电力电子电路非线性现象

电力电子电路非线性现象第一页，共85页。Contents1.NonlinearStudyonPENonlinearity–thedefaultpropertyofpowerelectronicscircuitsClassicexampleofnonlinearstudyRecentstudyfromabifurcationperspectiveCircuitshavebeenstudiedindetail,LiteraturereviewCurrentstateoffindingsBasicproblemsinanalysisWhatmodelstouse?Analysistechniques(Averaging,Discrete-timemap)SummaryofobservationsPossibleengineeringapplicationsSummaryofnonlinearstudyonPE2第二页，共85页。1、NonlinearStudyonPENatureofoperationTimevarying—differentsystemsatdifferenttimesANDNonlinear

—thetimedurationsarerelatednonlinearlywiththeoutputfunction3第三页，共85页。Nonlinearity–thedefaultpropertyofpowerelectronicscircuitsPowerelectronicsengineers/researchersaredealingwithnonlinearproblemsMuchofpowerelectronicsisaboutidentifyingnonlinearphenomenaand“taming”themtodousefulapplicationsClassicexamples:Averaging(R.DavidMiddlebrook,RichardBass)Discrete-timemodeling(HarryOwen,FredLee)Stabilityanalysis(GeorgeVerghese)Phase-planeanalysis/control(FredLee,BillBurns)Seriesapproximation(R.Tymerski)4第四页，共85页。ClassicexampleofnonlinearstudyBoostconverterwithcurrent-modecontrolIrefiLIrefiLd<0.5d>0.5Simpleanalysisrevealsachangeofstabilitystatusatacriticaldutycycleof0.5.Thecircuitisactually‘stable’beyondthecriticalpoint,thoughoperateswithalongerperiod.Thisperiod-doublingphenomenonwasobservedlongago.5第五页，共85页。RecentstudyfromabifurcationperspectiveIrefiLT/CR=0.125T/CR=0.625samplediLsampledIrefWiththehelpofcomputers,wecanstudythephenomenoninmoredetail.Bifurcationdiagrams(panoramicviewofstabilitystatus)WecanplotbifurcationdiagramsfordifferentsetsofparametersSampledvariableatsteadystateversusparameter,e.g.,iL(nT)vs.Irefnormalperiod-1operationbifurcationpoint6第六页，共85页。CircuitswhosebifurcationbehaviourshavebeenstudiedindetailVoltage-modeandcurrent-modecontrolledsimplebuckandboostconverters,andmanyothers7第七页，共85页。ViniLiLvoPeriod-doublingandchaosinvoltage-modecontrolledbuckconverterbifurcationdiagramchaoticattractorHamilletal.(1990)8第八页，共85页。Period-doublingandchaosincurrent-modecontrolledboostconverterbifurcationdiagramIrefiL(nT)9第九页，共85页。HopfBifurcationinparallelboostconvertersnormalperiod-1limitcycleoflongperiodquasi-periodicorbitbifurcationdiagram10第十页，共85页。LiteraturereviewHamillandJefferies[1988]-FirstanalysisofbifurcationandchaoticdynamicsinafirstorderPWMvoltage-modecontrolledconverter.DeaneandHamill[1990]-AnalysisofbifurcationinfirstorderandsecondorderPWMbuckconverters.Hamilletal.[1992]-Derivationofaniterativemaptoanalyzebifurcationinabuckconverterincontinuousmode.Deane[1992]-Firstreportonchaoticbehaviourinacurrent-controlledboostconverter.Tse[1994]-Derivationaniterativemaptodemonstrateperiod-doublingcascadesinaboostconverterindiscontinuousmode.

