电力系统边际发电成本的概率分布函数估计方法英文版.docx

电力系统边际发电成本的概率分布函数估计方法英文版.docx

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电力系统边际发电成本的概率分布函数估计方法英文版

电力系统边际发电成本的概率分布函数估计方法英文版

AnApproachtoEstimationofProbabilityDistributionFunction

ofMarginalGeneratingCostofPowerSystem

ZhangShaohua,YanMaosong,LiYuzeng

（ShanghaiUniversity）

AbstractUsingtheconceptofmarginalgeneratingunitandthetechnologiesofprobabilisticproductionsimulation,anefficientmethodwasdevelopedforestimatingprobabilitydistributionfunctionofmarginalgeneratingcostofpowersystem.Inthismethod,notonlytheuncertaintiesofgeneratingunitforcedoutageandloaddemandforecastareconsidered,buttheuncertaintyrelatedtofuelpricecanalsobeeasilyincorporated.Thepresentedmethodologyembodiesclearphysicalconceptssothatitcaneasilybetheoreticallyunderstoodandcomputationallyrealized.ApplicationofthismethodtoIEEE-RTSshowsitsreasonablenessandefficiency.Furtheralong,thismethodcanbeusefulinpricingstudiesofelectricitymarket.

Keywordspowersystem,probabilisticproductionsimulation,marginalgeneratingunit,marginalgeneratingcost,probabilitydistributionfunction

1Introduction

WithChina’spowerindustryrestructuring,aunifiedopenelectricitymarketwillbegradually

established.Peoplepaymoreattentiontotheelectricitypricingissues,whichisnowthecoreelementoftheelectricitymarket.Asatraditionalpricingmethod,marginalcostpricingtheoryhasbeenwidelyusedinthepowersystem,wherepowerismoresuitableforreal-timeoperationofthemarketmechanismwasundoubtedlypricingtheory,namelybasedonthepricingtheoryofthepowersystemshort-termmarginalcost.Short-runmarginalcostofelectricitysystemstatisticalestimationmethodsaremostlybasedonthepowersystemprobabilisticproductionsimulationtechnology,references[2]involvesbivariateprobabilitydistribution,thealgorithmistoocomplex;themethodofreferences[3]isonlyabouttheestimationofmarginalgenerationaveragecost,butitisnotthecompleteprobabilitydistributiondensityfunction.Existingrandomproductionsimulationtechniquesonlyconsidertheunitforcedoutageandtheuncertaintyofloadforecastwithouttakingintoaccounttheuncertaintyoffuelprices.

Thispaperpresentsamethodofsolvingthesystemshort-runmarginalcostofelectricitydistributionandprobabilitydensityfunction.First,giventheconceptofthemarginalgeneratingunitsandtheirprobabilitycalculationformula,fromwhichwecanobtainthemarginalcostofelectricitydistributionandprobabilitydensityfunction,andthewayofcalculatingtheuncertaintyofthefuelprice,andfinallypresentsthedataanalysisandconclusions.oftheadoptionofIEEE-RTS[4].

2Principles

Assumethatthesystemhasapowergenerationunit,,seekinganapproachtocalculationtheDistributionfunctionQs（x）andthedensityfunctionqs（x）attheconditionofbeinggivenloaddemandLinacertainmomentanditsmarginalcostofelectricityprobability（S）.

ThefirstapplicationisstochasticsimulationofpowersystemsZ-transformationmethod[5]ortheeffectivecapacityofthecumulativeamountofthedistributionmethod[6],afterthefirstipowerunitsareloaded,thesystemloadLfacedlossofloadprobabilityLOLPi,whereibelongsto{1,2,...,I}forthemosteconomicalloadsequenceofthepowergenerationunit.Bothmethodsusesystemtimingload.Z-transformmethodsuitableforsmallerscalesystemisanaccuratemethodtocalculatethevolume;cumulativemethodisaapproximationmethod,whichhasafastercalculationspeedandconsidertheuncertaintyofloaddemandforecasting.Itisalsosuitableforthelarge-scalesystemswithgoodaccuracy.ForZ-transformmethodandtheeffectivecapacityofthedistributioncumulativemethod,andthewaysofaddingtotheuncertaintyofloadforecastingisdetailedinthereference[5,6].

2.1Themarginalgeneratingunitofthesystemanditsprobability

Probabilisticmodelofgeneratinguniticanuseitseffectivecapacityxi（randomvariation）torepresent.

xi

withprobability:

1-FORii，withprobability:

FORi0，

FORiistherandomforcedoutagerateoftheuniti;Ciiscapacityofuniti（MW）.

Definition1AsfortheloadLinamoment,ifthesumoftheeffectivecapacityofthefirsti-1unitsislessthanorequaltoL,whilethesumoftheeffectivecapacityofthefirstiunitsisgreaterthanL,thei-thpowerunitcanbecalledthemarginalgeneratingunitofthesystematthattime.

Theorem1AsfortheloadLinamoment,themarginalprobabilitygeneratingunitofthei-thpowerunitmpiis

mpiLOLPi,1,LOLPi,i1,2,...I

mpI,1LOLIP

（1）WhereinLOLPiisthe

lossofsystemloadprobabilityLOLPafterthei-thpowerunitisloaded,firstI+1powerunitisanimaginarypowersupplythatcansupplypowerforthesystem.

