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1,Chapter4ModelingofNonlinearLoad,OrganizedbyTaskForceonHarmonicsModeling&SimulationAdaptedandPresentedbyPauloFRibeiroAMSCMay28-29,2008,Contributors:S.Tsai,Y.Liu,andG.W.Chang,2,Chapteroutline,IntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter,3,Introduction,Thepurposeofharmonicstudiesistoquantifythedistortioninvoltageand/orcurrentwaveformsatvariouslocationsinapowersystem.Oneimportantstepinharmonicstudiesistocharacterizeandtomodelharmonic-generatingsources.CausesofpowersystemharmonicsNonlinearvoltage-currentcharacteristicsNon-sinusoidalwindingdistributionPeriodicoraperiodicswitchingdevicesCombinationsofabove,4,Introduction(cont.),Inthefollowing,wewillpresenttheharmonicsforeachdevicesinthefollowingsequence:HarmoniccharacteristicsHarmonicmodelsandassumptionsDiscussionofeachmodel,5,Chapteroutline,IntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter,6,NonlinearMagneticCoreSources,HarmonicscharacteristicsHarmonicsmodelfortransformersHarmonicsmodelforrotatingmachines,7,Harmonicscharacteristicsofiron-corereactorsandtransformers,CausesofharmonicsgenerationSaturationeffectsOver-excitationtemporaryover-voltagecausedbyreactivepowerunbalanceunbalancedtransformerloadasymmetricsaturationcausedbylowfrequencymagnetizingcurrenttransformerenergizationSymmetriccoresaturationgeneratesoddharmonicsAsymmetriccoresaturationgeneratesbothoddandevenharmonicsTheoverallamountofharmonicsgenerateddependsonthesaturationlevelofthemagneticcorethestructureandconfigurationofthetransformer,8,Harmonicmodelsfortransformers,Harmonicmodelsforatransformer:equivalentcircuitmodeldifferentialequationmodelduality-basedmodelGIC(geomagneticallyinducedcurrents)saturationmodel,9,Equivalentcircuitmodel(transformer),Intimedomain,asinglephasetransformercanberepresentedbyanequivalentcircuitreferringallimpedancestoonesideofthetransformerThecoresaturationismodeledusingapiecewiselinearapproximationofsaturationThismodelisincreasinglyavailableintimedomaincircuitsimulationpackages.,10,Differentialequationmodel(transformer),Thedifferentialequationsdescribetherelationshipsbetweenwindingvoltageswindingcurrentswindingresistancewindingturnsmagneto-motiveforcesmutualfluxesleakagefluxesreluctancesSaturation,hysteresis,andeddycurrenteffectscanbewellmodeled.Themodelsaresuitablefortransientstudies.Theymayalsobeusedtosimulatetheharmonicgenerationbehaviorofpowertransformers.,11,Duality-basedmodel(transformer),Duality-basedmodelsarenecessarytorepresentmulti-leggedtransformersItsparametersmaybederivedfromexperimentdataandanonlinearinductancemaybeusedtomodelthecoresaturationDuality-basedmodelsaresuitableforsimulationofpowersystemlow-frequencytransients.Theycanalsobeusedtostudytheharmonicgenerationbehaviors,12,GICsaturationmodel(transformer),GeomagneticallyinducedcurrentsGICbiascancauseheavyhalfcyclesaturationthefluxpathsinandbetweencore,tankandairgapsshouldbeaccountedAdetailedmodelbasedon3Dfiniteelementcalculationmaybenecessary.Simplifiedequivalentmagneticcircuitmodelofasingle-phaseshell-typetransformerisshown.Aniterativeprogramcanbeusedtosolvethecircuitrysothatnonlinearityofthecircuitrycomponentsisconsidered.,13,Rotatingmachines,HarmonicmodelsforsynchronousmachineHarmonicmodelsforInductionmachine,14,Synchronousmachines,Harmonicsorigins:Non-sinusoidalfluxdistributionTheresultingvoltageharmonicsareoddandusuallyminimizedinthemachinesdesignstageandcanbenegligible.FrequencyconversionprocessCausedunderunbalancedconditionsSaturationSaturationoccursinthestatorandrotorcore,andinthestatorandrotorteeth.Inlargegenerator,thiscanbeneglected.Harmonicmodelsunderbalancedcondition,asingle-phaseinductanceissufficientunderunbalancedconditions,aimpedancematrixisnecessary,15,Balancedharmonicanalysis,Forbalanced(singlephase)harmonicanalysis,asynchronousmachinewasoftenrepresentedbyasingleapproximationofinductanceh:harmonicorder:directsub-transientinductance:quadraturesub-transientinductanceAmorecomplexmodela:0.5-1.5(accountingforskineffectandeddycurrentlosses)RnegandXnegarethenegativesequenceresistanceandreactanceatfundamentalfrequency,16,Unbalancedharmonicanalysis,Thebalancedthree-phasecoupledmatrixmodelcanbeusedforunbalancednetworkanalysisZs=(Zo+2Zneg)/3Zm=(ZoZneg)/3ZoandZnegarezeroandnegativesequenceimpedanceaththharmonicorderIfthesynchronousmachinestatorisnotpreciselybalanced,theselfand/ormutualimpedancewillbeunequal.,17,Inductionmotors,HarmonicscanbegeneratedfromNon-sinusoidalstatorwindingdistributionCanbeminimizedduringthedesignstageTransientsHarmonicsareinducedduringcold-startorloadchangingTheabove-mentionedphenomenoncangenerallybeneglectedTheprimarycontributionofinductionmotorsistoactasimpedancestoharmonicexcitationThemotorcanbemodeledasimpedanceforbalancedsystems,orathree-phasecoupledmatrixforunbalancedsystems,18,Harmonicmodelsforinductionmotor,BalancedConditionGeneralizedDoubleCageModelEquivalentTModelUnbalancedCondition,19,GeneralizedDoubleCageModelforInductionMotor,Stator,Excitationbranch,Attheh-thharmonicorder,theequivalentcircuitcanbeobtainedbymultiplyinghwitheachofthereactance.,mutualreactanceofthe2rotorcages,2rotorcages,20,EquivalentTmodelforInductionMotor,sisthefullloadslipatfundamentalfrequency,andhistheharmonicorder-istakenforpositivesequencemodels+istakenfornegativesequencemodels.,21,UnbalancedmodelforInductionMotor,Thebalancedthree-phasecoupledmatrixmodelcanbeusedforunbalancednetworkanalysisZs=(Zo+2Zpos)/3Zm=(ZoZpos)/3ZoandZposarezeroandpositivesequenceimpedanceaththharmonicorderZ0canbedeterminedfrom,22,Chapteroutline,IntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter,23,Arcfurnaceharmonicsources,Types:ACfurnaceDCfurnaceDCarcfurnacearemostlydeterminedbyitsAC/DCconverterandthecharacteristicismorepredictable,hereweonlyfocusonACarcfurnaces,24,CharacteristicsofHarmonicsGeneratedbyArcFurnaces,Thenatureofthesteelmeltingprocessisuncontrollable,currentharmonicsgeneratedbyarcfurnacesareunpredictableandrandom.Currentchoppingandignitingineachhalfcycleofthesupplyvoltage,arcfurnacesgenerateawiderangeofharmonicfrequencies,25,HarmonicsModelsforArcFurnace,NonlinearresistancemodelCurrentsourcemodelVoltagesourcemodelNonlineartimevaryingvoltagesourcemodelNonlineartimevaryingresistancemodelsFrequencydomainmodelsPowerbalancemodel,26,Nonlinearresistancemodel,simplifiedto,R1isapositiveresistorR2isanegativeresistorACclamperisacurrent-controlledswitchItisaprimitivemodelanddoesnotconsiderthetime-varyingcharacteristicofarcfurnaces.,modeledas,27,Currentsourcemodel,Typically,anEAFismodeledasacurrentsourceforharmonicstudies.ThesourcecurrentcanberepresentedbyitsFourierseriesanandbncanbeselectedasafunctionofmeasurementprobabilitydistributionsproportionofthereactivepowerfluctuationstotheactivepowerfluctuations.Thismodelcanbeusedtosizefiltercomponentsandevaluatethevoltagedistortionsresultingfromtheharmoniccurrentinjectedintothesystem.,28,Voltagesourcemodel,ThevoltagesourcemodelforarcfurnacesisaTheveninequivalentcircuit.Theequivalentimpedanceisthefurnaceloadimpedance(includingtheelectrodes)Thevoltagesourceismodeledindifferentways:formitbymajorharmoniccomponentsthatareknownempiricallyaccountforstochasticcharacteristicsofthearcfurnaceandmodelthevoltagesourceassquarewaveswithmodulatedamplitude.Anewvalueforthevoltageamplitudeisgeneratedaftereveryzero-crossingsofthearccurrentwhenthearcreignites,29,Nonlineartimevaryingvoltagesourcemodel,ThismodelisactuallyavoltagesourcemodelThearcvoltageisdefinedasafunctionofthearclengthVao:arcvoltagecorrespondingtothereferencearclengthlo,k(t):arclengthtimevariationsThetimevariationofthearclengthismodeledwithdeterministicorstochasticlaws.Deterministic:Stochastic:,30,Nonlineartimevaryingresistancemodels,Duringnormaloperation,thearcresistancecanbemodeledtofollowanapproximateGaussiandistributionisthevariancewhichisdeterminedbyshort-termperceptibilityflickerindexPstAnothertimevaryingresistancemodel:R1:arcfurnacepositiveresistanceandR2negativeresistanceP:short-termpowerconsumedbythearcfurnaceVigandVexarearcignitionandextinctionvoltages,31,Powerbalancemodel,risthearcradiusexponentnisselectedaccordingtothearccoolingenvironment,n=0,1,or2recommendedvaluesforexponentmare0,1and2K1,K2andK3areconstants,32,Chapteroutline,IntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter,33,Three-phaselinecommutedconverters,Line-commutatedconverterismostlyusualoperatedasasix-pulseconverterorconfiguredinparallelarrangementsforhigh-pulseoperationsTypicalapplicationsofconverterscanbefoundinACmotordrive,DCmotordriveandHVDClink,34,HarmonicsCharacteristics,Underbalancedconditionwithconstantoutputcurrentandassumingzerofiringangleandnocommutationoverlap,phaseacurrentish=1,5,7,11,13,.Characteristicharmonicsgeneratedbyconvertersofanypulsenumberareintheorderofn=1,2,andpisthepulsenumberoftheconverterFornon-zerofiringangleandnon-zerocommutationoverlap,rmsvalueofeachcharacteristicharmoniccurrentcanbedeterminedbyF(,)isanoverlapfunction,35,HarmonicModelsfortheThree-PhaseLine-CommutatedConverter,Harmonicmodelscanbecategorizedasfrequency-domainbasedmodelscurrentsourcemodeltransferfunctionmodelNorton-equivalentcircuitmodelharmonic-domainmodelthree-pulsemodeltime-domainbasedmodelsmodelsbydifferentialequationsstate-spacemodel,36,Currentsourcemodel,ThemostcommonlyusedmodelforconverteristotreatitasknownsourcesofharmoniccurrentswithorwithoutphaseangleinformationMagnitudesofcurrentharmonicsinjectedintoabusaredeterminedfromthetypicalmeasuredspectrumandratedloadcurrentfortheharmonicsource(Irated)Harmonicphaseanglesneedtobeincludedwhenmultiplesourcesareconsideredsimultaneouslyfortakingtheharmoniccancellationeffectintoaccount.h,andaconventionalloadflowsolutionisneededforprovidingthefundamentalfrequencyphaseangle,1,37,TransferFunctionModel,ThesimplifiedschematiccircuitcanbeusedtodescribethetransferfunctionmodelofaconverterG:theidealtransferfunctionwithoutconsideringfiringanglevariationandcommutationoverlapG,dcandG,ac,relatethedcandacsidesoftheconverterTransferfunctionscanincludethedeviationtermsofthefiringangleandcommutationoverlapTheeffectsofconverterinputvoltagedistortionorunbalanceandharmoniccontentsintheoutputdccurrentcanbemodeledaswell,38,Norton-EquivalentCircuitModel,ThenonlinearrelationshipbetweenconverterinputcurrentsanditsterminalvoltagesisI&VareharmonicvectorsIftheharmoniccontentsaresmall,onemaylinearizethedynamicrelationsaboutthebaseoperatingpointandobtain:I=YJV+INYJistheNortonadmittancematrixrepresentingthelinearization.ItalsorepresentsanapproximationoftheconverterresponsetovariationsinitsterminalvoltageharmonicsorunbalanceIN=Ib-YJVb(Nortonequivalent),39,Harmonic-DomainModel,Undernormaloperation,theoverallstateoftheconverterisspecifiedbytheanglesofthestatetransitionTheseanglesaretheswitchinginstantscorrespondingtothe6firinganglesandthe6endsofcommutationanglesTheconverterresponsetoanappliedterminalvoltageischaracterizedviaconvolutionsintheharmonicdomainTheoveralldcvoltageVk,p:12voltagesamplesp:squarepulsesamplingfunctionsH:thehighestharmonicorderunderconsiderationTheconverterinputcurrentsareobtainedinthesamemannerusingthesamesamplingfunctions.,40,Chapteroutline,IntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter,41,HarmonicscharacteristicsofTCR,HarmoniccurrentsaregeneratedforanyconductionintervalswithinthetwofiringanglesWiththeidealsupplyvoltage,thegeneratedrmsharmoniccurrentsh=3,5,7,istheconductionangle,andLRistheinductanceofthereactor,42,HarmonicscharacteristicsofTCR(cont.),Threesingle-phaseTCRsareusuallyindeltaconnection,thetriplencurrentscirculatewithinthedeltacircuitanddonotenterthepowersystemthatsuppliestheTCRs.Whenthesingle-phaseTCRissuppliedbyanon-sinusoidalinputvoltagethecurrentthroughthecompensatorisprovedtobethediscontinuouscurrent,43,HarmonicmodelsforTCR,HarmonicmodelsforTCRcanbecategorizedasfrequency-domainbasedmodelscurrentsourcemodeltransferfunctionmodelNorton-equivalentcircuitmodeltime-domainbasedmodelsmodelsbydifferentialequationsstate-spacemodel,44,CurrentSourceModel,bydiscreteFourieranalysis,45,NortonequivalencefortheharmonicpowerflowanalysisoftheTCRfortheh-thharmonic,Norton-EquivalentModel,Theinputvoltageisunbalancedandnocouplingbetweendifferentharmonicsareassumed,46,TransferFunctionModel,AssumethepowersystemisbalancedandisrepresentedbyaharmonicThveninequivalentThevoltageacrossthereactorandtheTCRcurrentcanbeexpressedasYTCR=YRScanbethoughtofTCRharmonicadmittancematrixortransferfunction,47,Time-DomainModel,Model1,Model2,48,Chapteroutline,IntroductionNonlinearmagneticcoresourcesArcfurnace3-phaselinecommutedconvertersStaticvarcompensatorCycloconverter,49,HarmonicsCharacteristicsofCycloconverter,AcycloconvertergeneratesverycomplexfrequencyspectrumthatincludessidebandsofthecharacteristicharmonicsBalancedthree-phaseoutputs,thedominantharmonicfrequenciesininputcurrentfor6-pulse12-pulsep=6orp=12,andm=1,2,.Ingeneral,thecurrentsassociatedwiththesidebandfrequenciesarerelativelysmallandharmlesstothepowersystemunlessasharplytunedresonanceoccursatthatfrequency.,50,HarmonicModelsfortheCycloconverter,Theharmonicfrequenciesgeneratedbyacycloconverterdependonitschangedoutputfrequency,itisverydifficulttoeliminatethemcompletelyTodate,thetime-domainandcurrentsourcemodelsarecommonlyusedformodelingharmonicsTheharmoniccurrentsinjectedintoapowersystembycycloconvertersstillpresentachallengetobothresearchersandindustrialengineers.,
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