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Ali Abur Accurate modeling and simulation of transmission line transients using frequency dependent modal transformations

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The frequency dependent line model (also known as the J. Marti model) which is currently used in most electromagnetic transient programs, is very efficient and accurate for most simulation cases. However, it makes an approximation in choosing the modal transformation matrix that is used to switch variables between the phase and modal domains at each simulation time step.
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  AccurateModelingandSimulationofTransmissionLineTransientsUsingFrequencyDependentModalTransformations AliAburOmerOzguu abur@ee.tamu.eduozgun@)ee.tamu.eduDepartmentofElectricalEngineeringTexasA&MUniversity(CollegeStation,TX778433128 Abstract: Frequencydependentlinemodel(alsoknownastheJ. Martimodel)whichiscurrentlyusedinmostelectromagnetictransientprograms[1],isveryefficientandaccurateformostsimulationcases.However,itmakesanapproximationinchoosingthemodaltransforruiitionmatrixthatisusedtoswitchvariablesbetweenthephaseandmodaldomainsateachsimulationtimestep.Thisapproximationmaynotholdtrueforcertaintowerconfigurationsrind/orconductortypeswherelineparametersvarydrasticallywithfrequency.Inthispaper,awaveletbasedalternativesohrtion,whichincorporatesfrequencydependenceoftransformationmatricesintothesimulationprocesswillbepresented.Keywords:Electromagnetictransientssimulations,frequencydependenttransmissionlineparameters,modaltrrmsfonnations,wavelettransform. I.INTRODUCTIONSimulationoflargeelectricpowersystemsduringsystemdis~bances,suchasshortcircuits,switchingof loads capacitorsorotherdevices,lineortransformerenergization,motorstarting,etc.hasbeenanactiveareaofresearchforthepastseveraldecadesfollowingtherapidimprovementsincomputertechnolob~.Powersystemscontaincomponentssuchastransmissionlineswhosemodelparametersvaryasafunctionoffrequencyandconsequentlylendthemselvesbesttofrequencydomainmodelingandsimulation.Ontheotherhand,therearedeviceswithtimevaryingandJornonlinearoperatingcharacteristicssuchassolidstaterectifiers,saturatedtransformers,surgearresters,metaloxidevaristors,etc.thatexistinpowersystemsandtheirmodelsaretypicallybestrealizedintimedomainduetotheirnonlinearcharacteristics.Reconcilingthesimulationandmodelingrequirementsofthesemixedsetofcomponentshasbeenoneofthechallengesfacedintheanalysisoftransientssofar.Thispaperaddressesthischallengebypresentinganalternativesimulationmethod,whichismotivatedbytheuniquepropertiesofthewavelettransform. FernandoH Magnago fernando.magnago@pca-corp.orgPCACorporation1921S.AlmaSchoolRd.207Mesa,Arizona85210UseofwavelettransformforsimulationofpowersystemtransientsisinvestigatedbyMeliopoulosandLeein[2],wherewaveletdomainequivalentcircuitsofR,LandCcomponentsareutilizedtocomputethetransientsinthewaveletdomainandrecoverthetimedomainsolutionviainversetransform.Applicationofwaveletdomainequivalentstocarryoutharmonicanalysisofnonlinearandtimevaryingloadsisreportedin[3]byZhengetal.Thesepapersdiscussthesimulationandmodelingoflumpedelements,whichcanbeusedtosynthesizecascadedpisectionstorepresentlines.Similarstudiescanalsobefoundin[4],[5]and[6],wherespatialdistributionofvoltagesalongnon-uniformmulti-conductortransmissionlinesissimulatedviathewavelettransformoftheresultingdifferentialequations.Theauthorsassumefrequencyindependentlineparametersinthesestudies.In[7],useofthewavelettransformforrepresentationoffrequencydependentparametertransmissionlines,withconstantmodaltransformationmatrices,isdiscussed.Modelingoflossytransmissionlineswithfrequencydependentparameterscanalsobeaccomplishedbydirectapplicationofthewavelettransform.Oneapproachistostartwiththegeneralformofthemulti-conductortransmissionlinepartialdifferentialequationsexpressedinthespatialdistancez,andtimet,forthevoltagesandcurrentsalongtheline.Then,usethewavelettransformtoconvertthemintolargesparsealgebraicequationswhosesolutionswillyieldcoefficientsofthewavelettransformofthevoltagesandcurrentsofinterest.Whilethisisaviableapproach,integrationofsuchacomputationalprocedureintoanexistingtransientssimulatormaynotbetrivialifpossibleatall.Instead,whatisproposedinthispaper,isafairlysimplemodificationofthewellknownconstantbutdistributedparameterlinemodel,orthesocalledtheBergeron’smodel[8],toincorporatefrequencydependenceoflineparametersusingthewavelettransform.Simulationoftransientsalongmultiphasetransmissionlineshasanadditionaldrawback,whichistherequirementthatthelineequationsoughttobedecoupledintoindependentmodalequations,sothateachonecanbesolvedeasilyintherespectivemodaldomain.Thisdecouplingisdonethroughalineartransformationmatrix,whichwillbeafunctionoffi-equencyifthecorrespondinglineparametersalsoare.Intimedomainsimulations,duetothelackofapracticalalternative,aconstanttransformationmatrixtypicallyevaluatedatachosenfrequencyisusedasanapproximation.  0-7803-6672-7/01/$10.00 (C) 2001 IEEE1443  So,evenwhenusingtheadvancedfrequencydependent(FD-methodwillbepresentedtoaccomplishthiswithoutaheringmodel)modelof[9],modaltransformationmatrixwillhavethebasicdiscretetimecircuitmodelofFig.1.tobeapproximated.Thesimulationmethod,whichwillbepresentedinthispaper,providesarathersimpleavenueto Ik(t)h(t)~ improvethisapproximationbyusingthewavelettransform. k- ThepaperisorganizedsuchthatareviewoftheFD-modelmandtheBergeron’sconstantparameter(CP-model)distributedlinemodelwillbepresentedfwst.Theproposedwavelet-basedsimulationandmodelingoftransmissionlineswith F TI k(t)ZOJt20Vtn(t) frequencydependentparameterswillbediscussednext.Simulationresultsofsomepowersystemtransientswillthen IkmImk beshownfollowedbytheirdiscussionandconclusions.II.REVIEWOFLINEMODELSMulti-conductortransmissionlinesusuallyrundistanceslongenoughtomaketheirlumpedparametermodelinginaccurate.Approximatemodelsthatcanfakethedistributednatureofthelineparameterscanbeobtainedbyusingseveralcascadedlumpedparameterpisectionmodels.Amoreaccuratemodel,whichisreferredtoastheconstantparameter(CP-model)linemodel,canbeobtainedbylumpingtheresistanceandmodelingtheremainingloss-lesspart,byusingthemethodofBergeron.Thismodelincorporatestravelingwavedelaysviaasimpleequivalentcircuitcontainingacurrentsourceandaconstantresistance(line’scharacteristicimpedance)ateachendoftheline.Thecurrentsourcesdependuponthevoltageandcurrentvaluesfromtheremoteendoftheline,withacertaintimedelaythatisdeterminedbythetravelingwavevelocityandthelinelength.ThismodelisshowninFig.1.VariationsoflineparameterssuchasR,LandCasafunctionoffrequency,aresimplyignoredwhenusingtheCP-modeloftheline.Inordertoaddressthisdeficiency,afrequencydependentlinemodel(FD-model)isdevelopedbyJ.Martiin[9].FD-modelessentiallyusesthesameequivalentcircuitastheCP-modelshowninFig.1,exceptforthefactthatthecharacteristicimpedanceZO,ateachendoftheline,arereplacedbyproperlychosennetworkequivalentsthathaveapproximatelythesameflequencyspectrumasthatofZO.Inaddition,thecurrentsourcevaluesarenolongersimpletimedelayedfunctionsofremotelineendvariables,butinvolvemorecomplicatedconvolutions[9].ProvidedthattherequiredaccuracyofthefittingfictionsthatapproximatethefrequencyresponseofZ.andthepropagationfimctionareattained,FD-modelofthelinecanbeusedintransientsimulationofsinglephaselinesverysatisfactorily.Whenmultiphaseconductorsareconsidered,oneisfacedwiththeadditionalburdenofdecomposingthelineequationsviaamodaltransformationmatrixT,,whichisitselffrequencydependent.InthecurrentimplementationofFD-model,T.iscomputedatasuitablefrequencyandmaintainedconstantthroughoutthesimulationperiod.Whileforsometowerconfigurationsandconductortypes,thisapproximationisquitevalid,certaincasesmayrequireaccurateincorporationoffrequencyeffectsonT,inthetimedomainsimulations.Inthenextsection,awaveletbased