power system analysis stevenson, grainger

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POWER SYSTEM ANALYSIS

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POWER SYSTEM ANALYSIS

John J Grainger Professor,' Department of EleClrica/ and Compl/{er Engineering

.Yonh larolina S{o{e Uniccrsi{y

WUliam D Stevenson, Jr

Lale Professor uf E/ec/m:o/ Engineering

Nonh Camfit/(J \{(J/c Ulllcersi{y

McGraw-Hill, Inc

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POWER SYSTEM ANALYSIS

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Library of Congress Cataloging-in-Publication Data

1 Electric power d i s tri b u t i o n 2 Electric power systems

I Stevenson, W i l li am D II Stevenson, William D Elements of

power system an al ys i s III Title

TK3001.G73 1994

When ordering this title, use ISBN 0-07-113338-0

Printed i1' Singapore

To THE MEMORY OF William D Stevenson, Jr

1912-1988

True friend and colleague

ABOUT THE AUTHORS

1"

'-1

John J Grainger is P r ofe s so r of El e ct r i cal and Computer Engineering at North Carolina State Unversity.He is a graduate of the ;'\ati o n a l University of Ireland and r ec e i v ed his M.S.E.E and Ph�D 'degrees at - the' L:"nivcrsity of Wisconsin­ Madison

Dr G ra ing e r is the founding Director of the El ect r ic Power Research Cent e r (It North Carolin,-l State University a joint uni\'er�ity/industry coopera­

tive research center in electric power systems engineering, He leads the Center's major research programs in transmission and distribution systems planning, design, (}uto01ation, and cont ro l areas as well as power system

dynamics_

Profes�or Grainger has aho «llight at the University of Wisconsin­ Madison, The Il l in ois Institute of Technology, 't>larquette University, and North Carolina State University His industrial experience has been with the Electric­ ity Supply Board of Ireland; Commonwealth Edison Company, Chicago; \Vis­

consin Electric Po\vcr Company, MilwaUKee; and Carolina Pmver & Light

Company, Raleigh Dr Grainger is an active consullant with the Pacific Gas and

El e ct ric Company, San Francisco; Southern California Edison Company, Rose­ mead; a n d mimy other power industry organizJtions His educational ane! technical involvements include the IEEE Povv'cr Engineering Society, The

Am e ri c a n Society of Engineering Education, the American Power Conference,

erRED, and CIGRE

Dr Grainger is the aut h or of numerous papers in the IEEE Power Engineering S ocie t y ' s Transactiotls and was recognized by the IEEE Transmis­

s i on and Distribution Committee for the 1985 Prize Paper Award

In 1984, P r of e ssor Grainger was chosen by the Edison Electric Institute for t he EEl Power Engineering E d u cator Award

William D Stevenson, Jr (deceased) was a professor and the A ssociat e Head of

the Electrical Engineering D e p a r t m ent of N o rt h Carolina State University A Fellow of the In stit u t e of Electrical and Electronics E nginee r s , he worked in private industry and taught at both Clemson Uni v e r s i ty and P r i nc e t on Univer­

sity Dr Stevenson also served as a consulting editor in electrical power

en?ineering for the McGraw-HilI Encyc!opedin of Science and Technology, He

was the recipient of several teaching and professional awards,

1.8 Vol t ag e anc1 Current in BalanccJ Three-Phase Circuits

1.9 Power in Balllnccu Three-Phase Circuits

1.10 Per-Unit Qualltities

1.11 C h a n ging the B,lse of Per-Unit OU(lntities

I.J2 Node Equations

1.13 The S i n gl e - L i n e or One-Line Diagram

1.14 Impedance and Reactance Dia g r am s

The Ideal Transf o rme r

Magnetically Coupled Coils

The Equivalent Circuit of a Single-Phase Transformer

Per-Unit Impedances in Single-Phase Transformer Circuits

Three-Phase Transformers

Three-Phase Transformers: Phase Shift and Equivalent Circuits

The Autotransformer

Per-Unit Impedances of Three-Winding Transformers

Tap-Changing and Regulating Transformers

The Advantages of Per-Unit Computations

Summary

P r o b l em s

1

.-, )

