DIGITAL COMPUTATION OF ELECTROMAGNETIC TRANSIENTS

1-1DIGITALCOMPUTATIONOFELECTROMAGNETICTRANSIENTSINPOWERSYSTEMS:CURRENTSTATUSJuanA.Martinez-VelascoDepartamentd'EnginyeriaElèctricaUniversitatPolitècnicadeCatalunya,SpainAbstract-Thisdocumentpresentsanintroductiontotime-domainsolutionofelectromagnetictransientsinpowersystemsusingadigitalcomputer.Currently,themostwidelyusedsimulationtoolstosolveelectromagnetictransientsarebasedonthetrapezoidalruleandthemethodofcharacteristics(Bergeron'smethod).Onlyworksrelatedtothissolutionalgorithmareconsideredinthisdocumentwhichcoverstwomaintopics:solutiontechniquesandmodelingofpowercomponents.Keywords:ElectromagneticTransients,Time-domainSimulation,TrapezoidalRule,NumericalOscillations,ControlSystems,Modeling.1.INTRODUCTIONTransientphenomenainpowersystemsarecausedbyswitchingoperations,faults,andotherdisturbances,suchaslightningstrokes.TheyinvolveafrequencyrangefromDCtoseveralMHz.Aroughdistinctionisusuallymadebetweenelectromechanicaltransients,traditionallycoveredbytransientstabilitystudies,andelectromagnetictransients.Thelattertypeoftransientscanoccuronatimescalethatgoesfrommicrosecondstoseveralcycles;theyareacombinationoftravellingwavesonlines,cablesandbuses,andofoscillationsinlumped-elementcircuitsofgenerators,transformersandotherdevices.Someelectromechanicaltransients,suchassubsynchronousresonance,forwhichdetailedmachinemodelsareneeded,areusuallyincludedinthisclassoftransients.Severaltoolshavebeenusedovertheyearstoanalyzeelectromagnetictransients.Atearlystages,miniaturepowersystemmodels,knownasTransientNetworkAnalyzers(TNA),wereused.Atpresent,thedigitalcomputeristhemostpopulartool,althoughTNAsarestillused;inaddition,thenewgenerationofreal-timedigitalsystemsareprobablythemostadequatetoolinsomeapplicationsforwhicheitheraveryhigh-speedorareal-timesimulationisrequired.Manytechniqueshavebeendevelopedtosolveelectromagnetictransientsusingadigitalcomputer.Theycanbeclassifiedintotwomaingroups:frequency-domainandtime-domaintechniques.Thesubjectofthisdocumentisthedigitalsimulationofelectromagnetictransientsinpowersystems,usingtime-domaintechniques.Presently,themostwidelyusedsolutionmethodisbasedontheapplicationofthetrapezoidalruleandtheBergeron'smethod,alsoknownasmethodofcharacteristics[1]-[6].Thisdocumenthasbeenarrangedasfollows.Section2dealswiththebasicsolutiontechniqueseitheralreadyimplementedorproposedforimplementationinelectromagnetictransientsprograms(emtps).Itcoversnotonlythealgorithmsaimedatsolvingthetransientsolution,butprocedurestoreducenumericaloscillationsproducedbythetrapezoidalrule,initializationmethods,andprocedurestosolvetheinterfacebetweenpowernetworksandcontrolsystems.Section3presentsasummaryofmodelingworksrelatedtothemostimportantpowercomponentstakingintoaccounttheirfrequency-dependentbehaviour.Duetodifficultiesfordevelopingpowercomponentmodelsaccurateenoughforawidefrequencyrange,muchworkhasbeendonetoprovidemodelingguidelinesfordigitalsimulationofeverytypeoftransientphenomenon.Section4summarizestheworkdoneinthisareaandreportsaboutworksstillinprogress.Sometopics,suchasparallelcomputationorreal-timeemtp-basedsimulationofelectromagnetictransients,whicharecloselyrelatedtothemainsubjectsofthisdocumentarenotcoveredhere.Aselectedbibliographyrelatedtotopicsofeachparthasbeenincludedattheendofthisdocument.2.SOLUTIONMETHODS2.1TRANSIENTSOLUTIONThestudiestosolvetravellingwaveproblemsbymeansofa1-2[G][v(t)][i(t)][I](1)t)][iA(t)][IA][GAB(2)vkmvkm(0)rthevikm(3)kmf(ikm,dikm/dt,t,...(4)digitalcomputerwerestartedintheearly1960'susingtwodifferenttechniques,theBewley'slatticediagram[7]andtheBergeron'smethod[8].Thesetechniqueswereappliedtosolvesmallnetworks,withlinearandnonlinearlumped-parameter,aswellasdistributed-parameterelements.TheextensiontomultinodenetworkswasmadebyH.W.Dommel[1].TheDommel'sschemecombinedtheBergeron'smethodandthetrapezoidalruleintoanalgorithmcapableofsolvingtransientsinsingle-andmulti-phasenetworkswithlumpedanddistributedparameters.ThissolutionmethodwastheoriginoftheElectroMagneticTransientsProgram(EMTP),whosedevelopmentwassupportedbyBonnevillePowerAdministration(BPA).Thetrapezoidalruleisusedtoconvertthedifferentialequationsofthenetworkcomponentsintoalgebraicequationsinvolvingvoltages,currentsandpastvalues.Thesealgebraicequationsareassembledusinganodalapproachwhere[G]isthenodalconductancematrix[v(t)]isthevectorofnodevoltages[i(t)]isthevectorofcurrentsources[I]isthevectorofhistoryterms.Veryoftenthenetworkcontainsvoltagesourcestoground,thentheequationissplitupintopartAwithunknownvoltagesandpartBwithknownvoltagesTheresultingconductancematrixissymmetricalandremainsunchangedastheintegrationisperformedwithafixedtime-stepsize.Thesolutionofthetransientprocessisthenobtainedusingtriangularfactorization.Oneofthemainadvantagesofthisprocedureisthatitcanbeappliedtonetworksofarbitrarysizeinaverysimplefashion.Bergeron'smethodcanbeefficientlyusedwithlosslessanddistortionlesslines.However,parametersofactualtransmissionsystemsarefrequency-dependent.Thefirstworksonfrequency-dependentmodelswereperformedfortelephonecircuitsin