Electric Systems, Dynamics, and Stability with Artificial Intelligence Applications

Electric Systems, Dynamics, and Stability with Artificial Intelligence Applications Jarnes A. Mornoh Howard University...

Author: James A. Momoh | Mohamed E. El-Hawary

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Electric Systems, Dynamics, and Stability with Artificial Intelligence Applications

Jarnes A. Mornoh Howard University Washington, D.C.

Mohamed E. El-Hawary Dalhousie University Hali&r, Nova Scotia, Canada

m M A R C E L

D E K K E R

MARCEL DEKKER, INC.

NEWYORK BASEL

ISBN: 0-8247-0233-6 This book is printed on acid-free paper.

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Series Introduction

Power engineering is the oldest and most traditional of the various areas within electrical engineering, yet no other facet of modern technology is currently undergoing a more dramatic revolution in both technology and industry structure. Among the most exciting changes are those where new solutions are being applied to classical problem areas. System dynamics and stability engineering have been crucial elements of power system engineering since early in the twentieth century. Smooth, continuous operation of modern power supply systems depends greatly on the accurate anticipation of interconnected equipment, dynamic behavior, and correct identification of the system’s operating limits. Proper engineering requires precise methods that can manage that knowledge and direct it to the design of economical and secure power systems. Artificial intelligence offers an exciting new basis for performing dynamic modeling and stability analysis, one that can provide considerable value and new insight to these often difficult aspects of power system performance. Electric Systems, Dynamics, and Stabiliv with Artificial Intelligence Applications is an exceedingly comprehensive and practical guide to both power system dynamics and stability concepts, and to the use of artificial intelligence in their modeling and engineering. Drs. Momoh and El-Hawary provide a comprehensive introduction to power system dynamics and stability, along with a thorough discussion of recently developed concepts such as transient energy func-

tions. Their book is rich in its appreciation of the intricate operating constraints and issues that real-world power system engineers and operators must face every day. But wtiat sets this book apart is its application of artificial intelligence to these long-recognized power system engineering challenges. Chapters 7-9 expI a i n h o 11ei i ra I tie t uwrk s, espert sy s t e 111 s us i in g know I edge - ba sed fra in e u'ork s. and t ' u ~ z ylogic can be applied to the solution of sotlie of the thorniest problems in power system dynamics. Like all books in Marcel Dekker's Power Engineering series, Elecstr-ic. SJ.Ytettr.v, ~ > j w t t i i c * st i, t i d Sttihilitjv Il'itli A rlificitrl Itit~~ligutic*ci App/ic.trliotis present? tnodern power technology i n ;I context of proven. practical applications: iiset'u ;is a reference book iis well iis for self-study and classroom use. Marcel Dekker':, Pourer Engineering series bill e\~entuallyinclude books co\w-ing the entire field of p o n w engineering, in all ot' its specialties and sub-genres, all ainned at provid ing practicing p o u w engineers u,ith the knou~ledgeand techniyiies they need to tneet the electric industrlf's challenges in the twenty-first century.

Preface

The intention of this book is to offer the reader a firm foundation for understanding and analyzing power system dynamics and stability problems as \yell as the application of artificial intelligence technology to these problems. Issues in this area are extremely important not only for real-time operational considerations, but also in planning, design, and operational scheduling. The significance of dynamics and stability studies grows as interconnected systems evolve to meet the requirements of a competitive and deregulated operational environment. The complexities introduced give rise to new types of control strategies based on advances in modeling and simulation of the power system. The material presented in this book combines the experience of the authors in teaching and research at a number of schools and professional developtnent venues. The work reported here draws on experience gained in conducting research sponsored by the Electric Power Research Institute. the National Science Foundation, the Department of Energy, and NASA for Dr. Momoh. Dr. EIHawary's work was supported by the Natural Sciences and Engineering Rcsearch Council of Canada and Canadian Utilities Funding. This book is intended to meet the needs of practicing engineers invol\.ed i n the electric power utility business, as well as graduate students and researchers. It provides necessary fundamentals, by explaining the practical aspects of artificial intelligence applications and offering an integrated treatment o f the evolution of modeling techniques and analytical tools.

Chapter I discusses the structure of interconnected power systems, foundations of system dynamics, and definitions for stability and security assessment. Chapter 2 deals with static electric network models and synchronous machine representation and its dynamics. Limits for operations of a synchronous machine and static load models are discussed as well. Chapter 3 deals with dynamic models of the electric network including the excitation. and prime mover and governing system models. The chapter concludes with a discussion of dynamic load models. Chapter 4 covers concepts of dynamic security assessment based on transient stability evaluation. This chapter includes both conventional and extended formulations of the problem. Chapter 5 , a complement to Chapter 4, treats the more recent approach of angle stability assessment via the transient energy function idea. Chapter 6 introduces the idea of voltage stability and discusses techniques for its assessment. Chapters 7 through 9 are devoted to an expose of artificial intelligence technology and its application to problems of system stability, from both the angle and the voltage sides. In Chapter 7, we introduce basic concepts of artificial neural networks, knowledge-based systems, and fuzzy logic. In Chapter 8. we deal with the application of artificial intelligence to angle stability problems. and the extension to voltage stability is presented in Chapter 9. Chapter I C offers conclusions and directions for future work in this field. In developing this book. we have benefited from input from many of 0111' students. colleagues, and associates. While they are too many to count, we wish to tnention specifically encouragement by H. Lee Willis, the editor of the PoweiEngineering Series for Marcel Dekker, Inc. The continual counsel and prodding o f B. J. Clark was extremely helpful. We acknowledge the able administrative support of Linda Schonberg and the assistance of our respective deans. We are grateful to Dr. Chieh for the great inspiration and generous contribu tions, and to many others, whose names are not included, in the development ot' this volume. Our students, both present and former. contributed their time and many valuable suggestions. Many thanks to them and especially to the young research assistants at the Center for Energy Systems and Control for putting up with the burdensome challenge of producing this book just in time. Finally. the book would not have been published without the help of our Creator and the support of our families.

Contents

Series 1titi.odiic.tiotz Prefiice

H. Lee Willis

1 Introduction 1 . 1 Historical Background 1.2 Structure at a Generic Electric Power System I .3 Power System Security Assessment

2 Static Electric Network Models Introduction 2.1 Complex Power Concepts 2.2 Three-Phase Systems 2.3 Synchronous Machine Modeling 2.4 Reactive Capability Limits 2.5 Static Load Models Conclusions

3 Dynamic Electric Network Models Introduction 3.1 Excitation System Model 3.2 Prime Mover and Governing System Models

10 10 11 14

21 31 32 35 36 36 36 40

3.3 Modeling of Loads COn c1LI s io n s 4

Philosophy of Security Assessment Introduction 4.1 The Swing Equation 4.2 Some Alternative Forms 4.3 Transient and Subtransient Reactances 4.4 Synchronous Machine Model in Stability Analysis 4.5 S U b t ran s i en t Eq u at i on s 4.6 Machine Models 4.7 Groups of Machines and the Infinite Bus 4.8 Stability Assessment 4.9 Concepts in Transient Stability 4.10 A Method for Stability Assessment 4.1 1 Matheinatical Models and Solution Methods i n Transient Stab i I it y Asse ss me n t for General Networks 4.12 Integration Techniques 4.13 The Transient Stability Algorithm Conc 1us ion s

S Assessing Angle Stability via Transient Energy Function 5.I 5.2 5.3 5.4

5.5 5.6

6

Introduction Stability Concepts System Model Description Stability of a Single-Machine System Stability Assessment for ri-Generator System by the TEF Method Application to ;I Practical Power System Boundary of the Region of Stability Conclusion

Voltage Stability Assessment Introduction 6. I Worhing Definition o f Voltage Collap\e Study Terms 6.2 Typical Scenario of Voltage Collapse 6.3 Time-Frame Voltage Stability 6.4 Modeling for Voltage Stability Studie\ 6.5 Voltage Collapse Prediction Methods 6.6 C las\ i fica t i o n c) f Vo 1t age Stabi 1 it y Pro b Ie111s 6.7 Voltage Stability As\es\ment Techniques

xi

Col 1 ter 1 f.\

6.8 Analysis Techniques for Steady-State Voltage Stability Studies 6.9 Parameterization 6.10 The Technique of Modal Analysis 6.1 1 Analysis Techniques for Dynamic Voltage Stability Studies Conclusion Modal Analysis: Worked Example ’

7

Technology of Intelligent Systems Introduction 7.1 Fuzzy Logic and Decision Trees 7.2 Artificial Neural Networks 7.3 Robust Artificial Neural Network 7.4 Expert Systems 7.5 Fuzzy Sets and Systems 7.6 Expert Reasoning and Approximate Reasoning Conclusion

8 Application of Artificial Intelligence to Angle Stability Studies Introduction 8. I ANN Application in Transient Stability Assessment 8.2 A Knowledge-Based System for Direct Stability Analysis Conclusions 9 Application of Artificial Intelligence to Voltage Stability Assessment and Enhancement to Electrical Power Systems Introduction 9.1 ANN-Based Voltage Stability Assessment 9.2 ANN-Based Voltage Stability Enhancement 9.3 A Knowledge-Based Support System for Voltage Collapse Detection and Prevention 9.4 Implementation for KBVCDP 9.5 Utility Environment Application Conclusion 10 Epilogue and Conclusions

135 IS 1

156 157

169 170 175 175 177 177 183 191

206 213 220

22 1 22 I 333

eh&

238 257

259 259 260 265 272 278 287 287

289 298 31 I 332 35 I

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Electric Systems, Dynamics, and Stability with Artificial Intelligence Applications

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1 Introduction

1.1

HISTORICAL BACKGROUND

Electric power has shaped and contributed to the progress and technological advances of humans over the past century. It is not surprising then that the growth of electric energy consumption in the world has been nothing but phenomenal. In the United States, for example, electric energy sales have grown to well over 400 times in the period between the turn of the century and the early 1970s. This growth rate was 50 times as much as the growth rate in all other energy forms used during the same period. Edison Electric of New York pioneered the central station electric pouter generation by opening of the Pearl Street station in 1881. This station had a capacity of four 250-hp boilers supplying steam to six engine-dynamo sets. Edison's system used a I 10-dc underground distribution network with copper conductors insulated with a jute wrapping. The l o b i t \ d t c i g e of tlic-, cii-mit limited the service area of a central station, and consequently central stations proliferated throughout metropolitan areas. The invention of the transformer, then known as the "inductorium." made ac systems possible. The first practical ac distribution system in the United States was installed by W. Stanley at Great Barrington, Massachusetts. in 1866 for Westinghouse, who acquired the American rights to the transformer from its

British investors Gaulard and Gibbs. Early ac distribution utilized 1000 V overhead lines. By 1895, Philadelphia had about twenty electric companies with distribution systems operating at 100 V and 500 V two-wire dc and 220 V three-wire dc: single-phase, two-phase, and three-phase ac; with frequencies of 60, 66. 125, and 133 cycles per second; and feeders at 1000-1200 V and 2000-2300 V. The consolidation of electric companies enabled the realization of economies o f scale in generating facilities. the introduction of a certain degree of equipment standardization, and the utilization of the load diversity between areas. Generating unit sizes of up to 1300 MW are in service, an era that w;i? started in I973 by the Cumberland Station of the Tennessee Valley Authority. Underground distribution o f Lwltages up to 5 kV was made possible by thc de\.elopment of rubber-base insulated cables and paper insulated, lead-co\.erec cables in the early 1900s. Since that time higher distribution voltages hi1L.e beer necessitated by load growth that would otherwise overload low-voltage circuit!, and by the requirement to transmit large blocks of power over great distances. Coninion distribution voltages in today's systems are in 5 , IS, 25, 35. and 6'1 kV \vltage classes. The growth in size of power plants and in the higher voltage equipment u ' a s accompanied by interconnections of the generating facilities. These interconnections decreased the probability of service interruptions, made the utilization of the most economical units possible, and decreased the total reserve capacity required to meet equipment-forced outages. This growth was ;11so accompanied by the use of sophisticated analytical tools. Central control of thc interconnected systems was introduced for reasons of economy and safety. Th,: advent of the load dispatcher heralded the dawn of power systems engineering. \+!hoseobjective is to provide the best system to meet the load demand reliablq. safe 1y , and economical 1y , U ti 1i zi ng state-of-the-art computer fac i 1i ties. Extra high voltage (EHV) has become the dominant factor in the transmi>sion o f electric power over great distances. By 1896, an 1 1 kV three-phase line M . ~ Stransmitting 10 MW from Niagara Falls to Buffalo over a distance of 2 3 miles. Today, transmission voltages of 230 kV, 287 kV, 345 kV, 500 kV, 735 k V , and 765 k V are commonplace, with the first I100 kV line scheduled fclr energization in the early 1990s. One prototype is the I200 kV transmission tower. The trend is possible. resulting in more efficient use of right-of-wa;!. lower transmission losses, and reduced environmental impact. The preference for ac was first challenged in 1954 when the Swedish Stale Power Board energized the 60-mile, 100 kV dc submarine cable utilizing I!. Larnm's Mercury Arc FAves at the sending and receiving ends of the world s first high-voltage direct current (HVDC) link connecting the Baltic island o f Gotland and the Swedish mainland. Today, numerous installations with voltages up to 800 kV dc have become operational around the globe. Solid-state technol-

ogy advances have also enabled the use of the silicon-controlled rectifiers (SCR) of thyristor for HVDC applications since the late 1960s. Whenever cable transmission is required (underwater or in a metropolitan area), HVDC is more economically attractive than ac. Protecting isolated systems has been a relatively simple task, which is carried out using overcurrent directional relays with selectivity being obtained by time grading. High-speed relays have been developed to meet the increased short-circuit currents due to the larger size units and the complex interconnections. For reliable service, an electric power system must remain intact and be capable of withstanding a wide variety of disturbances. It is essential that the system be operated so that the more probable contingencies can be sustained without loss of load (except that connected to the faulted element) and so that the most adverse possible contingencies do not result in widespread and cascading power interrupt ions. The November 1965 blackout in the northeastern part of the United States and Ontario had a profound impact on the electric utility industry. Many questions were raised and led to the formation of the National Electric Reliability Council in 1968. The name was later changed to the North American Electric Reliability Council (NERC). Its purpose is to augment the reliability and adequacy of bulk power supply in the electricity systems of North America. NERC is composed of nine regional reliability councils and encompasses \Tirtually all the power systems in the United States and Canada. Reliability criteria for system design and operation have been established by each regional council. Since differences exist in geography, load pattern, and power sources, criteria for the various regions differ to some extent. Design and operating criteria play an essential role in preventing major system disturbances following severe contingencies. The use of criteria ensures that, for all frequently occurring contingencies, the system will, at worst, transmit from the normal state to the alert state, rather than to a more severe state such as the emergency state or the in extreenzis state. When the alert state is entered following a contingency, operators can take action to return the system to the normal state.

STRUCTURE OF A GENERIC ELECTRIC POWER SYSTEM

1.2

While no two electric power systems are alike, all share some common fundamental characteristics including: 1.

Electric power is generated using synchronous machines that are driven by turbines (steam, hydraulic, diesel, or internal combustion).

2.

Generated power is transmitted from the generating sites o\rer long distances to load centers that are spread over wide areas. 3. Three phase ac systems comprise the main means of generation. transmission and distribution o f electric power. 4. Voltage and frequency levels are required to remain within tight tolerance lekrels to assure a high quality product. The basic elements of ii generic electric poww system are displayed i n Figure I . 1 . Electric power is produced at generating stations (GS) and transmitted to consumers through an intricate network of apparatus including transmission lines, transformers. and switching devices. Transmission network is classified as the folloufing: I . Transmission system 2. S i i bt ran smi ssi on system 3. Distribution system

The t i . i ~ i i ~ s i ~ i i . s . s i ~,s~~.strin )ii i n t erconnec t s a I 1 major genera t i ng st ;i t i c) n s ;I nd main load centers in the system. It forms the backbone of the integrated p o \ \ ~ r system and operates at the highest voltage levels (typically, 230 kV and ab0F.e The generator \vltages are usually i n the range of 11-35 kV. These are steppej up to the transmission cultage level, and power is transmitted to transmission hubstations where the voltages are stepped down to the subtransmission It.\,c.l (typically, 69 kV to I38 kV). The generation and transmission subsystems ;it e often referred to as the hiilk po\r.ei. systerii. The .viihtr.iiri.siiii,s.siori sy.stew transmits power at a lower voltage and in s m ii 1 le r q it a n t i t i es froi n the t ran siii is s io n sii bst a t i on to the d i s t ri bi i t i o n subs t ;Itions. Large industrial customers are commonly supplied directly from the s u b transmission system. In some systems, there expansion and higher \,ohage l e \ ~ l s becoming necessary for transmission, the older transmission lines are otten re1 2gat ed to sii bt ra n smi ssion function . The di.stri/mtioii .sj:steiri is the final stage in the transfer of' pourer to tlie individual customers. The primary distribution voltage is typically betwwn 4.0 k V and 34.5 kV. Small industrial customers are supplied by primary feedcrs at this \.oltage level. The secondary distribution feeders supply residential and commercial customers at I20/240 V. The function o f an electric power system is t o con\.ert energy from one of the naturally available forms to the electrical form and to transport it to the points of consumption. Energy is seldom consumed i n electrical form but is rather converted to other forms such as heat, light, and mechanical energy. The rid\rantage of the electrical form of energy is that it can be transported and controlled with relative ease and with a high degree of efficiency and reliability. A properly designed and operated power system should. therefore. meet lie fol low i ng fit nda men t a I reqii iremen t s: 1.

5

22 kV

--

500 kV

--

500kV

230kV

20 kV Tie line to neighboring system I

I

I

I

1 1

Transmission

Transmission system (230 kV)

I

Tie line

distribution

Subtransmission

-Residential

Figure 1.1 Basic elements of

1.

H

Commercial

power system.

The system must be able to meet the continually changing load demand for active and reactive power. Unlike other types of energy. electricity cannot be conveniently stored in sufficient quantities. Therefore, adequate “spinning” reserve of active and reactive power should be maintained and appropriately controlled at all times. The system should supply energy at minimum cost and with minimum 2. ecological impact. 3. The “quality” of power supply must meet certain minimum standards

6

with regard to the following factors: ( I ) constancy of frequency; (l ) constancy of voltage; and (3) level of reliability. Several levels of controls it~vol\~ing a complex array of devices are used to meet the above requirements. These are depicted in Figure I .2, which identifit*s the various subsystems of a power system and the associated controls. In this overall structure, there are controllers operating directly on individual system elements. In a generating unit these consist of prime mover controls and excihtion controls. The prime mover controls are concerned with speed regulaticm and control of energy supply system variables such as boiler pressures, temperitures, and tlows. The function of the excitation control is to regulate generatlir bultage and reactive power output. The desired MW outputs of the individual generating units are determined by the system-generation control. The primary purpose of the system-generation control is to balance the total system generation against system load and losses so that the desired frequency and power interchange with neighboring systems (tie flows) is maintained. The transmission controls include power and bdtage control devices. sw:h as static viir compensators, synchronous condensers, switched capacitors ;I id

Frequency

Tie flows

Generator power

Supplementary

--- ,Cmd-.-- - - - - - - - - - - - - - - ---------- - - - - - -- - - ,

---------I I I

I

I I

tI

I I

+

I I I

I I I I I

Voltage

I I I II I 1I

I I

I 1 II

Excitation System and control

I I

4-

I I I

speed/ power

i; i I

W Rime mover

i

Shaft Power

- Generator -

;

i

I

Power

I

;

*

’

Figure 1.2 Subsystems of a power system and associated controls.

Transmission Controls Reactive power arid Voltage control, INDC transmission and associated contro s

I n t roduc t iori

7

reactors, tap-changing transformers, phase-shifting transformers, and HVDC transmission controls. These controls described above contribute to the satisfactory operation of the power system by maintaining system voltages and frequency and other system variables within their acceptable limits. They also have a profound effect on the dynamic performance of the power system and on its ability to cope with disturbances. The control objectives are dependent on the operating state of the power system. Under normal conditions, the control objective is to operate as efficiently as possible with voltages and frequency close to nominal values. When an abnormal condition develops, new objectives must be met to restore the system to normal operation. Major system failures are rarely the result of a single catastrophic disturbance causing collapse of an apparently secure system. Such failures are usually brought about by a combination of circumstances that stress the network beyond its capability. Severe natural disturbances (such as a tornado, severe storm, or freezing rain), equipment malfunction, human error, and inadequate design combine to weaken the power system and eventually lead to its breakdown. This may result in cascading outages that must be contained within a small part of the system if a major blackout is to be prevented.

POWER SYSTEM SECURITY ASSESSMENT

1.3

The term Po\ivr. Sysfeni Stability is used to define “the ability of the bulk power electric power system to withstand sudden disturbances such as electric short circuits or unanticipated loss of system components.” In terms of the requirements for the proper planning and operation of the power system, it means that following the occurrence of a sudden disturbance, the power system will: 1.

Survive the ensuing transient and move into an acceptable steady-state condition, and 2. In this new steady state condition, all power system components are operating within established limits. Electric utilities require security analysis to ensure that, for a defined set of contingencies, the above two requirements are met. The analysis required to survive a transient is complex, because of increased system size, greater dependence on controls, and more interconnections. Additional complicating factors include the operation of the interconnected system with greater interdependence among its member systems, heavier transmission loadings, and the concentration of the generation among few large units at light loads. The second requirement is verified using steady state analysis in what is

Power Svstern Stablllty

.

4

+

+

I I

I

4

Small Disturbance (Contingency)

Sensitir it y

Eigenvaluc Analyv\

AMIYSIS

Frequency Domin Analysis

M

+I

Angle Stability

~

I

Unlned Approach

Enhancement (Prevcatlve Coatrol)

intelligent Support System

+

+

Small Disturbance (Quasi-Static)

