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Voltage Stability in Electrical Power Systems 1st Edition Farid Karbalaei

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VoltageStabilityin

ElectricalPowerSystems

JónAtliBenediktsson

AnjanBose

AdamDrobot

Peter(Yong)Lian

IEEEPress

445HoesLane

Piscataway,NJ08854

IEEEPressEditorialBoard

SarahSpurgeon, EditorinChief

AndreasMolisch

SaeidNahavandi

JeffreyReed

ThomasRobertazzi

DiomidisSpinellis

AhmetMuratTekalp

VoltageStabilityinElectricalPowerSystems

Concepts,Assessment,andMethodsforImprovement

FaridKarbalaei

ShahidRajaeeTeacherTrainingUniversity Tehran,Iran

ShahriarAbbasi

TechnicalandVocationalUniversityofIran Kermanshah,Iran

HamidRezaShabani

AalborgUniversity Esbjerg,Denmark

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LibraryofCongressCataloging-in-PublicationData:

Names:Karbalaei,Farid,author.|Abbasi,Shahriar(AssistantProfessor), author.|Shabani,HamidReza,author.

Title:Voltagestabilityinelectricalpowersystems:concepts, assessment,andmethodsforimprovement/FaridKarbalaei,Shahriar Abbasi,andHamidRezaShabani.

Subjects:LCSH:Electricpowersystemstability.|Electricpower systems–Control.

Classification:LCCTK1010.K3672022(print)|LCCTK1010(ebook)|DDC 621.319–dc23/eng/20220919

LCrecordavailableathttps://lccn.loc.gov/2022041340

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CoverDesign:Wiley

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Setin9.5/12.5ptSTIXTwoTextbyStraive,Pondicherry,India

Toourfamilies

Contents

AuthorBiographies xiii

Preface xiv

PartIConceptofVoltageStability,EffectiveFactorsandDevices, andSuitableSystemModeling 1

1HowDoesVoltageInstabilityOccur? 3

1.1Introduction 3

1.2Long-TermVoltageInstability 5

1.2.1ASimpleSystem 5

1.2.2VoltageCalculation 6

1.2.3IllustrationofVoltageCollapse 7

1.2.4TheReasonofVoltageCollapseOccurrence 8

1.2.5TheImportanceofTimelyEmergencyMeasures 11

1.3Short-TermVoltageInstability 13

1.3.1TheProcessofInductionMotorsStalling 14

1.3.2DynamicAnalysis 15

1.3.3StaticAnalysis 17

1.3.4TheRelationshipBetweenShort-TermVoltageInstabilityand LoadabilityLimit 19

1.4Summary 20 References 20

2LoadsandLoadTapChanger(LTC)TransformerModeling 21

2.1Introduction 21

2.2StaticLoadModels 24

2.2.1TheConstantPowerModel 24

2.2.2ThePolynomialandExponentialModels 25

2.3DynamicLoadModels 26

2.3.1ExponentialRecoveryModel 27

2.3.2InductionMotorModel 31

2.4TheLTCTransformers 35

2.4.1TheLTCPerformance 35

2.4.2TheLTCModeling 36

2.4.3TheLTCTransformerModel 38

2.5Summary 44 References 44

3GeneratorModeling 47

3.1Introduction 47

3.2SynchronousGeneratorModeling 47

3.2.1SynchronousMachineStructure 48

3.2.2DynamicEquations 48

3.2.3VoltageandCurrentPhasors 52

3.2.4Steady-StateEquations 54

3.2.5SimplificationofSynchronousMachineEquations 56

3.2.6SaturationModeling 57

3.2.7SynchronousGeneratorCapabilityCurve 59

3.2.8ExcitationSystemModeling 62

3.2.9GovernorModeling 63

3.2.10OverexcitationLimiter(OXL)Modeling 65

3.3WindPowerPlants 66

3.3.1Fixed-SpeedInductionGenerator(FSIG)-basedWindTurbine 69

3.3.1.1PhysicalDescription 69

3.3.1.2InductionMachineSteady-StateModel 69

3.3.1.3InductionGeneratorDynamicModel 72

3.3.2DoublyFedInductionGenerator(DFIG)-basedWindTurbine 75

3.3.2.1PhysicalDescription 75

3.3.2.2DFIGSteady-StateCharacteristic 76

3.3.2.3OptimumWindPowerExtraction[5] 77

3.3.2.4TorqueControl 78

3.3.2.5VoltageControl 80

3.4Summary 81 References 82

4ImpactofDistributedGenerationandTransmission–Distribution InteractionsonVoltageStability 83

4.1Introduction 83

4.2InteractionsofTransmissionandDistributionNetworks 83

4.2.1TheStudiedSystem 83

4.2.2StableCase(Case1) 85

4.2.3InstabilityDuetotheInabilityofTransmissionTransformer’sLTCto RegulateVoltage(Case2) 86

4.2.4InstabilityDuetotheInabilityofDistributionTransformer’sLTCto RegulateVoltage(Case3) 87

