Power System Analysis - McGraw-Hill Series Overview

Power System Analysis McGrawHill Series in Electrical and Computer Engineering SENIOR CONSULTING EDITOR Stephen W Director University of Michigan Ann Arbor Circuits and Systems Communications and Signal Processing Computer Engineering Control Theory Electromagnetics Electronics and VLSI Circuits Introductory Power Radar and Antennas Previous Consulting Editors Ronald N Bracewell Colin Cherry James F Gibbons Willis W Harmon Hubert Heffner Edward W Herold John G Linvill Simon Ramo Ronald A Rohrer Anthony E Siegman Charles Susskind Frederick E Terman John G Truxal Ernst Weber and John R Whinnery Power SENIOR CONSULTING EDITOR Stephen W Director University of Michigan Ann Arbor Chapman Electric Machinery Fundamentals Elgerd Electric Energy Systems Theory Fitzgerald Kingsley and Umans Electric Machinery Gonen Electric Power Distribution System Engineering Grainger and Stevenson Power System Analysis Krause and Wasynczuk Electromechanical Motion Devices Nasar Electric Machines and Power Systems Volume I Electric Machines Stevenson Elements of Power System Analysis Also Available from McGrawHill Schaums Outline Series in Electronics Electrical Engineering Most Outlines include basic theory definitions and hundreds of example problems solved in stepbystep detail and supplementary problems with answers Related titles on the current list include Analog Digital Communications Basic Circuit Analysis Basic Electrical Engineering Basic Electricity Basic Mathematics for Electricity Electronics Digital Principles Electric Circuits Electric Machines Electromechanics Electric Power Systems Electromagnetics Electronic Communication Electronic Devices Circuits Feedback Control Systems Introduction to Digital Systems Microprocessor Fundamentals Signals Systems Schaums Electronic Tutors A Schaums Outline plus the power of Mathcad software Use your computer to learn the theory and solve problemsevery number formula and graph can be changed and calculated on screen Related titles on the current list include Electric Circuits Feedback Control Systems Electromagnetics College Physics Available at most college bookstores or for a complete list of titles and prices write to The McGrawHill Companies Schaums 11 West 19th Street New York New York 100114285 2123374097 Power System Analysis Hadi Saadat Milwaukee School of Engineering WCBMcGrawHill A Division of The McGrawHill Companies Power System Analysis Copyright 1999 by The McGrawHill Companies Inc All rights reserved Printed in the United States of America Except as permitted under the United States Copyright Act of 1976 no part of this publication may be reproduced or distributed in any form or by any means or stored in a database or retrieval system without the prior written permission of the publisher Disclaimer The McGrawHill Companies make no warranties either expressed or implied regarding the enclosed computer software package its merchantability or its fitness for any particular purpose The exclusion of implied warranties is not permitted by some states The exclusion may not apply to you This warranty provides you with specific legal rights There may be other rights that you may have which may vary from state to state Trademark Acknowledgments IBM and IBM PC are registered trademarks of International Business Machines Corporation Microsoft and Windows are registered trademarks of Microsoft Corporation MATLAB and SIMULINK are registered trademarks of The MathWorks Inc This book is printed on acidfree paper 1 2 3 4 5 6 7 8 9 0 DOC DOC 9 4 3 2 1 0 9 PN 0122350 Set ISBN 0075616343 Publisher Kevin Kane Executive editor Betsy Jones Sponsoring editor Lynn Cox Marketing manager John Wannemacher Project manager Eve Strock Production supervisor Rich DeVitto Compositor York Graphic Services Inc Printer R R Donnelley Sons Inc Library of Congress CataloginginPublication Data Saadat Hadi Power system analysis Hadi Saadat p cm Includes bibliographical references and index ISBN 0070122350 1 Electric power systems 2 System analysis I Title TK1011S23 1999 62131dc21 httpwwwmhhecom PREFACE 1 THE POWER SYSTEM AN OVERVIEW 11 INTRODUCTION 12 ELECTRIC INDUSTRY STRUCTURE BALANCED FAULT 353 13 MODERN POWER SYSTEM 131 GENERATION SYNCHRONOUS MACHINE MODEL INCLUDING SALIENCY 467 132 TRANSMISSION AND SUBTRANSMISSION POWER SYSTEM CONTROL 527 128 INTRODUCTORY MODERN CONTROL APPLICATION 567 133 DISTRIBUTION B2 STABILITY 639 134 LOADS This book is intended for upperdivision electrical engineering students studying power system analysis and design or as a reference for practicing engineers As a reference the book is written with selfstudy in mind The text has grown out of many years of teaching the subject material to students in electrical engineering at various universities including Michigan Technological University and Milwaukee School of Engineering Prerequisites for students using this text are physics and mathematics through differential equations and a circuit course A background in electric machines is desirable but not essential Other required background materials including MATLAB and an introduction to control systems are provided in the appendixes In recent years the analysis and design of power systems have been affected dramatically by the widespread use of personal computers Personal computers have become so powerful and advanced that they can be used easily to perform steadystate and transient analysis of large interconnected power systems Modern personal computers ability to provide information ask questions and react to responses have enabled engineering educators to integrate computers into the curriculum One of the difficulties of teaching power system analysis courses is not having a real system with which to experiment in the laboratory Therefore this book is written to supplement the teaching of power system analysis with a computersimulated system I developed many programs for power system analysis giving students a valuable tool that allows them to spend more time on analysis and design of practical systems and less on programming thereby enhancing the learning process The book also provides a basis for further exploration of more advanced topics in power system analysis MATLAB is a matrixbased software package which makes it ideal for power system analysis MATLAB with its extensive numerical resources can be used to obtain numerical solutions that involve various types of vectormatrix operations 14 SYSTEM PROTECTION In addition SIMULINK provides a highly interactive environment for simulation of both linear and nonlinear dynamic systems Both programs are integrated into discussions and problems I developed a power system toolbox containing a set of Mfiles to help in typical power system analysis In fact all the examples and figures in this book have been generated by MATLAB functions and the use of this toolbox The power system toolbox allows the student to analyze and design power systems without having to do detailed programming Some of the programs such as power flow optimization shortcircuit and stability analysis were originally developed for a mainframe computer when I worked for power system consulting firms many years ago These programs have been refined and modularized for interactive use with MATLAB for many problems related to the operation and analysis of power systems These software modules are versatile allowing some of the typical problems to be solved by several methods thus enabling students to investigate alternative solution techniques Furthermore the software modules are structured in such a way that the user may mix them for other power system