OptimalGeneralizedOvermodulationforMultiphasePMSMDrives

PaulYoungandMatthiasPreindlDepartmentofElectricalEngineering,ColumbiaUniversity,NewYork,NY10027

Email:py2227@columbia.edu,matthias.preindl@columbia.edu

Abstract—MultiphasesystemshavethecapabilitytoachievesignificantlyhigherdrivevoltagesthanwhatispossiblewithsinusoidalPWMorlinear modulation/nthharmonicinjection.However,overmodulationbeyondthatofthezerosequenceyieldslargeharmoniccurrentsifnotimplementedproperly.Thispaperpresentsaschemethatenablesthisfullrangeoftheαβvoltagevectorwhile minimizingtheexcitationofthecomponentsinthehigher-orderdq(xy)plane(s).First,themaximumlimitsofovermodulationdrivevoltageamplitudesaredefinedforN>3phasesystemsusingvectorspaceinterpretationoftheDCbusvoltageconstraints.Next,thetheoreticallowerlimitsofhar-monicsequencesrequiredforovermodulationareestablishedforN>3phasesystemsbyformulationofaparametricquadraticprogrammingproblem. Withanovermodulationschemethatusesthisoptimizedsolution,adrivesystem’sspeedratingisimprovedduetotheincreasedvoltage,asitiswithnon-optimizedovermodulationschemes.However,sincetheharmoniccurrentsareminimized,thedrivevoltageimprovementislessrestrictedwhentakingintoaccountthesystem’scurrentratings.Theresultingincreasedrangeofoperationisdemonstratedfora9phasePMSMmodelwithRMScurrentconstraintsandisshowntobesupplementarytofluxweakening.

I.INTRODUCTION

Inthefieldofhighpowerelectricdrivesmultiphasema-chineshavebeenshowntoexhibitseveraladvantagesovertheir3phasecounterparts[2],[5],[12].Theincreasednumberofdevicesmakesthemmoreaptforhigherpower[2],[5].Theirredundancygivesthemaninherentfaulttolerancewhichiscrucialinhighreliabilityapplications,suchaselectricvehiclesandaerospaceapplications[2],[5],[12].Propercontroltechniquesallowforimprovedtorqueperformance[2],[5],[12].Inaddition,theincreasedamountofindependenthalf-bridgeslowerstheDCbusswitchingripple[2],[5].

Anotherimportantaspectisthepresenceofadditionaldegreesoffreedomthatallowamultiphasevoltagesourceinverter(VSI)togenerateahigherdrivevoltagethanthatallowedbysinusoidalPWMorlinearmodulationgiventhesameDCinputvoltage[2],[3].AsignificantamountofworkhasbeendoneonPWMandSVPWMstrategiesthatcapitalizeon DCbusutilizationor“overmodulation”capability,ingeneral[1]-[15].Linearmodulation,inparticular,iscoveredextensivelyinliteratureduetothefactthatitintroducesnoparasiticcurrentsforaneutral-isolatedsystem[11],[12].However,thefullpotentialdrivevoltageisnotattainedforphasenumbersgreaterthan3,andfurthermore,thebenefitsdiminishasthenumberofphasesincreases[3],[7],[13].Largevectormodulation,asdescribedin[2]and[3],enables

themaximumtheoreticaldrivevoltagetobeattainedforNphases,howeveraverylargeamountofharmoniccurrentsaregeneratedduetothelargeexcitationofvoltageharmonicsthatappearacrossleakageimpedances.Thisisaddressedin[3]toanextentfora5phasesystem,andtwomethodsarepresentedthatallowthissamevoltagecapabilitytobeachievedwithsignificantlylessharmoniccurrentsbyutilizingbothmediumandlargevectors.Intermsofatheoreticallimit,however,theoptimalwaytoutilizeallofthex,y,andγharmonicsequencesinanNphasesystemtoattainagivenovermodulationlevelisstillanopentopic.Thispaperpresentsageneralizedmethodforovermodula-tionofmultiphasedrivesthatachievesthefullvoltagerangebeyondthatoflinearmodulationandestablishesthetheoreticalminimumamplitudeofthexyharmoniccomponents.First,themaximumlimitsofdrivevoltagearedefinedforN>3phasesystems,anditisshownthatsomelevelofharmonicsequencesisnecessarytoattainhigherovermodulationlevels.Aparametricquadraticprogramisthenformulatedthatde-terminestheminimumrequiredmagnitudeoftheseinjectedcomponentsasafunctionofthetime-varyingαβvectorforanNphasesystem.Aparametricsolutionisobtainedforonlineimplementation.Finally,itisshownhowthisoptimalovermodulationmethodcanbeutilizedtoincreasethespeedcapabilityofaPMSMasasupplementtofieldweakening.

