'~ (Pdqo is orthogonal). 3.2 Given the following model v

= 10i +

d>'

dt' >.

= 0.05;

scale v, i, and>' as follows:

to get

V=RI

+ ~d1/J WB dt

'

"'=XI 'I'

Find R and X if VB = 10,000 volts, SB = 5 rad / see, and 1B = SB/VB, AB = VB/WB'

X

106 V A, WB

= 271"60

3.3 Using the Pdqo of Problem 3.1 with

hv cos(w,t + 0) h 2V eos(w,t + 0 h 2Ve08(w,t+O+ and Vd ,

2" 3) 2" 3 )

~Oshaft - w,t -~ , express the phasor V = Ve jB in terms of vq, and 6 that you get from using Pdqo to transform Va, Vb, V c into

6 D.

Vd, V q1 Va'

3.4 Neglect saturation and derive an expression for E' in the following alternative steady-state equivalent circuit:

62

CHAPTER 3. SYNCHRONOUS MACIIINE MODELING

1X'd

R, +

Writ e E' as a function of E~, id, I" , O. 3.5 Given the rnagnetizatioll curve shown in pu, compute X 771d and plot SG( 1/1) for

I I I I

I

u -

I I I I I

~,

I

1.11_

2 .0

1.0

3.0

I

3.6 R epeat t he derivation of the single-machine steady- st a te equivalent circuit using the following saturation function s: 5 d(2) --

S(2) - 5(2) - 5,(2) - 5,(2) ld

-

q

-

1q

-

5}~ = 5G(B~)

2q

-

5(2) - 0 0

-

3.8. PROBLEMS

63

where SG is obtained from the open-circuit characteristic as given in the text.

3.7 From Ref. [51], show that the saturation model of Problem 3.6 does satisfy all conditions for a conservative coupling field. 3.8 Given (3.122)-(3.124) and (3.125)-(3.127), together with (3 .64)-(3.70), find a circuit representation for these mathematical models . (Hint: Split Xd, Xfd , X'd , etc . into leakage plus magnetizing.) 3.9 Given the following nonlinear magnetic circuit model for a synchronous machine: 1/Id 1/Ifd 1/Iq

1/1,.

Xd( - Id)

+ X=dIfd -

Sd( 1/Id, 1/1 fd)

+ X fdIfd - Sfd( 1/Id, 1/1fd) X.( - Iq) + X=q!,q - Sq( 1/Iq, 1/1,.) X=.( -I.) + X'.!,q - S'q(1/Iq , 1/Ilq) X=d( - Id)

(a) Find the constraints on the saturation functions Sd, Sfd, Sq, S'q such that the overall model does not violate the assumption of a conservative coupling field. (b) If the other steady-state equations are Vd = - R.Id - 1/Iq Vq = - R.I.

+ 1/Id

Vfd

= RtdItd

0= R .!,q '

find an expression for E, where E is the voltage "behind" Rs+jXq in a circuit that has a terminal voltage

Chapter 4

SYNCHRONOUS MACHINE CONTROL MODELS 4.1

Voltage and Speed Control Overview

The primary objective of an electrical power system is to maintain balanced sinusoidal voltages with virtually constant magnitude and frequency. In the synchronous machine models of the last chapter) the terminal constraints

(relationships between Vd, fa, V q , f q , Vo, and fo) were not specified. These will be discussed in the next chapter. In addition, the two quantities Efa and TM were left as inputs to be specified. Efd is the scaled field voltage, which, if set equal to 1.0 pu, gives 1.0 pu open-circuit terminal voltage . TM is the scaled mechanical torque to the shaft. If it is specified as a constant, the machine terminal constraints will determine the steady-state speed. Specifying Efd and TM to be constants in the model means that the machine does not have voltage or speed (and, hence, frequency) control. IT a synchronous

machine is to be useful for a wide range of operating conditions, it should be capable of participating in the attempt to maintain constant voltage and frequency. This means that E fd and TM should be systematically adjusted to accommodate any change in terminal constraints. The physical device that provides the value of E fd is called the exciter. The physical device that provides the value of TM is called the prime mover. This chapter is devoted to basic mathematical models of these components and their associated control systems.

65

66

4.2

CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS

Exciter Models

One primary reason for using three-phase generators is t he constant electrical torque developed in steady state by the interaction of the magnetic fields produced by the armature ac currents wi th the field de curren t. Furthermore , for balanced three-phase machines, a de current can be produced in the field winding by a de voltage source. In steady state, adjustment of the field voltage changes the field current and, therefore, the terminal voltage. Perhaps the simplest scheme for voltage control would be a battery with a rheostat adjusted voltage di vider connected to the field winding through slip rings . Manual adjustment of the rheostat could be used to continuously react t o changes in operating conditions to maintain a voltage magnitude at some point. Since large amounts of power are normally required for the field excitation, the control device is usually not a battery, and is referred to as the main exciter. This main exciter may be either a de generator driven off the main shaft (with brushes and slip rings), an inverted ac generator driven off the main shaft (brushless with rotating diodes) , or a static device such as an ac-to-dc con verter fed from the synchronous machine terminals or auxiliary power (with slip rings). The main exciter may have a pilot exciter that provides the means for changing the output of the main exciter. In any case, Efd normally is not manipulated directly, but is changed through the actuation of the exciter or pilot exciter .

Consider first the model for rotating de exciter s. One circuit for a separately excited dc generator is shown in Figure 4 .1 [21J. Lal

rfI +

+

Saturation

+

-

K il l (r)\ 4>01

e on l l = Vfd

Figure 4.1: Separately excited dc machine circuit Its output is the unscaled synchronous machine field voltage small ral and L al , this circuit bas the dynamic model

ein1 Vfd

.

KalWI

For

dAjl

= 'in1rfl + & =

Vfd.

<Pal

(4.1) (4.2)

4.2. EXCITER MODELS

67

with the exciter field flux linkage related to field flux ¢fl by ( 43) Assuming a constant percent leakage (coefficient of di spersion (71)' the armature flux is . 1 ( 4.4) ¢al = -¢j1 <71

Assuming constant exciter shaft speed W"

Aj1 =

Nfl<7J

( 4.5)

'Uld KalWl

Now, the rela tionship between 'Ufd and iinl is nonlinear due to saturation of the exciter iron, as shown in Figure 4.2.

/sIOPOK

g,

---+~=

!(ro1 = constant) ~~iinl

Figure 4.2: Exciter saturation curve Without armature resistance or inductance, this curve is valid for opencircuit or loaded conditions. The slope of the unsaturated curve (air-gap line) is [55] KgI

=

( 4.6)

where L f1 us is referred to as the "unsaturated" field inductance. The saturation can b e accounted for by a saturation function fsat defin ed through Figure 4.2 as (4.7) !sat('Ilfd) = 6iinl / 'Ilfd In terms of these quantities,

\

All

=

L !IUS

K

gl

Vld

( 4.8)

68

CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS

and .

Vfd

'in1 = -K 91

+ f sat (Vfd )Vfd

( 4.9)

With these assumptions, the unsealed exciter dynamic model is (4.10 ) This equation must now be scaled for use with the previously scaled synchronous model. Since the armature terminals of the exciter are connected

directly to the terminals ofthe synchronous machine field winding, we must scale Vfd as before and use the same system power base. Thus, using (3.54), (3.59), (3.113), and (3.139), we define (4.11) (4.12) (4.13) ( 4.14) With these assumptions and definitions, the scaled model of a separately excited de generator main exciter is (4.15) The input VR is normally the scaled output uf the dIllpliflel" (or pilot exciter))

which is applied to the field of the separately excited main exciter. When the main de generator is self-excited, this amplifier voltage appears in series with the exciter field, as shown in Figure 4.3. This field circuit has the dynamic equation (4.16) with the same assumptions as above, and the new constant KEself defined as (4.17)

69

4.2. EXCITER MODELS

+

'i!

ra!

Saturation Lfl

La!

eoutl = vfd

Figure 4.3: Self-excited dc generator circuit

The scaled model of a self- excited de generator main exciter is, then, (4. 18 ) For typical machines, KEself is a small negative number. To allow voltage buildup, it would be necessary to specify VB to include the residual voltage that exists at zero iin1 , as shown in Figure 4.2. Also, if ein1 is replaced by a rheostat whose variable resistance is included in Tfl, KSself and SE would be functions of both the act ual field resistance and the rheostat resistance . This rheostat can be set to produce the required terminal voltage. In steady state with VB = Vres this requires

(4.19) Tha.t is ) since

incluues the rheo:stat :seLtiug) ii Ci:Lll be a.djulited fur a given steady-state condition. This would make T fl, rather than VR, a cont rol variable . In steady state, the exciter equation (written simply with K E) is TJl

(4.20) The input VR usually has a maximum VR mu , which produces maximum (ceiling) excitation voltage Efd m . . ' Since 'SE is a function of this excitation level, these quautities must satisfy

(4 .21 )

70

CHAPTER. 4. SYNCHRONOUS MACHINE CONTROL MODELS

When specifying exciter data, it is common to specify the saturation function through the number S Em .. and the number S ED.7S m.. , which is the sat uration level when Eld is 0.75 Efdm . . ' A saturation function is then fitted to these two points. One typical function is ( 4.22) When evaluated at two points, this function gives (4.23) ( 4.24)

SEO .75 m ll. x

For given values of K E , VR m u ) SETDu ) and SEO .75 mu ) the constants Ax and B. can be computed. Th.is is illustrated in the following example .

Example 4.1 Given : KE = 1.0, VR m . . = 7.3, SEm . . = 0.86, SED.75 m . . = 0.50 (all in pu) Find: Ax and Bx Solution: From (4 .20),

o=

-(1

+ 0.86)Efd

m ..

+ 7.3

0.86 = AxeB",EJdmu 0.50

=

AxeB;t:~EJdm ax

Solving these three equations gives Efd m . .

= 3.925

A. _ 0.09826 B. _ 0.5527

o References [55, 56J and [75J give additional information on these and other exciter models.

4.3

Voltage Regulator Models

The exciter provides the mechanism for controlling the synchronous machine terminal voltage magnitude. In order to automatically control terminal voltage, a transducer signal must be compared to a reference voltage and amplified to produce the exciter input VR. The amplifier can be a pilot exciter

4.3. VOLTAGE REGULATOR MODELS

71

(another de generator) or a solid-state amplifier. In either case, the amplifier is often modeled as in the last section with a Umiter replacing the saturation function.

dVR TATt=-VR+KAVin

(4.25)

Vmin R _< V R < _ vRmax

(4 .26)

where Vin is the amplifier input, TA is the amplifier time constant, and KA is the amplifier gain. The VR limit can be multivalued to allow a higher limit during transients . The steady-state limit would be lower to reflect thermal constraints on the exciter and synchronous machine field winding. Recall that VR is the scaled input to the main exciter. This voltage may be anything between zero and its limits if the main exciter is self-excited, but must be nonzero if the exciter is separately excited. We have assumed that the amplifier data have been scaled according to our given per-unit system. If the voltage Vin is simply the error voltage produced by the difference between a reference voltage and a conditioned potential transformer connected to the synchronous machine terminals, the closed-loop control system can exhibit instabilities. This can be seen by noting that the self-excited de exciter can have a negative KE such that its open-loop eigenvalue is positive for small satur ation SE. Even without this potential instability, there is always a need t o shape the regulator response to achieve desirable dynamic performance. In many standard excitation systems, t his is accomplished through a stabilizing transformer whose input is connected to the output of the exciter and whose output voltage is subtr acted from the amplifier input. A scaled circuit showing the transformer output as VF is shown in Figure

4.4.

Ideal

Figure 4.4: Stabilizing transformer circuit

If 1'2 is initially zero and L'2 is very large, then 1'2 must remain near zero.

72

CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS

With this assumption, an approximate dynamic model for this circuit is ( 4 .27) (4.28 ) where VF is a scaled output of the stabilizing transformer. Differen tiating VF and Efd gives

dVF dt

= N2 L tm ( N,

(dEfd _ Rtl N 2VF )) (Ltl + Ltm) dt L tm N , 1

(4 .29)

Using (4 .18) with general KE and defining

TF KF

+ L tm =t. Ln Rtl

(4.30 )

N, L

tm t. - -= N2 Rtl

(431 )

the dynamic model of the stabilizing transformer can be written as

_ TFdVF -- dt

-

V

F

+ }(F

(

-

KE

+ SE(Ejd)Efd + -VR) TE

TE

(4.32)

Another form of this model is often used by defining

Rf

KF =t. -EfdVF TF

(4.33)

With Rf (called rate feedback) as the dynamic state, (4.34 ) This form wjll be used throughout the remainder of the text. If the amplifier input Vin contained only the reference voltage minus the termina.l voltage minus VF, the voltage regnlator could still have regulation and stabilit y problems. There can be regulation problems when two or more synchronous machines are connected in parallel and each machine has an exciter plus voltage regulator. Since the synchronous machine field current has a major role in determining the reactive power output of the machine, parallel operation requires that the field currents be adjusted properly so

4.3. VOLTAGE REGULATOR MODELS

73

t~at

the machines share reactive power. This is accomplished through the addition of a load compensation circuit in the regulator, whia makes the parallel operation appear as if t~ere are two different terminal voltages even though both machines are paralleled to the same bus . This can be done by including stator current in the regulator input . To see ~ow this can be modeled, consider the scaled terminal voltages Va, Vi" and Ve found by transforming and rescaling Vd, V.' and Yo' Using (3.14) with Vo = 0 and (3.39 )

Va = h(Vdsin(W.t + .5) + V.cos(w.t + .5)) Vb = h Ve = h

(4.35)

2;) + V. cos (w. t+ .5 _ 2;)) (Vd sin (w,t+ .5+ 2;) + Vq cos (w,t +5 + 2;) ) (Vd sin (w,t + .5 -

(4.3 6)

(4.37)

The stat or line currents la, h , and Ie are related in the same way with Vd and Vq replaced by Id and l •. These expressions are valid for transients as well as steady state, and can be written alternat ively as Va = h/V]

Vb =

+ V.2 cos (w.t +.5 - ; + tan- 1 ~:)

( 4.38)

h/v:t + Vq2 cos (w.t +.5 _ ; + tan - 1 ~: _

2;)

(4.39)

+ .5 _ ~ + tan- 1 ~: +

2;)

( 4.40)

Ve = h/V] + V.2 cos (w,t

and similarily for la , h, and Ie . To see how load compensation can be performed, consider the case of an overexcited synchronous machine (serving inductive load), with the steady-state phasor diagram of Figure 4.5. Suppose that the sensed voltage is line-to-line RMS volt age Vae; then the uncompensated voltage is defined as

(4.41 ) Consider the compensated voltage deftned as

( 4.42)

74

CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS

,, -

Va,

Figure 4.5: Steady-state overexcited phasor diagram

where occomp is some positive compensation constant. The compensated terminal voltage will differ from the uncompensated terminal voltage by an amount proportional to occomp and the phase shift of lb. Wit h lb in the position shown in the phasor diagram, and if the reference voltage is near t he uncomp ensated value of Vi, the error sigual from the compens ated V, will tell the regulator to lower the terminal voltage (the reverse of what you might expect! ). When t he machine is undere..'(cited (serving capacitive load), the curr ent lb will lead V b, and the compensated voltage will be less t han the un compensated voltage. This voltage, when compared to a r eference voltage near the uncompensated voltage, will tell t he regulator t o r aise the terminal voltage (again, the reverse of wha t you might expect). When two generators are operating in parallel, if the neld current on one generator becomes excessive and causes "dTculating ll reactive power ) this reactive power will be an inductive load to the excessively excited machine and a capacitive load to the other . The compensation circuits then will cause the excessive excitation to be reduced and the other to be raised, thus balancing the reactive p ower loading . This type of bad compensation is called parallel droop compensation . Additional types of comp ensation that do not result in a drop in voltage under inductive load are also available.

In terms of the dq comp onents of V a, V c, and l b, we can define the compensated and uncompensated voltages as

4.3. VOLTAGE REGULATOR MODELS

75

(4.43)

This gives

( 4.44) Example 4.2

Va Ib

This is a power factor

v'uncomp

v'comp

1LO, Vb

= 1L -

120° , V c

= lLl20°

0.5Ll800 (all pu)

= cos 60° (lagging power factor load) -

1

y'3 y'3 = 1.0 1 _ I y'3 L _ 30°

y'3

"'camp 0.5Ll80

I

For cxcomp = 0.1

V'comp

= 1.025

If the reference voltage is 1.0, the voltage regulator will attempt to lower the voltage by lowering Efd .

o

76

CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS

Example 4.3 Let us use the same voltages, cos 30° (leading).

h

= 0.5L - 90 0 (pu). This is the power factor

V'uncomp

~ I V3 L -

30

0 -

0.05L - 90

0

I

0.986

If the reference voltage is 1.0, the voltage regulator will attempt to raise the voltage by raising Efd. This compensation will make two parallel generators share the total reactive power output .

o Realizing that the sensed voltage may be either compensated or uncompensated, we simply drop the subscript and write the final general expression for the amplifier input voltage using the stabili.zer feedback variable Rj, rather than VF, as

( 4.45)

In this model, we have not included any dynamics for the transducer, which could be a potential transformer, filters, and smoothers. There are many other fine details about excitation systems that may be important for some simulations. In keeping with the philosophy of this text as one on fundamental dynamic modeling, we conclude this section with a summary of a fundamental model of an excitation system :

4.4. TURBINE MODELS

T dEfd

E ----;J,"t -

77

- (KE

+ SE(Efd))Efd + VR

T dRf Pdt

Kp - Rf + -Efd Tp

T dVR Adt

-VR+KARf -

(4.4 7) KAKPE Tp fd

+KA(V;'ef - V,) Vmin R < _ ViR

< vRmax

_

(4.46 )

( 4.48) ( 4.49)

where KE may be the previously defined separate or self-excited constant, and V, may be either of the previously defined compensated or uncompensated terminal voltages. To be complete, the model should include an expression for 5 E and an algebraic equation for v,.

4.4

Turbine Models

The frequency of the ac voltage at the terminals of a synchronous machine is determined by its shaft speed and the number of magnetic poles of the machine. The steady-state speed of a synchronous machine is determined by the speed of the prime mover that drives its shaft. Typical prime movers are diesel engines, gasoline engines) steam turbines, hydroturbines (water wheels), and gas turbines. The prime mover output affects the input TM in the model of the last chapter. This section presents basic models for the hydroturbine and steam turbine. Hydroturbines Hydropower plants have essentially five major components. These are the storage reservoir, intake tunnel, surge t ank , penstock, and water (hydro) turbine. Precise nonlinear models of these components are not typically used in power system dynamic analysis. Alternatively, approximate linear models are used to capture the fundamental characteristics of the plant and its impact on the electrical system. Thus , the following models should not be used for studies where large changes in turbine power are expected. Since the turbine torque is the primary variable of interest, most models are made as

78

CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS

simple as possible while still preserving the turbine t orque and speed control characteristics. The power to the water turbine depends on the p osition of the gate valve at the bottom of the penstock. The power is derived from the water pressure that arises due to the water head (elevation). The p ensto ck is the water channel from the intake tunnel of the elevated reservoir down to the gate valves and turbine. The gate valve position then corresponds to a certain level of power PHV at rated speed. Using scaled parameters consistent with the last chapter, a simplified model of hydroturbine smallchange dynamics is [57J

+ A 12 f',wHT + A ,3 f',PHv = A 21 f',T H + Anf',wHT + A 23 f',PHV

f',F R = Anf',T H f',THT

( 4.50 )

(4 .51)

where f',F R is the change in water flow rate, f',T H is the change in turbine head, f',wHT is the change in hydroturbine speed, and f',THT is the change in hydroturbine output torque, all scaled consistently with the last chapter. One common model considers only two dynamic phenomena, the rate of change of flow deviation as [5 7J df',F R

dt

( 4.52)

and Newton's law for the turbine mass with scaled inertia constant HHT df',OHT

dt 2HHT df',wHT w, dt

(4.53 ) (4.54 )

where Tw is the water starting tinle in seconds , and TM is as previously defined for the synchronous machine shaft dynamics. The last equation allows for the case where the shaft connection between the hydroturbine output and the synchronous machine input is considered a stiff spring rather t han a rigid connection. In this case, the change in torque in to the synchronous machine would be (4. 55 ) where KH M represents the stiffness of the coupling between the turbine and the synchronous machine, and 0 is the machine angle, as previously defined. While it is clearly possible to keep all terms , it is customary to assume that A.,2 f',wHT and A 22 f',wHT are small compared to other terms.

79

4.4. TURBINE MODELS

For operation near an equilibrium point (denoted as superscript 0) with

(4.56) and

= W"

WHT

( 4.57)

the actual variables are THT = TYfT

TM

+ 6.THT

= TM + 6.TM

( 4.60)

+ 6.0 = o'HT + 6.0HT = w'HT + 6.wHT

(4.61 )

o= WHT

( 4.59)

+ 6.PHV

PHV = P'Hv

OHT

( 4.58)

0°

(4.62) ( 4.63)

The hydroturhine model written in actual per-unit torques (but valid only for small changes about an equilibrium point) is written with THT as a state: 1 T dTHT = -THT W dt All

+ dOHT

dt' 2HH1· dw HT W,

dt TN!

An + -All PHV

T (A 23

W

_ A13 A All

21) dPHV dt

(4.64 )

= WHT -W,

(4.65 )

= THT - TN!

( 4.66)

= -KHM(O -

( 4.67)

OHT)

When used with the synchronous machine model, the above TM expression must be used in the speed equation for the synchronous machine. Thus, the synchronous machine and turbine are coupled through the angles 5 and 0HT . The hydrovalve power PHV will become a dyn amic state when the hydrogovernor is added. For a rigid connection, THT becomes TM and the turbine inertia is simply added to the synchronous machine inertia H, giving only

T dTM_ W

dt

-

__1_1'

An P

All M + Au HV

+

_ A13 A Z1) dPHV

T (A W

23

All

dt

( 4.68)

80

CHAPTER 4. SYNCHRONOUS MACIDNE CONTROL MODELS

and TM remain s in tbe synchronous machine speed equation. For an ideal lossless hydroturbine at full load [57]

All = 0.5,

A21

= 1.5,

A13

= 1.0, A 23 = 1.0

(4.69)

and Tw ranges between 0.5 and 5 seconds. Additional hydroturbine models are discussed in references [57]-[59] . Steam turbines Steam plants consist of a fuel supply to a steam boiler that supplies a steam chest. Tbe steam chest contains pressurized steam tbat en ters a highpressure (HP) turbine thro ugh a steam valve. As in tbe hydroturbine, the power into the high-pressure turbine is proportional to the valve opening. A nonreheat system would then terminate in the condenser and cooling systems, witb the HP turbine sbaft connected to the synchronous machine. It is common to include additional stages, such as the intermediate (IP) and low (LP) pressure turbines. The steam is reheated upon leaving the highpressure turbine, and either reheated or simply crossed over between the IP and LP turbines. The dynamics that are normally represent ed are the steam chest delay, the reheat delay, or the crossover piping delay. In a tandem connection, all stages are on the same shaft. In a cross-compound system, the different stages may be connected on different shafts. These two shafts then supply the torque for two generators. In this analysis, we model the steam chest dynamics, the single reheat dynamics, and the mass dynamics for a two-stage (HP and LP) turbine tandem mounted . In this model, we are interested in the effect of the steam valve position (power Psv) on the synchronous machine torque TM. The incremental steam chest dynamic model is a simple linear single time constant with unity gaill, written in scaled variables consistellt with the last chapter as TCH

dClPCH

dt

=-

ClPCH

+ ClPsv

(4.70)

where ClPCH is the change in output power of the steam chest . This output is either converted into torque on the HP turbine or passed on to the reheat cycle. Let the fraction that is converted into torque be

(4. 71 ) and the fracti on passed on to the reheater be (l-KHP)ClPCH' The dynamics of the HP turbine mass in incremental scaled variables are

(4.72)

4.4. TURBINE MODELS

81 (4.73)

where !:;'THL is the incremental change in torque transmitted through the shaft to the LP turhine. This is modeled as a stiff spring:

(4.74) The reheat process has a time delay that can he modeled similarly as

(4.75 ) where !:;,PRH is the change in outpu t power of the reheater. Assuming that this output is totally converted into torque on the LP turbine,

(4.76) The dynamics of the LP turbine mass in incremental scaled variables are d!:;.hp

(4.77)

dt 29LP d!:;'w LP w, dt

( 4.78)

where the torque to the connection of the LP turbine t o the synchronous machine is assumed to be transmitted through a stiff spring as

(4.79) For operation near an equilibrium point (denoted by superscript 0) with PCH

= P~v, T HP = THL = KHPPCH = -KHL(O'LP - o'f.;p ), TLP = PRH = (1 - KHP )PCH , WLP = W WHP = 3 )

TM ==

PCB = -KLM(OO- 0LP)

w$ )

(4. 80)

the act ual variables are PCH

== PCB + !:;,PCB

(4 .81)

Psv

==

( 4 .82)

OHP

==

+ !:;.Psv 0HP + !:;'OHP P~v

(4.83)

82

CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS

= WHP + t:J.WHP PRH = PIm + t:J.PRH OLP = oLP + t:J.OLP WLP = WLP + t:J.WLP o = 0° + t:,.O TM = TM + t:J.TM

(4.84)

WHP

(4.85 ) ( 4.86) ( 4.87) ( 4.88) ( 4.89)

The steam turbine model written in actual per-unit torques (but valid only for small changes about an equilibrium point) is written as TCH

dPCH dt = - PCH

+ Psv

( 4 .90)

doHP dt

( 491)

2HHP dwHP w, dt = KHPPCH TRH

dPRR dt = -PRH

dhp

+ KHL(OLP -

+ (1 -

----;It = WLP - w,

KHP)PCH

OHP)

( 4.92) (4.93) (4 .94)

and TM in the synchronous machine speed equation must be replaced by ( 4.96) The steam valve position Psv will become a dynamic state when the steamgovernor equations are added . For rigid shaft couplings ( 4.97) and the two turbine masses are added into the synchronous machine inertia to give the following steam turbine model with TM as a dynamic state:

4.5. SPEED GOVERNOR MODELS

T

83

KHPTRH dTu To )PCH RH---;J.t - -Tu + (l CH

+ TCH

dPCH dt

-

KHPTRH P To sv CH

- PCH+ PSV

(4.98) (4.99 )

and Tu remains as a state in the synchronous machine model. It is possible to add more reheat stages and additional details. It is also possible to further simplify. For a nomeheat system, simply set TRH = 0 in (4.98)- (4.99), and the following model is obtained:

(4.100)

where again TM remains a state in t he synchronous machine model and Psv will become a state when the governor is added. The above non-reheat model is often referred to as a T ype A steam turbine model [59 , 60].

4.5

Speed Governor Models

The prime mover provides the mechanism for controlling t he synchronous machine speed and, hence, t erminal voltage frequency. To automatically control speed (and therefore frequency ), a device must sense either speed or frequency in such a way that comparison with a desired value can b e used to create an error signal to take corrective action. In order to give a physical feeling to tbe governor process , we will derive the dynamics of what could be considered a crude (and yet practical) mechanical hydraulic governor. This illustration and the derivation were originally given by Elgerd in [61]. Figure 4.6 gives a simple schematic of a flyball speed sensor witb ideal linkage to a hydraulic amplifier and piston for main valve control. Suppose t hat the distance of points a and e from a fixed higher horizontal reference are related to the per- unit values of a power change setting Pc and value power Psv , respectively. To see how the flyball function s for some

84

CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS Stearn

/

+~alve

4Lower Speed Changer

t Raise

'--

c b

d

To

Turbine

• e

Fly balJ Speed

Main

Govenor

Pisto n

, Figure 4.6: Mechanical-hydraulic speed governor [61]

fixed Pc, supp ose that a load is removed from the generator such that an excess of power is being supplied to the turbine through the valve. This excess power will cause a change in generator speed 6.w J which will increase the velo city of the lIyballs and hence lower point b. Lowering point b results in a lowering of point c since they are ass umed to be connecled by a rigid rod. Lowering point c must either lower d (if e does not change) or raise e (if d does not change ). If point d is lowered , the high- pressure lIuid will enter the hydraulic servo through the lower channel and exert a force on the main piston to move up point e. Thus , in any case, lowering c results in a raising of e and a corresponding decrease in Psv . The decrease in Psv will eventu ally stop t h e increase in speed that initiated the movement of point b. To model this action, we analyze t he linkages and note that any incremental change in the positions of points a, b, and c are related by

(4.101)

4.5 . SPEED GOVERNOR MODELS

85

Any incremental change in the position of points c, d, and e are related by

(4.102) The position of point a is related to the scaled value of Po so that

( 4.103) Neglecting the flyball dynamics, the position of point b changes in proportion to a. change in electrical speed as

(4.104) A change in the position of point d affects the position of point e through the time delay associated with the fluid in the servo. We assume a linear dynamic response for this time delay:

dAy. _ - K A dt

-

(4.105)

,Yd

Substituting the linkage relations in terms of the power change setting and the speed change:

dAy. dt

(4.106) Using t he proportionality between APsv and Ay. as

A

-

KdcKbo AP

y, - KKK de

be:

a

sv

( 4.107)

and deiining

DROOP "

( 4.108)

"

( 4.109)

Tsv

86

CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS

the incremental governor model is T sv dl:1Psv -_ dt

-

Ap SV

Ll

+ ApC Ll

-

(

Ws

27rdroop

)

l:1w

~ Ws

(4.110)

The quantity "droop" is a speed droop expressed in Hz/per-unit megawatts. Alternatively, we denne a speed regulation quantity RD as ( 4.111) For operation near an equilibrium point (denoted by superscript 0) with o posv+~ 1 Pc= Rn

(WO --1)

( 4.112)

Ws

the actual per-unit variables are

+ l:1Psv Pc = Pc + l:1Pc W= WO + l:1w

PSV = Psv

(4 .113) (4.114)

(4.115)

In these variables, the governor model including limits on the value position IS

dPsv = -Psv dt

Tsv~-

+ Pc -

o <:: Psv <:: PS'v x

1 (w - - 1) RD Ws

~

( 4.116) (4.117)

In addition to the limit on the valve position Psv, it may also be important to constrain the derivative of Psv as rate limits. If this is done, the above model corresponds to a General Electric type EH [59]. This model is not valid for large changes, but to illustrate the signincance of Pc and RD, suppose that the machine is unloaded with Psv = Pc = O. If Pc is left at 0 and the machine is loaded to its rating (Psv = 1), the full load speed would be (1- RD)w s . So, if the speed regulation is set for 5% droop (RD = 0.05), the change in speed between no load and full load would be 5% of the rated load. Thus, RD can be written as R D

_ % droop 100

(4.118)

4.6. PROBLEMS

87

The quantity Pc is a control input that Can be either a constant, or the output of an automatic generation control (AGC ) scheme. To provide zero steady-state error in speed (and therefore frequency), an integral control is needed. In multimachine power systems, this load frequency control (LFC) is used together with economic illspatch to maintain frequency at minimum cost on an areawide and systemwide basis. In this case, Pc would be the output of a load reference motor, which is driven by an AGC signal based on a unit control error. W1tile this control ideally would maintain rated frequency in steady state , the accumulated error during transients makes it necessary to have time corrections whenever the total accumulated time error passes a specified threshold [61). These controls involve fuel and boiler dynamics that are often considered slow enough to be constants .

4.6

Problems

4.1 Using the steady-state exciter model of (4.20) and (4.22) with K E = 1.0, VRmax 8.0, SErnax = 0.9 , SEO.75 max 0.5 (all pu), find Efdmax. A., and B •.

=

=

4.2 Using the exciter model of (4.46) with KE = 1.0, SE = 0, and TE 0.5 sec, compute the response of Efd for a constant input of VR = 1.0. Use an initial value of Efd = 0 (all pu).

=

=

Vres 0.05 , 4.3 Using the exciter model of (4.18) and (4.22) with VR find KEself so that Efd = 1.0 when A. = 0.1 and Bx = 0.6 (all pu) .

4.4 Using t he answer to Problem 4.3, compute the response of E fd for TE = 0.5 sec when it starts at zero. Tills requires the solution of a nonlinear diiferential equation. 4.5 Using the excit ation system models of (4.46)- (4.49), construct a block diagram in the Laplace domain that shows the control syst em with inputs Vrer and V" and output Efd.

4.6 Repeat Pr oblem 4.5 using VF as a dynamic state, rather t han Rf . 4 .7 Starting with the dynamic model of (4.46)- (4.49), derive the following dynamic model (a fast static exciter/ regulator):

88

CHAPTER 4. SYNCHRONOUS MACHINE CONTROL MODELS

4 .8 Using th e turbine/ governor model of (4.100) and (4 .116), wi th

Tsv = 2 sec

Pc

= 0.7 pu

Rv

= 0.05 pu

TCH =4sec w,=27r60r/s (a) Find the steady-state values of Psv and TM if w = 376.9

r/s.

(b) Find the dynamic r esponse of Psv an d TM if w ch anges at time zero to be 376.8 r I s.

Chapter 5

SINGLE-MACHINE DYNAMIC MODELS 5 .1

Te rm inal Constraints

Throughout Chapters 3 and 4, the constraints on 1d, 1q, 10 and Vd, Vq, Vo have been left unspecified. P erhaps the simplest terminal const raint that could be speci.fied is that of an ideal balanced three-phase resistive load (Rload in per unit). This terminal constraint is Vd

= 1dRIoad (Rload < 00)

(5.1)

Vq

= 1qRload (Rload < 00)

(5.2)

Vo

= 10RIoad (Rload < 00)

(5.3)

and 1d

= 1q = 10 = 0 (Rload = 00)

(5.4)

The most commonly used terminal constraint for a single machine is the notorious infinite bus. In most power engineering terminology, an infinite bus is an ideal sinusoidal voltage source with constant magnitude, frequency, and phase. In three-phase systems , this implies an ideal balanced symmetrical three- phase set such as

Va = v'2V. cos(wst + U.. ) Vb

= J2v, cos (w,t + Uv, _ 89

(5.5) 2;)

(5.6)

90

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS (5.7)

This is a positive phase sequence (ABC) set written in per-unit so that V, = 1.0 for rated voltage and w, = 2'1f f, for rated frequency f,. In many studies, 0., is arbitrarily selected as zero. It is useful to know how a synchronous machine dynamic model can be made into an infinite bus. We begin by using the dynamic models of (3.148)- (3 .159), (4.46)-(4.49), (4.100,) and (4.116)(4.117) with Vi defined through (4.43) and (4.44) . As discussed earlier, the "a phase" voltage of the machine during transients and steady state is

Va = v0.JVl

+ Vq2 cos (w,t + 8 -

;

+ tan- 1 ~:)

(5.8)

To qualify as an infinite bus, we must have

JVl

+ V;

=

Voo (a constant)

(5.9)

and 'If

8- 2

Vq = 0 + tan -1 -Vd"OO

( a constant)

(5.10)

Clearly, we must find parameter values that result in constants Vd, Vq, and 8. Considering the voltage magnit ude first, there are two ways in which JVl + Vq2 can he made a constant. The first involves an infinitely high-gain voltage regulator with an infinitely fast amplifier and exciter. This makes the field winding flux linkage infinitely fast so that V t is constant for all disturbances. This method , however, does not constrain Vd and Vq to be individually constant, and thus there is no way to force 0"00 to be constant. The second way to force the terminal voltage magnitude to be constant is actually the opposite of the high-gain regulator approach. Rather than force the field winding to be infinitely fast, we force it to be infinitely slow by letting Tdo equal infinity. In addition, we let T~o go to infinity as well as the machine inertia H . To complete the infinite bus specifications, we let R" X d, X~, T:lo' and T:~ be zero. To see the result, we write the model (3.148) -(3.159) with these parameters: 1 d"'doo

W,

dt

1 d·"'qoo

W,

dt

Woo -"'qoo W,

+ Vdoo

Woo --"'doo W.•

+ Vqoo

(5.11) (5 .12)

5.1. TERMINAL CONSTR AINTS

91

~ d..po= _ V W!l dt 000 dE~=

dt

dE~= dt

(5.13) (5 .14)

=0 =0

do=

dt =

Woo

(5.15) - w,

dw= = 0

dt

For w=(o)

(5.16) (5.17)

..pd= = E~=

(5 .18)

..pq= = - E~=

(5.19)

= w" this model requires E~= = E~=(O)

(5.20)

= Ed=(O) = 0=(0)

(5.21)

..pd= = E~=(O)

(5.23)

..pq= = - Ed=(O)

(5.24 )

Vd= = E~=(O)

(5.25 )

Vq= = E~=(O)

(5.26)

E~=

0=

(5.22)

which then gives

which satisfies the requirements of an infinite bus. Consider the model of a synchronous machine connected to an infinite bus through a balanced three-phase line in unsealed parameters:

(5.27) (5.28) (5.29)

92

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

with

(5.30)

Transformation and scaling consistent with Chapter 3 gIve the following terminal constraints in per-unit:

(5.31) w R e1q - _.1. d 'f't! W II

1 d..peq

+ v:

- - -d - t' W$

~ #'0 Y;.= Re 10 _ W dt II

cos (6 -

e•• )

(5.32) (5.33 )

with

..ped = Xep( - ld)

(5.34)

..peq = X,p ( - [. )

(5.35)

= Xeo( - 10)

( 5.36)

..peo

Note that V, is the RMS per-unit infinite bus voltage, and all quantities are scaled on the machine ratings. It is customary to set 0•• equal to zero, since one angle in any system can always be arbitrarily selected as a reference for all other angles. We will leave this angle as 0•• for now. While other terminal constraints can be specified, the infinite bus is the most widely used for single-machine analysis, partly because it has been traditional to study a single generator with the entire remaining network as a Thevenin equivalent impedance and voltage source. Such equivalents are clearly valid only for some steady-state conditions. Other uses of infinite bus models for machines have arisen recently as mechanisms for avoiding the problems associated with a reference angle and steady-state speed. Clearly, with an infinite bus in a system, the steady-state speed for synchronous machines must be w" and the r eference angle is conveillently specified. Because of their wide use, the remainder of this chapter is devoted to the analysis of single-machine infinite bus systems. It is also useful for illustrating several concepts of time scales in synchronous machines that will help in the extension to multimachine systems.

5.2. THE MULTI-TIME-SCALE MODEL

93

While the following sections are written to follow naturally, the Appendix gives an introduction to integral manifolds and singular perturbation, and provides the basic fundamentals used in the following sections to develop reduced-order models.

5.2

The Multi-Time-Scale Model

In this section, we study the special case of a single machine connected to an infinite bus. In part icular, we use the synchronous machine model of (3. 148)(3.159), the exciter / AVR model of (4.46)-(4.49) without load compensation, the turbine/governor mo del of (4.100) and (4.11 6)-( 4.117), and the terminal constraint of (5.3 1)- (5.33) and (5. 34)-(5.36). Before combining these, we introduce the following scaled speed and time constant Il

w, = T,(w - w,) T,

Il

(5.37)

J2R

(5 .38)

w,

and t he parameter Il

E =

1 w,

(5. 39)

-

where we assume that H is large enough so t hat

E«

T,

(5.40)

We also combine the machine and line flux linkages and parameters as follows: Il Il Il ,pde = ,pd + ,ped, ,pq, = ,pq + ,peq , ,po, = ,po + ,p,o,

Xde Il Xd

+ X,p,

=

+ X ep )

XqI e Il XqI

+ X,p, X"de = Il X" d + X epl X q,

Il

Xq

X~e ~ X~ + X ep , X"qe

Il X" = q + X ep l

Substituting (5 .31)- (5.33) into (3.148)- (3. 159) and adding t he other dynamic models give the following multi-time-s cale model:

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

94

(5 .43)

e d~; = R,eIa

(5.44)

, dE~ = - Eq'-( Xd - Xd' ) Tdadt

[I

- X3 )2(.}. d- (XX~-Xl. o/1d+ (X'd - X)1 l. d

d

- E~)l + E/d il d..p] d Td 0dt =

, dE~ Tqadt

.}. - 'n d

+ E'q -

= -Ed'+( Xq -

(X"d

(5.45) -

X l. )I d

X~ _- Xl,)2 X~' (.}.0/2. X.') [1q - (X~

(5.46 )

+ (X'q -

+Ed)]

X)1 l. q (5.47) ( 5.48 ) (5.49)

(5.50) (5.51) (5.52) (5.53)

..poe = - Xoe 1o dEJd TETt = - (KE dRf TPTt = - R/

+ SE(Efd))E/d + VR

+

Kp Tp Efd

dVR KAKp TATt = - VR + KAR/ Tp E/d+KA(Vref-V,)

(5.5 4)

(5 .55 ) (5. 56 )

< VR < _ wax R

(5.57)

JVl + V;

(5.58)

vmin R _

V, =

5.2. T HE MULTI-TIME-SCALE MODEL

R,ld

Vq

ReI, -

-

d,p,d Edt + V. sin( 6 - Ov.)

(1 + T,f) d,p" + V, cos( o - B w, ,pd - Edt

v ,)

(5.59) (5.60)

,p,d

-Xepl d

(5.61 )

,peg

- X,pI.

(5 .62)

- TM + Psv

(5.63)

dTM .'I'CHdt T

E) ,p" + ( 1 + 7~w,

Va

95

dPsv

sv ~

-

-Psv

0 < Psv < _

+ Pc pmax

sv

-

E

w,

R DTs

(5. 64 ) (5.65 )

This model could be put in closed form without algebraic equations by solving (5.51)- (5.53) and (5.6 1)- (5 .62 ) for fd , lq, 10 , ,ped, and ,p" and substituting into the rem aining differential equations, and by replacing the derivative terms in (5.59 ) and (5. 60) with their respective fun ctions, and substitution into (5.58) and (5.56 ). This single-machine/ infinite bus m odel has sever al classifications of dynam ic time scales. The stator t r ansients of ,pde and ,pqe are very fast relative to other dynamics . This shows up in the model as a small "time constant" ( multiplying their derivatives. The damper flnx linkages ,p'd and ,pz. are also quite fast, sin ce Td~ and T~~ typically are quite small. The volt age regulator states E fd and V/l te nd to be fast because TE and T A are typically small . T he field winding and rate feed hack states E~ , R f tend t o be slow because Tdo and TF can be relatively la rge. The t urbine /governor states TM , Psv tend to be slow becanse TCH and Tsv can be r elatively large . This can be cOllll tercd, however, by a small speed regulation· RD . The dampe r-winding flux linkage Ed can be either fas t o r slow, depending on T~o . These classifi cat ions should be considered as general rule-of-thumb gu idelines , and not absolute characteristics. The actual time-scale classification for a specific set of data can be quite different than that stated above, and can change between load levels. The t ime- scale characteristics of a m odel are importan t since they determine the step size r equired in a time simul ation. If a model contains fas t dynamics, a small step size is needed. If the phenomenon of interest in a t ime simulation is known to be a predonli nan tly average or slow response l it would be very helpful if the fast dynamics could· be eliminated. They !lIust

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

96

be eliminated properly, so that their effect on the slower phenomena of interest is still preserved. Only in rare cases can fast dynamics be "eliminated" exactly. One such case is given in the next section.

5.3

Elimination of Stator/Network Transients

We begin this section by considering the special case of zero stator and network resistance. For this special (although common) assumption, (5.42) - (5.44) and (5 .49) are E

E

(1+ ;, w,) ..pq. + V. sin( 6 - Bv,} d::- __ (1+ ;, W}Pd. + V, cos(o - Ov,)

d~:. _

• d..poe

= 0

(5.67) (5.68)

dt

do

T, dt =

(5.66)

( 5.69)

Wt

The first three differential equations have an explicit solution in terms of 6:

..pd, ..pq, ..poe

= =

V. cos( 0 - Ov.} -V. sin(o - Ov,}

= 0

(5 .70) (5.71) (5.72 )

This can be verified by substituting (5.70}-(5.72) into (5 .66}- (5.69) and observing an exact identity. If the initial conditions on ..pd" ..pq" ..po" and 6 satisfy (5.70}- (5 .72) , then ..pd., ..pq" and ..poe are related to 0 through (5.70)(5 .72) for all time. This makes (5.70}- (5.72) an integral manifold for ..pd" ..pq" and ..poe. If the initial conditions do not satisfy (5.70}-(5.72) (the system starts off the manifold), there is still an exact explicit solution for ..pd" ..pq" and ..poe as a function of 0 and time t . This interesting fact about synchronous machine stator transients has been explained in considerable detail in [63J through [65J. The result is as follows. For any initial conditions ..pd., ..p~e> ..p~, 0°, to and for any <, the exact solutions of the differential equations (5 .66 }- (5.68) are

(5.73)

5.3. ELIMINA'l'lON OF STATOR/ NETWORK TRANSIENTS

..poe = -Cl sin(w,t + 0 - e2) - V, sin(o - 0.,) "poe

=

97 (5.74) (5 .75 )

C3

with

C1 = [( ..pde -

_

V, cost 0° °

w,to + o _ _,.0

C2 -

+ tan

-1

0.,))2 + (..p;e +

V, sin ( 0° -

( ..p~. + V, sin(o" - 0.,)) _1.0 _TT ('0_0) 'fide.

Ys

cos

U

C3 -!Poe

0., ))2Jt

(5.76) (5 .77)

V.f

(5 .78)

Clearly, if the initial conditions are on the manifold (satisfy (5.70)-(5.72)), this solution simplifies to the manifold itself «5.70)- (5 .72)). We emphasize that this interesting solution is valid for any f, and reqnires only R .. = O. This solution does, however, show the importance of R,. . If the initial condition is off the manifold , the exact solution of (5.73)-(5 .75) reveals a sustained oscillation in ..pde and ..po., which means that the machine will never reach a constant speed equilibrium condition. Evidently, the resistance acts to damp out this oscillation in tbe actual model, keeping R ... When resistance R,. is not zero, an integral manifold for ..pd" ..po., and ..poe bas been sbown to exist only for the case in which f is sufficiently small [63]-[65J. A first approximation of this integral manifold (keeping R,e) can be found by setting f equal to zero on the left-hand side of (5.42)-(5 .65) and solving the algebraic equations on the right-hand side for ..pde, ..pq. and ..po. as functions of all remaining dynamic states. As a matter of consistency, f can also be set to zero on the right-hand side of (5.42)-(5.65). In either case, the resulting reduced-order model should approximate the exact model up to an "order f" error [66 , 67J . If tbe fin (5.64) is set to zero , the turbine / governor dynamics would no longer depend on shaft speed. It could be argued that since RD is usually also quite small, tbe ratio of f/ RD should be kept and , thus, keep governor dynamics. For f sufficiently small, this reduced-order mode! can be improved to virtually any order of accuracy (see Appendix). It is customary to set f equal to zero in (5.42)-(5 .44) and (5.59)-(5.60) to give the following approximation of the exact stator transients integral manifold written together with the original algebraic equations for currents:

o= o= o=

R"Id + ..pq, + V, sine 0 - 9., )

(5.79)

R"Iq - ..pde + V, cost 0 - 0.,)

(5.80)

R"Io

(5.81)

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

98

(5.82) (5.83) (5.84) These six algebraic equations clearly can be solved for the six variables ..pde, ..p.e, ..poe, I d, I. , and 10 as functions of 0, E~, Ed' ..p,d, and ..p2 •. If R .. is equal to zero, this approximation of the integral manifold becomes exact (compare (5 .70)-(5.72) and (5.79)-(5.84)). With the above results, the dynamic model has been reduced from a 14th-order model to an 11th-order model. The solution of (5 .79)-(5.84) for ..po, and 10 is trivial, and ..pd" ..p.e can be eliminated, leaving only two equa.tions to be solved for Id and I. as fun.ctions of 0, E~ , Ed' ..p,d, and ..p2.:

o = R.eId -

II (X~' - Xl.) I (X~ - X~') X.,I. - (X' _ X ) Ed + (X' _ X )..p2. q

l~

is

q

+V. sin(o - eva)

II

(5.85)

(X:J - Xl,)

(Xd - X:J)

I

o = R"I. + Xde 1d - (Xd _ Xl,) E. - (Xd _ Xl,) ..p1d (5.86)

+V, cos(o - Ov,)

These two real equations can be used to solve for fd and I.. They can also be used to make one complex equation and an interesting "dynamic" circuit. Adding (5.85) plus j times (5.86) a.nd mUltiplying by ,j(6-"./2) give the following "circuit" equation:

(X~I - Xt.) E' _ (X~ - X~') . 1. [ ( X'q _ X l!J ) d (X'q _ X t tl ) '1'2.

[

+ (X" _ •

XII)1] d.

+J. [(X:J - Xl,) E' + (Xd - X:J) .1. ]] ,j(6- "./2) (Xd - Xt.) • (Xd - Xl,) 'I'1d

= (R,

+ jX:J)(Id + j 1. ),j(0- "./2)

+(R, + jX,p)(Id + j1.)e j (0- "./2)

+ v"jO"

(5.87)

From (5.58)- (5.60) with < = 0,

V, =

JVl + V.2

(5.88)

5.3. ELIMINATION OF STATOR/ NETWORK TRANSIENTS

+ V, sin{ 6 R,Iq + X,p1d + V, cos( 6 -

99

Vd = R eId - X,pI.

Ov,)

(5.89)

V. =

Ov,)

(5.90)

These last two equations can be written as one complex equation.

(5.91) Equations (5.87) and (5.91) can be expressed as a "dynamic equivalent circuit" with a controlled source voltage behind X~ , as shown in Figure 5.1.

-

+]

(Xd - X b')

l] eX&-<12)

(Xd- X ;;);

. [ (Xi - X ,,) If. q+

(X~-Xb)

ld

Figure 5.1: Synchronous machine sub-transient dynamic circuit Thiscircuit does not imply anything about steady state, but merely reflects the consequence of setting < = 0 in the original full dynamic modeL Such a circuit is clearly not a unique representation of these algebraic equations. Many other similar circuits can be created by simply adding or subtracting reactances in the line and modifying the internal "source" accordingly [68]. An interesting and useful observation about this circuit is that the "real power from the internal source" is exactly equal to the electrical torque in the air gap (TELEC).

P from source

[

(X~I - Xt.) I (X~ - X~) (" II)] (X~ _ Xt.) Ed - (X~ _ Xl.) .,p2q + Xq - Xd Iq Id

+

(X~ - Xt.)

