IEEETRANSACTIONSONPOWERDELIVERY
1
MethodologyforDroopControlDynamicAnalysisofMultiterminalVSC-HVDC
GridsforOffshoreWindFarms
EduardoPrieto-Araujo,FernandoD.Bianchi,AdriàJunyent-Ferré,StudentMember,IEEE,and
OriolGomis-Bellmunt,Member,IEEE
Abstract—Thispaperaddressesthecontrolofmultiterminalvoltage-sourceconvertersathigh-voltagedirectcurrentinthecontextofoffshorewindfarms.Droopcontroliscommonlyusedtoregulatethedcvoltageinthiskindofgrid,anddroopparametersareselectedonthebasisofsteady-stateanalyses.Here,acontroldesignmethodologyisproposedbasedonthefrequency-responseanalysis.Thismethodologyprovidesacriteriontoselectthedroopgains,takingintoaccounttheperformancespecifications[i.e.,thedesiredvoltageerrorsandthemaximumcontrolinputs(currents)].Theapplicationofthemethodologyisillustratedwithafour-terminalgrid.
IndexTerms—Droopcontrol,high-voltagedirectcurrent(HVDC),multiterminal,offshorewindpower.
I.INTRODUCTION
HESEDAYS,thereisanincreasingnumberofoffshorewindfarms.Intheseoffshorefacilities,turbinescanbelocatedtensorhundredsofkilometersawayfromthecoastandconnectedtothemainpowergridbysubmarinecables.Inthesesituations,studieshaveprovedthatthemostconvenientpowertransmissionsystemsarethehigh-voltagedirect-current(HVDC)networks[1].Thesegridsconsistoftwoormorecon-vertersconnectedtoacommondcgrid[2].Themostcommontechnologyinthelastyearshasbeentheline-commutedcon-verters(LCCs)[3].However,thereisagrowingtrendtowardtheuseofvoltage-sourceconverters(VSCs)inoffshoreHVDCgrids[1],[4],[5].Thesepowerconvertersoffermorepossibili-tiesfortheoperationoftheoffshorewindfarms.VSC-HVDCs
ManuscriptreceivedOctober22,2010;revisedFebruary21,2011;acceptedApril10,2011ThisworkwassupportedbytheMinisteriodeCienciaeInno-vaciónunderProjectENE2009-08555.Paperno.TPWRD-00807-2010.
E.Prieto-AraujoandA.Junyent-FerréarewiththeDepartamentd’Enginy-eriaElèctrica,Centred’InnovacióTecnològicaenConvertidorsEstàticsiAc-cionaments(CITCEA-UPC),UniversitatPolitècnicadeCatalunya.ETSd’En-ginyeriaIndustrialdeBarcelona,BarcelonaPl.2.08028,Spain(e-mail:ed-uardo.prieto-araujo@citcea.upc.edu).
F.D.BianchiandarewiththePowerElectronicsandElectricPowerGridsDepartment,CataloniaInstituteforEnergyResearch(IREC),Barcelona08019,Spain.
O.Gomis-BellmuntiswithDepartamentd’EnginyeriaElèctrica,Centred’In-novacióTecnològicaenConvertidorsEstàticsiAccionaments(CITCEA-UPC),UniversitatPolitècnicadeCatalunya.ETSd’EnginyeriaIndustrialdeBarcelona,BarcelonaPl.2.08028,Spain.HeisalsowiththePowerElectronicsandElec-tricPowerGridsDepartment,CataloniaInstituteforEnergyResearch(IREC),Barcelona08019,Spain
DigitalObjectIdentifier10.1109/TPWRD.2011.2144625
T
permittheindependentcontrolofactiveandreactivepowerandcontinuousacvoltageregulation.Theypresentnocommutationfailure,black-startcapability,andthereisnoneedforvoltagepolarityreversaltoreversepower.Asadditionaladvantages,thefiltersaremorecompactandthecablesarelighter[6],[7].Ontheotherhand,thecostsandthecommutationlossesarehigherandtheyareabletohandleonlylimitedlevelsofvoltageandpower.ThefirstHVDCusingVSCtechnologyinwindfarms,calledBorWin1,wascommissionedin2010inGermany.Atotalof80windturbinesof5MWeachareconnectedbya75-kmun-dergroundcableand125-kmsubmarinecableat150kV[8].Inthenearfuture,therewillbealargeamountofoffshorewindfarmsconnectedwithVSC-HVDC.ItseemsreasonabletodeviseoffshoreVSC-HVDCgridsinterfacinganumberofsuchdifferentterminalswithdifferentacgrids,resultingintheso-calledmultiterminalVSC-HVDCsystem.MultiterminalVSC-HVDCstandsasaninterestingsolutiontoefficientlyconnectanumberofoffshorewindfarms,butalsoimpliesseveraltechnicalchallengesthatwillhavetobeaddressed,includingcontrol[9],operation[7],andprotection[6]issues.ThefirstmultiterminalusingLCC-HVDCtechnologygoesbacktothe1960s[10],[11].Itwasnotuntil2003thattheuseofthemultiterminalVSC-HVDCintheaggregationofoffshorewindpowerwasproposedby[12].Adetailedanalysisofdifferentsystemtopologiescanbefoundin[6].ImportantprojectsinvolvingHVDCmultiterminaltransmissionarecur-rentlyunderstudy,suchastheDesertecproject[13]andtheEuropeanoffshoreSupergrid[14].