analyses This book has more than 140 illustrative examples that use MATLAB to assist in the analysis of power systems Each example illustrates a specific concept and usually contains a script of the MATLAB commands used for the model creation and computation Some examples are quite elaborate in order to bring the practical world closer The MATLAB Mfiles on the accompanying diskette can be copied to the users computer and used to solve all the examples The scripts can also be utilized with modifications as the foundation for solving the endofchapter problems The book is organized into 12 chapters and 3 appendices Each chapter begins with a brief introduction describing the topics students will encounter Chapter 1 is a brief overview of the development of power systems and a description of the major components in the power system Included is a discussion of generating stations and transmission and subtransmission networks that convey the energy from the primary source to the load areas Chapter 2 reviews power concepts and threephase systems Typical students already will have studied much of this material However this specialized topic of networks may not be included in circuit theory courses and the review here will reinforce these concepts Before going into system analysis we have to model all components of electrical power systems Chapter 3 addresses the steadystate presentation and modeling of synchronous machines and transformers Also the per unit system is presented followed by the oneline diagram representation of the network Chapter 4 discusses the parameters of a multicircuit transmission line These parameters are computed for the balanced system on a per phase basis Chapter 5 thoroughly covers transmission line modeling and the performance and compensation of the transmission lines This chapter provides the concepts and tools necessary for the preliminary transmission line design 15 ENERGY CONTROL CENTER Chapter 6 presents a comprehensive coverage of the power flow solution of an interconnected power system during normal operation First the commonly used iterative techniques for the solution of nonlinear algebraic equations are discussed Then several approaches to the solution of power flow are described These techniques are applied to the solution of practical systems using the developed software modules Chapter 7 covers some essential classical optimization of continuous functions and their application to optimal dispatch of generation The programs developed here are designed to work in synergy with the power flow programs Chapter 8 deals with synchronous machine transient analysis The voltage equations of the synchronous machine are first developed These nonlinear equations are transformed into linear differential equations using Parks transformation Analytical solution of the transformed equations can be obtained by the Laplace transform method However MATLAB is used with ease to simulate the nonlinear differential equations of the synchronous machine directly in timedomain in matrix form for all modes of operation Thus students can observe the dynamic response of the synchronous machine during short circuits and appreciate the significance and consequence of the change of machine parameters The ultimate objective of this chapter is to develop simple network models of the synchronous generator for power system fault analysis and transient stability studies Chapter 9 covers balanced fault analysis The bus impedance matrix by the building algorithms is formulated and employed for the systematic computation of bus voltages and line currents during faults Chapter 10 discusses methods of symmetrical components that resolve the problem of an unbalanced circuit into the symmetrical components transformation and some applications The method is applied to the unbalanced fault which once again allows the treatment of the problem on simple per phase basis Algorithms have been developed to simulate different types of unbalanced faults The software modules developed for unbalanced faults include single linetoground fault linetoline fault and double linetoground fault 16 COMPUTER ANALYSIS Chapter 11 covers power system stability problems First the dynamic behavior of a onemachine system due to a small disturbance is investigated and the analytical solution of this linearized model is obtained MATLAB and SIMULINK are used conveniently to simulate the system and the model is extended to multimachine systems Next the transient stability using equal area criteria is discussed and the result is represented graphically providing physical insight into the dynamic behavior of the machine An introduction to nonlinear differential equations and their numerical solutions is given MATLAB is used to obtain the numerical solution of the swing equation of a onemachine system Simulation is also obtained using the SIMULINK toolbox A program compatible with the power flow programs is developed for the transient stability analysis of the multimachine systems Chapter 12 is concerned with power system control and develops some of the control schemes required to operate the power system in the steady state Simple models of the essential components used in control systems are presented The automatic voltage regulator AVR and the load frequency control LFC are discussed The automatic generation control AGC in singlearea and multiarea systems including tieline power control are analyzed For each case the responses to the real power demand are obtained The generator responses with the AVR and various compensators such as rate feedback and Proportional Integral Derivative PID controllers are obtained Both AGC and AVR systems are illustrated by several examples and the responses are obtained using MATLAB These analyses are supplemented by constructing the SIMULINK block diagram which provides a highly interactive environment for simulation Some basic materials of modern control theory are discussed including the poleplacement state feedback design and the optimal controller designs using the linear quadratic regulator based on the Riccati equation These modern techniques are then applied for simulation of the LFC systems Appendix A is a selfstudy MATLAB and SIMULINK tutorial focused on power and control systems and coordinated with the text Appendix B includes a brief introduction to the fundamentals of control systems and is suitable for students without a background in control systems Appendix C lists all functions script files and chapter examples Answers to problems are given at the end of the book The instructors manual for this text contains the workedout solutions for all of the books problem The material in the text is designed to be fully covered in a twosemester undergraduate course sequence The organization is flexible allowing instructors to select the material that best suits the requirements of a onequarter or a onesemester course In a onesemester course the first six chapters which form the basis for power system analysis should be covered The material in Chapter 2 contains power concepts and threephase systems which are usually covered in circuit courses This chapter can be excluded if the students are well prepared or it can be used for review Also for students with electrical machinery background Chapter 3 might be omitted After the above coverage additional material from the remaining chapters may then be appropriate depending on the syllabus requirements and the individual preferences One choice is to cover Chapter 7 optimal dispatch of generation another choice is Chapter 9 balanced fault The generator reactances required in Chapter 9 may be covered