[ ( Xd -

(X~ - X~)

]

Xl,) Eq + (Xd _ Xt.) .,pld Iq

From (5.44) and (5.82) to (5.83):

TELEC

I

100

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

These are clearly equal, since the reactances X:J. and X;. each contain X. p • The final form of this model that has eliminated the stator Jnetwork transients by setting. = 0 on the left and right sides of (5.42)-(5.65) (except for (5. 64), where we have chosen to keep the governor by arguing that RD can be small) can be written in terms of the original variable. as

Td" d!~= - E~ -

(Xd - X d)

Xd - X~ , [Id - (Xd _ Xu )2 (,p'd + (Xd - XI.)Id

-E~)l + Efd

(5.92)

T:;a d~:d = -1/;'d + E~ - (Xd - Xto)Id

T~a d!d

=

- Ed

+ (X. - X~)

(~~_ ::')2(1/;2, + (X~ -

[I, -

T~~ d~:q

=

(5.93)

XI.)I.

+ Ed)]

- 1/;2, - Ed - (X~ - XI.)I.

(5 .95)

do

(5.96)

- = w -w dt '

2H dw w,

dt

=

™-

(X~

- Xu), (Xd - X1) (Xd _ Xl.) E,!, - (Xd _ Xt.) 1/;'dI •

(X~'

- Xl.),

- (X q' - X t. ) EdId

(X~

-

dRj

TFdt

q

+ SE(Efd»Efd + VR

= -R f + -KF Ejd TF

X~')

+ (X' _ X t. )1/;2. I d

-(X~ - X:;)Id I, - TFW

dEfd TEaI"" = -( KE

(5.94)

( 5.97) (5.98) (5.99)

5.4. THE TWO-AXIS MODEL

dVR TATt

dTM

Tcw--d .t

101

= - VR+KARf -

KAKF TF Efd+KA(Vref-V,)

= - TM + Psv

(5.101)

dPsv = -Psv+Pc -- - 1 Tsv-dt

(5 .100)

RD

(w-

w,

-- 1

)

(5.102)

with the limit constraints

< - II:R -< vrnax R o :S Psv < P"Svx

vmin

R

(5.103) (5.104)

and the required algebraic equations , which come from the circuit of Figure 5.1, or the solution of the following for Id, I q :

o=

(R,

+ R.)ld -

(X~'

(X~/_X,,),

- ( X~ _ X,,) Ed

+ Xep)I. (X~_X~/)

.

+ (X~ __ Xl,) ..p2. + V, sm(o -

8.,)

(5.105)

V, cos(o - 8v,)

(5.106)

o = (R, + R,)l. + (XJ + X ,.)Id (X~ - Xl,)" (X~ - X3) - (Xd _ Xl,) E. - (Xd _ Xl,) ..p1d

and then substitution into the following equations for Vd , Vq :

+ V, sine 0 -- ev.) ReI. + X ,pld + v:. cost 0 - ev,)

Vd = ReId - X,plq

(5.107)

V. =

(5.108)

and finally

V, = JV1 + V:

(5.109)

The quantities Vref and Pc remain as inputs .

5.4

The Two-Axis Model

The reduced-order model obtained in the last section still contains the damperwinding dynamics ..pld and ..p2•. If T:Jo and T~~ are sufficiently small, there is

102

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

an integral manifold for these dynamic states. A first approximation of the fast damper-winding integral manifold is found by setting Td~ and T~~ equal to zero in (5.92)-(5.109) to obtain

o = -..p'd + E~ o = - ..p2q - E~ -

(X~ - Xl,)Id

(5.ll0)

(X~

(5.1ll)

- Xl,)I.

When used to eliminate ..pld and ..p2q in (5.107), the equations for hand Iq become

0= (R, +R.)Id -

(X~+Xep)Iq

-

o = (R, + R.)Iq + (X~ + X,p )Id -

E~+ V,sin(o -

ov,)

(5.112)

+ V, cos(o -

Ov,)

(5.113)

E~

As in the last section, these two real equations can be written as one complex equation:

[Ed

+ (X~ - Xd)Iq + jE~Jej(6-~) = (R, + jX~)(Id + jlq)ej(C-'!j) +(R, + jXep)(Id + jlq)ej(5- ~) (5. 114)

The equations for Vd and Vq remain the same as in the last section so that the circuit of Figure 5.2 can be constructed to reflect the algebraic constraints for this two-axis model. It is easy to verify that, as in the last section, the "real power from" the internal source is exactly equal to the electrical torque across the air gap (TELEc) for this model.

= Figure 5.2 : Synchronous machine two-axis dynamic circuit The final form of this two-axis model, which has eliminated the stator/network and fast damper winding dynamics , is obtained by substituting

5.4. THE TWO-AXIS MODEL

103

(5.l10) and (5.111 ) into (5.92)- (5.109) to eliminate 7fld and 7f2q: (5.115) (5.116)

(5.117) (5.118) dEfd

TE~ = -( K E

+ SE(Efd))Efd + VR

(5.119)

dRf KF TF = -Rf +-Efd dt TF dVR

Trdi = -VR+KARFdTM TCH'Tt

(5.120) KAKF TF Efd + KA(v;.,f - V,) (5 .121)

= -TM + Psv

Tsv dPsv Psv dt with the limit constraints

=-

+ Pc -

Vrnin < R _

o :::;

(5.122) -1 RD

(W-

- 1

)

(5.123)

W,

v:R <_ vrnax R

(5.124)

Psv :::; Pfv x

(5 .125)

and the required algebraic equations , which come from the circuit of Figure 5.2, or the solution of the following for Id, I q:

o= 0=

+ R,)Id - (X~ + X .p)Iq (R. + R,) I q + (Xd + X,p)Id -

(R,

+ V. sin(o - Ov.) E~ + V. cos(o - Ov.)

Ed

and then substitution into the following equations for Vd and

+ V. sin(" Relq + X .pId + V. cos(o -

(5.126) (5.127)

-v.:

Vd = R.I d - X.plq

Ov.)

(5.128)

Vq =

9v,)

(5. 129)

and finally

v, = JV} + Vq2 The quantities Vref and ·pc remain as inputs.

(5.130)

104

5.5

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

The One-Axis (Flux-Decay) Model

The reduced-order model obtained in the las t section still contains the damperwinding dynamics E~ . If T~o is sufficiently small, there is an integral manifold for this dynamic state also. A first approximation of this fast damper winding integr al manifold is found by setting T~o equal to zero in (5.115)- (5.130) to obtain (5.131) o = -E~ + (Xq - X~)Iq When used to eliminate Ed in (5.115)- (5. 130), the equations for Id and Iq then b ecome

+ R.)Id - (Xq + X.p)Iq + V.sin(5 - ev,) o = (R, + Re)Iq + (Xd + X,p)Id - E~ + V, cos(5 -

0= (R.

(5.132)

ev,)

(5 .133 )

As in the last sections, these two real equation s can be written as one complex equation:

+ jE~lej(6 -"/2 ) = (R, + jXd )(Id + jlq),(5-,,/2) +(R e + jX,p)(Id + jlq)e(6 -./2) + V,ejev>

[(Xq - Xd)Iq

(5. 134)

The equations for Vd and Vq r emain the same as in the last sections, so that the circuit of Figure 5.3 can be constructed to reflect the algebraic con straints for this one-axis model.

+

+

V efov,

s

Figure 5.3 : Synchronous machine one-axis dynamic circuit

It is easy to verify that, as in the last two sections, the "real power from" the internal source is exactly equal to the electrical torque across the air gap (TELEC) for this model.

5.5. THE ONE-AXIS (FLUX-DECAY) MODEL

105

The final form of this one-axis model, which has eliminated the stator/ network and all three fast damper-winding dynamics, is obtained by substituting (5.131) into (5 .115)- (5.130) to eliminate Ed: I

dE~

I

I

T dodt = -Eq - (Xd - Xd)ld

+ Efd

(5.135)

do

- = w-w dt '

(5 .136)

dw I -2H w, -dt = TM - E q 1g -

(

X q - Xd' ) la1q - TFW

(5.137)

TE d~t = -(KE + SE(Efd))Efd + VR dRf

(5.138)

KF

(5 .139) TF- = -Rf+ - Efd dt TF dVR KAKF TA""a;: = -VR + KARt TF Efd + KA(Vref - Vi) (5 .140) dTM TCW--it-- = --TM

+ Psv

Tsv dPsv = --Psv

dt with the limit constraints

(5.141)

+ Pc _ ..2.... (.':".. RD

vRrnin < _

o ::;

w,

1)

(5 .142)

VR < _ VRIDax

(5.143)

Psv ::; P!f'v x

(5.144)

and the required algebraic equations, which come from the circuit of Figure 5.3 , or the solution of the following equations for la, I q:

o = (R. + Re)la - (Xq + Xep)l. + V, sin(o -- ilv ,) o = (R, + Re)l. + (Xd + X,p)la -- E~ + V. cos(o -

(5.145) 8v ,)

(5.146)

and then substitution into the following equations for Vd, V.:

Vd = R.Id -- Xepl.

+ V. sin(o -- 8.,)

(5.147)

Vq = R.I. + Xepld

+ V, cos(o -- Ov,)

(5148)

and finally

Vi =

JVl + Vq2

The quantities Vref and Pc remain as inputs.

(5.149)

106

5 .6

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

The Classical Model

The classical model is the simplest of all the synchronous machine models, but it is the hardest to justify. In an effort to derive its basis, we first state what it is. In reference to aU of the dynamic circuits of this chapter , the classical model is also called the constant voltage behind the transient reactance (XdJ model. Return to the two-axis dynamic circuit (Figure 5.2) and the dynamic model of (5.115)-( 5. 130). Rather than assuming T~o = 0, as in the last section, assume that an integral manifold exists for Ed, E~, Eid, Rf, and VR, which, as a first approximation, gives E~ equal to a constant and (Ed (X~ - Xd)Iq) equal to a constant. For this constant based on the initial values Edo, I; , and E~o, we define the constant voltage

+

(5.150) and the constant angle

5'0

to ta

-

-1 (

n

E'o

) _ 1f

2

(5 .151)

R,

R,

jXd

go ef(o+l)'O)

q

E'o + (X'q _ X')Io d d q

+

Figure 5.4: Synchronous machine classical model dynamic circuit The classical model dynamic circuit is shown in Figure 5.4. Because the classical model is usually used with the assumption of constant shaft torque (initial TXt) and zero resistance, we assume TCH

=

00,

R.

+ R. = 0

(5 .152)

to

5 + 5'0

(5. 153)

and define

5classical

5.7. DAMPING TORQUES

107

The classical model is then a second-order system: d°classical dt

(5.154)

2 H rU,., E'°V: -d = TXt - (X' ; ) sin(odassical - Bu ,) - TFw (5 .155) w"

d+

t

e.p

This classical model can also be obtained formally from the two-axis model by setting X; = X~ and TCB = T~o = T~o = 00, or from the one-axis model by setting Xq = X~, T~o = 0, and TCB = T~o = 00. In the latter case, 0'° is equal to zero, so that 0classical is equal to 0 and E'o is equal to E;o.

5.7

Damping Torques

The dynamic models proposed so far have all included a friction windage torque term TFw. For opposition to rotation, such torque terms should have the form

(5.156) The literature often includes damping torque terms of the form To

= D(w -

w,)

(5 .157)

(w - w,)

(5.158)

or

D'

To = -

w.

These damping torque terms can be positive or negative, depending on the machine speed. Values of D' = 1 or 2 pu have been cited as a reasonable method to account for the turbine windage damping [62] . Such terms look like indu ction motor slip torque terms, and have often been included to model the short circuited damper windings. Thus, if damper windings are modeled through their differential equations, then their effects need not be added in TD . In this case, friction can be modeled through TFW, if desired. If damper windings have b een eliminated, as in the one-axis or classical models, all of the third damping effects have been lost. To account for their damping without including their differential equations, To can be added to the torque equation (added to TFW), with D appropriately specified to approximate the damper-winding action. We now justify this damping torque

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

108

term by returning to the two-axis model of (5.115)-(5 .130), which included one damper-winding differential equation. We would like to show that if Ed is more accurately eliminated, a damping torque term proportional to slip speed should automatically appear in the swing equation. To show this , we propose that the integral manifold for Ed should be found more accurately. To begin, we define the small parameter I" as r:, T'qo (5.159) p. = and propose an integral manifold for

Ed of the form

Ed =
+ 1"2
(5 .160)

where each
8<po

8q"

2

8
8q" 86

+ I"

) ( -Eq1 - ( Xd-Xdld+Ejd ') )/ 1 Tdo

I" ( 8EI+1"8EI+1"8EI+·· · q q q

+1" (

8q,0 86

8q,0 +1" ( OW

+ I"

2

8q,2 86

)

+ .. . (w - w.)

20
-E~Iq - (X~ - Xd)M. -

TFW) /

8q,0 +1' ( 8Ejd

8
+1'

o
2

(:~) )

)

+ I" 8Efd + I' 8Efd + ... (-(KE + SE)Ejd + VR / TE

(:~~ + I" :~~ + 1"2 :~~ + .. .) (-Rf + ~; Ejd) / TF

28q,2

8) (-VR+KAR f -

+ 1" 8VR + . ..

KAKF TF

Bfd

+KA(Vref - V,)) / TA 8¢o +1' ( 8TM

8"" + I"2 8TM 0¢2 + I" 8TM + . ..) (- TM + PSV) / TCH

+1" (o
8Ps v

8Psv

RD

(:. -1)) /Tsv = - (q,o + I'
(5.161)

5.7. DAMPING TORQUES

109

where hand Iq are

(R.

+ Re)(o + P.1 + p.22 + ... -

V.sin(.5 - 0•• )

61

+

(X~

+ Xcp)(E~ 6

V, cos(.5 - 0•• )) 1

-(Xd + Xep)(o + P.1

+ p.22 + ... -

V. sin(6 - 0•• ))

61

+

(R,

+ Re) (E~ 6

V. cos(.5 - 0•• )) 1

where (5.162) and Vd, Vq , and Vi are as before. This partial differential equation may seem difficult to solve; on the contrary, it is actually very easy to solve for 0' 1, 2 in sequence. To see how this is done systematically, suppose that we want the simplest possible approximation of the integral manifold . This is 0' and is found by neglecting all p. terms, giving (5.163) To find 0 as a function of the states, we must solve (5.162) and (5.163) neglecting all p. terms, giving

(Xq -

X~)(Xd

+ Xep)V. sin(6 -

0•• )

62

(Xq -

+

X~)(R,

+ Re)(E~ 62

V, cos(6 - 0•• ))

(5.164)

where

or (5.165) This is the first term of the series for the integral manifold for Ed' and is what would be obtained if T~o were simply set to zero . When substituted

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

110

in the remaining equations, it would not reflect any of the damping due to this da.mper-winding (this is what was done in Sections 5.5 and 5.6). If we want to recover some of the damping due to this damper-winding without resorting to the two-axis model, we can compute ¢ , by returning to (5.161) and keeping fl. te rms , but neglecting f1-' and higher powers of fl. . Note that ¢o is o nly a function of E~ and 6; all the partials of ¢o are zero except for two) giving

;~~ ( :- E~ -

(Xd -

X~)Id I,,~o +Efd)/T~o + a:ao(w -

w,)

q

_ _ '" _ (X _ X') ( (Xd -I- X.p)¢J 'f'I q q (R, -I- R.)2 + (Xd + X.p)(X~

-

+ X.p)

) (5.166)

where Id is evaluated at fl. = and W, written here as

o. This makes ¢, some fun ction of E~, Efd, 6, (5 167)

Stopping with this level of accuracy, the integral manifold for Ed is (5168) where

O(,u2) is

SOIne

error of ('order"

approximate expression for the damper-winding, is

Ed = fox(6)

E~,

fl.2.

In terms of th e above functions, an

which will capture some of the damping of

+ foR(E~, 6) -

-I- T~ohR(E~,Efd,6)

T~ohx(6)(w

- w,) (5.169)

Note that foR and hR are zero if R, -I- R. = o. Substitution of this approximation of the exact integral manifold for Ed into the two-axis model gives the circuit of Figure 5.5 and the following differential equations.

(5.170)

(5.171)

5.7. DAMPING TORQUES

111

+

-+ V, efl¥s

~

fox (0) + foR (Eq: 0) - Tqoftx (d) (00 - (Os) + T~ofiR (E~, Eld , 0)

+ (X; -

xJ) 1q +jEq1 ei(O-v2)

Figure 5.5: Synchronous machine one-ax .. dynamic circuit with damping term

+T~ohR(E~, EN, 0)) - E~Iq - (X~ - XWdIq

(5.172)

-TFW

dEjd TE--;{t

-(KE

dRj Tr-dF

-Rj

+ SE)Ejd + VR

(5.173)

KF

+ TF Efd

(5.174)

dVR KAKF TraF = -VR+KARj- TF Ejd+KA(Vref- V,) -TM

+ Psv

- Psv

+ Pc

(5.175)

(5.176) -

_1_ (-"'- - 1) RD

(5.177)

W6

with the limit constraints vmin R

< _ v:R < _ vRmax

o :s

Psv

(5.178)

:s PSyX

(5.179)

and the required algebraic equations for Id, I q , Vi, which come from the circuit of Figure 5.5 or the solution of the following equations for Id, I q:

0= (R,

+ Re)Id -

(X~

+ Xep)Iq -

fox(o) - foR(E~, 0)

+T~ohx(o)(w - w,) - T~ohR(E~, Efd, 0)

+V,sin(o-Bu,)

o=

(R,

+ Re)Iq + (Xd + Xep)Id -

(5.180) E~

+ V, cos(O -

Bu , )

(5 .181)

CHAPTER 5. SINGLE-MACHINE DYN AMIC MODELS

11 2

and then substitution into the following for Vd, Vq :

Vd = R,Id - X,plq V. = R, I.

+ V. sin( §

-

+ X,pld + V. oos( Ii -

Bu.)

(5.182)

0.. )

(5.183)

an d fmall y

V,

=

JV}

+ R, = 0 x - X' . (Ii Ed = X q +X • V.sm

+ y"2

(5. 184)

When R.

q

~

.0

Bu. ) - T'

( X' + X )(X - X') • (X 'P + X • )2 • q ~p

V. cos(b - Bu. )(w - w.)

(5. 185)

which gives an accelerating torque of

-

' ( (Xq - X~))') V2! cos 2(.U Tqo , X. + X'P -

-

B)( ,,-' w -

T~n

) (5.186)

- '1'Fw

T he

W,J

ter m reflects the da mping due to the damper-winding as a slip

torque term

- X~) V 2 2(1i B)( ) T D = 7 "qO( (Xq X q + Xe p)2 COS W - w.. 8

LI ."J

(5. 187)

This is often app roximated as TD "" D(w - w. )

(5 .188)

where D i, a d"mping const ant. The approximation of Equation (5.188 ) is normally cO!lsid ered valid only for small deviat ions about an operating point, but it has been used for large changes as well.

5.S. SYNCHRONOUS MACHINE SATURATION

5.8

113

Synchronous Machine Saturation

The dynamic models of (5.42)-( 5.65), (5.92)-(5.109), (5.115 )-(5.130), (5.135) -(5 .149), and (5.154)-(5.155) began with the assumption of a linear magnetic circuit ((3.148)-(3.159)). To account for saturation, it is necessary to repeat this analysis starting with (3.191)-(3 .203), rather than (3.148)(3.159). Without explicit information about the saturation functions S~Z), (Z) (z) (z) (z) (Z) d S(Z) . . . . Sq , So , Sid' S'd' S'q , an Zq' It IS not pOSSIble to ngorously show that these functions do not affect the fundamental assumptions about time-scale properties. We assume that the functions do not affect the time-scale property so that the stator plus line algebraic equations for this single-machine model become

o= o=

R«Id + ..pqe

+ V, sine 15- 0",) ..pde + V. cos( a- 0.,)

(5.189)

R,eIq -

(5.190)

o=

R,eIo

(5.191)

- Xl.) E' + (X~ - X%) "J • ..pde = -X"de I d + (X% (X~ _ Xl,) q (X~ _ Xl.) 'l-'Id - S~Z)(YZ)

(5.192)

11 (X~' - Xl.), (X~ - X~') ..pqe = -XqeJq - (X~ _ Xl.) Ed + (X~ _ Xt.) .,pZq

-S~Z)(YZ)

(5.193)

Vd = ReId - XepJq + V. sin(a - 0•• )

(5.194) (5 .195)

These equations can be written in circuit form as shown in Figure 5.6. With this assumption, the dynamic model with saturation included but stator/network transients eliminated is

X' X" d - Xt.)2 d ("'. - E q' - (X d- X')[I d d- (Xd'f'Id+ (X'd

-E~ + Sl~)(Yz))l- S)~(Yz) + Eid

-

X)I t. d (5.196)

(5.197)

CHAPTER. 5. SINGLE-MACHINE DYNAMIC MODELS

114

[[ (Xi-X,,) Ed-

(X , -X~)

+ (Xq -

+

(Xi -Xq )'!' (X,-X,, ) '"

x:;) I, .s~2) (Yzl]

+' [(X;;-X") E' + (X, - x:;) 'l' :::P)(Yzl J (Xd-X ls ) q (Xd-Xlr) 1d d

1] ",f>-"')

Figure 5.6: Synchronous machine subtransient dynamic circuit including saturation

I dEd I Tq0dt = -Ed

+

(

')[ X~ - X~' (0'. Xq - Xq Iq - (X~ _ X ,)2 'l'2q

I

+ (Xq' -

+Ed + S~!)(Y2))l + Sl!)(Y2)

)

Xl, Iq

(5.198)

Til d.,p2q qo dt

(5.199)

do

(5.200)

dt 2Hdw w, dt

_(X~' - X~)Idlq

+ S~2>tY2)Iq

- S~2)(Y2)Id - TFW dEfd

TS----;jt = - (Ks dRJ dt

(5.201)

+ SS(EJd))EJd + VR

TF- = -RJ

+ -KF EJd TF

dVR TA d t = - VR

+ KAR j

dTM T P l ' eHT! = - M+ sv

-

(5.202) (5.203)

KAKF TF EJd

+ KA(Vref -

V,)

(5.204) (5.205)

5.B. SYNCHRONOUS MACHINE SATURATION

115

dPsv = -Psv+Pc-1 (W Tsv---1) RD

dt

(5.206)

w,

with the limit constraints

vRrnin <_ V;R <_ vrnax R o :0; Psv :0; P?V'x

(5.207) (5.208)

and the required algebraic equations for Vi, Id, and I q , which come from the circuit of Figure 5.6:

o=

(

H,

+ He

/I )

Id -

(

Xq

+ Xep

(X~f_Xt.) f q - (X' _ X ) Ed

)1

q

I,

(X~ - X~) (2) . X ) ..p2q - Sq (Y2) + V.Slll(O - 0",) I,

+(X'q _

(5.209)

(5 .210) the following equations for Vd, Vq, Vi: Vd Vq

= RJd - X,pIq + V, sin(a - 0",) = R,Iq + X,pId + V. cos( a- 0.,)

V, = .JVJ

+ Vq'

(5.211) (5 .212) (5.213)

and finally the saturation function relations, which must be specified. The saturation functions, in general, may be functions of IdJ I q , and 10 , From (5.191), these functions would be evaluated at Io = O. Tfthe saturation functions include Id and I q , the nonlinear algebraic equations (5.209) and (5.210) must be solved for Id and Iq as functions of E~, Ed' 'I/J,d, .,p2q, and 6, and then substituted into (5.211)-5 .213) to obtain Vd, Vq, and Vi. If the saturation functions do not include I d , Iq) then the currents IdJ Iq can easily be solved from (5.209) and (5.210). To obtain a two-ms model with saturation included as above, we again assume that Td~ and T~~ are sufficiently small so that an integral manifold exists for .,p'd and .,p'q' A first approximation of this integral manifold is

116

CHAPTER 5. SINGLE·MACHINE DYNAMIC MODELS

found by setting T:Jo and T~~ equal to zero in (5.197) a nd (5.199) to obtain

( X~ - XI,)Id - si~)(Yz)

(5 .214)

o = - "'z. - E~ - (X~ - XI,)I. - S~~)(Yz)

(5.215)

o=

- ""d

+ E~ -

We now assume that sl~)( Yz ) and S~~\Yz) are such that these two equations can be solved for "'1d and "'z. to obtain "'1<1

= E~ - (X~ - XI,)I d

-

S;~)(Y3)

"'z. = - E~ - (X~ - XI. )I. - S~!) (Y3)

(5.216) (5.217)

where

(5.218) When used to eliminate and Iq become

o=

(R ,

"'1d

and "'z. in (5.196)-(5.213), the equations for Id

+ R,)Id -

(X~

+ X ,p) I. -

E~ - S~3)(Y3)

+V. sin(o -lJu ,)

o=

(R,

+ R,)I. + (X~ + X,p)Id -

V, cos(.5 - lJ u ,)

(5.219)

E;

+ S~3)(Y3) (5.220)

The saturation functions S~3)(Y3) and S~3)(Y3) arc a combina tion of S~2),

S~z) and s;~), sl~) as fou nd by substituting (5.216) and (5.2 17) into (5. 209) and (5.210) . These two equations can be represented by the circuit shown in Figure 5.7. The two· axis dynamic model with synchronous machine saturation is

(5.221 ) (5.222) (5.223 )

5.B. SYNCHRONOUS MACHINE SATURATION

[Ed+(Xq -Xd)lq+S~)(Y3)+ j(E; -S~

117

+

(y,»)].J(O-.n)

Figure 5.7: Synchronous machine two-axis dynamic circuit with saturation

(5.224) (5 .225) (5.226)

T

CH

dTM - -TM dt -

+ Psv

dPsv = -Psv Tsv-dt

(5 .228)

-1

+ Pc -

RD

(W-

w,

- 1)

(5.229)

with the limit constraints vg'in < R _

o ::;

v:R <_ vg'AX R

(5.230)

Psv ::; PF';x

(5.231)

and the required algebraic equations for Id, I q , which come from the circuit of Figure 5.7, or the solution of the following for Id , Iq:

o = (R, + R,)Id -

(x~

+ X,p)Iq -

E~ - S~3)(Y3)

+V, sin(o - iI•• )

o = (R. + R,)Iq + (X~ + X,p)Id - E~ + S~3\Y3) +V. cos(o - ou.)

(5. 232 )

(5.233)

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

118

then substitution into the following for Vd, Vq ,

Vi:

+ V, sin(o R,Iq + X,pld + V, cos(5 -

Vd = R,Id - X,plq

Ov,)

(5.234)

Vq =

Bv ,)

(5.235)

Vi

= VVd2

+ Vq2

(5.236)

If the saturation functions include a dependence on Id and I q , the nonlinear algebraic equations (5.232) and (5.233) must be solved for Id and Iq as functions of E~, E~, and 5 and then substituted into (5.234)- (5.236) to obtain Vd, Vq, and Vi. If the saturation functions do not include a dependence on Id and I q, then Id and Iq can easily be found from (5.232) and (5.233) . To obt ain a flux-decay model with saturation included, we again assume T~o is sufficiently small so that an integral manifold exists for E~. A first approximation of this integral manifold is found by setting T~o equal to zero in (5.222) to obtain

o = -E~ + (Xq - X~)Iq + S~!)(Y3) We now assume that to obtain

(5.237)

sl!)(y3 ) is such that trus equation can be solved for E~ (5 .238)

where (5.239) when used to eliminate Iq become

o = (R,

E~

in (5.221)-(5.236), the equations for Id and

+ R,)Id -

(Xq

+ X,p)Iq -

S~4)(Y4)

V. sin(o - Ov.)

o-

(R.

+ R,)Iq + (Xd + X,p)ld -

V. cost 0 - Ov. )

(5.240)

E~ + S~4)(y.) (5.241)

The saturation functions S~4)(Y4) and S~4)(Y4) are a combination of S~3),

S~3), and S~!) as found by substituting (5.238) into (5.2 19) and (5.220). These two equations can be represented by the circuit shown in Figure 5.8 .

5.8. SYNCHRONOUS MACIDNE SATURATION

[<Xq-X ti) l q+ S~4) { y 4)

119

+

li-v2) +j (E; -S~) (Y4»]eiC

Figu re 5.8: Synchronous machine one-axis (flux-decay) dynamic circuit with saturation

The one-axis (flux-decay) dynamic model with saturation is

(5.242) (5.243)

- S~4) ( Y4 ) Id - TFW - TD

dEjd TE----;[t dRf

= - (KE + SE ( Efd »Efd + VR

Tr-F =

- Rj

dTM Tcw -d(" = -TM

KF

(5.244) (5.245)

+ TF Ejd

(5 .246)

+ Psv

(5 .248)

1 Tsv dPsv = -Psv+Pc-dt

RD

(W- - 1 ) W,

(5.249)

with the limit constraints

vRrrun <_ o~

VR _<

v;max R

Psv ~

prJ x

(5.250) (5.251)

and the r equired algebraic equations , which corne from the circuit of Figure

CHAPTER 5. SINGLE-MACHINE DYNAMIC MODELS

120 5.8:

o=

(R,

+ Re)Id -

(X.

+ Xep)Iq

- S~4)(Y4)

+V, sin(6 - 0.,)

o=

(R.

+ Re)Iq + (X,i + Xep)Id -

(5 .252)

E;

+ S54)(Y4)

+V, cos(6 - 0•• )

(5.253)

then substitution into the following for Vd , Vq, Vi:

Vd = ReId - Xeplq

0.,)

(5.254 )

Vq =

+ V. sin( 6 ReI. + Xepld + V, cos(6 -

0•• )

(5 .255 )

Vi =

JVJ + Vi

(5 .256)

U the saturation functions include a dependence on Id and I q , the nonlinear algebraic equations (5.252) and (5.253) must be solved for Id and I q as functions of E~, 6 and then substituted into (5.254)-(5.256) to obtain Vd, v;" and V, . U the saturation functions do not include a dependence on Id and I., then Id and Iq can easily be found from (5.252) and (5.253). There is really no point in trying to incorporate saturation into the clas-

sical model, since it essentially assumes constant flux linkage.

5.9

Problems

5.1 Given the two-axis dynamic circuit of FigUIe 5.2, solve for Id and Iq in terms of the circuit parameters plus Ii, E~, Ed and the source V" 0.,. 5 .2 Given the one-axis dynamic circuit of Figure 5.3, solve for Id and Iq in terms of the circuit parameters plus Ii, E~ and the source V" 0••. 5.3 Using the two-axis dynamic model of Section 5.4, derive an expression for the voltage behlnd Xq that gives a circuit that is equally as valid as that of Figure 5.2. 5.4 Using the one-axis dynamic model of Section 5.5, derive an expression for the voltage behind Xq that gives a circwt that is equally as valid as that of Figure 5.3.

5.9. PROBLEMS

121

5. 5 Given a synchronous generator with a two-axis model:

+ jV.)e i (6 - ,,/2) (ld + j1.)ei (6-1< / 2)

(Vd

0.5L30° pu

= 1.0 pu,

Xd

(a) Find the steady-state values of 0,

E~,

R. = 0 pa, Xd

= 1.2 pu,

1LO pu

X.

= 0.2 pu, Ed.

X~ = 0.2 pu

(b) Find the inputs Efd and Tm. (c) Find E'o and 8'0 for the classical model. 5.6 Given the following two-axis dynamic model of a synchronous machine connected to an infinite bus, with the parameters shown (and R. = 0) : (a) If the machine is in a stable steady-state condition supplying zero real power and some reactive power Q, what can you say about the values of Id and I.? (b) Find the values of TM and Efd in part (a) if Q = 1 pu. (c) Describe, in as much detail as possible, the system response if Efd is suddenly changed to be 2.0 pu.

TJo = 5.0 sec, Xd = 1.2 pu, Xd = 0.3 pu T~o = 0.6 sec, X. = 1.1 pu, X~ = 0.7 pu w, = 2:rr60 r ad/sec, H = 6.0 sec, V = 1.0 pu ,

e = 0 rad

Chapter 6

MULTIMACHINE DYNAMIC MODELS Tills chapter considers the dynamic model of many synchronous mach.ines interconnected by transformers and transmission lines. For the initial ana.lysis, loads will be considered balanced symmetrical R-L elements. For notation, we adopt the following "number" symbols: m = number of synchronous mach.ines (if there is an infinite bus, it is machine number 1) n = number of system three-phase buses (excluding the datum or reference bus) b total number of macillnes plus transformers plus lines plus loads (total branches).

=

6.1

The Sy nchronous ly Rotating R e ference Frame

It is convenient to transform all synchronous machine stator and network variables into a reference frame that converts balanced three-pha.se sinusoidal variations into constants . Such a transformation is COSW8

t

- sin w"t 1

"2

cost w,t _ 2;)

_ sin(w,t _ 2;) 1

"2

123

cos(w,t + 23~) _ sin (w,t + 2;) 1

"2

1 (6.1)

124

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

with - sinw.t

- sin(w"t -

(6 .2)

23"'"

-sin(W.t+ 2;) Recalling the transformation and scaling of Section 3.3, we use the subscript i for all variables and parameters to denote machine i. We now define the scaled machine stator voltages, currents, and flux linkages in the synchronously rotating reference frame as

[ VDO VQi ]

e,.

=

- 1

Tdqo.Tdqoi

[ V,;

~:

I

VB ABC;

[ V.; ] Vbi

= Tqdo. VBDQi

Vci

VO i

=

['m] lQi

e,.

~TdqO' [ V.; ~: 1

["I

Tdqo.Ti.~i ~:

10.

=

=

i

T.

[

~TdqO' '.; ~: ]

= 1,. "J m

IBABCi

dqo.t

I

[,.; ] Ibi

BDQi

lei

BDQi

[•. ]

V>oi

=

[ v.; ] ~Tdqo. ~:

(6 .4)

i= 1" . . , m

/l BABC' [ 'D; ] e,. Tdqo.Ti.'m. [ •• ] _ T. V>Qi V>o' dq08 1\ V>Oi

(6.3)

;

V>bi

V>ci

i = 1,. "lm

(6 .5)

where Tdqoi is the machine i transformation of Section 3.3, and all base scaling quantities are also as previously defined . For 5; = ~ 0shaft; - w.t, it

6.1. THE SYNCHRONOUSLY ROTATING REFERENCE FRAME

125

is easy to show that sin 0, 1

T.dqo.J Tdqoi -

=

and

TdqoiT;;'~, =

[

:]

cos 0;,

ot. sin Of. 0 0

- cos

["..

co~ 0;

::

- cos 0; sin 0;

0

i=l, ... ,m

(6 .6)

i=l, ... ,m

(6.7)

= l, ... , m

(6.9)

This transformation gives

and i

We now assume that all of the m machine data sets and variables have been scaled by selecting a common system-wide power base and voltage bases t hat are related in accordance with the interconnecting nominal transformer ratings. Applying this transformation to the general model of (3.148)-(3.159) with the same scaling of (5.37)-(5.40) with E = l/w" the multimachine model (without controls) in the synchronously rotating reference frame is

d..pDi

E~

d..pQi

E~

= R.iID; +..pQi+ Vn;

i=l, . . . ,m

(6.10)

= R,;lQi -

i

= 1, ... , m

(6.1l)

d..pOi

E-;U:- = I

Td

R.Jiloi

~E~i - . E' -

·-

'" dt

-

-

qi -

..pOi

+ VOl

+VQ i

(Xdl· - X') di

+ (Xdi -

(6.12)

i = 1, .. . m I

[I .

dl -

Xt,i)Idi - E~i)l

(X~i - XX3J.)2 (,I. . 't'ldt

(X'

di. -

l .u

+ Efdi

i = 1, ... , m (6.13)

i= 1, . . . ,m

(6.14)

CHAPTER 6. MULTIMACillNE D YNAMIC MOD ELS

126

T' . dE~i qo'

~

d-t

E'

di

+ (X qi -

(X~i (X ' . _- XX~;) _)2(.1, Y' 2qi

X ' ) [I

qi -

qi

qt

i s).

i = I,. "Jm

Til . d'ifi2qi q01 dt T .d6 i " dt

i = I, . .. , m i

= Wti

=

_ XUI . d, d.

(6.18)

+ (X:J. - Xl,.) E' . + ( Xdi - X:JJ .1, . (X ' _ X .) (X' _ X .) 'l'ld, di

l.n

q.

di

In

i=l,,,.,m II - X.J --

q' q'

(6. 16)

(6 17)

1, . .. , m i = 1, .. . ,m

-

(6.15)

(6.19 )

( Xq/~'

- Xl,i) I (Xq', - Xq/~) E .+· . nh (X'ql. - X i S l.) d, (X'qt _ X In.) 'l'2qi ~

i= 1, ... ,m

(6.20) (6.21)

i = 1, . .. ,m

where

(VD.

+ jVQi) = (Vdi + jVqi) ej(Oi - ¥)

(ID.

+ jIQi) = (Idi + j Iqi)ej(oi- i)

('ifiDi

+ j'ifiQi)

= (,pd.

+ Nqi)ej(Oi- i)

= 1, ... , m i = 1, ... , m

(623)

i = 1, .. . , m

(6.24)

i

(6.22)

The terminal constraints for each machine are still unspecified . The next section gives a muJtimacrune set of terminal constraints, which can be analyzed in a manner similar to the last chapter.

6.2

Network .and R-L Load Constraints

Rather than intr oduce an infinite bus as a terminal constraint , we propose that all m synchronous machine terminals be interconnected by balanced symmetrical three-phase R-L elements . These R-L elements are e.ither transmission lines wher e capacitive effects have been neglected , or transformers. For now, we assume that all loads are balanced symmetrical three-phase R-L

6.2. NETWORK AND R-L LOAD CONSTRAINTS

127

elements, so that the multimacrune dynamic m odel can be written in a multitime-scale form , as in Chapter 5. We assume that all line, transformer , and load variables have been scaled by selecting the same common power base as the machines , and voltage buses that are related in accordance with the interconnecting nominal transformer ratings. The scaled voltage across the line, transformers, and loads is assumed to be related to the scaled current through them by

Vai

= -RJai + -W,1 -d"
i=m+l, . .. ,b

(6.25)

Vb 1.'

=

-RI; 1. 1.

+ -Ws1 -d"
i = m+1, ... ,b

(6.26)

-Ri1ci

#ci + -W,1 -dt

i =

m + 1, .. . ,b

(6. 27 )

Vci =

with

[--I "
Xem i

Xemi

Xemi

Xes i

Xemi

Xemi

Xemi

Xesi

[ X . ..

[ -1-1

I

- I;;

-lei i =

m+ 1, .. . ,b

(6.28)

To connect the machines, these constraints must also be transformed . The scaled line, transformer, and load variables in the synchronously rotating reference frame are defined as

m+ 1, ... ,b

(6.29)

= 1 + m, .· .. ,b

(6.30)

i=m+1, ... ,b

(6.31)

i =

i

The v'2 is needed because of (6.3)- (6.5). Applying this transformation to (6.25)-(6.28) gives the network and load constraints suitable for connecting the m machines:

128

CHAPTER 6. MULTJMACHINE DYNAMIC MODELS

d..pDi €-

R;IDi + ..pQi

-

dt

d..pQi €-

-

dt

-

d..pOi €-- -

dt

+ VDi

R;IQi - ..pDi + VQi Ri10i

+ VOi

+ 1, .. . , b

(6.32)

i=m + 1, ... , b

(6.33)

i = m

m+ 1, . . . ,b

(6.34)

+ 1, .. . ,b

(6.35 )

i=m +1, ... ,b

(6.36)

m+ 1, . .. , b

(6.37)

i = i = m

i =

with Xepi = Xe.6i - Xemi and Xeoi = ""'Ye"i + 2XeT11 i· The stator and network plus loads all have exactly the same form when expressed in t his r eference frame. The b sets of flux linkages and currents are not all independent .

6.3

Elimination of Stator/Network Transients

With this synchronous machine plus the R-L element model, it is possible to formally extend the elimination of stator/network transients of Section 5.3 to the multimachine case. To be proper, we should reduce our dynamic states to an independent set. For inductive elements, this is normally done by writing the dynamics in terms of an independent set of loop currents. To do this , we use several concepts from basic graph theory. Consider the three-node , four- branch directed graph of Figure 6.1. It is a directed graph because each branch (lab eled a , b, c, d) has an arrow associated with it. This arrow is the assumed direction of the branch current , as well as the assumed polarity of the voltage. d

o Figure 6.1: Directed graph

6.3. ELIMINATION OF STATOR/ NETWORK TRANSIENTS

129

For example, the four branch voltages (all written with the same rise / drop convention) must satisfy Kirchhoff's laws (the three loops shown in Figure 6.1): Va -

Va -

Vb -

'lie

= 0

(6 .38)

Vc -

'lid

= 0

(6 .39)

Vb -

Vd

= 0

(6.40)

or C!'11branch = 0

(6.41)

where Ca is the augmented branch-loop incidence matrix

a b Ca = c d

1

2

3

1 - 1

0 0

- 1

1

1 -1 0

0

-1

-1

( 6.42)

In general, the branch-loop incidence matrix is b X l, where l is the total number of loops in the graph (or circuit). Clearly, Ca is not unique, since we can define loops in any direction we choose. Furthermore, electrical engineers should recall that, for a given circuit (graph), there are at most b-n independent loop currents. For planar circuits, these are the mesh currents. In the graph shown, n = 2, since n was defined previously as the number of nodes excluding the reference node. Thus, since one of the nodes in Figure 6.1 must be the reference node , there are only two independent columns in C a (two independent rows of C!). For assumed arrows of the b branches and assumed directions of the lloops, the C a matrix can be written by inspection using the following algorithm:

C a ( i, j)

=

Ca(i , j)

=-1

cat i, j) =

1

if branch i is in the same direction as loop j, and branch i is in loop j if branch i is in the opposite direction of loop j, and branch i is in loop j

0 if branch i is not in loop j Since the branch directions are the assum ed branch current directions as well l (6.43)

130

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

which, for our example, gives

'a

= il ooP1

+ iloop3

(6.44 )

'b

- 'IOOPl - '100p3

(6.45)

'c

-ilooPl

+ iloop 2

(6.46)

.

.

td

= -'100p2 - ' 100p3

(6.47)

which can be easily verified by inspection. The b- n independent loops can be determined by partitioning C a in a special way. Begin with b branches all disconnected (as our multimachine model currently is). Connect n branches such that every node is connected , but no loop s are formed . The resulting graph is called a tree. The remaining b-n branches are called tree links, and are now added one at a time. Adding one tree link will create a loop . This loop should be oriented in the same direction as t he tree link . Adding a second tree link will create a second loop. This loop should also be oriented in the same direction as the second tree link, and must not include the previously added tree link. When this process is completed, there will be b-n loops, which correspond to the b-n tree links. Thus, the incidence matrix creat ed in this manner will have only b- n columns and will be structured as 1 1···('

Cb

=

n n+l

[: ]

(6.48)

b

T his b x l' (£' = b - n) branch loop incidence matrix is called a basic loop matrix, and does not have the subscript a . The columns of C a that do not appear in Cb are dependent . It should be clear that we still have the relations for Kirchhoff's laws: ' branch = Cbiindep. loop ctvbranch

=0

(649) ( 6.50)

This matrix will now be used with our b sets of stator/network/load equations. We define ..pDbranch ~ [..pDl·· · ..pDb]'

(6.51)

6.3. ELIMINATION OF STATOR/ NETWORK TRANSIENTS

131

In branch

L>

[IDl" . 1m]'

(6.52)

Vnbranch

L>

[VDl'" Vm]'

(6.53)

and similarily for Q and O. For a system with a basic branch loop incidence matrix describing the interconnection of these b branches as C b , we define loop flux linkages as '
L> C' = b1Pnbranch

(6.54)

'
L>

Ct'
(6.55)

' Ci'
(6.56)

and write the corresponding branch currents in terms of independent loop currents as

In branch = CbInloop

(6.57)

IQbranch = CbIQloop

(6.58)

= CbIoloop

(6.59)

Iobranch

By our choice of numbering, the first m branches are synchronous machines, and the last b'- m branches are transformers, lines, and loads. It might appear that this numbering scheme would not allow us to have two synchronous machines connected to the same bus, since the first n branches must form a tree. This clearly is not a real limitation, since the ordering of Ca can be changed arbitrarily after it is formed to suit a particular preference . We have not shown it formally, but the topology relationships between the abc branch voltages and branch currents are the same as for the dqo branch voltages and currents. We now write the stator/network/load transients of (6.10)-(6 .24) and (6.32)- (6.37) in vector/matrix form: f

d'
f

d'
f

d1Pobranch d.t = Rbranchlobranch

+ '
+ Vobranch

+ VQbranch

(6.60

)

(6 .61) (6.62)

where additional algebraic equations relating the flux linkages and currents could be written using (6.19)-(6 .24 ) and (6.35)- (6.37).

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

132

Multiplying these equations by q and using the properties of Cb from (6.50) and (6.57)-(6.59) gives the b-nindependent sets of stator/network/load transients E

d..pnloop dt

CgRbranchCbInloop

+ ..pQioop

(6.63)

E

d..pQioop dt

CgRbranchCbIQloop - ..pDloop

(6.64)

E

d..poloo p dt = CgRbranchCb1oloop

(6.65)

As in the single machine/infinite bus case of Chapter 5, the Rbranch = 0 case has a very interesting result. When resistance is zero, (6.63)-(6.65) have an exact integral manifold for ..pnloop' ..pQloop' and ..poloop' regardless of E: ..pnloop

Rbranch = 0

= ..pQloop

Rbranch = 0

= ..poloop

I =0 Rbranch = 0 (6.66)

This means that if ..pnloop' ..pQloop' and ..poloop are initially zera and Rbranch = 0, then ..pnlaap' ..pQloop' and ..poloop remain at zero for all time t regardless of the size of E. If ..pnloop' VoQl oop ' and Vool aap are not initially zero and Rbranch = 0, then Vonloap ' VoQloop oscillate forever and ..poloop remains at its initial value, according to (6.63)-(6.65). Unlike the single machine/infinite bus case, the Rbranch = 0 condition is not practical, since this means that the only possible load could be one or more synchronous machines acting as motors. For Rbranch not equal to zero but E sufficiently small, an integral manifold for Vonloop' VoQloop' and Vooloop still exists, but has not been faund exactly. An approximation of this integral manifold can be found by setting E to zero and solving all the algebraic equations for the stator/network/load flux linkages and currents. This requires the solution of the following equations:

o=

CgRbranchCblnloop

+ VoQloop

(6.67)

o=

CgRbranchCbIQloop - Vonloap

(6.68)

o=

CgRbranchCb1olaop

(6.69)

6.3. ELIMINATION OF STATOR/ NETWORK TRANSIENTS

133

with "'Dloop = 'Ci"'Dbranch

(6.70)

"'Qloop = Ci,;,Qbranch

(6.71)

= Ci"'abranch = CbIDloop = CbIQloop = CbIoloop

(6.72)

"'aloop I Dbranch

IQbranch Ia branch

(6.73) (6 .74) (6.75)

and (6.19 )-(6.24) plus (6.35)- (6.37) . Alternatively, we solve (6 .10)-(6.12), (6.19)-(6.24), and (6.32)-(6.37) evaluated at, = O. T he "zero" variables are decoupled and will not be carried further. Adding (6.10) plus j times (6.11) and using (6 .22)-(6.24) gives

0= R,i(Idi

+ jIqi)ej(oi - :V

- j(..pdi

+ Nqi)ej(o- ~) (6 .76)

Substituting (6.19) and (6. 20) gi ves the following complex equation, which can be written as the circuit of Figure 6.2.

( "Vdi +- J'Vqloj e Kii' "')

Figure 6.2 : Multimachine subtransient dynamic circuit (i

(X~i +J.(X' di. i

= 1,

.. . ,m)

Xl,i) E' . + . (X~i - X~;) . 1. oJ j(6i-~) Xlsi) q. J (X'di - Xlsi) 'l'ld. e

= l, .. . ,m

(6.77)

134

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

Adding (6.32) plus j times (6.33) and eliminating "'Di, "''Ii using (6 .35) and (6 .36) give the network/load equation , wruch can be written as the circuit of Fig'lre 6.3.

0 = (Ri + jXepi)(IDi

+ jfQ') + (VD. + jVQi )

(lOi + jIQi)

Ri

+ 1, ... ,b

i = m

(6.78)

jXepi

- (VDi + jVQi) +

Figure 6.3: Netw ork plus R-L load dynamic circuit (i

= m + 1,

... , b)

To convert from bran.ch subscript notation t o bus subscript notation, we make the following notational numbering of branches. Assume that loads are present at all n buses, and connected to the reference bus. T hese n branch voltages are t hen bus voltages denoted as i = l, ... , n

(6.79)

All branches in excess of these n have branch voltages that are either equal to bus voltages or equal to the difference between two bus voltages. With this bus notation , the multimacrune dynamic model is any connection of the circuits of Figures 6.2 and 6.3 into a network resulting in m macrunes, b branches, and n buses. Generalization of Network and Load Dynamic Models Before stating the final model, we now make a major assumption about the actual network and loads. We assume that an integral manifold also exists for all dynamic states associated with networks and loads that are not simple R-L elements . We ass ume that the integral manifold for the n etwork dynamic states can be approximated by the same representation as above (Figure 6.3 allowing X. P to be negative). We also assurne t hat the integral manifold for the load electrical dynamic states can be approximated by sets of two algebraic equations that can be written as sets of complex equations of the following general form, using the convention of Figure 6.4. (6.80)

6.3. ELIMINATION OF STATOR/ NETWORK TRANSIENTS

135

Bus i

r--'---, +

Figure 6.4: Generalized load electrical dynamic circuit where PLi( Vi) and QL,(Vi) may be nonlinear functions of the bus voltage magnitude Vi. Commonly used load models are

PL.(Vi)

PL~i

QLi(Vi)

QLoi + kQliVi + kQ2iVi2 + ...

+ kPh Vi + kP2i ViZ + . . .

= 1, ... , n

(6.81)

i = 1, ... , n

(6.82)

i

or any combination of terms involving any power of Vi. The constants PL oi a.nd QLoi represent a ('constant power" component , kPli and kQli represent a ['constant current magnitude at constant power factor" component, and ll k p2i and kQ2i represent a "constant impedance component. Note that PLi and QLi are given as "injected" powers, meaning that PL. will normally be negative for a passive load. With this generalized m odel for the network and loads, we write the algebraic equations for the interconnection of m machine circuits with all transformers, lines, and loads using the standard network bus admittance matrix defined through the bus voltages and net injected currents by

n

L YikejO:':hVke i8".

