控制電纜燒損事故之分析改善與預(yù)防對策

Chapter4

ModelingofNonlinearLoad

OrganizedbyTaskForceonHarmonicsModeling&SimulationAdaptedandPresentedbyPauloFRibeiro

AMSCMay28-29,2021Contributors:S.Tsai,Y.Liu,andG.W.Chang1ChapteroutlineIntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter2IntroductionThepurposeofharmonicstudiesistoquantifythedistortioninvoltageand/orcurrentwaveformsatvariouslocationsinapowersystem.Oneimportantstepinharmonicstudiesistocharacterizeandtomodelharmonic-generatingsources.CausesofpowersystemharmonicsNonlinearvoltage-currentcharacteristicsNon-sinusoidalwindingdistributionPeriodicoraperiodicswitchingdevicesCombinationsofabove3Introduction(cont.)Inthefollowing,wewillpresenttheharmonicsforeachdevicesinthefollowingsequence:HarmoniccharacteristicsHarmonicmodelsandassumptionsDiscussionofeachmodel4ChapteroutlineIntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter5NonlinearMagneticCoreSourcesHarmonicscharacteristicsHarmonicsmodelfortransformersHarmonicsmodelforrotatingmachines6Harmonicscharacteristicsofiron-corereactorsandtransformersCausesofharmonicsgenerationSaturationeffectsOver-excitationtemporaryover-voltagecausedbyreactivepowerunbalanceunbalancedtransformerloadasymmetricsaturationcausedbylowfrequencymagnetizingcurrenttransformerenergizationSymmetriccoresaturationgeneratesoddharmonicsAsymmetriccoresaturationgeneratesbothoddandevenharmonicsTheoverallamountofharmonicsgenerateddependsonthesaturationlevelofthemagneticcorethestructureandconfigurationofthetransformer7HarmonicmodelsfortransformersHarmonicmodelsforatransformer:equivalentcircuitmodeldifferentialequationmodelduality-basedmodelGIC(geomagneticallyinducedcurrents)saturationmodel8Equivalentcircuitmodel(transformer)Intimedomain,asinglephasetransformercanberepresentedbyanequivalentcircuitreferringallimpedancestoonesideofthetransformerThecoresaturationismodeledusingapiecewiselinearapproximationofsaturationThismodelisincreasinglyavailableintimedomaincircuitsimulationpackages.9Differentialequationmodel(transformer)Thedifferentialequationsdescribetherelationshipsbetween

windingvoltageswindingcurrentswindingresistancewindingturnsmagneto-motiveforcesmutualfluxesleakagefluxesreluctancesSaturation,hysteresis,andeddycurrenteffectscanbewellmodeled.

Themodelsaresuitablefortransientstudies.Theymayalsobeusedtosimulatetheharmonicgenerationbehaviorofpowertransformers.

10Duality-basedmodel(transformer)Duality-basedmodelsarenecessarytorepresentmulti-leggedtransformersItsparametersmaybederivedfromexperimentdataandanonlinearinductancemaybeusedtomodelthecoresaturationDuality-basedmodelsaresuitableforsimulationofpowersystemlow-frequencytransients.TheycanalsobeusedtostudytheharmonicgenerationbehaviorsMagneticcircuitElectriccircuitMagnetomotiveForce(FMM)Ni

ElectromotiveForce(FEM)E

FluxCurrentI

ReluctanceResistanceR

PermeanceConductanceFluxdensityCurrentdensityMagnetizingforceH

PotentialdifferenceV

11GICsaturationmodel(transformer)GeomagneticallyinducedcurrentsGICbiascancauseheavyhalfcyclesaturationthefluxpathsinandbetweencore,tankandairgapsshouldbeaccountedAdetailedmodelbasedon3Dfiniteelementcalculationmaybenecessary.Simplifiedequivalentmagneticcircuitmodelofasingle-phaseshell-typetransformerisshown.Aniterativeprogramcanbeusedtosolvethecircuitrysothatnonlinearityofthecircuitrycomponentsisconsidered.12RotatingmachinesHarmonicmodelsforsynchronousmachineHarmonicmodelsforInductionmachine13SynchronousmachinesHarmonicsorigins:Non-sinusoidalfluxdistributionTheresultingvoltageharmonicsareoddandusuallyminimizedinthemachine’sdesignstageandcanbenegligible.FrequencyconversionprocessCausedunderunbalancedconditions

SaturationSaturationoccursinthestatorandrotorcore,andinthestatorandrotorteeth.Inlargegenerator,thiscanbeneglected.Harmonicmodelsunderbalancedcondition,asingle-phaseinductanceissufficientunderunbalancedconditions,aimpedancematrixisnecessary14BalancedharmonicanalysisForbalanced(singlephase)harmonicanalysis,asynchronousmachinewasoftenrepresentedbyasingleapproximationofinductanceh:harmonicorder:directsub-transientinductance:quadraturesub-transientinductanceAmorecomplexmodela:0.5-1.5(accountingforskineffectandeddycurrentlosses)RnegandXnegarethenegativesequenceresistanceandreactanceatfundamentalfrequency15UnbalancedharmonicanalysisThebalancedthree-phasecoupledmatrixmodelcanbeusedforunbalancednetworkanalysisZs=(Zo+2Zneg)/3Zm=(Zo