4.3ImpactofDistributionGeneration(DG)Units 89

4.3.1ConnectingDGUnitstoMVDistributionNetworks 89

4.3.2ConnectingDGUnitstoHVDistributionNetworks 90

4.4Summary 93 References 95

PartIIVoltageStabilityAssessmentMethods 97

5TheContinuationPowerFlow(CPF)Methods 99

5.1Introduction 99

5.2TheCPFElements 100

5.3Predictors 101

5.3.1LinearPredictors 101

5.3.1.1TangentMethod 101

5.3.1.2SecantMethod 103

5.3.2NonlinearPredictors 104

5.4Parameterization 104

5.4.1LocalParameterization 105

5.4.2ArclengthParameterization 105

5.4.3LocalGeometricParameterization 106

5.4.4AlternativeParameterization 108

5.5Correctors 108

5.6DeterminingthePredictionStepSize 109

5.7ComparisonofPredictors 111

5.8SimulationofLocalGeometricParameterizationMethod 114

5.9SomeReal-worldApplicationsofCPF 115

5.10Summary 116 References 117

6PV-CurveFitting 119

6.1Introduction 119

6.2CurveFittingUsingThreePowerFlowSolutions 121

6.3CurveFittingUsingTwoPowerFlowSolutions 122

6.4CurveFittingUsingOnePowerFlowSolution 124

x Contents

6.5ComparisonofDifferentPV-CurveFittingMethods 128

6.6Summary 128

References 130

7Measurement-BasedIndices 131

7.1Introduction 131

7.2TheveninEquivalent-BasedIndex 132

7.2.1Background 132

7.2.2RecursiveLeastSquare(RLS)Algorithm 133

7.2.3Calculationof XTh Assuming ETh asaFreeVariable 135

7.2.4ReductionofParameterEstimationErrors 138

7.2.5Simulations 142

7.3IndicesBasedonReceivedPowerVariations 146

7.4EarlyDetectionofVoltageInstability 149

7.5IndicesforAssessmentFault-InducedDelayedVoltageRecovery (FIDVR)Phenomenon 152

7.5.1ConceptofFIDVR 152

7.5.2FIDVRAssessmentIndices 154

7.6SomeReal-WorldApplicationsofMeasurement-basedIndices 157

7.7Summary 157

References 158

8Model-BasedIndices 161

8.1Introduction 161

8.2JacobianMatrix-BasedIndices 161

8.2.1Background 161

8.2.2SingularityofJacobianMatrixattheLoadabilityLimit 163

8.2.3SingularValuesandVectors 164

8.2.4Simulation 165

8.2.5ReducedJacobianMatrix 168

8.2.6EigenvaluesandEigenvectors 169

8.2.7TestFunction 172

8.2.8TheMaximumSingularValueoftheInverseoftheJacobianMatrix 174

8.3IndicesBasedonAdmittanceMatrixandPowerBalanceEquations 176

8.3.1LineStabilityIndices 178

8.3.1.1StabilityIndex Lmn 178

8.3.1.2FastVoltageStabilityIndex(FVSI) 179

8.3.1.3StabilityIndexLQP 179

8.3.1.4LineCollapseProximityIndex(LCPI) 179

8.3.1.5IntegralTransmissionLineTransferIndex(ITLTI) 180

8.3.2BusIndices 180

8.3.2.1StabilityIndex L180

8.3.2.2ImprovedVoltageStabilityIndex(IVSI) 182

8.4IndicesBasedonLoadBusesVoltageandGeneratorsReactive Power 184

8.4.1ReactivePowerPerformanceIndex(PIV) 184

8.4.2ReactivePowerLossIndex(RPLI) 186

8.5IndicesDefinedintheDistributionSystem 189

8.5.1DistributionSystemEquivalent 190

8.5.2Indices 191

8.5.3Simulations 194

8.6Summary 197

References 197

9MachineLearning-BasedAssessmentMethods 199

9.1Introduction 199

9.2VoltageStabilityDetectionBasedonPatternRecognitionMethodsand IntelligentSystems 199