= 1, .. "m

(6.83)

i = m+l""ln

(6.84)

i

k~ 1

n

L

YikejClik Vk ej8k

k~l

where all quantities are as previously defined, and Yikejaik is the ikth entry of the network bus admittance matrix. This matrix is formed using all of

CHAPTER 6. MULT IMACHINE DYNAMIC MODELS

136

the branches of the form of Figure 6.3. It has the same formulation as the admittance matrix used for load- flow analysis . This representation gives the following dynamic model for the m machine, n bus power system after stator/ network and load electrical transients have b een eliminated, and using the load model of Figure 6.4 and (6.80) : I

dE~i

Tdo ·- , dt

= - E'qi ('1f;ld;

(X

X' )[1

di -

di

+ (X~i -

di -

(X~i_- XX:J;l_)2 (X' t.,n

di

XI,i)Idi - E~i)J

+ Efdi i :::: 1, . .. ) m

(6.85) ( 6.86)

T' . dEdi q OI dt = -

E'

+ (X qi -

di

('1f;2qi il . d'1f;2.i T qOl dt = - .,. 'f' 2qi

do; dt ::::

Wi -

.i

W6

E'di i

-

=

.

(X~i - X~~) qi - (XI. _ X .)2 lst

qt

+ (X~i -

[1

X' )

XI,i)I.i

(X'qi

+ Ed;)]

X )l L,,1. qi

-

i = 1, .. . , m i = 1, . .. , m

(6.87) (6 .88) (6.89)

1, ... , m

(6.90 ) T Ei

dEfdi dt -

i

dRfi TFi = - Rfi dt

' + -KF -' E f di TFi

i = 1, . .. , m

dVRi TAiTt = - VRi

+ KAiRfi -

KAiKFi TFi Efdi

+KAi(Vrefi - Vi)

- '1Mi

+ PSVi

i

i

= 1 , .. . ,m

= I, . .. , m

= 1, . . . , m

(6.91 ) (6.92 )

(6.93)

(6.94)

6.3. ELIMINATION OF STATOR/NETWORK TRANSIENTS T

SVi

dPSVi = -P SVi dt

+ PCi -

-1RDi

(W. --' - 1)

i = 1" ."m

137 (6.95)

Ws

with the limit constraints

(6.96) ~

0< PSVi

PS-vi

(6.97)

i = 1, .. . ,m

and the algebraic constraints

(X:Ji + J.(X' di -

+ , (X~i -

Xt,i) E' Xlsi) q'

J (X'di

-

X:J;) ./, .J j(5,-,;:) X ) 'rId, e lsi

i = l, ... ,m

(6.98)

n

I: ViVkYikej(O,-o.-<x,.)

i = 1, ... , m (6.99)

k=I n

L

ViVkY'ikej(8i-Bk-OCik)

k=l

i = m

+ 1, ..

'j

n

(6.100)

The voltage regulator input voltage is Vi, which is automatically defined through the network algebraic constraints. Also, fOT given functions PLi(Vi) and QLi(Vi), the n + m complex algebraic equations must be solved for Vi, Oi (i = l, ... ,n)) Jdi l Iqi (i = l, . . . ,m)in terms of the states 8;" E~i) 'l/J2qil E~i' ,p,di (i = 1, ... , m). The currents clearly can be explicitly eliminated by solving either (6 .98) or (6.99) and substituting into the differential equations and remaining algebraic equations. This would leave only n complex algebraic equations to be solved fOT the n complex voltages ViejO,.

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

138

The Special Case of "Impedance Loads" In some cases, the above dynamic model can be put in explicit closed form without algebraic equations. We make the special assumptions about the load representations:

PLi(Vi)

kp2i~2

i = 1, ... 1 n

(6.101)

QLi(Vi)

kQ2iVi2

i

(6.102)

=

1, .. . ,n

In this case, the loads and R,i + jX:J.. can be added into the bus admittance matrix diagonal entries to obtain the following algebraic equations: 1

( R . + 'X") $1.

_

dt

]

[(

II

")

Xqi - X di Iqi

(X~\-Xl'i),

+ (Xlq1 _

X

.) Edi

lSI

- X") (X"di - X lst·) E' . (X'· ql, qt ,,/. . +' ' X·) ] (X'di.- X·) q. (X qib t 'I'2q' lSl ( R 51.

+1J.X"di )

B v'e';

(6.103) 1

0= - ( R .n _

+ J·X"dt )

""

[ (Xqi - Xdi)Iqi

( X" - Xl 9·)1 ,

+ (X'qtqt _

X

.)Edi

l"",

- X") X·) (X'. qt ql 'IjJ . +' (X" di iS1 E' . (X~i - Xl,,) 2q. ] (X~i - Xl,i) q.

- X:J..) 0/' .] +]. (X~i ' - X .) 'I'ld. (X di

e j (8;-%)

iS1-

n

+ L(Gi" + jBik)VkejB.

i = 1, ... ,m

(6.104)

+ 1; ... , n

(6.105)

k=l n

0=

L(Gi" + jBiklVkejB.

i = m

k=l

where

Gi" + JBIl.:

is the ikth entry of the bus admittance matrix, including

all "constant impedance" loads and 1/(R,i + jX:J..) on the ith diagonal. Clearly, all of the VkejB. (k = 1, . . . , n) can be eliminated by solving (6.104) and (6 .105) and substituting into (6.103) to obtain m complex equa-

6.4. MULTIMACHINE TWO-AXIS MODEL

139

tions that are linear in Idil Iqi (i = l, ... ,m). These can, in principle , be solved through the inverse of a matrix that would be a function of 6i , E di , ,p2.i, E~i' ,pldi (i '" 1, ... , m). This inverse can be avoided if the following simplification is made:

X", qt = X"· dt

i = 1, .. . ,m

(6106 )

With this assumption, (6 .104) and (6.105) can be solved for Vke je • and substituted into (6.103) to obtain

where GO d re ik

+j B"re d is the ikth entry of an m X m admittance matrix (often ik

called the matrix reduced to "internal nodes") . This can easily be solved for ldi, lq, (i '" 1, ... , m) and substituted into the differential equations (6.85)(6.95) . Substitution of ld, l qi into (6.98) also gives Vi as a function of the states (needed for (6.93)), so the resulting dynamic model is in explicit form without algebraic equations.

6.4

Multimachine Two-Axis Model

The reduced-order model ofthe last section still contains the damper-winding dynamics ,pld, and ..p2q" If Tz'i and T~~, are sufficiently small, there is an integral manifold for these dynamic stat es . A first approximation of the fast damper-winding integral manifold is found by setting T:Jm and T~~, equal to zero in (6.86)- (6.88) to obtain

+ E~i -

o-

- 1jJldi

0 -

- .,p2qi -

(Xdi - Xlsi)Idi

i = 1, ... , m

(6.108)

Edi - (X~i - Xlsi )Iqi

i = 1, . . . ,m

(6.109)

When used to eliminate ..pld' and ,p2q" the synchronous machine dynamic circnit is changed from Figure 6.2 to Figure 6,5 , giving the following multimacrune two-axis model:

CHAPTER 6. MULTIMACIIINE DYNAMIC MODELS

140

Figure 6.5: 1, ... , m)

Synchronous machine two-axis model dynamic circuit (i

1 dE~i I I T doiTt - - Eq; - (Xdi - X di )1di

+ Efdi

' "dE~i T qm dt = - E'di + (X qi- X'qi )1qi d6, = dt

Wi

; = 1, .. . , rn

i= l , .. . ,m

(6.110)

(6.111) (6.112)

-w$

2Hi dWi w, dt

(611 3) T Ei

dEfdi dt -

dRfi TFiTt dVRi T Ai-----;It

i

- Rfi

+ -KF" T'Efdi

i

+ K AiRfi -

KA iKFiE T fdi

Fi

-

-

"V Ri

= 1, ... , rn

= 1, ... , rn

(6 .114 )

(6115)

Fi

(6.11 6) (6. 117) T SVi

dPSVi _ - PSVi dt

+ PCi

_ _

1_(Wi _ 1)

RDi

i = 1, ... , rn (6.118)

WfJ

with t he limit constraints

vmm:s; VRi ~ vmax o :::; PSVi ::; prvi and the algebraic constraints

i = 1, .. _,m

(6.119)

i = I, . .. , m

(6.120)

6.4. MULTIMACHINE TWO-AXIS MODEL

o=

141

+ (R'i + jXdi )(Idi + jlqi)ej(6'-~) -[Edi + (X~i - Xdi)Iqi + jE~iJej(6'- :V

ViejO,

i = 1, . . " m

(6 .121)

n

=

L

ViVk¥;kej(O,-Oh-"''')

i' = 1, . .. ,m

(6 .122)

k=l n

hi(Vi)

+ jQdVi) = L

Vi V kY ik ej (9,-O. - "".)

k=l

i = m

+ 1, ... ) n

(6. 123 )

As before, for given functions PLi(Vi) and QLi (Vi), the n+ m complex algebraic equations must be solved for Vi, 0i (i 1, .. . ,m) and Idi, Iqi (i 1, ... ,m) in terms of the states Oi, Edil E~i (i = 1, . . . ,m) . The currents can clearly be explicitly eliminated by solving either (6.121) or (6 .122) and substituting into the differential equations and remaining algebraic equations. This would leave only n complex algebraic equations to be solved for the n complex voltages ViejO,.

=

=

The Special Case of "Impedance Loads" In some cases, this two-axis dynamic model can be put in explicit closed form without algebraic equations. We make the special assumptions about the load representations

(6 .124)

PLi (Vi) = kp'iV;' QLi(Vi) = kQ2i Vi'

i

= 1, .. 'In

(6.125)

In this case, the loads and R,; + jXdi can be added into the bus admittance matrix iliagonal entries to obtain the following simplified algebraic equations for the two-axis model with "constant impedance" loads

. ·I qt.)j(6' · (Xr'q1.- X'dt.)I qt· (Id,+J e -¥ )-- ( R . 1 'X' ) [E'dt+ .n

+ jE~iJej(6'-¥) -

(

+J

di

I)

. 1. R .. + JXdi

Vi ejO,

i= 1, . .. , m(6.126)

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

142

· + J·E'qi1e j(6i-~) 0= - ( R . +1 'x' ) [E'di -'- (X'qi - X')I di ql J di I

,tt

n

+ L (G: k + jB;k)Vk eje•

i

=1

1 ,

"lm

(6.127)

k=] n

0 =

L(G:k + jB:k)Vkeje.

i = m

+ 1, ... , n

(6.128)

k=]

where G: k + j Elk is the ik th entry of the admittance matrix, including all "constant impedance" loads and I!(R,i + jX~;) on the ith diagonal. Clearly, all of the Vkejo. (k = 1, ... , n) can be eliminated by solving (6 .127) and (6.128) and substituting into (6.126) to obtain m complex equations that are linear in I di , lqi (i = I, ... ,m). These can, in principle, be solved through the inverse of a matrix that would be a function of the states Oi, E~i' Edi (i = I , ... ,m). This inverse can be avoided if the follo wing simplification is made:

i= I) .. '1m

(6.129)

With this a.ssumption, (6.127) and (6.128) can be solved for Vkeje. and substituted into (6.126) to obtain

(6.130)

i:::: 1) .. ' 1m where G' d + j B' d is the ikth entry of an m re Ie ik

X

m admittance matrix (often

ik

called the matrix reduced to "internal nodes"). This can easily be solved 1, ... , m) and substituted into the differential equations for 1di, 1qi (i (6.110)- (6.18) . Substitution of 1di, 1qi into (6 .121) also gives Vi as a function of the states (needed for (6.116)) , so the resulting dynamic model is in explicit form without algebraic equations.

=

6.5

Multimachine Flux-Decay Model

The reduced-aIder model of the last section still contains the damper-winding dynamics E~i . If T~oi for all i = 1, .. . , mare su:fficiently small, there is

6.5. MULTIMACHINE FLUX-DECAY MODEL

143

an integral manifold for these dynamic states. A first approximation. of the remaining fast damper-winding integral manifold is found by setting T~oi equal to zero in (6.111) to obtain

0 = -Edi

+ (Xqi -

X~i)Iqi

i = 1, . .. ,m

(6.131)

When used to eliminate E~i ' the synchronous machine dynamic circuit is changed from Figure 6.5 to Figure 6.6, giving t he following multimachine one-axis or flux-decay model: jxlr

R,l'i

(ldi + JIqi) t/('5j- xJ2)

.'

i(Xqi + Xdi') /qi+ ]'Eqi' J e ;(0;-><12)

Figure 6.6: Synchronous machine flux-decay model dynamic circuit (i=l,

... ,m) I dE~i Tdoi dt'

i = 1, ... , m

d6,· = dt

Wi-W.,

(6.133)

i =l , ... ,m

i = 1, . . . ,m

-(KEi -Rli

(6.132)

(6.134)

+ SEi(Ejdi))EJdi + VRi

+ Kr T 'Efdi

i = 1, ... , m

i = 1, .. . , m (6.135) (6.136)

Fi

-VRi+KAiRfi -

K~KJiEfdi Fi

KAi(Vrefi - Vi) dTMi

TCHi~

T

SVi

dPsvi dt

- -TMi

+ PSVi

= -PSVi + PCi -

i

i

= 1, . .. , m

= 1, .. . ,m

- 1-

RDi

(W' --' - 1 ) W.,

(6.137) (6.138)

i = 1, .. . ,m (6 .139)

144

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

with the limit constraints rrun VRi

< _

o :::;

v:R\. _< VRtmax

(6 .140)

i = I,. " lm

Psvi < P~'F

i = 1,. _.) m

(6.141)

and the algebraic constraints

o=

+ (R" + jX~i)(ld, + jlqi) ej(6,-~) - [(Xqi - X~;)Iqi + jE~,Jej(6,-~)

V,e j8 ,

(6 .142)

n

=L VtVkY'ikei(8i-6k - OC il,)

i

= 1, .. .

I

m

(6.143)

k=l n

h i(Vi) + jQLO(Vi )

= ~ ViVkYik ej (6,-8.-o< .. ) k=l

i=m+l, ... ,n

(6.144)

+

As before, for given functions PLi(Vi) and QLi(Vi ), the n m complex algebraic equations must be solved for Vi, Bi (i = 1, .. . , n), Idi. l Iqi (i = I, . .. I m) in terms of t he states Oi , E~, (i = 1, .. . , m ). The currents can clearly be explicitly eliminated by solving either (6.142) or (6.143) and substituting into the differential equations and remaining algebraic equations. This would leave only n complex algebraic equations to be solved for the n complex voltages Vie j8 ,. The Special Case of "Impedance Loads"

In some cases, this flux-decay dynamic model can be put in explicit closed form without algebraic equations . We make the special assumptions about the load representations hi(Vi)

= kp2'-V/

i

= 1, ... , n

(6 .145)

(6.146) = kQ2;V? i = 1, . .. , n R,i + j X di can be added into the bus admittance

QLi(Vi)

In this case, the loads and matrix diagonal entries to obtain the following simplified algebraic equations

6.5. MULTlMACHINE FLUX-DECAY MODEL

145

for the flux-decay model with "constant impedance" loads

[(X qi 1 (R b'\. + J'X') eli (

Rsi

-

X-'di. )Iqi + J'E'qi ]e j(6 ' 1

+1 jXdi') V;ej8i

o -- - ( R ' +1 ·x' ) [(Xq.· _ ~l J eli

i = 1, ... , m

:!! ) ,

(6.147)

Xl)I . ·E' .] j(6i - i) d. q. + J q. e

n

+ L(Gik + jBik)Vkej8.

i = 1, .. . , m

(6.148)

k=l n

0 = L(Gik+jB!k)Vkej8.

i=m + 1, ... ,n (6.149)

k=l

where G: k + jBik is the ikth entry of the admittance matrix, including all "constant impedance" loads and 1/(R si + jXd;) on the ith diagonal. Clearly, all of the Vkej8. (k = 1, ... , n) can be eliminated by solving (6.148) and (6.149) and substituting into (6.147) to obtain m complex equations that are linear in Jail Iqi (i = 1 , . . ", m). These can , in principle , be solved through the inverse of a matrix that would be a function of the states Oi , E~i (i = 1, ... ,m). This inverse can be avoided if the following simplification (usually not considered valid) is made:

Xqi = X di i = 1, ... ) m

(6.150)

With this simplification, (6. 148) and (6. 149) can be solved for Vkej8. and substituted into (6.14 7) to obtain ~

ld ,'

+ J'1qtL...J ' = "(G' d + J'B're d ) E'kej(6.-0i) q re k=l

where G' d re

ik

+ jB'red is ik

ik

an m

i = 1, ... , m

(6.151)

ik

X

m admittance matrix (often called the ma-

trix reduced to "internal nodes"). This can easily be solved for ldi, Iqi (i = 1, ... , m) and substituted into the differential equations (6 .132)- (6.139). Substitution of ldi, I.i into (6 .142) also gives Vi as a function of the states (needed for (6. 137)) so that the resulting dynamic model is in explicit form without algebraic equations . We emphasize that the simplification of (6 .150) is usually not considered valid for most machines .

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

146

6.6

Multimachine Classical Model

As in the single mach.ine/infinite bus case of Chapter 5, the derivation of the eJassical model requires assumptions that cannot be rigorously supported . Returning to the multimacrune two-axis model, rather than assuming T~o; = (i = 1, ... ,m), we assume that an integral manifold exists for Ed;' E~i' E,di, Rfi , VR ; (i = 1, . . . , m) that as a first approximation, gives each E~ ; equal to a constant and each (Ed; + (X~. - X d.)lq .) equal to a constant. For this constant based on initial values Ed'!' I;i) E~i, we define the constant voltage

o

E 1.'o ~ J(EI~ dl

+ ( X' . tp.

0 X'dt.)1ql .)2 + (E'~)2 q'

(6. 152)

and the constant angle

O'D

l;

• = tan

- I(

E'o q;

E'~ + (X'ql. _ X'd,.)1ql dt

0•

)

1f -

2

(6.153)

The classical model dynamic circuit is then shown in Figure 6.7.

dl

jX

R ,i

(/di + j1qi) ",~;-TrI2)

+

F igure 6.7: 1, .. . , m)

Synchronous machine classical model dynamic circuit (i

Because the classical model is usually used with the assumption of constant shaft t orque, we assume (6.154) TeH. = 00 The classical model

IS

then a 2m-order system (obtained from (6.110)-

(6. 123)) do;

i= 1,. "Im

dt

~TFWi

i=l, . . . ,m

(6. 155)

(6.156)

6.6. MULTIMACHINE CLASSICAL MODEL

147

and the algebraic constraints

i= l, ... ,m

z:: V; V

{6. 157}

n

kYik ei (8< - 8. - oc ;,)

{6.158}

k=l

PLi{V;}

+ jQLi{V;} =

z:: V;V n

ik k Yik ei (8; - 8, - oc )

k= l

i=m+l, ... ,n

{6.159}

As before , for given functions PLi {V; } and QLi{V;}, the n + m complex algebraic equations must be solved for V;, 0i {i = 1, ... ,n} and Idi, I. i {i = 1, ... ,m} in terms of the states Oi . The currents can easily be explicitly eliminated by solving either {6.157} or {6.158} and substituting into the differential equations and remaining algebraic equations. This would leave only n complex equations to be solved for the n complex voltages V;e i8;. The Special Case of "Impedance Loads" In some cases , this classical model can b e put in explicit closed form without algebraic equations . We make the special assumptions about the load representations

= kp2.v/

i

QLiCV;} = kQ2iV/

i

hi(V;}

= 1, ... , n = 1, ... , n

(6.160) (6.161)

In this case, the loads and H,i + j X di can be added into the bus admittance matrix diagonal entries to obtain the following simplified algebraic equations for the classical model with "constant impedance" loads

(6.162)

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

148

0i= 1, ... ,m

(6.163)

n

o=

+ jBik)Vkej9,

~(G:k

i = m

+ 1, .. "In

(6.164)

Ie=!

where G: k + jBik is the i kth entry of the admittance matrix, including all "constant impedance" loads and l/(R'i + jXdi) on the ith diagonal. Clearly, all of the Vkej9'(k = 1, .. . , n) can be eliminated by solving (6.163) and (6.164) and substituting into (6.162) to obtain m complex equations of the form ~

~) = '\'(G' +J· B' )E'oej(c.H~O) ( Id1.· +J·I)ej(ciql L-t red red k k=1

ik

where G' d + j B' d is the ikth entry of an m Ie

ik

(6.165)

ik

re

X

m admittance matrix (often

ik

called the matrix reduced to "internal nodes"). Defining belassical /; bi

+ b:o

i = 1, . .. ,m

(6.166)

the multimachine classical model with constant impedance loads is found by substituting (6.165) into (6.156): db e1assical =

dt

Wi -

W6

i = I I" " I m

(6167)

2Hi dWi = TM. - ~ E!O E£oG~ed cost belassical, - bclassical.) w, dt k=l

ik

m

- ~ E:oE~B~ed sin(bclassical, - bclassical.) k~l

ik

~TFWi

i=l, .. . ,m

(6.168)

The classical model can also be obtained formally from the two-axis model by setting X~i = X di , and TCHi = T~oi = Tdoi = 00 (i = 1, .. . , m), or from the one-axis model by setting X q• = X d., and T~oi = 0, TCH. = Tdoi = 00 (i = 1, ... , m). In this latt er case, b;o is equal to zero, so that belassical is equal to Oi and Eta is equal to E~i (i = 1, ... , m).

6.7. MULTlMACHINE DAMPING TORQUES

6.7

149

Multimachine Damping Torques

As in the single machine/ infinite bus case of Chapter 5, t he multimachine flux-decay and classical models do not have explicit speed-damping torques. The friction and windage torque term could be specified as (6.169) While this would provide some damping , the effect of all damper windings bas been lost in the reduction process . As in the single machine/ infinite bus case , it is possible to justify t be addition of speed-damping torques . The mathematical derivation of these damping torque t erms proceeds in a direct extension to the multimachlne case. For example, the integral manifold for each m achine E~i must be approximated m ore accurately, as done in Section 5.7. The resulting improved integral manifolds have the following general form: m

E~i - L Joik (Oi - Ok, E~i' E~k) k:::::l m

- T~oi L

Jlik( Oi - Ok, E~i' E~k ' E jdi, Efdk, Wi, Wk)

(6 .170 )

k=l

where Joik is the same as (6 .131) written after elimination of I qi . The functions hik contain all the speeds of all machines. Rather than use the complicated form of (6 .170) , it is customary to simply .recognize that this a dditional term involving speeds will contribute speed-danlping torques, which can b e approximated by m

TVi = LDik(Wk -w,)

(6.171)

k= !

with Dik treated as a cons tant. The swing equation for each machine would then be 2H · dw · w,' d: = TMi - TELEC, - TVi - TFwi (6.172) with TELEC, and all other dynamics evaluated using the simple integral manifold approximation for E~i in (6 .131). In order of accuracy, it is emphasized that the most accur ate model would keep all damp er-winding dynamics . The second most accurate model would use a good integral manifold ap proximation for the damper windings ((6.170»). The least accurate model would use the crude integral manifold for the damper windings (( 6.131», together with a constant Dik cross-damping torque term ((6.171».

150

6.8

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

Multimachine Models with Saturation

The dynamic models of (6 .10)-( 6.24), (6.85)-(6.100), (6.11 0)-( 6.123 ), (6. 132) - (6 .144), and (6.155)-(6.159) began with the assumption of a linear magnetic circuit. In order to account for saturation} it is necessary to repeat this analysis with the saturation functions included. For this case, we assume the saturation functions of [37], given in (3.204)-( 3.209). With these funct ions and the assumption that saturation does not significantly affect time scales, the stator/ network transients can still be formally eliminated to obtain the same dynamic circuits of Figures 6.2 to 6.4 and the following multimachine dynamic model:

EIqi

-

(Xdi - Xldi ) [Idi

-

+ (X~i -

-

(X~i_- XX:J..)_)2 (".If"ldi (X' di

l~'L

•

Xl,i)Idi - E~i)l- sj~ + Ejdi

i= l ) .. . , m

(6.173 )

-1/1'di + E~i - (Xdi - Xl,i)h E'

-

di

+

(1/12qi i

(X

qi -

+ (X~i -

Xl) qi

[I

qi -

Xl ,i)Iqi

i = 1, ... , m

(X~i -

X:;')

q1

ist

(XI , _ X

(6.174)

_)2

+ E~i)l + sgl

= 1,. "1m

(6.175)

Til . d1/12qi q'" dt d6i = dt

Wi -

2Hi dWi = T Ws

dt

(6.177)

w~

. _ (X:Ji - Xl,i) E ll . _ (Xdi - X:J,) M,

(X diI _ X in.) .'

q'

(X di ' _ X in.)'!f;ld;Iqi

(6. 178)

6.S. MULTIMACHINE MODELS WITH SATURATION

KF

151 (6.180)

i=l, .. . ,m

-RFi+ T lEfdi Fi

(6.181) -TMi

+ P SVi

Tsv i dP~Vi = -PSVi

+ PCi -

= 1, . . ", m

i

(6.182)

_1_ (Wi _ 1) RDi

with the limit constraint s rnlil < V . < Vrnax VRi Rl _ Rt

i = 1, .

,m

(6 .183)

Ws

i = 1, . .. ,m

(6.184)

i = 1 , . . . ,m

(6.185)

and the algebraic constraints

a=

Vie jB ,

+ (R" + jX~;)(h + jIqi )e j (5,-,!;)

_ [(XU _ XU)I . + (X~i - Xl,i) E' . _ (X~i - X~i) 7fJ . q' d, q' (X'. _ X ist.) d, (X'. _ X lS1.) 2q' ql ql

(X~i + J.(X'. _ dt

Xl,i) E' . + . (Xdi - X~J 01, .J i(Oi-:;') X .) q' J(x'. _ X .) '¥ld, e l Sl

dt

lst

(6.186)

n

L ViVkYikej(8.:-e",-CX:ik)

i

=

1, . .. 1 m

(6.187)

k=l

PL,(Vi) + jQLi(Vi) =

n

L

Vi VkYik ej (8,-8 k -O<;k)

k=l i=m+ l , ... , n

(6.188)

and finally the saturation function relations 01,11

(2) -" 1?b?1 '¥di S srni (I 'l/Ji·/11) S fdi (2) 6.

S'q'

=

'IjJ~JXqi - Xis£) 17fJ;'I(Xd' _ Xl,i)

. = 1, . .. , m

(6189)

1,

"

S,=,(I7fJi I)

i

= 1 , ... , m

(6.190)

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

152 where

(6.191) 01./1

'f'di

l>

i = 1, ... ,m

(6.192)

i = 1, . . . , m (6.193) and Ssrni are given nonlinear functions that are zero at

I'if'YI = 0 and increase

exponentially with I1/;YI. With our choice of saturation functions, the special case of impedance loads still leads to an explicit set of differential equations without algebraic equations. This should be clear, since the only added terms are S}>J., and

sl~l, which are assumed functions of dynamic states. This will not impair the elimination of currents and voltages needed to obtain the reduced admittance matrix formulation . If saturation of network transformers were considered, this elimination would involve nonlinearities in algebraic states that would make an explicit formulation difficult, if not impossible. The Multimachine Two-Axis Model with Synchronous Machine Saturation To obtain a two-axis model with saturation included as above, we again assume that Td~i and T~~i are sufficiently small so that an integral manifold exists for each 1/;'di and 1/;2qi' A first approximation of the fast damperwinding integral manifold is found by setting Td~i and T~~i equal to zero in (6.174) and (6.176) to obtain

+ E~i -

o=

-'1jJldi

o=

-'Ij;2qi -

(X~i

- Xlsi) I di

E~i - (X~i - Xlsi)Iqi

i = 1, ... , m

i

= 1,. -.

J

m

(6.194) (6.195)

When used to eliminate 1/;'di and 1/;2qi, the synchronous machine dynamic circuit is changed from Figure 6.2 to Figure 6.5, giving the following multimachine two-axis dynamic model with synchronous machine saturation:

i = 1, .. . ,m

(6.196)

6.8. MULTIMACHINE MODELS WITH SATURATION

' dEdi -Tqoi~ d5i dt 2Hi dwi --d w, t

TEi

-

X'qi )1qt' + S(3) lqi

+ (Xq -

E'di

.

,,=

= Wi - W$

+ SEi(Efdi))Efdi + VRi

= -YR ' + KA 'Rj ' ,

t

+ KAi(Vrefi dTMi

-TM

(6.198)

1, ... , m

dRji _ KFi T F' --Rji+TFiEjdi dt

TCHi~ =

(6.197)

" EdJdi - Eq;!qi

= TMi -

dEfdi dt = - (KEi

dVRi TA t'-dt-

i = 1, . . >,m

153

+ PSVi

TSVi d~Vi = - PSVi

+ pc. -

1

i = 1, ... , m (6.200)

(6.201)

i = 1, ... ,m KAiKFi TFi Eid't

Vi)

= 1, ... , m

i

(6.202) (6.203 )

i = 1, ... ,m _1_ RDi

(Wi - 1)

i = 1, ... , m (6 .204)

WIJ

with the limit constraints rnin VRt

< VRl. < i = _ Vrnax Rt

I, . "l m

(6.205)

i = 1, . .. ,m

(6.206)

_

o :s

PSVi

:s Pf&";x

and the algebraic constraints

o = Viei8; + (R •• + jXdi)(Idi + jlqi)ej(6;-~) (6 .207)

i = l , .. " m

n

L Vi Vk¥;k ei(8;- 8.-<X

ih )

i

= 1, . . . , m

(6 .208)

k=l

n

PL.(Vi)

+ jQLi(Vi) = L

ViVk¥;k ej (8; - 8. -

<x,,)

k= l

i=m+l, ... ,n

(6 .209)

154

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

and finally the saturation function relations (6.210) (6.211) where (6.212) We have made the assumption that when Wldi and W.qi are eliminated by the approximate integral manifold of (6.194) and (6 .195) , the saturation function can be approximated by using W~. "" E~i' W:;' '" - E di · This prevents Idi or Iqi from entering the saturation function explicitly. As before, the special case of impedance loads still leads to a dynamic model in explicit form without algebraic equations, because the saturation functions are assumed to be functions of only the dynamic states Ed. and E~i' The MuJtimachine Flux-Decay Model with Synchronous Machine Saturation To obtain a flux-decay model with saturation included, we again assume that T~oi is sufficiently smail so that an integral manifold exists for each E di . A first approximation of this integral manifold is found by setting T~o' equal to zero in (6.197) to obtain:

0= -Edi

+ (Xqi -

X~i)Iqi + s~!1

i = 1, ... , m

(6.213)

As in the single machine/ infinite bus case, there is now a problem, since (6.213) must be solved for each Edi (recall that each is a function of Ed. and E~;). While we could simply carry (6.213) along as another algebraic equation, we can again recognize that since we have already approximated the integral manifold (and, of course, neglected all off-manifold dynamics)

sl!l

sl!l ""

we may as well simplify further and set O. This gives the following flux-decay model with saturation only in the field axis.

i = 1) .. »m

(6.214)

6.8. MULTIMACHINE MODELS WITH SATURATION

155 (6.215)

(6.216)

dEJd, TEo dt

= -(KEd SE,(EJdi»

+ VRi

Efdi

i

= 1, ... , m

dRfi KFi Efd' i=l, ... ,m TF' - d= - Rf'+ TFi t dVRi KAiKFi T At" dt -- = -VR". . + KA 1 Rj 1 TFi. EfJa1· + KAi(Vrefi-V;) dTM TCHiTt = - TMi Tsvi dPSVi dt

+ PSVi

(0.218)

(6 .219)

i=l, .. . ,m

( 6.220)

i = 1, .. . , m

-PSVi + PCi - - 1 (Wi - - 1) RD, W,

=

(6.217)

i = 1, .. . ,m (6 .221)

with the limit constraints

min < V . < Vrnax V Rt _ R, _ Rt

o ~ PSVi

i = 1, . .. ,m

~ Pfftr'F

i

(6.222)

= l , ... ,m

(6 .223)

and the algebraic constraints

o = V;e;9, + (R'i + jXd,)(Idi + jlqi)e;(5i - ~)

n

L: ViVk~kej(9i-8/C - ()(ik)

i

=

1, ... I m

(6 .225)

k=l n

PLi(V;)

+ jQLi(V;) = L

V;VS;ke;(9i - 9'- ~i')

i

= 1, . .. , m

(6 .226)

k= l

and finally the saturation functi on relation (4) SJdi

--

S 8ml·(E'qi ) ,. -- 1

1'"

I

m

(6.227)

156

CHAPTER 6. MULTIMACHINE DYNAMIC MODELS

As before, the special case of impedance loads still leads to a dynamic model without algebraic equations, since the additional saturation terms are functions only of the dynamic states E~,. There is little point in trying to incorporate saturation into the multimachine classical model, since it essentially assumes constant flux linkage.

6.9

Frequency During Transients

The steady-state relationship between volta.ges and currents in loads clearly depends on frequency. For inductive loads, the reactance increases with frequency. For induction motors, the nominal speed increases with frequency. During transien ts, frequency does not have any meaning, since voltage and current waveforms are not pure sinusoids. It is, however, possible to denne a quantity during transients that reflects the concept of frequency and is equal to frequency in the sinusoidal steady state. T his can be done by considering the algebraic variab les VD" VQ, introduced earlier in the synchronously rotating reference frame together with their polar forms :

VD,

+ jVQ, = V;e j8 ,

i

= 1, ... , n

(6.228)

From the inverse t ransformation, the scaled abc voltages are (with Vo , = 0):

+ B,)

= 1, ... , n

Vai

=

v'zV; cos(w,t

Vb,

-

v'zV;cos (w,t+ B; _ 2;)

i

= 1, ... ,n

(6.230)

+ B, + 2;)

i

= 1, .. "

(6.231)

Vci =

v'zV; cos (w,t

i

(6.229)

n

We emphasize that these forms are valid for both transien t and steady-state analyses . In general, V; and B, will both change during a transient. A logical dennition of a "dynamic frequency" is l>

Wdi

=

Ws

+

dB; dt

i

= 1) . , .

I

n

(6.232)

If the multi machine system is in synchronism with all machines turning at a constant speed, the system frequency is equal to t his dynamic frequency (possibly above or below w,). During transients, each bus will have a dynamic frequency determined by dBd dt. We emphasize that this dennition is only one of many possible such quantities that have the property of being equal to true frequency in steady state .

6.10. ANGLE REFERENCES AND AN INFINITE BUS

157

Adding frequency dependence into the load model by using PLi (V;,Wd;), QL'(V;, Wdi) instead of h,(V; ), and QLi(V,) requires that Oi (i = 1, ... , n) become dynami c state variables. This is done by simply adding the following n additional different ial equations:

dOi dt =

Wdi - Ws

i

= 1, . .. , n

(6.233)

T he n dynamic frequencies Wdi aTe algebraic variables that must be eliminated together with aU other algebraic variables (1di, 1qi, Vi). Since this no rmally cannot be done easily, it is customary t o avoid making the Oi dynamic states by approximating the frequency dependen ce. This is done by keeping each Wd, constant over one time step of numerical integration (t.t) as

Wdi () t ""

Oi(t) - Oi(t - t.t) t.t

+ w.

(6 .234)

where t is the current time. In this method, the Oi remain as algebraic variables and are simply monitored at each time step to update the frequencydependent terms of PLi(Vi, Wd;) and QLi(Vi, Wd;) for the next time step.

6.10

Angle References and an Infinite Bus

The multimachine dynamic models of the last sections have at least one more different ial equation t han is needed to solve an m machine , n bus problem, because every rotational system must have a reference for angles. To illustrat e t his, we define the angles relative to machine 1 as

5~

l!J.

5i - hI

8.I~O.·-
i = 1, .. . ,m

(6.235 )

t· =

(6.236)

1 t"

'l n

with derivatives for the new dynamic states dol

_, = 0

(623 7)

dt dolt

dt = W i - Wl

i = 2, . .. ) m

(6.238)

Inspection of the algebraic equat ions that accompany all of the previous mo dels will reveal that all angles can be written in terms of 0; and 0;. This m eans that the full system models have the same form as before, with the following mo'difications:

158

CHAPTER 6. MULTIMACHJNE DYNAMIC MODELS • Replace Oi with 0; (and o( = 0). • Replace lJi with

IJ;.

• Replace w, in the time derivative of 0; with

Wi.

With this formulat ion, the order of the system is formally reduced by 1, since (6.239) While o( remains at zero for all time, Wi changes during a transient. The original 01 can be obtained by integrating Wi - W, over time. The original 0i (i = 2, . . . ,m) and 0i (i = 1, ... , n) can be recovered easily from 5I, Bi and 5; through (6.235)- (6.236), if desired. The dynamic system order can be further reduced either if Hi is set to i.niin.ity (machine 1 bas constant speed) or if speeds do not explicitly appear on the right-hand side of any dynamic equations (no speed-damping torques and no governor ac tion). This could be iormally done by defining , I:>

Wi. = Wi -

WI

i

= 1, ." , m

(6.240)

so that (6 241 )

w;

and replaces Wi as a dynamic state. In this case, there is no need to include either the angle or the speed equation for machine 1, since all other dynamics would depend only on 6; (i = 2, .. . ,m) and (i = 2, ... , m). This situation also arises when the only speed terms on the right-hand side appear in the swing equations with uniform damping (H;/ Di = Hk/ Dk i, k = 1, .. . , m). A common transformation used in transient stability analysis is the center-of-inertia (COl) reference. Rather than r eference each angle to a specific machine (i.e. , 6d, the COl reference transformation defines the COl angle and speeds as

w;

(6242)

"

WeDI =

(6.243)

where (6.244)

6.10. ANGLE REFERENCES AND AN INFINITE BUS and

. fJ. 2H, M ,-

w,

159

(6.245)

The CO l -referenced angles and speeds are

• n

Oi = 0, - OeD J Wi

6.

wi-weOl

i

= 1, . _. , m

(6.246)

i= l, ... ,m

(6 .247)

With the introduction of DeDI and WeDI, it is possible to use these as dynamic states together with any other m - 1 pairs of COl-referenced mechanical pairs. Choosing 2, ... , m, the new mechanical state-space would consist of OeDI , WeDI , 5i, Wi (i = 2, ... , m). This would require the elimination of 51 and W1 in terms of OeDI, WeDI and 5" Wi (i = 2, .. . ,m). The resulting system would have MT multiplying the time derivative of weD I. Since MT represents the total system inertia, it is usually quite large relative to any single M i . For this reason, it is often taken to be infinity, in which case the COl mechanical pair is eliminated, reducing the dynamic order by 2. It is important to emphasize that simply using COl-referenced variables does' Ilot , in itself, reduce the dynamic order. The reduction requires the use of COl-referenced variables together witb the approximation that MT is infinity. The resulting swing equations are complicated by the COl reference, since an inertia-weighted form of "system" acceleration is subtracted from each machine)s true acceleration.

Chapter 7

MULTIMACHINE SIMULATION In t his chapter, we consider simulation techniques for a multimachine power system using a two-axis machine model with no saturation and neglecting both the stator and the network transients . The resulting differentialalgebraic model is systematically derived . Both t he partitioned-explicit (PE) and the simultaneous-implicit (SI) methods for integration are discussed . The 51 method is preferred in both research gr ade programs and industry programs, since it can handle "stiff" equations very well. After explaining the 51 method consistent with our analytical development so far, we then explain the equivalent but notationally different method, the well-known EPRI-ETMSP (Extended Transient Midterm Stability Program) [70] . A numerical example to illustrate the systematic computation of initial conditions is presented.

7.1

Differential-Algebraic Model

We first rewrite the two- axis model of Section 6".4 in a form suita ble for simulation after neglecting the subtransient reactances and saturation. We also neglect the turbine governor dynamics resulting in TMi being a constant . The limit constraints on VRi are also deleted, since we wish to concentrate on modeling and simulation . vVe assume a linear damping term TF Wi = D i(Wi - w,). The resulting differential-algebraic equations follow from (6.196)-( 6.209) for the m machine , n bus system with the IEEE-Type I exciter as

161

CHAPTER 7. MULTIMACHINE SIMULATION

162

1. Differential Equations i=l, ... ,m f dEdi I Tqo, T t = - Ed,

+

(

dOi dt = Wi - w~

Xqi - XqiI ) I q,

Ei

T

(7.4)

i = l , ... ,m

dEfd, dt -

i = 1" "Im

dRfi = - Rfi

F,Tt

+ KTF,Fi Efd,

(7.2) (7.3)

i = 1, ... , Tn

- D,(w, - w,) T

i= 1, ... ,m

(7.1)

i= 1, . . . ,m

(7.5) (7.6)

dVR ' KA-KF' TA 1.'-dt-' = - VR ' + KA 1.'Rf1.' ' • Efdl' + KA t·(VIef 1 - v.) TFi t &

i

= 1, . .. ,m

(7.7)

Equation ( 7.4) has dimensions of torque in per-unit. When the stator transients were neglected, the electrical torque became equal to the per-unit power associated with the internal voltage source. 2. Algebraic Equations The algebra.ic equations consist of the stator algebraic equations and the network equations . The stator algebra.ic equations directly follow from the dynamic equivalent circuit of Figure 6.5 , which is reproduced in Figure 7.1. Application of Kirchhoff's Voltage Law (KVL) to Figure 7.1 yields the stator algebraic equations: (a) Stator algebraic equations 0= V;e ie , - [Edi

+ (R'i + jXd,)(Idi + jIq,)e i (6,-t)

+ (X~i -

i =l, ... ,m

Xd,)Iqi + jE~iJei(6'- f) (7.8)

7.1. DIFFERENTIAL-ALGEBRAIC MODEL

163

(b ) Network equations The dynamic circuit, together with the static network and the loads , is shown in Figure 7.2 . The network equations written at the n buses are in complex form. From (6 .208 ) and (6 .209), these network equations are jX~

R" +

[E" + (x~-X;)Jqi

(Vd' + jV,, ) ,)lk1rl2)

+ jE', ] ,Kb,-l
= Vi

"

Figure 7.1:

efJi = Vo; + jVQi

Synchronous machine two-axis model dynamic circuit (i -

1, .. . , m)

jX di i

m+ l Pl,m+l

'+ PL, (V,)+jQu(V')

jXd~

• • •

Network 1= YNV

• •

(Vm+l)

+}QL.Itt+/(v",+/)

R,~

m

+

?Lm

(Vm )

+ jQLm (Vm)

Figure 7.2: Interconnection of synchronous machine dynamic circuit and the rest of the network

164

CHAPTER 7. MULTIMACHINE SIMULATION

Generator Buses v;'ejei(h - jlqi)e -j(5i - ~}

+ PLi(v;') + jQLi(v;') =

n

L

V;.VkYikej(ei-ek-~ik}

k=l

i = I , . .. ,m

(7.9)

Load Buses n

PLi(Vi)

+ jQLi(V;) = L

V;VkYikej(ei - e. - ~,,} i = m

+ 1, ... , n(7.10)

k=l

In (7.9), V;.e j8i (h - jI.i)e -j(5i-~/2) !o. Fai + jQai is the complex power "injected" into bus i due to the generator . Thus, (7.9) and (7.10) are only the real and reactive power balance equation at all the n buses. Equation (7.9), which constitutes the power balance equations at the generator buses , shows the interaction of the algebraic variables and the state variables Oi, E~i' and E~i· We thus have 1. Seven differential equations (d .e.'s) for each machine, i.e., 7m d .e.'s

((7.1)- (7.7)). 2. One complex stator algebraic equation (7.8) (two real equations) for each machine, i.e., 2m real equations. 3. One complex network equation (7.9) and (7.10) (two real equations) at each network bus, i.e., 2n real equations. We have 7m + 2m + 2n equations with x = [xl ... x:"]' as the state vector where Xi = [ E~i E~i 0i Wi Efdi RJi VRil' as the state vector for each machine. y = [I,L. V' B' ]' is the set of algebraic variables where I d_q V

-

[Idl 1. , ... Id=I.= ]'

,

[VI· .. Vnl , B

= [8, ... Bn] ', -V = rv V 1 ... V nl '

Funct ionally, the refore, the differential equations (7.1)- (7.7)' together with the stator algebraic equations (7.8) and the network equations (7.9)-(7. 10), form a set of differential-algebraic equations of the form x

o

f(x,y,u) g(x,y)

(7.11) (7.12)

[ws Tmi VTefJ t as the input vector for each machine. We now formally put (7.1)- (7.10) in the form (7.11) and (7.12).

u = [ui ... u!nl t with Ui =

7.2. STATOR ALGEBRAIC EQUATIONS

7.2

165

Stator Algebraic Equations

There are several different ways of writing the stator algebraic equations (7.8) as two real equations for computational purposes. The idea is to express Idi)

Iqi in terms of the state and network variables. Both the polar form and the rectangular form will be explained.

7.2.1

Polar form

In this form, the network voltages appear in polar form. If we multiply (7.8) by e-j(Oi - ~) and equate the real and imaginary parts, we obtain E~i -

Vi

sin (Oi -

(}i) -

Rsi1di

+ X~ilqi

E~i - Vi cos (Oi - (}i) - Rsi1qi - X~ildi

o o

i = 1, . .. , m

(7 .13)

i = 1, . .. ,m

(7.14)

We define

-X'q' · Rsi

Then , from (7 .13) and (7.14):

Idi ] = [Zd_q,i]-l [ E fi - Vi sin(oi - ai) ] [ Iqi Eqi - Vi cos( 0, - a,)

i = 1, .. . , m

(7.15)

Equations (7.13) and (7 .14) are implicit in h, I qi , whereas in (7.15) they are expressed explicitly in terms of the state variables Xi and the algebraic varia.bles Vi, Bi . Thus i :::: I, ... , m

7.2.2

(7.16)

Rectangular form

This can be easily derived by recognizing the fact that

Vi = VDi

+ jVqi =

Vie j8 ,

=

Vi cos ai + jVi sin ai

(7.17)

By expanding (7 .13) and (7.14) and noting from (7 .17) that VDi = Vi cos Bi and VQi

= Vi sin Bi)

we obtain the implicit form in rectangular coordinates

as E~i - VDi sin Oi E~i

+ VQi cos Oi -

- VDt cos Oi -

VQi sin

+ X~ilqi

= 0

(7 .18)

Oi - RsiIqi - XdiIdl = 0

(7.19)

Rsi1di

166

CHAPTER 7. MULTIMACHINE SIMULATION

To obtain the explicit form , Id; , Iq; in (7.18) and (7.19) can be expressed in terms of Ed;' E~;, 0;, VD" and VQ;. Alternatively, the right-hand side of (7.15) is expanded as

Id' ] = [Zd _q,;t ' [ I qt

[E~; ] Eqi

_ [Zd_q ,,]-l [v.(Sino,.cos8; - cosO;SinO;J] v" (cos 0\ cos 01. + SIn Ot SIn 9 t )

(7 .20 ) Using the fact from (7.17) that VD; = V.cosO, and VQ , =

V. sin 0; , (7.20)

becomes

(7. 22) Note that (7.21) can be obt ained directly from (7.18) and (7.19). Symbolically, (7.16) or (7.22) can be expressed for all machines as

Id- q

hp(x, V, 0) or hr ( X,vD,vQ) h(x, V )

(7 23 )

Q

q

d

Figure 7.3: Graphical representation

7.2.3

Alternate form of stator algebraic equations

In much of the literature , a block diagram representation of stator equations is done through an "interface" block that reflects the machine- network transformation . The machine-network transformation is given by

7.3. NE TWORK EQUATIONS

167

- cos 0i ] [ FDi ] sm Oi FQi

i

= 1, .,., m

(7.24)

i

= 1, ... , m

( 7.25)

and

FDi ] [ FQt

=[

sin Oi . cos 0, -

cos 61

SIn 01

1[ Fdi F

]

qt

where F may be either I or V. Figure 7.3 is a graph.ical representation of (7.24) and (7.25) illustrated for the voltage Vi = Vie j8i . Using (7.24) in (7.21) , we obtain

(7 .26) Thus

Edi ' -- Vdi Vqt. [ E qi

] - [Z

-

.J [ Idi I qt·

d-q.,

1

i

= 1, ..

. ,m

(7.27)

The interface block in the block diagram in Figure 7.5 is now consistent with (7.24), (7. 25), and (7.26). Note that, in this formulation , algebraic equation (7.26) or (7 .27) is in machine reference only, whereas (7. 24) and (7.25) act as an Uinterface" between the machine and the network.

7.3

Network Equations

The network equations can be expressed either in power·balance or currentbalance form. The latter form is more popular with the industry software packages. We discuss both of them now.

7.3.1

Power-balance form

The network equations for the generator buses ( (7. 9» are separated into real and imaginary parts for i = 1, ... , m

CHAPTER 7. MULTIMACHINE SIMULATION

168

I d , Vi sine 0, - B,) + lq, Vi cos (0, - B,)

+ h,(Vi)

n

-L

ViVklik cos (Bi - Bk -

(Xik)

(7.28)

= 0

k=l

hVi COS(Oi

-

Bi) -lqiVi sin (Oi - B,)

+ QL,(Vi)

n

-L ViVklik sin (B, -

Bk- (Xik) = 0

(7.29)

k=l

For the load buses, a similiar procedure using (7 .10) gives for i . m+ 1, .. . 1 n n

PLi(Vi) -

L

ViVklik cos (B, - Bk- (X,k)

0

(7.30)

ViVklik sin (B, - Bk- (X,k) = 0

(7.31)

=

k=l n

QL,(Vi) -

L k=l

Note that load can be present at the generator as well as at the load buses. The network equations (7.28 )-(7.31) can be rearranged so that the real power equations appear first and the reactive power equations appear next) as follows. Real Power Equations

n

L

ViVklik cos (B, - Bk - (X,k) = 0

i = 1, ... , m(7.32)

k=l

n

hi(Vi) -

L

ViVklik cos (Bi - Bk -

(Xik)

=0

i = m

+ 1, ... , n(7.33)

k=l

Reactive Power Equations

n

L k=l

ViVkY,k sin (B, - Bk- (X,k) = 0

(7.34 )

7.3. NETWORK EQUATIONS

169

n

QLi(Vi) -

L

ViVk¥ik sin (Oi - Ok- CXik)

=0

i

= m + 1, ... , n (7.35)

k=l

Thus , the differenhal-algebraic equahon (DAE ) model is: 1. The differential equations (7.1)-(7.7)

2. The stator algebraic equations of the form (7.23) in the polar form 3. The network equations (7.32)-(7.35) in the power-balance form The differential-algebraic equatjons are no w written symbolica.lly as

x

(7. 36)

I d _ q = h(x, V)

o = 90(X, Id_q, V)

(7.37) (7.3 8)

Subshtution of (7.37) into (7.36) alld (7 .38) gives d;

= h(x, V, u)

0= 91(X , V)

(7. 39) (7.40)

Note that (7.40) is in the power-balance form. This is t he differelltialalgebraic equation (DAE) analytical model with the network algebraic variables in the polar form . We prefer this form, since in load-flow equations the voltages are geIlera.lly in polar fo rm. Simpli£ed forms of this model result from the reduced- order model of the synchronous machine as well as the exciter, which will be discussed later .

