Dynamic Analysis of Current Regulators for AC Motors Using Complex Vectors

DynamicAnalysisofCurrentRegulatorsforACMotorsUsingComplexVectorsF.Briz,M.W.Degnert.R.D.LorenzttDept.ofElectrical,ComputerandSystemsEngineeringUniversityofOviedoGijon33208,SpainTel:348-518-2289Fax:348-518-2068Email:fernando@hecate.etsiig.uniovi.esAbstract-Theanalysisanddesignofcurrentregulatorsforpolyphaseacloadsispresentedusingcomplexvectornotation.TheACmotorcurrentregulationproblemisanalyzedbystudyingboththecommandtrackinganddisturbancerejectioncapabilityofthecurrentregulator.Theuseofcomplexvectornotationandthegeneralizationofclassicalcontroltoolslikerootlocus,frequencyresponsefunctions,anddynamicstiffnessfunctionstocomplexvectorsprovideawayofcomparingtheperformanceofdiflerentcontrollertopologies.LimitationsintheperformanceofthesynchronousframePIcurrentregulatorareoutlinedandseveralwaysofimprovingitsperformancearesuggestedandinvestigated.I.INTRODUCTIONThesynchronousframePIcurrentregulatorhasbecomethestandardforcurrentregulationofpolyphaseacmachinesduetoitscapabilityofregulatingacsignalsoverawidefrequencyrange[1,2].Inareferenceframesynchronouswiththefundamentalexcitation,thefundamentalexcitationbecomesadcquantitythatiseasilyregulatedtothedesiredvalueusingaPIcontroller.EventhoughtheperformancecharacteristicsofthesynchronousreferenceframePIcurrentregulatormayseemintuitive,themultiple-input/multiple-outputnatureofthesystemmakesitsperformanceevaluationdifficult.Therepresentationofacmachinesandtheanalysisoftheircurrentregulatorscanbeapproachedusingbothscalarandcomplexvectornotation[3].Althoughbothnotationscanbeusedtoachievethesamefinalresult,themodelofthemachineusingeachnotationprovidesdifferentinsightintothecontrolproblemandsomesolutionscanbemoreintuitivelyseenusingonenotationortheother.Thestandardmatrixorscalarnotationdoesnoteasilylenditselftoclassicalcontroltools,likerootlocusorfrequencyresponsefunctions,otherthanallowingtheuseofmatrixalgebra.Theuseofcomplexvectornotationsimplifiesthemodelofanacmachinefromamultiple-input/multiple-outputsystemtoanequivalentsingle-input/single-outputcomplexvectorsystem.Theperformanceofthiscomplexvectormodelcanthenbeevaluatedusinggeneralizedformsofclassicalcontroltools,liketherootlocus,frequencyresponsefunction(FlZF)anddynamicstiffnessfunction(DSF),namelythecomplexvectorrootlocus,FRFandDSF.Complexvectorsaresystematicallyusedtostudytheperformanceofsynchronousreferenceframecurrentregulatorsinthispaper.Inductionmotormodelingispresentedfirst.AcomplexvectorbasedanalysisofthecommandtrackinganddisturbancerejectioncapabilitiesfortDept.ofMechanicalEngineering*Dept.ofElectricalandComputerEngineeringUniversityofWisconsin-MadisonMadison,WI53706Tel:608-262-0556Fax:608-265-2316Email:degner@cae.wisc.edu,lorenz@engr.wisc.eduthedifferentsynchronousframecurrentregulatorsisshowntoprovideincreasedinsightintheinductionmotorcurrentregulationproblem.Fromthisanalysis,severalimprovementsareproposed.11.INDUCTIONMOTORMODELING,CURRENTREGULATION,ANDBACK-EMFDECOUPLINGThenonlinearstateequationsgoverningtheelectricalandelectromagneticbehaviorofaninductionmotorusingcomplexvectornotation,withthestatorcurrentandtherotorfluxasthestatevariables,are(1)and(2),[3]:Thesuperscript'Is"denotesastationaryreferenceframeandpisthederivativeoperator.Fromacontrolperspectiveitisusefultotransformtheseequationstoanexcitationfrequencysynchronousreferenceframe.Thetransformationofagenericcomplexvectorquantity,betweenthestationaryandthesynchronousreferenceframe,denotedbythesuperscript"e",isdefinedby(3)forthecaseofasinglecomplexvectorquantityand(4)foritsderivative:(3)(4)Applying(3)and(4)to(1)and(2),thecomplexvectorequationsoftheinductionmotorinasynchronousreferenceframe(5)and(6)areobtained:Whenusingavoltagesourceinverter,controllingthestatorcurrent(seeFig.1)simplifiestheoveralldrivecontrolschemefrombothatorquecontrolandinverterdeviceprotectionperspective.Synchronousframecurrentregulatorshavebecometheindustrystandardforinvertercurrentregulation.Theyare1253preferredbecausealltheelectricalvariableshavedcsteady-statevalueswhenviewedinasynchronousreferenceframe.ThisenablesasimplePIregulatortoprovidezerosteady-stateerror,independentofthesynchronousfrequency.Inspiteofthisattractiveproperty,thedynamicresponseofthistypeofcurrentregulatorisfarfrombeingideal,showingadeteriorationasthesynchronomfrequencyincreases.Thedependencyoftheinductionmotor,(3,onthesynchronousfrequencyisseentocomefromasynchronousframecross-..............................."............................."...I..."~couplingterm,(7):iIja,L,hMotori......._...._....I................................................................I-Figure1:ComplexVectorBlockDiagramofaCurrentRegulatedInductionMotor,ShowninaSynchronousReferenceFrame.e-joeLosiidsoralternatelyviewedas-jweiqdrandfromtheelectromechanicalcross-couplingviathespeed(=synchronousfrequency)dependentback-emfvoltage:(7)Ifthesetwotermsaredecoupledfrom(3,thedependenceonthesynchronousfrequency(androtorspeed)disappearsandthestatorvoltageequationbecomesthatofanRLload,enablingsimple,fast,andaccuratecurrentregulation.Whenscalarinsteadofcomplexvectornotationisused,thetransformationofagenericcomplexvectorquantityffromcomplexvectortoscalarnotationcanbedonebytakingrealandimaginarypartsaftersubstitutingasshownin(9)[3]:fqd=fq-jfd(9)Applying(9)to(5),andassuming(ascommonlydoneforrotorfluxfieldorientation)thatthesynchronousframehasbeenalignedwiththerotorflux,(lo),theequivalentscalarnonlinearstateequations(11)and(12)correspondingtothecomplexvectorequation(5)areobtained:Differentrepresentationsoftheinductionmotorareoftenused[3,5]dependingonhowthequantitiesarerearranged.If,forexample,thefluxstateequation,(6),issolvedwiththecurrentstateequation(5)therotorspeedvariablecanbeeliminatedwiththeresultasshowninthescalarequations(13)and(14).~~s=Lospi~s+Rsi~s+Ls~eids-~e-ph,Lme(13)eRrThisrepresentationisnolongerasetofstateequations,sincetwostatederivativesappearineachequation.Nevertheless,theequationsdoallowsomeinsightifsomeassumptionsaremadeabouttherateofchangeofthestates.Iftherotorfluxdynamicsareassumedtobemuchslowerthanthestatorcurrentdynamicstherotorfluxderivativetermsaresmallandincertaincasescanbeneglected[3,5,7].Itshouldbenotedthatthetworotorfluxderivativetermshaveverydistinctcoefficients.Oneisnearlyconstantandoneislinearlydependentonexcitationfrequency.Thus,theassumptionismostvalidforlowexcitationfrequencies.Iftherotorfluxdynamictermscanbesafelyneglected,thentheresultingstatorvoltageequationsmaybeviewedasapproximatestateequationsforthestatorcurrent.Thesynchronousfrequencycross-couplingnowappearsasin(15)and(16),whichisofaverydifferentformthan(7)and(8):L,oieeqsIfthesetwotermsaredecoupledfrom(13)and(