II.ANALYSISOFMULTIPHASEVOLTAGELIMITATIONS

A.ClarkeTransformation

The N-Phase Clarketransformation,C,definedin[2](amplitude-invariant)asasquareN×N matrix,isusedtoconvertavoltage(orcurrent)vectorbetweenClarkedomainandphasedomaingivenby

Vclarke = CVphase

= VαVβVx1Vy1Vx2Vy2···Vγ+ Vγ−T

= VαβT Vxyγ

T T

(1)Vαβ consistsofthe2voltagecomponentsthatareactuatedbythemaincontrolloopstodrivethetorqueofthemachine[1]andisreferredtoasthe‘drivevoltage’inthispaper.TheN−2componentsinVxyγareauxillarysequencesthathavenegligibleeffectonthetorque[1]butcanbeappropriatelymodulatedtoincreasetheviablerangeofVαβamplitudes[13],aswillbeshownlaterinthissection.Notethatthispapertreats

theconditionofalinearPMSMmodelwhereonlytheα,βcomponentsproducetorque,asisthecaseforthemodelin[1].

B.VoltageConstraints

InordertodeterminethelimittowhichtheVxyγsequencesmaybeusedinthedrivesystem,thevoltagemodulationlim-itationsofthesystemmustbedefined.Thevoltageconstraintisbasedonthecapabilityofeachinverterlegtoindependentlygenerateanaveragevoltagethatcantakeonanyvaluebetween±VDC2 oftheinputDCvoltage.Thisisrepresentedbytheexpression

Vphase∈ −VDC2,VDC2

N

(2)

Vphase isrepresentedasavectorrestrictedtoan N-dimensional‘cube’withsidelengthofVDC.TransformingthisexpressionintoClarkedomainyields

Vclarke∈V=C −VDC2,VDC2

N

(3)

InClarkedomain,Vclarke isrestrictedto N-dimensional‘rotatedcube’orpolytopeV.ThetheoreticalmaximumsetofVαβ voltageswhichtheinverteriscapableofgeneratingisgivenbytheorthogonalprojectionofVontothe(two-dimensional)α,βplane,usingthedefinitionofprojectionfrom[7]

Vp=Proj1,2(V) (4)

AchievingtheentirerangeofVprequiresnontrivialsequencesofVxyγ componentsinordertokeepVclarkewithintheconfinesofpolytopeV(i.e.tokeepanyoftheinverterPWMsignalsfromsaturating). WhenrestrictingallVxyγcomponentstozero(i.e.standardsinusoidalPWM),thesetofinverteroperationisthenlimitedtotheintersectionoftheα,βplaneandpolytopeV,definedastheslicefrom[7]

Vs=Slice3,...,N(V) (5)

Linearmodulation,ornthharmonicinjection,iswhentheγ−

sequenceismodulatedbutxysequencesarerestrictedtozero,(i.e.3rdharmonicinjectionina3phasesystem).Thisisthemaximizeddrivevoltagewherenoharmoniccurrentsoccur[12].Thesetwhichthislevelofmodulationiscapableofreachingisachievedthroughacombinationoftwooperationsgivenby

Vn=Slice3,...N−1(Proj1,...N−1(V)) (6)

wheretheγ−componentisprojectedontotheα,βplane,andallthexysequencesaresettozero.Figure1generatedusingthepolyhedronlibraryofthe MultiParametricToolbox[7],displaysthe(normalized)Vs,Vn,andVpsetsforinvertersofphasenumberN=3through10inordertoillustratethelargerrangeofVαβenabledbyutilizingtheVxyγcomponents.NotethatforN=3phases,VnisidenticaltoVpduetothefactthattherearenoxysequencesina3phasesystem.AsNincreases,therangeofVndiminishestoVsforoddN.ForevenN,VnisalwaysequaltoVs.