Thestabilityofacpowersystemshasbeenwidelydiscussedintheliterature;see,forexample,[15]and[16].ThesestudiesalsoincludeHVDCsystemsandtheirpossiblecontributiontoimproveacsystemstability.Somedcgrid-managementstrate-giesbasedoncoordinatedclosed-loopdcvoltagecontrolanddcdroopcharacteristicswereproposedandsimulatedin[17].Liangetal.[9]addressedthemodelingandsimulationofmul-titerminalVSC-HVDCtransmissionsforoffshorewindpower.However,tothebestofourknowledge,thereisnostabilityanal-ysisnorsystematiccontroldesignprocedureformultiterminalVSC-HVDCgridsconnectingoffshorewindfarmstoacsys-tems.Thispaperinvestigatesthestabilityandthedynamicbe-haviorofmultiterminalHVDCgridsinoffshorewindfarmsap-plications.Adesignmethodologyofproportionalcontrolofthedcvoltagebasedonfrequency-responseanalysisisproposed.Thispaperisorganizedasfollows.ThenextsectionprovidesabriefdiscussionofthecontrolofVSC-HVDCmultiterminal
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Fig.1.TypicalHVDCmultiterminalnetwork.
networks.SectionIIIpresentsthemaincontribution,amodel-lingprocedureforcomplexVSC-HVDCmultiterminalsystems,andamethodologyfortheselectionofthedroopconstant.Theapplicationoftheproposedprocedureisillustratedinthecaseofafour-terminalgridinSectionIV.Finally,inSectionV,someconcludingremarksaredrawn.
II.MULTITERMINALGRIDCONTROL
Fig.1illustratesatypicalmultiterminalHVDCnetwork.Itconsistsofthedcgrid,themainacgrid(oracgrids),thewindfarmgrids,thewindfarmconverters(WFCs),andtheacgrid-sideconverters(GSCs).ThemultiterminalHVDCnetworkpermitsthetransferofpoweramongthedifferentunits,wheretheWFCsactaspowersourcesandtheGSCasloads.Inthispowertransmissionscheme,thesourcesinjectalloftheavailablepowerintothegridwhereasthecontroloftheGSCsseekstomaintainthedcvoltage.ThisalsoincludespowersharingamongthedifferentGSCs.Thenormaloperationmaybealteredwhensomeoftheconvertersreachthecurrentlimits.Thisusuallyoccursduringseverevoltagefaultsintheacgrid.Underthesecircumstances,theWFCsenterinvoltageregulationmodeandtheGSCsextractthemaximumpowerpossiblewithoutregulatingthedcvoltage.Inbothoperationmodes,someconvertersseektomaintainthedcvoltageandtheothersinjectorextractpowerwithoutcontrollingthevoltage[7].
Toregulatethedcvoltage,theso-calleddroopcontrolisem-ployed,whichisatechniquethatenablesthepowerdistributionamongdifferentterminalswithoutcommunications.Thecontrolofeachconverterisusuallyimplementedintwolevels:aninnerloopcontrollingthecurrentsandanouterloopregulatingthedcvoltage.Thedroopcontrolactsontheouterloopimposingacurrentreferencetotheinnerloop.Thecurrentand,thus,thepowerintheconverteraredirectlygovernedbythecurrentcontrolinaccordancewiththereferenceimposedbythevoltageloop.ThiscontrolschemeisshowninFig.2.Thecontrollawisgivenbythefollowingexpression:
(1)
whereisthedcvoltage,
isthereference,andisthedroopgain.Forthepresentstudy,thedynamicsofthecurrent
IEEETRANSACTIONSONPOWERDELIVERY
Fig.2.DroopcontrolschemeofaVSC.
loopcanbeconsideredmuchfasterthantheouterloop.There-fore,thedccurrentflowingthroughtheconverterwillbeas-sumedtobeequaltothereference.Theselectionofthegainforeachconvertermustbedone,takingintoaccounttheentiremultiterminalbehavior.Inadditiontothestaticconsiderationassociatedwiththedistri-butionofthepowersourcesandsinks,eachlocalcontrollercanaffecttheglobalstabilityandthedcvoltageinanotherter-minal.Forthesereasons,thedroopconstantselectionmustbeaddressedinthecontextofmultivariablesystemtheory.