11第十一页，共85页。Literaturereview(cont.)Chakrabartyetal.[1995]/FossasandOlivar[1996]-FurtherstudyofchaosinaPWMbuckconverter.Poddaretal.[1995]/Batlleetal.[1996]-Controlofchaosindc/dcconverters.TseandChan[1995]-Studyofbifurcationandchaosinafourthordercurrent-controlledCukconverter.ChanandTse[1996]-Studyofvariousroutestochaosinacurrent-controlledboostconverter.Banerjeeetal.[1997]-Examinationofbifurcationandchaosinthelightof“bordercollisionbifurcation”.Tse[1997]–Analysisofautonomousconvertersusingaveragedmodels.12第十二页，共85页。DiBernardoetal.[1998]-Studyofvarioussamplingandtheirapplicationsintheidentificationofbifurcationandchaos.ChanandTse[1998]–Proofofperiod-doublingindiscontinuousconvertersusingSchwarzianderivativesDiBernardoetal.[1998]-Analysisofthenon-smoothdynamics(suchasgrazing,skippingandsliding)ofdc/dcconverters.ElAroudietal.[1999]-Identificationofquasi-periodicityandchaosinaboostconverter.Mazumder,NayfehandBorojevich[1999]-Fast-andslows-scaleinstabilities.IuandTse[2000]-StudyofbifurcationinparallelconvertersLiteraturereview(cont.)13第十三页，共85页。ReviewpaperIEEEProceedingsVol.90No.5pp.768–781May2002Literaturereview(cont.)14第十四页，共85页。Twotypesofbifurcation*seeninpowerelectronicsStandardbifurcations(foundinothersystemsaswell)Period-doublingHopf(Neimark-Sacker)Saddle-nodeBordercollision(characteristicofpowerelectronics)

AbruptchangeofstabilitystatusduetosaturationCurrentstateoffindings

*Bifurcationreferstosuddenchangeofqualitativebehaviourofadynamicalsystemwhenacertainparameterisvaried.15第十五页，共85页。Standardbifurcations

Buckconverters(voltage-mode)–period-doublingBoostconverters(voltage-mode)–HopfDc/dcconvertersinDCM–period-doublingMostdc/dcconverters(current-mode)–period-doublingOthertypes–variety:saddle-node,crisis,etc.Bordercollision(characteristicofpowerelectronics)

Allstandardbifurcationsareinterruptedbybordercollision

Currentstateoffindings

Whowillgetwhat?16第十六页，共85页。Bordercollision–

signaturebifurcationofpowerelectronicscircuits

Bordercollision(Banerjee,PESC’97)Non-smoothphenomena“always”observedinpowerelectronicscircuits

boostconverterundercurrent-modecontrol17第十七页，共85页。Essence:standardbifurcationbeinginterruptedboostconverterundercurrent-modecontrolBordercollision18第十八页，共85页。Bordercollision–apracticalviewpoint

ExcessiveswingofcontrolsignalDutycyclesaturationOut-of-rangeoperation,preventingcontinuationofstandardbifurcationse.g.,buckconverterinvoltage-modecontrolperiod-doublingbordercollision19第十九页，共85页。Whatmodelstouse?Whattechniquestouse?Anopenquestion:

Whataretheapplications?(Whatdoweusetheresultsfor?)Basicproblemsinanalysis

20第二十页，共85页。Whatmodelstouse?AveragedmodelsDiscrete-timemodelsaveragedbehaviourttx(t)xnxn+1simple-detailsdestroyedcomplex-accurate21第二十一页，共85页。Whatmodelstouse?

Clue:Typicalnonlinearbehaviourobservedindc/dcconverters

Standardbifurcations

Hopfbifurcation,Period-doublingbifurcation,2TTtt22第二十二页，共85页。Whatmodelstouse?

Clue:Typicalnonlinearbehaviourobservedindc/dcconvertersPhasespaceview

Hopfbifurcation,Period-doublingbifurcation,viivlimitcyclelow-freq.orbit2Torbitvvii23第二十三页，共85页。Whatmodelstouse?

FastandslowscaledynamicsFastandslowscalesHopfbifurcation,Period-doublingbifurcation,limitcyclelow-freq.orbit2Torbitvviilow-frequencyphenomenon–averagedmodelsarecapablehigh-frequencyphenomenon–discrete-timemodelsareneeded24第二十四页，共85页。Whatmodelstouse?

Clue:Experimentsandcomputersimulationse.g.,Inacurrent-modeboostconverter,period-doublingisfound:bifurcationdiagramIrefiL(nT)25第二十五页，共85页。Whatmodelstouse?