14),respectively,andtherotorfluxderivativetermscanbesafelyneglected,thenapproximatelvdecoupledcontroloftheqandd-axiscurrentscanbeobtained.TheresultingapproximatestatorvoltageequationbecomesthatofanRLload,whichisanapproximationtotheexactdecouulingof(7)and(8).NotethattheresistanceoftheremainingRLloadin(13)and(14)doesnotcorrespondtotheresistancein(5)oncedecouplinghasbeencarriedout.Exactdecouplingof(7)and(8)canbeviewedasaddressingtwoseparatecross-couplingissues.Itisinstructivetofirstunderstandthatthecross-couplingin(7)resultssolelyfromthesynchronousframetransform.Thiscanbedemonstratedvia(17)and(18),whichrepresentathree-phase,symmetricIUloadinstationaryandthesynchronousreferenceframe,respectively:ItisseenthatbytransforminganRLloadtothesynchronousreferenceframeanidenticalcross-couplingtermtothatpresentintheinductionmotor,(7),iscreated.Thisterm,therefore,isacharacteristicofRLloadswhentransformed,andnotoftheinductionmotor,inparticular.Becausethecross-couplingcanbeexpressedas-jwe,appropriatelyformeddecouplingof(7)requiresnoparametersandcanbeperformedexactly.Incontrasttothis,thecross-couplingrepresentedby(8)istheeffectofback-emf,i.e.,rotorfluxandrotorvelocity,on1254thestatorcurrent.Thiselectromechanicalcross-couplingcouldbeviewedasadisturbanceiftheinductionmotorweremodeledasanRLload.Fromacontrolsystemsperspectivehowever,ifapproximatedecouplingofthiselectromechanicalcross-couplingcanbeachieved,theoverallsystemdynamicsareimprovedandcurrentregulatorpropertieswillbenearlyspeedinvariant.Becausetheapproximatedecouplingsolutionislessinsightfulinitstermsandisalsolimitedtolowexcitationfrequencies(and/orconstantrotorflux)itisconsideredlessattractivethantheexactdecouplingsolutionasaglobalcurrentregulationstrategy.Therefore,using(3,ortheequivalentscalarnotation(11)and(12),toapproachtheinductionmotorcurrentregulationproblemisconsideredmoreappropriateandisusedfortheremainderofthiswork.Thedifferentnatureandsourceofthecross-couplingin(7)and(8)suggestthattheybeconsideredseparatelyinthecontrollerdesign.Thisisdoneinthefollowingsections.111.EFFFCTSOFSYNCHRONOUSFRAMECROSS-COUPLINGONCURRENTREGULATIONIftherotorfluxdependentterm(back-emf),(8),isperfectlydecoupledfromthestatorvoltageequationoftheinductionmotor,(5),theequationreducestothatofasimpleFUload,(18).Usingthisassumptionitispossibletolookattheeffectsofthesynchronousframecross-couplingterm,(7),ontheperformanceofcurrentregulatorsforinductionmotorswiththesimplifiedmodelofanRLload[4].ThecomplexvectormodelofanRLloadinthesynchronousreferenceframe,(18),isseentohaveasingle,asymmetric,complexpolelocatedat-R/Ljwe.ThecomplexvectorblockdiagramoftheRLloadwithasynchronousframePIcurrentregulatorisshowninFig.2.Figure2:ComplexVectorBlockDiagramofanRLLoadwithaSynchronousFramePICurrentRegulator,ShownintheSynchronousReferenceFrame.TheperformanceoftheclassicalsynchronousframePIcurrentregulatorwasanalyzedbyapplyingittoa3-phaseFUloadwiththeparametersshowninTable1.TABLE1-RLLoadParametersR1.1aL3.7mHThecurrentregulatorwastunedbyselectingacontrollerzeroapproximatelyequaltothebreakfrequencyoftheRLload,i.e.,K#Kp=R/L.Thecontrollergainwasselectedtoachievearelativelylowbandwidthof200Hzsothatsystematictransienterrorswouldbemoreeasilyobserved.Anoverlayofthecommandedandexperimentalsystemresponseforamagnitudeandaphasestep,withconstantinputsynchronousfrequenciesof50and200Hz,isshowninFig.3.Seriousdegradationinthetransientperformanceisapparentasthesynchronousfrequencyincreases.Thecomplexvectorrootlocuscanalsobeplotted,asshowninFig.4forthreedifferentsynchronousfrequencies.20,,20,00.0500.0500.0500.05tim,(sec.)tim,(sec.)a)re=50Hzb)re=200HzFigure3:CommandedandExperimentalCurrentMagnitudeandPhaseStepResponsesforanRLLoadwithaSynchronousFramePICurrentRegulator-200-1000-200-1000-200-1000(0-zero,X-openlooppole,*-closedlooppole)Figure4:ComplexVectorRootLocusofanRLLoadwithaSynchronousFramePICurrentRegulator(200Hzbandwidth),ShownintheStationaryReferenceFrameV;=0,50,and200Hz).Thecomplexvectorrootlocus,asthescalarrootlocus,followsthemagnitudeandangleconditions.Nevertheless,becausetheinputsandoutputsarenolongerrealnumbersbutcomplexvectors,anditispossibletogetcomplex,asymmetricpolesandzeroes,i.e.,therootlocusdoesnothavetobesymmetricwithrespecttotherealaxis.TherootlocuswasobtainedusingthestandardrootlocusfunctionsintheMatlabcontrolssystemstoolbox.FromFig.4itisseenthatatlowfrequenciesthecontrollerzeroapproximatelycancelstheplantpole.Thisallowstheresponseofthesystemtobedominatedbythefasterclosedlooppole,placedatthedesired200Hzbandwidth.Forhighersynchronousfrequenciesthecontrollerzerointeractsmorewiththepoleaddedbythecontroller.Theresultingslowerrootmovesprogressivelyclosertotheimaginaryaxisawayfromthezero,withincreasingovershootexpected.Thecomplexvectortransferfunctiondescribingthesystemisgivenby(19):WZ)ThecomplexvectorFRF,showninFig.5,canbecalculated1255fromthistransferfunction.TheasymmetricrootlocusabouttherealaxisgivesrisetoanFRFasymmetricforpositiveandnegativefrequencies.ItisnotedthatalloftheFRF'sshowninFig.5haveaunitygainandzerophaseshiftatthesynchronousfrequency.However,atfrequenciesawayfromthesynchronousfrequencythereissignificantdistortionintheFRF.Sl&.s*9d1-2efEacross-couplingcausedbythetermjm&in(18).Theblockdiagramofthecross-couplingdecouplingformofthesynchronousframePIcurrentregulatorisinFig.6[4,5].Theeffectofthecross-couplingdecouplingistomovethepoleoftheplantfrom-R/L-jo,to-R/Linthesynchronousreferenceframe,whichmakesitpossibletodirectlycancelitusingtherealzeroaddedbythecontroller.Theresultingcomplexvectorrootlocusforthecross-couplingdecouplingsynchronousframePIcurrentregulatorisshowninFig.7forthreedifferentsynchronousfrequencies.-800-600-400-200020040060080090,f-800-600-400-2000200400600800kquency,Wz)Figure5:ComplexVectorFRFofanRLLoadwithaSynchronousFramePICurrentRegulator(200Hzbandwidth),ShownintheStationaryReferenceFrame(f,=0,50,and200Hz).ItisimportanttounderstandthemeaningoftheFRFatfrequenciesotherthanthesynchronousfrequency.Thesynchronousfrequencyisthesteady-statefundamentalcomponent.Bothdisturbancesandchangesinthecommandtrajectorysimultaneouslyexcitethesystemwithawiderangeoffrequencycontent.TheFRFshowshowthesystemrespondstothefrequencycontentthatisnotatthesynchronousfrequency.Fromthisanalysis,thetimeresponseinFig.3canbeexplained.Eventhoughthecommandedsynchronousfrequencyremainedconstant,magnitudeandphasestepsinthecommandedcurrentintroducedtransientcontentatfrequenciescenteredonthesynchronousfrequency.Thus,thetransientresponseofthecurrentregulatordependsonitscapabilitytoregulatebeyondthesynchronousfrequency.Iv.IMPROVEDCURRENTREGULATORDESIGNBYDECOUPLINGOFSYNCHRONOUSFRAMECROSS-COUPLINGAnidealsynchronousreferenceframecurrentregulatorwouldhaveatimeresponseindependentofthesynchronousfrequencywhenviewedinthesynchronousreferenceframe.SucharegulatorwouldhaveacomplexvectorFRFwithashapethatdoesnotvarywiththesynchronousfrequency.Instead,thecenteroftheFRFshapewouldjustshiftsothatitisalwayssymmetricaboutthesynchronousfrequency.Toachievethiswillrequiredecouplingtheeffectofthesynchronousfrequencycross-coupling.Therearetwopossibilitiesfordecoupling:a)statefeedbackdecouplingandb)symmetriccross-coupling.A.SynchronousFramePICurrentRegulatorwithCross-CouplingDecouplingviaStateFeedbackOnewayofmodifyingthesynchronousframePIcurrentregulatortoachievethedesiredresponseistodecoupletheFigure6:ComplexVectorBlockDiagramofanRLLoadwithaCross-CouplingDecoupling(viaStateFeedback)SynchronousFramePICurrentRegulator,ShownintheSnchronousReferenceFrame~200~p100~1~1ciF0-200-1000-200-1000-200-1000(0-zero,x-openlooppole,*-closedlooppole)Figure7:ComplexVectorRootLocusofanRLLoadwithaCross-CouplingDecoupling(viaStateFeedback)SynchronousFramePICurrentRegulator(200Hzbandwidth),ShownintheStationaryReferenceFrame(re=0.50,and200Hz).re&(Hz)Itshouldbenotedbyremovingtheplantcross-couplinginthesynchronousframe,thesysteminthestationaryframewillnowbecross-coupled.Thisisbecausecross-couplingoccurswiththeinversetransformofanuncoupledsynchronousframesystemtothestationaryframe.Thus,inthestationaryframerootlocusthecontrollerzeroappearstomovewiththeplantpoleandbotharetiedtothesynchronousfrequency.B.ComplexVectorSynchronousFramePICurrentRegulatorwithSymmetricCross-CouplingInsteadofmovingthepoleoftheplanttothelocationofthecontrollerzero,thecontrollerzerocanbemovedtothelocationoftheplantpolebymodifyingthecontrollerstructureasshowninFig.8tosymmetricallycross-couplethecontrollerwiththesynchronousfrequencyterm[4,61.ThisformofthesynchronousframePIcurrentregulatoriscalledthecomplexvector,synchronousframePIcurrentregulatorThisdesignisdirectlyanalogoustotheclassicalcontrolpole/zerocancellationmethodology,withtheonlydifferencebeingtheuseofcomplexvectorsallowstheplacementofthecontrollerzerooffoftherealaxis.TheresultingcomplexvectorrootlocusisshowninFig.9forthreedifferentsynchronousfrequencies.[41.12561Figure8:ComplexVectorBlockDiagramofanRLLoadwithaComplexVectorSynchronousFramePICurrentRegulator,ShownintheSynchronousReferenceFrame.Itshouldbenotedbysymmetricallycross-couplingthecontroller(andtheplant)inthesynchronousframe,boththecontrollerandtheplantwillbedecoupledinthestationaryframe.Thus,inthestationaryframerootlocusthecontrollerzeroappearsfixedwiththeplantpoleandneitheraretiedtothesynchronousli-equency.-#-200Hzr7-200-1000-200-1000-200-1000(0-zero,x-openlooppole,*-closedlooppole)Figure9:ComplexVectorRootLocusforanRLLoadwithaComplexVectorSynchronousFramePICurrentRegulator(200Hzbandwidth),ShownintheStationaryReferenceFrame(f,=0,SO,and200Hz).Itshouldfurtherbenotedthatthesymmetriccross-couplinghasnophysicalparameters,whichisconsistentwiththefactthattheoriginofthiscross-couplingissolelyfromthesynchronousframetransform.Iftheparameterestimatesarecorrect,andbothmodifiedcurrentregulatorshavethesamecontrollerPIgains,thecomplexvectorFRF'sforthetwocurrentregulatorsareidenticalandshowninFig.10[4].real,(Hz)1.51Is-800-m-100-2000200400m8001-800600-400-2000200400600800frequency,(Hz)Figure10:ComplexVectorFRFofanRLLoadforeitheraCross-CouplingDecouplingorComplexVectorSynchronousFramePICurrentRegulator(200Hzbandwidth),ShownintheStationaryFrame(fe=0,50,200Hz).TheshapeofthecomplexvectorFRFisindependentofthesynchronousfrequencyandsymmetricwithrespecttoit.Fig.11showsthestepresponseforthecomplexvectorsynchronousframePIcurrentregulator.