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

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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 un- dergoing a more dramatic revolution in both technology and industry structure.

Among the most exciting changes are those where new solutions are being ap-

plied to classical problem areas.

System dynamics and stability engineering have been crucial elements of

power system engineering since early in the twentieth century. Smooth, continu-

ous operation of modern power supply systems depends greatly on the accurate

anticipation of interconnected equipment, dynamic behavior, and correct identi-

fication of the system’s operating limits. Proper engineering requires precise

methods that can manage that knowledge and direct i t to the design of economi-

cal 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 Appli-

cations is an exceedingly comprehensive and practical guide to both power sys-

tem dynamics and stability concepts, and to the use of artificial intelligence in

their modeling and engineering. Drs. Momoh and El-Hawary provide a compre-

hensive introduction to power system dynamics and stability, along with a thor-

ough 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 ex-

pI a i n ho 11ei i ra I tiet uwrk s, espert sy s t e111s us i in g know I edge-based fra in eu'ork s.

and t ' u~zylogic 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.Y-

tettr.v, ~ > j w t t i i c * s , App/ic.trliotis present?t i t i d Sttihilitjv Il'itli A rlificitrl Itit~~ligutic*ci

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\~entually include books co\w-ing the entire field

of ponw engineering, in all ot' its specialties and sub-genres, all ainned at provid -

ing practicing pouw 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 understand-

ing 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 re-

search sponsored by the Electric Power Research Institute. the National Science

Foundation, the Department of Energy, and NASA for Dr. Momoh. Dr. EI-

Hawary's work was supported by the Natural Sciences and Engineering Rc-

search 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.

I t provides necessary fundamentals, by explaining the practical aspects of artifi-

cial intelligence applications and offering an integrated treatment of the evolu-

tion of modeling techniques and analytical tools.

Chapter I discusses the structure of interconnected power systems, founda-

tions 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 tran-

sient 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 func-

tion idea. Chapter 6 introduces the idea of voltage stability and discusses tech-

niques 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 artifi-

cial 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 Powei-

Engineering Series for Marcel Dekker, Inc. The continual counsel and prodding

of 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 H. Lee Willis

Prefiice

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 10

2.1 Complex Power Concepts 1 1

2.2 Three-Phase Systems 14

2.3 Synchronous Machine Modeling 21

2.4 Reactive Capability Limits 31

2.5 Static Load Models 32

Introduction 10

Conclusions 35

3 Dynamic Electric Network Models 36

Introduction 36

3.1 Excitation System Model 36

3.2 Prime Mover and Governing System Models 40

3.3 Modeling of Loads

COn c1LI s ions

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 SU bt ran s i ent Eq u at i ons

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

Stabi I ity A sse ssmen 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

Introduction

5 . I Stability Concepts

5.2 System Model Description

5.3 Stability of a Single-Machine System

5.4 Stability Assessment for ri-Generator System by the

TEF Method

5.5 Application to ;I Practical Power System

5.6 Boundary of the Region of Stability Conclusion

6 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 Clas\ i ficat i on c) f Vo1t age Stab i 1i ty ProbIe111s

6.7 Voltage Stability As\es\ment Techniques

Col1 ter 1f.\ xi

6.8 Analysis Techniques for Steady-State Voltage

Stability Studies 135

6.9 Parameterization IS 1

6.10 The Technique of Modal Analysis 156’

6.1 1 Analysis Techniques for Dynamic Voltage Stability Studies 157

Conclusion 169

Modal Analysis: Worked Example 170

7 Technology of Intelligent Systems 175

Introduction 175

7.1 Fuzzy Logic and Decision Trees 177

7.2 Artificial Neural Networks 177

7.3 Robust Artificial Neural Network 183

7.4 Expert Systems 191

7.5 Fuzzy Sets and Systems 206

7.6 Expert Reasoning and Approximate Reasoning 213

Conclusion 220

8 Application of Artificial Intelligence to Angle Stability Studies 22 1

Introduction 22 I 3 3 3 eh& 8. I ANN Application in Transient Stability Assessment

8.2 A Knowledge-Based System for Direct Stability Analysis 238

Conclusions 257

9 Application of Artificial Intelligence to Voltage Stability

Assessment and Enhancement to Electrical Power Systems 259

Introduction 259

9.1 ANN-Based Voltage Stability Assessment 260

9.2 ANN-Based Voltage Stability Enhancement 265

9.3 A Knowledge-Based Support System for Voltage

Collapse Detection and Prevention 272

9.4 Implementation for KBVCDP 278

9.5 Utility Environment Application 287

Conclusion 287

10 Epilogue and Conclusions 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. I t is not surprising then that the

growth of electric energy consumption in the world has been nothing but phe-

nomenal. 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. Edi-

son's system used a I 10-dc underground distribution network with copper con-

ductors insulated with a jute wrapping. The l o b i t \d tc ige 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. i n 1866

for Westinghouse, who acquired the American rights to the transformer from its

British investors Gaulard and Gibbs. Early ac distribution utilized 1000 V over-

head lines.

By 1895, Philadelphia had about twenty electric companies with distribu-

tion 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 econo-

mies 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 uni t 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 inter-

connections decreased the probability of service interruptions, made the utiliza-

tion of the most economical units possible, and decreased the total reserve ca-

pacity 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.

\+!hose objective is to provide the best system to meet the load demand reliablq.

safe1y , and economical 1y , U t i 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

kV, 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) l ink 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 trans-

mission is required (underwater or in a metropolitan area), HVDC is more eco-

nomically attractive than ac.

Protecting isolated systems has been a relatively simple task, which is car-

ried 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 interconnec-

tions.

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 cascad-

ing 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 ques-

tions 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 ade-

quacy 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 sys-

tem 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, trans-

mit 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.

1.2 STRUCTURE OF A GENERIC ELECTRIC POWER SYSTEM

While no two electric power systems are alike, all share some common funda-

mental 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 dis-

tances to load centers that are spread over wide areas.

3. Three phase ac systems comprise the main means of generation. trans-

mission and distribution o f electric power.

4. Voltage and frequency levels are required to remain within tight toler-

ance 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 transmit-

ted to consumers through an intricate network of apparatus including transmis-

sion lines, transformers. and switching devices.

Transmission network is classified as the folloufing:

I . Transmission system

2. Si i btransmi ss i on system

3. Distribution system

,s~~.strinThe t i . i ~ i i ~ s i ~ i i . s . s i ~ ) i i i n t erconnec t s a I 1 major genera t i ng s t ;i t i c) n s ;I nd main load centers in the system. I t forms the backbone of the integrated p o \ \ ~ r

system and operates at the highest voltage levels (typically, 230 kV and ab0F.e 1 .

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 I t . \ , c . l

(typically, 69 kV to I38 kV). The generation and transmission subsystems ;it e

often referred to as the hi i lk po\r.ei. systerii.

The .viihtr.iiri.siiii ,s.siori sy.stew transmits power at a lower voltage and in

s mii 1ler qitan t i t i es froi n the t ran s iii iss ion s i i bst at i on to the d i s t ri bi i t i on subst ;I-

tions. Large industrial customers are commonly supplied directly from the s u b

transmission system. I n some systems, there expansion and higher \,ohage l e \ ~ l s

becoming necessary for transmission, the older transmission lines are otten re1 2-

gated to s i i bt ra n smi ss ion 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

kV 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 to con\.ert energy from one of

the naturally available forms to the electrical form and to transport i t 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 i t 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 fitnda men ta I reqii ire men t s:

--

--

5

22 kV

- 500 kV 500kV 230kV

-20 kV

Tie line to neigh-boring system

I I 1 1 I Trans-

I Transmission system

I

mission (230 kV) Tie line

distribution

Subtransmission

Residential Commercial

Figure 1.1 Basic elements of H power system.

1. 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, ade-

quate “spinning” reserve of active and reactive power should be main-

tained and appropriately controlled at all times.

2. The system should supply energy at minimum cost and with minimum

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\~inga 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 excih-

tion controls. The prime mover controls are concerned with speed regulaticm and control of energy supply system variables such as boiler pressures, temperi-

tures, 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 Generator

flows power

Supplementary

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

I I I

I I It

I I Excitation I I I

I I I

+ System and control

4-

-

II

I I

I

I I I I I II

II

I

I

I

1I

I I1

I

W Rime mover

i

Voltage

Shaft

Power

-Generator

power speed/

-

Power

; i I *

i Transmission Controls

; I

; I

i ’Reactive power arid Voltage control, INDC

associated contro s

transmission and

Figure 1.2 Subsystems of a power system and associated controls.

I n troductio ri 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 sys-

tem 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 effi-

ciently 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 disturb-

ance 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 com-

bine 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.