Static (Power

Bifurcation T a p Changer

Singular Value Decompoiition

MaNmm Lnadability

Jacobian

Linear Rogramrmng

Vdtage Stability

~~~~-~~~~~~~

Lo 6

Frequencb Oscillatiw s, etc

Energy Functions

Y

Selective A MI y si s

O h e r Tvpes

Bifurcation Theory (Hopf. etc.)

Bifurcation

Techniques

+(SSR, T

Volruge (Slow)

Strwrural

Cntical

1

4

Use of the Energy Function and h e Bifurcalion Analytical Studies

Linear Rogrammng for Germation Rescheduling dnd

DecompositionOpimal

Load Control

Power H o w (OPR wth Transieru Energy Margin and Vdcagc Energy Margin d s

Control using Exper( System IFS) Pnmiizatim

Control Selections. output Classifier and Analyzer

Nonlinear Ragramming (NLP)with quadratic objccti~cfunctions and Linear Rogramrmng (LP) fu Generation and Load Control

Arttndal Neural N e t w o r b

(ANN),and/ a Expert Systems fasolution output

re l'er red to :is "s t ;i t i c sec ii r i t y iis se ssme n t . The t'i r st req U i re nit' 11t i s the h i i bjec t de ii 1t \v i t h i 11 "d y n am i c sec ii r i t y assess m e 11t . Dy n am i c sec 11 r i t y s t i i d i e s ;I re broadly classified ;is being either "angle stability studies" o r "voltage stabil;t),'' ;is depicted i n Figure 1.3. In "angle stability studies." problems are classified as either "large distitrbmce.' tix transient e\'aluiitioii, or "small disturbance" for steady state stability t'\,:ilurition. A similar classification for \soltage probleiix is indicated in Figure 1 ..3. Solution techniqites tor transient unglc stability e\,aluation incluck: "

"

Ti me doma i n si m ii 1at ion Direct methods Hybrid methods Pro babi I i st ic nie t hods. pat t e 1-11 recog n i t ion COnn pit t ;it iona 1 i n t e 1I i ge nce

Time domain simulation techniques invol\Ve judicious use of integration methods such as Runge-Kutta. trapezoidal rule of integration, and ~ ~ a \ . e t c ~ r m re 1axat i on. These met hods are part i c 41ar 1y use fu 1 for off- 1i ne transit‘11t st abi 1it y ana I y si s. On the other hand, in “voltage stability studies.” the problems are classified as “large disturbance” in some contingency cases, “small disturbance” i n quasistatic cases. and “static,“ which requires solutions to the general pouw tlo\v (algebraic) equations only. It is as a result of this classification that the solution techniques and requirements are derived. The bifurcation theory. Linear Programming applications. and the use of the energy fiinction are but ii few such tec h n i q Lie s. Ag ai n, the t i me domai n si m u I a t i on s t h ii t ;ire i 11\YII \red take ad \,;in t age of various numerical integration methods mentioned earlier. (The machine dj8namics lead to differential equations that are inherently nonlinear.) The unified approach as indicated on Figure 1.3 is aimed at encompassing the similarities and differences that distinguishes the Lrarious techniques used in assessing the stability of the electric power system. This is \4,hether or not the problems are a result of voltage or angle instability. The resulting enliancenient that is brought forth by this approach, measurable ;is a benefit-to-cost index. lies in the de\~lopmentand use of more robust tools for solLting present ~indlongrange problems. I n this light, various programming and optimization schemes that are applicable include decomposition Optiiutrl Po\t*or Flo\tp (OPF). Liiiorrr P i . o S r . c r i i i i i i i r i , ~ (LP), and Qir(rcltntic. P i . o R t . c r i i i i i i i i i S (QP), with the necessary and su f fi ci e n t sy stem and net work con st rai n t s. Finally. this book introduces three fundamental types of intelligence support systems that truly adds the rigor, value, and robustness to the desired enhancement schemes. These support systems include expert systems (ES), fuzzy logic (FL). and artificial neural network (ANN). Each have their unique characteristics (decision-support, classifiers, learning capabilities, etc.) and are ridaptable in providing viable solutions to a variety of voltage/angle instability problems associated with the electric power system. The discussion on this area of artificial intelligence applications to power system stability rind dynamics is presented i n the final few chapters of this book.

&

Static Electric Network Models

INTRODUCTION The power industry in the United States has engaged in a changing busincss environment for some time, by moving away from a centrally planned s y s t m to one in which players operate in a decentralized fashion with little knowleclge of the full-state of the network, and where decision-making is likely to be market driven rather than through technical considerations alone. This new environment is quite different from the one in which the system operated in the past. This leads to the requirement of new techniques and analysis methods for functions of system operation, operational planning, and long-term planning. Electrical power systems vary in size, topography and structural components. However, what is consistent is that the overall system can be diLiried into three subsystems, namely, the generation, transmission, and distribution subsystems. System behavior is affected by the characteristics of each of the major elements of the system. The representation of these elements by means of appropriate mathematical models is critical to successful analysis of sys ;em hehavior. Due to computational efficiency considerations for each diffe -ent problem, the system is modeled in a different way. This chapter describes some system models for analysis purposes. We begin in Section 2.1 by introducing concepts of power expressed as active, reactive, and apparent. This is followed in Section 2.2 by a brief reLiew

of three phase systems. Section 2.3 deals with modeling the synchronous machine from an electric network standpoint. Reactive capability curves are examined in Section 2.4. Static and dynamic load models are discussed in Section 2.5 to conclude the chapter.

2.1

COMPLEX POWER CONCEPTS

In electrical power systems one is mainly concerned with the flow of electrical power in the circuit rather than the currents. As the power into an element is basically the product of the voltage across and current through it, it is reasonable to exchange the current for power without losing any information. I n treating sinusoidal steady-state behavior of an electric circuit. some further definitions are necessary. To illustrate, we use a cosine representation of the sinusoidal waveforms involved. Consider an impedance element 2 = ZL$. For a sinusoidal voltage, v ( r ) is given by v(t) = tlC O S O t

The instantaneous current in the circuit shown in Fig. 2.1 is i(r)= /,,, cos(or - Q)

where the current magnitude is:

The instantaneous power is given by p(r) = ir(t)i(t)= XI I,,,[cos(or)cos(or- $)]

Using the trigonometric identity 1 coscl cosp = -[cos(a - p) + cos(cl + p)]

2

we can write the instantaneous power as

+ V

Z

Figure 2.1 Instantaneous current in a circuit.

The average power p ( , is seen to be

Since through 1 cycle. the a \ erage of co\( 2 o t - @ ) i \ zero. this term contri bnothing to the acerage of / I . I t i \ more convenient to use the effectiLre ( r i m ) calue\ of boltage and curt-cnt than the iiiiixiinuni ~ a l u e s S. ub\tituting xi = v% Ci,,,,). and I,,, = -\/'?(/,,,,,). \+e get ute\

Thus the p o ~ w entering any network is the product of the effectilre v:iIies of' terminal iaoltage and current and the cosine ol' the phase angle betLveen the \,oltage and ciirrent which is called the i ~ o ~ ~ i ~ ~ ) (PF). . , t ~ /This ~ . t oapplies ). to siniisoidal Lroltages and currents only. For ii purely resistive load. cos$ = 1. and I he current in the circuit is fully engaged in c o n \ q i n g power from the soiirce to the load resistance. When reactance (inductive o r capacitilte) as t+,ell ;is m i s [;incc are present, ii component o f the current in the circuit is engaged i n con\~r:~ ing energy that is periodically stored in and discharged from the reactance. This stored energy, being shuttled into and out of the rnagnelic field of ;it1 inductaiice o r the electric field of a capacitance. adds to the magnitude of the current in the circuit but does not add to the a\.erage power. The a\wage power in ii circuit is called i1ctiL.e power. and loosely speaking the po\+rer that supplies the stored energy in reacti\re elements is called reacliite po\i.er. Acti1.c pou.cr is denoted bjf P, and the rci1ctii.e pourer. is designattx ;i\ Q.Thej' are expressed as

I n both equations. 1' and 1 are rim calues of terminal Lfoltage and current. and Q, is the phase angle by mrhich the current lags the \,oltage. Both P and Q arc of the same dimension, that is in (Joules/s) Watts. Houeb'er, to emphasize the fact that Q represents the nonactive power, it is measiired i n reiictiLpe k~oltampereunits (\'at-).Larger and tiiore practical units arc k i 1 o r . m and megavars. related to the basic unit by

As\urnt. that 1'. 1' cos$, and 11 sin@,a11 shown in Fig. 2.2, are each multiplied bqr /. the r i m cralue of the current. When the components of iultage 1' ,:OS@

vv I3

Static' Electric N e t u v r k Models

Isin $

I

Figure 2.2 Phasor diagrams leading to power triangles.

and V sin$ are multiplied by current, they become P and Q respectively. Siniilarly, if I, I cos$, and I sin$ are each multiplied by V, they become V I , P. and Q respectively. This defines a power triangle. We define a quantity called the complex or apparent poufer, designated S. of which P and Q are orthogonal components. By definition, S = P + j Q = i/* = V/ cos@+ j V 1 sin@ = V/ (cos@+ , j sin@)

Using Euler's identity, we thus have

s = VIC"* or

s =VIL$ If we introduce the conjugate current defined by the asterisk

(*')

I* = l l ( L @

it becomes obvious that an equivalent definition of complex er is

s = VI"

01-

apparent

POW-

(2.5)

We can write the complex power in two alternati\,e f o r m by using the relationships Z7 and 7 = Y v This leads to

v=

s = ZI I"

=ZJIJ?

(2.6)

S=VY*V"= Y",VI"

(2.7)

or

V

I Figure 2.3 Series circuit o f n impedances.

Consider the series circuit shown in Fig. 2.3. Here the applied Lvltage i:, cqual to the sum of the voltage drops:

+ z , + . . +Z,:)

\'=I(&

Multiplying both sides of. this relation by /* results in

s = \'I* = I / * ( Z ,+ 22 + . . . + Z , ! ) or

uith

s,= ,II?z, being the individual element's complex power. Equation (2.8) is knourn as tke sunimiition rule for complex powers. The summation rule also applies to paral1i:l circuits. The use of the summation rule and concepts of complex pourer ai.e ai\rantageous i n solving problems of power system analysis. The phasor diagrams shown in Fig. 2.2 can be converted into compltmx pourer diagrams by simply following the definitions relating complex power .o tzoltage and current. Consider the situation with an inductive circuit i n urhic-h the current lags the voltage by the angle $. The complex conjugate of the current M i l l be in the first quadrant in the complex plane as shown in Fig. 2.3(a). Multiplying the phasors by V, we obtain the complex power diagram in Fig. 2 . 4 b). Inspection of the diagram as well as previous development lexis to ii relation for the power factor of the circuit: P

cos$ = __

1s I

2.2

THREE-PHASE SYSTEMS

A significant portion of all the electric power presently used is generated. trailsmitted. and distributed using balanced three-phase bultage systems. The single-

1.5

Static Electric N e t w w k Models

Figure 2.4 Complex power diagram showing the relationship among voltage. current, and power components.

phase voltage sources referred to in the preceding section originate in many instances as part of the three-phase system. Three-phase operation is preferable to single-phase because a three-phase winding makes more efficient use of generator copper and iron. Power flow in single-phase circuits is known to be pulsating. This drawback is not present in a three-phase system as will be shown later. Also, three-phase motors start more conveniently and, having constant torque, run more satisfactorily than single-phase motors. However, the complications of additional phases are not compensated for by the slight increase of operating efficiency when polyphase systems of order higher than three-phase are used. A balanced three-phase voltage system consists of three single-phase voltages having the same magnitude and frequency but time-displaced from one another by 120". Figure 2.5(a) shows a schematic representation where the sin-

\

120°

120 120°

Figure 2.5 diagram.

van

(a) A Y-connected three-phase system and (b) the corresponding phasor

gle-phase \,()Itage sources appear in ii wye o r Y-connection; a delta o r A configuration is also possible, as discussed later. A phasor diagram shouring each the phase voltages is also given in Figure 2.S(b). As the phasors rotate at the iiiigular t'reqiiency cu with respect to the refkrence line in the counterclockc~,isc (ciesignated as positive) direction, the positive maximum value first occurs t ' c r phase ( I and then in siiccession for phases h and ( * . Stated in ;I different ~t'aq'.to ;in ohser\~eri n the phasor space. the voltage of phase ( I arri\,es t i n t follo\zred 1'4, that o f h and then that of ( * . For this reitson the three-phase Lroltage of Fig. 2 5 i h said to have the phase sequcncc trbc. (order, phase sequence. o r rotation a11 mean the ss;itiie thing). This is important tor certain applications. For es~irnple. in three-phase induction motors. the phase seqitence dcterniines u,hether tlic i i i o t o r rotates clock\i.ise o r coiinterclock~ise.

2.2.1

Current and Voltage Relations

Balanced three-pha+e \y\tem\ c;in be \tudied u\ing technique\ de\reloped t o r $ingle-phaw circuit\. The arrangement o f the three \ingle-phaw Ioltagc\ i n t o ;I Y o r ;i A configuration require\ \ome modification\ in dealing U ith the o\er.iIl \>\tern.

2.2.2

Y -Connection

With reference to Fig. 2.6. the cotiimon terminal I I I \ called the neutral o r
This equatioii state\ that the ~ o l t a g eexisting from ( I to h i h equal to the coltage from ( i to 11 (i.e.. \,',) piu\ the voltage from I I to h. Thus Eq. ( 2 . 1 0 ) can be re~irittenas (.!.I

I

)

I7

= vp L -120O

Qn

Figure 2.6 Illustration of the phase and magnitude relation\ beiueen thc phaw and line coltage of' a Y-connection.

let

Since for a balanced system, each phase Lroltage has the same magnitude. set

UI

I K,! I =

'

&,I

I = I Yll I = 1;

(2.12)

where 1;: denotes the effective magnitude of the phase voltage. Accordinglj u e may write

K,, = y,o0

(2.13)

Kll = y - - 120"

(2.14)

yl,= y L - 230" = y L120"

(2.1s)

Substituting Eqs. (2.13) and (2.14) in Eq. (2.1 I ) yields

K,?= y

(1

-1

i

-

120")

= fip300

(2.16)

X( = fiy- - 90"

(3.17)

S i ni i 1arl y we obt ai n

(2.18)

The expressions obtained above for the line boltages shou that they constitute a balanced three-phase voltage system M’hose magnituctes are v‘? time\ thow of the phase voltages. Thus we write 1.: =

fiy

(2.111)

A current tlowing out of a line terminal ci (or 11 or i\ the same a\ thi\t tlon ing through the phase source Lroltage appearing between terminal\ 11 and ( I (or 1 1 and h or I I and c ) . We can thus conclude that for a Y-connected three phase wurce, the line current equals the phase current. Thu\ ( a )

(2.2 ) )

l, = f,,

In the aboce equation, l, denotes the effective \ d u e of the line current aid /,, denotes the effectike tfalue for the pha\e current.

2.2.3 A-Connection We n o w consider the case when the three single-phase sources are rearrangd to forrn ;I three-phase A-connection as shown in Fig. 2.7. It is clear from in5ptction of the circuit shown that the line and phase \oltage\ haw the wine magiii-

tude:

The phase and line currents. howec er are not identical, and the relation\liip betkveen them can be obtained by using Kirchhoff’s current lau at one of ‘he 1i ne term i nal s . In a manner similar to that adopted for the Y-connected source, let U\ con\icier the pha\or diagram shoun i n Fig. 2.8. Assume the phaw currents to t w

/ C

I

\b

n

bb’ ob’

Ibc

Figure 2.7 A A-connected three-phase s w r w .

I,,’

C’

Stn t ic Elect t-ic Net UY)t-k Mode Is

Figure 2.8 Illustration of the relation between phase arid line currents in a A-connection.

/
= I/,

I/,[= I,,i - 1 20" I,,, = I,,i 120"

The current that flows in the line joining given by

[I

to

ci'

is denoted by l$,,,, and is

As a result, we have

I,,,,, = I,,[I 1 20" - 1 i 01 L!

which simplifies to

I,,,,,= &,L

I 50"

Similarly.

Note that a set of balanced three-phase currents yields a corresponding set of balanced line currents whose magnitudes are fi times the rnagnitudes of the phase values:

20

\+hx l, denotes the m:ignitude 01' any o f the three line currents.

2.2.4 Power Relationships Arrume that ii three-phare generator is supplq ing a balanced loud \+,ith thc t h r w rintiroidal phase voltagcs: \' ( f 1 = \ '7 1;: \',(f)

\lIl(l)t

= \ 21,: \111((1)/ - 1 2 0 - )

\' ( I ) = \

21,: \lll((I)f

+ 120

)

~chcre@ is the phase angle bt.t\swn thc current iind voltage i n c d i ph;iw. ?'he total po\ser in the load is

This t i m s out to be expanded

L:sing

;I

US:

trigonometric identit,~, ive get

Yote that the last three terms in the abo\.e equation add tip to ~ r o l'hus . n'c obtain

When referring to the voltage level of' a three-phase s\'stem. by con\'ention. one in\.ariably tinderstands the line l-oltages. From the iibo\.e cliscussion thc relationship bet\\,t.cn the linc and phase \.oltages i n ;i Y-connectetl s ~ ~ s t e m s

The power equation thus reads in t e r m of line quantities: p10=

f i , l f l 11, l C O 4

We note that the total instantaneous power is constant. hairing a magnitude of three times the real power per phase. We may be tempted to assume that the reactiLre power is of no importance in a three-phase system since the Q terms cancel out. Houte\zer, this situation is analogous to the summation of balanced three-phase currents and \dtages that also cancel out. Although the sum cancels out, these quantities are still \very much in evidence within each phase. We extend the concept of complex or apparent power ( S ) to three-phase systems by defining SIO= 3y1;

where the L1ctik.e and reactijre powers are obtained from

In terms of line Lralues, we can assert that

sl,=

fiv1:

and

2.3

SYNCHRONOUS MACHINE MODELING

I n po\s'er system stability analysis. there are several types of models used for representing the dynamic behavior of the synchronous machine. These models are deduced by using some approximations to the basic machine equations. This section gives a brief introduction to synchronous machine equations. 2.3.1

Stator and Rotor Voltage Equations

In de\.eloping performance equations of a synchronous machine, the follou,ing assumptions are made:

I.

The stator urindings are sinusoidally distributed along the air-gap its far a s the mutual inductance effects with the rotor are concerned. 2. The stator slots cause no appreciable variation of the rotor inductances Lvith rotor position. 3. Magnetic hysteresis is negligible. 1. Magnetic saturation effects are negligible. Based on these assumptions, a synchronous machine can be represented b j six Lvindings as shown in Fig. 2.9. The stator circuit consists of three-phast;irniatiire uindings carrying alternating currents. The rotor circuit consists o .

field and iiniortisseur windings. The positive direction of ;I stittor winding cur. rent is rissumed to be into the machine. The Lwltage equations of the three-phase armature uindings are:

(),,

- N,,i,,

= tlt

Rot ation 0,

elec. rad/s

‘ c Figure 2.9 Stator mcf rotor circuits of ;I synchronous rnachinc. ci. h. c: Stator ph: w windings: jil: field cvinding: k d : d-axis armature circuit; k q : q-axis armature circuit; 1 = I . 2, . . . . r l ; I I = number o f xinature circuits; 0 = angle by Lvhich c l - a x i 5 leads the iiiagrietic axis o f phase ( I winding. electrical radiaii~;:CO, = rotor angular \‘elocitj. clectri1:al rad/x.

23

Static Electric Net\t*ork Models

where

L,,,,. L,,,,, L,, are self-inductance of nhc windings. L,,/,.L,,,, LI,,,, LI,,, L,,,,LIIlare mutual inductances between two stator winding (cih, hc, N C ' ) L,,,,/,L/,,,/,L,,,/ are mutual inductances between stator and fields windings L,lL,I,LlIAll, LlL,/are mutual inductances between stator winding and d-axis armature circuit L,,A,,, LhAll,L,A,lare mutual inductances between stator windings and q-axis armature circuit. The rotor circuit voltage equations are given by:

(2.24)

The rotor circuit flux linkages are given by

v/d= Lff,/;/,/ + L[?.diA[/- Ldd;/>-

LhAili/i -

L,k q i ,

vkd = Llldild + LALdilrl - LRAdili - L, vAq = LAAqikt, - LiiAyio - Lldqih - LcAq;c &iAdi,i

A,/;(

(2.25)

Equations (2.22) and (2.23) associated with the stator circuits together with equations (2.24) and (2.25) associated with the rotor circuits completely describe the electrical performance of a synchronous machine. The fact that mutual and self inductances of the stator circuits vary with rotor position q which in turn varies with time, complicates the synchronous machine Eqs. (2.22) to (2.25). The variations in inductances are caused by the variations in the permeance of the magnetic flux path due to nonuniform air gap. This is pronounced in a salient pole machine in which the permeances along the two axes are significantly different. Even in a round rotor machine there are differences between the two axes due mostly to the large number of slots associated with the field winding. The self and mutual inductances of the stator circuits are given by

I,, = I,

(,

+ 1,

! CO\

78

1, = I. ,, +

+

3 '

[:]=: 4!! K ,;,, -

tit

r.1.28,

where

All the inductances expressed as dqO components are seen to be constant, i.e.. they are independent of the rotor position. It is interesting to note that i,, does not appear i n the rotor flux linkage equation. This is because the zero sequence components of armature current do not produce net mmf :icross the air-gap. While the dqO transformation has resulted in constant inductances i n Eqs. (2.28) to (2.30). the mutual inductances between stator and rotor quantities are not. For example, the mutual inductance associated with the flux linking the field ufinding due to current i,, flowing in the d-axis stator winding from equation (2.30) is (3/2) L,,,,,. whereas from equation (2.29) the mutual inductance associated with flux in the d-axis stator winding due to field current is L,,,,,.This difficulty is overcome by an appropriate choice of the per unit sqrstcm for the rotor q u an t i t i es.

2.3.3 Per Unit Representation It is usually conivenient to use a per unit system to normalize system irariables. to offer computational simplicity by eliminating units and expressing sqrstem quantities as dimensionless ratios. Thus

A we 11-c h ose n per unit s y st eni can m i n i 111 i ze c om p i t ;i t i on a I effort , x i m p 1i f y e\'aluation. rind facilitate understanding of system characteristics. Some baw quantities may be chosen independently and quite arbitrarily, Lvhile others t'ol-

low automatically depending on the fundamental relationships between system \.ariables. Normally, the base values are chosen so that the principal ttariable? mrill be equal to one per unit under rated operating conditions. I n the case of a synchronous machine, the per unit system inay be used tcI remoF.e arbitrary constants and simplify mathematical equations s o that they rnq be expressed in terms of equivalent circuits. The basis for selecting the per uni system for the stator is straightforward, while it requires careful consideration for the rotor. The L,,,,-base reciprocal per unit system will be discussed here. The following base quantities for the stator are chosen (denoted by subscripts) f?,h,l,r.

= peak ~(alueof rated line-to-line voltage,

v

= peak value of rated line-to-line current, A ,f\.,,,< = rated frequency, Hz

The base \ d u e of each o f the remaining quantities ;ire automatically and depend on the above :is follows:

st't

271 .fh,l,c. electrical rad/s 2 o,,,,,.,,, = \clh.l,, - mechanical rad/s CO,,,~,,=

TFf '

~\t7.lW

Z+lre= -:, ohms 1\h,i\r.

The stator voltage equations expressed in per unit notations are g i \ m b j

27

The corresponding flux linkage equations may be written as __ - - -

w,/ = --&/id + LJ/
-

_-

-

L
-

= -L& + Ld,/i
wq

-

(2.32)

The rotor circuit base quantities will be chosen so as to make the flux linkage equations simple by satisfying the following: I . The per unit mutual inductances between different windings are to be reciprocal. This will allow the synchronous machine model to be represented by simple equivalent circuits. 2. All per unit mutual inductances between stator and rotor circuits i n each axis are to be equal. 3. The following base quantities for the rotor are chosen, i n view of the L,,,/-baseper unit system choose,

(2.33)

The per unit rotor flux linkage equations are given by

(2.34)

Since all quantities in Eqs. (2.31) to (2.34) are in per unit, we drop the overbar notation in subsequent discussion. If the frequency of the stator quantities is equal o the base frequency, the per unit reactance of a winding reactance is numerically equal to the per unit inductance. For example:

xl/= 2zf Ll/

(Q)

Dividing by Z\hd\c = 2nfhd,rL\hd\r., if f = f b d , c , then the per unit values of X,/ and L,, are equal. 2.3.4

Classical Representation of the Synchronous Machine

The per unit equations completely describe the electrical and dynamic performance of a synchronous machine. However, except for the analysis of very

\mall \ j \tern\, these equation\ cannot be used directly for \}stem \tability \tudI C \ . Some \implification\ and approxitiiations are required to reprewit the \ j nchronoii\ machine in stabilitj \tudie\. For large \y stem\. i t i \ nece\wrq to neg1ec.t the tr:in\tornier voltage term\ \ir<,arid \ir, atid the et'f'ttct o f \peed \ ariation\. Therefore, the rnachine equation tie\cribed bjf Eys. (2.33)and ( 2 . 3 3 )become

(2.36,

By defining the follou~ingvariables

the riiachine eyuationx becomes

( 2 . 37 )

Et: i \ the q-a\ii\ c-omponent of the \ oltage behind transient rcactanc\e I",, i \ the open-circuit tran\ient time con\tant. E, i \ the \folt:ige proportiond to I , , and olo i \ the ~ o l t a g eproportional to E,<,.S incc per u n i t \ = I_ , l'rom l!c~u~ltlon ( 2.37) m e hace M here

\'.

For studies i n \i.hich the period of analj'sis is sinall in comparison c+,itli I-,:, thc riiachine inodel ix often simplified bj' assuming that E'; is constant throiigho i i t the s t lidy period. T h i s ;is sii nip t ion e 1i m i na t e s t he on 1j , d i fte re n t i a 1 e y i i ;it i on asxociiited M i t h the electrical chxiicteristics o f the machine. A further approuirriation is to ignore transient saliency by assuniing that .v(; = .v(; and to :issiiriie that the t'lux linkage alw remains constant. With these assuinptions. the \,oIt;ige

behind the transient impedance R,,+ j x $ has a constant magnitude. The equi\.alent circuit is shown in Fig. 2.10. The machine terminal voltage phasor is represented by

q= E’i 6 - (R(,+.j.~;)i, The machine dynamic model is represented by T, @4! = M,,,- M , (It

P, = R , ( V , I )

v,= E’J

- (R,,+j x ; ) i,

(2.39)

where V , is the machine terminal voltage phasor and can be calculated from power flow considerations. Then ’ can be calculated. The machine scl?ingequation can then be solved. Equation (2.39) is the so-called classical model of the synchronous machine and is widely used in power system stability studies. This classical model is often used for three different time frames: subtransient, transient. and steadystate. Figure 2.1 I surnmarizes these three simple synchronous machine models. The subtransient and transient assume constant rotor flux linkages. and the steady-state model assumes constant field current. These models neglect saliency effects and stator resistance and offer considerable structural and computat i onal s i m pl i c i t y .

Figure 2.10 Eqiii\~alcntcircuit synchronous urith

x,;= xi:.

t

t

(a) Subtransient model

E’L6

(b) Transient model

EqL6

( c )Steady-state model Figure 2.11

Simple hjmchronous machine model.

Stcrtic Electric Netwwrk

2.4

Models

REACTIVE CAPABILITY LIMITS

It is important in voltage stability and long-term stability studies to consider the reactive capability limits of synchronous machines. Synchronous machines are rated in terms of maximum MVA output at specified voltage and power factor (usually 0.85 or 0.9 lagging) which they can carry continuously without overheating. The active power output is limited by the prime mover capability to a value within the MVA rating. The continuous reactive power output capability is limited by three considerations: armature current limit, field current limit, and end region heating limit. Figure 2.12 demonstrates a family of reactive capability areas for three different values of hydrogen coolant pressure. Note that the higher the pressure. the larger the capability curve. In Fig. 2.12, the region AB is the field current limited while the region BC is due to armature heating constraints.

t

Figure 2.12

0.6 p.f. lag t

Reactive capability curves of a hydrogen-cooled generator at rated \dtage.

2.5

STATIC LOAD MODELS

I v ed main 1y w it h gene rat o r st ii . bilit),. and little importance wits attached to loads. Recently, significant atteiitioii has been g i \ w to load modeling. Much ot' the domestic load and some industrial load consist o f heating and lighting. particularly i n the winter. and in earl)' loati 111ode 1s t hese \+'ere con s i de red iis constant i m pedances . Rot ii t i ng ecl LI i pine n t often inodeled ;IS ii siiiiple form o f synchronous machine and coniposite load u'ere simulated by a mixture of these two types o f load. A lot of uurk hits gon: i n t o the development ot' more accurate load models. These include some cornples niodels of specific large loads. Most loads. howe\ter. consist o f ;I I x g c c l i i a n t i t y of cli \wse eqii i pmen t of v;iry i ng I c \ ~sl atid coinposi t ion and s o m e cq 11 i va I e n I modc1 is iiece ssary . A static l o d model expresses the churacteristic o f the load at m y instarit ot' tirric in terms o f algebraic functions o f the bus \.()Itage niagnitude and t'wqi1cnc.y at that instant. The active power component P and the reacti\,c po\f.crr c 0111 pone 11t Q are c 011 si de red separate I y . ,A general load chxacteristic may be adopted such that the M V A loadii ~g Lit :I particulx bus is ;I function of' lroltage m d fi-eclucncj~: Con\re 11 t i onit 1 transient - stab i 1it y st L i d i e s were i n

Q = K,/\'" f A',, atid A'(, are constant\ \+hich depend upon the nominal :triable\ P and Q. For constant frequencq operation. u e write:

$1here

\

alue o f t ie

~+hm P and Q are acti\,e and reacti\re coinponetits of the load \+(hen the hiis \'oltage niagnitude is 1'. The siibscript 0 identifies the Fdiics o f the rcspccri\~ \ , x i ii b Ics ;it the i ii i t i 11 I or iiom i nal c i perat i tig cond i t ion. Static loads are reliiti\Jeljf iiniiffected by frecliicncj' change\, i.e.. ni,= = 0. and with constant impedance loads i i i , = 1 1 , = 2. The iniportance of' acciiiate 1 1 ,

load models has been deriionstrated f o r \coltage sensitive loads. Figure 2. I3 dtbiiioiistratcs the po~verand current characteristics 01' constant poLver, constant c iirrent, and constant inipedance loads. Manq reseiirchers identified the characteristic load parameters f:x \,iirioiis homogeneous loads. iypical \,alues are shoLvii i n Table 2.1 . These charac*terihtic*\

t

IVI Nominal voltage

Nominal voltage

(a)

(b)

Figure 2.13 Characteristic4 of different load models. ( a ) ActiLte and rcacti\c po\\er \ersus voltage. ( b ) Current \er\u\ \oltage.

inay be combined to @\re the overall load characteristic at a bus. For esample, a group of homogeneous loads, each with a characteristic of N.j , and a nominal power of P, inay be combined to give an overall characteristic of:

,=I

Table 2.1 Typical Values of Characteristic Load Parameters

Filament lamp FI Uoresc t'n t I amp Heater Induction motor half load Induction motor full load Reduction furnace A 1iiini nu m plant

I .6 1.2 2.0 0.2 0.I 1.9 I .8

0 3.o

0 1.6 0.6 2. I -.3 3

0

-I

.o

0 I .s 2.8

-03 -0.3

0 -2.8 0 -0.3 I .8 0 0.6

An iilternatii,e model which has been widely used to represent the Lroltagc: dependence of loads is the polynomial model:

This nwdel is corninonly referred to as the ZIP model. since i t is composed of constant impedance ( Z ) ,constant current (11,and constant power ( P ) coniponents. The parameters of the model are the coefficients P , to P , and Q , to which define the proportion of each component. When the load parameters m , and 1 1 , are less than or equal t o unity, a problem can occur when the voltage drops to a low value. As the voltage magnitilde decreases, the current magnitude does not decrease. In the limiting c;i!,e with zero cultage magnitude, ii load current flows m*hich is clearly irration: 1. gikren the nondynamic nature of the load model. From a purely practical poi i t of Lieut. then the load characteristics are only valid for a srnall L d t a g e deviation from nominal. Further, if the voltage is sinall. small errors in magnitude a i d phase produce large errors in current rnagnitiide and phase. This results in lo.;s of accuracy and with iterative solution methods of poor convergence. The ie effects ciin be overcome by using ii constant impedance characteristic to represent loads where the voltage is below some predefined ~ ~ a l u for e , example 0 . 8 p11. The parameters o f this model are exponents ci and h. With these exponerrts equal to 0, 1 . or 2, the model represents constant power. constant current, or constant i Inpe dan ce char;ict e r i st i c s, respect i c'e1y . For composite I oad s, thei r v i 1lies depend on the aggregate characteristics of load components. The frequency dependence of load characteristics is represented by niultiplying the exponential model or the polynomial niodel by ;I f'actor as follow>:

or

where Af is the frequency derivation (.f-.f;,).

CONCLUSIONS In this chapter we offered a review of power concepts for single and three phase systems. We also treated the fundamentals of synchronous machine models for stability evaluation including the idea of a reactive capability curk’e and static load models. The per unit system was reviewed and extended to quantities not frequently encountered, such as time and frequency. Also. some modeling aspects of static loads, such as frequency dependent loads in typical electric po\tw systems, were also discussed. The reader is referred to the bibliography section for references dealing with materials in the chapter and contributions made by many other pioneers in the field. Please note that an annotated glossary of terms is given to summarize the key definitions and terminology employed in this chapter.

3 Dynamic Electric Network Models

INTRODUCTION Chapter 2 focused on steady-state models to represent power system elemcr'is for the s o ~.alledstatic analysis studies. We nou turn O L I ~ attention to inodt Is for t ransi cii t o r d y niiiii ic operation a1 s t lidies. Th i s c ha pter describes soiiie sy s t e n i niodclh for analytical purposes. A model of an excitation system is studied i n Section 3.1 and Section 3.2 gikres ii discussion of a model of' the prime nio\er and go\~eriiorsystem. Dyii;iinic load models arc discussed i n Section 3.3 to conclude the chapter.

3.1

EXCITATION SYSTEM MODEL

The basic function of an excitation system is t o probride ;i direct current to lit. synchronous machine field Lvinding. I n addition, the excitation system performs control and protecti\,e functions essential t o the secure operation of the s>rstm by controlling the field Lmoltage and hence the field current 10 be within Licccpt;ib Ie le\.e1 s 11nder d i fferent operat i rig conditions. The control tunction~include the control o f iroltage and reiictiire po'i'er tl ow , there by en hanc i ng pourer s y ste 111 stabi I i t y . The protect i ve functions e n s i i re t h ;i t t he c ii pi1b i 1i t y 1i 111its o t the s y iic hronou s iii iic h i ne , e sc itat i o n s ~ 'tse 111, In d other ecliiipmcnt are not exceeded.

37

i

Power source (regulator)

Power i source ; (regulator) i

i

r

i

i

Power i Amplifier i

:

1

Reamplitier

+

I -

1 I

I I

I

L-

i j

i :

I

Power 'iource (exciter)

w

Excitation power SWrCe (exciter)

:

machine

I I

I

I I I I I

II I

4-1

7

I

I I

I I I I

I

I

-

:

I

I

I I

I

I ManuaL 'ii Conlrol:

Regulator

4

Excitation system

4

Figure 3.1

Excitation control system

E

~ S + S~y n c h, r o n o~u s ; machine

*

~

7

*:

Functional block diagram of a synchronous excitation control sy\tein.

Figure 3.1 shows the functional block diagram of a tj'pical excitation control system for ii synchronous generator. The following is ;i brief description of' the various subsysteins identified in the figure. The exciter pro\sides dc p o ~ ' e r to the synchronous machine field winding and constitutes the pouw stage of' the excitation system. Usually an exciter is modeled by the first-order system as shown in Fig. 3.2. The effect of saturation is considered by introducing S, and K,,, and thus the exciter model is given by E, =

1

1

+ S,, + K, + T , sv,

(3.1 1

The \.ohage regulator processes and amplifies the input control \ign;il\ to ii leiel and form appropriate for the control of the exciter. This include\ both regulating and excitation system stabilizing function (rate feedbach or lead-lag compensation). Nornially. the regulator is inodeled by a fir\t-order \ j stem a\ sho\+n i n Fig. 3.3. The regulator model is given by

~

Figure 3.2

Block diagram model of exciter

The terminal c oltage transducer senses the generator terminal voltage. rectifie\ and filters it t o dc, and compares it with a reference which repre\ent\ the h i r e d terminal c oltage. Figure 3.3 shows the terminal voltage tran\duc-er model given by:

The power system stabilizer provides an additional input signal to the r e p lator to further damp power system oscillations. Some commonly used i n w t signals iire rotor speed deviation, accelerating power, and frequency debiatirn. A pouw system stabilizer is modeled as shown in Fig. 3.5. The PSS model is gikwi by A V = K,,G'(s)Au+ Ki,G(.\)At;' U'here

Figure 3.3

Block diagram of the voltage regulator.

(.$.4)

V

vF

S

Figure 3.4

Block diagram of a terminal voltage transducer model.

Limiters and protective circuits ensure that the capability limits of excitor and synchronous generator are not exceeded. The full excitation system is modeled as shown in Fig. 3.6. The system is represented by the following set of linear differential equations: T,&V-Ij dt

?;C'VX=-V,-K(I::,-y+l(+y) dt

T

= V, - (S,+ K ,

dr dV T, --t = dr

-v+ &(& T

- (S,

KPSS

Figure 3.5

+ I ) E, + K , + 1 ) E,)

-

I + T5S

X2 l+T 2s

1+T1S

1+T2S

Block diagram of a typical Power System Stabiliter (PSS) model.

'"s

40

3.2

PRIME MOVER AND GOVERNING SYSTEM MODELS

The prinie soiirces of electrical energy supplied by utilities itre the kinetic enctg>' U ater and the thermul energ~'deri\wl frorn fossil fuels m d nucleiir fission. The prirnc rnoi.ers cot1Lw-t these soiirces of energj' into niechanicd entrgj' tl- at. in turn. is con\,erted to electrical form by the synchronous generator. The pri iiie m0 i . er go 1.ern i ng s y st e 111 prov i de s a means of c 011 t 1-0 I 1i ng pow e r and t'rccl iic II L' 1'. The !.Utict i o ti;i 1 re I iit i on s h i p betw ee 11 the biis i c e Ic me 11t s ;is soc i ii t ed w i t 11 PO\Ic I gcner:itioii and control i x shokkrn in Fig. 3.7. This section introduces the i i i o c cls I'oi- 111 draiilic turbines and go\,crning systems ;is well iis steam turbines atid tlicir go\~crnitigs>~stcms. of

3.2.1

Hydraulic Turbines and Governing System Model

The h>draulic turbine model describes the characteristics of' gate opening p :incl o 11 t p i t rncc ha 11i cal pow e r . I 11 pow e r system d y nam i c ;I 11a 1y s i s . the h >,drii LI I i c t LI rbiiic is ~isuallymodeled by an ideal lossless turbine along U ith the consideration ot' "wiitcr hiitiinicr" effect caused by the uwer inertia. is gic,en bl, 3.h) M

here T , )i \ \+raterstarting time. BCC;ILIWo f the "LI. ater h;itiiiiier" effect,

;I

change in gate position produce\

L

Energy Supply System t

Speed Governor

Valve or

1_1) Turbine

Generator

~

Speed Figure 3.7 Functional block diagram of pobver generation and control.

an initial turbine pouer change which is opposite that \shich I \ de\ired. For stable control performance, a large transient (temporari\ ) droop M it11 a long rcsetting time I \ therefore required. This is accomplisht.d by introducing a transient gain reduction compensation in the governing \i\ \tern. The coiiipen\ation retard\ o r limit\ the gate niovement until the mater flou and pouer output h a ~ c time to catch up. The gmerning system model i\ \ho\+n in Fig. 3.8.

Pilot Valve and

Figure 3.8 Bloch dingram of gojwning system for

max gate

;I

hjdraulic rurbinc.

42

The goirerning system model is given by

7(-,4! = -p tft

+ x,

(3.7)

= pilot v a l \ ~and ser\mi(itor time constant K , = servo gain 7;7 = main servo time constant R,, = permanent droop R , = temporary droop & = reset time ql,,,,, ,,,, = maximum gate opening rate q,ll,i, il,,rC = maximum gate closing rate p = gate position

3.2.2 Steam Turbines and Governing System Model A \team turbine convert\ stored energy of high preswre and high teniperatiire

jteani into rotating energy. The input of the steam turbine i\ control \,al\e pvsition (Ayl).nthile its output is torque (AT,). In power \tability analj\is. a order rnodel is used for \team turbine, i.e.,

r,,

ufhere = time constant Comparing the turbine models for hydraulic turbine and \team turbine, I t I \ clear that the re\ponse of a \teain turbine has no peculiarity wch a\ that extiib1tt.d by ;I hydraulic turbine d ~ i eto water inertia. The governing requiremen('t o f \team turbine\, in this re\pect, are more straightforward. There i\ no need for t riin 4 i e 11t droop coin pe n \a t ion. The go\ erning \ j \tern model i \ given by

max gate

band @

KS

'

-

r x l

S

-

1

l+sTG

Figure 3.9 Typical block diagram of a steam turbine

cx = K,

(U,

- U, - R,,X,)

(3.9)

lit

A typical governing model for steam turbine is shown i n Fig. 3.9.

3.3

MODELING OF LOADS

Load models are traditionally classified into two broad categories: static models and dynamic models. Earlier, we considered the static load models (Chap. 2). In this section, the dynamic load model is discussed.

3.3.1

Dynamic Load Models

Typically, motors consume 60 to 70% of the total energy supplied by a power system. Therefore, the dynamic effects due to motors are usually the most significant aspects of dynamic characteristics of system loads. Modeling of motors is discussed in this section. An induction motor can be represented by the equivalent circuit shomm i n Fig. 3.10. which accounts for quantities in one phase. In the equivalent circuit all quantities have been referred to the stator side. The directions of current shown are positive when operating as a motor, i n which case the slip s is positive. The rotor equation of motion is given by (3.10)

The torque

(r)is slip-dependent,

44

Figure 3.10

f:qiiiwlc.iit circuit

ol' ;I

ihrw-pli;isc.

iiiductioii in;icliiiic..

(3.1 I I

where /: is the nuinher o f poles atid ,<

s

is the slip. def'incd

;IS:

= !'C !I! 11,

being the synchronous speed of the machine. where P is the number of' pdes. aiicl,/'is the frequency. I n Eq. (3.I I ) we have

bvith 11,

U ,+,;

(K,+j .U,) x, = J - .U,,, K. +.; IX. + x,.., '

I t is noted that Ecl. (3.10)represents ;I stendy state pcrformaiice nioclcl 01 the intluction motor. with all qiiantities referrecl to the stator siclc. There ;ire inodcls that represent the trnnsicnt performance of the niotor th;it are bnsccl on I l u x liiikngcs. volt;iges. and torque variutions.

CONCLUSIONS In this brief chapter w e concentrated on models ibr power ~ystciiistahiliiy i n the tiiiie domain. We discussed electric excitation model as well as prime iiIover and governor system models. We concluded with a brief introduction to dynamic load model. For fiirther reading on the topic. ii reference list a r d an annotated glossary of teriiis is provided ;it the hack o f the book.

Philosophy of Security Assessment

IN TRO DUCTI 0N We are concerned with the implications of il major network disturbance such as a short circuit on a transmission line, the opening of a line or the switching 011 of a major load to name just a few. Here, we will consider the behaivior of the system immediately following such a disturbance. Studies of this nature are called transient stability analysis. The tendency of a power system to react to disturbances in such a manner as to maintain its equilibrium (stay i n synchronism) is referred to as stability. One way of classifying disturbances is through the categorization of small versus large. A disturbance is assumed to be small if the behavior of the system can be adequately represented through a linsarization of the nonlinear system of dynamic equations of the system. Stability considerations have been recognized to be among the essential tools in electric power system planning. The possible consequences of instabilitj, in an electric power system were dramatized by the northeast pobrer failure of 1965. This is an example of a situation that arises when il seLFere disturbance is not cleared away quickly enough. The blackout began with a loss of ;I transmission corridor. which isolated a significant amount of generation from its lo:id. More recently. a transmission tower in the Consolidated Edison sj~stemu ' a s hit by a severe lightning stroke in July 1977. The events that follo\s.ed led to the 45

shutdown o f power in New York City. Both events dramatized the consequences o f iiii instability in an interconnected electric power system. Our intention is to give ;in introduction to transient stability in electric power sjrstems. We treat the ciise of a single machine operating to supply iir infinite bus. The analysis of the tiiore complex problem of large electric po~vei' riet\\.orks 1t.ith the i titerconnections taken into consideration is treated ;is istell.

4.1

THE SWING EQUATION

I n the p o ~ er s \j'stem\ engineer'\ terminology, the dynamic equation relating t h t s inertial toryue to the net accelerating torque o f the s j nchrotious machine rotor i \ called the s~ ing equation. Thi\ \imply \tate\ (1% , - T',, tlt -

i4.1 t

The left-hand-side i \ the inertial torque uhich is the product of the inerti,i ( i n hg In') o f all rotating tii;i\se\ attached to the rotor shaft arid the angul: r :icccler:ition. The accelcrating torque T , , is in Nc\\ ton-meter\ and can be c'\ pres\ect as:

I n the abole, T,,,i \ the dritirig mechanical torque and i\ the retarding or load electrical torque. The angular po4tion of the rotor 0 may be eupre\\ed ii\ the f o l l o ~ing win of angle\

The angle cc is a constant Nhich is needed if the angle 0 is mea\ured from different from the iingular reference. The angle w,?t I \ the re\ult of the rotor angular motion at rated \peed. The angle 8 i \ time \ a r j ing and repre\erts de\ iation\ from the rated angular displacement\. This gi\ es the ba\i\ f-or our ~ i1se I-c1;it i on ;in

We find it iiiore conLtenient to substitute the dot notation

Therefore we have Jii' = T,,,- T,,

4.2

(4.5)

SOME ALTERNATIVE FORMS

Some useful alternative forms of Eq. (4.5) have been developed. The first is the power form which is obtained by multiplying both sides of by 03 and recalling that the product of the torque T and angular velocity is the shaft power. This results in Jog = P ,,, - P,,

The quantity J o ~is called the inertia constant and is truly an angular momentum denoted by M (Jdrad.):

Thus the power form is: M 8 = P,,,- P ,

(3.7)

A normalized form of the swing equation can be obtained by dividing Eq.

(4.5)by the rated torque THto obtain the dimensionless equation.

The left-hand side of the above equation can be further manipulated to yield a form frequently used. Recall the definition of the kinetic energy of a rotating body. This gives the kinetic energy at rated speed as 1 Wi = - J o ~

2

then L2 - W , TH o?,T,

We know further that the rated power is

-I8

Thus

Consequently we huvc

A constant which has proved very useful is c-noted by H. which is equ.11 speed divided by the rntcd power PK

to thc kinetic energy itt rated

The units of H ;ire in sec. As suing equation its

il

result we write the per unit or iiortiia1izi:d

Obscrving thiit .,,, = P,,,,, wc cat1 then write ZH 6 = p,,,- p ,

(4.0)

On

\vlicrc thc cquution is in p i i . 4.2.1

Machine Inertia Constants

Thc iltigttlitr iiionicntuiii inertia constant M as defined by Eq. (1.6)cu11be ohtainctl from mnnufacturcr supplicd machine datil. The machine kinetic ~11crgy. ,V. iiiay be written i n ter111sof M i1S follows:

where w K is the angular speed rclutccl to the frequency by

in electrical degrees per second. This i n tttrtl is

We c m therefore conclude that

The value of N is obtained from the moment of inertia of the machine Lisually denoted by WR‘ and traditionally given i n Ib-f’. The con\-ersion formula is:

The relation between H and M can be obtained using Eq. ( 3 . 8 )re\s
(4.14)

Here G is the machine rating. Thus (4.15)

The quantity H does not vary greatly with the rated power and speed of the machine, but instead has a characteristic value or set of \ d u e \ for each cla\s of machine. In the absence of definite information typical \ alues o f H may be used. The cur\,es in Figs. 3 . I . 3.2. and 4.3 give the general characteristic \ ariation of H for exi\ting and future large turbo generators.

5

--

#S

Figure 4.1 Inertia conatantb for large turbo generator rated 500 M V A and

below.

C'htrpter I 4-

3.5

--

A

CrJ

i . s 3

E

3--

2.5

--

2

--

I cn

2

-

3600 RPM FOSSIL

c

cn c 0

2

1.5

--

1

--

0.5

--

C -

-C- 1800 RPM NUCLEAR

0

,

Figure 4.2

1

1

I

1

, I

, -

Expected inertia constants for liitiire large turbo generators.

In sy stern studies U here several machines ha\ing different ratings are wed, the H constant for each machine, given to a base of the machine rating. rriu\t be conberted to the coiiimon system base by multiplying H in Eq. (4.14) b j the ratio (niachine base MVA/system base MVA).

4.3

TRANSIENT A N D SUBTRANSIENT REACTANCES

In order to understand the concept of transient and \ubtran\ient reactance\ of a \ynchronous generator, let U \ consider the transient behavior during a balariced t'ault. The dependence of the talue of the short circuit current in the ele:tric p o ~ e sr j \ t e m o n the in\tant i n the cycle at which the \hart circuit occur\ can be verified using ;I simple model. The inodel is ;I generaror uith wrie\ re\istance R and inductance L \ h o w n in Fig. 4.4. The xroltage o f the gene -ator I \ a\\uriied to Lary a\:

5I

4 -I

3 F

3.5

$

3--

3

z

I

--

2.5

--

v)

c.

c

2--

-A

450-514rpm

1.5

--

+B

2Ul-400rpm

+C

133-180rpm

1

--

+D

80-lXIrpm

0.5

--

tj 0

0

.E

r t -

0-

Figure 4.3

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Inertia constants of large water wheel generator\.

r ( t )= E,,, sin(cot + 00

With a balanced fault placed on the generator terminals at t = 0. then we can show that a dc term will in general exist. Its magnitude at t = O may be equal to the magnitude of the steady-state current term. The transient current i(r) is given by

where

The worst possible case occurs for the value of a g i \ m by:

jwL

R

In thi\ case the ciirrent magnitude ill approach titice the \tcadq -\talc II a \ alue iinmcdiiitely after the \hart circuit. The tran\ient current I \ pi\ -

iniiiiii \

Thus:

i ( t ) = -1E,,,

z

is It is clear that the maximum of i(r) is twice that of E,,,/Z.This n~a\~efortii shown in Fig. 4%

E,,,

For the case tana = or we have i ( t )= - sinot. This a.a\.eform is shourn i n

z

Fig. 3.Sb. It is clear from inspection of either the expression for the short circuit current or the response u'ajseform given in Fig. 4.6 that the reactance of the machine appears to be time varying. This is so if we assume a fixed Lfoltagesource E. For our power system purposes we let the reactance vary i n a step-wise. fashion X:, Xg;and X , , as shown in Fig. 3.6. The current history i ( t ) can be approximated in three time zones by three different expressions. In the first, denoted the subtransient inten-al. lasting up to 2 cycles. the current is I". This defines the direct axis subtransient reactancc

The second denoted the transient gives rise to

Figure 4.5

Short circuit current waLCeforms.

/

x;= EI' kchere I' is the transient current and X:, is the direct axis tran\ient reactance. T i e transient intenfa1 last5 for about 30 cycles. The steady-state condition gi\ses t 76' di rec t ii x i \ synchronous re ;ict ance .

Note that the subtran\ient reactance can be as low a s 7% o f t h e \ynchronc U \ re ;ic t ;i n ce .

S Y N C H R O N O U S M A C H I N E M O D E L IN STAB1LlTY ANALYSIS

4.4

A brief outline of equations to account for flux changes in a synchronous machine is given to define various electrical quantities and to construct phasor diagrams. The following approximations are involved in the models discussed:

I. 2.

3. 4.

5. 6. 7.

The rotor speed is sufficiently near 1.0 pi( and may be considered a constant. All inductances are independent of current. The effects due to saturation of iron are not considered. Machine winding inductances can be represented as constants plus sinusoidal harmonics of the rotor angle. Distributed windings may be represented as concentrated windings. The machine may be represented by a voltage behind an impedance. There are no hysteresis losses in the iron, and eddy currents are only accounted for by equivalent windings on the rotor. Leakage reactance only exists in the stator.

Under these assumptions, classical theory allows constructing a model of the synchronous machine in the steady-state, transient, and subtransient states. The per unit system adopted is normalized, although the term ”proportional” should be used instead of “equal” when comparing quantities. Note that one p i i field voltage produces 1 .O p i field current and 1 .O p i open-circuit terniinal voltage at rated speed.

4.4.1

Steady-State Equations

Figure 4.7 shows the flux and voltage phasor diagram for a cylindrical rotor synchronous machine ignoring all saturation effects. The following comments explain the construction:

I. 2.

3. 4.

5. 6.

The flux @, is proportional to the field current 1, and the applied field voltage and acts in the direct axis of the machine. The stator open-circuit terminal voltage E, is proportional to @, which is located on the quadrature axis. The voltage E, is proportional to the applied field voltage and may be referred to as E/. When the synchronous machine is loaded, a flux Q> proportional to and in phase with the stator current I and when added vectorially to the field flux @, gives an effective flux a,.. The effective internal stator voltage E, is due to @, and lags it by 90”. The terminal voltage V is found from the voltage E, by considering the

cf, Direct Axis

t I I I

I!

--b Quadrature Axis

\ oltage d r o p due to the Ieahage reiictance X and iiriiiatiirc rt" i+tatice K . 7 . B) +iniilaritj 01' triangle+. the dit'terencc betbeen E, and E i \ i n ph i\e U ith the / X \ oltage drop and i \ proportional to /. Theretim the \ o l ~ i g e difference n i a j bu treated ;i\ ;i koltage drop ;icro++ ;in iirniatiire rcictance .

x,

x>.

The 'riitii o f X,, a nd X i is termed the synchronous reactance, For the salient pole synchronous machine. the phasor diagram is more coinpies. Bec~iii'rethe rotor is syninietrical about both the d and c/ ;ixes i t is cot- \.unient to rusol\~e nianj' phasor quantities into coniponents i n these ;ises. 'The stator currunt niajc be trtxited i n this manner. Although a,, ujill bc proportional to I,:iind a,, w ill be proportional to /,,, because the iron paths i n the two :,xes ;ire diftcrent. the total iirniatiire reaction tlus 0 \+.ill not be proportional t o / iior necessarily be in phase \+rith i t . Retaining our earlier normalizing ;tsutiipti )ns.

it inay be assumed that the proportionality between !*, and @
4.4.2

Salient Pole Synchronous Machine

Figure 4.8 shows the phasor diagram of the salient pole synchronous machine. The d and q axes armature reactance4 are developed a4 in the c) lindrical rotor c a w . Direct and quadrature 4ynchronous reactance4 X,, and X,, can be establi\hed. i.e..

From the phasor diagram we have:

where V+ and V,. are the axial components of the terminal voltage V. In steady-state conditions it is reasonable to use the field voltage E, or tht! voltage equivalent to field current E, behind the synchronous reactances as thc machine model. In this case the rotor position (quiidrature axis) with respect t o the synchronously rotating frame of reference is given by the angular position of E,. As the cylindrical rotor model may be regarded as ii special case ofn salient machine (X,, = X,,), we will consider only the salient pole machine. 4.4.3

Transient Equations

For faster changes in the conditions external to the synchronous machine. the steady-state model is no longer appropriate. Due to the inertia of the flux l i n l ages these changes cannot be introduced throughout the whole of the modcl inimedintely. I t is essential to establish new fictitious voltages E,; and E,;. reprcsenting thc tlux linkages of thc rotor windings. These transient voltages call t-e shown to exist hehind the transient reactances X,; and X,;, E:, = v,,+ R,,I,,- la,x<;

E*;= V,,+ RJcl + l,,X;

The voltage E, is now considered as the sum of two voltages. E,, and E,, aid is the voltage behind synchronous reactance. In steady-state. current flows only in the field winding and hence, in that case, E,,= 0 and E, = 0. Allowing for the rotor tlux linkages change with time requires using t’ie following ordinary differential equations:

SE,;=

-,

-E

r,:..

The phasor diagram of the machine operating in the transient state is s h o w in Fig. 4.9.

Direct Axis

El

i

E,

Figure 4.9 Phasor diagram of a salient pole synchronous machine in the transient state.

4.5

SUBTRANSIENT EQUATIONS

Other circuits exist in the rotor, either intentionally, as in the case of damper windings, or unavoidably. These circuits are taken into account if a more exact model is required. The reactances and time constants involved are small and can often be practically disregarded. When required, the development of these equations is identical to that for transients and yields:

The equations are developed assuming that the transient time constants are large compared with the subtransient time constants. A phasor diagram of the synchronous machine operating in the subtransient state is shown in Fig. 4.10.

4.6

MACHINE MODELS

It is feasible to expand the model even further than the subtransient level but this is rarely done in multi-machine stability programs. Investigations using a

Direct Axis

generator inodel with up to seven rotor \+~indiiigs.h a\,e shown that using the \tandard niachine data. the I~IOI-e complex models do not necessu-ily j'ield acctiixte results. Ho~.t.\.er,improved results can be obtained if the data, cspeciall>. t tie t i nie constan t s. are appropriate 1y modified. The m o s t contwiient method of treating sq'nchronous machines of differir-g coniplcxit>' is to allow e x h machine the I ~ I ; I S ~ I I ~ possible LII~ number o f ec1u.ttioritr a i d then let the actual model used be cleterniined autoinatically accordir g to rhe data presented. Thus, fibre models are possible tot ii four-\s~indingrotor 4.6.1

Model 1

4.6.2

Model 2

4.6.3

Model 3

cl- and q-axis transient effects requiring two differential equations (SE:.and The follou~ingequations are used. A block diagram is shown in Fig. 3.1 1. E:, = v,,+ R,,I,,- l(/x:/

E:, = \'(,

+ RJ,, + 1,,x,;

U

Figure 4.1 1

Block diagram representation for model 3.

,SE).

4.6.4

Model 4

ti- and q-axis subtransient effects requiring three differential equation\ $E::, and \E:: ). The following equations are used.

4.6.5 t

(

Model 5

I- a i d (1- axi s s i i b t ra11s i e 11t e ffec t s req i i i ri n g f o u r cl i ffe re n t i a1 ey ii ;I t i o ri s are used.

.SE,:. .SE:;, and .SE:;). The follouing equations

( .;E

;.

The following mechanical equations need to be solFred for all these models

s6 = O - OH

4.7

GROUPS OF MACHINES A N D THE INFINITE BUS

Groups of synchronous machines or parts of the system may be represented by a single synchronous machine model. An infinite busbar, representing a large stiff system, may be similarly modeled as a single machine represented by model 1 , with the simplification that the mechanical equations are not required. This sixth model is thus defined as: 4.7.1

Model 0

Infinite machine-constant voltage (phase and magnitude) behind ci-axis transient reactance (X:,). Only the following equations are used.

4.8

STABILITY ASSESSMENT

In this section, we discuss the conventional approach to stability assessment applicable to a single machine against an infinite bus. The method leads to the equal area criterion. We concluded that a simple representation of the salient pole machine is offered by the model 0 given by:

U'e

A\

\+

;I

i I 1 make the to1lowi ng add i t ional a \ s

11111pt ions :

result: I:' = I' - I X' 0 = \'

+I

X'

The output po\ier of ttic niachine is g i \ w b j :

\ ' = I' c o d

I ' , = -\',

4iiii5

The phasor diagram is st1ou.n in Fig. 4.12. The electric P O M ' C ~output of the salient pole machine is thereti)re given b j :

The L ariation o f thc output for- salient pole rnachinc po\srer angle 6 is \ho\s,n in Fig. 4. 13.

L+

ith the torcliie o r

Network Imaginary Axis

Machine Quadrature Axis

Direct A xis

Real Axis

"d

Figure 4.1 2 Synchronous niachine and network frame\ o f rc!crence tor dcvclopiiig electric po\ver oii t pii t form u la .

= Pe

Figure 4.1 3

Power angle characteristics for a salient pole machint..

66

output Power, P

Angle,

S

(radians) Figure 4.14

Power angle characteristics for

I n the case o f

ii

;I

round rotor machine.

round rotor machine, u'e have Xf: = X ( ; and hence EV, . P , = -- sin6

x:,

The ~ariationof the output power for a round rotor machine uith the angle

6, (torquc or poww angle) is shown in Fig. 3.14. Ewmple 1 A \ynchronou\ machine is connected to an infinite bus through a transformer ha\ ing ii reactance of 0 . I p i and a double-circuit transmission line uith 0.45 pi' reactance for each circuit. The \ystem is shown in Fig. 4. IS. All reactance\ are giken to a ba\e o f the machine rating. The direct-axi\ transient reactance of t ie rnachine i\ 0. 15 p i t . Determine the \sariation o f the electrical pomw uith :ink le 6. Assullle v = I .o p. SoILitio17

An equitralent circuit o f the LtboLre system is shown in Fig. 3.16. From this ~ s ' e ha\-e the following: Xc(,= 0.475 p i .

.

Figure 4.1 5 System for example 1 ,

Changes in the network configuration between the t u o sides (sending and receiving) will alter the value of Xcyand hence the expression for the electric power transfer. The following example illustrates this point. Example 2

Assume for the system of Example 1 that only one circuit of the transmission line is available. Obtain the relation between the transmitted electric powrer and the angle 6. Assume other variables to remain unchanged. Solution

The network configuration presently offers an equivalent circuit as shoikw i n Fig. 4.17. For the present we have

Figure 4.16

Equiident circuit for example 1

EL6

Figure 4.17

Ecliii\ dcnt circuit tor cuainplc 2

Th c re to re.

0bserl.e that the ii1;ixiiiiiitii alue o f the corrc\ponding to the prec ious example.

4.9

i i m

clin e i \ l o ~ v trhan the

OI

c

CONCEPTS IN TRANSIENT STABILITY

111 order to gain ;in under\t;iiicling of the concept\ in\ ol\cd iri transient \tahility prediction. u e k v i l l concentrate or1 the \implil'icd netuorh con\isting of' ;i \eri:\ reactance connecting the machine and the int'inite h i \ . LTiider thew condition\ oiit. pou er cxprtxion recliice\ to

For .\iniplicity of notation u ' e \\'ill ;issiiiiie stcadJr-state \.alues. An iiiiport;int assumption that we adopt is that the electric changes iii\.olc,ed ;ire much faster th;in the resulting mechanical changes produced b), the gencrator/tirrbine spc ed coiitrol. Thus \i'e ;Issiiiiie that the rnechanical p o u w is ii constant tor the purpose o f triinsieiit stabilitj, cdculations. The functions P,,, iiiid P, are plotted i n Fig. -4.18.

The intersection o f the t L i ' o functions dcfincs t n ' o values for 6. The l o ~ r e r \~:ilueis denoted by 6,,.Consequently, the higher is n: - 6,,according to t h e \ >m iiietrj of the ciir\'e. At both points P,,,= P , , that is ti16/dt' = 0 and L\,C s a j ' that thc s)'stciii is in ecliiilibi.iitiii.

Power, P (P.u.)

?

(radians) Figure 4.1 8 Pou er-angle cun'e.

Assume that a change in the operation of the system occurs such that 6 is increased by a small amount A& Now for operation near 6,,. P, > P,,, and c126/c/t' becomes negatiLre according to the swing equation. Thus 6 is decreased. and the system responds by returning to its stable operating or equilibrium point. We refer to this as a stable operating point. On the other hand, operating at n 6,, results in a system response that will increase 6 and mo\re further from n 6,).For this reason, we call x - 6,)an unstable equilibrium point. If the system is operating in an equilibrium state supplying an electric pourer P, ,, with the corresponding mechanical power input P,,,,,,then the corresponding rotor angle is 6(,.Suppose the mechanical power P,,,is changed to P,,,, at a fast rate, which the angle 6 cannot follow as shown in Fig. 3.19. I n this case P;,l> P, and acceleration occurs so that 6 increases. This goes on until [he point 6, where P,,,= P,, and the acceleration is zero. The speed, houtvw, is not zero at that point, and 6 continues to increase beyond 6,. I n this region P,,,< P, and rotor retardation takes place. The rotor will stop at 6 where the speed is zero and retardation uill bring 6 down. This process continues on as oscillations around the new equilibriiini point 6,. This serves to illustrate what happens when the system is sub-jected to a sudden change in the power balance of the right-hand side of the suting equa-

70 Power, P (P.u.)

(radians) Figure 4.19

Po\scr angle curve.

tion. The \ituation described abo\fe will occur for 5udden change\ in P, ;i\ MI:II The \ j s t e m discussed in Examples 1 and 2 \er\re\ to illustrate thi\ point. nhich u e discii\\ further in the next example. E\i,?niplc. 3

The \j'sterii of example 1 i \ deliirering iin apparent power of 1 . I p i ( at 0.85 PF lagging with two circuits of the line in ser\ ice. Obtain the \ource voltage (ewit a t i o n koltage) E and the angle 6 under these conditions. With the \econd cir(*uit open a\ i n Example 2. ;i new equilibrium 1 1 1 angle can be reached. Shetch the power mgle curves for the t w ' o conditions. Find the angle 6,, and the electric POMer that can be tran\terred immediately follow ing the circuit opening, ;I\ \L ell ii) 6 , . A\\ume that the excitation \ oltage remain\ unchanged.

S o ILIt ion The power delivered is P,,= S cos $, P,, = I . I x 0.85 = 0.94 pi Using P = VI

COS

Thus we can

U

@, then the current i n the circuit is I =

rite

1

-~

1.1

cos

' 0.85

71

E=V+jXl

+ (1.1i - 31.79")(O.37SL90") = 1.28 + j0.44= I .3S i 19.20" p i r 1

Therefore, E = 1.35 P.u., 6,,= 19.20" The power angle curve for the line with two circuits according to Example 1 , is P,,,= 2.1 053 x 1.3Ssin6 = 2.83sin6

With one circuit open, the new power angle curve is obtained as in Example 2, thus

3

P,, = I .43 x 1.3Ssin6= 1.93sin6

The two power angle curves are shown in Fig. 4.20. From inspection of the curves, we can deduce that the angle 6,. can be obtained from P,,,= P,sinS, (curve B ) 0.93 = 1.93sin6,

6, = 29.15"

We can obtain the value of electric power corresponding to line open as

with one

P,.,o= 1.93sin19.2" = 0.63 pi4

4.10 A METHOD FOR STABILITY ASSESSMENT To predict whether a particular system is stable after a disturbance i t is necessary to solve the dynamic equation describing the behavior of the angle 6 immediately following an imbalance or a disturbance to the system. The system is said to be unstable if the angle between any two machines tends to increase without limit. On the other hand if under disturbance effects, the angles between e\.erq' possible pair reach maximum value and decrease thereafter, the system is deemed stable.

Angle. 13 (radians)

Assuming as we h w c :ilrcady done that the input is constant, ncglipi 3le doniping iuid coiist;itit source \dtagc behind the tnitnsient reactance. the angle hetwccn t w o machines either increase indefinitely or oscillates iif'tcr all disturh;iiicch h;i\.c occurred. Thcrcforc'. in the c;isc of tw:o machines, these \ \ i l l either f d l out of' step on the first swing or never. Here the obscrvution that the iiiachineh angular differetices stay constalt can be tiikt.tl ;IS ill1 indication 01' hystctii \tahility. A simple method lor determining htiibility knowii iis the c'cluiil-; re;i tiicthotl is nvnilable. We will discuss this here. The h\vitig ccluation for ;I machine connected to ;in infinite bus ci111he writtell a s

We obtain an expression for the variation of the ringirlar speed o w i t h ? . by noting the alternative form 0 tlo, =

Integrating, assuming

cc) = 0

P (16 M

and integrating the abo\,e equation,

\IT

obtain

or

The above equation gi\ves the relative speed of the machine M i t h r-eclpcct to a reference frame nim ing at a constant speed (by definition of the angle 6). If the \ystem is stable, then the speed must be zero when the acceleration i \ either zero or i \ opposing the rotor motion. Thus for a rotor n4iich i \ accelerating. the condition for stability i\ that a value of 6, exists such that

This condition is applied graphically in Fig. 4.2 I where the net area under the P,, - 6 cirr~~e reaches zero at the angle 6 as s h o w . Obsenve that at 6,,.P, is negative and conseqirently the system is stable. Observe that the area A equals A! as indicated. The accelerating power need not be plotted to assess stabilitj,. Instead. the same information can be obtained from a plot of electrical and mechanical pouters. The former is the power angle curve and the latter is assumed constant. 111 this case the integral iiiay be interpreted as the area between the P, c i r r \ ~and the c~rr~re of P,,,both plotted versus 6. The area to be equal to zero. must consist of a p0sitii.e portion A I.for which an equal and opposite negatiLre portion of ,4 must exist, for which P,,,< f,,. This explains the term equal-area criterion for stability. This situation is shown in Fig. 4.22. If the accelerating power reverses sign before the t ~ ’ oareas A and .4? ;ire equal. synchronism is lost. This situation is illustrated in Fig. 3.23. The areki A : is smaller than A , and as 6 increases beyond the value \!,here P(, re\’erxes sign again, the area A j is added to A , .

71

A

Figure 4.21

S t ;Ihi I it J condition for accelerating rot or.

Power, P

Pm

-

6,

0

Figure 4.22

Equal-area c*r.itcriori tor \tabilit!,.

b

Power Angle, 8 (radians)

75

Accelerating Power, P ,

Figure 4.23

I

Accelerating powrer as a function of the torque angle.

Example 4

Consider the system of the previous three examples. Determine whether the system is stable for the fault of an open circuit on the second line. If the system is stable, determine 6,, the maximum swing. Solution

From the examples given above we have 6,,= 19.2" 6r = 29.50'

The geometry of the problem is shown in Fig. 4.24. We can calculate the area A immediately: ?U IS

A I = 0.94 [29.15 - 19.20In 1.93sin6 t / S 1 *0 I0 2 0

Observe that the angles 6 , and 6,) are substituted for in radians. The result is: A I = 0.0262

The angle 6,. is

76

L

Power, P (P.u.

2.84

1.93

0.94

0

6 , = 1x0 - 6 , = 15o.xf;

This clcurly gives

;init

the system is stable. The iingl~.6 ,. is obtained hq solving tor A I = A , . Here we get:

77

This gives some algebra I .93 c o d , + 0.0 164 6, = 2.1376

The solution is obtained iteratively as 6, = 39.39" This example shows the application of the equal area criterion to the case of a generator supplying power to an infinite bus over tu'o parallel transmission lines. For the loading indicated above the system is stable. The opening of one of the lines may cause the generator to lose synchimism e\'en though the load could be supplied over a single line. The following example illustrates this point. Example 5

Assume that the system in Example 3 is delivering an actiLre power of 1.8 p i 1 using the same source, voltage E, as before. Determine urhether the sqrstem u?ill remain stable after one circuit of the line is opened. Solution

We have for the initial angle 6,, I .8 = 2.84 sin&,

6,,= 39.33" The angle 6,. is obtained from I .8 = 1.93 sin6,

6 , = 68.85"

The area A , is thus: A , = I .8(68.85 - 39.33jn

I 8o

-

OX

xs

31)

73

1 1.93sin6(16 = 0.13

The area A_.is obtained as: ii

A, =

It j 3.86\inti (16 - 1.816,- 6,1-180 = o.o6

We note that A I > A:, and the system is therefore unstable.

I f ii three-phase \hart circuit took place at ;i point on the extreme end o . the line. there i \ some impedance bet\veen the generator bus and the load ( i n f i tiite) h i \ . Therefore, \ome p o ~ e ri\ tran\mitted cvhile the fault i \ \ t i l l on. Tht. \ituation I \ \iiiiiIiir to the one\ anal) l e d abo\ e and \ie ii\e the follou ing e'iamplt: to illu\trate the point. Ewmple 6 .A generator is deli\,ering 25%- o f PI,,,,to an infinite bus through ;i transmission line. A f a u l t occurs such that the reactance b e t u w n the generator and the bii4 i \ incrcased to tuv times its prefuult value.

I. 2. 3.

Find the 6,, before the faiilt. Shoci, graphicallq, ci,hat happens when the f i i u l t is sustained. Find the niiisiriiiiin \raliie of 6 s\\ting in case of ;I sustained l'ault.

Solution Figure 4.25 illustrates the situation for this example. The amplitude of the po\i t r anglc c i i r ~ etttith the f;iiiIt su\tained is half o f the original \-alue. Before the t'itiilt \+e h a t e

At the

t'uiilt

instant, \4fe get

A \ before. the \tability condition yields

Hence 0.5

COb6,

+ --< K 6, = 0.5473 7'

Bq trial and error 6, = 46.3"

A

Power, P (P.u.)

0.50

0.25

Torque Angle, S (radians)

0 Figure 4.25

Power angle curves for example 6.

The following example illustrates the effects of short circuits on the network from a stability point of view. Example 7

The system of the previous examples delivers a power of 1 .O p i when subjected to a three-phase short circuit in the middle of one of the transmission circuits. This fault is cleared by opening the breakers at both ends of the faulted circuit. If the fault is cleared for 6, =50°, determine whether the system will be stable or not. Assume the same source voltage E is maintained as before. If the system is stable, find the maximum angle of swing. Solution

The power angle curves have been determined for the prefault network i n Example I and for the postfault network in Example 2. In Example 3 we obtained E = 1.35 plr

j0.225

Figure 4.26

NctLforh coni'iguration ciuring rhe faitlr.

There t'rm

During the fault the netnvrk offers ;I different cont'i~uratioti.L+ hic.t- i \ \ho\\ 11 i n Fig. 4.26. We \ + i l l need to reduce the netnorh in wch ii u ; i j ;I<, to obtain ;I clear path l'rotn the +oiirct' to the infinite h i \ . We do thi\ b j u\iiig ;I 1' - A tr~iii\t'ortnationii\ indicated in Fig. 4.27.

)(0.25)= I ,2 x = 0.45 + 0.25 + (0.45 0.235

and

friiilt

pourer angle curve is given by f = 1, I3 sin6

The three power angle curl'es are shown in Fig. 1.28. The initial angle i h gilren by the equation 1 .O = 2.84 sit&

6,,= 20.62" The clearing angle is 6, = SO"

Power

P (P.U.

2.a

/

I .93

\

1.13

I .a

y i

I

i i I

,

I I

.

I I

6f

Figure 4.28

Pre-fault. during fault, and post-fault poww angle

Torque ~ n g ~S e. (radians)

CLII'VL'S t o r t.xii11113lc

7

S2

The area A I can thus be crilculated as:

The maximum area A ? is obtained using the angle 6, I = 1.93 siii6:

6;'= 31.21" 6, = I80 - 6; = 148.79' NOW

We note that A I > A ? , and the system is theretore stable. T o calculate the angle o f iiiiisiiiiurn swing we ha\fe

Hence

Bjr trial and error

6, = 66.3''

4.1 1

MATHEMATICAL MODELS A N D SOLUTION METHODS IN TRANSIENT STABILITY ASSESSMENT FOR GENERAL NETWORKS

I t is coiiinioii practice to model static equipment in the transmission sj.! ten1 by lumped equicralent pi parameters independent o f the changes arising i n the gene rat i ng and load eq U i p me n t , Th i s approach i s e nip1oy ed i n m U 1t i m iic hi ne stabilitj programs because the inclusion o f time Lrarying parameters ~voulclpro-

duce major computational difficulties. Moreover, frequency, the most obtrious variable in the network, usually varies by only a small amount and thus, the errors involved are insignificant. Additionally. the rates of change of network variables are assumed to avoid the introduction of differential equations into the network solution. The transmission network can thus be represented in the same manner as in the load-flow or short-circuit programs, that is, by a square complex admittance matrix. The behavior of the network is described by the matrix equation: I,,, = YV

where I,,, is the vector of injected currents into the neturork due to generators and loads Y is the admittance matrix of the network, and V is the vector of nodal vol tages. Any loads represented by constant impedances inay be directly included in the network admittance matrix with the injected currents due to these loads set to zero. Their effect is thus accounted for directly by the network solution. 4.1 1.1

System Representation

Two alternative solution methods are possible. The preferred method uses the nodal matrix approach. while the alternative is the mesh matrix method. Matrix reduction techniques can be used if specific network information is not required. but this gives little advantage as the sparsity of the reduced matrix is usually very much less. Nodal Matrix Method

In this method, all network loads are converted into Norton equi\-alents of injected currents in parallel with admittances. The admittances can be included i n the network admittance matrix to form a modified admittance matrix which is then inverted, or preferably factorized by some technique so that solution at each stage is straight forward. The following solution process applies:

I.

2.

For each network load, determine the injected currents into the modified admittance matrix by solving the relevant differential and algebraic equations. Determine network voltages from the injected currents using the Zmatrix or factors.

As the network voltages affect the loads, an iterative process is often required, although good approximations can be used to avoid this. With the Nodal Matrix method, bus voltages are available directly and branch currents can be calculated if necessary.

4.1 1.2 Synchronous Machine Representation in the Network The equations representing ;I )syiichroiioiis miichine ;ire gi\ren in the form of The\,cnin voltages behind its impedances. This must be modified to ii currint source in parullel urith ;in admittmce using Norton's theorem. The x h i i t t m c e o f the machine thus formed may be added to the shunt admittance o f the 11i;ichine bus and treated as a netuwk parameter-. The \ ector /,!/ thus contains the Norton equi ktalen t c iirrcii t s of the sj'iichroiio us iiiach i nes , The sj'iichronous 11I achine equations are kvritten i n a frame OF reference rotating ivith its o\vn rotor. Thc real anci imaginary coniponenth o f the iietwork equations, iis giLrt.11 in Fig. 4.2 9. art' obrai 11ed !I-() III t h t' fo1I o c i ' i iig t ran sto r nia t i on

The in\,erse relation is

Machine direct axis

t

Network imaginary axis

Machine

The transformation also applies to currents. When saliency is accounted for, the subtransient and transient reactances i n direct and quadrature axes frames are different, and the Norton shunt admittance will have a different \ d u e in each axis, and when transformed into the netLivrh frame of reference. will be time varying. To circum\rent this difficult). a constant impedance is used while modifying the injected current. 4.1 1.3

Load Representation in the Network

To be suitable for representation in the overall solution method, loads niust be transformed into injected currents into the transmission netM-ork from kvhich the terminal voltages can be calculated. A Norton equivalent model of each load niust therefore be introduced. In a similar way to that adopted for synchronous machines. the Norton admittance may be included directly in the net1iu-k admittance matrix. A constant impedance load is therefore included in the network admittance matrix and its injected current is zero. This representation is extremely simple to implement. causes no computational problems, and impro\.es the accurricy of the netufork solution by strengthening the diagonal elements in the :idmittatice matrix. Nonimpedance loads may be treated similarly. I n this case the stead>,state values of voltage and complex power obtained from the load tlow are uscd to obtain a steady-state equivalent admittance (7,)) which is included i n the neturork admittance matrix [ Y ] . During the stability run, each load is s o l k ~ dsequentially along with the generators, etc., to obtain a new admittance i.e.:

(r).

The current injected into the network thus represents the de\siation of the load characteristic from an impedance characteristic.

By converting the load characteristic to that of a constant impedance, M hen the \toitage drops below some predetermined value ( I/,,,,,,),the irijected current i\ kept relatikrely small. An example of a load characteristic and its corresponding injected current is shown in Fig. 4.30. In an alternative model the low-voltage impedance is added to the net\sorh and the injected current compensates for the decriation from the actual characteristic. In this case, there is a nonzero injected current i n the initial steady-state operational condition.

86

I I) power

Figure 4.30 Load and injecled currents for a con4tant type load adju\tiiient. ( a ) Load current. ( h ) Injected current.

U ith

l o ~ iolta;e '

4.1 1.4 System Faults and Switching I n general most power system disturbances to be studied will be caused h y changes in the network, normally caused by faults and subsequent switchi ig action. Occasionally the effect of branch or machine switching will be considered. Although faults can occur anywhere in the system, it is much easier comrutationally to apply a fault to a bus. In this case, only the shunt admittance at the bus need be changed, that is. a modification to the relevant self-admittance o f the Y matrix. Faults on branches require the construction of ;i dummy bus at the fault location and suitable modification of the branch data unless the distar ce between the f w l t position and the nearest bus is small enough to be ignored

Philosophy of Secirr-ity Assessriierir

87

The worst case is a three-phase zero-impedance fault and this involves placing an infinite admittance in parallel with the existing shunt admittance. In practice, a nonzero but sufficiently low-fault impedance is used so that the bus voltage is effectively brought to zero. This is necessary to meet the requirements of the numerical solution method. The application or removal of a fault at an existing bus does not affect the topology of the network and where the solution method is based on sparsity exploiting ordered elimination, the ordering remains unchanged and only the factors required for the forward and backward substitution need be modified. Alternatively the factors can remain constant and diakoptical techniques can be used to account for the network change. 4.1 1.5

Branch Switching

Branch switching can easily be carried out by either modifying the relevant mutual- and self-admittances of the Y matrix or by using diakoptical techniques. In either case, the topology of the network can remain unchanged, as an open branch is merely one with zero admittance. While this does not fully exploit sparsity, the gain in computation time by not reordering exceeds the loss by retaining zero elements, in almost all cases. The only exception is the case of a branch switched into a network where no interconnections existed prior to that event. In this case, either diakoptical or reordering techniques become necessary. To avoid this problem, a dummy branch may be included with the steady-state data of sufficiently high impedance that the power flow is negligible under all conditions, or alternatively, the branch resistance may be set negative to represent an initial open circuit. A negative branch reactance should not be used as this is a valid parameter where a branch contains series capacitors. Where a fault occurs on a branch but very close to a bus, non-unit protection at that bus will normally operate before that at the remote end. Therefore, there will be a period when the fault is still being supplied from the remote end. There are two methods of accounting for this type of fault. The simplest method only requires data manipulation. The fault is initially assumed to exist at the local bus rather than on the branch. When the specified time for the protection and local circuit breaker to operate has elapsed, the fault is removed and the branch on which the fault is assumed to exist is opened. Simultaneously, the fault is applied at the remote bus, but in this case, with the fault impedance increased by the faulted branch impedance, similarly the fault is maintained until the time specified for the protection and remote circuit breaker to operate has elapsed. The second method is generally more involved but it is better when protection schemes are modeled. I n this case, a dummy bus is located at the fault

position. (even though it is close to the local bus) and a branch with a \U-:,! small impedance is inserted between the dummy bus and the local bus. Th: faulted branch then connects the dummy bus to the remote bus and the branc i shunt susceptance originally associated with the local bus is transferred to the dummy bus. This may all be done computationally at the time when the fault i s being specified. The two branches can now be controlled independently by sui ,able protection systems. An advantage of this scheme is that the fault durativn need not be specified iis part of the input data. Opening both branches et'!'el:ticely isolates the fault, \+rhich can remain permanently attached to the dunin bus. o r if auto-reclosing is required. it can be reiiiowd automatically after ;I su i tabIe de i on i ~at i on period . I f the network is not being solired by ii direct method, the second method will probably fail. During the iterati1.e solution of the network, slight volta,;e errors nrill ciiuse large currents to flow through ii branch with ;I 1w-y smill impedance. This will slow convergence and in extreme cases uill c~iusediwrgence. With ii direct method, based on ordered elimination, ;in exact solution o f the bus Lwltages is obtained for the iiijected currents specified at that particuhr iteration. Thus, pro\.ided that the impedance is not s o small that numerical problems occur L+,hencalculating the admittance, and the subsequent fwtors for 1 he forward m c l backward substitution, then convergence of the ow-all solut i oii betu.een machines m c l netL+x)rkwill be unaffected. The k-aluc.o f the lo\+!-impcdance branch between the cluriiriiy and local bus may be set at II fraction of the total branch impedance, sub-ject to a minimum value. I t this fraction is u n k r 0.001. the change i n branch impedance is \ w y small compared to the iiccur ic>r of the netuwk data input and it is unnecessary to niodify the iiiipedance of' the branch from the remote to the dummy bus.

4.1 1.6

Machine Switching

Machine su,itching may be considered, either as a network o r machine operation. It is ~i netu.ork operation if a duiiiiiiy bus is created to which the machine is connected. The dummy bus is then connected to the original machine hub b!, ;I IOU*- i mpedance branch. Alternati\rely. it may be treated as a machine operation by retaining the original n s t N v r k topology. When a machine is sw.itched out, it is necessai y to reniove its iri-jected currcnt froni the network solution. Also a n y shunt admittance included i n the network Y matrix, which is due to the machine mu!,t be re n10 v ed . Although a disconnected machine can play no direct part i n system stability, its response should still be calculated as before, nrith the machine stator current set to zero. Thus machine speeds, terminal \.oltages. etc.. can be obser\xd ci'eii

when disconnected from the system and in the ek'ent of reconnection, sensible results are obtained. When an industrial system is being studied many machines may be disconnected and reconnected at different times as the ~ ~ ~ l t ale\rel g e changes. This process will require many recalculations of the factors in\-ol\,ed i n the forumd and backward substitution solution method of the netmwk. However, these can be avoided by using the method adopted earlier to account for synchronous machine saliency. That is, an appropriate current is iiijected at the relt'\mt buses, which cancels out the effect of the shunt admittance.

4.12

INTEGRATION TECHNIQUES

Many integration methods have been applied to the power \ysteni tran\ient bility problem, and the principal methods are discus\ed nou. 4.1 2.1

\ta-

Predictor Corrector Methods

These methods for the solution of the differential equation .sY = F ( Y . X ) with Y ( 0 )= Y,, and X ( 0 ) = X , ,

hakre all been detreloped from the general k-step finite difference equation:

Basically the methods consist of a pair of equations, one being explicit

(PI,= 0) to give a prediction of the solution at t(rl + I ) and the other bein, 1111. plicit (PI,f 0) which corrects the predicted value. There are a great Larietj of 0

method\ aLrailable, wch as hybrid methods, the choice of which being made bq the requirements of the solution. It is usual for simplicity to maintain a constant step length with these method\ if X > 2. Each application of a corrector method inipro\e\ the accurac~of the method by one order, up to a maximum given by the order of acc~iracyof the corrector. Therefore, if the corrector is not to be iterated. it i\ common to use a predictor with an order of accuracy one less than that of the corrector. The predictor is thus not essential as the value at the preLrious step may be u\ed as a first crude estimate, but the number of iterations of the corrector may be 1arge. While for accuracy, there is a fixed number of rele\mt iteration\, i t is

desirable for stability purposes to iterate to some predetermined level of convergence. The characteristic root ( z , )of a predictor or corrector when applied to thc single variable problem

may be found from

Applying a corrector to the problem defined and rearranging gives:

the solution to the problem becomes direct. The predictor is now not necessaiy as the solution only requires information of 1' at the previous steps. i.e.. .it \v(ii = i + 1 ), for i = 1 , 2,. . . . . k . Where the problem contains two variables, one nonintegrable, such that:

then

where

Although tive.

and

~ i , , ,are ~

constant at a particular step, the solution is itera-

Strictly in this simple case, x ( i i + 1 ) could be eliminated but in the general multivariable case this is not so. The convergence of the method is now a fu ICtion of the nonlinearity of the system. Provided that the step length is sufficiently small. a simple Jacobian form of iteration gives convergence in only a few iterations. It is also possible to form a Jacobian matrix and obtain a solution by a Newton iterative process, although the storage necessary is much larger and the step length must be sufficiently small to ensure convergence.

For a multivariable system, the following two equations are coupled .sY = F( Y , X ) with Y ( 0 )= Y,) and X ( 0 ) = XII

and the solution of the integrable variables is given by the matrix equation. The elements of the vector L ' , , + ~are given by the vector form of

and the elements of the sparse

matrix are given by

The iterative solution may be started at any point in the loop, if Jacobian iterations are used. 4.12.2

The Euler Method

Consider the following ordinary differential equation:

Let x(t) be the state vector of this nonlinear differential equation, which is to be solved by an appropriate integration technique. The Euler method utilizes a predictor function based on the Taylor series expansion of .r(t + A[), uthere At is the step size. As such, by neglecting the higher order terms in the series expansion, we obtain the generalized Euler's formula: s ( t + A t ) = x ( t ) + x(t).Ar where

d.r(t ) i ( r ) = __ at

The method is not often used for real-time applications in power systems, as it is computationally burdensome. Also, the accuracy of this model for integration is sacrificed by the truncation of O[At)'] and higher order terms done in the Taylor series expansion. Figure 4.3 I outlines an example of Euler's integration technique to power system dynamic stability assessment. 4.12.3

The Modified Euler Method

Reconsider the following problem to be solved: i ( t )= f I ( s ) , t ]

Set Loop Control time: T,, & & Set Integration Step size: At

Flow Calculations and obtain P,(t), the machine injection in p.u. MW

Solve the system of differential equations for all machines: ?f

- 4"

w ( t ) = --(&

I

H

Set Fault Time: tf = t

Perform 'during fault'Power Flow Calculations

Figure 4.31 The transient stability algorithm

s=o-w

'

R'

w

R

=2nf

Q

@

Using EulerS Method, Solve the machine angles: 6 ( t )= u(t)-2 q 6(t + Ar) = &r)+ 8(f)Ar

Yes

I SaveDisplay Final Results:

I ~~

~

~

~

~

~

~

Set Fault Clearing Time: & = t

Obtain‘new’state of the machine andor the network perturbations i

Figure 4.31 Continued.

I

Torque Angle, Machine Speed, Generated Power, Times (t, tf. tmarretc.) and so on.

where x ( f ) and.f'(x, t ) are scalar or vector quantities. Expanding . d f ) to the riglit and left using the Taylor expansion yields:

If the O[(At)'Jand higher order terms are truncated for a small step size, it c;in be shown that the modified Euler formula is obtained as follows:

Alternatively, we may bfrite:

Note that the modified Euler inethod is not self*-starting and thus requires an initial prediction on the state variables in the \rector solution. 4.1 2.4

Trapezoidal Method

The trapezoidal rule is a nonself-starting integration technique that is somewhat related to the modified Euler method and is based on the geometric interpretation of the problem. The order of truncation in the Taylor series expansion of .v(t k Ar) is O[(A?)']. The trapezoidal formula is:

The merits of the trapezoidal method is evident in the accuracy brought forward iis a result of the truncation of the O[(At)'] and higher order terms. 4.1 2.5

Runge-Kutta Methods

Runge-Kutta methods are able to achieve high accuracy while remaining sir gle step methods. This is obtained by making further evaluation of the functions within the step. Here we present a class of self-starting prediction formulae, which are applicable in the assessment of transient analysis of rotating nachines, in which case we are solving the swing equation of the generators. The numerical integration of the ordinary differential equation given b).:

by a predictor generally involves calculation of xn + 1 as ii function of x,,, Y,, I, x,,-:, . . . fll. f,,-l, . . . , and tl,. If the predictor is self-starting, it must be free

.

fl,-.,

of terms in xn-l,x,,-?, . . . , f,, fll-l,fIl-?and so forth. That is, the next state solution sould be of the form: x,,+~= F(x,,, fl,, t"). Now, the equation below summarizes this requirement and is called the Runge-Kutta predictor formula.

i

111

where k ( i ) = h f r ( n ) + c,h, ! ( / I )

+ /=I

w, = 1

and the sum of all the weights equal to unity, i.e., I=

I

The coefficients are uniquely determined, giving rise to various orders and associated approximate or accurate models of the Runge-Kutta predictor formula. Table 4.1 summarizes the 2"d, 3rd,and 4Ih order Runge-Kutta formulae. Being single-step these methods are self starting and the step length need not be constant. If j is restricted so that j < i then the method is explicit and c I must be zero. W h e n j is permitted to exceed i, then the method is implicit and an iterative solution is necessary.

Table 4.1

Summary of the RK-2. RK-3, and RK-4 Predictor Forniulae

Runge-Kutta Predictor Formulae

Coefficients

RK-2 .Vil+]

= .Vll

+ h (k"' + k'?'] -

2

RK-3 .Vfl+l

= .Vfl

+ -It [k"' + 6

+ k'?']

k' ' = F(.vl,,t1,)

RK-3 = .vil

+

It

-

6

(k"' + 2k'!' + 2k'"

+ kf4']

k'?' = F!.Vlt

+ -h k " h + 2

-

2

i

Explicit Runge-Kutta methods have been used extensibdy in transient stability studies. They have the adkmtage that a packaged integration method is iisually available or quite readily constructed and the differential equations at-: incorporated with the method explicitly. It has only been with the introduction of no re detailed system component models with very small tiiiie constants, th:,t the problems of stability has caused interest i n other methods. Again. the advantage o f the Runge-Kutta techniques is that they are all seli'starting. The propagated error from one iteration to the next does not increa>e rapidly, thus the inethod is said to be stable. The exception is the solution 10 "stiff' prohleins, &(herethe solution may diverge unless a small step size is used. (The stiffness of the ordinary differential equations ot' the system can be iiie;isiirL d iis ii functiuon o f the ration between the smallest to the largest eigeindues of the linearized system or eigenvector analysis of the Jacobian matrix.) Ne\wtheless. they are not ~ e r ayttractive to pourer system engineers iis k poww flow calculatioris must be done at every iteration. where k is the order o f the Runge-Kutta forinula. The tollou,ing esnmple demonstrates the RK-4 application.

Exc?nip le

Application o f the 1"' order Runge-Kutta integration techniqiie to p o er~ \ J \tc in dynamic \tability asws\iiient. The algorithm is giL en belou. I. 2.

Start the RK-4 Subroutine. Initialize all R K - 3 dependencies.

COllIlt, d = 0: Step Size. At Time. t = 0 : M iix i mum Time, t Maximum iteration = itniax

,11,1,

3. Sol\e the initial pouer flow and the machine equation\, and obt.iin the generator torcllie angle (6(0)). terminal voltage\. bu\ in.jection\ P;"'(o).etc. 4. InitialiLe the RK-4 Coefficients: k , = I, = 0 for all i E { l , 4 } 5. Set the e\tirritrfc iirc/c.\ to: i = d + I . 6. Perform R K - 4 calculations (obtaining the estimate\).

97

7. Increment count: d = d + I . 8. Increment the estinicitr i1ii1e.v to: i = i + I .

9. Update the torque angle delta to: 6 ( t )= & ( I )

+ 5I k, and

sol\^ the pom'er

flow for I v , l,Py"(t), etc. 10. Compute the new coefficients.

I k, = ( ~ ( 0+)-I,-) 2

I, = w ( f i l i

-

2Ff)At

- f:"'(t))At

2H

I I . If d < 3. then go to step 7, otherwise continue. 12. Compute the final value of the power angle and the machine speed at t=t+At 1

6(t + At) = 6(t ) + -(ki 6 CO(t

+ 2k: + 2 k , + k,)

+ At) = CO( t ) + -(I 1, + 21, + 21, + 1 4 ) 6

13. Reset the count: d = 0 14. Compute the final power flow for this time interkal to get 1 V, 1. P?( t 1.

etc.

IS. Increment the timer ( t = t + At) and make the following comparisons: If

Elseif Else

then flag the user: "Maximum Number of iteration reached!" and go to step 16. ( t < t,,,,,,)and ( N < itmax) then increment the iteration count to: N = N + 1 and go to step 4. Flag the user: "Run time value of tmax reached!" and cont i n Lie,

( t < t,,,,,,) and ( N 2 itrncix)

16. Display all results, power angle, machine speed. generator p o ~ ~ egenr. erator terminal voltage, etc. 17. End of subroutine. 4.12.6

The Sampling Method (JamesJ. Ray; James A. Momoh, et al.)

Another promising development in this area is the sampling method. I t u'as developed at Howard University with collaboration from the Energy Systems

Network Laboratory (ESNL) in 1989. The sampling method, also known as the "theta" method. takes advantage of the expanded Taylor Series in its formulatior and approximations. It was incorporated in a stand-alone program capable oi' performing stability studies and fault analysis on a wide variety of electric, power systems. The sampling method has been tested against the classical integration methods. A 4: 1 speed advantage was observed, without sacrificing thr: accuracy brought forward in the results. Extended application of this method i r i power system stability studies is being developed for commercial purposes, both as a design tool and as a computational support system.

4.1 3

THE TRANSIENT STABILITY ALGORITHM

An overt'iew of the structure of a transient stability program is given in Fik. 4.32. Only the main parts of the program have been included, and as can be seen, the same system may have several case studies performed on it by repearedly specifying switching data when no further switching data is available. Cor trol returns to the start to see if another system is to be studied. With care, the program can be divided into packages of subroutines each concerned with only one aspect of the system. This permits the removal of component models when not required and the easy addition of new models whenever necessary. Thus for example, the subroutines associated with the synchronous machine, the AVRs, speed governors, etc., can be segregated from the network. Figure 4.32 shows ii more detailed block diagram of the overall structure where this segregation is indicated. The diagram is subdivided into the five sections indicated in Fi;. 4.33. While the block diagrams are intended to be self-evident several logic codes need to be explained. These are as follows: KASE

This is the case study number for a particular system. It is initially set to zero and increniented by I at the end of the initialization and at the end of each case study. KBlFA 7

The sparse vectored inverse of the nodal network matrix is obtained using three bifactorization subroutines. The first and second subroutines are integer routires which determine bus ordering and nonzero element location. The code KBIFA 1 is set to unity if it is necessary to enter these two subroutines, otherwise it is set to zero.

r+=?

Section 1

Read in steady-state system data

Section 2

Section 3

r

Either Solve for machines and network or recalculate nonintegrable variables (initially and when switching has occurred) Section 41. Print out results (if necessary) 2. Make power balance check of initial conditions (if necessary)

NO 1

YES

NO

Section 5,

-

Store initial

conditions

Figure 4.32 The transient stability algorithm.

Reset initial conditions

I00

Network calculations

,~ calcuyns ,~ Synchronou s machine

AVR calculations

A

,f

‘wed

,

governor calculjy~ns

KASE = 0 KBlFA I = 0 KBIFA3 = 0

input

YES

NO V

Figure 4.33a

Section I

KBIfA3

The elements o f the spar\e-vectored inverse are evaluated in the third bifactor-ization subroutine. The code KBIFA3 is set to unity if it i\ necessary to en er thi\ wbroutine, otherwise it i \ set to Lero. When KBIFA3 i \ unity, it indica e\ that a netiborh discontinuity ha\ occurred and hence it i\ a l w u\ed for t b i \ purpcx.

Time

The integration time.

H The integration step length. Like KBIFA3, it is also used t o indicate a discontinuity when it is set to zero.

4 Switching

Determine bus order and nonzero element location for bifurcation KBIFA3=1 TIME=O

+

Is KASE2I and is thae no

NO

switching a t T I M E = O ?

I

* KBIFm YES

K B I F G P sync machine *AVR initial Syncmachine t, conditions. initial conditions YES + Speedgov. 1 I T initial conditions

4

numerical part of bifurcation

Print out initial conditions when U S E > 1)

Figure 4.33b Section 2.

I 02

v

A

-7-T

Determine

switching ops. if branch change set KBIFA3 = 1 if new branch set KBIFAI = 1

I

‘ES

Determine bus order and non zero element location for bifactorization

1 KBIFA3 = 0

part of

KBIFA3 = 1

Figure 4 . 3 3 ~ Section 3.

Philosophy of Secirrity Axsessnierit

I

<>Solve for network

KBIFA3 = 1

E -

TIME = PRINT TIME YES Print out busbar and branch results if required

Print out AVR results

Print out sync. machine results

4 a

Set flag for step doubling

KBIFA3 = 0

I Figure 4.33d Section 4.

speed gov.

results if required

I04

v

A

YES

YES

r NO

1

Perform power balance check to confirm

NO

'End of case '

TIME = 0 KASE = KASE

+

1

1 KASE = 1 ?

r-?

Store initial steady - state)

(steady -state)

conditions

Figure 4.33e Section 5 .

PKlNTTlME

The integration time

:it

which the next printout ot' result\ is required.

MAXTIME

The predet'ined mixiniiiiii integration time for the case study

Lr' Start solution

Calculate constants

nonintegrable variables

,-,-,-,,-,---'

+I

speed gov. calc

-

I not required -----usually -- - - ----

a-

Evaluate integrable varaible using algebraic form of mpezoidal method 4

I

i

-1

Same for cach A b

Same for each speed gov.

[HALF= 0

Figure 4.34a

Section I

ITMAX

Maxiinuin number ot' iterations per step since last printout of results. Note that iiiany data error checks are required in a program of this they ha1.e been omitted from the block diagram for clarity.

tjp

but

106

I

Figure 4.34b

Section 2

A NO Re -evaluate conditions at beginning of step

H = H/2 IHALF = IHALF + I

NO

'Not converging '

TIME = MAXTIME I

Figure 4 . 3 4 ~ Section 3.

v YES

107

I ox

Strcictiire ot Machine '2nd Netivork Iterative SoI~iti017

The ~ - u c t u r of e this part of the program requires further description. T u o forris o f \olution are possible depending on whether an integration step is being etraluated or if the nonintegrable Lariables are being recalculated after ii discontinuitq, A bloch diagram is gi\en in Fig. 4.33. The ridditional logic codes uwd in this part of the program are: ERROR

The niiiXiniuin di t'ference betu een an) integrable 1 ariable from one iteration to ;in() t her. I TK

N u in ber of iterations reqii i red for solution. IHALF

Number ol' inimediate step halving required for the solution. TCILERANCE

Specified niiiYifnuni \ alue of ERROR for conc ergence. I f coil\ erpence ha\ not been achieved after a \pecified number of iterati on\ the ca\e study is tei-ininated. Thi\ is done by \etting the integration time cc ual to the iiia\iiiiuiii integration time. The latest rewlts are thu\ printed out ;iild ;i neu c;i\e \tudj i \ atternpted.

CONCLUSION This chapter dealt urith the philosophy of securit), assess~nentbased on frequency domain models arid equal area criterion concepts. In particular. we define the conventional ingredients for power systeni stability including app ications of the swing equation and its alternate forms. Frequency domain models of synchronous rnachines introduced the idca of subtransient. transient, and steady state reactances. Models for a salient pole and round wound synchronous machines were discussed. The equal area criteria and its applications were discussed. The chapter concludes with the treatnieiit of transient stability for the general network case, including ;I floMxhart of ii trans i e ri t st ii bi 1 i t 4' assess men t pro gra ni . Again, the reader is reminded that additional information inay be obti.ined from the list o f references and the annotated glossary of terms.

5 Assessing Angle Stability via Transient Energy Function

INTRODUCTION In the actual operation of an electric power system, the parameters and loading conditions are quite different from those assumed at the planning stage. As a result. to ensure power system security against possible abnorinal conditions due to contingencies (disturbances), the system operator needs to simulate contingencies i n advance, assess the results, and then take preventiLre control action if required. This whole process is called dynamic security assessment ( D S A ) and preventive control. Simulation studies (called transient stability studies) can take up to an hour for a typical system with detailed modeling for a 500-bus. 100-machine system. Since it takes a long time to conduct a transient simulation even for a single contingency, direct methods of stability assessment such as those based on Lyapunov or energy functions offer attractive alternatives. It should be noted that a transient stability study is often more than an investigation of whether the synchronous generators, following the ~ccurr-ence of disturbance, will remain in synchronism. It can be a general-purpose transient analysis. in which the "quality" of the dynamic system behavior is in\,estigated. The transient period of primary interest is the electromechanical transient. LISLIally lasting up to a few seconds in duration. If growing oscillations are of con-

cern, or if the behavior of special controls is of interest. a longer transient period may be covered in the study. For transient stability analysis, a nonlinear system model is used. The system is described by a set of differential equations and a set of algebraic eq 11at i o n s. Genera 11y , the differentia 1 equations are m ilch i ne equations , con t ro system equations. etc. The algebraic equations are system \,ohage equation:* involLting the network admittance matrix. The time simulation method and direc method are often used for transient stability analysis. The former method deter. mines transient stability by solving the system differential equation step by step. while the direct method determines the system transient stabiliry without explic. i t l y solving the system differential equations. This approach is appealing and has receiLved considerable attention. Energy-based methods are a special case o f the more general Lyapunokr’s second or direct method. the energy function being the possible Lyapiinov function. This chapter deals with transient stability by a specific direct method rnainl;,, the transient energy function (TEF) method. We begin by co\wing some basic concepts from the theory of nonlinear system stability.

5.1

STABILITY CONCEPTS

Consider an autonomous system described by the ordinary differential equatior .

where i =.t(t), and F ( A ) are n-vector\. F(.t) is generally a nonlinear function ( ~ f t . Stability in the sense of Lyapunob i \ referred to an equilibrium state of Eq. ( 5 . I ). The equilibrium \tate is defined as the stage i r at which t ( t ) remaii \ unchanged for all f . That is,

The solution for -tr from Eq. (5.2) is a fixed state since F(.t) is not an explicit tunction of t. For convenience, any nonzero i r is to be translated to the origin ( t = 0). That is. to replace t by .I + i t in Eq. ( 5 .I ) to have

which gives 15.3)

*fit /--“’

.. ................. I . .

........................ ...*............

“

....................... +

6 .

I Figure 5.1

It0 Illustration of local stability.

Note that the current s differs from the old one by 4. As can be seen later, from the definitions, this translation does not affect the stability of the system. Thus, the origin of Eq. (5.3) is always an equilibrium state. It should be noted that t in Eq. ( 5 . 3 ) may be any independent variable, including time. 5.1.1

Definitions and a Lemma

Stability (Local) The origin of the system described by Eq. (5.3) is said to be stable if for any given E > 0, there exists a 6 5 E such that ll.r,~ll< 6 implies ll.r(t)ll < E for all t where so is an initial state. The origin is called unstable if it is not stable. The concept is illustrated in Fig. 5.1 where the initial state x,) has a magnitude less than 6. and the trajectory of s remains within the cylinder of radius E . Asymptotic Stability (Local) The origin of the system described by Eq. (5.3) is said to be asymptotically stable if it is stable and also if given ~ ~ . <x 6l ~implies ~ ~ s -+ 0, (IS t -+00. The arrow is used to mean “approach,” Figure 5.2 demonstrates the idea of asymptotic stability where the trajectory tends to 0 as time tends to infinity.

Figure 5.2

Illustration of asymptotic stability.

Glolm11~~ Asjmptotic- Stnbilit)/ The origin o f the \ystem described by Eq. ( 5 . 3 ) is said to be globally asymptoticall> stable if i t is \table and also implie\ -+0 cis I + 00, for any .v,, in th? \v ho I e space . Positive Deti'nite Function

A uniquely defined. scalar and continuous function V(.v) is said to be positi\,c definite i n ii region K if V(.v) > 0 for .v # 0 and V ( 0 )= 0 . A space surt'ace formed by all .v satisfying W . v ) = 0 is called a contour. Ob\.iousl>r.contours with different \.alues cannot intersect one another. It' the> do. \'(.v) has tM'o \~al~ies at the intersection. We need the follou.ing lenimii qiiirecl for the proof o f ii Ljrapunov theorem. r t b -

L cYl117lC? There exists a sphere defined by l[.vll = N i n which V(.v) increases iiionotoiiical'4, along radical Lwtors emanating from the origin. That is, V( PI,) increases nionotonicully L b i t h p in 0 5 p I N for any unit \rector I I started from the origin. This ciin be shown using the assumption of positi\.e definiteness. First, coiltinuity. V(.Y)> 0 and V ( 0 )= 0, assiiiiie that V( pI1)increases monotonically \bri .h p i n an interval 0 Ip I pII and begins to decrease after p = pII. GiLfen ii I I .there i h ;in associated pIl Lbrhich may be unbounded pll(= 00). Let \ t - among all the U ' S that has the sniallest pi,. then IIp\, n*II= pill\1t.11 = PI, 5 pII. Since \'( PI,) increas$:\ monotonically nith p in the inter\,al 0 5 P I pi, 5 p,,. \+'eare able to identify the positi\,e number to be N = pi,.

L i ' c i p i ~ ~ i oTheoreni v There are three important theorems on 5tability de\ eloped by Lyapunov. V/e include thew here to form the theorem gi\ten below. In the theorem, V( i)i \ t i e total d u i \ atikc of \'( t ) on the trajectory \pecified by Eq. (5.3). That i\, \'(

\ )

= tll' -- = \'\(

\ )\

= 1' ( 1 I / ( t

tlt

i \ the row \ ector formed by the partial deri\ ati\res of V(.v).

U here \',( i)

Regi017~R, RI, R, A11 the regioris ;ire assuincd to contain the origin ;is an interior point. R: i < , \Libregion ot' K , \+,hicti is ;I siibi-egion of R : K: I K ,I K.

;I

Theorem

Let V ( x )be a positiise definite function with continuous partial deri\rriti\vesi n a region R, then The origin of the system described by Ey. (5.3) is stable if i'(.v) I 0 i n a subregion R , I R. The system is asymptotically stable in the region if it is stable and i'(.v) = 0 (identically zero) takes place only at the region in a subregion R.

I RI. The origin is globally asymptotically stable if the sqstem is asymptotically stable. R2 is the whole space and V(.v) + 00 as 11.v11 -+ 0.

Let be the srnaller one between a given E and N specified i n the lemma. That is. I - = Mill [ E . N I . Continuity of V(.\-)assures that there is a m i n i m u m of 1T.v) on the sphere ll.vll = t*. Let j*be among all the 11's that yield the minimum. V ( I - ,=) I U . then V(I*,,) 2 111 must hold for any 1 1 . Monotonicity tells that V( PI,) = 111 for a p i n the interval 0 I P I I - , and hence \'(.v = PI,)= i i i is enclosed by the sphere ~ ~= I", . rand~ also ~ it is a closed contour since I I is any unit vector. Let 6 be the minimum norm of the points on the closed contour \/(.I-) = 1 1 1 , then since ~ I x < , ~6~ is~ enclosed by the contour, V(.v,,) < 111 ~ O I I O M ~ S by monotonocity. Thus, any trajectory initiated from .v,) cannot possibly cross the sphere II.vll = t - 5 E due to the fact that \'(.v) is non-increasing and V(.v) 2 111 on the sphere II.YII = I - . I*

This completes the proof of part I of the theorem. Part 2 can be shown by observing V(.\-)can be identically zero only at the origin. Hence, V(.v) keeps on decreasing except at a countable number of points at which it stops decreasing momentarily. This implies that .v + 0 'L' I - + since V ( 0 )= 0 only when .v = 0 in R2. It seems ob\.ious t o have part 3 verified by the same reasoning used i n part 2. This is true except for the case when V ( x ) approaches a finite value as ll.v]l + 00 when .v,) is allowed to be any point in the whole space. The assumption that V(.v) 4 00 a s \l.vll + excludes this possibility which completes the proof of the theore ni . Except for relying on experience, there is no systematic method to find the Lyapunov function as required by the theorem. It has been shown that any stable and constant linear system has ii V ( x )but no one has yet shonm its existence i n ge ne ra 1 for non I i near sy stem s. L \

-

I14

Chciptrr 5

It can be hhown that instability and asymptotic stability of the system dt:scribed by Eq. (5.3) are the same as its linearized system at the origin, which is

where A =J;(O) is a constant and n-square matrix. That is, the system describd by Eq. (5.3) is unstable if at least one eigenvalue of A has ii positive real p x t and is asymptotically stable if all eigenvalues have negative real parts. Asymptotic stability as judged from the linearized system is simple but 01 less practical use. It is valid only in a sufficiently small region which is not easily known. The Lyapunov function contains more information on stability i n the regions R , and R2. For instance. global asymptotic stability tells us that the trajectory initiated from anywhere in the whole space, converges to the origiii. 5.1.2

Application of Lyapunov's Method to the Simple Pendulum

We consider the dynamics of a pendulum as a prototype for exploring stability of an electric power system. The motion of a pendulum with friction is described by

where -n: < n: < rl is the angle, CI is the damping constant and dh is the undamped angular velocity. We regard the problem as a mathematical one without rest-iction on 8. To convert the system to the standard form of Eq. (5.3). we detine that .vI = 8 and .i, = .Y? to get:

First, we want to check if it is possible to find a L Y ~ ~ U Ifunction W V for the problem. The matrix A for the linearized system is:

Both eigenvalues of A have negative and real parts when N > 0. This sugge;ts a possibility of finding a Lyapunov function. The Lyapunov function is sometimes referred to as a generalized energy function. The name comes from the fact that it has an initial positive kralue and does not increase as time goes on. This is the case of physical systems that iiiok'e without interference from the outside such as the pendulum system. To see this,

let M . J , p and I be the mass. inertia, damping constant and the length of the pendulum, then one dynamic equation becomes J

6 + pi 6 + M

~ sine I =o

Multiply by vi and then integrate with respect to t :

The total energy of the pendulum (sum of kinetic and potential energjr) is

The time derivative is

Therefore, the proposed Lyapunov function is given by

with V ( x )= -ari

Thus, the Lyapunov function is actually the total energy per unit inertia of the pendulum system. Now. we have V(.r) = h( 1 - cosx,) + - .rj 2 1

7

in the region R = { -271: < x < 271:;free .r2)

The derivative of

11 is

The origin is stable since the condition of part 1 is satisfied by choosiiig R I = R. This is also true for the choice ( I = 0. which is the case o f ii frictionless pendulum. :0. This makes .i-, = 2. V(.v) = -(i.vi = 0 implies that .v2 0 and hence .i= -u.v2 - h sinv, = 0 which yields sinvl = 0 o r .vl = /in.Therefore. \f(.v) = 0 only at the origin by choosing R, = { - n < .vl < n: ,ftw. v 2 ) . Thus, t ie origin is asymptotically stable since the condition o f part 2 of the thcorein is satisfied in R2. 3. Part 3 is not applicable for the follcnving t M ' o reasons: ( i ) \/(.I-) = 0 at .vl = 2 1 m i n the whole space, ( i i ) .vl + 00 Mfith.v2 = 0 makes V(.v) = h( I - cos.^,) 5 2 b which does i r o t approach infinity iis required. 1.

For demonstration purposes, M'L' employ the proposed \'(.v) to find ;I 6 5 E required by the definition of stabilit}!. To this end. let I ( = [ l r , , I ( : ] ' be any u n i t i w t o r started from the origin, then

iis

3

+ .v> = 1.7

7

o r \l.vll = I - clV/t/.v, = h siiul - .I I < 0 for h I I and .vl > 0. Since \/(.I) is a decreasing function, it ha\ a niinuinum at . t l = +r- and .I = 0. Therefore, ui = h( I - CO\/-).The expression for the minimuin becomes coirplicated for h > 1 . on the circle .I

- h sin.\l)> 0 for - .vl > 0 and h I I . Since the minimuin occ~trslit .vl = 0 and .v2 = &U, w e h a i e the rninitwrn norm

on the contour \'(.v) = 111. c//A/.vI = 2(.vl

The inequality holds for the reason that

117

For the case b > I , we may choose an independent \variable a = wf Thus. the original dynamic equation becomes

\iith CO'

= 17.

where 0 = &/da and CI' = cdw. Hence, the results of ( b ) and ( c ) are \Valid because h = I . Probably. there exists a better V(.t-) to yield the same results of ( b ) and (c) without considering h 5 1 and h > 1 separately. Although different Lyapunov functions may serve the same purpose on stability. they may be different from other points of view such as estimating 6 and other control applications. 3. The condition of part 2 guarantees that the closed contour

keeps on decreasing in R,, as time goes on until C = 0 as a l i m i t . Being nonnegatiLte. each term of V(.r) must be zero siniultaneouslj \idien C = 0. This suggests that xI 40 and x2+ 0 as C = 0 and hence asjiniptotic stability for the origin is established.

5.2. 5.2.1

SYSTEM MODEL DESCRIPTION Real Power Supplied by a Generator

For a power network consisting of 12-generators connected together by mutual admittances as shown in Fig. 5.3, we may write in matrix form:

Figure 5.3 A general rt-Generator system.

I 18

where [ I ] is the injected current vector, [ E ]is generator internal voltage vector, and [ V ] is the system admittance matrix in which the generator impedances art: included.

for all i, j = 1.2. . . n. The matrix element and complex voltage are specified by ,

-

Y,, = G,, + jR,, and E, = E, 6,

(5.f 1

The real power supplied by generator i to the network is

(5.7,

where PI,= C,,sin(6, - 6,) + D,, cos(6,- 6 , ) with C,,= E, E, H , , and Ill =,E, E, G,,

The power P,, is the real power delivered by generator i to j ; it may 3e positive or negative. P,,= E ; G,, is the power delivered to the local load at gencrator I . From Eq. (5.8) it is clear that f,,depends on difference\ between phiw angles rather than individual phase angles. This result suggests that one may choose an arbitrary reference for the angles without affecting the resulting q , . Indeed, we will first choose a reference rotating at synchronous speed to cle\tribe rotor dynamic\ and then a center of inertia to minimize the kralue of rotor hinetic energy.

5.3 STABILITY OF A SINGLE-MACHINE SYSTEM Consider ii generator connected to an infinite bus with voltage V L 0" throirgh a pure reactance X as shown in Fig. 5.4. If the internal reactaxe and EMF of

Assessitig Aiigle Stcihility \in Transiertt Eiiergy Fiirictiori

I I9

2

1

v L0 Infinite Bus

Figure 5.4

Single generator and infinite bus system.

the generator are given by X , and E L 6 respectively, we have from Eq. (4.7) that P,. = A ( s ) sin6

with EV

A(x)= -

x,+ x

Multiply on both sides (4-7) by w6t = d6 gives M

O = (P,,,- PJd6

(5.9)

Let S,,= (6,,,0), S, = (&, a,) and S,,, = (S,,,, 0) be the start. fault clearing and maximum states of the fault respectively as shown in Fig. 5.5. Let S = (6, w) be any state on the P-6 curve generated by X , then integration of Eq. (5.9)from S to S, yields 1

-M

2

1

d - - MO$= P,,,(6- 6,) - A(X)(COSG,- ~0.46) 2

(5.10)

(a) During the hult: X = X , For S = S,,,we obtain from the above equation 1

- Mw,! = PJ

2

6,)- 6,) - A(X,)(COSG,- COS^,,) A I

(b) After the clearance: X = X , For S = S,,,,we have from Eq. (5.1 I ) that

(5.1 1 )

I30

Power P (P.U.: A - Prefault

B- Post-fault

C - During fault

p , !i Ii

--Lx I I I I I

I

I

I

n

+ Torque Angle,

S

(radims)

Thus, ~ . ' conclude e froin ( a ) and ( b ) that

Therefore. 6,,,can be found by \olving the nonlinex algebraic Eq. ( 5 .I2 1 o r graphically from Fig. 5.5 to judge stability. Thi\ i \ hno\\n ;is the eyual-.iic.a criterion for \tability \ t i d y of power system\. I t i\ important to note that the excessive energy A , created during fault to con\eer-teci to rolor kinetic energy a l clearance. Thi\ re\ult in\pire\ the ii\e o the transient t.~it.rg~f function (TEF)after clearing. That i\. to obtain f r o m Eq. (f . 1 2 ) U ith X = X , '

I'I where E = -1 MO2 + P J 6 2

- 6,)

(5.13)

and V = A ( X , COS^,

5.3.1

- COS^)

Transient Swing

Let II = do/clt denote the acceleration of the system, then Eq. (5.13) together with Fig. 5.5 show that CI < 0 above the P,,, line and that ( I > 0 belour i t after friult clearance. There are two angles: 6, = sin-' P,,,/A(X,) and 6,,= TI: - 6, that correspond t o II = 0 . Figure ( 5 . 5 ) reveals that 6 increases from 6,,with O) = 0 due t o II > 0 until reaching 6,,,< 6,, at which o = 0. Since ( I < 0 at 6,,,,6 begins to decrease until o = 0 at an angle less than 6, and then comes back because ( I > 0. As such. the poclrer angle swings back and forth around 6,. This is the case for E - 1' < 0 at 6,, because 6,, is unreachable (0is imaginary). Ho\be\,er. delayed fault clearance may result in large A , which makes 6 cross 6,, uith 03 2 0. Then. 6 increases further without return due to CI > 0. This is thc case for ( E - V ) 2 0. We will consider the transient to be stable if the p o u w angle s\\Iings around 6, and is otherwise unstable. This definition makes it possible to ;isscss the stability by means of the (TEF) as follows: ( a ) The transient is stable if ( E - V) < 0 at 6,,; large magnitude yields better stability. ( b ) The transient is unstable if ( E - V) 2 0 at 6,,.

5.4 STABILITY ASSESSMENT FOR IFGENERATOR SYSTEM BY THE TEF M E T H O D Consider a power system consisting of ri-generators. The dynamics of each is described by Eq. (5.1 1). We have

de, = (It

for i = I , 2, . . . , 1 1 . The dynamics are second order and nonlinear differential equations coupled together through the phase angles contained i n the expression

I22

Chcipter .i

e,.

of Although the numerical solution can be determined, a closed form of solution is impossible to obtain. We need not discuss the solution of the system of equations as stability that does not require the exact solution. Another reference with velocity CO,,is to be chosen to minimize the integralsquare error

for any t , and t:. The necessary and sufficient condition for this purpose is M,

M,o,,

= 1-

with M , =

I

c 1-

M,

I

which indicates that the reference is the center of inertia (COI).The CO1 has a phase angle 6,,satisfying

With respect to 6,,, all generators have phase angles 0, = 6, - 6,)

In terms of 8,, Eq. (5.14) is now expressed by M , (10 ~- + M U! ! !,! ( tlt

tlt

= U,,,, -

c ,= p, - P:,

(5.15)

with

and

where j f i signifies summation o f j from 1 to in P,, may be replaced by 8,,= 8, - 8, because

ti

except i. All the angles 6, - 6,

Multiply Eq. (5.1 1 ) by 8, and then sum i from 1 to

U:

I23 (5.17)

The second term of Eq. (5.17) is zero because

The states at clearance angle 6, and unstable equilibrium 6,, are to be specified by 11 pairs of (angle, velocity), that is.

Integration of Eq. (5.15) with clt from S, to S,, gives (5.18)

where

and

c I1

1'

v = j P:, e, dt ,=I

(5.30)

,

c' and 14 denoting S, and &. We know from Eq. (5.19) that E is the total energy input plus kinetic energy of rotors. The electrical energy stored and dissipated in the system is V given by Eq. (5.20). The energy here is referred to power integration with respect to phase angle (not time). The stored energy in V is path independent but the dissipated energy i n V is path dependent. To show this, we multiply Eq. (5.17) by 6, and then sum from i = 1 to 11 to obtain:

with

(5.2 I )

The preceding equation can be verified by carrying out the summation. Substitutions of 6, from Eq. (5.8) gives

c,e,+ ~ , , 6=,C , sine,, e,,+ D,, cos(8,+ 8,)

(5.22)

Therefore, the integration of Eq. (5.2

)

results in

-

8 , )- c o \ ( 8 - 8/11+ I

(5.33)

We can see from Eq. (5.23) that the first part is the stored energy and I \ independent of the path of integration. But, the second part denoted by I dcpenci\ OII the path of 8,. Sonie kind of approximation has to be used to evaluatc sincc O , ( t ) cannot be found analytically. Lct 0, be qqxoximated for all i = I . 2,. . . . 1 2 , by

behere C, and K, are constants but may change with i andf7t) is the only one f ) r all I . For con\mience, we use a parameter I / to make

where t, and

I,,

are the times at the clearance and unstable equilibrium. Then 0, = C = c'

+K +K

f ( t )=

C'

+ K, f l t + ( I , - t ) ' I

g(")

for t, I r It,, and 0 5 I I I I . The two constants are required to meet the bound:q c ond i t ion s

Solving the twro constants gives

Based on this approximation, we haw

and hence it follows that

with

This result makes i t possible to integrate I as

It is interesting to note thatfit) or g(rr) need not be known and that anj J t ) yields the same approximation for I as indicated by Eq. (5.28).Thus. one may perceive.f(r) to be the Col:

which naturally minimizes the integral-square error /M,(O,- f' fdt This is probaI,

bly the best choice of theJs. For example, if g ( u ) is chosen as the combination:

then g ( 0 ) = g , g( 1 ) = a + p and

In concept, one may regard a system as stable if the kinetic energy accuniulated at the instant of clearance can be absorbed by the electrical components of the system. Thus. i t is the kinetic energy that determines the stabilitjr. It is usually during the fault that some generators are affected and tend to separate from the rest that are coherent with the system Col. So far as the kinetic energqr is concerned. the gross motion of the separating generators (say the first k ) may

be considered as a single generator with inertia and velocity the same as that o f their COI. That is, i

M, =

i

C M , and MR = I

I

M , 6, I-

I

To be coherent with the system's COI, the rest of the generators must ha1 e zero velocity; 8, = 0 for all i = k + I , k + 2, . . . 1 1 . Coherence suggests that the rest of the generators rnay be regarded as an infinite bus. As such, the whole system beha\res like a single generator and an infinite bus and hence the restilt obtained before may be applied t o the multiple generator system. We modify Eqs. (5.18) and (5.19)according to the conclusion and updi te of Eq. (5.19) below. E

-V

1

= - M,n:,

-

where

and

Note that R,, a,,,(€I,,), and (€I,,),, denote the velocities and angles at the clearance and unstable state. Comparing Eq. (5.26) with Eq. (5.1 I ) enables u s to draw the same con( lu+ion as being made for single generator systems. That is. the system is stab12 if E < V and unstable if E > V. It is inconclusive to talk about E = V \ince n e have made approxiniations in evaluating E and V.

5.5

APPLICATION T O A PRACTICAL POWER SYSTEM

The application of the direct method to actual power systems is quite difficult. A number of simplifying assumptions are necessary. To date, the analysis has been mostly limited to power system representation with generators represented

by classical models and loads modeled as constant impedances. Recently. there have been several attempts to extend the method to include more detailed load models. In a multi-machine power system, the energy function V describing the total system transient energy for the postdisturbance system is given by:

(5.30)

where = angle of bus i at the postdisturbance SEP J, = 2 H , o , ,= per unit moment of inertia of the it’’generator (3:

The transient energy function consists of the following four terms:

1/2 Z 1,~:: change in rotor kinetic energy of all generators in the CO1 reference frame 2. C P,:,(e,- (3:): change in rotor potential energy of all generators relatiLre to CO1 3. CC C,/(cos@,- cos0,;): change in stored magnetic energy of all branches 4. ZZ D,, cos0,,d(O,+ (3,): change in dissipated energy of all branches 1.

The first term is called the kinetic energy (En,,)and is a function of only generator speeds. The sum of terms 2, 3, and 4 is called the potential energy (E,,,,)and is a function of only generation angles. The transient stability assessment procedure involves the following steps:

Step 1 Calculation of Step 2 Calculation of ing Vd Step 3 Calculation of stability index

the critical energy V,,. the total system energy at the instant of fitult-clearstability index: V,,- V,,.The system is stable if the is positive.

Time-domain simulation is run up to the instant of fault clearing to obtain the angles and speeds of all the generators. These are used to calculate the total system energy (V,,)at fault clearing. The flowchart of TEF for transient stability analysis is shown in Fig. (5.6).

5.6

B O U N D A R Y OF THE R E G I O N OF STABILITY

The calculation of the boundary of the region of stability, V ( ?is, the most difficult step in applying the TEF method. Three different approaches are briefly described here,

b I Input system data

