Fuzzy logic in Load forecasting.docx
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Fuzzy logic in Load forecasting.docx
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FuzzylogicinLoadforecasting
ShortTermLoadForecastingwithFuzzyLogicSystems
Introduction:
SeveralpapershaveproposedtheuseofFuzzyLogicforshorttermloadforecasting.Atpresentapplicationoffuzzymethodforloadforecastingisintheexperimentalstage.Forthedemonstrationofthemethodafuzzyexpertsystemsthatforecaststhedailypeakload,isselected.
FuzzyExpertSystems:
Thefuzzysystemisapopularcomputingframeworkbasedontheconceptsof‘fuzzysettheory’,‘fuzzyifthenrules’and‘fuzzyreasoning’.Thestructureoffuzzyinferenceconsistsofthreeconceptualcomponents,namely:
RuleBasecontainingaselectionoffuzzyrules.
Databasedefiningthemembershipfunctions.Theseareusedinthefuzzyrules.
Reasoningmechanismthatperformstheinferenceprocedureupontherulesandgiven
factsandderivesareasonableoutputorconclusion.
Sometimesitisnecessarytohavecrispoutput.ThisrequiresamethodcalledDe-fuzzification,toextractacrispvaluethatbestrepresentsthefuzzyoutput.Withsuchcrispinputsandoutputs,afuzzyexpertsystemimplementsanon-linearmappingfromtheinputspacetotheoutputspace.Thismappingisaccomplishedbyanumberofif-thenrules,eachofwhichdescribesalocalbehaviorofthemapping.
Toillustratethisletusconsider:
X:
asetofdataorobjects.(Example.Forecasttemperaturevalues).
A:
anothersetcontainingdata(orobjects)
x:
anindividualvalueofthedatasetX.
isthemembershipfunctionthatconnectsthesetXandA.Themembershipfunction
DeterminesthedegreethatxbelongstoA.
Itsvaluevariesbetween0and1.
Thehighvalueof
meansthatitisverylikelythatxisinA.
Themembershipfunctionisselectedbytrialanderror.Therearefourbasicmembershipfunctionsnamely:
Triangular.
Trapezoidal.
Gaussian.
Generalizedbell.
TheMATLABm-file“disp_mf.m”displaysallthesemembershipfunctionsasinfigure1.
Figure1.Membershipfunctions
Thetriangularfunction“triangle(x,a,b,c)”isdefinedas:
Ithasthreeparameters‘a’(minimum),‘b’(middle)and‘c’(maximum)thatdeterminetheshapeofthetriangle.
Figure2showsthetriangularfunctionoftriangle(x,20,60,80):
Figure2.Triangularmembershipfunction
Atrapezoidalmembershipfunctionisspecifiedbyfourparametersgivenby:
A=trapezoid(x,a,b,c,d)
Thefunctionisdescribedas:
Theplotofthefunctiontrapezoid(x,10,20,60,95)isshowninfigure3:
Figure3.Trapezoidalmembershipfunction
Similardefinitionsforgaussianandgeneralizedbellcanbegiven.Howevertriangularandtrapezoidalfunctionsaresimpleandmostfrequentlyused.Themembershipfunctionsarenotrestrictedtothesefour.Onecanhavetheirowntailor-madefunctions.Thefunctionsaboveweremereonedimensionalinnature.Inprincipleonecanevenhavemulti-dimensionalmembershipfunctions.ComingbacktooursetsAandX,wecandefinethefuzzysetAinXasasetoforderedpairsgivenby:
Forexampleinthetriangularmembershipfunctionshownonthelefthandside,weseethatforx=40(x-axis)belongstoA=0.5(y-axis).Theco-ordinatesforthistriangleare:
x1=20(Lmin);y1=0orA(x1)=0.
x2=60(Lmid);y2=1orA(x2)=1.
Theslopeofthemembershipfunctionbetweenx1andx2isthendefinedas:
Thustheequationoftheraisingedgeofthetriangleis:
Theoutsideregionisdescribedby:
Thecombinationoftheaboveequationswouldresultinthetriangularmembershipfunctionequation:
FuzzySetsandFuzzyOperations:
ConsidertwofuzzysetsAandB,asshowninfigure4,withmembershipfunctionsA(x)andB(x)respectively.Thesetwofuzzysetscanbecombinedindifferentwaysasbelow:
UnionC=AB.
IntersectionC=AB.
SumC=AB.