1.3 POWER SYSTEM SECURITY ASSESSMENT

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 require-

ments for the proper planning and operation of the power system, i t 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 depen-

dence 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 1 . 4

T+ + + Strwrural Volruge (Slow) Oher Tvpes (SSR, Lo 6

I I Frequencb Oscillatiw s, etc

I 4 + +

Small Small Static Disturbance Disturbance (Power

(Contingency) (Quasi-Static)+-Cntical Sensitir it y Bifurcation Bifurcation Eigenvaluc AMIYSIS I Theory (Hopf. Analyv\ e tc . )

Bifurcation T a p Changer Singular Value Decompoiition

Techniques Frequency M Y Energy D o m i n Functions Analysis Angle

Stability Linear Selective Rogramrmng

Vdtage M a N m m Jacobian A M I y si s Stability Lnadability

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

4 Unlned Use of the Energy Function and h e Bifurcalion Analytical Studies Approach I

Linear Rogrammng for DecompositionOpimal Nonlinear Ragramming Germation Rescheduling dnd Power H o w (OPR wth (NLP)with quadraticEnhancement

(Prevcatlve Load Control Transieru Energy Margin and objccti~cfunctions and

Coatrol) Vdcagc Energy Margin ds Linear Rogramrmng (LP) fu

Generation and Load Control

Control using Exper( System Control Selections. output Arttndal Neural Networb IFS) Pnmiizatim Classifier and Analyzer (ANN), and/ a Expert

intelligent Systems fa solution outputSupport System

re l'erred to :is "s t ;i t i c sec i i r i ty iisse ss men t ." The t'i rs t reqUi renit' 11t i s the hii bjec t

deii 1t \v i t h i 11 "d y n amic sec i i r i ty assessme11t ." Dy nam ic sec11 r i ty st i id ies ;I re

broadly classified ;is being either "angle stability studies" or "voltage stabil; t),''

;is depicted i n Figure 1.3.

In "angle stability studies." problems are classified as either "large distitrb-

mce.' tix transient e\'aluiitioii, or "small disturbance" for steady state stability

t'\,:ilurition. A similar classification for \soltage probleiix is indicated i n Figure

1 ..3. Solution techniqites tor transient unglc stability e\,aluation incluck:

Ti me domai n s i mii 1at ion

Direct methods

Hybrid methods

Probabi I i st ic nie t hods. pat te1-11 recog n i t ion

COnn pit t ;it iona1 i n te 1I igence

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 ion. These met hods are part ic41ar1y usefu 1 for off- 1i ne transit‘ 11t s t abi 1i t y

ana I y s i s.

On the other hand, in “voltage stability studies.” the problems are classified

as “large disturbance” in some contingency cases, “small disturbance” i n quasi-

static cases. and “static,“ which requires solutions to the general pouw tlo\v

(algebraic) equations only. I t is as a result of this classification that the solution

techniques and requirements are derived. The bifurcation theory. Linear Pro-

gramming applications. and the use of the energy fiinction are but ii few such

tech n i qLies. Agai n, the t i me domai n s i m u I at i ons t h ii t ;ire i 11\YII \red take ad \,;in t age

of various numerical integration methods mentioned earlier. (The machine dj8-

namics 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\~lopment and use of more robust tools for solLting present ~ind long-

range problems. I n this light, various programming and optimization schemes

that are applicable include decomposition Optiiutrl Po\t*or Flo\tp (OPF). Liiiorrr

Pi .oSr . c r i i i i i i i r i ,~(LP), and Qir(rcltntic. Pi.oRt .cr i i i i i i i i iS (QP), with the necessary and

su f fici en t sy stem and net work constr ai n ts. Finally. this book introduces three fundamental types of intelligence support

systems that truly adds the rigor, value, and robustness to the desired enhance-

ment schemes. These support systems include expert systems (ES), fuzzy logic

(FL). and artificial neural network (ANN). Each have their unique characteris-

tics (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 artifi-

cial intelligence applications to power system stability rind dynamics is pre-

sented 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 sys tm 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 mar-

ket driven rather than through technical considerations alone. This new environ-

ment 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 func-

tions of system operation, operational planning, and long-term planning.

Electrical power systems vary in size, topography and structural compo-

nents. 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 ma-

chine from an electric network standpoint. Reactive capability curves are exam-

ined 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)= tlCOSOt

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)= i r ( t ) i ( t )= XI I,,,[cos(or)cos(or- $)]

Using the trigonometric identity

1coscl 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\( 2ot - @) i \ zero. this term contri b-

ute\ nothing 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 .Sub\tituting xi = v% Ci,,,,). and I , , , = -\/'?(/,,,,,). \+e get

Thus the p o ~ wentering 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 ~ ~ ) . , t ~ / ~ . t o ) .(PF). This applies to siniis-

oidal Lroltages and currents only. For ii purely resistive load. cos$ = 1. and I he

current in the circuit is fully engaged in con \q ing power from the soiirce to

the load resistance. When reactance (inductive or 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

or 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 i n ii circuit is called i1ctiL.e power. and loosely speaking

the po\+rer that supplies the stored energy i n 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. Hou-

eb'er, to emphasize the fact that Q represents the nonactive power, i t is measiired

i n reiictiLpe k~oltampere units (\'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 multi-

plied bqr /. the r i m cralue of the current. When the components of iultage 1' , :OS@

vvI 3Static' 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. Sinii- larly, 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 @

i t becomes obvious that an equivalent definition of complex 01- apparent POW-

er is

s = VI" ( 2 . 5 )

We can write the complex power in two alternati\,e fo rm by using the relationships v=Z7 and 7 = Y v

This leads to

s = ZI I" = Z J I J ? (2.6)

or

S=VY*V"= Y" ,V I " (2.7)

V

I Figure 2.3 Series circuit of n impedances.

Consider the series circuit shown in Fig. 2.3. Here the applied Lvltage i:,

cqual to the sum of the voltage drops:

\'=I(& +z,+ .. . +Z,:)

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:

Pcos$ = __

1s I

2.2 THREE-PHASE SYSTEMS

A significant portion of all the electric power presently used is generated. trails-

mitted. and distributed using balanced three-phase bultage systems. The single-

Static Electric N e t w w k Models 1.5

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 gen-

erator copper and iron. Power flow in single-phase circuits is known to be pul-

sating. 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 compli-

cations 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 volt-

ages 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°

van120Pnce120°

Figure 2.5 (a) A Y-connected three-phase system and (b) the corresponding phasor

diagram.

gle-phase \,()Itage sources appear in ii wye o r Y-connection; a delta o r A configu-

ration 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 i n ;I different ~t'aq'.to

;in ohser\~erin 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 t r b c . (order, phase sequence. or 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 io tor 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-Connec tion

With reference to Fig. 2.6. the cotiimon terminal I I I \ called the neutral o r <tar

(k') point. The ~ol tagesappearing between any t u o of the line terminal\ 1 1 . h,

m d ( h a b e different relation\hips in magnitude and phaw to the \oltage\ apptar-

ing bet\$een any one line terminal and the neutral point 1 1 . The \et o f colta;e\

\', . 1; , ,itid itre called the line coltage\, and the \et of coltage+ li,. and \'\'(

c i ~ creferred to ii\ the phaw c oltage\. Con\ideration o f p h a w r diagram\ pro\ ide\

t he rcq ii I red re I ;it ionsh i p\.

The etfectike kaliie\ o f the phase \wlt:ige\ ;ire \hohn III Fig. 2.6 i i \ \,',, \;,.

and l'!.Each ha\ the wile magnitude, and each 15 di\placed 120" from the olher

tuo ph;i\or\. To obtain the magnitude and phare angle o f the line \oltnge froin

( I to h (i.e . \ ' , , I . L+C apply Kirchhoff'\ voltage l a ~ :

This equatioii state\ that the ~o l t ageexisting from ( I to h i h equal to the

coltage from ( i to 11 (i.e.. \,',) piu\ the voltage f rom I I to h. Thus Eq. (2 .10 ) can

be re~irittenas

(.!.I I )

I7

Qn = vp L -120O

Figure 2.6 Illustration of the phase and magnitude relation\ beiueen thc phaw and

line coltage of' a Y-connection.

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

let UI set

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")

= f i p 3 0 0 (2.16)

S i ni i 1arly we ob t ai n

X( = f iy- - 90" (3.17)

(2.18)

The expressions obtained above for the line boltages shou that they consti- tute 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( a )

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\

l, = f,, (2 .2 ) )

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 now consider the case when the three single-phase sources are rearrangd to forrn ;I three-phase A-connection as shown in Fig. 2.7. I t is clear from in5ptc- tion 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 nals.

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 tw

/ n \b bb’ ob’

C Ibc

I,,’ C’

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

I

Stntic Elect t-ic NetUY)t-k Mode Is

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

/ < I / , = I/ ,

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

I ,,, = I,, i 120"

The current that flows in the line joining [ I to ci' is denoted by l$,,,,and is

given by

As a result, we have

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

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

\+hxl, 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;: \ l I l ( l ) t

\ ' , ( f ) = \ 21,: \111((1)/ - 120- )

\' ( I ) = \ 21,: \ l l l ( ( 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 i n the load is

This t ims out to be expanded US:

L:sing ;I 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 tem s

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

p 1 0 = 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 , , t l t

Rotation

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 of xinature circuits; 0 = angle by Lvhich cl-axi5 leads the iiiag-

rietic axis of phase winding. electrical radiaii~;: ( I CO,= rotor angular \‘elocitj. clectri1:al

rad/x.

Static Electric Net\t*ork Models 23

where

L,,,,.L,,,,,L,, are self-inductance of nhc windings.