(Thecross-couplingdecouplingcontrollerhasnearlyidenticalcharacteristicsandthusisnotshown).Thetimeresponseisseennowtocorrespondtothetunedbandwidthindependentofthesynchronousfrequency.00.0500.0500.0500.05tim,(sec.)tim,(sec.)a)f,=SOHzb)f,=200HzFigure11:CommandedandExperimentalCurrentMagnitudeandPhaseStepResponseforanRLLoadwithaComplexVectorSynchronousFramePICurrentRegulator(200Hzbandwidth),ShownintheSynchronousFrame.ThedecreasedparametersensitivityofthecomplexvectorformofthesynchronousframePIcurrentregulatorincomparisonwiththecross-couplingdecouplingviastatefeedbackwasdemonstratedin[4].Themainreasonforthisdifferenceisthebeneficialimpactofsymmetriccross-couplinginachievingpole-zerocancellationindependentofthesynchronousfrequency.v.DYNAMICSTIFFNESSANALYSISFORSYNCHRONOUSFRAMEPICURRENTREGULATORSThesimplificationoftheinductionmotortoanRLloadcouldalsobedonebyconsideringtheback-emfterm(8),asadisturbancetothestatorvoltageequation,(5).asshowninFig.12,insteadofassumingperfectdecoupling.l.".l".."........"I".-.I.I.....-I."...Reg.Figure12:RepresentationoftheInductionMotorbyanRLLoadwithDisturbanceInputShowninaSynchronousReferenceFrameInthiscasetheback-emfwouldbeadisturbancetothesystemthatwouldneedtobecompensatedforbythecurrentregulator.ThissuggeststhatastudyofthedisturbancerejectioncapabilityofthedifferentcurrentregulatordesignsbymeanoftheDSFgeneralizedtocomplexvectorswillprovideincreasedinsightsinanalyzingtheeffectofback-emf.Substitutingthe"CurrentRegulator"blockinFig.12withthedifferentcurrentregulatorstransferfunctions,thefollowingcomplexvectorDSF's,i.e.disturbancevs.outputtransferfunctions,areobtainedfortheclassical(20),cross-couplingdecoupling(21),andcomplexvector(22)currentregulators.Separatingthephysicalparametersinsteadoftheusualpole-zerorepresentationwasconsideredtoprovidemoreinsight.1257GdKi+=LS+R+K+-Ps-jw,6dAKi-=Ls+R+Kp-j0,L+-ss-jwe+=Ls+R+‘DdKi‘qdItisnotedthatthedynamicstiffnesshasunitsofimpedancewhere,inthiscase,highimpedancewithrespecttotheback-emf“disturbance”voltagewouldbepreferred.ThesefunctionsarerepresentedinFig.13(onlymagnitudesareshown).204ClassicalP10810020b)Cross-DecouplingCoupling020C)810Complex0Vector-800-600-400-2000200400600800frequency,(Hz)figure13:DynamicStiffnessAnalysisforanRLLoadwitha)Classical,b)Cross-CouplingDec.andc)ComplexVectorSynch.PICurrentRegulators(200Hzbandwidth),ShownintheStat.Frameve=0,50,and200Hz).AswasthecaseforthecomplexvectorFRF,thedisturbanceinputisnotasinusoidalsignal,butarotatingvector,andbecausepositiveandnegativefrequenciesareneeded,alinearscalehastobeusedforthefrequencyaxis.Fors=jwe,i.e.thedisturbancevoltagevectorrotatingatthesynchronousfrequency,allthethreecurrentregulatordesignsprovideinfinitedynamicstiffness,thereforethecurrentregulationisnotaffectedbythedisturbanceinput,whichagreeswiththezerosteady-stateerrorproperty.Atfrequenciesotherthanthesynchronousfrequency,theDSFshowshowmuchthecurrentregulationwillbeaffected.Theminimumineachcurveshowsthefrequencyforwhichthecurrentregulatorwillbeweakest(lowestimpedance),andhowweakitwillbe.BycomparingthethreeDSF‘sonecanconclude:Forwe=0(dcexcitation),al