Fang　Jun-long,　Xing　Yu,　Fu　Yu,　Xu　Yang,　and　Liu　Guo-liang

College　of　Electrical　and　Information,　Northeast　Agricultural　University,　Harbin　150030,　China

Rural Power System Load Forecast Based on Principal Component Analysis

FangJun-long,XingYu,FuYu,XuYang,andLiuGuo-liang

CollegeofElectricalandInformation,NortheastAgriculturalUniversity,Harbin150030,China

Powerloadforecastingaccuracyrelatedtothedevelopmentofthepowersystem.Thereweresomanyfactorsinfluencing thepowerload,buttheireffectswerenotthesameandwhatfactorsplayedaleadingrolecouldnotbedeterminedempirically.Based ontheanalysisoftheprincipalcomponent,thepaperforecastedthedemandsofpowerloadwiththemethodofthemultivariate linearregressionmodelprediction.Tooktheruralpowergridloadforexample,thepaperanalyzedtheimpactsofdifferentfactorson powerload,selectedtheforecastmethodswhichwereappropriateforusinginthisarea,forecastedits2014-2018electricityload,and providedareliablebasisforgridplanning.

load,principalcomponentanalysis,forecast,ruralpowersystem

Introduction

Loadforecastingisthebasisfornetworkplanning (Yanget al.,2013)andhasabiginfluenceonthe powersystem.Finishingtheworkofthepowerload forecastingisanimportantsafeguardtorealizethe securityofthegridandtheeconomicaloperation(Niu et al.,1999;JiandWang,2001;Ranaweeraet al., 1997).Nowadays,therearenumerousstudiesabout theloadforecastingmethods,butthereisnowayto determinewhichoneisthebestandthemostaccurate method(Luoet al.,1997).Inordertomakeanaccurate predictionofthepowerloadduringthenextfew years,theprincipalcomponentanalysismethodwas introducedintothepowerloadforecastingproblems. Throughtheexamples,themethodgotahighaccuracy predictivevalue,andprovidedsomereferencestothe ruralpowergridplanning.

Materials and Methods

Mathematical model of principal component analysis

Supposedanobservedobjectwhichcouldbemeasuredbyp indicatorsX1,X2,…,Xp,pindicatorsthen constitutedonep-dimensionalrandomvector,denoted asthefollowing:

SetthemeanoftherandomvectorXwasthe covariance matrix was, and made a linear transformationofpindicator:

Fromai=(a1i,a2i,…,api),then

Theso-calledprincipalcomponentanalysis(XuandWang,2006),thelinearcombinationofF1,F2,…,FpwasunrelatedandthevarianceofVar(Fi)=aiT∑aiwas maximized.Informedbymatrixtheory,coefficient vectorsai=(a1i,a2i,…,api),i=1,2,…,pineachequation wastheeigenvaluethateigenvectorofthecovariance matrix ∑ of the matrix Xcorrespondedto,whichwas makingVar (F1)tothemaximum.Thismaximum reached the first eigenvalue of the covariance matrix ∑correspondedtothefeaturevector.Andbythisanalogy, Var (Fp)reachedthemaximumatthefeaturevectorthat pcharacteristicvaluecorrespondedto.

Fromformula(2),

Principal component analysis of calculation process

(1)Assumednobservedobjects.Notingpindicators' predictivevalueof iobservedobjectwasxi1,xi2,…,xip, theneachpindicator'sobservedvaluesofnobjectscould beexpressedasthematrixformasthefollowing:

Including,thenumberofobservedobjectswasn,and thenumberofindicatorsorvariableswasp.

Fromtheformula

Thevarianceofindicatorsorvariableswas1,and meanvaluewas0afternormalization.

(3)Determinedthecorrelationcoefficientmatrix consistedoftheobservations,asformula(8):

Among,

(4)Seekedmnon-negativeeigenvaluesλ1,λ2,…,λmofthecharacteristicequation|R–λI|=0andeigenvectors correspondedtotheeigenvalueλiaccordingtocorrelationmatrixR:

(5)Principalcomponents.Mprincipalcomponents composedoftheeigenvectorswereasthefollowing:

ThemaincomponentsF1,F2,…,Fmwereunrelated, andtheirvariancesweredecreasing.

(6)Selectedm(m＜p)principalcomponents.Ifthe sumofthefirstmprincipalcomponentF1,F2,…,Fmvariancesofthetotalvariancesclosedto1(ingeneral, onlyreached85%),thenselectedthefirstmprincipal componentsF1,F2,…,Fm.Thesumofmprincipal componentvariancesreached85%ofthetotalvariances meantreservedoriginalindicatorsorvariablesX1, X2,…,Xp'sinformationbasically,thus,thenumberof theindicatorsorvariableswasreducedbyptom,and thenplayedaroleinscreeningindexorvariables.

(7)Combinedwithexpertise,weexplainedthe selectedprincipalcomponents.

Computing eigenvalues and eigenvectors

Theeffectdegreesofeachfactorontheelectricload weredifferent.Ingeneral,thevariablesexistedcertain correlationbetweenthem,thereby,theinformation providedbythevariableoverlapedtoacertainextent (Zou,2008).Therefore,theanalysesofthevariables werequiteessential,usingaminimumofmaincore factorstoreflectthevastmajorityoftheoriginalvariable informationasfaraspossible.Principalcomponent analysiswasamethodforprocessinghigh-dimensional data.Analyzedthevariablestoachievethepurposeof findingoutmainfactorsbythemethodoftheprincipal componentanalysis(ZhangandLiu,2011).