L,,/,. L,,,, LI,,,, LI,,, L,,,,LIIlare mutual inductances between two stator winding (cih, hc, NC')

L,,,,/,L/,,,/,L,,,/ are mutual inductances between stator and fields windings

LlIAll,L,lL,I, 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,kqi ,

vkd = Llldild + LALdilrl - &iAdi , i - LRAdili - L,A,/;(

vAq = LAAqikt, - LiiAyio - Lldqih - LcAq;c ( 2 . 2 5 )

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 ,;,, r.1.28, tit

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 f lux linkage equation. This is because the zero sequence com-

ponents 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 f lux linking the field ufinding due to current i,, flowing in the d-axis stator winding from equa-

tion (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 un i t sqrstcm for the

rotor qu an t i t i es.

2.3.3 Per Unit Representation

I t is usually conivenient to use a per uni t system to normalize system irariables.

to offer computational simplicity by eliminating units and expressing sqrstem

quantities as dimensionless ratios. Thus

A we11-chosen per unit sy steni can m i n i 111 ize comp i t ;i t i on a I effort, x imp1i 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 t c I

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 sub-

scripts)

f?,h,l,r. = peak ~(alue of 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 st't

and depend on the above :is follows:

CO,,,~,,= 271 .fh,l,c. electrical rad/s

2 o,,,,,.,,, mechanical rad/s = \clh.l,, -

TFf ' ~ \ t 7 . l W

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<lL,/~L/

- _ - - -

wq = -L& + Ld,/i</ -WO = -MO (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 uni t 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,,,/-base per 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)

= 2nfhd,rL\hd\r.,Dividing by Z\hd\c 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 perfor-

mance of a synchronous machine. However, except for the analysis of very

\mall \ j \tern\, these equation\ cannot be used directly for \}stem \tability \tud-

I C \ . Some \implification\ and approxitiiations are required to reprewit the \ j n-chronoii\ machine i n stabilitj \tudie\. For large \ y stem\. i t i \ nece\wrq to neg-

1ec.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 .33 )become

( 2 . 3 6 ,

By defining the follou~ing variables

the riiachine eyuationx becomes

( 2 . 3 7 )

M here 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 ageproportional to E,<,.Sincc per u n i t \ = I_ , l'rom

l!c~u~ltlon(2.37) m e hace

For studies i n \i.hich the period of analj'sis is sinall i n comparison c+,itli I-,:, thc riiachine inodel i x often simplified bj' assuming that E'; is constant throiigh-

o i i t the st lidy period. Th i s ;issiinipt ion e 1imi nates t he on 1j , d i ftere n t i a1 ey i i ;it ion

asxociiited M i t h the electrical chxiicteristics o f the machine. A further approui-

rriation 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,,+jx$ has a constant magnitude. The equi\.a-

lent circuit is shown in Fig. 2.10. The machine terminal voltage phasor is repre-

sented 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,,+ jx ; ) 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?ing equa-

tion 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 steady-

state. 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 sali-

ency effects and stator resistance and offer considerable structural and computa-

t i onal s i mpl i ci t y .

Figure 2.10 Eqiii\~alcntcircuit synchronous urith x,; = xi:.

t t

(a) Subtransient model

E’L6

(b) Transient model

EqL6

(c )Steady-statemodel

Figure 2.11 Simple hjmchronous machine model.

Stcrtic Electric Netwwrk Models

2.4 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 over-

heating. 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 0.6 p.f. lag t

Figure 2.12 Reactive capability curves of a hydrogen-cooled generator at rated \dtage.

2.5 STATIC LOAD MODELS

Con\re11 t ionit 1 transient -stab i 1i ty s t L i d i es were i n I v ed main 1y w it h gene rat o r s t 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 i n earl)' loati

111ode 1s t hese \+'erecons i dered iis constant i mpedances . Rot ii t i ng eclLI 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 of load. A lot of uurk hits gon:

i n t o the development ot' more accurate load models. These include some corn-

ples niodels of specific large loads. Most loads. howe\ter. consist o f ;I I x g c

c l i i a nt i t y of cli \wse eqiii pmen t of v;iry i ng I c \ ~ ls atid coinposi t ion and some

cq11 i va I en Imodc1 is iiecessary .

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'w-

qi1cnc.y at that instant. The active power component P and the reacti\,c po\f.crr

c0111 pone 11t Q are c011 s i dered separate I y . ,A general load chxacteristic may be adopted such that the M V A loadii ~g

Lit :Iparticulx bus is ;I function of' lroltage m d fi-eclucncj~:

Q = K,/\'" f

$1here A',, atid A'(, are constant\ \+hich depend upon the nominal alue o f t ie

\ :triable\ P and Q. For constant frequencq operation. u e write:

~+hmP 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 iiib 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\Jeljfiiniiffected by frecliicncj' change\, i.e.. ni, = 1 1 , =

0. and with constant impedance loads i i i , = 1 1 , = 2. The iniportance of' acciiiate

load models has been deriionstrated f o r \coltage sensitive loads. Figure 2. I3 dtbiii-

oiistratcs the po~verand current characteristics 01' constant poLver, constant c iir-

rent, 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 I .6 0 0 0

FI Uoresc t' n t I amp 1.2 3.o - I .o -2.8

Heater 2.0 0 0 0

Induction motor half load 0.2 1.6 I .s -0.3

Induction motor fu l l load 0.I 0.6 2.8 I .8

Reduction furnace 1.9 2. I -03 0

A 1iiini nu m plant I .8 -.-3 3 -0.3 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 )conipo-

nents. 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 magni-

tilde 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 Ldtage deviation

from nominal. Further, i f 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 rep-

resent loads where the voltage is below some predefined ~ ~ a l u e , for 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 In pe dan ce char ;ic te ri s t i cs , respect i c'e1y . For composite I oads , thei r v i 1-

lies depend on the aggregate characteristics of load components.

The frequency dependence of load characteristics is represented by n iu l t i -

plying the exponential model or the polynomial niodel by ;I f'actor as follow>:

or

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

CONCLUSIONS

I n 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 as-

pects 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 ransiciit or dy niiiii ic operation a1 st l idies. Th i s cha pter describes soiiie sy s te 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-

;ibIe le\.e1s 11nder d i fferent operat i rig conditions.

The control tunction~ include the control o f iroltage and reiictiire po'i'er

tlow , thereby enhanc i ng pourer sy ste111 stabi I i ty . The protect i ve functions e n s i i re

t h ;i t t he cii pi1b i 1 i t y 1i 111its o t the sy iichronou s iii iich i ne , e sc itat i on s ~ ' st e111, In d

other ecliiipmcnt are not exceeded.

37

i

Power Power i i Power source

(regulator) source

(regulator) ; i

i j

'iource (exciter)

w 1 r i i Excitation 4 - 1

Reamplitier Power i : power

Amplifier i SWrCe 7 machine

I + i (exciter) - 1 I : I : I

I I I L - I

I

III

I

I I I

I

I I I

I

I II

I -I I II : I I

I

~ , ~ 74 Regulator i M a n u a L E ~ S + S y n c h r o n o u s ~ ~'iConlrol: ; machine 4 Excitation *

system Excitation

control system *:

Figure 3.1 Functional block diagram of a synchronous excitation control sy\tein.

Figure 3.1 shows the functional block diagram of a tj'pical excitation con-

trol 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

1 (3.1 1E, =

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

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. recti-

fie\ and filters it t o dc, and compares i t 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;' (.$.4)

U'here

Figure 3.3 Block diagram of the voltage regulator.

V S v F

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 - I j dt

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

T = V, - (S,+ K , + I ) E,dr

dVT,--t = -v+ &(& - (S , + K , + 1 ) E,)

d r T

X 2 1+T1S '"s -KPSS I + T5S l+T 2s 1+T2S

Figure 3.5 Block diagram of a typical Power System Stabiliter (PSS) model.

40

3.2 PRIME MOVER AND GOVERNING SYSTEM MODELS

The prinie soiirces of electrical energy supplied by utilities itre the kinetic enctg>'

o f 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.

i n tu rn . is con\,erted to electrical form by the synchronous generator. The pri iiie

m0 i .er go1.ern i ng sy ste111 prov i des a means of c011 t 1-0 ng pow er and t'rccl iicIIL'1'.I 1i

The !.Utict i oti;i 1 re I iit ion sh i p betwee11 the biis i c eIcme11t s ;issoc i iit ed wi 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\~crnitig s>~stcms.

3.2.1 Hydraulic Turbines and Governing System Model

The h>draulic turbine model describes the characteristics of' gate opening p :incl

o11 t p i t rnccha 11i cal power . I 11 power system dynam i c ;I 11a 1y s i s . the h >,drii LI I i c t LI r-

biiic 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 Valve 1_1) Turbine Generator

Governor or

~

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 rc-

setting time I \ therefore required. This is accomplisht.d by introducing a tran-

sient gain reduction compensation in the governing \i\ \tern. The coiiipen\ation

retard\ o r limit\ the gate niovement unti l 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 max gateand

Figure 3.8 Bloch dingram of gojwning system for ;I hjdraulic rurbinc.

