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Electric
Power
Systems
Research
jou rn al h om e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / e p s r
A
novel
VSC-HVDC
link
model
for
dynamic
power
system
simulations
Luis
M.
Castro
a,∗,
Enrique
Acha
a,
Claudio
R.
Fuerte-Esquivel
b aTampereUniversityofTechnology,DepartmentofElectricalEngineering,Tampere,FinlandbUniversidadMichoacanadeSanNicolásdeHidalgo,FacultyofElectricalEngineering,Morelia,México
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received23September2014 Receivedinrevisedform9April2015 Accepted7May2015
Availableonline29May2015 Keywords:
VSC HVDC FACTS
Newton–Raphsonmethod Dynamicpowersystemsimulations
a
b
s
t
r
a
c
t
ThispaperintroducesanewRMSmodeloftheVSC-HVDClink.Themodelisusefulforassessingthe steady-stateanddynamicresponsesoflargepowersystemswithembeddedback-to-backand point-to-pointVSC-HVDClinks.TheVSC-HVDCmodelcomprisestwovoltagesourceconverters(VSC)linkedby aDCcable.EachVSCismodelledasanidealphase-shiftingtransformerwhoseprimaryandsecondary windingscorrespond,inanotionalsense,totheACandDCbusesoftheVSC.Themagnitudeandphase angleoftheidealphase-shiftingtransformerrepresenttheamplitudemodulationratioandthephase shiftthatexistsinaPWMconvertertoenableeithergenerationorabsorptionofreactivepowerpurelyby electronicprocessingofthevoltageandcurrentwaveformswithintheVSC.Themathematicalmodelis formulatedinsuchawaythattheback-to-backVSC-HVDCmodelisrealizedbysimplysettingtheDCcable resistancetozerointhepoint-to-pointVSC-HVDCmodel.TheNewton–Raphsonmethodisusedtosolve thenonlinearalgebraicanddiscretiseddifferentialequationsarisingfromtheVSC-HVDC,synchronous generatorsandthepowergrid,inaunifiedframe-of-referenceforefficient,iterativesolutionsateach timestep.ThedynamicresponseoftheVSC-HVDCmodelisassessedthoroughly;itisvalidatedagainst theresponseofadetailedEMT-typemodelusingSimulink®.Thesolutionofarelativelylargepower
systemshowstheabilityofthenewdynamicmodeltocarryoutlarge-scalepowersystemsimulations withhighefficiency.
©2015ElsevierB.V.Allrightsreserved.
1. Introduction
Continuousincreases in electrical energy consumption have encouraged a great deal of technological development in the electricalpowerindustry.Inparticular,thedevelopmentofnew equipmentforpowertransmissionthat enablesamore flexible powergridaimedatachievinghigherthroughputs,enhancing sys-temstabilityand reducing transmissionpowerlosses,hasbeen highontheagenda[1,2].TheVSC-HVDClinkisthelatestequipment developedinthearenaofhigh-voltage,high-powerelectronicsand itsintendedfunctionistotransportelectricalpowerinDCform,as wellastoenabletheasynchronousinterconnectionofotherwise independentACsystems [3], and toprovide independent reac-tivepowersupport.ThetechnologyemploysInsulatedGateBipolar Transistors(IGBTs),drivenbypulsewidthmodulation(PWM) con-trol.Thisvalveswitchingcontrolpermitstoregulatedynamically, inanindependentmanner,thereactivepowerateitherterminalof theACsystemandthepowerflowthroughtheDClink[4].
∗ Correspondingauthor.Tel.:+358465496289.
E-mailaddress:luismiguelcg@hotmail.com(L.M.Castro).
TheVSC-HVDCmodelputforwardinthispapercomprisestwo VSCmodelslinkedbya cableonitsDC sides.Inturn,eachVSC modelis madeupofanidealphase-shiftingtransformerwhich synthesisesthephase-shiftingandscalingnatureofthePWM con-trol.Theidealphaseshifteristakentobetheinterfacebetween theACandDCcircuitsoftheVSC.Themodelmakesprovisionsfor therepresentationofconductionlossesandswitchinglosses.Since bothconvertersarecapableofindependentlycontrollingthe reac-tivepowerexchangedwiththeACpowergridattheirrespective ACnodes,theVSC-HVDCdynamicmodelusestwo independent dynamicvoltageregulators.Bothcontrolloopsareaimedat pro-vidingtherequiredreactivepowersupportattheirrespectiveAC nodestomaintainpre-setvoltages,byregulationoftheiramplitude modulationcoefficients.Likewisethemodelcorrectlyaccountsfor thedynamicsoftheDClink.Thisiscarriedoutbyusingacontrol blockthatactsupontheDCcurrenttoadjusttheDCvoltageofthe VSC-HVDClink.
ItshouldbementionedthatinanearlymodeloftheVSC-HVDC system,thetwo VSCare emulatedby idealisedvoltagesources
[5–8].Alternatively,theVSCshavealsobeenrepresentedby equiv-alentcontrolledcurrentsources[9,10],wherethecurrentstobe injectedintotheACgridsarecomputedbytheexistingdifference betweenthecomplexvoltages oftheVSCterminalsandtheAC http://dx.doi.org/10.1016/j.epsr.2015.05.003
systemnodesatwhichtheVSC-HVDCisembedded.Morerecently, theconceptofdynamicaveragemodellinghascaughtthe atten-tionofthepowersystemcommunitysinceitallowsthemodelling ofVSCsinamoredetailedmanner[11].Inthedynamicaverage modellingapproach,theaveragevalueoftheoutputvoltage wave-formiscalculatedateachswitchinginterval,avaluethatchanges dynamicallydependingonthevalueofthereferencewaveform.The VSCisrepresentedbyathree-phasecontrolledvoltagesourceon itsACsidesandasacontrolledcurrentsourceonitsDCsides[12]. However,thisapproachmaybetimeconsumingwhenrepetitive simulationsstudiesarerequired,suchasinpowergridexpansion planningandinoperationplanning.Thesolutiontimeisalwaysan importantpointtokeepinmindandinthismethod,increasingtime stepsisalwaysatemptationbutcautionneedstobeexercisedwhen usingthedynamicaveragingmethodsince,asreportedin[12],the useoflargetimestepsmayaffecttheaccuracyoftheresults.Itis worthmentioningthatifharmonicsorelectromagnetictransients arethestudysubject,suchahighlevelofmodellingdetailis nec-essary,where,forinstance,thePWMcontrolneedstobemodelled explicitlytoachievemeaningfulresults.
