A Virtual synchronous machine implementation for distributed control of power transformers in SmartGrids

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Electric Power Systems Research

jou rn a l h om ep a g e :w w w . e l s e v i e r . c o m / l oc a t e / e p s r

A Virtual Synchronous Machine implementation for distributed control of power converters in SmartGrids

Salvatore D’Arco

^a

, Jon Are Suul

^a,b,∗

, Olav B. Fosso

^b

aSINTEFEnergyResearch,7465Trondheim,Norway

bDepartmentofElectricPowerEngineering,NorwegianUniversityofScienceandTechnology,7495Trondheim,Norway

a r t i c l e i n f o

Articlehistory:

Received1February2014

Receivedinrevisedform18October2014 Accepted3January2015

Availableonline2February2015

Keywords:

Distributedgeneration Energyconversion Inertiaemulation Powerelectroniccontrol Small-signalstability VirtualSynchronousMachine

a b s t r a c t

Theongoingevolutionofthepowersystemtowardsa“SmartGrid”impliesadominantroleofpower electronicconverters,butposesstrictrequirementsontheircontrolstrategiestopreservestabilityand controllability.Inthisperspective,thedefinitionofdecentralizedcontrolschemesforpowerconverters thatcanprovidegridsupportandallowforseamlesstransitionbetweengrid-connectedorislanded operationiscritical.Sincethesefeaturescanalreadybeprovidedbysynchronousgenerators,theconcept ofVirtualSynchronousMachines(VSMs)canbeasuitableapproachforcontrollingpowerelectronics converters.ThispaperstartswithadiscussionofthegeneralfeaturesofferedbytheVSMconceptin thecontextofSmartGrids.AspecificVSMimplementationisthenpresentedindetailtogetherwithits mathematicalmodel.Theintendedemulationofthesynchronousmachinecharacteristicsisillustrated bynumericalsimulations.Finally,stabilityisassessedbyanalysingtheeigenvaluesofasmall-signal modelandtheirparametricsensitivities.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Theincreasingpenetrationofpowergenerationfromrenewableenergysourcesandthetransitionfromacentralizedpowerproduction modeltodistributedgenerationareexpectedtoposeseriouschallengestothedevelopmentandoperationoffuturepowersystems.This tendencyisastrongmotivationbehindtheparadigmshiftfromthetraditionalpowersystemarchitecturetowardsanapproachensuring moreflexibilityandcoordinationbetweenthegenerationunitsandloadsthatispromisedby“SmartGrids”[1].Atthesametime,theshare oftheelectricpowertransferredthroughthepowersystemwhichisprocessedbyatleastonepowerelectronicconversionstageinthe pathfromprimaryenergyconversiontofinalconsumptioniscontinuouslyincreasing.Alreadyin2007,itwasestimatedthatthisshare wouldreach80%around2015[2],andevenifthedevelopmenthasbeenslightlyslower,suchahighshareofpowerelectronicconversion isexpectedtobeexceededduringthecomingyears.Thus,powerelectroniccontrolwillhaveacrucialroleintheemergingSmartGrid scenario,asthepresenceofpowerconvertersinthepowersystemandtheirimpactonglobalstabilityandcontrollabilitycontinuesto increase.

AlthoughtheongoingSmartGriddevelopmentspointtowardsanincreasinglevelofcommunicationandintegrationbetweenvarious elementsofthepowersystem,distributedarchitectureswithlocalprimarycontrolofconverterscombinedwithcentralizedsecondary controlseemtobeanappropriateapproachforoptimizingsteady-stateoperationwhileensuringimmediateresponsetotransientevents.

Thus,converterunitsshouldbeabletoreactautonomouslytoabruptchangesinthepowersystemoperatingconditions,whilecomplying onalongertimescalewiththeset-pointsandservicerequirementsrequestedbythesystemoperatorthroughexternalcommunication.

Inclassicalpowersystems,theSynchronousMachine(SM)withspeedgovernorandexcitationcontroloffersfavourablefeaturesto supportthesystemoperationwithinadistributedcontrolscheme.Indeed,SMscontributetothesystemdampingthroughtheirinertia, participateintheprimaryfrequencyregulationthroughthedroopresponseofthespeedcontroller,andprovidelocalcontrolofvoltageor reactivepowerflow.Thesecapabilities,andespeciallytheinertialanddampingresponsecommontoallSMs,arenotinherentlyofferedby thepowerelectronicsinterfacescommonlyadoptedfortheintegrationofrenewableenergysources.Adistributedmodelforproduction

∗Correspondingauthorat:SINTEFEnergyResearch,7465Trondheim,Norway.Tel.:+4795910913;fax:+4773594279.

E-mailaddress:Jon.A.Suul@sintef.no(J.A.Suul).

http://dx.doi.org/10.1016/j.epsr.2015.01.001

0378-7796/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

andlocalcontrolisalsoopeningthepossibilityofislandedoperation,whichisinherentlyfeasiblewithoneormorecontrollableSMsin theislandedarea.Suchislandingoperationisusuallymorecomplextoachievewithpowerconverterinterfacesdesignedforintegration withalarge-scalepowersystem.

Powerfrommanytraditionallarge-scalegenerationfacilitiesiscurrentlybeingreplacedbydistributedgenerationcapacityfromwind powerandphotovoltaics.Thetraditionalcontrolstructuresimplementedinthepowerconvertersfortheseapplicationsrelyonthesyn- chronizationtoastablegridfrequencysupportedbylargerotatinginertiasandarenotinherentlysuitableinaSmartGridcontext.Thus, fromanimplementationperspective,significantresearcheffortsarestilldevotedtowardsdevelopmentofcontrolschemesforpower electronicconvertersexplicitlyconceivedtoaddresstheconditionsemerginginfutureSmartGrids.GiventheinherentbenefitsoftheSMs outlinedabove,acaptivatingapproachisthecontrolofpowerelectronicconverterstoreplicatethemostessentialpropertiesoftheSM andbythatgainequivalentfeaturesfromafunctionalpointofview.Thus,severalalternativesforprovidingauxiliaryserviceslikereactive powercontrol,dampingofoscillationsandemulationofrotatinginertiawithpowerelectronicconvertershavebeenproposed[3–8].Some ofthesecontrolstrategiesareexplicitlydesignedtomimicthedynamicresponseofthetraditionalSM,andcanthereforebeclassifiedin broadtermsasVirtualSynchronousMachines(VSM).