Fig.1:PolytopeprojectionsnormalizedtoVDC2 :innerpoly-gons(Vs)arethemaximumdrivevoltagerangesattainablewithstandardsinusoidalmodulation,theintermediatepoly-gons(Vn)aretherangesenabledbylinearmodulation(i.e.3rdharmonicinjectioninthecaseofa3phasesystem),andtheouterpolygons(Vp)arethetheoreticallimitsforNphases.

N Vs Vn Vp3 1.00 1.155 1.1554 1.00 1.00 1.005 1.00 1.051 1.2316 1.00 1.00 1.1557 1.00 1.026 1.2528 1.00 1.00 1.2079 1.00 1.015 1.26010 1.00 1.00 1.231∞ 1.00 1.00 1.273

Fig.2:Tableofinscribedradii(normalized),correspondingtothemaximumpossiblesinusoidalα,βamplitudeswhenoper-atinginstandardsinusoidalmodulation(Vs),linearmodula-tion/nthharmonicinjection(Vn),andthemaximumtheoreticallimit(Vp)forNphases

SincethispaperonlyaddressesthecaseofsinusoidalVαβwaveforms,thesteadystatetrajectoryontheα,βplanewillbeassumedtobecircular.Therefore,thelargestpossibletrajectoryofVαβ isdefinedbytheradiusoftheinscribedcircleoftheconstrainedrangeontheα,βplane.Figure2showsthemaximumsteadystateamplitudesofVαβforsetsVs,Vn,andVpfromFigure1.Thesenumericalresultsareconfirmedbytheworkdonein[6]forthelinearmodulationcasesofoddNandtheworkdonein[3]forVpofN=5.

III.QUADRATICPROGRAM WITHVOLTAGECOSTFUNCTION

A.Formulation

AswasshowninFigures1and2,thereissignificantdrivevoltagecapabilitytobegainedbyutilizingtheVxyγcomponents,especiallyasthenumberofphasesincreases.

However,sincetheVxyγvoltages,minustheγ− component,

appearacrossleakageimpedancesandgenerateparasiticcur-rents[1],[11],itiscrucialtokeeptheiramplitudesaslowaspossible.Inthissection,aquadraticprogrammingproblemwillbeformulatedthatminimizes||Vxy||asafunctionofVαβwhilemeetingthevoltageconstraintsoftheinverter.Thesystemvoltageconstraintsfrom(2)arerepresentedasasystemof2Ninequalitiesby

Vphasei = [TVclarke]i≤VDC2

−Vphasei = −[TVclarke]i≤VDC2

(7)

whereT=C−1andsubscripti=1toNdenoteseachele-mentofVphase.WhenVclarkeissplitupintoitspreviously-definedconstituentsections,Vαβ andVxyγ,theleftsideoftheinequalitiesin(7)canbeexpressedas

TVclarke=T1,2Vαβ+T3−N Vxyγ (8)

whereT1,2isamatrixconsistingofthefirsttwocolumnsofT,andT3−N isamatrixconsistingofcolumns3throughNofT.InthisanalysisthecomponentsofVxyγarethedecisionvariables,whereasVαβ areknownquantitiesdefinedbythecontrolloops.Thus,termsin(8)associatedwithVαβcanbemovedtotherightsideoftheinequalitiesin(7).Thisyields

[T3−N Vxyγ]i ≤ VDC2 −[T1,2Vαβ]i

[−T3−N Vxyγ]i ≤VDC2 +[T1,2Vαβ]i

(9)

Thissystemofinequalitiescannowbeusedtodefinetheconstraintsofaparametricquadraticprogrammingproblemoftheform

minimize 12xTHx+fTx

s.t.Ax≤b+Fθ(10)

asdescribedin[16]where

A=T3−N−T3−N

b=VDC2 F=

−T1,2T1,2

x=Vxyγ θ=Vαβ

(11)

Minimizationof Vxy2isrepresentedinthequadraticpro-

grammingcostfunctionwith

H=I(N−3)×(N−3) 0

0 0f=0 (12)

ThetermsalongthediagonalofHcorrespondtoVxi2and

Vyi2coefficients.ThelastdiagonalentryinHis0because

itcorrespondstoV2γ−.Thesolutionforxthatminimizesthecostfunction(asafunctionofθ)isreferredtoasx∗.