9.2.1TheIntelligentSystemsTrainingApproaches 200

9.2.2TheIntelligentSystemsTypes 201

9.2.2.1ArtificialNeuralNetworks(ANNs) 201

9.2.2.2DecisionTrees(DTs) 205

9.2.2.3Support-VectorMachines(SVMs) 207

9.3Summary 214

References 214

PartIIIMethodsofPreventingVoltageInstability 219

10PreventiveControlofVoltageInstability 221

10.1Introduction 221

10.2DeterminationofLM 222

10.2.1StaticAnalysis 222

10.2.2DynamicAnalysis 224

10.3DeterminationoftheOptimalValueofControlActions 227

10.4ComputationofSensitivities 229

10.4.1ComputationofSensitivitiesBasedontheComputationof MLP 229

10.4.2ComputationofSensitivitiesWithouttheComputationofMLP 230

10.5DeterminationoftheMostEffectiveActions 233

10.6Summary 235

References 235

11EmergencyControlofVoltageInstability 237

11.1Introduction 237

11.2LoadShedding 238

11.2.1UVLSagainstLong-termVoltageInstability 239

11.2.1.1CentralizedRule-basedController 239

11.2.1.2DistributedRule-basedController 241

11.2.1.3Two-levelRule-basedController 243

11.2.2UVLSAgainstBothShort-andLong-termVoltageInstability 243

11.2.3LoadSheddingBasedonIncrementalValueofGeneratorReactive Power 244

11.2.4AdaptiveLoadSheddingBasedonEarlyDetectionofVoltage Instability 245

11.3DecentralizedVoltageControl 247

11.4TheuseofActiveDistributionNetworksinEmergencyVoltage Control 250

11.5CoordinatedVoltageControl 255

11.5.1ModelPredictiveControl 255

11.5.2PredictionofTrajectoryofVariables 256

11.5.2.1SimplificationsRequiredforEmergencyVoltageControlinthe TransmissionNetwork 256

11.5.2.2EulerStatePrediction(ESP) 259

11.5.2.3Two-PointPredictionMethod 259

11.5.2.4PredictionUsingTrajectorySensitivity 260

11.5.3CostFunction 263

11.6Summary 264 References 264

Index 267

AuthorBiographies

FaridKarbalaei receivedBScdegreeinpowerengineeringfromK.N.ToosiUniversityofTechnology,Tehran,Iran,in1997,andMScandPhDdegreesinpower engineeringfromIranUniversityofScienceandTechnology,Tehran,in2000and 2009,respectively.Currently,heisanassociateprofessorinShahidRajaeeTeacher TrainingUniversity.Hisresearchinterestsarepowersystemdynamicsandcontrol,voltagestabilityandcollapse,reactivepowercontrol,windpowergeneration, andoptimizationmethods.

PostalAddress:FacultyofElectricalEngineering,ShahidRajaeeTeacher TrainingUniversity,Lavizan,Tehran,Iran.Phone:+989124445325,Email: f_karbalaei@sru.ac.ir(Correspondingauthor)

ShahriarAbbasi receivedBScandMScdegreesinpowerengineeringfromShahidRajaeeTeacherTrainingUniversityTehran,Iran,in2008and2011,respectively.HereceivedPhDdegreesinpowerengineeringfromRaziUniversity, Kermanshah,Iran,2018.Currently,heisanassistantprofessorinTechnical andVocationalUniversityofIran,KermanshahBranch.Hisresearchinterests arepowersystemplanning,uncertaintymodeling,voltagestabilityandcollapse, windpowergeneration,andoptimizationmethods.

PostalAddress:TechnicalandVocationalUniversityofIran,KermanshahBranch, Kermanshah,Iran.Phone:+989169848928,Email:shahriarabasi@gmail.com

HamidRezaShabani receivedtheMSdegreeinpowerelectricalengineering fromShahidRajaeeTeacherTrainingUniversity(SRTTU),Tehran,Iran,in 2014.Also,hereceivedthePhDdegreefromIranUniversityofScienceandTechnology(IUST),Tehran,Iran,in2021.HisthesistitleinthePhDprogramwas “evaluationoflarge-disturbancerotorangleinstabilityinthemodernpowersystems, withhigh-Penetrationofwindpowergeneration.” Hecurrentlyworksasa postdoctoralresearcheratAalborgUniversity(AAUEnergy)intheEsbjergEnergy Section.Hismainresearchinterestsincludepowersystemstability,powersystem dynamicandcontrol,andrenewableenergies.

PostalAddress:TheFacultyofEngineeringandScience(AAUEnergy), AalborgUniversity,6700Esbjerg,Denmark.Phone:+4552780690,Email: hmdrzshabani94@gmail.com

Preface

Voltageinstabilityhasbeenconsideredabout60yearsagoandisstillamajorcause ofblackoutsinelectricalpowersystems.Sofar,extensivestudieshavebeenconductedonthistopicandtheresultofwhichispublicationofthousandsofarticles andafewnumberofbooks.Thearticlescoverawiderangeofsubjectsrelatedto voltagestability,includingpropersystemmodelingforvoltagestabilitystudies, onlineandofflinevoltagestabilityassessmentmethods,andmethodstoprevent voltageinstability,whichincludetwosetsofpreventiveandemergencymethods.

Beingfamiliarwiththeallabove-mentionedsubjectsisnecessaryforpower engineersbecauseeffectivepreventionofvoltageinstabilitynecessitatestimely detectionofit.Timelydetectionalsorequiresknowledgeofthemechanismofvoltageinstabilityandpropermodelingofthesystem.Sincevoltageinstabilityisa localphenomenon,contrarytorotorinstability,manymethodsofdetectingand preventingvoltageinstabilityareperformedlocally.Therefore,inadditionto theengineersworkinginthesystemcontrolcenter,allpowerengineersinlocal controlcentersshouldbefullyfamiliarwiththisfield.