Duringthelastdecade,severalconceptsforVSMshavebeenpresentedwithdifferentnamesanddifferentpracticalimplementations [4,8–12].Thefirstreviewstudiesprovidinganoverviewofimplementationshavebeenrecentlypublishedin[10,13],withanattemptto defineaclassificationframeworkpresentedin[10].Thereviewin[10]alsohighlightshowsomeimplementationsofferonlypartiallythe benefitsoftheSMswhileonlyafewcanensurefeaturesasislandoperationofsingleormultipleunits.

MostpreviousstudiesofVSM-basedcontrolstrategieshavepresentedparticularimplementationschemeswhichhavebeenverified bytime-domainsimulationsand/orlaboratoryexperiments.Afirststudythatincludeddetailedmodellingandsmall-signalstabilityof aparticularVSMimplementationwaspresentedin[14].However,thismodelwasmainlydevelopedfortuningoftheconvertercontrol loopsanddidnotconsidertheprimarypower-frequencycontrolorthedynamicsofthegridfrequencydetectionneededtoensurean implementationoftheVSMdampingeffectthatadaptstovariationsinthegridfrequency.AVSMsystemmodeladdressingalsothese issueswasrecentlypresentedin[15].

ThispaperincludesacomprehensivetreatmentofaparticularVSMimplementation,startingfromadiscussionofthecomparative advantagesofferedbytheVSMconceptinthecontextofSmartGridsinabstractterms.ThenabriefoverviewoftheVSMdevelopmentstatus isoffered,withthepurposeofidentifyinggeneralpreferencesforselectingspecificimplementationsforfutureSmartGridapplications.

TheselectedimplementationisbasedonaninternalrepresentationoftheSMinertiaanddampingbehaviourthroughareducedorder swingequation,togetherwithcascadedvoltageandcurrentcontrollersforoperatingaVoltageSourceConverter(VSC),basedonthe generalschemefrom[15].Thepaperderivesstep-by-stepadetailednonlinearmathematicalmodelforthisVSMimplementation,and acorrespondingsmallsignalmodelinordertoapplylinearanalysistechniquestothesystemintheperspectiveofstabilityassessment andcontrollertuning.Theeffectofsystemparametersonthepolesofthelinearizedsystemmodelisalsoanalyzedbycalculatingthe parametricsensitivitiesofthesystemeigenvalues.ThefeaturesandperformanceoftheinvestigatedVSManditslinearizedsmall-signal modelisverifiedwithreferencetoafewselectedcasesbynumericalsimulations.

2. ApplicationofVirtualSynchronousMachinesintheSmartGridcontext

Powergenerationfromdistributedrenewableenergysourceslikewindandphotovoltaicpowerplantsisusuallyconnectedtothe powergridthroughactivelycontrolledpowerelectronicconverters,andsimilarinterfacesareappliedforenergystoragesystemsandan increasingshareofcontrollableloads.Theconventionalschemeforsuchgridconnectedpowerconvertersisbasedoncurrentcontrolled VoltageSourceConverters(VSCs),whicharesynchronizedtothemeasuredgridvoltagethroughaPhaseLockedLoop(PLL)[16].This approachusuallyrequiresarelativelystronggridwiththepresenceofunitsthatcanmaintainandstabilizethegridfrequencyandvoltage.

EvenifauxiliaryserviceslikefrequencyandvoltagesupportcanbeprovidedbycurrentcontrolledVSCs,thisfunctionalitymustbeadded throughadditionalouterloopcontrollerswhicharenotinherentlyapplicableforoperationinislandedmode[17].Althoughthisapproach canbesuitableforarelativelylowpenetrationofgridconnectedconverters,itdoesnotseemsustainableforoperationinalongterm SmartGridperspectivewiththeexpecteddominantpresenceofpowerelectronicconversionunitsandahighdegreeofflexibilityinthe networkconfigurations.

2.1. ChallengesforpowerconvertercontrolinfutureSmartGrids

Inthelastdecade,severalalternativeconceptsandapproachesforcontrolandoperationofpowerconvertersdistributedinthepower systemhaveemerged.Anoticeableexamplefromtheoverallsystemoperationpoint-of-viewistheconceptofVirtualPowerPlants(VPPs) thataimstoaggregategenerationresources,energystoragesandloadsintoclustersthatcanbecontrolledbythedistributionsystem operatorinasimilarwayastraditionalpowerplants[18,19].SuchVPPsshouldcoordinatethecontrollableunitswhileensuringsupplyto theuncontrolledloadsinthesystem,butmustalsobeabletosupplyauxiliaryserviceslikecontrolofvoltageorreactivepowerflowand supportthefrequencyregulationofthesystem.Insmallisolatedpowersystems,orincasepartsofthedistributionsystemshouldbeable tooperateinislandedmode,thegenerationunitsaggregatedtogetherinoneVPPmustalsoensureasufficientsysteminertiatokeepthe systemstablewhilemaintainingthepowerbalancewithoutlargefrequencydeviations.TheVPPconceptscurrentlyunderdevelopment arecapableofprovidingfrequency-activatedpowersupporttothesystemwithinafewseconds,andcanthereforeensurethesteady-state powerbalance[20].However,afasterresponseisrequiredforensuringaninertia-basedpower-frequencybalancethatwillbeableto keepthesystemstableintransientconditions.Asmentioned,suchapower-frequencyresponseisanaturalfeatureoftraditionalSMs whichisnotinherentlypresentinthecurrentcontrolledVSCsusuallyappliedforintegratingrenewablesourcestothegrid.SMsofferalso additionaladvantagesasautomaticsynchronizationandpowersharinginresponsetochangesintheoperatingconditions.

FromtheseconsiderationsitappearsthataSmartGridcanrepresentachallengingenvironmentforpowerconvertercontrolschemes, especiallyduetothepossiblelargepenetrationofpowerelectronicsandthevariabilityoftheoperatingconditions.Indeed,alargepen- etrationofconverter-interfacedunitswillcorrespondtoalowerlevelofphysicalinertiathanintraditionalpowersystemdominatedby

largesynchronousmachines,andincertainconditionsitcanevenbenecessarytooperatepurepowerelectronic,inertia-less,systems.

ConsideringthataflexibleSmartGridframeworkcanresultinmorefrequentreconfigurationsofthepowersystem,withcorresponding variationsinequivalentgridimpedanceandthepossibilitytooperatepartsofthesystemasgroupsofelectricalislands,thevariabilityof theconditionscanintroduceafurtherdimensionofcomplexity.Thiscanalsoleadtolargerandmorefrequenttransientsandcorresponding requirementsforthecontrolsystemstomaintainindividualstableoperationaswellascontributingtothesystemstabilityinawiderange ofoperatingconditions.Thedesignofpowerelectroniccontrolschemesshouldcopewiththesechallengesandpreferablymitigatetheir effects.