4 2

The goirerning system model is given by

7(-,4! = -p + x, t f t

( 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 pvsi-

tion (Ayl).nthile its output is torque (AT,). In power \tability analj\is. a

order rnodel is used for \team turbine, i.e.,

ufhere r,,= 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 extiib-

1tt.d by ;I hydraulic turbine d~ ie to 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 e11t droop coin pe n \a t ion.

The go\ erning \ j \tern model i \ given by

- - max gate

band 1r x l -

' @ KS - l + s T G

S

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 sig-

nificant 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,

bv i th 1 1 , 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

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

I t is noted that Ecl. (3.10)represents ;I stendy state pcrformaiice nioclcl 0 1 the intluction motor. with all qiiantities referrecl to the stator siclc. There ;ire inodcls that represent the trnnsicnt performance of the niotor th ; i t 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 ~ystciii stahiliiy 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 dy-namic load model. For fiirther reading on the topic. ii reference list ard an annotated glossary of teriiis is provided ;it the hack o f the book.

Philosophy of Security Assessment

INTRODUCTI 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 synchro-nism) 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 linsariza-

tion 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 transmis-

sion corridor. which isolated a significant amount of generation from its lo:id.

More recently. a transmission tower in the Consolidated Edison sj~stem u ' a s hit

by a severe lightning stroke in July 1977. The events that follo\s.ed led to the

4 5

shutdown of 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 ~ ser \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',, i4.1 t tlt -

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

;in 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

1s~ i e I-c1;it ion

We find i t iiiore conLtenient to substitute the dot notation

Therefore we have

Jii' = T,,,- T,, (4.5)

4.2 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 mo-

mentum 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

1Wi = - J o ~

2

then

L 2 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 to thc kinetic energy itt rated speed divided by the rntcd power PK

The units of H ;ire in sec. As il result we write the per unit or iiortiia1izi:d suing equation its

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 oh-tainctl from mnnufacturcr supplicd machine datil. The machine kinetic ~11crgy. ,V. iiiay be written i n ter111sof M i1S follows:

where wK is the angular speed in electrical degrees per second. This i n tttrtl is rclutccl to the frequency by

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 formu-

la is:

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

H = -N (4.14)

G

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 \ due \ 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 3 - -

s 3 2.5 - -E I cn 2 2 -- -3600 RPM FOSSIL c

cn c 0 -C-1800 RPM NUCLEAR

1.5 - -2 -C

1 - -

0.5 - -

, 1 1 1 , , -0 I I

Figure 4.2 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 rs 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\:

5 I

4 - -I

3 3.5 - -F

$ 3 - -3 z I 2.5 - -v)c. c

tj 2 - - -A 450-514rpm0 0

.E 1.5 - - +B 2Ul-400rpm

r +C 133-180rpm -t

1 - - +D 80-lXIrpm

0.5 - -

. . . . . . . . . . . . . . . . . . . . 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:

j w L R

In t h i \ case the ciirrent magnitude ill approach titice the \tcadq -\talc II a \ -

i n i i i i i i \ alue iinmcdiiitely after the \hart circuit. The tran\ient current I \ pi\ -

Thus:

E,,,i ( t )= -1z

I t is clear that the maximum of i (r) is twice that of E,,,/Z.This n~a\~efortii is

shown in Fig. 4%

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

Fig. 3.Sb.

I t is clear from inspection of either the expression for the short circuit cur-

rent or the response u'ajseform given in Fig. 4.6 that the reactance of the ma-

chine 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.

/

Ex; = I'

kchere I' is the transient current and X:, is the direct axis tran\ient reactance. Tie

transient intenfa1last5 for about 30 cycles. The steady-state condition gi\ses t 76'

di rect 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;ict;i nce.

4.4 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

A brief outline of equations to account for flux changes in a synchronous ma-

chine is given to define various electrical quantities and to construct phasor

diagrams. The following approximations are involved in the models discussed:

I . The rotor speed is sufficiently near 1.0 pi( and may be considered a

constant.

2. All inductances are independent of current. The effects due to satura-

tion of iron are not considered.

3. Machine winding inductances can be represented as constants plus si-

nusoidal harmonics of the rotor angle.

4. Distributed windings may be represented as concentrated windings.

5. The machine may be represented by a voltage behind an impedance.

6. There are no hysteresis losses in the iron, and eddy currents are only

accounted for by equivalent windings on the rotor.

7. 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 ”propor-

tional” 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 ternii-

nal 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 . The flux @, is proportional to the field current 1, and the applied field

voltage and acts in the direct axis of the machine.

2. The stator open-circuit terminal voltage E, is proportional to @, which

is located on the quadrature axis.

3. The voltage E, is proportional to the applied field voltage and may be

referred to as E/.

4. 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,.. 5. The effective internal stator voltage E, is due to @, and lags i t by 90”.

6. 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 /. Theret im the \ o l ~ i g e

difference n i a j bu treated ;i\ ;i koltage drop ;icro++;in iirniatiire rcict-

ance x, .

The 'riitii o f X,,and X i is termed the synchronous reactance, x>. For the salient pole synchronous machine. the phasor diagram is more coin-

pies. Bec~iii'rethe rotor is syninietrical about both the d and c/ ;ixes i t is cot- \.u-

nient to rusol\~enianj' 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,,will be proportional to /,,, because the iron paths i n the two :,xes

;ire diftcrent. the total iirniatiire reaction tlus 0 \+. i l l not be proportional t o / iior

necessarily be i n phase \+rith i t . Retaining our earlier normalizing ;tsutiipti )ns.

i t inay be assumed that the proportionality between !*, and @</,is i i n i t y but tlic

proportionality between and @(, is less than uni ty and is a function of the

saliency.

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 caw. Di-

rect 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 i t 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 f lux 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,;. reprc-senting thc t lux 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 t lux linkages change with time requires using t’ie following ordinary differential equations:

-E ,SE,;= -r,:..

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

Direct Axis E l

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

I t 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.ha\,e shown that using the

\tandard niachine data. the I~IOI-e complex models do not necessu-ily j'ield accti-

ixte results. Ho~.t . \ .er , improved results can be obtained if the data, cspeciall>.

t tie t i nie constan ts. 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 ~ L I I ~possible number o f ec1u.t-

tioritr 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~indingro tor

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 ,SE). The follou~ing equations are used. A block diagram is shown in Fig. 3.11 .

E:,= v,,+ R,,I,,- l(/x:/

E:,= \'(, + RJ, , + 1,,x,;

U

Figure 4.1 1 Block diagram representation for model 3.

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 Model 5

t I- a id (1- axi s s i i b t ra11s i e11t effec t s reqi i i ri n g f o ur cl i fferen t i a1 ey ii ;I t ioris ( .;E;. .SE,:. and .SE:;). The follouing equations are used..SE:;,

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 \+i I 1 make the to1lowi ng add i t io nal a \ s 11111pt ions:

A\ ;I 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 ' , = -\ ' , 4iii i5

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 L+ i th the torcliie o r

po\srer angle 6 is \ho\s,n in Fig. 4. 13.

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 formu la.

= Pe

Figure 4.1 3 Power angle characteristics for a salient pole machint..

66

output Power, P

Angle, S (radians)

Figure 4.1 4 Power angle characteristics for ;I round rotor machine.

I n the case of ii round rotor machine, u'e have Xf: = X ( ; and hence

E V , .P , = -- sin6

x:,

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

6, (torquc or poww angle) is shown i n 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 of the LtboLre system is shown in Fig. 3.16. From this ~ s ' e

ha\-e the following: X c ( ,= 0.475 p i .

.

Figure 4.15 System for example 1 ,

Changes in the network configuration between the tuo 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

E L 6

Figure 4.17 Ecliii\ dcnt circuit tor cuainplc 2

Thcretore.

0bserl.e that the ii1;ixiiiiiitii alue o f the i i m clin e i \ l o ~ v rthan the OI c

corrc\ponding to the prec ious example.

4.9 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 \ \ ' i l l ;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 tn 'o values for 6. The lo~re r

\~:ilueis denoted by 6,,.Consequently, the higher is n: - 6,,according to the \>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 4.18 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.

I f the system is operating in an equilibrium state supplying an electric

pourer P,,, with the corresponding mechanical power input P,,,,,,then the corre-

sponding 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 un t i l [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\ M I : I I The \ js tem discussed in Examples 1 and 2 \er\re\ to illustrate th i \ 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 (ewi-

t 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

i i ) 6 , . A\\ume that the excitation \ oltage remain\ unchanged.

S oILItion

The power delivered is P,,= S cos $, P,,= I . I x 0.85 = 0.94 pi 1

Using P = VI COS @, then the current i n the circuit is I = -~ cos ' 0.85 1 . 1

Thus we can Urite

71

E = V + j X l

1 + (1.1i - 31.79")(O.37SL90")

= 1.28 +j0.44 = I .3S i 19.20" p i r

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,,,= 2.1053 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 with one

line open as

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 immedi-

ately 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 hwc :ilrcady done that the input is constant, ncglipi 3le doniping iuid coiist;itit source \dtagc behind the tnitnsient reactance. the angle hetwccn two 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 iiia-

chineh 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 writ-tell a s

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

P0 tlo, = (16

M

Integrating, assuming cc) = 0 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). I f

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~~ereaches 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 pout-

ers. 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. I f the accelerating power reverses sign before the t ~ ’ oareas A and .4? ;ire

equal. synchronism is lost. This situation is illustrated i n 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 i t J condition for accelerating rotor.