Clue:Experimentsandcomputersimulationse.g.,Inparallelboostconverters,asfeedbackgainincreases,aseriesofchangesareobserved…Stableperiod-1 quasi-periodic limitcycle26第二十六页，共85页。Analysistechniques

ThebasicquestionsofpracticalimportanceareWhereandwhenithappens?

Locationofboundaryofoperation–bifurcationpointHowithappens? Identificationofthetype

ofbifurcation27第二十七页，共85页。Technique–Averagingapproach

Averagingapproach

Deriveasetofcontinuousaveragedequations:

ExaminetheJacobianmatrixJ(XQ)andfindthelocioftheeigenvalueswhenabifurcationparameterisvaried.

Identifytheconditionfortheeigenvalue(s)movingacrosstheimaginaryaxisinthecomplexplane:

e.g.,apairofcomplexeigenvaluesmovingacrosstheimaginaryaxisimpliesaHopfbifurcation.s28第二十八页，共85页。Advantages:Widelyusedandwellknown.Relativelyeasytoderivethecontinuousaveragedequation.Limitation:Onlyabletopredictlow-frequencyslow-scalebifurcationbehavioursuchasHopfbifurcation.Technique–Averagingapproach

29第二十九页，共85页。Discrete-timemapapproach

Deriveadiscretetimemap(iterativefunctionf):ExaminetheJacobian,J(XQ)andfindthelocioftheeigenvalueswhenabifurcationparameterisvaried.

Identifytheconditionfortheeigenvalue(s)movingouttheunitcircleinthecomplexplane:

e.g.,oneoftheeigenvaluesmovingoutthrough-1thenimpliesaperiod-doubling.Technique–Discrete-timemapapproach

30第三十页，共85页。Advantages:Providearelativelycompletebehavioralinformation.Abletopredictstandardbifurcationssuchasperiod-doublingbifurcation,Hopfbifurcationandsaddle-nodebifurcation.Limitation:Derivationoftheiterativemapismorecomplicatedcomparedtothecontinuous-timeaveragedequation.

Technique–Discrete-timemapapproach

31第三十一页，共85页。Basicprinciple:Itusuallyinvolveschangeoftheformofthequalitativemodelbeforeandafterthecollision.Results(reportedinPhysicsliterature):Therearetheoreticalpublications(C.Grebogi,H.E.Nusse,S.Banerjee,M.diBernardo)onthetypeoftransitionatthecollision.Thesetransitionsareusually“sudden”changes,e.g.,

fromperiod-1tochaos fromperiod-1toanotherperiod-1

Analysisofbordercollision

32第三十二页，共85页。Ofengineeringinterestisthelocationoftheborder.Analysisinvolvesidentifyingthemechanismthatcausesthecollision,e.g.,controlsignalswingstofarupordownOnecananalyzethesituationandderiveformulaforthecriticalpoint.Analysisofbordercollision

33第三十三页，共85页。SummaryofobservationsStandard

bifurcationssuchasperiod-doublingandHopfbifurcationarecommonlyobservedindc/dcconverters.Bordercollisionbifurcationcomesintoplaytodisruptthegrowthofstandardbifurcations.—practicallymoreimportantbecausestandardbifurcationisalwaysthefirstbifurcationnexttotheusualstableoperation

—signaturephenomenoninpowerelectronics34第三十四页，共85页。Possibleengineeringapplications

Prediction/BetterunderstandingSystematiccollectionofresultsconcerningbifurcationtoformausefuldesignguide.

operationboundary instabilityfeatures—

[e.g.,recentfindingonPFCconverter]DesignUseofchaoticoperationtoadvantage.

transientspeed? EMC?35第三十五页，共85页。Applicationexample

–bifurcationanalysisinPFCboostconverter

PFCboostrectifier=typicallyaboostconverterundercurrent-programmingcontrolTheinputcurrentisforcedtotracktheinputvoltagewaveshapeNote:essentiallyslopecompensation36第三十六页，共85页。Applicationexample