OneofthegeneralcharacteristicsoftheemergingSmartGridscenarioisthepresenceofacommunicationinfrastructurethatcanincrease thevolumeofsignalinteractionbetweencontrollableunitsinthepowersystemandfacilitatetheircoordination.Thiscanleadtoawide rangeofoptionsforcentralizedcontrolschemes,likethementionedconceptofVPPs,wherereferencesaredeterminedbyacentralized controlunitanddistributedtotheindividualconverters.However,itshouldbenotedthatreducingthenecessityofcommunicationbetween theunits,especiallyduringtransients,canincreasetherobustnessofthesystemandreducetherisksintheeventoftemporaryunavailability ofthecommunicationinfrastructure.Thus,distributedcontrolconceptswhereindividualunitscanautonomouslydefinetheirtransient responsebasedonlocalmeasurementsarestillrelevant.Moreover,decentralizedschemeswhereonlysteady-statereferencesorset-points aredistributedfromacentralizedsystemcontrollercanlowertherequirementsintermsofbandwidthandlatenciesforthecommunication infrastructure,resultinginlowerinstallationandoperationcosts.Multipleexamplesofpossiblesolutionsfordecentralizedcontrolofpower electronicconvertercanbefoundinthelargeliteratureonisolatedMicroGrids[17,21,22]althoughanexhaustiveanalysisisbeyondthe scopeofthispaper.However,notalloftheseschemesaresuitableforSmartGridapplicationswheretheconverterareexpectedtooperate mostofthetimeingridconnectedmode.Itshouldalsobementionedthatmostcontrolschemesthatallowforbothgridconnectedand stand-aloneoperationwhilealsomaintainingsomedecentralizedcontrolfeatures,tendtobefairlycomplicatedsincetransitionsbetween thesetwomodesusuallyrequireareconfigurationofthecontrolstructure.

2.2. GeneralcharacteristicsofVSMs

IntheemergingSmartGridcontext,theVSMconceptcanofferabasisforrealizingflexibledecentralizedconvertercontrolschemesthat canoperatebothingridconnectedandislandedconditions,andthatcanalmostseamlesslyswitchbetweenthecorrespondingoperating modes.FurthermoretheinherentinertialcharacteristicoftheVSMcanprovideservicesasfrequencysupportandtransientpowersharing asprimarycontrolactions.Theseareindeedbasedonlyonlocalmeasurementanddonotdependonexternalcommunicationsasintypical alternativeschemes.Still,thereisnoconflictbetweenthislocalcontrollabilityandtheabilitytooperateinahierarchicalstructurewhile followingexternalreferencesandset-pointsprovidedbyacentralizedcontrollerforoptimizingthesystemoperation.Moreover,afurther advantageoftheVSMapproachliesinitsconceptualsimplicity,duetotheimmediateandintuitivephysicalinterpretationofitsbehaviour withanalogytothecorrespondingbehaviourofaphysicalmachine.

ThedominantbehaviourofSMsintermsofinertiaresponseanddampingcanbemodelledbythetraditionalswingequation[23].

Consideringthesegeneralcharacteristics,severalcontrolstrategieshavebeendevelopedforallowingpowerelectronicconvertersto providesyntheticorvirtualinertiatothepowersystem,andhavebeenproposedforavarietyofapplicationslikeforinstancewind turbines,energystoragesystemsandHVDCtransmissionschemes[3–14,24–26].Someofthesecontrolmethodsprovideasynthetic inertialresponsetovariationsinthegridfrequencyandonlyafewaimstoexplicitlyreplicatethefeaturesofthetraditionalSMs.However, emulationoftheinertiaanddampingeffectsrequiresanenergybufferwithsufficientcapacitytorepresenttheenergystorageeffectof theemulatedrotatinginertiaavailable.Thus,theamountofvirtualinertiathatcanbeaddedtothesystembyasingleVSMunitwillbe limitedbytheDC-sideconfigurationandbythecurrentratingoftheconverter.

ThecurrentstateoftheartontheVSMconcepthasbeenpresentedin[10,13].Thesereviewsidentifythatsomeoftheimplementations proposedasVSMsdonotexploitthefullpotentialoftheconceptbecausetheystillrelyonaPLLfordetectingthegridvoltagephaseangle, thegridfrequencyanditsderivative,thusrequiringthepresenceofrotatinginertiainthegrid.Otherproposedimplementationsofthe VSMconceptarebasedonthesimulationofaninternalmathematicalSMmodelinsidethecontrolsystemprovidingavoltagereference outputforthePWM[9].However,directopenloopPWMsignalgenerationfromthesevoltagereferencespreventsthepossibilityto explicitlyembedthelimitationsandcontrolledsaturationsofvoltagesandcurrentsthatarenormallyrequiredasprotectivefunctionsfor safeoperationofpowerelectronicconverters.Theseprotectivefunctionscanbeeasilyincludedinacascadedcontrolscheme[10,17,27,28]

wheretheoutputfromtheVSMinertiaemulationisusedasreferenceforavoltagecontrolloopcascadedwithaninternalcurrentcontrol loop.Numerically,thisapproachissufficientlyrobustforpracticalimplementationsandwillinthefollowingbeassumedasthereference VSMscheme,elaboratedfrompreviousstudiesin[14,15].ItcanalsobenotedthattheimplementationofaVSMbasedontheswing equationprovidingreferencesforoperationoftheconverter,undercertainconditionshasbeenshowntobeequivalenttothefrequency- droop-basedcontrolstrategiesfirstdevelopedforUninterruptablePowerSupply(UPS)systemsandMicroGrids,asdemonstratedin[10,29].

However,theinterpretationoftheparametersinaVSMapproachseemstobesimplerandmoreintuitivethantheequivalentparameters inthecommonlyappliedMicroGridschemesand,thus,preferable.

3. MathematicalmodeloftheVSMreferenceimplementation

ThissectiondescribesthecontrolschemefortheselectedVSMreferenceimplementation,consideringeachfunctionalblock,andderives thecorrespondingmathematicalmodel.

3.1. Controlsystemoverview

AnoverviewofthestudiedVSMconfigurationisshown inFig.1,wherea VSCisconnectedtoa gridthroughanLCfilter.Inthe following,theswitchingeffectsoftheVSCareneglectedandanidealaveragemodelisassumedformodellingtheconverter.Furthermore, noapplication-specificconstraintsoftheDC-sideoftheVSCareconsideredand,thus,modellingandcontroloftheenergysourceorstorage ontheDCsideoftheconverterisnotfurtherdiscussed.