Ontheotherhand,inlarge-scalepowersystemapplications, itlooksattractivetorepresenteach VSCasacontrolledvoltage sourceowingtoitsmuchreducedcomplexity.However,itsinternal variablesmaynotbereadilyavailable.Incontrast,thenewmodel introducedherecapturesverywellthekeyoperational character-isticsoftheVSCsmakinguptheHVDClink.Thisisdonebyusing explicitstatevariablesthatencapsulatetheactualperformanceof theACandDCcircuitsforboth,thesteady-stateanddynamic oper-atingregimes.Furthermore,thenewVSC-HVDCmodelpossesses thefourdegreesoffreedomfoundinactualVSC-HVDC installa-tions,characterisedbyhavingsimultaneousvoltagesupportatits twoACterminals,DCvoltagecontrolattheinverterconverterand regulatedDCpowerattherectifierconverter.
ThenumericalimplementationoftheVSC-HVDCmodelis car-riedoutusingaunifiedframeworkwhichsuitablycombinesthe algebraicanddiscretiseddifferentialequationsoftheVSC-HVDC linkmodel,thesynchronousgeneratorsandthenon-linear alge-braic equations of thepower grid.This iterativesolution takes advantageoftheNewton–Raphson(NR)methodthusfacilitating theefficientsolutionofthenon-linearequations.Thediscretisation ofthedifferentialequationsiscarriedoutusingtheimplicit trape-zoidalruleofintegrationwhichhasbeenproventobenumerically stableandaccurate[13,14].Inthispaper,specialattentionispaid tothenewdynamicVSC-HVDCmodel,emphasisinghowthe alge-braicanddiscretiseddifferentialequationsareassembledtogether inthisframework.
2. VSC-HVDCmodelfordynamicanalysis
2.1. Keyphysicalcharacteristics
IftwoVSCstationsarelinkedasshowninFig.1,aVSC-HVDC systemisformedandtermedpoint-to-pointconfiguration.Inthis arrangement,electricpoweristakenfromonepointoftheAC net-work,convertedtoDCintherectifierstation,transmittedthrough theDClinkandthenconvertedbacktoACintheinverterstation
Fig.1. SchematicrepresentationofaVSC-HVDClink.
Fig.2.VSCequivalentcircuitfortheinverterstation.
andinjectedintothereceivingACnetwork.Inadditiontotransport powerinDCform,thiscombinedsystemisalsocapableofsupplying reactivepowerandprovidingindependentdynamicvoltage con-trolatitstwoACterminals.Itisworthmentioningthatbysetting thecableresistanceRDCtozero,therepresentationreducestothat
oftheso-calledback-to-backVSC-HVDCconfiguration.Pleaserefer totheAppendixAforthesymbolsusedinallequationsandfigures. 2.2. VSC-HVDCsteady-statemodel
Fig.2depictstheequivalentcircuitoftheVSCcorrespondingto theinverterstation;asimilartopologycanbeformulatedforthe rectifierstation.Itssteady-staterepresentationreliesonanideal phase-shiftingtransformerwithcomplextaps,aseriesimpedance onitsACsideaswellasanequivalentvariableshuntsusceptance BeqI,andashuntresistoronitsDCside[15].
TheseriesreactanceX1IrepresentsdeVSC’sinterface
magnet-icswhereastheseriesresistorR1Iisassociatedtotheohmiclosses
whichareproportionaltotheACterminalcurrent squared.The shuntresistor(withaconductancevalueofGswI)producespower
losstoaccountfortheswitchingactionoftheconverter valves. Thisconductanceiscalculatedaccordingtoratedconditionsand ensuresthattheoperatingconditionsonswitchinglossesare repre-sentedbyscalingthequadraticratiooftheactualterminalcurrent IItothenominalcurrentInom: GswI=G0I
II/Inom
2.Notethatthe squaringofthisratioistogivetheswitchingconductanceterman overallpowerperformance.Thefollowingassumptionsaremade inthemodel:(a)thecomplexvoltageV1=k2maIEDCejIisthe
volt-agerelativetothesystemphasereference;(b)thetapmagnitude maIoftheidealphase-shiftingtransformercorrespondstotheVSC’s
amplitudemodulationcoefficientwherethefollowingrelationship holdsforatwo-level,three-phaseVSC: k2=
3
⁄
8;(c)theangle IisthephaseangleofvoltageV1;(d)EDC istheDCbusampli-tudevoltagewhichisarealscalar.Bearingthisinmind,thenodal powerflowequationsfortheseriesbranchoftheVSCrepresenting theinverterstationarederivedfromthenodaladmittancematrix developedinAppendixB.Aftersomearduousalgebra,theactive andreactivepowersexpressionsforthepowersinjectedatboth endsoftheVSC,nodesvIand0vI,arearrivedat:
PvI=Vv2IG1I−k2maIVvIEDCI
G1Icos vI−I +B1Isin vI−I (1) QvI=−Vv2IB1I−k2maIVvIEDCI G1Isin vI−I −B1Icos vI−I (2) P0vI =k22m2aIEDCI2 G1I−k2maIVvIEDCI G1Icos I−vI +B1Isin I−vI +PswI (3) Q0vI =−k22m2aIEDCI2 B1I−k2maIVvIEDCI G1Isin I−vI −B1Icos I−vI +QeqI (4)where, PswI=EDCI2 G0I
IvI/Inom 2 (5)QeqI=−k22m2aIEDCI2 BeqI (6)
Likewise,asimilarsetofequationsmaybeobtainedfortheVSC correspondingtotherectifierstation.Toobtainthesteady-state equilibriumpoint,thesetofmismatchpowerflowequationsthat mustbesolvedtogetherwiththosearisingfromallthenetwork’s nodesis: PvR=−PvR−PvR,load−PvRcal=0 (7) QvR=−QvR−QvR,load−QvRcal=0 (8) PvI=−PvI−PvI,load−PvIcal=0 (9) QvI=−QvI−QvI,load−QvIcal=0 (10) E0R=EDCR2 −EDCIEDCR−PschRDC=0 (11) P0vR=−Psch−P0vR=0 (12) P0vI=Psch− (EDCR−EDCI)2R−1DC−P0vI=0 (13) Q0vR=−Q0vR=0 (14) Q0vI=−Q0vI=0 (15)
where,inthisparticularcase,PvRcal,QvRcal,PvIcalandQvIcal,standfor
thepowersflowingfrombusvRtokandvItom,respectively.They aregivenby PvRcal=Vv2RGRR+VvRVk
GRkcos vR−k +BRksin vR−k (16) QvRcal=−Vv2RBRR+VvRVk GRksin vR−k −BRkcos vR−k (17) Similarequationsmaybeobtainedforthepowerflowingfrom busItombysimplyexchangingsubscriptsin(16)and(17).Itshould beremarkedthat(12)ensuresthatthepowerflowleavingthe rec-tifierstationbekeptatthescheduledvaluePsch.Giventhatthe inverterstationischosentokeeptheDClinkvoltageataconstant valueEDCIthenEq.(11)allowsthecomputationoftheDCvoltageintherectifier’sside,EDCR.Sincetheobjectiveistoregulatethe
volt-agemagnitudeatbothACsidesoftheVSC-HVDCwhilekeeping theDCvoltagefixed,VvI,VvRandEDCIarenotpartofthesetofstate
variablesthatneedtobecomputed.Thus,vR,maR,vI,maI,EDCR,R,
I,BeqRandBeqI,constitutethesetofstatevariablesthatmustbe
calculatedbysolving(7)–(15)withtherestoftheequations aris-ingfromthenetwork.ToguaranteethateachVSCoperateswithin feasibleoperatinglimits,alimitcheckingofthemodulationratio andterminalcurrentmusttakeplace,thatis,ma≤1andI≤Inom.