7.3.2

Current-balance form

Instead of the power· balance form of (7.32)- (7.35) , one can have the currentbalance form, which is essentially the nodal set of equations

(7.41) where Y N is the n X n bus admittance matrix of t he network with elements Y ik = ¥ikej",,, = Gik +j Bik , I is the net injected current vector and V is the bus voltage vector. Depending on how I is expressed, it can take different

CHAPTER 7. MULTIMACHINE SIMULATION

170

forms , as discussed below. Equation (7.41) can also be derived from (7. 9)(7.10) by dividing both sides of the equation by ViejO, and then taking the complex conjugate as follows.

i =l , ... ,m

h(Vi) - jQLi(Vi) = 9

Vie

j i

(7.42)

n

L

YikejlXilo Vk ei910

i =

m+ 1, . .. ,n

(7.43)

k=l

These equations are the same as (6.83) and (6.84). Equations (7.42) and (7.43) can be symbolically denoted in matrix form as (7.44) The other algebraic equation is

I d _ q = h(x, V)

(7.45)

Substitution of (7.45) in (7.36) and (7 .44) leads to the DAE model

x

h(x, V , u) (7.46)

Example 7.1 We illustrate the DAE models discussed in the previous section with a numerical example. We consider the popular Western System Coordinating Council (WSCC) 3-machine, 9-bus system [731 shown in Figure 7.4. This is also the system app earing in [74] and widely used in the literature. The base MVA is 100, and system frequency is 60 Hz. The converged load-flow data obtained using the EPRI-IPFLOW program [751 is given in Table 7.l. The Ybus for the network (also denoted as Y N ) can be written by inspection from Figure 7.4 and is shown in Table 7.2. The machine data and the exciter data are given in Table 7.3. The exciter is assumed to be identical for all the machines and is of the IEEE-Type 1. Denne 2H, ~ Mi. Assume w. that ~

= 0.1,

~

= 0.2, and j}, = 0.3 (all in pu)

7.3. NETWORK EQUATIONS

171

Gen 2

IS. OKV

Gen2

1.0'25 pu

230 ICY

Go",

Go" ,

S tation C 230KV

13.8 ICY

1.025 pu

230KV

'"""o,;;"","",o.",w,,,o'+-'7-+_-{ _

}-_t---E_t-z"~",o",oo"",,:;:·:;;o,,,n'--l~z

y", O.+jO.l045

y ", 0.+ jO.m4~

163MW

Tap '"

ill

""

(2) Z = jO.0625 (!) i

'0

O.+jO.O

StatiQII A

"

@

e e"

~

• ~

" e0

230KV

l ~MW

lOOMW

35MVAR

Tap=W

, " e" e" i ~

®

85MW

®

y ", n.+jO.O

o::.~ c ~

SlJ.tionB 230KV

®

®

90MW 30MVAR

'" O.Ol+jO.083 Z= O.017+jO.092 y .. O.+}).079

50MVAR

Z", jO.0586

Y = O.+jO.088

Tap "

~

Z .. O_+jO .0576

Go" ,

J'", O.+jO.O

_

+_16.5KV

CD

1.04 po

I

Slack Bus

Figure 7.4: WSCC 3-machine, 9-bus system; the value of Y is half the line charging (Copyright 1977. Electric Power Research Institute. EPRI EL0484 . Power System Dynamic Analysis, Phase 1. Reprinted with Permission.).

Table 7.1: Load-Flow Results of the WSCC 3-Machine, 9-Bus System Bus # 1 2 3 4 5 6 7 8 9

(swing) (P-V) (P-V) (P-Q) (" ) (" ) (" ) (" ) (" )

Voltage (pu) 1.04 1.025L9.3' 1.025L4.7' 1.026L - 2.2' 0.996L-4.0' 1.013L - 3.7' L026L3.7' L016LO.7' L032L2.0'

PG

QG

-PL

- QL

(pu)

(pu)

(pu)

(pu)

0.716 1.63 0.85 -

0.27 0.067 -0.109 -

-

-

-

-

-

-

-

-

-

-

-

-

1.25 0.9 -

LOO -

0.5 0.3 0.35 -

172

CHAPTER 7. M ULTIMACHINE SIMULATIO N

Table 7.2: Y N for the Network in Figure 7.4

2 1 2

, ,

4

, , , 9

- j11 . 361

0

"

j 1 7 . 361 0

" 0 0

"

0 -j 16

" " 0

"

j16

"

"

,

•

0

j l7 .36 1

"

"

-j1 7 .0 6 !;

"

""

~.307

-1.365

_ )39 , 3 0 9 - l. 3SS

til1.60t

+il1.60{

-j1 7 .338

- 1.9 402

0

0

"

, "

0

2.553

+i 10.lil1 0 0

; 17 .0 156

" 0

"

,

7 0 ; 16 0

0

"0 - l.I1142 +ilO.H I 0

8

0

" 0

0

0

- 1.1U + ;5 .975

0

0

0

3.22. - i l li.IIU

-1. 186 +j5.915

" "

0

"

2.801i - ;36. 44 60 - 1.8 l7

t i 1 3.1398

- 1.21!2 +j6.688

"

- 1 .6!. 7 +; 13 .59 8 2.172 -;23.3 0 3 - l.lI;S

t i i. 7at

Table 7.3: Machine and Exciter Data Machine Data Parameters

M/C 1

M/ C 2

M/C 3

H(secs)

23.64 0.146 0.0 608 0.09 69 0.0969 8. 96 0.31

6.4 0.8958 0.1198 0.8645 0.1969 6.0 0.535

3.01 1.3125 0.1813 1.2578 0.25 5.89 0.6

Parameters

Exciter 1

Exciter 2

Exciter 3

KA TA (sec) J(E TE(sec ) KF TF(sec)

20 0.2 1.0 0.314 0.063 0.35

20 0.2 1.0 0.314 0.063 0 .35

20 0.2 1.0 0. 314 0.063 0.35

Xd(pu) Xd( pu) X,(pu) X;( pu) Td,(sec) T;,(sec)

Exciter Data

SEi (E jd ;)

= 0.003ge1.555EJ"

i=1 ,2,3

,

"" " "

j17.055

- 1. 282 +;5.5 8 8

" _ 1.155 +j9.784 2.437 - ;32. 154 0

173

7.3. NETWORK EQUATIONS

The differential equations corresp onding to (7.1)- (7.7) are

.

E~i

E' .

Edi

E~i

0,

Oi

Wi

'

[A;]

+ Ri(E~i) Edt) EIdi) Idi, Iqil Vi ) + Gi U i

Wi

j;; fdi

Ejd;

Rf;

Rf;

VRi

VRi

i=I,2,3 (7.4 7) where - 1

T~oi

0

A;

0

0

0

-1

0

0

T~Di

1

0

0

0

0

0

T~oi

0

0

0

1

0

0

0

0

0

0

- Di

0

0

0

0

0

0

0

-~ Ei

0

TEi

0

0

0

0

W Tp .

- 1 Tpi

0

0

0

0

TAiTpi

~ Ai

Mi

- K.·k E ,

1

0 -1 TAi

(7 .48)

i = 1,2,3

- (Xdi-X~i)Id;

l'~oi

(Xqi-X;.:>lqi

,m

T' .

o

2H; [( EdJdi + E~J.;) + ( X~, -

R; -

SBi{ E ldi) TEj

l

o -,IfAiAi V-t

Xdi)hI.;]

CHAPTER 7. MULTIMACHINE SIMULATION

174 0 0 -1 l2.i.

Ci

0 0 0 1

M,

M,

0 0 0

0 0 0

0 0 0 0 0 0 fui

Ui

;~, ]

= [

i=1,2,3

(7.49 )

Vrefi

TAi

Substituting the numerical values, we obtain

A,

=

Az =

A3 =

-0.112

0

0

0

0.112

0

0

0

-3 .226

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

-0.1

0

0

0

0

0

0

0

-3 .185

0

3.185

0

0

0

0

0.514

-2.86

0

0

0

0

0

-18

100

- 5

-0.167

0

0

0

0.167

0

0

0

-1.87

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

-0 .2

0

0

0

0

0

0

0

- 3.185

0

3.185

0

0

0

0

0.514

-2.86

0

0

0

0

0

-18

100

-5

-0 .17

0

0

0

0.17

0

0

0

-1.67

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

-0.3

0

0

0

0

0

0

0

-3.185

0

3.1 85

0

0

0

0

0.514

-2.86

0

0

0

0

0

-18

100

-5

(7.50)

175

7.3. NETWORK EQUATIONS

R,

- 0.0095 Idl

- 0 .13Id2

o o

1.25Iq2

- 8( Ed1Idl

o

+ E~, Iq,)

-29.5 (Ed2Id2

+ E~2Iq2 )

- 0.29Idl I ql 0.0124e1.555Efd>

- 2.27Id2 I q2 0.0124 e1.555E fd2

o

o

- 100V,

- 100V2

- 0.1 9Id3 1. 7Iq3

o - 62 .6(Ed3Id3 + E~3Iq3 ) - 4. 3Id3 I q3 o.0124e1.555E fd3

(7. 51 )

o - 100V3

0 0 0 8 0 0 0

0 0 0 0 0 0 100

C, -

0 0 - 1 0.1 0 0 0

C3 =

0 0 0 0 0 0 - 1 0 0 0.3 62. 6 0 0 0 0 0 0 0 0 0 100

, C2 =

0 0 -1 0.2 0 0 0

0 0 0 29.5 0 0 0

0 0 0 0 0 0 100

(7.52)

The stator algebraic equations corresponding to (7.13) and (7.14) (assuming R" ::= 0) are Ed, - V, sin(a, - 8, ) + 0.0969Iq1 = 0

CHAPTER 7. MULTIMACHINE SIMULATION

176

E~l - Vl cost bl - Bl ) - 0.0608Idl = 0

Ed2 - V2 sin(b2

-

B2 ) + 0.1969Iq2 = 0

E~2 - V2 co;( b2

-

B2) - 0.1l98Id2 = 0

Ed3 - V3 sin( 03 - B3 ) + 0.2500 I q3 = 0 E;3 - V3 cos(b3

-

B3 ) - 0.1813IdJ = 0

(7.53)

The network equations are (with the notation Bij = Bi - Bj ) as follows. The constant power loads are treated as injected into the buses . Real Power Equations

IdlVlsin(b l - Bl

)

+ IqlV1COS (b1 -

Bl )

- 17 .36Vl V, sin B14 = 0

Id2V2sin(02 - B2)

+ Iq 2V2 cos(b2 -

B2)

- 16.00 V2 V7 sin B27 = 0

Id3 V3 sin(b3 - B3 )

+ Iq3V3 COS(b3 -

B3 )

- 17.06V3Vg sin 039 = 0 -17.36V4VlsinB41 - 3.31Vi

+ 1. 36V,V5 cos B45

+ 1. 942V,V6 cos 8.6 -1.25 + 1.36VsV, cos Os.

l1.6Vs V. sin BS4

- 1l.6V,V5 sinB45

10.51 V. V6 sin B46

=0

+ 1.19Vs V7 cos 857

- 5.97Vs V7 sin OS7 - 2.55Vl = 0 -0 .9

+ 1.94V6V 4 cos 064 - 16V7V 2 sin Bn

10.51 V6 V, sin B64 - 3.22V.,"

+ 1.28V6V9 cos 869 - 5.59V6V9 sin 069 = + 1. 19V7V5 cos 075 - 5.98V7V5 sin 875 - 2.8vl + 1.62V7V. cos 87• - 13.7V7V. sin 07• = 0

+ 1.62V.V7 cosB. 7 - 13.7V.V7 sinO. 7 + 1.16V.Vg cos 8.g - 17.065V9 V3 sin 893 + 1.28VgV6 cos B96 -

-1

2.77V.2 9.8Vs V g sin B. 9 = 0 5.59VgV6 sin B96

0

7.3. NETW ORK EQUATIONS

177

+ 1.15V9V 8 cos li98 - 9.78V9V8 sin 1i98

(7.54)

- 2.4Vi = 0

Reactive Power Equations

Id' V, cos ( 0, - 9,) - I., V, sin( 0, - 0, ) + 17.36V, V. cos li14 - 17 .36V,'

=0

Id'V, cos( 0, - 0,) - I q, V, sin ( 0, - 0,)

+ 16V,V7 cos li 27 -

l6V,' = 0

IdaVa.COs(oa -Oa ) - I qaV3sin(oa - Oa)

17 .36V4 V, cos Ii"

- 0.5 + 1.37V 5V 4 sin li54

- 0.3+ l.94V6 V.sinIi6•

16V7V, cos 1i72

:;.

+ 17 .07V3V 9 cos 039 -17.07V; = 0 - 39 .3V42 + l.36V.V5 sin 1i. 5 + 11.6V.Vs cosli. 5 + 1.94V.Vs sin li4S + 10 .52V. Vs cos 1i.6 = 0 + 1l.6V5V. cos lis. -

17.34V s2

+ 1.19V5V7 sin 1i57 + 5 .98Vs V7 cos li57 = 0 + 10.51V6 V4 cos1i64 - 15 .84V; + l.28V6 V9 sin1i69 + 5.59V6V9COS069 = 0 + 1. 19·V7V 5 sin 975 + 5.98V7V5 cos 075 - 35.45Vi + 1.62V7V8 sin 978 + I3 .67V7Vs cos 97S = 0

- 0.35+ 1.62Vs V7 sin 987 + 13 .67VsV7 cos OS7 - 23.3Vi + 1. 15VsV9 sin 98 9

=0 + l.28V9V6 sin996 + 5.59V9V6 cos 1i96 + 1. 16V9V8 sin li98 + 9.78V9V8 cos 998 -

+ 9.78V8V9 cos 989 l7 .065V9V3cos1i93

32 .15Vi = 0

(7.55) It is easy to solve (7.53) for Idi , Iqi (i = 1,2,3 ), substitute them in (7 .47) and (7.54)- (7.55 ), and obtain the equations :i: h(x, V,u) and 0 g,(x, V). T his is left as an exercise for the reader.

=

o

=

CHAPTER 7. MULTIMACHINE SIMULATION

178 Example 7.2

In this example, we put the DAE mo del wi th the network equations in the current-balance form. The differential equations (7.47) and the stator algebraic equations (7.53) are unchanged . The network equations are (7.56) where (ld'

+ j 1., )e3(6 , -,
(Id2

+ jI.,)e;(6,-1f/ 2)

(Id3

+ jI.3 )e;(5, - " / 2)

0+ jO

I o (ld_q,

X,

(- 1.25 + jO.5) /Vs (- 0.9 + jO.3)/17;

V) =

(7.57)

0+ jO (- 1+ jO .35)/'V; 0+ jO It is an easy exercise to substitute Id_q from (7 .53) in (7.57) to obtain t he network equations for the for m I,(x, V) = Y NV .

o

7. 4

Industry Mo del

We now present an equivalent but alternative formulation that is used widely in commer cial power system simulation packages [72J. The principal difference is in terms of suitably rearranging the equations from a programming point of view. This will be referred to as the industry model. From (7.15) we have

1d. ] [ 1qt.

= [Zd- q" .J-1 [ E' E~i - Vi sin(6, - Oi )] . -v'.cos(6-0)

AJso from (7.21):

q1'

1t

.

,=l , ... ,m

(7 .58)

7.4. INDUSTRY MODEL

- [Zd_q,.]-l [ i

179

;r: ]-

[Zd_q,;j-l [

~~~ ~:

- cos 0. ] sin Oi

[ ~~: ]

= 1,. "lm

(7.59)

Hence, Idil lqi are functions of either (E~i,E~ilOilVi,Oi) or (Edil E~i l Oi, VDi) VQ .). For ease in programming, the electric power output of machine i in (7.4) is defined as

Pei ~ Edi1di

+ E~ilqi + (X~i -

Xdi)Idi1qi.

= Pepi( Edt' E~il Oi, Vi, 81. ) or Peri (Edi

I

E~i) Oi, VDi , VQi )

(7.60)

if we substitute for Id. and Iq. from (7.58) or (7.59) . The terminal voltage Vi is

(7.61) We define two vectors

( 7.62) and

(763)

E is a subset of the state vector x and W is a vector of algebraic variables . With these definitions, we can express the differential equations (7.1)- (7.7) as

:i: = F(x , W, u)

(7 .64)

The stator algebraic equations (7 .58) or (7.59), together with (7 .60) and (7.61), are expressed as

W = G(E , V)

(7.65)

CHAPTER 7. MULTIMACHINE SIMULATION

180

The network equations from (7.46 ) are

(7.66) smce E is a subset of the vector:z:. Thus, assembling (7 .64), (7.65), and (7.66) results in

i; =

W =

12 (E, V)

F(:z:, W, u) G(E,V)

= YNV

(7 .67) (7.68) (7.69)

Equation (7.67) has the structure i;

= A (x)x

+ BW + Cu

(7.70)

The only dependency of A(:z:) on x comes through the saturation function in the exciter . A(x) , B , and C are matrices having a block structure with each block belonging to a machine. G contains the stator algebraic equations and intermediate equations for and Vi in terms of (E, V) . This formulation is easy to program. Alternatively, we can substitute (7.68) in (7 .67) to obtain

p.,

x = j,(x,V,u) II(x , V) = YNV

(7.71)

which is precisely equal to (7.46) . The DAE model with the network equations in the current-balance form is the preferred industry model, since the network-admittance matrix has to be refactored only if there is a network change. Otherwise , the initial factorization will remain. The DAE model with network equati ons in powerbalance form (discussed in Section 7.3.1) has the advantage that the Jacobian of the network equations contains the power flow Jacobian, a fact usefnl in small-signal analysis and voltage collapse studies , as discussed in Chapter 8 . Equations (7 .67)-(7 .69) can be interpreted as a block diagram, as shown in Figure 7.5.

7.4 . INDUSTRY MODEL

181

" v,

i,

-J= ~\,v

r--'Th OlherMa.:;

c-+-

Vj

1-- - - - - - - - - - - - - - - - - - - - - - - - - - - --

,, , ~D;', VQi

-+,

Vrd ,i

1

,

I

Exciter

v...,, V'll

- ',

! EN'

Mac)l1ne

Mt,ci ,ille.

Int.uf",,,

(~triC N)

,

IDi.IQI :

Idj , l qi

, ,,, ,,

J Pei Swmg

,

6,

~ ---- - ----- - ----- -------- ----- --

Figure 7.5: Block diagram conceptualization of (7.67)-(7.69) Example 7.3 We will express the 3-machine system in Example 7.1 in the form of (7.6 7)(7.69). Thus ( 7.67)- (7.68) become

[:: ]

[ 11

0

0

A,

0

0

A3

][ :: ] + [

~I

~

0

B2 0

] [ : : ] +Cu

B3

W3

(7.72) where -I

0

0

0

- I

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

- Di M,

0

0

0

0

0

0

0

0

0

0

K Fj

T]' .

TF j

0

0

0

0

-K';!'E;

K"

0 0 A-,

-

1

0

Ti,i

'f'/. '0'

Td oi

-(K

E' +S.,(E [dill TE i

TpiTAi

0 - 1

TAi

1

i = 1,2,3

TEi

0 - I TAi

(7 .73)

CHAPTER 7. MULTIMACIDNE SIMULATION

182

-(Xdi -Xdcl T~oi

0 (Xqi - X~;)

0

Bi=

T~o1

0

0

0

0

0

0

0

0

0

0

-1 Mi

0

0

0

0

0

0

0

0

0

0

0

0

_f,M

;=1,2,3

Ai

Idi

Wi=

lqi

; = 1, 2,3

Pei 11;

(7.74)

Cis Diag(Ci) and 11, = Diag(ui), where Ci and Ui are given by (7.49). The stator algebraic equations and the intermediate equations corresponding to (768) are

][~: ]

Wi -

;=1,2, 3

(7.75)

If Idi and lqi in Pei are expressed from the first two equations, then Wi = Ci(Ei, Vi). A similar substitution for iJ;, Iq , in (7.56) and (7.57) leads to

l,(x,v) = YNV

(7.76)

Substitution of numerical values in the above equa tions is trivial, and is left as an exercise for the r eader. o

Before we discuss the important issue of in.itial condition computation, we consider two simplifications that yield models far simpler than the two-axis model. They are also amenable to a network theoretic approach. Using these simplifications, we develop later (1) the structure-preserving flux-decay model, (2) the structure-preserving classical model, and (3) the internal node model using the classical machine model and constant impedance loads . Both (1) and (2) can have nonlinear load representations.

7.5. SIMPLIFICATION OF THE TWO-AXIS MODEL

7.5

183

Simplification of the Two-Axis Model

Two simplifications can be made in the two-axis model-one regarding Xd and X~ , and the ot her regarding the nature of loads. These can be done independently or together, resulting in simplified models. Simplification #1 (neglecting transient saliency in the synchronous machine) Transient saliency corresponds to different values of X di and X di , then the stator algebraic equation (7.8) is simplified as

X~i.

If X'· =

••

o = V;e j8 ; + (R,i + jXdi)(h + jI.;)ej(5; - ~/ 2) -(Edi

+ jE~Jej(5; -"/2)

(7.77)

The equivalent circuit for machine i is shown in (Figure 7.6).

+

Figure 7.6: Equivalent circuit with X di =

X~i

The machine-network transformation gives

+ j E~i)ej(5;-,,/2) (h + jIqi)e j (5;-,,/2)

(Edi

E Di '

+ J·E Qi '

ID i + jIQi

(7.78) (7.79)

Thus, with all the quantities in the synchronous reference frame, we have the equivalent circuit (Figure 7.7). Writing the KVL equation for Figure 7.7, we get (En; + jEQJ (R'i + jXdi)(IDi + jIQi) + V;e j8;, which is equivalent to (7. 77). In the differential equations, the expression for electric power P" given by (7.60) is simplified as

=

(7.80)

184

CHAPTER 7. MULTIMACHINE SIMULATION

+ Vi~i

Figure 7.7: Equivalent circuit (all quantities in network reference frame) It can be verlfied that the right-hand side of (7.80) is also equal to EbJDi + EQJQi by taking the complex conjugate of (7.79), multiplying it by (7.78), and equating the real part. This assumption of X di = X~i is often called "neglecting transient saliency." The advantage is that, in the equivalent circuit, all variables are in the network reference frame, and it is particularly useful in the current-balance form of Section 7.3.2. In (7 .46) , the algebraic equations that are equivalent to (7.42) and (7.43) can be directly expressed in terms of IDi, IQ i' and Vi using (Id; + j I q;)e i (5 i - ,,/ 2) = IDi + jIQ;.

Simplification #2 (constant impedance load in the transmission system) Here, the loads are assumed to be of the constant imp edance type, i.e., PLi(V;) = kp2'-V!

(7.81)

Y/

(7.82)

Q Li(V;) = k Q2

Then (7.83) Since (PL;

+ jQL;) = VJ~i'

where fL; is t he injected current, (7.84)

But V? = ViV: and conjugating (7.84), we obtain

(k P2i

'k)

- J Q2i

= hi Vi

-

= -Yii

(7.85)

7.5. SIMPLIFICATION OF THE TWO-AXIS MODEL + Iii

185

=[Wi + jlLQi Yii

Figure 7.8: Load admittance representation Because of the orientation of Vi and I Li, there is a negative sign in front of the load admittance y" in (7.85). From Figure 7.8, we can verify that

h,(V;) ;?QLi(V,)

,

=h, = -Yii V;

(7.86)

Hence

This is consistent with the fact that PL, and QL, are injected loads. Yii is the complex admittance due to the loads. Since (Id, + jIq ,)e j (6; - "/2) = In; + JIQ;, for ; = 1, ... , m and using (7.86), the network equations (7.42) and (7.43) become n

(In,

+ jIQ,)

LVkej8kYik ej",,"

+ Y,i V;e j8 ;

;= 1, ... ,m (7.87)

k=l n

L Vk

o

ejo

;. Ytke

jocilo

+ Yii V i ej8i

i

= m + 1, .. . ,n

k=l

n

IDi

+ J"IQi = L Vk ej8k Yi"e joci ;.

i

= I, . . . , m

k= l n

o

L Vkej8"'1'ikej cxik k=l

i =

m+ 1, ... ,n

(7 .88)

186

CHAPTER 7. MULTIMACHINE SIMULATION

where

Equation (7.88) can be written as

11

v,

[Y']

(7. 89)

o wher e I" ... ,lm are the complex injected generator currents at the generator buses. Let the modified Y bus denoted as y' be partitioned as m

[V'] =

m

n - m

n-m

(Yl

(7.90)

Y3

Since there are no injections at buses m to get

+ I, ... , n, we can eliminate

them

(7.91)

-

-

-

-

-1 -

where Yred = (Y 1 - Y 2Y. Y 3 )· Note that we can make either or both of the above two simplifications . If we make only the constant impedance approximation, then we can obtain a passive reduced network, as in Figure 7.9, but the source will be a dependent one, as in Figure 7.1. On the other hand, if we make the assumption X di = X~i only, then we obtain the source representation, as in Figure 7.9, but the network equations will remain in (7.42) and (7.43). If we make both assumptions, then we obtain a passive reduced network, as in Figure 7.9 , and the voltage E: = Ebi + j E Qi ·

7.5. SIMPLIFICATION OF THE TWO·AXIS MODEL

187

R,

jX,!

2

R,

E;

Ei

Y",d

jXdzu

Rm

'"

Figure 7.9: Network reduced at generator nodes and X/u =

X~i

Example 7.4 Based on the load·flow results in Table 7.1, convert the loads as constant admittances, and then obtain a Y red representation (all in pu). Solution Note that the load specined in Table 7.1 is the negative of the injected load. Hence, at buses 5, 6, and 8, Yii is given by Y55

=

1.25-jO.5 (0 .9956)'

'0 "04 = 126 . - J.o Y88 -

,

-Y66 --

0.9-jO .3 - 0 877 ( 1.013)' - . -

I - jO.35 - 0 969 (1.016)' - . -

J'0292 .

'0339

J .

These elements are now added to the Y 55, Y 66, and Y 88 elements of the Y bus matrix in Table 7.2, resulting in y' as

, • 2

• -y , = , 7

, ,

,•

-il7 ·00G

• ,,

0

( 3 .30 14

0 0 ( - 1.3 562

0 0 0 ( - 1.U:22

-j3IU O S~)

t i tUlO")

t il0.li l 07 )

3

0

0 0 jl1 .361

-j 16

0 0

0 0

0

0

0

0

0

0

till .eot. )

(, .au - ; I1.au)

( -1.9"2

0

{ -1.366

ti t a .fill ) 0

; 16

,

jl1 . ~ I51.

2

- jl7 .SSt

0

0

0

0

0

; 16 0 0

0

i 17 .066

0

0 0 ; 17 .06 b 0 0

0

(4 . 102

0

0

0

0

0

0

( - 1 .282

+;6 .688)

( _ 1 .282

+ ;6.686 )

-;111.133) ( - 1 ,188

•

0 0 0

( -1. 188 +;6 .916)

( :2 .a 0 5 - ;36 .4 64)

( _ 1.6 1.,

+ ;13 .1198) 0

•0

0

+ ;6 .976) 0

7

0

0

(-1. (11 7

0

fila .siS) (3.Hl -j23 .I!H:2 ) { - 1.u.r;

+jrIl.7(4)

( - 1.Ui6

+i ll.184) (:l .t37 -;3'2 ..1156 )

188

CHAPTER 7. MULTIMACillNE SIMULATION

Yred is obtained by eliminating the buses 4 to 9 from y'. The resulting Y red is given by 1.105 - j4.695 0.096 + j2.253 0.004 + j2.275 ] Y r ed = 0.096 + j2.257 0.735 - j5.114 0.123 + j2 .826 [ 0.004 + j2.275 0.123 + j2.826 0.721 - j5.023 With this simplification, we have I = Y red V.

o

7.6

Initial Conditions (Full Model)

The initial conditions of the state variables for t he model in Section 7.1 are computed by systemat ically solving the lo a d·flow equations of the network first, and then computing the other algebraic and state variables. The loadflow equations are part of t he network equations, as shown below. Load-Flow Formulation We now revert to the formulation of the network equ ations that were written as real and r eactive power-balance equation s at the nodes, i. e., (7.32)- (7.35). We reproduce them below by writing real power equations first, followed by the reactive power equations.

n

-L

VYkYik cos (0; - Ok- OC;k)

=0

= 1, .. . , m(7. 92 )

i

k=] n

PL;(Vi) -

L

ViVkYik cos (0; - Ok- OC;k)

=0

i = m + 1, ... ,n (7. 93 )

k= l

n

- L V;VkYik sin (0; -

Ok- O(;k ) = 0

i

= 1, . .. , m

k=l

(7.94 )

7.6. INITIAL CONDITIONS (FULL MODEL)

189

n

QLi(Vi) - ~ ViVkYik sin (8i

-

8k - CXik) = 0

i = m

+ 1, .. . , n

(7.95)

k=l

It follows from the dynamic circuit (Figure 7.10) that

Fai

+ jQai

= VJ~i = ViejO, (Idi - jIq;)C j (5' - "/2) = Vi(cos8; +jsin8;)(Id;- jlq;)(sino;+jcoso;) (7.96)

N ow we equate the real and imaginary parts along with the use of trigonometric identities . It can be shown from (7 .96) that Fa; =

h Vi sin( 0; - 8;)

+ Iq;Vi cos( 0; -

8;)

(7.97)

and (7.98)

(7.99) the injected generator current at the generator bus in the synchronous reference frame. Figure 7.10 explains the equations at the generator bus.

IS

\ +

[Edi + (Xq! + jE

-x;J Iqi

'.J ,,1("-"")

+

Pli (Vi) + j QLi. (Vj)

q'

Figure 7.10: Synchronous machine dynamic circuit We further define net injected power at a bus as

(7.100)

CI-IAPTER 7. MULTIMACI-IINE SIMULATION

190

Thus, t h e power-balance equations at the buses 1, ... , n are n

Pi (Oi,Idi,Iqi, Vi , Oi) =

I: V; VkYik cos (Oi -

Ok - ()(ik)

k=1

i == 1, ..

(7.101 )

"m

n

PLi(V;) =

I: V;VkYik cos (Oi -

Bk -

OCik )

k=1

i == m+ l, .. . , n

(7 .102)

n

Qi( Oi ,Idi ,Iqi, V;,e.) =

I: V;VkY,k sin (Oi -

Bk - ()(,k)

k=l

i

=

l, ... ,m

(7. 103)

n

Q£i(V;)

= I: V;VkYik sin (Bi -

Bk - ()(ik )

k= ]

i = m

+ 1, ..

'J

(7. 104)

n

Standard Load Flow Load flow has b een the traditional mechanism for computing a proposed steady-state operating point . We now define standard load flow as follows, using (7 .101)-( 7.1 04). Loads are of the constant p ower t yp e. 1. Specify bus voltage magnit udes numbered 1 to m.

2. Specify bus voltage angle at bus number 1 (slack bus) . 3. Specify net injected real power Pi a t buses numbered 2 to m . 4. Specify load powers PL, and QLi at buses numbered m

+ 1 t o n.

The following equations result from (7.101), (7.102), and (7.104) chosen accorcling to criteria (3) and (4). These are known as the load-flow equations. Thus, we have

o=

- Pi +

n

I: V;VkYik cos (B, k= 1

i = 2, .. .,Tn

Bk- OCik)

(PV buses)

(7.105)

7.6. INITIAL CONDITIONS (FULL MODEL) n

o = - PLi + ~ V;VkYik cos (Oi - Ok- <Xik)

191 i=m + l , ___.n

(PQ buses)

(7 .106)

k=1 n

o=

-QLi

+ ~ V;VkYik sin

i=m + l, ... ,n

(Oi - Ok- <Xi.)

(PQ buses)

(7.107)

k=1

where Pi(i = 2, ... ,m), V;(i = 1, .. . ,m), PLi(i = m+ 1, ... ,n), QLi(i = m + 1, ... , n), and 0, are specified numbers. The standard load flow solves (7 .105)-(7.107) for 02, . .. ,Om V=+l,. '" Vn . After the load-flow solution, we compute the net injected powers at the slack bus and the net injected reactive power at the generator buses as

P,

+ jQ,

n

-

L

V, VkYlkej(61-6, - ~,,)

(7.108)

k=l n

Qi = LV,vkYiksin(Oi - Ok- <Xik)

i = 2, ... , m

(7.109)

k=l

The generator powers are given by PG1 = P , - PL1 and QGi = Q. - Q Li (i = 1, . . . ,m). This standard load flow has many variations, including the addition of other devices such as tap-changing-under-Ioad (TCUL) transformers, switching VAR sources, HVDC converters, and nonlinear load representation . It can also include inequality constraints on quantities such as QG. at the generators, and can also have more than one slack bus. For details, refer to [15, 16, 18]. One important point about load flow should be emphasized. Load flow is normally used to evaluate operation at a specific load level (specified by a given set of powers). For a specified load and generation schedule, the solution is independent of the actual load model. That is, it is certainly possible to evaluate the voltage at a constant impedance load for a specific case where that impedance load consumes a specific amount of power. Thus, the use of ((constant power" in load-flow analysis does not require or even imply that the load is truly a constant power device. It merely gives the voltage at the buses when the loads (any type) consume a specific amount of power. The load characteristic is important when the analyst wants to study the system in response to a change, such as contingency analysis o.r dynamic analysis. For these purposes, standar d load flow is computed on the basis of constant PQ loads and usually provides the "initial conditions" for t he dynamic system.

192

CHAPTER 7. MULTIMACHINE SIMULATION

Initial Conditions for Dynamic Analysis We use the model in the power-balance form from (7.36)- (7.38).

fo(x,Id-q, V, u)

(7 .110)

h(x , V)

(7.1l1) (7.112)

It is necessary to compute the initial values of all the dynamic states and the fixed inputs TMi and Vrefi (i = 1, ... , m). In power system dynamic analysis, t he fixed inputs and initial conditions are normally found from a base case load-flow solution. That is , the values of Vrefi are computed such that the m generator voltages are as specified in the load flow . The values of TMi are computed such that the m generator real power outputs PGi are as specified and computed in the load flow for rated speed w,. To see how this is done, we assume that a load-flow solution (as defined in previous section) has been found, i.e., solution of (7.105)-(7.107). The first step in computing the initial conditions is normally the calculation of generator currents from (7.96), as IGi = IGie],; (PGi - jQGi)/V: .

=

Step 1 Since PGi = Pi - PLi and QGi = Qi - QLi:

(7.113) This current is in the network reference frame and is equal to (Idi + jlq i)e i (h; - " /2). In steady state, aU the derivatives are zero in the differential equations (7 .1)- (7.7) . The first step is to calculate t he rotor angles 6i at all the machines. We use the complex stator algebraic equation (7.8) and the algebraic equation obtained from (7.2) b y setting Edi = o. From t he latter, we obtain

E~i = (Xqi - X~.JJgi

i = 1, .. . J m

(7.114)

7.6. INITIAL CONDITIONS (FULL MODEL)

193

Substitution of (7.114) in (7.8) results in

Vie ie,

+ (R,. + jXd,) (1di + j1.. )ei (o' - "/2) [( Xq; - X;.)1q• + (X;, - Xdi)1q• + j E~ilei(6' - "/ 2) =

0 (7.115)

Vie ie ,

+ R,.(1d. + j1.. )ei (6'-"/2) + j Xdi (lJi + j I q;)ei (/i'-1C/2) - [(Xqi - Xd.) 1q;+ jE~iJej(·' -"/2) = 0

i = 1, ... , m (7.116)

i.e .,

Vie ie,

+ R,.( Idi + j1qi)e i (/i,-,, / 2) + jXdJdiei(6, - "/2) - X qi1qiei(/i, - " / 2) -

]·E'q'·eilli,- " / 2) = 0 ,

•,, = 1 , . . . , m

(7.117)

Adding and subtracting jXqJdiei(6'- "/2) from the left-hand side of (7.117) results in

Vie ie ,

+ -

+ jXqi)(Id. + jlqi)e j (6'-"/2l j [(Xqi - Xdi)1di + E~iJ ei(6'-"/2) = (R"

0

i

= 1, ... ,m

(7.118)

Replace (1di + j 1.. )ei (6,-,,/ 2) by 1Giei~', whlch is already calculated in (7.113):

Vieie,

+ (R'i + jXqi)1Gieh = ((X .. - Xd;)1di + E~i)eili'

i = 1, ... , m

(7.119)

The right-hand side of (7.119) is a voltage behlnd the impedance ( R'i+ jX. i ) and has an angle ai . The voltage has a magnitude (E~+ ( Xqi - XdJ1di) and an angle ai = angle of (Vie ie , + (R,i + jXqi)1Gieh,). The complex number representation fOT computing ai from (7.119) is shown in Figure 7.11. Thls representation is generally known as "phasor diagram" in the literature and "locating the q axis" of the machine.

194

CHAPTER 7. MULTIMACmNE SIMULATION

Q

di [Oi

jX .i f Gi

D d

Figure 7.11: Representation of stator algebraic equations in steady state Step 2

Oi is computed as O. = angle on (V;e 19,

+ (R,; + jX.;)Icie h ,).

Step 3 Compute I d;, 1.;, Vd; , Vq; for the machines as

Id.

+ jIq;

V.d l

+ J' v. . ~ ql

= Ic;e1b, - 6,+,,/2) --

, (9,-o,+,,/2) v.·e 1

i= I, . .. ,m

i = 1, .. 'Im

(7.120) (7.121)

Step 4 Compute Ed; from (7.27) : (7.122) which from (7.114) is also equal to (Xq; the calculations.

X~;)Iq;.

This serves as a check on

Step 5 Compute E~i from (7.27) E~; = Vq;

+ R,;lq; + XdJd;

i = 1, ... , m

(7 123)

7.6. INITIAL CONDITIONS (FULL MODEL)

195

Step 6 Compute Eldi from (7.1) (aiter setting the derivative equal to zero): i = l, ... ,m

(7.124)

Step 7 With the field voltage Eldi known, the other variables Rli, VR. and Vrefi can be found from (7.5)- (7 .7) (aiter setting the derivatives equal to zero): i = 1, ... , m

i= l , ... , m

(7.125) (7.126)

i = l, ... ,m

(7.127)

Note that if the machine saturation is included, the calculation for E~. and E~i may be iterative. The mechanical states Wi and TMi are found from (7 .3) and (7 .4) (after setting the derivatives equal to zero):

(7 .128)

i= I , . . "m

TMi

= EdJdi + E~Jqi + (X~i -

Xd;)hIqi

i

= 1, ... , m

(7 .129)

This completes the computation of all dynamic-st ate initial condi tions and fixed inputs. Thus, we have computed x(o)'-1!(o) , and u from the load-flow data. For a given disturbance, the inputs remain fixed throughout the simulation . If the disturbance occurs due to a fault or a network change, the algebraic states must change instantaneously. The dynamic states cannot change instantaneously. Thus, it will be necessary to solve all the algebraic equations inclusive of the stator equations with the dynamic states specified at their values just prior to the disturbance as initial conditions to determine the new initial values of the algebraic states. From the above description, it is clear that once a standard load-flow solution is found , the remaining dynamic states and inputs can be found in a systematic manner. The machine rotor angles O. can always be found, provided that: i = 1" . " m

(7.130)

CHAPTER 7. MULTIMACIDNE SIMULATION

196

IT control limits are enforced, a solution satisfying these limits may not exist . In this case, the state that is limited would have to be fixed at its limiting value and a corresponding new steady-state solution would have to be found. This would require a new load flow by specifying either different values of generator voltages , different generator real powers , or possibly specifying generator-reactive power injections , thus allowing generator voltage to be a part of the load-ftow solution. In fact , the use of reactive power limits in load flow can usually be traced to an attempt to consider excit ation system limits or generator capability limits . Example 7.5 For Example 7.1, compute the iuitial conditions . The solved load-flow data are given in Table 7.1 (all in pu). Machine 1 Step 1

PGI - jQGI

0.716 - jO.27

V-I

1.04 LOo

= o. 736 L -

20 .66°

Step 2

+ (R" + jXqdIGle;~') Angle of (( 1.04LO° + jO.0969) (0.736 L - 20.66°))

51 (0) = Angle of (V1e j8 ,

=

= 3.58° Step 3

Id1 + jIql = IGle h 'e- ;(o,-"/ 2)

= (0.736 L - 20 .66°)(1 L86.42°)

= 0.302 + jO.671 Idl = 0.302

1,1 = 0.671

7.6. INITIAL CONDITIONS (FULL MODEL)

197

= ( 1.04LOO)( l L86.42°) = 0.065

+ j1.038

VdI = 0.065 , V qI = 1.038

Step 4 E~,

= (XqI - X~,)IqI = (0.0969 - 0.0969)(0.67l) = 0

It can be verified that this is also equal to VdI

+ R"Idl

- X~, I qI .

Step 5

E~, = Vql

+ R"Iql + X~, Id'

= 1.038 + 0 + (0.0608)(0302) = 1.056 Step 6 E JdI = E~l

+ (XdI -

= 1.056

X~l)Idl

+ (0.146 -

00608)(0.302)

= 1.082 Step 7 VRI = (KEI RJl

+ 0.003ge1.55SEJdl)EJdI =

= -TKFl Efdl FI

Vrefl = V,

V

Rl + -KAI

(1.021)(1.082) = 1.105

= (0.063) -0 - 1.082 = 0.195 .35

= 1.04

1.105

+ -20

= 1.095

The mechanical input TMI i5 computed as follows : TUI

= E~, Idl + E~,Iql + (X~, - X~l)IdlIql = (0)(0302) + (1.056)(0.671) + (0.0969 = 0.7l6

0.0608)(0302)(0.67l)

CHAPTER 7. MULT1MACHINE SIMULATION

198

Similarly, for other machines, we can compute the state and the algebraic variables as

oz( 0) Id2 1q2 V d2 V q2

E~2 E~2 Efd2 Rf2 VRZ V"f2 TM2

= 61.1" = 1.29 = 0.931 = 0.805

Iq3

= 54.2" = 0.562 = 0.619

Vd3

= 0.779

03(0) Id3

= 0.634 = 0.622 = 0.788 = 1.789

E~3

= 0.666 = 0.624

E~3

= 0.768

Efd3

= 1.403

Rf3

= 0.252

Vq3

= 0.322 = 1.902

VR3

= 1.12 = 1.63

V re j3 TM3

= 1.453 = 1.09 = 0.85

0

Angle Reference, Infinite Bus , and COl Reference As explained in Section 6.10, by taking one of the angles as a reference, the order of the dynamic system can be reduced from 7m to 7m-1. Furthermore, if the inertia constant on this reference angle machine is infinity, the order of the system can be reduced to 7m-2. This is also possible if the machines have zero or uILiform damping. Finally, we can also use center-of-angle formulation instead of relative rotor angle formulation. T he center-of-angle formulation is discussed in Chapter 9.

7.1

Numerical Solution: Power-Balance Form

The number of algorithms that have been proposed for the numerical solution of t he DAB syst em of equations is very large. There are basically two approaches used in p ower system simulation packages. 1. Simultaneous-implicit (51) method 2. Partitioned-explicit (PE) method

7.7. NUMERICAL SOLUTION: POWER-BALANCE FORM

199

The S1is numerically more stable than the PE method. It is also t he method used in the EPRl 1208 stability program known as the ETMSP (Extended Transient Midterm Stability Program) program [70J. We illustrate the S1 method on the WSCC system with a two-axis model for the machine and with the exciter on all the three machines .

7.7.1

S1 Method

We illustrate with t he differential-algebraic model in power-balance form.

x= [d- q

!o(X,[d-q,

(7.131)

V,u)

= h(x, V)

0=

(7 .132)

(7133)

90(X,[d-q, V)

All the initial conditions at t = 0 have been computed. In the S1 method, the differential equations in (7.131) are algebraized using either implicit Euler's method or a trapezoidal integration method . These resulting algebraic equations are then solved simultaneously with the remaining algebraic equations (7.132)- (7.133) using Newton's method at each time step. Review of Newton's Method Let !( x) = 0 be the set of nonlinear algebraic equations, i.e.,

ft(x" ... ,xn ) = 0 fz(Xl, . .. , x n

)

= 0

(7 .134)

(0) E xpan d the equations . 't'al A ssurne an lill 1 guess X(0) l)"" X n . series and retain only the linear term.

III

a Taylor

(7.135) Solving for x results in an improved estimate for x:

(7 .136)

CHAPTER 7. MULTIMACHINE SIMULATION

200

In general , (7.137) where k is the iteration count and [l] ~ ~ is called the Jacobian. In an expanded version:

[l ] =

~ ""

'ilL a.,

B/,

a/, a""

(7.138)

&1

Define (7.139) Equation (7.137) can be recast as follows: [l(k)].6",(k)

= - f(",(k »),

k

= 0,1,2,. . .

(7.140 )

Equation (7. 140) is a linear one and has to be solved for .6",(k). Then from (7.1 39 ) ",(Hi)

=

",(k)

+ .6 ",(k) ,

k = 0,1,2,...

(7. 141)

With an initial value of ",(0), steps corresponding to (7 .140) and (7.141) are repeated and at the end of each iteration compute mre Ih(",(k+1») I. IT this is < f where f is th e specified tolerance, the Newton iterates have converged. Numerical Solution Using SI Method Let the subscripts nand n + 1 denote the time instants tn and t nH , respectively. Then, integrating the differential equations in (7 .131) from tn to tn+l using tbe trapezoidal rule and solving the resulting algebraic equations with the remaining algebraic equations at tn+! , obtain ( 7.142)

7.7. NUMERICAL SOLUTION: POWER-BALANCE FORM

201

i.e ., tlt

xn+l =

Xn

-

+ T[ !o(Xn+b1d- q,n+l) V n +1 ) U n +l)

+!D( Xn , Id _ q,n, V n , un)]

(7.143)

o=

Id-q ,n+1 - h( x n+1, V n+tl

(7.144)

0=

9o(Xn +l,Id- q,n + l , V n + 1 )

(7.145)

where tlt = tn+l - tn is the integration time step. Rearranging (7 .143)(7 .145),

(7.146) t. Id-q ,n+l - h ( x n+1 , V n +1) F 2(x n+1,!d-q,n+1, V n +,)

=

=0

(7.147)

t. gD(X n +l,!d-q,n+l, V n +,) = F3 (x n +1,!d-q,n+1, V n +,) = 0

(7.148)

At each time step, (7 .146)-(7.148) are solved by Newton's method . The Newton iterates are 7m

2m

2n

7m ( 1, 2m J. 2n Jr

J2 Js J.

JJ6

(k)

3

[

)

J9

(k) x n +1

(k+l)

x n+ 1 I(k+1)

d-q,n + l

V(k+1) n+l

tlXn+1 tl1d-q,n+1 L'. V n+l

=-

+ L'.x (k) n +1

-

I(k) d-q ,n+1

-

V(k)

71+1

f) [~: f)

(7.149)

(7.150) (k )

+ L'.Id _ q,n+1

k) + L'. vin+1

(7.151) (7.152)

X!:'2,

is the converged where k is the iteration number at time step t n +1' value Xn at the previous time step. The iterations are continued until the norm of the mismatch vector [F F 2 , F3 ]' is close to zero. This completes " the computation at time step tn+1'

CHAPTER 7. MULTIMACHINE SIMULATION

202

If there is a change i.n reference input Vrefi' or TMi, the DAE (differential algebraic equation) model can be integrated with known values of aU variables. But if there is a disturbance in the network , a different procedure has to be adopted, as explained below. Dist urbance Simulation The typical disturbance corresponds to a network disturbance , such as a fault where the parameters in the algebraic equations change at t = O. The algebraic variables can change instantaneously, whereas the state variables do not. Hence, at t = 0+, witb the network disturbance reflected in the network equations, we solve the set of algebraic equations for Id _q(O+ ), V(O+) as

(7153) (7.154) where the superscript f indicates that the algebraic equations correspond to the fault ed state. With the value of Id_.(O+), V(O+) so obtained, the trapezoidal method is then applied. Note that the initial guess for the vector [x' IJ_q V'l' at time instant tntI is the converged value at the previous time insta.nt, i.e., Xntl

Jd-q ,n+l [

1

(0)

Vntl

=

[

Xn

Id-q ,n

I

,n = 0,1,2, . ..

(7 .155)

Vn

If there is a change in generation or load, this is simply taken care of by changing PGi and QGi or PLi and QLi in (7.153) and (7.154), and computing Id_q and V at t = 0+. If there is a short circuit at bus i, then set Vi - 0 and delete the P and Q equations at bus i from 90(X, Id-q , V) to obtain

7.7 .2

PE method

1. Incorporate the system disturbance and solve for V(O+), Id_q(O+) as in (7.153) and (7.154). 2. Using the values of Id_.(O +), V(O+) integrate the differential equations :i:

= fo(x,Id-q, V)

, x(O)

=

",0

(7 .156)

7.8. NUMERICAL SOLUTION: CURRENT-BALANCE FORM

203

to obtain x(1). 3. Go to Step 1 and solve for Id _q(1), V(1) again from the algebrai c equations .

(7 .157) (7.158) 4. Integrate the differential equations to obtain x(2), and again solve the algebraic equation to obtain Id_q(2), V(2). 5. This procedure is repeated until t = Tdesired or there is a change in the configuration. In the second case, a similiar procedure is followed. The PE scheme , although conceptually simple, has numerical convergence problems such as interface errors , etc. [73].

7.8

Numerical Solution: Current-Balance Form

In this section, we explain the industrial approach to implement the SI method forming the structured approach to the problem. It is the basis of the well-known ETMSP program of EPRI [70]. The current- balance approach is favored, since the bus admittance matrix is easily formed and factorized. The DAE model from (7.36), (7 .44), and (7.45) is :i: = !o(X,!d-q, V, u)

Id- q = h(x, V) Io(Id-q,x,v ) = YNV

(7.159) (7.160) (7.161)

The use of the SI method to solve these equations yields the set of algebraic equations: Fl(Xn+1 , Id- q,n+ l,Vn+lIUn+ l,Xn,Id-q ,n,Vn , 1tn) = 0

(7.162)

F 2(x n+l,!d-q,n+1> V n+,) = 0

(7.163)

(7.164)

CHAPTER 7. MULTIMACHINE SIMULATION

204

Instead of treating x and Id_q as separate vectors, we form a new vector X = [X, .. . X~Jt . Associate Id-q,; with the respective x; so that X; [xl I,L q;]' . Equations (7.162) and (7.163) are replaced by

=

Also, we replace (7.164) by its rectangular equivalent oj·(X n +! , V.:'+l) y& V.:'+l, where e stands for expanded form, i.e.,

r -

[1m IQI ... IDn IQnJ'

V' - [VDI VQ' ... VDn VQnJ' and Y& consists of 2 X 2 square matrices with real numbers . With this, the application of Newton 's method yields the following equations.

6.X, B GV1

AGGJ

AGGm CVGl

RGJ

CVGm

6Xm 6.V{

BGvm

Y' N

-

RGm

Rh

+ Y'L 6.V·n

RVn (7. 165)

The various elements of the Jacobian are defined as

aI'

i = 1, 2, . .. , m

and

•_ YL - -

aI' aV.:'+l

For ease of understanding, Vi' can be considered as a vector (VDi, VQ;)', and Ii (IDi,lQ;)t. Y& becomes a 2n X 2n matrix of real elements. The right-hand sides of (7.165) are the residuals . In Newton's method for solving f ix) = 0, the residuals at any iteration k are - f(x(k»). YI is computed as follows (the suffix n + 1 is dropped for ease of notation) :

=

Y'L ~ _ [aI' av. ]

(7.166)

7.B. NUMERICAL SOLUTION: CURRENT-BALANCE FORM

Yl consists of 2

X

205

2 blocks of the type

-

[

aVQi ~ '!..!.ru.] 81

(7.167)

~

91

aVVi

aVQi

The solution method to solve the linear equation (7.165) is as follows. We recognize that the nonzero columns in BGVi correspond to t. V(, and that those in CGVi correspond to t.Xi. Thus, from (7.165) i = l, ... ,m

(7.168)

From (7.165),

[YN + Yll t.V'

+ L:)CVGit.Xi) =

[Rv-]

(7.169)

i=l

Substituting t.Xi from (7.168),

lye

+ Yl- E(CVGiAG~iBGV;)]

e [t.v ] =

[R~]

(7.170)

1 . h R V,' -- ReV - "m were L..Ji=l CVG1.·AGGi R G1.·

Thus, the algorithm first solves for t.V e in (7.170) and then solves (7.169) iteratively to convergence for t!1Xi (i = 1, . . . ,m). The same computations are then repeated at the next time instant and so on. Some Practical Details [70J The Jacobian in (7.165) is expensive to compute at each iteration. Let J;:), be the Jacobian evaluated at t = tn+1 and k represent the iteration count at that time instant. Solving (7.169) and (7.170) results in

1 X(k+ )] [ V e(k+1)

[X(k)] Ve(k)

[t.X(k)]

+ t. Ve(k)

k = 0,1,2, ...

(7.171)

The very dishonest Newton method (VDHN) holds the Jacobian in (7.165) fixed for a period of time. Thus (0) ] [ [ I n +!

t.X(k) ] _ [ (k)] - Rl

t. V,(k)

(7.172)

206

CHAPTER 7. MULTIMACHINE SI MULATION

where the time instant l ? n + l. This means that the initial Jacobian at t = t n +1 is held constant for some time steps after t,,+I. This reduces the overall cost of comp utation . The choice of when to r eevaluate the Jacobi an is based on exper ience. The Jacobian must be reevaluated at any major system change . Between time st ep s, it is reevaluated if the pr evious time-step iteration was considered too slow (took t hree or more iterations). Within a time step , a m aximum of five iterations are taken using the same Jacobian . Prediction [70J Whenever a Jacobian is evaluated at of taking the converged values of the ear prediction for generator variables voltages. Thus, at time instant tnH, of the Jacobian would be

t he beginning of a time step, instead previous ti me step , we can use a linand geometric prediction for network the initial estimate in the evaluation

(7. 173) where X n , X n _ 1 are the previous converged values at the previous t ime steps. For the network variables, each voltage initial guess is v (o) _ n+l -

7.9

(Vn )2 V.

(7 .174)

n- I

Reduced-Order Multimachine Models

While many types of reduced-order multimachine models a re possible, we discuss three specific Dues.

1. Flux-decay model with a fast exciter . T he network st ructure is preserved. 2. Structure-preserving model with a classical m achine m odel. 3. Classical m odel with network nodes eliminated.

7.9. REDUCED-ORDER MULTIMACHINE MODELS

7.9.1

207

Flux-decay model

This model is widely used in eigenvalue analysis and power-system stabilizer design. If the damper-winding constants are very small, then we can set them to zero (i.e., there is an integral manifold for these states, as discussed in Section 6.5) , and we obtain from (7.2):

0=

- Ed, + (Xq, -

X~,)Iq,

i

= 1, .'"

(7.175)

m

We eliminate Ed, from (7.4) and (7.8) using (7 .175). The synchronous machine dynamic circuit is modified as shown in Figure 7.12. It is also common, while using the flux-decay model, to have a simplified exciter with one gain and one time constant, as shown in Figure 7.13. The complete set of differential· algebraic equations (7 .1)-(7.10) become (assuming no governor and no damping): jXdi

+ [(Xqi - X di') lqi

" ] + JEqi

3 (0r' 1Cl2) ~

+

( Vd'r +J'Vqr.) ,j(O;- 1Cl2) -- v I ,Jll1

Figure 7.12: Synchronous machine flux-decay model dynamic circuit

Vi

Figure 7.13: Static e:cciter (one gain- one time constant) Generator Equations I

dE~i

i = l)".,m

Tdo,Tt d8, dt

=Wi -Ws

i=l, . .. ,m

(7.176) (7.177)

CHAPTER 7. MULTIMACHINE SIMULATION

208 2Hi dWi

----

w, dt

(7.178) T

dEfdi Ai

dt

-Efdi

+ KA;(Vref; -

Vi) i

= 1, . .. , m

(7.179)

Stator Equations Assuming R,; = 0, and substituting for E~; from (7.175), we obtain the stator equations (7.13) and (7.14) in polar form as Vi sin( 0; - 0;) - Xq;Iq; Vi cost 0; - 0;)

+ X~;h -

E~;

o o

i= l, ... ,m

(7.180)

i= 1, ... ,m

(7.181)

Network Equations The network equations can be written in the power-balance form as in Section 7.3.1, or in the current-balance form as in Section 7.3.2. Initial Conditions The steps to compute the initial conditions of the multimachine flux-decay model DAE system are given below. Step 1 From the load flow, computeiGieh, as in (7.113) as (PGi -jQGi)/V;e- j8 , Step 2 Compute 0; as angle of [V;e j8, Step 3 Compute Wi

= w, from

+ jXq;IG;e h ,] .

(7.177).

Step 5 Compute E~i from (7.181) as E~i = ViCOS(Oi - Oi) Step 6 Compute E fdi from (7.176) as E fdi = E~i Step 7 Compute Vref; from ("(.179) as Vref ,; =

+ (Xdi -

f/:: +

Vi,

+ X~ih. X~i)Idi'

209

7.9. REDUCED-ORDER MULTIMACHINE MODELS

Step 8 Compute TMi from (7.178) as TMi = E~Jqi

7 .9.2

+ (Xqi -

X~i)hIqi '

Structure-preserving classical model

To obtain the classical model, we set Tdoi = 00 and Xqi = X di in (7 .176) and (7.178), respectively. This results in E~i being a constant equal to the initial value E~i.. Ignoring (7.176) and (7 .179) , the resulting differential equations are dOi dt 2Hi dwi

i = 1, . . . ,m

---d = t

W,

TMi -

E~i.Iqi

i = 1, .. . ,m

(7.182) (7.183)

These are known as the swing equations in the literature. The stator algebraic equations (7.180) and (7 .181) are added after multiplying the former by -j. We replace Xqi by X di and E~i by E~i.. Thus

+ jIqi) + Vi(COS(Oi - Bd Xdi(Idi + jIqi) + Vie- j (O;-8;j

E~i = Xdi(Idi =

jsin(oi - Bill (7.184)

Equation (7.184) can be rearranged as

+ jIqi)ej(o; -~/2) + Vie j8; jXdi(IDi + jIQi) + Viej8;

E~iejo; = jX~;(Idi

=

(7.185)

This represents a voltage Ei = E~iejo~ of constant magnitude behind a transient reactance X~i (Figure 7.14). Henceforth, we will denote E~i = Ei.

+

g? LOi 'l'

+

= EjL.oj

Figure 7.14: Constant voltage behind transient reactance This forms the basis of the structure-preserving model which, is discussed next.

210

CHAPTER 7. MULTIMACHINE SIMULATION

Structure-Preserving Model with Constant Voltage Behind Reactance T his model is widely used in structure-preserving transient energy function and voltage collapse literature that uses energy functions. It consists of the swing equations (7.182) and (7.183) and the network algebraic equations at the nodes 1, ... , n. In Figure 7.15, the n bus transmission network is shown augmented by the constant voltage behind r eactances at the generator buses 1, . . . , m. The generator internal nodes are denoted as n + 1, .. . , n + m . The complex power output at the ith internal node in Figure 7.15 can be expressed as

E;I;

Eiejo'((Ei - Vi)jjX d;)' = Eiejo'(Eie-jo; - Vie - je, )/( -jXdi ) =

= jxE;' di

- j EXi;' (( cos(O; - ei)

+j

sin( Oi - Bi))

di

_ EiV; sin(o; - B;) X'di

. (El _ EiV; COS(Oi - Bi) ) X'di

+ J X'di

(7.186)

Figure 7.15: Structure-preserving model with constant voltage behind reactance

7.9. REDUCED-ORDER MULTlM.4.CHINE MODELS

211

Alternatively, Eiej6'(h - jlqi)e-j(6'-~/2)

EiI;

EJqi

+ jEih

(7 .187)

Comparing (7.186) and (7.187) , the real and reactive power output at the generator internal nodes are given, respectively, by Real power EJqi = Reactive power EiIdi =

Ei Vi sine 6i - B;) X di E2

£-v.

X di

di

(7.188) (7189)

: - X',' COSCOi - Oi)

This shows that rcal power P is associated with lq and rea.ctive power is

associated with ld. At the generator buses 1, ... , m, we can express t he complex power due to the generator s as FCi

+ jQCi = v;17 v;1': - ViejO,

[( Ei - Vi)

fjX~ir

_ ViejO, (Eie - j6i - Vie-j8,) I( -jX~i )

= £-v.. (cos (B · 1.

,

I

.

8) + J. sin (0 - 0)) _J_ 1. X' 1.

\

y2 ~ X'

-

di

EiVisin(Oi -oi)

-

X'

~

.I(· _V;>

+ J '- )f1 + ~

di

EiV;COS(Oi-Oi))

X,

~

(7.190) Therefore

(7.191)

Fc; -

QCi

=

(7.192)

The structure-preserving model consists of the swing and the network equations (neglecting resistance) as

0,: 2Hi. w,

- - Wi

= Wi - Ws

= T Mi-

i = l, .. . ,m

(7.193)

EiVi sin(Oi - Oi) X di

(7.194)

CHAPTER 7. MULTIMACHINE SIMULATION

212

n

PLi(V;)

+ POi = L

V;VjBij sin(Oi - OJ) i

= 1 , . .. , n

(7.195)

;=1 n

QLi(V;)

+ QOi

= - L:V;VjBijCOS(Oi - Oj) ;=1

i= l , ... ,n

(7 .196)

where POi and QOi for i = 1, ... , m are given by (7.191 ) and (7.192), and POi = QGi = 0 for i = m + 1, . . . , n. The right-hand sides of (7.195) and (7.196) are the sums of real and reactive power on the lines emanating from bus i under the assumption of negligible transmission line resistance. This is shown as follows. If we neglect transmission line resistances, then the network admittance matrix is Y N = [jBij], and the total complex power in the network transmission lines from bus i is given by

(7 .197)

Expanding the expression (7.197) results in

Vi. j8 ,

n

L: -j BijVje- j8j ;=1

n

= L: -jV;VjBij(COS(O; j=l n

= L:V;VjB;jsin(O; j=l

OJ)

+j

sin(Oi - OJ))

n

OJ) -jL:V;VjB;jcos (O; - OJ )

(7198)

i =1

The rea.l and imaginary parts of (7 .198) are the right-hand sides of (7.195) and (7. 196) , respectively. The model given by (7.193)- (7.196) is called the structure-preserving classical model. An interesting observation from (7.189) and (7.192) regarding reactive power is the following. The reactive power absorbed by X di is obtruned by substracting QOi from EJdi, resulting in

E,Idi - QGi = (EI - 2EiV; cas{6, - 0,)

+ V!)/X~;

(7.199)

An alternative form of the structure-preserving model is to consider the augmented Y matrix obtruned by including the admittance corresponding

7.9. REDUCED-ORDER MULTIMACHINE MODELS

213

to the transient reactances of the machines. Thus , with proper ordering, n+l. . . n+m

l. . . m

m + l. .. n

- jj

o

n+l n+m 1

Yaug =

(7 .200)

- jj

m

m+l

o n

where

Y= DiagC~J

i=l, . .. ,m

and

-YNl = -YN + [Y0 0] 0 Considering Yaug = [jBij], (7 .193)-(7.196) can be written more compactly as (denoting , temporarily, 6/s as Oi'S and E/s as Vi 'S). Bi

=

Wi - w~

2H .

-

i

=n +

(7 .201)

l , .. 'In+ m

n+71l

'Wi = TMi -

"',

I: V;Vj Bijsin(6i -

OJ)

J' ; 1

(7.202)

i=n + l, ... ,n+m n+~

PLi(Vi)

=

L

Vi V j B ijsin(6i- 8j)

i= 1"

.. ,n

(7,203)

j= l n+~

QL.(Vi)

=-

I: ViV;Bij COs(8.-6j )

i= 1, .. . ,n

( 7.204)

j= 1

It can be verified that, in (7,202), the second term on the right-hand side is only EiVi sin(o. - 6.) /X~i' It is easy to verify t he equivalence of (7,203)

CHAPTER 7. MULTIMACHINE SIMULATION

214

and (7.204) with (7 .195) and (7 .196), respectively. Note that, in this model , we are allowed to have nonlinear load representation. This model is used in voltage stability studies by means of the energy function method [11 7].

7.9 .3

Internal-node model

This is a widely used model in first-swing transient stability analysis. In this model, the loads are assumed to be constant impedances and converted to admittances as

_

-(PLi - jQLi )

YLi=

V.2

,

(7.205)

i=l",.,n

There is a negative sign for YLi) since loads are assumed as injected quantities. Adding these to the diagonal elements of the Y NI matrix in (7.200) makes it Y N2 = Y NI + Diag(lhi). The modified augmented Y matrix becomes

m+

L .. m

n+ L .. n+m

L .. n

n+l

o

-y n+m

1

Y new aug

(7.206)

-y

~

m

m+ 1

YN2

o n

The passive portion of the network is shown in Figure 7.16. The network equations for the new augmented network can be rewritten as m

n

m (YA n Yc where YA =

y, YB [-y I 0], Yc

(7.207)

[~y],

and YD

YN2.

The n

network buses can be eliminated, since there is no current injection at these buses. Thus

7.9. RED UCED·OR DER MULTIMACHINE MODELS

IL

+ 1 ...._ fY''fY"''-_ _

-

+:::'" • •

YLI

In

215

+-- - - ,

+ 1 __

m +2

"Network

• jX dm 11 +111

- r--.

• • ,, --+--- -..,

Y Lm

= F igure 7.16: Augmented Y matrix with constant impedance

(yA - y By;;lYC)EA (7.208) Yint EA where t he elements ofI A and EA are , respectively, Ii = (Idi+ jIqi)ej(c. - "/2) = ID, + j IQi a nd Ei = EiLoi· T he elements of Yint are 1'" = Gij + jBir Sin ce the network buses have been elimin ated, we may renumber t he internal nodes as 1, . . . J m for ease of notation. m

I ; = LY;jE,

(7.209)

i= 1, . . . ,m

j=l

Real electrical power out of the internal node i in Figure 7.1 6 is given by

F"

= Re[E;J:J

[

Y,jE j '6' ~-"~]

- Re E;eJ

,

Re [EiejC.

~

J=l

f (G;, - jBi,)Eje-jC;] J=l

=

Re

[~(G'j -

jB,j)E,Ej[coS(Oi - OJ)

+j

sin(oi - OJ )J ] (7.210)

CHAPTER 7. MULTIMACHINE SIMULATION

216 Define

(7.211 ) Then

p., = L= EiEj( Gij cos 5,j + B,j sin 5ij ) ;=1

m

E'G ·· t U

+""(C ~

· sin 5·· 'J

IJ

+D ·

1,3

cos 5) 1,J

(7.212)

j= l

#'

where

= E,EjB,j

(7.213)

Dij = EiEjGij

(7.214)

Cij

Thus , the classical model is

d5. = Wi - W 4 dt 2Hi dw, _ T . _ p. . Ws

dt -

M t

t:1.

(7.215) . '1.=

11 ••• l m

(7.216)

where p., is given by (7.212). Since p •• is a function of the 5;'s, (7.215)(7.216) can be integrated by any numerical algorithm.

7.10

Init ial Conditions

Step 1 From the load flow, compute!, = ID. + jIQi as ( Pc. - jQc,) / V,e- j6 , . Step 2 Using (7.185), compute E" 5i as E;L5. = Vie j8 ,

+ jX~i(IDi + jlq,).

Step 3 From (7.193) or (7.215), Wi = w, . Thus, computation of initial conditions is simple when a classical model is used.

7.10. INITIAL CONDITIONS

217

Example 7.6 Compute EiL6i (i = 1,2,3) and Yint for Example 7.1 using the classical model (all in pu) . We firs t compute Ii (i = 1,2,3) 0.716 - jO.27 = 0.7358L - 20.66° 1.04LO° 1.63 - jO.067 = 1.5916L6.947° 1.025L - 9.3° 0.85 + jO.109 = 0.8361Ll2.0° 1.021 - 4.7°

i\ 12

-

13

-

1.04LO° + jO.068]]

E 1 L61 E 2 L02 E3 L63

= 1.054L2.267° 1.025 L9.3° + j O.1198I2 = 1.050 L19. 75° - 1.0214.7° + jO.181313 = 1.017 Ll3.2° 10, 11, 12

1, 2,3

4 ... 9

Y

-y

o

10 11 12

- - --

ynew

aug -

1 2 3

-y -

---

Y N2

4

o 9 where

y=

-

-

and Y N2 = Y Nl

[

+ [y0

o

- j16.45 0

-j8.35

o

o

-J52 ]

CHAPTER 7. MULTIMACHINE SIMULATION

218

Yint is obtained by eliminating nodes 1 .. . 9 as in (7. 208 ), and is given by

Yint

=

0.845 - j2.988 0.287 + j1.513 0.210 j1.226 ] 0.287 + j 1.513 0.420 - j2 .72 4 0.213 + j1.088 [ 0210 + j1.226 0.213 + j 1.088 0.277 - j2 .368

+

o

7.11

Conclusion

This chapter has discussed the formulation of a multimachine system model wi t h t he two-axis model, as well as the reduced-order flux-decay model and the classical model. The computation of in itial conditions and solution methodology for the two-axis model using t he simultaneous-implicit method has been discussed. Th e methodology of the ETM SP program of EPRI has also b een discussed. The reduced-order flux-decay model and the classical model, as well as the computation of initial conditions, are discussed. We also discussed the structure-preserving classical model.

7.12

Problems

7.1 Perform a load flow for the 3-machine system of Example 7.1 by varying the load at bus 5 in increments of 0.5 until PI, = 4.5 pu. Plot the voltage magnitude at bus 5 as a fun ction of the load . This is called a PV curve. 7.2 In Problem 7.1 , the increased load at each increment is allocated to slack bus 1. In practice, the AGe system allocates it t hrough the area control error, etc, As an alternative, consider allocating the increased load in proportion to the inertias of the machine, i.e., 6.PL5 is allocated to buses 2 and 3 as = P~2 + Ji~ 6.PLs and Pgl =

Ji;

pgl

P~3 + 6.PI,5 , where P~2 and P~3 are the generator powers befo re the load is increased and HT = Hl + H2 + H 3 . Again draw t he PV curve. 7.3 Repeat Problems 7.1 and 7.2 for a contingency of lines 5 to 7 being o utaged. Draw the PV curves for the pre-contingency state and the contingent case on the same graph.

7.12. PROBLEMS

219

7.4 This problem and Problem 7.5 can be done using symbolic software. As explained at the end of Example 7.1, Idi , Iq i can he eliminated from (7.47) as well as (7.54) and (7 .55) using (7.53). Express the resulting equations in the form :i:

= h(x,v , u )

0= gl(X, V) This is the DAE model with the algebraic equations balance form ((7.39) and (7.40)). 7.5 Using (7.53), substitute for DAE in the form

h,

Iqi

:i:

11 (x, V)

III

the power-

(i = 1,2 ,3) in (7.57) to express the

= i(x , V , u) = YNV

7.6 Derive 11 (x, V) = Y NV directly in Problem 7.5 from 0 computed in Problem 7.4. (Hint: See Section 7.3.2.) 7.7 Express Example 7.1 in the form :i: = A(x)+BW+Cu

W

= G(E,V)

o=

gl(X, V)

and Example 7.2 in the form :i:

W

11 (x, V)

= A(x)+BW + Cu = G(E,V) = YNV

7.8 In Example 7.1, loads (both real and reactive) at buses 5 and 6 are increased by 50 percent. With the new load -flow compute the initial conditions for the variables, Id., I.i, Vdi, V.i, Vrefi and TM. (i = 1,2,3). 7.9 Under nominal loading conditions of Example 7.1, there is a three-phase ground fault at bus 5 that is self-clearing in six cycles. Do the dynamic

220

CHAPTER 7. MULTIMACHINE SIMULATION simulation using the SI method or any available commercial software for 0-2 sec . Use the two-axis model and the IEEE- Type 1 exciter data in Example 7.1. You may use the model obtained in Problems 7.4, 7.5 , or '7.7.

7.10 Repeat Problem 7.8 for the flux-decay model and a fast exciter with KA = 25 and TA = 0.2 sec. Do the simulation as in Problem 7.9. 7.11 In Problems 7,9 and 7.10, find the gain KA at which the system will become unstable. (Instability occurs when relative rotor angles diverge.) 7.12 In Problems 7.9 and 7.10, find by repetitive simulation the value t cr , the critical-clearing time at which the system becomes unstable. 7.13 Repeat Problems 7.9 and 7.10 for the fault at bus 5 followed by clearing of lines 5 to 7. Assume tel = six cycles. Also find the critical-clearing time tcr . 7.14 For Example 7.1, write the structure-preserving model in the form of (7.201 )-(7 .204). 7.15 Obtain Y int for the system in Example 7.1 for both the faulted state and post-fault states for (a) self-clearing fault at bus 5 and (b) fault at bus 5 followed by switching of the lines 5 to 7. Write the equations in the form (7.215)- (7.216). 7.16 Find tcr in Problem 7.15 for both cases by repetitive simulation, and compare the results with Problems 7 .12 and 7.13, respectively.

Chapter 8

SMALL-SIGNAL STABILITY 8.1

Background

In this chapter , we consider linearized analysis of multimachine power systems that is necessary for the study of both steady-state and voltage stability. In many cases, instability and eventual loss of synchronism are initiated by som~ spurious d.isturbance in the system resulting in oscillatory behavior that, if not damped , may eventually build up . This is very much a function of the operating cond.ition of the power system . Oscillations, even if undamped at low frequencies, are undesirable because they limit power transfers on transmission lines and, in some cases) induce stress the mechanical shaft. The source of inter-area oscillations is difficult to diagnose. Extensive research has been done in both of these areas. In recent years, there has been considerable interest in dynamic voltage collapse. As regional transfers vary over a wide range due to restructuring and open transmission access , certain parts of the system may face increased loading conditions . Earlier, this phenomenon was analyzed purely on the basis of static considerations , i.e., load-flow equations. In this chapter, we develop a comprehensive dynamic model to study both low-frequency oscillations and voltage stability using a two-axis model with IEEE-Type I exciter, as well as the flux-decay model with a high-gain fast exciter. Both the electromechanical oscillations and t heir damping , as well as dynamic voltage stability, are discussed. The electromechanical oscillation is of two types :

in

221

CHAPTER 8. SMALL-SIGNAL STABILITY

222

1. Local m ode, typically in the 1 to 3-Hz range between a remotely located power station and the rest of the system .

2. Inter-area oscillations in the range of less than 1 Hz . Two kinds of analysis are possible: (1) A multimachine linearized analysis that computes the eigenvalues and also finds those machines that contrihute to a particular eigenvalue (both local and inter-area oscillations can be studied in such a framework ); (2) a single-machine infinite-bus system case t hat investigates only local oscillations.

Dynamic voltage stability is analyzed by monit oring the eigenvalues of the linearized system as a power system is progressively loaded. Instability occurs when a pair of complex eigenvalues cross to the right-half plane. This is referred to as dynamic voltage instability. Mathematically, it is called Hopf bifurcation. Also discussed in this chapter is the role of a power system stabilizer that stabilizes a m achine with respect to the local mode of oscillation. A brief review of the approaches to the design of t he stabilizers is given. For detailed design procedures, it is necessary to r efer to the literature. References [77J and [78J are the basic works in this area.

8.2

Basic Linearization Technique

A unified framework is presented in this section for the linear analysis of multimachine systems. The nonlinear mo del derived in Chapter 7, with a two-axis model with IEEE-Type I exciter or flux-decay model wi t h a static exciter , is of t he form :i;

= f(x ,y,u)

0 = g(x,y)

(8.1) (8.2)

where the vector y includes both the Id_q and V vectors. Thus, (8.1) is of dimension 7m, and (8.2) is of dimension 2(n + m). Equation (8.2) consists of the stator algebraic equations and the network equations in the power-balance form . To show explicitly the traditional load-flow equat ions and t he other algebraic equations, we partition y as y - [IL. OJ V, . .. V=

- [y; I ytl'

I O2 . . . an V=+l . . . VnJ' (83)

8.2. BASIC LINEARIZATION TECHNIQUE

223

Here, the vector Yb corresponds to the load-flow variables, and the vector Y. corresponds to the other algebraic variables. Bus 1 is taken as the slack bus and buses 2, ... , m are the PV buses with the buses m + 1, ... , n being the PQ buses. The dimension of x is 7m. Linearizing (8.1) and (8.2) around an operating point gives

~ !:lx

B

A

[ aa ]- [

C

Dn D2!

Dl2

hF

] [

~~: ] + E[~uJ

(8.4)

Define

JAE=[Dll D2!

D!2 ]

(8.5 )

hF

Eliminating ~Y. , ~Yb we get ~:i; = AsysL'.x where A sys ( A - BJAke). hF is the load-flow Jacobian. The model represented by (8.4) is useful in both small-signal stability, analysis and voltage stability, since hF is explicitly shown as part of the system differential-algebraic Jacobian. We use the above formula tion for a multimachine system with a twoaxis machine model and the IEEE-Type I exciter (model A) and indicate a similiar extension for the flux-decay model with a fast exciter (model B). The methodology is based on [79J and [80J-

8.2.1

Linearization of Model A

The differential equations of the machine and the exciter are the same as in Chapter 7, except that the state variables are reordered . The synchronous machine dynamic circuit and the IEEE-T ype I exciter are shown in Figures 8.1 and 8.2, resp ectively. The differential and algebraic equations follow .

+

+ [E~i +(X ~i-X;;) lq; + }. 1::''Ii

Figure 8.1: 1, ...

,m)

I d('6-1f1v I

_

(Vd; + j V'li ) ~,-nr..) = V,ei9i = Vo; + j VQi

Synchronous machine two-axis model dynamic circuit (i

CHAPTER 8. SMALL-SIGNAL STABILITY

224

+

v

Figure 8.2: IEEE- Type I ezeiter model Differential Equations

do. dt

dw.

dt

=

(8.6)

Wi - W,

TMi

-- -

[E~. - XdihJlqi

[Ed.

M; M. D.(w. - w.) Mi

+ X~JqiJldi Mi (8.7)

dE~i

(8.8)

dt dEdi

(8.9)

dt dEfdi

(8.10)

dt dVRi

(8 .11 )

dt dRFi

(8.12)

dt Stator Algebraic Equations The stator algebraic equations in polar form are

Ed. - Vi sin (5. - Oi) - R.Jdi

+ X~Jqi

= 0

(8.13)

8.2. BASIC LINEARIZATION TECHNIQUE E~i -

225

Vi cos (b, - Bi) - R.i1qi - X:lildi

=0

(8.14)

for i = 1, . .. ,m

Network Equations The network equations are lJ;V;sin(o; - 0;)

+ Iq;V;cos(o; -

0;) + Pc,;(V;)

n

- L: V;VkYik cos(O; -

Ok -

Uik)

= 0

(8.15)

k=l

Id;lIicos(o; - 0;) - Iq;lIisin (oi - Oil

+ QLi(V;)

n

- L: lIiVkYik sin(Oi -

Ok - U;k)

= 0 i = 1,2, . .. , m (8.16)

k=l

n

(8.17) k= l n

QLi (lIi) -

L: lIiVkYik sin (Oi k=l

for i

Ok - Uik) = 0

(8.18)

= m + 1,.,. , n

Equations (8. 6)-(8 .18) are linearized analytically, as explained below. The linearization of the differential equations (8.6)-(8.12) yields (8. 19) db.w;

dt

db.E~i

dt db. Ed; dt db.Efdi dt

CHAPTER 8. SMALL-SIGNAL STABILITY

226

+ KAi 6R F"-

__ 6VRi TAi

TAt.

..

KAi Ai 6.RFi KFi 6E - - TFi + (TFiJ2 fdi fori =l, .. . ,1n

+-T 6.Vrefi

(8 .23 ) (8.24 )

a

where f,i(Efdio) = - KBi + Efdi.a SE~~t·)+ S"(Eld;.) where the symbol stands for partial deTivative. Writing (8.19) through (8.24) in matrix notation, we obtain 0 0 0

f).6i

6Wi

