power system (the electric power engineering)

Dorf The Electric Power Engineering Handbook, Second Edition, Leonard L.. Per-Unit Scaling Extended to a General Three-Phase System 2 Sy mmetr ical Components for Power System Analysis T

Electric Power Engineering Handbook

Second Edition

Edited by

Leonard L Grigsby

Electric Power Generation, Transmission, and Distribution

Edited by Leonard L Grigsby

Edited by James H Harlow

Edited by John D McDonald

Power Systems

Edited by Leonard L Grigsby

Power System Stability and Control

Edited by Leonard L Grigsby

The Electrical Engineering Handbook Series

Series Editor

Richard C Dorf

University of California, Davis

Titles Included in the Series

The Handbook of Ad Hoc Wireless Networks, Mohammad Ilyas

The Biomedical Engineering Handbook, Third Edition, Joseph D Bronzino

The Circuits and Filters Handbook, Second Edition, Wai-Kai Chen

The Communications Handbook, Second Edition, Jerry Gibson

The Computer Engineering Handbook, Second Edtion, Vojin G Oklobdzija

The Control Handbook, William S Levine

The CRC Handbook of Engineering Tables, Richard C Dorf

The Digital Avionics Handbook, Second Edition Cary R Spitzer

The Digital Signal Processing Handbook, Vijay K Madisetti and Douglas Williams The Electrical Engineering Handbook, Third Edition, Richard C Dorf

The Electric Power Engineering Handbook, Second Edition, Leonard L Grigsby The Electronics Handbook, Second Edition, Jerry C Whitaker

The Engineering Handbook, Third Edition, Richard C Dorf

The Handbook of Formulas and Tables for Signal Processing, Alexander D Poularikas The Handbook of Nanoscience, Engineering, and Technology, Second Edition,

William A Goddard, III, Donald W Brenner, Sergey E Lyshevski, and Gerald J Iafrate

The Handbook of Optical Communication Networks, Mohammad Ilyas and

Hussein T Mouftah

The Industrial Electronics Handbook, J David Irwin

The Measurement, Instrumentation, and Sensors Handbook, John G Webster

The Mechanical Systems Design Handbook, Osita D.I Nwokah and Yidirim Hurmuzlu The Mechatronics Handbook, Second Edition, Robert H Bishop

The Mobile Communications Handbook, Second Edition, Jerry D Gibson

The Ocean Engineering Handbook, Ferial El-Hawary

The RF and Microwave Handbook, Second Edition, Mike Golio

The Technology Management Handbook, Richard C Dorf

The Transforms and Applications Handbook, Second Edition, Alexander D Poularikas The VLSI Handbook, Second Edition, Wai-Kai Chen

Electric Power Engineering Handbook

Second Edition

POWER SYSTEMS

Edited by

Leonard L Grigsby

CRC Press

Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300

Boca Raton, FL 33487-2742

© 2007 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S Government works

Printed in the United States of America on acid-free paper

10 9 8 7 6 5 4 3 2 1

International Standard Book Number-10: 0-8493-9288-8 (Hardcover)

International Standard Book Number-13: 978-0-8493-9288-7 (Hardcover)

This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use

No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any informa- tion storage or retrieval system, without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For orga- nizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for

identification and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication Data

Power systems / editor, Leonard Lee Grigsby.

p cm.

Includes bibliographical references and index.

ISBN-13: 978-0-8493-9288-7 (alk paper)

ISBN-10: 0-8493-9288-8 (alk paper)

1 Electric power systems I Grigsby, Leonard L

Table of Contents

Preface

Editor

Contributors

I Power System Analysis and Simulation

1 The Per-Unit System

Charles A Gross

2 Symmetrical Components for Power System Analysis

Tim A Haskew

3 Power Flow Analysis

Leonard L Grigsby and Andrew P Hanson

4 Fault Analysis in Power Systems

Charles A Gross

5 Computational Methods for Electric Power Systems

Mariesa L Crow

II Power System Transients

6 Characteristics of Lightning Strokes

17 Transmission Plan Evaluation—Assessment of System Reliability

N Dag Reppen and James W Feltes

18 Power System Planning

Hyde M Merrill

19 Power System Reliability

Richard E Brown

20 Probabilistic Methods for Planning and Operational Analysis

Gerald T Heydt and Peter W Sauer

The generation, delivery, and utilization of electric power and energy remain one of the most ging and exciting fields of electrical engineering The astounding technological developments of our ageare highly dependent upon a safe, reliable, and economic supply of electric power The objective ofElectric Power Engineering Handbook, 2nd Edition is to provide a contemporary overview of this far-reaching field as well as to be a useful guide and educational resource for its study It is intended todefine electric power engineering by bringing together the core of knowledge from all of the many topicsencompassed by the field The chapters are written primarily for the electric power engineeringprofessional who is seeking factual information, and secondarily for the professional from otherengineering disciplines who wants an overview of the entire field or specific information on one aspect

challen-of it

The handbook is published in five volumes Each is organized into topical sections and chapters in anattempt to provide comprehensive coverage of the generation, transformation, transmission, distribu-tion, and utilization of electric power and energy as well as the modeling, analysis, planning, design,monitoring, and control of electric power systems The individual chapters are different from mosttechnical publications They are not journal-type chapters nor are they textbook in nature They areintended to be tutorials or overviews providing ready access to needed information while at the sametime providing sufficient references to more in-depth coverage of the topic This work is a member ofthe Electrical Engineering Handbook Series published by CRC Press Since its inception in 1993, thisseries has been dedicated to the concept that when readers refer to a handbook on a particular topic theyshould be able to find what they need to know about the subject most of the time This has indeed beenthe goal of this handbook