xii CONTENTS

4.5 Flux: Linkages between Two Points External to an Isolated

4.7 Flux Linkages of One Conductor in a Group 153

5.2 The Potential Differen ce between Two Points Due to a Charge 172

5.4 Capacitance of a Three-Phase Line with Equilateral Spacing 177

5.6 Effect of Earth on the Capacitance of Three-Phase Transmission

CONTENTS xiii

6.3 The Medium-Length Line

6.4 The Long Transmission Line: Solution of the D i ffere n t i a l

Eq ua t i on s

6.5 The Long Transmission Line: lnterpretation of the Equations

6.6 The Long Transmission Line: Hyperbolic Form of the Equations

6.7 The Equivalent Circuit of a Long Line

6.8 Power Flow through a Transmission Line

6.9 Reactive Compensation of Transmission Lines

6.1 0 Transmission-Line Transients

6.1 1 Transient Analysis: Traveling Waves

6.12 Transient Analysis: Rct1ections

6.13 Direct-Current Transmission

6.14 Summary

Problems

7 The Admittance Model and Net wor k Calculations

7.1 Branch and Node Admittances

7.2 Mutually Coupled Branches in Y hu,

7.� An Equivalent AdmittZlncc i'-Jc(wnrk

7.4 Modifie,l(ion or VioL"

7.5 The Network Incidence Matri\ and Y QUI

7.6 The Method of Successive Elimination

7.7 Node Elimination (Kron Reduction)

7.8 Triangular F,lctoriz(ltioll

7.0 Sparsity ancl Ncar-Optilll(li Or(lcring

7.10 Summar),

Prohlcms

8 The Impedance Model and Network Calcu lations

8.1 The Bus Admittance and Impedance Matrices

8.2 Thcvcnin's Thcorcm and Zbus

8.3 Mpclification of (Ill Existing Zhu:>

8.4 Direct Determination of Zou:>

8.5 Calculation of Zous Elements from YbU5

8.6 Power Invariant Transformations

8.7 Mutually Coupled Br a n c h e s in Zbus

8.8 Summary

Problems

9 Power-Flow Solutions

9.1 The Power-Flow Problem

9.2 The Gauss-Seidel Method

9.3 The Newton-Raphson Method

9.4 The Newton-Raphson P ow e r -Flo w Solution

9.5 Power-Flow Stu d i es in System Design and Operation

10.2 Internal Voltages of Loaded Machines under Fault Conditions 383

10.4 Fault Calculations Using ZhU5 Equivalent Circuits ]95

11.2 The Symmetrical Components of U nsymmetrical P h asors 418

11.4 Power in Terms of Symmetrical Com ponents 427 11.5 Sequence Circuits of Y and tl I mpedances 429 11.6 Sequence Circui ts of a Symmetrical Transmission Lin e 435 11.7 Sequence Circuits of the Synchronous M achine 442

13.1 Distribution of Load between U n i ts within a Plant 532

13.5 Classical Economic Dispatch with Losses 555

13.8 S o l v ing the Unit Commitment Problem

13.9 Summary

Problems

14 Zbus Methods in Contingency Analysis

14.1 Adding and Removing Multiple Lines

14.2 Piecewise Solution of Interconnected Systems

14.3 Analysis of Single Contingencies

14.4 Analysis of Multiple Contingencies

14.5 Contingency Analysis by dc Model

14.6 System Reduction for Contingency and Fault Studies

14.7 Summary

Problems

15 State Estimation of Power Systems

15.1 The Method of Least SqU;:HCS

15.2 Statistics, Errors ,wd Estimates

1 5.3 Test for Bad Data

15.4 Power System State Estimation

1 is Thc Structure «(Ill! r:orm;ltion of H \

1 5.6 Sumnnry

Problems

16 Power System Stability

16.1 The Stability Problem

16.2 Rotor Dynamics ;lnd the Swing Equation

16.3 Further Considerations of the S w i ng Equation

16.4 Thc Power-Angle Equ(ltion

16.5 Synchronizing rowcl Cocflicicnls

16.6 EClual-Area Criterion of S t abil i ty

16.7 Further Applications of the Equal-Area Criterion

16.8 Multimachine Stability Studies: Classical Representatio!1

16.9 Step-by-Slep Solution of the Swing CunT

16.10 Computer Programs for Transient Stability Studies

16.11 Factors Affecting Transient Stability

16.12 Summary

Problems

Appendix A

A.i Distributed Windings of the Synchronous Machine

A.2 P-Transformation of Stator Quantities

Appendix B

B.l Sparsity and Near-Optimal Ordering

B.2 Sparsity of the Jacobian

687

688

69S

695 60S

The aim here is to instill confidence and understanding of those concepts

of power system analysis that are likely to to be encountered in the study and practice of electric power engineering The presentation is tutorial with empha­ sis on a thorough understanding of fundamentals and underlying principles The approach and level of treatment are directed toward the senior undergraduate and first-year graduate student of electrical engineering at technical colleges and universities The coverage, however, is quite comprehensive and spans a

wide range of topics commonly encountered in electric power system engineer­ ing practice In this regard, electric utility and other industry-based engineers will find this textbook of much benefit in their everyday work

Modern power systems have grown larger and more geographically expan­ sive with many interconnections between neighboring systems Proper planning, operation, and control of such large-scale systems require advanced computer­ based techniques, many of which are explained in a tutorial manner by means of numerical examples throughout this book The senior undergraduate engineer­ ing student about to embark on a career in the electric power industry will most certainly benefit from the exposure to these techniques, which are presented here in the detail appropriate to an introductory level Lik�wise, electric utility engineers, even those with a previous course in power system analysis, may find that the explanations of these commonly used analytic techniques more ade­ quately prepare them to move beyond routine work

Power System Analysis can serve as a basis for two semesters of undergrad­ uate study or for first-semester graduate study The wide range of topics facilitates versatile selection of chapters and sections for completion in the semester or quarter time frame Familiarity with the basic principles of electric

XVII

xviii PREFACE

circuits, phasor algebra, and the rudiments of d ifferential equations is assumed The reader should also have some understanding of matrix operations and notation as they are used t hroughout the text The coverage includes newer /

h

tOPlCS suc as state estima tion a n d unit commitment, as wel l as more detailed presen tations and newer approaches to traditional subjects such as transform­ers, synchronous machines, and network fau lts Where appropriate, summary tables allow quick reference of i mportant ideas Basic concepts of computer­based algorithms are presented so that students can implement their own compu ter programs

Chapters 2 and 3 a re devo ted to the tra nsforme r and sync hrono us ma­

chine, respectively, and should complement material covered in other electric circuits and machines courses Transmiss ion-line parameters and calcula t ions are studied in Chapters 4 through 6 Networ k m odel s based on the admittance

and impedance representations are developed in Chapters -; and 8, which also introduce gaussian elimination, Kron reduction, triangular factorization, a nd the

Zbus building algorithm The power-flow problem, symm e trical com ponents, and unsymmetrical faults are presented in Chapters 9 through 12: \\hereas Chapter

