英文原文.docx
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英文原文.docx
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英文原文
Newdigitalfilterforunbalancedistortedcurrentandvoltage
estimationinpowersystems
Abstract
Thispaperpresentsanovelalgorithmforidentifyingandmeasuringthesymmetricalcomponentsofdistortedthreephasevoltageorcurrentwaveformsinelectricalpowersystems.Theproposedalgorithmiscapableofestimatingthesymmetricalcomponentsaswellastheharmoniccontentsofagivenunbalanceddistortedsignal.Theproposedtechniqueisbasedonstochasticestimationtheorem.Theproblemisformulatedasanestimationproblemandpresentedinstatespaceform.Theproposedalgorithmusedtoestimatethepositive,negativeandzerocomponentsofunsymmetricalwaveformsaswellastheharmoniccontentofagivendistortedsignal.Applicationoftheproposedalgorithmhasbeenconductedonvarioustestcases.AmongwhichapracticalsimulatedpowersystemhasbeenimplementedusingEMTP.Variousscenariosarecarriedouttosimulaterealisticsituationsofunsymmetricalwaveforms.Effectsofbaddataonthesolutionaccuracyarealsostudied.Thespeedofconvergenceisexaminedbychangingtheestimatorinitialconditions.Resultsobtainedshowthattheproposedtechniquecanestimateandtrackthesymmetricalcomponentsofnon-stationarythreephaseunbalancevoltageorcurrentwaveformsinnoisyenvironments.Fastaccuratesolutionsareguaranteedregardlessoftheinitialconditions.Itisalsoshownthatbadmeasurementshavenoeffectsonthefinalaccuracyoftheestimation.
1.Introduction
Themethodofsymmetricalcomponentsisaveryimportanttoolfortheanalysisofthreephaseelectricalsystems.Protectionofpowersystemneedsaccurateidentificationofsymmetricalcomponentsofthemeasuredsignals.Duringunbalanceddisturbances,symmetricalcomponentschangetheirvaluessignificantly.Itisthusveryimportanttotracksymmetricalcomponentson-line.Severalmethodshave,sofar,beenproposedtocalculatethesymmetricalcomponentsofavoltageorcurrentsignals,manyofthesemethodsarebasedonstaticstateestimation,othersarebasedondynamicstateestimationtechniques.Referencepresentsareviewonsomeofstaticestimationtechniquesandproposedtheuseofrecursiveleasterrorsquarestechnique.Referenceproposedadigitalfilteringalgorithmfortheestimationofsymmetricalcomponents.Thealgorithmisbasedonusingtwodigitalfiltersworkingtogetherforfastestimation.Theinputthreephaseunbalancedsystemistransformed,first,intoα–βtransformation.Theleasterrorsquareestimationtechniqueisthenusedtoidentifythemagnitudeandphaseangleofeachsequencecomponent.Asknown,theleasterrorsquareestimationsolutionisaffectedbythepresenceofbaddata.InreferenceafastefficientmethodbasedonfastFouriertransform(FFT)ispresented.TherearebasicassumptionsembeddedintheapplicationoftheFFT.Theseassumptionsare(a)thesignalisstationary;(b)thesamplingfrequencyisequaltothenumberofsamplesmultipliedbythefundamentalfrequencyassumedbythealgorithm;samplingtheoremissatisfied.MisapplicationofFouriertransformalgorithmwouldleadtoinaccurateresults.Referencepresentsamethodbasedonnon-recursiveNewtontypealgorithm.Thealgorithmisnotsensitivetopowersystemfrequencychangesandtoharmonicdistortionofinputsignals.DynamicstateestimationtechniquessuchasKalmanfiltering(KF)andweightedleastabsolutevaluedynamicfiltersarealsopresentedinmanyRefs.Thealgorithmpresentedinreferencedetectsthepositiveandnegativesequencecomponentsafterfilteringthezerosequencecomponents.Althoughthismethodgivesgoodresults,itwasnottestedinthepresenceofbaddatapoints.Referencepresentsanapplicationofadynamicfilterforon-lineestimationofsymmetricalcomponents.Themethoddetectsthesequencecomponentsofpuresinewaveforms.Sequencefiltersarealsofrequentlyusedinpowersystemprotectionforidentifyingvoltageorcurrentsymmetricalcomponentsduringabnormaloperations.Sequencefiltersoutputcanbeaffectedbysaturationthatmayhappentothefiltercoilelements.Inreferenceanotheron-linemethodforestimatingsymmetricalcomponentsispresented.Themethodisbasedontheuseofasetofenhancedphase-lockedloopsystems.Referenceintroducedanadaptivelinearcombinerforsymmetricalcomponentsestimation.Thetechniqueiscapableofestimatingboththesymmetricalcomponentsaswellastheharmoniccontentsofmeasuredunsymmetricaldistortedsignals.ThealgorithmisbasedonKalmanfilteringtechnique.Approachesbasedonartificialintelligencetechniquessuchasgeneticalgorithms,fuzzylogicandneuralnetworksarealsoproposedtodetectandidentifyvoltageandcurrentsequencecomponents.Manyothertechniquesaresuggestedandimplementedinthetimedomain.Thispaperpresentsanoveltechniquebasedonrecursivealgorithmwhichcanbeusedfordigitalidentificationofthesymmetricalcomponentsoftheharmoniccontaminatedvoltageorcurrentwaveformsinelectricalpowersystems.Thealgorithmisanoptimaldynamicestimatorbasedonstochasticestimationtheory,whichisapplicableforestimatingandtrackingthenon-stationarysignals.UnlikeKalmanfilter,whichminimizestheerrorsquare,theproposedestimatorgainmatrixisderivedinsuchawaytominimizetheabsoluteerrorintheestimationprocess,thustheestimatorisnameddynamicleastabsoluteestimator(DLAVE).Themethodallowsaveryfastdeterminationandisolationofthefundamentalandharmoniccomponentsandconsequentlythedesiredparametersofsymmetricalcomponentswhenachangeinthepowersystemoccurs.Resultsobtainedshowthattheproposedtechniqueefficientlyestimatesthesymmetricalcomponentsofthreephaseunbalancevoltageorcurrentwaveformsunderdifferentcircumstances.