briefly from Section 87 without covering Chapter 8 in its entirety After reading the book students should have a good perspective of power system analysis and an active knowledge of various numerical techniques that can be applied to the solution of large interconnected power systems Students should 2 BASIC PRINCIPLES find MATLAB helpful in learning the material in the text particularly in solving the problems at the end of each chapter I would like to express my appreciation and thanks to the following reviewers for their very helpful comments and suggestions Professor Max D Anderson University of MissouriRolla Professor Miroslav Begovic Georgia Institute of Technology Professor Karen L Butler Texas AM University Professor Kevin A Clements Worcester Polytechnic Institute Professor Mariesa L Crow University of MissouriRolla Professor Malik Elbuluk University of Akron Professor A A ElKeib University of Alabama Professor F P Emad University of Maryland Professor L L Grigsby Auburn University Professor Kwang Y Lee Pennsylvania State University Professor M A Pai University of IllinoisUrbana Professor E K Stanek University of MissouriRolla My sincere thanks goes to Lynn Kallas who proofread the early version of the manuscript Special thanks goes to the staff of McGrawHill Lynn Cox the editor for her constant encouragement Nina Kreiden editorial coordinator for her support and Bev Strock senior project manager for her attention to detail during all phases of editing and production I wish to express my thanks to the Electrical Engineering and Computer Science Department of Milwaukee School of Engineering and to Professor Ray Palmer chairman of the department for giving me the opportunity to prepare this material Last but not least I thank my wife Jila and my children Dana Fred and Cameron who were a constant and active source of support throughout the endeavor 21 INTRODUCTION 22 POWER IN SINGLEPHASE AC CIRCUITS CHAPTER 1 THE POWER SYSTEM AN OVERVIEW 11 INTRODUCTION Electric energy is the most popular form of energy because it can be transported easily at high efficiency and reasonable cost The first electric network in the United States was established in 1882 at the Pearl Street Station in New York City by Thomas Edison The station supplied dc power for lighting the lower Manhattan area The power was generated by dc generators and distributed by underground cables In the same year the first waterwheel driven generator was installed in Appleton Wisconsin Within a few years many companies were established producing energy for lighting all operated under Edisons patents Because of the excessive power loss Rl2 at low voltage Edisons companies could deliver energy only a short distance from their stations With the invention of the transformer William Stanley 1885 to raise the level of ac voltage for transmission and distribution and the invention of the induction motor Nikola Tesla 1888 to replace the dc motors the advantages of the ac system became apparent and made the ac system prevalent Another advantage of the ac system is that due to lack of commutators in the ac generators more power can be produced conveniently at higher voltages 23 COMPLEX POWER The first singlephase ac system in the United States was at Oregon City where power was generated by two 300 hp waterwheel turbines and transmitted at 4 kV to Portland Southern California Edison Company installed the first threephase system at 23 kV in 1893 Many electric companies were developed throughout the country In the beginning individual companies were operating at different frequencies anywhere from 25 Hz to 133 Hz But as the need for interconnection and parallel operation became evident a standard frequency of 60 Hz was adopted throughout the US and Canada Most European countries selected the 50Hz system Transmission voltages have since risen steadily and the extra high voltage EHV in commercial use is 765 kV first put into operation in the United States in 1969 For transmitting power over very long distances it may be more economical to convert the EHV ac to EHV dc transmit the power over two lines and invert it back at the other end Studies show that it is advantageous to consider dc lines when the transmission distance is 500 km or more DC lines have no reactance and are capable of transferring more power for the same conductor size than ac lines DC transmission is especially advantageous when two remotely located large systems are to be connected The dc transmission line acts as an asynchronous link between the two rigid systems eliminating the instability problem inherent in the ac links The main disadvantage of the link is the production of harmonics which requires filtering and a large amount of reactive power compensation required at both ends of the line The first 400kV dc line in the United States was the Pacific Intertie 850 miles long between Oregon and California built in 1970 The entire continental United States is interconnected in an overall network called the power grid A small part of the network is federally and municipally owned but the bulk is privately owned The system is divided into several geographical regions called power pools In an interconnected system fewer generators are required as a reserve for peak load and spinning reserve Also interconnection makes the energy generation and transmission more economical and reliable since power can readily be transferred from one area to others At times it may be cheaper for a company to buy bulk power from neighboring utilities than to produce it in one of its older plants 24 THE COMPLEX POWER BALANCE The bulk generation of electricity in the United States is produced by integrated investorowned utilities IOU A small portion of power generation is federally owned such as the Tennessee Valley Authority and Bonneville Power Administration Two separate levels of regulation currently regulate the United States electric system One is the Federal Energy Regulatory Commission FERC which regulates the price of wholesale electricity service terms and conditions The other is the Securities and Exchange Commission SEC which regulates the business structure of electric utilities The transmission system of electric utilities in the United States and Canada is interconnected into a large power grid known as the North American Power Systems Interconnection The power grid is divided into several pools The pools consist of several neighboring utilities which operate jointly to schedule generation in a costeffective manner A privately regulated organization called the North American Electric Reliability Council NERC is responsible for maintaining system standards and reliability NERC works cooperatively with every provider and distributor of power to ensure reliability NERC coordinates its efforts with FERC as well as other organizations such as the Edison Electric Institute EEI NERC currently has four distinct electrically separated areas These areas are the Electric Reliability Council of Texas ERCOT the Western States Coordination Council WSCC the Eastern Interconnect which includes all the states and provinces of Canada east of the Rocky Mountains excluding Texas and HydroQuebec which has interconnects with the northeast These electrically separate areas import and export power to each other but are not synchronized electrically The electric power industry in the United States is undergoing fundamental changes since the deregulation of the telecommunication gas and other industries The generation business is rapidly becoming marketdriven This is a major change for an industry which until the last decade was characterized by large vertically