13 p rovides a self-contained developmen t of economic dispatch and the basics

of unit commitment Con t i ngency analysis and external equivalents are the subjects of Chapter 14 Power system state estimation is covered in Chapter 15,

while power system stability is introduced i n Cha p t e r 16 Homework problems and exercises are provi ded at the end of each chapter

I am most pleased to acknowledge the assistance given to me by a number

of people with whom I have been associated within the Department of Electri­cal a n d Computer Engineering at North Carolina State University Dr Stan

S H Lee, my colleague and friend for m any years, has always willingly given his time and effort when I needed help, advice, or suggestions at the various stages

of d evelopment of this textbook A n u mber of the homework problems and solutions were contributed by h i m and by Dr Gamini Wickramasekara, one of

my former graduate students at Nor th Carolina State University Dr Michael 1

Gorm a n , another of my recent gradu;lle sLucJents, gave ullstintingly or himselr in

d eveloping t he com p u ter-based figures and solutions fo r many of the nu merical

examples throughout the various chapters of the text Mr W A drian Buie, a recen t graduate of the Department of Electrical and Computer Engineering, -un dertook the challenge of committing the e ntire textbook to the computer and

p roduced a truly professional manuscript; i n this regard, Mr Barry W Tyndall was also most helpful in the early stages of the writin g My loyal secretary, Mrs Paulette Cannady-Kea, has a lways enthusiastically assisted in t he overall pro­

ject I am greatly indebted and extremely grateful to each and a l l of these individuals for their generous efforts

Also within the Department of Electrical and Computer Engineering at

North Carolina State University, the successive leadership of Dr Larry K

Monteith (now Chancellor of the University), Dr Nino A Masnari (now Director of the Engineering R esearch Center for Advanced Electronic Materi­als Processing), and Dr Ralph K Cavin III (presently Head of the Depart�ent),

PREFACE xix

along with my faculty colleagues, particularly Dr AJf re d 1 Goetze, provided an

environment of suppor t that I am very p l eased to record

The members of my family, especially my wife, Barbara, have been a great source of patient understanding and encouragement during the preparation of this book I ask each of them, and my f ri e nd Anne Stevenson, to accept my

sincere thanks

McGraw-Hill and I would like to thank the following reviewers for their

many help f u l comments and suggestions: Vernon D Albertson, University of Minnesota; David R Brown, University of Texas at Austin; Mehdi

Etezadi-Amoli, Un i versi ty of Nevada Reno; W Mack Grady, l)nivcrsity of Texas at Austin; Clifford Grigg, Rose-Hulman Institute of Technology; William

H Kersting, Ne"v Mexico State University; Kenneth KsuempeI, Iowa State

University; Mangalore A Pai, Unin'rsiry of Illinois Urbana-Champaign; Arlin

G Phadke, Virginia Polytechnic Institute and State Uni\'ersily; B Don Russell,

Texas A & M University; Peter W Sauer, Uni\'ersit) of Illinois, Urbana­

Champaign; <lnd Ernie L Stagliano Jr Drexel Uni\'ersity

John 1 Grainger

CHAPTER

1

BASIC CONCEPTS

Normal and abnormal conditions of oper a tion of the sys tem are the concern of the power system engineer who mu st be very familia r with steady-state ac circuits, part icu l arly three-phase circuits The purpos e of thi s chapter is to

review a f ew of the fund3mental i d e Cls of such circuits; to est ablish the notation used throughout the book; and to introduce the expression o f va lues of vol t age, current, impedanc e , a nd power i n per u n i t Modern power system ana lysis rel ies

a lm ost exclusively on n odal n erwork represen tatio n wh ich is introduced in the

form o f the b u s admittance a nd the b u s impedclnce matrices

phasors (with appropriate subscripts where necessary) Vertical bars enclosing V

and I, that is, I VI and III, designate the magnitudes of the phasors Magnitudes

of complex numbers such as impedance Z and admittance Ya re a lso indic a ted

by vertical bars Lowercase letters generally indicate instantaneous v alues Where a generated voltage [electromotive force (emf)] is specified, the lett e r E

rather than V is often used for voltage to emphasize the fact that an e m f rat her

tha n a general potentia l difference between two points is being considered

1

2 CHAPTER 1 BASIC CONCEPTS

If a voltage and a current are expressed as functions of time, such as

I VI = 1 00 V and III = S A

These are the values read b y the ordinary types of voltmeters a n d ammeters Another n a me for the rms value is the effective value The average power expended in a resistor by a current of magnitude III is 1I12R

To exp ress these quantities as p hasors, we e mploy Euler's identity SiG = cos e + j sin e, which gives

cos e = Re{8JO} = Re{cos e + j sin e} (1 1 ) where Re m eans the real part of We now write

If the current is the reference p hasor, we h ave

I = 58 jO° = 5 LQ: = 5 + j 0 A

and t he voltage which leads the reference phasor by 30° IS

V = 100£J30° = 1 00 � = 86.6 + j50 V

Of course, we might not choose as the refe rence phasor either the vol tage

or the curren t whose instantaneous expressions are v and i, respectively, i n which case their phasor expressions would i nvolve other angles

In circuit d iagrams it is often most convenient to use polarity marks in the form of p lus and minus signs to i ndicate the terminal assumed positive when specifying voltage An arrow on the diagram specifies the direction assumed positive for the flow of current In the single-phase equivalent of a three-phase circuit single-subscript notation is usually sufficient, but double-subscript ,nota­tion is usually simpler when deal i ng with all three p hases

