Distribution System Modeling and Analysis ThThee E ELLEECCTTRRIICC PPOOWWEERR EENNGGIINNEEEERRIINNGG SSeerireises sseerrieiess eeddiittoorr LLeeoo GGrriiggssbyy Published Titles Electromechanical Systems, Electric Machines, and Applied Mechatronics Sergey E. Lyshevski Electrical Energy Systems Mohamed E. El-Hawary Electric Drives Ion Boldea and Syed Nasar Distribution System Modeling and Analysis William H. Kersting Linear Synchronous Motors: Transportation and Automation Systems Jacek Gieras and Jerry Piech Forthcoming Titles Induction Machine Handbook Ion Boldea and Syed Nasar Power System Operations in a Restructured Business Environment Fred I. Denny and David E. Dismukes Power Quality C. Sankaran Distribution System Modeling and Analysis William H. Kersting New Mexico State University Las Cruces, New Mexico CRC Press Boca Raton London New York Washington, D.C. 0812_frame_FM.fm Page iv Tuesday, July 31, 2001 10:49 AM Library of Congress Cataloging-in-Publication Data Kersting, William H. Distribution system modeling and analysis / William H. Kersting p. cm. -- (Electric power engineering series) Includes bibliographical references and index. ISBN 0-8493-0812-7 (alk. paper) 1. Electric power distribution–Mathematical models. I. Title. II. Series. TK3001 .K423 2001 621.31—dc21 2001035681 CIP 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. 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Visit the CRC Press Web site at www.crcpress.com © 2002 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-0812-7 Library of Congress Card Number 2001035681 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper 0812_frame_FM.fm Page v Tuesday, July 31, 2001 10:49 AM Contents 1 Introduction to Distribution Systems...........................................1 1.1 The Distribution System......................................................................2 1.2 Distribution Substations......................................................................2 1.3 Radial Feeders.......................................................................................5 1.4 Distribution Feeder Map.....................................................................6 1.5 Distribution Feeder Electrical Characteristics..................................8 1.6 Summary................................................................................................9 2 The Nature of Loads.....................................................................11 2.1 Definitions............................................................................................11 2.2 Individual Customer Load................................................................13 2.2.1 Demand...................................................................................13 2.2.2 Maximum Demand................................................................13 2.2.3 Average Demand....................................................................14 2.2.4 Load Factor.............................................................................14 2.3 Distribution Transformer Loading...................................................15 2.3.1 Diversified Demand...............................................................16 2.3.2 Maximum Diversified Demand...........................................17 2.3.3 Load Duration Curve............................................................17 2.3.4 Maximum Noncoincident Demand....................................17 2.3.5 Diversity Factor......................................................................18 2.3.6 Demand Factor.......................................................................19 2.3.7 Utilization Factor....................................................................19 2.3.8 Load Diversity........................................................................20 2.4 Feeder Load.........................................................................................20 2.4.1 Load Allocation......................................................................20 2.4.1.1 Application of Diversity Factors...........................21 2.4.1.2 Load Survey..............................................................21 2.4.1.3 Transformer Load Management............................25 2.4.1.4 Metered Feeder Maximum Demand....................25 2.4.1.5 What Method to Use?.............................................27 2.4.2 Voltage-Drop Calculations Using Allocated Loads..........27 2.4.2.1 Application of Diversity Factors...........................27 2.4.2.2 Load Allocation Based upon Transformer Ratings................................................31 2.5 Summary..............................................................................................32 Problems..........................................................................................33 0812_frame_FM.fm Page vi Tuesday, July 31, 2001 10:49 AM 3 Approximate Methods of Analysis.............................................39 3.1 Voltage Drop........................................................................................39 3.2 Line Impedance...................................................................................41 3.3 “K” Factors..........................................................................................43 3.3.1 The K Factor.......................................................................43 drop 3.3.2 The K Factor........................................................................46 rise 3.4 Uniformly Distributed Loads...........................................................47 3.4.1 Voltage Drop...........................................................................48 3.4.2 Power Loss..............................................................................50 3.4.3 The Exact Lumped Load Model..........................................52 3.5 Lumping Loads in Geometric Configurations..............................55 3.5.1 The Rectangle..........................................................................55 3.5.2 The Triangle............................................................................60 3.5.3 The Trapezoid.........................................................................65 