integrated monopolies The implementation of open transmission access has resulted in wholesale power markets In the future utilities may possibly be divided into power generation transmission and retail segments Generating utilities would sell directly to customers instead of to local distributors This would eliminate the monopoly that distributors currently have The distributors would sell their services as electricity distributors instead of being a retailer of electricity itself The retail structure of power distribution would resemble the current structure of the telephone communication industry The consumer would have a choice as to from which generator they purchase power If the entire electric power industry were to be deregulated final consumers could choose from generators across the country Power brokers and power marketers will assume a major role in this new competitive power industry Currently the ability to market electricity to retail end users exists but only in a limited number of states in pilot programs Extensive efforts are being made to create a more competitive environment for electricity markets in order to promote greater efficiency Thus the power industry faces many new problems with one of the highest priority issues being reliability that is bringing a steady uninterruptable power supply to all electricity consumers The restructuring and deregulation of electric utilities together with recent progress in technology introduce unprecedented challenges and opportunities for power systems research and open up new opportunities to young power engineers 25 POWER FACTOR CORRECTION The power system of today is a complex interconnected network as shown in Figure 11 page 7 A power system can be subdivided into four major parts Generation Transmission and Subtransmission Distribution Loads Generators One of the essential components of power systems is the threephase ac generator known as synchronous generator or alternator Synchronous generators have two synchronously rotating fields One field is produced by the rotor driven at synchronous speed and excited by dc current The other field is produced in the stator windings by the threephase armature currents The dc current for the rotor windings is provided by excitation systems In the older units the exciters are dc generators mounted on the same shaft providing excitation through slip rings Todays systems use ac generators with rotating rectifiers known as brushless excitation systems The generator excitation system maintains generator voltage and controls the reactive power flow Because they lack the commutator ac generators can generate high power at high voltage typically 30 kV In a power plant the size of generators can vary from 50 MW to 1500 MW The source of the mechanical power commonly known as the prime mover may be hydraulic turbines at waterfalls steam turbines whose energy comes from the burning of coal gas and nuclear fuel gas turbines or occasionally internal combustion engines burning oil The estimated installed generation capacity in 1998 for the United States is presented in Table 11 Steam turbines operate at relatively high speeds of 3600 or 1800 rpm The generators to which they are coupled are cylindrical rotor twopole for 3600 rpm or fourpole for 1800 rpm operation Hydraulic turbines particularly those operating with a low pressure operate at low speed Their generators are usually a salient type rotor with many poles In a power station several generators are operated in parallel in the power grid to provide the total power needed They are connected at a common point called a bus 26 COMPLEX POWER FLOW Today the total installed electric generating capacity is about 760000 MW Assuming the United States population to be 270 million Installed capacity per capita 760 x 109 270 x 106 2815 W To realize the significance of this figure consider the average power of a person to be approximately 50 W Therefore the power of 2815 W is equivalent to 2815 W 50 W 56 power slave The annual kWh consumption in the United States is about 3550 x 109 kWh The asset of the investment for investorowned companies is about 200 billion dollars and they employ close to a half million people With todays emphasis on environmental consideration and conservation of fossil fuels many alternate sources are considered for employing the untapped energy sources of the sun and the earth for generation of power Some of these alternate sources which are being used to some extent are solar power geothermal power wind power tidal power and biomass The aspiration for bulk generation of power in the future is the nuclear fusion If nuclear fusion is harnessed economically it would provide clean energy from an abundant source of fuel namely water Table 11 Installed Generation Capacity Type Capacity Percent Fuel MW Steam Plant 478800 63 Coal gas petroleum Nuclear 106400 14 Uranium Hydro and pumped storage 91200 12 Water Gas Turbine 60800 8 Gas petroleum Combined cycle 15200 2 Gas petroleum Internal Combustion 4940 065 Gas petroleum Others 2660 035 Geothermal solar wind Total 760000 10000 Transformers Another major component of a power system is the transformer It transfers power with very high efficiency from one level of voltage to another level The power transferred to the secondary is almost the same as the primary except for losses in the transformer and the product VI on the secondary side is approximately the same as the primary side Therefore using a stepup transformer of turns ratio a will reduce the secondary current by a ratio of 1a This will reduce losses in the line which makes the transmission of power over long distances possible 27 BALANCED THREEPHASE CIRCUITS The insulation requirements and other practical design problems limit the generated voltage to low values usually 30 kV Thus stepup transformers are used for transmission of power At the receiving end of the transmission lines stepdown transformers are used to reduce the voltage to suitable values for distribution or utilization In a modern utility system the power may undergo four or five transformations between generator and ultimate user 132 TRANSMISSION AND SUBTRANSMISSION The purpose of an overhead transmission network is to transfer electric energy from generating units at various locations to the distribution system which ultimately supplies the load Transmission lines also interconnect neighboring utilities which permits not only economic dispatch of power within regions during normal conditions but also the transfer of power between regions during emergencies Standard transmission voltages are established in the United States by the American National Standards Institute ANSI Transmission voltage lines operating at more than 60 kV are standardized at 69 kV 115 kV 138 kV 161 kV 230 kV 345 kV 500 kV and 765 kV linetoline Transmission voltages above 230 kV are usually referred to as extrahigh voltage EHV Figure 11 shows an elementary diagram of a transmission and distribution system High voltage transmission lines are terminated in substations which are called highvoltage substations receiving substations or primary substations The function of some substations is switching circuits in and out of service they are referred to as switching stations At the primary substation the voltage is stepped down to a value more suitable for the next part of the journey toward the load Very large industrial customers may be served from the transmission system The portion of the transmission system that connects the highvoltage substations through stepdown transformers to the distribution substations is called the subtransmission network There