Bifurcation-behavior-in-parallel-connected-boost-c

IEEETRANSACTIONSONCIRCUITSANDSYSTEMS—I:FUNDAMENTALTHEORYANDAPPLICATIONS,VOL.48,NO.2,FEBRUARY2001233BifurcationBehaviorinParallel-ConnectedBuckConvertersH.H.C.IuandC.K.TseAbstract—Thispaperdescribesthebifurcationphenomenaofasystemofparallel-connecteddc/dcbuckconverters.Theresultsprovideusefulin-formationforthedesignofstablecurrentsharinginamaster–slaveconfig-uration.Computersimulationsareperformedtocapturetheeffectsofvari-ationofsomechosenparametersonthequalitativebehaviorofthesystem.Thesearesummarizedinaseriesofbifurcationdiagrams.Inparticular,itisfoundthatwhilevariationofthevoltagefeedbackgainsleadstostandardperiod-doublingbifurcation,variationofthecurrentsharingratioleadstobordercollisionbifurcation.Analysisispresentedtoestablishthepossi-bilityofthebifurcationphenomenaandtolocatethecurrentsharingratioatwhichbordercollisionoccurs.IndexTerms—Bifurcation,buckconverter,paralleldc–dcconverter.I.INTRODUCTIONParallelingpowerconvertersallowshighcurrenttobedeliveredtoloadswithouttheneedtoemploydevicesofhighpowerrating.Themaindesignissueinparallelconvertersisthecontrolofthesharingofcurrentamongtheconstituentconverters.Ifadc/dcconverterisre-gardedasavoltageregulatorthatprovidesverystiffvoltagetoaload,thenitistheoreticallyimpossibletoputtwosuchconvertersinparallelfeedingthesameloadandsharingequalcurrent,unlessthetwocon-vertersareperfectlyidentical.Inpractice,mandatorycontrolisneededtoensurepropercurrentsharing,andmanyeffectivecontrolschemeshavebeenproposedinthepast[1]–[4].Onecommonapproachistoem-ployanactivecontrolschemetoforcethecurrentinoneconvertertofollowthatoftheother.Theessenceofthiscontrolapproachistomon-itorthedifferenceoftheoutputcurrentsintwoconstituentconverters(i.e.,currenterror)andincorporatethisinformationinthemainvoltagecontrolloop.Specifically,forthecaseoftwoconvertersconnectedinparallel,oneconvertersimplyhasavoltagefeedbackcontrolwhiletheotherhasanadditionalinnercurrentloopthatprovidesthecurrenterrorinformationwhichisusedinturnto“adjust”thevoltagefeedbacklooptoensureequalsharingofcurrent.Suchaschemeiscommonlyknownasthemaster–slavecurrent-sharingscheme[1],[3].Nonlineardynamicsandbifurcationbehaviorareimportanttopicsofinvestigationinpowerelectronics[5]–[12].Asparallelconvertersys-temsgainpopularityinpowerelectronicsapplications,thereisastrongmotivationforbetterunderstandingoftheirnonlineardynamicsandbi-furcationbehavior.Inthispaper,weattempttoprobeintosomenon-linearphenomenaofasystemofparallel-connectedbuckconverterscontrolledunderamaster–slavecurrent-sharingscheme.II.MASTER–SLAVECONTROLLEDPARALLEL-CONNECTEDDC/DCCONVERTERSThesystemunderstudyconsistsoftwodc/dcconverterswhichareconnectedinparallelfeedingacommonload.ThecurrentdrawnbyManuscriptreceivedAugust16,1999;revisedMarch2,2000.ThisworkwassupportedbytheResearchGrantsCouncil,HongKong,underresearchGrant(PolyU5131/99E).ThispaperwasrecommendedbyAssociateEditorM.D.Bernardo.TheauthorsarewiththeDepartmentofElectronicandInformationEngi-neering,HongKongPolytechnicUniversity,HongKong,China(e-mails:[e-mail];[e-mail]).PublisherItemIdentifierS1057-7122(01)01388-5.Fig.1.Blockdiagramofparallel-connecteddc/dcconvertersunderamaster–slavecontrol.Fig.2.Pulse-widthmodulation(PWM)showingrelationshipbetweenthecontrolvoltageandthePWMoutput.theloadissharedproperlybetweenthetwobuckconvertersbytheactionofamaster–slavecontrolscheme,asmentionedbrieflyintheprecedingsection.Fig.1showstheblockdiagramofthismaster–slaveconfiguration.DenotingthetwoconvertersasConverter1andConverter2asshowninFig.1,theoperationofthesystemcanbedescribedasfollows.Bothconvertersarecontrolledviaasimplepulse-widthmodulation(PWM)scheme,inwhichacontrolvoltagevconiscomparedwithasawtoothsignaltogenerateapulse-widthmodulatedsignalthatdrivestheswitch,asshowninFig.2.ThesawtoothsignalofthePWMgeneratorisgivenbyvramp=VL+(VUVL)tmodTT(1)1057–7122/01$10.00©2001IEEE234IEEETRANSACTIONSONCIRCUITSANDSYSTEMS—I:FUNDAMENTALTHEORYANDAPPLICATIONS,VOL.48,NO.2,FEBRUARY2001Fig.3.Twoparallel-connectedbuckconverters.whereVLandVUaretheloweranduppervoltagelimitsoftheramp,andTistheswitchingperiod.ThePWMoutputis“high”whenthecontrolvoltageisgreaterthanVramp,andis“low”otherwise.ForConverter1,thecontrolvoltageisderivedfromavoltagefeed-backloop,i.e.,vcon1=VosetKv1(vVref)(2)whereVosetdcoffsetvoltagethatgivesthesteady-statedutycycle;Vrefreferencevoltage;Kv1voltagefeedbackgainforConverter1.ForConverter2,anadditionalcurrenterrorsignal,whichispropor-tionaltotheweighteddifferenceoftheoutputcurrentsofthetwocon-verters,determinesthecontrolvoltage.Specificallywewritethecon-trolvoltageforConverter2asvcon2=VosetKv2(vVref)Ki(i2mi1)(3)whereKv2voltagefeedbackgainofConverter2;Kicurrentfeedbackgain;mcurrentweightingfactor.Underthisscheme,theoutputcurrentofConverter2willfollowthatofConverter1ataratioofmto1,wherem0.Whenm=1,weexpectequalcurrentsharing.Inmuchoftheliterature,Converter1isreferredtoasthe“master”whichoperatesindependently,andConverter2the“slave”whichimitatesthemaster’scurrentvalue.III.STATEEQUATIONSFORTWOPA