3.6 Summary..............................................................................................71 References.....................................................................................................71 Problems..........................................................................................71 4 Series Impedance of Overhead and Underground Lines.........77 4.1 Series Impedance of Overhead Lines..............................................77 4.1.1 Transposed Three-Phase Lines.............................................78 4.1.2 Untransposed Distribution Lines........................................79 4.1.3 Carson’s Equations................................................................81 4.1.4 Modified Carson’s Equations...............................................83 4.1.5 Primitive Impedance Matrix for Overhead Lines............85 4.1.6 Phase Impedance Matrix for Overhead Lines..................86 4.1.7 Sequence Impedances............................................................89 4.2 Series Impedance of Underground Lines.......................................95 4.2.1 Concentric Neutral Cable.....................................................96 4.2.2 Tape-Shielded Cables...........................................................101 4.3 Summary............................................................................................105 References...................................................................................................105 Problems........................................................................................105 5 Shunt Admittance of Overhead and Underground Lines......109 5.1 The General Voltage-Drop Equation.............................................110 5.2 Overhead Lines.................................................................................111 5.3 Concentric Neutral Cable Underground Lines...........................115 5.4 Tape-Shielded Cable Underground Lines....................................119 5.5 Sequence Admittance.......................................................................121 5.6 Summary............................................................................................122 References...................................................................................................122 Problems........................................................................................122 0812_frame_FM.fm Page vii Tuesday, July 31, 2001 10:49 AM 6 Distribution System Line Models.............................................125 6.1 Exact Line Segment Model.............................................................125 6.2 The Modified Line Model...............................................................132 6.3 The Approximate Line Segment Model.......................................136 6.4 Summary............................................................................................141 References...................................................................................................141 Problems........................................................................................141 7 Regulation of Voltages...............................................................145 7.1 Standard Voltage Ratings................................................................145 7.2 Two-Winding Transformer Theory................................................147 7.3 The Two-Winding Autotransformer..............................................152 7.3.1 Autotransformer Ratings....................................................156 7.3.2 Per-Unit Impedance.............................................................158 7.4 Step-Voltage Regulators...................................................................162 7.4.1 Single-Phase Step-Voltage Regulators..............................163 7.4.1.1 Type A Step-Voltage Regulator............................163 7.4.1.2 Type B Step-Voltage Regulator............................164 7.4.1.3 Generalized Constants..........................................167 7.4.1.4 The Line Drop Compensator...............................168 7.4.2 Three-Phase Step-Voltage Regulators...............................174 7.4.2.1 Wye-Connected Regulators..................................175 7.4.2.2 Closed Delta-Connected Regulators...................180 7.4.2.3 Open Delta-Connected Regulators.....................183 7.5 Summary............................................................................................193 References...................................................................................................194 Problems........................................................................................194 8 Three-Phase Transformer Models .............................................199 8.1 Introduction.......................................................................................199 8.2 Generalized Matrices.......................................................................200 8.3 The Delta–Grounded Wye Step-Down Connection....................201 8.3.1 Voltages..................................................................................202 8.3.2 Currents.................................................................................206 8.4 The Ungrounded Wye–Delta Step-Down Connection...............212 8.5 The Grounded Wye–Grounded Wye Connection.......................222 8.6 The Delta–Delta Connection...........................................................224 8.7 The Open Wye–Open Delta Connection......................................236 8.8 The Thevenin Equivalent Circuit...................................................242 8.9 Summary............................................................................................245 Problems........................................................................................245 0812_frame_FM.fm Page viii Tuesday, July 31, 2001 10:49 AM 9 Load Models.................................................................................251 9.1 Wye-Connected Loads.....................................................................252 9.1.1 Constant Real and Reactive Power Loads.......................252 9.1.2 Constant Impedance Loads................................................253 9.1.3 Constant Current Loads......................................................253 9.1.4 Combination Loads..............................................................254 9.2 Delta-Connected Loads...................................................................257 9.2.1 Constant Real and Reactive Power Loads.......................257 9.2.2 Constant Impedance Loads................................................258 9.2.3 Constant Current Loads......................................................258 9.2.4 Combination Loads..............................................................258 9.2.5 Line Currents Serving a Delta-Connected Load.............259 9.3 Two-Phase and Single-Phase Loads..............................................259 9.4 Shunt Capacitors...............................................................................259 9.4.1 Wye-Connected Capacitor Bank........................................259 9.4.2 Delta-Connected Capacitor Bank......................................260 9.5 The Three-Phase Induction Motor.................................................261 References...................................................................................................266 Problems........................................................................................266 10 Distribution Feeder Analysis.....................................................269 10.1 Power-Flow Analysis.....................................................................269 10.1.1 The Ladder Iterative Technique.....................................270 10.1.1.1 Linear Network................................................270 10.1.1.2 Nonlinear Network..........................................271 10.1.2 The General Feeder..........................................................274 10.1.3 The Unbalanced Three-Phase Distribution Feeder.....276 10.1.3.1 Series Components..........................................276 10.1.3.2 Shunt Components..........................................278 10.1.4 Applying the Ladder Iterative Technique....................279 10.1.5 Putting It All Together.....................................................279 10.1.6 Load Allocation.................................................................289 10.1.7 Summary of Power-Flow Studies..................................289 10.2 Short-Circuit Studies......................................................................290 10.2.1 General Theory..................................................................290 10.2.2 Specific Short Circuits......................................................293 10.3 Summary..........................................................................................298 References...................................................................................................299 Problems........................................................................................299 Appendix A..........................................................................................303 Appendix B..........................................................................................307 Index.....................................................................................................309 0812_frame_FM.fm Page ix Tuesday, July 31, 2001 10:49 AM Preface In the last 40 years many papers and textbooks devoted to the computer modeling and analysis of large power system networks have been written. For the most part the models and analysis techniques have been developed for large interconnected transmission systems and synchronous generators. Little, if any, attention was devoted to the distribution system and its major components. As a result, the distribution engineer has not had the same number of tools as the systems engineer to analyze the distribution system under steady-state (power-flow) and fault (short-circuit) conditions. Without these tools the distribution engineer has been left in the dark (no pun inten- ded) as to the operating characteristics of distribution feeders. A lot of “seat of the pants” engineering has had to take place in order to keep the lights on. In recent years more attention has been devoted to the computer modeling and analysis of distribution systems. Computer programs are now available so that the distribution engineer can develop a real feel for how the distri- bution system is operating. With the tools, power-flow studies can be run to simulate present loading conditions and to help with the long-range plan- ning of new facilities. The tools also provide an opportunity for the distri- bution engineer to do such things as optimize capacitor placement in order to minimize losses. Different switching scenarios for normal and emergency conditions can be simulated, and short-circuit studies provide the necessary data for the development of a reliable coordinated protection plan for fuses, reclosers, and relay/circuit breakers. In short, the distribution engineer now has the needed tools. So what is the problem? “Garbage in, garbage out” is the answer. Armed with a commercially available computer program, it is possible for a user to prepare incorrect data that will lead to results that do not make any sense. Without an understanding of the models and a general “feel” for the operating characteristics of a distribution system, serious design errors and operational procedures may result. The user must fully understand the models and analysis techniques of the program. Most power systems textbooks and courses are limited to the modeling and analysis of balanced three-phase systems. The models and analyses assume a balance so that only a single-phase equivalent model is required. While this works fine for interconnected systems, it is not sufficient for the modeling and analysis of a distribution system. A distribution system is inherently unbalanced, and therefore three-phase models of all the compo- nents must be employed. There is a significant difference between the com- puter programs developed for interconnected system studies and the programs developed for distribution systems. The data requirements for the