III.FREQUENCY-RESPONSEANALYSISFOR
DROOPGAINSELECTION
Inthissection,amethodologyforthedroopconstantselec-tionbasedonmultivariablefrequency-responseanalysisispre-sented.Previoustoproposingthismethodology,asystematicprocedureisintroducedtoobtainalinearrepresentationofcom-plexmultiterminalHVDCnetworks.A.MultiterminalHVDCNetworksModeling
Fromtheviewpointofadcgridanalysis,themultiterminalcanberepresentedastheinterconnectionofnodesandbranches.AnexampleofthisrepresentationisshowninFig.3.TheWFCsinjectingpowerintothegridarethepower-inputnodesandtheGSCs.Thepoweroutputnodesextractpowerfromthegrid.Thecablesinterconnectingthenodesarethebranches.Therearealsonodeswhereonlycablesconverge,thosearecalledtheintermediatenodes.ThegeneralmultiterminalsetupdepictedinFig.3consistsofpower-inputnodes,power-outputnodes,andintermediateconnectionnodes,andbranches.Thislastnumberdependsontheparticularinterconnectionpattern.Next,themodelingofeachtypeofnodeisbrieflyexplained.
InputandOutputPowerNodes:ThewindfarmsandtheacsystemsareconnectedtotheHVDCgridthroughHVDCpowerconverters.Forthepresentanalysis,itissufficienttoconsidertheaveragedynamicbehavior.Inthissituation,theacsideoftheconvertersaremodeledasthreevoltagesourcesandthedcsideasacurrentsourceandacapacitor[18].Usingthissimplifiedrepresentation,eachwindfarmandeachacsystemaremodeledasdccurrentsources,asillustratedinFig.4.Attheconverterdcside,thepowerflowinthenodeisrepresentedbyacurrentcomingfromasourceofvalue
(2)
where
istheincomingpowerandisthedcvoltageatthenode.Itwillbeassumedthatthevoltageremainscloseto
thenominalvalues
.Underthisassumption,thecurrentcanbeassumedproportionaltothepower.
Branches:Thecablesbetweennodesaremodeledby-equivalentcircuits,seeFig.5.Whenthesecircuitsconverge
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PRIETO-ARAUJOetal.:METHODOLOGYFORDROOPCONTROLDYNAMICANALYSIS
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Fig.3.NodeandbranchschemeofamultiterminalHVDCnetwork.
Fig.4.Equivalentrepresentationofthewindfarmandtheacgridconvertersfordcgridanalysis.
Fig.5.-circuitmodelingabranchelement.
toinputoroutputnodesandtoother-circuits,thereareseveralcapacitorsinparallel.Inthesecircumstancesandwiththeaimofkeepingthenumberofvariablesasminimumaspossible,thetotalcapacitancescanbereducedtoanequivalentonegivenby
(3)
IntermediateNodes:Thecablesinthedcgridmayjointwoormoreterminalsatintermediatepoints.Thesenodeswillbedenotedasintermediatenodes,thenodemarkedwithin
Fig.3isanexampleofthistypeofnode.Again,thenumberofcapacitancescanbereducedbyreplacingthecapacitancesofthe-equivalentcircuitsandtheinputandoutputnodesbyatotalcapacitancesgivenby(3).
Anequivalentcircuitcanbeobtainedfromtheinterconnec-tionofthenodesandbranchesaftertheaforementionedsimpli-fications.Then,usingcircuitslawsandaftersomevariablema-nipulations,itispossibletofindasetoffirst-orderdifferentialequationsdescribingthedynamicbehavioroftheentiremulti-terminalHVDCgrid.Thesedifferentialequationsareknownasthestate-spacerepresentationandcanbeexpressedinthefol-lowingcompactform:
(4)
whereisthestatevector,
andaretheinputs,andaretheoutputs,and,andarematricesofsuitabledimensions.Thesematricesareobtainedafterarrangingthevariablesandapplyingmatrixcomputationlaws.
Thestatevectorconsistsofinternalvariablesthatcharac-terizetheentirestateofthesystem.Inanelectricalsystem,thecurrentsintheinductorsandthevoltagesinthecapacitorsarecommonlyselectedasstates.Therefore,inthecaseofthemul-titerminalHVDCnetworkinFig.3,thestatevectorisgivenby
Eachnodehasonecapacitorandeachbranchhasoneinductor;
therefore,thetotalnumberofstatesis
.Theinputsaredividedintotwovectors,thevectorgathersthevariablesthatcanbeusedtocontrolthesystemandaredisturbances(i.e.,externalvariablesthatarenotpossibletoma-nipulate).InthecaseofthemultiterminalHVDCnetworks,theinputsofthesystemarethecurrentinjectedorextractedbytheconverters,therefore
wherecorrespondstothesetofindicesofthenodeswhere
theconvertersinjectorextractpowerwithoutvoltagecontrol
and
denotesthesetofindicesofthenodeswherethedroopcontrolisapplied.Noticethattherelationmustbeheld.