~~~~~~~

~

P o w e r n o w calculation

C on ting en cy ap ecifica tion

Form during fault Y matrix and reduced Y matrix Form post-fault Y matrix and reduced Y matrix C alcglate post fault SEP C alcolate critical energy V

Calculate .yatem total energy V, at clearing time

H es

2

.I

1.

The Closest Unstc?bleEquilibrium Point (UEPI Approxh

Early papers on the application of the TEF method for transient stability analysis used the following approach to determine the smallest V , ,

Step 1

Step 2

Determine all the UEPs. This is a c h i e i d by solving the postdist u rbance system steady -st at e equations w i t h di ffe re n t i n i t i a1 va 1lie s of bus angles. Calculate system potential energy at each of the UEPs obtained i n step 1 . The critical energy V ( ris gi\fen by the system at the UEP. which results in the minimum potential energy.

This approach computes the critical energy by implicitly assuming the fault location. hence, the results are very conservative. 2.

\I

or\t

The Controlling UEP Approach

The degree of conservatism introduced by the closest UEP approach is such that the results are usually of little practical value. The controlling UEP approach removes much of this conservatism by computing the critical energy depending on the friult location. This approach is based on the obser\ation that the system trajectories for all critically stable cases get close to those UEPs that are closelj, related to the boundary of system separation. The UEPs are called the controlling or relevant UEPs. The essence of the controlling UEP method is to use the constant energy surfrice through the controlling UEP to approximate the rele\mt part of the stability boundary (stable manifold of the controlling UEP) to which the fiiulton trajectory is heading. For any fault-on trajectory q ( t ) starting from a point p ~ A ( . v ,with ) V ( p )< V(.i-), if the exit point of the fault-on trajectory lies in the stable manifold of .i-, the fault-on trajectory must pass through the connected constant energy surfiice AV((.?) before it passes through the stable manifold of .t(W’(.i-)) (thus exits the stability boundary AA(-t,J). Therefore, the connected constant energy surfuce A\’, (.P) can be used to approximate the part of the stability bondary AA(,?,)for the fault-on trajectory .v,( I). The computation process in this approach consists of the following steps:

Step 1 Determine the controlling UEP, A-,,, for the fault-on trajectory .v,(I). Step 2 The critical energy V, is the value of the energy function V ( * )a t the controlling UEP, that is, V, = V(x,,,). Step 3 Calculate the value of the energy function \I(.) at the time of fiiult clearance (say, I,,) using the fault-on trajectory I{, = V(.v,(I(,)). Step 4 If V, < V , , then the postfault system is stable. Otheritrise, it is unstable.

The key element of the controlling UEP method is how to find the controllinp UEP for a fault-on trajectory. Much of the recent work in the controlling UEP method is based on heuristics and simulations. A theory-based algorithm to find the controlling UEP for the classical power system model with transfer conductance G, is presented now. The energy function is of the form:

JI

where M , = eration.

M,, .Y' = (6',0) is the stable equilibrium point (SEP) under consij'=I

Algorithm to Find the Controlling UEP The reduced system is

The algorithm for finding the controlling UEP consists of the following steps: From the fault-on trajectory ( 6 ( f ) , w ( r )detect ) , the point 6* at which the projected trajectory 6(r) reaches the first local maximum :)t E,,(*).Also, compute the point 6- that is one step ahead of 6" along 6(r).and the point 6' that is one step after 6*. Step 2 Use the point 6" as initial condition and integrate the postfault

Step 1

I1

reduced system Eq. (5.31) to find the first local minimum of I say at 6:. Step 3 Use 6- and 6' as initial conditions and repeat Step 2 to find the corresponding points, say 6, and 6; respectively. Step 4 Compare the values of If(6)i at 6,, 6:, and 6;. The one with the smallest lralue is used a s the initial guess to solve Eq. (5.3 I ),.f; (6) = 0, say the solution is 6,,,. Step 5 The controlling UEP with respect to the fault-on trajectory is (6,(#. 0).

I.f;(s>),

I:

The proposed algorithm finds the controlling UEP Lria the controlling UEP of the reduced system Eq. (5.31) with respect to the projected frlult-on trajectory 6(r).Steps 1-4 find the controlling UEP of the reduced system and step 5 relates the controlling UEP of the reduced system to the controlling UEP of the original system. Theoretical justification of the proposed algorithm can be found in work done by Chiang. 3. The Boundary of Stability-Region-Based Controlling UEP (BCU) Method

Earlier UEP methods faced serious convergence problems when solving for the controlling UEP, especially when the system is highly stressed or highly unstressed, or when the mode of system instability is complex. These problem\ usually arise if the starting point for the UEP solution is not sufficiently close to the exact UEP. Some of the convergence problems can be otm-come by the BCU method which has the capability of producing a much better \tarting point for the UEP solution.

CONCLUSION One of the major innovations in stability assessment is based on the energy function concept. which is an offshoot of Lyapunov stability criteria. This chapter introduced the fundamental Lyapunov stability thought and the procedure of constructing Lyapunov stability function. The main thrust of this chapter is to utilize concepts of system modeling to evaluate system stability using the energy function method. The reader is referred to the list of references and the annotated glossary of terms for further information on the subject matter.

Voltage Stability Assessment

INTRODUCTION Voltage stability studies evaluate the ability of a power system to maintain acceptable voltages at all nodes under normal conditions and after being subject :cl to contingency conditions. A power system is said to have entered ii state of voltage instability when a disturbance causes a progressive and uncontrollable decline in voltage values. Inadequate reactive power support from generators. reactive sources, and transmission lines ciiri lead to k d t a g e instability o r e ~ w i \.oltage collapse, which have resulted i n several major system fiiilures (b1ac.kouts) such as:

I. 2. 3. 3. 5.

September I9 10, New York Power pool. Northern Belgium System and Florida System disturbances of I982 Swedish system disturbance in December 1983. French and Japanese system disturbance5 in 1987. Recently, i n the late nineties, i n the U.S. and other parts of the wor d.

The literature and background studies reLriewed indicate that \roltage i n
disturbance may either be small or large changes in essential load. The consequence of voltage instability may, however. have uidespread impact on the system. Power system voltage stability involves generation. transmission, and distribution. Voltage stability is closely associated with other aspects of poM-er system steady-state and dynamic performance. Voltage control, reacti1.e power compensation and management, rotor-angle stability, protective relaying. and control center operations all influence voltage stability. Voltage stability studies involve a wide range of phenomena. Because of this, voltage stability means different things to different people. It is a fast phenomenon for engineers involved with the operation of induction motors. air conditioning loads, or HVDC links. It is a slow phenomenon (iri\.ol\.ing, for example. mechanical tap changing) for others. (Appropriate analysis methods have been discussed with the debate centering on \%!hetherthe phenomena of' voltage stability are static or dynamic). Voltage instabilitjr or collapse is a dqrnamic process. The term "stability" implies that a dynamic system is being discussed. A power system is a dynamic system. In contrast to rotor angle ( s j w chronous) stability, the dynamics involve mainly the loads and the means for voltage control. Voltage stability has alternatively been called load stability. The loss of lines or generators can sometimes cause \.ohage quality degradation. This phenomenon has equally been attributed to the lack of sufficient reactikre reserve when the power system experiences a he;\\,} load or a se\.ere contingency. Thus, voltage instability is characterized in such ;i I J that ~ \dtage magnitude of the power system decreases gradually and then rapidly near the collapsing point. Voltage stability is classified as either static voltage stability or dynamic voltage stability. The latter is further classified into small signal stability and large disturbance stability problems. A unified frame\%.orkrelated to voltage stability problems will be shown in a proceeding section. In dynamic voltage stability analysis, exact models of transformers. Static Voltage Compensating devices (SVCs), induction motors and other types ot' loads are usually included in problem formulations in addition to models 01' generators, exciters. and other controllers. Small signal voltage stabiliti, problems are formulated as a combination of differential and algebraic equations that are linearized about an equilibrium point. Eigen analysis methods are used to analyze system dynamic behavior. Small signal analysis can provide uset'ul information on modes of voltage instability and is instructik'e in locating V A R compensations and in the design of controllers. Large disturbance \wltage stability is approached mainly by using numerical simulation techniques. Since s ~ r s tem dynamics are described by nonlinear differential and algebraic equations that cannot be linearized in nature. Voltage collapse is analyzed based on ii center manifold voltage collapse model.

I24

WORKING DEFINITION OF VOLTAGE COLLAPSE STUDY TERMS

6.1

Voltage stability has been viewed as a steady-state “viability” problem suitable for static analysis techniques. The ability to transfer reactive power from production sources to consumption sinks during steady operating conditions is a major aspect of voltage stability. The following definitions are often used i n voltage stability studies. Voltage collapse incidents in the U.S., Europe, and Japan ha\x led to different explanations. interpretations. and concerns. To achieve a unifying franrework, a working definition for detection and prevention of Ldtage collapse I-iis been constructed (EPRI RP 2473-36).

6.1.1

Classification of Voltage Collapse Detection

Detection of VC is based on determination of imminence of power system voltage violating its limits which value may lead to system instability and consequently voltage collapse. The phenomenon in many instances is due to a def cit in reactikte power generation, loss of critical lines or degradation of control on key buses.

6.1.2 Classification of Voltage Collapse Prevention Voltage collapse prevention is any action taken to reduce the likelihood of power system degradation due to the violation of operating limits. Prevention scheme includes the use of optimal power tlow strategy and other measure:* to minimize voltage deviations. Other definitions of voltage collapse or stability are documented in the li terature. For example, according to the IEEE working group in voltage stability. voltage collapse/stability is defined as follows: Voltage stability is the cihility of a system to maintain kroltage so that when load admittances are increased, load power will increase, so that t o t h power and voltage are controllable. Voltage collapse is the pr.oc.t..s.s by which voltage instability leads to 10s.; of voltage in a significant part of the system. Voltage degradation may also lead to “angle stability” as well, especially if the preventive measures are not enforced quickly enough. Sometimes only careful post incident analysis can discover the primary cause of voltage collapse.

Voltcige Stcrhility A.s.ses.sment

1-35

6.2 TYPICAL SCENARIO OF VOLTAGE COLLAPSE Assume that a power system undergoes a sudden increase of reactive power demand following a system contingency, the additional demand is met by the reactive power reserves carried by the generators and compensators. It is possible, because of a combination of events and system conditions. that the additional reactive power demand may lead to voltage collapse, causing a major breakdown of part or all of the system. A typical sequence of events leading to a voltage collapse can be as follows: 1. The power system is experiencing abnormal operating conditions with large generating units near the load centers being out of service. Some EHV lines are heavily loaded and reactive power resources are low. 2. A heavily loaded line is lost which causes additional loading on the remaining adjacent lines. This increases the reactive power losses in the lines causing a heavy reactive power demand on the system. (Reactive power absorbed by a line increases rapidly for loads above surge impedance loading). 3. Immediately following the loss of the line, a considerable reduction of voltage takes place at adjacent load centers due to extra reactive power demand. This causes a load reduction, and the resulting reduction in power flow through the lines would have a stabilizing effect. The generator AVRs would, however, quickly resolve terminal voltages by increasing excitation. The resulting additional reactive power flow through the inductances associated with generator transformers and lines would cause increased voltage drop across each of these elements. At this stage, generators are likely to be within the limits of P-(2 output capabilities, i.e., within the armature and field current heating limits. The speed governors would regulate frequency by reducing MW output. 4. The EHV level voltage reduction at load centers would be reflected into the distribution system. The ULTCs of substation transformers would restore distribution voltages and loads to prefault levels in about 2 to 4 minutes. With each tap change operation, the resulting increment in load on EHV lines would increase the line XI’ and RI’ losses, which in turn would cause a greater drop in EHV lines. If the EHV line is loaded considerably above the SIL, each MVA increase in line flow would cause several MVAs of line losses. 5. As a result, with each tap-changing operation, the reactive output of generators throughout the system would increase. Gradually. the generators would exceed their reactive power capability limits (imposed by maximum allowable continuous field current) one by one. When the first generator reached its field current limit, its terminal voltage would drop. At the reduced terminal voltage for a fixed MW output, the armature current would increase. This may further limit reactive output to keep the armature current within allowable limits. Its share of reactive loading would be transferred to other generators, leading to overloading of more and more generators. With fewer generators on automatic

excitation control, the system would be much more prone to voltage instability. This \.+wildlikely be conipounded by the r:duced effectiLmess of shunt conipensators at low voltages. The process will eventually lead to voltage collap ;e o r avalanche, possibly leading to loss of synchronism of generating. Sometimes. the term 'voltage security' is used. This means the ability of ;I system not only to operate i n a stable inode but also to remain stable follohiiig credible contingencies o r load increases. It often means the existence o f a considerable margin from an operating point to the voltage instability point fc)llo\\,ing credible contingencies.

6.3

TIME-FRAME VOLTAGE STABILITY

Voltage instability incidents in \.arious places around the world has been c.cscribed iis ii function o f time. The time ranges from seconds to tenths of minutcs. Three time frames and sconarios described by Carson Taylor are sumniari;red ;is follo\+fs: I . Transient cultage stability: This occurs bet\+.een 0 to 10 seconds. 2. Classical Voltage Stability: This occurs betureen I to 5 minutes. This i s t he c 1iiss i c scc nar i o i n 1'0 I 1,i ng aii t 0111 at ic on-I oad tap change r , d i st r i bu t ion \,( 1I t age re gi i 1at ion . xicl gene r;i t o r c U rre 11t I i m i t i n g . T h i s sce n ar io i n \'oI \re s high 1o;i J s . high p o ~ v e r imports from ii remote generator, and a large distribution. Tap changcr action is also significant, i t it can be beneficial o r detritnental depend,ng on load characteristics and location of the tap changer. Collapse in one area c-an at'tect ;I much larger area, thus, leading to a mtijor black-out. This occurrenct' i \ tj.pical on the East Coast. the pacific West, and in Japan. 3. Long-term Voltage Stability: This in\rol\res se\feriil minutes. Se\.t ral VC incidents (the Tokyo I977 blackout. the Shieden blackout) are cxperien:cs o f VC. The factors may include overload time, limit of transmission, loss load di\,ersitJ,due to low Lroltage (theoretically controlled loads), timeliness of appI>'ing reuc*ti\.epower. and other operating interventions such iis load shedding. Reported idtage i ns tabi I i t y i tic iden t s \v i t h and L+ii t hoii t vo I tage are su 111I narixd in Figure 6. I . For classical \,()Itage instability. the phenomenon \+rill occur at the onset o f the \,oltagc collapse. For long-term stabilitjr, the shorter-t. me frame phenoniena w i l l occur once \ d t a g e begins to sag leading to \ultage c:ollapse.

6.4

MODELING FOR VOLTAGE STABILITY STUDIES

Voltage stability studies involve the solution to algebraic and differential equations that map the system behavior under steady-state and transient state\. Belo\v are tjpical \ ector\ encountered and the notation used.

---

1 1

Classical (Large Disturbance)

(Large Disturbance)

Long-Term (Load Buildup)

i ;-yGeneratorExc. Dynamics

LTCs

Rime Mover Control

0

0

Load DiversityJrhermostat 0

Max. Exciter Limiter

0

Mechanically Switched Capacitors

Linnransformer Overload

0

Inertia Dynamics

0

F-

Gen. ChangdAGC

Boiler Dynamics

1

0

-

Operation Intervention

DC Converter LTCs

I

I

I1

1

1

1

1

I

1

I

Gas Turbines

1

I min 1

4

I I

Ihr

10 min I

I

1 I1

I

I

Time in seconds

Figure 6.1 Time-iramc of voltage stability (courtesy of C a r ~ nTa)rlor). 1.

Dynamic state \rector. ~ ( t )

\+!here 8 ( t ) = Rotor angle E’(t ) = Voltage components of synchronous machine S ( r ) = Dynamics in load bus p ( t ) = Other dynamic states (exciter, governor) 2.

Algebraic state vector,

y(t)

where ~ ( t )6 .( t )= Bus voltage niagnitude and angle

I 1

Chtrprrr 6

Q(t)

q(r) 3.

= Nonscheduled reactive power = Other algebraic variables

Parameter vector, p ( t )

where = Turbine shaft power P,(r 1 P , ( t 1, Q r ( t )= Scheduled load power = Controlled voltage or set points 6 d (t ) = Other similar parameters CT(t 1

VOLTAGE COLLAPSE PREDICTION METHODS

6.5

The framework for voltage stability studies can be simplified to fit the time span of the analysis. The categories of interest are as follows: 6.5.1

Static Stability

Assume all time derivatives equal zero at some operating point H ( . v , j - . p )= 0 6.5.2

Dynamic Stability

At some operating point, small perturbations (local)

6.5.3

Extended Stability

Simulation through time (up to hours)

6.6 CLASSIFICATION O F VOLTAGE STABILITY PROBLEMS Voltage problems are distinguished in three categories: 1.

Primary phenomena related to system structure. These retlect the autonoinous response of the system to reactive supply/demand imbalanccs.

Voltuge Stcrhility Assessrtierit

139

Secondary phenomena related to control actions. These reflect the counterproductive nature of some manual or automatic control actions. 3. Tertiary phenomena resulting from interaction of the above. 2.

This classification of voltage quality problems implies that the problems involve both static and dynamic aspects of system components. Voltage collapse dynamics span a range in time from a fraction of a second to tens of minutes. Time frame charts are used to describe dynamic phenomena which show time responses from equipment that may affect voltage stability. The time frame chart is shown in Table 6. I , where .q is a state vector representing transient dynamics. I,is a state vector representing long-term dynamics. Y is a state vector representing very fast transient dynamics related to network components and P is a system parameter vector. Then the time frames to be considered become very fast transient, transient, and long term. The main characteristics of the three time frames are as follows: 1. A very fast transient voltage collapse involLres network RLC components having very fast response. The time range is from microseconds to milliseconds. 2. A transient voltage collapse involves a large disturbance and loads ha\.ing a rapid response. Motor dynamics following a fault are often the main concern. The time frame is one to several seconds. 3. A long-term voltage collapse usually involves a load increase or a power transfer increase. Within this time frame, a voltage collapse shows load restoration by tap-changer and generator current limiting. Manual actions by system operators may be important. The time frame is usually 0.5 to 30 minutes. Since voltage stability is affected by various system components in a wide time range, in order to tackle this problem, one must consider proper modeling and analysis methods. Currently, voltage stability approaches mainly include static and dynamic, i.e., transient voltage collapse and long-term voltage collapse.

Table 6.1 Time Frame and Relevant Models in Voltage Stability Assessment Voltage stability models and time scale Micro to milli seconds

A few seconds

Minutes

I40

6.7

VOLTAGE STABILITY ASSESSMENT TECHNIQUES

The loss o f lines o r generators can sometimes cause degradation in voltage. This phenomenon hiis equally been attributed to the lack of sufficient reacti\,e reset ve when the pouer system experiences a heavy load o r severe contingency. Thus. \-oltage stability is characterized in such a way that voltage magnitude o f the p o ~ w system decreases gradiially and then rapidly in the neighborhood o f I he collapsing point. Voltage stability is classified as static voltage stability and dq'naiiiic \.oltage stability. The latter is further di\ridt.d into srnall signal stabilitJr ancl large disturbance stability problems. I n d y n a i n ic v o 1t age st 11bi I it y ana 1y si s, e x ac t t node I i ng of t ran s fo r mc rs , SVCs. induction motors, and other types of loads are usually included in problem formulations i n addition to models of generators, exciters. and other controllers. Small signal voltage stability problems are fimiiulated as a combination o t differential rind algebraic equations that are linearized about an equilibri .in1 point. Eigen analysis methods are used to analyze system dynamic beha\rior. Small signal rinalysis can provide useful information on modes of \voltage in:,tability and is instructikre i n locating VAR compensations and in the design o f controllers. On the other hand, large disturbance voltage stability is mainly dorilt \+,ith bj. numericul simulation techniques. since system dynamics are descrilxx! by nonlinear differential and algebraic equations that caniiot be linearized i n nature. The mechanism o f \,()Itage collapse has been explained as saddle node bifurcation i n some literature. Voltage collapse is anrilyzed based on ii ceriter manifold \ d t a g e collapse model. Static Lroltage stability analysis is based on po\+w system load tlow e q u tions. Indices characterizing the proximity of ;in operating state to the collapse point are cie\~eloped.The degeneracy of the load tlo\ss Jacobian matrix has bcen used ;is ;in iritlex of p o \ + w system steady-state stabilitjr. I:nder certain coiiditions, ;I change i n the sign o f the determinant o f the Jacobian niatris diiiing c*ontinuoiis \rariations o f pnrameters means that ;I real eigeiivalue o f the liii1:ari d sb+ring equations crosses the imaginary axis into the right half o f the coinples plane and stability is lost. Various researchers ha\.e considered th;it ;I change i n the sign o f the Jacobian matrix may probubljf not indicate the losj of steady-state stabilitj, urhen e\wi niirnber eigen\Aues \+!hose real part cross the imaginury axis. Voltage stability is also related to iiiiiltiple load flow soluti~ms. A proximity indicator t o r Lroltage collapse (VCPI) was defined for ;I bus, ;in area. o r the complete system ;is ii Lector of ratios o f the incremental gener.ited reactive powcr at a generator to a given reactive load d e n i d increase. A dil'terent indicator ( L index) is calculated from normal load tlow results with reaxm;Ib I e coin p u t ii t ions. The 111 i n i in u 111 si n g u 1ar va 1u e of the J iicob i an was p r o p ) sed ;is ii \coltage security index. since the magnitude o f the rninimiim singular \ d u e coincides Lsrith the degree of Jacobian ill-conditioning and the proximity to col-

lapse point. Based on a similar concept, the condition number of the Jacobian is also applied as an alternative voltage instability indicator by pioneers in the field. Bifurcation theory is used to analyze static stability and voltage collapse. Static bifurcation of power flow equations were associated Lvith either di\wgenttype instability or loss of casualty. Researchers ha\.e described necessary and sufficient conditions for steady-state stability based on the concept of feasibility regions of power flow maps and feasibility margins but with high computational efforts. A security measure is derived to indicate system \~ulnerabilityto Lwltage collapse using an energy function for system models that include idtage \,ariation and reactiire loads. It is concluded that the key to applications of the energy method is finding the appropriate T\pe-l low voltage solutions. In addition to the above methods for direct coinputation of stability index. some indirect approaches, based on either the continuation method or optimization methods haire been developed to compute the exact point of collapse. I n app I y i n g the con t i n u at ion methods , assumptions about 1oad c h a n g i 11g pat t e 1-17s are needed. In summary, the methods for static voltage instability analysis are based on multiple load flow solutions (voltage instability proximity indicator [ VIPI], energy method), load flow results ( L index, VCPI), or eigen\Aues of the Jacobian matrix (iiiinimum singular value and condition number). While studies on djrnamic voltage collapse shed light on control strategy design (off-line applications), static \.oltage stability analysis can provide operators with guideline information on the proximity of the current operating state to the collapse point (on-line applications ). In this case, an index, which can @\re ad\wice utarning about the proximity to the collapse point, is useful. The next discussion will be that of the formulation of selected \x)lta,c’e stability indices. Of the wide range of techniques available. we shall discuss the VIPI method. a method based on singular value decomposition. condition nuinber of the Jacobian, and the method based on the Energy Margin.

6.7.1

Voltage Instability Proximity Indicator Method

The \.oltage instability proximity indicator (VIPI) was developed by Y. Tamura et al. based on the concept of multiple load flow solutions. A pair of load f l o ~ solutions .vl and .v2 are represented by two vectors ci and 17 as followfs:

which are equivalent to:

where .vl is the normal (high) power tlow solution and .v2 its corresponding IOYA. voltage power tlow solution; LI is a singular vector in the space of node voltages and h is a margin vector in the same space. We now define two other vectors Y , and Y(ci). called singular vectors in t k space of node specifications. The relationship between these \rectors is shou n in Figure 6.2. VIPI is defined by the following equation:

where vector Y,, consists of bus injections computed with respect to .vl but the injection kAues corresponding to reactive powers of PV buses are replaced by the squared values of voltage magnitudes. Y(cc)consists of bus injections with respect to vector ( I : I( .v ( 1 is the /,-norm of vector, .v. The computation of VIPI is easy once the relevant low voltage power tlow solutions are obtained. Generallj, speaking. finding all the relevant low voltage solutions are time-consuming tor practical size systems. 6.7.2

Minimum Singular Value (U,,,,,,) Method

When an operating state approaches the collapse point, the Jacobian matrix of the power tlon equations ( J ) , approaches singularity. The minimum singu ;ir

Figure 6.2 Concept o f VIP1 in the node specification space.

value of the Jacobian matrix expresses the closeness of Jacobian singularity. The singular value decomposition method is used to solve the minimum singular value for static voltage stability analysis. According to the theory of singular value decomposition. power flow Jacobian can be decomposed as: (6.11 )

J = LEV'

-

where: J E R'J1*'J1 is the power flow Jacobian matrix; C = diag(o,, (J?, , oJJ) with (J,,,,, = (J, 2 o22 2 (J,~= (J,,,,,,2 0. If matrix J has rank ) - ( I " 5211).its singular values are the square roots of the I" positive eigenvalues of A7A (or A A ' ) . U and V are orthonormal matrices of order 211, and their columns contain the eigenvectors of AA' and ATA respectively. From Eq. (6.9). i t can be obtained that A V, = CJ,~(,

(6.13) A ' U , = (3,V,

(6.13)

E, = u,V:

(6.14)

We define

Then Eq. (6.I 1 ) can be written as:

(6.16)

then, as far as the /?-norm of the J matrix is concerned, J' is a matrix of rank - 1 nearest to the J matrix of rank n . This means that the smallest singular value of a matrix is a measure of the distance between matrices J and J'. As for the power flow equations, its minimum singular value expresses the proximity of the Jacobian to singularity. It can be used as an index for static voltage stability. 11

6.7.3 Condition Number of the JacobianMethod The condition number is used in numerical analysis to analyze the propagation of errors in matrix A or vector 6 in solving variable vector x for the linear equation Ax = h. If matrix A is ill-conditioned, even very small \xiations i n vector 17 (or A ) may result in significant changes in solution \rector .v.

For the Iinearized load flow equations, the condition number of the Jacobi.in matrix can be iised to measure its conditioning and whether any small variations in i'ector 11 ( o r A ) inay result in significant changes in solution vector .r. For the Iinearized load tlow equations, the condition number of the Jacobi,in matrix can be used to measure its conditioning and whether any small variations in loads niay lead to large changes i n bus voltages. If the condition number is greater than a specified threshold, this will Iiieiin that the current operating state is close to the collapsing point. A precise measure of the sensitivity of a linear sq'stem solution with resptxt to matrix A o r vector h ciiii be defined as:

For pobrer tlocb Jacobian iiiatrix J . the \ alue of Cond, ( J ) can gile an indication of the condition o f J "with respect to inversion." A small ~ ~ a l uof'e Cond2( J ) ( 1 10) refer\ to a bell-conditioned Jacobian matrix (relatitrely large voltage stability margin): a large value of Cond2(J ) ( >100) mean\ that the opi*rating 5tute is \cry close to the point of Jacobian \ingiilarity and has a ION colt ige \tiibilitj iiiargin. The extreme condition is that J is singular and Cond,(Jr is infinite. Hence. the condition nuniber Cond?(J ) ciin be used to iiieii\ure the proxiiiiitj of the operating \tates to voltage collap\e.

-

6.7.4

Energy Margin-Based Method

The energy method uses an energy function, dericed from a clo\ed form \o:tor integration of the real iiiid reactive mismatch equations betu een the oper;ihle pou er flou \olution and ;i I w oltage power tlou \olution, to prokide ii yuiintitatice tiieasiire ol' I ~ O M clo\e the \y\tem i \ to toltage instability. The point of t oltage in\tability correspond\ to the \addle node bifurcation point definec hq ;i \iiigiiliir potver tlou Jacobian N i t h ~ e r o c nergj margin. The encrgq function t'or boltage \tabilit> analysi\ i \ defined as:

1-45

(6.19)

with real and reactive mismatches defined as:

where: .I-' = ( a ' , V ' )is the normal operable power tlow solution ( o r the stable equilibrium point, SEP): xf' = (a",V")is the relevant low Yoltage po\\.er ilow solution with respect to x ' (or unstable equilibrium point, UEP). A large energy value indicates a high degree o f \rollage stabilitjr ndiilc ;i small \yalue indicates a low degree of voltage stability. I n applj'ing the c n e r g ~ ~ method. the key is finding the relevant UEPs. Since the number o f rele\rant l o ~ i p \.o~tagepower tlow solutions is very large ( Y '- I for a practical syteni. the exhausti\ve approach is not feasible. There is an i m p r o \ 4 technique to compute all the Type- I UEPs based on the results that sho\i' for tjpical po\i'er sj'stems. the system always loses steady-state stability by a saddle node bifurcation bctureen the operable solution and a Type- 1 low-voltage solution. That condition restricts the computation of relevant UEPs only corresponding to sj'sterii PQ buses, or practically PQ load buses. After finding all thc rele\mt UEPs. the buses corresponding to which the energy function has the lowest \ d u e s arc buses \sulnerable to voltage instability. Similar to the VIP1 method. the energy methods depend on the low-voltage power flow solutions. urhere the Ne\s,tonRaphson method with the optimal multiplier can be used.

6.8 6.8.1

ANALYSIS TECHNIQUES FOR STEADY-STATE VOLTAGE STABILITY STUDIES Introduction to the Continuation Method

I n its early stages, \dtage collapse studies were mainly concerned \\!it11 stead>.state voltage behavior. The voltage collapse is often described ;is a problcm that results when a transfer limit is exceeded. The transfer l i m i t of an elcctrical pom'er network is the tnaximal real or reactive power that the system c;in deli\w from the generation sources t o the load area. Specit'icallqr. the transfer l i m i t i h

136

Clttipter 6

the maximal amount of power that corresponds to at least one power-tlow sol J tion. From the well-known P-V or Q-V curves, one can observe that the volta;;e gradually decreases as the power transfer amount is increased. Beyond the maximum power transfer limit, the power-tlow solution does not exist, which implies that the system has lost its steady-state equilibrium point. From an analytical point of view, the criteria for detecting the point of voltage collapse is the point where Jacobian of power-flow equations become singular. The steady-state operation of the power system network is represented ‘ ~ y power-tlow equations given in equation (6.20).

where 8 represents the vector of bus voltage angles and V represents the vec or of bus voltage magnitudes. h is a parameter of interest we wish t o b a y . In general the dimension of F will be 211,,, + t i / ) u , where ups and / I / ’ \ are the number of PQ and PV buses, respectively. From equation (6.20) one obtains the fundamental equation of sensitii i t ) an a I y s i s

Let .v’ = [e,V]’.From Eq. (6.2 1 ), one can obtain an ODE system

( 62 2 )

For a specific variation of the parameter h, the corresponding variatioIi t o the solution x is calculated by evaluating the Jacobian (dF/d.4-].It should be emphasized that the singularity of the power flow Jacobian dF/d.v is necesaary but not a sufficient condition to indicate voltage instability. The method proposed to observe the voltage instability phenomenon is closely related to tn111tiple power flow solutions. which are caused by the nonlinearity of power flow solutions. The drawback of the method is that it relies on the Newton-Raphson method of power flow analysis, which is unreliable in the vicinity of the vollage stability limit. As such, researchers have developed a technique knourn as the con t i n u ati on met hod.