Furthermore,toenablegoodstartingconditions,theNRalgorithm isinitialisedasfollows:theamplitudemodulationratios,maIand
maR,andtheangles,IandR,aresetat1and0,respectively.
2.3. VSC-HVDCdynamicmodel
TheVSC capacitor’s dynamics play an importantrole inthe behaviouroftheHVDClinkwhensubjectedtovoltageandpower variationscomingfromtheexternalACnetwork.Ontheotherhand, inthisVSCapplication,DCvoltagecontrolisatargetinorderto pursueastableoperationoftheDClink.Theinverterconverteris theonethattakesonsuchatask,wherethefollowing relation-shipholdsatitsDCterminals:ic=−IDCR−IDCI,beingIDCRandIDCI
thecurrentsinjectedatrectifier’sDCbusandatinverter’sDCbus,
Fig.3.VSC-HVDCdynamiccontrollerfortheDCvoltage.
respectively.Substitutingthiscurrentrelationshipintothe expres-sionthatallowscalculatingthecapacitor’scurrentic=CDCdEdtDC,we
getthedifferentialequationwithwhichtheDCvoltagedynamics arerepresented. dEDCI dt = −IDCR−IDCI CDC (18) IDCR= P0vR EDCR (19) ThevalueofCDCisestimatedfromtheamountofenergystored
inthecapacitor:Wc= 1
⁄
2CDCE2DC.TheelectrostaticenergystoredintheDCcapacitorcanbeassociatedwithanequivalentinertia con-stantHc[s]asWc=HcSnom,whereSnomwouldcorrespondtothe
ratedapparentpoweroftheVSC.Thistimeconstantissmallandit maybetakentobeHc≈5ms[16].Hence,theper-unitvalueofthe
capacitorwouldbeCDC=2SnomHc/E2DC.
Arguably,thecurrentbalanceshownin(18)isakintothepower balanceinsidetheHVDClinkforsteady-stateoperatingconditions whenthederivativetermbecomeszero.Thus,wheneverthe cur-rent/powerbalanceisdisturbed,voltagevariationswillappearin theDClink.AsshowninFig.3,thedynamiccontroloftheHVDC’sDC voltageiscarriedoutbyusingtheDCcurrententeringtheinverter converter,IDCI,asthecontrolvariable.Theerrorbetweentheactual
voltageEDCIandEDCInomisusedbyaPIcontroller,withgainsKpedc
andKiedc,toobtainnewvaluesofDCcurrentIDCI.
Thedifferential andalgebraic equationsarising fromtheDC voltagedynamiccontrollerare
dIDCIaux
dt =Kiedc(EDCI−EDCInom) (20)
IDCI=Kpedc(EDCI−EDCInom)+IDCIaux (21)
Simultaneously,therectifier converter must ensurethat the activepowerleavingthisstationbekeptatthescheduledvaluePsch.
FromFig.1,itcanbeinferredthatthepowerenteringtheinverter stationisPschminusthepowerlossincurredbytheDCcable
resis-tor.Iftheinverterstationisselectedtoperformthecontrolofthe DClinkvoltageEDCIthentheDCvoltageattherectifier’sside,EDCR,
canbecomputedatanytimebyapplyingKirchhoff’svoltagelaw intheDCcircuit,asfollows:
EDCR=EDCI−RDCIDCR (22)
Theangularaperturebetweenthephase-shiftingangleofthe rectifier R and the voltage angle vR is related to the power
exchangeoccurringatanytimebetweenthenetworkandthe rec-tifier’sDCbus.Hence,theangulardifferenceR=vR−R isalso
akeyparameterthatrequiresproperregulationwiththeaimto achievethescheduledactivepowertransferPschfromtherectifier
stationtowardstheinverterstation.Then,thepursuedpower bal-anceontheDCsidewillnowbegivenbythefollowingexpression: P0vR+Psch=0,asshowninFig.4.
The equations that allow the assessment of the dynamic behaviourforthescheduledpowercontrollerare
dRaux
Fig.4. DC-powertransfercontrollerfortheVSC-HVDClink.
R=Kppdc(Psch+P0vR)+Raux (24)
TheACvoltagedynamiccontroloftheVSC-HVDCcallsfortwo controlloops,asshowninthefirst-ordercontrolblocksofFig.5. ThemodulationindicesmaIandmaRareresponsibleforeither,
con-trollingthevoltagemagnitudesattheACsidesoftherectifierand inverterstations ofatthescheduledvalues, VvI0 andVvR0,orto
exertthefixedreactivepowersetpoint:Qref
I andQRref.Thecontrols
aredesignedinsuchawaythatthemodulationindicesmaIand
maRarereadjustedateverytimestepaccordingtothevoltageor
reactivepowercommands.