This volume of the handbook is devoted to the subjects of power system analysis and simulation,power system transients, power system planning, and power electronics If your particular topic ofinterest is not included in this list, please refer to the list of companion volumes seen at the beginning ofthis book

In reading the individual chapters of this handbook, I have been most favorably impressed by howwell the authors have accomplished the goals that were set Their contributions are, of course, most key

to the success of the work I gratefully acknowledge their outstanding efforts Likewise, the expertise anddedication of the editorial board and section editors have been critical in making this handbookpossible To all of them I express my profound thanks I also wish to thank the personnel at Taylor &Francis who have been involved in the production of this book, with a special word of thanks to NoraKonopka, Allison Shatkin, and Jessica Vakili Their patience and perseverance have made this task mostpleasant

Leo GrigsbyEditor-in-Chief

Leonard L (‘‘Leo’’) Grigsby received his BS and MS in electrical engineering from Texas Tech Universityand his PhD from Oklahoma State University He has taught electrical engineering at Texas Tech,Oklahoma State University, and Virginia Polytechnic Institute and University He has been at AuburnUniversity since 1984 first as the Georgia power distinguished professor, later as the Alabama powerdistinguished professor, and currently as professor emeritus of electrical engineering He also spent ninemonths during 1990 at the University of Tokyo as the Tokyo Electric Power Company endowed chair ofelectrical engineering His teaching interests are in network analysis, control systems, and powerengineering

During his teaching career, Professor Grigsby has received 13 awards for teaching excellence Theseinclude his selection for the university-wide William E Wine Award for Teaching Excellence at VirginiaPolytechnic Institute and University in 1980, his selection for the ASEE AT&T Award for TeachingExcellence in 1986, the 1988 Edison Electric Institute Power Engineering Educator Award, the1990–1991 Distinguished Graduate Lectureship at Auburn University, the 1995 IEEE Region 3 Joseph

M Beidenbach Outstanding Engineering Educator Award, the 1996 Birdsong Superior Teaching Award atAuburn University, and the IEEE Power Engineering Society Outstanding Power Engineering EducatorAward in 2003

Professor Grigsby is a fellow of the Institute of Electrical and Electronics Engineers (IEEE) During1998–1999 he was a member of the board of directors of IEEE as director of Division VII for power andenergy He has served the Institute in 30 different offices at the chapter, section, regional, andinternational levels For this service, he has received seven distinguished service awards, the IEEECentennial Medal in 1984, the Power Engineering Society Meritorious Service Award in 1994, and theIEEE Millennium Medal in 2000

During his academic career, Professor Grigsby has conducted research in a variety of projects related

to the application of network and control theory to modeling, simulation, optimization, and control ofelectric power systems He has been the major advisor for 35 MS and 21 PhD graduates With hisstudents and colleagues, he has published over 120 technical papers and a textbook on introductorynetwork theory He is currently the series editor for the Electrical Engineering Handbook Seriespublished by CRC Press In 1993 he was inducted into the Electrical Engineering Academy at TexasTech University for distinguished contributions to electrical engineering

Rensselaer Polytechnic Institute

Troy, New York

Charles A GrossAuburn UniversityAuburn, AlabamaAndrew P HansonPowerComm EngineeringRaleigh, North CarolinaTim A HaskewUniversity of AlabamaTuscaloosa, AlabamaGerald T HeydtArizona State UniversityTempe, Arizona

Alireza KhotanzadSouthern Methodist UniversityDallas, Texas

Stephen R LambertShawnee Power Consulting, LLCWilliamsburg, Virginia

Juan A Martinez-VelascoUniversitat Politecnica de CatalunyaBarcelona, Spain

Thomas E McDermottEnerNex CorporationPittsburgh, Pennsylvania

Niskayuna Power Consultants, LLC

Schenectady, New York

Jose´ Rodrı´guezUniversidad Te´chnica Federico Santa Marı´aValparaı´so, Chile

Francisco de la RosaDistribution Control Systems, Inc

Hazelwood, Missouri

Peter W SauerUniversity of Illinois at Urbana-ChampaignUrbana, Illinois

Gerald B Sheble´

Portland State UniversityPortland, OregonMahesh M SwamyYaskawa Electric AmericaWaukegan, Illinois

I Power System

1 The Per-Unit System Charles A Gross 1 -1

Impact on Transformers Per-Unit Scaling Extended to Three-Phase Systems Per-Unit Scaling Extended to a General Three-Phase System