Fig.1.OverviewofinvestigatedsystemconfigurationandcontrolstructurefortheVirtualSynchronousMachine.

TheVSM-basedpowercontrolwithvirtualinertiaprovidesfrequencyandphaseanglereferencesωVSMandVSMtotheinternalcontrol loopsforoperatingtheVSC,whileareactivepowercontrollerprovidesthevoltageamplitudereferencevˆ^r∗.Thus,theVSMinertiaemulation andthereactivepowercontrollerappearasouterloopsprovidingthereferencesforthecascadedvoltageandcurrentcontrollers.APLL detectstheactualgridfrequency,butthisfrequencyisonlyusedforimplementingthedampingtermintheswingequation.Thus,the operationoftheinnerloopcontrollersdoesnotrelyonthePLLasinconventionalVSCcontrolsystems,butonlyonthepower-balance-based synchronizationmechanismoftheVSMinertia.

3.2. Modellingconventions

InFig.1,uppercasesymbolsrepresentphysicalvaluesoftheelectricalcircuit.Thecontrolsystemimplementationandthemodelling ofthesystemarebasedonperunitquantities,denotedbylowercaseletterswherethebasevaluesaredefinedfromtheapparentpower ratingandtheratedpeakvalueofthephasevoltage[30].

Themodelling,analysisandcontroloftheelectricalsystemisimplementedinSynchronousReferenceFrames(SRFs).Thetransformation fromthestationaryreferenceframeintotheSRFsarebasedontheamplitude-invariantParktransformation,withthed-axisalignedwith avoltagevectorandtheq-axisleadingthed-axisby90^◦[30].Thus,themagnitudeofcurrentandvoltagevectorsatratedconditionsis 1.0pu.

Wheneverpossible,SRFequationsarepresentedincomplexspacevectornotationas:

x=x_d+j·xq (1)

Thus,activeandreactivepowerscanbeexpressedoncomplexorscalarformas:

p=Re(v·i)=vd·i_d+vq·iq

q=Im(v·i)=−v_d·iq+vq·i_d

(2) ThecurrentdirectionsindicatedinFig.1resultinpositivevaluesforactiveandreactivepowersflowingfromtheconverterintothe grid.

3.3. Systemmodelling

Inthefollowingsub-sections,theimplementationofeachfunctionalblockoftheVSM-basedcontrolandthemathematicalmodelsof allsystemelementsfromFig.1arepresentedasabasisfordevelopinganon-linearmodelofthesystem.Thissystemmodelwillalsobe usedtoestablishalinearizedsmall-signalstate-spacerepresentation.

3.3.1. VSMinertiaemulationandactivepowerdroopcontrol

Theemulationofarotatinginertiaandthepower-balancebasedsynchronizationmechanismsofthisvirtualinertiaisthemaindifference betweentheinvestigatedVSMcontrolstructureandconventionalcontrolsystemsforVSCs.TheVSMimplementationinvestigatedinthis caseisbasedonaconventionalswingequationrepresentingtheinertiaanddampingofatraditionalSM[10,14].Theswingequationused fortheimplementationislinearizedwithrespecttothespeedsothattheaccelerationoftheinertiaisdeterminedbythepowerbalance accordingto:

dω_VSM dt =p^r∗

Ta − p Ta−p_d

Ta

(3) Inthisequation,p^r*isthevirtualmechanicalinputpower,pisthemeasuredelectricalpowerflowingfromtheVSMintothegrid,and p_disthedampingpower,whilethemechanicaltimeconstantisdefinedasT_a(correspondingto2HinatraditionalSM).Theperunit mechanicalspeedω_VSMofthevirtualinertiaisthengivenbytheintegralofthepowerbalancewhilethecorrespondingphaseangle_VSMis givenbytheintegralofthespeed.AblockdiagramshowingtheimplementationoftheVSMswingequationisshownontherightinFig.2.

Fig.2.VirtualSynchronousMachineinertiaemulationwithpower-frequencydroop.

TheVSMdampingpowerp_d,representingthedampingeffectofatraditionalSM,isdefinedbythedampingconstantk_dandthedifference betweentheVSMspeedandtheactualgridfrequency.Thus,anestimateoftheactualgridfrequencyisneededfortheVSMimplementation.

Asindicatedinthefigure,thefrequencyestimateisinthiscaselabelledasω_PLLandisprovidedbyaPLL.

Anexternalfrequencydroop,equivalenttothesteady-statecharacteristicsofthespeedgovernorforatraditionalsynchronousmachine, isincludedinthepowercontroloftheVSMasshownintheleftpartofFig.2.Thispower-frequencydroopischaracterizedbythedroop constantkωactingonthedifferencebetweenafrequencyreferenceω^∗_VSMandtheactualVSMspeedω_VSM.Thus,thevirtualmechanical inputpowerp^r*totheVSMswingequationisgivenbythesumoftheexternalpowerreferenceset-point,p^*,andthefrequencydroop effect,asshownontheleftofFig.2.

FormodellingtheVSMinaSRF,thephaseangleoftheVSMingridconnectedmodeshouldbeconstantundersteady-stateconditions andshouldcorrespondtothephasedisplacementbetweenthevirtualpositionoftheVSMinternalvoltageandthepositionofthegrid voltagevector.SinceonlythedeviationoftheVSMspeedfromtheactualgridfrequencyshouldbemodelledtoachievethis,anewset ofvariablesrepresentingthespeeddeviationıω_VSMandthecorrespondingphaseangledifferenceı_VSMisintroduced.Thus,thepower balanceoftheVSMinertiacanbeexpressedby(4),whiletheVSMphasedisplacementisdefinedby(5):

dıω_VSM dt =p^∗

Ta− p

Ta−k_d(ω_VSM−ω_PLL)

Ta −k_ω(ω_VSM−ω^∗)

Ta (4)

dı_VSM

dt =ıω_VSM·ω_b (5)

SincetheVSMspeedinsteadystatewillbecomeequaltothegridfrequencyωg,thefrequencydeviationıω_VSMwillreturntozerounder stablegridconnectedoperation.

TheactualperunitspeedoftheVSMshownintheblockdiagramofFig.2canbeexpressedfromthespeeddeviationıωVSMresulting from(4)andthegridfrequencyωgasgivenby(6).ThecorrespondingVSMphaseangle_VSMisthendefinedby(7)

ωVSM=ıωVSM+ωg (6)

d_VSM

dt =ωVSM·ω_b (7)

Thephase angle_VSMwillthenbecomeasaw-toothsignalbetween0and2␲,whichisthephaseanglethatwillbeusedforthe transformationbetweentherotatingreferenceframedefinedbytheVSMinertiaandthethree-phasesignals,asindicatedinFig.1.

3.3.2. Reactivepowerdroopcontroller

Thedroop-basedreactivepowercontrollerappliedinthiscaseissimilartothecontrollerscommonlyappliedinmicrogridsystems [17,27].Thevoltageamplitudereferenceˆv^r∗usedfortheinnerloopvoltageandcurrentcontrolisthencalculatedby(8)whereˆv^∗isthe externalvoltageamplitudereferenceandq^*isthereactivepowerreference.Thegainkqisthereactivepowerdroopgainactingonthe differencebetweenthereactivepowerreferenceandthefilteredreactivepowermeasurementqm.Thestateofthecorrespondingfirstorder lowpassfilterappliedinthiscaseisdefinedby(9),whereω_fisthecut-offfrequency.Ablockdiagramoftheresultingcontrolstructureis showninFig.3:

vˆ^r∗₌vˆ^∗₊kq(q^∗−qm) (8)

dq_m

dt =−ω_f·qm+ω_f·q (9)

3.3.3. Referenceframeorientations

ThesynchronizationoftheVSMcontrolsystemtothegridisbasedonthephaseangleorientationofthevirtualrotoroftheVSM,and thephaseangleVSMisusedinthetransformationsbetweenthestationaryreferenceframeandtheVSM-orientedSRF.Thus,thepower

Fig.3. Reactivepowerdroopcontroller.

α β

vg

δθ

VSM

vo

q

ω

VSM⋅

ω

b , d

gd

v

θ

VSM

θ

PLL

δθ

PLL

, gq

v

ω

g⋅

ω

b

Fig.4. VectordiagramdefiningtheSRFandvoltagevectororientations.

balanceoftheVSMswingequationwillensurethesynchronizationtothegridvoltagewithouttheneedforatraditionalPLL.SincetheVSM- orientedSRFinsteadystaterotateswiththesamefrequencyasthegridvoltage,thisphaseanglewillbecontinuouslyincreasingbetween 0and2␲,asindicatedinthevectordiagramshowninFig.4.Accordingtoitsdefinition,thephaseangleı_VSMisinsteadrepresentingthe phasedifferencebetweentheVSMSRForientationandtherotatinggridvoltagevector,asalsoindicatedinFig.4.

TheVSM-orientedSRFisusedforbothcontrolandmodellingofthesystem,andtherefore,alsothemodeloftheelectricalsystem willberepresentedinthisreferenceframe.Thishassignificantadvantagesforthemodellingofthesystem,sincemultiplereference frametransformationsbetweenalocalSRFforcontrollerimplementationandaglobalSRFforelectricalsystemmodellingcanbeavoided.

Consideringtheamplitudeoftheequivalentgridvoltagevˆgtobeknown,thevoltagevectorvgintheVSM-orientedSRFcanthenbe expressedby(10):

vg=ˆvge^−jıVSM (10)

Bythepower-balance-basedsynchronizationeffectoftheVSMswingequation,thecontrolsystemdefinesitsownreferenceframe orientationwithrespecttothegridvoltage.Inprinciple,noadditionalreferenceframesareneededtomodelthesystemfromFig.1.

However,sinceanestimateforthegridfrequencyisusedtoimplementtheVSMdampingeffect,aPLLoperatingonthemeasuredvoltage voatthefiltercapacitorsisimplementedaspartofthecontrolsystem.Thus,thisPLLwillestablishitsownSRFalignedwiththevoltage vectorv_o.ThephaseangledisplacementofthisPLLwithrespecttothegridvoltagecanthenbedefinedası_PLLinasimilarwayasfor thephaseangledisplacementoftheVSM.ThedetailedimplementationofthePLLwillbepresentedinthefollowingsub-section,butthe definitionofitssteadystatephasedisplacementıPLLwithrespecttothegridvoltage,andthecorrespondingphaseanglePLLbetween therotatingPLL-orientedSRFandthestationaryreferenceframe,isshowninFig.4.

AccordingtothedefinitionsindicatedinFig.4,thephaseanglebetweentheVSMandPLLorientedSRFswillbedefinedbythedifference betweentheVSMand PLLangles.FormodellingofthePLLinitsownreferenceframe,thevoltagevoatthefiltercapacitorscanbe transformedfromtheVSM-orientedreferenceframetothePLL-orientedreferenceframeby:

v^PLL_o =v^VSM_o e^{−j(ıPLL−ıVSM)} (11)

3.3.4. Phaselockedloop

ThePhaseLockedLoop(PLL)appliedinthiscasefortrackingoftheactualgridfrequencyisbasedon[31,32]anditsstructureisshown inFig.5.ThisPLLisusingfirstorderlow-passfiltersontheestimatedd-andq-axisvoltagecomponentsandaninversetangentfunctionto calculatethephaseangleerrorofthePLL.Thisphaseangleerrore_PLListheinputtoaPIcontrollertrackingthefrequencyofthemeasured voltage.Forthepracticalimplementation,theestimatedfrequencyω_PLListhenintegratedtoobtaintheestimateoftheactualinstantaneous phaseanglePLLusedfortransformationofthevoltagemeasurementsintothePLL-orientedSRF.FormodellingofthePLL,thevoltagevector v_ointheVSM-orientedSRFmustbetransformedintothePLL-orientedSRFaccordingto(11).

Fig.5.Phaselockedloop.

Thestatesoftheappliedfirstorderlow-passfiltersinthePLL,definingthefilteredvoltagevPLL,canbeexpressedby(12),wherethe cut-offfrequencyoftheappliedlowpassfiltersisgivenbyωLP,PLL:

dvPLL

dt =−ω_LP,PLL·v_PLL+ω_LP,PLL·voe^{−j(ıPLL−ıVSM)} (12)

Theintegratorstateε_PLLofthePIcontrollercanthenbedefinedby:

dεPLL

dt =tan⁻¹

V_PLL,q V_PLL,d

(13) InthesamewayasexplainedfortheSRFmodellingoftheVSMswingequation,aspeeddeviationıωPLLwithrespecttothegridfrequency isdefinedforthePLLaccordingto(14).Thecorrespondingphaseangledisplacement,ıPLL,ofthePLListhendefinedby(15):

ıω_PLL=k_p,PLL·tan⁻¹

V_PLL,q V_PLL,d

+k_i,PLL·ε_PLL (14)

dı_PLL

dt =ıωPLL·ω_b (15)