Fig 1 CP-modelofaloss-lesstransmissionlineIII.WAVELET-BASEDFDLINEMODELWavelettransformfacilitatestimedomaindecompositionofsignalsintoasub-bandoffrequencyranges.Thisimpliesthattheentiresimulationcanbedecomposedintosub-bandseachofwhichcanbecalculatedindependentoftherestatagivensimulationtimestep.Theadvantageofthisapproachwillhoweverbethatlineparametersaswellasthemodaltransformationmatricesusedinaparticularsub-bandoffrequenciescanbeproperlychosenastheonescorrespondingtothatfi-equencyband.Thesebandsoffi-equenciesarereferredtoasscalesofthewavelettransformduetotheirspeciallogarithmicstructure[1O].FrequencydependenceofthelineparametersR,LandCaswellastheresultingtransformationmatriceswhicharefimctionsoftheseparameters,canbeapproximatedbysubstitutingrepresentativevaluescalculatedforeachwaveletscale.Thiswillresultinasmanylinemodelsasthenumberofchosenscalesforagivenlineineachmode.FollowingtheCP-modelofFig.1,thelinemodelforscalekandmodei,willlookidenticaltothecircuitinFig.1,exceptforthefactthatallvariables,parametersandcurrentsourcevalueswillcorrespondtothatmodeandscale.Thetransformationmatrixusedtoobtaintheterminalcurrentsandvoltagesforthismodewillbedifferentforeachscale.Whilethisisstillanapproximationduetothechoiceofdiscretelyratherthancontinuouslychangingmatricesfromonewaveletscaletothenext,properchoiceofscalesbasedontheobservedvariationsinthelineparameterswillimprovethisapproximationdrastically.Thus,thefollowingitemizedprocedureisproposedforsimulatingtransientsinvolvinglineswithfkequencydependentparameters:.Calculatethelineparametersasafimctionoffi-equencyandselectflequencyranges(scales)toproperlydiscretizetheparameters.   CalculatethecharacteristicimpedanceZi”,thetraveldelayjkandthemodaltransformationmatrixT,katscalekandmodeiforallmodesandallchosenscales.ThesevaluesarecalculatedatIiequencieswithineachscalek  0-7803-6672-7/01/$10.00 (C) 2001 IEEE1444    andareassumedconstantforthewholerangeoffrequenciesdefriedbythatscale.Usethediscretewavelettransform (DWT)todecomposethreephaseterminalinputsignalsintothechosenscalesinthewaveletdomain.LetoftheDWTofthethreephasesendingendvoltagev, ~,ct),begivenas:WV ~,j(n)=DWT{v, ~,.t)}wherenrepresentsthenumberofdiscretetimestepsatscalek.Applymc~daltransformationsusingthecorrespondingTvkmatrixforscalek,andcalculatethemodalvoltages [1 vak=T kWV Wvck Solvethediscretetimelineequationsateachscaleforeachmode,andupdatethecurrentsources.Convertallmodalvoltagesineachscale,intothephasedomain,usingtheinverseofT,k.Reconstructthemulti-phaseterminalvoltagesignalsfromtheirdiscretewaveletdomaincomponents~a,b,,kineach scaJe k,,byinversewavelettransform.Wavelettransformanditsinverseareaccomplishedcomputationallyquiteefficientlyviasparsematrixoperations.Nextsectionillustratessomepracticalcaseswherethisapproachprovedviableasevidentfromthecomparisonofresultswiththoseofthewell-establishedFD-model.IV.SIMULATIONSThesamplepowersystemusedintransientsimulationsisshowninFig.2.A50miletransmissionlinewhoseconductordataandtowergeometryareshowninAppendixI,isusedforthestudy.TheLineConstantsauxiliaryroutineofATP[11]isusedtocalculatethelineparametersateachfrequencylevel.kZ(w) rl/   r ‘? ZloadFig.2.StudiedpowersystemSeverallineenergizationcasesareconsidered,includingdirectcurrentandalternatecurrentsourcesforbothbalancedandunbalancedloads.Samplingrateischosenas5psecforallcasesandthenumberofwaveletscalesischoseninordertocapturetheentirefrequencyspectrumfromhighesttothesteadystatefrequencyrange.WaveletdecompositionisdonebyusingDaubechieswaveletasthemotherwavelet,basedonourpreviousexperience[7].Openendedlineenergizationtransientsaresimulatedfirst.Incase1,receivingendvoltagesignalsinthreephasesfora100Voltssinglestepenergizationaresimulated.TheresultsareshowninFig.3forbothproposedandexistingFD-model. ReceivingEndVoltageWaveform~3m~I 2 :0SolidWaveletModel-:-I(SI-dottedFDModelO0.0320.0040.0060.