Zneg)/3

ZoandZnegarezeroandnegativesequenceimpedanceaththharmonicorderIfthesynchronousmachinestatorisnotpreciselybalanced,theselfand/ormutualimpedancewillbeunequal.16InductionmotorsHarmonicscanbegeneratedfromNon-sinusoidalstatorwindingdistributionCanbeminimizedduringthedesignstageTransientsHarmonicsareinducedduringcold-startorloadchangingTheabove-mentionedphenomenoncangenerallybeneglectedTheprimarycontributionofinductionmotorsistoactasimpedancestoharmonicexcitationThemotorcanbemodeledasimpedanceforbalancedsystems,orathree-phasecoupledmatrixforunbalancedsystems17HarmonicmodelsforinductionmotorBalancedConditionGeneralizedDoubleCageModelEquivalentTModelUnbalancedCondition18GeneralizedDoubleCageModelforInductionMotorStatorExcitationbranchAttheh-thharmonicorder,theequivalentcircuitcanbeobtainedbymultiplyinghwitheachofthereactance.mutualreactanceofthe2rotorcages2rotorcages

19EquivalentTmodelforInductionMotorsisthefullloadslipatfundamentalfrequency,andhistheharmonicorder‘-’istakenforpositivesequencemodels‘+’istakenfornegativesequencemodels.20UnbalancedmodelforInductionMotorThebalancedthree-phasecoupledmatrixmodelcanbeusedforunbalancednetworkanalysisZs=(Zo+2Zpos)/3

Zm=(Zo

Zpos)/3

ZoandZposarezeroandpositivesequenceimpedanceaththharmonicorderZ0canbedeterminedfrom21ChapteroutlineIntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter22ArcfurnaceharmonicsourcesTypes:ACfurnaceDCfurnaceDCarcfurnacearemostlydeterminedbyitsAC/DCconverterandthecharacteristicismorepredictable,hereweonlyfocusonACarcfurnaces23CharacteristicsofHarmonicsGeneratedbyArcFurnacesThenatureofthesteelmeltingprocessisuncontrollable,currentharmonicsgeneratedbyarcfurnacesareunpredictableandrandom.Currentchoppingandignitingineachhalfcycleofthesupplyvoltage,arcfurnacesgenerateawiderangeofharmonicfrequencies

24HarmonicsModelsforArcFurnaceNonlinearresistancemodelCurrentsourcemodelVoltagesourcemodelNonlineartimevaryingvoltagesourcemodelNonlineartimevaryingresistancemodelsFrequencydomainmodelsPowerbalancemodel25NonlinearresistancemodelsimplifiedtoR1isapositiveresistorR2isanegativeresistorACclamperisacurrent-controlledswitchItisaprimitivemodelanddoesnotconsiderthetime-varyingcharacteristicofarcfurnaces.modeledas26CurrentsourcemodelTypically,anEAFismodeledasacurrentsourceforharmonicstudies.ThesourcecurrentcanberepresentedbyitsFourierseriesanandbncanbeselectedasafunctionofmeasurementprobabilitydistributionsproportionofthereactivepowerfluctuationstotheactivepowerfluctuations.Thismodelcanbeusedtosizefiltercomponentsandevaluatethevoltagedistortionsresultingfromtheharmoniccurrentinjectedintothesystem.27VoltagesourcemodelThevoltagesourcemodelforarcfurnacesisaTheveninequivalentcircuit.Theequivalentimpedanceisthefurnaceloadimpedance(includingtheelectrodes)Thevoltagesourceismodeledindifferentways:formitbymajorharmoniccomponentsthatareknownempiricallyaccountforstochasticcharacteristicsofthearcfurnaceandmodelthevoltagesourceassquarewaveswithmodulatedamplitude.Anewvalueforthevoltageamplitudeisgeneratedaftereveryzero-crossingsofthearccurrentwhenthearcreignites28NonlineartimevaryingvoltagesourcemodelThismodelisactuallyavoltagesourcemodelThearcvoltageisdefinedasafunctionofthearclengthVao:arcvoltagecorrespondingtothereferencearclengthlo,k(t):arclengthtimevariationsThetimevariationofthearclengthismodeledwithdeterministicorstochasticlaws.Deterministic:Stochastic:29NonlineartimevaryingresistancemodelsDuringnormaloperation,thearcresistancecanbemodeledtofollowanapproximateGaussiandistribution