1.2 SINGLE SUBSCRIPT NOTATION

1.2 SINGLE-SUBSCRIPT NOTATION 3

Figure 1.1 shows an ac circuit with an emf represented by a circle The emf is

Eg, and the voltage between nodes a and 0 is identified as v, The current in

the circuit i s lL and the voltage across ZL is VL To specify these voltages as phasors, however, the + and - markings, called polarity marks, on the diagram and an arrow for current direction are necessary

In an ac circuit the terminal marked + is positive with respect to the terminal marked - for half a cycle of voltage and is negative with respect to the other terminal during the next half cycle We mark the terminals to enable us to say that the voltage between the terminals is positive at any instant when the terminal marked plus'is actually at a higher potential than the terminal marked minus For instance, in Fig 1.1 the instantaneous voltage V, is positive when the tenninal marked plus is actually at a higher porential t h a n the terminal marked with a negative sign During the next half cycle the pos i t i v e ly marked terminal is

actua)ly negative, and v, is negative Some authors use an arrow but must specify whether the arrow points toward the t e rm i n a l which would be labeled plus or toward the terminal which would be labeled m i n us In the convention described above

The current arrow p e r for m s a similar function The su b s c r i pt , in this case

L, is not necessary llnles� other currents arc �rcsenl Obviollsly, the actual direction of current flow in an ac circuit reverses each half c ycl e The arrow points in the direction which is to be called positive for current \Vhen the

current is actually flowing in the dircciion opposi te to that of the arrow, the

current is negative The phasor ,'urrenl is

Since certain nodes ill t h e circuit h a v e been assigned letters, the voltages

may be designated by the single-letter subscripts identifying the node whose voltages are expressed with respect to a reference node In Fig 1.1 the instantaneous voltage Va and the phasor voltage v" express the voltage of node

a with respect to the reference node 0, and Va is positive when a is at a higher

FIGURE 1.1

An ac circuit with emf E� and load impedance Z[

4 CHA PTER 1 BASIC CONCEPTS

potential than o Thus,

T h e use of polarity marks for vol tages and d i rect ion arrows for currents can be avoided by double-subscript notation The u n derstanding of th ree-phase circuits

is considerably clari fied by adop ting a system of double subscripts The conven­

t ion to be followed i s quite simp le

In denoting a current the order of the subscripts assigned to the symbol for current d efines the d irection of the flow of current when the current is considered to be positive In Fig 1 1 the arrow pointing from a to b defines the positive direction for the curren tIL associated with t he arrow The i nstanta­neous current i L is positive when the current is actuall y in the direction from a

to b, and in double-subscript notation this current is iab' The current iab is equal to -iba'

In double-subscript notation the letter subscripts on a voltage indicate the nodes of the circ uit between which the voltage exists We shall follow t he convention which says that the first subscript denotes the voltage of that node with respect to the node identified by the second subscript This means that the instantaneous voltage Vab across Z A of the circuit of Fig 1 1 is the voltage of node a with respect to node b a n d that vab is positive during that half cycle when a is at a higher potential than b The corresponding phasor voltage is V,,/J'

which is related to the current fal) flowi ng from node a to node b by

where 2/1 is the complex impedance (al so called 2(/h) and Y/1 = 1/2/1 is t he complex admittance (also called Y:II)

Reversing the order of the subscripts of either a current or a voltage gives

a cur rent or a voltage 1 800 out of p hase with the original; that is,

v bu = V ab c j 1800 = V ab / 1800 = - V ab

The relation of single- and double-subscript notation for the circui t of Fig

1.1 is summ arized as follows:

1.4 POWER 1:-" SINGLE,PHASE AC CIRCUITS 5

In writing Kirchhoff's voltage law, the orde r of the subscripts is the order

of tracing a closed p ath around the circuit For Fig 1.1

Nodes n a nd 0 are the same in this cir c u i , and 11 has been introduced to

i denti fy the path more precisely , Replacing VaG by - Van and noting that Vab = IobZA yield

and so

1.4 PO\VER IN SINGLE-PHASE AC CIRCUITS

( 17)

Although the [unciamcntJI theory o[ the transmissicn of energy ciescIihc� the

travel of energy in terms of the interaction of cleuri - Hnd nngnctic field" tlie

power system engineer is usually more concerned \\ltll llc.'>cribing the r�!IC o[

change of energy with respect to time (v-,:hich is the dcilnilion of /)()l\'t'!') in term"

of voltage anel current The unit of power is a H'(I/[ The pmvcr in Wd!l" being

absorbed by a lond at any instant is the product of [he inst,1f11<ll1coU\ \Ull,q;e

drop across the 10,1(.1 in volts and the il1\(,llltar1Cnu: (linenl inlo the I(}�!d ill amperes 1f the terminals of the IO,ld arc designated Ci ,Inel fI, and il· the vlllL!gc

a nd current are expressed by

and ion = I,,, l'l)s( (ut - 0)

the instantaneous power is

( J .S) The angle e in these e qu ati on s is positive for current lagging the vult<\gc and

negative for leading current A positive value of fJ expresses the r,lte at which energy is being absorbed by the part of the s yst e m between the poi nt s {l and II

The i n s t a n ta n e ou s power is obviously posit ive when both Vall and illn arc

positive and becomes negative when Vall and i,," are opposite in sign Figure 1.2

illustrates this point Pos i tiv e power calculated as v'lIi all results when current is

flowjng in the direction of a vo l t age drop and is the rate of t r ns fe r of energy to

the load Conversely, nega tiv e power ca lcu l a ted as vaniall res u lt s when current is

flowing in the d irection of a voltage rise and means energy is being transferred from the load into the system to which the load is connected I f Van and ian are

jn phase, as they are in a purely r esist ive load, the instantaneous power will

never become negative If the current and voltage are out of p h a se by 90"', as in

a p u rely inductive or purely capacitive idea l circuit element, the instantaneous