2.Mathematicalmodeling
Theideahereistopresenttherelationshipbetweentheunsymmetricalwaveformsandthesymmetricalcomponentsinstatespaceformwhichissuitablefortheproposedalgorithm.Thisisdoneviatwosteps.Inthefirststep,thedistortedsignalisdecomposedtoitsfundamentalandharmoniccontents.Inthesecondstep,thefundamentalcomponentisresolvedtoobtainitssymmetricalcomponents.
2.1Distortedwaveformmodeling
Assumethatwehaveanon-sinusoidalwaveformgivenas
where
isthedccomponentofthecurrent;jequals1forfundamentalandequals2,3,...forharmonics;
isthemaximumorderofharmonicconsidered;
isthe
maximumvalueofthecurrentcomponent;isthephaseangleofthejthcurrentcomponent
Eq.
(1)canbeexpandedas
Withoutloosinggenerality,itcanbeassumedthatthesignalsarecontaminatedwithharmonicsoforders3rd,5thand7thinadditiontothedcoffsetcomponent,sowecanwrite
Where
Ifthecurrentsignalissampledatapre-selectedrate,T,thenmsampleswouldbeobtainedatt1,t2,...,tm.Inacompactmatrixformwecanwrite
Inacompactmatrixform,Eq.(5)canberewritteninstatespaceformas
whereZ(k)ism×1measurementvectorofthecurrentsamples;H(k)ism×9measurementmatrix;X(k)is9×1statevectortobeestimated;e(k)ism×1measurementerrorvector,tobeminimized.ItisassumedtobewhitesequencewithknowncovarianceR(k).Thestatetransitionequationwillbeintheform
whereΦ(k)isthe9×9statetransitionmatrixgivenasaunitdiagonalmatrix(assumingrotatingreference);(k)isthe9×1errorvectorofthestateassumedtobeawhite(uncorrelated)sequencewithknowncovariancematrixQ(k).
Oncethestatevectorisidentified,theRMSvaluesoffundamental,dcandharmoniccurrentsandtheirphaseanglescanbecalculatedas
wherekequals1forfundamentaland3,5,7fortheotherharmonics.Xkstandsforthevariablenumber,i.e.X1,X2,...
Oncethefundamentalcomponentisseparatedfromtherestofthesignal,thesymmetricalcomponentscanbeestimatedusingthefollowingmodel.
2.2.Puresinusoidalwaveform
Now,considerathreephaseunsymmetricalfundamentalvoltagesorcurrentsgivenas
Wecanwritethesymmetricalcomponentsas
wherea=exp(j2_/3)=1∠120
Substitutesin(a+b)=sin(a)cos(b)+cos(a)sin(b)cos(a+b)=cos(a)cos(b)−sin(a)sin(b).
Wewillendupwiththefollowingequations:
Theseequationscanbewritteninacompactmatrixformas
wherethevaluesofthetimevaryingequationsare
The(h)coefficientsareconsideredastheremainingpartsofEq.(12).Ifthethreephasecurrentsaresampledatarateof(fs)withinthespecifiedwindowsize,msampleswouldbeobtainedforeachcurrentatt1,t2,...,tm,wheret1istheinitialsamplingtime.InastatespacematrixformwecanwritethissystemasinEq.(6).
whereZ_(t)3m×1vectorofcurrentmeasurements,H_(t)3m×6connectionmatrix,X_(t)6×1statevectortobeestimatede_(t)3m×1errorvectortobeminimized.
Thecomputationalburdenofthealgorithmisrelatedtothesizeofthewindowandthesamplingfrequency(fs)usedfortheanalysis.Basedonthesamplingtheory(thesamplingfrequencyshouldbeatleasttwicethehighestharmonicfrequencyconsidered),thereforeinourcase,thesamplingfrequencyshouldnotbelessthan250Hz.Thatmeansfivesamplespercycle.Extensiverunsshowthatthealgorithmneedsfrom50to100samplestoreachthesteadystatesolution.Thereforeinsomecasesweused50–100samplesperonecycleandinothercasesweusedfivesamplespercycle(withoverallof50samplesin10cycles).Inconclusionthecomputationalburdendependsmainlyonthenumberofsamples(simplythenumberofequationsconsidered(m)).Increasing
thenumberofsamplesincreasesthecomputationalburden.Thereforebymaintainingthenumberofsamplessmallwhilesatisf
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