The differentialequations representing the dynamics of the modulationindiceswhenvoltagecontrolisselectedare
d (dmaR) dt = KmaR(VvR0−VvR)−dmaR TmaR (25) d (dmaI) dt = KmaI(VvI0−VvI)−dmaI TmaI (26)
3. Dynamicframeofreference
Inthis papertheinterest isin assessingtheeffectiveness of thenewVSC-HVDCmodeltoregulatevoltagemagnitudeateither terminaloftheACsystem,followingachangeinthepower net-worksuchasa stepchangeinsystemloadorthetrippingof a transmissionlineortransformer.Hence,thesolutionmethod pre-sented in [13],is selected toimplement theVSC-HVDC model developedinSectionII.Thisapproachcombinesthesetofalgebraic equations(27)representingthepowernetworkwiththesystem ofdifferentialequations (28)describingthedynamic behaviour ofthesynchronous generatorsand theircontrols,toobtainthe solutionasafunctionoftimein aunifiedframeofreference.It usestheimplicit trapezoidalmethod(seeAppendixC)which is knowntobenumericallystable,preservingareasonableaccuracy
[13,14],
0=f (X,Y ) (27)
˙y=g (X,Y,t) (28)
Fig.5.AC-busvoltagecontrollers:(a)rectifierstationand(b)inverterstation.
whereXandYarevectorsofvariablesthatarecomputedatdiscrete pointsintime.
Theseequations areefficiently solved usingtheNR method. In this case, theconventional power flow Jacobian matrix, J,is enlargedtoaccommodatethepartialderivativesthatarisefrom thediscretiseddifferentialequationsanditscontrolvariables.The NRmethodprovidesanaccuratesolutiontothesetofequations givenbyF(Z)=0,bysolvingforZinthelinearisedproblemJZ=− F(Z),inarepetitivefashion.InthiscaseZisavectorthatcontains thenetwork’sstate variables andthe statevariables pertaining tothesynchronousgeneratorsandtheircontrolsor,indeed,any othercontroldevicesuchastheVSC-HVDClink.Inanexpanded form,
⎡
⎣
QP F(y)⎤
⎦
=−⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
∂
P∂
∂
P∂
V∂
P∂
y∂
Q∂
∂
Q∂
V∂
Q∂
y∂
F(y)∂
∂
F(y)∂
V∂
F(y)∂
y⎤
⎥
⎥
⎥
⎥
⎥
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⎦
⎡
⎣
V y⎤
⎦
(29)where P and Qare theactive and the reactivepower mis-match vectors, respectively; F(y) is a vector that contains the discretiseddifferentialequationsofeachmachineorcontrolling device; , V and y represent the vectors of incremen-tal changes in nodal voltage angles and magnitudes, as well as the state variables arising from each differential equation. TheNR methodstartsfroman initialguessfor Z0 and updates
the solution at each iteration i, i.e., Zi+1=Zi+Zi, until a
pre-defined toleranceis fulfilled. In this unified solution,all ofthe statevariables areadjustedsimultaneouslyinordertocompute the newequilibrium point of the power systemat every time step.
3.1. DiscretisationandlinearisationoftheVSC-HVDCequations fordynamicsimulations
Toenableasuitablerepresentationinthisunifiedframeof ref-erence,theVSC-HVDCdifferentialequations arediscretisedand expressedintheformofamismatchequationinthesameformas thatofthenetwork’sactiveandreactivepowermismatch equa-tions: FEDCI=EDCI,t−t+ t 2 ˙EDCI,t−t−
EDCI,t− t 2 ˙EDCI,t =0 (30) FIDCIaux=IDCIaux,t−t+ t 2 ˙IDCIaux,t−t −
IDCIaux,t− t 2 ˙IDCIaux,t =0 (31) FRaux=Raux,t−t+ t 2 ˙Raux,t−t−
Raux,t− t 2 ˙Raux,t =0 (32) FdmaR=dmaR,t−t+ t 2 d ˙maR,t−t−
dmaR,t− t 2 d ˙maR,t =0 (33) FdmaI =dmaI,t−t+ t 2 d ˙maI,t−t−
dmaI,t−t 2 d ˙maI,t =0 (34) where, ˙EDCI,t=CDC−1 −IDCR,t−IDCI,t (35)
˙EDCI,t−t=CDC−1
−IDCR,t−t−IDCI,t−t(36)
˙IDCIaux,t=Kiedc
EDCI,t−EDCInom
(37)
˙IDCIaux,t−t=Kiedc
EDCI,t−t−EDCInom (38) ˙Raux,t=Kipdc Psch+P0vR,t (39) ˙Raux,t−t=Kipdc Psch+P0vR,t−t (40) d ˙maR,t=TmaR−1 KmaR VvR0−VvR,t −dmaR,t (41) d ˙maR,t−t=TmaR−1 KmaR VvR0−VvR,t−t−dmaR,t−t (42) d ˙maI,t=TmaI−1 KmaI VvI0−VvI,t −dmaI,t (43) d ˙maI,t−t=TmaI−1 KmaI VvI0−VvI,t−t −dmaI,t−t (44) TheEqs.(30)–(44)governthedynamicbehaviourofthe VSC-HVDCmodel.ThefirsttwoequationscapturetheDCvoltageand currentperformance oftheDC linkwhentheenergybalanceis perturbedowingtoadisturbanceintheACnetwork.Likewise,the equationinvolvingtheangular apertureR (32)deals withthepowerunbalancepresentintheDClink.Also,theEqs.(33)and
(34)enablethecomputationofthenewvaluesofthemodulation indiceswithwhichthetargetACvoltagesareacquiredfortheactual ACnetwork’soperatingconditions.InordertolinktheVSC-HVDC’s controlvariableswiththegrid’sstatevariablesatnodesvRandvI, thealgebraicpowermismatchEqs.(7)–(10)mustbeused. How-ever,tocompletethemodelfordynamicsimulationpurposes,two morealgebraicequationsareneeded.Oneforcalculatingthe volt-ageEDCRattherectifierstation’sDCbus(45)andanothertoenable
theHVDCtoachievethepowerbalanceattheinverter’sDCbus
(46).
E0R=EDCR−EDCI+RDCIDCR (45)
P0I=EDCIIDCI−P0vI (46)
Eqs. (7)–(10), (45)–(46) and (30)–(34) constitute the set of mismatchequations that mustbeassembled togetherwiththe equationsofthewholenetwork,synchronousgeneratorsandtheir correspondingcontrollers.ThelinearisedformoftheVSC-HVDC mathematicalmodelisgivenby,
F=−
J11 J12 J21 J22 z (47)F=
PvR QvR PvI QvI E0R P0I FEDCI FIDCI,aux FRaux FdmaR FdmaI Tz=
vR VvR vI VvI EDCR I EDCI IDCI,aux Raux dmaR dmaI TwhereJ11comprisesthefirst-orderpartialderivativesofthepower
mismatchequationsandinnerVSC-HVDC’smismatchequations withrespecttothenetwork’sandVSC-HVDC’sstatevariables. Like-wise,J12containsthefirstorderpartialderivativesarisingfromthe
algebraicmismatchequationswithrespecttothecontrolvariables oftheVSC-HVDClink.ThematrixJ21consistsofpartialderivatives
oftheVSC-HVDC’sdiscretiseddifferentialequationswithrespect totheACvoltagesandangles,thephase-shiftingangleIandthe
DCvoltageEDCR.Lastly,J22isamatrixthataccommodatesthe
first-orderpartialderivativesoftheVSC-HVDC’sdiscretiseddifferential equationswithrespecttotheirowncontrolvariables.