137

Voltcige Stcthility Assessriierlt

6.8.2

Continuation Method and Its Application to Voltages Stability Assessment

Consider the power flow equation defined in Eq. (6.20). The vector function F consists of \I scaler equations defining a curve in the I I + 1 diniensional (.v,h) space. Continuation means tracing this curve. For a convenient graphical representation of the solution (.v.h)of Eq. (6.20) we need a one-dimensional measure of x. The frequently used measures are: (i) (ii)

1 .v 1 = E','=,xf (square of the Euclidean j s I = max 1 .I-,1 (maximum norm), I=I 1 s 1 =xi for some index k , 1 I k I n .

norm),

If

(iii)

In power systems generally we use the measure of (iii). As can be seen from Fig. 6.3 we have a type of critical solution for h = k*, where for h > h* there are no solutions. For h < h* we have two solutions (one is the high voltage

state variable

parameter

Figure 6.3 The fold type curve including predictor-corrector step.

solution and the other is the low voltage solution). When h approaches h ' ~ 0<. h+).both solutions merge. At this point the Jacobian of the power f l o ~solution , is singular. In the mathematical literature these points are called turning points. fold points. or bifiircation points. An algebraic featiire of the turning point is given by F , below

F , ( . V ' ~ ' .is~ singular '~) for rank < 11. F ( p:,A:!:)/F.\( , .v* .A:]: ) h iis ;I full rank conditions. 1 -

cl:

II

and satisfies some nondegenericitj,

Several techniques ha\ e been proposed to calculate these point\. Thew methods based their iinaIy\is on t\vo approache\ referred to a\ direct and indii cct inet hods . 6.8.3

Detection of Voltage Collapse Points Using the Continuation Method

Direct Methods This approach tries to find the maximum allowdAe wriation of h: that is. an operating point (.P,A';:) of the equation:

such that the Jiicobian at this point is singulur. I t solves the tolluct~ing\>stem o f eq 11at i on s

This procedure basically augments the original set of pobrer tlow eqiiat ions F(.v,h)= 0 by F,(.v,)L)/I = 0 where h is ;in u-\fector n,ith / I , = 1 , The disadewmges of this approach are: The dimension of the nonlinear set of equations to be solcved is tnice that for the concmtional power flow. The approach requires good estimate of the vector h. The advantage is that. convergence of the direct method is \,ery fast i the initial operating point is close to the turning point. The enlarged system is sol\wl in such ii way that it requires the solution of four I I x / I ( 1 1 is the dimensicln o f the Jacobian F,(.v,h))linear systems, each w i t h the same matrix. requiring only one LU decomposition. '

indirect Method (Continuation Methods)

Assuming that the first solution (.q,,&,)of F(.u.h)= 0, is available. the continuation problem is to calculate further solutions, ( . ~ , . h ,(.v2.h>). ), until one reaches a target point. say at h = A*. The ith continuation step starts from an approsimation of (x,,h,)and attempts to calculate the next solution. However, there is an intermediate step in between. With predictor, corrector type continuation, the step i 3 i + I is split into two parts. The first part tries to predict a solution. and the second part tries to make this predicted part to coni'erge to the required solution:

Continuation method\ differ among other thing\, i n the follo\J ing: ( 1 ) choice of predictor, (2) type of the parameterization \trategy. ( 3 ) type of corrector method. ( 3 ) step length control. All four aspects uill be explained through the formulation of the power flow equations. I n order to apply the continuation method to the poner tlou problem, the power flow equation\ must be reformulated to include a load parameter (A). This can be done by expressing the load and the generation at a bus as ;I function of the load parameter ( h ) .The general form of the ne\+ equation\ a\sociated with each bus i is:

where the subscripts L,, G,, and T, denote bus load, generation, and power out of a bus respectively. The voltage at bus i is V, 8, and Y,/ a,/ is the (i,j)th element of the system admittance matrix [ Yut,5).P,l(h) and Q,,(h ) terms depend on the type of load model. For example for the constant power load:

(6.26)

For the nonlinear model

I so

In addition, for any type of load model. the active power generation term can be modified to obtain

where the following definitions are made PIr < , Ql,,,= Original load at bus i, active and reactive respectively = Multiplier to designate the rate of load change at bus

i as h changes

= Power factor angle of load change at bus i = Apparent power which is chosen to provide appropriate scaling of

h

= Active generation at bus i in the base case = Constant to specify the rate of change in generation as = Initial voltage at the bus

h varies

= Frequency dependent fraction of active power load = Voltage exponent for frequency-dependent active power load = Voltage exponent for nonfrequency-dependent active power load = Ratio of uncompensated reactive power load to active power load = Voltage exponent for uncompensated reactive power load = Voltage exponent for reactive power compensation

Now if F is used to denote the whole set of equations, then the problem can be expressed as il set of nonlinear algebraic equations given by Eq. (6.20). The predictor, corrector continuation process can then be applied to those equations. The fir\( task in the predictor \tep is to calculate the tangent vector. Tlii\ tector can be obtained from factorizing Eq. (6.21).i.e.,

(6.18)

On the left side of the equation is a matrix of partial derivatives multipl ed by vector of differentials. The former s the conventional power flow Jacobian

151

Voltage Stcihility Assrsstnerit

augmented by one column ( F j , ) ,while the latter t = [de,dV,dh]' is the tangent vector being sought. A normalization has to be imposed in order to give t a nonzero length. One can use for example (6.29)

e:t = tL = I

where ek is an appropriately dimensioned row vector with all elements equal to zero except the kfh one, which equals one. If the index k is chosen properly. letting tk = + I .O imposes a nonzero norm on the tangent vector which guarantees that the augmented Jacobian will be nonsingular at the point of maximum possible system load. Thus the tangent vector is determined as the solution of the 1i near sy s tem

Once the tangent vector has been found by solving Eq. (6.30), the prediction can be made as follows:

[E][;] [i] =

where "*" denotes the predicted solution and step size.

6.9

(6.31 )

+U

(3

is a scalar that designates the

PARAMETERIZATION

The branch consisting of solutions of Eq. (6.20) forming a curve in the (.LA) space has to be parameterized. A parameterization is a mathematical way of identifying each solution on a branch. A parameterization is a kind of measure along the branch. There are many different kinds of parameterization. For instance, by looking at a PV curve, one sees that the voltage is continually decreasing as the load nears maximum. Thus, the voltage magnitude at some particular bus could be changed by small amounts and the solution is found for each given value of the voltage. Here the load parameter would be free to take on any value it needed to satisfy the equations. This is called local parameterization. I n local parameterization the original set of equations is augmented by one equation that specifies the value of one of the state variables. In equation form this can be expressed as follows:

1-52

=o,

\%=[q

(6.31 I

where q is an appropriate \~aluefor the kth element of J*. N o w once ii suitablc index k and the value o f 11 itre chosen. a slightly modified Ne~\'ton-Rapliioii (N-R) power flow method (altered only in that one additional equation and oiic iiddirional state sariable are involved) can be iised to sol\.e the set of equatior-s. This pro\.ides the corrector needed to modify the predicted solution found i n tlic prc \,iou s sect ion. The algorithm tor static assessnient is shourn i n Figure 6.4. We ciin u s e ;I simple example to explain the static ~inulysisproccdure.

6.9.1

Static Assessment: A Worked Example

Consider

U

;i

\j'steiii is represented by

here h is ;I irariation parameter from h,,= 0 to h, = A,,, To begin, s,ol\.e the s j stem equations at h = 0 , ~e ha\re

1

Input System Data

f Select Contingencies

*

1

Select Continuation Parameter &

.

Solve base load flow

I

I

Choose stepsize

o

I

A p

OT

* Calculate Stability index

Figure 6.4 The algorithm for static assessment.

*Q

[

2-r - 1 0 2 I -11 0 1 0

d.r

[&

=0

clh

Since s,,= 1. substituting into the above equation we have:

[

2 -I 0 c1.r 2 1 4][" = O 0 1 0 dh

and

[ ."];[ t1.r

0.5

dh

2.0

1.0

Therefore,

Choosing

0 = 0. I ,

one gets .?, = .c*= 1 .o + 0.05 = 1 .OS

+ 0.1 = -I .9 h, = h* = 0 + 0.2 = 0.2 f, = j'" = -2

where .t,y, and are the approximated solutions. In order to find the solution of F(.u,~.,h) = 0. we need to solve the equation

where r\ is an appropriate value of y. Choose r\ = y* = - 1.9. we have the solution of

Voltcige Stcihility Assessriieiit = 1 .OS =-1.9 h, = 0.2

XI '1

Based on the solution of (x,y,L), we can get the solution of (.I-~.~!~,X~), we have

Choose

CJ = 0. I ,

one gets

, ,[

; I:i: + 11 =

0.2 + 9 21

Choose q = -1.8, we have the solution of XI

h, = 2

=

fl

= -1.8 0 - 1.8

Using the same procedure until the target system is reached. The modal analysis procedure is given in the following. System linearization equation is given by

where p represents the variation parameter. At .Y,~= I , y,, = -2, pi,= 0, the above equation can be reduced to

I56

6.10 THE TECHNIQUE OF MODAL ANALYSIS The inodal o r eigen\ralue analy\is method i\ a kind ot' sensitivity analy\is but the modal \eparation provides additional insight. The \ystem partitional niatri-, equations of the Newton-Raphson method can be reu ritten as

(6.33)

M here the partitioned Jacobian retlects ;I solb~edpower tlow condition and include\ enhanced de\.ice modeling. By letting AP = 0. we can write

here J K , is a reduced Jacobian matrix of the system. J K directly relates the h i \ voltage magnitude and bus reactive power injection. Let h, be the ith eigenwlue of J , with and q, being the corresponding col urn n right eiget ntor~ and ~ row left e i gen\.ect or, re spec t i \re1y . The ith modal reactise power Lwiation is U

cl

where A':

cci,

= I with

c,l the j t h element o f 5,.The corresponding ith mocal

~ vtage l \wiit t ion is

The magnitude of each eigenvalue h, determines the weakness o f the coriesponding modal voltage. The smaller the magnitude of h, the uteaker the c o r ~esponding modal voltage. If h, = 0. the ith modal voltage will collapse because any change in that modal power will cause infinite modal voltage irariation. If all eigenkralues are positive, the system is considered voltage stable. This is a dift'erent dynamic system where eigenvalues with negative real parts i re stable. The relationship between system bdtage stability and eigen\ralues of the J , matrix is best understood by relating the eigen\dues with Q-\' sensiti\.ity o f each bus. J , can be taken as ;t syminetric matrix and therefore the eigenvuli es

of J , are close to being purely real. If all the eigenc~aluesare positiLe J K is positi\re definite and the V-Q sensitivities are also positiLre. indicating that the system is voltage stable. The system is considered voltage unstable if at least one of the eigenvalues is positive. A zero eigenvalue of J , means that the system is on the \yerge of voltage instability. Furthermore, small eigenvalues of J , determine the proximity of the system to be voltage unstable. The participation factor of bus k to mode i is defined as

For all the small eigenvalues, bus participation factors determine the areas close to cdtage instability. In addition to the bus participations, modal analysis also calculates branch and generator participations. Branch participations indicate which branches are important in the stability of a g i \ m mode. This proLides insight into possible remedial actions as well as contingencies. which inay result in loss of voltage stability. Generator participations depict which machines niust retain reactit,e reserves to ensure stability of a given mode. Figure 6.5 depicts the technique static voltage stability assessment using modal analysis. For a practical system with several thousand buses it is impractical and unnecessary to calculate all the eigenvalues. Calculating only the minimum eigenvalue of J K is not sufficient because there are usually more than one \ \ ~ a k modes associated with different parts of the system. and the mode associated with the minimum eigenvalue may not be the most troublesome mode as the system is stressed. The I I I smallest eigenvalues of J , are the I I I least stable modes of the system. If the biggest of the vz eigen\dues, say mode I I I . is a strong enough mode, the modes that are not computed can be neglected because they are known to be stronger than mode ni. An implicit inixrse lopsided simultaneOLIS iteration technique is used to compute the I I I smallest eigenvalues of J , and the associated right and left eigenvectors. Similar to sensitivity analysis, modal analysis (see the worked example at the end of this chapter) is only valid for the linearized model. Modal analj~sis can, for example. be applied at points along P-V cur\.es or at points i n lime of a dynamic si mU I at i on.

6.1 1 ANALYSIS TECHNIQUES FOR DYNAMIC VOLTAGE STABILITY STUDIES I t is only recently that the effects of system and load dynamics are being in\.estigated in the context of voltage collapse. The dynamics that are being considered are:

1.58

+-

r

Obtain system architecture and network data

t I . Solve Base Case power flow 2. Do Contingency Analysis 3. Select a desired set

+

Detailed Analysis?

I . Fonn Full Jacobian Matrix. J

Determine weaker voltage areas based on eigenvalues

I

Perform participation factor analysis

I I

Compute

A V = g/\-'qAQ Plot P-V and P-QCurves

L--l_--l Figure 6.5

Static voltage stability assessment using modal analysis.

1.

Machine and excitation system dynamics including power system stabilizer (PSS). 2. Load dynamics. 3 . Dynamics of SVC controls and FACTS devices. 4. Tap-changer dynamics. 5. Dynamics due to load frequency control. AGC, etc. While 1, 2, and 3 involve fast dynamics, 4 and 5 represent slow dynarrics. A classification process of dynamic voltage stability vis-a-vis static stability is shown in Figure 6.6. Here "load" implies demand and "U" represents set pcints of LFC, AGC, and voltage/VAr controls at substations. .Y, represents the slow

Volfuge SfcJhi1if.Y Assessriierif

1

,

Subsystems: x&(x *x , u * I - d ) k $ ( x *XF"U*Load)

N

I

Subsystem F k f + x $xFut Load)

(Voltage Collapse) Type. 11 Instability

Both Subsystems S and F are Stable

I

Figure 6.6 Classification of voltage instabilities.

variables such as the state variables belonging to tap-changing transformers, AGC loop and center of angle variables in the case of a multi-area representation. .rf. represents the fast variables belonging to the generating unit including PSS and governor, induction motor load dynamics, SVC dynamics, and so forth. The overall mathematical model is of the form: (6.38) (6.39) (6.40)

(6.411

Ignoring the more slower AGC dynamics and the faster network transients (60 Hz) we can categorize the variables appearing in Eq. (6.38)-(6.41). x\ = [ n , ]

where

i = 1, . . . ,I?

11,

= transfortner tap ratio

rectangular \ ariables o f ith

/, =

bii4 \

oltage o r [ O f ]

\:j

niachinc terminal currents in machine rcference franie i = I , . . . . 111 de\ired real power of ith generator cle\ired \wltage at ith gcneratoi bu\ de~tredI oltagc at the bu\ controlled b j tap-changer

i

i = 1, 2. . . . . 111 i = 1, 2.

. . . . III

pr = c'ector of load parameters to be defined.

The state \w-iables of the static VAr system ( S V C ) control and induct ,011 motor \ + r i l l appear in .v, if included i n the o\rerall model. As an example \%re giife belour the equations for ;I I I I machine 11 bus s q w m halting p tap-changing t ra n sforme rs . On Iy the sy tic h ronoii s mac h i ne 11nd t a p-c hanger d y nani i c s arc i n c 1U ded .

6.1 1.1

Equations of Slow and Fast Subsystems

For an ur-rnachine, 11-bus system ha\ing I - tap-changing transformerh. the follou~ing equations are applicable Slo \. v Subsystem

f a s t Subsystem

~,,,c'%

tit

= -E;, - (

x(/,- x;,) + E,,,,

T\,'& = -vRl+ K rlt

\J,, -

T / , S = - RI, + ~KIE , , , , tlt TI I

Tl

i

=

.

I ....

E,,,, + K ,,( v,,.,,-

v,)

i = I . . . . . 111

I

i = I , . . . . 111

(6.43)

I

The algebraic equations for the stator and network can be used to andjrze the system

6.1 1.2

load Flow and Equilibrium Point

The equilibrium point is calculated for a given set of reference points, \',',, ,.Tl,,,,l',, and a given demand P I ,and QL,and then solving the follonfing equations for the \w-iables 8?, .8,,. V,,,+], ,V,,.

I62

Cticipter- ri

We may alternatively combine Eq. (6.44) in a compact way as ,I

P:"' = ~ v , v , Y , , c o s-( 8, ~ ,- a,k)= o

i = I , . . . , II

i-I (1

Q:"' =

CV,V,Y,,~~ - e, - a,h) ~(O, =o

i=

I ,

. . . , 11

(6.3: )

i-I

and

The parameter vector p l can be defined in terms of PI,,,,Q,l,l,tt,,,, IZ,,,, etc. The equilibrium point is calculated for a given set of reference points V , , , ,.T,,,. V,,,, and a given demand P I , and Ql,.The load flow equations are extracted from Eqs. (6.45) and (6.46)as follows 1 . Specify bus voltage 2. Specify bus voltage 3. Specify net injected bered 111 + 1 to 1 1 . Solve the following

magnitudes numbered 1 to m. angle number 1 (slack bus). real power P'",'=P I , and Q'"/'=Q l , at all buses numequations for the variables

e?,. . . ,elf. v,,,,,.. . . . v,,.

The standard load-flow Jacobian matrix involves the linearization of E q. (6.47) with respect to 6:. . . . . 611rVlll+lr . . . , V,,. After the solution using Newtcm's method. compute

(6.48)

In the above load flow problem one can include inequalities on Q generation at P-V buses, switching Var sources, etc. From the load flow solution, the initial conditions of state variables in Eq. (6.48) can be computed systematically. The initial value of V, is V,,,. L inearization Define 9' = [ 9:0;] corresponding to generator and load buses. Also define .v' = [s: 1

.4= [.v;,.Y;, . . . J,!]

where s:=

[HI,

. . . , ll,,]

and .I-;

= [G,,o,,E;,.E,,,E I,,,, VR,*R,,]i = I. . . .

1

111

and the algebraic variables as I,, Vq, V,. Also let

s,:= (Pi,,( V,l.Qi,( V , ) ) The linearized equations corresponding to Eqs. (6.32)-(6.43) can be expressed as

A,,

0 0 0

0

In Eq. (6.48) the variations corresponding AVl in the nonlinear load characteristic is contained in ASLeand ASL,.

6.1 1.3 Static Stability (Type I Instability) In Eq. (6.46), suppose that both Ais= AtI.= 0. Then we have a static situation with all equations being algebraic. Let all the voltage deviations in AO? and At!,

be denoted by AP. then the rest of the algebraic variables can be eliminated (assuming con\tant power load) to express A P = J I HAp,. If det (1,) -+0 ;I\ load is increased it is referred to a s Type I static in\tability. i.e., the \y\tem i \ not able to handle the increased load.

6.1 1.4

Dynamic Stability (Type II Instability)

Eliminating the algebraic ~ariablesi n Eq. (6.39)atid assuming expressed a s

6.1 1.5

Ail E

0 . it can bt.

Slow Instability

Theoretically i t shoiild be possible to eliminate AY,in Eq. (6.50)using the singiilar perturbation theory and obtain the linearized slow system as At, = A,A.\.,. The time scale o f the phenomena is so large that linearized results may ii8.)t retlect the true picture. For such a time intensi\re phenomena. nonlinear siii1ul.ition is recolnmended. 6.11.6

Fast Instability

Fint u e rearrange the Lrar.iable\ [Al,.Av,,A$',] a r [AZ,.&.AV,, . . . .AV,,l 1 A02.AOj. . . . .AO,,,AV,,,+,, ... = [A:.Avl. Next b e a\\ume I , a\ con\tant and load par:inieter\ a\ con\tant which implie\ Ap, = 0 . We get

For the constant power case, both AS, and AS2 are = 0. Otherwise. AS, = AS,,(V,) and AS2,= AS,,(V ,). For a given voltagedependent load, AS,, and can be computed. Only the appropriate diagonal elements of B:, C,, and C; M i l l be modified and we obtain the system

Now

cqis the load flow Jacobian J L I and

B2

I

8'

c. c,

is obtained as

= J , / . The system matrix. A

Atl = A,,,Avl + E h

Using drastic assumptions about voltage control and load characteristic\ that the steady-state stability associated with the system matrix, A,,, can be determined by examining the load flow Jacobian, J , / . 6.1 1.7

Voltage Stability Assessment

The algorithm for Lwltage collapseholtage stability assessment includes static and dynamic assessment. The algorithm for dynamic stabilitl- assessnient is shown in Fig. 6.7. 6.1 1.8

VSTAB-Voltage

Stability Assessment (EPRI)

A more promising method with the trade name VSTAB. uses po~7el-tlow and modal analysis techniques. It provides assessment of the proximity to Lvltage instability and determines the mechanism of voltage instability. I n this method. the proximity to \voltage instability is evaluated by conducting a series of p o ~ ' e r tlow solutions with load increase until load tlow diLwgsiice is encountered. When load flow divergence is encountered, the step size for load increase ix reduced and the power flows are continued. The voltage stability limit is considered to have been reached when the step size reaches the cutoff due specified by the user. The load level at this point is the maximum loadabilitjf.This procedure is carried out simultaneously for the intact system as well as for contingencies. Load increase can be carried out with or without generation scaling. The slack bus generation is not scaled. Loading can be by area or by zonc. The mechanism of voltage instability is studied in VSTAB by using modal analysis. Modal analysis employing V-Q sensitivities can identifqr areas that have potential problems and provide information regarding the mechanism of Lroltage collapse. The method is briefly discussed as follows. The usual power tlow equations can be expressed in the linearized form.

Ael

A \I

where AP = incremental change in bus real power

(6.53)

I66

* Select Contingencies

+

Select Continuation Parameter P(Q)

*

Sdve base load flow rk Increase P(Q)

+

Run the power flow I

<

Converge? Yes

I

I Compute initial conditions of state variable

I

-

Calculate eigenvalue

A

1

Figure 6.7 Summary of the dynamic voltage stability assessment technique.

AQ = incremental change in bus reactive power injection A 0 = incremental change in bus voltage angle A V = incremental change in bus voltage magnitude The system dynamic behavior can be expressed by the first order diffei-ential equation,

167

Voltage Stability As.se.ssineiit

where X = state vector of the system V = bus voltage vector For the steady-state condition X = 0, using the enhanced device models used in Eq. (6.54). Equation (6.53) can be rewritten as: (6.55)

where

Af,,= incremental change in device real power output AQ,, = incremental change in device reactive power output AV,, = incremental change in device voltage magnitude A0(,= incremental change in device voltage angle and AZ2represent a modified form of J,w. J\>,, JVH. The terms A l l , AI?, J,-, in the terms associated with each device. We can study the Q-V sensititrity while keeping f constant. For this analysis we can substitute Af = O in Eq. (6.53) to give us upon simplification, AV = Jk

A0

(6.56)

where

By analyzing the eigenvalues and eigenvectors of the reduced Jacobian J R . we arrive at 1) =

A-’ x q

(6.58)

or 1

v, = -9,

A,

where

h is the iIh modal voltage.

(6.59)

I hS

U\ing modal analy\is. these relative bu\ participation and branch particip.1(ion factor\ c m be computed for the i"' mode. The complete procedure for \tatic \ oltage \tabilit> a\se\snient L ia rnodal analy\i\ is outlined in Fig. 6.8. 6.1 1.9

Preventive Control of Voltage Stability

There are t ~ let o els of lroltage stability enhancement, the fir\t le\rel uith del ic:e ba\eci control. the \econd level i \ i n the form of operation-based control. The

Obtain base case

Set for pre-contingency

solution Solve the load flow

0 : cont ingenc

Generate QV curves

-

New load level or change

Figure 6.8 Thc VSTAB algorithm (ClEPRI).

1 ,+b

voltage stability is improved by optimal system operation conditions. The static analy\is method is used for the determination of prcbmitive control scheme. System operation conditions are determined by F ( 8 . 1.’ h ) = 0. The design of a broltage stability preventive control scheme includes the \teps outlined i n Fig. 6.9.

CONCLUSION Pourer sy st e 111 i.01tag e st abi I i t y i n vol ves genera t i on, t ran s ni i ssion and d i st ri h i tion. So to maintain the voltage stabilitr is crucial tor thc normal operatioti

Input System Data

4 Select Critical Contingencies

1 Use Optimal Power Flow to do Contingency

4 Identify and Rank contingencies with low stability limit using VSTAB

I

+

Select the first contingency

1

Any other contingencies ?

Select the next contingency from the list

4

Output results

Incorporate the selected contingencies in the Contingency Constrained OPF for expanding the lowest stability limit

c Adjust control parameters to reflect optimized values

I

a

Figure 6.9 V o I t ag e

5t

abi 1 it y pre \.ent i v e cont ro I \c he iii e .

I 70

Chccprer 6

of a power system. In adequate reactive power support from generators arid transmission lines lead to voltage instability or voltage collapse which halie resulted in several major system failures (blackouts) such as the massive Tokyo blackout in July 1987. In order to prevent the stability limit being reached or exceeded during a given contingency, remedial actions need to be recommended. It is well knov,Tn that in all cases, voltage instability is caused by inadequate transmission capacity at a given operating condition due to a contingency, which the system cannot withstand. Based on contingencies that occur, the distribution of plant gene1 ators, transmission tlows and load to meet given stability criteria is usually done by using effective/economical control actions. Future work in the determination of adequate remedial measures for stability enhancement have been proposed in past publications, where the correctiLre control action is handled as an optimization problem. The two-stage formulati .In to achieve the desired stability enhancement utilizes the concepts of Chapta S and this chapter. The first stage handles voltage stability enhancement while [he second-stage optimization scheme deals with angle stability enhancement. The process will lead to a unified index. Hence, when carrying out stability enhancement based on a selected list of contingencies, only enhancement of the app-opriate problem (either voltage or angle) needs to be carried out, thus saving labor and computational time. Future work in unifying the indices while inc,.>rporating the irarious available controls is still a challenge. The reader is intrir ed to research further literature in selected references located at the end of .he book. Also. the annotated glossary of terms supports the chapter.

M O D A L ANALYSIS: WORKED EXAMPLE Consider the SO0 kV. 322 km (200 miles) lines transmission system shown in Fig. I O(a) below supplying power to a radial load from a 'strong' power system represented by an infinite bus. The line parameters, as shown in Fig. 10(b). are expressed in their respective per unit values on a common system base of 00 M V A and 500 kV. 1 . 1 Compute the full admittance matrix of the two-bus system and write the power flow equations from the sending end to the receiving end in the form:

I .2 Hence or otherwise, write down the expressions for the four (4) sub-matrices of the Jacobian in the linearized load tlow equations as defined by:

Voltage Stcihility Assswiierit

I7i

(a) Infinite Bus

Load Bus Bus 2

Bus I

pZ-JQz,

Transmissicm Line

(3c1

Load Load

Shunt Qsh -7

(b)

j

Infinite Bus Bus I

v,= 1 .OLOO

Load Bus Bus 2

Y, = 2.142 -J24.973

"2

=I

v, I

Figure 6.10 The SS0 kV, 370 km (230 miles) line tranwii\\ion \ystem \uppl>4ng a radial load: (a) schematic diagram of the transmission system and (b) the equi\alt.nt WYE circuit repmentation of the transmission line.

1.3 When P2 = 1500 MW, calculate the eigenvalues of the reduced Q-V Jacobian matrix and the V-Q sensitivities with the following different reactive power injections for each of the corresponding two voltages on the Q-V curve. a. Q, = 500 MVAR. b. Q,=400 MVAR. c. Values of Q, close to the bottom of the V-Q curve 1.4 Determine the voltage stability of the system by computing the eigenvalues of the reduced V-Q Jacobian matrix for the following cases:

a.

b.

P = 1500 MW, Q, = 450 MVAR. P = 1’300 MW. Ql = 950 MVAR.

(Aswnie that the reactiLne pouw Qi i \ wpplied by

ii

\hunt capacitor).

Solution

From the figure. the admittance matrix of the 2-bus system is 2.142 - j22.897 -2.142 -2.142 + i24.973 2.142

+ -

j24.973 j23.897

The expression tor P and (2 at any bus k are gibfen by:

where

Hence.

are interested in only P1 and Q2 b v i t h V, = 0 p i i .

Hence the expressions tor the Jacobian terms are give by:

J,ll =

a P+= -2.132cose + 24.973sine + 4.284\’> a v,

(a) The linearized power flow equations are

with

The expression for J,,. J,,, Jpo. and Jl,, were given before. For this simple system, JK is a I x 1 matrix. The eigenvalue lambda of the matrix is the same as the matrix itself. The Q-V sensitivity is equal to the inlverse of the eigenkdue. For each of the Q,s, there are two solutions for the recei\,ing end \*oltage. Table 6.2 summarizes the V, 8, h, and dV/dQ with P = 1500 MW and Q = 500. 400, 306, and 305.9 MVAr. For each case the eigenvalue and W Q sensiti\,ity are both negatikre at the low voltage solutions, and are both p0sitii.e at high jroltage solutions. With Q = 305.9 MVAr close to the bottom of the Q-\’ c~11-k~ tlVIdQ is large and h is very small.

Table 6.2

Results for Modal Analysis Worked Example

Lou Voltage Solution

High Voltage Solution

500.0 400.0 306.0 305.5

1.024 0.956 0.820 0.184

-37.3 -40.1

-48.2 -48.7

17.03 12.4 I 0.52 0.02

0.059 0.08 1 1.923 50.10

0.67 1 0.706 0.812 0.815

-66.7 -60.3 -48.8 -48.6

-39.87 -20.96 -0.9SO

-0.700

-0.02S

-0.048 -I .OS3 -1.134

( b ) With the shunt capacitor connected at the receiving end of the line, the self admittance is: Y,, = 2.142 - j(22.897

-

H,)

with P = IS00 MW and A 450 MVAr reactive shunt capacitor. V, = 0.98I , 8 = -39.1 degrees. Since B, = 4.5 pu., then Y2:= 2.132 j(22.897 - 4.5) = 2.132 -j18.397. With this new value of Y2:, the r t s duced Q-V Jacobian matrix is J , = 5.348, and J , is positive indicating that the system is voltage stable. with P = I900 MW and a 950 MVAr reactive shunt capacitor, V, = 0.995, 8 = -52.97 degrees. Since B, = 9.5 pi'.. then YJ2= 2.142 j(22.897 - 9 . 5 )= 2.142 -jl.397. With t h i h new value of Y?:. the reduced Q-V Jacobian matrix is J , = - 13.683, and J , is negative indicating that the system is voltage iinstable.

7 Technology of Intelligent Systems

INTRODUCTION The previous chapters on voltage and angle stability assessment represent rich techniques for the analytical solutions of large interconnected systems. Several important and notable successes have been achieved in solving general simulation based calculations of stability for off-line studies. There remains a large amount of problems in power systems that are largely solved by human experts either in conjunction with results from numerical analysis and decision support systems. The following features frequently characterize these problems: Inadequate model of the real world. Complexity and size of the problems which prohibit timely computation. 3. Solution method employed by the human is not capable of being expressed in an algorithm or mathematical form. It usually involves many rules of thumb. 4. The operator decision making is based on fuzzy linguistics description. 5 . Analysis of security such as voltage or angle is based on human judgement and experience. 1.

2.

These drawbacks have motivated the power system community to seek alternative solutions techniques through the use of artificial intelligence systems and variants of its techniques. In this chapter, we present a brief summary of I75

such techniques including expert systems decision trees. artificial neural t i c a t works, fuzzy logic systems, and their hybrids. These rechniqiies have been einployed for solving various p o ~ ' e system r operation and planning probleins and especially tor the assessment o f transient and i d t a g c stability prediction, inslab i 1i t y pre v e ii t i on . and con t i i i ge nc y ran k i n g , et c . Soiiie cornriional it y exi s t s among i i i t e 1I i ge 11 t s y s t e m s approaches. The ~ 1 ' s t e m req 11 ire me iit s for de ve1opi ng or ;issessi rig i ii te I 1i gen t systems approaches L re ;Is tollou~s: 1 . Ability to identify system states. 2. Selecticity o f controls. 3. Learning ability to update knowledge. 4. Coordination of tasks. 5 . Flexibility. 6. Ability to handle uncertainty.

An expert system (ES). also referred to ;is Knowledge Based System. embodies human expertise in ii narrow t'ield o r domain i n a machine implement:ition form. It utilizes elements of human knowledge to provide decision support at ;I l e ~ dcomparablc to the human expert and is capable of justify in,(7 Its reasoning. I t separates the inference mechanism f'rom the kno~vletlgeand uses one o r niore knowledge structures such as production rules frames. semantic nets. p r x l icatc calculus. and objects to represent knowledge. The function o f the expert system consists of its ability t o collect uicl store an expert's ability to solve a problem so that a lionexpert can use it. The fuiictional components of an expert system are represented and defined here. '

I . Knou.ledge Base: Contains all relc\iant informaticm about the s>'slem 11nder study. 2. User intertxe: Input/output or so called nian-machine interface gi\,es the necessary information and decision rules to the user. 3. Inference Engine: Analyzes the system using i t then rules based on go;i I /cl 11t ;I o I.-i cn t ed s t rii t egy cii 11ed foru.ard/back \+(ard ch ;i i 11i n g.

Other modules such iis control iiiechanism and niodification loop lire iisiifro tn ex pert systems to iic hiebt robustness. I n the knowledge base, for example i n a power system. the database g ves the operating characteristics of the system through state estimation. The re5ults from off-line studies are iist'd ah sets o f rules in the knoufledge base. Three special ti'atures distinguish Expert Systems from traditional pvu er s st e m an ii I y s i s : ii I 1y t:xc 1iided

>r

1.

The expert system allon~seasy tlexibility ot' mrinipulation ot' doi-iain specific k n o ~ ~ l e d gwithout e hai.ing to \\.:itch for the impact o f chaiigeh on the u ' a ~n'e ;ire reasoning it.

2. The expert system is concerned with manipulating symbolic informat i o n rather than the numerical information directly. 3. The expert system addresses problems where knowledge may be deterministic and more imprecise and allows for certainty in reasoning.

FUZZY LOGIC AND DECISION TREES

7.1

Fuzzy set theory deFdoped by Prof. Zadeh i n 1965 as a mathematical means of describing vagueness in linguistics. A fuzzy set is ;i generalization of' ordinary sets that allows assigning ii degree of membership for each element to range from (0, I ] iiiterLral. A fuzzy set differs from a crisp set which has a unique binary membership fraction in that the fuzzy set has an infinite number of membership function3 that inay represent it. Fuzzy reasoning offers a way to understand system beha\,ior through in t erpo I at i on approx i in ate1y between i n pu t and o i l t pi1t s i t u a t ions. Fuzzy logic is based simply on the way the brain deals urith inexact intimnation. A fuzzy system combines and applies sets with fuzz). rules to problcnis \+,ith overall complex nonfinear behavior. As a structured normal estiniator, i t expresses fuzzy if the rules of some expert knowledge. Fuzzy set theory offers a new method for niodeling the inexactness and u ncertai n t y concern i n g deci si on - ma ki ng . Fuzzy I ogic i m pro ~ ' e tshe potent i a 1 to r mode I i ng h u in an reason i n g and for present i n g and u t i 1i z i n g 1i n g U i st i c desc ri pt i on s i n a corn pu t e r i zed i n fere nc i ng e n v i roil men t . The two methods of deiveloping fuzzy models are based on: 1.

Laws of cause and effect which use rules of relations described hqr reasoning in variable sets theory. 2. Laurs of transition. which use ordinary equations to express. ciiiisc' and effect re I at i on s hi p . Fuzzy set theory uses the concept of possibility defined as a number bet u w n one (completely possible) and zero (totally iriipossible) instead of probab i I i t y w h i ch nieas u res the appropriate u ncert ai n t y of avai 1a b 1e s t a t i s t ic;i 1 i n formation. Where probability fails without statistical data. fuzzy set theorj, does a better job than other intelligent systems such as neural nets and expert systems.

7.2

ARTIFICIAL NEURAL NETWORKS

While f i i u ~ expert , 5yrtem. and deci\ion tree technique\ are rule-bii\ed. there are certain AI techniques that use Artificial NeLird Netuforh ( A N N ) pattern rccognition clu5tering \trategies. An ANN is considered ii machine that i \ dc\igiicd to model the M orh the brain performs in dealing ufith a particular t a 4 of intcr-

est. The neural network is usually implemented using electronic components or simulated in a software computation. A neural network performs computatim through learning. It is a massively parallel distributed computation that is trained with the appropriate data and reproduces trained strategies and adapts according to charging situations. Pattern recognition techniques on the other hand, act as classifiers depeiiding on the pattern of interest. Clustering is also one of the unsigned ANN lea-ning techniques. Instead of selecting a training set, we construct a set of unlabelcd trial factors. These techniques use Euclidean distance to estimate the membership in a pattern group or to classify according to some prevailing rates. Of the three intelligent systems approaches discussed here neural netwo *ks are best suited to problems where voluminous data and a complex nonlin:ar relationship exists between the input and output patterns, such as instability assessment and load forecasting problems. Artificial neural network schemes share the ability to identify the model structure of unknown systems with input/ output data and efficiently solve combinational problems, which define their relationships. An overview of artificial neural networks and some significant neural learning algorithms is provided in this chapter. The evaluation of intelligent systems has shown that neural-based approaches are most beneficial for power system stability assessment, due to their ability to generalize and "learn" from historical information. A basic otferv e w of neural computing practices and methods is protided in the next section I n comparison to neural nets, expert system approaches are rather system specific and rigid. Fuzzy set theory is relatively new as a power system analysis tool. Further validation of membership functions and the use of network opera1 ion support using real fault situations warrants further investigation of fuzzy logic as a power system analysis tool. A review of the salient points defining the three intelligent systems discussed here is provided in Table 7.1. ANNs are parallel distributed processing systems composed of nonliiiear processing elements that perform in a manner similar to the most elementary functions of biological neurons. For instance, ANNs possess the ability to Icarn from experience, generalize from previous examples to new ones, and to absf ract pertinent information from examples containing irrelevant or incomplete data. ANNs are not suited to simple mathematical tasks, such as computing tvl tage drops along ;I feeder. However, ANNs are beneficial in solving a great number of pattern recognition problems, that are either computationally burdensome or impossible for conventional iterative programs to solve. Neural networks offer the following advantages: ANNs have the ability to learn and construct a complex nonlinear mapping through a set of input/output examples. The network architec-ture allows for easy training without a need for a structured model.

Table 7.1 Features of Intelligent Systems Approaches Intelligent systems

System components

Artificial neural networks

Neurons Weighted connections between neurons Learning algorithm

Expert systems

Knowledge base User interface Inference engine

Fuzzy logic

Crisis sets Membership functions Fuzzy sets

Functionality Learning a mapping (relationship) from examples Se I f-org an i za t ion i n t o classes or clusters Associatilre memory Optimization General Learning from inference Data structure Linkage to external applications Distributed kno\+.ledge Specific Handling of human subjectivity Human Knokvledge in "thinking" machine Handling of human ambiguity in communication

A pp I ica I i on areas

C1assi fi c at i on Estimation

Es t i mat i (3 i i Dec i s i (3 n making

Decision making i n cases of inexact or amb i g U ou s data sets

Input variables can be easily added or deleted. Correlated or uncorrelated data can be utilized. Neural networks have a superior noise rejection capability that can effectively deal with uncertainties of the actual process. Neural networks execute very quickly. Most of the calculation overheads occur during the initial off-lines training. Neural networks consist of a large number of parallel processing units which can be implemented using general or special purpose hardware. Hence neural networks can relieve the burden of computation from the Energy Management System computers. Before neural networks can gain the necessary recognition as useful problem solving tools in the power industry. certain fundamental issues have to be addressed. Some are associated with neural network fundamentals and others are problem dependent and are listed below:

Determining the proper consistency of the training and testing data slits including. the number of patterns. input dimensions. and statistical properties. in order to pro\.ide adequate generalization and kiiowlec ge retention. Neural network decomposition in order to find the optimal stage or location for the use of neural netm'orks. This is an important consideration in dealing with large-scale systems. The use o f feature selection and clustering techniques for data preprocessing. This can help achieire reduction in both dimensional and c( mbi na t ori a I coin p I e x i t y. Neural n e t u w k must not be allowed to niemorize the training data nor be c oiii e sat i i r at ed . A N N s ha\re been developed i n numerous configurations. Despite their dikrersity, some cominonality exists among the various network paradigms. In the fo 11o\v i ng sections so ni e fii n damen t a 1s o 1' net U'o rk c (1in pU t i rig are pre se n t e d ,

7.2.1

Fundamentals of ANN

We review ;I \pecific neural network architecture, it\ actik ation function and element\ of training. Thi\ is followed by LI discu\sion o f some of the factor\ of 1earn i ng ii 1gori t h nis i tic 1lid i rig bac h - propag a t i on. coii n t er pro pag a t ion and c 1L \ t e r I earni ng ;i 1gor i t h m s . Architecture

The artificial neuron, also known as the processing element, is riiodeled to reflect the configuration of the biological neuron (Fig. 7.1 ). A set of input? are applied. each representing the output of another neuron. Each input to a nc tiron is multiplied by ;I weight, corresponding to the input's connection strenglh o r importance to that neuron. The relationship described above is illustrated in Fig. 7.2. Here, ;i sct of inputs, are applied to the neuron. The inputs, collectively defined ;is the \rector X, are niultiplied by their associated weights (W,,)before being applied t o the suriiriiation block, E. The inputs X, now multiplied by the weights. collectiitely defined as the vector W. are summed at the summation block. Z to product. the activation level A, of the./"' neuron in the next layer. subject to a threshold \ alue, 8. This relationship c m be stated i n vector notation as follows:

Activation Functions

The activation level, A,, is further processed by an acti\,ation function. F. to produce the output signal of the j " ' neuron, Out,. The activation functior rnay be ii simple linear function, such a s a gain. K :

Figure 7.1 Component4 of biological neurons.

Figure 7.2 Components of an artificial neuron.

IN2

Out, = KA,

Where, K is a constant. Or it can be a threshold function, such as: Out = 0 i t A , < 8

Where 8 is a constant threshold value. Alternately, F' may be a function that more accurately simulates the nonlinear characteristics of a biological neuron. and permits more general network functions. One such function is the sigmoid activation function. which is the activation of choice for the majority of conteniporary neural systems.

The sigmoid function is known as a "squashing function," since it conpresses the range of A,, so that the value of Out, never exceeds some low boun 3aries. The boundaries are defined as 0 and 1 in the case of the sigmoid function. For a large value of A,, the denominator in Equation 7.4 would approach I , and thus produce an output close to 1. Likewise, for a value of A, equal to --oo. the denominator in Equation 7.4 would approach +=. and thus produce an output approaching zero. In order to provide a clearer understanding of' the role of the activation function in neural computing, we think of the activation function as defining a nonlinear gain in an analog electronic system. The gain is calculated by cotnpirting the ratio of the change in Out, to a small change in A,. Thus. the gain is the slope of the curve at a specific activation level. The gain may vary front a IOU, value at large negative activation levels, to a high \.due at the zero point, and back to a low value at large positive levels. The central high gain region o f the sigmoidal activation function solves the problem of processing small sign: Is, while its regions of lower gains at positive and negative extremes are app-opriate for large activation lelrels. Thus a neuron ciin perform adequately olrer ;I wide range of input levels. Training the Neural Network

A neural network is trained so that a given stimulus (set of inputs) will produce a desired response (output set). Each input or output set is referred to a; a vector. Training is the process by which input vectors are continuously applied to a network, while adjusting the unknown network components. until e;ich input vector produces the desired output vector.

Training algorithms can be classified as supervised or unsupervised. Supervised training requires pairing each input vector with a desired output, or target vector, together these are called a training pair. Usually, neural networks are trained with a large number of these training pairs. A rule-of-thumb is to have at least 100 such training pairs for each input neuron, in order to represent a wide variety of system conditions. In the supervised training process an input vector is applied, the actual output of the network is determined and compared to the target value. The difference between the actual and target values is the error that is then fed back through the network to update the interconnection weights, and thus, minimize the error. The entire process is repeated until the error for the entire training set has met an acceptable tolerance level. For unsupervised training, the training set consists solely of input vectors. These types of training processes group similar input sets into classes according to the statistical properties of the training set. There is no way to predetermine which output classes will be produced by a given input vector. Therefore, unsupervised training generally requires some postprocessing of data to transform the output classes into some meaningful format. Usually, all that is required is a simple identification of the input-output relationships defined by the netu-ork. Some significant training algorithms are presented and discussed in the next section.

7.2.2 Back Propagation Learning Algorithm Of the existing neural network paradigms, the back propagation learning algorithm is an interesting approach to the solution of classification problems. The back propagation learning algorithm is a technique for optimizing the interconnection weights in an ANN by minimizing the global error between the desired and actual output of all the cases evaluated during the training session. The procedure begins with the presentation of the input stimuli and the desired output of all the training cases and the initialization of all the interconnection weights to random values. The input stimuli are fed forward through the network to compute the actual output values. The error between the desired and actual outputs are computed and propagated back through the network to adjust the weights. This process is repeated until the global error satisfies a prespecified tolerance value.

7.2.3 Counter Propagation Learning Algorithm The counter propagation network is not as robust as the back propagating techniques, but it can provide quick solutions for applications that cannot tolerate long training sessions. The network functions as a look-up table capable of generalization. The training process associates input vectors with their corre-

sponding output vectors. that may be either, in binarjf or continuous form. Tlic genera I i zat ion capa bi 1it y of the cou ri t e r propagation network a 1lows it to ex t r;i 2 t ;i desired output response e \ w i urhen presented u!ith purtially incomplete o r incorrect input data. This makes the counter propagation network useful fl )r pat t e rn re cognition , pat t e 1-11CO ni p 1e t i on, and sign a 1 enhance men t ripp I i c at i o n s . I ii its simplest form the counter propagation network is ;i combination of two MY Iknown training algorithms. the Kohonen, and the Grossbet-g Outstar. The Kohonen layer functions in a winner take all fashion. That is, for ;I giiteii input vector only one output neuron in the Kohonen layer is acti\rat :cl (oiitputs ;i logical one), all of remaining outputs of the Kohonen layer are h c Id at zero. The net output of each Kohonen neuron is the cveighted sum of' the inputs. therefore, the net output with the highest \ d u e is the "winner." The Grossberg layer functions in a similar manner with its net output eqiial to the sum o f the u,t.ighted values from the Kohonen layer. Since there is oril~, oiie nonzero term from the Kohonen layer the calculution is simple. In fact, Ihc only action of each iieuron in the Grossberg layer is to output the value of rlie weight that connects it to the s i ng 1e lionze r o k oh o ne n ti e U ro 11. The limitations of the counter propagation network mike i t inttrior to b i c h propagating techniques for most map ping net "ark applications. Ho\ve\rer, 1lic counter propagation IictLvork has been found to be useful for r;ipid prototj'p iig of systems that can e~wituallyyield to the greater accuriicy and rohustncss ot' bac k propag a t i 11g it I gor i t h m s. 7.2.4

Clustering learning Algorithm

A clustering-based A N N M ' ~ S implemented for applications to pou.er sys1ciii problems. The clustering A N N is similar in nature to the ART1 A N N developed by Carpenter and Grossberg and Kohonen's self-organizing map. The algoril h m i iic 1ides 11 feature ext riic t ion scheme for stu t i s t ical cleri vii t i on of ii moi-e rot 111st set of tcaturc \rariables than those that could be prokrided through purely het ristic methods. The clustering ANN utilizes both the superczised and unsuper\sised traiiiing methodologies. The supervised module was used to synthesize the mapping hct c c w n the input stimuli atid the desired output clusters forriicd bq' the tinsuper\,ised inodule. Continuous handshaking het\\.t.cn the tcvo modules rakes place until a prcspecified tolerancc is satisfied.

7.3

ROBUST ARTIFICIAL NEURAL NETWORK

Back propagation ANNI iire well suited to enhancement of the conc~entional Tran\icnt Energj Function (TEF) proccdure Iijr many re;isoii\. First. the) can

be trained by the results of off-line TEF studies. Second, they can be quickly called on-line to classify a system subject to disturbance ;IS secure or potentidly insecure. Third, they can generalize from cases that MXI used i n training to classify cases that were not previously evaluated. Fourth, they are robust enough to handle a variety of systems. And finally, the multi-layer perceptron net\zurks ciin be used to overcome nonlinearly separable problems. The de\.elopment of the robust ANN invol\.es the selection of ;in appropriate network architecture to determine how the neurons ure connected. and the selection of an appropriate learning algori thin to determine how neurons proccss their information and how the connection strengths betnfeen layers \\we optimized. and the application of training data acquired from off-line.

7.3.1

Network Configuration

The ba\ic neuron model for the back propagation algorithm contain\ input\. t /. that are adjusted by weights, and summed. After ufhich, a bia\ or thrcshold value. 0, is added. and the total is passed through the acti\ation function F( i 1 ;is an output. Mathematically this process can be rcprcwnted by: \ t y l ,

Where I!$, is the weight applied between i"' neuron of input layer and,j"' ne~iro~i of hidden layer and represents the total number of input neurons. Before starting the training process, all of the Lveight must be initializcd to small random numbers (-0.5 < \t'/,, < + O S ) . This ensures that the net\s.ork is not satiirated by large values of the weights. The random nature of the initial Lireights also allows the network to learn unequal final values of the \+!eightsif that is what is required for optimal performance. /,11,1,

7.3.2

Overview of Training

Training the robust artificial neural network requires the following step-bj -step procedure :

I . Apply all the input vectors from the training set to the netuwh. 2. Calculate the output of the network for each input \rector. 3. Compute the error between the actual network output and the desired output for each input vector. 4. Calculate the global error as the averaged sum of the squared errorh for the entire training set.

Clttrpter 7

IN6

5. Adjust the weights and thresholds in a manner that minimizes the error. 6 . Repeat steps 1-5 until the global error is acceptably low. Upon completion of the training procedure the network is used for recognition and the weights are not changed. It may be seen that steps 1 and 2 constitute ii "forward pass" through the network. in that the signal propagates from the network input to its output. Steps 3-5 are a "reverse pass," where the globiil error propagates backwards through the network and is used to adjust the weights and thresholds. The first two steps in the training procedure can be expressed as follow;: an input vector, X,is applied and an output vector, Out. is produced. The inputtarget vector pair, X and T, comes from the training set. The output vector, Out. is produced as a result of the application of X to the network. Calculations n multi-layer neural nets are performed on a layer by layer basis, beginning wi h the hidden layer closest to the input layer. The weighted values of the inp.it vector are summed in the form:

Where A, = active lektel of neuron j in the hidden layer X, = iIh input from the input vector X W,,,= interconnected weight applied between i"' neuron of input layer and j"'

neuron of hidden layer. The nonlinear sigmoidal activation function is applied to the activati .In level, A , in Eq. (7.7). to dri\.e the value of hidden neurons toward one of the two states (0 or I ; stable or unstable).

(7.7)

Where: H , = output of neuron j in the hidden layer 8, = dynamic threshold value of neuron j in the hidden layer The addition of the threshold value, 8, increases training efficiency. The same process is applied for networks with additional hidden layers with the output of the preceding layer serving as inputs to the next layer in the netwcrk. In the output layer the output nodes are defined as the weighted sum of the outputs from the preceding layer. No activation function is applied at the output node. The output neurons are computed as follows:

Where: Out, = output of kIh neuron in the output layer W$,,a = interconnected weight applied between j"' neuron in the hidden layer and k t hneuron of the output layer. Therefore, the calculation of the outputs of the final layer requires the application of Eq. (7.6) to each layer, from the network's input to its output. In the reverse pass, defined by steps 3-5, two tasks must be accomplished. First, the weight of the output layer must be adjusted. And second, the weight and thresholds of the hidden layers must be adjusted. In order to adjust the weights in the output layer an update factor for each weight must be computed. First, the global error, E, for the entire training set is computed using Eq. (7.9) as the average sum of the squared differences betureen the actual network outputs { OutL]and the target, or desired values, { T,].

The error signal is multiplied by the derivative of the signioid function [Out,( I - Out,)] for the kIh neuron of the output layer, thereby producing the 6 value for the neuron.

the 6 is then multiplied by the output value from the previous layer that corresponds to the weight being adjusted. = 6, H,

(7.1 1 )

where: A H , ~=, update ~ factor for interconnection weight between the j"' neuron in the hidden layer and the k"' neuron in the output layer. An identical process is performed for each weight proceeding from a neuron in the hidden layer to a neuron in the output layer. Adjustment of the weights between the input layer is slightly different, since the hidden layers have no target vectors. Therefore, the undate process described above cannot be used for this set of weights. Equation (7.1 1 ) is used for all of the weights, regardless of location, however, the 6 term for the weight between the input layer and hidden layer must be calculated without the benefit of the target vector. Instead, the 6 values from the output layer are used to adjust the 6 values

i n the first hidden layer :incl the 6's are propagated backisrurds to :ill of thi: preceding layers. Consider a neiiron i n the hidden layer immediately behind the output layer. During the for\\w-d pass the output values of each layer are processed forward to the next layer via the interconnection weights, in ternis of the 6. values ar: passed in the re\'erse direction from the output layer to the hidden layer, during the training period. Each weight between the hidden and output layers is multiplied bjr the 6 \ d u e of the output neuron to which it is attached. The ~ ~ ~ l(+!' iie 6 for the weights in the hidden layer preceding the output l a j w i s produced bj, summing all of the products from the output layers and niultiplying them by the deri\.ati\res of the sigmoidal function, Lit that particular hidden node. Thus,

The 6 values are calculated for each neiiron i n ;i given hidden laqrer, and i I 1 of the ueights i n that l a ~ w are atljusted according11 uhing the expression belou :

for ;i multi-layer networh this process is repeated layer by layer. until the input layer is reached and 2111 of the unknowns ha1.e been adjusted. 7.3.3

Summary of Learning via Robust Back Propagation ANN

In SLII11111iiry. the learning period consists of ii gibm network uith ;I random w t of \ifeight \falues being simulated by rill of the inpur vectors in a training set. A global system error is computed, which is iisually quite large initially. thus -equiring rici-jijustmentof the interconnection weights via a process known iis training. A irariety of training algorithms exist. of which ii Robust Back propagation has significant advantages o \ ~ classical r Back propagation algorithms. Using the robust procedure, update factors for all of the weights in the network .ire computed based on the global system error between the actual netuork outputs and the target values. A gradient descent method is used to track the global error bvithin an acceptable tolerance level by optirnizing the interconnection weights. The conjugate gradient procedure possesses the ad\,antages of simp licity and minimal storage requirements in mapping ii given input to its coi-responding output state. A successt'iil learning exercise will yield a stable sel 01' utights. which exhibit only minor fluctuations in the \alue if further learning is attempted. Precautions should be taken against "over training," uhich occurs tirhen the network has become too specific to the training data and is no longer ablc to

generalize to previously unseen data. To pre\,ent this occut-rence the networL is periodically presented with test data during the training procedure. If the global system e r r o r increases from one checkpoint to the next then the training proce\+ was halted. A comparison of the robust tool to classical Back propagation paradigms is pre\ented in Table 7.2.

Table 7.2 Classical Back Propagation Versus Robust ANN Fe ;i t i i re Update scheme

Cla\s1cal bach propagation Arbitrary threshold

Data pre xe n t ation

St oc h a st ic present at ion

Error function

Local error function

Speed

Neural Ware Professional II/Plus: Over 3000 iterations for the XOR pro bl e 111. Rumelhart, Hinton & Williams 2-15 iterat i o n for XOR pro b I eni

Ad\ antage\ of

Robu\t A N N

robu \ t AN N

Dynamic t h reshold

S treilm I i nes Ieiir11it1g ~ I * o cess by updating dqm-

namic threshold during e\.ery iteration of learning session I nd i \,idual ci>wam i c threshold f o r eacli liidden node increases ticc Cl r3c 4' Batch present ;I- Batch dat a present at ion order can remain setion quential \ahicli is much less computation all>, i n tens i \,e than s t ocliil~ t ic data pre se n t at ion. I hat must be random Global error tunclion repGlobal error function resents true s)'steiii wide error for each iteration i n the learning sex s ion Weights ilrc llpdiited f o r entire training data at each iteration 152 iteration for Significant speed enh anc eine n t over c I a s s i the XOR procal methods blem

The benefits in speed and accuracy of the Robust ANN over classical Ba:k propagation methods can be traced back to three main characteristics. First, the Robust ANN utilizes a dynamic (adaptive) threshold value for each hidden node during the training session that is updated along with the interconnecti,)n weights as opposed to the arbitrary thresholds that allow for greater individual ty in the learning process for each hidden node and thus have a positive impact on the accuracy of the final output. The second and third beneficial traits of the Robust A N N are coupled. Sir ct' the Robust ANN utilizes a global error function that updates the weights and thresholds based on the average error of the entire data set, the training data can be presented in an identical sequential order for each pass. In contrast, the cla>sical Back propagation method utilizes a local error function that updates the data after each training vector and thus the data must be presented in a differxit random order for each pass, which increases the computational complexity m d thus, the required time for convergence.

7.3.4 Conclusions Several different types of neural nets have been discussed in this section. This is by no means an exhaustive list. A selection has been made based on appl cability to power system problems. While these have been inspired by biological neurons. the issue is not their ability to exactly model biological systems. hut their importance a s another solution paradigm for power system problems. The area of artificial networks is a very active field of research. For instance, it includes neural smithing for optimal architecture and learning design. Pruning techniques are used to determine the optimal number of neurons; to solve a specific problem. and adaptive learning algorithms to avoid retrairiing for large data sets. Training with noise can be used i n order- to overcome o\wfi(ting problems. There are continuing efforts to combine and enhance A N N methods *.s,ith regularization theory. stochastic, Bayesian, or other statistical and pattern ret.wgnition techniques in order to define optimality criteria for ANN perforinancc. I n fact, ANNs might be considered a set of highly adaptive statistical tools. I t itormation-theoretic concepts are introduced i n order to measure how ANNs gcnt'ralize, i.e., how they will perform on unknown test data. Finally, the combination of neural networks mrith other AI techniques like symbolic structures from expert systems, fuzzy logic, and genetic algorithr-is is being explored for many technical applications. Here complex problem: are described with fuzzy techniques or expert rules and the subsequent classitication, regression, or optimization tasks are solved with ANNs.

7.4

EXPERT SYSTEMS

It is not easy to give a precise definition of an eqwrt system. because the concept of the expert system itself is changing as technological advances in computer systems take place and new tasks are incorporated into the old ones. In simple words, it can be defined as a computer systetn that models the reasoning and action processes of a human expert in a given problem area. Expert systems, like human experts, attempt to reason within specific knowledge domains. An expert system allows the knowledge and experience of one or more experts to be captured and stored in a computer. This knowledge can then be used by anyone requiring it. The purpose of an expert system is not to replace the experts, but simply to make their knowledge and experience more widely available. Typically there are more problems than there are experts aF&lable to handle them. The expert system permits others to increase their productivity. improve the quality of their decisions, or simply to solve problems when an expert is not available. Valuable knowledge is a major resource and it often lies with only a fen experts. It is important to capture that knowledge so others can use it. Experts retire, get sick, move on to other fields, and otherwise become unavailable. Thu\ the knowledge is lost. Books can capture some knowledge. but they lea1.e the problem of application to the reader. Expert systems provide direct means of applying expertise. An expert system has three main components: a knowlecigc~Imw. an irfkrerice erigirie, and a rrtnri-machine iriterfclce. The knowledge base is the set of rules describing the domain knowledge for use in problem solving. The prime element of the man-machine interface is a working memory which serves to store information from the user of the system and the intermediate results of knowledge processing. The inference engine uses the domain knowledge together with the acquired information about the problem to reason and provide an expert solution. 7.4.1

A Working Definition

An expert system is an artificial intelligence (AI) program incorporating a knowledge base and an inferencing system. It is a highly specialized piece of software that attempts to duplicate the function of an expert in some field of expertise. The program acts as an intelligent consultant or advisor in the domain of interest, capturing the knowledge of one or more experts. Nonexperts can then tap the expert system to answer questions, solve problems, and make decisions in the domain. The expert system is a fresh new, innovative way to capture and package

knowledge. Its strength lies i n its ability to be put to practical use when ;in expert is not :i\xilable. Expert systems riiake knowledge more widely a\,ailable and help overcome the age-old problem of translating knowledge into practical results. It is one more way the technology is helping us to get a handle o n 11-e o\.ersupply of information. All AI software is knowledge-based ;IS it contairrs useful fricts, data. and relationships that are applied to a problem. Expert systems, howe\rer, are LI special type of knowledge based syste ii they contain heuristic knowledge. Heuristics are primarily from real world exp8:rience. not from textbooks. It is knowledge that comes directly from those people-the experts-who have worked for many years within the domain. It is knowledge derived from learning by doing. It is perhaps the most useful kirid o f knowledge, specifically related to everyday problems. I t has been said that knowledge is power. Certainly there is truth in that but in a more practical senw. knonrledge becomes power only when it is applied. The bottom line in any ficld of endeavor is RESULTS, some positive benefit o r outcome. Expert systems i re one niore \+'ay to achieFre results fiister and easier.

7.4.2 Characteristics of Expert Systems One advantage of the expert system is its permanence. Human expertise c ;in quickly fade. regardless o f whether it involves mental or physical activity. In expert must constantly practice and rehearse to rnaintain proficiencjc in so no probleni area. An significant period o f disuse can seriously affect the expel 1's perforrnance. Once expertise is acquired i n an expert system. it is around forever, barring catastrophic accidents related to memory storage. Its perinanelice is not related to its use. Another ad\.antage of the expert system is the ease c+.ith which it ciin be transferred o r reproduced. Transferring from one human to mother is the labi)rio u s , lengthy. and expensiLVe process called education ( o r in some ciises. knoufledge engineering). Transterring an expert system is the tri\.ial process of cop>'ing o r cloning a program o r data file. An expert system is also much easic. to document. Documenting human expertise is extremely difficult and time c o n suming. Documenting :in expert sj'stem is ;I straight forwud mapping bet\+ecn the way in that the expertise is represented i n the system and the naturiil 1;ingU age de sc r i pt ion of that r epre sen t iit i on . An expert system produces niore consistent. reproducible results than does the human expert. A human expert may make different decisions in idenl ical si tiiations because o f emotional factors. For exaniple. 11 human m a ~ f' orgc t to use a n important rule in ii crisis situation because of time pressure o r stress An expert system is not susceptible to these distractions. A final advantage of the expert system is its low cost. Human experts, especially the highly skilled ones, itre very sc;irce and hence very expena9i\fo.