2 Sy mmetr ical Components for Power System Analysis Tim A Haskew 2 -1

Fundamental Definitions Reduction to the Balanced Case Sequence Network Representation

in Per-Unit

3 Power Flow Analysis Leonard L Gr ig sby and Andrew P Hanson 3 -1

Introduction Power Flow Problem Formulation of Bus Admittance Matrix Formulation of Power Flow Equations P –V Buses Bus Classifications Generalized Power Flow Development

Solution Methods Component Power Flows

4 Fault Analysis in Power Systems Charles A Gross 4 -1

Simplifications in the System Model The Four Basic Fault Ty pes An Example Fault Study

Further Considerations Summar y Defining Terms

5 Computational Methods for Electr ic Power Systems Mariesa L Crow 5 -1

Power Flow Optimal Power Flow State Estimation

Three-Phase System 1-11

In many engineering situations, it is useful to scale or normalize quantities This is commonly done inpower system analysis, and the standard method used is referred to as the per-unit system Historically,this was done to simplify numerical calculations that were made by hand Although this advantage hasbeen eliminated by using the computer, other advantages remain:

. Device parameters tend to fall into a relatively narrow range, making erroneous valuesconspicuous

. The method is defined in order to eliminate ideal transformers as circuit components

. The voltage throughout the power system is normally close to unity

Some disadvantages are that component equivalent circuits are somewhat more abstract Sometimesphase shifts that are clearly present in the unscaled circuit are eliminated in the per-unit circuit

It is necessary for power system engineers to become familiar with the system because of its wideindustrial acceptance and use and also to take advantage of its analytical simplifications This discussion

is limited to traditional AC analysis, with voltages and currents represented as complex phasor values.Per-unit is sometimes extended to transient analysis and may include quantities other than voltage,power, current, and impedance

The basic per-unit scaling equation is

Per-unit value¼actual value

The base value always has the same units as the actual value, forcing the per-unit value to bedimensionless Also, the base value is always a real number, whereas the actual value may be complex.Representing a complex value in polar form, the angle of the per-unit value is the same as that of theactual value

Consider complex power

Suppose we ar bitrarily pick a value Sbase, a real number wi th the units of volt-amperes Div idingthroug h by Sbase ,

SffuSbase¼

Vffa I ff  bSbase :

We fur ther define

Vbase I base ¼ Sbase : (1: 3)Either Vbase or Ibase may be selected arbitrarily, but not both Substituting Eq (1.3) into Eq (1.2), we obtain

SffuSbase¼

Vffa Iff  bð ÞVbase IbaseSpuffu ¼ Vffa

Vbase

Iff  bIbase

Spu¼ Vpuffa Ipuff  b 

The subscript pu indicates per-unit values Note that the form of Eq (1.4) is identical to Eq (1.2).This was not inevitable, but resulted from our decision to relate VbaseIbaseand Sbasethrough Eq (1.3)

If we select Zbaseby

Zbase¼VbaseIbase ¼

V2 base

V=IZbase

Zpu¼V=VbaseI=Ibase ¼

VpuIpu:

Observe that

Zpu¼ ZZbase¼

Rþ jXZbase ¼

RZbase

þ j XZbase

Thus, separate bases for R and X are not necessary:

Zbase¼ Rbase¼ Xbase

By the same logic,

Sbase¼ Pbase¼ Qbase

Example 1.1(a) Solve for Z, I, and S at Port ab in Fig 1.1a.

(b) Repeat (a) in per-unit on bases of Vbase¼ 100 V and Sbase¼ 1000 V Draw the corresponding unit circuit

per-Solution(a) Zab¼ 8 þ j12  j6 ¼ 8 þ j6 ¼ 10 ff36.98 V

I¼Vab

Zab¼

100ff010ff36:9¼ 10ff36:9amperes

Vbase¼

1000

100 ¼ 10 AVpu¼100ff0

¼ 0:8 þ j0:6 puConverting results in (b) to SI units:

For power system applications, base values for Sbase and Vbase are arbitrarily selected Actually, inpractice, values are selected that force results into certain ranges Thus, for Vbase, a value is chosen suchthat the normal system operating voltage is close to unity Popular power bases used are 1, 10, 100, and

1000 MVA, depending on system size

Consider the total input complex power S

S¼ V1I1*þ V2 I2*þ V3 I3*

¼ V1I1*þN2

N1 V1I2*þ

N3N1 V1I3*

¼V1N1½N1I1þ N2 I2þ N3I3*

Arbitrarily select two base values V1baseand S1base Require base values for windings 2 and 3 to be:

V1pu ¼ V 2pu ¼ V3pu (1: 17)

Div ide Eq (1.9) by N1

I1 þN2

N1 I2 þ

N3N1 I3 ¼ 0

Now div ide throug h by I1 base

I1I1baseþN2 =N1

Equations (1.17) and (1.18) suggest the basic scaled equivalent circuit, shown in Fig 1.3 It iscumbersome to carr y the pu in the subscript past this point: no confusion should result, since allquantities w ill show units, including pu