6 CHAPTER 1 BASIC CONCEPTS

o

� �� -� �� -+ -FIGURE 1.2

Current, voltage, and power plotted versus time

power will h ave equal positive a n d negative half cycles and its average value wi ll

a lways be zero

By using trigonometric identities the expression of Eq ( 1.8) is reduced to

1 3( b) is the phasor diagram The component of ian in phase with Van is iR, and from Fig 1 3(b), 11 R \ = Ilan \ cos 8 If the maxi mu m value of ian is Imax, the maximum value of iR is Imax cos 8 The instantaneous current iR must be in phase with Vall For Vall = Vmax cos wt

1 4 POWER IN SINGLE·PHASE AC CIRCUITS 7

o --.-= -+-1 -

-� , -"" t = 0

FIGURE 1.4

Vol t age, curreot in ph ase wit h the vo lt age, and t he re s u l ting power plone d , ersus ti me

f m ax sin 8 Since i x must lag v ll ll by 90° ,

i x = J m:c( sin 8 sin 6J (

max i"

Then,

v m;!; ".1''' cos e ( 1 + cos 2 6J [ )

Jr.-(1.11)

( 1 1 2)

which is the i nsta n t aneous p ower i n the res is t a nce a n d t he first term In Eq

(1.9) Figu re 1 4 shows V,"J!? plotted ve rsus (

Simil arly,

( 1 13)

which is the instantaneous powe r in the inductance a n d th e second term in

Eq (1.9) Figure 1 5 shows Van ' ix a n d their prod uct plotted versus t

Examination of Eq (1.9) shows that t he term co ntaining cos f) IS always positive and has a n a v er a g e value of

8 CHAPTER 1 BAS I C CONCEPTS

O

r -�� -� -� -_7 -FIGURE 1 5

Voltage, curre nt laggi ng the voltage by 90° , anu t h e re s ul t i ng power plot ted ve rsus t i m e

P is the quantity to which the word power refers when not modified by an adjective identifying it otherwise P, the average power, is also ca lled the real or

active power The fundamental u nit for both instantaneous and average power

is the watt, but a watt is such a small unit in relation to power system quantities that P is usually measured in kilowatts or megawa tts

The cosine of the phase angle e between the voltage and the current is called the power factor An inductive circuit is said to have a lagging power factor, and a capacitive circuit is said to h ave a leading power factor In other words, the terms lagging power faclOr and leading power factor i ndicate, respectively, whether the current is l agging or leading the applied vol tage The second term of Eq ( 1 9), the term containing sin e, is altern ately positive and negative and has an average value of zero This component of the instantaneous power p is called t he instantaneous reactive power and expresses the flow of energy alternately toward the load and away from the load The maximum value of this pulsating power, design ated Q , is ca lled reactive power

or reactive IJo/lampcres and is very use ful in describi n g the operation of a power system, a s b ecom cs i ncrcasi ngly cvi d c n t i ll ru rl h c r d iscuss i o n The re act ive

1 4 POWER IN SINGLE·PHASE AC CI RCU ITS 9

designate the units for Q a s vars (for voltamperes r e a c tiv e } The more practical

unit s for Q a r e kilovars or megava r s

In a simpit: s e r i es circuit whe r e Z is equal to R + )X we can substitute Ill iZI for IVI in Eqs (1 1 5) and ( 1 1 7) to obtain

( 1 1 9)

Recognizing that R = IZlc o s e and X = I Z l sin e, we then find

E q u ,\ t i o n s ( 1 1 5 ) il n d ( 1 1 7 ) r rov i d e anot h e r m e t ho d of co m p u t i ng the

[Jowe r ractor : i n ce we s e c [ i l , L ( () II > c [ ; t i l (J T h e powe r L l c l o r is t i l e r'dore

or from £qs 0 1 5) and (1.18)

If the instan taneous power expressed by Eq ( 1 9) i s the power in a pred omina n t ly ca paci tive c i r c uit with the same impressed voltage, e b e c om e s negative, m a k i n g sl n e a n d Q negative If capacitive and inductive circ u i ts are in paral icl , t h e instanta neous reactive po w e r for the RL c i r c u it is 1 800 o u t of phase wi t h the i ns t a n t a neous reactive power of the RC circu i t The net reactive

pO\l, e r is the difference be tween Q fo r t he RL ci rcu i t and Q for the RC ci rcu i t

/\ I.� ()s i t ivc va lue is ass i gn ee1 t o Q drawn by a n i n d u ct ive 1 0,l cl ;1 n (1 (l n ega t ive sign

to Q d rawn b y a capacit ive loa o

P o w e r system engin eers u s u a l ly t h i n k of a capacitor as a g e nerator o f

p m i t i ve react ive pow e r ra the r th a n a l o a d requ i ri n g n ega tive reactive power