~~~i

-

6.E~i

6.E.!di

0

0

0

0

-~ f;

_ I"i.

0

0

0

1 T~.. i

0

0 0

0

0

0

l .i( E /d,.)

1

0 "

M;

- T~ .. i

0

0 0 0

f).R p i

0

D" -=

Mi

0

f). VR'

1

0

1

- =T., ...

0

0

0

0

0 0

0

0

o

o

1 olio (X~ .: ~X ;;)M,

IS..".;.

TA •

(TF~~ :I

0

0

!Bl

K <1iK E~ ";tiT"

0

T ",;;

1 -

f).o , 6.wi ~E~i 6.E~i f).E/ di f). VRi

f).Rp,

TFi

E;."

o +

o o o o 0 0

,

T

0

o o

o

0

0

0

0 0 0

1 Mi

0 0

0 0 0

0 _ !£U

0

0

[ M, ] f). V;

0 0 0

+

0 0

TA•

0 0 IS..".;. T A .:

0

i= l ""lffi

(8.25) . D enotmg

[6h 1 .

61 . q.

68i AI gi, 6V; = L> [ t.

1= 6.V

gi ,

and

[ 6V 6TMi 1= 6u., (8.25) ren

8.2. BASIC LINEARJZATION TECHNIQUE

227

can be written as

For the m-machine system, (8.26) can be expressed in matrix form as (8.27) where A B B 2 , and E, are block diagonal matrices. " linearize " We now the stator algebraic equations (8 .1 3) and (8.14):

(8.28) !':.E~i

- cos( 6io - Oio )!':. Vi + Vio sin( 6io - Oio)!':.6, - Vio sin( 6io - O'o)!':.O, i = 1, ... , m ( 8.29) - R,,!':.lq' - Xd,!':.h = a

Writing (8.28) and (8.29) in matrix form, we have

!':.Oi 6Wi

o a 100 o 100 0 + [-R~i X~i] -Xdi

-R"

i

!':.Edi !':.Efdi !':. VRi !':.RF'

[ Md; ] !':.lq.

+ [ V;ocos(6i~ -_Oi~) - V,osm(6.0

~]

!':.E~,

0. 0 )

- sin (6io - Ow) ] [ !':.Oi. ] = - cos(6.o-0. o) !':.V,

= l"",m

a ( 8.30)

Rewriting (8.30), we obtain (8.31) In matrix notation, (8.31) can be written as (8.32)

CHAPTER 8. SMALL-SIGNAL STABILITY

228

where C D and D, are block diagonal. Linearizing the network equations " (8.16), " (8.15) and which pertain to generators, we obtain Via sin(Oio - Bio)fl1di

+ Jdio sin(5io - Bio).6..Vi + IdioV'io C08(Oio - (}io).6..0i

-IdioV'iocOS(Oio - Bio)!:l(}i

+ VioCOS(Oio

- Bio).6..Iqi

+Iqiocos(6io - Bio)6Vi -IqioViosin(oio - ( 10 ).6..0i

+Iqio Vio sin( 0'0 - e'o)t,.ei -

[f

VkoYik cost eio - eko - aik)] t,. Vi

k=l

n

k=l

+

[V. ~ V"y;,

"o('"

,~ 0,,1] -

M,

- Vio L [VkoYik sm(e,o - eko - aik)] t,.ek + n .

ah(Vi) all: t,. Vi = 0

k~l

(8.33)

t

ie'

Via cos( Oio - Bio).6..Idi

+ ldio COs( Oio - 8io).6.. Vi - ldio Via sin( 0';0

-

8io ).6..5,;

+Jdio Via sin{ 6io -

Bio).6..Bi - Via sin( 6io - Bio).6..Iqi -Iqiosin(oio - (}io).6.Vi - IqioViocos(6io - 8io)1:1 0i

+IqioViocos(oio - eio)t,.ei -

[f

VkoYiksin(eio - eko - aik)] t,.Vi

k=1

n

k=l

- [v.. ~ V~y" m,('. - '" - =1]

l'.O,

+ Vio L [VkoYik cos(eio - eko - "ik)] Mk n

+

aQLi(Vi) aVi t,. Vi = 0

10== 1

ie' i=1,2, ... ,m

(8.34)

B.2. BASIC LINEARIZATION TECHNIQUE

229

Rewriting (8 .33) and (8.34) in matrix form, we obtain

DSl ,=+l

+[

:

(8.35)

D Sm ,Tn+ l

where the various submatrices of (8.35) can be easily identified . In matrix notation, (8.35) is (8.36) where

for the non-generator buses i = m + 1, . . . ,n. N ote that C 2 , D3 are block diagonal, whereas D 4 , Ds are full matrices. Linearizing network equations (8.17) and (8.18) for the toad bnses (PO buses) , We ob tain

n

-ViD L [Y;k COS(Oio - Oko - Uik)) t>Vk k= l n

- Vio L [VkoY;k sin(Oio - Oko - Uik)] 6. Ok 1.=1

;
(8 .37)

CHAPTER 8. SMALL· SIGNAL STABILITY

230

n

- ViD

I: [Yik sin( BiD -

BkD - Ciik)] L':. Vk

k=l n

+ViD I: WkDYik sin( BiD -

BkD - Ciik)] L':.Bk

10=1

# i=m+l, ... ,n

(8.38)

Rewriting (8.37) and (8.38) in matrix form gives

(8.39)

where, again, the various submatrices in (8.39) can be identified from (8.37) and (8.38). Rewriting (8.39) in a compact notation ,

(8AO) where D 6 , D7 are full matrices. Rewriting (8.27), (8.32), (8.36), and (8AO) together ,

+ B1L':.Ig + B 2 L':. Vg + E,L':.u o = C1L':.x + D,L':.Ig + D 2 L':. Vg 0= C 2 L':.x + D3L':.Ig + D4L':.Vg + DsL':.Vl o = D6L':. Vg + D 7 L':. VI

L':.± = A,L':.x

where X

t t = r.xl· , xm

It

(8.41) (8.42)

(8.43 ) (8.44)

8.2. BASIC LINEARIZATION TECHNIQUE Xi

=

[<5£ Wi E~i

Edi

231

Efdi VRi

Rjil t

Ig = [Id' Iq, .. .Idm Iqml ' Vg = [0, V, ... Om

Vml'

VI = [Om+' Vm+,.·. On Vnl' U

=

, .. U ,m I' [U1,

u; = [TMi Vref.l'

This is the linearized DAE m odel for the multimachine system. Equations (8.41)-(8.44) are equivalent to the linear model of (8.4), except that in these equations the dependence of loads on the voltages has not been specified, whereas in (8.4) the loads were assumed to be of the constant power type. This model is quite general and can easily be expanded to include frequency or V dependence at the load buses. The power system stabilizer (PSS) dynamics can also be included easily. In the above model, 6 I g is not of interest and, hence, is eliminated from (8.41) and (8.43) using (8 .42) . Thus, from (8.42),

6Ig = - D:;' C,6x - D:;' D 2 6 Vg

(8.45 )

Substituting (8.45 ) into (8 .43) , we get C 2 6x

+ D 3 ( - D:;'C,6x -

D:;I D 2 6 Vg) + D 46 Vg

+ D s6Vl =

0

(8.46) Let

and

[C 2

-

D 3 D:;'C, ]

Note that Dl ' involves taking a series of 2 1, . .. , m). Equation (8.46) is now expressed as

K 2 6x

(8.47)

f1 K2

2 inverses of Zd_q,i(i =

X

+ K,6Vg + D s 6Vl =

(8 .48)

0

If (8.45) is substituted in (8.41 ) to eliminate 6Ig , the new overall differentialalgebraic m odel is given by

6';' = (A, - B,D,'C 1 )6x

o= 0=

+ ( B2 -

+ K,6 Vg + D56 Vi D66Vg + D76Vi

K , 6x

B,D:;' D 2 )6 Vg

+ E I 6u

(8 .49)

(8.50) (8.51)

232

CHAPTER 8. SMALL-SIGNAL STABILITY

Equations (8.49) - (8.51) can be put in the more compact form

[ where 6 VN

~x ] _ [~: ~:]

[

:I

N

]

+ [ ~' ] 6u

(8.52)

= [ ~ ~ ]. Reorder the variables in the voltage vector and de-

=

=

fine 6Vp [6y~ 6yt J' [68, 6Vl . .. 6Vm 1682" . 68" 6Vm +1" .6V"J'. 6Yb is the set of load-flow variables, and 6yc is the set of other algebraic variables in the network equations. Note that we have eliminated the stator algebraic variables . In any rotational system, the reference for angles is arbitrary. The order of the dynamical system in (8.52) is 7m, and can be reduced to (7m -1) by introducing relative rotor angles. Selecting 01 as the reference, we have

0;

0;

0·iI

= Oi -

0,

i = 2, . . . ,m

Wi

i

= 0

= Wi -

5; = 0 8; = 8, - 0,

= 2, ... }m

i = 1, . . , ,n

Since the angles 0, (i = 1, ... , m) and 8i (i = 1, ... , n) always appear as differences, we can retain the notation with the understanding that these angles are referred to 0" Tlus implies that the differential equation corresponding to 01 can be deleted from (8.52) and also from the column corresponding to 60, in A' and B' . The differential equations 66, Wi (i 2, ... , m) are replaced by 65; = 6w, - 6w, . This means that in the A' matrix we place minus 1 in the intersections of the rows corresponding to 66i(i = 2, ... , m) and the column corresponding to 6w,. This process is analytically neat, but is not carried out in linear analysis packages . Since angles appear as differences, they are computed to within a constant. Hence, a zero eigenvalue is always present. Recognizing this fact, we retain the formulation corresponding to (8 .52). We rewri te the differential-algehraic (DAE) system (8.52) after the reordering of algebraic variables as

=

6x ] [

o o

=

[AICf q

=

(8 .53 )

8.2. BASIC LINEARIZATION TECHNIQUE

233

For voltage-dependent loads, only the appropriate diagonal elements of Dj, and D~2 will be affected. Now, D~2 is the load-flow Jacobian hF modified by the load representation and

Dj, [ D'21

Dj2] = J' D'22

AE

(8.54)

the network algebraic Jacobian with voltage dependencies of the load included. Note that, compared to the algebraic Jacobian JAE of (8.4), the stator algebraic variables have been eliminated to obtain J'i!E. The system Asys matrix is obtained as 6.± = Asys6.x

+ E6.u

(8.55 )

where

(8.56) This model is used later to examine the effects of increased loading on the eigenvalues of Asys and the determinants of J'i!E and J LF for (1) constant power case,

(2)

constant current case, and (3) constant impedance case for

the models A and B. We now show that the model in (8.53) is consistent with the development from the nonlinear model in power balance form from Chapter 7. These are (7.36)-(7.38): x = !o(X,!d_q, V,u)

I d _ q = h(x, V)

o = go(X,!d-q, V)

(8 .57) (8.58) (8.59)

By substituting Id_q from (8.58) in (8.57) and (8.59), we obtain

± = f,(x, V, u) 0= g,(X,V) Linearizing (8.60) and (8.61) results in 6.± = of, 6. ax x

+ aVp of, 6. V. + of, 6. p au u

o = a9' 6. + ag, 6. V. ax

x

aVp

p

(8.60) (8.61)

CHAPTER 8. SMALL-SIGNAL STABILITY

234

where Vp = [0, V , ... Vm I O2 ... On Vm+l ... Vn ]', then we can identify

A'

ofl -"""B;-

ail - [B'1H' 2]

)~ -

~ = [ gi ], ~ = [~t: ~t: ] , *= [ 8 .2.2

l' ]

Linearization of Model B

As discussed in Chapter 7, the multimachine model with the flux- decay model and fast exciter (Figures 8.3 and 8.4) is given by the following set of differential-algebraic equations : jXa, +

Figure 8.3: Dynamic circuit Jor the fhn-decay model

Vref

+

• L..J

v Figure 8.4: Static exciter model Differential Equations

do.

-dt -- w·. w• dw. dt dE~.

dt

(8.62)

_ E~;Iq' _ (Xqi - X~;) I .1 . _ D.(w. - w.) - TM. Mi Mi dt qt Mi Mi

(8.6 3)

E~i _ (Xd. - X d.) I d . + Efdi - _ T' T'doi • T'doi doi

(8 .64)

B.3. PARTICIPA TION FACTORS

235

dEfd' = _ Efdi + KA·(V . _ Vi) T Ai T Ai ref,t dt for i = 1) ... m

(8.65)

I

Stator Algebraic Equations The stator algebraic equations are

Vi sin( 8;

- 0;)

+ R,;Id;

- Xq;Iq;

0

E~i - Vi cos( 5i - fh) - R5i1qi - X~ildi - 0

(8.66) (8.67)

for i = 1, ... ) m

Network Equations The network equations are the same as (8.15)- (8. 18). The linearization of this model is done in the same manner as in model A and, hence, is not discussed .

8.3

Participation Factors

Due to the large size of the power system, it is often necessary to construct reduced-order models for dynami c stability studies by retaining only a few modes. The appropriate definition and determination as to which state variables significantly participate in the selected modes become very important. This requires a tool for identifying the state variables that have significant participation in a selected mode. It is natural to suggest that the significant state variables for an eigenvalue Ai are those that correspond to large entries in the corresponding eigenvector v;. But the entries in the eigenvector are dependent on the dimensions of the state variables which are, in general, incommensurable (for example, angle, velocities, and flux). Verghese et al. [811 have suggested a related but dimensionless measure of state variable participation called participation factors. Participation factor analysis aids in the identification of how each dynamic variable affects a given mode or eigenvalue. Specifically, given a linear system

x = Ax

(8.68 )

CHAPTER 8. SMALL-SIGNAL STABILITY

236

a participation factor is a sensitivity measure of an eigenvalue to a diagonal entry of the system A matrix. This is defined as

8Ai

Ph = - 8akk

(8.69)

where Ai is the ;th system eigenvalue, all is a diagonal entry in the system A matrix , and Pki is the participation factor relating the kth state variable to the ;th eigenvalue. The participation factor may also be defined by (8 .70 ) where Wki and Vki are the Ic th entries in the left and right eigenvector associated with the ;th eigenvalue. The right eigenvector Vi and the left eigenvector Wi associated with the ; th eigenvalue Ai satisfy AVi

=

(8.71)

AiV":

w:A = W~A. i

(8.72)

It is n ot obvious that the definitions given in (8.69) and ( 8.70) are equivalent . We establish the equivalence as follows. Consider the system [A - AJ]Vi = 0

(8.73)

wHA - AJ] = 0

(8.74)

where Vi and Wi need not be normalized eigenvectors. It is our goal to examine the sensitivity of the eigenvalue to a diagonal element of the A matrix. From (8.68), assuming that the eigenvalues and eigenvectors vary continuously with respect to the elements of the A matrix, We write the perturbation equation as (A

+ 6A)(Vi + 6vi) = (Ai + 6Ai)(vi + 6vi)

(8.75 )

Expansion yields [Avi] + [6 Avi

+ A6vi] + [6A6 vi] =

+ [6Aivi +Ai6vi] + [6Ai6 v;] [AiVi]

(8 .76)

Neglecting the second-order terms 6A1l.vi and 6Ai6vi and using (8.71), we obtain [A - AJ]6vi

+ 6Avi = 6Aivi

(8.77)

8.3. PARTICIPATION FACTORS Multiply (8.77) by the left eigenvector

237

wl

to give

wHA - AiI]6.vi + w;6Avi

= w;.6.).i'lJi

(8.78)

The first term in the left-hand side of (8.78) is identically zero in view of (8.74), leaving w;l>.AVi = w;l>.Aivi

(8.79)

Now, the sensitivities of Ai with respect to diagonal entries of A are related to the participation factors, as follows. Assume only that the kth diagonal entry of A is perturbed so that 0 l>.A =

[

...

0

0

l>.akk

o

0

(8 .80)

Then, in (8.79), the left-hand side can be simplified, resulting in w;.6.Avi

= Wki6akk'lJik = W;6Ai'lJi

(8 .81)

Solving for the sensitivity gives the participation factor as = Pki

(8 .82)

Equation (8 .82) thus shows that (8.69) and (8 .70) are equivalent . An eigenvector may be scaled by any value resulting in a new vector) which is also an eigenvector. We can use thiB property to choose a scaling that simplifies the use of participation factors, for instance, choosing the eigenvectors such that W;Vi = 1 simplifies the definition of the participation factor. In any case) since 2:k=l Wki'lJik = W~Vi' it follows from (8.82) that the sum of all the participation factors associated with a given eigenvalue is equal to 1, i.e., n

LPki = 1

(8.83)

k=l

This property is useful, since all participation factors lie on a scale from zero

to one. To handle participation factors corresponding to complex eigenvalues, we introduce some modifications as follows. The eigenvectors corresponding to a complex eigenvalue will have complex elements . Hence, Pki is

defined as (8.84)

238

CHAPTER 8. SMALL-SIGNAL STABILITY

A further normalization can be done by maldng the largest of the participa.tion factors equal to unity. Example 8.1 Compute the participation factors of the 2 X 2 matrix :i; = Ax , where

The eigenvalues are

), 1

n

[~

A=

= 5 , ),2 = -2.

The right eigenvectors are

and

=

The left eigenvectors are com puted as wi [3 4J and w~ = [1 - 1J. Verify that W~"l = W j " 2 = O. Also, WiVl = 7 = WiVl' Letting i == 1 and k == 1,2 , successively, we obtain the participation of t he state variables Xl> X2 in the mode ),1 == 5 as . Pu

-

P2l =

=

,

- ( (3)?) )

,

=

W ll'Un W1'UI W2 1 1112

C4~(1)) -

W 2 'U2

=

3

-

7

4 7

Letting i 2 and k 1,2, we obtain the participation of the state variables Xl , x, in the mode ),2 == -2 as

P22

4

,

-

,n

-

W12'U21

Pl2

7

W 2 'U2

=

W22 11

W 2V2

3 7

The participation matrix is t herefore p == [

Pll P21

P12 P22

3

4

]- [ ] 'i 4

'i

'i 3

'i

8.3. PARTICIPATION FACTORS

239

Normalizing tbe largest participation factor as equal to 1 in eacb column results in

Pnorm = [ 0'175

o The ith column entries in tbe P or Pnorm matrix are the sensitivities of the ith eigenvalue with respect to the states. Example 8.2 Compute the participation fact ors corresponding to t he complex eigenvalue of

-01.4

A=

0

0

[ - 1.4 9.8

The eigenvalues are )., = ·0.6565 and ).2,3 = 0.1183 ± jO.3678. The right and left eigenvectors corresponding to the complex eigenvalue ).2 = 0.1183 + jO.3678 are

V2

0.0138 - jO.0075 ] = - 0.0075 - jO.04 , [ - 0.9918 - jO.1203

W2

[0.838 - jO.0577 ] = 0.4469 jO.307 -0.0061 + jO.0205

+

Using the formula ill the previous section , we obtain

P21 = 0.2332; P22 = 0.3896 , Pn = 0.3772 Note P21 + Pn + P23 = 1. We can normalize with respect to Pn by making it unity, in wh.ich case P21(norm) 0.598, Pn(norm) 1 and P23(nOrm) 0.968.

=

=

o

=

240

CHAPTER 8. SMALL -SIGNAL STABILITY

Example 8.3 The numerical Example 7.1 is used to illustrate the eigenvalue computation. Compute the eigenvalues, as well as the participation factors, for the eigenvalues for the nominal loading of Example 7.1. The damping Di '" a (i 1,2,3) . The machine and exciter data are given in Table 7.3 . Loads are assumed as constant power type.