Example 1.2The 3-w inding sing le-phase transformer of Fig 1.1 is rated at 13.8 kV= 138kV =4.157 kV and 50 MVA =40MVA =10 MVA Terminations are as follow ings:

13.8 kV w inding: 13.8 kV Source

138 kV w inding: 35 MVA load, pf ¼ 0.866 lagging

4.157 kV w inding: 5 MVA load, pf ¼ 0.866 leading

Using Sbase ¼ 10 MVA, and voltage ratings as bases,

(a) Draw the pu equivalent circuit

(b) Solve for the primar y current, power, and power, and power factor

Solution(a) See Fig 1.4

All values in Per-Unit Equivalent Circuit:

1.2 Per-Unit Scaling Extended to Three-Phase Systems

The extension to three-phase systems has been complicated to some extent by the use of traditionalterminology and jargon, and a desire to normalize phase-to-phase and phase-to-neutral voltage simul-taneously The problem with this practice is that it renders Kirchhoff ’s voltage and current laws invalid insome circuits Consider the general three-phase situation in Fig 1.5, with all quantities in SI units.Define the complex operator:

a¼ 1ff120The system is said to be balanced, with sequence abc, if:

Vbn¼ a2VanVcn¼ aVan

−

General Source

General Load

c n

If the load consists of wye-connected impedance:

The equivalent delta element is:

ZD¼ 3ZY

To convert to per-unit, define the following bases:

S3fbase¼ The three-phase apparent base at a specific location in a three-phase system, in VA.VLbase ¼ The line (phase-to-phase) rms voltage base at a specific location in a three-phase system, in V.From the above, define:

(a) Determine all bases

(b) Determine all voltages, currents, and impedances, in SI units and per-unit

Solution(a) Sbase¼S3fbase

100

3 ¼ 33:33 MVAVbase¼VLbaseffiffiffi

37

3

7 VbnVanVcn

26

37

37

I0¼ 0 kA 0 puð ÞI1¼ 2:510ff 36:9 kA 0:6ff  36:9ð puÞI2¼ 0 kA 0 puð Þ

Inclusion of transformers demonstrates the advantages of per-unit scaling

Example 1.4

A 3f 240 kV :15 kV transformer supplies a 13.8 kV 60 MVA pf¼ 0.8 lagging load, and is connected

to a 230 kV source on the HV side, as shown in Fig 1.6

(a) Determine all base values on both sides for S3fbase¼ 100MVA At the LV bus, VLbase¼ 13.8 kV.(b) Draw the positive sequence circuit in per-unit, modeling the transformer as ideal

(c) Determine all currents and voltages in SI and per-unit

Solution(a) Base values on the LV side are the same as inExample 1.3 The turns ratio may be derived fromthe voltage ratings ratios:

N1N2¼240= ffiffiffi3p15= ffiffiffi3

p ¼ 16

[ (Vbase)HV side¼N1

N2(Vbase)LV side¼ 16:00(7:967) ¼ 127:5 kV(Ibase)HV side¼ Sbase

(Vbase)HV side¼

33:330:1275¼ 261:5 A

Load

Low Voltage (LV) Bus

High Voltage (HV) Bus

Results are presented in the following chart.

S3fbase VL base Sbase Ibase Vbase Zbase

LV 100 13.8 33.33 4.184 7.967 1.904

HV 100 220.8 33.33 0.2615 127.5 487.5

(b) VLV¼7:967ff0

7:967 ¼ 1ff0

pu

S1f¼60

3 ¼ 20 MVA

S1f¼ 20

33:33¼ 0:6 pu

The positive sequence circuit is shown as Fig 1.7

(c) All values determined in pu are valid on both sides of the transformer! To determine SI values onthe HV side, use HV bases For example:

Van¼ 1ff0ð Þ127:5 ¼ 127:5ff0kV

Vab¼ 1:732ff30ð Þ 127:5ð Þ ¼ 220:8ff30 kV

Ia¼ 0:6ff  36:9ð Þ 261:5ð Þ ¼ 156:9ff  36:9A

Example 1.5Repeat the previous example using a 3f 240 kV:15 kV D

SolutionAll results are the same as before The reasoning is as follows

The voltage ratings are interpreted as line (phase-to-phase) values independent of connection (wye ordelta) Therefore the turns ratio remains:

N1N2¼240= ffiffiffi3p15= ffiffiffi3

p ¼ 16

As before:

Van

ð ÞLV side¼ 7:967 kVVan

1.3 Per-Unit Scaling Extended to a General Three-Phase System

The ideas presented are extended to a three-phase system using the following procedure

1 Select a three-phase apparent power base (S3ph base), which is t y pically 1, 10, 100, or 1000 MVA.This base is valid at ever y bus in the system

2 Select a line voltage base ( VL base ), user defined, but usually the nominal rms line-to-line voltage

at a user-defined bus (call this the ‘‘reference bus’’)

3 Compute

Sbase ¼ S3ph base 

= 3 (Valid at every bus) (1:23)

4 At the reference bus:

Vbase ¼ V L base = ffiffiffi

3

p

(1:24)Ibase ¼ Sbase =Vbase (1:25)Zbase ¼ V base =I base ¼ V2

5 To determine the bases at the remaining busses in the system, star t at the reference bus, which we

w ill call the ‘‘from’’ bus, and execute the follow ing procedure:

Trace a path to the next nearest bus, called the ‘‘to’’ bus You reach the ‘‘to’’ bus by either passingover (1) a line, or (2) a transformer

(1) The ‘‘line’’ case: VL base is the same at the ‘‘to’’ bus as it was at the ‘‘from’’ bus Use Eqs.(1.2), (1.3), and (1.4) to compute the ‘‘to’’ bus bases

(2) The ‘‘transformer’’ case: Apply VL baseat the ‘‘from’’ bus, and treat the transformer as ideal.Calculate the line voltage that appears at the ‘‘to’’ bus This is now the new VL baseat the ‘‘to’’bus Use Eqs (1.2), (1.3), and (1.4) to compute the ‘‘to’’ bus bases

Rename the bus at which you are located, the ‘‘from’’ bus Repeat the above procedure until youhave processed every bus in the system

6 We now have a set of bases for every bus in the system, which are to be used for every elementterminated at that corresponding bus Values are scaled according to:

per-unit value¼ actual value=base value

where actual value¼ the actual complex value of S, V, Z, or I, in SI units (VA, V, V, A); basevalue¼ the (user-defined) base value (real) of S, V, Z, or I, in SI units (VA, V, V, A); per-unitvalue¼ the per-unit complex value of S, V, Z, or I, in per-unit (dimensionless)

Finally, the reader is advised that there are many scaling systems used in engineering analysis, and, infact, several variations of per-unit scaling have been used in electric power engineering applications.There is no standard system to which everyone conforms in every detail The key to successfully using anyscaling procedure is to understand how all base values are selected at every location within thepower system If one receives data in per-unit, one must be in a position to convert all quantities to

SI units If this cannot be done, the analyst must return to the data source for clarification on what basevalues were used

Symmetrical Components for Power System

Analysis

Tim A Haskew

Universit y of Alabama

2.1 Fundamental Definitions 2 -2Voltage and Current Transformation Impedance

Transformation Power Calculations System Load Representation Summar y of the Symmetrical Components

in the General Three-Phase Case2.2 Reduction to the Balanced Case 2 -9Balanced Voltages and Currents Balanced Impedances

Balanced Power Calculations Balanced System Loads

Summar y of Symmetrical Components in the Balanced Case2.3 Sequence Network Representation in Per-Unit 2 -14Power Transformers

Modern power systems are three-phase systems that can be balanced or unbalanced and w ill have mutualcoupling between the phases In many instances, the analysis of these systems is performed using what isknow n as ‘‘per-phase analysis.’’ In this chapter, we wil l introduce a more generally applicable approach tosystem analysis know as ‘‘symmetrical components.’’ The concept of sy mmetrical components was firstproposed for power system analysis by C.L For tescue in a classic paper devoted to consideration of thegeneral N-phase case (1918) Since that time, various similar modal transformations (Brogan, 1974)have been applied to a variet y of power t y pe problems including rotating machiner y (Krause, 1986;Kundur, 1994)

The case for per-phase analysis can be made by considering the simple three-phase system illustrated

in Fig 2.1 The steady-state circuit response can be obtained by solution of the three loop equationspresented in Eq (2.1a) throug h (2.1c) By solvi ng these loop equations for the three line currents, Eq.(2.2a) throug h (2.2c) are obtained Now, if we assume completely balanced source operation (theimpedances are defined to be balanced), then the line currents w ill also form a balanced three-phaseset Hence, their sum, and the neutral current, wi ll be zero As a result, the line current solutions are aspresented in Eq (2.3a) throug h (2.3c)

Va IaðRSþ jXSÞ  IaðRLþ jXLÞ  InðRnþ jXnÞ ¼ 0 (2:1a)

Vb IbðRSþ jXSÞ  IbðRLþ jXLÞ  InðRnþ jXnÞ ¼ 0 (2:1b)

Vc IcðRSþ jXSÞ  IcðRLþ jXLÞ  InðRnþ jXnÞ ¼ 0 (2:1c)

If one considers the introduction of an unbalanced source or mutual coupling between the phases inFig 2.1, then per-phase analysis will not result in three decoupled networks as shown in Fig 2.2 In fact,

in the general sense, no immediate circuit reduction is available without some form of reference frametransformation The symmetrical component transformation represents such a transformation, whichwill enable decoupled analysis in the general case and single-phase analysis in the balanced case

2.1 Fundamental Definitions

2.1.1 Voltage and Current

Transformation

To develop the symmetrical components, let

us first consider an arbitrary (no tions on balance) three-phase set of voltages

assump-as defined in Eq (2.4a) through (2.4c) Notethat we could just as easily be consideringcurrent for the purposes at hand, but volt-age was selected arbitrarily Each voltage isdefined by a magnitude and phase angle.Hence, we have six degrees of freedom tofully define this arbitrary voltage set

We can represent each of the three given voltages as the sum of three components as illustrated in Eq.(2.5a) throug h (2.5c) For now, we consider these components to be completely ar bitrar y except for theirsum The 0, 1, and 2 subscripts are used to denote the zero, positive, and negative sequence components ofeach phase voltage, respectively Examination of Eq (2.5a-c) reveals that 6 degrees of freedom exist on theleft-hand side of the equations while 18 degrees of freedom exist on the rig ht-hand side Therefore, for therelationship between the voltages in the abc frame of reference and the voltages in the 012 frame ofreference to be unique, we must constrain the rig ht-hand side of Eq (2.5).