InaccordancewiththedefinitionsintroducedfortheVSMswingequation,theactualperunitfrequencyωPLLdetectedbythePLLisgiven by(16).ThephaseangleusedintheimplementationofthePLL,fortransformationofthemeasuredthree-phasevoltagemeasurements intothePLL-orientedSRF,isthendefinedby_PLLaccordingto(17):

ωPLL=ıωPLL+ωg (16)

d_PLL

dt =ωPLL·ω_b (17)

3.3.5. Virtualimpedanceandvoltagecontrollers

AsindicatedinFig.1,thevoltageamplitudereferencevˆ^r∗resultingfromthereactivepowerdroopcontrollerinFig.3ispassedthrough avirtualimpedancebeforeitisusedasa referenceforcontrollingthevoltagevoatthefiltercapacitors.Thisvirtualimpedancecan beconsideredasanemulationofthequasi-stationarycharacteristicsofthesynchronousimpedanceina traditionalSM.Thevirtual impedancewillinfluencethesteady-stateanddynamicoperationoftheVSM,anditwillbeshownhowitcanbeusedtoshapethe dynamiccharacteristicsofthesystem.Sincepowerflowingthroughthevirtualinductancewillcauseaphaseangledisplacementbetween thegridvoltageandthevirtualinertiapositionoftheVSM,itwillalsoreducethesensitivityoftheVSMtosmalldisturbancesinthegrid.

Theinfluencefromthevirtualresistancer_vandinductancel_vonthecapacitorvoltagereferencevectorv^∗_oisdefinedonbasisofthecurrent ioaccordingto[33,34]:

v^∗_o=vˆ^r∗−(r_v+j·ω_VSM·l_v)·i_o (18)

Theresultingd-andq-axisvoltagecomponentsv^∗_o,dandv^∗_o,qareuseddirectlyasreferencesforthedecoupledSRFPIvoltagecontrollers asshownintheleftpartofFig.6.

ThedetailedstructureoftheSRFPIcontrollersforthefiltercapacitorvoltageisshowninthemiddleofFig.6,andisproducingthe referencevaluesi^∗_cvfortheconvertercurrents[27].Thesecurrentreferencescanbeexpressedby(19),wherethePIcontrollergainsare definedbyk_pvandk_iv.Againfactork_ffithatcanbesetto1or0isusedtoenableordisablethefeed-forwardofmeasuredcurrentsflowing intothegrid.ItshouldalsobenotedthatthedecouplingtermsofthevoltagecontrollerarebasedontheperunitspeedoftheVSMinertia asdefinedin(6).Thestates␰aredefinedtorepresenttheintegratorsofthePIvoltagecontrollersasgivenby(20):

i^∗_cv=kpv(v^∗_o−vo)+k_iv␰+j·c₁·ω_VSM·vo+k_ffi·io (19) d␰

d =v^∗_o−v_o (20)

Thecurrentreferencesfromthevoltagecontrollersshouldbelimitedtoavoidover-currentsincaseofvoltagedrops,faultconditions orotherseveretransients.Thisalsoimpliesthatthevoltagecontrollersmustbeprotectedfromwindupconditionsincasethecurrent referencesaresaturated.However,therequiredlimitationsandanti-winduptechniquesfortheinvestigatedVSMschemearesimilarto whatisneededinconventionaldroop-basedcontrolschemeswithcascadedSRFvoltageandcurrentcontrollers,asforinstancediscussed in[28].Sincetheselimitationsarenotinfluencingthedynamicsofthecontrolschemewithinthenormaloperatingrange,furtherdetails willnotbediscussedhere.

3.3.6. Currentcontrollersandactivedamping

TheappliedinnerloopcurrentcontrollersareconventionalSRFPIcontrollerswithdecouplingterms[27,35],asshownintherightside partofFig.6.Theoutputvoltagereferencefromthecontrollerisdefinedby(21),wheretheresultingvoltagereferencefortheconverter isdenotedbyv^∗_cv.TheproportionalandintegralgainsofthePIcontrolleraredefinedbykpcandk_ic,andagainfactork_ffvisusedtodisable orenablethevoltagefeed-forwardintheoutputofthecurrentcontrollers.Thestates␥aredefinedtorepresenttheintegratorsofthePI controllersaccordingto(22):

v^∗_cv=k_pc(i^∗_cv−i_cv)+k_ic·␥+j·l₁·ω_VSM·i_cv+k_ffv·v_o−v^∗_AD (21) d␥

dt =i^∗_cv−icv (22)

In(21),thevoltagereferencefortheconverteralsoincludesanactivedampingtermv^∗_ADdesignedforsuppressingLCoscillationsinthe filter[36].TheimplementationoftheactivedampingalgorithmisshowninFig.7,andisbasedonhighpassfilteringofthemeasuredvoltage vo,obtainedfromthedifferencebetweenvoandthelowpassfilteredvalueofthesamevoltage.Theresultinghighpassfilteredsignalis thenscaledbythegaink_ADaccordingto(23)andsubtractedfromtheoutputofthecurrentcontrollerstocanceldetectedoscillationsin thecapacitorvoltages:

v^∗_AD=kAD(vo−␸) (23)

Thecorrespondinginternalstates␸ofthelowpassfiltersusedfortheactivedampingaredefinedby(24),whereωADisthecut-off frequency:

d␸

dt =ω_AD·vo−ω_AD·␸ (24)

ForthepracticalimplementationoftheVSCcontrolsystem,thevoltagereferencev^∗_cv resultingfromthecurrentcontrollerandthe activedampingisdividedbythemeasuredDC-linkvoltagetoresultinthemodulationindexmasshowntotherightofFig.6.Neglecting theswitchingoperationoftheconverterandanydelayduetothePWMimplementation,theinstantaneousaveragevalueoftheperunit converteroutputvoltageisgivenbytheproductofthemodulationindexandtheactualDC-voltage.Underthisassumption,theoutput convertervoltagewillbeapproximatelyequaltothevoltagereferenceassummarizedby(25)[37]:

m= v^∗_cv vDC

, vcv=m·vDC→vcv≈v^∗_cv (25)

Thus,theACsideoperationoftheconverterwillbeeffectivelydecoupledfromanydynamicsintheDCvoltage,anditisnotnecessary tofurtherdiscussormodeltheDCsideoftheconverterforachievinganaccuraterepresentation ofthedynamicsontheACside.It shouldbenotedthattheactualsourceorstorageunitconnectedtotheDClinkoftheconvertermightstillimposerestrictionsonthe allowablepowerexchangeduringvariousoperatingconditions.However,tomaintaingeneralityandavoiddetaileddiscussionofparticular application-specificlimitationsitwillbeassumedthatthepowerrequestedfromtheACsideisalwaysavailableattheDClinkofthe converter.