J11=
⎡
⎢
⎢
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⎢
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⎣
∂PvR ∂vR ∂PvR ∂VvR 0 0 ∂PvR ∂EDCR 0 ∂QR ∂vR ∂QvR ∂VvR 0 0 ∂QvR ∂EDCR 0 0 0 ∂PvI ∂vI ∂DPvI ∂VvI 0 ∂PvI ∂I 0 0 ∂QvI ∂vI ∂QvI ∂VvI 0 ∂QvI ∂I ∂E0R ∂vR ∂E0R ∂VvR 0 0 ∂E0R ∂EDCR 0 0 0 ∂P0I ∂vI ∂P0I ∂VvI 0 ∂P0I ∂I⎤
⎥
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⎦
, J12=⎡
⎢
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⎣
0 0 0 ∂PvR ∂dmaR 0 0 0 0 ∂QR ∂dmaR 0 ∂PvI ∂EDCI 0 0 0 ∂PvI ∂dmaI ∂QvI ∂EDCI 0 0 0 ∂QvI ∂dmaI ∂E0R ∂EDCI 0 0 ∂E0R ∂dmaR 0 ∂P0R ∂EDCI 0 0 0 ∂P0R ∂dmaI⎤
⎥
⎥
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⎦
J21=⎡
⎢
⎢
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⎢
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⎢
⎣
∂FEDCI ∂vR ∂FEDCI ∂VvR 0 0 ∂FEDCI ∂EDCR 0 0 0 0 0 0 0 ∂FRaux ∂vR ∂FRaux ∂VvR 0 0 ∂FRaux ∂EDCR 0 0 ∂FdmaR ∂VvR 0 0 0 0 0 0 0 ∂FdmaI ∂VvI 0 0⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, J22=⎡
⎢
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∂FEDCI ∂EDCI 0 0 0 ∂FEDCI ∂dmaI ∂FIDCI,aux ∂EDCI ∂FIDCI,aux ∂IDCI,aux 0 0 0 0 0 ∂FRaux ∂Raux ∂FRaux ∂dmaR 0 0 0 0 ∂FdmaR ∂dmaR 0 0 0 0 0 ∂FdmaI ∂dmaI⎤
⎥
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⎦
(48)The steady-state conditions that are employed to start the dynamic simulation are calculated through the conventional Newton–RaphsonpowerflowalgorithmincludingtheVSC-HVDC linksteady-statemodel,asdiscussedinSection2.2.Sucha solu-tionwillprovideadequatestartingconditionstoensurereliable dynamicsimulations.
Fig.6.TestsystemusedtovalidatetheproposedVSC-HVDCmodel.
4. StudyCases
4.1. ValidationofthenewVSC-HVDCmodel
TheprowessofthenewVSC-HVDClinkmodelisdemonstrated bycarryingoutacomparisonagainstthewidely-usedEMT-type simulationsoftwareSimulink®.Itshouldbementionedthatboth types of simulation tools enable dynamic assessments of elec-tricalpower networks but they takea fundamentally different approach.Simulink® representseverycomponent ofthepower gridbymeansofRLCcircuits andtheircorresponding differen-tialequationsrequirediscretisationatrathersmalltimesteps,in theorderofmicro-seconds,toensureastablenumericalsolution. Conversely,thesolutionoftheRMS-typemodelintroducedinthis paperrequiresonlyonephaseofthenetwork(positivesequence), usingfundamental-frequencyphasorsofvoltagesandcurrentsas opposedtothethree-phaserepresentationalong with instanta-neouswaveformsofvoltagesandcurrentsusedinanEMTtoolsuch asSimulink®.
TheVSC-HVDCmodelcomparisoniscarriedoutusingarather simplepowersystemcomprisingtwoindependentACnetworks (2000MVA,230kV,50Hz)which areinterconnectedthrough a VSC-HVDClink(200MVA,±100kVDC)withaDCcablelengthof 75km,asshowninFig.6.Bothconverterstationscompriseeach astep-downtransformer,ACfilters,converterreactor,DC capaci-torsandDCfilters,wherethechangesofthetransformers’tapare notsimulated.Themodelofthepowersystemincludingthe VSC-HVDClinktogetherwithitsparameterscanbefoundinthesection of‘demos’inSimulink®as:VSC-BasedHVDCTransmissionSystem (DetailedModel),whereastheparametersofthenewVSC-HVDC modelareshowninAppendixD.Toensureareliablenumerical solution,theEMT-typesimulationpackagediscretisesthepower systemandthecontrolsystemwithasampletimeof7.406sand 74.06s,respectively,whereasforthedevelopedRMS-typemodel, anintegrationstepof1msisused.Alltheresultsshowninp.u. valuesarebasedontheHVDCstation’srating.
Initiallytherectifierstationissettocontroltheactivepower transmissionatPsch=200MW(1p.u.),theinverterisresponsible
forcontrollingtheDCvoltageatEDCInom=200kV(1p.u.).The
rec-tifierandinverterstationsaresettocomplywithafixedreactive powercommandof0p.u.and−0.1p.u.,respectively.Inorderto reachthesteady-stateequilibriumpointinSimulink®,the simula-tionisrunuptot=1s.Atthispoint,theactivepowertransmission isreducedfrom200MWto100MW,thatis,a−50%stepisapplied tothereferencescheduledDCpower.Furthermore,att=3s,astep changeof−5%isappliedtothereferenceDCvoltageoftheinverter, i.e.,theDCvoltageisdecreasedfrom1p.uto0.95p.u.