Th ts concept is ve ry l og ic a l , for a ca p a c i t o r d rawi n g negarive Q in parallel w i t h

a n ind uct ive loa u red u ces t he Q which w o u l d o t h e rw i s e h ave to b e supplied b y

t h e system ( 0 t h e i nduct ive loa d I n o t h er words, t h e capacitor supplies t h e Q required by t h e i n d u c t iv e loa d Th i s is the same as considering a capacit o r as a

d ev i ce that delive rs a lagg i n g cu rrent r a t h e r t h a n a s a device w h ic h d raws a

l eadi n g currenL as shown i n Fig 1 6 An a dj ustable capacitor in p a r a ll el wi t h an

inductive l oad, for instance, can be adj u s ted so that the leading curre n t to the

c a p a c i t o r i s exact l y e q u a l in m ag n i t u d e t o the c o m p onent of current in the

i n d uct ive load which is l a gg i ng thc v o l t a g e by 90° Thus, the res u l tant cu r r e n t is

i n ph ase with the v o l t a g e Th e i n d uc tivc circ u i t stil l requires positive reactive

power, b ut the net reactive power is zero I t is for t h i s reason that the power system engineer finds i L conve n i e n t to con s i u e r t h e capaci tor to be supplying rea ctive power to the inductive load W hen t he words positive and n egative a re

not used , positive reactive power is assumed

If the phasor expressions for voltage and cu rrent are known , the cal c u l ation of real and reactive power is accomplished conven iently in complex form If the vol tage across and the curren t i n to a certain load or part of a circuit a re expressed by V = IVI� a nd I = 1 1 1 il , respectively, the product of voltage times the conjugate of current in pol a r for m is

VI* = I v!cja X 1 1 1 £ -jt3 = I VI I l l c i(a -t3) = I VI 1 11 / a - f3 (1 22)

This q u antity, called the complex power , is usually designated by S In rectangu­

Reactive power Q will be positive when the phase angle a - f3 between voltage

a n d cu rrent i s positive, that is, when a > f3 , wh ich means that current i s lagging the vol tage Conversely, Q will be negative for f3 > a, which indicates that current i s leading the voltage This agrees with the selection of a positive sign for t he re act ive power of an ind u ctive c i rcu i t and a n e ga tive si g n for t he reactive

power of a capacitive circuit To obta i n the proper sign for Q, it is necessary to calculate 5 as VI* rather than V * I , which would reverse the sign for Q

1.6 THE POWER TRIANGLE

Equ ation ( 1 24) suggests a graphical m ethod of obtaining the overall P, Q, and

p hase a ngle for several loads in parallel since cos e is P / 1 5 1 A power trian gle

_ _ .c a n be drawn fo r a n i n ductive load, as shown i n Fig 1.7 For several loads i n

s

Q

e

FIGURE 1.7

1 7 D I R ECTION OF POWER FLOW 1 1

p Power t ri a ngle for a n induct ive load

For ,H i i n d u c t iv e l o a d Q w i l l be d rawn v e r t i ca l ly u p w a r d s ince i t i s positive A

capa c i t i ve load \vil l have negative react ive pow e r, a nd Q wi l l be vertical ly downward, Figure 1 8 i l lustrates the powe r tri a ngle co m posed of PI' Q\ , and Sl for a lagg : ng power-factor load ha v ing a p hase angl e e l co m b i ned with the

pow e r t r i a n g l e composed of P2 ' Q7" a n d S2 ) which is fo r a c a p acitive load w i th a

ll c g Zl t ive e : , These t w o l oads in p ar a l l e l result in the tri angl e h a v i ng sides

is CUII�i l k n.: J , The q u cs t i o n i n voives t h e d i r e c t i o n o f n ow of powe r, t h a t is,

w h e t h e r p o w e r i s be i n g gel/era /ed o r absorhed when a vo l t age a n d a cu rre n t are spec ifie d

The quest ion o f d e l ive ring p o w e r to a ci rcu i t o r a b s o r bi n g power from a circuit i s r a t h e r o b v i o u s fo r a d c sys t e m Cons i d e r the current a n d v ol tage of

Fig 1 9( 0 ) where de c u rr e n t 1 is flo w i ng thr o ugh a battery, If the voltmeter V m and the a m m e t c r A m bot h read upsc a l e t o show E = 1 00 V and 1 = 1 0 A, the

b a t te ry is be ing charge d ( a b s o r b i n g e n e rgy) at the rate given by the product

E1 = 1000 \V, On the other hand, if t he ammeter connection s have to be

12 C H A PTER 1 BASIC CONCEPTS

( 6 )

E

Con n ections of: (a) ammeter and vol t meter to measure de current I and voltage E of a battery ;

( b ) watt meter to measu re real power absorbed by ideal a c volt age sou rce E

reversed i n order that i t reads u pscale with t he current arrow still in the

d irection shown, then 1 = - 1 0 A and the product EI = - 1 000 W; that is, the battery is d i scharging (delivering energy) The same considerations apply to the ac circuit relationships

For an ac system Fig 1 9( b ) shows within t he box an ideal voltage source

E (consta n t magnitude, constant frequency, zero impedance) with polarity marks to i ndicate, as usual, the terminal which is positive during the half cycle

of positive instantaneous vol tage Similarly, the arrow indicates the direction of current 1 i n to the box during t he positive half cycle of current The wattmeter

of Fig 1 9(b) has a current coil and a voltage coil corresponding, respectively, to the ammeter A m and the voltmeter yO) of Fig 1 9( a ) To measure active power the coils m ust be correctly conn ected so as to obta i n an upscale reading By definition we know that the power absorbed inside the box is

