Table Of Content
Modern Power Systems Analysis
Xi-Fan Wang Yonghua Song Malcolm Irving
l l
Modern Power Systems
Analysis
123
Xi-FanWang YonghuaSong
Xi’anJiaotongUniversity TheUniversityofLiverpool
Xi’an Liverpool
People’sRepublicofChina UnitedKingdom
MalcolmIrving
BrunelUniversity
Middlesex
UnitedKingdom
ISBN978-0-387-72852-0 e-ISBN978-0-387-72853-7
LibraryofCongressControlNumber:2008924670
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Preface
Thepowerindustry,acapitalandtechnologyintensiveindustry,isabasicnational
infrastructure.Itssecurity,reliability,andeconomyhaveenormousandfar-reaching
effects on a national economy. An electrical power system is a typical large-scale
system. Questions such as how to reflect accurately the characteristics of modern
electricalpowersystems,howtoanalyzeeffectivelytheiroperatingfeatures,andhow
toimprovefurthertheoperatingperformancearealwaysattheforefrontofelectrical
powersystemsresearch.
Electricalpowersystemanalysisisusedasthebasicandfundamentalmeasureto
study planning and operating problems. In the last century, electrical power
researchers have undertaken a great deal of investigation and development in this
area, have made great progress in theoretical analysis and numerical calculation,
andhavewrittenexcellentmonographsandtextbooks.
Overthelast20years,thechangesinelectricalpowersystemsandotherrelevant
technologieshavehadaprofoundinfluenceonthetechniquesandmethodologiesof
electricalpowersystemanalysis.
First, the development of digital computer technology has significantly im-
proved the performance of hardware and software. Now, we can easily deal with
loadflowissueswithovertenthousandnodes.Optimalloadflowandstaticsecurity
analysis,whichwereonceconsideredhardproblems,haveattainedonlinepractical
applications.
Second, the applications of HVDC and AC flexible transmission technologies
(FACTS)haveaddednewcontrolmeasurestoelectrical powersystems,andhave
increasedpowertransmissioncapacity,enhancedcontrolcapability,andimproved
operating characteristics. However, these technologies bring new challenges into
theareaofelectricalpowersystemanalysis.Wemustbuildcorrespondingmathe-
matical models for these new devices and develop algorithms for static and
dynamicanalysisofelectricalpowersystemsincludingthesedevices.
In addition, the rapid development of communication technology has enabled
online monitoring of electrical power systems. Therefore, the demand for online
softwareforelectricalpowersystemanalysisbecomesmoreandmorepressing.
Furthermore, worldwide power industry restructuring and deregulation has
separated the former vertically integrated system into various parts, and the once
v
vi Preface
unified problem of power system dispatching is now conducted via complicated
bilateral contracts and spot markets. New issues such as transmission ancillary
serviceandtransmissioncongestionhaveemerged.
Inrecentyears,severalpowerblackoutshavetakenplaceworldwide,especially
the‘‘8.13’’blackoutontheeasterngridofUSAandCanadaandtheblackoutsthat
occurredsuccessivelyinothercountrieshaveattractedagreatdealofattention.
All ofthese aspectsrequirenew theories, models,andalgorithms for electrical
power system analysis. It is within such an environment that this book has been
developed.Thebookiswrittenasatextbookforseniorstudentsandpostgraduates
aswellasareferencebookforpowersystemresearchers.
Weacknowledgethesupportfromvariousresearchfundingorganizations,their
colleagues, and students, especially, the special funds for Major State Basic Re-
searchProjectsofChina‘‘ResearchonPowerSystemReliabilityunderDeregulated
EnvironmentofPowerMarket’’(2004CB217905).Weexpressourspecialgratitude
to Professor Wan-Liang Fang and Professor Zheng-Chun Du for providing the
original materials of Chaps. 5 and 6, and 7 and 8, respectively. We also express
our sincere gratitude tothe following colleagues for their contributions tovarious
chaptersofthebook:ProfessorZhao-HongBieforChaps.1and3;ProfessorXiu-Li
Wang for Chaps. 2 and 4; Dr. Ze-Chun Hu for Chap. 3; Dr. Xiao-Ying Ding for
Chap.4;Dr.LinDuanforChaps.5and6;ProfessorDe-ChiangGangforChap.7;
andProfessorHai-FengWangforChaps.6and8.