is no clear delineation between transmission and subtransmission voltage levels Typically the subtransmission voltage level ranges from 69 to 138 kV Some large industrial customers may be served from the subtransmission system Capacitor banks and reactor banks are usually installed in the substations for maintaining the transmission line voltage 28 YCONNECTED LOADS FIGURE 11 Basic components of a power system Thermal Station Fossil Transmission 115 765 kV Nuclear Station Hydro Station Switching Station Very Large Consumers HV Substation HV Substation HV Substation Large Consumers 69 138 kV Gas Turbine Medium Consumers Distribution Substations 4 345 kV Distribution Transformers Residential Consumers 240120 V 29 ΔCONNECTED LOADS dred feet in length then deliver power to the individual consumers The secondary distribution serves most of the customers at levels of 240120 V singlephase threewire 208Y120 V threephase fourwire or 480Y277 V threephase fourwire The power for a typical home is derived from a transformer that reduces the primary feeder voltage to 240120 V using a threewire line Distribution systems are both overhead and underground The growth of underground distribution has been extremely rapid and as much as 70 percent of new residential construction is served underground 134 LOADS Loads of power systems are divided into industrial commercial and residential Very large industrial loads may be served from the substationship network and small industrial loads are served from the primary distribution network The industrial loads are composite loads and induction motors form a high proportion of these loads These composite loads are functions of voltage and frequency and form a major part of the system load Commercial and residential loads consist largely of lighting heating and cooling These loads are independent of frequency and consume negligibly small reactive power The real power of loads are expressed in terms of kilowatts or megawatts The magnitude of load varies throughout the day and power must be available to consumers on demand The dailyload curve of a utility is a composite of demands made by various classes of users Smaller peaking generators may be commissioned to meet the peak load that occurs for only a few hours In order to assess the usefulness of the generating plant the load factor is defined The load factor is the ratio of average load over a designated period of time to the peak load occurring in that period Load factors may be given for a day a month or a year The yearly or annual load factor is the most useful since a year represents a full cycle of time The daily load factor is Daily LF average load peak load 11 Multiplying the numerator and denominator of 11 by a time period of 24 hr we have Daily LF average load 24 hr peak load 24 hr energy consumed during 24 hr peak load 24 hr 12 The annual load factor is Annual LF total annual energy peak load 8760 hr 13 210 ΔY TRANSFORMATION Generally there is diversity in the peak load between different classes of loads which improves the overall system load factor In order for a power plant to operate economically it must have a high system load factor Todays typical system load factors are in the range of 55 to 70 percent There are a few other factors used by utilities Utilization factor is the ratio of maximum demand to the installed capacity and plant factor is the ratio of annual energy generation to the plant capacity 8760 hr These factors indicate how well the system capacity is utilized and operated A MATLAB function barcycledata is developed which obtains a plot of the load cycle for a given interval The demand interval and the load must be defined by the variable data in a threecolumn matrix The first two columns are the demand interval and the third column is the load value The demand interval may be minutes hours or months in ascending order Hourly intervals must be expressed in military time Example 11 The daily load on a power system varies as shown in Table 12 Use the barcycle function to obtain a plot of the daily load curve Using the given data compute the average load and the daily load factor Figure 12 211 PERPHASE ANALYSIS 12 AM 2 AM 6 2 6 5 6 9 10 9 12 15 12 PM 2 PM 12 2 4 14 4 6 16 6 8 18 8 10 16 10 11 12 11 12 AM 6 The following commands data 0 2 6 2 6 5 6 9 10 9 12 15 12 14 12 14 16 14 16 18 16 18 20 18 20 22 16 22 23 12 23 24 6 P data3 Dt data2 data1 Column array of demand interval W PDt Total energy area under the curve Pavg WsumDt Average load Peak maxP Peak load LF PavgPeak100 Percent load factor barcycledata xlabelTime hr ylabelP MW result in Pavg 115417 Peak 18 LF 6412 212 BALANCED THREEPHASE POWER 14 SYSTEM PROTECTION In addition to generators transformers and transmission lines other devices are required for the satisfactory operation and protection of a power system Some of the protective devices directly connected to the circuits are called switchgear They include instrument transformers circuit breakers disconnect switches fuses and lightning arresters These devices are necessary to deenergize either for normal operation or on the occurrence of faults The associated control equipment and protective relays are placed on switchboard in control houses 15 ENERGY CONTROL CENTER For reliable and economical operation of the power system it is necessary to monitor the entire system in a control center The modern control center of today is called the energy control center ECC Energy control centers are equipped with online computers performing all signal processing through the remote acquisition system Computers work in a hierarchical structure to properly coordinate different functional requirements in normal as well as emergency conditions Every energy control center contains a control console which consists of a visual display unit VDU keyboard and light pen Computers may give alarms as advance warnings to the operators dispatchers when deviation from the normal state occurs The dispatcher makes judgments and decisions and executes them with the aid of a computer Simulation tools and software packages written in highlevel language are implemented for efficient operation and reliable control of the system This is referred to as SCADA an acronym for supervisory control and data acquisition 16 COMPUTER ANALYSIS For a power system to be practical it must be safe reliable and economical Thus many analyses must be performed to design and operate an electrical system However before going into system analysis we have to model all components of electrical power systems Therefore in this text after reviewing the concepts of power and threephase circuits we will calculate the parameters of a multicircuit transmission line Then we will model the transmission line and look at the performance of the transmission line Since transformers and generators are a part of the system we will model these devices Design of a power system its operation and expansion requires much analysis This text presents methods of power system analysis with the aid of a personal computer and the use of MATLAB The MATLAB environment permits a nearly direct transition from mathematical expression to simulation Some of the basic analysis covered in this text are Evaluation of transmission line parameters Transmission line performance and compensation Power flow analysis Economic scheduling of generation Synchronous machine transient analysis Balanced fault Symmetrical components and unbalanced fault Stability studies Power system control Many MATLAB functions are developed for the above studies thus allowing the student to concentrate on analysis and design of practical systems and spend