We begin by forcing the a0, b0, and c0 voltages to have equal magnitude and phase This is defined in

Eq (2.6) The zero sequence components of each phase voltage are all defined by a single magnitude and asingle phase angle Hence, the zero sequence components have been reduced from 6 degrees of freedom to 2

Second, we force the a1, b1, and c 1 voltages to form a balanced three-phase set w ith positive phasesequence This is mathematically defined in Eq (2.7a-c) This action reduces the degrees of freedomprov ided by the positive sequence components from 6 to 2

Vb1 ¼ V1 ff uð 1  120Þ ¼ V1  1ff 120 (2 :7b)Vc1 ¼ V1 ff uð 1 þ 120Þ ¼ V1  1ffþ 120 (2 :7c)

And finally, we force the a2, b2, and c 2 voltages to form a balanced three-phase set w ith negative phasesequence This is mathematically defined in Eq (2.8a-c) As in the case of the positive sequencecomponents, the negative sequence components have been reduced from 6 to 2 degrees of freedom

transformation can be defined as indicated in Eq (2.12) The over tilde ( ) indicates a vector ofcomplex numbers.

Va

Vb

Vc

26

37

3

7 VV01

V2

26

37

~Vabc ¼ T ~

V0V1V2

26

37

3

7 VbVaVc

26

37

~V012 ¼ T 1 ~

Equations (2.13) and (2.14) define an identical transformation and inverse transformation for current

IaIbIc

26

37

3

7 I0I1I2

26

37

~IIabc ¼ T ~

II 012 (2: 13)I0

I1I2

26

37

3

7 IaIbIc

26

37

Va  V a ¼ jX aaIa þ jXabIb þ jX caIc (2:15a)

37

37

5 ¼ j

Xaa Xab XcaXab Xbb XbcXca Xbc Xcc

26

3

7 IaIbIc

26

37

5 (2:16)

~Vabc  ~VV abc 0 ¼ Zabc ~

IIabc (2:17)

Multiply ing both sides of Eq (2.17) by [T ] 1 y ields Eq (2.18) Then, substituting Eq (2.12) and(2.13) into the result leads to the sequence equation presented in Eq (2.19) The equation is w rittenstrictly in the 012 frame reference in Eq (2.20) where the sequence impedance matrix is defined in

Eq (2.21)

T

 1 ~Vabc  T 1 ~

V

V abc 0 ¼ T 1

Zabc

 ~IIabc (2:18)

~V012  ~VV 012 0 ¼ T 1

Zabc

T

 ~II012 (2:19)

~V012  ~VV 012 0 ¼ Z012

 ~II012 (2:20)

26

37

5 (2:21)

2.1.3 Power Calculations

The impact of the sy mmetrical components on the computation of complex power can be easily derivedfrom the basic definition Consider the source illustrated in Fig 2.4 The three-phase complex powersupplied by the source is defined in Eq (2.22) The algebraic manipulation to Eq (2.22) is presented,and the result in the sequence domain is presented in Eq (2.23)

in matrix form and in Eq (2.24) in scalar form

S3f¼ VaIa *þ VbIb *þ VcIc *¼ ~VabcT ~IIabc* (2:22)

3

7 11 1a 1a2

1 a2 a

26

37

37

37

Note that the nature of the sy mmetrical component transformation is not one of power invariance, asindicated by the multiplicative factor of 3 in Eq (2.24) However, this w ill prove useful in the analysis ofbalanced systems, which w ill be seen later Power invariant transformations do exist as minor variations

of the one defined herein However, they are not t y pically employed, althoug h the results are just asmathematically sound

2.1.4 System Load Representation

System loads may be represented in the sy mmetrical components in a variet y of ways, depending on the

t y pe of load model that is preferred Consider first a general impedance t y pe load Such a load isillustrated in Fig 2.5a In this case, Eq (2.17) applies w ith eVV abc 0 ¼ 0 due to the solidly grounded Yconnection Therefore, the sequence impedances are still correctly defined by Eq (2.21) As illustrated inFig 2.5a, the load has zero mutual coupling Hence, the off-diagonal terms w ill be zero However,mutual terms may be considered, as Eq (2.21) is general in nature This method can be applied for anyshunt-connected impedances in the system