=

Solution Following the linearization procedure results in a 21 X 21 sized Asys matrix_ Because of zero-damping, two zero eigenvalues are obtained. The eigenvalues are shown in Table 8.1. The participation factors associated with the eigenvalues are given in Table 8.2. Only the participation factors greater than 0.2 are listed . Also shown are the state variables and the machines associated with these state variables. From a practical point of view, this information is very useful. Table 8.1: Eigenvalues of t he 3-Machine System -0.7209 -0.1908 -5.4875 -5.3236 -5.2218 -5.1761 -3. 3995 -0.4445 -0.4394 -0.4260 -0.0 000 -0.0000 -3.2258

±J·12 .7486 ±j8.3672 ±j7.9487 ± j7.9220 ±j7.8161

± j1.2104 ±jO.7392 ±jO.496 0

8.4. STUDIES ON PARAMETRIC EFFECTS

241

Table 8.2 : Eigenvalues and Their Participation Factors Eigenvalue -0.7209 ± jI2.7486 -0.1908

±

;8.3672

-5.4875

±

j7.79487

-5.3236

±

j7.9220

-5.2218

±

j7.8161

-5.1761 -3 .3995 -3.2258 -0.4445 ± j1.2104

-0.4394

±

jO .7392

-0.4260

±

jO.4960

0.0000 0.0000

Machine Number 3 2 2 1 2 2 3 3 1 1 2 3 3 2 1 2 3 1 2 3 3 2 1 2 1 2

Machine Variable

PF

Ii,w 6,w 6,w 6,w

1.0, 1.0 0.22, 0.22 1.0 , 1.0 0.42, 0.42 1.0, 0.98 0.29 1.0, 0.98 0.29 1.0, 0 .97 0.31 1.0 0.92 1.0 0.89 1.0, 0.74 0.67, 0.48 0.38, 0.28 1.0, 0.78 0.78, 0.60 0.22 1.0, 0.83 0.43 , 0.33 1.0, 1.0 0.26, 0.26 1.0, 1.0 0.26, 0.26

VR,Ejd RJ VR,Ejd RJ VR , Ejd RJ E'd

E 'd E'd E'd E~)RJ

E~ l RJ E~ , Rf E;,Rj E;,Rj

,

E'

E~lRJ E~,Rf

6,w .b,w 6,w 6,w

o

8.4

Studies on Parametric Effects

In this section, the effect of various parameters on the small-signal stability of the system is studied .

CHAPTER 8. SMALL·SIGNAL STABILITY

242

8.4.1

Effect of loading

The WSCC 3-machine, 9-bus system of Chapter 7 is considered. The real or reac tive loads at a particular bus / b uses are increased continuously. At each step, the initial conditions of the state variables are computed, after running the load flow, and linearization of the equations is done. Ideally, the increase in load is picked up by the generators through the economic load dispatch scheme. To simplify matters, the load increase is allocated among the generators (real power) in prop ortion to their inertias. In the case of increase of reactive power, it is picked up by the PV buses . The Asys matrix is formed, and its eigenvalues are checked for stability. Also dethF and detJ~E are computed. The step·by.step algorithm is as follows: 1. Increase the load at bus/buses for a particular generating unit model. 2. If the real load is increased , then dis tribute the load among the various generators in proportion to their inertias . 3. Run the load flow . 4. Stop, if the load flow fails to converge. 5. Compute the initial conditions of the state variables, as dIscussed in Chapter 7. 6. From the linearized DAE model, compu te the various matrices. 7. Compute dethF, detJ~E' and the eigenvalues of Asys. 8. If Asys is stable, then go to step (1). 9. If unstable, identify the st ates associated with the unstable eigen· valuer s) of Asys using t he participation factor method, and go to step (1). The above algorithm is implemented for mo dels A and B. Nonuniform damping is assumed by choosing Dl = 0 .0254 , D2 = 0.0066, and D3 = 0.0026. The resnlts are summarized in Tables 8.3 and 8.4. It is observed that for constant power load, with model A and the IEEE· Type I slow exciter, it is the voltage control mode that goes unstable at PL5 = 4.5 pu. Examination of the participation factor indicates t hat t he pair of state variables E~, Rj of machine 1 in t he e.."'{citation system is responsible for this m o del. In the case of model B, the mode that goes unst able is due to the electromechanical

243

8.4. STUDIES ON PARAMETlUC EFFECTS

variables 6, w of machine 2 at a load of 4.6 pu. A value of K A = 45 is assumed . The point at which the eigenvalues cross over to the right-half plane is called the Hopf bifurcation point , and the point at which the detJAE changes sign is the singularity-induced bifurcation. These are discussed in the literature in detail [82, 85}.

Table 8.3: Eigenvalues with Model A, KA = 20 "gn(detJLp )

.•gn(detJ/4.E}5

4.3 4.4 4.5 (A)

+ +

4.6 4.7 4.8

+ + +

+ + +

Load at Bus

4 .9

5.0 (8) 5.1 5.2 5.3 5.35

+ +

+ + +

+ -

· 0. 16]8

0.4 446 ± 1.0825 ± 2 .605 1 ± 17.568,

+ +

-

5.45

± j1.9769

-0.0522 ± j2 .11 02 0.1268 ± ,2.2798

+ +

-

Criti cal Eigenvalue{s)

,gn(deU~R)

+ + + + + + +

+

5.0 (8) 5.1 5.2 5.3 5.4 5.5

+

+ + + -

+

-

Eft E,l E', I

0 .3505

"gn(dellLF)

4 .9

,2'

Ed2 1Eql

E~ ll 61 ,02

0 .0496 -0.1454 LF d oes not con verge

4.3 4.4 4.5

+ + + +

E~1' E~2' E:a E:n, E;l ,E E~3

1.0 526 0 .6553

L06d a.t Bus 5

4.6 (A) 4.7 4.8

E;l ' RJ1 Eq1.lE~2

;2 .4911 ;2 .7064 ,2.439 2 1. 7849

Table 8.4: EigenvaLues with Model B, KA

+

Associated SLa.tes E ,Rf1 11 E'll,RJl

""

= 45

Critical Eigenvalue(s) -0.1119 ± ,8.8738

± ,8.8401 ± j8 .81 83 0.0901 ± j8 .8421 0 .1587 ± ;8.9371

.0.0729 -0.0035

0.1292 ± j9.0538 0 .7565 ± j20.1162 0.0471 ± j9.09 02 14 .7308 7.1144 4.2567 2.3120

-0 .0 597 ± jS.7819 LF does not con verge

Associated States 52 , CoI 2 62 ,W2 02,'-'2

5 2 ,'-'2 (h,w2

6'2 , toT2 E~l l E~l 62,W2

E~l' E~l Ej l E" E~l 6'2 . w :I

244

8.4.2

CIIAPTER 8. SMALL-SIGNAL STABILITY

Effect of KA

It was found that, for m odel A, the increase in KA alone did not lead to any instability. The stabilizing feedback in the IEEE-Type I exciter was removed, and then an increase in K A led to instability for this model, as well. For model B, a sufficient increase in KA led t o instability even for a nominal load .

8.4.3

Effect of type of load

Appropriate voltage-dependent load modeling can be incorporated into the dynamic model by specifying the load functions. The load at any bus i is given by

= 1, .. 'In

(8 .85)

i = 1, .. _, n

(8.86)

i

where PLio and QLio are the nominal real and reactive powers, r espectively, at bus i, with the corresponding voltage magnitude Vio, and npi' no' are the load indices . Three types of load are considered. 1. Constant power type (np

= no = 0)

2. Constant current type (n p

= no =

3. Constant impedance ty pe (np

1)

= no = 2)

The step-by-step procedure of analysis for a given generating-unit model is as follows: 1. Select the type of the load at various buses (i.e., choose values of and nq at each bus) .

np

2. Compute the system ma.trix. 3. Compute the eigenvalues of Asys for sta.bility analysis . For the three types of loads mentioned earlier, the eigenvalues of model A for increased values of load PLo at bus 5 are listed in Tables 8.5 to 8.7. First of all, the relative stability of constant power , constant current , and constant impedance-type load has been shown for a nominal operating point

8.4. STUDIES ON PARAMETRIC EFFECTS

245

= 1.5 pu and QLo =

0.5 pu (Table 8.5) . We observe that the system is dynamically stable for all types of loads. For an increased value of load at bus 5 (PLo 4.5 pu, QLo 0.5 pu), the eigenvalues are listed in Table 8.6. From Ta.ble 8.6 We observe that, for this increased load at bus 5, the system becomes dynamically unstable if the load is treated as a constant power type, whereas for the other two types of loads the system remains stable. 0.5 pu), in which we Finally, we take another case (PLo = 4.6 pu , QLo show that the constant impedance type load is more stable than the constant current type. To demonstrate this condition, we take model B with a high gain of the exciter (KA 175). The eigenvalues for various kind of loads are listed in Table 8.7. Both the constant power and constant current cases are unstable, whereas the constant impedance type is stable . These results corroborate the observation in the literature that constant power gives poor results as far as network loadability is concerned [84]. PLo

=

=

=

=

Table 8.5: Eigenvalues for Different Types of Load at Bus 5 for model A (PLo = 1.5 pu ; QLo = 0.5 pu): (a) constant power; (b) constant current; (c) constant impedance Constant P ower (a) -0.7927 ± j12.7660 -0.2849 ± j8.3675 -5.51 87 ± j7 .9508 -5.3 325 + j7.9240 -5.2238 ± j7 .8156 -5. 2019 -3.4040 -0 .4427 ± j1.2241 -0 .4404 ± jO.7413 -0.0000 -0.1975 -0.4276 ± jO.4980 -3.2258

Constant Current (b) -0. 7904 ± j12 .7686 -0.2768 ± j8 .3447 -5.5214 ± j7.9516 -5 .3335 ± j7.9247 -5. 2273 ± j7 .8259 -5.2030 -3.4462 -0 .4537 ± j1.1822 -0 .4412 ± jO.7416 -0.0000 -0.1974 -0.4276 ± jO.4980 -3 .2258

Constant Impedance (c) -0 .7887 ± jI2.7706 -0.2703 ± j8.3271 -5 .5236 ± j7 .9523 -5.3344 ± j7. 9253 -5.2301 ± j7 .8337 -5 .2039 -3.4801 -0.4617 ± j 1.1439 -0.4419 ± jO.7418 -0.0000 -0.1973 -0.4277 ± jO.4980 -3.2258

CHAPTER 8. SMALL-SIGNAL STABILITY

246

Table 8.6: Eigenvalues for Different T ypes of Load at Bus 5 for Model A (PLo = 4.5 pu; QLo = 0.5 pu): (a) constant power; (b ) const ant current ; (c) constant impedance Constant Power (a) -0 .7751 ± j12 .7373 -0.2845 ± j8.0723 -6.7291 ± j7 .8883 -5.6034 ± j7.9238 -5.2935 ± j7.6433 -5.2541 0.1268 ± j2 .2798 -2.5529 -0.4858 ± jO .7475 -0.0000 -0.5341 ± jO .5306 -0.1976 -3.2258

Constant Current (b) -0.7335 ± j12 .7842 -0.2497 ± j8.0650 -6. 7669 ± j7.9730 -5 .6287 ± j7.9557 -5.2812 ± j7 .8419 -5.2715 -3 .5296 -0 .5020 ± j 1. 253 1 -0.0000 -0.4910 ± jO.7561 -0 .5360 ± jO.7561 -0.1972 -3.2258

Constant Impedance ( c) -0.7285 ± j 12.7936 -0.2444 ± j8 .0659 -6.7760 ± j7.9895 -56338 ± j7 .9639 -5 .2938 ± j7.871 2 -5 .2790 -3.8 105 -0 .5303 ± j1.0434 -0 .4950 ± jO .7653 -0 .5371 ± jO .5336 -0.0000 -0.1970 -3.2258

Table 8.7: Eigenvalues for Different T ypes of Load at Bus 5 for Model B with KA = 175 (PLo = 4.6 pu, QLo = 0.5 pu): (a) constant power; (b) constant currentj (c) constant imp edance Constant Power (a) -1.96 10 ± j19.3137 -0.1237 ± j 15.5812 0.4495 ± j9 .1844 -3 _1711 ± j8 .2119 -2 .7621 ± j7.1753 11 -0.0000 ·0 .1987

Const ant Current (b) -0.2039 ± j15.5128 -2.2586 ± j12 .4273 0.1441 ± j9 .0636 -3.1359 ± j8.1099 -2 .7599 ± j7 .1594 -0.0000 -0.1989

Constant Impedance

(e) -0 .2054 -2 .1834 -0 .0607 -3 .0930 -2 .7575 -0 .0000 -0.1990

± j 15.5285 ± j11.1128

± j9 .1218 ± j8 .0364 ± j7 .1462

8.4. STUDIES ON PARAMETRIC EFFECTS

8.4.4

247

Hopf bifurcation

For model A, when the load is increased at bus 5, it is observed that the critical modes for the unstable eigenvalues are the electrical ones associated with the exciter , and are complex (Table 8.3). At a load of 4.5 pu, the eigenvalues cross the jw axis. This is known as Hopf bifurcation (point A). When the load is increased from 4.8 pu to 4.9 pu, the complex pair of unstable eigenvalues split s into real ones that move in opposite directions along the real axis. The one moving along the positive real axis eventually comes back to tbe left-half plane via +00 when the load at bus 5 is increased from 4.9 pu to 5.0 pu (point B). This is the point at which detJAE changes sign. This is also known as singularity-induced bifurcation in the literature [85J . The other unstable real eigenvalue moves to the left, and is sensitive to the varia.ble E~, of the exciter. This eigenvalue returns to the left-half plane at a loading of approximately 5.4 pu, and the system is again dynamically stable (poin t C). For the load at bus 5 = 5.5 pu, the load flow does not converge . It is possible through other algebraic techniques to reach the nose of the PV curve or the saddle node bifurcation. This phenomenon is pi ctorially indicated in the PV curve for model A and also is the locus of critical eigenvalue(s) in the .-plane (Figures 8.5 and 8.6). 1.2

~

;.

A B

c

0.8

0 .6

0 .4

0.2

0 0

2

3

4

5

Real Power at Bus 5

Figure 8.5: PV curve for bus 5 with model A

6

248

CHAPTER 8. SMALL-SIGNAL STABILITY

I I

IA

,1 ... - .... - .. , I

I

CI

'\

\ B

o ----- - - U-- ~ l A' I

o Real Part

Figure 8.6: The qualitative behavior of the critical modes of Asys as a function of the load at bus 5 (model A) At point A, Hopf bifurcation occurs , and it has been shown to be subcritical, i.e. , the limit cycle corresponding to the E~ - R f pair is unstable [86J. However , load-flow solution still exists. In this region, E~ and Rf state variables are clearly dominant initially. As the eigenvalues become real and positive, other state variables st art participating subst antially in the unstable eigenvalues, as indicated in Table 8.3. For model B , which has the fast static exciter with a single time constant, the modes that go unstable are the electromechanical ones (Table 8.4). When the Hopf bifurcation phenomenon in power systems was first discussed in the literature for a single-machine case, the electromechanical mode was considered as tbe critical one [78, 87J. It was called low-frequency oscillatory instability. In studies relating to voltage collapse, it was shown that the exciter mode may go unstable first [82J . From Tables 8.3 and 8. 4, it is seen that both the excit er modes and tbe electromechanical modes are critical in steady-state stability and voltage collapse , and that t hey both participate in the dynamic instability depending on t he machine and exciter models . Hence, decoupling the QV dynamics from t he PC dynamics as suggested in the li terature may not always hold. It may be t rue for special system configuration / operating conditions . Load dynamics, if included, can be considered as fast dynamics, and the phenomenon of detJ~E changing sign will still exist. In conventional bifurcation-t heory terms, one can think of solving g( x , y) = 0 for y = h( x) in (8.2) and substituting this in the differential equation (8.1) to get x = lex, h(x)). The change in sign of detJ~E (which generally agrees with the sign of JAE in (8 .5)) is the instant at which

B.5. ELECTROMECHANICAL OSCILLATORY MODES

249

the solution of y is no longer possible. This is also tied in with the concept of the implicit fun ction theorem in singular perturbation theory.

8.5

Electromechanical Oscillatory Modes

These are the modes associated with the rotor angles of the machines. These can be identified through a participation factor analysis in the detailed model. In the classical model with internal node description , we have only the rotor angle modes. We now discuss the computation of these modes as a special case . The equations have been derived in Chapter 7 and are reproduced below. db;

dt 2H.

i=l, ... ,m

-

Wi-W"

-

TMi-Pei

(8.87)

dWi

w, dt

(8.88)

'i= l, ... ,m

Linearization around a n operating point

0

gives

~6.0

(8.89)

dt ' d dt 6.w;

(8.90)

Because TMi = constant , and 6.P,. =

2::7=1

a:';i 6. OJ , we get (8.91)

6. W i

( 8.92) where J

In matrix form we ha.ve

=,

(8.93)

250

CHAPTER 8. SMALL-SIGNAL STABILITY

(8.94 )

The elements of Aw as

aTe given by 2~: aC:;:.i. The matrix Aw can be written

-w,

2H,

of,,) a5,

-w . apel

2Hl 80m.

Aw =

(8.95) -w.~ 2H~

85 ,

~!2Lm. 2H~

a5~

0

The elements of Aw can also be expressed in a polar nota.tion by noting that

and

Hence, in (8.93)

(8.96) Therefore

ap,; ---

a5

J

J

=,

(8 .97)

8.5. ELECTROMECHANICAL OSCILLATORY MODES

251

For the 3-machine case

211 (B)B21'12 si n{ O:12 -6f2 ) +8 1 B 3 1'13 .in(a:13 -6

Aid

13 )]

~Elg2Y12

sin{<X12 - 6f2)

~r81B3Y13.in(1X13 - 6fall

= "'. 2d31 83Bl Y 13 .in(oc31 +E2 82 Y23

- oj!)

'in ( ""32 -6~2»)

(8.98) although (Xij = (Xji_ However, the sum of the three columns = 0, hence, the rank of Aw = 2. This means that one eigenvalue of Aw = O. Aw can be expressed as Aw = M-l K, where M' - [diag(frt)] and K = [K;j ]: Aw is not symmet ric , SInce O;,j

=

- Oij1

=

I:

Kij

EiEj Yij sin( ex;,j - 6?) '1

J = ,

(8.99)

j :::::; ly!.i

EiEjYtj sin (CXij _6°) '1

jf.i

(8.100)

Ki/S are called the synchronizing coefficients, and are equivalent to those in

(8.93). Eigenvalues of A and Aw Case (a): Transfer conductances are neglected, i.e.,

Gij =

O.

This implies (Xij = ~ and Kij = Kji by inspection from (8.99) and (8.100). Hence, the matrix]( is symmetric. It can be shown mathematically that (1) the eigenvalues of

~~-]

A = [-M'--:=O:"1<",(+1

(8.101)

are the square roots of t he eigenvalues of Aw = M-l K. (2) The eigenvalues of Aware real, including one zero eigenvalue . The zero eigenvalue is due to the fact that a reference angle is necessary, and the angles appear only as differences. Hence, A has a pair of zero eigenvalues. H the real eigenvalues of Aware negative, then A has complex pairs of eigenvalues on the imaginary

CHAPTER 8. SMALL-SIGNAL STABILITY

252

=

axis. The nonzero eigenvalues ±jwk of A( k 1,2, . . . , m - 1) define t he electromechanical modes. If Aw has a positive real eigenvalue, then A has at least one positive real eigenvalue, and, hence, the system is unstable. Case (b): Gi;"# O(i "# j) . K is not symmetric. However, Aw will still have a zero eigenvalue. Hence, A will also have two zero eigenvalues. It appears that the eigenvalues of A are also the square roots of the eigenvalues of Aw.

Example 8.4 Compute the electromechanical modes with and without transfer conductances for the 3-machine system. In Example 7.6 , the voltages behind transient reactances and t he reduced-order model at the internal buses around the operating point have been computed . They are reproduced below.

E, = 1.054 L2.27°

E 2 = 1.05L19.75° E3 = 1.02Ll 3.2°

The Y int mat rix is given b y 0.287 + j1.513 0.210 + j1.226 0.287+ j1.513 0.420 - j2. 724 0.213 + j1.088 [ 0.210 + j1.226 0.213 + jl.088 0.277 - j2.368 0.845 - j2.988

Yint

=

1

The elements of Aware calculated as follows. W6

Ws

WS

- H = 7.974, - H = 29.453, - H = 62.625 2, 22 23

C 12 = 1.678 , C13 = 1.321 , G ll = 0.845,

D12 = 0.318 , D13

=

1

E,

1= 1.054

0.226

C 21 = 1.678 , C 23 = 1.165 , G 22 = 0.420 ,

1E2 1= 1. 05

D21 = 0.318 , D23 = 0.228 C 31 D31

0'2

= 1.313 , C 32 = 1.165 , C 33 = 0.277 , 1E3 1= 1.02 = 0.226 , D32 = 0.228 = - 17.46",0 13 = -10.91", 0~3 = 6.55"

B.S. ELECTROMECHANICAL OSCILLATORY MODES

253

We next obtain the following quantities for use in computing elements of

Aw· C '2 cos 0~2 - D'2 sin or2 = 1.696 C'3 cos 0~3 - D 13 sin 6fa

= 1.332

C 21 cos 6~, - D21 sin 6~, = 1.506 C23cos6~3

- D23sin6~3 = 1.139

C 31 cos o~,

- D31 sin o~,

= 1.246

C 32 cos 032 - D32 sin 632 = 1.191 From (8.93), we can now calculate the elements of Aw. Case (a): A", for Gij

01

0 is

1

-24.209

13 .521

44.356

- 77.902

10.688 33.546

78 .031

74.586

-152.6 17

A", = [ Case (b): With Gij = 0

A", = [

-23.83

13.15

45.64

- 79.04

10.68 33.40

78.48

74.28

- 152.76

The eigenvalues of A are obtained as Case (a): 0 , 0,

± j8.7326, ± j13.393

0, 0,

± j9.807, ± j 13.435

Case (b):

Notice the difference by neglecting G;j.

o

1

254

8.6

CHAPTER 8. SMALL-SIGNAL STABILITY

Power System Stabilizers

So far we have discussed the linearized model, eigenvalues, and its application to voltage stability in the context of a multimachine power system. We now discuss a stabilizing device, called the power system stabilizer (PSS), used to damp out the low-frequency oscillations. Although considerable research is being done in designing P SS for a multimachine system, no definitive res ults have b een applied in the field. The design is still done on the basis of a single machine infinite bus (SMIB) system. The parameters are then tuned on-line to suppress the modes, both the local and inter-area modes. We explain the basic approach based on control theory, and illustrate it with an example.

8.6.1

Basic approach

The differential-algebraic nonlinear model of a single machine connected to an infinite bus is given by

x

f(x,y,u)

(81 02 )

o

g(x,y)

(8 103)

where x is the state vector, y is the vector of algebraic variables, and 9 consists of the stator algebraic and the network equations. Let the operat ing point be xO, yO, uo. The p erturbed variables are x = XO + 6x, y = yO + 6y, and u = UO + 6u. Linearization of (8.102) and (8.103) leads to 6:i:

o

-

+ B6.y + E6.u C6.x + D6.y A6x

( 8.104) (8.105 )

If D is invertible, (8.106) Therefore 6.:i:

= (A -

BD - 1 C)6.x

+ E6.u = Asys6.x + E6.u

(8.107)

A , B , C , D , and E are appropria te Jacobi ans of (8. 102 ) and (8. 103 ) evaluated at the operating point. We illustrate t he procedure for a single-machine system . Equation (8.103) consists of the two stator algebraic equations and the network equations in either the power-balance or the current-balance form.

8.6. POWER SYSTEM STABILIZERS 8.6.2

255

Derivation of K1-K6 constants [78,88]

Historically, when low-frequency oscillations were first investigated analytically, they took a two-step approach. A single machine connected to an infinite bus is chosen to analyze the local (plant) mode of oscillation in the 1- to 3-Hz range. A ·flux-decay model is linearized with Efd as an input, and the model so obtained is put in a block diagram form. Then a fast-acting exciter between t:,. Vi and t:,.Efd is introduced in the block diagram. In the resulti ng state-space model, certain constants called the 1(1-1(6 are identified. These constants are functions of the operating point. The state-space model is then used to examine the eigenvalues, as well as to design supplementary controllers to ensure adequate damping of the dominant modes . The real and imaginary parts of the electromechanical mode are associated with the damping and synchronizing torques, respectively. We now outline this approach, which is a special case of the general development discu ssed in the earlier sections of this chapter. The single machine connected to an infi.n.ite bus through an external reacta.nce Xe and resistance Re is the widely used configuration with a fluxdecay model and stator resistance equal to zero. No local load is assumed at the generator bus. Figure 8.7 shows the system. The flux-decay model of the machine only is given by (8.62)-(8.64), with Efd being treated as an input; the equations are given below.

Figure 8.7: Single-machine infinite-bus system

E~

= -

T~ (E~ + (Xd -

6= w -

Xd)Ir Eid)

(8.108)

do

Ws

(8.109) (8.110)

CHAPTER 8. SMALL-SIGNAL STABILITY

256

The stator algebraic equations are given by (8. 66) and (8.67), and we assume R, = O. We use Vi t o denote t he magnitude of the generator terminal voltage . The equations are

XqIq - Vt sin (0 - 8) = 0

(8. 111)

=0

(8 .112)

E~ - Vt cos

(/j - 8) - X~Id

Now

Hence

(8.11 3) Expansion of the right-hand side results in

Vd

+ jV. = Vi

sin (.5 - 8) + j Vt cos (.5 - 0)

(8.114)

Hence, Vd = Vi sin(o - 8) and V. = Vi cos(.5 - 0) . Subst ituting for Vd and V. in (8.111) and (8.112), we get the st ator algebraic equations

=0

(8.11 5)

E~ - V. - XdId = 0

(8.116)

XqIq - Vd

The network equ ation is (assuming zero phase angle at the infinite bus)

(Vd

+ jV.)ei (6 - "/2) R,

(Vd

+ jVq ) -

VooL O°

+jX, V oo e- i (5 - ,,/2)

Re + jXe

(8.117) (8 .118)

Cross-multiplying and separating into real and imaginary parts, /j

(8. 119)

V 00 cos /j

(8 .120)

ReId - XeIq = Vd - V00 sin XeId

+ ReIq = Vq -

We thus have, for the single-machine case, the differential equat ions (8.108 )(8.110) and the algebraic equations (8.115 ), (8.116), (8 .119), and (8.120). We linearize them around an operating point, and eliminate the algebraic variables I d , I q , 0, Vd, Vq as follows.

B.6. POWER SYSTEM STABILIZERS

257

Linearization

Step 1 Linearize the algebraic equations (8.115) and (8.116): (8 .121) Step 2 Linearize the load-flow equations (8.119) and (8.120): (8 .1 22) Step 3 Equate the rigb.t-hand sides of (8 .121) and (8.122), and simplify

R, [ (Xc

[

+ X~)

0

f'>.E~

]

+

[- V"" cos 6° ] V"" sin 60

f'>.6 (8.123)

Now

R. [ (X, +X~)

- (Xc +Xq) ] - ' Rc

=~[ f'>.

R. - (Xc + X~)

(Xc

+ Xq)

]

R, (8 .124)

where the determinan t f'>. is given by

f'>. = R~

+ (Xc + Xq) (Xc + X d)

Solve for f'>.Jd. f'>.Iq in (8 .123). to get (after simplifi.cation)

-R.V"" cos 5° + V"" sin 5°(Xq + Xc) ] R.V"" sin 50 + Voo cos 5°(X d + Xc) (8 .125)

CHAPTER 8. SMA LL-SIGNAL STABILITY

258 Step 4

Linearize the differential equations (8.108)- (8.110). We introduce the normalized frequency v = .!!!.. so that the linearized differ ential equations become w.

+

[T~o ~ 1

(8. 126)

2H

Step 5 Substitut e for 6.Id , 6.1. from (8. 125) into (8. 126) to obtain

,K.

., 1 6.Eq = - K T' 6.E q 3

-

do

- , Tdo

6.6

1

+ T'do 6.Efd

(8. 128)

6.5 = w.6.v

.

K2

(8.127)

,K,

6.v = - 2H 6.E. - 2H 6.6 -

Dw.

2H 6.v

1

+ 2H 6.TM

(8 129)

where 1

K3 =

1+

(Xd -

X~)(Xq

+ X ,)

(8. 130)

6.

K.

= Voo(Xd6.- X~) [( X.. + X. ) SIn.

K2

= ~ [1;6. -

5° - R, cos 5°1

(8.131)

1;(Xd - Xq )(Xq + X,) - R,(Xd - X.)ld + R,E~oJ (8. 132)

8.6. POWER SYSTEM STABILIZERS tVoo{( X~

259

- Xq)I~ - E~O}{(X~ t X,) cos 50 t R, sin aD}]

(8.133 ) Since

Vi

JV} t Vq'

=

v:'t --

V'd t V'q

2V,t:. V, = 2VJ t:. Vd

+ 2V; t:. V,

VO

VO

V,

V,

t:. V, = ~ t:. Vd t ...'L t:. Vq

(8.134)

Substituting (8. 125) in (8.121) r esults in

[~~:] = ~ [-:~ ~,] X, t X, [ R,

t:.t:.~~ ] + [ t:.:~ 1

[

- 6. _

- R, Voo cos 00 t Voo(X, + X,) sin 5°) ] R,Voosin 5° t Voo cos 5°(X~ + X,)

1 [

X,R, -X~(X, t X,)

Xq ( R eVoo sin 00 + Voo cos aO(X~ + X, ) - X~( - ReVoo cos 00 + Voo(Xq t Xe)sin oo )

[ ~~~ 1t [ 6.:~ 1

1 (8.135)

Substituting (8.135) in (8.134) gives t:. Vi = Ks6.o t K6t:.E~

(8.1 36)

where

(8.1 37) (8.1 38)

CHAPTER 8. SMALL-SIGNAL STABILITY

260

The constants th at we have derived are called the K1-K6, developed by Heffron -Phillips [88], and later by DeMello-Concordia [78], for the study of local low-frequency oscillations. Example 8.5

In Figure 8.7 , assume that R e = 0, X e = 0.5 pu, V,LB = 1 L15° pu, and Voo LO° = 1.05 LO° pu. The machine data are H = 3.2 sec, T/u, = 9.6 sec, KA = 400, TA = 0.2 sec , R, = 0.0 pu, Xq = 2.1 pu, Xd = 2.5 pu , X~ = 0.39 pu, n 0, and w, 377. Using the flux- decay model, fin d (1) the initial values of state and algebraic variables , as well as Vref , TM , and (2) Kl-K6 constants.

=

=

1. Computation of Initial Conditions

The technique discussed in Section 7.6 is followed. The superscript on the algebraic and state variables is omitted. IGeh = (Id

0

+ jIq )ej (0 - "/2) = lLl5° ~ 1.05 LO° = 0.5443Ll8° JO.5

6(0)

= angle of E

= V,e j8 + (Rs + jXq)IG eh 1Ll5° + (j2.1)(0.5443 Ll 8°)

where E E =

.

= 1.4788L65.52°

Therefore 6(0) = 65 .520. Id + jIq = Ia ei'r e -j(6-.. / 2) = 0 .5443 L42.48°, Id = 0.4014, and Iq = 0.3676 . Vd

+ jV

q

= Ve j8 ,-j(6-,,/ 2) = 1L39.48°

Hence

Vd

= 0.77185,

Vq = 0.6358 1

From (8. 116): E~ = Vq +X~Id

= 0.63581 +

(0.39 )(04014) = 0.7924

8.6. POWER SYSTEM STABILIZERS

261

From (8.62)-(8.65), setting derivatives = 0,

Efd = E~

+ (Xd -

= 0.7924

Efd

Vref = V,

+ KA

w, = 377,

I'M

XWd

+ (2.5 -

0.39)0.4014 = 1.6394

1.6394 = 1 + 400 = 1.0041 =

E~lq

+ (Xq -

X~)ldlq

= (0.7924)(0.3676)

+ (2.1 -

0.39)(0.4014)(0.3676)

= 0.5436 This completes the calculation of the initial values .

2. Computation of K1-K6 Constants The formulas given in (8.130)-(8.134) and (8.137)-(8.138) are used. /::,. = R~

+ (Xe + Xq)(Xe + X~)

= 2.314 ~ = 1 + (Xd - X~)(Xq K3 /::,.

+ X e)

= 3.3707

K3 = 0.296667 K 4 = Voo(Xd6.- X~) [(X q

cO + X)· e SIn u -

Re

COS

CO]

U

= 2.26555 1

K, = /::,. [I;/::" - I~(X~ - Xq)(Xq

+ Xe) -

Re(X~ - Xq)I'd

= 1.0739

Similiarly, K

"

K s , and K6 are calculated as K, = 0.9224

Ks = 0.005 K6 = 0.3572

o

+ ReE~oJ

262

8.6.3

CHAPTER 8. SMALL·SIGNAL STABILITY

Synchronizing and damping torques

To the single. machine infinite· bus system of Figure 8.7 , we add a fast exciter wbose state· space equation is (8 .13 9) The linearized form of (8.139) is (8 .140 ) Then the machine differential equations (8.127)-(8.129), the exciter equation (8.140), and the algebraic equation (8.136) can be put in t he block diagram form shown in Figure 8.8. Both the normalized frequency 1/ and w in rad/ sec are shown.

(fad/sec) ~1'

-

A

~CO

r--,

!.

1-:;+_ 115 (fad)

o !:.E'q

~

I + SfA

Figure 8.8: Block diagram of the incremental flux.decay model with fast ex· citer (dotted portion represents the exciter) System loading, as well as the external network p arameter X., affect the parameters Kl -K 6. Generally these are > 0, but under heavy loading, K s might become negative, contributing to negative damping and instability, as we explain b elow .

8.6. POWER SYSTEM STABILIZERS

263

Damping of Electromechanical Modes There are two ways to explain the damping phenomena: 1. State-space analysis [89J

2. Frequency-domain analysis [78J 1. State-space analysis (assume that ClTM == 0)

Equations (8 .127)-(8.129) are rewritten in matrix form as

I [-

[

ClE~ Cl6 = Clv

K,~,;. 0

-If·

0

0

w,

---/if'

-,Iii

_ Dw . 2H

T""

I[~: I tI +[

U" (8.141)

Note that, instead of Clw, we have used the normalized frequency deviation Clv = Clw / w, . Hence, the last row in (8. 141) is Clv = _ K 2 ClE _ Kl Cl6 _ D w, Clv ' 2H q 2H 2H

(8.142)

ClEfd is the perturbation in the field voltage. Without the exciter, t he

machine is said to be on "manual control." The matrix generally has a pair of complex eigenvalues and a negative real eigenvalue. The former corresponds to the electromechanical mode (1 to 3-Hz range), and the latter the fiuxdecay mode. Without the exciter (i.e., K A = 0), there are three loops in the block diagram (Figure 8.8), the top two loops corresponding to the complex pair of eigenvalues, and the bottom loop due to ClE~ through K., resulting in the real eigenvalue. Note tha.t the bottom loop contributes to positive feedback . Hence , the torque-angle eigenva.lues tend to move to the left-half plane , and the negative real eigenvalue to the right . Thus , with constant Eld, there is "natural" damping. With enough gain , the real pole may go to the right-half plane . Tills is referred to as monotonic instability . In the power literature, this twofold effect is described in more graphic physical terms. The effect of the lag associated with the time constant T~o is to increase the dam ping torque but to decrease the synchronizing tor que. Now, if we add the exciter through the simplified representation, the state-space equation

CHAPTER 8. SMALL-SIGNAL STABILITY

264

will now be modified by making 6.E f d a state variable . The equation for 6.Efd is given by (8. 140) as . 1 KA 6.Efd =~ -6.Efd+ -(6.V f ~ 6.v,)

TA

TA

re

=~ ~6.Efd ~ KA K 5 6

K .4 K 66.E' TA 6. TA q

TA

6.V + KA TA ref

(8. 143)

Ignoring the dynamics of the exciter for the moment, if Ks < 0 and KA i5 large enough, then the gain through T,i" is approximately - (K4 + KA K 5)K 3 . Thi5 gain may become positive, resulting in negative feedback for the torqueangle loop and pushing the complex pair to the right-half plane. Hence, this complicated action should be studied carefully. The overall state-space model for Figure 8.8 becomes - 1 K3 T~Q

6. E'q 6.6 6.i;

~

6. Efd

.-p,

1

0

T'do

6.E'q

do

0

0

w,

0

6.6

-;11

-if!'

_Dw~

0

6.11

- K ~K ,

-K~K.

0

-1

6.Efd

1A

1A

2H

TA

0

+

0 0

6. Vref

~ (8 .144)

The exciter introduces an additional n egative real eigenvalue. Example 8.6 For the following two test syst ems whose K, - K6 constants and other paramet ers are given, find the eigenvalues for K A = 50. Plot the root locus for varying K A. Note t hat in system 1 K 5 > 0, and in system 2 K5 < o. Test System 1

Kl = 3.7585 K2 = 3.6816 K3 = 0.2162 K4 = 2.6582 K s = 0.0544 K6 = 0.3616 T~o = 5 sec H = 6 sec TA = 0.2 sec

8.6. POWER SYSTEM STABILIZERS

265

Test System 2

Kl = 0.9831 K3 = 0.3864 Ks = - 0.1103 T!w = 5 sec

K2

= 1.0923

K. = 1.4746 K6 = 0.4477 H = 6 sec TA = 0.2 sec

The eigenvalues for KA = 50 are shown below using (8 .144). Test System 1 -0.353 ± j10.946 -2.61 ± j3.22

Test System 2 0.015 ± j5.38 -2 .77 ± j2 .88

Notice that test system 2 is unstable for this value of gain. The root loci for the two systems can be drawn using MATLAB. An alternative way to draw the root locus is to remove the exciter and compute the transfer function AA~l(J) = H(s) in Figure 8.8. Note t hat H(s ) includes all the dynamics except that of the exciter. With the G(s) = feedback transfer function , as in Figure 8.9. /). Vret +

1:0""

we can view H(s ) as a

KA

I + sTA

-'-

l1Efd

l!.V,

If(s)

Figure 8.9: Small-signal model viewed as a feedback system

H (s ) can be computed for each of t he two systems as System 1

+

H s _ ( ) -

S3

0.0723s 2 7.2811 + 0.9251s 2 + 118.0795. + 47.74

CHAPTER 8. SMALL-SIGNAL STABILITY

266 System 2

2

H(s) _ -

S3

+

+

0.0895s 3.5225 0.5176s 2 30.886s 2

+

+ 5.866

The closed-loop characteristic equation is given by 1 + G(s)H(s) = 0, where G( s) = The root locus for each of the two test systems is shown in Figure 8.10. System 1 is stable for all values of gain, whereas system 2 becomes unstable for K A = 22 .108.

1.:'0·2..

10

5

- 10

- IS L-Jc,-"""--'--'-:-':-~-':--'-:--'---,~+-, - $ -4 .5 --4 -3 .S _3 - l.S - 2 -1 .5 ~1 -O.S 0 Real Axis

(. )

, 6

, 2

~• 11

0

-, .A

.;;

...

-5

-J

_2

-.

o

(»)

Figure 8.10: Root loci for (aJ test system 1 and (b) test system:2

o

8.6. POWER SYSTEM STABILIZERS

267

2. Frequency-doma.in analysis through block diagram

For simplicity, assume that the exciter is simply a high constant ga.in [(A, i.e., assume TA = in Figure 8.8. Now, we can compute the transfer function I1E~(s) / 115(s) as

°

I1E~(s )

(8 .145)

M (s)

This assumes that 11 v"ef = 0. The effect of the feed.back around TJo is to reduce the time constant. If Ks > 0, the overall situation does not differ qualitatively from the case without the exciter, i.e., the system has three open loop poles , with one of them being complex and positive feedback . Thus, the real pole tends to move into the right-half plane . If [(s < and, consequently, [(, + KAKS < 0, the feedback from 115 to I1T. changes from positive to negative, and, with a large enough gain K A , the electromechanical modes may move to tbe right-half plane and the real eigenvalue to the left on the real axis . The situation is changed in detail, but not in its general features , if a more detailed exciter model is considered. Thus , a fast-acting exciter is bad for damping , but it has beneficial effects also . It minimizes voltage fluctuations , increases t he synchronizing torque , and improves transient stability. With the time constant T A present ,

°

_ -[(K,(1 + sTA ) + KA[(S)]K3 115(5) - KAK6K3 + (1 + K3Tdos)(1 + sTA )

I1E~ ( 5 )

(8.146)

The contribu tion of this expression to the torque-angle loop is given by

I1T, (5) = K I1E~(5 ) ~ H( ) M(s)

2

M(s)

s

(8.147)

Torque-An gle Loop

=

Letting I1TM 0, the torque-angle loop is given by Figure 8.11. The undamped frequency of the torque-angle loop (D '" 0) is given by the roots of the characteristic equation 2H 2 5 w,

-

+K, Sl,2

-

(8.148)

°

±./K,W, J

2H

Tad/sec

(8.149)

CHAPTER 8. SMALL-SIGNAL STABILITY

268

,

K,

IV-

;/ Te -,,;.

,

/

ro,

1

,

2Hs

s

/

D

dO)

I

, /

1

- dT, )

211 ,

-.~ -+

W.

D5 + K 1

Figure 8.11: Torque-angle loop

With ahlgher synchronizing torque coefficient K , and lower H , 51 ,2 is hlgher. K, is a comp li cated expression involving loading conditions and external reactances . The value of D is generally small and, hence, neglected. We wish to compute the damping due to E~ . The overall block diagram neglecting damping is shown in Figure 8.12. From this diagram and the closed-loop t ransfer function, it can be verified th at the characteristic equation is given by

To rque-angle loop

I \..

/

dT,

1

,

0

I

2II s~ +K I

"'.,

I' H(s)

Figure 8.12: Torque-angle loop with other dynamics added

269

8.6. POWER SYSTEM STABILIZERS

(8.150)

H ( s )6.ij therefore con tributes to both the synchronizing torque and the damping torque. The contributions are now computed approximately. At oscillation frequencies of 1 to 3 Hz , it can be shown that K4 has negligible effect . Neglecting the effecl of K4 in Figure 8.8, we get from (8.147)

(8.151) Let s = jw. Then

H (jw) =

- K2KA K S

-c-:--------:-":-~~---:-,.,-----:-:-

(i, + KAKe -

+ jW(it, + T~o)

w2T~oKA)

- K2KAKs(x - jy)

(8 .152)

+ y2

",2

where X

=

1

2

I

Ka + KAKe - w TdoTA

Y = w

(~ + T~o)

(8. 153) (8 .154)

From (8 .150), it is clear that, at the oscillation frequency, if Im[H (jw) ) > 0, p ositive damping is implied, i.e., the roots move to the left-half plane. If I m [H (jw)] < 0, it tends to make the system unstable, i.e., negative damping results . Thus

' ) ] = - K 2KA K SX" ' . = contn'b' utlOll to t he synch rOlllzmg R e [H (JW ",2 + y2 torque component du e to H (s)

(8155 )

K sY" ' )] = + K 2KA ' . I m [H (JW 2 2 = contn'butlOn to d amplllg x

+y

torque component due to H(s)

(8 .156)

CHAPTER 8. SMALL-SIGNAL STABILITY

270 Synchronizing Torque

For low frequencies, we set w "" O. Thus, from (8.155): -K2 K S

Ks

for high KA

(8.157)

Thus , the to tal synchronizing component is Kl - Kk~' > O. Kl is usually high, so that even with Ks > 0 (low to medium external impedance and low-to-medium loadings), K, > o. With Ks < 0 (moderate to high external impedance and heavy loa.dings), the synchronizing torque is enhanced posit ively.

£:K.

Damping torque

(8. 158) This expression contributes to positive damping for Ks > 0 but negative damping for Ks < 0, which is a cause for concern . Further, with Ks < 0, a higher KA spells trouble (see Figure 8.10). This may offset the inherent machine damping torque D. To introduce damping, a power system stabilizer (PSS) is therefore introduced. The stabilizing signal may be Ll.v, tJ.Paoo , or a combination of both. We discuss this briefly next. For an extensive discussion of PSS design, the reader is referred to the literature [77, 78].

8.6.4

Power system stabilizer design

Speed Input PSS Stabilizing signals derived from machine speed, terminal frequency, or power are processed through a device called the power system stabilizer (PSS) with a transfer function G(s) and its output connect ed to the input of the exciter. Figure 8.13 shows the PSS with speed input and the signal path from tJ.v to the torque-angle loop.

8.6. POWER SYSTEM STABILIZERS

271

tl.TpSS

PSS

K2 +

K3

KA

7d;

I +.r TA

1+ s K3

+ ~Vrcf

Figure 8.13 : Speed input PSS Frequency-Domain Approach [78] From Figure 8.13, the contribution of the PSS to the torque-angle loop is (assuming ~Vref 0 and ~5 0)

=

=

~Tpss

(f,- + KA K 6) + s (ft + Tdo) + s2TdoTA = C(s)CEP(s)

(8 .159)

For the usual range of constants [78], the above expression can be approximated as (8. 160) For large values of KA (high giUn exciter) , this is further approximated by ~Tpss

~v

K2 C(s) = K s[I + s(T.io I KAK6) ][ I + sTA ]

(8.161)

If this were to provide pure damping throughout the frequency range, then C (.) should be a pure lead function with zeros, i.e., C(.) = Kpss[1 + s(Tdol KAK6)](1 + .TA) where Kpss = gain of the PSS. Such a function is not physically realizable. Hence, we have a compromise resulting in what

CHAPTER 8. SMALL-SIGNAL STABILITY

272

is called a lead-lag type transfer function such that it provides enough phase lead over the expected range of frequencies. For design purposes, G(s) is of the fo rm

G(s) = K

(1

+ .IT,) (1 + sT3 )

pss (1 + sT2 ) (1

sTw

+ sT.) (1 + sTw)

= K

pss

G (s) 1

(8.162)

The t ime constants T" T 2 , T 3 , T. should be set t o provide damping over the range of frequencies at which oscillations are likely to occur . Over this range they sho uld compensate for the phase lag introduced by the machine and the regulator. A typical technique [77J is to compensate for t he phase lag in t he absence of PSS such that the net phase lag is: 1. Between 0 to 4S 0 from 0.3 to 1 Hz

2. Less t han 90 0 up to 3 Hz Typical values of the parameters are:

K pss is in t he range of 0.1 to SO T, is T 2 is T.l is T. is

the the the the

lead time constan t, 0.2 lag time constant, 0.0 2 lead time constant, 0.2 lag time constant, 0.02

to to to to

1.S sec 0.15 sec 1.S sec O.IS sec

The desired sta.bilizer gain is obtained by fir st finding the gain at which the system becomes unstable. This m ay be obtained by act ual test or by root locus study. Tw , called the washout time constant, i s set at 10 sec. The purpose of this constant is to ensure that there is no st eady- st ate error of voltage r eference due to sp eed deviation . Kpss is set a t K pss, where Kpss is the gain at which the system becomes unstable [77J . It is important t o avoid interaction between the PSS an d the tor sional modes of vibration . Analysis h as revealed that such interaction can occur on nearly all modern excitation systems, as they have rela ti vely high gain at high frequencies. A stabilizer-torsional instability with a high-resp onse excit ation system m ay res ult in shaft damage, p articularly at light generator loads where the inherent mechanical damping is small . Even if shaft damage does not occur, such an instabilit y can cause saturation of the stabilizer output, causing it to be ineffective , and possibly causing saturation of the voltage regulator, resulting in loss of synchronism and tripping the unit . It is imperative that stabilizers do not induce torsion al instabilities. Hence,

3

8.6. POWER SYSTEM STABILIZERS

273

the PSS is put in series with another transfer function FILT(s) [77] . A typical value of FILT(s) '" 570;;~'+ " . The overall transfer function of PSS is G( s )FILT( s). Design Procedure Using the Frequency-Domain Method The following procedure is adapted from [90]. In Figure 8.13, let

6.Tpss = GEP(s)G(s) 6.v

(8.163)

where GEP(s) from (8.159) is given by

GEP(s) =

K2KA K 3 KAK3KS + (1 + sTdoK3)(1

+ sTA )

(8.164)

Step 1 Neglecting the damping due to all other sources, find the undamped natural frequency Wn in rad/sec of the torque-angle loop from

2H _s2 W,

+ Kl = 0

(8.165)

) l.e.

Step 2 Find the phase lag of GEP(s) at s = jWn in (8. 164). Step 3 Adjust the phase lead of G(. ) in (8.163) such that

LG(s) I.=iwn +LGEP(s)

I.=iw= n

0

(8.166)

Let

G(s) =

J(

pss

(1 ++ ST1)k 2 1

81

(8.167)

ignoring the washout filter whose net phase contribution is approximately zero . k = 1 or 2 with Tl > T 2 . Thus, if k = 1: Ll

+ jwnTl

= Ll + jwnT2 - LGEP(jwn )

(8.1 68)

274

CHAPTER 8. SMALL-SIGNAL STABILITY

Knowing Wn and LGEP(jwn ) , we can select T,. T, can be chosen as some value between 0.02 to 0.15 sec. Step 4 To compute Kpss, we can compute Kpss , Le., the gain at which the system becomes unstable using the root locus, and th en have Kpss = ~ Kpss . An alternative procedure that avoids having to do the root locus is to design for a damping ratio ~ due to PSS alone . In a second-order sys tem whose characteristic equation is 2H _5' + Ds + K, w,

= 0

(8.1 69 )

The dampi ng ratio is ~ = ~ D / VM K, where M = 2H /w,. This is shown in Figure 8.14. ~ ---- I ,

I

'

,

, / / /

/

/ / / / /

I /

X

Figure 8. 14: Damping ratio The charac teristic roots of (8.169) are

- it ± J(it)' -1fI 81,2

2

2~d ± J~ - C~I)' if -~wn ±jw V e

-

j

n

1-

4K, M

B.6. POWER SYSTEM STABILIZERS

We note that

Wn

=

J1fi..

~=

275

Therefore

D

2M Wn

= _D_

I._M_

2M 'V K,

=

D

C7 ;;2..; rK 'i7,=;M

(8.170)

Verify that

To revert to step 4, since the phase lead of G( s) cancels phase lag due to GEP(s) at the oscillation frequency, the contribution of the PSS through GEP(s) is a pure damping torque with a damping coefficient Dpss . Thus, again ignoring the phase contribution of the washout filter , (8.171) Therefore, the characteristic equation is 2

5

i.e .,

52

+ 2(Wn 5 + W~ =

+ Dpss + K, M 5 M

= 0

(8.172)

O. As a result, (8 .173)

We can thus find Kpss , knowing for ( is between 0.1 and 0.3.