TheDCvoltagesattheconverters’DCterminalsareshownin
Fig.7correspondingtocaseswherestepchangesinthereference DCpowerandDCvoltageareapplied.Asexpected,some differ-encescanbeseenfromtheresultsobtainedusingbothsolution techniques.Thedynamicperformance oftheDC voltagesofthe RMS-typemodel followswellthedynamic pattern obtainedby theswitching-basedHVDCmodelsimulatedinSimulink®.Avery considerabledifferenceexistsbetweenthetwoapproachesatthe start ofthesimulation (0.5sof thesimulation), a factthat can beexplainedbytheverydifferentmannerinwhichbothpower
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.85 0.9 0.95 1 1.05 1.1 Time [s] DC voltage [p.u] EDCR EDCI EDCR EDCI Proposed model
Simulink model Step change in DC power
Step change in DC voltage
Fig.7. DCvoltageperformancefortheproposedandSimulinkVSC-HVDCmodel.
systemsimulationsareinitialised;ourproposedVSC-HVDCsystem usesanaccuratestartingconditionfurnishedbyapowerflow solu-tionwhereastheSimulink®modelstartsfromitscustomaryzero initialcondition,i.e.,thecurrentsandvoltagesoftheinductorsand capacitors,respectively,aresettozeroatt=0s.
Similarconclusionscanbedrawnwhenanalysingthedynamic responseof theDCpowerfollowing theapplicationofthestep changesinDCpowerandDCvoltage,asshowninFig.8.Asforthe changeintheDCpowerreference,itcanbeseenthatthepower sta-bilisesinnomorethan0.5s;thisshowstheratherquickresponse androbustnessaffordedbythedynamiccontrolsoftheVSC-HVDC linkevenintheeventofadrasticchangeinthetransmittedDC power.Ontheotherhand,thestepchangeintheDCvoltage ref-erencecausesmomentarypowerflowoscillationsintheDClink whicharealsodampedoutquiterapidly.
ThedynamicbehaviouroftheHVDC’smodulationindicesare depictedinFig.9.ThenegativestepchangeintheDCpower refer-enceyieldsaverynoticeablevariationinthemodulationindices; thedynamicperformanceofthemodulationindicesascalculated bybothSimulink®andtheproposedHVDCmodel,followthesame trendalthoughanexact matchwasnotexpected.Afterthefirst disturbance,a steady-stateerrorof0.87%and1.67%is obtained forthemodulationindices of therectifierand inverter, respec-tively.Similarly,oncetheoscillationsdue tothestepchangein thereferenceDCvoltagehavebeendamped,thedifferencesinthe modulationindicesstandat0.07%and1.16%,respectively.These relativelysmallvariationsmaybeexplainedbytheverydifferent modellingandsolutionapproachesusedbythetwoquitedifferent simulationtechniquesusedforthecomparison,theinitial steady-statevaluesoftheconverters’indicesandmostimportantlydueto
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.4 0.6 0.8 1 1.2 Time [s] DC power [p.u] Simulink Proposed model
Step change in DC power
Step change in DC voltage
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.7 0.8 0.9 1 1.1 Time [s] Modulation index maR maI maR maI Proposed model
Step change in DC power
Step change in DC voltage Simulink model
Fig.9. ModulationindicesperformancefortheproposedandSimulinkVSC-HVDC model.
Fig.10.RelevantareaoftheNewEnglandtestsystem.
theDCvoltagebehaviourasthishasastrongimpactonthe per-formanceofthemodulationindices.Moreimportantly,theresults furnishedbythetwosoftwaresimulationsfollowthesametrend.
Forthesakeofcomparison,Table1showstheVSC-HVDCresults asobtainedbythenewmodelandtheSimulink®modelat differ-entpointsintimeofsimulation.Table1alsoshowsthecomputing timesrequiredtosimulatethetestsystemusingboththeRMS-type VSC-HVDC modeland theEMT-typesimulation tool Simulink®, withthenewmodelbeingapproximatelyninetimesfasterthanthe EMTsimulation.Thesignificantcomputationaltimesavingwithout jeopardisingtheaccuracyoftheresultsmakesthedeveloped VSC-HVDClinkmodelasuitableoptionforlarge-scalepowersystem simulations,specificallyinstudiesthatrequirelongersimulation timessuchasthoseinvolvingsynchronousgenerators’frequency variationsandlong-termvoltagestabilityissues.
4.2. NewEnglandtestsystemwithembeddedVSC-HVDClink TheNew England test system[17] is modified, asshown in
Fig.10,toincorporatethemodelofaVSC-HVDCwiththe parame-tersshowninAppendixD.Thetransmissionlineconnectingnodes 4and14isreplacedbyaVSC-HVDClink.TheDCcableresistanceis assumedtobe0.24%ontheVSCs’base:Snom=300MVA,resulting
Fig.11.Voltageperformanceatdifferentnodesofthenetwork.
inthesameresistancevalueasthatofthereplacedtransmission lineforthesystem’sbase:0.08%.Therectifierandinverterstations, VSCRandVSCI,exertvoltagecontrolattheirrespectiveACterminals
atVvR=1.01p.uandVvI=1.03p.u,respectively.Forthesteady-state
conditions,thehigh-voltagesideoftheLTCtransformers,which correspondtonodes4and14,areheldfixedatthesamevoltage levelasthosefortheconverters’terminals,VvRandVvI;underthese
conditions,theLTC’stapsarecomputedthroughthesteady-state powerflowalgorithm;theirvaluesarekeptconstantduringthe dynamicsolution.Inadditiontoprovidingreactivepowercontrol, theHVDClinkperformsactivepowerregulationattherectifier station’sDCbus atPsch=100MW,whichimplies thattheactive
powerisdrawnfromnode4andinjectedtonode14,asdepicted inFig.10.Theactiveandreactivepowerspresentedintheanalysis forthesteady-stateanddynamicoperatingregimesaregivenat thehigh-voltagesideofeachLTCtransformer.Allresultsshownin p.u.valuesarebasedontheHVDCstation’sratings.
Duringsteadystate,therectifierstationisdelivering153.359 MVArtothenetworksoastoupholditstargetvoltagemagnitude withamodulationratioof0.8282,whereastheinverterstation operateswithamodulationratioof0.8423,injecting27.306MVAr tothegrid.InthecaseoftheactivepowerflowingthroughtheHVDC system,thepowerenteringtherectifierstationstandsat101.159 MWandthepowerleavingtheinverterstationtakesa valueof 99.641MW.Itisclearthatthedifferencebetweenthesetwo pow-ersisthetotalpowerlossincurredbytheHVDCsystemincluding thatproducedbytheDClinkcable.Takingasareferencethe nomi-nalapparentpowerforeachconverterSnom,thetotalpowerlosses
standat0.504%ofwhich0.386%correspondstotherectifierstation and0.112%totheinverterstationwhilstthepowerlossproduced byJoule’seffectintheDCcablestandsat0.006%,recallingthatits magnitudeisdependentonthelengthoftheDCtransmissionline.