less time on programming 3 GENERATOR AND TRANSFORMER MODELS THE PERUNIT SYSTEM 12 1 THE POWER SYSTEM AN OVERVIEW PROBLEMS 11 The demand estimation is the starting point for planning the future electric power supply The consistency of demand growth over the years has led to numerous attempts to fit mathematical curves to this trend One of the simplest curves is P P0eαtt0 where α is the average per unit growth rate P is the demand in year t and P0 is the given demand at year t0 Assume the peak power demand in the United States in 1984 is 480 GW with an average growth rate of 34 percent Using MATLAB plot the predicated peak demand in GW from 1984 to 1999 Estimate the peak power demand for the year 1999 12 In a certain country the energy consumption is expected to double in 10 years Assuming a simple exponential growth given by P P0eαt calculate the growth rate α 31 INTRODUCTION 13 The annual load of a substation is given in the following table During each month the power is assumed constant at an average value Using MATLAB and the barcycle function obtain a plot of the annual load curve Write the necessary statements to find the average load and the annual load factor Annual System Load Interval month Load MW January 8 February 6 March 4 April 2 May 6 June 12 July 16 August 14 September 10 October 4 November 6 December 8 32 SYNCHRONOUS GENERATORS CHAPTER 2 33 STEADYSTATE CHARACTERISTICSCYLINDRICAL ROTOR BASIC PRINCIPLES 34 SALIENTPOLE SYNCHRONOUS GENERATORS 21 INTRODUCTION The concept of power is of central importance in electrical power systems and is the main topic of this chapter The typical student will already have studied much of this material and the review here will serve to reinforce the power concepts encountered in the electric circuit theory In this chapter the flow of energy in an ac circuit is investigated By using various trigonometric identities the instantaneous power pt is resolved into two components A plot of these components is obtained using MATLAB to observe that ac networks not only consume energy at an average rate but also borrow and return energy to its sources This leads to the basic definitions of average power P and reactive power Q The voltampere S which is a mathematical formulation based on the phasor forms of voltage and current is introduced Then the complex power balance is demonstrated and the transmission inefficiencies caused by loads with low power factors are discussed and demonstrated by means of several examples Next the transmission of complex power between two voltage sources is considered and the dependency of real power on the voltage phase angle and the dependency of reactive power on voltage magnitude is established MATLAB is used conveniently to demonstrate this idea graphically Finally the balanced threephase circuit is examined An important property of a balanced threephase system is that it delivers constant power That is the 35 POWER TRANSFORMER power delivered does not fluctuate with time as in a singlephase system For the purpose of analysis and modeling the perphase equivalent circuit is developed for the threephase system under balanced condition 22 POWER IN SINGLEPHASE AC CIRCUITS Figure 21 shows a singlephase sinusoidal voltage supplying a load Let the instantaneous voltage be vt Vm cosωt θv 21 and the instantaneous current be given by it Im cosωt θi 22 The instantaneous power pt delivered to the load is the product of voltage vt and current it given by pt vtit VmIm cosωt θv cosωt θi 23 In Example 21 MATLAB is used to plot the instantaneous power pt and the result is shown in Figure 22 In studying Figure 22 we note that the frequency of the instantaneous power is twice the source frequency Also note that it is possible for the instantaneous power to be negative for a portion of each cycle In a passive network negative power implies that energy that has been stored in inductors or capacitors is now being extracted It is informative to write 23 in another form using the trigonometric identity cos A cos B 12 cosA B 12 cosA B 24 Q VIsin θ 29 36 EQUIVALENT CIRCUIT OF A TRANSFORMER which results in pt 12 VmIm cosθv θi cos2ωt θv θi 12 VmIm cosθv θi cos2ωt θv θv θi 12 VmIm cosθv θi cos2ωt θv cosθv θi sin2ωt θv sinθv θi The rootmeansquare rms value of vt is V Vm2 and the rms value of it is I Im2 Let θ θv θi The above equation in terms of the rms values is reduced to pt VI cos θ 25 where θ is the angle between voltage and current or the impedance angle θ is positive if the load is inductive ie current is lagging the voltage and θ is negative if the load is capacitive ie current is leading the voltage The instantaneous power pt can be decomposed into two components The first component of 25 is pRt VI cos θ 1 cos2ωt θv 26 The second term in 26 which has a frequency twice that of the source accounts for the sinusoidal variation in the absorption of power by the resistive portion of the load Since the average value of this sinusoidal function is zero the average power delivered to the load is given by P VI cos θ 27 This is the power absorbed by the resistive component of the load and is also referred to as the active power or real power The product of the rms voltage value V and the rms current value I is called the apparent power and is measured in units of volt ampere The product of the apparent power and the cosine of the angle between voltage and current yields the real power Because cos θ plays a key role in the determination of the average power it is called power factor When the current lags the voltage the power factor is considered lagging When the current leads the voltage the power factor is considered leading The second component of 25 is pXt VI sin θ sin 2ωt θv 28 vt Vm cosωt it Im cosωt 60 37 DETERMINATION OF EQUIVALENT CIRCUIT PARAMETERS V Vθ I Iθi 38 TRANSFORMER PERFORMANCE The above equation defines a complex quantity where its real part is the average real power P and its imaginary part is the reactive power Q Thus the complex power designated by S is given by S V I P jQ 210 The magnitude of S S P² Q² is the apparent power its unit is voltamperes and the larger units are kVA or MVA Apparent power gives a direct indication of heating and is used as a rating unit of power equipment Apparent power has practical significance for an electric utility company since a utility company must supply both average and apparent power to consumers The reactive power Q is positive when the phase angle θ between voltage and current impedance angle is positive ie when the load impedance is inductive and I lags V Q is negative when θ is negative ie when the load impedance is capacitive and I leads V as shown in Figure 24 In working with Equation 210 it is convenient to think of P Q and S as forming the sides of a right triangle as shown in Figures 23 and 24 39 THREEPHASE TRANSFORMER CONNECTIONS From 213 the impedance of the complex power S is given by Z V² S 214 24 THE COMPLEX POWER BALANCE From the conservation of energy it is clear real power supplied by the source is equal to the sum of real powers absorbed by the load At the same time a balance between the reactive power must be maintained Thus the total complex power delivered to the loads in parallel is the sum of the complex powers delivered to each Proof of this is as follows I V FIGURE 25 Three loads in parallel For the three loads shown in Figure 25 the total complex power is given by S V I V I₁ I₂ I₃ V I₁ V I₂ V I₃ 215 Example 22 In the above circuit V 12000 V Z₁ 60 j0 Ω Z₂ 6 j12 Ω and Z₃ 30 j30 Ω Find the power absorbed by each load and the total complex power 310 AUTOTRANSFORMERS S₁ V I₁ 1200020 j0 24000 W j0 var S₂ V I₂ 1200040 j80 