S = VI * = P + jQ = I V l l i l cos () + j l V l l i l sin e ( 1 2S )

where 8 is the phase a ngle by which 1 lags V Hence, if the wattmeter reads

u pscale for t he connections shown i n Fig 1 9( b ), P = I Vi l l i cos 8 is positive and real power is being a bsorbed by E If t he wattmeter tries to deflect downscale, t hen P = I VI II l cos 8 is n egative and reversing the connections of the curre nt coil or the voltage coil, but not both, causes the meter to read

upscale, indicat ing that positive power is be i ng supplied by E inside the box This is e quivalent to saying that negative power is being absorbed by E If the wattmeter is replaced by a varmeter , similar considerations apply to the sign of the reactive power Q absorbed or supplied by E In general, w e can determine the P and Q absorbed or supplied by any ac circuit simply by regarding the circuit as enclosed in a box with entering current 1 and voltage V h aving t he polarity shown in Table 1 1 Then, the numerical values of t he real and imaginary p arts of the p roduct S = VI* determi ne the P and Q absorbed or , suppli ed by the enclosed circuit or network When current 1 lags voltage V by

1 7 D I RECTION OF POWER FLOW 13

an <1 n g l c 0 be tween 0° and 90" , we nnd t h tlt P = I vi I l lcos D and Q =

I VI Jl sin 0 <I n.: bOlh positive, indica t ing \vatts a l1u vars a rc bei ng absorbed by

t h e ind uct ive c i rcuit inside t he box When I leads V by an anglc between 0°

and 90° , P i s s t i l l posit ive b u t e a n d Q = I VI I I I s i n e are bot h nega t ive,

i nd ica ting t hat n e gative 'lars are being absorbed or positive vars are b e i n g

s u p p l i e d by the capaci tive ci rcu it insid e t he box

Exa m ple 1 1 Two i d e a l vol tage sourccs d esign ated 3 S mach ines 1 an d 2 are

co n n e c t e d , 3S shown i n Fig 1 1 0 If E I = 100ft V, £'2 = 1 00/300 V, and

Z = 0 + j5 fl , determ i n e (a) wh e t h e r C el ch mach i n e is ge n e rating or consu m i ng

r e a l power a n d t h e cl mou n t , (b) whether each mach ine is receiving or su pplying

re().ctive powc r a n d t h e a m o u n t , and (d the P a n d Q absorbed by the i m p e d a nce

14 CHAPTER 1 BASIC CONCEPTS

The current entering box 1 is - I and that entering box 2 is f so that

268 val' The mac h i nc i s ,lct U ll l l y � 1 I l l u t or

Machine 2, expected to be a motor, h a s n e g ative P 2 and nega t ive Q2 '

Therefore, this machine generates energy at the rate of 1 000 W and supplies reactive power of 268 var The machine is actua lly a generator

Note that the supplied reactive power of 268 + 268 is equal to 536 var, which is required by the inductive reactance of 5 n Since the impedance is purely reactive, no P is consumed by the impedance, and al l the watts generated by machine 2 are transferred to machine 1

THREE-PHASE CIRC UITS

Electric power systems are supplied by t hree-phase generators Ideally, t h e 'generators are supplying balanced three-phase loads, w hich means loads wit h identical impedances i n a ll three phases Lighting loads and small motors are, of course, single-phase, but distribution systems are designed so that overall the

p hases are essentially balanced Figu re 1 1 1 shows a Y-connected generator with neutral m a rked 0 su p p ly i n g a b a l a nced- Y load with neutral m arked 11 I n discussing this circuit , w e assume t h a t t h e impedances o f t h e connections between the terminals of the generator and the load, as wel l as the impedance

of the direct connection between 0 and n , are negligible

The equivalent circuit o f the three-phase generator co nsists of an emf i n each o f the three phases, as indicated by circles on the diagram Each emf i s i n series with a resistance and inductive reactance composing the im ped ance Zd'

Points a', b', a nd c' are fictitious since the generated emf cannot be separated from the i mpedance o f each phase The termin als of the m achine a re the points

a , b , a nd c Some a ttention is given to this equivalent circuit in Chap 3 In the generator the emfs E(lIOJ Eb1o ' and £c'o are equal in magnitude a nd displaced from each other 1 200 in phase If the magn i tude of each is 1 00 V with Ea10 a s reference,

Ea'o = 100LQ: V E/,'o = 100/2400 V

Ul VOLTAG E A N D C U R R E NT IN BALANCED TH R E E - P HASE CI RCU ITS 1 5

Ci rc u I t d i �l gralll o f �I !'-co n n<:ctnl bCJlCra t o r (,OIl I I C C 1 C U [ 0 a h,t l a n cc d - Y l o ;,d

provided t h e p h ase seq u e n c e is abc , which m e a n s t h a t Eo'o leads EI/o b y 1 2 00

and £b'o in t u rn l e a d s L,'n by 1200 _ The circu i t d i agram gives no ind ication of

phase se que nce , b u t Fi g 1 1 2 s hows these emfs w i t h p h ase seque nce abc

A t the gene rator te rminals (and at the l o a d i n t h is case) t h e termi nal

vol l ages t o neu tral a r e

V - E - I Z ao - U f O u n d

v c o -- F -" ( ' U - I Z ( 1 / -I e!