Xi’an,China Xi-FanWang
Liverpool,UK YonghunaSong
London,UK MalcolmIrving
Contents
1 MathematicalModelandSolutionofElectricNetwork ................. 1
1.1 Introduction ............................................................ 1
1.2 BasicConcepts ......................................................... 2
1.2.1 NodeEquationandLoopEquation ............................ 2
1.2.2 EquivalentCircuitofTransformerandPhase
ShiftTransformer .............................................. 9
1.3 NodalAdmittanceMatrix ............................................. 13
1.3.1 BasicConceptofNodalAdmittanceMatrix .................. 13
1.3.2 FormulationandModificationofNodal
AdmittanceMatrix ............................................ 17
1.4 SolutiontoElectricNetworkEquations .............................. 22
1.4.1 GaussEliminationMethod .................................... 22
1.4.2 TriangularDecompositionandFactorTable ................. 27
1.4.3 SparseTechniques ............................................. 34
1.4.4 SparseVectorMethod ......................................... 38
1.4.5 OptimalOrderingSchemesofElectricNetworkNodes ...... 43
1.5 NodalImpedanceMatrix ............................................. 48
1.5.1 BasicConceptofNodalImpedanceMatrix .................. 48
1.5.2 FormingNodalImpedanceMatrixUsingAdmittance
Matrix ......................................................... 50
1.5.3 FormingNodalImpedanceMatrixbyBranchAddition
Method ........................................................ 56
2 LoadFlowAnalysis ......................................................... 71
2.1 Introduction ........................................................... 71
2.2 FormulationofLoadFlowProblem .................................. 73
2.2.1 ClassificationofNodeTypes ................................. 73
2.2.2 NodePowerEquations ........................................ 76
2.3 LoadFlowSolutionbyNewtonMethod .............................. 79
2.3.1 BasicConceptofNewtonMethod ............................ 79
2.3.2 CorrectionEquations .......................................... 83
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viii Contents
2.3.3 SolutionProcessofNewtonMethod .......................... 88
2.3.4 SolutionofCorrectionEquations ............................. 89
2.4 FastDecoupledMethod .............................................. 101
2.4.1 IntroductiontoFastDecoupledMethod ..................... 101
2.4.2 CorrectionEquationsofFastDecoupledmethod ............ 104
2.4.3 FlowchartofFastDecoupledMethod ....................... 107
2.5 StaticSecurityAnalysisandCompensationMethod ................ 113
2.5.1 SurveyofStaticSecurityAnalysis ........................... 113
2.5.2 CompensationMethod ....................................... 114
2.6 DCLoadFlowMethod .............................................. 119
2.6.1 ModelofDCLoadFlow ..................................... 120
2.6.2 OutageAnalysisbyDCLoadFlowMethod ................. 122
2.6.3 N-1CheckingandContingencyRankingMethod ........... 123
3 StochasticSecurityAnalysisofElectricalPowerSystems ............ 129
3.1 Introduction .......................................................... 129
3.2 BasicConceptsofProbabilityTheory ............................... 130
3.2.1 ProbabilityofStochasticEvents ............................. 130
3.2.2 RandomVariablesanditsDistribution ...................... 132
3.2.3 NumeralCharacterofRandomVariable .................... 133
3.2.4 ConvolutionofRandomVariables ........................... 135
3.2.5 SeveralUsualRandomVariableDistributions .............. 136
3.2.6 MarkovProcess .............................................. 138
3.3 ProbabilisticModelofPowerSystems. .............................. 140
3.3.1 ProbabilisticModelofLoad ................................. 140
3.3.2 ProbabilisticModelofPowerSystemComponents ......... 141
3.3.3 OutageTableofPowerSystemComponents ................ 142
3.4 MonteCarloSimulationMethod .................................... 145
3.4.1 FundamentalTheoryofMonteCarloSimulation
Method ....................................................... 145
3.4.2 SamplingofSystemOperationState ........................ 148
3.4.3 StateEvaluationModel ...................................... 150
3.4.4 IndicesofReliabilityEvaluation ............................. 151
3.4.5 FlowchartofCompositeSystemAdequacyEvaluation ..... 152
3.4.6 MarkovChainMonteCarlo(MCMC)Simulation
Method ....................................................... 156
3.5 ProbabilisticLoadFlowAnalysis .................................... 161
3.5.1 CumulantsofRandomDistribution .......................... 162
3.5.2 LinearizationofLoadFlowEquation ........................ 168
3.5.3 ComputingProcessofProbabilisticLoadFlow ............. 171
3.6 ProbabilisticNetwork-FlowAnalysis ................................ 178
3.6.1 Introduction .................................................. 178
3.6.2 Network-FlowModel ........................................ 180
3.6.3 LowerBoundaryPointsofFeasibleFlowSolutions ........ 186
3.6.4 ReliabilityofTransmissionSystem .......................... 188
Contents ix
4 PowerFlowAnalysisinMarketEnvironment ......................... 193
4.1 Introduction .......................................................... 193
4.1.1 TransmissionOwner ......................................... 193