Expert systems, in contrast, are relatively inexpensi\re. They are costl}! to develop but relatively inexpensive to operate. Their operating cost is .just the nominal computer cost of running the program. Their high de\relopment cost ( j'ears of effort of high-priced knowledge engineers and domain experts) is offset b!, their low operating cost and the ease of making new copies of the system. Although expert systenis tend to perform well, there are important are;is i n which hurnan expertise is clearly superior to the artificial kind. This does not reflect a fundamental limitation of expert systems or A I , just the ci~rrentstateof-t h e- art . One such area is creativity. People are much more creative and inno\.ati\~ than even the smartest programs. A human expert can reorganize information and use it to synthesize new knowledge. while an expert system tends to beha\-e in a somewhat uninspired routine manner. Human experts handle unexpected ei'ents by using iniaginati\re and novel approaches to problem sol\.ing, including draw i n g analog i e s to s i t u at i ons i n com ple te 1y d i ffe ren t pro b 1e 111 dom ;i i n s . Programs have had little success in doing this. Another area where human expertise excels is learning. Human experts adapt to changing conditions: they adjust their strategies to conform to n e ~ ' situations. Expert systems are not particularly adept at learning new concepts o r rules. probably because it is a very difficult task that has ;il\+ays been made i n dekreloping progrms that learn, but these programs tend to ~ + w r ki n cxtremely simple domains and do not do well when confronted \+.it11 the complex it!^ and de tai Is of re a1- wo r I d pro b1ems. Human experts can make direct use of complex sensory input, Likther it be sight. sound. taste, or smell. But expert systems nianipulrite sqwibols that represent ideas and concepts, so sensory data must be transformed into s ~ m b o l s that can be understood by the system. Quite a bit of information niay be lost in translation, especially when visual scenes are mapped into sets of ob-jects and the relations between them, Human experts can look at the big picture, examine all aspects of ;i problem. and see how they relate to the central issue. Expert systems. on the othcr hand. tend to focus on the problem itself, ignoring issues rele\mt to. but sepiirate from. the basic problem. This happens because it takes ;i huge amount of expertise just to handle the basic problem, and it would take almost ;is much expertise to handle each of the hundreds of tangential probleins that could arise. In the future, when faster and cheaper techniques for acquiring expert k n o ~ v l edge are developed. this situation may change. 7.4.3 Applications in Power System Operations The focus of expert system research in the power systems operations area has been to help the system operator to function more effecti\~ely.Specific ob-jectives include:

Chcr],te,. :7

Understanding the interface requirements to integrate databases, vari ous computer architectures, full graphics systems, and applications software expected in future control centers; Defining the appropriate applications of expert systems (e.g., as a routine aid to operators, providing action plans during emergencies, or the rebuilding of a system after a major disturbance); Building domain specific knowledge for general purpose expert systerns (e.g., rules and techniques used by today's experienced operators to handle the daily operation of a power system): Developing and evaluating specific expert systems (e.g., replacing tcdious jobs done by humans); Demonstrating expert systems that aid the operators in the most co>t effective areas. Control center operators benefit from using expert systems because suc n systcrns not only identify a problem. but they can also provide the underlyins reasoning used to define the problem, and a set of recommended actions t:) ameliorate the problem. The best applications of expert systems have been i 11 diagnostic situations where information about the system being studied is held in static tables. For application in power systerns, information about the systern can be taken from the real time database or a simulator.

7.4.4 Rule-Based System There is a lot of confusion about terminology in the artificial intelligence literature because there are few standardization efforts. So in the following paragraph we present the definitions used in this chapter. A knowledge-based system is ' a program in which the domain knowledge is explicit and separate from the program's other knowledge." It consists of the following fi\re components: tt e knowledge base, the knowledge acquisition module, the inference engine, the explanation module. and the user interface (Fig. 7.3). The Knowledge Base Contains Facts and Rules Like any database, the knowledge base contains the data, called facts, whkh describe fixed properties of the problem. In addition, a knowledge base contairis instructions describing how to process these data. If these instructions are given in an IF . . . THEN . . . rule format, then we are talking about a rule-based sy st e in. For the rule IF A THEN B, A is called the left-hand side (LHS) and B the right hand side (RHS) of the rule. If A and B are logical expressions which ciln be true or false. A is called the premise and B the conclusion. But as we w I I

-

KNOWLEDGE ACQUISITION

KNOWLEDGE BASE FACTS & RULES

ENGINE

t

Figure 7.3 Knowledge-based system.

see later, we will also include actions like closing an element in the LHS or the RHS of a rule. These actions will change the initial facts. They cannot be considered as simple logic expressions. These facts and instructions represented as rules may already hakre been formulated and written down explicitly because they are common inkvariable knowledge of the domain. In our case there exist manuals describing the possible actions which can be performed on different elements and normally ekreryone familiar with these manuals can manipulate these elements. In particular these manuals describe the problem completely. Nevertheless this work i n control centers is done by operators who are certainly experts i n this domain and who know their domain by heart. In other cases, there is a human expert who uses heuristic or complex knowledge acquired during years of work. Often this kind of knowledge is incomplete and the reasoning mechanism applied by the human expert cannot be replaced by pure logic rules. For instance, intuition or so called common sense cannot be formalized this way. The main reason why we call our systcm a rulebased system and not an expert system is due to the fact that in principle our knowledge is complete. It can easily be formulated as rules. Also we do not want to contribute the misunderstanding that in any domain where a human expert explains his knowledge by I F . . . THEN , . . rules, this reasoning can be efficiently implemented by an expert system. This misunderstanding leads to hundreds of unsatisfying feasibility studies on expert system applications which can be found in the literature today. In short, a rule-based system is a special case of a knowledge-based system.

I96

Knorvledge Base Acquisition Module 2nd I/O Intertke
The knowledge base is built up using a program called a knowledge acquisition component. Depending on the nature of facts and rules this component may b: a sophisticated rule editor generating ASCII o r binary files or ii program extracting relevant data from another database or e\wi l'rom a real-time process a n J storing it in the predefined format on nuss storage medium. The knowledge acquisition module as well as the user interface should bt. user t'riendly . Menu-driven and/or mouse-driwi dialogue, graphic output multi nrindoLving ull help to achieve this goal. These techniques are already n'c , I k non'11. The Interference Engine Provides Control

Whereas rules describe how to process data, the instructions of the inferencc engine describe how to process these rules. They fix for instance the order in Lvhich turo rules concerning the same element are executed or fired. One possibility concerns the enumeration of the rules, where the numbers f'ix the prioril), o f the rule. Another possibility is to process the youngest mentioned rule fir>t. The processing protocol may be specific for the problem. Then it is called metalknowledge with meta-rules. In order to keep the inference engine ;IS general ;is possible it is sometirnes convenient to store this tneta-knowledge in the knom 1edge base. The instructions also define the general problem solving strategy. There are two important techniques. backward chaining and forward chaining. The go: 1dri\.en strategy backward chaining tries to prove a goal, kvhile data-dt-ihrencontrol forward chaining deduces new data from existing facts. The Expl,iri,ition Module Explnnntion ot the Inference Chain

The explanation module provides comment on the reasoning process.

;I

trace facility for the end user in order to

Shells 2nd Progrminiing Languages are not ldentic-a1

An "empty" hnowledge-ba\ed system which provides control and data structut e\ but no hnowledge is called a shell. Any shell is written in a certain programming language for in\tance Parcal, LISP, OPS83, o r PROLOG. AI1 these langua: e \ h a b e their ad\ antage\ and di\advantage\. We note that OPS83 provide\ the d;ita type "rules" and the forward chaining control strategy, whereas PROLOG I \ e\pecially \uited to implement backward chaining control, because of i t \ ba1.htraching facilities. Neither procedural languages like Pa\cal nor functional oiie\ lihe LISP pro\ ide thew facilities that therefore habe to be programmcd explic-

itly. Although they may be more efficient for a final inipleiiientatioii they are not suited for rapid prototyping in order to test concepts and methodologies. We should emphasize however that PROLOG does not fit in our definition of a knowledge-based system shell because neither a control strategy nor an explanation facility are not provided but have to be programmed. Using PROLOG facts as knowledge base facts and PROLOG predicates as rules does not permit the explicit separation of the inference engine and knou.1edge base. Although it may be tempting to quickly implement a prototype i n pure PROLOG in order to test the concept, this prototype cannot be enhanced without changing the code of the program (except for those cases u'here only pure predicate calculus is used and no extra control is needed i n order t o enhance efficiency). Note however that the PROLOG trace facility does not automatically provide an explanation module. Depending on the PROLOG-interpreter, the output of this facility is more or less unreadable for nonpro~rr\mniers (e.g., variables are simply denoted by numbers). Furthermore, predicates used for building up strategies have to be hidden to the trace procedure because showing them will produce a lot of redundant information. The user-interf;ice provided by standard CPROLOG is rather poor for our application. Similar arguments concerning control. explanation, and user-friendliness are iralid f o r OPS83. Let us have a look at the reasoning process of the human substation operator. The operator asks whether i t is possible to close an element o r to s ~ i p p l ~ ' power to a certain load and how to do it. He therefore p r o ~ ~ !as hqpothesis instead of looking at the substation configuration and collecting all possible conclusions and actions based on the initial state of the station. For this re;ison. many power oriented rule base systems have been written i n PROLOG because this goal-driven strategy reflects the reasoning process of the end user better.

7.4.5

Reasoning with Uncertainty in Rule Based Systems

One of the important features of expert systems is their ability to deal u-ith incorrect or uncertain information. There will be times when an expert system, in gathering initial inputs, will ask a question for which there is no iinsnw. I n such a case, simply say so. Expert systems are designed to deal urith cases such as this. Because one may not have a particular fact. the search process \{,ill undoubtedly take a different path. it may take longer to conic up \\rith an ans~ver. but the expert system will. Traditional algorithmic software simply cannot deal with incomplete information. If one leaves out a piece of data, one may not receive an ans\ver at all. If the data is incorrect, the answer will be incorrect. This is u9here artificial in t e 11i ge nc e pro gra in s, partic u 1ar 1y e x pe rt systems. are part i c 111ar 1y U sefu I \\.hen the inputs are ambiguous or completely missing. the progi-am may still find a

solution to the problem. The system may qualify that solution, but at least it is an answer that can in many cases be put to practical use. This is consistent with expert level problem solving where one rarely has all the facts before making ii decision. Corninon sense or knowledge of the problem tells us what is importmt to know and what is less important. Experts almost always work with incoinplete or questionable information. but this doesn’t prevent them from solving the problem. Thus, increasingly in the design of expert systems, there been a focus on inethods of obtaining an approximate solution to a problem when there is no clear conclusion from the given data. Logically, a s expert system problems hecome inore complex, the difficulty of reaching a complete conclusion certainty increases, s o in some cases, there must a method of handing uncertainty. Researchers report that a classical expert system gave incorrect results due to ;he sharpness of the boundaries created by the if-then rule of the system: howe\er, once a method for dealing with uncertainty (in these two cases fuzzy set theory) was used. the expert system reached the desired conclusions. The successful performance of expert systems relies heavily on human 2xpert knowledge derived from domain experts based on their experience. ‘:he other forins of knowledge include causal knowledge and information from case studies, databases, etc. Knowledge is typically expressed in the form of high level rules. The expert knowledge takes the form of heuristics, procedural rules. and strategies. It inherently contains vagueness and imprecision because expxts are not able to explicitly express their knowledge. The process of acquiring knowledge is also quite imprecise, because the expert is usually not aw;irv of all the tools used in the reasoning process. The knowledge that one reasons \b,ith inay itself contain uncertainty. Uncertain data and incomplete information are other sources of uncertainty in expert systems. Uncertainty i n rule based expert systems occurs in two forms. The first form is lingiiistic uncertainty which occurs if an antecedent contains vague statements such as the level is “high” or “the value is near 20.” The other fori-i of uncertainty. called evidential uncertainty. occurs if the relationship betfieii an obserkration and a conclusion is not entirely certain. This type of uncertain1y is most corninonly handled using conditional probability which indicates the likelihood that a particular observation leads to a specific conclusion. The study o f making decisions under either of these types of uncertainty will be referred to iis plausible or approximate reasoning. Several inethods of dealing with uricertainty in expert system have been proposed, including: Subjectikre probability Ce rt a i n t y facr ors Fuzzy measures Fuzzy set theory

The first three methods are generally used to hand evidential uncertainty, while the last method, fuzzy set theory is used to incorporate linguistic uncertainty. As expert assessments of the indicators of the problem may be imprecise, fuzzy sets may be used for determining the degree to which a rule from the expert system applies to the data that is analyzed. When applying a method of reasoning with uncertainty to a rule based expert system. there must be a method of combining or propagating uncertainty between rules. A method of propagating uncertainty for the method of reasoning with uncertainty will be discussed next. Subjective Probability and Statistics One method of dealing with uncertainty is to use conventional statistics and probability. For example, with the use of statistics, sufficient data may be available to compute the mean (average), median, and standard deviations. These new figures derived from original data provide additional knowledge which will help in making a decision. Recall that probability is simply a ratio of the number of times that a particular action will occur for a given number of attempts. It is really a ratio as shown below: P(.u) = Nuniber of occurrence of an event/total number o f e\'ent\ that take place

The probability of ,v occurring, stated as P ( x ) , is the ratio of the number of times s occurs to the total number of event that takes place. For example, i n rolling a standard die, the probability is one-sixth that any one of numbers, one through six, will come up. This may also be expressed as a fraction 0.16667, or as a percentage, 16.67%. In many knowledge representation cases. the probability for a certain condition or action may be known or can be estimated. For the probability of a certain even taking place is 70%, then one may initiate some action if the probability is equal or greater than 70%. If the probability is less than 70%. then perhaps an action may not be taken. For example, the production rule below uses the probability: IF the stone is clear, without color THEN it is diamond (probability 60%)

An example will illustrate this. Suppose we ask ten engineers whether they can program in the BASIC language. Out of the ten, three say they can. We can use these figures to compute the probability: P(BAS1C) = -J 10 = 0.3

What this \ays is that the probability of a n engineer to be able to progra 11 i n BASIC is 0.300. W e c;in also expres\ thi\-a percentage by \imply niultipl:, ing the probubilitj b j 100. We \ay that the probabilitj o f engineer\ being able to prograni in BASIC i \ 3O'k. Probability figure\ lihe thi\ can be used to determine rule strength if the) fit the problem. Multiple probability lralues U i l l occur i n manj \jf\tem\. For example, a riile may haLe three part\ t o i t \ antecedent, each \+ith a probabilitj crilue. The o ~ e r d l probability o f the c alue then become\ the product o f the indii idual probabilitic.\, i f the part\ o f the antccedent are independent o f one another. I n ;I three p;irt antecedent, the probabilitie\ may bc 0.90. 0.70, and 0.65. The o\ erall prohabilit) i \

The combined probability i\ about 4 I c/c . But thi\ i\ true only if the indi\ idual part\ of' the antecedent do not affect o r depend c m one another. Sometime\ one rule reference\ another. Here the indiL idual rule probabilit i t ' \ can propagate from one to another. There i\ ;i need to e ~ a l u a t ethe tc t i l l probabilitj o f ii ~eq11enceof rules o r a path through the \earth tree to determine i t ii \pecific rule fires. Or one maj be able to use the combined probabilitj to prcclict the be\t path through the warch trce. I n other ~ + o r d \ t,he probabilirit.\ become the "cost\" of' the indiL idual arc\ i n the tree. Thcre are niitiierwi\ methods o f computing combined probabilitie\. If' thc r ~ i ahre independent, ;I \iiiiple product can be uwd ;I\ d e w i b e d before. HOMe\cr, mo\t ekent\ iind rule\ are dependent upon one another. I n that cii\(. ;I \peci;il procedure ciilled Baye\' RLIICo r Theorem c;in compute the probabilitj of' e \ ent A occurring gi\ en that e\ent B Iia\ alrcadq occurred. Thi\ i \ e \ p r e ~ \ e d ;i\ P ( A 111). B a j e \ ' Theorem i\:

not attempt to explain thi\ rule, but one \hould be ;i~+are o t the manq expert \y\tems iise Baye\' Theorem in\tead o f certaint) fac tot-\ to deal N i t h uncertainty. Se\ era1 iiiii-ior expert \j \tern c k \ elopment tool\ ii\e B a) e\ ia n proba bi 1i t y .

We

Mill

t'ac%tthat

hlcmitres ot Beliet ,ind Disbeliet

Mca\ure\ o f belief' and disbelief arose from the desire that ekvidence \hmld incrementally increa\e the belief o r di\belief in ii h j pothe\i\. The formal det'inition o t the tiie;i\Lire o f belief w;i\ b a w l on the idea that if ;I prior probabilitJf, P ( / I )i \ defined, then the maximum amount of belief that can be added 10 P ( / I )

from a new piece of e\fidenceis 1 - P(I1).If a piece of e\sidence confirms R I 1 10) . then this would amount to. adding P ( h ic) - P ( h ) to the pre\.ioii\ belief, so the belief i n 11 has been increased by Ap(11) =

All e ) - All)

I

- p(11)

The mea\ure of increased di\belief can be defined \iniilarly. No\! u r i t h t h i \ idea, let the measure of increase belief ( M B ) giken sonic t'\ idence about ;I hypothesis / I be defined as: (1

and let the measure of increased disbelief ( M D ) be defined as

Note that when evidence P is assigned to a hypothesis I,. only one ot thc MII or M B functions will be greater than zero so that ;I single piece 01' c.\.idencc cannot be iised as both a measure of the confirmation and negation of ;I Iijpttiesis. As the nieasures of belief and disbelief were used in the design of' ;in expert system. it \vas found that a representation of the uncertainty i n terms o f ;I single me as U re nw U I d be n i ore con ve n i e n t i n in aki n g coin pari son s of d i fferc n t hypot li eses. Certnint~/Fnctors

There are se\-eral methods of dealing with uncertain information. I n rule based expert s y s t e m . numerical factors indicating the truth o r probability o f a premise o r conclusion are used as a measure for uncertainty. These numerical factors are kno\vn as certainty factors (CF) and probability. In ii high percentage of expert system rules, there L i ' i l l be no ambifuit!. or know with confidence whether o r not ;I p;trticular preniisc uncertainty. We o r conclusion is true or false. If the information is not kno\i.n at all. then ttic m r i l l

rule requesting it will not fire. In cases where there is the possibility that tie information is not known, special rules can be created to deal with this problem. The rule might state that if a particular piece of information is not available, then a certain action will be initiated. Still, there are many cases where the information is known but we hakfe less than IOO% confidence i n its truthfulness. Just iis weather forecasters L I S C a number to predict the likelihood of rain, so can a confidence number be used with production rules. A forecaster may say that there is a 80% probability of rain. They are saying that they don’t know for sure whether or not it is going to rain. On the other hand, they have enough information to be able to say that 80% of the time under similar circumstances it rains. While a certaiiity or confidence factor is not really a probability, it is a number that helps to represent the uncertainty. A certainty factor is simply a measure of the corifidencc that ii particular fact or rule is true or not true. It is irsually a numl)t.r between 0 and I , where zero indicates no confidence and one means full or complete confidence. We also hear certainty factors called confidence factors or rule strength. Certainty factors are used with both the premise (IF) and conclusion (THEN) portions of ii rule. The two examples gikren below show hou, corrfidence factors are used. IF THEN

the patient has the fever, CF = 0.6 prescribe rest i n bed

IF AND AND THEN

the patient is sneezing has a runny nose has watery eyes the patient has a cold, CF=O.S

Figure 7.4 shows several ways to use certainty factors. The scale is up to the progranimer. In Fig. 7.4A, a scale of 0 to I is used where 1 =absolute certainty; that is. 100% truthfulness or validity of the premise or conclusior of a rule. The 0, of course. indicates absolute uncertainty or falsity. Intermed ate \palues have varying degrees of truthfulness or uncertainty. A scale of 0 to I 0 or 0 to 100 can also be used with the same result. The + and - scale shourri in Fig. 7.4B is another approach. A + 5 indicates absolute certainty while ;I - 5 indicates 100% contradiction. The 0 in the center of the scale indicates unknown. One could also use a -I/O/+I scale as well. Determining whether a particular rule is to fire requires the inference eng.ine t o look at the confidence factor and evaluate it. For example, if one is using the 0 to 1 scale. we might want the rule to fire if the confidence factor is abow ii certain threshold level. say at 02. you may assign a threshold of + I or - I

CONFIDENCE

0 FALSE

(8)-5

.25

.5

.75

1

UNKNOWN

0

TRUE

+5

Figure 7.4 Confidence and certainty factor scales.

depending upon the circumstances at the minimum acceptable level for determining whether something is true or false. Other levels may be set depending upon the problem. In rules with compound premise clauses connected by AND or OR, each clause may have its own CF. For such situations, there must be a way to compute a CF for the rule. This is done by using the minimum CF of all clauses connected by AND or the maximum CF of all clauses connected by OR. Some examples will illustrate this.

RULE 1: IF X(0.4) AND Y(0.75) THEN Z Composite CF = 0.4

RULE 2: IF D(0.3) AND E(0.8) THEN F Composite CF = 0.8

I f each rule in ;I reasoning chain has a CF. each will. of course, affect rhe other. The outcome has to be decided based upon some composite e\'aluation. One way this is done is with a special formula.

This \ay\ that the CF tor rule X is added to the CF for rule Y and from that is wbtracted the product o f the CF\ for rule\ X and Y. Belou are tuu rules that illustrate this point:

RULE 3: IF P AND Q T H E N R(O.65)

RULE 4: IF R T H E N S(O.20) The ccmiposite CF is then: 0.65

+ -2.00

- (0.65)(0.20) = 0.85 - 0 . I3

= 0.72

Of coitrse. there will usua11y be more than two rules in ;I chain. The a h e formula can be u\ed by taking the composite CF of two rules and combiniiig i r Lvith the CF of ;I third rule. That ne\v compo\ite CF is then combined th ;I fourth, and s o on. Fuzz\., 1ogic

F u / q \et-ba\ed techniques can provide an excellent tramworh tor \jf\teinatitally representing the impreci\ion inherent i n an expert'\ hno\t ledge. W t . U W the fo I IOW i 11g ex ample to i I I U \ t ra te,

IF the teriiperature is high (0.8) A N D system i \ operating in hea\!y load period (0.9) THEN system i\ high stressed The parameters in the premise and consequent temperature, load pcriod. and \y\tetii \tress can be represented using simple fuzzy inembership function\. In thi\ case. the f i i z ~ jinformation is contained in the terms high arid heal j .

Although, there are situations where membership grades probabilities can take on similar t'alues, they are not the same. One distinguishing factor bet\b.een probability fuzzy membership grades is that the summation probabilities on a finite universal set must equal 1 . The main drawback of nonfuzzy methods i n dealing with uncertainty is their handling of linguistic terms. Fuzzy theory provides a natural frame\sa-k for dealing with linguistic terms used by experts. Imprecision in numeric data can be easily dealt with by expressing i t as a fuzzy number. Fuzzy sets can be conveniently incorporated in expert systems to better deal uith uncertainty imprecision. Structure of the Expert StaDility Assessment Systeni

The fuzzy expert system structure is shown in Fig. 7.5. Its database contains the power system topology. The knowledge base of the fuzzy expert system contains all the data of the contingencies. The information is based on known statistics of protection performance used in the system. If these data are not a\railable \ t k n a fiiult occurs, the fuzzy expert system asks the dispatcher to provide them and then saves them in a database for future use. Models for estimation of possible contingencies, and heuristic rules about the relay characteristics for actual fiiult determination are also include.

I Dispatcher interface

On-line Data from power system

Database

Network identifier

A

-+

Knowledge Base

Hypotheses and Calculations

Inference Engine

Determination of states

Figure 7.5 Fuzzy expert system structure.

206

7.5

FUZZY SETS AND SYSTEMS

In 1965, L. A. Zadeh laid the foundation of fuzzy set theory a s a method to deal with the imprecision of practical systems. Bellman and Zadeh write: "Much decision making in the real world takes place in an environment in which the goals. the constraints and the consequences of possible actions are not precisely known." This "imprecision" or fiizziness is the core of fuzzy sets or fuzzy logic:. Fuzzy sets were proposed as a generalization of conventional set theory. Pal.tially a s a result of this fact, fuzzy logic remained the purview of highly specialized and mathematical technical journals for many years. This changed abruptly in the late 1980s. 7.5.1

Fuzzy Sets

In ;I conventional (nonfuzzy, hard, or crisp) set, an element of the universe eithcr belongs to or does not belong to the set. That is, the membership of an element is crisp-it is either yes (in the set) or no (not in the set). A fuzzy set is ;I generalization of an ordinary set in that i t allows the degree of membership for each element to range over the unit interval [ O , I ] . Thus, the membership function of a fuzzy set maps each element of the universe of discourse to its range of space. which, in most cases, is assumed to be the unit interval. One major difference between crisp and fuzzy sets is that crisp sets a1wa:;s hi1t.e unique mernbership functions, whereas ebvery fuzzy set has an infinite nur-iber of possible membership functions that may represent it. This enables fuz;!~; systems to be adjusted for maximum utility to a given situation.

7.5.2

Fuzzy Systems, Complexity, and Ambiguity

Zadeh's principle of incompatibility was given in 1973 to explain why there is a need for ;I fwzy systems theory. The principle states. in essence, that as tie complexity of ;I system increases, our ability to make precise and yet significant statements about its behavior diminishes until ;I threshold is reached beyorid mrhich precision and significance (or relevance) become almost mutually exclusive characteristics. This suggests that complexity and ambiguity (imprecision are correlated: "The closer one looks at a real-world problem, the fuzzier t'econies its solution." It is characteristic of the way a human thinks to treat problems involvi,ig complexity and ambiguity i n a subjective manner. Complexity generally steins from uncertainty in the form of ambiguity: these are features of most social. technical, and economic situations experienced on a daily basis. In considering a coniplex system, humans reason approxirnately about its behatrior (a capabil rty that computers do not have) and thus maintain only ;I generic understanding o f

the problem. This generality and ambiguity are adequate for a human to perceive and understand complex systems. As one learns more and more about a system, its complexity decreases and understanding increases. As complexity decreases, the precision afforded by the computational methods becomes more useful in modeling the system. For less complex systems, thus involving little uncertainty. closed-form mathematical expressions offer precision descriptions of the system’s behavior. For systems that are slightly more complex but for which significant data exist, model-free methods, such as computational neural networks, pro\.ide powerful and effectiLre means to reduce some uncertainty through learning based on patterns i n the available data. Basic statistical analysis is founded on probability theory or stationary random processes, whereas most experimental results contain both random (typically noise) and nonrandom processes. One class of random processes or stationary processes exhibits the following three characteristics: 1.

The sample space on which the processes are defined cannot change from one experiment to another, that is, the outcome space cannot change. 2. The frequency of occurrence, or probability, of an event uithin that sample space is constant and cannot change from trial to trial or experiment to experiment. 3. The outcomes must be repeatable from experiment to experiment. The outcome of one trial does not influence the outcome of a pre\ious or future trial. However, fuzzy sets are not governed by these characteristics. The fundamental difference between fuzziness and probability is that fuzziness deals with deterministic plausibility, while probability concerns the likelihood of nondeterministic, stochastic, events. Fuzziness is one aspect of uncertainty. It is the ambiguity (vagueness) found in the definition of a concept or the meaning of a term such as confortable tenipemtirrc’ or \ i d 1 cwokod. Hob ever, the uncertainty of probability generally relates to the occurrence of phenomena, as symbolized by the concept of randomness. In other word5, ;1 statement is probabilistic if it expresses some kind of likelihood or degree of certainty or if it is the outcome of clearly defined but randomly occurring events. For example, the statements “There is a 50-50 chance that he will be there.” “It will be sunny tomorrow,” and “Roll the dice and get a six” demonstrate the uncertainty of randomness. Hence, fuzziness and randomness differ in nature: that is, they are different aspects of uncertainty. The former conveys ”subjective” human thinking. feelings, or language and the latter indicates an “objective“ statistic in the natural sciences.

From the modeling point of view. fuzzy riiodels and statistical models al ;o possess philosophically different kinds of information: Fuzzy memberships represent similarities of objects ot imprecisely defined properties, while probabilities convey relative frequencies. The quest for a method of quantify nonrand(~ri1 uncertainty (imprecision, vagueness, fuzziness) i n physical processes is the ha:,ic premise of fuzzy system theory, for to understand uncertainty i n ;I system is to understrind the system itself. As understanding impro\ses, the fidelity in mod~:li ng i iiipro LW. Whene\fer precision is elrident, for example, fuzzy systems are less efficient than iiiorc precise algorithins in offering ;I better understanding o f the problem. Re q 11i r i ng p re c i sion i n c ng i ne e r i n g mode 1s and prod 11 c t s t ra n s1ate s to re y 11i r i 11g high cost and long lead times i n production and dc\~elopment.For other tl- ;in siinple systeins. expense is proportional to precision: More precision entiiils higher cost. When considering the use of fuzzy logic for ;I giLm problc 111. ;in engineer o r scientist should ponder the need for exploiting the tolerance for imprecision. Not only does high precision dictate high costs but it also ent;iils low tractability in ;I problem. On the other hand. t ' u u y systems can focus on modeling problems char iicterized by iinprecise o r ;inibiguoiis information. The following iire situatir is \\,here i t is appropriate to formulate system problems cvithin ;I fuzzy sj'slcm fra llle\4fork:

I . In processes involbing hunian interaction (e.g.. hunian descripti\.c o r i n t 11i t i ve t h ink i rig ) 2. When a n expert is iivailable w'ho can specify the rules underljring the sj'steiii beha\.ior and the fuzzy sets that represent the characteristic, o f eac h \# ar i able 3. When ii mathematical model of the process does not exist, o r exists but is too difficult to encode. o r is too complex t o be e\~aluatedfast eiic.ugh for real-tinie operation, oi. in\wlves too much niemory oil the design .itecl chip architecture 4. In processes concerned w'ith continuous phenomena (e.g.. one o r i-iore of the control \w-iablcs are contin~ious)that are not easily broken d o u ~ i in t o discre t e segments 5 . Whcn high arribient noise leifels must be dealt t+.ith o r it is i m p o t i n t to iise iiicxpensi\~csenwrs and/or lou~-precisionmicrocontrollers. The ability to use f u ~ z q 's ystcni tools \+.ill allow one to addresh the \rast iiiii.jori t j , of' pro b Ie ni s that ha\ c t he precedi ng c harac t er i st ic s . Fuzzy for mu I ;I . ions ciiii help to achic\,e tractabilitj,, robustness, anti l o w w solution costs. Any field c;in be fu//,i tied ancl hence gencralized by replacing the coiicept o f ;i crisp set in the target field bj. the concept ot' ;i l'iizzj' set. Therefore. WO c;iii t'iiuif soiiic basic fields such as graph theor),. arithiiictic. and probabilitj, tl icory

to develop fuzzy graph theory, fuzzy arithmetic, and f u n y probability theor). respectiLrely, moreo\'er, we can also fuzzify some applied field\ such a\ neural networks. pattern recognition, and mathematical programiiiing to obtain f u ~ q n e u ra I net ur ork s , fuzzy pattern rec og n i t i on , and fuzzy mat lie ni at i ca 1 prog rani m i n g , re spec t i v e I y . The ad v ant ages of fu zz i fca t i on i n c I LI d e great e r ge n e ra 1 i t y . higher expressive power, an enhanced ability to model real-world problems, and a methodology for exploiting the tolerance for imprecision.

7.5.3 Fuzzy Theory in Electric Power Systems With the remarkable and successful penetration of fiizqr sj'stenis into manuflicturing, appliances, and computer products, their applications in power systenis are beginning to mature and receive wider acceptance i n the electric po\+w community. The application of fuzzy set theory to po~j'ersystems is a rela[i\~el~, new area of research. Concepts of fuzzy set theory were first introduced i n solLing po\sw sj'steni long-range decision-making problems in the late 1970s. Houe\ter. substantial interest in its applications to power areas is fairly recent. While con\~entional analytical solution methods exist for many problems i n po\s.er system operation, planning. and control. their formation of real-world problems suffers from restricti ve as su in p t i on s. Even U'i t h these ass u nip t i on s. so1v i ng 1arge- sca I e po\\.er system probleins is not trivial. Moreover. many uncertainties exist i n a signit'icant number of problems because power systems are large. coniplex. \J-idelJr spread geographically. and influenced by unexpected events. These factors make it difficult to deal effectively with many power system problems through strictl), conventional approaches alone. Therefore, areas of computational (artificial) intelligence emerged in recent years in power systems to complement convcntional mathematical approaches and proved to be effecti\.e when properly coupled together. I n conceptualizing power system problems, the expert's empirical kno\\.ledge is generally expressed by language containing ambiguous or t'uzq descriptions. As ii result, classical Boolean logic may not be a \ d i d [ool to represent such expertise. Fuzzy logic, on the other hand, is a natural choice for this pu~-pose. The growing number of publications on applications of fuzzy-set-based approaches to power systems indicates its potential role i n sol\ing po\\.er system problems. Results obtained thus far are promising, but fuzzy set theory is not widely accepted. The reasons for its lack of acceptance include the follouins: Misunderstanding of the concept Exce\si\ e claims of some researchers Lach of implemented and available systems I t \ {tatus ;I\ ii n e t~heory

Unexpected events and their uncertainties are traditionally represented by probability. However, it has recently been made clear that some of the uncertain facts are intrinsically of a fuzzy nature and are difficult to manage properly using probabilistic approaches. There are problems in power systems that contain conflicting objectives. In power system operation, economy and security, maximum load supply, and minimum generating cost are contlicting objectives. The combination of thcse objectives by weighting coefficients is the traditional approach to such problems. Fuzzy set theory offers a better compromise and obtains solutions t nat cannot be easily found by weighting methods. The benefits o f fuzzy set theory over traditional methods are as follows: Provides alternatives for the many attributes of objectiLVes selected Resolves contlicting objectives by designing weights appropriate to a sclected objective Prokiies the capability for handling ambiguity expressed in diagnostic processes, which involves symptoms and causes Power system components have physical and operational limits that are usually described as hard inequality constraints in mathematical formulaticms. Enforcing minor violations of some constraints (practically acceptable) increases the computational burden and decreases the efficiency and may even prel ent finding a feasible solution. In practice, certain slight violations of the inequality constraints are permissible. This means that there is not a clear constraint boundary and the constraints can be made soft. Traditionally. this problem has b x n managed by modifying either the objective function or the underlying iterai icte process. The fiizzy set approach inherently incorporates soft constraints and thus si m p 1i fie s i mp 1e nie n t at i on of such considerations. The following steps are recommended when fuzzy set theory is used to solve power system problems: Decription of original problem: The problem to be solved should firs1 be \tated mathematically and linguistically. Defining thresholds for variables: For a given variable, there is a specific \ alue with the greatest degree of \atisfaction evaluated from empirical hnoh ledge. and a certain deviation is acceptable with decreasing degree of sati\faction until there is a value that is completely unaccqtable. The two \alues corresponding to the greatest and lea\t degrc: of sati s tic t i on are termed threshold \. Fuzzy quantizittion: Based on the threshold values already determined, proper foriiis of member\hip functions are constructed. The funct on\ 4hould reflect the change in degree of satidaction with the change I I I \ ariables e\duated by experts.

Selection of the fuzzy operations: In terms of the practical decision-making process by human experts, a proper fuzzy operation is selected so that the results obtained are like those obtained by experts. The interpretation of results using fuzzy systems is based on domain experts' reasoning. Therefore, at this level a hybrid fuzzy set-expert system scheme is desirable. It helps to remove any ambiguity that may occur i n problem solving. 7.5.4

Membership Functions: Definitions

Let X be a set of objects, called the universe, whose element are denoted A-. Membership in a subset A of X is the membership function, MA from X to the real interval (O,I]. The universe is all the possible elements of concern in the particular context. A is called the fuzzy set and is a subset of X that has no sharp boundary. p,,!is the grade of membership .Y in A . The closer the LAue of p.\is to 1 , the more .r belongs to A . The total allowable uni\rerse of values is called the doinniri of the fuzzy set. The domain is a set of real numbers. increasing monotonically from left to right where the values can be both positive and negative. A is completely characterized by the set of pairs

Support of a fuzzy set A in the universal set X is the crisp set that contains all the elements of X that have a nonzero membership grade in A . That is Supp A = ( x E XIp,,(s)> 0 )

(7.15)

Fuzzy sets with a finite support assume that .Y, is an element of the support of fuzzy set A and that p 1 is its grade of membership in A . Then A is written as (7.16)

When X is an interval of real numbers a fuzzy set A is expressed as (7.17)

An empty fuzzy set has an empty support which implies that the membership function assigns 0 to all elements of the universal set. A technical concept closely related to the support set is the alpha-level set or the "a-c-irt." An alpha-level is a threshold restriction on the domain of the

'' I

C'llt1ptc.I'

7

fuzzy set based on the membership grade of each domain value. This set, A . is the a-cut of A which contains all the domain \falues that are part of the f ~ i ; ~ set at a minimum membership value of a . There are two kinds of a-cuts: n ~ a k and strong. The weak a-cut is defined as A,, = ( X . p,,(.r( 2 U } and the strong acut as A,, = (X,p,., (.Y( > a}. Also, the alpha-level set describes a p o i + w o r strength function that is used by fuzzy models to decide whether or not ;i truth ~ ~ a l ushould e be considered equi~~alent to zero. This is an important facilit). that controls the execution o f fiizzy rules ;is well a s the intersection o f niultiple fii I L J ' sets.

The degree of membership is known iis the membership or truth function since i t establishes ;I one-to-one correspondence between a11element in the domain and ii truth value indicating its degree o f membership i n the set. I t takes the forrn,

The triangular membership function is the most freqiiently iised funciion and the most practical. but other shapes are also used. One is the trapei,oid ivhich contains more information than the triangle. A fuzzy set can d s o be represented bq' a quadratic equation (invol\ring squares. I I -. or numbers to the second power) to product. ;I continuous curve. Three additional shapes ncliich are named for their appearance ;ire: the S-function, the PI-function, and tht: I_func t io 11. 7.5.5

Set Operations

Union m d Intersection of F L I Z Z ~Sets

The cluxsical union ( U ) and intersection (n)of ordinary subsets of X iire t'stended by the following formulas for intersection. A n B and union. A U H :

w,here p t C I N and p

are respectively the rnembership functions of A U B m d

AnB.

For each element .v in the ~iniversalset, the function in Eq. (7.19) t a h ;is its iirgiiment from the pair consisting of the element's menibership grades i 1 set A and in set B and yields the membership grade o f the element in the set constituting the union of A and B . The disjunction or union of two sets means that any element belonging to either of the sets is included in the partnership which expresses the maximum \ d u e for the two fuzzy sets in\wlved.

y

The argument to the function in Eq. (7.20) returns the membership grade of the element in the set consisting of the intersection of A and B. A conjunction or intersection makes use of only those aspects of set A and set B that appear in both sets which expresses the minimum value for the two fuzzy sets in\.ol\-ed. 7.5.6

Complement of a Fuzzy Set

The complement of A , - A , which is the part of the domain not i n a set, can also be characterized by Not-A. This is produced by inverting the truth function along each point of the fuzzy set and is defined by the membership function

v.v E

x.p \ (.v) = I - p, (.\)

(7.71)

The complement registers the degree to which an element is complementary to the underlying fuzzy set concept. That is, how compatible is an element‘s value I.\-] with the assertion, x is NOT J , where .\-is an element from the domain and .v is a fuzzy region. A fuzzy complement is actually a metric. It measures the distance between two points in the fuzzy regions at the same domain. The linear displacement between the complementary regions of the fuzzy regions determines the degree to which one set is a counter example of the other set. We can also view this as a measure of the fuzziness or information entropj’ i n the set. Defining Fuzzy Sets The steps below gi\re general guidelines i n defining fuzzy sets. 1.

2.

3.

De)te)i-iiiiiwthe type of j k y r?iecisicreiiieiit. Fuzzy sets can define: Orthogonal mappings between domain values and their member\hip in the set (“ordinary fuzzy set”); Differential surfaces which represent the first derivative of some action, degree of change between model states, or the force of control that must be applied to bring a system back to equilibrium: A proportional metric which reflects a degree of proportional compatibility between a control state and a solution state; A proportionality set which reflects a degree of proportionality between a control state and a solution state. Choosc~the slzcipe (or su$cice i w i - p i i o l o g ~of~ )tlw .fii:qq set. The shape maps the underlying domain back to the set membership through a correspondence between the data and the underlying concepts. Some possible shapes are triangular, trapezoidal, PI-curve, bell-shaped. Scurves, and linear. Every base fuzzy set must be normal. Selwt ciii cippropricite degree of m-erlap. The series of indi\ idual

Chtrprrt- 7

-1.

7.6 7.6.1

fuzzy sets, associated with the same solution variable, are converted into one continuous and smooth surface by overlapping each fuzzy set with its neighboring set. The degree of overlap depends on the concept modeled and the intrinsic degree of imprecision associated with the two neighboring states. Eii.siirt~thcit the clorritiiris m i o r i g the f i r x j suts cissoc-ieitetl btvitli tlic s t i r r i p .sol i r t ior i \Uricih1e.s J her re tli iJ scitiie irt i i\vr-sr of' cliscoi r I-se.

EXPERT REASONING A N D APPROXIMATE REASONING Fuzzy Measures

The fuzzy measure assigns a value to each crisp set of the universal set signifling the degree of evidence or belief that a particular element belongs in the set. For example, we might want to diagnose an ill patient by determining whethttr this patient belongs to the set of people with, pneumonia. bronchitis, emph1.sema, or a common cold. A physical examination may provide us with helpfril yet inconclusive evidence. Therefore, we might assign a high value, 0.75. to our best guess, bronchitis, and a lower value to the other possibilities, such as 0.45 to pneumonia. 0.30 to a common cold, and 0 to emphysema. These values rcflect the degree to which the patient's symptoms provide evidence for one disease rather than another, and the collection of these values constitutes a fuz;ty measure representing the uncertainty o r ambiguity associated with several welldefi ne d a I t e rn at i v e s. A fuzzy measure is a function

p C P(x) is a family of subsets of X such that: I . $ E p a n d X ~p. 2. If A E p then E p. 3. I f p is closed under the operation of set union, that is if A E p arid B E p. then also A U B E p. The set p is called a Bore1 field, since A U B 2 A and A U B 2 B . We ha1 e where

max[g(A). , q ( B ) ]5 g(A UB) due to the required monotonicity. Similarly, since n B C ,4 and A f l B C B. we have (?(A fl B ) 5 min[g(A),g(B)]. Two large classes of f u z y measures are referred to as belief and plausibilitj nieasures which are complementary (or dual) in the sense that one of them cqin be uniquely derived from the other. Given a basic assignment 111. a belief measure and ii plausibility measure are uniquely determined by the formulas

A

215

which are applicable for all A E p(x). Also m(A) refers to the degree of evidence or belief that a specific element of X belongs to the set A alone. The beliyf i?ze~i.sure, Bel(A), represents the total evidence or belief that the element belongs to the set A as well as to the various special subsets of A. The p l a i t s i h i l i f ~ nieasur-e, P l ( A ) , represents not only the total evidence or belief that the element in question belongs to the set A or to any of its subsets but also the additional evidence or belief associated with sets that overlap with A . There are also three important special types of plausibility and belief measures, prohcihility i ? ~ ~ w . w i - e . s and a pair of complementary measures referred to as possihility criicf riecx).ssir!, measures. 7.6.2

Approximate Reasoning

The root mechanism in a fuzzy model is the proposition. These are statements of relationships between mode variables and one or more fuzzy regions. A series of conditional and unconditional fuzzy associations or propositions is evaluated for its degree of truth and all those that have some truth contribute to the final output state of the solution variable set. The functional tie between the degree of truth in related fuzzy regions is called the method of implication. The functional tie between fuzzy regions and the expected value of a set point is called the method of defuzzification. Taken together these constitute the backbone of approximate reasoning. Hence an approximate reasoning system combines the attributes of conditional and unconditional fuzzy propositions, correlation methods, implication (truth transfer) techniques, proposition aggregation, and defuzzification. Unlike conventional expert systems where statements are executed serially, the principal reasoning protocol behind fuzzy logic is a parallel paradigm. I n conventional knowledge-based systems pruning algorithms and heuristics are applied to reduce the number of rules examined, but in a fuzzy system all the rules are fired. 7.6.3

The Role of Linguistic Variables

Fuzzy models manipulate linguistic variables. A linguistic variable is the representation of a fuzzy space which is essentially a fuzzy set derived from the evaluation of the linguistic variable. A linguistic variable encapsulates the properties of approximate or imprecise concepts in a systematic and computationally useful way. The organization of a linguistic variable is:

Clltr/’t‘~l7

L,

,/

t ((1, .

. . . . . . q,,}( I l l . . . . . . . I f , } f ;

(7.2:’ I

where predicate q represents usuality o r frequency yualifiers, I i represents ;I hedge and j : is the core fuzzy set. The presence of qualifier(s) and hedge(s) i s optional. Hedges change the shape of fuzzy sets in predictable ways and func:tion in the saine fashion as adverbs and adjectiires in the English languitgc. Frequency and usuality qualifiers reduce the deri\red fuzzy set by restricting t t e truth niembership function to ;I range consistent uith the intentional meaning of the qualifier. Although 11 linguistic variable may consist o f many separate ierm\. i t is consictered ii single entity in the fuzzy proposition.

7.6.4 Fuzzy Propositions A f u z q model consists of‘ii series o f conditional and unconditional tuzzy prop^ sitions. A proposition o r statement establishes a relationship between a \-alue I I I the underlJring domain and ;I f’iizzy space. Propo.sitiori is one that is qualified an “it” statement. The proposition following the if term if. is the antecedent predicate m d is an arbitrary fuzzy proposition. The proposition follom~ingtlie t l i o r i term is the consequent and is also any arbitriiry fuzzy proposition l)r

interpreted ii\ t i\ a member of Y to the degree that \I’ is ii member o f 2. A n unconditional f u z q proposition is one that i\ not yualified by an if \tatement. X i s k’

c\.herc X is ii sciilar frwii the domain and Y is ;I linguistic variable. Ll nco nd i t i onal state men t s are always app I ied within the inode 1 and de pen d ing on h o w they ;ire applied. serve either to restrict the output space or to defiiie ;t clefnult solution space. We interpret an unconditional fuz7y proposition ;is .Y is the minimum subset o f Y m.hen the output fuzzy set X i \ empty. then X is restrictcd to 1’. othercvisc. for the clotiiain o f 1; X becomes the nziri(X, k’). The solutioii fuzzy space is updated by taking the intersection of the solution set ancl the target fuzzy set. If ii model contains ;i mixture of conditional o r unconditional propositioiis, then the order o f execution becomes important. Unconditional propositions are gent.rally used to establish the defnult support set for a model. If none o f the conditional rules executes, than ii b-alue for the solution krariable is deterrnincd from the space bounded by the cinconditionals. For this reason they must he executed bcfore any o f the conditionals. The effect o f ei.:iluating ii fuzz>.p r o l ~ ) sition is ;I clegree o r grade o f iiieiiibership deric,ecl from the trmsfer functioti

where .r is a scalar from the domain and Y is ;I linguistic \variable. This is the essence of an approximate statement. The derived truth membership \Value establishes a compatibility between .r and the generated fuzzy space. This truth value is used i n the correlation and implication transfer functions to create or update fuzzy solution space. The final solution fuzz), space is created by aggregating the collection of correlated fuzzy proposition. 7.6.5

Fuzzy Implication

The rrzoriotoriic method is a basic fuzzy implication technique for linking the truth of two general fuzzy regions. When two fuzzy regions are related through a simple proportional implication function. if.\- is Y then :is 1.1'

functionally represented by the transfer function

then under a restricted set of circumstances, a fuzzy reasoning sq'stem can de\relop an expected \value without going through composition and decomposition. The value of the output is estimated directly from a corresponding truth membership grade i n the antecedent fuzzy regions. While the antecedent fuzzy expression might be complex, the solution is not produced by any formal method of defiizzification. but by a direct slicing of the consequent, fuzzy set at the antecedent's truth level. Monotonic reasoning acts as a proportional correlating function between two general fuzzy regions. The important restriction on monotonic reasoning is its requirement that the output for the model be a single fuzzy variable controlled by a single fuzzy rule (with an arbitrary complex predicate 1. The multiplication space generated by the general composition r-irlc..~ of' irzfi~r-rric-e is derived from the aggregated and correlated fuzzy spaces produced b), the interaction of many statements. In effect all the propositions are run in parallel to create an output space that contains information from all the propositions. Each conditional proposition whose evaluated predicate truth is abo\re the current a-cut threshold contributes to the shape of the output solution Lrariable's fuzzy representation. There are two principal methods of inference i n fiizzy systems: the min-max method and the fuzzy additive method. These methods differ in the way they update the solution variable's output fuzzy representation. For the riziti-1w.v irzjkr-rrzcr method the consequent fuzzy region is restricted to the minimum of the predicate truth. The output fuzzy region is updated by

2 IN

Chtrprer 7

taking the maximum of these minimized fuzzy sets. The consequent fuzzy s c t is modified before it is used to set each truth function element to the minimum of either the truth function or truth of the proposition’s predicate. The solution fuzzy set is updated by taking, for each truth function value, the maximum c ~ f either the truth value of the solution fuzzy set or the fuzzy set that was correlated to produce the minimum of consequent. These steps result in reducing the strength of the fuzzy set output to equal the maximum truth of the predicate and then, using this modified fuzzy region, applying it to the output by using the OR (union) operator. When all the propositions have been evaluated, the output contains a fuzzy set that reflects the contribution from each proposition. The fuzzy additive compositional inference method updates the soluticn variable’s fuzzy region in a slightly different manner. The consequent fuzzy region is still reduced by the maximum truth value of the predicate, but the output fuzzy region is updated by a different rule, the bounded-sum operatioi. Instead of taking the ma~(p,~)[.r,], p,,[y,]) at each point along the output fuzi.4. set, the truth membership functions are added. The addition is bounded by [ I , 01 so that the result of any addition cannot exceed the maximurn truth value o f a fuzzy set. The use of the fuzzy additive implication method can provide a better representation of the problem state than systems that rely solely on the min-max inference scheme. 7.6.6

Correlation Methods

The process of correlating the consequent with the truth of the predicate s t e m from the observation that the truth of the fuzzy action cannot be any greater than the truth of the proposition’s premise. There are two principal methods ot restricting the height of the consequent fuzzy set: correlation minimum and correlation product. The most common method of correlating the consequent with the premise truth truncates the consequent fuzzy region at the truth of the premise. This is called correlation minimum, since the fuzzy set is minimized 3y truncating i t at the maximum of the predicate’s truth. The cw-rlelcrtionn i i n i v i i ! u i mechanism usually creates a plateau since the top of the fuzzy region is sliced by the predicate truth value. This introduces a certain amount of informatim loss. If the truncated fuzzy set is multi-modal or otherwise irregular, the surfrice topology above the predicate truth level is discarded. The correlation method, however, is often preferred over the correlation product (which does preser c-e the shape of the fuzzy region) since it intuitively reduces the truth of the conxquent by the maximum truth of the predicate, involves less complex and faster arithmetic, and generates an aggregated output surface that is easier to defuzzify using the conventional techniques of composite moments (centroid) or composite maximum (center of maximum height). While correlation minimum is the most frequently used technique, the tor-

relation product offers an alternative and, in many ways, better method of achieving the correlation. With correlatioti product, the intermediate fuzzy region is scaled instead of truncated. The truth membership function is scaled using the truth of the predicate. This has the effect of shrinking the fuzzy region while sill retaining the original shape of the fuzzy set. The correlation product mechanism does not introduce plateaus into the output fuzzy region. although it does increase the irregularity of the fuzzy region and could affect the results obtained from composite moments or composite maximum defuzzification. This lack of explicit truncation has the consequence of generally reducing information loss. If the intermediate fuzzy set is multimodal, irregular, or bifurcated i n other ways this surface topology will be retained when the final fuzzy region is aggregated with the output variable’s under generation fuzzy set.

7.6.7 Aggregation The evaluation of the model proposition is handled through an aggregation process that produces the final fuzzy regions for each solution variable. This region is then decomposed using one of the defuzzification methods. Methods of Defuzzification

Using the general rule of fuzzy inference, the evaluation of a proposition produces one fuzzy set associated with each model solution variable. Defuzzification or decomposition involves finding a value that best represents the information contained in the fuzzy set. The defuzzification process yields the expected value of the variable for a particular execution of a fuzzy model. In fuzzy models, there are several methods of defuzzification that describe the ways we can derive an expected value for the final fuzzy state space. Defuzzification means dropping a “plumb line” to some point on the underlying domain. At the point where this line crosses the domain axis, the expected value of the fuzzy set is read. Underlying all the defuzzification functions is the process of finding the best place along the surface of the fuzzy set to drop this line. This generally means that defuzzification algorithms are a compromise with a tradeoff between the need to find a single point result and the loss of information such as a process entails. The two most frequently used defuzzification methods are composite moments (centroid) and composite maximum. The crtitrwicl or center or graLrity technique finds the balance point of the solution fuzzy region by calculating the weighted mean of the fuzzy region. Arithmetically, for fuzzy solution region A , this is formulated as (7.25)

where tl is the /'" domain value and u(t1) is the truth membership value for that domain point. A centroid or composite moment defuzzification finds a point representing the fuzzy set's center of grak4y. A iiiti.viiiiiiiii c~~c.onipositit,ri find.; the domain point with the maximum truth. There are three closely related kind\ of composite maximum techniques: the average maximurn, the center of maxiniurii, and the simple composite maximum. If this point is ambiguous (that i>. it lies along a plateau), then these methods employ a conflict resolution approac I such as averaging the values or finding the center of the plateau. Also there are other techniques for decomposing a fuzzy set into an expected value. The ti\'ei-ti,qe of' iiiii.u'iiiiiiii i ~ i l i r c )defuzzit'ication method finds the mean niaxiniuni value of the fuzzy region. If this is a single point, then this \Aue is returned; otherwise, the value of the plateau is calculated and returned. 7'110 ti\vi-iigtl of' the ~iori:ororegion is the same as taking the average of the support set for the output fuzzy region. The ,fiii- t i i i t l iitwr e ~ l ~ qof' e the siippo 't w t technique selects the value at the right fuzzy set edge and is of most w e when the output fuzzy region is structured as a single-edge plateau. Tlir c w i t t ' r ( ? ~ ' i ~ i t i . v ; i i i i itechnique, ii~.~ in a inultirnodal or multiplateau fuzzy region, finds tt t' highest plateau and then the next highest plateau. The midpoint betnmeen ttlc centers of these plateaus is selected.

CONCLUSION In this chapter, we offer the reader ii glimpse of the main set of tools from the emerging area of intelligent systems. While there are excellent books and oth:r forms of literature on the subject treated, the attempt of the coverage is to oftrr a brief summary of the ingredients of artificial neural networks (A"). exptart systems (ES). and fuzzy logic (FL) systems. The coverage is intended to provide the necessary background for the following t u u chapters. Once again, the reader is inirited to consult the list ot' references and the annotated glossary of terms in the back of' the textbook. ;I+ it source of further i n forma t i on.

Application of Artificial Intelligence to Angle Stability Studies

INTRODUCTION The computational requirements associated with dynamic security analysis (DSA) by conventional methods are two or three orders of magnitude more than the requirements for static security analysis. Therefore comprehensi\~eon-line DSA is infeasible in present power system control centers. The exploitation of novel techniques to solve the problem of DSA is essential for on-line implementation. The interest in application of artificial intelligence ( A I ) to dynamic security analysis has been increasing steadily. The following advantages are addressed through using these methods: I.

2. 3. 4.

5.

A decomposition of the on-line DSA problem into manageable subprob 1e ins. The need for an integrated environment for expert systems, neural networks and conventional programs working together. Identification of the subproblem for which the neural networks are most suitable. Feature variables of the power system and its dynamic security attributes to be used as the training inputs to the neural network. Procedure to select appropriate contingency and check security of system using expert system. 22 I

8.1

A N N APPLICATION IN TRANSIENT STAB1LlTY ASSESSMENT

The comprehensive on-line DSA is infeasible in present power system control centers. The exploitation of novel techniques to solve the problem of DSA is essential for on-line implementation. Neural network ( N N ) is one of the stateof-the-art methods used to perform dynamic security analysis in power systeins. The interest in N N methods has been increasing in \carious fields because they can work well where rule-based expert systems and conventional algorithms .ire inadequate. For example, quantitative rules based on subjective inputs can be better implemented as ;I neural network classifier which can be trained on precisely defined training examples. 8.1.1

Robust ANN-Based Transit Energy Function

Earlier in this book, the formulation of the dynamic stabilitjr problem for m u l t i power system was presented. As previously stated, the process by which o p r n tors and planners monitor the behavior of power systems, screen the imious random and scheduled contingencies that may occur, and recommend correct ive control action( s ) to maintain or regain system stability, is known as dynamic security assessment (DSA). A hybridized tool, that incorporates the beneficial aspects of a structured preserving energy function arid a robust artificial neriral network. has been implemented to solve the contingency screening problen I of DSA. It is more accurate than conventional transient energy function ( T 5F) methods. since the load buses will be modeled as nonlinear voltage dependent components instead of being absorbed into the admittance matrix. while m;iintaining the robust ANN properties of both speed and tlexibility. Developncent of the ANN-enhanced TEF tool involves the construction of two modules, corresponding to the trainingksting mode of one operation and the recall modt: of the other operation respectively. An implementation scheme for both modules of the integrated tool is provided here. Specific emphasis is placed on the training of the robust ANN, the handling of the "over-training" issue and the evaluation of network performa ice. Description ot Learning Process tor the Robust ANN-Based TEF Every neural system is subject to a learning process, through which a relationship is established between a set of known inputs and 11 known result. The learning algorithm for the robust ANN tool is shown in Fig. 8.1. The sunis o f the weighted inputs at each node of the hidden layer of the net were compi ted. The sigmoidal activation was then applied to the sum of each hidden node in order to represent a better representation of the effects of the input stimuli. The weighted sum of the hidden nodes was then computed as the initial output of

Atiglr Smbi1it.v Stirdies

0 Apply input stimuli X

Sum weighted inputs at hidden layer

A = C X WI

Apply sigmoidal activation hnction at hidden layer

H = 1 /(I +

Calculate the actual output

out =

c

HWH

Compute the global error for N training cases

E = ( 1 / 2 ~ ) C ( ~ a r g-0ut)2 et

Adjust the weights and thresholds in a manner that minimizes E

E < Tolerance?

Figure 8.1

Learning algorithm for robust ANN tool.

the ANN. The error for each case was computed as the sum of the squares of the differences between the desired and the actual outputs. For each interation, the global error was computed as the average of the sum of the errors for the cases evaluated.

The mininiization routine, shown in Fig. 8.2, was then used to determitie the gradient of the unknowns (weights and thresholds) with respect to the global error. These \Aues were then used to update unknowns and the input stimuli u'ere prescnted to the network over and over again. until the tolerance l i m i t u'as satisfied. The final values ot' the weights and thresholds were then used by t'ie recall module of the ANN enhanced energy function (TEF) tool to map t ie relationship between components of the energy of the system and the stabililqr/ instability of the system. The data generated by the TEF program was then presented to the roblist ANN, u n t i l the specified criteria \+riis satisfied. Approximately I000 cases 1t.t re ewluated. \ f r i t h close to 60% of those dedicated to training the ANN tool. The reiis011 for the \.List number of training cases was t o ensure that ii dikrerse repi-esentation of the sqsteni, under \.arious conditions. u'as prokrided to the ANN. Again. the learning algorithm utilized a conjugate gradient search procediir-t. t o compute the minimum value o f the error function and thus optimize the interconnection weights and threshold values. The initial ureights and the thrt':,hold values ~ ' e r tset ' to random \,alues between -03 and +O.S. The goal of any A N N application is to pro\side an accurate response to ii given stimulus i n ii timely and efficient t1i;inner. Therefore, simple netnrork architecture is utilixd :is a base case and more complex topologies \+!eree\ aluated i n turn. The accuracy of the training session was defined by a user-spccificd tolerunce o f 10.'. Once the conLwgence criteria had been realized the retwork uas tested using data not presented during training. The results o f h e testing phase demonstrated the ability of the A N N to generalize. An additional routine was included to prek'ent the network from "o\ ertraining" o r becoming too specific to the training set. This \+';isaccomplislied t y t.\~aluatingthe global error function with respect t o the test data (data not used for training) subject to the weights and thresholds at ei'erq' t\i.cntj,-t't'th iteration of the learning process. If the global error of the test data incre;id from one check point to the next then the training process \ v x stoppcd.

K c ~ c a l lModule For the recall mode of operation the TEF program and the robust A N N \\ere integrated into a unified program to compute the security state of the g i \ w sj'stern for ii selected contingency. The initial prefriult conditions ere cotiipt,tet! using ii generic Newton-Raphson load tlow method, as before. The clearing energy were calculated in the traditional niiinner to represent the condition:; of the pouw system at f r i u l t clearing time. These values u ~ then e applied to the Robust ANN, using the weights and thresholds obtained in the traininig session. t o determine the security state of the power system. A description of the ;I gorithiii for the recall riiodule o f the robust ANN-based TEF tool is presentet1 in Fig. 8 . 3 .