Goodunderstandingofissuesrelatedtovoltagestabilityrequiresreadinghundredsofarticles.Duetothefactthatarticlesdonothaveeducationalpurpose,itis difficulttofullyunderstand,summarize,andrelatethemtogether.Therefore,a bookthatpresentsalltheabovesubjectsinacomplete,arrangedandcomprehensiblewayisneeded;sothatitfirstexplainsthenecessaryfundamentals(whichis notdoneinarticles)thenpresentstherelatedsubjectsinanappropriateclassificationandsequence.Theaimofthisbookistopresentallthevoltagestability subjectssothatreaderswiththelevelofbachelorinformationcanuseitwell. Ofcourse,thestate-of-theartonvoltagestabilityisintroducedinittobeuseful foruniversityprofessors,masteranddoctoralstudents.

Thisbookconsistedofthreeparts.Thecontentsofthesepartscanbesummarizedasfollows:

ThePartI: ConceptofVoltageStability,EffectiveFactorsandDevices,andSuitable SystemModeling consistedoffourchapters.InChapter1,theconceptofvoltage instabilityanditstypesarefirstdescribed.Thenhowlong-terminstabilityand voltagedropoccurduetotheactivitiesoftapchangersandthermostaticloads

areillustrated.Theoccurrenceofshort-termvoltageinstabilityduetothepresence ofinductionmotorsisalsodescribed.Inthischapter,bysimulatingonasimple system,theimportanceofloadabilitylimitincreaseinmaintainingvoltagestabilityisshown.Theconceptofexitingfromattractionregionandtheimportanceof timelyperformingofemergencymeasuresarealsoexplainedbysimulation.The purposeofthischapteristofamiliarizethereaderquicklyandingeneral(notin fulldetail)withtheconceptandcausesofvoltageinstabilityaswellashowtopreventitfromoccurring.

InChapter2,differentdynamicandstaticloadmodelsusedinreferencestoanalyzevoltagestabilityareintroduced.Thesemodelsrepresentthebehaviorofintegratedloadsseenfromdifferentbussofthepowersystem.Inshort-termvoltage stabilityanalysis,thedynamicbehaviorofloadissimulatedasthedynamicmodel ofinductionmotor.Hence,apartofthethirdchapterisdevotedtopresentingthe algebraicanddifferentialrelationsofinductionmotor.Anotherpartofthischapter introducesthetypesoftapchangersandmodelingtransformerswithvariabletap. Inreferences,therearetwomodelstorepresentvariabletaptransformers.Thedifferencebetweenthesetwomodelsisinthesidethattheequivalentimpedanceof transformerisseenfromit.Thesimulationsverifythatthesetwomodelsleadto differentvaluesforthesystemloadabilitylimit.Giventheimportanceofdeterminingthecorrect(real)loadabilitylimitinvoltagestabilitystudies,selectionofthe propermodelisveryimportant.Thisisdiscussedinthischapter.

Chapter3dealswithmodelingofsynchronousgeneratorandtwotypesofdistributedgenerationsources(FSIG-andDFIG-basedwindturbines).Themodeling degreeshouldbechosenaccordingtotheintendedtypeofstudy.Inthischapter, suitablemodelsforstudyingeachtypeofvoltageinstability(longtermandshort term)arepresented.

Chapter4explainstheimportanceofconcurrentmodelingofdistributionand transmissionnetworksinassessingvoltagestability.Thisisshownthatsometimes, separatemodelingofdistributionandtransmissionnetworkscausesasignificant errorindeterminingthevoltagestabilitylimit.Alsointhischapter,theeffectofthe presenceofdistributedgeneration(DG)sourcesonvoltagestabilityisinvestigated. Thesesources,whicharemainlyconnectedtodistributionnetworks,uptothe conditionhavedifferenteffectsonvoltagestability.

ThePartII: VoltageStabilityAssessmentMethods,includestheChapters5–9. ThisPartofthebookisdedicatedtovoltagestabilityassessmentmethods.Voltage stabilityassessmentisperformedforseveralpurposes.Oneofthemisdeterminationofthevoltagestabilitymargin,whichiscalculatedforboththecurrent(nocontingency)systemandprobablecontingencies.Whatisimportantincalculating thevoltagestabilitymarginisspeedandaccuracyofthecalculation.Sincethe determinationofthestabilitymarginmustberepeatedeveryfewminutes,itscalculationforalargenumberofprobablecontingenciesispossibleonlyifthecalculationtimebeveryshortwhilemaintainingtherequiredaccuracy.Forthis,the

methodsofcontinuationpowerflow(CPF)andPV-curvefittingarepresented, whicharediscussedinChapters5and6.