Table2showsthemainVSC-HVDCresultsasgivenbythe steady-statepowerflowsolutionwhichservesthepurposeofinitialising thedynamicsimulation.
Hence,usingvaluesfromthesteady-statepowerflowsolution, itiseasytoproceedwiththecalculationoftheinitialvaluesforthe controlvariablestakingpartinthedynamicsoftheVSC-HVDClink; theseareemployedtoinitialisethedynamicsimulationofthetest network,whensubjectedtothedisconnectionofthetransmission lineslinkingnodes25-2,2-3and3-4,att=0.1s.
Table1
ComparisonofVSC-HVDCvariablesfortheproposedmodelandtheSimulink®model.
Time(s) Proposedmodel Simulink®model
EDCR EDCI maR maI PDCI EDCR EDCI maR maI PDCI
t=1− 1.0105 1.0000 0.8553 0.8296 0.9895 1.0007 0.9968 0.8499 0.8301 0.9897
t=3− 1.0053 1.0000 0.8389 0.8172 0.4974 1.0044 0.9996 0.8476 0.8005 0.4977
t=5 0.9556 0.9500 0.9329 0.8711 0.4971 0.9554 0.9494 0.9336 0.8827 0.4954
Table2
ComputedVSC-HVDCvariablesbythepowerflowsolution.
Qgen EDC ma Beq LTC’stap Ploss
VSCR 153.359MVAr 1.0002p.u. 0.8282 −2.6655◦ 0.5182p.u. 1.0252 1.1595MW
VSCI 27.306MVAr 1.0000p.u. 0.8423 6.8416◦ 0.0918p.u. 1.0044 0.3383MW
Fig.12.Dynamicbehaviouroftheconverters’modulationindices.
Fig.13.ReactivepowergeneratedbybothconvertersmakinguptheHVDClink.
Fig.11showsthevoltagemagnitudesatvariousnodesfollowing achangeinthenetwork’stopology.Duringthetransientperiod,the targetvoltagesetpointisachievedveryquicklybytheactionofthe AC-busvoltagecontrollersthatregulatetheconverters’modulation indicesmaRandmaI,asshowninFig.12.Theratherpromptaction
ofbothcontrollersleadstoveryrapidreactivepowerinjectionat bothconvertersACterminals,ascanbeseeninFig.13,resultingin theveryeffectivedampingofthevoltageoscillationsandenabling asmoothvoltagerecoverythroughoutthegrid.
AssoonasthedisturbancetakesplaceintheACsystem,the energybalanceintheDClinkisbroken;thevoltagesagsthattake placeatbothconvertersACterminalsreducetheactivepowerbeing transferredthroughtheDClink;thecurrent-IDCRthatflowsfrom
therectifierstationVSCR towardstheinverterstationVSCIdrops
abruptlyfrom0.166p.uto0.146p.u,asillustrated bytheblue lineinFig.14.Amomentarymismatchbetweenbothconverters DCcurrentsisthenproducedbecausetheDCcurrentcannotbe instantlyre-establishedduetothetimeconstantsinvolvedinthe
Fig.14.DCcurrentbehaviourfortherectifierandinverter.
Fig.15.DCvoltagebehaviourfortherectifierandinverter.
Fig.16.VSC-HVDC’sACactivepowerandDC-powertransferbehaviour.
currentcontrolleroftheinverterstation;asaresult,DCvoltage deviationstakeplace,asshownin Fig.15,reachingaminimum valueof0.978p.u.duringthetransientevent.Nevertheless,once thiscontrollerstartsrespondingtotheDCvoltagevariations,the currentIDCI,depictedbythegreen lineinFig.14,startstracing
theDCcurrentoftherectifierIDCRtocompensateforthevoltage
drop,enablingaspeedyrecoveryoftheDClinkvoltage.Itshould beremarkedthatinviewofthefactthatthecableresistanceis relativelysmall,soisthevoltagedropalongtheDCtransmission line,resultinginquitesimilarmagnitudesand,ofcourse,dynamic behavioursofthevoltagesatbothDCbuses,EDCRandECDI.
ThesimulationresultsfortheactivepowerandDC-power trans-ferfollowingthedisconnectionofthetransmissionlinescloseto theHVDClinkareillustratedinFig.16.Thebluelinerepresentsthe activepowerenteringthehigh-voltagesideoftheload-tapchanger transformer coupledto the rectifier station whereas the green linecharacterizestheactivepowerperformanceattheinverter’s LTC’shigh-voltageside.Thepowerdifferencerepresentsthepower lossesincurredbytheVSC-HVDClink,includingthoseproducedby theDCcable.TheDC-powertransferPschconsistingoftheproduct
ofvoltageEDCRandcurrent−IDCR,isalsoshowninthesamegraph.
Sincethevoltageandcurrentcontrolshavebeenshowntooperate efficiently,asillustratedinFig.14andFig.15,thenafastpower recoveryis achievedinspiteoftheseveredisturbanceoccurred inthenetwork.Giventhatthepowerflowing fromtherectifier towardstheinverterstationhasbeenbroughtbacktoitsinitial targetpowertransferof0.333p.u,thedeviationofthepowerangle Rsuffersameremarginalincrease,ascanbeseeninFig.17,only
toagreewiththenewreachedsteady-stateconditionswhere dif-ferentcurrentsand,therefore,activepowerlossesareproduced.
Fig.17.DynamicperformanceofvariousanglesinvolvedintheVSC-HVDC dynam-ics.
therectifierandinverterconverters,RandI.Asexpected,these
anglesfollowthesamepatternasthoseobtainedbythenetwork’s voltageangles,atthenodeswheretheVSC-HVDCsystemis con-nected.
5. Conclusions
AnewVSC-HVDCmodelforRMSdynamicsimulationsof large-scalepower systemshasbeen introducedin thispaper. Thisis anall-encompassingmodelthatfacilitatestheback-to-backand point-to-pointrepresentationoftheVSC-HVDCbysimply mod-ifyingthe DC cable resistancevalue. Themodel possesses four degreesoffreedom,acharacteristicthatconformstoactual VSC-HVDClinks,i.e.,itexertssimultaneousvoltagecontrolonitstwo ACterminalsandatitsDCbusandtransmittedpowerthroughthe DClink.