Wn

and the desired ( . A reasonable choice

Step 5 Design of the washout time constant is now discussed . The PSS should be activated only when low-frequency oscillations develop and should be automatically terminated when the system oscillation ceases . It should not interfere with the regular function of the excitation syst em during steadystate operation of the system frequency. The washout stage has the transfer function

Gw(s) =

sTw

1 + sTw

(8 .174)

CHAPTER 8. SMALL-SIGNAL STABILITY

276

Since the washout filter should not have any effect on phase shift or gain at the osciilating frequency, it can be a chieved by choosing a large value of Tw so that sTw is much larger than unity. (8. 175) Hen ce , its phase contribution is close to zero . The PSS wiil not have any effect on the steady state of the system since, in st eady state,

6.v = 0

(8.176)

Example 8.7 The purpose of this example is to show that the int roduction of the P SS will improve the damping of the electromechanical mode. Without the PSS, the A m atrix, for example, 8.5 , is calculated as - 0.3511

o - 0.1678 - 714 .4

-0.236 0 -0.144 -10

0 0.104 377 0 0 0

o

-5

Tb.e eigenvalues are >'1,2 = -0 .0875 ± j7 .11 , >'3,4 = -2.588 ± j8 .495 . The electromechanical mode >' 1,2 is poorly damped. Instead of a two-stage lag lead compensator , we will have a single-stage lag-lead PS S. Assume th at the damping D in the torque-angle loop is zero. The input to t he stabilizer is 6.v . An extra state equation will be added. The washout stage is omitted , since its objective is to offset only the de steady-state error. Hence, it does not play any role in the design . The block diagram in Figure 8.15 shows a single lag-lead stage of the PSS . The added state equation due to the PSS is 1

6.iJ = -T- 6.y 2

+ Kpss 6.v + Kpss -TJ 6.v. T2

T2

- 1 - -6.y + Kpss 6.v T2

T2

(-K + Kpss -TJ - - 2 6.E , T2 2H q

-Kl 6.0 ) (8.177) 2H

The new A matrix is given as - 1 K3T~

1.ffd.'

0

0

0

377

0

0

-.J!'

-;ftl

0

0

0

K1o;;'

- KpK,

0

-J

TA

{f,:

- K,T, (~)

~

0

-K, T, 12

(4Jr) 2

1..4.

1~

2

1

Tdo

0

-1

T,

8 . 7. CONCLUSION

277 rd

( 1+57, J

uv

KpSS

--

( IH T2)

Il)"

f' , +

K"

+''/ -

1+ t'J :<\

PSS ~ VI

F igure 8.15: Exciter wi th PSS vVit h a choice of Kpss

= 0.5, T,

-0.3511 0 -0 .1678 - 714.4 0 .12

= 0.5, T2

- 0.236

0 0 377 - 0.144 0 - 10 a - 0.36 5

= 0.1, the

0.104

0

0

a

0

0 2000 - 10

-5

°

new A matrix is

a nd the ~i genvalue, are A' ,2 = -0 .8612 ± j 7.7012 A:<,4 = - 1.6314 ± .i8.55 01 , Ar, = -10 .3661. Not e t he improvement in damping of the electromechanical mode A [ ,2 '

o

8.7

Conclusion

In t his c:ha pter 1 we have discllssed li near models of single and multiInachine

system s with different degrees of mac hin e and load modeling. The effect of different ty pe, of loading on t he st eady-st at e stabili ty was discu ssed for t he lll u it imachine case ; Hopf bifu rcatio n in the cun t ex t of voltage collapse Wit, discussed . Th e design of a powe r- syst em st a bilizer f or dam ping the local mode of osci llation was discussed for the single-machine infinite-bus case .

8.8

Problems

8.1 The single line diagram fo r t he two-area system is gi ven in Figu re 8.1 6. Th e transmission line data , machine data , ex cit ation system_data, and lo ~rl-fl o w resillts arc given in T ables 8. 8 t o 8.11.

278

CHAPTER 8. SMALL-SIGNAL STABILITY

Area 1

1

e-H

;::

101

102

Area 2

3

13

112

111

1J

~ H-e

~ ~

~-.-J

12

2

Figure 8.16: Two-area system

Table 8.8: Transmission Line Data on 100 MVA Base

•

I

To Bus

From Bus Numb er

Number

Series Resistance (R.) pu

Series Reactance (X.) p u

Shunt Sus ceptance (B) pu

1 2 3 3 3 3 3 11 12 13 13 101 101 111 111

101 102 13 13 13 102 102 111 112 112 112 102 102 112 112

0.001 0.001 0.022 0.0 22 0.022 0.002 0.002 0.001 0.001 0.002 0.002 0.00 5 0.005 0.005 0.005

0.012 0.012 0.22 0.22 0.22 0.02 0.02 0.0 12 0.012 0.02 0.02 0.05 0.05 0.05 0.05

0.00 0.00 0.33 0.33 0.33 0.03 0.03 0.00 0.00 0.03 0.03 0.075 0.075 0.0 75 0.0 75

8.8. PROBLEMS

279

Table 8.9: Machine Data

I, Variable

Machine at Bus 1 0.022 0.00028 0.2 0.033 8.0 0.19 0.061 0.4 54.0 0.0

X, (pu ) R . (pu) Xd (pu) X~

(p u)

T~o (sec)

Xq (pu) X~ (pu) T~o (pu ) H (sec) D (pu)

Machine at Bus 2 0.022 0.00028 0.2 0.03 3 8.0 0.19 0.061 0.4 54 .0 0.0

Machine at Bus 11 0.022 0.00028 0.2 0.033 8.0 0.19 0.061 0.4 63.0 0.0

Machine at Bus 12 0.022 0.00028 0.2 0.033 . 8.0 0.19 0.061 0.4 63 .0 0.0

Table 8.10: Excitat.ion Sys tem Dat a Variable KA (pu) TA (pu)

at Bus 1 200

Machine at Bus 2 200

at Bus 11 200

Machine at Bus 12 200

0.0001

0.0001

0.0001

0.000]

Machine

Machine

Table 8.11: Load-Flow Results for the System I

I

VQl!a.gc

Angle

Real

M&f;~)Ud e

(degrees)

P ower

1.03 1.01

l.OJ 1- 01 1. 0108 0.9876

8.21 ~4. -1. 00H 0.0 -lIJ. 20t.l 3 . 16616 - 6 .2433

PQ

1.0096 O. 98S0

-U .9 H3

PQ

0.9 76 1

PQ

0 .9116

Bua Bua Type N u m b er !

I

I I

, t

II 12 lOt

10' III ll2

, "

P PV S Wing PV

PQ PQ PQ

Re active P o wer

G e n. ··(~u)

Go n.·I~ u ) 1.3386

- 14 . H94

7.0 7.0 7 .217:2 7.0 0 .0 0. 0 0. 0 0.0 0 .0

-:.13 . 2 \,12 2

0.0

0. 0

-4. .69 71

1.5920 1 .H66 1.!O83 0.0 0. 0 0 .0 0. 0 0 .0

Real

Reao;live

P o;"(~ru) Lo&d pu

Lo&d"(;u)

0.0 0.0 0.0 0.0 0 .0 0 .0 0. 0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0 .0 0.0

11.5,," lO.n

2.12

P o w "r

2 . lI!!

(a) Using the two-axis model for the generator and constant power load representation , obtain eigenvalues of the linearized

CHAPTER 8, SMALL-SIGNAL STABILITY

280 system ,

(b) Repeat (a) wit h one tie line out of service, Any comments? 8.2 With three tie lines in ser vice, add a PSS a t bus 12 with the following parameters, Kpss = 25, Tw = 10 sec, T, = 0,047 sec, T2 = 0,021 sec, T3 = 3.0 sec, and T4 = 5.4 sec, What are the new eigenvalues? 8,3 Repeat Problem 8,2 with two tie lines in service, 8.4 Consider the single machine connected to an infinit e bus in Figure 8,7, Assume that V"" = LO. The parameters are as follows: Line: R,

= 0,0, X,

= 0.4 pu

Generator: Xd = L6 sec, Xq = 1.55 pu, X:, H = 3.0 sec

= 0.32

Injected power into the bus: P = 0.8 pu , Q

= 0.4

Exciter: KA

= 50, TA

pu, Tdo

= 6,0 sec,

pu

= 0.05 sec

(a) Compute the KI-K6 constants. (b) Compute the eigenvalues, (ans: -14.5662, -5,7351, -,0808 j8,55)

±

8.5 In a single-machine-infinite-bus system (Figure 8,17), there is a local load at the generator bus, The parameters are Line: RE = -,034 pu, X E = 0,977 pu Generator: Xd = 0.973 pu, Xq = 0,550 pu, X:, = 0.230 pu, Tdo = 7.76 sec, H = 4.63 sec lnjected power: P

= LO pu, Q = 0,015 pu

Generator terminal voltage: V, = L05 pu Local load (constant impedance): G = 0.249 pu, B = 0.262 pu

= 50, TA = 0,05 sec Assuming V, = V,LO°, compute V""

Exciter: KA

(a) = V"" L{3, (b) Compute the Theverun equivalent looking into the external network from the generator bus as V th = V/x, L{3' in series with an impedance Zth' Show that

R

'X _

e+ J

e -

RE + jXE

1+ (RE

+ jXE)(G + jB)

281

8.8. PROBLEMS

-

V,

I

Figure 8.17: Single-machine infinite-bus case (localiDad) and

(c) Compute 6(0). (d) To apply the results of Section 8.6.2, we set (3' = 0 and replace 6(0) by 6(0) - {3" This makes the infinite bus a reference bus with phase-angle zero. (e) Compute the KI-K6 constants and the eigenvalues. -10.316 ± j3.2644, -0.2838 ± j4.9496)

(ans:

8.6 For each of the two test systems below (Example 8.6), whose Kl-K6 constants and other parameters are given:

(a) Write the state space model in the form (8.144). Assume D O. (b) Plot the root locus as KA is varied from a small to a high value. At what value of KA does instability occur and what are the unstable eigenvalues. (c) Find the eigenvalues at KA = 50 . Test System 1

Kl K6

= 3.7585, K2 = 3.6816, K3 = 0.2162, K4 = 2.6582, = 0.3616, Tdo = 5 sec, H = 6 sec, TA = 0.2 sec.

Ks

= 0.0544,

Ks

= -0.1103,

Test System 2

Kl K6

= 0.9831, K2 = l.0923, K3 = 0.3864, K4 = l.4746, = 0.4477, Tdo = 5 sec, H = 6 sec, TA = 0.2 sec.

CHAPTER 8. SMALL-SIGNAL STABILITY

282

8.7 A single machine with a flux-decay m odel and a fast exciter is connected to an infinite bus through a reactance of jO.5 pu. Th e generator t erminal voltage is 1L15° and the infinite bus voltage is 1.05LOo The para.meters and initial conditions of the state variables are given below. Parameters

= 3.2 sec, T~o = 9.6 sec, K A = 400, TA = 0.2 sec = 0.0017 pu , Xq = 2.1 pu, Xd = 2.5 pu, Xd = 0.39 pu H

R,

D '" 0, W, = 377 r ad/sec Initial conditions using the flux-decay model and the fast exciter

6(0)

-

= 0.7719 , Vq(O) = 0.6358

0.3999,1q(0) = 0.3662

Id(O) E~(O)

Vref

65.52°, Vd(O)

0.7949, Efd(O) -

= 1.6387,w(0) = 377 rad /sec

1.0041 , TM = 0.542

(a) Comput e the Kl-K6 constants and the undamped natural frequency of the torque-angle loop . (b ) Compute the eigenvalues . 8.8 Find the particip ation' factors of the eigenvalues for the following systems, where x = Ax.

(a)

(b)

Chapter 9

ENERGY FUNCTION METHODS 9.1

Background

In this chapter, we discuss energy function methods for transient stability analysis. In transient stability, we are interested in computing the critical clearing time of circuit breakers to clear a fault when the system is subjected to large disturbances. In real-world applications, the critical clearing time can be interpreted in terrus of meaningful quantities such as maximum power transfer in the prefault state. The energy function methods have proved to be reliable after many decades of research [93, 96] . It is now considered a promising tool in dynamic security assessment .

9.2

Physical and Mathematical Aspects of the Problem

The ultimate objective of nonlinear dynamic simulation of power systems is to see whether synchronism is preserved in the event of a disturbance . This is judged by the variation of rotor a~gles as a function of time. If the rotor angle O. of a machine or a group of machines continues to increase with respect to the rest of the system, the system is unstable. The rotor angle O. of each machine is measured with respect to a fixed rotating reference frame that is the synchronous network reference frame . Hellce, instability of a machine means that the rotor angle of machine i pulls away from the rest

283

CHAPTER 9. ENERGY FUNCTION METHODS

284

of the system . Thus, relative rotor angles rather than absolute rotor angles must be monitored to test stability/instability. Figure 9.1 shows the rotor angles for the cases of stability and instabilit y. Figure 9.1(a) shows that all relative rotor angles are finite, as t -. 00. In Figure 9.1(b), the rotor angle of one machine is increasing with respect to the rest of the system; hence, it is a single-machine instability. Figure 9.1(c) is a group of two machines going unstable with respect to the rest of the' system.

Rotor Angles

t J

8, (a) S table

I'

Rotor

Angles

i

Rotor

Angles

I);

i

8,

(b)

Un.~ta.ble

(c) Unstable •t

Figure 9.1: Behavior of rotor angles for the (aJ stable, and (bJ, (c) the unstable cases In its simplest form , a power system undergoing a disturbance can be described by a set of three differential equations:

-

f[(x(t))

- oo
(9 .1)

x(t) -

fF (x( t))

O < t~tcl

(9.2)

itt)

f(",(t))

tcl
(9.3)

x(t)

-

x(t) is the vector of state variables of the system at time t. At t = 0)

a

9.2. PHYSICAL AND MATHEMATICAL ASPECTS

285

fault occurs in t he system and the dynamics change from fl to fF . D uring o < t < tel, called the faulted period, t he system is governed by the fault-on dynamics fF A ctually, before the fault is cleared at t = tel, we may have several switchings in the network, each giving rise to a different fF For simplicit y, we have t aken a single fF, in dicating that there are no struct ural changes bet ween t 0 and t = tel . When the fault is cleared at t = tel , we have t he postfault system with its dynamics f (x( t )). In the prefault p eri od - 00 < t :S 0, t he system would h ave settled down to a steady state , so that x(o) = :to is known. Therefore, we need not discuss (9 .1) . We then have ouly

=

x(t) = fF(x(t))

0 < t :S tel

x(o)=xo

(9.4)

and

x(t) = f (x(t))

t> tel

(9.5)

with the initial condition x(tel) for (9.5) provided by the solution of the faulted system (9.4) evaluated at t = tel. Viewed in ano ther manner, t he solution of (9.4) provides at each instant of time the possible initial conditions for (9.5). Let us assume that (9.5) has a stable equili brium point x,. The question is whether the trajectory x(t) for (9.5) with the initial condition x( tel) will converge to x, as t ..... 00. Th e largest value of tel for which this holds t rue is called the critical clearing time t" . From this discussion , it is clea r that if we have an accurate estimate of the region of attraction of the p ostfault stable equilibrium point (s.e.p) x., t hen t er is obt ained when the trajectory of (9.4) exits the region of attraction of (9.5) at x = x * . Figure 9.2 illustrates this concept for a two-dimensional syst em. The computation of the r egion of attraction for a general nonlinear dynamical system is far from easy. It is not, in general, a closed region. In the case of power systems with sim ple-machine models , the characterization of t his region has been discussed theoretically in t he literature. The stability region consists of surfaces passing throngh the unstable equilib rium p oints (u.e .p's ) of (9.5). For each fault, the mode of instability (i.e., one or more machines going unstable) may be different if the fault is not cleared in time . We may des crib e the interior of the region of attraction of the postfault system (9.5 ) thr ough an inequality of the type Vex) < VeT> where Vex) is

CHAPTER 9. ENERGY FUNCTION METHODS

286

the Lyapunov or energy functi on for (9.5). V(x) is generally the sum of the kinetic and potential energies of the post faul t system. The computation of Vcr , called t he critical energy, is different for each fault and is a diffi cult st ep . There are currently three basic methods, with a number of variations on each m ethod . M, jX';

&H

•

j X, In fmile Bus

Figure 9.2: Region of attraction and computation of tcr These methods are: 1. Lowest energy u.e. p method [99]

Vcr = V(x=) , where x uc is the unstable equilibrium p oint (u.e.p) resulting in the lowest value of Va among the u.e.p's lying on the stability boundary of (9 .5). This requires the computation of many u.e .p 's of the p ostfault system and, hence, is not comput ati onally attractive. Moreover, it gives conservative resul ts. Reference [99J was the first systematic application of Lyapunov's m etho d for transient st ability. 2. P otential Energy Boundary Surface ( PEBS ) method

=

Vcr maximum value of the p otential energy component of V(x) along the trajectory of (9 .4). Tb is is kno wn as the potential energy boundary surface (P EBS ) method [lOOJ. 3. Con trolling u.e.p method

Vcr = V(XU), at whi ch

is the relevant or controlling u.e.p, i.e. , th e u.e.p closest to the point where t he fault-on traj ectory of (9. 4) exits the region of attraction of (9.5). This is called the controlling n.e. p method first proposed in [97]. The boundary-controlling n.e.p (BCU ) method [98J (also called the exit -p oint method) is an efficient technique to compute this controlling u.e .p. XU

We will discuss the PEB S method (2) and t he controlling u.e.p method (3) in detail. The computation of tcr involves the following steps.

9.3. LY.4.PUNOV'S METHOD

287

1. Computing x" the stable equilibrium point of t he postfault system

given by (9.5) . 2 . Formulating V( x) for (9.5). This is not a difficult step . Generally, V(:z:) is the sum of kinetic and p otential energies ofthe postfault system, i.e., V(x) = VK E + VPE· 3. Computation of Va-. In the P EBS method (2), Vcr is obtained by integrating the faulted t raject ory in (9.4) until the potential energy part VPE ofV(x) reaches a maximum Vf!'Eax.. This value is taken as Va- in the PEBS method. In the cont rolling u.e.p method (3) we integrate (9.4) for a short interval followed by either a minimization problem to get the controlling u .e.p, or integration of a r educed· order postfault system after the PEBS is reached ( BC U method) [98] to get the controlling u.e.p x U . V CT is given by V(x u ) = VPE(X u ), since VKE is zero at an u.e.p. 4. Calculating the time instant t eT when V(x) = Va- on t he faulted tra,jectory of (9.4). The faulted trajectory has to b e integrated for all the three methods to obtain t CT • In the PEBS method (2), the faulted tra· jectory is already available while computing Vcr. It is also available in the BCU method. The comput ation time is least for the PEBS method (2). T he relative merits of t hese various m ethods and their variations are discussed extensively in the literatu re [101].

9.3

Lyapunov's Method

In 1892, A . M. Lyapunov, in his famous P h.D . dissertation [118], proposed that the stability of the equilibrium point of a nonlinear dynamic system of dimension n

:i:

= f(x)

, f (O)

=0

(9.6)

can be ascertained without numerical integration. He said that ifthere exist s a scalar function V( x) for (9.6 ) that is positive· definite, i.e., V( x) > around the equilibrium point "0" and the derivative V( x) < 0, then the equilibrium is asymptotically stable. V(",) is obtained as ~i=1 ~~:i:i = ~i=1 ~~f;(x) \7VT . f(x) wher e n is the order of the system in (9.6). Thus, f(x) enters directly in the computation ofV(x). The condition V(x) < 0 can be relaxed

°

=

CHAPTER 9. ENERGY F UNCTION METHODS

288

to Vex) < 0, provided that Vex) does not vanish along any other solution with the exception of x = o. Vex) is act ually a generalization of the concept of t he energy of a system . Since 1948, when the results of LyapunoV' appeared in the English language together with potential applica.tions, there has been extensive liter ature surrouncling this topic . Application of the energy function metho d to power system stability began with the early work of Magnusson [102J and Aylet t [103], followed by a formal application of the m ore general Lyapunov's method by EI-Abiad and N agappan [99J. Reference [99J provided an algorithmic procedure to compute the criti cal clearing time . It us ed t he lowest energy u.e.p metho d to compute V= . Although many different Lyapunov fun ctions have been t ried since t hen , the first integral of motion, which is the sum of kinetic and potential energies, seemed to have provided the bes t result. In the power literature, Lyapunov's method has become synonymous with t he transient energy funct ion (TEF) method and has been applied successfully [93 , 98J. Today, this technique has proved to be a pract ical tool in dy namic security assessment . To m ake it a practical tool , it is necessary to compute the region of stability of the equilibrium p oint of (9 .5). In physical sys tems , it is finite and not the whole state-space . An estimate of the region of stability or attraction is characterized by an inequality of the type V(x) < Vcr. The computation ofV= remained a form id a ble barrier for a long time. In the case of a multimachine classical model with loads b eing treated as constant impedances, there are well-proved algorithms. Extensions to multimachine systems with detailed m odels have b een m ade [104, 106J .

9.4

Modeling Issues

In applying the TEF technique, we must consider the model in two time frames, as follows: 1. Faulted system

(9.7) 2. Postfault system :i;

= f(x(t)) , t > tel

(9.8)

In reality, the model is a set of differenti al-algebraic equations (DAE), i.e.,

9.4. MODELING ISSUES

289

x

fF(X(t),y(t))

o

gF(X(t), y(t)) , 0 < t ::: tel

(9.9) (9.10)

and

x

f(x(t), y(t))

(9.11)

o

g(x(t), y(t)), t > tel

(9.12)

The function 9 represents the nonlinear algebraic equations of the stator and the network, while the differential equations represent the dynamics of the generating unit and its controls. In Chapter 7, the modeling of equations in the form of (9.9) and (9.10) or (9 .11) and (9.12) has been covered extensively. Reduced-order models , such as a flux-decay model and a . classical model, have also been discussed. In the classical model representation , we can either preserve t he network structure (structure-preserving model) or eliminate the load buses (assunling constant impedance load) to obtain the internal-node modeL These have also been discussed in Chapter 7. Structure-preserving models involve nonlinear algebraic equations in addition to dynanlic equations, and can incorporate nonlinear load models leading t o the concept of structure-preserving energy function (SPEF) V(x, y), while models consisting of differential equations lead only to closedform types of energy functions V(x). The work on SPEF by Bergen and Hill [104] has been extended to more detailed models in [105]-[108]. It is not clear at this stage whether a more detailed generator or load model will lead to more accurate estimates of t er . What appears to be true , however) from extensive simulation studies by researchers is that , for the so-called first-swing stability (i.e., instability occurring in 1 to 2 sec interval ), the classical model with the loads represented as constant impedance will suffice. This results in only differential equations, as opposed to DAE equations. Both the PEBS and BCU methods give satisfactory resnlts for this modeL We first discuss this in the mult imachine context. The swing equations have been derived in Section 7.9.3 (using P=, = TMi) as

(9.13)

290

CHAPTER 9. ENERGY FUNCTION METHODS

where

m

Poi = EtGii

+ 2) Gij sin Oij + Dij cos Oij)

(9.14)

;==1 f.i

Denoting 2H; 11,.1,

II Jv[.1

d2/j. Jv[;

dt 2'

and p'.1.

do .

II

Pml. - E~G " we get \ Ul

m

+ D; dt' = Pi - 2::=

(C,j sin oii

+ Dij cos Oij)

(9.15)

;=1#

which can be written as

(9.16) be the rotor angle with respect to a fixed reference. Then 0, = D:i -Wat. 6; =
D:,

(9.17)

Equations (9.17) and (9.18) are applicable both to the faulted state and the posHault state, with the difference that Poi is different in each case, because the internal node admittance matrix is different for the faulted and postfault system. The model corresponding to (9 .17)-(9.18) is known as the internal-node model since the physical buses have been eliminated by network reduction.

9. 5. ENERGY FUNCTION FORMULATION

9.5

291

Energy Function Formulation

Prior to 1979 , there was considerable research in constructing a Lyapunov function for the system (9.15) using the state-space model given by (9.1 7) and (9.18) [109] - [11 2]. However, analytical Lyapunov fun ctions can be constructed only if the transfer conductances are zero, i.e., D ,j ;: O. Since these terms have to be accounted for properly, the first integrals of motion of the system are constructed , and these are called energy functions . We have two op tions, to use either t he relative rotor angle formulation or the center of inertia formulation . We use the latter, since there are some advantages. Since the angles are referred to a center of inertia, the resulting energy functi on is called the transient energy function (TEF). In this formulati on , the angle of the center of inertia (COl ) is used as the reference angle, since it represents the Hmean moti on" of the system. Although the resulting energy function is iden tical to V( 5, w) (using relative rotor angles), it has the advantage of being more sy=etric and easier t o handle in terms of the path-dependent t erms . Synchronous st ability of all machines is j udged by examining the angles referenced ouly to COl instead of relative rotor angles. Modern literature invariably uses the cor formulation. The energy function in the COl notation, including D ,j terms (transfer conductances) , was first proposed by Athay et a1. [97]. We derive the transient energy fun ction for the conservative system (assuming Di = 0). The cent er of inertia (COl) for t he whole system is defined as 1

60

1

m

= -- I:: Mi 6i and MT .=1

the center of speed as Wo

m

= - - I:: MiWi

(9.1 9)

MT i=1

where MT = 2::Z;1 Mi . We then transform the variables 6i , Wi to the variables as (Ji = 6i - 00 l Wi = Wi - Wo' It is easy to verify

cor

ei = 6. - 50

The swing equations (9. 15) with Di

Pi 10.

t

. J::::l:;>!i

= fire)

i

= 0 become (omitting the algebra):

(Cij sin eij

+ Dij cos eij) -

= 1, ... , m

MM; Pea J T

(9. 20)

292

CHAPTER 9. ENERGY FUNCTION METHODS

where m

m

Pi=Pmi - E!G,i ; Peo/=L Pi - 2 L i=l

m

L

DijCOSOij

i = l j=i+ l

If one of the machines is an infinite bus, say, m whose inertia cons tant Mm is very large , then tJ~ Peol '" 0 (i '" m) and also 00 ' " Om and Wo '" "'m . The COl variables become Oi = Oi-Om and Wi = Wi - Wm' In the literature where the BeU method is discussed [98], Om is simply taken as zero. Equation (9 .20) is modified accordingly, and there will be only (m - 1) equations after omitting the equation for machine m. We consider the general case in which all MIs are finite . Corresp onding to the faulted and t he postfault states, we have two sets of differential equations ,

MdWi = fF(O) I dt 1

dO. = dt

Wi ,

O < t ~ tel

i = 1,2 , . .. , m

(9.21)

and

dw ' M ,' -dt' = li (O) dOi dt

= Wi

J

t > tel i = 1,2 , . "lm

(9 22)

Let the p ostfault system given by (9 .22) have the stable equilibrium point at 0 = 0' , w = O. 0' is obtained by solving the nonlinear algebraic equations

Ii(O) = 0 , i = 1, . . . , m

(9 .23)

Since E:', M;O; = 0, Om can be expressed in terms of the other Ois and substituted in (9.23), which is then equivalent to

h(O" . . . , Om-i) = 0 , i = 1, . .. , m - 1

(9.24)

The basic procedure for computing the critical clearing time con sists of the following steps : 1. Construct an energy or Lyapunov function V( O,W) for the system (9.22) , i.e., t he postfault system.

9.5. ENERGY FUNCTION FORMULATION

293

2. Find the critical value of V(O,w) for a given fault denoted by Vcr . 3. Integrate (9.21) , i.e., the faulted equations, until V(O,w) = Vcr . This instan t of time is called the critical clearing time tor .

While this procedure is common to all the methods , they differ from one another in steps 2 and 3, i.e., -finding Vcr and integrating the swing equations . There is general agreement that the first integral of motion of (9.22) constitutes a proper energy function and is derived as follows [94J. From (9.22) we have, for i =1 , ... ,m

dt

= Midwi = dOl = M 2dW 2 = d0 2 = .. . Mmdwm = dO m h(0)

WI

fl(O)

W2

fm (B)

Wm

(9.25)

Integrating the pairs of equations for each machine between (0:, 0), the postfault s.e.p to (0., Wi) results in

I1;(B,..;:,) =

~Miwl - ].8 2

6~,

i

MB)dB;, i

= 1, .. . , m

(9.26)

This is known in the literature as the individual mach.ine energy functon [113]. Adding these functions for all the machines, we obtain the first integral of motion for the system as (omitting the algebra):

(9.27) 1

m

- "2 L M;w; 8.+8 . ].

L

p.( fJ; - Oil -

'

J

8/+9;

= VKE (W)

m

L L

[C;j(cos B;j - cos 0lj )

t= l j=i+ 1

i=l

i= l

_

m-l

Tn

]

D '].. cos 0·,·d(fJ , 1· + fJ ]·)

+ VPE(fJ)

SInce

M . ].8i L -' PCO[dO; = 0 m

i==l

MT 81

(9.28)

(9.29)

CHAP TER 9. ENERGY FUNCTION METHODS

294

Note that (9.28) contains path-dependent integral terms. In view of trus, we cannot assert that Vi and V are positive-definite. If D ,j _ 0, it can be shown that V(B,w) constitutes a proper Lyapunov fu nction [93, 109,110).

9.6

Pot e ntial Energy Boundary Surface (PEBS)

We first discuss the PEBS method because of its simplicity and its natural relationship t o the equal-area criterion. Ever since it was first proposed by Kakimoto et al. [100] and Athay et al. [97), the m ethod has received wide attention by researchers b ecause it avoids computing the con trolling (relevant) u .e.p and requires only a quick fault -on system integration to compute Vcr. We can even avoid computing the postfault s.e.p., as discussed in Section 9.6 .5. In trus section, we will first motivate the method through application to a single-machine infinitebus system , establish the equivalence between t he energy function and the equal-area criterion, and , finally, explain the multimacrune PEES method.

9 .6.1

Single-machine infinit e-bus system

Consider a single-machine infinite-bus system (Figure 9.3). Two parallel lines each having a reactance of XI connect a generator having t ransient reactance of Xd through a transformer wjth a reactan ce of X, to an infinite bus whose voltage is EzLO". A three-phase fault occurs at t he ruiddle of one of the lines at t = 0, and is subsequently cleared at t = tet by opening the circuit breakers at both ends of the faulted line. The prefault, fault ed, and post fault configurations and their reduction to a two-machine equivalent are shown in Figures 9.4, 9.5, and 9.6. The electric p ower P e during and postfault states are E,X ~" sin 6' X E, E, ,in, and E, E, .in' faulted P refaultl ' P X I r espectively. The computation of Xl for t he prefault system and X for the postfault system is straightforward, as shown in Figures 9.4 and 9.6. 1

M, j Xd

•

H-Di-- -....:J_' x---','----_n-~___4_

~<<::>--4 ~

I <: ;? jXt

~

jX,

1-Cl---....:..,.'---o-4 \ F

Inflnite Bus

Figure 9.3: Single-machine infinite-bus system

9. 6. P OTENTIA L ENERGY BO UNDA RY SURFACE (PEBS)

295

=

Ih. LO'

=

=

Figure 9.4: Prefault system and its two- machine equivalent

jXl

j

xi

jX, )

&2

E1Lli

. Xl 2

j-

E2LO ~

F

--

"

(0)

jXl

j(Xd + X,) Xl j2

ElLa

"

"

::

)

. Xl 2

1:;2

::

::

(b)

(0)

=

Figure 9.5: Faulted system and its two-machine equivalent

LO·

296

CHAPTER 9. ENERGY FUNCTION METHODS

Figure 9.6 : Pastfault system and its two-machine equivalent Example 9.1 Compute XF for th e faulted system in Figure 9.3. XF is t h e reactance between the internal node of the machine and the infinite bus. It can be cornpuleu fur this lletwork by p~rfunIlillg a Y - 6 trans[uIIIlation in Figure 9.5(b) . It is more instructive to illustrate a general method that is applicable to the multimachine case as well. Figure 9.5 ( a) is redrawn after lab eling the various nodes (Figure 9.7) . The point at which t he fault occurs is labeled node 4. Xg = Xd + X, .

2

4

Figure 9.7: Fault ed system There are current injections at nodes 1, 2, and 4, and none at node 3. The nodal equation is Yl l

-

a

a

Y22

Y31 0

Y32 Y42

Y13

a

Y23 Y 24 Y33 Y34 Y4 2 Y44

9.6. POTENTIAL ENERGY BOUNDARY SURFACE (PEBS) Since the fault is at node 4 with the fault impedance equal to zero , V. Hence, we can delete row 4 and column 4, resulting in

297

= O.

Node 3 is eliminated by expressing V3 from the third equation in terms of E] and E2 and substituting in t he first two equations. This results in (omitting the algebra):

XF can be computed from the off-diagonal entry as

1 jXF

Now

-1 ) (-1 ( -jXg jXg

1 +-1 ) + -jX. j~

- 1 (

- 1) -jX]

1 j(Xl

+ 3Xg)

Hence

We do n ot require the other elements of the reduced Y matrix.

o

298

9.6.2

CHAPTER 9. ENERGY FUNCTION METHODS

Energy function for a single-machine infinite-bus system

The energy function is always constructed for the postfault system. In the SMIB case, the postfault equations are

d'o

Mdt-2 = P

Tn

- pmax sin 0 e

(9.30)

where p;;uax = Ef', 0 is the angle relative to the infinite bus, and ~~ = w is the relative rotor-angle velocity. The right-hand side of (9.30) can be written as

-aVpR

ali

J

where

(9.31) Multiplying (9.30) by ~:, it can be rewritten as

d [M 2 (dO)' dt + VPE(o) 1= 0

dt l.e.

J

~ [~MW2 + VPE(O)] dt 2

= 0

l.e. ,

d -[V(o,w)] = 0 dt

(932)

Hence J the energy function is

(9.33) It follows from (9.32) that the quantity inside the brackets V( 0, w) is a constant. The equilibrium point is given by the solution of 0 = p= - p;;uax sin 0,

i.e ., 0' = sin -1 (p~x). This is a stable equilibrium point surrounded by

9.6. POTENTIAL ENERGY BOUNDARY SURFACE (PEBS)

299

two unstable equilibrium points OU = 1f - 0' and 5u = -1f - 0'. If we make a change of coordinates so that VPE = 0 at 0 = 0', then (9.31) becomes

VPE( 0,6') = -Pm ( 0 - 0') - p,;uax( cos 0 - cos 0')

(9.34)

With this, the energy function V( 0, w) can be written as 1

V(o,w) = 2Mw2-Pm(0-6')-P,;uax(coso-coso') = VKE

+ VPE( 6,0')

(9.35)

where VKE = ~MW2 is the transient kinetic energy and VPE(O, 0') = -Pm(o0') - p,;uax( cos 6 - cos 0') is the potential energy. From (9.32) it follows that V(6,w) is equal to a constant E, which is the sum of the kinetic and potential energies, and remains constant once the fault is cleared since the system is conservative. V( 0, w) evaluated at t = tel from the faulted trajectory represents the total energy E present in the system at t = tel . This energy must be absorbed by the system once the fault is cleared if the system is to be stable. The kinetic energy is always positive, and is the difference between E and VPE(6, 6'). This is shown graphically in Figure 9.8, which is the potential energy curve. VpE

t ----------------

, ~---

,'(b ),, ,I , , ocl

I

8U=n_o s

--'0

Figure 9.8: Potential energy "well" At 0 = 0', the postfault s.e.p, both the VKE and the VPE are zero, since w = 0 and 6 = 6' at this point. Suppose that, at the end of the faulted period t = tel, the rotor angle is 6 = 6el , and the velocity is wei. Then

el + Vel = VKE PE

(9.36)

300

CHAPTER 9. ENERGY FUNCTION METHODS

This is the value of E. There are two other equilibrium points of tbe system (9 .30 ), namely, OU = 7r - 0' and Ju = -7r - 0'. Both of these are unstable and, in fact, are Type 1 (saddle-type) e.p.'s . T ype 1 e.p.'s are characterized by the fact that the linearized system at that e.p. has one real eigenvalue in the right-half plane. The potential energy is zero at 0 = 0' and has a relative maximum at AU and 0 Ju. At the point (a), and wei are known from the faulted trajectory ; hence, V(ocl,wei) = E is known. This is shown as point (b). If E < VPE(Ou) , t hen since the system is conservative, the cleared system at point (a ) will accelerate until point (b ), and then start decelerating. If E > VPE(OU), then the cleared system will accelerate beyond 0" and, hence, the system is unstable. VPE( ou) is obtained from (9.34) as -P=(7r - 20') + 2P,;nax cos 0'. If 0 decreases due to deceleration for t > 0, then the system is unstable if E > V(6 U). The points OU and Ju constitute the zero-dimensional PEBS for the single-machine system. Some researchers restate the above idea by saying that if the VPE is initialized to zero at oct, VtfE represent s the excess kinetic energy injected into the system. Then stability of the system is determined by the ability of the postfault system to absorb this excess kinetic energy (i.e. , the system is stable if VPE(OU) - VPE (Ocl) > VtfE) ' Most of the stability concepts can be interpreted as if the moment of inertia M is assumed as a particle that slides without friction within a " hill" with the shape VPE(O ). Motions within a p otential "well" are bounded and, hence, stable. It is interesting to relate the p otential "well" concept to the stability of equilibrium points for small disturbances. Using (9.31), (9.30) can be written as

a=

oct

=

d2 0 M-= dt 2

(9.37)

We can expand the right-hand side of (9.37) in a Taylor series about an equilibrium point 0', i.e. , 0 = 0* + /';0 , and retain only the linear term. Then

(9.38)

s· i.e. ,

9.6. POTENTIAL ENERGY BOUNDARY SURFACE (PEES)

301

/':,0 = 0

(9.39)

If 8':,~B < 0, the equilibrium is unstable. If 8':,~E > 0, then it is an oscillatory system, and the oscillations around 0' are b ounded. Since there is always some positive damping, we may call it stable. In the case of (9 .30), it can be verified that 0' is a stable equilibrium point and that both OU and 6u are unstable equilibrium points using this criterion . The energy function, Lyapunov function, and the PEBS are thus all equivalent in the case of a single-machine infinite-bus system. It is in the case of multimachioe systems and oonconservative systems that each method gives only approximations to the true stability boundary! In the case of multimachine systems, the second derivative of Vp E is the Hessian matrix.

15'

15'

Example 9.2 Consider an SMIB system whose postfault equation is given by d25 . do 0.2= 1 - 2 Sill 0 - 0.02dt 2 dt The equilibrium points are given by 5' 7r -

~

= 56'11", gu = -

71'" -

~

=

= sin- 1 (n. Hence,

0'

=

~, 5u =

-~7r . Linearizing around an equilibrium point

"0" results in

d2 /':,0 0 d/':,o 0.2 - 2- = - 2 cos 0 /':,0 - 0.02 - dt dt

This can be put in the state-space form by defining /':,0 , /':,w state variables.

o - lOcos5°

/':,5

as the

1 - 0.1

For 5° = 0' = ~, eigenvalues of this matrix are A1,2 = - 0.05 ± j2.942. It is a stable equilibrium point called the focus. For 5° = 5u or 6u , the eigenvalues

CHAPTER 9. ENERGY FUNCTION METHODS

302

are )" = 2.993 and )" = -2 .8 93. Both are saddle points. Since there is only one eigenvalue in the right-half plane , it is called a T ype 1 u.e.p.

o Example 9.3 Construct the energy function for Example 9.2. Verify the stability of the equilibrium points by using (9.39) . T he energy function is constructed for the undamped system, i.e., the ~oefficient of ~~ is set equal to zero . M = 0.2, P= = 1, p:;nax = 2, 0' = ~. The energy function is 7r

1

V(o,w) = 2 (0 .2)w' - 1(0-7r/6) - 2(cosO-cos"6) = O.lw' - (0 -

~) - 2(coso - (0.866) )

i)- 2(coso ° 0' °< o.

VPE (O,O') = - (0 2

8 VPE(0 , 0')

80

= 2 cos

At 0 = 0' = 7r / 6, 2 cos 0 > 0; hence = 0" = S6~ or g" = -~" , 2 cos equilibrium points.

0.866 )

°

is a stable equilibrium point . At Hence , both 0" and gu are unstable

o 9.6.3

Equal-area criterion and the energy function

The prefault , faulted, and postfault power angle curves P e for the singlemachine infinite-bus system are shown in Figure 9.9. The system is initially at = 0° . We shall now show that the area A, represent s the kinetic energy injected into the sy stem during the fault, which is the same as V!fE in Figure 9.8. A, represents the ability of the postfault system to absorb th.is energy. In terms of Figure 9.8, A2 represents VPE(O")- VpE (6cl). By the equal-area criterion, the system is stable if A, < A 2 . Let the faulted and postfault equations, respectively, be

°

2

M ddt 0 = Pm. 2

-

pF·

c e Sllu

(9.40 )

9.6. POTENTIAL ENERGY BOUNDARY SURFACE (PEBS) /

303

Pre-fault

P,max

Pos l-fault

Faulted

8U

Figure 9.9: Equal-area criterion for the SMIB case and

(9. 41 ) where

and

pmax = E,E2 e

X

The area A, is given by

(9.42)

CHAPTER 9. ENERCY FUNCTION METHODS

301

Hence, AJ is the kineti c energy injected into the system due to the fault. Area A2 is gi ven by

- P~(b"

_ VPE(a U )

-

ael )

-

VPE(a el )

from (9 .34). If we add area A3 to both sides of the criterion AJ < A 2 , the result is

(9.43 ) Now

A3 =

1"'·'

= - Pm Changing bel, wel to any

A,

+ A3

( P;'"' sin a-

P~)do

(ocl _ b') - r;'ax (cosa el a, w iUld ad ding A,

cosb')

(9 .44 )

to A 3 , gives

1

= 2Mw2 - P~(o - 6·') - p;nax( coso - coso' )

(9.45 )

This is the same as V(b , w) as in (9.35) . Now, frolll Figure 9.9:

A2 I- A3 = r - "(P;naXsino - Pm)db

1"

=:

2P;laXcos6s - Pm(7r -

2o~)

(946)

The right-hand side of (9.46 ) is also verified to be the sum of the areas A2 and A 3 , for which analytical express ions have been derived. It may be verified from (9.35) that

V(8,w)

I,.," w:..:O

(9.4 7)

9.6. POTENTIAL ENERGY BOUNDARY SURFACE (PEBS)

305

+ A2

< A2 + A 3 ,

Thus, the equal-area criterion Al < A2 is equivalent to Al which in turn is equivalent to V(o,w)
(9.48)

where Vcr = VPE(OU). Note that li,w are obtained from the faulted equation. Example 9.4 For Example 9.2, (1) state analytically the equal-area criterion. (2) If the faulted system is given by 0.2d2 o

do

dt

dt

"---,-:;;-- = 1 - sin 0 - 0.022

compute tcr using the results of part (1) . Note that the energy function is for a. conservative system, while the faulted system is not conservative.

The energy function is given by

V(O,w) =

~(0.2)W2 -

1(6 -

~) -

7r

2(coso - cos

+ 1.732

= 0.1w 2

-

(6 - -) - 2cos6 6

= 0.1w 2

-

li - 2 cos 6 + 2.256

V(6 ,0) = -P=(7r - 20') U

_ (7r _ = -2.09

~)

+ 2P;:oaxcos6'

2;) HCos~

+ 3.464 =

1.374

Hence, the equal-area criterion is

V(6,w) < 1.374 where 6, ware calculated from the fault-on trajectory. To obtain t~, the faulted equations are integrated and V(6,w) are computed at each time point . When V(6,w) = 1.374, the time instant is t~.

o

306

CHAPTER 9. ENERGY FUNCTION METHODS

9.6.4

Multirnachine PEBS

In the previous section, we mentioned t h at the points aU and gu were the zero-dimensional PEDS for the SMIB system. In the case of multimachine systems, the PEDS is quite complex in the rotor-angle space. A number of unstable equilibrium points surround the stable equilibrium point of the postfault system. The potential energy boundary surface therefore constitutes a multidimensional surface passing through the u.e.p 's. The theory behind the characterization of the PEDS is quite detailed, and is dealt with in the literat ure [93,94]. We can extend the concept of computing Vcr using the PEDS method for a multimachine system as follows.

In the previous section, we showed that in the SMID case Vor = VPE(a

U

),

i.e., V(5 ,w) is evaluated at the nearest equ.ilibrium point (SU, 0) if the ma-

chine loses synchronism by accelerat ion. aU is therefore not only the nearest, but also the relevant (or controlling), u.e.p in this case. In the case of the multi machine system, depending on the location and nature of the fault, the system may lose synchronism by one or more machines going unstable. Hence, each disturbance gives rise to what is called a mode of ins tability (MOl) [114]. Associated with each MOl is an u.e .p that we call the controlling u .e.p for that particular disturbance. A number of u.e.p's surround t he s.e.p of the postfault system. Mathematically, these are the solutions of (9.23). From the prefault s.e.p, if the faulted system is integrated and cleared critically, then the postfault trajectory approaches a particular u.e.p depending on the mode of instability. This u.e.p is called the cont rolling u.e .p for that disturbance. In the multimachine PEDS, we can visualize a multidimensional potential "well" analogous to Figure 9.8 for the SMID case. For a three-machine system, one such "well" 15 shown.in Figure 9.10 where the axes are the COl-referenced rot or angles 01 , O2 of two machines. The vertical axis represents VPE(O). Equipotential contours are shown, as well as the three nearby u.e.p's U1 , U2 , U3 . The dotted line connecting these u.e.p's is orthogonal to the equipotential curves and is called the PEDS. If at the instant of fault-clearing the system state in the angle space has crossed the PEDS, the system will be unstable. If the fault is cleared early enough, then the postfault trajectory in the angle space will tend to return to equilibrium event ually because of the damping in the system. The critical clearing time t er is defined to be the time instant such that the postfault trajectory just stays within the "well." It is a conjecture that the critically cleared traject ory passes "very close" to the controlling u.e.p. This is called the "first swing" stable phenomenon. To find the critical value of V (a,w), the fault -on trajec-

9.6. POTENTIAL ENERGY BOUNDARY SURFACE (PEBS)

307

tory is monitored until it crosses the PEBS at a point 0'. In many cases,

ou,

the controlling u.e.p, is close to 0', so that VPE(Ou) '" VPE(O') " V~. This is the essence of the PEBS method. A key question here is the detection of the PEBS crossing. This crossing is also approximately the point at which VPE(O) is maximum along the faulted trajectory. Hence, V~ can be taken as equal to VP'iX( 0) along the faulted trajectory. It can be shown [97] that the PEBS crossing is also the point at which fT(O) . (0 - OS) = O. f(O) is the accelerating power in the postfault system . That this is the same point at which VPE( 0) is maximum has been shown to be true for a conservative system [93J. In recent years , this PEBS crossing method has been the basis of improved algorithms such as the "second kick" method [ll51. We now explain the basic PEBS algorithm in detail. It will help in understanding the newer algorithms.

-4

-, .

_s· 200

F igure 9.10: The potential energy boundary surface (reproduced from [97])

CHAPTER 9. ENERGY FUNCTION METHODS

308

The energy function given by (9.29) is repeated here:

1

V(O,w)

m

m

= -2 ~M;w; . 1.=1

m

m

~P;(Oi - On - ~ ~ [C;j(COSOij - COSOfj)

.

.

..

l=13=1.+1

t=l

(9.49) The last term on the right-hand side of (9.49) denoted by Vd(O) is a pathdependent term . This can be evaluated using trapezoidal integration as m-l

m

Vd(O) = ~ ~ I;j

(9.50)

i=l j=i+l

where at the kth step

1 . Iij(k) = I;j(k - 1) + -D;j[cos(Oi(k) - OJ(k)) 2

[Oi(k)

+ OJ(k) -

+ cos(O;(k -

1) - 8j (k - 1))]

Oi(k - 1) - OJ(k - 1)]

(9.51)

with

Iij(O) = 0 This evaluation of Iij(O) is correct when the postfault system is the same as the prefault system, but is somewhat inaccurate if there is line-switching. This is explained in the next section. Equation (9.49) is rewritten as

(9.52) where (9.53)

Vp ( 0) = - ~ Pi( 0; - Oil - ~ ~ Cij ( cos Oij - cos O:j) i=l

i=l j=i+l

9,6. POTENTIAL ENERGY BOUNDARY SURFACE (PEBS)

309

and Vd(O) is given by (9.50). It can be shown [97] that the points 0 on the PEBS are defined by l:~, f;(O)(O. - On = O. This is the characterization of the PEBS. In vector form, this can be written as fT(O) . (0 - 0') = O. Denoting 8-8' = ii, we can show by analogy to the zero transfer conductance case that inside the PEBS F(O).{) < 0, and outside the PEBS it is > 0 [93]. In the absence of transfer conductances, f(8) = aV~:(8). When 8 is away from 0', within the potential multidinlensional "well," aV',;~(8) (which is the

0'»

gradient of the potential energy function) and iJ (i.e., (8 are both > O. Hence, fT(O) . iJ < 0 inside the "well." Outside the "well," 0 - 0' is > 0 and av (e) a~ < 0, resulting in f T (0)·0• > O. On the PEBS, the product f T (0)·0• is equal to zero. The steps to compute t~ using the PEBS method are as follows: 1. Compute the postfault s.e.p 0' by solving (9.23) .

2. Compute the fault-on trajectory given by (9.21). 3. Monitor fT(O) . ii and VPE(O) at each time step. The parameters in f( 0) and VPE( 0) pertain to the postfault configuration. 4. Inside the potential "well" fT(O). {) < O. Continue steps 2 and 3 until fT(O). (} = O. This is the PEBS crossing (t·,O·,w*). At this point, find VPE(O*). This is a good estimate of V~ for that fault. 5. Find when V(e,w) = Vcr from the fault-on trajectory. This gives a good estimate of ter. One can replace steps 3 and 4 by monitoring when VfJ'iX(O) is reached, and taking it as Ver. There will be some error in either of the algorithms.

9.6 .5

Initialization of VPECB) and its use in PEBS method

In this section , we outline a further simplification of the PEBS method that works well in m any cases, particularly when 0' is "close" to 80 • While integrating the faulted trajectory given by (9.21), the initial conditions on the states are 0;(0) = Of and Wi(O) = O. In the energy function (9.29), the reference angle and velocity variables are 0: and W;(O) = O. Thus, at t = 0, we evaluate VPE(O) in (9.29) as

CHAPTER 9. ENERGY FUNCTION METHODS

310

The path-dependent integral term in (9.54) is evaluated using the trapezoidal rule:

~Dij

[,j(O) =

[cos( Oi' - OJ)

+ cos(O; -

Oil] [(Oi'

+ OJ) -

(Oi

+ OJ )]

(9.55)

If the postfanlt network is the same as the prefault network , then K = O. Ot herwise, this value of K should be included in the energy function. If one uses the potential energy boundary surface (PEBS) method, then when the postfault network is not equal to the prefault network, this term can be subtracted from the energy func tion , i.e. ,

V (O,w)

=

VKE (W)

+VPE(O) -

VPE(OO)

(9.56)

Hence, the potential energy can be defined, with 0° as the datum, as

VPE(O)

6

VPE (O) - VPE (OO)

=-

[t, t

= -

L 18'' f,(O)dO, m

i=l

J;(O)dO, -

t, l;r

f,(O )dO,] (9.57)

8° i

If the path-dependent integral term in (9.57) is evaluated, using trapezoidal integration as in (9.51), [,j(O) = O. At the PEBS crossing 0* , VPE(O*) gives a good approximation to Vcr' The PEBS crossing has been shown as approximately the p oint at which the p otential energy VPE reaches a maximum value. Hence, one can directly monitor VPE and thus avoid having

9.6.