Similarly,theoutputispartitionedintotwovectors.Thevectorcontainsthevariablesthatcanbeusedinthecontrolofthedcvoltage.Ontheotherhand,standsforthevectorofvariablesthatarenotavailabletobeusedbythecontroller.InthemultiterminalHVDCscheme,thecontrollerscanonlyusetheinformationprovidedbythevoltageatthenodeswheredroopcontrolisapplied.Theremainingvoltagesmustalsobemaintainedclosetotheratedvaluesbuttheycannotbefedbacktothecontrollers.Hence
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Thetransfermatrixofthesystem
(5)
isobtainedfromthestate-space(4),where
Thetransfersmatricesandrelatethecurrentsimposedbythecontrollerwiththecontrolledandnondirectlycontrolled
voltages,respectively,whereasthetransfermatrices
andconnectthecurrentnotusedinthecontrolwiththecon-trolledandnondirectlycontrolledvoltages,respectively.B.DroopGainSelection
Inamultiterminalscheme,thedistanceamongconvertersisusuallylargeandthecommunicationsarenotreliableenoughtobeusedinthedcvoltagecontrol.Asaconsequence,eachcon-trollermustcomputethecontrolvariablesfromtheinformationprovidedbythevoltageattheownnode.Inmatrixterms,themultivariablecontrollerhasanexpressionoftheform
..
.
..
.
(6)
where
isascalarparametertobedeterminedandarethe
constantsobtainedfromasteady-statestudy[17].Theseconstantsareassociatedwiththeresistancevaluesofthelineandtheamountofpowerincomingoroutgoingfromeachterminal.Theseconstantsarepositiveinthecaseofpower-outputnodesandnegativeinthecaseofpower-inputnodes.
ThedroopcontrolschemeisdepictedinFig.6.Itcanbeob-servedthatonlythevariableisfedbackintothecontroller.Theobjectiveofthedroopcontrolistomaintainthedcvoltagewithindesiredlimitswhenthesystemisdisturbedbythevaryingcurrentsofthenodeswithoutvoltagecontrol.Thecontrolinputmustalsobekeptunderthelimitsimposedbythemaximumcurrentsintheconverters.Therefore,theselectionofthegain
musttakeintoaccounttheseperformancespecificationsbe-sidesguaranteeingclosed-loopstability.FromFig.6,itiseasytoprovethatthevariablesofinterestaregivenbythefollowingexpressions:
(7)(8)(9)
IEEETRANSACTIONSONPOWERDELIVERY
Fig.6.Droopcontrolschemeinamultiterminalgrid.
whereand
arethesensitivitytransferfunctionwiththeidentitymatrixofdimension.
Theeffectofthegain
onthestabilitycanbeanalyzedbycomputingtheeigenvaluesoftheclosed-loopmatrix
.Re-placing
inthestatespace(4),theclosed-loopmatrixisgivenby.Then,forclosed-loop
stability,thegain
mustensurethatalleigenvaluesofhavenegativerealpart.Asimplepoweranalysisrevealsthatthe
closed-loopsystemisstableforany
.Infact,sincethecontrollawmakesthecurrent(with)proportional
tothevoltage
atthesamenode,thegaincanbein-terpretedasapassiveadmittance.Thatis,thedroopcontrolissimilartoaddenergydissipationtothesystemand,therefore,
theclosed-loopsystemwillbealwaysstablefor
.Therelationbetweenthegain
andtheperformanceob-jectivescanbeanalyzedwiththehelpofthefrequencyresponseofthesystem.Thisanalysisconsistsinevaluatingthetransfer
functionin
andinanalyzingthesingularvaluesoftheresultingcomplexmatrixfunctionsof.Thesingularvalues
ofthefrequencyresponseof
aredenotedaswheredenotesthetheigenvalueofthematrix.Thesin-gularvaluesprovideinformationabouthowavectorofsinu-soidalsignalsoffrequencyisalteredbythesystem.Inmulti-inputmultioutputlinearsystems,avectorofsinusoidalsignalssuffersnotonlyachangeinitsmagnitudeandphase,butalsoachangeinitsdirection.Themaximumamplificationthatthevectorcanexperienceisgivenbythemaximumsingularvalue
andtheminimumamplificationbytheminimumsin-gularvalue
.Thisanalysiscanbeinterpretedastheextensionofthepopularsingle-inputsingle-outputfrequencyresponseanalysistomultivariablesystems.Here,themagnitudeofthefrequencyresponseisreplacedbythesingularvalues(see[19]foramoredetailedexplanation).