48000 W j96000 var S₃ V I₃ 1200020 j20 24000 W j24000 var The total load complex power adds up to S S₁ S₂ S₃ 96000 W j72000 var Alternatively the sum of complex power delivered to the load can be obtained by first finding the total current I I₁ I₂ I₃ 20 j0 40 j80 20 j20 80 j60 1003687 A and S V I 120001003687 1200003687 VA 96000 W j72000 var 311 THREEWINDING TRANSFORMERS 25 POWER FACTOR CORRECTION It can be seen from 27 that the apparent power will be larger than P if the power factor is less than 1 Thus the current I that must be supplied will be larger for PF 1 than it would be for PF 1 even though the average power P supplied is the same in either case A larger current cannot be supplied without additional cost to the utility company Thus it is in the power companys and its customers best interest that major loads on the system have power factors as close to 1 as possible In order to maintain the power factor close to unity power companies install banks of capacitors throughout the network as needed They also impose an additional charge to industrial consumers who operate at low power factors Since industrial loads are inductive and have low lagging power factors it is beneficial to install capacitors to improve the power factor This consideration is not important for residential and small commercial customers because their power factors are close to unity Example 23 Two loads Z1 100 j0 Ω and Z2 10 j20 Ω are connected across a 200V rms 60Hz source as shown in Figure 27 a Find the total real and reactive power the power factor at the source and the total current I1 2000 A 20 A I2 2000 A 4 j8 A S1 VI1 20002 j0 400 W j0 var S2 VI2 20004 j8 800 W j1600 var 312 VOLTAGE CONTROL OF TRANSFORMERS Total apparent power and current are S P jQ 1200 j1600 20005313 VA I SV 20005313 105313 A Power factor at the source is PF cos5313 06 lagging b Find the capacitance of the capacitor connected across the loads to improve the overall power factor to 08 lagging Total real power P 1200 W at the new power factor 08 lagging Therefore θ cos108 3687 Q P tan θ 1200 tan3687 900 var Qc 1600 900 700 var Zc V2 Sc 2002 j700 106 5714 Ω C V2 2π605714 4642 μF The total power and the new current are S 1200 j900 15003687 I SV 15003687 753687 A Note the reduction in the supply current from 10 A to 75 A 313 THE PERUNIT SYSTEM FIGURE 28 Circuit for Example 24 An inductive load has a lagging power factor the capacitive load has a leading power factor and the resistive load has a unity power factor For Load 1 θ1 cos1028 7374 lagging The load complex powers are S1 1257374 kVA 35 kW j120 kvar S2 10 kW j40 kvar S3 15 kW j0 kvar The total apparent power is S P jQ S1 S2 S3 35 j120 10 j40 15 j0 60 kW j380 kvar 1005313 kVA The total current is I SV 1000005313 71435313 A The supply power factor is PF cos5313 06 lagging b A capacitor of negligible resistance is connected in parallel with the above loads to improve the power factor to 08 lagging Determine the kvar rating of this capacitor and the capacitance in μF 314 CHANGE OF BASE Total real power P 60 kW at the new power factor of 08 lagging results in the new reactive power Q θ cos¹08 3687 Q 60 tan3687 45 kvar Therefore the required capacitor kvar is Qc 80 45 35 kvar and Xc V² Sc 1400² j35000 j56 Ω and C 106 2π6056 4737 µF and the new current is I S V 60000 j45000 1400 14000 53573687 A Note the reduction in the supply current from 7143 A to 5357 A 4 TRANSMISSION LINE PARAMETERS The complex power S12 is given by S12 V1I12 V1 Z Z Y δ1 V2 Z Y δ2 V1² Z Y V1V2 Z Y δ1 δ2 Thus the real and reactive power at the sending end are P12 V1² Z cos γ V1 V2 Z cosγ δ1 δ2 Q12 V1² Z sin γ V1 V2 Z sinγ δ1 δ2 Power system transmission lines have small resistance compared to the reactance Assuming R 0 ie Z X90 the above equations become P12 V1V2 X sinδ1 δ2 Q12 V1 X V1 V2 cosδ1 δ2 41 INTRODUCTION For maintaining transient stability the power system is usually operated with small load angle δ Also from 219 the reactive power flow is determined by the magnitude difference of terminal voltages ie Q V1 V2 Example 25 Two voltage sources V1 1205 V and V2 1000 V are connected by a short line of impedance Z 1 j7 Ω as shown in Figure 29 Determine the real and reactive power supplied or received by each source and the power loss in the line I12 1205 1000 1 j7 313511002 A I21 1 j7 31356998 A S12 V1I12 3762 10502 975 W j3633 var S21 V2I21 31356998 1073 W j2945 var Line loss is given by SL S1 S2 98 W j688 var 42 OVERHEAD TRANSMISSION LINES 43 LINE RESISTANCE 50000 1953349 1855084 98265 100000 3433715 3280828 152923 150000 4909938 4666382 243566 200000 6370676 6001201 369475 250000 7804848 7275125 529723 44 INDUCTANCE OF A SINGLE CONDUCTOR a Positive or ABC phase sequence b Negative or ACB phase sequence 45 INDUCTANCE OF SINGLEPHASE LINES E1 inputSource 1 Voltage Mag a1 inputSource 1 Phase Angle E2 inputSource 2 Voltage Mag a2 inputSource 2 Phase Angle R inputLine Resistance X inputLine Reactance VAn Vp0 5 LINE MODEL AND PERFORMANCE Vbn Vp120 51 INTRODUCTION Vcn Vp240 52 SHORT LINE MODEL The relationship between phase and line currents can be obtained by applying Kirchhoffs current law at the corners of Δ Ia Iab Ip10 1240 3Ip30 Ib Ibc Ip1120 10 3Ip150 Ic Ica Ip1240 1120 3Ip90 The relationship between the line currents and phase currents is demonstrated graphically in Figure 215 If the rms of any of the line currents is denoted by IL then one of the important characteristics of the Δconnected threephase load may be expressed as IL 3Ip30 Thus in the case of Δconnected loads the magnitude of the line current is 3 times the magnitude of the phase current and with positive phase sequence the set of line currents lags the set of phase currents by 30 53 MEDIUM LINE MODEL The phasor diagram in Figure 217 shows the relationship between balanced phase and linetoline voltages From this phasor diagram we find Vab Vac 3Van30 3Van30 3Van Substituting in 233 we get Ia 3Van ZΔ Van ZΔ 3 Ia Now for the Yconnected circuit we have Van ZY Ia Thus from 236 and 237 we find that ZY ZΔ 3 54 LONG LINE MODEL Since the neutral carries no current a neutral wire of any impedance may be replaced by any other impedance including a short circuit and an open circuit The return line may not actually exist but regardless a line of zero impedance is included between the two neutral points The balanced power system problems are then solved on a perphase basis It is understood that the other two phases carry identical currents except for the phase shift We may then look at only one phase say phase A consisting of the source VAN in series with ZL and ZP as shown in Figure 218 The neutral is taken as datum and usually a singlesubscript notation is used for phase voltages 6 POWER FLOW ANALYSIS where Vp and Ip are the magnitudes of the rms phase voltage and current respectively The total instantaneous power is the sum of the instantaneous power of each phase given by p3φ vania vbnib vcnic 242 Substituting for the instantaneous voltages and currents from 240 and 241 into 242 p3φ 2VpIpcosωt θvcosωt θi 2VpIpcosωt θv 120cosωt θi 120 2VpIpcosωt θv 240cosωt θi 240 Using the trigonometric identity 24 p3φ VpIpcosθv θi cos2ωt θv θi VpIpcosθv θi cos2ωt θv θi 240 VpIpcosθv θi cos2ωt θv θi cos2ωt θv θi 480 p3φ VpIpcosθv θi cos2ωt θv θi cosθv θi cos2ωt θv θi 240 VpIpcosθv θi cos2ωt θv θi p3φ 3VpIpcos θ 244 Thus the complex threephase power is S3φ P3φ jQ3φ 246 S3φ 3V pIp 247 Equations 244 and 245 are sometimes expressed in terms of the rms magnitude of the line voltage and the rms magnitude of the line current In a Yconnected load the phase voltage Vp VL3 and the phase current Ip IL 61 INTRODUCTION In the Δconnection Vp VL and Ip