Power, P

P m

-0 6,

b Power

Angle, 8 (radians)

Figure 4.22 Equal-area c*r.itcriori tor \tabilit!,.

75

Accelerating

Power, P , I

Figure 4.23 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 re-

sult 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,. i s obtained hq solving tor A I = A, . Here we get:

77

This gives some algebra

I .93 c o d , + 0.0164 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:

OX x s

A , = I .8(68.85- 39.33jn - 1 1.93sin6(16 = 0.13 I 8o 31) 7 3

The area A_.is obtained as:

ii

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

180

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 ed 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 betuwn the generator and the bii4

i \ incrcased to tuv times its prefuult value.

I . Find the 6,,before the faiilt.

2. Shoci, graphicallq, c i , h a t happens when the f i i u l t is sustained.

3. Find the niiisiriiiiin \raliie of 6 s\\ting i n 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 ~ e tttith the f;iiiIt su\tained is hal f o f the original \-alue.

Before the t'itiilt \+e h a t e

At the t 'u i i l t instant, \4fe get

A \ before. the \tability condition yields

Hence

0 . 5 C O b 6 , + --<6, = 0.5473K

7'

Bq trial and error

6,= 46.3"

A

Power, P (P.u.)

0.50

0.25

Torque

0 Angle, S (radians)

Figure 4.25 Power angle curves for example 6.

The following example illustrates the effects of short circuits on the net-

work 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. I f the system

is stable, find the maximum angle of swing.

Solution

The power angle curves have been determined for the prefault network i n Exam-

ple 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 ,2x = 0.45 + 0.25 + (0.45 0.235

and fri i i l t pourer angle curve is given by

f = 1, I3 sin6

The three power angle curl'es are shown i n 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 II I I I , I I . Torque

6 f ~ n g ~ e .S (radians)

Figure 4.28 Pre-fault. during fault, and post-fault poww angle 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 of the changes arising i n the

generat ing and load eq U i p men t , Th i s approach is enip1oyed i n mU 1t i miichi ne

stabilitj programs because the inclusion o f time Lrarying parameters ~voulcl pro-

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 com-

plex 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 in-

jected 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 . For each network load, determine the injected currents into the modi-

fied admittance matrix by solving the relevant differential and algebraic

equations.

2. Determine network voltages from the injected currents using the Z-

matrix or factors.

As the network voltages affect the loads, an iterative process is often re-

quired, 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

of the machine thus formed may be added to the shunt admittance o f the 11i;i-

chine bus and treated as a netuwk parameter-. The \ ector /,!/ thus contains the

Norton equi ktalent c iirrcii t s of the sj'iichroiious iiiach i nes , The sj'iichronous 11I a-

chine 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 i n Fig.

4.29. art' obrai 11ed !I-( )I I I t h t' fo1I o c i ' i iig t ranstornia t i on

The in\,erse relation is

Network

t imaginary axis

Machine Machine

direct axis

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 con-

stant impedance is used while modifying the injected current.

4.11.3 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 admit-

tance 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 net-

urork admittance matrix [ Y ] . During the stability run, each load is s o l k ~ dse-

quentially along with the generators, etc., to obtain a new admittance (r).i.e.:

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 correspond-

ing 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 character-

istic. 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 U ith l o ~ 'iolta;e

adju\tiiient. ( a ) Load current. ( h ) Injected current.

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 consid-

ered.

Although faults can occur anywhere in the system, i t is much easier comru-

tationally 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

fau l t 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

87 Philosophy of Secirr-ity Assessriierir

The worst case is a three-phase zero-impedance fault and this involves plac-

ing an infinite admittance in parallel with the existing shunt admittance. In prac-

tice, 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 imped-

ance 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 protec-

tion 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 i t is better when protec-

tion schemes are modeled. I n this case, a dummy bus is located at the fault

position. (even though i t 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 al l 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. i t can be reiiiowd automatically after ;I

su i tabIe dei 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~iuse diwr-

gence. 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 prob-

lems occur L+,hen calculating the admittance, and the subsequent fwtors for 1 he

forward m c l backward substitution, then convergence of the ow-al l solut ioii

betu.een machines m c l netL+x)rk will be unaffected. The k-aluc. o f the lo\+!-impcd-

ance 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 i t 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 opera-

tion. I t 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 I O U * - i mpedance branch.

Alternati\rely. i t 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 admit-

tance included i n the network Y matrix, which is due to the machine mu!,t be

re n10ved. 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 discon-

nected and reconnected at different times as the ~ ~ ~ l t a g e le\rel 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 \ta-

bility problem, and the principal methods are discus\ed nou.

4.12.1 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

( P I ,= 0) to give a prediction of the solution at t(rl + I ) and the other bein,0 1111-.

plicit ( P I ,f 0) which corrects the predicted value. There are a great Larietj of

method\ aLrailable, wch as hybrid methods, the choice of which being made bq

the requirements of the solution.

I t is usual for simplicity to maintain a constant step length with these meth-

od\ 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~iracy of

the corrector. Therefore, if the corrector is not to be iterated. i t 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 conver-

gence. 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 and ~ i , , , ~are constant at a particular step, the solution is itera-

tive.

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 f u IC-

tion of the nonlinearity of the system. Provided that the step length is suffi-

ciently 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 expan-

sion, we obtain the generalized Euler's formula:

d.r(t)s ( t + A t ) = x ( t )+ x(t).Ar where i ( r ) = __

at

The method is not often used for real-time applications in power systems, as i t

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 w ( t ) = --(& - 4"

H

I Set Fault Time: tf = t s = o - w R'

' w R

=2nf

Perform 'during fault'Power Flow Calculations

Figure 4.31 The transient stability algorithm

@ 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 I Torque Angle, Machine Speed, Generated

Obtain‘new’state of the machine Power, Times (t,

andor the network perturbations tf. tmarr etc.) and i so on.

Figure 4.31 Continued.

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, i t 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 interpreta-

tion 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 i n the assessment of transient analysis of rotating na-

chines, 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, fl,-., . . . , and tl,. I f the predictor is self-starting, i t must be free

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.

where k ( i )= h f ir ( n ) + c,h, ! ( / I ) + 111

/=I

and the sum of all the weights equal to unity, i.e., w, = 1 I= I

The coefficients are uniquely determined, giving rise to various orders and

associated approximate or accurate models of the Runge-Kutta predictor for-

mula. 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. When 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"' + + k'?']

6

RK-3 k' ' = F(.vl,,t1,)

= .vil + -I t (k"' + 2k'!' + 2k'" + kf4'] h6 k'?'= F!.Vl t + - k " h + - i2 2

Explicit Runge-Kutta methods have been used extensibdy in transient sta-

bility 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. I t 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 1 0

"stiff' prohleins, &(here the 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 yattractive to pourer system engineers iis k poww flow calculatioris

must be done at every iteration. where k is the order of the Runge-Kutta forinula.

The tollou,ing esnmple demonstrates the RK-4 application.

Exc?niple

Application o f the 1"' er \ J \ tc inorder Runge-Kutta integration techniqiie to p o ~

dynamic \tability asws\iiient. The algorithm is giL en belou.

I . Start the RK-4 Subroutine.

2. Initialize all R K - 3 dependencies.

COllIlt, d = 0: Step Size. At

Time. t = 0: M iix i mum Time, t ,11,1,

Maximum iteration = itniax

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 RK-4 calculations (obtaining the estimate\).

97

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

I9. Update the torque angle delta to: 6( t )= & ( I ) + 5 k , and sol\^ the pom'er

flow for I v , l,Py"(t), etc.

10. Compute the new coefficients.

Ik , = ( ~ ( 0 )+ - I , - ) - 2Ff)At

2

I , = w ( f i l i - f : " ' ( t ) )At2H

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 + A t

16(t + At)= 6(t )+ -(ki + 2k: + 2 k , + k, )

6

ICO(t + At) = CO( t )+ -( 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 ( t< t,,,,,,) and ( N 2 itrncix) then flag the user: "Maximum Number of iteration reached!" and go to step 16.

Elseif ( t< t,,,,,,)and ( N < itmax) then increment the iteration count

to: N = N + 1 and go to step 4.

Else Flag the user: "Run time value of tmax reached!" and con-

t i n Lie,

16. Display all results, power angle, machine speed. generator p o ~ ~ e r .gen-

erator terminal voltage, etc.

17. End of subroutine.

4.12.6 The Sampling Method (James J.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 inte-

gration 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.

4.13 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 i t by repear-

edly 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 i t is necessary to enter these two subroutines, otherwise i t is

set to zero.

r+=? Read in steady-state system data Section 1

Section 2

Section 3

r

Either Solve for machines and network

or recalculate nonintegrable variables

(initially and when switching has occurred) Section 4-

1. Print out results (if necessary)

2. Make power balance check of

initial conditions (if necessary)

NO 1

YES NO

Section 5,

- Store initial Reset initial

conditions conditions

Figure 4.32 The transient stability algorithm.