Proof:

FromthedefinitionofLOLP,LOLPirepresentstheprobabilityofwhichthesumoftheeffectivecapacityofthefirstiunitsislessthantheloadL,then

LOLPi,1,LOLPiProp（xjL）,Prop（xjL）

j1j1i,1i

Prop（xjL）,Prop（xjL）,1

j1j1i,1i

（2）

Wherethefirstandsecondoftherightsideformulais,respectively,theprobabilitiesofwhichthesumoftheeffectivecapacityofthefirsti-1unitsislessthanorequaltotheloadLandwhichthesumoftheeffectivecapacityofthefirstiunitsislessthantheloadL.Accordingtothetheoryofprobability,sinceatwoeventsmustoccur,theaboveformularepresentstheprobabilityoftwoeventsoccursimultaneously,namelyLOLPi-1-LOLPirepresentstheprobabilitiesthattheloadLis

greaterthanorequaltothesumoftheeffectivecapacityofthepreviousi-1unit,butlessthanthesumoftheeffectivecapacityofthefirstiunits,thatis,whentheloadisL,itindicatestheprobabilitiesofmarginalpowergenerationunitoftheuniti,then

PPi1,2...,ImpiLOLi,1,LOLi,

WhereLOLP0indicatesLOLP,withoutunitsloading,aslongasL>0,LOLP0mustbe1.

TherecursiveformulaalsoappliestotheI+1thhypotheticalpower,becauseLOLPI+1equalto0,thenmpI,1equaltoLOLPI.

2.2Theprobabilitydensityanddistributionfunctionofthemarginalcostofgeneratingsystem

Withoutconsideringtheuncertaintyofpowerunitfuelprice,namelyoperatingcostsofeverypowergenerationunitisaconstant.Becausethecostofeverymarginalgeneratingunitisthecostofpowergenerationsystem,soaccordingtotheabovetheorems,whenloadisL,themarginalprobabilitydensityfunctionofthemarginalgenerationcostis,

qs（x）mp1（x,b1）,mp2（x,b2）,,mpI（x,bI）,mpI,1（x,bI,1）（3）Wherebiisthe

operatingcostofuniti,i=1,2,...,I;bI,1istheoperatingcostofhypotheticalpower,whichequalstothelossescostofthesystem;（x）isaunitimpulsefunction.Fromtheformula

（1）,itiseasyto

verify:

I,1

j1mpj1namely,the

probabilitydensityoffunctionisareasonabledensityfunctioninmathematicsoftheformula（3）.Thusprobabilitydistributionfunctionofthemarginalcostofelectricitycanbe:

Qs（x）mp1u（x,b1）,mp2u（x,b2）,,mpIu（x,bI）,mpI,1u（x,bI,1）（4）Whereu（x）istheunit

stepfunction.

TheexpectedmarginalcostofelectricityE[S]andvarianceVar[S]canbeexpressedas

E[S]mpjbj

j1

I,1I,1（5）2,[E[S]]Var[S]mpjb2（6）j

j1

2.3Theinclusionoftheuncertaintyoffuelprice

Asthesamewithgeneratorsrandomlyforcedoutage,theuncertaintyoffuelpriceofthefuturegenerationalsohasanimportantinfluenceontheestimationofthemarginalcostofelectricity.Thestrictlyaccuratemethodistodescribetheunitoperatingcostofeachpowergenerationunitasarandomvariationbi,thatis,usetheprobabilitydensityfunctiontodescribeeachrandomvariablebi.Astheformula（3）showsthat,thedensityfunctionofthemarginalcostofelectricityshoulddescribeastheprobabilityoftakingseveralrandomvariablevalues（ratherthanthedeterminedamount）,thedensityfunctionofeachrandomvariablebiis

qbi（x）bp（x,b）,i1,2,,I（7）ijij

j1IBi

Wherein,bijisthej-thstatevalueoftheoperationcostbiofuniti;bpijisthej-thstateprobabilityoftheoperationcostbiofuniti,bpij1;IBiisthestatevalueofthe

j1IBi

totaloperatingcostsbiofuniti.

Astheuncertaintyoffuelpriceandrandomnessofgeneratingunitforcedoutagearemutuallyindependent,theprobabilitydensityfunctionofthemarginalcostofelectricitygenerationsystemcanbeexpressedas

qs（x）

Impbp（x,b）,mpiijiji1J1IBi

iijIIBiI,1（x,bI,1）（8）i1J1

Thatis,thedensityfunctionformula（8）isreasonable,aprobabilitydistributionfunctionofthemarginalcostofelectricityisexpressedas

Qs（x）mpibpiju（x,bij）,mpI,1u（x,bI,1）（9）

i1J1IIBimpbp,mpI,11

Thustheexpectedvalueandvarianceofthemarginalcostofelectricitymayalsobeobtained.

3numericalexample

TheabovemethodsweretestedbyIEEE-RTSdata[4].PowerunitdataisshowninTable1,inordertoavoidtheissueofwaterandelectricity,ittakestheplaceofthesamecapacityofthermalpowerunit.thesystemcostofshortagelossesshouldbetakenasthevalueofperkWhelectricity,whichiscloselyrelatedtosociallaborproductivity,andcanbeapproximatedasGDPdividedbythetotalgeneratingcapacity,（itisassumedtobe120%ofthemaximumunitoperatingcostsofpowergeneratingunit）.

Table1dataofgeneratingunit

cellnumberofunit（MW）forcedoutageoperatingcostcumulativenumbertheunitcapacityrate,/MWhcapacity（MW）

ThecurveofexpectedvalueofmarginalcostofpowergenerationsystemsisshowninFigure1,representsthemachinewithoutconsideringforcedmethodofcumulation.,whichgivesthecalculativeresultofZ-transformationandaccumulativemethod,forthecalculationofthepresentsystem,theresu