( 1 26)

S ince 0 and / I a rc a t t h e s<.nn e pot en t i (t i , V;w, ViI() ' a n d V;,o a re equal to v'w' Von'

a nci V:'r!' respect ively, and t h e l i n e c urre nts (which arc a lso t h e phase c urrents

FI G URE 1 1 2

Pha�or d i agram o f t h e emfs o f t he c i rcllit shown i n Fig 1 A 1

16 CHAPTER 1 BASIC CONCEPTS

for a Y connection) a re

FIGURE 1 13

P hasor d i agram of c u rr e n t s in a b a l anced t h re e - p h as e loa d :

(a) phasors d rawn from a common po i n t ; ( b ) a d d i t i o n of t h e

p h a sors form i n g a closed t r i a n g l e

Zd + ZI< Zu lbll Eh, o Vb II

( 1 27)

Z" + Z" Za fCII Ec'o - VOl

Fig 1 1 1 b e tween the neutrals of the gen erator a nd loa d Then, the connection between lZ and 0 may have any i m p e d a n ce, or eve n bc open, and Il and 0 will remain a t the s a m e pot e n t i a l H t h e l o a d i s n o t b a l anced, t h e s u m or t h e

currents will n o t be zero a n d a c urren t w ill flow between 0 a nd !1 For the unbalanced condit ion 0 and n w i l l not be at the same poten tial u n less they a re con nected by zero impedance

Because of the phase d i spl acement of the voltages and currents in a balanced three-phase system, i t is conveni e n t to have a shorthand method of indicating the rot ation of a phasor t hrou gh 1 20° The result of the mu l t iplica­tion of two complex numbers is the p roduct of t he ir mag n i tudes and the sum of their angles If the complex n umber expressing a phasor is m ultipl ied by a complex n u mber of unit magnitude a n d angle e , the resu lting complex n umber represents a phasor equal to the original p hasor displaced by the angle e The complex n u mber of u n it magn itu d e and associ ated angle e is an opera tor that rotates the p hasor on which it operates through the angle e We a re a lready familiar with the operator j, which cau ses rota tion through 90° , and the operator - 1 , wh ich causes rotation t h rou g h 1 800 • Two successive appl ications

of the op erator j cause r ota tion t hrough 90° + 90° , w h i c h leads US to the

1 8 VOLTAG E A N D CUR RENT IN BALANCED THREE-PHASE C I R C U ITS 17

conclusion that j x j causes rotation through 1 800 l and thus we recognize t hat

j2 is equal to - 1 O t h e r powers of t h e operator j are found by similar analysis

The l et ter a is commonly used to designate the operator that causes a rotation of 1 200 in the counterclockwise direction Such an operator is a

com plex number of unit magnitude with an angle of 1 200 and is defined by

18 CHA PTER I BAS I C CONCE PTS

Fl G URE 1 15

Phasor diagram of l i n e-to- l i n e vo lt ages i n relation to l i ne-to- n eutral voltages i n a bal anced t h ree­ phase circuit

The line-to-line voltages in the c i rcuit of Fig 1.11 a re Vab , VbC' and Vca

Tracing a path from a to b through n yields

( 1 28) Although £a'o and Van of Fig 1 1 1 are not in phase, we could decide to use �I/l

rather than £a'o as reference i n defining the voltages Then, Fig 1 1 5 shows the

p hasor d iagram of voltages to neutral and how Vab is found In terms of operator a we see that Vbll = a 2 VUII ' and so we have

Figure 1 14 shows that 1 - a2 = 13 �, which means that

Vab = 13 Vl1r1 t )JO° = /3-Vall �

Q

T -b

1 /l V O LTA G E A N D C U R R ENT IN BALANCED TH R EE-PHAS E C I R CU ITS 1 9

FIGURE 1 1 6

c Alt erna tive method of d rawing the phasors of Fig 1 15

the magnitude of ba la n ced l i n e-to- l ine vol tages of a t h ree-phase circuit is a lways

e q u al to !3 time s the m agn itude of the l i ne-to-neutral voltages is very impor­

t a n t

Figure 1 1 6 shows another way o f d isplaying the line-t o-line and line-to­

n e u tral vo l tage s The line - to- l ine vUl tage phasors are d rawn to form a close d

t r i a n g l e ur i e n t e d t o agre e w i t h t h e chos en refe rence in t h i s case �/fl' The

v e rt ices o f t h e t r i a n g l e ,\ IT labeled so t h a t ea ch phasor begins and ends a t the

v e r t ic e s correspon d i n g t o t h e o rd e r o f t he s ubsc ri p ts of that phasor voltage

Linc- to-Ileutral vol tage phasors are d rawn to the cen ter of the triangle O nce

t h i s p h asor d iagram is understood, it w i l l be found to be the simplest way t o

d etermine the va rious vol t ages

The orde r in which t he vertices a, b, and c of the t ri angle fol low each

other w h e n the triangle is ro tated counte rclockwise about n i ndicates the p h ase

s e q u :ncc T h e i mpor tance o f p h ase sequence becomes clear when we discuss

t r a ns form ers and \vh en symmetrical components are used to a nalyze unbalanced

fau Its on power systems

A s e p a r a t e cu rre n t d i agram can be d rawn to re late each curre n t properly

wi t h re spect t o its phase \·ol t agc

EX<1mple 1 2 I n a b a la nced t hree-phase circu i t t he vo ltage V;/b i s 173.2!!! V

D c t l.::rm ine 3 1 1 the vol t ages and t h e cu rre n t s in a Y-con nceted load h aving ZL =

1O.L20" D Assu me t h a t t h e pha:;c seque nce is ahc

Solution V/i t h r�J!) ;1 <; r e f ere n ce , t h e phasor d i agram of vol t agcs is d rawn as shown

in Fig 1 1 7 from w h i ch it i s d e t c r m i n e d t hat

1.' v b e 1 73 2 1 _ / 2.400 V l/ c o = 1 n 'j · L / 1 2()O V _ _

v (lit = I (lo/I - 3 (t V

_

v h ,, -- 100 � / 2 1 0° V VOl = 1 OO� V Each curre n t lags the \ o l t age across i t s :load im pedance by 20° and each current

magn i t ude is 1 0 A Figure 1 1 8 is the phasor d iagram of the currents

J I , " 1 == 1 0 � / 1 90° A