–bifurcationanalysisinPFCboostconverter

WeobserveasymmetricalslopecompensationinthisPFCboostrectifier:+veslopecompensationin[0,π/2]-->lessstable–veslopecompensationin[π/2,π]-->morestableBifurcationanalysiscanrevealinterestingphenomenon…wherejustforconvenience.Discrete-timemap:37第三十七页，共85页。Applicationexample

–bifurcationanalysisinPFCboostconverter

BifurcationanalysisSupposetheinputvoltageisRelatingthecompensationslopemc

withtheinputvoltagevariationforthePFCcase:mc=–Question:Wouldtherebefast-scaleinstability(e.g.,period-doubling)?Atwhatphaseanglewoulditoccur?38第三十八页，共85页。Applicationexample

–bifurcationanalysisinPFCboostconverter

BifurcationanalysisJacobian:Characteristicmultiplier:Period-doublingoccurswhenl=–1.Criticalphaseangle(algebraomitted):39第三十九页，共85页。Applicationexample

–bifurcationanalysisinPFCboostconverter

Fastscaleinstabilityatphaseangle40第四十页，共85页。Applicationexample

–bifurcationanalysisinPFCboostconverter

Thisformulaallowsustodoanumberofthings:e.g.,Predictingfast-scaleinstability41第四十一页，共85页。Applicationexample

–bifurcationanalysisinPFCboostconverter

Definingstabilityboundaries42第四十二页，共85页。Applicationexample

–bifurcationanalysisinPFCboostconverter

region2region343第四十三页，共85页。SummaryofnonlinearstudyonPEPowerelectronicsengineers/researchersaredealingwithnonlinearproblemsMuchofpowerelectronicsisaboutidentifyingnonlinearphenomenaand“taming”themtodousefulapplicationsRecap:Forpowerelectronics(nonlinearsystemsingeneral), –“stability”referstooperationintheexpectedregime –avarietyofwaysthesystemcanbecomeunstable (togetawayfromtheusualoperation) –anumberofaffectingparameters —BIFURCATIONANALYSIS44第四十四页，共85页。SummaryofnonlinearstudyonPEEngineersfrequentlyask:

Whataretheapplicationsofchaosandbifurcationstudies?Topicsoffutureresearch:

ReorganizingresultsintermsofpracticaloperatingconditionsandparametersDevelopingdesign-orientedbifurcationproceduresIdentifyingnewphenomenainpracticalcircuits45第四十五页，共85页。ToprobefurtherBooksC.K.Tse,ComplexBehaviorofSwitchingPowerConverters,NewYork:CRCPress,2003or2004.S.BanerjeeandG.C.Verghese(Eds.),NonlinearPhenomenainPowerElectronics:Attractors,BifurcationsandNonlinearControl,NewYork,IEEEPress,April2001.Reviewpaper

C.K.TseandM.diBernardo,“Complexbehaviorofswitchingpowerconverters,”ProceedingsoftheIEEE,vol.90,no.5,pp.768–781,2002.JournalsIEEETransactionsonCircuitsandSystemsPartIInternationalJournalofBifurcationandChaosInternationalJournalofCircuitTheoryandApplications46第四十六页，共85页。Contents2.TypicalExamplesIntrouductionFocusontheissuesmorepracticalThesystemsunderstudyFast-andslow-scaleinstabitiesMainparametersPreviousstudiesQuickglimpseattheresultsofsimulationandexperimentQuestionofpracticalimportanceDesign-orientedchartsAnalysis(judgingrules,trackingeigenvaluemovements)Conclusionandfutureworks47第四十七页，共85页。3、TypicalExamplesCoexistingfast-Scaleandslow-scaleinstabilityincurrent-ModecontrolledDC/DCconverters:analysis,simulationandexperimentalresults