POTENTIAL ENER.GY BOUNDARY SURFACE (PEBS)

311

to m onitor the dot product fT (0) . (0 - 0') as in step 4 of the previous sectio n. This leads t o the importan t advantage of not having to compute Ir. In fas t screening of contingencies , this coulcl result in a significant saving of compu tation . On large-scale system s, this has not been investigated in the literature so fa r. Example 9.5 Compute the Yint for Example 7.1, usin g the classical mo del for the fault at bus 7 follo wed clearing lines of 7- 5. Using the PEES method, comput e ler. Use fT(O) . iJ as the criterion for PEBS crossing.

Y int wit h fault at bus 7 (faulted system )

=

In the y,:'ue; m a tri x of Example 7.6, si nce Vs 0, we delete the row and column corresponding t o b us 5 . Then eliminate aU buses except the internal no des 10 , 11, and 12. The resul t is

yfnt

0.6588 - j3.8 175 0.0000 - jO.OOOO = 0.0000 - j O.OOOO 0.0000 - j5.4855 [ 0.0 714 -j jO.6296 0.0000 - jO.OOOO

0.0714 + jO.6296 0.0000 - jO.OOOO 0.1750 - j2 .7966

J

Y int with lines 7- 5 cleared (postrault system) Ybll s = Y N is firs t computed with lincs 7- 5 removed, and the rest of the steps a re as in Example 7.6 . Buses 1 to 9 are eliminated, resulting in

+

1.1411 - - j2.2980

0.1323 jO.7035 Yint = 0.1323 + jO.7035 0.3810 - j2 .0202 [ 0. 1854 + j 1.0611 0.1965 + j 1. 2031

=

0.1854 -I· j1.0611 0. 1965 + j1.2031 0.2723 - j2.3544

=

J

The initial rotor angles are 51 (0) 0.0396 rad , 52 (0) 0.344 rad , and 63 (0) = 0.23 rad. The COA is calculated as 50 = T 2:l=l M;6;(0) = 0.116 rad, where MT = Ml + M2 + M 3 · Hence, we have 01 (0) = 61 (0) - 60 = -0.0764 rad, O2 (0) 52 (0) - 50 0.229 Tad, and B3 ( 0) = 53(0) - 50 0.114 rad, W, (0) = W2(0) = W3(0) = O. The postfault s.e.p is calculated as OJ = -0.1649, O2 = 0.4987, B~ = 0.2344 . The steps in computing ter are given below.

=

=

J

=

CHAPTER 9. ENERGY FUNCTION METHODS

312

1. From the entries in Yint for faulted and postfault systems, the a.ppropriate C.; and D,;'s are calculated to pu t the equations in the form of

(9 .21) and 19.22).

2. V(O,w) is given by VKE+ VPE(O), where VKE = tM,w; and VPE(O) is given by (9.49) . The path-integral term is evaluated as in (9.51), with 1,;(0) = 0, and the term (9.55) is added to VPE(O) . 3. The faulted system corresponding to (9 .21) is integrated and at each time step V(O, w) as well as VPE(O) are computed . Also the dot product p '(U) ·8 is monitored. The plots of V(O,W) an d VPE(O) are shown in Figure 9.11. Figure 9.12 shows the plot of jT(O) . O.

4. Vf!kax = 1.3269 is reached at approximate by 0.36 sec. Note from Figure 9.12 that the zero crossing of JT(O) . 0 occurs at approximately the same time. 5. From the graph for V(O,w),

tCT

= 0.199 sec when V(O,w) = 1.3269.

I

: . . '-" ...._.i S ......_..... ...! ...... ......•.•..... - .--"'---'-" -'" . - " ' ,i ..- - -,.-

. 4 _ .. _-_.,/_ ...,

3

.... ... ...... : ...

H

••• _

' ! j

i .-'····

,.-(

i ,..-_. -t···__·-.. . . L .............. ~........ ·······"i'··:~;/:.: . . 1···-· .

iI

r

.1'.'

H

•

I'

· ··r·_··+· . ·,·······- r)::.··/+· · ··--1-.- . I

/

.

······t-_····· ·······T ...... ....; ...... _-- .......... :.•",~'+. . . .-··. _·t-_· .-.- -1-"'" . _-, :

2

;

i,

1

I

I

' ,

:.--i--~l,~t~~ -t~. Time(scc)

Total and potential energies: (a) V(6, w) (dashed line); (b) VPE(O) (solid line) Figure 9.11:

9.7. THE BOUNDARY CONTROLLING U.E.P (BCU) METHOD

313

3

...... . + .. .. . . . . . . . .... . .- --j

2.5 2

1.5

.... !

T' ............_. ". -----..-

'? . . _.-

•••• _H"

... _, ._.

.....

.§

j ~

til

"

j "

0.5

~

.

-..

0

- ._-

-0.5

.... .;. ....

",.

-

•......... _.. .....•.~ .

. . ... .

-;-. ..... .

~

..... t; ... ,' ...........: .... ,..

-1

M

• • • • • ••

- 1. 5 ':----:-:.-;:---.,..,...---::-':-:---::'::--~:_;_-_:_:-_:__::_:_-_;:

o

0.05

11.1

0.1 5

0.2

0.25

0.3

0.35

(),4

Tillle(sec)

Figure 9.12: The monitoring of the PEBS crossing by fT(O) . iJ

o

9.7

The Boundary Controlling u.e.p (BCU) Method

This method [98] provided another breakthrough in ap plying energy function m ethods to stability analysis aft er the work of Athay et al. [97], which originally proposed the con trolling u.e.p method. The equations of the postfault system (9.22) can be put in t he state-space form as

() = Wi

Mi ~i = fi (O)

aVPE(O) aOi

i = 1) .. " m

Now

.

av·

av .

V(O,w) = ao 8 + aw w

(9.58 )

314

CHAPTER 9. ENERGY FUNCTION METHODS

= -

m

m

i= ]

i=l

2: fi(IJ)Oi + 2: M,w/;)i

m

(9.59)

=0

Hence, V( 0, w), is a valid energy function. The equilibrium points of (9.58) lie on the subspace IJ,w such that e.Rm, w o. In the previous section, we have qualitatively characterized the PEBS as the hypersurfaces connecting the u .e.p's. We make it somewhat more precise now. Consider the gradient system

=

. IJ

=

-iWPE(O)

(9.60 )

8IJ

Note that the gradient system has dimension m, which is half the order of the system (9.58 ). It has been shown by Chiang et al. [98J that the region of attraction of (9.58) is the union of the stable manifolds of u .e.p's lying on the stability boundary. If this region of attraction is projected onto the angle space, it can be characterized by

(9.61) where IJF is an u.e.p on the stability boundary in the angle space. The stable manifold W'(IJr" ) of Or" is defined as the set of trajectories that converge to IJr" as t --> +00. Since the gradient of VPE(O) is a vector orthogonal to t he level surfaces VPE(O) = constant in the direction of increased values of VPE (O), t h e PEES in the direction of decreasing values of VPE(O) can be described by the differential equations iJ aV~~(8) f(IJ) . Hence , when the fault -on trajectory reaches the PEBS at e = 0* corresponding to t = t*, we can integrate the set of equations for t > t* as

=

iJ

=

frO), B(t*) = IJ*

=

(9 .62)

where f(IJ) pertains to the postfault system . This will take IJ(t) along the PEBS to the saddle points (u.e .p 's U1 or Uz in Figure 9.10 depending on

9.7. THE BOUNDARY CONTROLLING U.E.P (BCU) METHOD

315

0') . The integration of (9.62 ) requires very small time steps since it is "stiff." Hence , we stop the integration until II f(O) " is minimum. At this point , let 0 =0 0ltpp ' If we need the exact 0" , we can solve for f (O) =0 0 in (9.23) using Oltpp as an initial guess . The BCU method is now explained in an algorithmic manner. Algorithm 1. For a given contingency that in volves ei ther line switching or load/ generation change, compute the postdisturbance s.e.p . 0' as follows. The s.e.p and u.e.p 's are solutions of tbe real power equations

MO) Since Om

=0

=0

M. p. - p•• (O) - - Peor (O)

M~

MT

=0

0 i

=0

1, .. . , m

(9.63)

z=:::;;:1 M.O. , it is sufficient t o solve for 1;(0) = 0 i=ol, ... , m-l

(9.64)

witb Om being substituted in (9 .64) in terms of 01 , ... ,Om- 1. Generally, the s.e.p. 0' is close to 0° the prefault e.p . Hence, using 0° as the starting point , (9.64) can be solved using the Newton-Raphson method. 2. Next , compute the controlling u.e.p . 0" as follows:

(a) IntegTatethe faulted system (9 .21 ) and compute V(O,W) = VKE(W) + VPE (O) given in (9.52) at each time step. As in the PEBS algori thm of the previous section, determine when the PEBS is crossed at 0 =0 0' corresponding t o t =0 t'. Tills is best done by ftnding when fT(O) . (0 - 0') =0 O. (b) After the PEBS is crossed, the faulted swing equations aTe no longer integrated . Instead, the gTadient system (9 .62) of the postfault system is used. This is a reduced-order system in that ouly the 0 dynamics are considered as explained earlier, i.e., for t > t'

iJ = f(O) ,

ott')

=0

o·

(9 .65)

CHAPTER 9. ENERGY FUNCTION METHODS

316

Equation (9.65) is integrated while looking for a minimum of m

E l /i(B) 1

(9.66)

i=l

=

At the first minimum of the norm given by (9.66), B Bapp and VPE(B app ) Vcr is a good approximation to the critical energy of the system. The value of /1':i p p is almost the relevant or the controlling u.e.p.

=

=

(c) The exact u.e.p can be obtained by solving I(B) 0 and using Oapp as a starting point to arrive at B" . Note that since I( B) is nonlinear, some type of minimization routine must be used to arrive at B". Generally, Bapp is so close to B" that it makes very little difference in the value of Vcr whether B" or Bapp is used .

3. Vcr is approximated as Vcr = V(O",O) = VPE(B") . Because of the path.dependent integral term in VPE, this compu· tation also involves approximation. Unlike computing VPE(B) from the faulted trajectory where B was known, here we do not know the trajectory from the full system . Hence, an approximation has to be used. The most convenient one is the straight·line path of integration. VPE(B") is evaluated as [97]:

m

rn

m-l

-E

Pi(Or -

On - E

E

[eij cos (Bij - cos Bfj)

i=l j=i+l

( B1' - 0» + (B" - B') ( )] (Br - Bn _ (OJ _ OJ) Dij sm Bij - sm °ij 1.

1.

' ]

'U'6

(9 .67)

We derive the third term of (9.67) as follows. Assume a ray from Of to and then any point on the ray is Bi = Bi + p(B; <:: p < 1). Thus, d(Oi + OJ) dp(B)' - Bf + Bj - Bj). The path· dependent term Vd(B) in (9.49) is now evaluated at 0" as Vd(B) L~ll LT~i+l Iij.

or

onto

=

Iij = [(B)' - Bn

=

+ (OJ -

Oil] [

Di; cos[(Bi - OJ)

9.7. THE BOUNDARY CONTROLLING U.E.P (BCU) METHOD

317

+p((O? - on - (OJ - OJ))]dp (Or - on -/- (OJ - OJ) (Or-Ot) - (Oj-Oj)

D [. (0' 0') ij

sm

+p ((Or - on - (OJ - OJ))

on

on

i -

j

]1:::

(Oi + (0'1 [. nu (Or - on _ (OJ _ OJ) Dij sm Vij

.

-

,]

sm Bij

(9.68)

4. To compute t eT> we go back to t he already-computed value of V(B , w) from the fault-on trajectory in the case of a fault, or the postdisturbance trajectory in the case of load/generation change, and find the time when V (0, w) = Ver. In the case of a fault this time instant gives ter and the system is stable if the fault is set to clear at a time t < ter . In the case of load/ generation change, the system is stable if V(O, w) < Ver for all t. Example 9.6 Compute t cr using the BCU method for Example 9.5. Use Oapp instead of the exact controlling u.e.p OU to compute Vcr. The PEBS crossing is computed as in Example 9.5. t' is obtained as 0 .3445 sec and OJ' = -0 .7319 rad, 0, = 2.413 rad and 0; = 0_618 rad. The gradient system for t > t' is given by

iJ = flO), O(t' ) = 0' where I ( 0) is given by (9 .30) with the parameters Cij and Dij pertaining to the postfault system. The equations are now integrated until II flO) II = L~~ l I MO) I is minimum (Fig. 9.13). The minimum obtained is II I II 0.985 . At this point, OaPPI = -0.9894 rad, eaPP2 = 2.124 rad, and 0~pp3 = 1.164 rad. With this, Vcr = VPE(Oapp) is calculated using (9.67) as 1.3198. Hence , from the fault-on trajectory in Figure 9_11 , ter = is between 0 .196 and 0.199 sec . In this case, ter by both the PEBS and BCU method is about the same. The effect of loading, as well as the closed-form approximation to the p ath-dependent integral on tCT> is discussed in [120].

=

o

CHAPTER 9. ENERGY FTJNCTION METHODS

318

J.:! ].1 5

11

.-

.~ . -------_ .. .......

..L.

... , ...

lJJ.~

.-.................

~

0

E 0

"/.

.1..., ,

o.<JS

.1

I). !J

!

.. _,

OX:;

.j....

n , /{

0..:;:

1t.7

0.6

tl.l<

I.l

0.9

1.2

1.1

1..

1.5

TillLC(SCC)

Fi gure 9.13: Plot

9.8

II

II

frO)

as a function of time in the gradien t system

Structure-Preserving Ene rgy Functions

The structure-preserving model (7.20 1)-(7.204) is reproduced below .

flt =

Wi. -

(9.69 )

W!I

n + :rn

Mi W; = TM i-

L

V;VjB; jsin(Oi - Oj)

j~l

i=n+ l, .. . ,n +m

(9 70)

n+m

L

V;VjBijsin(B;-Oj ) i= 1, . . ,n

(9. 71 )

j=l

n+m

QdVi)

=-

L

V;VjB;j COS(Oi - OJ) i

= 1, .. . , n

(9.72)

j= l

Note that the Iotor angles 8i 's are also uenoted as Oi'S. Here, we are assuluing constant real pO'Yver loads so that PLi (Vi ) = PLi, and reactiv e power as

9.9. CONCLUSION

319

nonlinear voltage-dependent loads. If the angles are referred to the COl 00 = 2:::1 Migi, then the angles referenced to COl become ei = gi - 00 (i = 1, ... , n + m). It can be shown [94] that the energy function is given by

xi:r

(9.73) where

(9.74)

~t

Bii(V/ - (lit?)

(9.75)

1

n+m,-l n+17l

I: I: i=l

Bij(ViV, cos eij - V:V; cos eij)

(9.76)

j=1. 11

n

- I:h(ei - en

(9.77)

1

This energy function has been used for both transient-stability and voltagestability analyses. In the second case, only the P E term is used, along with the concepts of high- and low-power flow solutions [117].

9.9

Conclusion

In this chapter, we have discussed

detail the transient energy function method for angle stability. The basis of the method has been shown to be the famous Lyapunov's direct method [118]. The equivalence of the energy function to the equal-area criterion has been shown for the single-machine case. For the multimachine case, the PEBS and the BCU have been explained in detail. The TEF method can be used to act as a filter to screen out contingencies in a dynamic security assessment framework [119] . The BCU method is known to be sensitive with respect to the PEBS crossing g*. III

CHAPTER 9. ENERGY FUNCTION METHODS

320

The method can be made more robust by tracking the stable manifold to the controlling u.e .p using what is called "shadowing" method [121J. Finally, the structure-preserving energy function has been derived. The TEF area is an active area of research.

9.10

Problems

9.1 A single machlne connected to an infinite bus has the following faulted and postfault equations. Faulted d 2 /j 0.0133 dt 2 = 0.91

0

:s tet

Postfault d2 6

.

0.0133 dt 2 = 0.91 - 324 sm 6

t

> tet

The prefault system is the same as the postfault system. (a) Find V(o , w) and Vcr using the u.e .p formulation. (b) Explain stability t est of (a) using the equal-area criterion . Sketch the areas A A 2 , A 3 . " (c) Find tcr using Vcr. (d) Find te< using PEBS method . 9.2 For the 3-machine system of Example 7.1, a fault occurs at bus 7 and is cleared at tet with no line switching. (a) Based on prefault load flow and using the classical model, compute the voltages behind tra.nsient reactances and the rotor angles at t = 0-. (b) Find Yint for the faulted and the postfault cases . (c) Write the energy function V(O,w), assuming Gij = O(i

i

j).

(d) Write the faulted equations in COl notation , together with the initial conditions . (e) Compute tor using the PEBS method. (f) R epeat (c) , (d) , and (e), assuming G ij

i o.

9.10. PROBLEMS

321

9.3 For Problem 8.1 using the classical model, compute Yint for prefault and Yint for a fault at bus 112. 9.4 Compute for Problem 8.1 the voltages behind transient react ances and the rotor angles at t = 0-. (a) Using results of Problem 9.3, write the energy funct ion V(O,W) . Assume that the fault at bus 112 is self-clearing. (b) Using the PEBS method, compute tor' 9.5 Use the BCU method to compute tor for Problems 9.2 and 9.4. Do this nrst for G i j = 0 (i i- j) and then for Gij i- O. Compare the results with the PEBS method.

Appendix A

Integral Manifolds for Model Reduction A.1

Manifolds and Integral Manifolds

The teTm "manifold" in this chapter refeTs to a functional relationship between variables. FOT example, a manifold fOT z as a function of x is simply another term for the expression

z

= h(x)

(A. I)

When x is a scalar, the manifold is a line when plotted in the z, x space. When x is two-dimensional, the manifold is a surface that might appear as in Figure A.I To define an integral manifold, we have to intToduce a multidimensional dynamic model of the form

dx dt

= f(x, z)

x(o) = xo

(A.2 )

dz dt

= g(x, z)

z(o)

= Zo

(A.3)

An integral manifold for z as a function of x is a manifold

z

= h(x)

which satisfies the differential equation for z. manifold of (A.2)- (A.3) if it satisfies

8h 8xf(x,h) = .9(x,h) 323

(A.4) Thus, h(x) is an integral

(A.5)

324

APPENDIX A. INTEGRAL MANIFOLDS FOR MODEL

z

Figure A.l: Geometrical interpretation of a manifold If the initial conditions on x and z lie on the manifold (ZO = h(xO)), then t he integral manifold is an exact solution of the differential equation (A.3), and the following reduced-order model is exact:

dx dt = f(x , h(x))

A.2

x(o) = XO

(A.6)

Integral Manifolds for Linear Systems

The concept of integral manifolds is illustrated in this section through a series of examples . Consider the oversimplified system

dx dt dz dt

x(o)=xo

- x +z

-

- 10z + 10

z(o) = ZO

(A.7) (A.B)

We say that z is predominantly fast because of the 10 multiplying the right side. We say that x is predominantly slow because it has only 1 multiplying the right side. For comparison later, let's solve for the exact response. Because it is a linear time-invariant system, the solution will be of the form

x(t) =

cle), "

z( t) =

C4e),"

+ c,e)," + C3 + C5e)," + C6

(A.9) (A.I0 )

A.2. INTEGRAL MANIFOLDS FOR LINEAR SYSTEMS

325

where Al and A2 are the two eigenvalues . These are found from the systemstate matrix as follows:

(A.11)

.

A

The eigenvalues are the solutions of -1 A + 10

A+l

determinant [AI - A] =

o

=0

(A.12)

or (A + l)(A + 10) - 0 = O. The roots are Al = -1, A2 = -10. The constants C3 and C6 are the steady-state solution

0= -x" + z" } 0= -10z88 + 10

Zss =

1

Xss =

1

(A.13)

so x

= cle- t

+ C2e-10t + 1

Z

= C4e-'

+ C5e-10' + 1

(A.14) (A.15)

The other constants are found from the initial conditions:

x( 0) = dx dt

10

dz dt

(A.16)

= -C, - 10c2 = -xo

z( 0) = -

+ C2 + 1 = XO

C,

1

-

Cl =

X

0-

+ C5 + 1 =

C4

+ ZO

(A .18)

ZO

10c5 = -10zo

-C4 -

(A.17)

+ 10

(A.19)

Solving

C3

O

ZO

+

= ZO -

1

-10 9 C4

=

C2

=

ZO -

1

(A.20)

9

a

(A.21)

the exact solution is

X=

x

°

10 --+ 9

ZO

e

9

'-v-'

-, -

ZO -

1

9

'-v--'

small if is not large Zo

small if .0 is not large

e- 10'

+1

(A.22)

326

If

APPENDIX A. INTEGRAL MANIFOLDS FOR MODEL

is not large , the major part of x is slow . The fast variation of z contributes only small amounts to x. To see how we can develop a reduced-ord er model, look for an integral manifold of the form ZO

z = hx

+e

(A.23)

where hand c are const ant s . Substituting into the z differenti al equation , dx

hdt

= -10(hx + c) + 10

(A.21 )

or

h( - x

+ hx -I- c) =

- lOhx- IDe + 10

(A 25)

One solution is h = D, c = 1. This means that z = 1 is an exact integral manifold for the z variable . That is , if z = 1 a t any t ime , then z remains equal to one for all time. Or, more properly stated: "lfth e initial condit ions start on the manifold, then t he system remains on the manifold. " If this integral man ifold is substituted into the x different ial equat ion , the reducedorder model (valid only for ZO = 1) is dx

- = -x dt

+1

x(o) =

XO

(A26)

which clearly exhibits the exact slow eigenvalue and the following solution

x = (X O

-

l)e - '

+1

(A.27)

Compare this to the exact solution of ( A.22) . If ZO is equal to 1.0 , then (A.26) gives the exact resp onse of x for any xo . If ZO is not equal to 1.0, then (A .26) will no t give the exact response of x . For this case, we define the off-manifold variable 71 as 6.

1)=2 - 1

( A.2S)

which has the dynarn.ics

d1)

- = -101) dt

1)(o) = zO-l

(A.29)

Note that 1) = 0 is an exact int.egral manifold because if 1) = 0 at any time, 1) = 0 for all time. The exact solution, when 1)(0) 'I 0, is

7J(t ) = (ZO - 1)e- 1o ,

( A.30)

A.2. INTEGRAL MANIFOLDS FOR LINEAR SYSTE}[S

327

The integral manifold plus off-manifold solution for z as a function of x and t is z = 1 + (ZO ~ 1)e- 10 ' (A.31) This gives the exact reduced-order model (for any zo) 0 -dx = -x + 1 + (ZO ~ 1 )e- 1 ' dt

x( 0)

= XO

(A.32)

While this may have been obvious from the beginning, the steps that led to this result are important for cases where it is not obvious. The fast time-varying input into this slow subsystem is somewhat undesirable. An interesting approxiroation can be made to eliminate this term as follows. Since the time-varying term decays very ra,p idly (e - lOt ), this term enters the slow differential equation almost as an impulse. It is ZO ~ 1 at tiroe zero and essentially zero for t > O. Using the following impulse identity,

Im p (t)

i:> =

lim a

1 -~ - e a

(A.33)

a

-+ 0

Equation (A.32) can be approximated by

~:

""

~x + 1 + (zOl~ l)Imp(t)

x(o) = XO

(A.34)

This impulse can be eliminated by recognizing that its integral can be included in the initial condition on x

x(t) = XO

+

' l °

(~x

+ l)di +

or

x(t) = XO

+ ZO -

1+

l'(zO-l) °

l'

Imp(i)di

(A.35)

10

(- x

.

+ l)dt

(A.36) 10 ° This gives the approximate reduced-order model using the exact integral manifold for z and accounts for the z initial condition off-manifold dynamics through a revised initial condition on x: dx

- "" -x dt

+1

x ( 0) = x °

+(

ZO

1)

'--c-:-::-..!.

10

(A.37)

To see the impact of this approximation, consider the example in which XO = 1 and ZO = O. The exact response of x is the solution of dx

(a) - = dt

~x

+1_

e-

10t

x(0) = 1.0

(A.38)

328

APPENDIX A. INTEGRAL MANIFOLDS FOR MODEL

The a.pproxima.te response of '" using only the exact integral manifold for z is the solution of dx (A.39) (b) - '" -x + 1 x(o) = 1.0 dt The improved approximate response of x accounting for the off-manifold dynamics is the solution of dx '" -x dt

(c) -

+1

x(0)=0.9

(A.40)

A comparison of these solutions is given in Figure A.2. 1.0 i ' - - - - - -

1.0

1.0

0.5

0.5

0.5

0 <------;-';;---.,. 1.0 0

0 °

°0

1.0 (b)

(a)

1.0 (c)

Figure A.2: Comparison of exact and approximate solutions It is important to observe that the basic phenomenon was captured by the slow manifold of (b). The correction for the fast initial condition was a second-order effect approximated fairly well. Before leaving this example, we return to (A.25) and observe that another integral manifold solution is h

= -9, c= 10

(A.41)

which gives another exact integral manifold ,

z = -9x

+ 10

(A.42)

If this solution is used in the original differential equation for x, the redueedorder model is dx dt = - lOx

+ 10

x(o) =

",0

(A.43)

The solution of this is x

=

(Xo -

1)e- 10 '

+1

(A.44)

t

A.2. INTEGRAL MANIFOLDS FOR LINEAR SYSTEMS

329

Compare this to the exact solution of (A.22). It is exact if ZO = -9x O + 10. This means it is exact for X O = 1, ZO l. But, if X O 0.5, ZO 1 (not significantly djfferent), there will be a large error in this reduced-order model, which is good only for a very limited range of initial conditions. We now extend this concept to the more general case with coupling in both equations as:

=

dx - = -x

=

=

+z

dt

(A.45)

dz = -lOx -10z dt

-

(A.46)

As before, we say that z is predominantly fast and x is predominantly slow. This is due to the 10 multiplying the right side of the z djfferential equation. The eigenvalues of this system are found from the state matrix

A=

- 1

(A47)

[ - 10

A+l

PI - AI

= (A +

10

1)(A + 10) + 10

=0

(A.4S)

The roots are

= -2 .3

A1

= -S.7

.\2

(A.49)

so the exact solu tion is

x = Z

=

+ C2e-S.7t + C3 C4e-2.3t + cse- S .7t + C6 cle - 2 .3t

(A50) (A.51)

The steady-state values of x and z are Xss

=

C3

= 0

Zss =

C6

(A.52)

= 0

Focusing on x, let's solve for c, and C2.

x(O) = dx

-

dt

C1 +C2 =

X

O

I0 = - 2.3c, - 8.7c2

(A.53)

= - x° + z

0

(A.54)

Solving Cl

=

7.7x O + ZO 6.4

, C2

l.3x o + ZO = 6.4

(A.55)

APPENDIX A. INTEGRAL MANIFOLDS FOR MODEL

330 so:

x=

7.7 ° x + 6.4

-

ZO

e- 2_3t

_

6.4

XO

° -1.3 x+ 6.4

----

6.4

e- 8 "7' (A.56)

----

sma.ll if ~o is not large

small if ~o is not large

As before, if ZO is not"large, the x response is dominated by the slow component. If we are interested only in capturing the mode with eigenvalue -2 .3, we propose that this mode is associated with the state x, and seek to eliminate z from (A.45)-(A.46) through an integral manifold of the general linear form

(A .57) Substituting into (A.46) (and using (A.45)) gives he - x

+ hx + cJ =

-lOx - 10hx -10c

(A.58)

For arbitrary x, this has the solutions

c= 0 c

= -1.3

(A. 59)

h= - 7.7

(A.60 )

h

=0

Thus, there are two integral manifolds for z as before: 1M 1:

- 1.3:1:

(A.61)

1M 2: z = - 7.70:

(A.62)

Z=

Substitution of 1M 1 into (A.45) gives dx dt = - 2.30:

(A.63)

and substitution of 1M 2 into (A.4S) gives the faster subsystem dx dt = -8.70:

(A.64)

Tills second result is somewhat disturbing, since we had originally proposed that the variable x was associated with the -2.3 mode. Several important

A.2. INTEGRAL MANIFOLDS FOR LINEAR SYSTEMS

331

points are illust rated her e. First, integral manifolds can be used to decouple systems and find eigenvalues. Second, integral manifolds are not unique (one for each eigenvalue, it appears!). Third, the choice of the integral manifold determines the phenomena retained in the red uced-ocaer model. To understand more about this technique, consider the same sys:em wit h the intro duction of the small parameter < (1/ 10 in the above example), and the inclusi on of initial conditions dx dt = - x

dz

f-

dt

+z

x(o)

= XO

(A. .65 )

= - x- z

z(o)

=

(A..66)

ZO

Again we seek a linear manifold z = h(E)X

+ C(E)

(A.6 7)

where we presume that hand c would normally depend on the parameter Substitution of (A.67) into (A.65)-( A.66 ) gives

+ h(E)X + C(f)) =

- x - h(f)x - C(E)

E.

(A .68)

Again, for arbitrary x, t he solutions are

C(f) = 0, C(E)

= 0,

For positive

f,

J(1 - E)2 2f 2< 1- < - -1 J(1 - E)2 h(f) = - -- E h(f) = -1-+ -1

2E

2E

=-

4f,

h(o)

4f,

h(o) = - oo(A.70)

1 (A.69)

these solutions exist only for

o < E :'0 0.1715

(A.71)

This makes sense, since the original system has complex eigenvalues for f less t han 1 but gr eater t han 0 .1715 . It would be impossible to capt u.re a complex mode from a single-state equation. This simple example illu.strates that integral manifolds may not exist and may not be unique . Hwe are interested in the "slow mode," which we propose is associated with the variable x, we need a systematic way to compute the correct integral manifold . If z is infinitely fast (f = 0), th e integr al manifold of interest is z = -x (h = -1) . When < is near zero, we propose that t he

APPENDIX A. INTEGRAL MANIFOLDS FOR MODEL

332

integra.! manifold of interest should be near -1. To systematically compute h for E nea.r zero, we expa.nd h( f) in a power series in ( as

(A.72) and return to the example by substituting in to (A.66) and (A.67) (using

c«)=O)
+

+ <2h2 + .. .)( - 0: + (ho +
For arbitrary"" we solve for h o, hI, h 2 , ... by equating coefficients of powers of £: 0= - 1 - ho or ho = - 1

(A74)

h o( -l + ho)= - h, oIh, = h o - h~ =- 2

(A. 75 )

etc. Stopping with these terms,

(/,76 )

h«) '" -1 - 2< This approximates the exact integral mitnifold of (A.65)-(A.66) as

z'"

-( 1 + 2<)",

(A .77)

Using this in (A.65 ) gives the approximitte redu ced-order model (valid when the initial conditions satisfy ZO ~ h( <)0: 0 ) dx

-

dt

'" -(2 + 2<)x

",(o) = xo

(A.78)

This model could be improved to any degree of accuracy by in cluding additiona.! terms of h( f). Since these terms were computed from a power series near the integral manifold of interest (slow manifold ), the correct m ode has been captured. Note that it is necessary to consider only t he off-manifold dynamics due to the initial conditions if the reduced o rder model is going to be used in a simulation . If only eigenvalues are of interest, the off-manifold dynamics due to initial conditions are not relevant. For < = 0.1, as in the previous example, the slow eigenvalue (- 2.3 ) is approximated in (A.78) by

A.2. INTEGRAL MANIFOLDS FOR LINEAR SYSTEMS

333

-2.2. As a warning, it is important to note that in this example the integral manifold always exists when E is actually zero . When E is small but not zero, the infinite series of (A .72) can always be computed, but it s convergence and, hence, its validity depend on the size of the actual E. In tms example, we would expect that tms series converges only when E is less t han 0.1715 . Thus , whlle terms of the integral manifold series can be computed for any E, it should be used only when E is less than 0.1715. While computation of the integral manifold is the primary task in model reduction, it may not make sense to find a good approximation and then ignore the possibili ty that z does not start on the manifold (zo t h(E)"'o). As in the earlier example, it is possible to approximate tms impact on '" by computing the off-manifold correction. To do tms, we define the off-manifold variable as (with c( E) = 0)

71

Cl.

z - h(E)'"

(A .79)

which has dynamics (recalling (A.fi8) with c( E) = 0)

d71 Edt

= - (1 + Eh(E))71

71(0)

= ZO -

h(E)"'O

(A.80)

Clearly, Ti = 0 is also an integral manifold because if 71 = 0 at any time, then 71 = 0 for all time. The exact off-manifold dynamics are the solution of (A.80):

71(t) =

(ZO -

h(E)",O)e- ~) • t

(A .81)

If h( E) is known exactly, then the exact slow subsyst em is dx

- (1- h(E))'"

dt

+ (ZO -

",(o) = XO

h(E)xO)e (

1+_ h ( .))

•

t

(A .82)

As before, tms fast input can be approximated by an impulse and eliminated by modification of the initial condition to obtain

d",

dt "'(0)

'" - (1 - h(E))x (A.83)

Clearly, if ZO = h(E)"'O (on the slow manifold), tms result is exact when h(E) is exact. In our example , heEl was never found exactly. Using the approximation h(E) '" -1 - 2E with E = 1/ 10 gives the following slow subsystem

334

APPENDIX A. INTEGRAL MANIFOLDS FOR MODEL

approximate model (approximate for two reasons : h( E) is no t exact and the off-manifold dynamics are not exact): dx dt

~

- 2.2:z:

x () 0 =

100

-:2:

88

0

10 0 + -z

88

(A.84)

The basic result of all this is summarized as follows. Consider a linear system in standard two-time-scale form dx dt dz Edt

Ax

+ Bz

(A.85)

Cx

+ Dz

(A.86 )

where A, B , C , C are of order 1 (not big) and D is nonsingular (D- l exists). A first-order approximation of the slow dynamics of x is obtained by simply setting E = O. dx

Ax + Bz

(k8?)

+ Dz

(A.88)

dt 0= Cx

which gives the first app roximation of the slow manifold (ho) (A.89) and the first approximation of the slow dynamics of x (A.90) Using this approximate model will introduce two errors. One is due to the fact that the integral manifold is not exact (ho is only the largest part of h(E)). The second error is due to the fact that the initial condition on z may not satisfy the integral manifold. It can be shown that if the initial condition on z is not big, then the to tal error of these two approximations is small. The term "not big" means t h at ZO should be on the order of magnitude of 1 or less. The term "small" means on the order of magnitude of E. References [66] and [67] discuss this in detail. In most cases, the first-order approximati on of the slow manifold coincides exactly with the steady-stat e relationship between z and :2: in the z differential equation, because the result of setting E to zero is the same as

A.3. INTEGRAL MANIFO LDS FOR NONLINE.-I.R SYSTD fS

335

the res ult of setting ~~ to zero. There is, however, a p rofo-CIa theoretical difference. Setting ~~ = 0 implies z = const ant. This would be contradicted by the change of z when x changes (z = hox). Setting e = 0 does not imply z = constant, and thus is not contradicted by t he change of z ",he:: I changes.

A.3

Integral Manifolds for Nonline ar Systems

\!Vhile there are many reduction techniques that can be applied to linear systems , the primary advant age of the integral manifold approach is it.s straightforward extension to nonlinear systems . We begin by considering the general form

dx

= I(x,z)

x(o)

= Xo

(k9 1)

dt = g(x,z)

z(o)

= Zo

(A.92)

dt

dz

To analyze the dynamics of x, it is necessary to also compute the dynamics of z. A reduced-order mo del involving only x requires the elimination of z from (A.9Il- An integral manifold for z as a function of x has t he form z

= h(xl

(A.93)

and must satisfy (A.92)

fJh fJxl(x,h) = g(x,h)

(A .94)

While , in general, it is very difficult to find such an integral manifold, t here are several ver y important cases in which h can be either found exactly or approximated to any degree of accuracy. We begin with a generic example, which closely resembles the single synchronous machine connected to an infinite bus with stator transients

dx,

-

dt

dx,

= x, - 1

dt = I(X1 , X"Z1,Z,)

dZ 1

dt dz, dt

(A .95)

(A.96)

(k9?) (k98)

336

APPENDIX A. INTEGRAL MANIFOLDS FOR MODEL

We suppose that only manifold of the form

XI

and

X2

are of interest, a nd look for an integral

(A.99)

ZI = hl(XI, X2) Z2

=

(A .IOO)

h2(xI' X2)

This two-dimensional integral manifold must satisfy

(A.I OI)

(A.I02) These partial differen ti"l equations can be solved by nrst assuming that hi and h2 are independent of X2 and then equating coefficients of '"2 to give Bhl _ -

BX,

h

(A.I03)

2,

(A.I04) Eliminating the partials gives the solution

(A.I05) h2

where tan ex=

= V cos ex sin( ex

- xtl

(A.I06)

Thus, if the initial conditions on ZI, Z2, and XI satisfy (A.I05)- (A. I06) , then substitution of (A.I05)- (A.I06) into (A.96) gives an exact reduced-order model. When the initial con ditions on ZI, Z2, and X I do not satisfy (A_I05)-(A.I06), it is necessary to introduce the "off-manifold " variables IJ.

'71 '72

t. = ZI t. = Z2 -

V cos ex cost ex

-Xl)

(A.IO"r)

V cos ex sin( ()(

- Xl)

(AI08)

A.3. INTEGRAL MANIFOLDS FOR NONLINEAR SYSTEMS

337

These off-manifold variables have the following dynamics: dr/1 dt = - Ir'll

d1), dt

=-

+ X,'7,

17'72 -

(A.109) (A.11 0)

X2'71

This system has the explicit solution 1)1

'72 =

where

+ Xl - (2) -cle - '" sin(t + X, - C2)

= cle-

q

,

(A.lll)

cos(t

(A. 112)

c, and C2 are found from initial conditions by solving z)- V

cos ex cos(cx: - x~) =

Cl cos(x~

- C2)

(A.1l3) (A. 1l4)

This result leads to an exact reduced-order model in following in (A.96):

X,

and

X2

Z, = V cos ex cos(ex -x,) + cle-
= V cos ()( sin(ex

-X,) -

e,e- q , sin(t +

Xl -

(2)

by using the

(A.1l5) (A.ll6)

Such exact integral manifolds and exact off-manifold solutions are rare in dynamic systems. The synchronous machine stands out as a unique device with this property [64J. A very broad class of systems in which integral manifolds can often be found or approximated to any degree of accuracy is the class of two-timescale systems of the form dx dt = J(x,z) dz E

dt = g(x,z)

x(o) =

XO

(A.1l7)

z(o) =

ZO

(A.1l8)

These systems are called two-time scales because when f is small , the z variables are predominantly fast and the x variables are predominantly slow. This is clear b ecause the derivative of z with respect to time is proportional to 1/ E, which is large for small E. As in the linear case of the last section, we propose an integral manifold for z as a function of X and EO

z = h(x, f)

(A. 1l9)

338

APPENDIX A. INTEGRAL MANIFOLDS FOR MODEL

Vve assume that

f

is sufficiently sma.IJ so that the manifold can be expressed

as a power senes In e:

(A. 120) Substitution into (A.119) and then (A.118) gives (A.121 ) Expanding

f and

g about

f

= 0,

f(x, h) = f(x, ho)

+
g(x,h) = g(x,ho)

+
h, ."

ag

(A .122 ) (A.123)

the partial differential equation to be solved is

(A.124) Equating coefficients of powers of < produces a set of algebraic equations to be solved for ha, hi, hz ... : 0 = g(x, ho) (;1 :

aha ag ax f(x, ha) = az

(A.125)

Iz=ho hi

(A. 126)

etc. Clearly, the most important equation is (A.125), which requires the solution of the nonlinear equation for ho. Once this is found, the solution for h, simply requires nonsingular agjaz. Normally, if (A.125) can be solved, the nonsingularity of ag j az follows . As in the linear case, the use of an integral manifold in a reduced-order model can give exact results only if it is found exactly and if the initial conditions start on it. IT the initial conditions do not start on the manifold (do not satisfy (A.1l9)), an error will be introduced. To eliminate this

A.3. INTEGRAL MANIFOLDS FOR NONLINEAR SYSTEMS

339

error, it is necessary to compute the off- manifold dynamics . This is done by

introducing the off-manifold variables t>

1)=Z - h(X,E)

(A.127)

with the following dynamics dr} E-

dt

=

g(x, 1) + h) -

8h E"

uX

f(x,

1)

+ h)

(A.128)

and the initial condition

1)(0)

= ZO -

h(xO, E)

(A.129)

These off-manifold dynamics normally are difficult to compute because they require x. As a first a.pproxirnation, (A.128) could be solved using x as a constant equal to its initial condition . This is a reasonably good approximation because the off-manifold dynamics should decay (if they are stable) before x changes significantly. A geometric illustration of the integral manifold and the off-manifold dynamics is shown in Figure A.3. z

Figure A.3: Off-manifold dynamics

1£ z starts off the surface z = h( x, E), it should deca.y rapidly to the surface (if it is stable), as shown in the solid line. The dotted line shows the trajectory of z if off-manifold dynamics arc neglected and z is forced to begin on the surface. It can be shown that, if z is stable, llsing ho as an approximation for h

and neglecting off-manifold dynamics only introduces "order E'l eTIor into the slow variable x response. If further accuracy is desired and h is approximated by ho + Eh" there will still be "order E" error if the off-manifold dynamics

340

APPENDIX A . INTEGRAL MANIFOLDS FOR MODEL

are neglected. To reduce the error to "order E2," it is necessary to include hI and approximate 1) to order £2. This can be done by approximating the off-manifold dynamics as

(Al30) with

(A.131) and ho, hI, aho/ax evaluated at x = x o. Additional illustrations of integral manifolds and off-manifold dynamics are given in [63]- [69J.

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Index Admittance matrix, 169 Angle reference, 157,232 Automatic generation control, 87 A utomatic voltage regulator, 70

electromechanical modes, 263 negative, 262, 269, 270 positive, 270 Damping coefficient, 275 Damping ratio, 274 Damping torque, 107, 112, 255, 262, 263, 269, 270, 275 multimachine, 149 Dc steady-state error, 276 DeMello-Concordia constants, 260 Differential-algebraic equations industry model, 178 Differential-algebraic model, 161 Directed graph, 129 DQO transformation, 26 Droop, 86 droop, 86 Dynamic security assessment, 283

Bergeron 's method, 10, 21 Bifurcation, 243 Hopf, 243, 247 saddle node, 247 singularity-induced, 243 Boundary-controlling u.e.p method, 286, 313 Branch currents, 131 Branch voltages, 129 Branch-loop incidence matrix, 129 Center of inertia, 158, 291 Characteristic equation , 266 Characteristic impedance, 10 Classical model, 106 multimachine , 146 Compensated voltage, 73, 74, 76 Conservative system, 307 Controlling u.e.p, 286 Coupling field, 29 Critical clearing time, 283, 285 Critical energy, 286

E'q,E'd,Efd, 40 Eigenvalue, 235 Eigenvectors, 235 Electromagnetic transients , 7 EMTP , 7 solution methods, 13 transmission line models, 8 Electromechanical modes, 249,267 Energy kinetic, 286 , 287 lowest u.e.p, 286 potential, 286, 287

D-axis, 25 Damper winding, 24, 95 Damper-winding, 112 Damping 353

INDEX

354 Energy fun ction methods, 283, 29 1, 301 b oundary-controlling u .e.p, 313 controlling u.e.p, 286 lowest-energ y u.e.p, 286 potential energy b oundary surface , 286 structure preserving , 318 EPRI, 6, 170, 171 , 199, 203 Equal-area criteria, 303 Equilibrium points saddle type, 300 st able, 28 7 type 1, 300 ETMSP, 199, 203 Euler's metho d, 22 Excess kinetic energy, 300 Exciter, 65 IEEE-T ype I, 170 saturation, 70 static, 234 Exit-point method , 286 E xplicit integration, 161 Faulted trajectory, 300 Field current, 41, 51 First-swing transient stability, 214 Flux linkages, 35 , 39, 43 , 95 Flux-decay model, 104, 154, 207 multimacrune, 142 Frequency, 156 Frequency cont rol, 83 Frequency response, 54 Frequency-domain analysis , 263,267 Generator notation, 24 Governor , 83 , 84 model, 86 Gradient system, 314

Heffron-Phillips constants, 260 Hessian matrix, 301 Hopf bifurcation, 247 HVDe , 191 Hydraulic speed governor, 84 Hydroturbines , 77 IEEE-Type I exciter, 170,22 1, 224 Implicit integration, 161 Inductance, 36 Inertia constant, 25, 33, 90 Infinite bus , 89, 157 Initial conditions, 188, 192, 208, 260 classical model, 216 full model, 188 Instability monotonic, 263 torsional , 272 Integral manifolds, 323 Integration schemes, 14 Euler's forward, 22 explicit, 161 implicit, 14 trapezoidal rule , 14, 15 Internal-node model, 214 Jacobian, 205, 223, 233 algebraic equation, 223 load flow , 223 KI-K6 constants , 255 Laplace domain, 57 Lead-lag transfer function, 272 Leakage r eactance, 38-40 Linear magnetic cirellit, 35, 41 Linearizat ion, 222, 234, 257 Load flow, 188 , 190 Load models, 126 , 134, 244

INDEX impedance, 138, l41, 144,147, 184 Low-frequency oscillations, 254, 255 Lowest-energy u.e.p method, 286 Lumped parameter modeling, 13 Lyapunov's method , 286, 287, 301 Magnetic circuit, 30, 35, 39 constraints, 30 linear, 35, 39 . nonlinear, 43 Magnetizing reactance, 40 Manifolds integral, 323 Mode of instability, 285, 306 Model reduction, 323 Modeling, 8 Bergeron's method, 14 classical model, 106, 146 exciter, 66 flux-decay model, 104, 142 governor , 83, 86 loads, 126, 134 multi-time-scale, 93 multimachine, 123 one-axis model, 104 synchronous machine, 23 tra.nsmission line, 8 turbine, 77, 83 two-axis model, 101, 139 voltage regulator, 70 Motor notation, 24 Multi-time-scale modeling, 93 Multidimensional potential well, 306 Multimachine m odels classical model, 146 flux-decay , 207 internal node model, 214 red uced order, 206

355 structure preserving, 209 two-axis model, 152 Multimachine PEBS, 306 Natural frequency undamped,267,273 NERC, 5 Network equations, 167 current-balance form , 169 power-balance form, 167 Newton's method, 199 very dishonest, 205 Nonlinear magnetic circuit, 43,4·6 Numerical solution current-balance, 203 power-balance, 198 One-axis model, 104 Open-circuit voltage, 51 Operational impedance, 54, 60 Park 's transformation, 24, 26 Participation factors , 235 Partitioned-explicit, 198 Path-dependent integral, 310 PEBS, 286 Per-unit scaling, 24,31,33,37 Phase lag, 272, 275 Phase lead, 273, 275 Phase velocity, 10 Phasors, 35, 51 Polar form, 165 Polarity, 24 Potential energy boundary surface , 286 , 294, 301 Potential energy well, 299 Power balance , 27,28 Power system stabilizer, 254, 270 design, 270 frequency domain design, 273

356

INDEX

gain, 271, 272 speed input, 270 stabilizing signals, 270 torsional instability, 272 washout stage, 272 Prime mover, 83 Q-axis , 25, 193 Reactive power, 54 Rectangular form, 165 Reduced-order modeling, 206 Reference angle, 157, 251 Reference frames center of inertia, 158, 198 machine, 26 synchronously rotating, 123 Reference node, 129 Region of attraction, 285 , 286, 314 Root locus, 266, 274 Rotor angle absolu te, 284 relative, 284 Saddle-node bifurcation, 247 Saturation, 43, 45, 48, 49, 67, 150 constraints, 30

exciter, 70 synchronous machine, 113 Self-excited de exciters, 69 Separately excited de exciters, 66 Shaft dynamics , 78 Shaft stiffness , 78 Simulation, 161 Simultaneous-implicit, 198 Single-machine steady state, 50 Singularity-induced bifurcation, 247 Speed governor, 83 Stability, 221, 283 small-signal, 221

transient, 283 Stability methods, 286 Stable equilibrium point (s.e.p), 285 Standstill test, 57 State-space analysis, 263 Static exciter, 207, 234 Stator algebraic equations, 165, 165 Stator /network transients, 96, 128 Steady state, 50 Steam turbine, 80 Structure-preserving model, 209 Structures, 3 physical, 3 political, 5 time-scale, 4 Sub-transient reactance, 39, 40

Synchr onous machine power balance, 28 Synchronizing coefficient, 268 Synchronizing torque, 255, 262, 263, 267- 270 Synchronous machine, 23 assumptions, 35 conventions and notation, 23 exciter , 65 inductance, 36 linear magnetic circuit model, 41 modeling, 23 nonlinear magnetic circuit model, 46 parameters, 39 Park's t ransformation, 24, 26 per-unit scaling, 24, 31, 37 saturation, 43, 45, 48, 49, 113 schematic, 25 steady-state circuit, 51 terminal constraints, 89

INDEX three damper winding model, 24 transformations and scaling, 26 Synchronously rotating reference frame, 123, 283 T'do,T'qo, 39 Til do ,T"qo , 39 Tap-changing-under-load (TC UL) transformers, 3, 191 Terminal constraints , 89 Time constants, 48, 54 Topology, 129 Torque of electrical origin, 30, 52, 102 Torque-angle loop , 264, 267, 276 Torsional modes, 272 Transfer conductances, 251 Transformations center of inertia, 158 machine-network, 166 Park's, 24 synchronously rotating, 123 Transient algebraic circuit, 99, 102, 104, 106, 139, 146 Transient energy function, 288 Transient reactance, 39, 40 Transient stability, 283 first-swing, 214 Lyapunov's method, 287 Transmission line models, 8 pi equivalent, 13 receiving end, 12 sending end, 12 Trapezoidal rule, 13-15, 200 Turbine, 77 hydro, 77 model, 83 steam, 80

357 Two-axis model, 101, 102, 152 multimachine, 139 Uncompensated voltage, 74, 76 . Unstable equilibrium point (u.e.p), 285 Voltage collapse, 248 Voltage regulator, 70 Voltage stability, 222 Voltage/current polarity, 24 Washout filter, 273,275, 276 WSCC 3-machine system, 171 X'd, X'q, 39 X"d ,X"q , 39 Xad per-unit system, 24 Xd, Xq, 39 Ybus, 170 Ybus , 163, 169, 186 Yreduced, 186 Zero eigenvalue , 232 Zero sequence, 50