B.Implementation

Forsimulationin MATLAB,theoptimalsolutioncanbecalculatedinreal-timebysolvingtheoptimizationproblemin(10)ateverytimestepasafunctionofθ(Vαβ)usinganonlinequadraticprogrammingfunctionfrom[14].ThisisverycomputationallyexpensivewithrespecttopowerelectronicssamplingtimesandnotpracticalforDSPimplementation.However,itisknownthatthesolutionofaparametricopti-mizationprobleminthisformisacontinuous,piecewiselinear

functionofVαβ[16].Therefore,theoptimalovermodulationsolutioncanbepredeterminedandimplementedwithalookuptableandasetoflinearoperations.ThesolutionovertheinputrangeofVαβ∈Vpisdividedintoanumberof‘criticalregions’asshowninFigure3,whichwasgeneratedusingtheMultiParametricToolboxOp-timizationInterface[7].Ineachcriticalregion,thesolutionx∗

(Vxyγ∗)isanaffinefunctionofVαβ.TheN−2elementVxyγ

∗

iscalculatedbymultiplyingVαβbyan(N−2)×2JacobianmatrixandaddinganN−2elementoffsetvector.Thesetermsweredeterminedfromsensitivityanalysisobtainedfrom[16].Fortheonlinealgorithm,asetofinequalitiesisfirstusedtodeterminethecriticalregioninwhichtheVαβ vectorlies,utilizingtheradialsymmetryofthesetofpolygonregions.TheJacobianmatrixandoffsetvectorelementscorrespondingtotheselectedcriticalregionarethenretrievedfromalookuptableandusedtocalculateVxyγ

∗.Figure4,alsogeneratedusingtheOptimizationInterfaceof[7],displaystheVxiandVγ− componentsofthesolutionVxyγ

∗asafunctionofVαandVβfora9phasesystem. WhenVαβ followsacirculartrajectoryoutsideofVsorVnontheα,βplane,theharmonicspresentineachcomponentcanbeobservedintheseimages.Figure5showssimulated(switchingcycleaverage)phasevoltagewaveformsfora5phaseVSIwithasteadystateVαβamplitudeof1.15p.u.using3separateovermodulationmethods:thelargevectormethodfrom[2]and[3],method1from[3]thatutilizesbothmediumandlargevectors,andtheoptimalsolutionmethod.Figure6comparesthex1voltagesofthese3conditionsasthefigureofmeritandshowsthattheoptimalsolutionmethodwillgenerateanx1sequencewithroughly30%lessRMSthanmethod1from[3]atthisVαβamplitude.

IV.ANALYSISOFVIABLERANGEOFSPEED

A.RMSCurrentConstraint

Inordertoquantifytheimprovedspeedratingenabledbytheincreaseddrivevoltage,aconstraintmustbeplacedontheRMScurrentthattakesintoaccounttheparasiticcurrentsgeneratedbytheovermodulationscheme.Thislimitationwillensurethatthepowerdissipationratingsintheinvertersolidstateswitches,thewindings,andtherotorarenotviolated[8]whichisgivenby

IphRMS ≤Irating (13)

BasedontheClarkeTransformationfrom[2]ofphasecur-rents,assumingallphaseshavethesameRMScurrentundersteadystateconditions,eachphasecurrentcanbeexpressedas

IphRMS =IαRMS

2+IβRMS2+ IxyRMS

2

2(14)

whereIαandIβaretheonlycomponentsthatcontributetothetorqueofthemachine,andIxyareparasiticcurrentsgeneratedbytherespectiveVxycomponents[1].Asaresult,givenanoperatingtorque,Iratingputsanupperboundontheamount

(a)N=5Phases

(b)N=7Phases

(c)N=9Phases

Fig.3:Criticalregionsofparametricquadraticprogrammingsolutionoverfullrange(Vαβ∈Vp)for5,7,and9phases.Thesolutionineachpolygonregionislinear.Thelargeregionineachcentercorrespondstostandardsinusoidalmodulation(Vs)whereVxyγ

∗=0.

ofIxyand,therefore,Vxythatmaybegenerated.ThisinturnlimitsVαβ toasmallersetthanVp.Thus,minimizingVxymodulationisimportantbecausetheharmoniccurrentstendtobethelimitingfactorofovermodulationlevel.