Theperformancespecificationsaretominimizetheeffectofthedisturbancesonthedcvoltagesandtomaintainthecontrolinputunderreasonablelimits.Thesespecificationscanbeex-pressedintermsofthesingularvaluesinordertodeterminethe
constraintson
.Forexample,themaximumenergyoftheerrorcausedbyanyinputofboundedenergyisgivenby
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PRIETO-ARAUJOetal.:METHODOLOGYFORDROOPCONTROLDYNAMICANALYSIS
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Fig.7.Four-terminalgridusedtoillustratethedroopconstantselectionmethodology.
wheredenotesthe2-normof.Therefore,tominimizetheeffectsofthedisturbanceonthevoltageerrorandon,canbeinterpretedasminimizing
andtomaintainthecontrolinputunderreasonablelimitscanbeexpressedasensuringthat
isboundedinthefrequenciesofinterest.Ingeneral,largevaluesofachieveasmallervoltageerror
butmayalsodemandlargecontrolinputs.Theoptimal
isacompromiseamongalloftheseobjectives.
IV.FOUR-TERMINALGRIDEXAMPLE
Asimplefour-terminalHVDCgridisusedtoillustratethedroopselectionmethodologypresentedinprevioussections.Thefour-terminalgridisdepictedinFig.7andconsistsoftwooffshorewindfarmconvertersWFC1andWFC2andtwoonshoregrid-sideconvertersGSC1andGSC2.ThevaluesoftheparametersarelistedinTableI.Thefour-terminalHVDCgridhastwopower-inputnodes,twopower-outputnodes,andthreebranchesrepresentingthecableslinkingtheconverters.Thecapacitorsaretheresultofcombiningthecapacitancesofthenodesandthecorrespondingbranchside,asexplainedinSectionIII-A.
Twoscenariosareanalyzed.Inthefirstcase,droopcontrolisappliedtobothgrid-sideconverterswhereasthewindfarmcon-vertersinjectallofthewindpoweravailable.Inthesecondsce-nario,duetoafaultintheacgrid,bothwindfarmconvertersreg-ulatethedcvoltages,andthegrid-sideconvertersextractpowerfromtheHVDCgridattheirmaximumcapacity.
TABLEI
PARAMETEROFTHEFOUR-TERMINALEXAMPLE
A.Case1:DroopControlintheACGridSide
Applyingcircuitlawstothefour-terminalgridinFig.7,thefollowingdifferentialequationscanbeobtained:
(10)(11)(12)
(13)(14)
andthefollowingalgebraicequations:
(15)(16)(17)(18)
Therearefourcapacitorsandthreeinductors;therefore,thevari-ables,andaresufficienttocom-pletelydefinethestateofthissystem,i.e.,
AstheWFCsinjectallavailablepowerintothegridandtheGSCsareregulatingthedcvoltages,theinputandoutputresultdividesinto
Thepurposeofthedroopcontrolappliedtotherightsideofthefour-terminalgridinFig.7istomaintainthedcvoltagestabilitywhenthecurrentscomingfromthewindfarmconvertersWFC1andWFC2change.Therefore,thevectorofthesecurrentsisthe
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Fig.8.Eigenvaluesoftheclosed-loopmatrixA
forseveralvaluesofK
(Case1).
disturbanceandthecontrolinputisthevectorofthecurrentsoftheGSCsand.Thevoltagesmeasuredandfedbacktothecontrollerarethevoltagesandwhereasthevoltagesandarenotavailableforthecontrollerbutaredesirabletomaintainthemclosetotheratedvalue.
Afterthepreviousdefinitions,substitutingthecurrentsinthecapacitorsin(10)–(11)bytherelations(15)–(18)andreorga-nizingthedifferentialequations,thematricesinthestate-spacerepresentation(4)result
Thedroopcontrollerinthecaseoftwoinputsandtwooutputsissimply
Theconstantsandhavebeensetin1becauseallofthelinesareofthesamelongitudeanditisdesirabletoextractthesameamountofpowerfromeachterminal.
IEEETRANSACTIONSONPOWERDELIVERY
InFig.8,theeigenvaluesoftheclosed-loopmatrix
canbefoundforseveralvaluesofgain
.Noticethattherealpartsoftheeigenvaluesbecomemorenegativeforhighervaluesofgain.Thisstabilizingeffectisinaccordancewiththefactthatanincrementinthedroopconstantissimilartoincrementingtheenergydissipationinthesystem.
AsmentionedinSectionIII-B,thedroopconstantisselectedinaccordancewithaperformancecriterionmeasuredintermsofthe2-normofthevoltageerrorofthevoltagenotmeasuredandofthecontrolinput.