ThemodelsolutioniscarriedoutusingtheNRmethodwhich solvessimultaneouslythealgebraicanddifferentialequationsat eachtimestep.Thepoint-to-pointVSC-HVDCmodelcomprisestwo series-connectedVSCstationsandaDCcable.EachVSCmodeluses anidealphase-shiftingtransformerasitscoreelement.The con-ductionlossesandtheswitchinglossesoftheHVDCconvertersare wellcapturedinthemodel.Furthermore,theVSC-HVDCdynamic modelisfittedwithindependentcontrollersfortheACandDC cir-cuitstorepresentthequitedistinctdynamicperformancesofthe twocontrolcircuits.
Thepoint-to-pointVSC-HVDCmodelintroducedinthispaper has been validated using the EMT simulation tool Simulink®, wheretheresultsobtainedfromthetwofundamentallydifferent approachesagreedquitewellwitheachother.Nevertheless,itwas shownthatthegreaterlevelofdetailneededinanEMTsolution comeswithanonerousprice-tagtopayintermsofavery consider-ablecomputationaltimecomparedtothecomputingtimeincurred bythenewRMS-typeVSC-HVDCmodel.
TheperformanceofthenewVSC-HVDCmodelwastestedina largernetworkwhichiswidelyusedinacademiccircles, compris-ing39nodes.TheVSC-HVDClinkmodelperformedwellinterms ofitsmodellingflexibilityandinattainingthesetcontroltargets.It wasshownthattheDCcurrentcontrollerandtherectifier’sangular aperturecontrolleroperateefficientlytostabilizeboththeDC volt-ageandtheDCpower,respectively.Likewise,theACvoltagecontrol wasquicklyachievedduetotheproperactionofthecontrollers actinguponthemodulationindicesoftheconverters.
AppendixA.
EDC:DCvoltage.IDC:DCcurrent.CDC:DCcapacitance.RDC:DC
transmissionlineresistance. Inom: VSC’s nominalcurrent.Snom:
VSC’snominalpower.Psch:Scheduledactivepower.ma:VSC’s
mod-ulationindex.ϕ:VSC’sphase-shiftingangle.k2=
3
⁄
8:Constantforatwo-level,three-phaseVSC.G0:Shuntresistorwhichaccounts
fortheswitchinglossesoftheVSC.Beq:Equivalentvariableshunt
susceptance.Y1:VSC’sseriesadmittanceassociatedwith
conduc-tion losses and interface magnetics.V: Complex nodal voltage. I:Complexcurrentinjection.S: Complexnodalpowerinjection. Kpedc,Kiedc:ProportionalandintegralgainsfortheDCvoltage
con-trol.Kppdc,Kipdc:ProportionalandintegralgainsfortheDCpower
control.Kma,Tma:Proportionalgainandtimeconstantforthe
mod-ulationindex control.Subscripts Rand Istandfor rectifier and inverter,respectively.
AppendixB.
InconnectionwithFig.2,thevoltageandcurrentrelationships intheidealphase-shiftingtransformerare:
V1 EDCI = k2maI∠I 1 and k2maI∠−I 1 = I2 I1 (A.1) ThecurrentthroughtheimpedanceconnectedbetweenvIand 1is:
I1=Y1(VvI−V1)=Y1VvI−k2maI∠IY1EDCI=IvI (A.2)
whereY1=(R1I+jX1I)−1.
Atnode0vI,thefollowingrelationshipholds, I0vI=−I2+GswIEDCI=−k2maI∠−IY1VvI+k22m2aIY1EDCI
+jBeqIk22maI2EDCI+GswIEDCI (A.3)
RearrangingEqs.(A.2)and(A.3)yields:
IvI I0vI = Y1 −k2maI∠IY1 −k2maI∠−IY1 k22m2aI Y1+jBeqI +GswI VvI EDCI (A.4) Thereforethepowerinjectionswouldbe, SvI S0vI = VvI EDCI × Y1∗ −k2maI∠−ϕIY1∗ −k2maI∠ϕIY1∗ k22m2aI Y1∗−jBeqI +GswI Vv∗I EDCI (A.5) AppendixC.Asafirststep,theimplicittrapezoidalmethodcallsfor alge-braizingany differentialequation ˙y bymeansof expressingits step-by-stepsolutionasanintegralform,
˙y(t)−F(X(t),Y (t))=0 (B.1) Y (t)−Y (t−t)− t
t−t F(X(t),Y (t))dt=0 (B.2)AssumingthatallfunctionsF(·)varylinearlyoverthetime inter-val[t−t,t],theareaundertheintegralcanbeapproximatedbya trapezium;thedifferentialalgebraicequationgivenintheformof amismatchequationisthen,
FY=Yt−t+t2 ˙Yt−t−
Yt− t 2 ˙Yt =0 (B.3)
AppendixD.
ParametersusedinSection4.1:(i)Theparametersofthepower systemcanbefoundinthesectionof‘demos’inSimulink®as: VSC-BasedHVDCTransmissionSystem(DetailedModel).(ii)VSC-HVDC databased onthe HVDC rating Snom=200MVA: RDC=0.042704
p.u.; EDCInom=1.0 p.u.; G0I=G0R=2e-3 p.u.; R1I=R1R=2e-3 p.u.;
X1I=X1R=1e-3 p.u.; Hc=0.014; Kpedc=0.6; Kiedc=35; Kppdc=0;
Kipdc=5;KmaI=KmaR=25;TmaI=TmaR=0.02;ZLTCtransf=0.005+j0.15
p.u.
Parameters used in Section 4.2: (i) Synchronous genera-tors are equipped with exciter, automatic voltage regulator, speed governor and hydro turbine. Generators and network data are available in [17]. (ii) VSC-HVDC data based on the HVDC rating Snom=300MVA: RDC=0.0024 p.u.; EDCInom=1.0
p.u.;G0I=G0R=2e-3p.u.;R1I=R1R=2e-3p.u.;X1I=X1R=0.01p.u.;
Hc=0.007; Kpedc=0.05; Kiedc=1.0; Kppdc=0.002; Kipdc=0.075;
KmaI=KmaR=25.0; TmaI=TmaR=0.02; XLTCtransf=0.05 p.u. (iii)
Load models: PL=PL0
0.2+0.4V/V0 +0.4V/V0 2 and QL=QL0
0.2+0.4V/V0 +0.4V/V0 2
,where,PL0andQL0are
thenominalactiveandreactivepowersdrawnbytheloadatrated voltageV0.
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