B.TheoreticalRangeofOperationforVs,Vn,andVp

-0.2

1

0

X1

1

-

0

0.2

,

0-1 -1

BelowaretheequationsdescribingtheoperationofanNphasePMSMwithsinusoidally-distributedwindings,obtainedfrom[1],[4],[10],and[11]

(a)Vx1

-0.4

-0.2

1

0

X2

1

0.2

-

0

,

0.4

0-1 -1

Solution

(b)Vx2

-0.2

1

0

X3

1

-

0

0.2

,

0-1 -1

Solution

(c)Vx3

-0.1

1

0.-

1

-

0

,

0.1

0-1 -1

Solution

(d)Vγ− Solution

Fig.4:Vxi∗andVγ−

∗componentsoftheparametricquadraticprogrammingsolutionoverVαβ∈Vpfora9phasesystem.NotethatasVαβ followsacirculartrajectory,eachcompo-nentfollowsthecontoursofitsrespectivesurface,andtheharmonicspresentcanbevisualized.

Vd = Lddiddt+Rsid−wLqiq

Vq = Lqdiqdt+Rsiq+wLdid+wψ

Te = N4p(ψiq+(Ld−Lq)idiq)

Vxy = LldIxydt +RsIxy

(15)

Inaddition,VxyγasafunctionofVαβisestablishedbytheparametricsolutiontothequadraticprogrammingproblemin(10).

Vxyγ∗=MPQP(Vαβ) (16)

Figure7showsthetheoreticalsteadystatespeedasafunctionoftorquefora9phasePMSMmodelbasedonequations(14),(15),and(16)andtheconstraintsfrom(2)and(13).ThismodelusesthesameparametersandVDC astheBLPMmachinefrom[2].Curvesareshownfor6differentscenariosincludingwithandwithouttheuseoffluxweakening[8].Fromthegraph,itisshownthattheoptimalovermodulationallowssignificantspeedenhancementoverlinearmodulation(whenoperatingbelowmaximumtorque)duetotheincreaseddrivevoltagecapability.Thisistrueregardlessiffluxweaken-ingisused,implyingthatthisoptimalovermodulationcanbeusedtosupplementthebenefitsoffluxweakening[8].Notethatasthetorqueincreases,IαandIβincrease,leavinglessroomforIxy.Atfulltorque,theoptimalovermodulationrangeconvergestoVn.

0 : 2: 3:

3

-1

0

1

Phase A

0 : 2: 3:

3

-1

0

1

Phase A

(a)LargeVector

0 : 2: 3:

3

-1

0

1

Phase A

(b)MediumandLargeVectorMethod1from[3]

(c)OptimalSolution

0 : 2: 3:

3

-0.5

0

0.5

X1 Voltage

Fig.5:Phasewaveformsof3methodstoachieve5phaseovermodulationatradius=1.15p.u.

Fig.6:Harmoniccurrent-generatingx1 voltagesofthe3methodsfromFigure5toachieve5phaseovermodulationatradius=1.15p.u..BlueisLargeVector Method,RedisMediumandLargeVector Method,andYellowisOptimalSolution.y1iscomparabletox1andthus,isnotshown.