IEEETransactionsonCircuitsandSystemsI,55(10):3335-3348，Nov.2008

48第四十八页，共85页。AboutthePaper-IntroductionSystemofinvestigationClosed-loopcurrent-modecontrolledDC/DCconverterscomposedofaninnercurrentfeedback(fast)loopandanoutervoltagefeedback(slow)loop.TwotypesofbifurcationfoundpreviouslyFast-scalebifurcation(period-doubling):innerloopinstabilitySlow-scalebifurcation(Hopf):outerloopinstabilityQuestion:Canthetwobifurcationshappensimultaneously?Ourresearchshows:thetwobifurcationscanhappensimultaneouslytheoperatingboundariesinparameterspaceofvariousregionsincluding

stable,slow-scaleunstable,fast-scaleunstableandcoexistingfast-andslow-scaleunstableregions.49第四十九页，共85页。Focusontheissuesmorepractical

Systemunderinvestigation:basiccurrent-modecontrolledconverters:Boost,BuckandBuck-boost,whereBoostisstudiedindetail,themethodcanbeextendedtotheothers.Modelforpracticalconvertersystem.system’smodelshouldtakeintoaccounttheeffectsoftheparasiticresistancesofinductor,outputcapacitor,diodeandtransistor.Importantparameters.FocusontheeffectsofvaryinginputvoltageEandloadresistanceRonthesystem’sbehaviors,astheycanvarycontinuouslyduringoperation.Methods:Simulation,Analysisbasedonasimplifiedanddesign-orientedmodel,ExperimentalconfirmationDiscussion:

limitationofanalysismethodCanthemethodbeextendedtomoregeneralsystemthatincludestwokindsoftime-scale,fastandslow?50第五十页，共85页。ThesystemsCurrent-modecontrolledBoostconverterinnercurrentloop(fast)outervoltageloop(slow)Currentwaveform51第五十一页，共85页。Peak-currentmodecontrolledboostconvertersCM-PWMcontroller:anoutervoltageloopandaninnercurrentloop.Outerloop:EAandPIcontroller,itsoutputprovidesreferenceforinnerloop.Innerloop:acurrenttransformerandacurrentsensingamplifier(IA),Acompensationrampistostabilizetheconverterifawiderangeofoutputvoltageisrequired.52第五十二页，共85页。flip-floplatchissetperiodicallybyclocksignal,turningontheswitchST.iLgoesuplinearly,andiscomparedwithareferencelevel(vk-Vp),Whenitspeakreachesthereferencelevel,theoutputofthecomparatorresetstheflip-flop,therebyturningofftheswitch.switchisoff,iLfallsalmostlinearlyiftheoutputcapacitorissufficientlylarge.Thecyclerepeatswhentheflip-flopissetagainbytheclock.wheremcistheslopeofthecompensationrampsignal,Misthesamplinggainoftheinductorcurrent,andm1andm2aretherisingandfallingslopesoftheinductorcurrentwiththeswitchonandoff,respectively.Operationofcurrentmodecontrolledconverter53第五十三页，共85页。Current-modecontrolledBuckandBuck-boostconverters54第五十四页，共85页。Fast-scaleandslow-scaleinstabilitiesFast-scalebifurcation(period-doubling)Slow-scalebifurcation(Hopf)ttTTiLiLtTiLtTiL55第五十五页，共85页。MainparametersAffectingfast-scalebifurcation(innerloopinstabilityproblem)Risingslopeofinductorcurrentm1=E/L(HereLischosentobeconstant)CompensationslopemcAffectingslow-scalebifurcation(outerloopinstabilityproblem)VoltagefeedbackgaingFeedbacktimeconstantfAffectingboth

Fromthesimulationandexperiment,E

andRwillaffectboththefastandslow-scalestabilityofthesystems,asEwillaffectthesystem’sduty-cycle,andRwillaffectthesystemsnaturalfrequency.56第五十六页，共85页。PreviousstudiesThetwokindsofbifurcationwerestudiedandreported