If the load is D-connected, then it should be converted to an equivalent Y-connection prior to thetransformation (Ir w in, 1996; Gross, 1986) In this case, the possibilit y of unbalanced mutual coupling

w ill be excluded, which is practical in most cases Then, the off-diagonal terms in Eq (2.21) w ill be zero,and the sequence networks for the load wil l be decoupled Special care should be taken that the zerosequence impedance w ill become infinite because the D-connection does not allow a path for a neutralcurrent to flow, which is equivalent to not allowing a zero sequence current path as defined by the firstrow of matrix Eq (2.14) A similar argument can be made for a Y-connection that is either ungrounded

or grounded throug h an impedance, as indicated in Fig 2.5b In this case, the zero sequence impedance

w ill be equal to the sum of the phase impedance and three times the neutral impedance, or,Z00 ¼ ZY þ 3Zn Notice should be taken that the neutral impedance can var y from zero to infinit y.The representation of complex power load models w ill be left for the section on the application ofbalanced circuit reductions to the sy mmetrical component transformation

2.1.5 Summary of the Symmetrical Components in the General

Three-Phase Case

The general sy mmetrical component transformation process has been defined in this section Table 2.1 is

a short form reference for the utilization of these procedures in the general case (i.e., no assumption of

balanced conditions) Application of these relationships defined in Table 2.1 will enable the powersystem analyst to draw the zero, positive, and negative sequence networks for the system under study.These networks can then be analyzed in the 012 reference frame, and the results can be easilytransformed back into the abc reference frame.

Example 2.1The power system illustrated in Fig 2.6 is to be analyzed using the sequence networks Find thefollowing:

Transformation Equations Quantity abc ) 012 012 ) abc

3

5 VVab

Vc

2 4

3

5 VVab

Vc

2 4

3

5 ¼ 11 1a 2 1a

1 a a 2

2 4

3

5 VV01

V2

2 4

3 5

3

5 IIab

I c

2 4

3

5 IIab

I c

2 4

3

5 ¼ 11 1a 2 1a

1 a a 2

2 4

3

5 II01

I 2

2 4

3 5

(a) three line currents

(b) line-to-neutral voltages at the load

(c) three-phase complex power output of the source

SolutionThe sequence voltages are computed in Eq (2.25) The sequence impedances for the feeder and the loadare computed in Eqs (2.26) and (2.27), respectively The sequence networks are draw n in Fig 2.7

V0

V1

V2

26

37

3

7 255ff 0

250ff120277ff 130

26

37

5 ¼

8: 8ff 171267:1ff 324: 0ff 37

26

37

37

375V (2: 26)

37

375V (2: 27)

The sequence currents are computed in Eq (2.28a-c) In Eq (2.29), the sequence currents andsequence load impedances are used to compute the zero, positive, and negative sequence load voltages

Positive Sequence Network

Negative Sequence Network

V1

V2

26

37

3

7 2:1ff 144

79: 6ff 247:2ff 64

26

37

¼

6:6ff162251:7ff622:8ff46

26

375V

26

37

3

7 2: 1ff 144

79:6ff247:2ff64

26

37

5 ¼

83: 2ff 2773: 6ff 14782:7ff 102

26

37

37

3

7 6:6ff162

251:7ff622:8ff46

26

37

5 ¼

263:0ff9232:7ff129261: 5ff120

26

375V (2:31)

S3f¼ 3 V 0I0 * þ V1I1 * þ V2I2 *

¼ 57:3 þ j 29:2 kVA (2:32)

2.2 Reduction to the Balanced Case

When the power system under analysis is operating under balanced conditions, the sy mmetrical ponents allow one to perform analysis on a sing le-phase network in a manner similar to per-phaseanalysis, even when mutual coupling is present The details of the method are presented in this section

com-2.2.1 Balanced Voltages and Currents

Consider a balanced three-phase source operating w ith positive phase sequence The voltages are definedbelow in Eq (2.33) Upon computation of Eq (2.12), one discovers that the sequence voltages that resultare those shown in Eq (2.34)

~Vabc ¼

Va ffua

Va ff ua  120ð Þ

Va ff ua þ 120ð Þ

26

37

5 (2:33)

~V012 ¼

0

Va ffu a0

26

37

5 (2:34)

In Eq (2.35), a source is defined w ith negative phase sequence The sequence voltages for this case arepresented in Eq (2.36)

~Vabc¼

VaffuaVaff uað þ 120ÞVaff uað  120Þ

26

3

~V012 ¼

00

Va ffua

26

37

5 (2: 36)

These results are par ticularly interesting For a balanced source w ith positive phase sequence, only thepositive sequence voltage is non-zero, and its value is the a-phase line-to-neutral voltage Similarly, for abalanced source w ith negative phase sequence, the negative sequence voltage is the only non-zerovoltage, and it is also equal to the a-phase line-to-neutral voltage Identical results can be shown forpositive and negative phase sequence currents

2.2.2 Balanced Impedances

In the balanced case, Eq (2.16) is valid, but Eq (2.37a-b) apply Thus, evaluation of Eq (2.21) results inthe closed form expression of Eq (2.38a) Equation (2.38b) extends the result of Eq (2.38a) toimpedance rather than just reactance

Xaa ¼ Xbb ¼ Xcc  Xs (2: 37a)Xab ¼ Xbc ¼ Xca  Xm (2:37b)

37

37

37

37

5 (2:38b)