Voltagestabilitymarginshowsthelevelofsystemstabilityinthefaceofvarious contingencies.Itdoesnotdirectlyprovideinformationaboutthevulnerablepoints ofsystemaswellastheimportantelementsininstabilityoccurrence.Thisinformation,whichhelpsoperatorstodecideabouttakingthevoltageinstabilitypreventivemethods,isobtainedbyvoltagestabilityindices.Anumberofvoltage stabilityindicesarecalculatedbasedonthesystemmodel.Theseindiceshelpa lotindefiningandrankingthecriticalcontingencies,aswellasindetermining thenecessaryactionsaftereachcontingencyoccurrence.Anothersetofindicesuses onlyvariablesmeasuredatdifferentpointsofsystemanddoesnotrequirethesystemmodel.Theseindicescanbeusedtoquicklyidentifythecurrentstatusofsystem andearlydetectionofvoltageinstability.Also,whenvoltagestabilityassessment requiresdynamicanalysisandtimesimulation,thevoltagestabilityindicescan beusedtoreducetherequiredsimulationtime.Theseindicesareallintroduced inChapters7and8,andtheadvantagesandapplicationsofeachonearestated.

InChapter9,themachinelearning-basedmethodstoassessandmonitorvoltage stabilityofpowersystemareintroduced.Generaltopologiesofthesemethodsand theircapabilitieswereintroduced.Thesemethodsarecategorizedinfivemethods. Foreachmethod,atableincludinginput(s),output(s),usedtechnique,andcase studyispresented.

InthePartIIIofthisbook: MethodsofPreventingVoltageInstability,themethods topreventvoltageinstabilityarediscussed.Allactionsusedtopreventvoltagecollapsearedividedintotwocategories:preventiveandemergency.Thepurposeof preventiveactionsistoincreasethevoltagestabilitymarginofthepowersystem. Increasingthevoltagestabilitymarginisconsideredforboththecurrent(no-contingency)systemandprobablecontingencies.Therefore,theseactionsareapplied whenthesystemisstable,butthereisasmalldistancebetweenthecurrentoperatingpointandthevoltagestabilitylimit.Thepurposeofdeterminingpreventive actionsistoimprovesystemstabilitywiththeleastmeasures(especiallywiththe minimumloadshedding).

Theemergencyactionsareperformedwhenthesystembecomesunstabledueto oneormorecontingencies,andiftheseactionsarenotapplied,avoltagecollapse willoccurinafewmomentsorminutes.Indeterminingtheemergencyactions, thespeedofcalculationsisveryimportantbecausethelatertheseactionsare applied,voltagestabilitymaintenanceispossiblewithmoreactions.

Someofthevoltagestabilitystudiesaredevotedtomethodsfordetermining preventiveandemergencyactions,whicharediscussedinChapters10and11.

FaridKarbalaei,ShahriarAbbasiand HamidRezaShabani

HowDoesVoltageInstabilityOccur?

1.1Introduction

Thephenomenonofvoltageinstabilityisoneofthemajorproblemsoftoday’s powersystems.AccordingtotheInstituteofElectricalandElectronicsEngineers (IEEE)/TheInternationalCouncilonLargeElectricSystems(CIGRE)definition, voltagestabilityistheabilityofapowersysteminmaintaininganacceptable steady-statevoltageatallbuseswhensubjectedtoacontingency.Theconsequence ofvoltageinstabilityisvoltagecollapse.Unlikerotorangleinstability,whichis morerelatedtogeneratoroperation,voltageinstabilitydependsontheamount andcharacteristicofloads.Forthis,voltageinstabilityisalsocalledloadinstability [1].Voltageinstabilityisdividedintotwocategories;longtermandshortterm.In thelong-termtype,voltagecollapseoccursduringaprocessofafewtensofsecondsorafewminutes,butintheshort-termvoltageinstability,thevoltagecollapseoccursrapidlyandwithinafewseconds.

Ingeneral,thereasonofvoltageinstabilityisthepresenceofdeviceswhose powerconsumptionisnotmuchdependentonvoltage.Voltagedropinitially reducestheinputpowerofthesedevices,butafterafewmomentsorminutes, thereactionofthesedevicescausestheirreceivingpowertoincreasetoavalue closetothevaluebeforethevoltagedrop.Powerrecoverymaybedoneforactive poweronlyorforbothactiveandreactivepowers.Figure1.1 conceptuallyshows theprocessofrecoveringtheactivepowerofadevicewhenvoltagedrops.Itis observedthatatfirstthepowerisreducedbutinthesteadystateitsvaluebecome closetotheinitialvalue.Therefore,itissaidthatthisdevicehasthecharacteristic ofconstantsteady-statepower.Inthisfigure,itisassumedthataconstantreduced voltageisappliedtothedevice.Also,thefluctuationsofpowerareignoredwhenit isrecovering.Inpractice,thevoltageofadevicedecreasesasitspowerconsumptionincreases.Insteady-stateconditions,thisvoltagedropissmall,butwhenvoltageinstabilityoccurs,powerrecoverycausesaseverevoltagedrop.Anecessary

VoltageStabilityinElectricalPowerSystems:Concepts,Assessment,andMethodsforImprovement, FirstEdition.FaridKarbalaei,ShahriarAbbasi,andHamidRezaShabani.

(notsufficient)conditionformaintainingvoltagestabilityisthatthetransferofthe requiredpowertoconstantpowerconsumersispossible.Otherwise,long-orshorttermvoltageinstabilitywilloccur,dependingontothepowerrecoverytime constant.