I 0 0

Synchronou s machine AVR calculations ‘wed governorNetwork calculations calcuyns,~ ,~A ,f c a l c u l j y ~ n s ,

KASE =0

KBlFA I = 0

KBIFA3 = 0

input

YES

NO

V

Figure 4.33a Section I

KBIfA3

The elements of the spar\e-vectored inverse are evaluated in the third bifactor-i-

zation subroutine. The code KBIFA3 is set to unity if i t i \ necessary to en er

thi\ wbroutine, otherwise i t i \ set to Lero. When KBIFA3 i \ unity, i t indica e\

that a netiborh discontinuity ha\ occurred and hence it i \ a lw 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 disconti-

nuity when i t is set to zero.

4 Switching

Determine bus order

and nonzero element

location for bifurcation

KBIFA3=1 TIME=O

++ Is KASE2IIs KASE2I NO NO

and is thae noand is thae no

switching a t T I M E = O ?switching a t T I M E = O ?

YESYES ** II K B I F G P syncmachine *AVR initialKBIFmSyncmachine

t, conditions.initialinitial

YES conditionsconditions + Speedgov. 1 T initialinitial

conditionsconditions

I 4 numerical part

of bifurcation

Print out initial

conditions

when U S E > 1)

Figure 4.33b Section 2.

I 02

v A -7-TDetermine

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 bifactorization1 KBIFA3 = 0

part of

KBIFA3 = 1

Figure 4 . 3 3 ~ Section 3.

Philosophy of Secirrity Axsessnierit

I Solve for network

KBIFA3 = 1<>-TIME =

E- PRINT TIME

YES Print out

Print out busbar Print out AVR results

and branch results sync. machine

if required results

4 speed gov.

a results if required

Set flag for

step doubling

KBIFA3 = 0

I

Figure 4.33d Section 4.

I04

A v YES YES

r 1

NO 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

Start solution Lr'

Calculate constants

nonintegrable variables ,-,-,-,,,,-,,---' +I speed gov. calc

I not usually requireda- ------- - - ---- - - -1

Evaluate integrable Same for cach A

varaible using algebraic b Same for each

form of mpezoidal method 4 speed gov.

I

[HALF= 0

Figure 4.34a Section I

ITMAX

Maxiinuin number ot' iterations per step since las t printout of results.

Note that iiiany data error checks are required in a program of this t j p but

they ha1.e been omitted from the block diagram for clarity.

i

106

Figure 4.34b Section 2

I

107

v A YES

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.

I ox

Strcictiire ot Machine '2nd Netivork Iterative SoI~iti017

The ~ - u c t u r e of 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 etralu-

ated 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~nent based on fre-

quency domain models arid equal area criterion concepts. In particular. we de-

fine the conventional ingredients for power systeni stability including app ica-

tions 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 tran-

s i eri t s t ii bi 1i 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 con-

tingencies i n advance, assess the results, and then take preventiLre control action

if required. This whole process is called dynamic security assessment (DSA)

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 i t takes a long time to conduct a transient simulation even for a single

contingency, direct methods of stability assessment such as those based on Lya-

punov 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. LISLI-

ally 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

eq11at ion s. Genera 11y , the differentia 1 equations are m ilch i ne equations , cont 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 ir 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)

I

*fit /--“’

I.... ................................. ........................ ...*........................ “ ....................... +

6 .

It0

Figure 5.1 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. I t 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 i t is not stable.

The concept is illustrated in Fig. 5.1 where the initial state x,) has a magni-

tude 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 l ~ ~ ~< 6 implies s -+ 0, ( I S 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- Stnbil it)/

The origin o f the \ystem described by Eq. ( 5 . 3 ) is said to be globally asymptoti-

call> stable i f i t is \table and also implie\ -+0 cis I + 00, for any .v,, in th?

\v ho I e space.

Posit ive 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. I t ' the>

do. \ ' ( .v) has tM'o \~al~iesat the intersection. We need the follou.ing lenimii r t b -

qiiirecl for the proof o f ii Ljrapunov theorem.

LcYl117lC?

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 niono-

tonicully L b i t h p i n 0 5 p IN for any unit \rector I I started from the origin.

This ciin be shown using the assumption of positi\.e definiteness. First, coil-

t i nu i ty . V(.Y)> 0 and V ( 0 )= 0, assiiiiie that V(pI1)increases monotonically \bri .h

p i n an interval 0 Ip I pIIand 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 i n 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 o vTheoreni

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

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

Regi017~R, RI, R,

A11 the regioris ;ire assuincd to contain the origin ;is an interior point. R: i < , ;I

\Libregion ot' K , \+,hicti is ;I siibi-egion of R : K: I K , I K.

Theorem

Let V ( x )be a positiise definite function with continuous partial deri\rriti\ves i n a region R, then

The origin of the system described by Ey. (5.3) is stable if i ' ( .v) I0 i n

a subregion R, I R.

The system is asymptotically stable in the region if i t is stable and i'(.v) =0 (identically zero) takes place only at the region i n a subregion R.

I RI. The origin is globally asymptotically stable if the sqstem is asymptoti-

cally stable. R2 is the whole space and V(.v) + 00 as 11.v11 -+0.

I *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 ~ ~ . r ~ ~= I",and also i t is a closed contour since I I

is any uni t 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,,) < 1 1 1 ~ O I I O M ~ S by monotonocity. Thus, any trajectory initiated from .v,) cannot possi-

bly cross the sphere II.vll = t - 5 E due to the fact that \'(.v) is non-increas-

ing and V(.v) 2 1 1 1 on the sphere II.YII = 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 i t stops decreasing momentarily. This implies that .v +0 L \'L' I - + -

since V ( 0 )= 0 only when .v = 0 in R2. I t 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 a s \l.vll +- excludes this possibility which completes the proof of the00

the ore 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 stems.

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 px t

and is asymptotically stable if all eigenvalues have negative real parts. Asymptotic stability as judged from the linearized system is simple but 0 1

less practical use. I t 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, C I is the damping constant and dh is the undamped

angular velocity. We regard the problem as a mathematical one without rest-ic-

tion 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 i t is possible to find a L Y ~ ~ U I W V function 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 i t 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 ~ Isine = o

Multiply by iv 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

1 7

V(.r) = h( 1 - cosx,) + - .rj2

in the region

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

The derivative of 11 is

1 . 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.

2. V ( . v ) = -(i.vi =0 implies that .v2 0 and hence .i:= 0. This makes .i-,=

-u.v2 - h sinv, = 0 which yields sinvl = 0 or .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 thco-

rein 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.

For demonstration purposes, M'L' employ the proposed \'(.v) to find ;I 6 5 E

iis 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

3 7 7

on the circle .I + .v> = 1.- 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, i t ha\ a niinuinum at .t l = + r - and .I = 0.

Therefore, ui = h( I - CO\/-). The expression for the minimuin becomes coirpli-

cated for h > 1 .

on the contour \'(.v) = 1 1 1 . c/ /A/ .v I = 2(.vl - h sin.\l)> 0 for- .vl > 0 and h I I .

Since the minimuin occ~trslit .vl = 0 and .v2 = &U, we haie the rninitwrn

norm

The inequality holds for the reason that

117

For the case b > I , we may choose an independent \variable a = wf \ii t h CO' = 1 7 .

Thus. the original dynamic equation becomes

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 unt i l 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. SYSTEM MODEL DESCRIPTION

5.2.1 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 ; i t may 3e

positive or negative. P,,= E ; G,, is the power delivered to the local load at gencr-

ator 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

21

v L 0

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

E VA ( 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 d - -1 MO$= P,,,(6- 6,)-A(X)(COSG,- ~0.46) (5 .10)

2 2

(a) During the hult: X = X ,

For S = S,,,we obtain from the above equation

-1

Mw,! = PJ 6,)- 6,) -A(X,)(COSG,- COS^,,) A I (5 .1 1 )2

(b) After the clearance: X = X ,

For S = S,,,,we have from Eq. (5.1 I ) that

I30

Power

P (P.U.:

A -Prefault

B- Post-fault

C -During

fault

I III

Ip , i i --Lx! I I

I II II + TorqueTorque

n Angle, S (radims)

Thus, ~ . ' econclude 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~ffunction (TEF)after clearing. That i \ . to obtain f r o m Eq. (f . 12)

U ith X = X ,

I'I

where

E = -1 MO2 + P J 6 - 6,) (5.13)

2

and

V = A ( X , COS^, - COS^)

5.3.1 Transient Swing

Let II = do/clt denote the acceleration of the system, then Eq. (5.13) together

with Fig. 5.5 show that C I < 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 corre-

spond 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 de-

crease 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 (0 is 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 C I > 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 i t 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. (5.11 ) . 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

of e,.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 integral- square error

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

M , = M,o,, with M , = cM , 1- I 1- 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 , ~-+ M ,(10 (!!!!U = U,,,, -c ,= p, - P:, (5.15)

t l t t l t

with

and

where j f i signifies summation o f j from 1 to t i except i. All the angles 6,- 6, in P, , may be replaced by 8,,= 8,- 8, because

Multiply Eq. (5.11 ) 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 speci-

fied by 11 pairs of (angle, velocity), that is.