I

Begin minimization routine

+

I

Input length of vector, total # of cases, and initial data &

Define maximum # of iterations

I

Define the current error

I

+

I I

Compute the gradient of the error function

+

Save the current value of the gradient

Initialize the counter, set n=l

1 Search along a line defined by the gradient to find the minimum value of the error function for the current unknown vector, as the gradient is updated with respect to displacement

Yes

I

Global error <previous

Compute global error

Evaluate test data with current unknown

t N= 1

Update the gradient with a function of the training momentum

Stop; Return

Figure 8.2

Minimization roil t ine.

The results obtained by the training procedure were presented to the robust ANN as training data, with the clearing energy functioning as the input stimuli and the stability assessment computed by the TEF method functioning as the desired output value. The input stimuli were weighted and sunimed at the hidden layer of the net. The sigmoidal activation function "as then applied to each

.

226

7 Begin

Step I

Step 2

Compute steady state condition

Step 3

Compute pre-fault system conditions w . t . CO1 reference frame

Step 4

Compute components of energy at fault clearing time

Step 5

4

Provide stimuli (components of clearing energy) to Robust ANN

Step 6

Apply weights & thresholds determined via training session

Step 7

4 Compute stability index of power system

step 8

S1ep 9

Figure 8.3 Recall Module of the robust ANN-based TEF tool.

hidden node, and the resulting values at each hidden node were weighted ard summed as the stability index of the system. A block diagram of the recall module 3 of the robust ANN-based TEF tool is shown in Fig. 8.3. Steps 1-4 of Fig. 8.3 follow the procedure for computation of the clearir g energy. In step 5 the six components of the clearing energy were applied to the input layer of the robust ANN as shown in Fig. 8.4. The inputs were then multiplied by the interconnection weights computed during and summed at the hidden layer. The sigmoidal activation function was then applied to summation at each hidden node, subject to the threshold value determined by the training session. The final values of each hidden node were then weighted and summt:d to compute the stability index of the power system for the given contingency

Angle Sttrhi1it.v Stiidic).,

22 7

Figure 8.4 Robust ANN-based TEF network architecture.

The number of iterations required for training and the percentage of correct/ incorrect classifications were the contributing factors i n the determination of the optimal network architecture and significant input features for the solution of the robust ANN-based TEF. 8.1.2

Results

The ANN-enhanced TEF program defined in Sec. 8.2.2 was tested on the IEEE 39-bus I 0-generator power system. As previously stated, simulation by the ANN-enhanced TEF program involved two stages of work. In the first stage simulation studies were conducted on the TEF program to collect data defining the effects of close to 1000 different contingencies on the IEEE 39-bus test system. A robust learning algorithm was then used to synthesize the relationship between the faulted systems, caused by the various contingencies. and the postfault system. by optimizing the interconnection weights and thresholds i n the network. In the second stage the robust ANN was integrated into the TEF tool to alleviate the computational burden caused by the deterinination of the critical energy. In this section the ability of a robust ANN-based TEF tool to sol\re the DSA problem for electric power systems is demonstrated. The results of studies on the IEEE 39-bus test system using robust ANN-based TEF tool, "ere Lralidated via comparison to results from an industrial grade TEF program and conventional time domain simulation studies. The comparison of results is preceded by a description of the IEEE 39-bus system.

The analysis of the results presented here clearly demon\trates the abilit) of the ANN-enhanced scheme to significantly decrease the time necessary f o . dynamic security assessment. Input Data and System Diagrcm tor the IEEE 39-Bus Power System In order to verify the accuracy of the TEF tool, the results of case studies 011 the IEEE 39-bus system using the robust ANN-based TEF were compared to results generated by an industrial grade program on the same system. The pouretsystem generation and load data for the IEEE 39-bus network are presented in Tables 8. I and 8.2. A schematic diagram of the 39-bus power system is prokrideci i n Fig. 8.5. Con7pDarison ot RoOust A NN-Based TEF to Classicnl TEF The robu\t ANN-based TEF procedure &as used to \ o l ~ ethe contingerick w-eening problem for DSA. Validation of the security a\w\mient determine J by the robwt ANN-bawd TEF & a s achieved through compariwn to the rewlt\ generated u\ing the indu\try grade TEF3.0 program, del eloped by Ontario H j -

Table 8.1

I 2 3 4 5 6 7 8 9 I0

39-Bus S p t m Load Data

3 4 7 8

12 IS 16

I8 20

21

II 12 13 I4

23 24

1s

27 '8 29 31 39

16 17 I8 I9

'5 26

322.00 5 ()O.oo 2 33.80 522.00 8.50 320.00 329.40 I 58.00 68 0 .oo 2 74.00 247.50 308.60 '44.00

139.00 28 1 .oo 206.00 283.30 9.20 1 lO4.00

2.40 I84.00 84.00 176.00 88.00 153.00 32.30 30.00 IO3.00 1 15.00 84.60 -92.20 47.20 17.00 75.50 27.60 26.90 4.60 250 .oo

Table 8.2 39-Bus System Generator Data Gen no.

Bus no.

p, (MW)

IV I (Volt)

I 2 3

39 31

IOOO.00 N/A * 650.00 632 .oo 5 08 .oo 650.00 5 60.00 5 40.00 830.00 2 5 0.00

I .0300 0.9820 0.983 1 0.9972 I .o 123 1.0493 1.0635 I .0278 1.0205 1.0475

32 33 34 35 36 37 38 30

1 5 6

7 8 9 I0

I

J

I

26

t

1

28

29

38

2

s7 18

I 8

I 17

a4 8

I

La

I

14

I’ 21 56

6

12 10

+a

10

Figure 8.5 Diagram o f 39-bus power system.

dro under EPRI project RP2206- 1, and classical time domain simulation studies using the trapezoid a1 n u merica 1 i n t e gra t ion tec hn i q lie. A CO mpara t i v e an a 1y :; i s of the results of simulation tests on the IEEE 39-bus system with respect to accuracy is presented here. A three-phase fault occurring at generator 30, was used as a basis for cornparsion. The fault clearing time and assessment for each case are listed below in Table 8.3. Evaluation of the results in Table 8.3 clearly demonstrate that the security assessments proc,ided by the robust ANN-based TEF tool are generally in good agreement with the security assessiiients provided by the results of con\mtiotial energy function and time domain simulation methods. Testing ot Robust ANN-Based TEF on IEEE 39-Bus System

In order to generate enough trainingksting data for the robust ANN the effe:ts of okw 1000 contingencies were tested on the IEEE 39-bus system and ecduated using the TEF method. Close to 60% of these cases were randomly seleced for training with the remaining 40+% reserved for testing. The contingencies represented three-phase faults ;it the terminal of each generator in the system. with fnult clearing times Igarying between .000I seconds and 0.5 seconds. Each input file contained the energy components at clearing time and the desired fi ial output for each case. In complex pattern classification problems, such as this one, the success 01' the training session is highly dependent on the initial weights and thresholds and the topology of the network. For all of the studies conducted the weights and thresholds were initialized at random values between -03 and +0.5. Since there is no general rule for the determination o f the optimal number of node:' in the hidden layer many topologies were evaluated. Table 8.3 Comparison of Results for 3 9 Fault at Bus 3 0 Cr i t I c aI clearing tinie, t, 0.1 0. I2

0. I4 0.16 0.18

0.20 0.22 0.24

TEF 3.0 AW

TEF 3.0 a\\e\\ment

Robust ANN-bad TEF A W

Ro bu \ t ANN-ba\ed TEF ;i\\e\\iiient

5.9047 5.3756 4.7045 3.8742 2.8703 1.683I 0.1702 - I .2373

Stable Stable Stable Stable Stable Stable Stable U 11st able

0.102538 0.102539 0.102540 0.102542 0. I02543 0.102543 0 .I02544 0.867565

Stable Stable Stable Stable Stable Stable Stable U 11 stable

Table 8.4 Description of Test Cases

Case

No. of variables in input vector

A

6

B C

5

D

5

E

5

F

5

G

5

Description of test case Input vector contains all six variables Kinetic energy component excluded from the input vector Component of potential energy w.r.t. position excluded from the input vector Component of potential energy consumed in the real pober load excluded from the input vector Component of potential energy conwned in the reactive power load excluded from the input vector Component of potential energy consumed in the generator reactance excluded from the input i'ector Component of potential energy consumed in the tranmission line excluded from the input vector

5

Selection of Relevant Features of Robust A NN-Based TEF

In all, twenty-nine different network architectures were trained and tested to determine the optimal topology for the ANN-enhanced TEF tool. In order to determine which of the six components of the energy of the faulted system featured significantly in the assessment of system security, each of the variables was alternately deleted from the traininghesting set. The studies are defined as cases A-G for simplicity (see Table 8.4). The training performance (number of presentations of input data) and accuracy (testing error percentage) were used to evaluate the studies. A decrease in the number of presentations for training coupled with similar or improved classification of the test data would give a good indication that ii certain component did not feature prominently in the learning process, and thus in the assessment of security. The traininghesting results (Tables 8.5-8.1 1 ) clearly showed that the netTable 8.5 Case A Architecture

No. of presentations

6-2- 1 6-3- I 6-4-1 6-5- 1

504 18096 252 5014

Testing error c/c

0.35 5.86 19.14 4.95

Table 8.6

CW B

5-7- I 5-3- I 5-4- 1 5-5- 1

387 600()O 2464 I 40669

0.68 0.68 3.83 -0

~'orkw s ith tLi'o neurons i n the hidden layer generally converged with the fe\vcst number o f presentations. In niost cases. testing accuracy uras significantlq, ; i f tectecl bj, the exclusion o f any one o f the input \winbles. Ilourecw. u,hen the kinetic energy component was excluded from the training set (Case B [ Ta1)le 8.61) there it'iis ii clear inipro\wiient in the speed of the network. while testing iiccur;icy u';is maintained at close to 100%.Thus, the optimal network architccture l i the ~ IEEE 39-bus system \\';is the configuration \irith the kinetic eneIg4' coniponerit deleted from the training set and only tn'o neurons in the hidden I aq'er . Therefore. it ~viisestablished that the kinetic energy component "as not ;I niiijor contributor to the ;issessnient of the dynamic stabilitj of the 39-bus S : ~ ' S tom. Lkrhen using the robust ANN-based TEF tool. E\Auation of the results of the clearing energy module on a 3-bus 2-generator system. as well as the -Wbus I O-generator IEEE test system provided sonie interesting obser\ations. I n the 2-generator system the kinetic energy term "as one o f three components that dominated the SPEF at f i i u l t clearing time. For ;I 3-phase fault, lasting for ii duration of 0.2 seconds. on the first machine of the system the magnitude of' the kinetic etiergy term comprised close to 2XCd o f the magnitude of the total energy. I n comparison, for ;I 3-phase fault lasting 0.2 seconds on the largest machine in the 39-bus system the magnitude o f the kinetic energy componcnt of the SF'EF at fault clearing time o n l y constituted 0.002% of the magnitude of the total clearing energy term (Table 8.12). This inay be partially due to the Iact that the kinetic energy component is computed a s ii function of the ratio o f the

Table 8.7 Cilse C ,Arc-hitecturc

No. o f prcwntations

Testing error '4

5-2- I 5-3- I 5-4- I 5-5- I

754 3960 1359 136'15

17.79 19.14 9.91 -0

Table 8.8 Case D Architecture

No. of presentations

Testing ei-ror (1:

5-2- I 5-3- I

846 36247 3 I20 573

16.44 30.05 6.53 6.98

Architecture

No. o f presentations

Testing error (,l

5-2- 1 5-3- 1 5-3- I 5-5- 1

I069 I678 I0373 756

16.22 5.63 13.5I 19.82

Architecture

No. of presentations

T e d n g error C/r

5-2- 1 5-3- 1 5-4- 1 5-5- I

60000 60000

5-4- 1 5-5- I

Table 8.9 Case E

Table 8.10 CW F

A0000 588 10

3.15 3.38 5.86 -0

Table 8.11 Case G Architecture

N o . of presentations

Testing error r/c

5-2- 1 5-3- I 5-4- I 5-5- I

1304 60000 410 3 I33

23.77 20.05 12.16 0

Table 8.12 hliignitude of Components of TEF at Fault Clearing Time System

W,

W'J

3-bu\

8.72 E-2

2-generator 3 9-bu \ 10-generator

1.90 E-5 3 . S O E-5 9.54 E-2 2.38 E-?

0.2001

9.78 E-6 2.73 E-2

1.26 E-3 1.79 E-4

0.9338

6.35 E-2

moment of inertia of the critical generator, MA,to the moment of inertia of ihe entire \y\tem, M,. Therefore, for systems containing a greater number of generator\, the hinetic energy term will decrease in magnitude. The dominance of each component in the TEF term had a significant effxt on the \election of the feature variables for an assessment of system \tablit>. Therefore. a niore exten\i\e \tudy is needed, with detailed machine models L nd a ~arietyof te\t systenis, before a general statement can be made on the featlire \ ariables for DSA of electric power \ystenis by energy function technique\.

8.1.3

ANN-Based Critical Clearing Time Assessment

Critical clearing time (CCT) - t, is defined as maximal fault duration for which the system remains transiently stable. Mathematically, CCT is a complex function of prefault system conditions (operating point, topology, system pararneters), fault structure (type and location) and postfault conditions that themselves are dependent on the protective relaying policy employed. It would be highly desirable to define this relation analytically but the diversity of variables in~ d v e dmakes this task extremely complicated. I n practice, CCT can be obtained in one of two ways: ( 1 ) by trial .ind error analysis of system postdisturbance equations or (2) by integrating faull-on equations and checking the \!due of the Lyapunov or energy function unt I i t reaches it previously determined critical level V,,. I n either approach, a numerical integration process is involved. From the point of view of on-line impleri-enrations of CCT assessment, this presents a major difficulty. Application of ANNs is a promising alternative. The capabilities of neuralnets enable them to readily synthesize the complex mappings that transform the niinierous input attributes or features of a power system into the single valued space of CCTs. We examine the generalization capabilities of neural nets, fixusing on their ability to deal with a large range of operating conditions and changes in netvwrk topology. For CCT assessment, each synchronous generator is representec. by three features derived from the measurable parameters of the po\+,ersystem.

The first feature, A I , ,is the rotor angle relative to the inertial center of angle (COA) measured at the instant of fault initiation.

The second feature, A?,, represents the acceleration parameter of the synchronous generator during the fault. Because the mechanical output i n the period immediately following the disturbance remains unchanged. the machine fault dynamics may be described by:

where P,, is the output of the i'h generator during the fault, when it rernains almost constant. The third feature, A.,,. is proportional to the kinetic energy each generator stores during the fault. (8.3)

It is important to observe that all three attributes can be e\duated without solving any differential equations. They are expressed in terms of the operating parameters of the system prior to the fault and include the structure of the fault. Selected features are related to the fault on dynamics, however, and thus can provide a good representation of system behavior during the first transient swing. The example four-machine power system shown in Fig. 8.6 will be used to illustrate an ANN approach to CCT assessment. Three-phase short-circuit faults are simulated at line 2-3 near bus 3. The fault-clearing policy is to restore the prefault system topology. Two different system topologies are used to generate the training set patterns: topology 1 with all seven lines in seririce: topology 2 with the line 3-4 removed. Fifteen different loading conditions are selected for each topology, all with the loading level of the system in the range 0.6-2.0 relative to the nominal operating point. The power factors of the loads are maintained at their nominal values. The training set consists of 30 twelve-dimensional patterns (4 generators times 3 features) labeled with corresponding CT values. The latter are obtained by numerically integrating the posdisturbance system equations. The neural network used for this study is shown in Fig. 8.7. It consists of an input layer with twelve units and a single hidden layer with six units. After training, the mapping learned by the net is represented by the weights and

2.M

3

7

Figure 8.6 One-line diagram o f the four-machine pouer sy\tem.

Input Pattei

LJ Figure 8.7 Multi-lajwxl neural-net bawd a\wswient of critical clearing time.

thresholds; the least squares error was reduced to 2 x IO-' after SO10 presentations of the training set. The other parameters of the learning process \%.erethe learning rate r\ = 0.3 and the momentum 5 = 0.4. The performance of the neural network in new test cases. i.e., cases not seen during the training session, is illustrated in Table 8.13. Examples 1-7 correspond to power system topology 2, a topology used i n the learning session. Examples 8-20 correspond to a new system topology obtained by rertmring lines 2-5. The critical clearing times estimated by the neural netn-ork agree with the analytical values. Provided that a few different topologies appear i n the training session, the network is able to operate generally, handling new topologies as well as new operating conditions. This performance impro\.es o\'er con\mtional nonadaptive pattern recognition in which every new topology requires ii completely new set of discrimination rules.

Table 8.13 Compari\on of CCT5 Obtained M ith Nunierical Integration of Differential Equation\ (Actual CCT) and E\timated U ith a Neural Net (E\timated CCT) Ex ample 1

3

3 4

Load 1 ~ 3 ~ 1 Actual CCT /"( sec

0.65 0.85 0.9s

5

1.15 I .45

6 7 8 9

1.65 1.95 0.80 0.90

10

1 .oo

II 12 13 14

1.10 1.20 1.30

1s

1.40 1 .so

16 17 18 19 20

I .60 I .70 1.80 I .90 2.00

0.59 0.49 0.46 0.39 0.33 0.29 0.25 0.53 0.49 0.46 0.43 0.40 0.38 0.36 0.34 0.32 0.3 I 0.29 0.28 0.23

Estimated CCT sec

0.59 0.49 0.45

0.39 0.33 0.29 0.25 0.53 0.19 0.46 0.43 0.40 0.38 0.36 0.34 0.32 0.3 1 0.29 0.25 0.22

8.2 A KNOWLEDGE-BASED SYSTEM FOR DIRECT STAB ILlTY ANALYSIS Identifying the safe operating regimes requires analysis of a large number of combinations of outages and the associated derivation and n-ianipulation of th? massive data and related sensitivity information. It is an extremely difficult a d time consuming process when conventional computation methods are used. Such methods are implemented in the form of traditional computer programs uith no direct provision for exploiting the operator knowledge of the system and without utilizing proper reasoning techniques and inferring logics (expel-t systems) to process such “operator knowledge.” The use of expert systems i n TEF method applications would therefore enable the security assessment prctgrains to perform the required cognitive tasks. The expert system methodology appears to be uniquely suited to take aclvantage of the following important attributes of the TEF method: ( I ) The abilitjr to generate, organize, store. and retrieve a large amount of information required for ponver system transient analysis to be used for planning and operating ii stability-limited power network, and (2) to use this information for intellige~it decision-making to assess the dynamic security status of a given power network. 8.2.1

Expert System Development in DSA

In order to d e c ~ l o pa TEF-expert sytem, the following information is needed. ( a ) General formation and application concepts ( b ) TEF knowledge base structure and configuration ( c ) Application rules In dynamic security assessment. the knowledge base and applicaticm rules should include the contingency data and operating security information. Therefore, the rules can be constructed as follows: ( 1 ) Rules derived based on theoretical concepts ( 2 ) Rules and data constructed from detailed off-line computation ( 3 ) Rules established from operator knowledge and experience ( 4 ) Rules and data collected from previous on-line computed seen ari o s ( 5 ) Rules established from analyst knowledge and experience in the analysis of multiple TEF cases and comparisons with cornparal. le numerical integration cases. 8.2.2

Role of the Expert in DSA

At present. most of the murk in DSA is performed off-line. It requires an exten\ive amount of interaction froni the expert who use\ a time-domain sirnulatil)n

program to identify the response of the power system to a number of postulated disturbances. The role of the expert involves: (a) Significant amount of human interaction (b) Use of human intuition, judgement and expertise and cognitive skills repetitively (c) Routine actions by the engineer or system operator Select the Base Case

In the operations planning environment, the base system configuration is obtained from real-time operating conditions or by manual adjustments to a full system base case. The need for the study is identified by the system operator whenever the system configuration reaches a state for which operating limits are not prespecified by previous studies. The operator may also initiate a study before approving scheduled maintenance on major equipment. Of all the possible study cases, the engineer selects only a few as base cases worthy of detail study. The base case selected represents a credible worst case scenario during the period of interest. Selecting the Relevant Contingencies

For a base case obtained in the previous section, the engineer identifies appropriate credible outages which are potentially dangerous to the system based on past experience. and system configuration deviations from the full system and base case power flow solution. In general, the contingencies most commonly studied involve the following: Loss of a single generator Three-phase-to-ground fault cleared by the primary protection by opening the faulted line Single-phase-to-ground fault cleared by backup protection by delayed opening resulting in multiple equipment outages Three-phase-to-ground fault cleared by the primary protection by opening multiple transmission lines sharing the same transmission towers.

It is impossible to enumerate all possible outages. Therefore, only the most severe credible outages of important components of the system are considered. The dynamically important components in the system are identified by the explicit or implicit use of rules. Operator Actions

After examining the results for a number of selected case studies and contingencies the engineer determines the operating constraints for the system such that

they will be vaild for ;i number of operating conditions not explicitly studied. The system operators are required to operate the system within the sccuriij constraints. In the cases o f minor security Lriolations for which timely postcontingenc. correctiLre actions are a\xilable, the operator may choose to continue the oper.ition o f the system in its current condition. In the cases of other security t.io1.itions, the operator may choose to do the following: Reschedule generation Reschedule interchanges Arm o r disarm remedial actions such as generation rejection. load shedding. a 11d s j 'st e iii se para t i o i i . 8.2.3

Rule Base Structure

Bawd on the acquired hnowledge, i t i \ found that xolution oi the dynamic W C L I a\w\wietit problem fit\ naturally with ;I rule-bawd \y\tem combined M i t h procedure\. Therefor-e, the rule-based technique is applied to demonstrate t ie tea\ibilitj o f ;in expert \ J \tern approxh. I n [he proce\\ o f A n o ~ l e d g eba\e ceLelopiiient. the attempt u;i\ made to iii;iirit;iiri generalit4 o f rule+ U hile i i i i r i i i i i 1tng the \)\tern \peciftc feature\.

ritj

hfodules ot K u l t ) B ~ s eStructure There iire f i modules ~ ~ in the de\ eloped rule base. Each mociule contain< ii number o f rule\ performing ii yxcific task in the dynamic wcurity a\se\\iiicnt. For con\ enience, each module o f the rule haw \ + i l l be r e f e r i ~ ~toI a s a "tash. '

First Task: Monitor System Conditions AtialyLe s>.steiii operation conditions for g i \ w system status. Check i t I he changes in sj'steiii conditions since the last xsessiiierit are sigiit'icant enough t o u w r a n t ii new ;isscssiiieiit. It' there ;ire significant changes then DSA is necdccl. The decision processes iii\wl\d here ha\.e to relj, completely on the prccontingency steady-state condition of the pow'er sj'steiii and its relationships \+(it11 pre\,ioitsly riiialq~zedbase case(s). Second Task: Contingency Analysis and Selection Select contingencies rc'leLrant to the current system state. Rules in this task inclucle esiiiiiiiiirig the iiiipoi-tmce o f each comporient to the dynamic securitj o f the sj'stem m c l selecting the outage o f the inore important components for t tirther studj,. Check \\.hether [he current sq'steiii slate is secui.e ic,ith respect [ ( ) ;i g i vc' 11 coii t i nge tic y . Th i s t ;is li iii ay use i 1;ib 1e i ii form at ioil regard i ng t he b .ise c ;ise . gi \,en cont i ii ge tic y . and the t ran s ie 11t condition of' the sy st e m i m med i ii t e1 j ti, I I o\+'i 11 g l'au 1t -c I ear i ii g. :i\r:i

Third Task: Acceptability Judgement Using results of security and invulnerability assessment, rules i n this task can make a judgement on the overall acceptability of the operating conditions. Another function of rules in this task is to peform a direct judgement of acceptability based on qualitatikre assessment of the sysem operating point (generation. load, and transmission), the external condition (weather, etc.), and disturbance (outages. etc.), and reactions (line tripping, etc.). Rules for the direct acceptability judgement i n the first task provide a sufficient condition for acceptability. Fourth Task: Security and Invulnerability Assessment If the direct acceptability assessment task does not give a positive mslver. then the next task will be initiated. This task contains rules. which conipare security margins to the specified thresholds. The security margins are obtained bjf comparing the current operating value of a critical parameter such as line tlow to the maxi mal f l ow \,a1ue ii t which instab i 1 it y occ LIrs . I n \'U 1nerab i I i t y ;is se sstnc n t is also performed i n this task. At present, invulnerability refers to it "sensiti\'c" operating condition that injdves a severe drop i n security margin i n ternis of' the transient energy function (TEF) if a perturbation o f the operating condition occ LI rs . Fifth Task: Remedial Actions This is a collection of rules, which can be used to suggest change i n thc operating conditions in order to regain system security. If the security margin drops to a negative value (or below a specific threshold), then controls must be chosen to eliminate the abnormality. The selection of control is based on sensiti\,it>fof the control with respect to the security margin and the amount of a\,ailable controls. The controls necessary to fully correct the insufficiency of security margin are suggested. I n addition to the tasks mentioned above, i t is necessary to modit'). the securi t y knowledge base through adding significant new information about the contingency and the initial operating conditions to the existing security :irchi\re in the knowledge base. Modification occurs only if the present analysis requires ii change in the security region of the system as implied by the knoutledge already a\sailable i n the knowledge base. The structure of DSA F i i t h rule base is presented in Fig. 8.8.

Task 1: System State Monitor

This task consists of rules I .. I . I , and 1.2. It senres as a fast monitor of' the s ~ ~ s t ebase m state. This task can only give a "yes" annvcr or ;t "not-sure" ;inswer. If the answer is "not-sure," further analysis is needed, such ;is iinaIj,sis of Tasks 2 and 3.

No significant change c

r

No, analyze next outage t

System State Monitor (System condition change?)

-

Contingency Analysis and Selection (Is important outage?)

Security & Invulnerability Assessment

Remedial Actions

v

v

*

Modify security knowledge base

1

Figure 8.8 DSA structure uith rule base.

1.

IF the system S is in the base case. THEN S certainly satisfies the dynamic security requirement. Data: System base case data. Out p U t : S-d y n am i c security . I . 1 IF system condition has only a small change around the base cme. THEN system can maintain dynamic security.

Data: System base case data with small change. Output : S-dy namic security . 1.2 IF system condition has significant change. THEN system may not satisfy the dynamic security criterion. and further analysis is needed. Data: System base case data with big change. Output: System may have the problem of dynamic security.

Task 2: Contingency Analysis and Selection

This task also consists of rules I., 1 . 1 , and 1.2. This task is in\.ol\Ved if the system condition analysis in Task 1 gives a “not-sure” answer under the significant change of system condition, especially under the outage cases. It offers a fast analysis of the system contingency scenarios. Only the important contingency scenarios are selected for further dynamic security analysis.

I.

IF the outage scenario OC is not important. THEN OC will not be selected. Data: System data nd outage parameters. Output : OC-neglected. 1 . I IF the contingency performance index (PI) is less than the threshold (PI,) for a given outage scenario OC. THEN OC will not be selected. Data: system data, outage parameters, PI. and PIT. Output : OC-neglec ted. 1.2 IF the contingency performance index (PI) is greater than the threshold (PI,) for a given outage scenario OC. THEN OC will be selected for further dynamic security analysis. Data: system data, outage parameters. PI, and PI7. Output : OC-selected.

Task 3 : Acceptability judgement

This task can be considered as having 2 subtasks. One subtask consists of the rule numbered 2 and its subordinates (i.e., rules 2.1, 2.1.1 and 2.1.2, 2.2 and 2.2.1). It serves as a faster analyzer of the system conditions (e.g., weather. load trend, etc.), and tries to draw a conclusion about acceptability. This task can only give a “yes” answer or a “not-sure” answer. If the answer is “not-sure,” the other subtask (represented by rules 1 ., I . 1 , and I . 1 . 1 ) will be imroked. This task gives a “yesho” answer to the acceptable or unacceptable question depending on the output of Task 4.

I.

IF e ~ w oyperating condition in C satisfies the security and n v ii 1ne rab i 1ity constraints and policies, THEN C is acceptable. Data: security and vulnerability euluation results (satisf.:d- or-not 1, all cases in C. policy descriptions Output: C-acceptabili ty

I. I

2.

IF SF(C) = yes and CF-SF(C) is greater than a threshold for every operating condition in C. THEN the system satisfies the security and in\~ulnerabilityc( Inst rii i n t s. Data: SF(C), CF-SF(C) for all cases i n C. threshold. Output : sec 11r i t y , i n \.ii 1nera bi 1it y constraints sat i sfied or not . 1.2 IF every operating condition in C is secure and in\ulneral)le THEN SF(C)= yes. Data: "seciire" o r "\rulnerability" status of all cases in C. Output: SF(C). IF (Mode o f Operation M is NORMAL, and Disturbance D is mediiim o r large). o r ( M is High-Risk and Disturbance is Small). THEN C is acceptable. Data: Mode of Operation, Disturbance. Output : C accept abi I i t y .

2.1

IF System S is SUSCEPTIBLE and External condition E is THREATENING THEN the Mode o f Operation M is High-Risk. Othenvise, M is NORMAL. Data: S, E. Output: M. 2.1.1

2.1.2

IF LoaD (LD) is STEADY and Transmi\sion (TR) i \ NORMAL, or LoaD is PICK UP and GeneratioN (GN 14 LOW and TR i\ Normal, o r LoaD is STEADY and GeiieratioN (GN) is LOW and TR i \ WEAK and Generation Inertia (GI) i \ LARGE. THEN the System S i \ ROBUST. 0the r w i se, i t i s S U SC EPTI B LE. Data: LD. TR, GN, GI. output: s. IF WEather (WE) is OF CONCERN, o r WEather is OF NO CONCERN and INterconneciion ( I N ) ha\ INSUFFICIENT SUPPLY. THEN External Condition E i\ THREATENING. Data: WE. IN. Output: E.

2.2

IF the UPSET is MINOR and the REACTION is ADEQUATE. THEN the Disturbance is Small. Date: UPSET, REACTION. Output: D. IF the UPSET is SEVERE and the REACTION is ADEQUATE. THEN the Disturbance D is Medium. Data: UPSET, REACTION. Output: D. IF the UPSET is MINOR and the REACTION is INADEQUATE. THEN the Disturbance D is Medium. Data: UPSET, REACTION. Output: D. IF the UPSET is SEVERE and the REACTION is INADEQUATE. THEN the Disturbance D is Large. Date: UPSET. REACTION. 2.2. I IF the Corrective Action is LINE TRIPPING and the TIMING is not FAILED, or the Correctii-e Action is GENERATION REJECTION and TIMING is ON TIME. THEN the Reaction R is ADEQUATE. Otherwise, it is INADEQUATE. Data: Corrective Action, Timing. Output: R.

Task 4: Security and Vulnerability Assessment

Rules for Security Assessment This task is invoked if the system condition analysis in Task 3 gi\m :I "notsure" answer. As the name implies, this task consists of two parts. I n one the security is assessed based on the use of energy margin (or security margin). I n the other part. the sensitivity of the energy margin with respect to parameter bw-iations is considered, i.e., how vulnerable the system is to parameter changes. 1.

IF a family of regimes in a class have not crossed zero at a parameter \alue I. and the total probability of this family is higher than or equal to p. THEN the system is secure with probability p for this class of contingencies as long as the parameter is bounded by the value I . Data: probabilities of the given class of contingencies, I, p. frtmily of regimes. Output: secure w Prob p. 1 . 1 IF an operating regime crosses the parameter (horizontal) asi\ at

246

a value IMAX. THEN the system operating point is secure as long as the parameter is less than IMAX. Data: IMAX, zero-crossing of the given regime. Output: Operating point secure or not.

2.

IF the normalized energy margin (NEM) is greater than a threshold (zero). or the Strength SG is HIGH and Stress is LOW or MEDIUM, or the Strength SG is Medium and Stress is LOW. THEN the operating condition C is secure. Otherwise, it is insecure. Data: NEM. Threshold, SG, Stress. Output: C secure or not.

3. IF (Security Margin > Threshold). THEN (System is Secure). Otherwise, System is Insecure. Data: Security Margin, Threshold. Output: Secure (YESINO). 3.1

IF (Decision on updating Security Margin is YES). THEN (Call Procedure Update Security Margin). (see Task 5 for procedure). Data: Decision on updating Security Margin, Mismatches between base case and current generationdinterchanges, sensitivil y of each generationhnterchange to Security Margin. base case Scciirity Margin Output: (current) Security Margin.

3. I . 1

IF (current generations are NOT equal to base case generations) AND (current tie interchanges are NOT equal .o base case interchanges). THEN (Update = YES). Data: current (real-time) generation/interchange profile. base case generation/interchange profile. Output: Decision on updating Security Margin ( YES/NCl).

Rules for Vulnerability Assessment 1.

IF the degradation of energy margin (EEM) is larger than a threshold for any of the specified variations v, or the IMPRESSIONABILITY (IM) is HIGH, or the IM is LOW and UNSETTLENESS (US) is LARGE. THEN the operating condition C is vulnerable. Otherwise, it is i n l r u l nerable.

Angle Stcihilitj* Stiidirs

24 7

Data: DEM, Threshold, “Variations”, IM, US. Output : C-vul nerable-or-not . I . I IF Sensitivity of Energy Margin (SEM) is higher than a threshold. THEN IMPRESSIONABILITY (IM) is high. Otherwise, it is LOW. Data: SEM, Threshold. Output: IM. IF Delta Parameter Change (DPA) is larger than a threshold. THEN the UNSETTLENESS is LARGE. Otherwise, it is SHALL. Data: DPA, Threshold. output: us. Task 5: Remedia/-Actions

This Task is invoked only if the security assessment of Task 4 gives a negative answer. The objective here is to adjust the available controllers in such a way that the system becomes secure. 1.

IF (Corrected-Security-Margin < Threshold) AND (there are available control act ions). THEN (Select next control action in priority list) AND (Determine amount of threshold violation). Data: Corrected-Security-Margin, Threshold, available controls ordered in priority. Output: Highest priority control c, Amount of Security Margin Violation. 1.1

IF (control action c is available) AND (control action c has the highest priority). THEN (Determine the necessary amount of c for correction) AND (Calculate the Corrected-Security Margin). Data: availability of control c, the current control c selected from rule 2. Output: amount of control c, needed, corrected-security-margin. I .2 IF (control action c is at is limit). THEN (set status of c to UNavailable) AND (remove c from the priority list). Data: control c and its limit. Output: updated availability status of control c

1.3 IF (control action c is the current remedial control) AND (necessary amount of control c is not at its limit). THEN (no more remedial action is needed)

Data: control c. necessary amount of c. Ou t pu t : stop c hoos i rig re med i a I iict ions.

2.

IF (Security Margin < Threshold) AND (there are NO a\railable control ;ic t ion s. THEN (no remedial action is possible).

8.2.4 Summary I n this section fundamental concepts and basic requirements for ;i dynamic security ;issessiiieiit expert system halte been introduced tor a stability-limited po\s er net\\x)rk. The computerized procedure uses power system data pro\Vidcd bq, t ic T E F method ;is input data to the expert system. For ii g i c m contingency the T E F method prolrides energy margins and ciicrgy margin sensitiLrities to operating parameters. The energy margins pro\.i:Ic ;I measure o f the system security. The energy margin sensiti\,ities proifidt. ;I ~iie:isure of system \fulnerability t o changes i n operating paranieters. This data is suitable for pro\-iding the logic needed by an expert system for dynaniic security assessment. A fraiiie\sw-k for identifying and organizing the knowledge base recltii Y tiients for the \.artoils tasks incul\vd in dynarnic security ;issessiiieiit has hccn presented. A multilevel tree structure for the TEF-expert sqrstem is proposecl 'or the analq'siis associated u i t h decisions concerning the pouw system's securitj/ \xilnerability status. It is important to note that the procedure outlincd ahc 1i.c pi-ocrides ;i suitable franienwk in Lvhich to implcliicnt ii much needed dqwt-iic contingency classification and selection algorithm.

8.3

FUZZY-LOG IC-BASED POWER SYSTEM STAB1LlZATlON

De\relopment o f ii fuuy-logic-based power system stabilizer to maintain stal~ili t y and enhance closed-loop perfi)rriiance of ii power system is described in this chapter. Simulation studies on ;i single-machine inf'inite-bus system and o i ;I 111ti I t i-machi lie pouvr s)'s tern model show itcry sat i s factory performance. The fii~zq'-logic-based PSS (FLPSS) h:is been implemented on ;I lo\i--co\t micro controller and tested in the laboratory on a physical model ot' ;I single11i;ich i tie i ni t'i11 it e bus s i r s t em . Experi men t ~ i ltest s and resu I t s ;ire a1so desc ri bcd . 8.3.1

FLC Structure

In concentional control. the m o u n t of control i \ determined in relntion t o ;I number 01' data inputs using a \et of equation\ to expre\\ the entire control proces*\.E\pre*\\ing huriian experience i n the torni ol' ;I mathematicril t'ormula I \

a very difficult task, if not an impossible one. Fuzzy logic pro\lides a simple tool to interpret this experience into reality. Fuzzy logic controllers are rule-based controllers. The structure of the FLC resembles that of a knowledge-based controller except that the FLC utilizes the principles of fuzzy set theory in its data representation and its logic. The basic configuration of the FLC can be represented simply in four parts. as sho\s.n i n Fig. 8.9: Fitz,ifi(utioii riiotlirlt), the functions of which are. first. to read.

me^'w e . and scale the control variable (e.g., speed, acceleration) and. second, to transform the measured numerical values to the corresponding linguistic (fuzzy) \wiables with appropriate membership \Aues. Kiio,i,lcdgc~htrsc~,which includes the definitions of the fuzzy niembersliip functions defined for each control variable and the necessary rules that specify the control goals using linguistic \.ariables. Ii;fi.i-cvic.c) i i i d i m i s i i i . which is the kernel of the FLC. It should be capable of simulating human decision making and influencing the control xtions based on fuzzy logic. Dc.fir,7,7;~i’c’titioiii ~ i o d i r l c ,which converts the inferred decision from the linguistic \,ariables back to numerical values.

8.3.2

Fuzzy logic Controller (FLC) Design

The design process of an FLC may be split into the five steps described belo\ir. Selection o i Control Variables

The selection of control variables (controller inputs and outputs) depends on the nature of the controlled system and the desired output. It is more coninion to

I

Knowledge Base

Controller input

Crisp

b Fuzzifier

+

Interface

Defuzzifier -+

Figure 8.9 Schematic diagram of the FLC building blocks.

Controller output -b

use :he output error ( e ) and the rate o i derivative of the output (e’) as controller inputs. Some investigators have also proposed the use of error and the integral of error as an input to the FLC. Mendx~shipFunction Detinition

Eacn of the FLC input signals and output signals. fuzzy variables ( X , = ( e . e ‘ . has the real line R iis the universe of discourse. In practice. the universe cf. discourse is restricted to a comparatively sinall interval (Xmin,, Xmax,l . The universe of discourse of each fuzzy variable can be quantized into a number c f overlapping fuzzy sets (linguistic variables). The number of fuzzy sets for each fuzzy b.ariable varies according to the application. A common and reasonable nutnber is an odd number (3. 5. 7 . . .). Increasing the number of fuzzy S C I S results in ;I corresponding increase in the number of rules. A niernbership function is assigned to each fuzzy set. The membership functions map the crisp values in!o fuzzy values. A set of membership functior s defined for seven linguistic variables N B . N M . KS, 2. PS. PM. and PB, which stand for Negative Big, Negative Medium, Negative Small. Zcro, PositiLc Small. Positive Medium. and Positive Big. respcctively. Membership functiom can be of a variety of shapes, thc most unusual being triangular, ii-apezoidal. or a bell shape. The triangular shape is used for the controller. For simplicity, it is assumed that the membership functions are symmetricail :md each one overlaps with the adjacent functions by 50%. In practice. t t t membership functions are normalized in the interval [-L. L ] ,which is symmetrical around zero. Thus. control signal iimpli tudes (Iuzzy variables) are expressc d in !ernis of controller parameters (gains). U)).

Rule Creation and Inference

In general. fuzzy systems. iis functim estimators. map an input fuzzy set to iln output fuzzy set S: I’“ - l”. Fuzzy rules are the relations between the fuzzy sets. They usually are in the form “if A. then B,” where A is the rule antecedent a!!d B is the rule consequence. Each rule defines a fuzzy patch in the Cartesian product A x B (system state space). The antecedents of each fuzzy rule descri!w a fuzzy input region in the state space. This enables one to effectively quanti.w continuous state space so that it can cover a finite number of these regions. In terms of associative memory definition (FAM), each rule represents an association (Al: Bl). Thc structure proposed firing all the rules at the same time (analogous to neural networks). This enables easier, faster. and very large scale intcgrated (VLSI) analogue and digital designs. A fuzzy system using t v v f o antecedents and one consequence (A,, B,. C,) is shown in Fig. 8.10. The association (Al9 B,. C,) or the rule of A;, and B, then C, maps inputs 4.

25 I FAMrule 1

Figure 8.10 FAM system architecture for two fuzzy antecedent and one consequence.

B, to C’, a partially activated version of C. The corresponding output fuzzy set C combines the partially activated sets C”, . . . , C””,that is,

Fuzzy Inference The well-known inference mechanisms in fuzzy logic are the correlation-minimum encoding and the correlation-product encoding. Consider fuzzy sets A and B to be fuzzy subsets of X , Y. The geometric set-as points interpretation of finite fuzzy sets A and B as points in unit cubes allows the representation of the sets of vectors. Thus, A and B can be represented by numerical fit vectors A = ( ( I , . . . . , q,)and B = ( h , ,. . . , b,,,),\there a l = rnA (x,) arid b, = i l l 8 ( U , ) . Let the relation between A and B be governed by the FAM rule ( A , B ) . Using the definitions. the correlation-minimum encoding and the correlation-product encoding can be expressed follows. 1.

The correlation minimum encoding is based on the fuzzy outer product notation. Expressed in matrices, the pairwise multiplication is replaced by the pairwise minima and the column sums with the column maxima. This max-min composition relation is denoted by the composition operator 0. Thus the fuzzy outer product of the fit row vectors A and B , which forms the FAM matrix M , can be given by

252 M=A’OH

Where

and the cap operator indicates the pairwise minima. 2. The correlation-product encoding use\ the \tandard mathematical o product of the fit vector\ A and B to form the FAM matrix M : A

urhere

Correlation-miriitnum encoding produces a matrix of clipped B sets, while corr:lation-product encoding produces ii matrix of scaled R sets. Det~izziticationStrn tegy

Defuuification is a process of converting the FLC inferred control actions from f i w y \values to crisp values. This process depends on the output f i 1 7 ~ y\t-t. Mhich is generated from the fired rules. The output furLy \et I \ formed bq either ;I corre 1at i o11-mini mU in encoding o r the corre 1ati on- produc t e ncod i n g . 8.3.3

Fuzzy Rules

Fuzzy rules play a mujor role in the FLCs and have been increstigated extensicrely. However, rules usually can be generated using knoufledge and operati ig experience with the system or through the understanding of system dynaiiiic?. In most cases, fuzzy rules map two input fuzzy variables, for example. the error e and the derivative of error 6’ * , into one output fuzzy cw-iable the control signal u. In such a system each fuzzy Lwiable can be easily quantized t c ;I number o f fuzzy sets. 8.3.4

FLC Parameter Tuning

Parameter tuning for the FLC plays an important role in achieving the controllt‘r goals. Prectious experience with the controlled system is helpful in selecting he initial c dues o f the FLC parameters. If sufficient information is not a\.aila > I t . about the controlled system. the selection of witable FLC parameters can Iw1

come a tedious trial-and-error process. Some efforts have been reported i n the literature to automate the tuning of the FLC parameters at the design stage to get an optimal or near optimal system performance. Another algorithm to tune the FLC parameters off-line is proposed i n this section. The objective of the proposed parameter tuning algorithm is to change the controller gains in an organized manner to achieve desired system response. The tuning algorithm tries to minimize three system performance indices (PIS). These indices are the system overshoot (OS) and the performance indices J , , J:, are gi\-en as (8.8)

where r is the system reference, J is the system output, P is the system error, and t is the time. An FLC with two inputs and a single output has two input parameters K , . K,,, ’ and one output parameter K,,. The three parameters Kc, K:, and K,,, are tuned using the guided search algorithm. The algorithm changes the three parameters in overlap loops, simulates the system with the new parameters, and calculates the performance index. It also detects if one of the parameters degrades the performance indices or leads to instability. I n this case i t stops incrementing this parameter. If the desired performance indices are achieved, the search stops. Otherwise, i t continues ()\!er the specified search range of the FLC parameters. In case of a complete lack of information about the parameters. the search for the best parameters may require a large number of iterations in searching for a proper minimum. Using some practical information about signal l e ~ ~ lits , is easy to set an operating range to the FLC parameters. 8.3.5

Automatic Rule Generation

I n some cases the dynamic behavior of the controlled system is unpredictablc and difficult to understand. This situation imposes the need to automate the rule generation process. A lot of effort has been devoted to achieve this goal. I n some cases an upper hierarchical set of rules has been chosen to generate the controller rules. This technique requires some understanding of the system dynamics to build these types of supervisory rules. In some other cases, the rules were generated using an artificial neural network. The generation of rules is achieved by training the neural net\vorh

using sampled data sets. Although effective. this method requires sonis new iil network background and long training time. In this section two effective automatic rule generation ( ARG) methods arc proposed. These methods are very similar to the neural network technique. Th,:), also use sampled data ensernble but employ ii fuzzy system to generate the rules instead o f a neural network. The rules can be obtained easily from thc kno\r-n desired perforrnance (input-output data pairs) of the system when contro1l:d by another well-designed controller. for example. ;in adaptive controller 01 ;i proportional-integral-derivative (PID). I n the case o f a two-input. one-output FLC. the controller requires t\r o f u z ~ y\*ariables. and using ;I fuzzy mapping function (fuzzy rules). i t gencraic.3 one-output t u ~ z ycontrol ac*tion.Instead. the ARG is ;I re\erse process. I t ii?cs the t h r w f u ~ z y\.ariables (c. 0'. 1 1 ) and generates the miipping function ( t i i i ~ y rules). ARC' Using Highest Mdtch Method

The highest niatch ( H M ) algorithm properly tuzzifies the three input variabIc.5 (c. 1 1 ) . Each fuzzy \-ariable matches in two tuzzy subsets. The algoritliiii considers only the subsets with highest match (highest membership value). Tt~us i t c iiiatches i n both X and PS with membership values 0.75 and 0.25. resplxtively. it will consider that o matches only i n the 2 subset. Similarly the 0' ;.lid 1 1 subsets will be PS and N S . respectively. Thus tor this sample data. the r.ilc hii\*ingthe antecedents o = Z and Y' = PS will be assigned to ;r consequence I ' = NS: that is. the generated rule will be: (1'.

This process is repeated for the whole set of saiiipled data ensemble. At the end of the abo\,c-mcntioned process. the saine rule may be assigned to differciit coiiscquences. To rcsolvt. this problem. the final rule consequence is generated by \\,cighing all thc consequences ;issigned to this rule by thcir number ot o c ~ u r re nce s. A RG Using FLIZZ)' 1 1 1 i ~ ~ r ~ ~Revcrse t1ce Engineering

The irlgorithni tiientioned in Sec. 5.6. I focuses the inforniation derived from the input-output data set on ;I single rule (the highest niatch) and ignores the e f t c t on the other fired rules. This may lead to ;I considerable number of unassig led (empty) rulcs. Thc f u z ~ yinference reverse cngineering (FIRE) technique i i \ ~ ~ i c I s this drawback considering the aggregated effect o f all fired rules.

2.55

Aiigle Stability Stirdirs

8.3.6

Fuzzy-Logic-Based Power System Stabilizer

A FLPSS with a set of FAM rules and continuous membership functions as discussed earlier has been designed. An organized method is followed to generate the rule base and to tune the parameters of the FLPSS. The performance of the FLPSS is first studied using a simulation model of a synchronous machine connected to an infinite bus. The FLPSS has been tested on the power system under various fault and load disturbances to ensure its effectiveness, robustness, and reliability as a power system stabilizer. Single-Machine Power System Model

A power system model consisting of a synchronous machine connected to a constant voltage bus through a double-circuit transmission line is used in the simulation studies. A schematic diagram of the model is shown in Fig. 8.1 1 . The system is represented by a ninth-order nonlinear model including the governor and the automatic voltage regulator (AVR) exciter. The state equations representing the power system and the synchronous machine, governor. and AVR model parameters are given in the referenced literature. The solution of the model differential equations was obtained using a fourth-order Runge-Kutta method with a simulation time step of 1 ms.

I

Governor Transmission Lines

Figure 8.11 A schematic diagram of a synchronous machine model with stabilizers.

The control signal generated by the PSS is injected a s a supplementary \tabiliing signal to tile the A V R siimming point. The sy\teni model has bee 1 designed to support both the CPSS and the FLPSS m d can be ea\ily extendel to support other types of PSS. FLIZZ,V Logic Power System Stnbilizer Llsing the description given in Sec. 8.4.3 for FLC design, a power system stablizer based o n the FLC algorithm has been developed. Since the goal of this application is t o stabilize and itnpro\fe the damping of the synchronous machitic:. speed de\iition Ao and actiLre power de\riation A P , , ha\re been selected as thc controller inputs. The controller output is then injected i n t o the AVR siiiiiming poi nt . This configuration implies that the FLC has twfo input parameters. K,,, anti K,,, atid one output parameter, K,,, iis seen in Fig. 8.12. The selection o f thew parameters is usually subjectiire and requires p r e \ k i s knowledge of the fuzi>control \cu-iubles (input and output signals). Also pre\,ious experience of' tlic controlled system dyn~iniicsis corninonly iised In the creation of the fuzzy control riilt's. Ho~+re\rer,i i n organized approach ;is described in the next section hiis been adopted for the generation o f rules and tuning of parameters for the FLPS 5. FL PSS/RLIIc) Generclt io n 'I n d PJ rcimeter Tuning

U\ing the FIRE A R C inethod and ;I sampled data \et generated by using the CPSS, ii proper \et o f rule\ was obtained. The rule\ used in all the tollo~titig

Fuzzy mapping (rules)

is

Figure 8.12

Schematic diagram of the FLPSS.

25 7 Active Power

r NB

1 Active Power

NM

NS

Z

PS

PM

PB

NB NB NM NM NS Z PS

NB NM NS NS Z PS PM

NB NM NS Z PS PM PB

NM NS Z PS PS PM PB

NS Z PS PM PM PB PB

Z PS PM PM PB PB PB

deviation

NM

I PM I PB Figure 8.13

I NS Iz

FLPSS rules generated by fire algorithm.

experiments are shown in Fig. 8.13. The correlation-product inference niechanism is used to generate the output fuzzy set for the FLPSS. The defuzzification process is based on the center-of-gravity method. Once the proper rules are obtained, the proper parameter tuning should be d in order to achiekre good performance. Tuning of FLPSS parameters can be a tedious trial-and-error process if not enough information is a\railiible about the range of controller ~w-iablesand how they change mrith different disturbances. The objective of the off-line tuning algorithm is to determine the controller parameters that provide the desired system response. The tuning algorithm tries to minimize three system PIS by \.aryiiig the FLPSS parameters. In the present case, the output parameter is set t o gi1.e the maximum allou~blecontrol action, while the other two parameters K,,,, K,,. \%'ere tuned using the guided search algorithm. The selected values of the FLPSS parameters are those that minimize the use of the guided blind search. Once the FLPSS had been tuned, the parameters were kept unchanged throughout subsequent studies. The tuned parameters of the FLPSS are K,,,= 100

K,, = 0.2 1

K,, = 0.1

CONCLUSIONS Unlike the classical design approach, which requires a deep understmding of the system, exact mathematical models, and precise numerical values. ;i basic feature of the fuzzy logic controller is that a process c;in be controlled urithout the knowledge of its underlying dynamics. The control strategy learned ttiro~igh experience can be expressed by a set of rules that describe the behaj-ior of the

2-58

Chccprer- Y

controller using linguistic terms. Proper control action can be inferred from thirS rule base that emulates the role of the human operator or a benchmark contrcd action. Thus, fuzzy logic controllers are suitable for nonlinear, dynamic prccesses for which an exact mathematical model may not be available. Using the principles of fuzzy logic control, a PSS has been designed t.1 enhance the operation and stability of a power system. Results of simulation and experimental studies look promising. The reader is reminded that further information on the contents of this chapter can be found in the referenced articles and transaction papers. at the end c f this volume.

Application of Artificial Intelligence to Voltage Stability Assessment and Enhancement of Electrical Power Systems

INTRODUCTION Most of the voltage collapse related incidents are believed to be related to hearily stressed systems where large amounts of real and reactive power are transported over long EHV transmission lines while appropriate power resources are not available to maintain normal voltage profiles at receiving end buses. I n some cases, however, voltage profiles show no abnormality prior to undergoing \yoltage collapse because of load variations. Operators may observe no ad\Fance warning signals until sudden significant changes in the voltage magnitude result in actions of automatic protective equipment to crash the network. This phenomenon is caused by large and small disturbances. Large disturbances consist of the loss of generators, transmission lines, and transformers. Small disturbances on the other hand consist of slow variation in the system load. The common analytical techniques used for voltage stability assessment during these disturbances are:

I.

2.

Minimum singular value decomposition which determine the singularity of the Jacobian matrix of the system under study; The concept of multiple load flow solutions which are used for determining the proximity of a particular voltage collapse point. This method derives the variation of load change for specified bus Froltage degradation;

3. The concept of energy margin developed as an indicator to voltaglr stability by computing the system stable equilibrium points, SEPs, and ii n st ;I b 1e eqi i i 1i bri u In poi n t s, UEPs : 4. The condition number of power flow Jacobian matrix used to estimat: the voltage collapse point; and 5. The continuation method, which computes the neighborhood of wltag: collapse for variation in load. The scheme is based on the solution rf the load flow equations by suitable modification of the Jacobian using ii continuation parameter to avoid singularity of the Jacobian near tht. collapse point. There are a lot of reservations about the results, accuracy. difficulties, and the computational burden involved i n using these techniques. Therefore, a tool Lvhich can probide timely e\raluation of voltage stability of the system i i n d t ~ diirersified operating conditions would be very useful. In recent years, efforts to irnprokre on speed, accuracy, and ability to handle stressed / i I 1-con d i t i one d sy st e ni s have 1ed to the de v e 1opine n of i n t e 11i ge n t sy :itenis-based tools. The potential application of ANN. and ES as alternati\,e approaches for solving certain difficult power system problems, where the con\'t'titional techniques ha\re not achiekred the desired speed, acciiracy, or efficicnc y is quite promising. This chapter deals with the application of ANN and ES to \,oltage stabili-y iissess men t.

9.1

ANN-BASED VOLTAGE STABILITY ASSESSMENT

The purpose of ANN-based voltage stability assessment is to obtain the n i a ~i niiini MW loading of the system for a given contingency without conductilig P1' studies. The proposed scheme for ANN-based wltage stability assessment iitilizes a multi-layer feed forward network employing back propagation algorithni for the training process. Details of constructing the network architecture, preparation of the training data, and testing process N f i l l be done in the follou.iiig sect ions. 9.1.1

Mechanism for Generating Training Data Based on Modal Analysis

The mechanism used for the production of the data needed for training the proposed network is based on modal analysis, employing V Q sensitivities that have the ability to identify areas that have potential problems and pro\ride information regarding the mechmism of ivltage collapse. The method can be tlescri bed as follows:

The system dynamic behavior can be expressed by the first order differential equation.

x =,fcx.v, where X = state vector of the system V = bus Lroltage Lfector Under the steady state condition X = 0, using enhanced device models. the linearized power tlow equations can be written as:

where: AP,/: Incremental change in device real power output: AQ,/: Incremental change in the device reactive power output; A@,/:Incremental change in the device voltage angle; A Incremental change in the device voltage magnitude;

v/:

J,*. J,,l,J,,, and J,, represent a modified form of the ponfer tlow Jacobian elements in the terms associated with each device. We can study the QV sensitivity while keeping P constant. and substitute A P = 0 in the linearized power flow equations to give after some simplification.

where

We examine the eigen values and the eigen vectors of the reduced Jacobian .Ih'. to arrive at:

where A - ' is a diagonal matrix with entries l/h,,h, is the i"' modal tvltage. \! = q A V and q = qAQ. The bus participation factors determine the contribution of each h, to 1'0 sensitivity at bus k . They can be expressed in terms of the left and right eigcn \rectors of J K , as:

,362

The branch participation factors P,, which give the relative participation )f branchj in mode I are given by: p,, =

AQloss for b r a n c h j max A Q for all branches

The A Q loss can be found by calculating the A V and A 0 change at both ends of the branch. The generator participation factors c,,,that gic-e the relative participation of machine in mode i are given by: 1~ er,,= rnaxA AQ Qforformachine all machines

The machine participation factors can be used to determine the generatclrs that supply the most reactive power on demand. The reactive reserve at these generators can contribute heavily to voltage stability. The modal analysis algorithm is utilized in the EPRI VSTAB program. 9.1.2

Suggested Neural Networks Architecture

The ANN-based voltage stability assessment proces\ is designed to predict the critical sy\tem loading, expressed in MW, for a given contingency without ccmducting PV sirnulations. The network architecture, as shown in Fig. 9.1, is a\ fol10ws : The input layer consist\ of 12 neurons that repre\ent the 12 input \,ariabl:\. namely: 1. 3

L .

3. 3.

5. 6. 7. 8. 9. 10. 11.

Q,totallQ,installe~(ignoring slack bus reactive generation). Q, at most critical generator/Max Q , at that generator. The mo\t cr tical generator is the one with the lowest voltage. The Q, reserve at generator with least Q, re\erve. The number of generators \iting at limiting Qq. Lowejt voltage at base case loading. Number of buses with voltage below I p . Total active power demand in MW. Total reactive power demand in MVAR. Total active power loss in MW. Total reactive power losses in MVAR. Ratio of the mo\t critical branch MVA flow\, to total MVA denxnd (from the power tlow solution for the intact \ystem).

pected M a x i m u m

Th Wa

Hidden L a y e r

Figure 9.1 Suggested network design.

12. Ratio of the most critical branch MVA flows to the maximum total MVA demand before collapse (intact system only). The number of the neuron in the hidden layer was obtained experimentally i.e., this number was determined from studying the network behavior during the training process taking into consideration some factors like convergence rate, error criteria, etc. In this regard, different configurations will be tested and the best suitable configuration will be selected based on the accuracy level required. The output layer consists of one neuron representing the predicted maximum MW loading in MW. 9.1.3

Training Process

The data needed for the training process is obtained from the simulation results using EPRI VSTAB. Next the results from VSTAB are analyzed and processed for training and testing. The training data exploited about 75% of the simulation results. Finally, the data sets, set aside for testing, were presented for the trained network for the testing process. The data was modified to satisfy the input requirements of the neural network for better discrimination between data sets. One thousand cases were produced by the VSTAB program for the training and testing process. The accuracy during the training process was set to 1 .OE-03. The configuration with 5 neurons achieved a convergence tolerance below 1 .E-03 in about 1000 iterations. The

configuration with 7 Iieiirons i n the hidden layer achiewd the same accuracy le\,el with ii higher number of training iterat,ons. The configuration with 3 neiirons i n the hidden layer did not achieye ;in accuracy below 1 .OE-3 in aboiit 600(1 it t:rii t i on s pos s i b I y reac h i ng a I oca I ni i n i mu m . The t rai ii i ng error s honrt.d tiiore icariation Liith change in the architecture than in the previous studies. I t \+mobserved that the critical branch loading makes a significant contribution tonz'ard Ifoltage collapse. The lower the total Q, reserc'e. the lower the alloudAe niasimum M V A loading on the system. The Q, reserve at the nio.;t c ri t i c;i I ge ne r;i t o r is \'cry i n st 1-11men t ii I i 11 main t ;i i n i ng high mas i in i i m I oada hi I i t yv,. Con\.crselj., iiiipro\,etiients in the voltages and/or reactive reserve via compens.ition at such buses c;m increase the stability margin. The cases with low cwltagcs during contingencies at base case loading hakre a lower maximum M V A loadit ig before colliipse. The total acti\.e and reacticre losses for the contingencies at tlic base ciise loading signif)( the degree of change i n the net\\w-k t1on.s due .o cont i ngcncies.