Oneofthemostimportantdevicesthatcreateconstantpowercharacteristicis loadtapchanger(LTC)transformer.Inmostcases,thevariabletapislocatedon thehighvoltage(HV)sideofthetransformer.ThisisduetolesscurrentintheHV side,whichmakesiteasiertochangethetap.Anotherreasonisthehighnumberof windingturnsintheHVside,whichmakesthevoltageregulationmoreaccurate [1].Thetapcontrolsystemusuallycontrolsthevoltageoflowvoltage(LV)side, whichhaslowershort-circuitlevel.ThesedevicesfixthevoltageofLVside,independentoftheHVsidevoltage.Bykeepingthisvoltagefixed,thepower

Figure1.1 Powerrecoveryaftervoltagedrop.

1.2Long-TermVoltageInstability 5

consumptionattheLVsidealsoremainsconstant.Hence,assumingthetransformerlossesremainconstant,thepowerreceivedfromtheHVsideisindependentofitsvoltage.Therefore,theloadseenfromtheHVsideofthetransformerhas thecharacteristicofconstantsteady-statepower.Ifthesystemisstable,voltage andpowerrecoverywillbedonebytapchangerinafewtensofseconds.Inadditiontotapchanger,thermostaticallycontrolledheatloads(TCLs)canalsocreate constantpowercharacteristic.Whenvoltagedrops,theseloadswillremaininthe circuitlongerbecausetheymustproducethenecessaryheatenergy.Therefore,as thevoltagedecreases,theimpedanceofasetoftheseloadsdecreases.Asaresult, whenthevoltagedrops,thepowerconsumptionbytheseloadsdoesnotchange muchinthesteadystate.Itisclearthatintheearlymomentsofvoltagedrop,TCLs haveanimpedancecharacteristicandaconstantpowercharacteristiciscreated duringaprocessofseveralminutes.

Themainreasonofshort-termvoltageinstabilityisthepresenceofinduction motorloads.Speedreductionandstallingofthesemotorsleadtosuddenincrease intheirreactivepowerconsumptionbythemandvoltagecollapse.Afteravoltage reduction,initially,thepowerconsumptionofaninductionmotordecreases,but duetodecreasingspeedandincreasingslip,theactivepowerconsumedbythe motorgraduallywillincrease.Theamountofactivepowerincrementdepends onthetypeofmechanicalloadsuppliedbythemotor.Thepowerrecoveryin inductionmotorsisdonequicklyinafewseconds.Motorstallinghappenswhen themotorisunabletosupplyitsconnectedmechanicalload.Inthischapter,using simulationonsimplenetworks,theprocedureofvoltageinstabilityoccurrence duetotheabove-mentionedfactorsisillustrated.

1.2Long-TermVoltageInstability

Inthissection,usinganexample,theprocedureofvoltageinstabilityduetooperationofLTCsisshown.Also,thereasonofthisoccurrenceisillustrated.

1.2.1ASimpleSystem

Tosimulatetheoccurrenceofvoltageinstability,thesimplesystemofFigure1.2 is used.Inthissystem,ageneratorsuppliesastaticloadwithvoltage-dependent characteristicthroughtwoparallellinesandanLTCtransformer.Fortheload, theexponentialmodelaccordingtoEqs.(1.1)and(1.2)isused.Intheseequations, P0 and Q0 are,respectively,thedemandedactiveandreactivepowersofthisloadat thevoltage1pu.Acompletediscussionaboutloadmodelingispresentedin Chapter2.

Figure1.2 Single-linediagramofasimplesystemwithLTCtransformer.

Thetransformerismodeledasaseriesleakageimpedanceandanidealtransformer.Nexttotheload,thereisacompensatingcapacitorwiththeadmittance 0.45pu.Inthisexample,thegeneratorismodeledasaconstantvoltagesource, andthevoltage V1 isassumedtobe1pu.Therefore,thedynamicbehaviorofthis systemisonlyduetotheoperationofLTC.ItisassumedthattheLTCisinstalledat theHVsideanditsvalue(a)canvaryfrom0.85to1.15puinthestepsof0.005pu (0.5%).ThedutyoftheLTCismaintaining V3 between0.99and1.01pu.Thisrange iscalledtheLTCdeadband.ThedeadbandisalwaysconsideredlargerthanLTC steps,usuallytwiceofLTCsteps.Otherwise,theLTCchangeswillnotconvergeto asteady-statevalue.

1.2.2VoltageCalculation

Tocalculate V3 fordifferenttapvalues,theequivalentcircuit π isusedtomodel LTCtransformer[2].Bydoingso,theequivalentsingle-phasecircuitofthissystem isasshowninFigure1.3.Bychangingthetapvalue,theimpedanceoftheequivalentcircuitbranchesofthetransformerchanges.Inthissystem,thebus1isslack

Figure1.3 Single-phaseequivalentofthesystemofFigure1.2.