Thevoltageerrorisgivenby(7),whereweareinterestedin
theparticularinput
.Inthissituation,thetransfer
hasatransmissionzeroat0forthepartic-ulardirectionof
(e.g.,for1)Thisalsoholdsforanyothervalueof.Asaconsequence,
thevoltageerrorreducesto
(19)
ThesingularvaluesofcanbeseeninFig.9.Itis
clearthatthelargerthe
is,thesmallertheerror.Inparticular,at
0andforthemaximumvoltageerrorof10%(kV)andtheratedcurrent667A
ThisconstraintcanbeextendedtotheremainingfrequenciesresultingintheshadowareainFig.9.Theconstraintontheerrorisrelaxedinhighfrequenciessinceitisimpossibletosatisfyauniformlimitwithoutviolatingthebandwidthlimitationsofthe
converters.Thetransferfunctions
ofwhichtheirsingularvaluesareinsidetheshadowareainFig.9satisfytheerrorconstraints.Inacaseofa22transfermatrix,itisnotpossibletofindthat
therefore,.Inmoregeneralcases,thelimitonthegaincanbefoundnumerically.
Theeffectof
ontheoutputisgivenby(8).Theobjectiveistomaintainthedcvoltageinthenondirectlycontrolledter-minalvoltageclosetoaratedvalue.Again,theparticularinput
isconsidered.Forthisparticularinput,
theoutputof
isindependentofandresultsareequaltotheinput,i.e.,
Therefore,itispossibletoanalyzethedeviationfromtheratedvaluebydefining
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PRIETO-ARAUJOetal.:METHODOLOGYFORDROOPCONTROLDYNAMICANALYSIS
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Fig.9.MaximumsingularvaluesofthefunctionS(s)G(s)forseveralvaluesofK(Case1).Thesingularvaluesinsidetheshadowareasatisfytheerrorconstraint.
Fig.10.MaximumsingularvaluesofthefunctionG(s)+Ginside(sthe)KSshadow(s)Garea(s)satisfyforseveraltheconstraintvaluesofonKe(Case.
1).ThesingularvaluesFig.10showthesingularvaluesofthistransferfunction.It
canbeobservedthatforhighervaluesof
,themaximumsingularvaluesof
becomesmallerinlowfrequencies.However,inhighfrequencies,as
Fig.10shows,anincrementof
mayproducetheoppositeeffectincertaincases.Itcanbeseenthatfor
,thesingularvaluesareinsidetheshadowareaandfulfilledthecon-straintsonthevariable.Inparticular,at
0and1/22.5,thevoltageinthewindfarmnodesresult
thatis,a10%errorinthevoltage.
Thecontrolinputisgivenby(9).Again,sincewearein-terestedintheparticularinput
,thissignalresultsgovernedbythetransfer
.Fig.11showsthemaximumsingularvaluesofforsev-eralvaluesof.Theshadowareaindicatesthesingularvalues
thatsatisfytheperformancespecifications.Noticethatthecon-straintdecreasesinhighfrequenciestoconsiderthelimitsonthebandwidthoftheconverters.Itcanbeobservedthatthelow-frequencycomponentsofthecontrolinputareindependent
ofthevalueof
.However,inhighfrequencies,thistransferpresentsresonancepeaksthatforsomevaluesof
,violatetheconstraintsindicatedbytheshadowarea.Thisconstraintim-posesanupperlimitonthegain
.Inparticular,fromFig.11,itcanbeconcludedthat
fulfilstheconstraintonthecontrolinput.
Fromthepreviousanalysis,itcanbeconcludedthatthegainbettersuitstheperformancespecificationsis.
Inordertoevaluatethedroopgainpreviouslyselected,simu-lationswerecarriedoutinMatlab–Simulink.Theanalyzedsce-nariocorrespondstotwosimultaneousandequalchangesinthepowerinjectedintothedcgridbytheWFCs.Thepowersin-jectedbythetwoconverterschangefrom0MWtotheratedvalueat0.05sandreturnto0MWat0.20s.Fig.12(a)showsthepowerflowateachconverter.ThesolidlinescorrespondtothepowerinjectedbytheWFCsandthedashedlinestothepowerextractedbytheGSCs.Itcanbeobservedthatthepoweron-goingfromtheGSCsisalmostcoincidentduetotheselection
ofthepowerdistributionfactors
.Asaconse-quence,bothGSCsextractapproximatelythesameamountofpower.Thepowerlossesofthedcgrid,atratedpowertransmis-sion,arearound375kW.TheevolutionoftheterminalvoltagescanbeseeninFig.12(b).Thedcvoltagesremainat145kVduringtheperiodwherethepowerflowiszerosincethereisnovoltagedropinthegridresistances.Oncethepowerinputincreases,thedcvoltagesmovetowardanewvoltageequilib-rium.Noticethatduringnonzeropowerflow,therearediffer-encesbetweenthevoltageatthewindfarmterminalsandthevoltagesinthegrid-sideterminalsduetothepower-flowdirec-tion.Fig.12(c)showsthecurrentsflowingthrougheachVSC.Bothpowerandcurrentevolutionsaresimilar,exceptforascalefactor,whichindicatesthattheinitialapproximationofconsid-eringthecurrentproportionaltothepowerhasbeenreasonable.Itcanalsobeobservedthatthecurrentsneverexceedthecon-verterlimits.