V.RESULTS

Figure8showstheblockdiagramofamultiphasedrivesystemwith20kHzswitchinganda9phasePMSMmodel.Thismodelisbasedonthemachinefrom[2],aswell.Theouterspeedloopandinneridandiqcurrentloopsdeter-mineVαβ todrivethetorqueofthemachine.TheoptimalovermodulationblockthencalculatestheremainingVxyγ

∗

componentsinrealtimeusingtheLUTandVαβ

0

50

100

150

200

250

300

350

0 2 4 6

w [rad/s]

Torque [Nm]

Vαβ ϵ Vs

Vαβ ϵ Vn

Vαβ ϵ Vp(optimal solution)

.CurrentmeasurementsaretransformedintoParkdomaininorderto

0

50

100

150

200

250

300

350

0 2 4 6

w [rad/s]

Torque [Nm]

Vαβ ϵ Vs

Vαβ ϵ Vn

Vαβ ϵ Vp(optimal solution)

(a)withoutfluxweakening

(b)withfluxweakening

Fig.7:Maximizedspeedof9phasePMSMequationsunder6conditions:Vαβ∈Vs(sinusoidalPWM),Vαβ∈Vn(linearmodulation),andVαβ ∈Vp(withoptimalsolution),withandwithoutfluxweakening.AllcasesmeettheconstraintofIphRMS ≤3A

processthecomponentsseparately:idandiqarefedbackintothetorque-drivingcurrentloops,andixycomponentsarefedintotheoptimalovermodulationblockinordertomakeanyoffsetcorrections.

Threesteadystateconditionswithaloadtorqueof5NmandIphRMS limitof3AaresimulatedandwaveformsareshowninFigure9(nofluxweakeningwasused).Theoptimalover-modulation(middlerow)strategyallowedamaximumspeedimprovementof6.8%overlinearmodulation(toprow)duetotheincreaseinavailabledrivevoltage.Sincetheoperationwasnotatthefulltorquerating,thesmallamountofharmoniccurrentscouldbetolerated.ThebottomrowofFig9showsthesamemodulationindexandspeedastheoptimalsolutiononlyimplementedusingtheLargeVectorMethodfor9phasesfrom[2]asabenchmarkreference(Theimprovedlowerharmonicmethodsfrom[3]weredevelopedfor5phases).Thismethodhastheavailabledrivevoltagebutproducesharmoniccurrents

αβxy0 

abc

Optimal Overmodulation CalculationsPIPI

PI

dq

αβ 

∑ 

∑ 

encoder 

d/dt∑ ω* 

Id* 

Vα 

Vβ  

Vd 

Vq 

Iq* Ixy 

Vxyɣ  

VPH 9 ɸ PMSM

9 phaseVSI

ɵ 

αβxy0 

abc

IPH 

dq

αβ 

Iα 

Iβ 

ɵ 

ɵ 

Fig.8:BlockDiagramof9phaseVSIdrivesystemwithoptimalovermodulationblock

severaltimestheamplitudeofthatgeneratedbytheoptimalsolution.Notethatthecurrentharmonicsareonlyduetothexysequencesand,thus,havenegligibleaffectonthetorquequalityinthecaseofsinusoidally-distributedwindings.

VI.CONCLUSION

Thispaperpresentsageneralizedmethodforovermodu-lationthatestablishthetheoreticallowerlimitsofharmonicvoltagesequencesbyformulationofaquadraticprogrammingproblem.Usingthisoptimalsolution,theincreasedspeedratingduetoincreaseddrivevoltagecapabilityanddecreasedcurrentstressisdemonstratedina9phasePMSMsystemmodel.Itisshownthatthisoptimalovermodulationcanbeusedtosupplementthebenefitsprovidedbyfluxweakening.

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60 80 100

time[msec]

-50

0

50

Volts

60 80 100

time[msec]

-5

0

5

Amps

,24-26June2009.

60 80 100

time[msec]

-50

0

50

Volts

60 80 100

time[msec]

-5

0

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Amps

(a)Maximizedlinearmodulationatm=1.01andw=248

60 80 100

time[msec]

-50

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50Volts

60 80 100

time[msec]

-20

0

20

Amps

(b)Overmodulationusingoptimalsolutionatm=1.075andw=265

(c)OvermodulationusingLargeVectorMethodatm=1.075andw=265

Fig.9:Simulated9phasePMSMwaveformsfor3conditions:(top)maximizeddrivevoltageforlinearmodulation(Vαβ∈Vn),(middle)maximizedovermodulationforIphRMS ≤3Ausingoptimalsolution,and(bottom)overmodulationusingLargeVectorMethodasbenchmarkreference

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