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

(5.18)

where

and

I 1 1'

v = c j P:, e,dt (5.30) ,=I ,

with 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:

(5.2 I )

The preceding equation can be verified by carrying out the summation. Substitu-

tions 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 , )- co\ (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 + K f ( t )= C' + K, f l t + ( I , - t ) ' I = c' + K g( " )

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

Solving the twro constants gives

Based on this approximation, we haw

and hence i t follows that

with

This result makes i t possible to integrate I as

I t 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 proba-

I ,

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 accuniu-

lated 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. I t 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 i

M , = C M , and MR = M , 6, I I 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.

1E - V = - 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 TO 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

(3: = angle of bus i at the postdisturbance SEP

J, = 2 H , o , ,= per unit moment of inertia of the it’’ generator The transient energy function consists of the following four terms:

1 . 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

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 the critical energy V,,.

Step 2 Calculation of the total system energy at the instant of fitult-clear-

ing Vd Step 3 Calculation of stability index: V,,- V,,.The system is stable if the

stability index 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 diffi-

cult step in applying the TEF method. Three different approaches are briefly

described here,

Input system data b~~~~~~~ ~I Power now calculation

Con ting ency apecifica 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 .I

Calculate .yatem total energy V, at clearing time

Hes 2

1 . The Closest Unstc?ble Equilibrium 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 Determine all the UEPs. This is ach ie id by solving the postdis-

t u rbance system steady -s tat e equations wi t h di ffe re n t i n i t i a1 va 1lies

of bus angles.

Step 2 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 \I or\t

fault location. hence, the results are very conservative.

2. 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 control-

ling 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 f i i u l t -

on 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, i t is u n -

stable.

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

eration. ' = I

Algorithm to Find the Controlling UEP The reduced system is

The algorithm for finding the controlling UEP consists of the following steps:

Step 1 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 I1

reduced system Eq. (5.31) to find the first local minimum of I : II.f;(s>),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. (5.3I ),.f;(6) = 0, say the solution is 6,,,.

Step 5 The controlling UEP with respect to the fault-on trajectory is (6,(#. 0).

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 un-

stressed, 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 chap-

ter 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 re- ferred 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 ac-

ceptable 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 kd tage instability o r e ~ w i

\.oltage collapse, which have resulted i n several major system fiiilures (b1ac.k-

outs) such as:

I . September I9 10, New York Power pool.

2. Northern Belgium System and Florida System disturbances of I982

3. Swedish system disturbance in December 1983.

3. French and Japanese system disturbance5 in 1987.

5 . 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 <ta-

bility or collapse are characterized by a progressike fall of idtage which can

take several forms. The main factor is the inability of the network to meet the

demand of reactive power. The process of instability may be triggered by seine

forni o f di\turbance, resulting i n changes in the reactikre poucr requirement. '-he

disturbance may either be small or large changes in essential load. The conse-

quence of voltage instability may, however. have uidespread impact on the

system.

Power system voltage stability involves generation. transmission, and distri-

bution. Voltage stability is closely associated with other aspects of poM-er system

steady-state and dynamic performance. Voltage control, reacti1.e power compen-

sation 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. I t is a fast phe-

nomenon for engineers involved with the operation of induction motors. air

conditioning loads, or HVDC links. I t 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 \%!hether the phenomena of'

voltage stability are static or dynamic). Voltage instabilitjr or collapse is a dqr-

namic 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 degra-

dation. 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 i n 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\%.ork related

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 i n addition to models 01'

generators, exciters. and other controllers. Small signal voltage stabiliti, prob-

lems 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 VAR

compensations and in the design of controllers. Large disturbance \wltage stabil-

ity 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

6.1 WORKING DEFINITION OF VOLTAGE COLLAPSE STUDY TERMS

Voltage stability has been viewed as a steady-state “viability” problem suitable

for static analysis techniques. The ability to transfer reactive power from pro-

duction 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 differ-

ent explanations. interpretations. and concerns. To achieve a unifying franre-

work, 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 volt-

age violating its limits which value may lead to system instability and conse-

quently 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 l i ter-

ature. 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 mea-

sures 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 possi-

ble, because of a combination of events and system conditions. that the addi-

tional 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 ab-

sorbed 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 addi-

tional 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 al-

lowable 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 \.+wild likely be conipounded by the r:duced effectiLmess of shunt coni-

pensators 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. I t often means the existence o f a con-

siderable margin from an operating point to the voltage instability point fc)l lo\\ ,-

ing credible contingencies.

6.3 TIME-FRAME VOLTAGE STABILITY

Voltage instability incidents in \.arious places around the world has been c.c-

scribed 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\+.een0 to 10 seconds.

2. Classical Voltage Stability: This occurs betureen I to 5 minutes. This

i s t he c1iiss ic sccnario i n 1'0 I 1,i ng aiit0111 at ic on-I oad tap change r , d i st r i bu t ion \,(1I t -

age regi i 1at ion . x ic l gener;i t or c Urre 11t I i mi t i n g. Th i s scen ario i n \'oI \re s high 1o;i Js. high p o ~ v e rimports from ii remote generator, and a large distribution. Tap

changcr action is also significant, i t i t 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 nstabi I i t y i tic ident s \v i t h and L + i i t hoii t vo I tage are su 111I na-

r ixd i n Figure 6. I . For classical \,()Itage instability. the phenomenon \+r i l l 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 \dtage begins to sag leading to \ultage c:ol-

lapse.

6.4 MODELING FOR VOLTAGE STABILITY STUDIES

Voltage stability studies involve the solution to algebraic and differential equa-

tions that map the system behavior under steady-state and transient state\. Belo\v

are tjpical \ ector\ encountered and the notation used.

--- -11

Classical Long-Term (Large Disturbance) (Large Disturbance) (Load Buildup)

GeneratorExc. Dynamics LTCsi-Rime Mover Control Load DiversityJrhermostat

0 0 0

Max. Exciter Limiter 0

Mechanically Switched Capacitors Linnransformer Overload -;-y- Gen. ChangdAGC 0 4

Inertia Dynamics Boiler Dynamics 0 0 1

Operation Intervention I

DC Converter LTCs Gas Turbines F- - 1

I min 10 min I h r 1 I I I 1 1 1 1 1 I 1 I I I I 1 I1 I I I 1

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

Chtrprrr 6

Q ( t ) = Nonscheduled reactive power

q ( r ) = Other algebraic variables

3. Parameter vector, p ( t )

where

P,(r 1 = Turbine shaft power

P,( t 1, Q r ( t )= Scheduled load power

6 d ( t ) = Controlled voltage or set points

CT(t 1 = Other similar parameters

6.5 VOLTAGE COLLAPSE PREDICTION METHODS

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 auton-

oinous response of the system to reactive supply/demand imbalanccs.

139 Voltuge Stcrhility Assessrtierit

2. 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.

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 net-

work 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 compo-

nents having very fast response. The time range is from microseconds to milli-

seconds.

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 con-

cern. 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 ~ wsystem 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 dy n a i n ic vo1t age st 11bi I i t y ana1y si s, exac t t node I ing of t ran sformc rs ,

SVCs. induction motors, and other types of loads are usually included in prob-

lem formulations i n addition to models of generators, exciters. and other control-

lers. 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:,ta-

bility 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 coiidi-

tions, ;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:ar-

i d sb+ring equations crosses the imaginary axis in to the right hal f o f the coin-

ples 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'ter-

ent indicator ( L index) is calculated from normal load tlow results with reaxm-

;IbIe coinput iit ions. The 111 i ni in u 111 s i n g u 1ar va1ue of the J iicob i an was prop )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 i n the

field.

Bifurcation theory is used to analyze static stability and voltage collapse.

Static bifurcation of power flow equations were associated Lvith either di\wgent-

type 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 \~ulnerability to Lwltage

collapse using an energy function for system models that include idtage \,aria-

tion and reactiire loads. I t 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 optimiza-

tion methods haire been developed to compute the exact point of collapse. I n

a p p I y i n g the co n t i n u at ion methods , assumptions about 1oad ch an g i 11g pat t e1-17s

are needed.

In summary, the methods for static voltage instability analysis are based on

multiple load flow solutions (voltage instability proximity indicator [ VIPI], en-

ergy method), load flow results ( L index, VCPI), or eigen\Aues of the Jacobian

matrix (iiiinimum singular value and condition number). While studies on djr-

namic voltage collapse shed light on control strategy design (off-line applica-

tions), static \.oltage stability analysis can provide operators with guideline in-formation 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 sta-

bility indices. Of the wide range of techniques available. we shall discuss the

VIPI method. a method based on singular value decomposition. condition nuin-

ber 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(c i ) .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 Jacob-

ian can be decomposed as:

J = LEV' (6.11 )

where: J E R'J1*'J1is 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 singu- lar 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)

We define

E, = u,V: (6.14)

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

11 - 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.

6.7.3 Condition Number of the Jacobian Method

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

i n 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 or vector h ciiii be defined as:

For pobrer t locb Jacobian iiiatrix J . the \ alue of Cond, ( J ) can gile an

indication of the condition of J "with respect to inversion." A small ~ ~ a l u eof'

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*ra-

ting 5tute is \ c ry close to the point of Jacobian \ingiilarity and has a I O N 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 yuiinti-

tatice 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 ocnergj 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~tage power tlow solutions is very large (Y'- I for a practical syteni. the

exhausti\ve approach is not feasible. There is an impro \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 bc-

tureen 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 \ dues 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,ton-

Raphson method with the optimal multiplier can be used.