2.2.3 Balanced Power Calculations

In the balanced case, Eq (2.32) is still valid However, in the case of positive phase sequence operation,the zero and negative sequence voltages and currents are zero Hence, Eq (2.39) results In the case ofnegative phase sequence operation, the zero and positive sequence voltages and currents are zero Thisresults in Eq (2.40)

S3f¼ 3 V 0I0 *þ V1I1 *þ V2I2 *

2.2.4 Balanced System Loads

When the system loads are balanced, the sequence network representation is rather straig htfor ward We

shall first consider the impedance load model by referring to Fig 2.5a, imposing balanced impedances,

and allowing for consideration of a neutral impedance, as illustrated in Fig 2.5b Balanced conditions

are enforced by Eq (2.41a-b) In this case, the reduction is based on Eq (2.38) The result is presented in

Eq (2.42) Special notice should be taken that the mutual terms may be zero, as indicated on the figure,

but have been included for completeness in the mathematical development

37

37

5 (2:42)

The balanced complex power load model is illustrated in Fig 2.8 The transformation into the

sequence networks is actually defined by the results presented in Eqs (2.39) and (2.40) In positive

phase sequence systems, the zero and negative sequence load representations absor b zero complex

power ; in negative phase sequence systems, the zero and positive sequence load representations absor b

zero complex power Hence, the zero complex power sequence loads are represented as shor t-circuits,

thus forcing the sequence voltages to zero The non-zero sequence complex power load turns out to be

equal to the single -phase load complex power This is defined for positive phase sequence systems in Eq

(2.43) and for negative phase sequence systems in Eq (2.44)

2.2.5 Summary of Symmetrical Components in the Balanced Case

The general application of sy mmetrical components to balanced three-phase power systems has been

presented in this section The results are summarized in a quick

reference form in Table 2.2 At this point, however, power

trans-formers have been omitted from consideration This w ill be rectified

in the next few sections

Example 2.2Consider the balanced system illustrated by the one-line diagram in

Fig 2.9 Determine the line voltage magnitudes at buses 2 and 3 if the

line voltage magnitude at bus 1 is 12.47 kV We wi ll assume positive

phase sequence operation of the source Also, draw the zero sequence

network

SolutionThe two feeders are identical, and the zero and positive sequence

impedances are computed in Eqs (2.45a) and (2.45b), respectively

The zero and positive sequence impedances for the loads at buses

1 and 2 are computed in Eq (2.46a-b) throug h (2.47a-b),

respect-ively The D-connected load at bus 3 is converted to an equivalent

power load model.

Y-connection in Eq (2.48a), and the zero and positive sequence impedances for the load are computed

in Eq (2.48b) and (2.48c), respectively

Z00feeder ¼ Zs þ 2Zm ¼ j 6 þ 2 j 2ð Þ ¼ j 10 V (2: 45a)Z11feeder ¼ Zs  Zm ¼ j 6  j 2 ¼ j 4 V (2:45b)

TABLE 2.2 Summary of the Symmetrical Components in the Balanced Case

Transformation Equations Quantity abc ) 012 012 ) abc

Voltage Positive Phase Sequence: Positive Phase Sequence:

3

5 ¼ V0a

0

2 4

3

V c

2 4

3

5 ¼ T  VV01

V 2

2 4

3

5 ¼ a V2 V11

aV 1

2 4

3 5 Negative Phase Sequence: Negative Phase Sequence:

3

5 ¼ 00

V a

2 4

3

V c

2 4

3

5 ¼ T  VV01

V 2

2 4

3

5 ¼ aVV22

a 2 V 2

2 4

3 5

Current Positive Phase Sequence: Positive Phase Sequence:

3

5 ¼ I0a

0

2 4

3

I c

2 4

3

5 ¼ T  II01

I 2

2 4

3

5 ¼ a I21I1

aI 1

2 4

3 5

Negative Phase Sequence: Negative Phase Sequence:

3

5 ¼ 00

Ia

2 4

3

I c

2 4

3

5 ¼ T  II01

I 2

2 4

3

5 ¼ aII11

a 2 I 1

2 4

3 5

3 5

Power S3f¼ V a Ia* þ V b Ib* þ V c Ic* ¼ 3V a Ia*

S 3f ¼ V 0 I* 0 þ V 1 I* 1 þ V 2 I 2

¼ 3V1 I 1* positive ph : seq 3V2I * 2negative ph : seq

Z00bus 1¼ Zs þ 2Z m þ 3Z n

¼ 166 þ j 55ð Þ þ 2 0ð Þ þ 3 j 20ð Þ ¼ 166 þ j 115V (2:46a)Z11bus 1¼ Zs  Z m ¼ 166 þ j 55ð Þ  0 ¼ 166 þ j 55V (2:46b)Z00bus 2¼ Zs þ 2Zm þ 3Z n

¼ 140 þ j 105ð Þ þ 2 0ð Þ þ 3 0ð Þ ¼ 140 þ j 105V (2:47a)Z11bus 2¼ Zs  Z m ¼ 140 þ j 105ð Þ  0 ¼ 140 þ j 105V (2:47b)