9.1.4

Case Study: New England 39-Bus System

The Lroltage stability studies were conducted on the New England 39-bus sqsstem. shoum i n Fig. 9.2. wthich is slightly adapted to the study. The existing base c;ise evas used to create ;i \.ariety of cases with different load distributions Lzmith Iiear-base case loading. different generation patterns, generator voltage patten IS. and transformer tap ratio settings. This was achieved through randomly Lw-yiiig relecmt pat-meters about the original base case values. In some cases econon- ic dispatch and loss minimization were used to make the solution comply Lkcith L W 1t age and flow 1i i n i t s. Con t i nge nc i e s consist i rig of d i ffere n t b rant h oii t ag es and different generator outages giving rise to different Q, reserves were implcmenteci on all o f the ;ibo\re data sets. In total, a few thousand different c;ises \\'ere generated. For each case the 12 parameters used as inputs to the Ab N u w c produced cc'ith the corresponding maximum system loadability. 7574 of' tlic sirnulation results tirere used to train the network ~ , h i l the e remaining 25% ~ v re c iised for the testing process. Based o n the training error statistics, there were 5 neurons in the hiddun layer architecture for the testing process. A sample test result for the back propagation iilgorithrn using the selected ANN architecture is shown in Fig. 9.3. Figure 9.3 shours the difference between the desired and computed outitit for the test data expressed in MW. The global nortnalized test error w i s 2.07E03, which correspond to SO M W . The minimum error is less than 100 M'N. The difference between the maxinium load at collapse and base load is in Ihe neighborhood of 1000 MW indicating an accuracy better than 90% in the A N N t es t re s LI 1t s.

-

Area 1

-

3 7

-30

25

+I

2

27

18 17

I

+

,.(.....................................................

1

?

t

,_,.....

.......................

1t

1

28

,. ........... . . . .

9 f'" dJJ

/;

.....,

24

I

6 t

2

12 1

11

6

"\.

"

........

.

-35

I "'i

23

I

13

10

--a*

Area 2

29 -38

22

T .......................

8

9

-

-

15 ...

'-I-:

7 -

I

!,

3

.................................................. ....._,.P.II

5

t

+

16

-..,

9.2

1

26

I+

!

/

L 4 i 2 i 2 ;

3

-

-36

+-

j

f

Area 3

ANN-BASED VOLTAGE STABILITY ENHANCEMENT

The ~ ' o r kof Sec. 9. I is extended to accommodate voltage stability enhancemcnt using switchable shunt compensators. The buses sensitiire to reactiive compcns;ition are identified Lria modal analysis and used as candidate buses for q~plj.ing compensation. Key physical parameters contributing toward voltage collqxc such as reactij'e generation reserve and reactive compensation at base load are used ;IS key input Lrariables. The ANN output provides the enhanced ni;isiniuni demand at collapse and the reactive compensation needed to achieixe the enhanced stability margin. The New England 39-bus system is used for the study. 9.2.1

Voltage Stability Enhancement Using Shunt VAR Compensation

The \ T O It age I* t a b i 1it j r enhance men t us i ng shunt V A R cornpc n \a t ion in odi tic \ the nt'tworh power flow equation\ at base load to the form gi\ en bqr:

,366

100

80 60 40 20

0 -20 -4 0 -60 -80 -100 J Contingency Numbor

Continwncy Number Figure 9.3 A \ample

test

result.

26 7

P:-P~,+P,,,=o

i = 1 , 2 , . . . . NI?

;E

gen

~ = o .. I . . . . ~k

The collapse state power flow equations are:

+ P,;,= o i = I , 2, . . . . NI? i E Q:k- Qkf - (2:: + Q(;,= 0 P:” -

gen

= 0,

I ,

. . . , ~k

The symbol (’) represents the state collapse point, Q( represents the shunt VAR compensation, and k represents the contingent cases. One form of shunt VAR compensation is via static VAR compensators (SVC). The optimal Lralue of SVC at collapse point can be obtained through optimization where the generator voltages and transformer tap ratios can be optimized. An alternative. using VSTAB, employing power flow without optimization is used in the current study. 9.2.2

Suggested Neural Networks Architecture

The ANN-based voltage stability enhancement is designed to predict the enhanced maximum demand at collapse and the reactive compensation needed to achieve the enhanced stability margin. The network architecture. as shoufn in Fig. 9.4, is as follows:

er

Hidden Layer

Figure 9.4

Suggested artificial neural network (ANN) design.

,368

The Iiipcit Layer

V o 1t age stab i 1 i t 4' ;is se s s me n t req u i re s i de n t ifi c;it i c) n of the c()1I ap se po i n t b;i 51: cl o n load bwiation coitiinonly employing ii property such iis singularity of t i e load tlow Jacobian matrix at the collapse point. The load at buses participatiiig in collapse. the reactive compensation at base load at selected buses, and the reactit.e reserve at each generator tire used ;is input to the neural network. The nurnber of neurons in the hidden layer is determined bj,: A,, + N, + N,

tz8her-e iV, i \ the number of buse\ participating in the collap\e;

N , i \ the number of VAR site\; and N , i \ the number of generator\. Hidden Layer

I n the \tidy, 1-2 hidden layers uith multiple hidden node are tested. For eich proposed nettzwh architecture. the number of the hidden neuron\ varie\ from 2-30. OcitpLI t L7' !/er

I n this study ~e asses\ rc'actiLte compensation at collapse and the maxirnim poner demand at the c o l l i p e point. Hence the number o f the output node. i \ determined by: N , + I .

9.2.3 Studies Conducted For the New England 39-bus system, S VAR sites were chosen based on the participation factors. Voltage stability studies were then carried out Lvith t h x buscs treated as VAR sites containing snitchable shunts. The load at buses participating in the collapse and the magnitude of the shunts required to m:.intain the voltage within an acceptable range (0.9S-I .OS p i ) at base load and at the collapse point mw-e noted along with the Q , reserve at base load and the total rnaximum power demand at collapse. The load at buses numbered 12. IS. 16. 20, 2 I , 23, and 24 were used as part of the input while buses numbered 12. 15. 20, 2 1, and 24 were selected a s VAR sites based o n their high participation. A few hundred load configurations were generated using ii random perturbat ion around the base load and the preceding procedure u.as repeated for all different 1oad con fi g Urii t ions. Different ANN urchitectures urere constructed uith hidden nodes ranging from 2-20, and hidden layers from 1-2. The different ANN were trained lY,ith

the 85% of the simulated data results. The stability margin was computed as the ratio of the maximum power demand at collapse to the base load providing an index in the range of 1.0-2.0. All other data was comrerted to per unit during training and testing. The ANN training statistics are presented in Table 9.1. This table shoi4.s the approximate number of presentations required to achie\.e an error below the tolerance I .0E-04. The training process is terminated if convergence was not reached with 100,000 presentations. The following remarks are made:

I.

The different ANN architectures with one hidden layer and nith less than 6 hidden nodes did not converge beloMy the tolerance error for o\.er 100.000 presentations. 2. All architectures with one hidden layer and with less than 6 hidden nodes did not converge below the tolerance error for oi'er 100.000 present at i on s. 3. All architecture with 1 hidden layer and ufith more than 5 hidden nodes converged below the tolerant error in less than 4000 presentations. 4. Within the scope of the architectiires tested, those with 1 1 . 14. and 30 hidden nodes converged with the least number of presentations. i.e.. 600-700 presentations.

Table 9.1 Training Statistics for Different ANN Architectures ANN architecture Number of hidden layers 1

Number of hidden nodes

3-5 6 7 8-9 I0 II 12-13 14 15 20

7

L

30 2-5; 2-5 6; 6 10; 10

Number of iterations for convergence

> 100,000 3800 I600 2200 I000 600 1500 600 1700 1100 700 > 100,000 10.000 3600

5. The addition of an extra hidden layer did not improve the performance of the ANN. 6. With 2 hidden layers and with less than 6 hidden nodes in each laye1. the ANN did not converge below the tolerance within 100.000 present a t ion s. 7. For architectiires comprising 6 hidden nodes i n each layer, the number of presentations required for convergence was about 10.000. 8. For 10 hidden nodes in each layer. the ANN converged in about 3600 presentations. A comparison of the test results using the different architectures uith 1 hidden layer are shown in Tables 9.1-9.3. The errors are expressed in MW and MVAR for better readability. Fror-i these tables the following remarks are made: The maximum error is in its maximum \Aue for the network with 3 and 4 hidden nodes. 2. The error starts decreasing at 5 hidden nodes and stabilizes for 6 hidden nodes and higher. 3. The lowest average error is with 7 hidden nodes or uith 30 hidden nodes. 4. Since a smaller architecture is naturally preferred, the architecture w i t h 7 hidden nodes is recommended for the study. 1.

Table 9.2

Maxitnum Test Error for Various ANN Architectures

Maximum error in reactiLre compen\ation at collap\e expre\\ed in MVAR Number o f hidden node\

Q, 12

Qc15

20

Q , 2I

Q(24

3 4 5 6 7 8

37.9 33.7 39.7

31.3 23.4 16.I 15.3

1 1 .o

10.0

10

13.5 9.7 12.0 10.0

66.7 33.2 15.1 9.1 8.2 10.9 7.8 9.0 8.2 10.9 8.6

58.6 35.3 10.1 10.5 13.0

9

86.0 47.4 28.7 17.9 13.6 28.0 13.9 22.2 28.8 17.4

16.9 14.8 15.2 16.5 10.9

II 12 30

11.2 9.2

8.4

18.8

8.2 8.5 10.5 11.1 8.1

11.9

11.4

M ax imu m error in e\timated inaxi nium demand MW

121.6 113.1 34.8 29.’ 26.9 25.5 32 9 28.9 35.0 30.3 29 8

Table 9.3

Average Test Error for Various ANN Architectures

Average error in reactive compensation at collapse expressed in MVAR Number of hidden nodes

Q, 12

Ql 15

Q,20

Q,21

Q,24

3 4 5 6 7 8 9 I0 11 12 30

19.6 15.3 12.0 5.1 3.9 4.8 5.1 5.2 4.1 5.5 4.9

31.9 18.3 8.9 8.7 8.3 12.2 7.9 9.5 13.1 8.7 9.7

20.9 11.0 5.7 4.8 5.5 5.2 3.9 3.7 4.0 3.8 2.6

19.0 14.1 3.9 5.1 6.3 4.5 4.2 4.3 5.6 5.0 3.5

12.2 10.7 8.3 5.2 5.3 6.3 5.7 6.2 6.9 6.2 4.9

Average error in e \ t i mated niax i in u m demand MW 39.6 39. I 15.9 16.0 9.8 12.8 11.1

10.3 10.0 11.5

7.9

A sample set of results for the chosen architecture (i.e., 1 hidden layer and 7 hidden nodes) is shown in Tables 9.4 and 9.5. Table 9.4 shows the difference between the desired and the computed output for the test data expressed i n MW and MVAR. Table 9.5 shows the desired output for reactive power compensation and Table 9.4

Number of hidden nodes 1 2 3 4

5 6 7 8 9 10

Sample ANN Test Error for Architecture that Has 7 Hidden Nodes

Q, 12

Q, 15

Q,20

Q,21

Q,24

Maxi mum error in estimated maxini u ni demand MW

0.28 9.18 3.3 1 0.46 0.80 4.89 1.85 5.8 2.47 8.79

1.35 10.85 4.28 13.6 5.32 2.26 9.8 1 11.52 10.63 13.813.18

1.77 5.8 6.4 1 7.32 5.2 1 6.13 8.16 7.16 5.04 2.31

1.52 2.18 6.95 12.98 12.03 3.18 6.99 0.84 9.75 6.47

2.76 6.86 2.77 0.85 7.74 11.87 4.58 10.53 1.34 3.82

2.01 2.92 16.46 1.24 26.92 11.96 17.89 16.36 0.66 1.93

Error in estimated reactive compensation at collapse expressed in MVAR

Table 9.5

Desired Output for Te\t Cases ( f r o m VSTAB) De4ired rtlactic e compcnwtion at collapw expre\\ed in M V A R Q 15

Q 20

0.28

I .35

9.18 4.31

10.85

1.77 5.8

ca\e

Q I3

I 2 3 4

0.46 0.80 4.89 1.85 5.8 2.17 8.79

5 6

7 8 9 I0

4.28 13.6 5.32 2.26 9.8 I 11.52 t 0.63 13.813.I8

6.41

7.32 5.21 6.13 8.16 7. I6

5.04 -.. I31

Q 21

Q, 24

1.52

2.76 6.86 2.77 0.8.5 7.74 11.87 4.58 10.53

2.18 6.95 12.98 12.03 3.18 6.99 0.84 9.75 6.47

I .31 3.82

De\i red demand at collap\e in h1M'

2.0 I 2.92 16.46

1.24 26.92 11.96

17.89 16.36 O.t>6

I .93

the pon er demand at collapse. For the reacticfe compen\ation. we see that I he error ranges from about 0.28 MVAR to 13.6 MVAR. From Table 9.3 m e w e that the acerage error is in the neighborhood of 3-8 MVAR. For the e\tima cd cicmand at collapse. the error ranges from about 0.66 MW to about 26.9 M W . The de\ireci demand being i n the range of 10,000 MW. Hence, the relat ce percentage error i \ Ies\ than ;ipprouitnatelj 0.3(A indicating good A N N prec'ict io n .

9.3 A KNOWLEDGE-BASED SUPPORT SYSTEM FOR VOLTAGE COLLAPSE DETECTION AND PREVENTION 9.3.1

Voltage Collapse Using Expert System Technology

Thc \theme 1i)r expert \y\tetn-based coltage collapse detection and prediction i 11c I 11de \ t h e dec i s i on proced i i re, h n oN!1edge iicc1LI i \ i t ion a bo LI t h o M.' to \c I ec t ,i n d ranh indicators affecting \ oltage collapse (VC), prec entic t' mea\iire\ to hancile c ariou\ c iolations. and \election ot appropriate models for croltage collap\e dctection and p r e e~ntion. Specificallq . it \hould be able to handle the t'ollocving lash\: I. 2. 3. 4.

Selection for performance index for contingencie\: Selection of' optitniution models for preiention; Recommendation o f pre\ ention niea\iire\ for cwltage problem\; and Selcction ot. procedi~i*etor conductiiig c\pet-itnent\ that lead to \ olt;ige col I:Ip\e.

9.3.2

Knowledge Acquisition and Presentation

The knowledge about the system performance variables and sensiti\?ityparaiiieters may be acquired off-line. The knowledge that needs to be acquired is BS fo 11oW h :

I . Load history: 2. System configurations and critical outages; 3. Control variables: generator taps, VAR, etc.: 4. List of facts on generator data, bus data, capacitor data. line data, and interchange data: and 5 . Solution models and techniques. The identification variables involve the relevant parameters associated \vith load history, critical line outages, control variables, and other niodeling techniques. From the voltage collapse detection schemes reviewed, i t was found that load history is not taken into account, the selection of critical lines depends on the system conditions and that prioritization of control parameters affects system behavior. Therefore, currently used mathematical predictive techniques cannot be used to ascertain adequate voltage collapse detection and preLrenti\.e measures. The proposed work reported here basically considers the static model using multiple-solution and divergence evaluation of power flow calculation to predict VC and the dynamic model for studying the effects of governors. exciters. etc. The preventative measures used are the VAR planning technique and optimum power flow (OPF) for correcting violations of voltages and flows at an optimum cost. It is intended that the proposed expert system should be able to identify the type of data encountered, choose the best model for voltage collapse detection and use the appropriate model of OPF for prevention. The knowledge/esperience regarding the determination of the best protection for given impact parameters should be appropriately acquired off-line before implementation. 9.3.3 Structural Design of KBVCDP Scheme The estimation of voltage collapse phenomena is performed in three major tasks. namely ( 1 ) power flow base established or divergence control, (2) selection of detection schemes, and (3) correction of violations. This framework de\.eloped in numerical methods is designed and enhanced by using KB strate,'c'les to achieve optimal performance. Figure 9.5 depicts the implementation logic of interdependent models required to build the knowledge base voltage collapse detection and prevention (KBVCDP) scheme. It consists of a detection and prevention scheme enhanced by the KB support, as described in the proceeding sections.

c' Input Data

x

Planning System Expansion

I

Operational Run Power Flow to obtainBaseCase Solutions Result Evaluation by Knowledge-Based Decision-Making

t

4 Yes

-

I

4

Automatic Display of Detection: P-V, Q-V Curves Parametered causes Operational & Planning

r Knowledge-

of Measures of Preventing V.C.

1

Knowledge-Based Model and Method Selection for Sophistic ate

Knowledge-based support for screening of possible indication which will cause Voltage Collapse Contingency - Selection - Load Condition EventAnalysis Prioritization

-

I

>

4

-

I

Planning for Siting

Selection of VCD Numerical Methods

I I I

Perform Optimal Power Fli)w

4 Transient Stability Model

4

Power Flow Model

.

Dynamic Stability Model

Result Evaluation; Model & Method Selection Suggestions

Figure 9.5

Result Evaluation

-

B

F l o ~ ~ c h aof r t the KBVCDP implementation.

-

'7

Automatic Display of F'reveiition Results Effectiveness Reliability Economics

2 7-5

Power Flow Divergence Monitoring (KBPFDM)

The program modules utilize heuristic rules to identify cases of load flow run that may diverge due to bad data, telemetering error, method of solution. etc. Suggested corrective measures are also given by the expert system. Indicator Selection (KBIS)

For an off-line type of study, VC estimation may be detected by evaluating the impact of different indicators on the system that will predict limit \.iolations, thereby causing VC. The KBIS provides prioritized parameters for detection schemes, which can be any of the numerical methods or improved power flow techniques such as an automatic power flow technique. The automatic power flow technique is a self-contained power flow approach that gives the sensitivity of voltage measurement with respect to parameter changes and make Q V and PV plots to identify the knee point. Other selectable detection schemes developed within KBVCDP include other numerical techniques such as Barbier's and Schlueter's. KBVC Prevention Scheme (KBVCP)

The identified critical VC points and measures for preventing VC are handled effectively by the KBVCP. It selects appropriate controls as constraints for the optimization algorithm, minimizes relevant objective functions and includes other conditions and models in seeking an optimum corrective measure that is economical and effective. The display of VC knee point and the indicators are given in graphical form. Using the KB system, the corrective measures and the rules fixed are tabulated in matrix forms. 9.3.4

Structural Design of Expert System

A preliminary prototype of the expert system used to verify the feasibility of

the KBVCDP has been designed in PROLOG. Its structure is shown in Fig. 9.6. It consists of the following parts: knowledge base, blackboard, inference engine and user interface, and application programs. Knowledge Base A knowledge base in the expert system (ES) is the principal source of knowl-

edge used for the KBVCDP implementation. It consists of a fact base and a rule base. The first base consists of fact statements and contains the basic description and record of all power system components, status and configurations and knowledge needed for describing power system states, conditions for selecting measures and methods for interpretation of results. The knowledge contained i n

HUPFP

I

Man-Machine Interface

Figure 9.6

Data Bridges (C)

%

KnowledgeBased Support Systems (OPS 83; PROLOG)

Prototj pe w-ucture o f the hnob ledge-based wpport \y\tein.

the rule ba\e I \ u\ed to de\cribe the cau\e-effect relation\hip between fact\ i n the tact baw and perform the ba\ic deci\ion mahing that bill lead t o a good deci\ion \upport \chenie. These rule\ are wed in forming the KBPFDM, KBIS, KBMS, KBCM, etc. B/xkDoC?rd

The blackboard is designed for database inanagemetit and communication t c t u w n the expert system and other units. Messages to prompt new rules or st.irt new evaluation, terminate a study, select relewit data, or improve qualitati\-e reasoning are stored uithin the blackboard. With results reasoning comple1t.d output modules and user access modules are fetched to complete the study. Inkrenc-e Engine

The inference engine is dedicated to making effecti\,e use o f domain know lec ge in the hnowledge base and to perform message interaction on the blackboard. based o11 d ii t 11 d r i c'e11, foruwd - ch a i n i n g , and tii otiU I ar proced U re s.

Rules for Detecting Power Flow Divergence The rules for detection are deceloped based on the experience of the operator\. literature review, and interviews with experts. The rules detect bad system d;ita, chech errors i n telemetering, and evaluate the erroneous results due to nurnerical methods l i d . Forming those rules and their combination\ i \ designed to i t dicate \pecific load tlow runs that lead to divergence. RLi/es for Se/ecting VC PCirm?eters(indicators)

Rules decelopment for prioritizing parameters [hat may lead to VC itre con\tructed. They consider the impact o f ec'ents \uch a s line outage, generation

outage, load transformer taps, excitation effects, and other combinations along with other sy\teni conditions. The severity of indicators causing VC :ire ranhed. Rules for Selecting Models and Approaches for VC Detection

Rule development for selecting VC detection methods are based on choice of models, desired options (speed, accuracy, and display), and other niajor e\'ents causing VC. The expert system selects the indicators that ha\,e a higher priority for VC. Several system parameters, limits, and violations are stored i n the database. The detection system condition and events causing VC are based on the skills and experience of the operator. Very often the operator's decision \+rill change according to the severity of the contingency or other factors. The detection and prevention of voltage collapse (VC) based on the K B approach is fast and accurate. The system reads the data, performs manipulations, prioritizes the indicators causing VC and selects appropriate measures. models, and methods for detection and correction.

9.3.5

Knowledge Organization for KBVCDP System

The de\dopment of a KB support system for KBVCDP requires the determination of a knowledge organization scheme for the VC detection and pre\ ention. Using a matrix format, the heuristic rules describing detection strategies are formalized. The rules are divided into each of the subtasks. With appropriate data, the prototype KB system will be tested. Rules for Preventive Measures Selection

Rules for selecting preventive measures are developed based on the follo\+ring aspects: 1.

2.

Rules for ranking corrective measures in t e r m of the economic evaluation. The effects of system conditions such as loading, generation. length of lines, and seasons of the year are employed to select appropriate measures. Rules for evaluating reactive power, LTC, and others constrain effectiveness. These rules evaluate the effectiveness of LTC, and other controls as constraints for the optimization algorithm. The role of LTC as detrimental or beneficial is evaluated by using system conditions. The selection of appropriate objective functions is identified by using system conditions and limit violations based on the type of \kdation. ranking of indicators causing the violations. and other system conditions.

9.4

IMPLEMENTATION FOR KBVCDP

The algorithm described in Fig. 9.5 was programmed on a VAX I1/780 and IBM PC compatible computer. The numerical programs are coded i n FORTRAN, while the symbolic computation programs are coded in PROLOG. The functional descriptions of each of the KB modules are discussed. The data files and communication between these modules are also displayed in Fig. 9.7. The program runs on the IEEE 57-bus test system using the rules developed for each of the KB modules. These rules are simulated based cn predetermined experiments. When the experience and knowledge aspects are filled in the matrix accorcling to different selections, different rules can then be developed. The rule matri s approach makes rule development both easier and faster. At the same time, the rule matrix method is also very useful for the new rule developed while the neA' experiences have been gained. Each column of the rule matrix is a knowledge u n i t for the rule condition or conclusion and suggestions that are identifiell. Tables 9.6-9.9 summarize some important results based on experience. Other combinations of system conditions are checked and results will cla+ sify divergence of the power flow as critical, sek'ere, or normal for the po"l:r flow divergent part of KBVCDP. Other parts of KBVCDP are developed by the same approach. Rules are capable of prioritizing causes of power flow (PF) divergence. multiple solutions, and extraneous solutions. Other associated knowledge bases for kultage collapse detection include the development of rules that study the indicators for detecting voltage collapse and selecting the performance index fq.lr \!oltage collapse studies, and rules for selection of measures for \voltage collap ;e peri od s. 9.4.1

Limitation of the Operator Assisted Expert System

When the human experts' heuristics are successfully encoded in the KB, a reipiew of the literature indicates that the ES was comparable in performance with human expert to a certain extent. Creating the KB for the VCDP or for any ;lf the operational problems (e.g., security-assisted. and voltage control) is ;1 dit'icult problem. One must first define the problem to be solved and then encode it in a set of rules. However. the KB cannot solve the problem by itself. A human expert uses past experience together with the background of the whcle problem to make the correct decision on causes of VC and preventive niethoh needed. Hence, in order to design an ES that can utilize the expertise of t i e human, we must find a way of integrating the human expert's empirical knovrledge i n one domain. Successfully encoding the human expert's empirical knowledge into a x t