1.2Long-TermVoltageInstability 7

andthebuses2and3arethePQtype.Tocalculatethevoltages,theloadflow equationsarewrittenandsolvedwithvoltage-dependentloadatbus3.Assuming P0 and Q0 are,respectively,equalto0.5and0.2pu,thesteady-statetapvalue becomes1.00andthemagnitudeofvoltage V3 isequalto0.992(avaluebetween 0.99and1.01pu).

Assumingtheoutageofoneoftheparallellines,thevoltageofbus2andconsequentlythevoltageofbus3decreases.Afterthisvoltagedrop,theLTCreduces thetapvalueinstalledattheHVsidetorecoverthevoltage.TheLTCandvoltage changesareshowninFigure1.4.Itcanbeseenthatbyafewchangesoftapvalue, thevoltageofbus3isrecoveredtothedesiredrange.

1.2.3IllustrationofVoltageCollapse

Nowwiththetwoparallellines,thevaluesof P0 and Q0 areincreasedto0.83and 0.33pu,respectively.Inthiscondition,insteadystate,thetapvalueandvoltageof bus3,respectively,convergeto1.05and0.991pu.Now,withthesevaluesfor P0 and Q0,andoutageofoneoftheparallellines,theLTCcontrolsystemwillstart reducingitstapvalueagaintorecoverthevoltageofbus3.Butinthiscase,as showninFigure1.5,voltagerecoveryisnotachieved.Atthefirstfewsteps,the voltageofbus3increasesbyanyreductioninthetap.Butaftertheseinitialsteps, anydecreaseinthetapleadstoavoltagedrop,whichmeansthatthevoltageinstabilityhasoccurred.

Voltage at bus 2

Voltage at bus 3

Tap level

Figure1.4 Voltagerecoveryafterlineoutage.

8 1HowDoesVoltageInstabilityOccur?

at bus 2

1.2.4TheReasonofVoltageCollapseOccurrence

Thereasonofimpossibilityofvoltagerecoveryisthatvoltagerecoverymeansthe recoveryofactiveandreactivepowerstovaluescloseto P0 and Q0 (0.83and0.33pu inthisexample).Therefore,voltagerecoveryisachievedonlywhenafteraline outage,thesystembeabletodelivertheseamountsofpowertothebus3.Tobetter understandthissubject,thetransformerimpedanceisshiftedtoitsprimarysideas showninFigure1.6.Bydoingso,thepowerdeliveredtotheprimarysideofthe idealtransformeristhesameasthepowerdeliveredtotheload.Now,duetothe tapoperation,theloadseenfromtheprimarysideoftheidealtransformerhasthe characteristicofconstantsteady-statepower.Because,regardlessoftheprimary sidevoltage,thesecondarysidevoltageandconsequentlyitspowerconsumption insteadystateisalmostconstant.(Ofcourse,thesteady-statevoltagemayvaryin thedeadbandrange,causingslightchangesinthesteady-statepowerconsumption.)Itisobviousthattheconstantpowercharacteristicwillbeobtainedduringa fewtensofsecondstoafewminutesprocess,dependingonthetimedelayofthe tapchanges.Inadditiontothesteady-statecharacteristic,therearealsosometransientcharacteristicsthatindicatethemomentaryrelationshipbetweenpowerand voltageintheprimarysideoftheidealtransformer.Thesecharacteristicsare accordingtoEqs.(1.3)and(1.4).

Figure1.5 Voltageinstabilityafterlineoutage.

1.2Long-TermVoltageInstability 9

Aloadthathasatransientcharacteristicinadditiontothesteady-statecharacteristiciscalledadynamicload.Intheabove-mentionedexample,theloadseen fromtheprimarysideofthetransformerisadynamicload.Thepowervariations ofdynamicloadsduetovoltagechange,inadditiontothesteady-stateterm,havea transientterm.Now,duetotheconstantsteady-statepowercharacteristic,astable operatingpointisachievedonlywhenitispossibletodeliverpowertothe demandedload.Forthisreason,inmanyreferences,themaximumloadability ofapowersystemisintroducedasthelong-termvoltagestabilitylimit.

Occurrenceofvoltageinstabilityimpliesthattheactiveandreactivepowersof 0.83and0.33puaredefinitelymorethanthemaximumpowersthatcanbetransferredtotheprimarysideoftheidealtransformer.Figure1.7 showsthevoltage variationsversustheactivepowervariationsreceivedattheprimarysideofthe idealtransformerbeforeandafterthelineoutage.Indrawingthesecurves,the variationofthetransformerseriesimpedanceduetothetapchangesisneglected. Theerrorofthisapproximationisverysmallsincethetransformerimpedanceis connectedinserieswiththelineimpedance.Thesecurves,whicharewidelyused inpowersystemvoltagestabilitystudies,arecalledpowervoltage(PV)curves[3]. Usually,thesecurvesaredrawnwithconstantpowerfactors.Thepowerfactorin Figure1.7ischosenbasedontheactiveandreactivepowersoftheloadandthe reactivepowerproducedbythecapacitoratvoltage1pu.Itcanbeseenthatthe maximumtransferableactivepowerafterlineoutageisequalto0.75pu,whichis lessthanthepowerrequiredtovoltagerecovery(0.83pu).InFigure1.7,alongwith