(080.010.0120.0140.0160.0180.02-303,, : ’”~ O0,0U20.004O.OW0.13W0.010.0120.0140.0160.0180.02 ~ -imIoo,m20.00400C60,008O,of0.0120.0140,0160.0180.02 Time(Seconds) Fig.3. Case1:Singlestepenergizationofanopenendedtransmissionline. Incase2,asetofunbalancedresistiveloadsgivenbyR,=2KQ,Rb=3K~,and~=2KQ,isconnectedatthereceivingendoftheline.Fig.4showsthesinglestepenergizationtransientsforthiscase. ReceitingEndVoltageWaveform~2001I   solid:WaveletmodelzQ..IrndottedJ.MartimodeliO0.0020.0040.0360.0080.010,0120.0140.0160.0180.02  :~ 0.0020,0040,0060.0080.010.0120.0140.0160.0180.02-‘203, ~  lml O0.0020.0040.8560.0080.010.0120.0140.0160.0180.02lime(Seconds) Fig.4.Case2:Singlestepenergizationtransients  0-7803-6672-7/01/$10.00 (C) 2001 IEEE1445  ThedifferencesbetweentheproposedwaveletmodelandexistingFDmcldelsimulationsinbothcases,areduetothefactthat,FDmodelusesaconstanttransformationmatrixT,whereasthewaveletmodelincorporatesfrequencydependenceofitintothesimulations.TherestofthesimulationsarecarriedoutbyusingthreephaseACvoltagesource,whereallthephasesareenergizedsimultaneously.Fig.5showstheresultsofcase3,wherethesamelineandloadconfigurationasincase2,isnowenergizedbyabalancedthreephasesinusoidal source. ReceivingEndVoitageWsveform ~l=~ f .mo~—~ 0.010.02 0.030.040.050.060.070.08 o 0.02 0 030.040.050.Q30.07O.oa o  1 0s2 0.030.040.C50.060.070.08 Time(Seconds) Fig.5.Case3:SinusoidalACvoltagesourceenergizationtransientsIncase4,theloadresistanceischosenclosetothecharacteristicimpedanceofthetransmissionlinetoreducereflectionsfiornthereceivingend.SimulationresultsaredisplayedinFig.6wherereflectionsaresignificantlydiminishedwhencomparedtoFig.5,consistentwiththeexpectationsfic~mthismodel. ReceivingEndVoltageWaveformi:Fa o0.010.020.030.040.050.060.070.08 i:trd 0.01 0.020 030.040.050.060.070.08 o 1 2 o.m0.040.050.060.070.08 Time(Seconds) Fig 6 Case4: AC voltagesourceenergizationtransientsafterchangingloadvalue In case5,theeffectofusingfrequencydependentmodaltransformationmatrix(Tv)isfurtherillustrated.Inordertoaccomplishthis,thetowergeometrychosenforthelineusedinthepreviouscases,isslightlymodified(seeFig.A.11inAppendixI).Initially,themodaltransformationmatricesareintentionallykeptconstantwhilesimulatingthetransientswiththewaveletbasedmodel.TheresultsofthiscasearecompqedwiththoseoftheFD-model.AsshowninFig.7,theymatchedquitewell.Thisisexpected,sinceT,matrixisassumedtobeconstantbytheFD-modelaswell. RecehingEndVoltageWweformforConstantTMatricesnrI i.l:~ ,“IDISolidWeveletmodel o 1 2 3 4 5 0.06 0.070.08 r’~j o 1 0.020.030.040.050.060.070.08~“lm=500 : -Imo0.010.020.03O.M0.050.060.070.08 Time(Seconds) Fig.7.UntransposedlinesimulationwithconstantTvmatrix. ReceivingEndVoltagaWaveform1 1 0.050 0 0060 0650.07 Tims(Seconds) Fig 8 EffectsoffrequencydependentModalTransformationMatrices. Next,modaltransformationmatricesarecalculatedforeachfrequencylevel,andthewaveletbasedlinemodelisimplementedpersection3.TheeffectofthismodelingimprovementisevidentfromtheresultsshowninFig.8wherebothFD-modelandwaveletbasedmodelsimulationsarepresentedtogether.Itisinterestingtonotethataratherslightperturbationofthetowerconfigurationmayleadto  0-7803-6672-7/01/$10.00 (C) 2001 IEEE1446
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