IL3 Substituting for the phase voltage and phase currents in 244 and 245 the real and reactive powers for either connection are given by P3φ 3VLILcos θ 248 and Q3φ 3VLILsin θ 249 A comparison of the last two expressions with 244 and 245 shows that the equation for the power in a threephase system is the same for either a Y or a Δ connection when the power is expressed in terms of line quantities When using 248 and 249 to calculate the total real and reactive power remember that θ is the phase angle between the phase voltage and the phase current As in the case of singlephase systems for the computation of power it is best to use the complex power expression in terms of phase quantities given by 247 The rated power is customarily given for the threephase and rated voltage is the linetoline voltage Thus in using the perphase equivalent circuit care must be taken to use perphase voltage by dividing the rated voltage by 3 Example 27 A threephase line has an impedance of 2 j4 Ω as shown in Figure 219 62 BUS ADMITTANCE MATRIX b The line voltage at the combined loads c The current per phase in each load d The total real and reactive powers in each load and the line a The Δconnected load is transformed into an equivalent Y The impedance per phase of the equivalent Y is Z2 60 j45 3 20 j15 Ω The phase voltage is V1 20785 3 120 V The singlephase equivalent circuit is shown in Figure 220 The total impedance is Z 2 j4 30 j4020 j15 30 j40 20 j15 2 j4 22 j4 24 Ω With the phase voltage Van as reference the current in phase a is I V1 Z 1200 V 24 5 A The threephase power supplied is S 3V1I 3120050 1800 W b The phase voltage at the load terminal is V2 1200 2 j450 110 j20 1118103 V 63 SOLUTION OF NONLINEAR ALGEBRAIC EQUATIONS The line voltage at the load terminal is V2ab 3 Z130 V2 3 1118197 19364197 V c The current per phase in the Yconnected load and in the equivalent Y of the Δ load is I1 V2Z1 110j2030 j40 1 j2 2236634 A I2 V2Z2 110j2020 j15 4 j2 44722656 A The phase current in the original Δconnected load ie Iab is given by Iab I2330 44722656330 25825656 A d The threephase power absorbed by each load is S1 33I12 311181032236634 450 W j600 var S2 33I22 3111810344722656 1200 W j900 var The threephase power absorbed by the line is SL 3RL jXLI2 32 j45² 150 W j300 var 631 GAUSSSEIDEL METHOD The phase voltage at the load terminals is V2 381053 2200 V The singlephase equivalent circuit is shown in Figure 221 The total complex power is SR3φ 56010707 j0707 132 528 j396 660873687 kVA With the phase voltage V2 as reference the current in the line is I SR3φ3V2 6600003687322000 1003687 A The phase voltage at the sending end is V1 22000 04 j271003687 24017458 V The magnitude of the line voltage at the sending end of the line is V 3V1 324017 4160 V b The threephase power loss in the line is SL3φ 3RI² 304100² j327100² 12 kW j81 kvar c The threephase sending power is SS3φ 33I1 3240174581003687 540 kW j477 kvar 632 NEWTONRAPHSON METHOD The peak amplitude Vm and the phase angle θv of the sinusoidal supply vt Vm cosωt θv The impedance magnitude Z and the phase angle γ of the load The program should produce plots for it vt pt prt and pxt similar to Example 21 Run the program for Vm 100 V θv 0 and the following loads An inductive load Z 12560Ω A capacitive load Z 2030Ω A resistive load Z 250Ω a From prt and pxt plots estimate the real and reactive power for each load Draw a conclusion regarding the sign of reactive power for inductive load b Using phasor values of current and voltage calculate the real and reactive power for each load and compare with the results obtained from the curves c If the above loads are all connected across the same power supply determine the total real and reactive power taken from the supply An inductive load consisting of R and X in parallel feeding from a 2400V rms supply absorbs 288 kW at a lagging power factor of 08 Determine R and X Two singlephase ideal voltage sources are connected by a line of impedance of 07 j24 Ω as shown in Figure 224 V1 5001626 V and V2 58500 V Find the complex power for each machine and determine whether they are delivering or receiving real and reactive power Also find the real and the reactive power loss in the line A balanced delta connected load of 15 j18 Ω per phase is connected at the end of a threephase line as shown in Figure 225 The line impedance is 1 j2 Ω per phase The line is supplied from a threephase source with a linetoline voltage of 20785 V rms Taking Van as reference determine the a Current in phase a b Total complex power supplied from the source c Magnitude of the linetoline voltage at the load Load 2 A balanced resistive load that draws a total of 6 kW Load 3 A Yconnected capacitor bank with a total rating of 16 kVAR a What is the total system kW kVAR power factor and the supply current per phase b What is the system power factor and the supply current per phase when the resistive load and induction motor are operating but the capacitor bank is switched off 315 Three loads are connected in parallel across a 1247 kV threephase supply Load 1 Inductive load 60 kW and 660 kVAR Load 2 Capacitive load 240 kW at 08 power factor Load 3 Resistive load of 60 kW a Find the total complex power power factor and the supply current b A Yconnected capacitor bank is connected in parallel with the loads Find the total kVAR and the capacitance per phase in µF to improve the overall power factor to 08 lagging What is the new line current CHAPTER 3 GENERATOR AND TRANSFORMER MODELS THE PERUNIT SYSTEM 31 INTRODUCTION Before the power systems network can be solved it must first be modeled The threephase balanced system is represented on a perphase basis which was described in Section 210 The singlephase representation is also used for unbalanced systems by means of symmetrical components which is treated in a later chapter In this chapter we deal with the balanced system where transmission lines are represented by the π model as described in Chapter 4 Other essential components of a power system are generators and transformers their theory and construction are discussed in standard electric machine textbooks In this chapter we represent simple models of generators and transformers for steadystate balanced operation In the analysis of power systems it is frequently convenient to use the perunit system The advantage of this method is the elimination of transformers by simple impedances The perunit system is presented followed by the impedance diagram of the network expressed to a common MVA base 32 SYNCHRONOUS GENERATORS Largescale power is generated by threephase synchronous generators known as alternators driven either by steam turbines hydroturbines or gas turbines The armature windings are placed on the stationary part called stator The armature windings are designed for generation of balanced threephase voltages and are arranged to develop the same number of magnetic poles as the field winding that is on the rotor The field windings require a relatively small power 023 percent of the machine rating for its excitation placed on the rotor The rotor is also equipped with one or more shortcircuited windings known as damper windings The rotor is driven by a prime mover at constant speed and its field circuit is excited by direct current The excitation may be provided through slip rings and brushes by means of dc generators referred to as exciters mounted on the same shaft as the rotor of the synchronous machine However modern excitation systems usually use ac generators with rotating rectifiers and are known as brushless excitation The generator excitation system maintains generator voltage and controls the reactive power flow