1 . I PONER

SYSTEM

TASK MANAGER 4

l n f m Engine

COMMlrZJlCATlO

CHANNEL

Operaton

environment ~

~~~

~

NUMERICAL. PROGRAMS

Dispty Intcrprewh

of Resutu and SUggCStCd

actions

on

screen alum

R artincly recording

-

FORTRAh'. C

2x0

Table 9.6a Matrix o f Rules for Voltugc Collapw: Method Selcction

of production rules might also pro\fe to be a difficult task. The knowledge baw is likely to be incomplete and inay be inconsistent. These limitations iire jus .ified in VCDP since the cause of' VC are system-dependent. The model uscd in\'ol\,es the interaction of system parameters, system conditions, etc. Thc EiS designed may have the problem of portability m c i inconsistent knowledge. The t rad i t ion a I non k no w 1edge - ba sed approaches for v o 1t age c o 11ii p se det c c tion and prevention can be used to s o l k ~the probleiiis in VCDP. At the s;iiiic time, the following are common probleriis: The di\ ergence reawn for the power tlwf,automatic series run power tlon . and optimal pouer tlow program\ are alu ay\ difficult to identify especially for the les\ experienced power engineer\. Even the judgement of experienced engineers can be wrong because \x)luminous input and 01 i t put data are in\wl\ ed. At the wme time, the divergence of the iteration procedure m a y be cau\ed by reawn\ in the program other than in the data.

28I

Table 9.6b Matrix of Rules for Voltage Collapse: Method Selection

I

Events

1

I

IF

Table 9.7 Corrective Measures and Economical Ranking Then

If No.

Measures

I

Capacitor dispatch Dispersed generation control Demand-\ide management Voltage reduction for load relief LTC and distribution voltage regulator Load shedding Series conipensat ion Parra I lel compensation Network expan4on New plants

3

3 4 5 6 7 8 9 I0

Econoin i c a1 condition number

3

1 6

5 4

7 8 3 9 I0

Then

I

Table 9.8 Matrix o f Rules tor Poiver Flow Di\wgence Indicators (Solutions Methods) Then

If

Rulu niir~iber

I

Mis-

I11itial

I11 ;It c he h

v ;I I 11es

Iricr-

Dec.

Ye5

No

For Newt o riR i ~ p h s o t ~ For GLitlssINR) Seidel (CS) divergence clivcrgencc cliie to cltic to

X

Concl us ion

I I I-condition

X

problem 7

X X

&

3 J

X

5

X

X

Data re pre\e n t at i on -

X

X X

X

>Zero

Deficit

s

X X

_________-

X

X

x N X

-

X

X X

x

Initial 1 d u e \

Surplus

Beneficial

X X

X

X X

X

X

x X I:

The \voltage collapse indicator selection is \.cry complicated. I n po\i'er sq'stern \ d t a g e collapse studies. what causes the \roltage co1l:ipse is the most important study aspect of the stable problem. Most of the time. there may be more than one cause, but deteriiiining the original and most important ones are difficult. Because of the selection indicators, the pre\'entive iiieiisure selection is also quite complicated. The selection of pre\rentivc measures iii\wl\.es considera t ion of econom ic s, plan n i n g s t ra t eg i e s. opera t i oiia I sec U r i t j, and system teasibility for the measures. At this time. there :ire no acceptable strategies thr this problem. For certain study problems, the selection of studj, models and nuiiicrical methods is ;I complex one even for researchers \ 1 7 h o 1iai.e been stucijing voltage collapse phenomena for ;i long time. Interpretation ot' the numerical program results usually require the -judgeiiieiit of ;in engineer. To solve the abo\,e problems for the nonkno\l..lcdgc-hased approaches. the k n OM'1edge - based ;I pp roach has t he fo11o w i n g ad Y an t uge s : The data bridge can transfer the quantitatiLre data into qualitati\re data directly. Thus, the KB scheme can directly use the data for logical reasoning for determining any possible errors in the numericd data anti making suggestions for correcting them. This procedure n,ill s i i \ ~;I considerable m o u n t of time for indicator identification. The k 11o LV I edge - based v o 1t age co11ap se i nd i c at o r se 1ec t i on . st idy inode I . and method selection scheme can make more sophisticated and faster choices for the decision-making procedure. The selected indicators and study model is more suitable for the study. This is the siiiiic ;is i i i preventive measure selection, the KBVCDP gives more choice of iiitxisures rather than the right judgement at due time. The KBVCDP also gives heuristic and clear guidance on how to detect aiiti pre\'ent \,olt:ige collapse in a more con\renient ~ z a q ' . The approach, using knowledge-based technology, is suggested to enhance the detection and prevention scheme for voltage collapse. The suggested approach has been prototyped as an expert system program both on the IBM PC/ AT using the PROLOG language and on the VAX I1/780 using OPS83 iind C languages. The data bridge concept has been introduced. and data bridges tor d i ffere n t 1111meri ca 1 programs have been designed and i m p I eine n ted . S h o rt e I' IW soniiig time and more accurate results by using KBVCDP are \.eritiecf 1 ~ the ~ 8 results and the explanation.

9.4.2 Examples of KBVCDP Output The KBVCDP developed has been tested on different \j'\[eiiis under oper;itional conditions. The program was initially deLreloped under the VAX 1 I /780 and

Table 9.10a Test Results Number I : Rule Matrix for Power Flom Divergence (57-Bus System) Test case 11U niber

Rule fired 1

3

x I2

IS 16

20 21

V-6 position nuniber

Tp

In limits Out I i mi ts Out I i in i t s In limits Out I i m i t s Out I i mi t s In limits Out I iini t s

1

3

I 3 12

3

39 1

Line\

PV position number

R/X ratio

number

System condition

12, 39 12, 39, I 27, 36, 42 27, 36. 42 I , 39 27, 36, 42 I , 12 12, 39

Low High High High High Low High L,ow

No No 1-5 No No No 1-5 No

Se\rere Severe Critic a I Critical Critical Critical Severe Severe

105s

IBM PC/AT environment using PROLOG, PASCAL. and FORTRAN. Later, the program was modified using OPS83, C, DCL, and FORTRAN language\ entirely under the VAX I1/780 environment. Tests of KBVCDP on the 57-Bus System

9.4.3

To demon\trate the capability of the KBVCDP, a series o f tests were performtd on the 57-bus system. Tables 9.10a-9.10f give the rewlts of the rule matrix fq)r

Table 9.10b Test Results Number 2: Rule Matrix for Power Flow Divergence Indicator\ (Solution Method) Te\t ca\e Ruled number fired Mimatch 9

I

10

2

Increase bus n U m be r

Initial \ :due\

Data preci\ion

Tolerance

NewtonRaphson method

Ok

Double

Not reached 11I -condi t ional data

Ok Ok

Single Double

Not reached Satisfied Data precision

12

II

12

Decrease 4 Increase bus number 29 S Increase bus number 29

Bus number Double 33 hea\'ily loaded

Satisfied

Initial \,alue

Ga I\\Seidcl me h o d

28.5

Voltcigr Stcihilitj~As.srssriieiit/Elec.tr.ic.al Portvr Sj~stenis

Table 9 . 1 0 ~ Test Results Number 3: Rule Matrix for VC Indicator Selection Test case number 13 14 15

Rule fired Event load I Event load 3 Generation event 3 LTC event 2 Compen5ation event 5

16 17

Line outage number

LTC condition

Compensation

Rank

1

1-15 1-15 1-15 1-15

Off limit 4-18 Off limit 4-18 In limits

Serie5 and parallel on Serie\ and parallel on Serie5 and parallel on

I - 15

Off limit 4-18 Off limit 4- 18

Serie5 and parallel on Serie5 on

I 2

2 7

L

Table 9.10d Test Results Number 4: Rule Matrix for Detection Method Selection Test case number

Model selection

Speed Accuracy

Display ~

18 19 20 21 22

23 24 25

X

Dynamic Dynamic Static and dynamic Static Static Static Static Static

Table 9.10e

Selected method ~

X X X X

X X X

~

~~

Tamura and Barhier Tamiira Ta mu ra Hu and Schlueter Hu and Schlueter Schlueter Hourard Uni BPA

Test Results Number 5: Rule Matrix for Preventive Measures Selection

Test ca\e number

Model selected

Q limits

26 27 28 29

Dynamic Dynamic Dynamic Static and dynamic

out out Out

V limits out

out

out out

Flow over

out out out out

Q(,> Q(,,,l,

Yes Yes

Correct I \ e meawre ranh

5 3 1 3

Table 9.10f Te\r Re\ults Number 5 : Rulc Matrix for OPF

~~

30 31

X X

~

X X

X X

X X

X X

X X

X X

X X

Voltage and c w t

Q,,,\,c,

the power flow divergence indicator with respect to power flow data. Test cases number I and number 2 are with correct slack bus and control bus position. The RX ratio and data format results in the same system condition. that is. classified as "severe."

9.5

UTILITY ENVIRONMENT APPLICATION

The developed KBVCDP is numerical program-dependent. For different nuinerical programs, the data bridges from numerical program to knowledge-based program vary. The application of KBVCDP to different utility companies needs more implementation of the data bridges for special progranis.

CONCLUSION The use of artificial neural networks for voltage stabilitj' assessment m d enhancement has been presented. The method utilizes modal analysis to obtain suitable buses for use as VAR sites. The ANN-based maxiniurn loadabilitj, of the power system was demonstrated and the ANN-based maximum loadabilit>r of the power system with VAR compensation was demonstrated ;IS ~ s ~ e l l . The maximum loadability of the power system and rei1ctiL.e po~i'erconipensation at collapse used for training the ANN haLre been obtained using the EPRI VSTAB program. The sensiti\rity of the accuracy of the predicted output from the ANN has been investigated with different ANN architectures. The trained ANN has been tested with input data not previously seen by ANN. It is concluded that the back propagation algorithm can be used Lvith reasonable accuracy for providing an estimate for the reactive compensation at collapse as well as a measure of the voltage stability margin in the presence of sis.itchable shunts. The appropriate ANN architecture needs to be determined based on the re1ev an t sy st ein data. The trained ANN provides reasonable results in an extremely short time, almost instantaneously. when compared with other existing met hods i i t i I izing su cce ssi v e power fl ow or opt i m i zat i on . The inputshutputs for training are easily obtained kria successi\re po~i'er tlows and do not require extensive computation when compared \s,ith certain other energy-based methods utilizing ANN for assessment of the Lroltage stability limit. The design, analysis, and development of a knois.ledge-based expert system to detect and prevent voltage collapse has been demonstrated. The detection scheme selects parameters that cause voltage collapse and also evaluates a\,ailable models of the power system. It uses rule-based techniques for prioritizing

indicators and choosing appropriate numerical methods. The prevention schemc identifies measures needed to correct violations of bus voltages and flows due to a loss of generating units or critical lines. Selected measures are under-load tap changers, capacitors, line switching, load shedding, and generation adjustment. The knowledge-based system ranks these measures in terms of their effectiveness in corrections. Subject to equipment and network constraints. a specialized optimal power flow scheme based on a reactive power model of a power system minimizes cost and power loss. The expert system was tested on medium-sized power systems. Bonneviell: Power Administration's Puget Sound Area, and New England's Boston Are: . Results reveal the need for adequate use of under-load tap changers and other control devices as a short-term measure to prevent voltage collapse. Such devices can otntiate the need for the installation of a new generation of capacity and the expansion of transmission systems.

10 Epilogue and Conclusions

This book treated broad issues in power system dynamics with special emphasis on the twin subjects of angle and voltage stability. Both are essential tools for dynamic security assessment. Dynamic security characterizes a power-system’s ability to withstand disturbances and ensure continuity of service. In operational planning, dynamic security analysis encompasses a large class of problems. such as finding the security status of a network, the power transfer limit in a transmission corridor, the worst contingency in a specific area, or performing some complex sensitivity analyses. In practice, dynamic security is often measured in terms of a dynamic security limit, defined as the maximum power transfer for which the network cannot only survive the worst possible normal contingency, but also guarantee an acceptable level of service quality without loss of load. Dynamic security analysis is dominated by the use of algorithmic software for off-line evaluation and control of load flow, transient stability, and other network characteristics. Even though simulations are readily performed, assessing and ensuring dynamic security across all possible topologies and contingencies remains a formidable challenge. Power system networks currently operate more and more in a stressed state where conservative operation often results in significant financial consequences. The process for an on-line dynamic security assessment requires a reasonable level of built-in intelligence to detect and determine the following:

1.

2. 3. 4.

Assess the system dynamic performance Determine the degree of stability o r instability (margin) Determine the sensitivity of the margin to key \w-iables Iteratively repeat the preceding steps to obtain the stability limit.

In order to d o ii credible job of security assessment, a large number of coti1bin.Ltions o f nctccwk configurations under contingency must be done. This is bejvricl the capability of the typical operational planning unit i n iiii electric pokcw- s y tem. There iire three reasons for this: Determining the power transfer limit is ;i complex. iterative process requiring the execution o f many stability simulations and considerable expertise. T o perform a single power transfer limit search, one begi is by executing ;I power tlow for ;I g i k w topology and carrying out nimual modifications to the input data until ;I satisfactory steady-state C;ISC is tound. This is then used to initialize the t1etu.or-k for a transieiitstability simulation, also manually initiated by the user. Once the simulation is complete. m'e may resort to yet another tool to apply transietitstability criteria, determining what the next step u.ill be (i.e., incre;i se o r decrease power transfer in the faulted corridor), to modify local-flow software inputs accordingly and re-enter the process. This is repeated until the required limit is found. 2. The process o f computing transient stability simulation remains higttl), time-intensi\~edue to expertise-related tasks. This is true for data cq)lIect ion, i n pi1t \'a1 idat ion, oii t pu t post process i ng . an cl resii I t anal y s i s -eyuiring ii considerable amount of time. 3. Additional expertise is reyuired to scale the problem clou~iin such ;I \+ray that the dynamic security limit of only ii small set o f topologies need to be found explicitly. This reduced set forms ii basis for cstimating the needed dynamic security limits. The price to pay is ii lack of precision that translates into inore conser\xti\re security limits. I.

'

In order to improve the efficiency and accuracy of off-line intelligence S ~ C L Ii tI y analysis. the application of artificial intelligence techniques has been proposxi. Alternatively, expert system. fuzzy logic and ANN usually in combination U ith energy based methods o r simplified algorithms may be used. An interest mg recent concept involves the L I S of ~ a generalized shell for the purpose o f mec x i nizing processes in dynamic security assessment. MWXLLLI et al. ( 1993) detii~)nstrate that semantic networks combined nrith frame-based structures can eftectiirely model the process and provide a framework for constructing a his tilj powerful and user-friendly en\rironment. Their research carried out on a production level prototype shows that it is possible to mechanize routines traditiondly carried out by hunian experts. This mechanization greatly enhances the reali/at i o n of complex processes.

The expectation of dramatic increases in the interconnection and complexity of power systems has raised concerns about the performance, security. and control of transmission and distribution networks. These increases are expected to result primarily from two developments: New legal requirements allowing greatly expanded Miheeling of p o \ \ ~ r over existing networks, and 2. The wide-spread application of power-electronics-based acti\re s\vitching and control devices to raise the real power capacity of existing lines k i i local control of idtage, impedance, or phase angle. 1.

We have already suggested that power systems are already becoming increasingly stressed. Unexpected behavior has been observed i n many netnarks under 11 n LI sual stress , beh akri or suggesting that some i m portan t s y ste in d y n aiii i c s are not yet well understood and hence are not accounted for i n simulations conducted for control and security purposes. Enhanced fundamental knontledge about system stability and nonlinear dynamics may be necessary i n order to control existing and future power systems reliably and continuously iinder all contingencies. In addition to ensuring power network security under normal and exceptional stress. such improved understanding could yield substantial economic benefits by allouring secure operation closer to performance margins than is currently considered safe. Interconnected power networks can be described as stressed. highly nonlinear, noncontinuous systems. Systems of this type are extremely difficult t o :~cciirately model, either mathematically or conceptually. Thus, current p o ~ ' e sqstem r simulations generally aim to model the slow, quasi-continuous dynamics most significant to system dispatchers, such as excessive voltage and apparent pouw variation with respect to changing demand. The details of transient beha\ior, with which most computational complexity is associated, are for the most part neglected. Ty p i c a I network si m u 1at i on s are consi de ra b 1y s i inp 1i fied and app 1-0x i in ;i t e (although they still incorporate hundreds of complex equations). I n addition. even the most comprehensive models include fundamental approximations to account for anticipated behavioral nonlinearity or noncontinuity and depend on critical assumptions about performance because we lack ;I complete understanding of the physical behavior of complex power system. For example, to fncilitate computation for conventional simulations, parameters like load \.alues are held constant, even though these parameters are known to \ m y slightly. And the modeling of systems that incorporate active control devices requires the use of assumptions because complete descriptions of device behaLrior are not yet av ai I ab I e . Observing that all models do not use the same approximations and assuniptions. their predictions of system behavior may differ strikingly, especially for

conditions of stress. Utility experts thus find it difficult to assess the validity ( ~ f a given model's results. Nonetheless, as long as predictions indicate that control measures will maintain generation-load imbalances within a normal operating range, dispatchers assume that the system will remain stable, and they are almost always correct. Recent studies of historical voltage collapse events suggest that traditional load simulations (such as those assuming constant load angle, constant impedance, and constant current) may not capture important voltage dynamics. For example, a voltage collapse may be initiated by the oscillatory behavior of gerierator exciters, but models that consider only the slow dynamics of the system ignore exciter dynamics and other transient behaviors. As a result, unsteady system states may be incorrectly assessed as stable, with continued operaticn possibly leading to local or systemwide failure. Chaos theory, the study of nonlinear dynamics, may provide new approaches for understanding the transient behavior of complex power systetiis and for ensuring network stability. Mathematical research during the past 2 0 years indicates that any system involving feedback can exhibit chaos. It can tie shown that chaos is present in simple power system models over a range of loading conditions. Recent research results suggest that events such as voltage collapse, low-frequency electromechanical oscillations, and transient stabilii y may be linked to chaotic behavior. It has been established that there is a relationship between voltage collapse and chaos-related bifurcations in voltage-reactive power solutions. Significa i t work has been carried out aimed at increasing our understanding of chaos arid transient stability. In particular, researchers examined in detail certain unusual, possibly chaotic behaviors observed in small power systems and are looking at tools for evaluating system stability under parametric or structural variations. Efforts focus on improving the characterization of bifurcations in simple arid complex power systems. Bifurcations, which represent qualitative transformations in a system's operational behavior (changes from stable to unstable states, for example, or the onset of multiple allowable solutions), can lead to chac-s. Because fundamental power flow equations have multiple solutions (swing d ynamics), bifurcations in power system behavior are always possible. A three-bus model was enhanced to include generator dynamics, revealing several additional bifurcations, some of which lead to chaotic oscillation through period-doubling cascades. For these cases, an infinite number of power flew solutions occur in response to only a small change in reactive load, causing rapid tluctuations in voltage and load angle that could produce power flow oscillations exceeding the thermal limits of transmission lines and lead to system collapsc!. Simulations focusing on these dynamic-generator-case bifurcations also indicate that voltage collapse may take place before the reactive power demand is increased to the system's steady-state operating limit, the point at which tie

static-case, saddle-node bifurcation occurs. Thus, the model system, and probably actual power systems as well, may be less stable under fluctuating realworld conditions than under the steady-state cases assumed in conventional stability calculations. Structural stability is a relatively new concept in power system analysis. Broadly speaking, a system is structurally stable if small variations in the model do not change qualitatively the set of trajectories originating from all initial conditions in the state space. For a given dynamic model. we examine system behavior subject to small disturbances in the domain of parameter space. This is different from Lyapunov stability where we wish to know if a perturbation in the state-of-the-system results in the trajectory returning to the equilibrium asymptotically. A dynamic system which is unstable in the sense of Lyapunov is structurally stable since the trajectories do not change for small changes i n parameter values. A complete characterization of structural stability for two-dimensional systems can be related to the nature of the equilibria and limit circles. For higher order systems such a complete characterization is not possible. Bifurcation theory is directly related to structural stability. Research work has been conducted that aims to provide dispatchers with a better understanding of the liinits of all significant structural parameters (such as load angle, current, impedance, and reactive power demand) so that power systems can be operated to maximize cost-effectiveness without initiating harmful chaotic behavior. Structural stability is important for power system security because key parameters are known to fluctuate slightly. For a system to be assessed as structurally stable, its behavior must return to its steady-state if perturbed. For example. a power network is considered structurally stable if slight changes in current or load angle do not appreciably affect the interactions between voltage and other system variables. (This differs from the more common notion of Lyapunov stability, which assesses stability on the basis of the initial values of system \,xiables.) Structural stability defines the parameter range within which system behavior is qualitatively similar; multiple ranges may exist for a given parameter. Little work has been done in this area for nonlinear models of power systems. Research has revealed that for systems characterized by more than two parameters, no complete description of structural stability is possible. But parametric limits of structural stability can be related conceptually and mathematically to bifurcations. Both indicate qualitative changes in system behavior and can be seen in phase portraits of a system. Thus, bifurcation analysis of system behavior can partially characterize structural stability ranges. It was observed that if control actions are taken into account, the critical values of parameters that influence voltage collapse may be lower than those predicted by static criteria. They postulate that the range over which a dynamic system is structurally stable is always smaller than the range for the same system

considered under static conditions only. Thus, parameter margins identified by con\vntional, static-case pourer system models to ensure safe system beha\,ior m a ~be ' obre 1.1 y perm i ss i ve for d y n ;i m i c , real - wor1d networks . Results suggest that knoLvledge of dynamic stability limits is key for relih i e control o f modern pourer systems. These margins must be satisfactory < i t all system buses for the bwying load demands typical over ;i 23 h r period. 1'0 identify the iniirgins. an enhanced understanding of active pouw netMwks is req U i red. Part i c U Iar I y i i n port ant are p h y sic a1 and iliat he nia t i ciil cle sc ri p t i on 5 ( ) 1' the beha\,ior o f i1ctii.e control de\,ices that fully account for their l'ast-s\t.itchiI g cI y niini ics. There is a need for better analytic capabilities i n both the areas of syste ii planning and operations. Promising techniques include the use o f energy t'un:tion methods. which are based upon the use of LyapunoLP's direct method f1)r assessing the stabilitjr of systenis of nonlinear differential equations. After inarrj' j'ears of research. eriergjf functions ;ire n o w successfully used to iissess systeiii transient stability i n the planning context. and are on the ~ ' c r g eof being uscd on-line to determine the transient stability limits. More recently. energy metholls 1iac.e also been employed thr assessing the system's quasi-static Lroltage securitJ . The relationship between the use of energy functions in these tct'o different contexts. ~ b ' i i sexamined by Overbye et al. ( I99 1 ). The key points for energy function usage in the transient stability problem is that the energy is used to determine whether the postfault is b,ithin the region o f iittrxtion of the postfault stable equilibrium point (SEP).arid that the deterniination o f the critical boundary of stability depends o n the f;iulted system trii-jector)'. In contrast. for the voltage security problem the system state is assumed to \rarl' cliiasi-statically (i.e., not subject to a large disturbance) in responw to t he slots, (with ;I time scale o f min to hr) Lwiation i n sq'stem parameters, such ;is the aggregate lorids. Loss of the system stability i n this context \\wild occur I- o t through ii large disturbance pro\-fidingthe system state with enough "energ)." to escape the SEPs energy "cvell." but rather through the coalescing of' the S I P wfith one of the tj'pe-one UEPs contained on the boundary through ii sxltlle node bifurcation iis the energy "u~ell"tlattens out. Because o f the netxi to model nonimpedance loads and to retain the network topology. the SPM has been used e xc I U s i v e 1y i n volt age security assess men t . I t has been proposed that the UEP energy, which is ;i measure of the clistancc betu.een the SEP and UEP, be used ;IS a measure for the sj'steni cliiirsistatic \.oltage security. Numerical testing has shown that the energy iiieasu~w tend to change i n a manner proportional to the change in the system's maximum steady loadability. System steady-state voltage security could thus be asses>ed bq, moni tori rig these energjr iiie;isiires, with decreasing energq' \ralues i ncliciiti ng

increased vulnerability to voltage instability; a \ d u e of energy approaching zero would indicate an imminent loss of system operating point. There is, ho\lfeirer, a significant difference i n the use of these UEP energies from their transient stability use. In transient stability the goal is to determine it .siri,qlc' cr-itictrl oiiorgj* W I I ~ M . which depends upon the fault-on system trti-jectory. I n contrast with the steady-state Lroltage problem one is concerned urith maintaining system voltage security while facing an imprecisely kno\bfnfuture system parameter \*ariation. While a future voltage collapse ~ . o u l dbe characterized by the coalescing of the SEP with a particular UEP, there is no need to determine precisely this UEP a priori. Rather the energies associated Lvith ;I number of UEPs would be monitored. It is shown that each UEP energ>r can be used to quantify the voltage stability in a particular portion of the system and that only the small subset of UEPs with low associated energies need to be calculated. This is because i n ii large system, there is a need to simiiltaneoiisl~~ monitor the voltage security in a number of separate areas. T u v major similarities in the application of energ}' methods to the transient and voltage problems is that the same energy formulation can be us\c.d i n both. and that both make use of the energy of the UEPs on the boundary of the region of attraction. In general for a three bus system, it is shokvn that the po~verflo\v equations have an SEP solution and another type-one UEP solution. Energj. methods could then be used for transient analysis by determining if the energ~' after a fault is cleared is less than that of a specific threshold. Whether the system possesses other solutions depends on the load model used. If the load is modeled as constant impedance then there are no other solutions to the power flow equations, since the network is linear and could be replaced by a Thevenin equivalent impedance. However. if the load is modeled as constant power, then the system has two additional solutions. Phase portraits can be used to illustrate a number of the mechanisms that could cause the system to lose its stable operating point: and that \s~ould thus need to be considered in security analysis. For the transient analysis. stability would be assessed by determining if the energy at the time the fault is clemxi is less than 19". The \Aue of IY'depends on the exit point (i.e.. the point \ b h x the critically cleared trajectory would cross the stability boundary & ) . Since this value would typically be by a large generator angle. the exit point \vould be in the Lricinity of the transient stability UEP. A new use of the energy function would be for dynamic voltage security assessment. As in the case of transient analysis, stability would be assessed by determining if the energj' after some disturbance is less than some critical value. Howe\.er. the distiirbance here u~ouldbe one that utould tend to cause the system trajectory to exit the SEPs UEP. Such a disturbance would be characterized by Io\v \.oltage niagnitu~ies(;is opposed to large generator angles), and may hmre ;i substantialljr longer time

frame than the transient stability problem (on the order of tnin as opposed to > ). Examples of such disturbances are the effects of LTC transformers, generators hitting reactive power limits, and load dynamics. Quasi-steady voltage security could be monitored by noting the variation in the UEP energy as the system state changes in response to the slow (on the order of min to hr) variation in system parameters, such as the aggregate loac s. The second segment of this book introduces a relatively new technolopy and its direct applications and future to power system engineers. Over the p a t few decades, the topic of artificial intelligence (AI) has gained the interest of researchers and scientists in many fields. The applications span many disciplines including electrical engineering extending its way to electrical power system analysis, protection, and enhancement. The resulting advantages will inherently depend on the specific applications of the existing methods that are available to the engineering sector. As discussed in prekfious chapters of the book, the primary branches i n artificial intelligence are Fuzzy Logic, Artificial Neural Nerworks, and Expcrt Systems. The obvious benefits of using Fuzzy Logic and/or Fuzzy Logic Expcrt Systems include the enhancement of the assessment criteria used in the study of large power systems, and also as decision support systems. This mode of 41 application as a support tool to the existing technique provides clearer solution alternatives to problems, in which case there are many variants with multi-levcls of uncertainties. Therefore the assessment of the power system is impro\,ed and the appropriate control actions to be exercised in the event of contingencies are more efficiently allocated, especially in a utility where the resources are either limited or expensive to use. Furthermore, the use of Fuzzy Logic and/or Expxt Systems i n power system stability studies can significantly reduce computational burden in some algorithms, as the human or engineering judgment is exploited in determining redundancies of choice. The classic example is cont ingency analysis of a power system, whereby it may not be necessary to consider (111 possihle cases, but rather cill (sufficiently)pr-ohcihle cases. The area of Artificial Neural Networks (ANN) has its place as a planning and operation tool or support tool also, especially i n light of some of the more recent restructuring developments that has occurred in the power industry. Thcse changes, such as deregulation of the power industry, have been a result of changes in the markets for electricity both as a corninodity and service. The challenge in the application of ANN lies in the unanswered questions that u'e have come face to face with as both engineers and power system tools and policy makers. For instance, what are the impacts of artificial intelligence support tools on power system reliability and stability'? Can the technology impro\.e the desired goals of the classical integration techniques (Runge-Kutta, Modified Euler, Trapezoidal, and so forth) making them better assessment methods'? How

can we perform detailed security assessnient of the p o u ~ system r isit11 the introduction of this enhancement method? I n conclusion. we have realized the capabilities and limitations of the existing methods utilized i n performing coniprehensi\re analysis on the ponrer s j ' s tem. both in the areas of steady state and transient stability studies. Maintaining power system stability is one challenge, but it is the llip-side of the coin that motivates us to go to work, which are the problems associated \\!it11 p o n w sjrstern instability. The sekrerity of the faults resulting from Lvltage and/or angle instability has led to the de\celopment of many methods of monitoring and pre\renting these unwanted results. Such methods. some of nthich u w e highlighted and discussed in detail in this book, are currently being enhanced \ k i the application of state of the art technologies. The cost-benefit analysis of such tools is yet to be done for large-scale power system applications. for both planning and operational purposes. In this light, continued research and the use of artificial intelligence as an enhancement to the computational and decision tool nil1 introduce much benefit to the power industry.

Glossary Accelerating torque: The difference between the input torque to the rotor aid the sum of the load and loss torques; the net torque available for accelerating the rotating parts. Activation function: Hidden units that are needed to introduce nonlinearity into a neural network. Active power: The average power in an electric power circuit is referred to as the active power. Admittance: The term admittance (symbol Y ) is used in a transform network (s-domain or complex, for sinusoidal steady state operation) to denote the inverse of the impedance parameter. The current through an element is the product of the element's admittance and the voltage across it. Angle stability: Stability of an electric power system evaluated on the ba:;is of the analysis of the dynamics of the rotor angle dynamics and the couplcd electric network. Angular acceleration: The rate of change of angular rotor speed. Angular speed: The rate of change of the rotor angle. Approximate reasoning: A computational rnodeling of any part of the pi.ocess used by humans to reason about natural phenomena. Artificial Neural Network (ANN): Physical cellular systems that can be used to acquire, store. and utilize experiential knowledge. Back propagation learning algorithm: The back propagation algorithm modifies synaptic connection strengths with nonlocal error information. The algorithm propagates the instantaneous squared error backward from 1 he output through the hidden layers to the input at each iteration. Back propagation or counter propagation neural networks: A neural netLvork training method that uses the generalized delta rule that uses a gradient descent inethod to minimize the total squared error of the output coinputed by the net. Capacitor bank: An assernbly at one location of capacitors and necess;irl' accessories required for a complete operating installation.

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Capacitive reactance: The imaginary part of the impedance of a predoniinantly capacitive element. Certainty factors: Guesses or projections by the domain expert about the relevance of evidence to support the recommendations passed by the inference engine of the rule-base system to the user. Clearing angle: The angle corresponding to the time elapsed from the beginning of overcurrent to the final circuit interruption. Conductance: The following are not equivalent but are supplementary: ( 1 ) The real part of admittance and (2) the physical property of an element that is the factor by which the mean square voltage must be multiplied to gi\.e the corresponding power lost as heat or as other permanent radiation or loss of energy from the circuit. Continuation power flow: The Jacobian matrix of the power flow computation becomes singular at the voltage stability limit. Continuation pourer flow overcomes this problem by using locally the parametrized continuation method involving predictor and corrector steps. Convergence: For an iterative process, convergence means that the successive differences between subsequent values of the unknown values ha\*eprogressively become closer and are below a certain predefined convergence tolerance threshold. Counter propagation: A neural network means that combine an unsupenised Kohonen layer with a teachable output layer. Crisp sets: The fuzzy set term for traditional set theory. That is. whether or not an object belongs to a set. Critical clearing angle: For a system of one machine connected to an infinite bus. and for a given fault and switching arrangement, the critical clearing angle is that switching angle for which the system is at the edge of instability. Current: A generic term used where there is no danger of ambiguitjt to refer to any one or more currents specifically described. The use of certain adjectives before "current" is often convenient, as in conducting current. Database: A collection of clauses with each clause on a database representing something that is known to be true. Decision trees: Tools for making number-based decisions where a lot of coinplex information needs to be taken into account. Direct method: The direct method is usually referred to as Lyapunoi. second method of stability assessment, which attempts to determine stabilitj b j using a suitable Lyapunov function. Direct-axis: The axis that represents the direction of the plane of symmetry of the no-load magnetic flux density, produced bq' the main field \+inding. normally coinciding with the radial plane of symmetry of a field pole.

Direct-axis component of armature current: The component of the arniatitre current that produces a magnetomotive force distribution that is sy 11metrical about the direct axis. Direct-axis component of armature voltage: The component of the armatlire i.oltage of any phase that is in time phase with the direct axis coniponc!nt of current in the same phase. A direct axis component of tdtage may be produced by: ( I ) rotation of the qiiadrature axis component of rnagnctic flux, ( 2 ) w-iation (if any) of the direct axis component o f magnetic fliix. and ( 3 ) resistance drop caused by the tlow of the direct axis component of armature current. Direct-axis component of magnetomotive force: The component of rnagiietomotit'e force that is directed along the direct axis. Direct- axis current : The c Urren t that prod Uce s d i rect -ax is magneto mot I \.c force. Direct-axis magnetic-flux component: The magnetic-flux component directed along the directed axis. Direct-axis subtransient impedance: The magnitude obtained by the vector addition of the value for the armature resistance and the value tor diicct axis subtransient reactance. The resistance ~ a l u eto be applied in this c;iw will be ;I function of frequency depending on rotor iron losses. Direct-axis subtransient reactance: The quotient of the initial \Aue of ;I sudden change in that the fundamental alternating currcnt coniponent of armature cmoltage, that is produced by the total direct-axis priiiiary flux. vind the \,alue of this simultaneous change i n fundumental alternating curt cnt component o f direct axis armature current, the machine running at r; ted speed. Direct-axis subtransient voltage: The direct-axis component of the term nal \,oltagc \+rhich appears iinmediately after the sudden opening of the extoi'nal circuit when the machine is running at a specified load, before any i'lux variation in the excitation and damping circuits has taken place. Direct-axis synchronous impedance: The magnitude obtained by the \w:tor addition of the \ d u e for armature resistance and the \lalue for direct ;\xis sy tic hronous reac t iince. Direct-axis synchronous reactance: The quotient of a sustained \Aue of that fundamental alternating current component of armature \dtage that is produced by the total direct axis flux due to direct axis armature current and the vdue of fundamental alternating current component of this current. the machine running at rated speed. Unless otherwise specified, the V;IIUC of synchronous reactance will be that corresponding to rated armature curient. For most machines, the arrnature resistance is negligibly small compared to the synchronous reactance. Direct-axis transient impedance: The inagnitude obtained by the \rector xi-

30 I

dition of the value for armature resistance and the value for direct axis transient reactance. Direct-axis transient reactance: The quotient of the initial due of a sudden change in that fundamental alternating current component of armature \,()Itage, which is produced by the total direct axis flux, and the value of the simultaneous change in fundamental alternating current component of direct axis armature current. The machine is assumed to be running at rated speed. Direct-axis transient voltage: The direct axis component of the armature voltage that appears immediately after the sudden opening of the external circuit when running at a specified load. neglecting the components that decay in the first few cycles. Direct-axis voltage: The component of voltage that would produce direct axis current when resistance is limited. Distribution system: The distribution system is the final stage in the transfer of power to the individual customers. Dynamic response: An output expressed as a function of time resulting from the application of a specified input under specified operating conditions. Dynamic security assessment: The process of using tools of transient stability analysis in predicting the vulnerability of a power system to contingencies. Electric power system: The function of an electric ponw system is to convert energy from one of the naturally available forms to the electrical form and to transport it to the points of consumption. Equal area criterion: For a rotor that is accelerating. the condition for stability is that a maximum value of the rotor angle exists and that the area under accelerating power versus rotor angle curve is zero up to that maximum rotor angle. Excitation control: The function of excitation control is to regulate the generator voltage and reactive power output. Excitation system: The equipment providing field current for a synchronous machine, including all power, regulating. control, and protective elements. Expert systems (ES): A computer program that processes problem-specific information in the working memory with a set of rules contained i n the knowledge base system, using an inference engine to infer new information. Fault: A physical condition that causes a device, a component, or an element to fail to perform in a required manner. for example, a short-circuit. a broken wire, an intermittent connection. Feedback: The return of a fraction of the output to the input. Feeder: A set of conductors originating at a main distribution center and supplying one or more branch circuit distribution centers. or any combination of these two types of equipment. Fictitious voltage: Voltage terms introduced in transient equations to explain

the flux inertia linkages created by rapid changes in conditions external to the machine (see Transient equations). Flux linkages: The sum of fluxes linking the terms forming the coil. Forward chaining: The process used in an expert system for deriving new information from known information. In forward chaining of rules, all rules whose pre-conditions are fulfilled by the data set of the given observations and derived intermediate results are eventually triggered. Frequency: The number of complete cycles of sinusoidal variation per unit time. Frequency response: A characteristic, expressed by formula or graph, which describes the dynamic and steady-state response of a physical system in terms of the magnitude ratio and the phase displacement between a sinuso dally krarying input quantity and the fundamental of the corresponding 011 put quantity as a function of fundamental frequency. Fuzzy expert system: An expert system that uses a collection of fuzzy m e w bership functions and rules, instead of Boolean logic, to reason about datii. Fuzzy logic: Fuzzy logic deals with ambiguity in defining various variabks involved in describing the operation of a system. Fuzzy logic systems (FLS): These are systems which mathematically modl:l complex relationships that are usually handled in a vague manner by 1a11g U age or 1i n g u ist ics. Fuzzy quantization: A mathematical means of describing vagueness in l i i i guistics. The membership functions defined on the input variables are applied to their actual \ d u e s in order to determine the degree of truth for the premise of each rule. Fuzzy reasoning: A fomal mathematical procedure for the representation ~ ) f uncertainty that is used in the management of real systems. Fuzzy set theory: A mathematical formulation developed by Dr. Lotfi Zadvh of UUBerkeley in the 1960s as a means to model the uncertainty of natural language. The theory is used to solve problems that contain vagueness 'or uncertainty in the representation of knowledge and Pactual statements. Fuzzy subset: Fuzzy subset F of a set S can be defined as a set of ordercd pairs. each with the first element from S, and the second element from the interval [O. I ] . with exactly one ordered pair present for each element of S. Governor: The assembly of fluid, electrical, or mechanical control equipment used for regulating the tlow of water, steam. or other medium to the p r i m mover for such purposes as starting, holding speed or load, or stopping. IF-THEN-ELSE Rule: A basic knowledge representation by means of w-hich directional relationships between objects can be represented using a pr econdition and an action that is often supported by the certainty of the actic'n. Impedance: The term impedance (symbol 2 ) is used in a transform netnwrk (s-doinain or complex, for sinusoidal steady state operation) to relate cur-

rent through an element and the voltage across it, via the extended Ohm's law v = ZI. Inductance: The property of an electric circuit by virtue of which a varying current induces an electromotive force in that circuit or a neighboring circuit. Induction motor: An ac motor in which a primary winding on one member (usually the stator) is connected to the power source and a polyphase secondary winding, or a squirrel cage secondary winding on the other member (usually the rotor) carries induced current. Inertia constant: The energy stored in the rotor when operating at rated speed expressed as kilowatt-seconds per kilovoltampere rating of the machine. Inference engine: The brain of the expert system as it contains the knowledge that is used to decide on how to apply the rules to infer new knowledge. It controls the firing of the rules in the expert system. Infinite bus: A voltage source that maintains constant voltage and frequency regardless of the load connected to it. Intelligent systems: These are man-made systems that emulate functions of living creatures and human mental faculties. Iterations: Describes a process, which repeatedly executes a sequence of steps until some condition is satisfied. Knowledge-based system (KBS): A sub-program for solving problems, in which the problem-solving methods and the knowledge are kept separate, which makes it amenable to modification by exchanging the knowledge with the same problem-solving method and contributes to the explainability by enabling the knowledge used for deriving the problem solution to be given. Lag: The delay between two events. Laplace transform: The quantity obtained by performing the operation:

where s = 0 + jm. Learning algorithm: These are computational procedures that are generally aimed at ill-defined and time-varying processes and use heuristics, adaptation and pattern recognition techniques to construct neural networks and rule-based systems. Linear system: A system or element with the properties that if is the response to .rI, and y2 is the response to x2, then ,!( + p ) is the response to (.r, +.r2), and k y , is the response to ks,.

Load: The electric power used by d e ice\ ~ connected to an electrical generating sytetii. Load angle: The angular di\placement. at a specified load. of the center lire of ;I field pole froni the axi\ of the armature magnetoriioti\ e force pattern. Load angle curve: A characteri\tic curve giving the relation\hip between tt e rotor di\placement angle and the load, for con\tant value\ of wmature I oltage, field current. and power factor. Load center: A point at which the load of ;I gi\en area i \ a\\iinied to tx concentratcd. Load model: A mathematical representation of the \ ariation of the acti\ e ar~d reactiLe power ~-equirement\o f the load at ii certain point in the \j {te ii ttrith Loltage a n d frequency at that point. Lyapunov function: A \calar differentiable function V( 1 defined in \oiiic open region including X , such that in that region:

Man-machine interface: A niedium through which the user of the expert s j stem is able to communicate ufith the computer allouring data transfer i n t ic foriii of i n pu t data, re po r t s, s y s t e tii tii ai n t e n ;i 11ce , and re s U 1t s . hlechanical power: The power imparted by the prime m o ~ wto the synchionous generator. hlembership function: I t is a graphical representation ot' the magnitude ot participation of each input to the fuzzy system and by computing thc 1ogic.d product of the membership weights for each acti\re rule, a set o f fuc./j' 011 t pu t res po ti se iiiag n it iide s are produced. hleta-knowledge: The k~iou~ledge that ;I system has about its o ~ . ' ikiiou.lcdgc i that characterizes its ou'ti workings. This could include probabilities ot' wc h i t ec t U r;i I moduIe , e f fic i e ticie s o r de pe ndabi I i tie s. hlonotonic reasoning: A deduction in the theory of classical logic that c;in be proven to be valid under all conditions for all time. Neural network: Neural networks consist of numerous, simple processing units o r "iieurons" that we can globally program for computation. Nc~ir;il tie t uvrks are t rii i tied to store, recognize. and ;is soc i a t i \re I y re t ri e\re pat t eims o r database entries, to sol\.e problems in\~olvingestimating sampled t'uiictions when we cio not know the form of the functions. Neuron: The neuron is the fundamental unit of the n e r \ w s system, particularlj. the brain. The neuron is ii simple processing u n i t that receikw ;nd c w it1bi ties s ig ti;i I s fro 111 inanq' other tieu roils t h roiigh fi I ii me 11t arj' i 11pit t path s.

Newton's method: An iterative method for finding the solution to a set of nonlinear equations using first order derivatives. Nonlinear system: A system described by a set of nonlinear equations. I n this case the superposition principle does not apply. Perception learning rule: An iterative learning procedure that usually converge to the correct weights that will ultimately give the correct output for all input training patterns. Phase angle: The measure of the progression of a periodic nave i n time or space from a chosen instant or position. Positive definite: A quadratic form V ( x )= .Y' Q X is positive definite if \'(.v) > 0 for all values of .Y different from zero. Power factor: The cosine of the angle between the current through ;in electric circuit element and the voltage across it is referred to as the element's power factor. Power flow equations: The system of equations relating the active and tmctive power injections at each bus in an electric network i n terms of voltage magnitudes and phase angles. Power system stability: The term is used to define the ability of the bulk power electric system to withstand sudden disturbances such as electric short circuits or unanticipated loss of system components. For further details see Sec. 1.3. Prime mover: The machine used to develop mechanical horsepomw necessary to dri\ve ;I generator to produce electrical pourer. Prime mover control: The function of prime mover control is to regulate the speed and control energy supply variables such boiler pressures, temperatures, and tlows. Quadrature axis transient impedance: The operator expressing the relation between the initial change in armature voltage and a sudden change in quadrature axis armature current component, under the following assumptions: ( 1 ) Only the fundamental frequency components considered for both voltage and current. (2) No change in the voltage applied to the field Liyinding. ( 3 ) The rotor running at steady speed. (4) Considering only the slokt'est decaying component and the steady state component of the \voltage drop. If no rotor winding is along the quadrature axis and/or the rotor is not made out of solid steel, this impedance equals the quadrature-axis synchronous impedance. Quadrature axis: The axis that represents the direction of the radial plane along which the main field winding produces no magnetization. nornially coinciding with the radial plane midway between adjacent poles. The positive direction of the quadrature axis is 90" ahead of the p0sitii.e direction of the direct axis, in the direction of rotation of the field relati1.e to the armature.

Quadrature axis component of armature current: The component of the armature current that produces a magnetomotive force distribution that is symmetrical about the quadrature axis. Quadrature axis component of armature voltage: The component of the armature voltage of any phase that is in time-phase with the quadrature axis component of current in the same phase. A quadrature axis component of voltage may be produced by: ( I ) rotation of the direct axis component of magnetic tlux, ( 2 ) variation (if any) of the quadrature axis component of magnetic tlux, (3) resistance drop caused by the flow of the qiiadratuie axis component of armature current. Quadrature axis current: The current that produces quadrature axis magnctomotive force. Quadrature axis magnetomotive force: The component of a magnetomotive force that is directed along an axis in quadrature with the axis of tt-e poles. Quadrature axis subtransient impedance: The operator expressing the relation between the initial change in armature voltage and a sudden change . n quadrature axis armature current, under the following assumptions: ( I ) On y the fundamental frequency components considered for both voltage and current. ( 2 ) N o change in the voltage applied to the field winding. (3) The rotor running at steady speed. If no rotor winding is along the qiiadratu-e axis and/or the rotor is not made out of solid steel, this impedance equals the quadrature axis synchronous impedance. Quadrature axis subtransient reactance: The ratio of the fundamental coriponent of reactive armature voltage, due to the initial value of the fundamental quadrature axis component of the alternating current component q.>f the armature current, to this component of current under suddenly applicd balanced load conditions and at rated frequency. Unless otherwise stated, the quadrature axis subtransient reactance is that corresponding to ratcd armature current. Quadrature axis subtransient voltage: The quadrature axis component of the terminal voltage that appears immediately after the sudden opening of the external circuit when the machine is running at a specific load, before any tlux variation in the excitation and damping circuits has taken place. Quadrature axis synchronous impedance: The impedance of armature winding under steady state conditions where the axis of the armature current and magnetomotive force coincides with the quadrature axis. In large synchronous machines where the armature resistance is negligibly small. t ne quadrature axis synchronous impedance is equal to the quadrature axis synchronous reactance. Quadrature axis synchronous reactance: The ratio of the fundamental coinponent of reactive armature voltage, due to the fundamental quadrature axis

307

component of armature current, to this component of current under steadystate conditions and at rated frequency. Unless otherwise stated, the \,alue of quadrature axis synchronous reactance will be that corresponding to the rated armature current. Quadrature axis transient reactance: The ratio of the fundamental component of reactive armature voltage, due to the fundamental quadrature axis component of the alternating current component of the armature current, to this component of current under suddenly applied load conditions and at rated frequency. The value of current is determined by extrapolating the envelope of the alternating current component of the current wave to the instant of the sudden application of the load, neglecting the high-decrement current during the first few cycles. Note that usually the quadrature axis transient reactance equals the quadrature axis synchronous reactance except in solid-rotor machines, since in general there is no really effective field current in the quadrature axis. Quadrature axis transient voltage: The quadrature axis component of the armature voltage that appears immediately after the sudden opening of the external circuit when running at a specified load, neglecting the components that decay in the first few cycles. Quadrature axis voltage: The component of voltage that would produce quadrature axis current when resistance is limited. Reactance, effective synchronous: An assumed value of synchronous reactance used to represent a machine in a system study calculation for a particular operating condition. Reactive capability curve: A curve that identifies the reactive capability limits of a synchronous machine based on three considerations: ( 1 ) Armature current limit; ( 2 ) field current limit; and (3) end region-heating limits. Reactive power: The reactive power Q is defined as the square root of the square of the apparent power S minus the square of the active power P.

Reactive power is developed when there are inductive, capacitive, or nonlinear elements in the system. It does not represent useful energy that can be extracted from the system but it can cause increased losses and excessiire voltage peaks. Region of stability: A region in the state space, where the system is stable. Regulator: An electric machine regulator that controls the excitation of a synchronous machine. Regulator, excitation system: A regulator that couples the output variables of the synchronous machine to the input of the exciter through feedback and

foruwd controlling elements for the purpose o f regulating the synchronoi~~ machine output variables. Resistance: The physical property of an element, deLfice,branch, netuwk. o . system that is the factor by urhich the mean square conduction current must be multiplied to give the corresponding power lost by dissipation as heat o r as other permanent loss of energy from the circuit. Simulation: The representation of the functioning of one system by another-. for example to represent a physical system by ;i mathematical model. Slip: The difference between the synchronous speed and the actual speed of , I rotor to the synchronous speed. Soft computing: A term coined by Zadeh to refer to the emerging area of’ coinputational intelligence such as fuzzy logic, neural networks. geneti,: a I go r i t h 111s. and e x pe rt sy ste 111s. Stability: An aspect of system behavior associated with sj,stems ha\!ing th: general property that bounded perturbations result in bounded perturbation?, in the output. Stability, absolute: Global asymptotic stability maintained for all gikren not’linearities. Stability, Lyabunov: For ii solution $(.\-(t , ) ) , t ) ,Lyabunov stability means that for every gi\.en E > 0 there exists a 6 > 0 such that liAv(t,,)liI 6 implies I~ASI]~ 5 E for t 2 t,,. Stability, power system: In a system of two or more synchronous machint s connected through an electric network, the conditions in which the diffe ence o f the angiihr positions of the rotors of thc machines either rernair I?, constant w hi 1e not subjected to a di st Ur batic e , or becomes con st an t fo1I ov ing an aperiodic disturbance. Stabilizer: An excitation system stabilizer is an element or group of elenienis that modify the forward signal by either series or feedback compensatic 11 to improve the dynamic performance of the excitation control system. Stabilizer, power system: An element or group of elements that provide ;.II additional input to the regulator to improve pouw system djrriamic perfori i i ~nce i . Static: Refers to a state in which a quantity exhibits no appreciable chanlre within an arbitrarily long time interval. Security assessment: I n an operational environment, security assessment co 1sists of predicting the vulnerability of the system to possible disturbance!,. Steady state: That in m!hich some specified characteristic o f a condition such as value, rate. periodicity, or mplitude exhibits only negligible change over :in arbitrarily long time. Steady state contingency analysis: Steady state contingency analysis predic t x power tlows and bus voltage conditions following events such as transmi ssion line outages. transformer outages and generator outages. *-

1

Steady state stability: A condition that exists i n a power system if it operates with stability when not subjected to an aperiodic disturbance. I n practice, a variety of relatively small aperiodic disturbances may be present without any appreciable effect upon the stability. This is \.did as long as the resultant rate of change in load is relatively slow in comparison m i t h the natural frequency of oscillation of the major parts of the system or \,it11 the rate of change in field flux of the rotating machines. Steady state stability limit: The maximum power tlow possible through sotne particular point in the system when the entire system or the part of the sjfstem to which the stability limit refers is operating with steady state stability. Subtransient current: The initial alternating component of armature current following a sudden short-circuit. Subtransient Reactance: The reactance of a generator at the initiation of a fault. Subtransmission system: In contrast to the transmission system. the subtransmission system transmits power at a lower voltage and in sinaller quantities from the transmission system to the distributions substations. Susceptance: The imaginary part of admittance. Synchronous generator: A synchronous alternating current machine that transforms mechanical power into electrical po~ver. Synchronous machine: A machine in which the a\rerage speed of normd operation is exactly proportional to the frequency of the system to which i t is connected. Synchronous reactance, effective: An assumed due o f synchronous reactance used to represent a machine in a system study for a particular operating condition. Synchronous speed: The speed of rotation of the niagnetic t l u s produced bjr or linking the primary winding. System generation control: The function of system generation control is to balance the total system generation against system load and losses so that the desired frequency and power interchange with neighboring systems is maintained. Three phase system: A combination of circuits energized by alternating electromotive forces which differ in phase by one third of a cycle (120"). Time delay: A time interval which is purposely introduced in the pert'ormance of a function. Torque: The vector product of force and moment arm and is uridelj. designated by the unit Newton meter. Transient current: The current under nonsteady conditions. Also. the alternating component of armature current immediately follonting a sudden short circuit, neglecting the rapidly decaying component present during the first few cy c I e s.

Transient energy function: This is a Lyapunov function used in the direct method for transient stability analysis. Transient equations: Used whenever conditions in the modeled system change faster than a static representation of voltage and tlow can acconim.)date. Under such conditions. tlux inertia means these rapid changes cannot establish themselves throughout the machine immediately, and two new fictitious \dtages, usually referred to a s E(; and E(; must be introduced to represent the tlux linkages in machine rotor winding, and used in the model equations. Transient reactance: The reactance of a generator between the subtransient and synchronous states. Transmission control: These include power and voltage control devices s1iC-h as st a t ic V A R co111 pen sat or s, sy nchronou s con de n sers. swi t c he d capac i t c r s and reactors, tap-changing transformers, phase shifting transformers. a id H VDC transmission controls. Transmission line: A line used for power transmission. Transmission system: A transmission system interconnects all major gencrating stations and main load centers in an electric power system. Turbine-generator unit: An electric generator with its driving turbine. Voltage: Voltage is synonymous with potential difference between two wnductors. Voltage collapse: It is the process by which the sequence of events accompanying voltage instability leads to a low iinacceptable voltage profile ir ;I significant part of the power system. Voltage regulator: A synchronous machine regulator that functions to maintain the terminal voltage of a synchronous machine at a predeterniircd \ d u e . or to vary it according to a predetermined plan. Voltage stability: A voltage stability study evaluates the ability of a pov,w system to maintain acceptable voltages at all nodes under normal conditions and after being subjected to contingency conditions. Voltage stability indexing: Many indices characterizing the proximity of an operating state t o the voltage collapse point have been developed. The 11egeneracy of the load flow Jacobian matrix is used a s an index of the po\b'er system steady state stability. 1

Appendix: Chapter Problems

PROBLEMS FOR CHAPTER 2 Problem 2.1

A single-phase source is connected to a two-terminal, passive circuit with equiLtdent impedance measured between the terminals given by:

z= (2.0 3 2 . 0 ) R The source current is i(t) = 4 cos o tkA. Determine: a. The instantaneous power, b. The real power, and reactive power delivered by the source, and c. The source power factor. Problem 2.2

Consider a single-phase load with an applied voltage: \,(I) =

and load current

120 cos

(Wt

+ 10") volts

i(r)

= 8 c o s (cot - S0")A.

Determine the power triangle. Find the pourer factor and specify Lvhether it is lagging o r leading. Calculate the reactive poufer supplied by capacitors i n parallcl u'ith the load that correci 4 the pourer flictor to 0.9 lagging. Problem 2.3 A circuit consist5 of t u o impedances, Z , = 2 0 30"Q and Z; = 30 - 4Soi2 i n parallel, supplied by ;i source boltage V = 120 - 6O"\olts. Determine the p o c ~ c r

triangle for each of the impedances and for the wiircc. Problem 2.4

An inclustrial plant consisting primarily of induction motor loads absorbs 1000 hW at 0.75 power factor lagging. Compute the required k V A rating of ii shunt capacitor t o improkre the poner factor to 0.9 lagging. b. I f a synchronou\ motor rated 1000 h p u i t h 90% efficiency operating at rated load and at unity p o ~ e frrictor i \ added to the plant instead of' t'ie capacitor, c ;I I c LI1;i tc t hc re s 111t i ng pourer I'ac t or. A \ \ 11me con \ t ;in t \ o It age. ( 1 h p = 0.746 k W ) ;i.

Problem 2.5

The real power delicered by ii soiirce to two impedances. Z, = (3.0 + j S . O ) i2 and Z: = 10.0 i2 connected i n parallel, is IS00 W. Determine: ii.

b.

The red power absorbed by each of the impedances. The source current.

Problem 2.6 A \inglt.-phaw \oiirce ha5 a terminal voltage V = 120 Lwlts rind a current 1 = 25 W A , Lvhich Iea\e\ the positibe terminal of the \ ~ i i r c e .Determine the r:al L

-

and reacti\ e power, and \tate

1% hether

the source i \ delivering o r absorbing u : h .

Problem 2.7 A source supplies p ~ ~ toe the r following three loads connected in parallel: ( 1 ) load draLving I0 kW, ( 2 ) an induction motor drauing 10 k V A at 0.90

;I lighting

power factor lagging, and (3) a synchronous motor operating at 10 lip, 8Sck efficiency and 0.95 power factor leading ( 1 hp = 0.746 M). Determine the real. reactive, and apparent poufer delivered by the soi~rce. Also, draw the source power triangle. Power 2.8

Three identical impedances of (26 + j l 5 ) 52 are connected in wye t o a 460 V balanced three-phase source. Determine: a. The magnitude of the line currents. b. The total power dissipated for the three phases. Problem 2.9

Three identical impedances of (18 + j 2 2 . 5 ) 52 are connected in balanced three-phase source. Determine: a. b. c. d.

The The The The

NIT

to a 550 V

magnitude of the line currents. total power dissipated for the three phases. total reactive power, and power factor.

Problem 2.10

Repeat Problem 2.9 for the case where the three impedances are connected i n delta. Problem 2.11

Current, voltage, and power to a balanced three-phase circuit are meawred and found to be 20 A, 550 V, and 10.5 kW. respectively. Determine equi\.alent circuits per phase as follows: a.

Wye-connected, series combination of resistance and reactance i n each phase. b. Delta-connected, parallel combination of resistance and reactance in each phase. Problem 2.12

Voltage, apparent power, and power to a balanced three-phase circuit are measured and found to be 460 V, 50 kVA, and 48.5 kW respectively. Determine equivalent circuits per phase as follows:

a.

Wye-connected, parallel combination of resistance and reactance i i i each phase. b. Delta-connected, series combination of resistance and reactance in each phase.

Problem 2.13

The current, voltage. and power factor of a balanced three-phase circuit are measured and found to be IS A, 440 V, and 0.75 lagging respectively. Determinlz equivalent series-connected resistance and reactance circuits per phase if th: phases are: a. b.

Wye-connected. De 1t a-con nec t ed .

Problem 2.14

The voltage, apparent power, and power factor of a balanced three-phase circuit are measured and found to be 600 V, 150 kVA, and 0.9 leading respectivelj. Determine equivalent parallel connected resistance and reactance circuits per phase if the phases are:

a. Wye-connected. De 1t a-con ne ct ed .

b.

Problem 2.15

Three impedances are connected in delta to a balanced 208 V, three-phase source of sequence abc-. The impedances are Z,,ll= I0 + j 20

n

z,,, = 20 -jl0 R z,,, = 20 + j 10 R What are the three-phase voltages'? Calculate the three-phase currents. c. Calcualte the three-line currents.

a.

b.

Problem 2.16

Three impedances are connected in delta to a balanced 460 V, three-pha?e source of sequence cdx. The impediances are

3 I5

Z(,/,= 2s +j 1 5 R Z,, = 17 -j18 R z,,, = 20 + j 2 0 R a. What are the three-phase voltages? b. Calculate the three-phase currents. c. Calculate the three line currents. Problem 2.17

Two loads are connected to a 460 V, three-phase balanced source. One is a three-phase motor connected in delta and running such that the power is 25 kW with a line current of 35 A. The power factor is known to be lagging. The other is a single-phase 10 kW heater that takes a unity factor current of 22 A when connected between lines b and c. Using V,,,,as a reference. determine the three line currents. Problem 2.18

Two loads are connected to a 460 V, three-phase balanced source. One is a three-phase balanced load connected in wye and haLting a line current of 2 0 A with a power factor of 0.9 (lagging). The other is a single-phase load which has a current of 15 A at a power factor of 0.7 (leading) when connected between as a reference, determine the three line currents. lines a and c. Using Problem 2.19

Two loads are connected to a 208 V, three-phase balanced source. One is a three-phase motor connected in delta and running such that the line current is 10 A with a power factor of 0.866 (lagging). The other is a single-phase heater which takes a current of 15 A at a power factor of 0.98 (lagging) ufhen cotinected between lines a and b. Using Vllbas a reference, determine the three line currents. Problem 2.20

Two inductive loads are connected to a 460 V, three-phase balanced source. One is a three-phase balanced load of 50 kW connected in wye and having a line current of 125 A. The other is a single-phase load of 5 kW and 10 kVA connected between lines a and c. Using V,,,?as a reference. determine the three line currents.

Problem 2.21

A three-phase induction motor is connected to a balanced S S 0 V, 60 Hz supplq For a particular mechanical load the input is 100 kVA and 80 kW. The pouer factor is to be increased to 0.95 (lagging) by means of a delta-connected capaci tor banh connected to the motor terminal\. Determine the capacitance per pha\c. required. Problem 2.22

When a certain three-phase induction motor is operated at its rated load thc current, voltage, and power are 70 A, 550 V. and SO kW respectively. A seconcl motor, when connected to the same source, takes a current of SO A and a powerof 30 kW. Normally both motors operate simultaneously. Assuming that thc system frequency is 60 Hz. Determine the delta-connected capacitance per phase required to raiw the power factor to 0.95 (lagging). b. With thi\ value of capacitance remaining i n the circuit. determine thc re\ulting power f'actor M hen the second motor is disconnected.

;I.

Problem 2.23 A certain inductikre, balanced three-phase load dissipates 60 hW with a currer t o f 66 A L+,henconnected to ;I S S 0 V, 60 Hz supply.

Obtain the parameters o f the equivalent wye-connected circuit in whic h the reactance and resistance are connected in sereies. b. A set o f three capacitors, each 500 CIF,is connected in \cries with the load. Determine the current, voltage, and pouw of the original load. c. Obtain the ow-all power factor. ;I.

Problem 2.24 A sq nchronous machitie has the following inductance\ associated uith the xtator w i nd i ng s :

f., = 3.3 + 0.05

CO\

L,,,,= - I .6 - 0.05

26 mH

CO\

[ 217 + x3 1 I

~ H

Use equations 2.26 and 2.30 to determine L,, and L,, in Henrqrs

Problem 2.25

A 500 MVA, 24 kV, 60 Hz three-phase synchronous machine has the f o l l o ~ ~ i n g inductances in Henrys: L,, = 5 mH L,, = 4.5 m H

Determine the base impedance and inductance. and then find the per u n i t value of L,, and L,,. Problem 2.26

A synchronous machine serves a load with:

v,= I

i

10" /"I

1,= 0.5 ,-2O"

/"I

Assume that X,, = 1.2 and neglect the armature re4stance. Find the \aIue of E' and 6 based on the equivalent circuit of Fig. 2.10. Problem 2.27

Consider a static load represented by the model of Eq. (2.40). It is hno\\n that ri 10% increase i n voltage magnitude results i n a 15% increaw i n p o er. ~ Predict the percentage change in power for a 10% decrease i n Ldtage. Problem 2.28

The model of Eq. (2.41) represents the dependence of the actiLre po\s-er of ;1 static load on voltage. The following measurements are available for incre;ises in the voltage magnitude and the corresponding increase i n active ponter:

AV%

AP%

10 15

58.5 68.13 78.00

20

Predict the change in active power for a 10% decrease i n \>oltagcmagnitude.

318

PROBLEMS FOR CHAPTER 3 Problem 3.1

Consider the exciter model given by Eq. (3.1) with K, = 1.00, S, =0.8, and T, = 0.5 s. Assume that the voltage reference is ;i unit step. Find the exciter voltage output Y , a s a function of time. Assume zero initial conditions.

Problem 3.2

Find the steady state \Aue of the exciter voltage for Problem 3.1.

Problem 3.3

Use the block diagram of Fig. (3.5) for a typical stabilizer to establish a state space model of the system.

Problem 3.4

ASSLIIII~ that the input steps for the stabilizer of Fig. (3.5) are given by AO = 0.1 and A P ! . Find the steady state \ d u e of AV,. Assume that: T , = 0.I

7;= 0.05

K,, = 0.6

K,, = 0.8

7 ;= 0 . 2

Problem 3.5

Consider the model of Fig. (3.8) represented by Eq. (3.7). Write a state space model for the system in terms of four states. Define the control input as: Ldt) = CO,,,- CO,

Assume that: T,, = 0.2

T, = 5

K, = 0.4

q,= 0.05

K, =5

K,,= 0.04

Find the response to a unit step input.

Appendix

Problem 3.6 Repeat Problem 3.5 using a three state model involving x I ,s2,and q.

Problem 3.7 Consider the model of Fig. (3.9). Write a state space model for the system in terms of the two states. Define the control input as: u ( r ) = a,,, - 0,

Assume that: T,; = 0.2

K, =4

R,, = 0.05

Find the eigenvalues of the system and the response to a unit step input.

PROBLEMS FOR CHAPTER 4 Problem 4.1 Given a 60 Hz, four-pole turbo generator rated 20 MVA, 13.2 kV, with an inertia constant of H = 7 kWs/kVA: a. Calculate the kinetic energy stored in the rotor at synchronous speed. b. Find the acceleration if the net mechanical input is 26,800 hp and the electric power developed is 16 MW. c. Assume that the acceleration in part b is constant for a period of 10 cycles. Find the change in 6 in that period. If this generator is delivering rated MVA at 0.8 PF lag when a fault reduces the electric power output by 50%, determine the accelerating torque at the fault time.

Problem 4.2 A 60 Hz alternator rated at 20 MVA is developing electric power at 0.8 power factor lagging with net mechanical input of 18 MW. Assume that acceleration is constant for a period 15 cycles, in which 6 attains a value of IS" electrical from zero initial conditions. Calculate the inertia constant H for the machine.

Problem 4.3 Show that the speed of a generator subject to a constant decelerating power of 1 p i 4 will be reduced from rated value to zero in 2 H seconds.

320

Problem 4.4 A 20 MVA. 13.8 k V , 60 HL. two-pole, Y-connected three phase alternator ha\ an armature w inding re4stance of 0.07 ohms per phase and a leakage reactanc c of 1 .9 ohms per phase. The ;irni;itiire reaction EMF for the machine i \ relatcd to the ;irtii;itiire current by E,,, = -jlO.O 1 I ( / . Assume that the generated EMF i \ related to the field current by E, = 60 I,. a.

b.

Coiiipiite the field current required to establish rated kwltage across the terminal\ of a load \+!henrated armature current i \ delikwed at 0.8 F'F I agg i ng . Conipiite the field current needed IO prob idc rated terminal \ d t a g e o ;i load that draw\ 10074 of rated current at 0.85 lagging.

Problem 4.5

A 10 MVA. 13.8 k V , 60 H L . two pole, Y-connected synchronous generator i \ deli\wing rated current at rated voltage and unity PF. Find the armature resi+ tance a1id \ynchro1ioii\ reactance gicren that the field excitation ~oltage i + 1 1035.41 V m d lead\ the tcrminal cxdtage by an angle 47.96".

Problem 4.6 A 1500 hVA. three-phase, Y-connected, 1I60 V. 10-pole, 60 Hz synchronoti\ generator has iin armatlire re4stance of 0 . 126 ohms per phaw and ii synchronou\

reactance of 3 ohms per pha\e. Find the full load generated kroltage per phaw at 0.8 PF lagging.

Problem 4.7

The

nchronoii\ reactance o f ;i cylindrical rotor synchronou\ generator i\ 0 . 9 0 I f the machine i \ deli\ering actiire poww o f I .OO p i r to ;in infinite bus M IN \c koltage is 1 .OO p i , at unit) PF, calculate the excitation vollage and the pov er ang I e .

p i .

Problem 4.8

A cylindrical rotor niachine is delivering actiLte ~ O er M of 0 . 8 0 pr and reactiLe pouter of 0.60 pi( at ;I terminal i ~ l t a g eo f 1 .OO p i r . If the power angle is 2 2'. co mpu t e the e YC i t ;i t ion b v l t age and the mac h i tie ' \ \y nchronoii s reactance .

Problem 4.9 A cylindrical rotor machine is delivering active power of 0.80 p i and reacti\,c poiver of 0.60 pi when the excitation voltage is 1.20 p i and the p o \ + ~angle r is 25". Find the terminal voltage and synchronous reactance of the machine. Problem 4.10

The reactances ~v,/and .v,/ of a salient-pole syiichronou\ generator :ire 0.95 and 0.70 per unit. respectiLrely. The armature resistance i \ negligible. The generator delivers rated kVA at unity PF and rated terminal Ltoltage. Calculate the c'witation iroltage. Problem 4.11

The reactances and .v,/ of a salient-pole synchronous generator are 1 .OO and 0.60 per u n i t respectii.ely. The excitation voltage is I .77 pi' and thc infinite bu\ \ d t a g e is maintained at 1 .OO p i . For a power angle of 19.4".compute the acti\ c and reacti\'e power supplied to the bus. Problem 4.12

A \alient pole machine wpplies a load of I .20 pi' at i i n i t i \ PF to an infinite bu\. The di re c t a x i \ and q11adr a t U re a x i \ synchronous re ;K t ;in c tj.\ are : t , / = 0.9283

\,,

= 0.4384

The pourer angle 6 is 25". Evaluate the excitation and terminal Yoltages. Problem 4.13

Consider the ca\e of an electric machine connected to an infinite bu\ through ;i reacti\e electric net\horh wch that the magnitude of the po\ser anglc c i i n e i \ unity. A change in the networh results in a new power angle ciine ~ i t nh iagnitude t . Suppow the machine is delivering a power p before the chiingc oc'c~ir\. then \how that the maximum value of p such that the \ystem remain\ \table sati s fi es :

Verify that for .v = 0.5, the inaxirnum value of prefiiult po\srer p is approximatel> 0.4245.

Problem 4.14 A generator is delivering 0.60 of P,,,',, to an infinite bus through a transmission line. A fault occurs such that the reactance between the generator and the bus

is increased to three times its prefault value. When the fault is cleared, the maximum power that can be delivered is 0.80 of the original maximum valul:. Determine the critical clearing angle using the equal-area criterion. Problem 4.15 A generator is delivering 0.50 of P,,,,, to infinite bus through a transmission 1in;t.

A fault occurs such that the new maximum power is 0.30 of the original. Whc n the fault is cleared, the maximum power that can be delivered is 0.80 of the original maximum value. a. b.

Determine the critical clearing angle. If the fault is cleared at 6 = 75". find the maximum value of 6 for whicsh the machine swings around its new equilibrium position.

Problem 4.16

The power angle curk'es for a single machine against an infinite bus system is P = 2.8 sin 6. Under fault conditions, the curve is described by: P = 1.2 sin 6

Assume that the system is delivering a power of 1.0 p i i prior to the fault atid that fault clearing results in the system returning to the prefault conditions. I f the fault is cleared at 6, = 60°, would the system be stable'? Find the maximum angle of swing 6, if the system is stable. Problem 4.1 7

A generator is delivering 0.55 of P,,,,, to an infinite bus through a transmissicm line. A fault occurs such that the reactance between the generator and the bus is increased to three times its prefault value. When the fault is cleared. tie maximum power that can be delivered is 0.75 of the original maximum valutb. Determine the critical clearing angle using the equal-area criterion. Problem 4.18

The 60 Hz synchronous machine shown in Fig. I is generating 235 MW and 3 0 MVAr o f power. The voltage at the infinite bus y is 1 .O +jO.O p i , and the line

Bus p

Bus q

Figure 1. Impediance diagram of sample power system.

reactance is 0.065 1 x 1 on a 100 MVA base. The machine transient reactance is 0.22 1711 and the inertia constant is 3.78 per unit on a 100 MVA base. a. Solve the initial power flow of the system. b. Using the Euler integration technique, calculate the changes in phase angle and speed of the generator for a three-phase fault at bus 11. which is cleared after 0.06 second. Use a time step size of 0.02 second and a total time of 0. I8 seconds.

P,: generator real terminal power Q,: generator reactive terminal power f,,y:real power flow Ql,ll: reactive power flow E': voltage behind the transient reactance V/,: voltage at bus p V(/: voltage at bus q sI,': generator transient reactance .I-/,(/: transmission line reactance Problem 4.19

Draw a detailed flow chart of the Modified Euler technique, as applied to power system stability studies. b. For the problem given in No. 1, using the Modified Euler integration technique. calculate the changes in phase angle and speed of the generator for a three-phase fault at bus p , which is cleared after 0.06 second. Use a time step size of 0.02 second and a total time of 0.18 seconds.

a.

Problem 4.20

The 100 MVA, 13.8 kV, 60 Hz synchronous machine has a transient reactance of 0.02s p i r and an inertia constant of 4.00 per unit on its own base. It is

supplying ii poufer of 95 M V A at ;I pourer factor o f 0.895 lagging to an infinite bus. The kroltage at the infinite bus is I .O +jO.O 1711 and the net reactance of tlie parallel lines i \ 0.30 ~ I on I a I00 M V A base. Drau ;I \ingle line diagram of the \y\terii and \olie i t \ initial p o ~ z c i tlou . b. U\ing the R K - 3 (Runge-Kutta) integration technique to calculate tlie change\ in phase angle and speed of the generator for a three-phaw 1';iuIt at the generator bu\, which i \ cleared after 0.06 \econd. U\c ;I tinit' step \ize ot 0.02 \econd and ii total time of 0.18 second\.

;I.

Problem 4.21

For the system given in Problem 4.3, consider the loss of one of the parallel branche\ M how impedance is 0.06 p i . If thi\ fault occurred at r = 0.0 \econd m d M ;i\ cleared through breaher reclosurc at r = 0.50 second, calculate the chungc\ in phase angle and speed of the generator from t = 0.00 to t = 0 . I2 \econds. U\e the Euler integration technique uith a time \tep \i7e of 0.02 \e c o n d.

Problem 4.22

where T,

r,,

6

= electrical torque,

= mechanical torque, = angular speed of the machine. anti

K,.K2= constants that are functions of the network.

a.

D e ~ ~ l oirnplementation p algorithms for solving the swing equation by any two numerical methods. b. Comment on the computations efficiency of the two algorithms i n ;i. especially for large multi-machine power systems.

PROBLEMS F O R CHAPTER 5 Problem 5.1

Define and briefly discuss the following terminologies, as applied to the topic of dynamic stability assessment (DSA) in electric polver sjtstenis. a. Angle stability b. Center of inertia, CO1 c. Stable equilibrium point, SEP d. Unstable equilibrium point, UEP e. Critical energy V,, f. Energy margin Problem 5.2

Consider a simple power system as shown in Fig. 2. consisting of a generator delilrering power to a large system represented by an infinite bus through tlvo transmission circuits. The single generator represents a thermal generating plant consisting of four 600 MVA, 24.5 kV. 60 Hz units supplying pouw to the infinite bus. Bus B is the infinite bus, which can be represented by a \dtage source of constant \,ohage inagnitiude and constant frequency. The initial operating conditions of the system. with quantities expressed i n pi( on ;i 2200 MVA. 24.5 kV base. are as follows: P = 0.953 p i r E, = I .o /"I at 15.50"

(2 = 0.532 p i E,, = 0.935 p1r

ill

O.OO@

The classical model of the generators has unitized parameters lumped to that of a single equivalent generator given as: X,: = 0.325 p i and the machine constant, H = 3.06 MW.s/MVA.

Bus A

Bus B j0.45

j0.83

Figure 2.

Single line diagram of the power system for problem 5.2.

EB

If a solid three-phase fault occurs at point F as shown in Fig. 2, and i!, subsequently cleared by i . s o l d q q the faulted circuit, then: a. b. c. d.

e.

Write the dynamic equations for the postfault system referred to tht. center of inertia. COI. Write the expression for the \ystem energy function. Calculate the postfault \ystem stable equilibrium point (SEP). unstablt: equilibrium point (UEP), and the critical energy V , , . Calculate the energy at fault clearing with t , = 0 . 0 8 , 0.12. and 0.10 \ec on d. Determine the system stability for each of the three fault durations.

In this problem. the netuork reactances shown are on the resistances are assumed to be negligible.

ii

2200 MVA base and

Problem 5.3

Re-visit the data given i n Problem 5.2. Consider a loss of line contingency th: t u.as created due to an open circuit fault on the branch of the parallel transmi>sion \ystem whow reactance is 0.83 pi(. If the friult was subsequently cleareJ by controlled breaker reclosure, then: Calculate the postfiiult system stable equilibrium point (SEP), unstable equilibrium point (UEP), and the critical energy V,,. b. Calculate the energy at fault clearing with t, = 0.10. 0.20, and 0.30 second. c. Hence or otherwise. determine the system stability for each of the thrte fiiu It d iirat ion s.

a.

I n this problem, the netuork reactances shown are on a 2200 MVA base ard the resistances iire assumed to be negligible.

Problem 5.4

Discuss Lyapunov's stability criteria, as applied to .stdilirj9m.ses.snio 'it studies performed on electrical power systems. b. Explain how the generalized expression of the transient energy function for ii multi-machine power system can be used to determine the stability o f the system. c. What are the challenges faced by engineers and researchers in attempting to obtain fast and iiseful solution to the transient energy function ,)f part b? Also cornrnent on the desired accuracy of the solution.

a.

32 7

PROBLEMS FOR CHAPTER 6 Problem 6.1

Consider the 550 kV, 370 km (230 miles) line transmission system shown in Fig. 3a below supplying power to a radial load from a 'strong' power system represented by an infinite bus. The line parameters, as shown in Fig. 3b, are expressed in their respective per unit values on a common system base of 100 MVA and 55 kV.

(a)

Infinite BUS Bus I

Load Bus Bus 2

Transmission Line

Shunt

Load

Qsh

(b)

Infinite Bus Bus I

Load Bus Bus 2

Figure 3. The 550 kV. 370 km (230 miles) line transmi\sion system \upplying a radial load. ( a ) The whematic diagram. (b) The equivalent U ye circuit reprewntation of the tranmission line.

Compute the full admittance matrix of the two-bus system and write the pou'er tlou' equations from the sending end to the receiving end in the torm:

6.1.1

6.1.2 Hence or otherwise, write down the expressions for the four ( 3 )\ubm;itrices of the Jacobian i n the l i n e a r k d load tlow equation\ ;I\ defined by:

6.1.3 When P1= 1600 MW. calculate the eigen\alue\ of the reduced Q-\' J Icobian niatrix and the \/-@ sen\iti\ itie\ M ith the tollon ing different reactil c p o er~ iii-jcctions for each of the corre\ponding t L j o L ()Itage\ on the Q- I'cur\ c.

;I. b.

c.

QI= 5 10 MVAR (II = 305 MVAR Values of Q, close to the bottom o f the Y-Q curve.

6.1 .-I Determine the coltage stability of the systeni by computing the eigenLiiIlie\ of the reduced \"-Q Jacobian matrix for the follou iiig ca\es: ;I.

b.

P = 1600 MW. Q , = 3 6 0 MVAR P = 1890 MW. @, = 965 MVAR

Assumc that the rerictikre pourer Q, is supplied by

;I

shiint capacitor

Problem 6.2 Thi\ problem demonstrate\ the concept of parameterization. C o n d e l - ;I \ i niplc 2-bus power \y\tem ;I\ \ h o u n in Fig. 4, M here re are intere\ted in the beh:t\ ior on the bus \ oltage, V to the amount 01' reactiLc' p o w r demand at the bu\ unclcr prc \ pec i fied conditions.

Bus q

Bus p

-

Total Load 3.6-t i s

GiLren that s is the parameter of interest, which represents reacti\-e po\j er. a n d the fo1 1owi n g p h y s ica I 1y - in e an i n gfu I para m ete ri za t ion eq u a t ions : 0 = 3.6 - V

o = s + 10 V'

-

8 + 10 V sin 8 v sin e - 10 v c o s 8

COS

. . ..(i) . . . .(ii)

6.2.1 Solve the parameterization equations in Sec. 6.2. I using \ A w s of .s from 0.00 to .s,,,',, in increments of 0.100 1x1 MVAr.

6.2.2 What is the value of .s,,,,,, at which point no real solutions are obtainable? Plot the resulting lV( vs. s and 0 vs. s traces on the same grid. 6.2.3 How safe is it to deduce the maximum reacti\Te loridabilitj l i m i t of the given 2-bus power system'?

Problem 6.3 This problem demonstrates the basic concepts of Modril nnalysiis. a s upplicd to voltage stability studies in power systems. A 4 B u s p v e r system is shown i n Fig. 5 and the operating conditions are summarized i n Table 1 .

6.3.1 Compute the full and reduced Jacobian matrices of the system. 6.3.2 Calculate the eigenvalues and eigenvectors of the reduced Jacobian matrix.

Bus I

Bus 3

5 212

=O.OOO+j0.095

234

Bus 2 ~2~

=O.OOO +j0.055

=O.OOO+jO.3 15

I Bus4 L

Figure 5 . A -+-Bus p m ' e r \ p t e m for Modal anaiy\i\.

Table 1 Operating Conditions for the +Bus System Complex bus voltages, V,

Bus no. i

Voltage magnitude 'V,I ( p )

Voltage angle 8, (deg)

I .ooo I .000 0.655 0.795

-25.890 -38.980 -38.980

+o.ooo

6.3.3 Determine the modal reactive power variation and modal voltage variition. 6.3.4 Determine the bus, branch, and generator participants. Problem 6.4

Using the algorithm for static assessment, solve the following system of algebraic equations represented by: I . I . ~ ' - ] . I ? = 3.2 2.2s + 0 . 9 = ~ -k

. . ,. (i) . . . . (ii)

where k in a parameter varying from k = 0 to k,,,,,. Problem 6.5

The synchronous machine shown in Fig. 6 is generating 250 MW and 85 MVAr of power. The machine transient reactance is 0.200 p i . and the line reactance i \

Bus I

Figure 6. The impedance diagram of sample powet system.

Bus 2

0.045 p i r , both on a 100 MVA base. The value of the voltage at the infinite bus (node 2) is I .OOO +jO.OOO pi! 6.5.1 Solve the power flow of the system for both low and high voltage power solutions.

6.5.2 Construct the necessary vectors required by the VIPI Method and compute this proximity index. 6.5.3 Consider a bolted three-phase fault occurring i n the middle of the transmission line. Calculate the multiple power flow solutions (as in part 6.5.1). and the voltage instability proximity index. 6.5.4 Comment on the VIP/ values obtained in 6.5.2 and 63.3.

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Index

cihc-Phase sequence, IS, I6 Accelerating power. 72 Accelerating torque, 36 ActiLtation functions, 182 Activation level, 182 Active load, 5 Actibre power. 12, 2 I AGC, IS8 AI (see Artificial Intelligence) Alpha cut, (a-cut), 21 1 Angular momentum, 38 ANN ( . s m Artificial Neural Network) ANN-Based TEF, 222 algorithm, 223 learning process, 222 Antecedents, 250 Apparent power, 13 Approximate reasoning. 2 14-2 15 ARG (.see Automatic Rule Generation) Armature windings. 22 Artificial Intelligence, 22 1, 259, 296 Artificial Neural Networks, 9, 177, 178 architecture, I80

Artificial neuron (see Neuron) Automatic Rule Generation. 253 Automatic Voltage Regulators ( . s w AVR) Average power, 12 AVR, 98. 255, 1.35 Axon, 181 Back propagation, 183. 188 Balanced 3-pha\e sy\tems, 15 Base value. 26 Belief measure. 2 IS Bellman. 206 Benefit-to-cost index, 9 Bifurcation point\, 148 Bifurcation theory. 9, 141 Branch switching. 87 Bulk power system, 3 Carpenter, 184 CCT (see Critical clearing time) Cell body, 181 Center of inertia, 122 3s 1

Certainty lxtors. 20 I Chuos theory. 292 Clustering. I84 Coriiplcx potvcr. I I . I3 CO1 ( S C T Centcr o l incrtiu) Condition nunihcr. I43 Confidence scnles (.w,Certainty l~ictors) Conjugate citrrent. I3 Contiriti:itioii nicthod. I 4s. I47 Control \ alvc position. 42 Corrector. 90. I49 Counter propagation. I X4 Critical clearing time. 234 Cylindric;il rotor synchronous ni;ichinc. 55

Dcl'w/.ificurion. 2. IS. 2 19. 249 I)elt~i-conncctioti. I X 1)ciidritc.s. I8I Direct axis. 56 Direct axis synchronot~srciict;tncc. 54 1)istrihution systctii. 4 JqO tnasloriiintion. 24 Dynaiiiic security. 289 I1yi;tinic security ;isscsc~iicnt( DSA). 8. 22 I Dynamic stability. 164 Dyiuiiiic voltogc stnhility. IS7 nlgorithlii. I hh cquilihriitni point. I h l I'tist subsystetns. I6 I linc:iri/at ion. I62 lo;d Ilou. I h I \IOW sllhsystclns, 160 Edison Electric. I EHV (.we Exwct high t~olt~tgc) Eigcnvnlucs. 114. 156 Elcctric;il torque. 46

Electric poucr systeni. 3 l w / y theory. 209 E I t ~ ~ t r o ~ i i c ~ ~ hIriiii~icJil. i t ~ ~ ~ c ; ~I 09 l Energy titnction. 127. I30 Energy Systcnis Network Lnhoratory. 97 E q i d ;irc;i criteria. 73-75

Error signal. I87 tS ( s w Expert systeni) ESNL ~ s w Energy systctiis network lahoratory) Euclidctiii distiince. I78 Euclidean nortii. I47 Excitation control. 6 Excitation control systcni. 37 Excitation systeni. 40 nlodrl. 36 Excilcr niodel. 37 linear differential cquntions. 39 Expert rcnsoning. 2 I4 Expert systeni. 9. 176. 191 characteristics. I92 ilcfinition. I91 DSA. 238 iufcrciicc engine. I9 I kno\vlcdgc acquisition. 273 knowledge base. I I riinn-niachinc intcrl'clce. I91 nieiisiircs. 200 prohuhility. I99 rule-b;iscd structure. 740 subjective proh;ihility. 199 E?ctrn high wltnge. 2. 135. 2.59

FACTS. 1% FIRE ( . s w F u i i y Inlcrcncc Reverse En piticering) FL ( . s w F u z ~ ylogic) FL,C ( s c o F u u y Logic Controllcr) FLPSS (sw Fii/./.y Logic Power Systeiii Stiihilizer) Flux. 55 Flux linkages. 23 Four-rnxhinc power ry~ccrii.239 Frcqueiicy dependent lond. 34 Fuui ficnt ion. 2-10 F u z ~ yinlcrcncc. 2s I Fui/y Inlcrcnce Reverse Engineering. 254 Fuzxy logic. 9. 177. 204 correlatioti niethod. 2 I8 linguistic viiriahle. 2 IS iiionotonic nicthod. 2 I7

IFu//y logic I

paraiiie t e r t 11n in g . 25 2 power \y\teni \tabili/er, 248, 255 rule\, 252 Fu//) Logic Controller. 248, 349 F w / y propmition, 2 I6 F w z y \et\. 206 definition, 2 I3 operation\, 2 I2 F w / y \et theorj. I77 Gate opening, 40 Generator real poiaer. 1 I8 re ac t an ce \. 5 4 \hart circuit current, 53 Generator model, 52 Generator voltage, 6 Generic electric po\ver \ptriii. 4 Global e r i w function. 189 Gro\\berg Outstar algorithm, I84 Hidden layer. 187 Hidden node. 226 High Voltage DC linh\, 133 Hi gh - VoI t age Di rect Current , 2 HVDC tran\nii\\ion control, 7 HVDC ( $ 0 1 ) High-Voltage Direct Current) Hqdraulic turbine model. 40

IEEE 5 7 - b ~ te\t \ \ Y \ ~ C I T I .278 IEEE 39-buj system, 227. 229 IEEE \sorhing group, 134 I I1 du ct a nce m u t u a l , 23 \elf, '3 Induction motor. 43 equi\ dent circuit. 44 \hp. 44 torque. 43 Inertia, 46 Inertia coii\tant, 38, 47-50 Inference chain, 196 Inference engine. 175. 196. 276 Inference. rule\, 3 I7

Infinite bu\. 63. 68, I 8 In\tantaneou\ current. I I I n\tnntaneou\ PO\\ er. I I I1t egral -\y 11are error, 25 Integration method\. 9 I n t egra t I on t ech n I qiic\ Euler method. 91 modified Eulrr method. 9 I predictor corrector. 89 R U nge- Ku t ta met hod. 9. 94 4 ;in1p I I ng inc t hod, (1 7 trape/oidal method. 9. 94 U a\ eform rt'la\iallon. 9 Intelligent \ j \tern\, 173 appronche\. I79 Jacobian matri\, I4 I . I43 reduced, I56 KB ( ~ C OKIN)\\ledge haw) KBVCDP ( $ O O Kilo\+Icdgc-Ba\ed Voltage Collap\c, Detection and PrcLention) h- fi n i te difference eq LI;I t I on \, 89 Kinetic energj, 17-48 Kirchoff'\ lau. 16 Knowfledge bnw. 176, 194. 249 Kllouledge-bi\\ed \> \teiIi. 195. 338 Knowledge-Ba\ed Voltage Collap\e. Detection and Prei ention. 373 i m p lemcn t at I on, 2 74. 378 5tructural de\lgn. 375 Kohonen laqer. I84 Learning algorithm\. global error. I83

L Index, 140 Linear prograinming. 9 Linear sy\tem sen\iti\ it). I44 Line \ d t a g e . 16 Load characteri\tic\. 3 1 Load model\ con\tant po\+er. I49 dynamic. 43. I58 linear model. 150 Load paran1etcrs. 33

3.54

Load representation. 85 Local stability. I 1 1 LP (.we Linear programming) LTC, 277 LU decomposition, 148 Ly ap u no v function, 115 theorem. I12 Machine models, 59 Machine rating, 39 Machine switching, 88 Marceau, 290 Maximum norm. 147 Mechanical torque. 46 Membership function, 2 1 I , 250 Mininiiini singular value. 142 Min-niax inference. 2 I7 Mismatch, I45 Modal analysis, 156 Models exciter, 38 generator, 52 gobw-ning system, 40, 42 hydraulic turbines, 40 PSS, 39 static electric network, 10 static loads. 32 synchronous machine, 2 I terminal voltage transducer, 39 \ d t a g e regulator, 38 Modified Eider method. 91 Mutual inductance, 23 Necessity measure. 2 15 NERC (,we North American Electric Reliability Council) Neural networks, 22 1 learning algorithms, 183 training, 183 Neuron, I80 New England 39-bus power jystem, 264 Newton-Raphson. 145. I56 New Yorh power pool, 132 n-Generator System Model, 1 17 Nodal matrix method, 83

Normalization (see per unit representation) North American Electric Reliability Council. 3 OPF (see Optimal power flow) Optimal p o u w tlow, 9 Parameterization, 15 1 Participation factor, 157, 262 Pendulum, I14 kinetic energy, 115 potential energy, I15 Per unit representation, 25 PF (see power factor) Phase diagram. 13. 56-60 Phase-shift transformers, 7 Phase voltage, 16 PI D (.see Proportional-ln tegral Deri vative) Plausibility measure, 2 15 Post-fault curve, 8 1 Power angle cur\re, 70, 120 wound rotor machine, 66 Power factor, 1 1 . 12. I4 Power failures, 132 Power tlou,, 165 linearired model, 165 Power interchange. 6 Power relationships. 20 Power system basic elements, 5 complex power concepts, 1 1 security assessment, 45 three-phase systems, 14-2 I Power system stability, 7 Power system stabilixrs, 38, 158 Power triangles, I3 Predictor, 149 Predictor-corrector step, 147 Pre-fault curve, 8 1 Prime nioiw, 40 Prime mover controls, 6 Probability measure, 2 15 Proportional -Integral Deri vati \,e. 254 Proposition, 2 16 PSS ( s c v Power system stabilizers)

1iide.r

p.u. (see Per unit representation) P-V curve, 146, IS1 QP (.see Quadratic programming) Quadratic programming, 9 Quadrature axis, 56 Quality of power, 5 Quantization, 2 10 Q-V curve, 146 Q-V sensitiktity. 1.56, 167 Ray. Jame\ J., 97 RBS ( W O Rule-based system) Reactive capability limits, 3 1 Reactive load, 5 Reactice power, 6, 13. 21 Reactors, 7 Robu\t ANN. 222 Rotor voltage, 21 Rule-based system, 194 Rule\ divergence. 276 \electing modeh for VC detection. 277 selecting VC parameters, 276 Runge-Kutta. 9, 95 Saliency effects, 65 Salient pole synchronous machine, 56, 57 Self inductance. 23 Semitivity analysi5. 146 SEP ( t e e Stable equilibrium point) Sigmoidal activation function. 225 Sigmoidal function, 182 SIL. 13s Speed regulation, 6 Squashing function, 182 S [ability angle, 8 BCU. 131 boundary, I27 definitions, 1 1 1 equal area method, 72 lemma, 112 Lyapunov's theorem, 1 12 positice definite function, 112

3-55

[Stability] SEPs, 260 simple pendulum, I14 single-machine system, I 18 UEP, 129 voltage, 8 Stability assessment. 63. 7 1-82 fuzzy expert systems, 205 TEF method, 121 voltage, 132 Stable equilibrium point, 69, 130. 260 Star connection (.see WY E connection 1 Static assess men t algorithm, IS3 Static load model, 32 Static security assessment, 8 Static stability. I63 Static var compensators, 6 Static voltage compensating deiricea, 133, 158 Static voltage stability assessment modal analysis. IS8 Stator voltage, 21 Steam turbine. 42, 43 Steam turbine model. 42 Sub-transient equations 59 Sub-transient reactance, SO. 53 Sub-transmission system, 4 SVCs (see Static voltage compensating devices) Swing equation, 46, 72 Switched capacitors, 6 Synapse, 181 Synchronous condensers. 6 Synchronous machine. 3 classic model, 27 dynamic model, 29 equivalent circuits, 29-30 rnodeling, 2 1-30 network representation, 84 rotor circuit, 22 stator circuit, 22 steady state model. 30 subtrans ient niode 1. 30 transient eq u a t i on s. 5 8 transient model. 30 voltage equations, 22

S j iichronoii\ machine mockling. 2 1 Sy\tttni haw. 26 S ) \tern t-lllilt\. 86

Tamiira, Y . I 4 1 Tangent \ ector. 1 5 0 Tap-changing tran\toriner\. 7 Target \ alue. I87 Ta! tor eup,in\ion. 93 TEF ( wt' Tran\ient encrgj tiinction) Three-phaw pofier, 20. 2 I Three-phaw \) \tern. 14 Thre\hold tiinction. 182 Time traiiie\ rllodel\. I39 RLC component\, 139 Ti aining algorithin\. 184 Gro\\bci g Out\tar. I84 Kohonen. I84 Training proce\\. 263 output ~ector.I83 Traii\ient Jroop. 4 I Ttaii\ient energy tunetion, I 10. 120, 185. 212 tlouchart. I28 inte~ral-\qiiarederror ([SE). I22 w t e d energj. I23 Trim 4I e11t cci i i a t ion\. 5 8 Trnn \ it' n t I e;lc t ance , 53. SO Tramient 4tabillt). 68. 109 algorithm. 92-93. 98- I01 concept\. 68 Tran\ien t \ t 'i bi I i t j ;i4\e\\iiien t. 222 Tran\ient \U ing. I2 1 Trun\nii\\ton networh, 4 Tr,ipe/oid:iI iule, 0 Trllth tLinctloti\ ( S O P MeilIber\hip tliiiction 1 Turbo generator high pre\\iire, 49 Ion prt.\\ure. 49

Unxtablc equilibrium point. 69. 130, 260 UnxiiperL i d training. I83 VAK, 265 VC ( WO \ oltage collapw ) VIPI. I41 Voltage collap\e. 134. I35 cla\\i t ication\, I 34 deflnl(1on. I33 detection, 1-18. 272 pietliction method\. 138 predictor \tep, I SO prevention. 272 time-frame. I36 Voltiige iii\t~ib~l~tj, 159 ( Voltage \t;ibilitj problem\) cla\\itication, 159 Voltage iii\tabilit) indicator\, 140 Voltage regulator. 37 Voltage \tabilitj ;i\\e\\iiicnt. 132. 165. 259 ANN-ba\ed, 260 technique\. 140 time-traiiie. 139 Vo It age \ t a bi I i t y Inodellng. I36 pit.\cnti\c control, 168 tiiiie-franie (Cnr\on Taj lor). I37 Volt,ige \tabilitj problem\, 138 cla\\ilication. I38 Voltage tr,in\duccr. 38 V-Q \ C i i \ i t t \ i t ) , I57 VSTAB, (EPRI). 105. 263 \('(I

U'ale r ham mer e ffe c t , 4()

Water 4tarting tiine. 40 W i i \ f~orin re I ;ix at i on 9 Wjv-conncction. I6 Y-connection, IS ( W JWye-connectic 11) Y-Bus or admittance matrix. 85. 118

UEP (,we Unxtable equilibrium point) LlLTCs. 135 Llncertainty. I97

Zadeh. 177 ZIP load iiiodel, 34

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