6.8 ANALYSIS TECHNIQUES FOR STEADY-STATE VOLTAGE STABILITY STUDIES

6.8.1 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 maxi-

mum 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 upsand / 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 sis

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-].I t 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 pro-

posed to observe the voltage instability phenomenon is closely related to tn111ti-

ple power flow solutions. which are caused by the nonlinearity of power flow

solutions. The drawback of the method is that i t 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

cont 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 repre-

sentation of the solution (.v.h)of Eq. (6.20) we need a one-dimensional measure

of x. The frequently used measures are:

( i ) 1 .v 1 = E','=,xf (square of the Euclidean norm),

( i i ) j s I = max 1 .I-, (maximum norm), 1 I=I I f

( i i i ) 1 s 1 =xi for some index k , 1 I k I n .

In power systems generally we use the measure of ( i i i ) . 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 ful l rank I I and satisfies some nondegenericitj,1 - cl: )

conditions.

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

eq11at i ons

This procedure basically augments the original set of pobrer tlow eqiiat ions

=F(.v,h)= 0 by F,(.v,)L)/I0 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 i t requires the solution of four I I x / I ( 1 1 is the dimensicln of

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 continua-

tion problem is to calculate further solutions, ( .~ , .h , ) , unti l one reaches a(.v2.h>). target point. say at h = A*. The ith continuation step starts from an approsima-

tion 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 correc-

tor 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

PI , Ql,,,= Original load at bus i, active and reactive respectively r <

= 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 h varies

= Initial voltage at the bus

= 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 equa-

tions.

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

Voltage Stcihility Assrsstnerit 151

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

e:t = tL= I (6.29)

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 possi-

ble 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:

(6.31 )

=[E] [ ;]+ U [i] where "*" denotes the predicted solution and (3 is a scalar that designates the

step size.

6.9 PARAMETERIZATION

The branch consisting of solutions of Eq. (6.20) forming a curve i n 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 in-

stance, by looking at a PV curve, one sees that the voltage is continually de-

creasing as the load nears maximum. Thus, the voltage magnitude at some par-

ticular 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 i t needed to satisfy the equations. This is called local parameteriza-

tion. 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

\%=[q =o , (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 section.

The algorithm tor static assessnient is shourn i n Figure 6.4. We ciin use ;I

simple example to explain the static ~inulysis proccdure.

6.9.1 Static Assessment: A Worked Example

Consider ;i \j'steiii is represented by

U 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

Input System Data 1 f

Select Contingencies

* Select Continuation Parameter 1

&

Solve base load flow .

I I Choose stepsize o I

* Calculate Stability index A p OT * Q

Figure 6.4 The algorithm for static assessment.

2-r -1 0 d.r

2 I - 1 1 [& = 0[ 0 1 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[ ."];[ 1.0

dh 2.0

Therefore,

Choosing 0 = 0. I , one gets

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

f, = j'" = -2 + 0.1 = - I .9

h, = h* = 0 + 0.2 = 0.2

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

'1

Voltcige Stcihility Assessriieiit

XI = 1 .OS

=-1.9

h, = 0.2

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

X I = fl = -1.8

h,= 2 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 i n -

clude\ enhanced de\.ice modeling. By letting AP = 0. we can write

Uhere 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 cl and q, being the corresponding

col urn n right eiget n ~ ~tor and row left e i gen\.ect or, re spect i \re1y .

The ith modal reactise power Lwiation is

where A': cci,= I with c,l the j t h element o f 5,. The corresponding ith mocal

~ v ltage \wiit t ion is

The magnitude of each eigenvalue h, determines the weakness o f the corie-

sponding modal voltage. The smaller the magnitude of h, the uteaker the c o r ~e-

sponding 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. I f all the eigenc~alues are 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 indi-

cate 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 ei-

genvalue 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 simultane-

OLIS 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 s i mU I at i on.

6.11 ANALYSIS TECHNIQUES FOR DYNAMIC

VOLTAGE STABILITY STUDIES

I t is only recently that the effects of system and load dynamics are being in\.esti-

gated 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

I . Fonn Full Jacobian Matrix. J Analysis?

Determine weaker voltage areas

based on eigenvalues I Perform participation factor

analysisI 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 stabi-

lizer (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

1

Volfuge SfcJhi1if.Y Assessriierif

, Subsystems: Subsystem F x&(x *x , u * I - d ) k f + x $xFut Load)

k $ ( x *XF"U*Load)

N (Voltage Collapse) Type.11

Instability

Both Subsystems S and F are Stable I 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 representa-

tion. .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 , ] i = 1, . . . ,I?

where

1 1 , = 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. . . . . I I I

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, i f 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 ran s formers. On Iy the sy tich ronoii s mach i ne 11nd t ap-c hanger d y nani i cs arc i n -c1Uded .

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 ' % = -E;, - ( x(/,- x;,) + E,,,, i = I . . . . . tit

-T\,'& = -vRl+ K \J,, E,,,,+ K ,,( v,,.,,- v,) i = I . . . . . 111

rlt Tl I

KI I T / , S= - RI,+ ~ E , , , , i = I , . . . . 111 (6.43)

tlt TI I

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

the system

6.11.2 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 , . . . , I I

i-I

Q:"'

(1

= CV,V,Y, ,~~- e, - ~(O,a,h)= 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 magnitudes numbered 1 to m. 2. Specify bus voltage angle number 1 (slack bus).

3. Specify net injected real power P'",'= P I ,and Q'"/'= Ql , at all buses num-

bered 111 + 1 to 1 1 .

Solve the following equations 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 genera-

tion 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, . . . , l l , , ]

and

.I-; = [G,,o,,E;,.E,,,EI,,,, 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 ex-

pressed as

A , ,

0

0

0

0

In Eq. (6.48) the variations corresponding AVl in the nonlinear load charac- teristic 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,. I f det (1,)-+0 ;I\

load is increased i t 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 ~ariables i n Eq. (6.39)atid assuming Ail E 0. i t can bt.

expressed a s

6.1 1.5 Slow Instability

Theoretically i t shoiild be possible to eliminate AY,in Eq. (6.50)using the singii-

lar 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.i-

t ion 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 pa-

r: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 8' = J , / . The system matrix. A c. c,I

is obtained as

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. I t 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 i x

reduced and the power flows are continued. The voltage stability limit is consid-

ered 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 proce-

dure is carried out simultaneously for the intact system as well as for contingen-

cies. 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.

( 6 . 5 3 )A \IAel

where

AP = incremental change in bus real power

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-en-

tial 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

The terms A l l , AI? , and AZ2represent a modified form of J,w. J\>,, JVH.

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,

(6.59)

where

h is the iIhmodal voltage.

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 ~ olet 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: contingenc

Generate QV curves 1 b+,

- New load level or change

Figure 6.8 Thc VSTAB algorithm (Cl EPRI) .

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 s t e111 i.01tag e st abi I i t y i n vol ves generat i on, t ran sni i ss ion and d i s t 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

a I

Figure 6.9 VoI t age 5 tabi 1it y pre\.ent i v e cont ro I \c he iiie.

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. I t is well knov,Tn

that in all cases, voltage instability is caused by inadequate transmission capac-

ity at a given operating condition due to a contingency, which the system cannot

withstand. Based on contingencies that occur, the distribution of plant gene1 a- tors, 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 stabil-

i ty 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 enhance-

ment based on a selected list of contingencies, only enhancement of the app-o-

priate problem (either voltage or angle) needs to be carried out, thus saving

labor and computational time. Future work in unifying the indices while inc,.>r-

porating 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

MVA 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-matri-

ces of the Jacobian in the linearized load tlow equations as defined by:

Voltage Stcihility Assswiierit I7i

(a) Infinite Bus Load Bus Bus I Bus 2

pZ-JQz,

Transmissicm Line

(3c1Load Shunt Load

Qsh -7 j

(b) Infinite Bus Load Bus Bus I Bus 2

v, = 1 .OLOO "2 =I v, IY, = 2.142 -J24.973

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 Jacob-

ian 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 eigenval-

ues of the reduced V-Q Jacobian matrix for the following cases:

a. P = 1500 MW, Q,= 450 MVAR.

b. 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 + j24.973

-2.142 + i24.973 2.142 - j23.897

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

where

Hence. are interested in only P1 and Q2b v i t h V, = 0 p i i .

Hence the expressions tor the Jacobian terms are give by:

a P+J,ll = = -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, J K 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

High Voltage Solution Lou Voltage Solution

500.0 1.024 -37.3 17.03 0.059 0.67 1 -66.7 -39.87 -0 .02S

400.0 0.956 -40.1 12.4 I 0.081 0.706 -60.3 -20.96 -0.048 306.0 0.820 -48.2 0.52 1.923 0.812 -48.8 -0.9SO - I . O S 3

305.5 0.184 -48.7 0.02 50.10 0.815 -48.6 -0.700 -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 re- duced Q-V Jacobian matrix is J , = -13.683, and J , is negative indicat-

ing that the system is voltage iinstable.