3.3.7. Electricalsystemequations

TheelectricalsystemincludedinthemodelaccordingtoFig.1consistsofasetoffilterinductorsconnectedtotheconverter,ashunt capacitorbankrepresentingthecapacitanceoftheLCfilter,andaThéveninequivalentofthegrid.Thissimplestructureisassumedto achieveasimplemodelthatmainlyincludesthedynamicsoftheconvertercontrolsystemanditsinteractionwiththeequivalentgrid voltage.However,amorecomplexACgridtopologycanbeeasilyincludedinthemodelforbothsimulationsandanalysis.Consideringan instantaneousaveragemodeloftheconverter,theSRFstatespaceequationsoftheelectricalsystemcanbeestablishedasgivenby(26) [27,35]:

dicv dt =ω_b

l_f vcv−ω_b l_f vo−

_r

l_fω_b

l_f +j·ωgω_b

icv dvo

dt = ω_b c_f icv−ω_b

c_f ig−j·ωgω_b·vo

dio

dt =ω_b lg vo−ω_b

lg vg−

rgω_b

lg +j·ωgω_b

io

(26)

Intheseequationsicvisthefilterinductorcurrent,vcvistheconverteroutputvoltage,voisthevoltageatthefiltercapacitors,igisthe currentflowingintothegridequivalentandv_gisthegridequivalentvoltage.Theinductanceandequivalentresistanceofthefilterinductor isgivenbyl_fandr_lf,thefiltercapacitorisc_f,whilethegridinductanceandresistancearegivenbylgandrg.Theperunitgridfrequencyis givenbyωg,whilethebaseangulargridfrequencyisdefinedbyω_b.Itshouldbenotedthatthestatespacemodelfrom(26)canrepresent theelectricalsysteminanySRF,butinthiscasethesystemwillalwaysbemodelledintheSRFdefinedbytheVSMswingequation.

Itshouldalsobenotedthatthepresentedmodelonlyrepresentsthecaseofgrid-connectedoperation,whiletheinvestigatedVSM schemeisinherentlysuitableforstand-aloneoperation.Inthiscasetheoperationalfrequencywillonlybedeterminedbytheactualload inthesystem,thepower-frequencydroopgainandthepowerandfrequencyreferencesfortheVSM.Furtherdetailsonmodellingofthe investigatedVSMimplementationinislandedoperation,andcorrespondinganalysisofthedynamiccharacteristicsinstand-alonemode canbefoundin[38].

3.4. Non-linearsystemmodelingrid-connectedoperation

AllequationsneededfordetailedmodellingoftheVSMconfigurationingrid-connectedoperationhavebeenpresentedintheprevious sub-sections,andcanbereducedtoamodelonstate-spaceformwith19distinctstatevariablesand6inputsignals,withthestatevector xandtheinputvectorudefinedby(27).Theresultingnon-linearstate-spacemodeloftheoverallsystemisgivenby(28):

x=

vo,d vo,q i_cv,d icv,q _d q i_o,d io,q ϕ_d ϕq...

...vPLL,d vPLL,q ε_PLL ı_{VSM d} q qm ıω_VSM ı_PLL

T

u=

p^∗ q^∗ ˆvg ˆv^∗ ω^∗ ωg

T

(27)

dvo,d

dt =ω_bω_gvo,q+ω_b

c_f i_cv,d−ω_b c_f i_o,d dvo,q

dt =−ω_bωgv_o,d+^ωb

c_f icv,q−ω_b c_f io,q

di_cv,d

dt =ω_b(k_ffv−1−k_AD−k_pck_pv)

l_f vo,d−ω_bc_fk_pc

l_f ωgvo,q−ω_b(k_pc+r_f)

l_f i_cv,d+ω_bk_ic

l_{f d}+ω_bk_pc(k_ffi−k_pvr_v)

l_f i_o,d+ω_bk_pck_pvl_v l_f ωgio,q

+ω_bk_AD

l_f ϕ_d+ω_bk_ivkpc

l_{f d}−ω_bkpckpvkq

l_f qm−ω_bicv,qıωVSM+ω_bkpckpvl_v

l_f io,qıωVSM−ω_bc_fkpc

l_f vo,qıωVSM+ω_bkpckpvkq

l_f q^∗+ω_bkpckpv l_f ˆv^∗ dicv,q

dt = ω_bc_fkpc

l_f ωgv_o,d+^{ωb(kffv−1−kAD−kpckpv)}

l_f vo,q−ω_b(kpc+r_f)

l_f icv,q+ω_bk_ic

l_f q−ω_bkpckpvl_v

l_f ωgi_o,d+ω_bkpc(k_ffi−kpvr_v) l_f io,q

+ω_bk_AD

l_f ϕq+ω_bk_ivk_pc

l_f q+ω_bi_cv,dıω_VSM−ω_bk_pck_pvl_v

l_f i_o,dıω_VSM+ω_bc_fkpc

l_f vo,dıω_VSM d_d

dt =−kpvvo,d−c_fωgvo,q−i_cv,d+(k_ffi−kpvr_v)i_o,d+kpvl_vωgio,q+k_ivd−kpvkqqm+kpvl_vio,qıω_VSM−c_fvo,qıω_VSM+kpvkqq^∗+kpvˆv^∗ d_d

dt =c_fωgvo,d−kpvvo,q−icv,q−kpvl_vωgi_o,d+(k_ffi−kpvr_v)io,q+k_ivq−kpvl_vi_o,dıω_VSM+c_fvo,dıω_VSM di_o,d

dt =ω_b

lg vo,d−ω_br_g lg

i_o,d+ω_bωgio,q+ω_bˆvgcos(ı_VSM) lg

dio,q

dt = ω_b

lg vo,q−ω_bωgi_o,d−ω_brg

lg

io,q+ω_bvˆgsin(ı_VSM) lg

dϕ_d

dt =ω_ADvo,d−ω_ADϕ_d dϕq

dt =ω_ADvo,q−ω_ADϕ_q dvPLL,d

dt =ω_LP,PLLv_o,dcos(ı_PLL−ı_VSM)+ω_LP,PLLvo,qsin(ı_PLL−ı_VSM)−ω_LP,PLLv_PLL,d dvPLL,q

dt =−ω_LP,PLLvo,dsin(ı_PLL−ı_VSM)+ω_LP,PLLvo,qcos(ı_PLL−ı_VSM)−ω_LP,PLLvPLL,q

dεPLL

dt =tan⁻¹

vPLL,q

v_PLL,d

dıVSM

dt =ω_bıω_VSM dı_d

dt =−vo,d−r_vi_o,d+l_vωgio,q−kqqm+l_vio,qıω_VSM+kqq^∗+ˆv^∗ dıq

dt =−vo,q−l_vωgi_o,d−r_vio,q−l_vi_o,dıω_VSM dqm

dt =−ω_fio,qv_o,d+ω_fi_o,dvo,q−ω_fqm

dıωVSM

dt =−1 Ta

i_o,dvo,d− 1 Ta

io,qvo,q+k_dkp,PLL

Ta

tan⁻¹

vPLL,q

v_PLL,d

+k_dk_i,PLL Ta

ε_PLL−k_d+kω

Ta

ıω_VSM+ 1 Ta

p^∗+kω

Ta

ω^∗−kω

Ta

ωg

dı_PLL

dt =ω_bkp,PLLtan⁻¹

vPLL,q

vPLL,d

+ω_bk_i,PLLεPLL

(28)