Throughstatisticaldataandthecombinationof subjectiveandobjectiveanalyses,foundthefactors affectingtheelectricalloadinmanyaspects.Tookthe ruralpowergridforexample,andlistedtherelevant factorsaffectingthepowerloadasTable1.

Firstly,calculatedthecorrelationcoefficientmatrix Rasformulas(7)and(8)inTable1,asthefollowing:

Table 1　Related factors to Jiamusi power load

Startedfromthecorrelationcoefficientmatrix R,computedtheeigenvalues,contributionrate andcumulativecontributionrateofeachprincipal component,andtheresultswereshowninTable2.

Then,wegottherelationsbetweenprincipalcomponentsandthestandardizedvariableswere:

Results

Result analyses and main index selection

FromTable2,thefirstandsecondmaincomponent eigenvalueswerelargerthanothers.Thecumulative contributionratehadreached99.38%.Amongthem, fromformula(12)thatx1,x2,x5andx6informulaF1hadahigherloadfactor,madeadescriptionofthefirst principalcomponentF1havingabigrelevancewith thefirstindustrialpower,thesecondindustrialpower, populationandGNPoftheregions;fromformula (13),x3andx4informulaF2hadahigherloadfactor, anditcouldbeconcludedthatthesecondprincipal componentF2hadabigrelevancewiththetertiary industryelectricityconsumptionandresidential electricityconsumption.Thus,thefactorsabovewere themainfactorsaffectingtheloadforecasting.

Easytoseefromequation(12),ruralpowergrid loadwasinfluencedmainlybyindustry,agriculture,populationandGNPoftheregions.Industry andagriculturewereregardedasthemainproductivi-tiesinrural.Suchaconclusionwasinlinewithrural powergridactualsituation.Fromformula(13),the commercialandresidentialelectricityconsumption wasonlytobethesecond,continuingdevelopthe economyservicesmadethebusinessservicesoccupya highstatusinthisregion'spowerloadgradually.

Table 2　Eigenvalues, contribution rate and cumulative contribution rate of each principal component

Discussion

Multiple linear regression models

Thepowerloadwasinfluencedbymultiplefactors. Multiplelinearregressionmodelshaduniversal significanceinmulti-elementanalysissystem. Therefore,thechoiceofthemultiplelinearregression modelanalysis,commonpredictingbyoptimal combinationofmultipleindependentvariablesor estimatingthedependentvariableweremorerealistic. Assumingadependentvariablewasinfluenced bykindependentvariablesx1,x2,…,xk,andthenitsN setsofobservedvaluewereyα=x1α,x2α,…,xkα,α=1,2,…, n,thegeneralformofmultiplelinearregressionmodel was:

Intheformula,β0,β1,β2,…,βkwereundeterminedcoefficients,andεαwasrandomvariables(Bin, 2010).

Multiple linear regression model to test

Takinganindexhavingagreatercorrelationofthe mainfactorstothefirstprincipalcomponentF1was μ1.Takinganotherindexaccordingtohaveagreater correlationofthemainfactorstothefirstprincipal componentF2wasμ2.

Accordingtotheprincipleoftheleastsquare method,obtainedthefollowinglinearregression predictionmodelusingMatlabsoftware.

Including,R2=0.3932,F=106.64,Sig=0.0000meant themodelwasvalid.Thespecificdataisshownin Table3.

AsitisshowninTable3,theaveragerelativeerror betweentheactualandpredictivevalueoftheload, during2004-2013wasonly0.918%,whichreached arelativelyhighpredictionprecision.Bythedeep dataanalysesinTable1,itmeantthefirstindustrial electricityconsumption,electricityconsumptioninthe secondindustry,residentialelectricityconsumption, population,GNPintheareaandmaximumload utilizationhoursweresteadilygrowing,steady growth,lowvolatility,sowecouldpredicttheseseveral factorsaboveinthenextfiveyearsbythemethod ofaveragegrowthrate.However,thegrowthrate ofthetertiaryindustryelectricityconsumptiontook amonotonicallyincreasingtrend,usingthemethodoflinearregressionequationstopredict.Last, finishedthepredictionofthepowerloadinthisarea during2014-2018byfittingfunctioncalculatingthe maximumload,specificresultsisshowninTable4.

Table 3　Rural power grid load fitting relative error

Table 4　Predictive value of 2014-2018 power load

FromTable4,powerloadofthisareaduring 2014-2018wasincreasing.Itwouldincreaseto 1468.78MWby2018.Alsofindingouttheincreasing agriculturalpowerconsumptionwasamajorfactorto increasethepowerload,inlinewiththeruralpower gridelectricityactualsituation.

Conclusions

Bytheprincipalcomponentanalyses,thepaperpredictedanimportantpowerindicator,themaximum powerload.Firstly,bythethoughtofreducingthe dimension,analysesofthemainfactorsamongso manyeffectfactorsfoundtherightweightofdifferent affectingfactors.Then,wegotthemaximumpowerload predictivevaluebycalculatingthemaineffectfactors. Fromtheaboveexample,themethodnotonlysimplified theregressionmodel,butalsohadahighfitting accuracy,andprovedthefeasibilityofthismethod.

References

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TM926Document code: AArticle ID: 1006-8104(2015)-02-0067-06

11December2014

SupportedbytheScienceandTechnologyResearchProjectFundofProvincialDepartmentofEducation(12531004);ProjectofHeilongjiangLeading TalentEchelonTalented(2012)

FangJun-long(1971-),male,professor,supervisorofPh.Dstudent,engagedintheresearchoflocalpowersystem.E-mail:junlongfang@126.com