B.Case2:DroopControlintheWindFarmSide
Inthesecondcaseofstudy,itisassumedthatasimultaneousfaultinbothacgridsforcestheGSCstoenterincurrentlim-itationmode.Inthiscircumstance,theWFCsareresponsibleforregulatingthedcvoltage.Hence,thecontrolinputsarethecurrentsinjectedbytheWFCs,andthedisturbancesarethecur-rentsextractedbytheGSCs,i.e.,
Ontheotherhand,themeasuredvariablesarethewindfarm-sidevoltagesandthenondirectlycontrolledvariablesarethe
gridsidevoltages,i.e.,
Thestatespacemodelhasthesamematrixbuttheinputand
outputmatricesarenowgivenby
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Fig.11.MaximumsingularvaluesofthefunctionKS(s)G(s)forseveralvaluesofK(Case1).Thesingularvaluesinsidetheshadowareasatisfytheconstraintonthecontrolinput.
Fig.12.Simulationscorrespondingtoachangeinthepowerinjectedintothegridbythewindfarmconverters(Case1).
Thedroopcontrollerinthecaseis
IEEETRANSACTIONSONPOWERDELIVERY
Fig.13.Eigenvalueoftheclosed-loopmatrixAforseveralvaluesofK
(Case2).
Fig.14.MaximumsingularvaluesofthefunctionS(s)G(s)forseveralvaluesofK(Case2).Thesingularvaluesinsidetheshadowareasatisfytheerrorconstraint.
Fig.15.MaximumsingularvaluesofthefunctionG(s)+Ginside(sthe)KSshadow(s)Garea(s)satisfyforseveraltheconstraintvaluesofonKe(Case.
2).Thesingularvaluessincethedroopcontrolisappliedtothewindfarmside.
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PRIETO-ARAUJOetal.:METHODOLOGYFORDROOPCONTROLDYNAMICANALYSIS
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Fig.16.MaximumsingularvaluesofthefunctionKS(s)G(s)forseveralvaluesofK(Case2).Thesingularvaluesinsidetheshadowareasatisfytheconstraintonthecontrolinput.
Fig.13showstheeigenvaluesoftheclosed-loopmatrixforseveralvaluesofgain.Noticethattherealpartsoftheeigenvaluesbecomemorenegativeforhighervaluesofgain.Alsointhisscenario,thecaseisconsideredwhere
.Therefore,theperformanceis
associatedonlywiththeinput
.InFigs.14and15,themaximumsingularvaluesofthetransfers
,andcanbe
seenforseveralvaluesof.Theresonancepeaksarelighterdampedinthiscase.Forthisreason,inordertofulfilthelowfrequencieserror,largervaluesmustbeacceptedinhighfrequencies.NoticeinFig.16thattheconstraintonthecontrolinputhasbeenrelaxedinthehighfrequencyforthesame
reason.Asaconsequence,thegain
hasbeensetat1/20.Thesystemhasalsobeenevaluatedbysimulations.Inthesce-narioconsidered,bothWFCsinjecttheratedpowervaluewhiletwovoltagesagsareappliedintheacgrid.Athree-phasevoltagesagof10%ofthenominalacvaluesisappliedtotheacgridconnectedtotheGSC1.Atthesametime,anothervoltagesagof20%isappliedtothegridconnectedtotheGSC2.Bothsagslast0.2s.Thethree-phasevoltagesineachacgridareshowninFig.17whereasthecorrespondingthree-phasecurrentscanbeseeninFig.18.
Fig.19presentstheevolutionofthevariablesinthedcgrid.Itcanbeobservedthattheacgridfaultprovokesanincrementofthealldcvoltages[Fig.19(b)].TheseincrementsareduetothefactthattheGSCsoperateincurrentlimitationmodetoavoidthedisconnectionbyovercurrentsduringthegridfault.WhentheWFCsvoltagesexceed160kV,thecorrespondingconvertersstarttoapplydroopcontrolinthedcgrid,reducingthepowerinjectedtothegridfrom100to20MW[Fig.19(a)].Thedcvoltagelimitis164.5kV.ThedccurrentalsodecreasesduringthevoltagesagduetothepowerreductioncausedbythedroopcontrolintheWFCs[Fig.19(c)].Noticethatthedisconnectionofthesystemduetoovervoltagewasavoidedduringthefault.V.CONCLUSION
AdesignmethodologyfordroopcontrolinmultiterminalHVDCgridshasbeenpresented.Themethodologyincludesa
Fig.17.Simulationscorrespondingtoavoltagesagintheacgrids(Case2).(a)Three-phasevoltagesinthegrid1.(b)Three-phasevoltagesinthegrid2.(a)Voltage(inkilovolts).(b)Voltage(inkilovolts).