Thesteadystateoperatingpointofthesystemunderanycombinationsofinputsignalscanbefoundbysolvingthisnonlinearsystem modelwithderivativetermssettozero.

3.5. SmallsignalmodelofthereferenceVSM

Sincethestate-spacemodelfrom(28)isnonlinear,classicalstabilityassessmenttechniquesbasedoneigenvaluesarenotdirectly applicable.Thus,inthissection,acorrespondinglinearizedsmall-signalstate-spacemodelisderivedintheformgivenby:

x˙=A· x+B· u. (29)

wheretheprefix denotessmall-signaldeviationsaroundthesteady-stateoperatingpoint[30].Thevaluesofthestatevariablesatthis linearizationpointaredenotedbysubscript‘0’whentheyappearinthematrices.Forconvenienceofnotation,thedynamicmatrixAis expressedthroughfoursub-matricesaccordingto:

x˙₁

˙ x₂

=

A₁₁ A₁₂ A₂₁ A₂₂

·

x₁ x₂

+B· u (30)

whereA₁₁,A₁₂,A₁₃andA₂₂aregivenby(31)–(34),whiletheBmatrixisgivenby(35):

A11=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎣

0 ωbωg,0 ωb

cf 0 0 0 −ωb

cf 0 0 0

−ωbωg,0 0 0 ωb

cf 0 0 0 −ωb

cf 0 0

ωb(kffv⁻¹⁻kAD−kpc kpv⁾

lf −ωbcf kpcωg,0

lf −ωb(kpc+rf)

lf 0 ωbkic

lf 0 ωbkpc(kffi−kpv^rv⁾ lf

ωbkpckpv^lvωg,0 lf

ωbkAD

lf 0

ωbcf kpcωg,0 lf

ωb(kffv⁻¹⁻kAD−kpc kpv⁾

lf 0 −ωb(kpc+rf)

lf 0 ωbkic

lf −ωbωg kpckpv^lv lf

ωbkpc(kffi−kpv^rv⁾

lf 0 ωbkAD

lf

−kpv −cf ωg,0 −1 0 0 0 kffi−kpv^rv kpv^lvωg,0 0 0

cf ωg,0 −kpv ⁰ −1 0 0 −kpv^lvωg,0 kffi−kpv^rv ^{0 0}

ωb

lg 0 0 0 0 0 −ωbrg

lg ωbωg,0 0 0

0 ωb

lg 0 0 0 0 −ωbωg,0 −ωbrg

lg 0 0

ωAD 0 0 0 0 0 0 0 −ωAD 0

0 ωAD 0 0 0 0 0 0 0 −ωAD

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎦

(31)

A₁₂=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 ω_bk_ivkpc

l_f 0 −ω_bkpckpvkq

l_f

ω_b(−l_fi_cv,q,0+kpckpvl_vi_o,q,0−c_fkpcvo,q,0)

l_f 0

0 0 0 0 0 ω_bk_ivkpc

l_f 0 ω_b(l_fi_cv,d,0−k_pck_pvl_vi_o,d,0+c_fk_pcvo,d,0)

l_f 0

0 0 0 0 k_iv 0 −kpvkq kpvl_vi_o,q,0−c_fvo,q,0 0

0 0 0 0 0 k_iv 0 −kpvl_vi_o,d,0+c_fv_o,d,0 0

0 0 0 ω_bvˆg,0sin(ı_VSM,0)

lg 0 0 0 0 0

0 0 0 ω_bvˆg,0cos(ıVSM,0) lg

0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

(32)

A₂₁=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎣

ωLP,PLLcos(ıPLL,0−ıVSM,0) ωLP,PLLsin(ıPLL,0−ıVSM,0) 0 0 0 0 0 0 0 0

−ω_LP,PLLsin(ı_PLL,0−ı_VSM,0) ω_LP,PLLcos(ı_PLL,0−ı_VSM,0) 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

−1 0 0 0 0 0 −r_v ωgl_v 0 0

0 −1 0 0 0 0 −ωgl_v −r_v 0 0

−ω_fio,q,0 ω_fi_o,d,0 0 0 0 0 ω_fvo,q,0 −ω_fv_o,d,0 0 0

−i_o,d,0

Ta −i_o,q,0

Ta

0 0 0 0 −v_o,d,0

Ta −v_o,q,0 Ta

0 0

0 0 0 0 0 0 0 0 0 0

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎦

(33)

A₂₂=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

−ωLP,PLL 0 0 −ωLP,PLL

vo,q,0cos(ıPLL,0−ıVSM,0)

−vo,d,0sin(ıPLL,0−ıVSM,0)

0 0 0 0 ωLP,PLL

vo,q,0cos(ıPLL,0−ıVSM,0)

−vo,d,0sin(ıPLL,0−ıVSM,0)

0 −ω_LP,PLL 0 ω_LP,PLL

vo,d,0cos(ıPLL,0−ıVSM,0) +vo,q,0sin(ıPLL,0−ıVSM,0)

0 0 0 0 −ω_LP,PLL

vo,d,0cos(ıPLL,0−ıVSM,0) +vo,q,0sin(ıPLL,0−ıVSM,0)

0 1

vPLL,d,0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 ωb 0

0 0 0 0 0 0 −k_q lvi_o,q,0 0

0 0 0 0 0 0 0 −lvi_o,d,0 0

0 0 0 0 0 0 −ωf 0 0

0 kdkp,PLL

T_avPLL,d,0

kdki,PLL

T_a 0 0 0 0 −kd+kω

T_a 0

0 ωbkp,PLL

vPLL,d,0

ω_bk_i,PLL 0 0 0 0 0 0

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

(34)