4.1.2 IndependentOperator ........................................ 194
4.1.3 PowerExchange .............................................. 194
4.1.4 AncillaryService ............................................. 195
4.1.5 SchedulingCoordinator ...................................... 195
4.2 OptimalPowerFlow ................................................. 196
4.2.1 GeneralFormulationofOPFProblem ....................... 196
4.2.2 ApproachestoOPF ........................................... 198
4.2.3 InteriorPointMethod(IPM)forOPFProblem .............. 202
4.3 ApplicationofOptimalPowerFlowinElectricityMarket .......... 217
4.3.1 Survey ........................................................ 217
4.3.2 CongestionManagementMethodBasedOnOPF ........... 223
4.4 PowerFlowTracing .................................................. 228
4.4.1 CurrentDecompositionAxioms ............................. 230
4.4.2 MathematicalModelofLossAllocation .................... 232
4.4.3 UsageSharingProblemofTransmissionFacilities .......... 234
4.4.4 MethodologyofGraphTheory .............................. 238
4.5 AvailableTransferCapabilityofTransmissionSystem ............. 241
4.5.1 IntroductionToAvailableTransferCapability .............. 241
4.5.2 ApplicationofMonteCarloSimulationinATC
Calculation ................................................... 245
4.5.3 ATCCalculationwithSensitivityAnalysisMethod ........ 246
5 HVDCandFACTS ....................................................... 255
5.1 Introduction .......................................................... 255
5.2 HVDCBasicPrinciplesandMathematicalModels ................. 258
5.2.1 HVDCBasicPrinciples ...................................... 258
5.2.2 ConverterBasicEquationsNeglectingLc ................... 261
5.2.3 ConverterBasicEquationsConsideringLc ................. 267
5.2.4 ConverterEquivalentCircuits ............................... 273
5.2.5 MultipleBridgeOperation ................................... 276
5.2.6 ConverterControl ............................................ 279
5.3 PowerFlowCalculationofAC/DCInterconnectedSystems ....... 281
5.3.1 ConverterBasicEquationsinperUnitSystem .............. 282
5.3.2 PowerFlowEquations ....................................... 283
5.3.3 JacobianMatrixofPowerFlowEquations .................. 286
5.3.4 IntegratedIterationformulaofAC/DC
InterconnectedSystems ...................................... 289
5.3.5 AlternatingIterationforAC/DCInterconnectedSystems ... 294
5.4 HVDCDynamicMathematicalModels ............................. 299
5.5 BasicPrinciplesandMathematicalModelsofFACTS ............. 301
5.5.1 BasicPrincipleandMathematicalModelofSVC ........... 302
5.5.2 BasicPrincipleandMathematicalModelofSTATCOM ... 308
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5.5.3 BasicPrincipleandMathematicalModelofTCSC ......... 313
5.5.4 BasicPrincipleandMathematicalModelofSSSC .......... 319
5.5.5 BasicPrincipleandMathematicalModelofTCPST ........ 322
5.5.6 BasicPrincipleandMathematicalModelofUPFC ......... 325
6 MathematicalModelofSynchronousGeneratorandLoad .......... 333
6.1 Introduction .......................................................... 333
6.2 MathematicalModelofSynchronousGenerator .................... 335
6.2.1 BasicMathematicalEquationsofSynchronous
Generator ..................................................... 336
6.2.2 MathematicalEquationsofSynchronousGenerator
UsingMachineParameters ................................... 343
6.2.3 SimplifiedMathematicalModelofSynchronous
Generator ..................................................... 351
6.2.4 Steady-StateEquationsandPhasorDiagram ................ 354
6.2.5 MathematicalEquationsConsideringEffect
ofSaturation .................................................. 357
6.2.6 RotorMotionEquationofSynchronousGenerator .......... 360
6.3 MathematicalModelofGeneratorExcitationSystems ............. 363
6.3.1 MathematicalModelofExciter .............................. 365
6.3.2 VoltageMeasurementandLoadCompensationUnit ....... 375
6.3.3 Limiters ....................................................... 376
6.3.4 MathematicalModelofPowerSystemStabilizer ........... 377
6.3.5 MathematicalModelofExcitationSystems ................. 377
6.4 MathematicalModelofPrimeMoverandGoverningSystem ...... 381
6.4.1 MathematicalModelofHydro-Turbine
andGoverningSystem ....................................... 382
6.4.2 MathematicalModelofSteamTurbine
andGoverningSystem ....................................... 389
6.5 MathematicalModelofLoad ........................................ 393
6.5.1 StaticLoadModel ............................................ 395
6.5.2 DynamicLoadModel ........................................ 397
7 PowerSystemTransientStabilityAnalysis ............................ 405
7.1 Introduction .......................................................... 405
7.2 NumericalMethodsforTransientStabilityAnalysis ................ 407
7.2.1 NumericalMethodsforOrdinaryDifferentialEquations ... 408
7.2.2 NumericalMethodsforDifferential-Algebraic
Equations ..................................................... 425
7.2.3 GeneralProcedureforTransientStabilityAnalysis ......... 427
7.3 NetworkMathematicalModelforTransientStabilityAnalysis ..... 430
7.3.1 TheRelationshipBetweenNetworkandDynamicDevices . 431
7.3.2 ModelingNetworkSwitchingandFaults .................... 439