separately.Itisshownthatfast-scalebifurcation

cannotbepredictedbyconventionalaveragedmodel.Whileconventionalaveragedmodel

isenoughtoexplain

Slow-scalebifurcation

Discrete-timemodel

oraveragediscrete-timemodelcanbeexpectedtopredictboth.57第五十七页，共85页。QuickglimpseCycle-by-cyclesimulationofthesystemwiththeexactpiecewiseswitchedmodel.Circuitcomponentsareasfollows:58第五十八页，共85页。Quickglimpseatchanginggandfstablefast-scaleunstablesaturationcoexistingfast-andslow-scaleunstableslow-scaleunstableslow-scaleunstable59第五十九页，共85页。Quickglimpse:simulatedresultsforboostatchangingEorRwithdifferentg60第六十页，共85页。Quickglimpse:simulatedresultsforbuckatchangingtheslopeofcompensationramp61第六十一页，共85页。Quickglimpse:simulatedresultsforbuck-boostatchangingtheEandR62第六十二页，共85页。Schematicdiagramofexperimentalboostconverterundercurrent-modecontrol63第六十三页，共85页。ConfirmationfromExperimentalresults(forboostconverter)Experimentalresultswithfs=25kHz,mc=3.5~4mV/us,rL=0.04,rC=0.03,rd=0.01;Rs=0.08R=30mc=0.431V/40us;Ca=8.2nF,Ra=38.98K.ComparedwiththesimulatedresultsundersameconditionsExp.E=3.10VSim.E=3.14VClose-upview64第六十四页，共85页。Exp.E=4.369V,fast-scalebifurcationSim.E=3.75V,fast-scalebifurcationSim.E=4.848V,stableExp.E=4.848V,stable65第六十五页，共85页。Exp.E=2.834V,saturatedSim.E=3.0V,saturated66第六十六页，共85页。ConfirmationfromExperimentalresults(cont.)Ra=47k,Ca=6.8nF,(1)R=25,E=3.239V,slowbifur.Sim.E=3.45

67第六十七页，共85页。ConfirmationfromExperimentalresults(cont.)Ra=47k,Ca=6.8nF,(1)R=39.89,E=2.562V,interactingbifur.

Sim.E=2.80

68第六十八页，共85页。ConfirmationfromExperimentalresults(cont.)withthecompensationslopemc=0andRa=6.8k,Ca=15nF

Interactingbifur.Atfs=27kHz,R=40.32,E=2.682VInteractingbifur.Atfs=27kHz,R=25.05,

E=3.34VInteractingbifur.Atfs=58kHz,R=45,E=2.437V69第六十九页，共85页。QuestionofpracticalimportanceUnderwhatparameterrangesthesystemwillbifurcationinto

fast-scaleunstableregion?slow-scaleunstableregion?interactingfast-scaleslow-scaleunstableregion?70第七十页，共85页。Design-orientedcharts-Operating

boundariesforboostin(R,E)parameterspace,withdifferentfeedbackgain71第七十一页，共85页。Design-orientedchartsOperatingboundariesin(g,E)parameterspacewithdifferenttimeconstant72第七十二页，共85页。Design-orientedchartsOperatingboundariesin(g,E)parameterspacewithdifferenttimeconstant73第七十三页，共85页。Analysis-DerivationofStateEqu.State1：STonandSDoff,fornTs≤t≤(n+d)TsState2：SToffandSDon,for(n+d)Ts≤t≤(n+1)Ts(1)74第七十四页，共85页。Analysis-DerivationofStateEqu.(1)(2)(3)Wherek0=Vref(1+gkd),andk1=[0–g/(1+kc)-1].Note:weonlyneedtoconsidertheexpressionofcontrolsignalvkduringstate175第七十五页，共85页。Analysis-DerivationofStateEqu.76第七十六页，共85页。Analysis-DerivationofSimplifiedDiscrete-timeModelTheformofthediscrete-timemapweaimtogetisthestateequationforthecircuitinanyswitchstatecanbewrittenintheformofalineardifferentialequation:Att=nTs,thesystementersstate1,andatt=(n+dn)Ts,goestostate2.77第七十七页，共85页。AnalysisReferthepublishedpaperfordetails.Summary:Derivethediscrete-timeiterativemapthatdescribesthedynamicsoftheentiresystem:DerivetheJacobian:Examinetheeigenvalues.78第七十八页，共85页。Analysis-judgingrulesAlltheeigenvaluesinsidetheunitcircleindicatesstableoperation.Slow-scalebifurcationoccurswhenapairofcomplexeigenvaluesmoveoutofthe