Figure1.6 Transformerimpedancetransferredtotheprimaryside.

decrement

Load transient characteristic with different tap levels

thePVcurves,thetransientcharacteristicsoftheactivepowerseenfromtheprimarysideoftheidealtransformerareplotted.Thesecharacteristicsareobtained fordifferenttapvalues.Theintersectionpointsofthetransientcharacteristicswith thePVcurvesaretheactivepowerandvoltagevaluesfordifferenttapvalues.As canbeseen,asthetapdecreases,theintersectionpointmovestowardthepointof maximumtransferablepower(thenoseofPVcurve).Untiltheintersectionpoint reachesthenoseofPVcurve,anytapreductionwillincreasethereceivedpowerby theload.Duetothevoltage-dependentcharacteristicoftheload,increasingthe receivedpowermeansincreasingthevoltage V3,althoughthevoltage V 3 decreases.AftercrossingthenoseofPVcurve,thepowerandvoltage V3 will reduceaftereachtapreduction.Therefore,thenoseofthePVcurveisalsocalled thecollapsepoint.

Figure1.8 showsthePVcurveswithdifferentpowerfactorsoftheload.Itis shownthatthemoreleadcharacteristicoftheload,i.e.themorereactivepower compensation,thehighermaximumloadability,andalsothehighercollapsepoint voltage.Ifthereactivepowercompensationlevelisnothigh,thevoltageofcollapsepointwillbelow(evenabout0.5pu).However,ifthereactivepowercompensationlevelishigh,thecollapsepointvoltagemayevenreachvalueshigher than0.9pu.Giventhattheoperatingvoltageofthesystemisusuallyintherange of0.95–1.05pu,ifthecompensationlevelislow,thentherewillbealargedistance betweentheoperatingrangeandthecollapsepoint.Undertheseconditions,the probabilityoflong-termvoltageinstabilityisverylow.Butifthelevelof

Post-fault PV curve

Figure1.7 ThePVcurvesseenfromtheprimarysideoftheidealtransformerofFigure1.6.

1.2Long-TermVoltageInstability 11

PF = 0.8 lead

PF = 0.9 lead

PF = 1.0

PF = 0.9 lag

PF = 0.8 lag

compensationishigh,theoperatingrangeisclosetothecollapsepoint.Hence,itis foundthatthelong-termvoltageinstabilityistheproblemofpowersystemsthat havebeenhighlycompensated.

1.2.5TheImportanceofTimelyEmergencyMeasures

TheFigure1.7 illustratesthattheinabilityofLTCinmaintainingvoltageappears whenitisnotpossibletotransfertheneededpowertoabuswhosevoltageiscontrolled.Therefore,asthiswillbeshowninfuturechapters,thebasisofallvoltage instabilitypreventingmethodsisincreasingthemaximumloadabilitylevelofthe powersystem.Thisincreaseisusuallydonebymeasuressuchasincreasingthevoltageofthegenerators,connectingthecapacitors,andremovingthereactors.If increasingtheloadabilitylevelbenotsufficient,thelastmeasureistoreducethe systemload.Theimportantpointisthattheabovemeasurescanbeeffectivewhen appliedinatimelymanner.Thelaterthemeasures,themoremeasuresmustbe takentomaintainvoltagestability.Toexplainthis,letusassumethatthereis anothercapacitorwithsusceptance0.20puinbus3thatconnectswhenneeded (Figure1.9).Byconnectingthiscapacitor,itispossibletomaintainthevoltagestabilityofthesystem.Because,asshowninFigure1.10,themaximumtransferable powerofthesystemwhenthecapacitorisconnectedandoneoftheparallellines isremovedbecomesmorethanthedemandedpower.Inthisfigure,similarto Figure1.7,thedashed-linecurvesaretheloadtransientcharacteristicsseenfrom

Figure1.8 ThePVcurveswithdifferentpowerfactorsofload.

Before line outage

After line outage

After line outage and capacitor connection

theprimarysideofthetransformer,whichhavechangedduetothetapchanges.To illustratetheimportanceoftakingtimelymeasures,twoscenariosareconsidered.In thefirstone,thecapacitorisconnectedwhentheintersectionoftheloadtransient characteristicsandthenewPVcurve(thePVcurveafterthecapacitorconnection)is atthepointM.Inthismoment,thereceivedpowerbyloadismorethan P0,which impliesthatthevoltageofbus3ishigherthan1pu.Thiscausesthetapvalueto increaseandtheloadcharacteristicstomovetowardthepointA,whichistheequilibriumpointaftercapacitorconnection.Inthisscenario,thecapacitorisconnected intimeandconsequentlythevoltagestabilityismaintained.

Inthesecondscenario,itisassumedthatthecapacitorisconnectedwitha longerdelaysothattheloadtransientcharacteristicscollidewiththenewPV curveatthepointN.Inthismoment,theloadactivepowerislessthan P0,which

Figure1.9 Capacitorconnectionasanemergencymeasure.

Figure1.10 Attractionregionconcept.