isthevariancewhichisdeterminedbyshort-termperceptibilityflickerindexPstAnothertimevaryingresistancemodel:R1:arcfurnacepositiveresistanceandR2negativeresistanceP:short-termpowerconsumedbythearcfurnaceVigandVexarearcignitionandextinctionvoltages30Powerbalancemodelr

isthearcradiusexponentnisselectedaccordingtothearccoolingenvironment,n=0,1,or2recommendedvaluesforexponentmare0,1and2K1,K2andK3areconstants31ChapteroutlineIntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter32Three-phaselinecommutedconvertersLine-commutatedconverterismostlyusualoperatedasasix-pulseconverterorconfiguredinparallelarrangementsforhigh-pulseoperationsTypicalapplicationsofconverterscanbefoundinACmotordrive,DCmotordriveandHVDClink33HarmonicsCharacteristicsUnderbalancedconditionwithconstantoutputcurrentandassumingzerofiringangleandnocommutationoverlap,phaseacurrentis

h=1,5,7,11,13,...Characteristicharmonicsgeneratedbyconvertersofanypulsenumberareintheorderofn=1,2,···andpisthepulsenumberoftheconverterFornon-zerofiringangleandnon-zerocommutationoverlap,rmsvalueofeachcharacteristicharmoniccurrentcanbedeterminedbyF(

,

)isanoverlapfunction34HarmonicModelsfortheThree-PhaseLine-CommutatedConverterHarmonicmodelscanbecategorizedasfrequency-domainbasedmodelscurrentsourcemodeltransferfunctionmodelNorton-equivalentcircuitmodelharmonic-domainmodelthree-pulsemodeltime-domainbasedmodelsmodelsbydifferentialequationsstate-spacemodel35CurrentsourcemodelThemostcommonlyusedmodelforconverteristotreatitasknownsourcesofharmoniccurrentswithorwithoutphaseangleinformationMagnitudesofcurrentharmonicsinjectedintoabusaredeterminedfromthetypicalmeasuredspectrumandratedloadcurrentfortheharmonicsource(Irated)Harmonicphaseanglesneedtobeincludedwhenmultiplesourcesareconsideredsimultaneouslyfortakingtheharmoniccancellationeffectintoaccount.

h,andaconventionalloadflowsolutionisneededforprovidingthefundamentalfrequencyphaseangle,

136TransferFunctionModelThesimplifiedschematiccircuitcanbeusedtodescribethetransferfunctionmodelofaconverterG:theidealtransferfunctionwithoutconsideringfiringanglevariationandcommutationoverlapG

,dcandG

,ac,relatethedcandacsidesoftheconverterTransferfunctionscanincludethedeviationtermsofthefiringangleandcommutationoverlapTheeffectsofconverterinputvoltagedistortionorunbalanceandharmoniccontentsintheoutputdccurrentcanbemodeledaswell37Norton-EquivalentCircuitModelThenonlinearrelationshipbetweenconverterinputcurrentsanditsterminalvoltagesisI&VareharmonicvectorsIftheharmoniccontentsaresmall,onemaylinearizethedynamicrelationsaboutthebaseoperatingpointandobtain:I=YJV+INYJistheNortonadmittancematrixrepresentingthelinearization.ItalsorepresentsanapproximationoftheconverterresponsetovariationsinitsterminalvoltageharmonicsorunbalanceIN=Ib-YJVb(Nortonequivalent)38Harmonic-DomainModelUndernormaloperation,theoverallstateoftheconverterisspecifiedbytheanglesofthestatetransitionTheseanglesaretheswitchinginstantscorrespondingtothe6firinganglesandthe6endsofcommutationanglesTheconverterresponsetoanappliedterminalvoltageischaracterizedviaconvolutionsintheharmonicdomainTheoveralldcvoltageVk,p:12voltagesamples

p:squarepulsesamplingfunctionsH:thehighestharmonicorderunderconsiderationTheconverterinputcurrentsareobtainedinthesamemannerusingthesamesamplingfunctions.39ChapteroutlineIntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter40HarmonicscharacteristicsofTCRHarmoniccurrentsaregeneratedforanyconductionintervalswithinthetwofiringanglesWiththeidealsupplyvoltage,thegeneratedrmsharmoniccurrents

h=3,5,7,···,

istheconductionangle,andLRistheinductanceofthereactor

41HarmonicscharacteristicsofTCR(cont.)Threesingle-phaseTCRsareusuallyindeltaconnection,thetriplencurrentscirculatewithinthedeltacircuitanddonotenterthepowersystemthatsuppliestheTCRs.Whenthesingle-phaseTCRissuppliedbyanon-sinusoidalinputvoltagethecurrentthroughthecompensatorisprovedtobethediscontinuouscurrent42HarmonicmodelsforTCRHarmonicmodelsforTCRcanbecategorizedasfrequency-domainbasedmodelscurrentsourcemodeltransferfunctionmodelNorton-equivalentcircuitmodeltime-domainbasedmodelsmodelsbydifferentialequationsstate-spacemodel43CurrentSour