Fig.18.Simulationscorrespondingtoavoltagesagintheacgrids(Case2).(a)Three-phasecurrentsingrid1.(b)Three-phasecurrentsingrid2.(a)Current(inamperes).(b)Current(inamperes).
Fig.19.Simulationscorrespondingtoavoltagesagintheacgrids(Case2).(a)Power(inmegawatts),(b)voltage(inkilovolts),and(c)current(inamperes).
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systematicproceduretoformulatealinearmodelofthemul-titerminalgrids.Basedonthismodelandfrequency-responseanalysis,acriterionisprovidedtoselectthedroopgain,takingintoaccountthedynamicsoftheentiremultiterminalHVDCsystem.Thelimitationofdcvoltageerrorsandtheconvertercurrentsdefinesarangeonthedroopgainsthatachieveabettercompromisebetweenthespecifications.Eachlocalcontrollercanaffecttheglobalstabilityandthedcvoltageinotherter-minals.Forthesereasons,thedroopconstantselectionmustbeaddressedinthecontextofmultivariablesystemtheorytoconsiderthedynamicbehavioroftheentiremultiterminalgrid,bothinnormaloperationandinfaultconditions.
Afour-terminalgridexamplehasbeenusedtoillustratetheapplicationoftheuseofthemethodology.Nevertheless,theprocedureisapplicabletoanyothermultiterminalHVDCgridswithmoreinputsandoutputs.Thecomplexityofthemodelin-creaseswiththenumberofnodesandbranchesbutthecom-putationofthesingularvaluesdoesnotinvolveaseriouslimi-tationwiththecurrentalgorithms.Therangeofdroopgainsisobtainedonlyfromthemaximumsingularvalues;therefore,itscomputationisindependentofthecomplexityoftheparticularmultiterminalgrid.
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AnalysisandDesing.Hoboken,NJ:Wiley,2007.EduardoPrieto-AraujowasborninBarcelona,Spain,in1986.HereceivedtheIndustrialEngi-neeringdegreefromtheUniversitatPolitècnicadeCatalunya,Barcelona,Spain,in2011.
Currently,heiswiththeDepartamentd’Enginy-eriaElèctrica,Centred’InnovacióTecnològicaenConvertidorsEstàticsiAccionaments,UniversitatPolitècnicadeCatalunya.ETSd’EnginyeriaIndus-trialdeBarcelona,Barcelona,Spain.Hisareasofinterestarethemodelingandcontrolofelectricalmachines,powerconverters,andHVDCgrids,
relatedtorenewablegenerationsystems.
FernandoD.BianchireceivedtheB.S.andPh.D.degreesinelectronicengineeringfromtheNationalUniversityofLaPlata(UNLP),Argentina,in1999andin2005,respectively.
From1999to2006,hewasaPh.D.studentandaPostdoctoralFellowattheLaboratoryofIndustrialElectronic,ControlandInstrumentation(LEICI,UNLP,LaPlata,Argentina).From2006to2010,hewasaPostdoctoralResearcherattheTechnicalUniversityofCatalonia,Barcelona,Spain.In2010,hejoinedthePowerElectronicsandElectricPower
GridsGroup,CataloniaInstituteforEnergyResearch(IREC),Barcelona,asaScientificResearcher.Hismainresearchinterestsincluderobustcontrolandlinearparameter-varyingsystemsandtheirapplicationstothecontrolofrenewableenergyconversionsystems.
AdriàJunyent-Ferré(S’09)wasborninBarcelona,Spain,in1982.HereceivedtheIndustrialEngi-neeringdegreefromtheUniversitatPolitècnicadeCatalunya,Barcelona,Spain,in2007,whereheiscurrentlypursuingthePh.D.degreeinelectricalengineering.
Hisareaofinterestisthecontrolofpower-elec-tronicconvertersfortheoperationofrenewablegen-erationsystemsunderdifferentgrid-faultconditions.
OriolGomis-Bellmunt(S’05–M’07)receivedtheIndustrialEngineeringdegreefromtheSchoolofIndustrialEngineeringofBarcelona(ETSEIB),TechnicalUniversityofCatalonia(UPC),Barcelona,Spain,in2001andthePh.D.degreeinelectricalengineeringfromtheUPC,in2007.
In1999,hejoinedEngitrolS.L.,wherehewasProjectEngineerintheautomationandcontrolindustry.In2003,hedevelopedpartofhisPh.D.thesisintheDLR(GermanAerospaceCenter),Braunschweig,Germany.Since2004,hehasbeen
withtheElectricalEngineeringDepartment,UPC,whereheisLecturerandparticipatesintheCITCEA-UPCResearchGroup.Since2009,hehasalsobeenwiththeCataloniaInstituteforEnergyResearch(IREC).Hisresearchinterestsincludethefieldslinkedwithsmartactuators,electricalmachines,powerelectronics,renewableenergyintegrationinpowersystems,aswellasindustrialautomationandengineeringeducation.
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