Thedifferencebetweenthesumandtheunionoperationmaybewellunderstoodfromfigures6and7.Theaimistodeterminetherightcombinedfunctionoftwosetssuchthatthedesiredoutputisobtained.Theunionandintersectionoftwomembershipfunctionsisillustratedinthefigures5and6respectively:
Figure4.MembershipfunctionoffuzzysetsAandB
Figure5.UnionoffuzzysetsAandB
TheUnionoftwofuzzysetpoints,whichlieinAandB,isgivenby:
Figure6.IntersectionoffuzzysetsAandB
TheIntersectionoperationisdefinedbytheequation:
Similarlythesumofthetwofuzzysetscanbegivenintheformoftheequationgivenbelow:
Figure7.SumoffuzzysetsAandB
LoadForecastingUsingFuzzyLogic.
TheFuzzyInferencesystems,unlikeneuralnetworks,areappliedtopeakloadandthroughloadforecastingonly.Theproposedtechniqueforimplementingfuzzylogicbasedforecastingis:
Identificationoftheday.(Monday,Tuesdayetc.,)Letssayweselect‘Tuesday’.
ForecastmaximumandminimumtemperaturefortheupcomingTuesday
Listingthemaximumtemperatureandpeakloadforthelast10-12Tuesdays.
Fortheselectedhistoricaldatawefitapolynomial.
Letusconsideranumericalexample.Wehavetheloadandtemperaturedataasinthetablebelow:
Load
10200
10500
10180
10700
10680
10850
11100
11030
11100
Temperature
31
31.57
32.4
32.6
32.67
33.1
33.6
33.81
34.23
Nowwefitastraightlineforthisdata.Theresultofthiscurvefittingisshowninfigure8.
Figure8.Polynomialcurvefittingonhistoricaldata
Thedataisfittedbyalinearregressioncurve.Theactualdatapointsarespreadovertheregressioncurve.ThisregressioncurveiscalculatedusingthesimulationtoolssuchasMATLABorMathCAD.Theresultofthisregressionanalysisresultsintheequationofastraightline:
Where,
Lp:
Peakload.
Tp:
Forecastmaximumdailytemperature.
gpandhp:
Constantsderivedfromtheleastsquarebasedregressionanalysis
Forthedatapresentedabovethegpandhpwerecalculatedas300.006and871.587respectively.AsanexampleiftheforecasttemperatureTp=35,thentheexpectedorforecastpeakloadiscalculatedtobe:
Thisregressionmethodhascertainamountofstatisticalerror,whichisevidentbythespreadofthedatapointsaboutthecurve.Thiscanbeimprovedbyaddingaregressiontermtotheequation.Thismodifiedequationisshownbelow:
Where,Lpistheerrorco-efficient
Determinationoftheerrorco-efficientiscarriedoutbythefuzzymethod.Theregressionerrorco-efficienthasthreecomponents,namely:
Statisticalmodelerror
Temperatureforecastingerror
Operators’Heuristicrule
StatisticalModelError:
Thestatisticalmodelerrorisdefinedasthedifferencebetweeneachsamplepointandtheregressionline.Indescribingthiserrorasafuzzymodel,weassigndifferentmembershipfunctionsforeachdayoftheweek.Anexpert,usingtrialanderrormethod,determinesthesefunctions.Atriangularmembershipfunctionisthenassigned.Thefunctionhasamembershipvalueof1whentheloadis0anddecreasesto0ataloadvalueof2.Thisiscalculatedusingtheformulagivenbelow:
MW
Where,
Lpiisthepeakload.
Tpiisthemaximumtemperature.
nisthenumberofpointsselectedfortheday.
Inourexampleis450MWandthevariablesofthetriangularmembershipfunctionF1(L1),inthisexampleare:
L1_min=–450MW,L1_mid=0MW.
Thesubstitutionofthesevaluesgivesusthefinalmembershipfunction:
With=450MWandL=-1500MWto500MW,themembershipfunctionisshowninfigure9.
Figure9.MembershipfunctionofF1(L1)
ThevaluesforthetriangleareL1_min=–450MW,L1_mid=0MWandL1_max=450MW.ThusF1(L1)describesthestatisticalerrormodel.
Temperatureforecastingerror:
Theforecasttemperatureiscomparedwiththeactualtemperatureusingstatisticaldataavailableforthepreviousyears.Theaverageerrorandthestandarddeviationarecalculatedfromthisdata.Inourexampletheerrorislessthan4degrees.Thetemperatureforecastingerrorproduceserrorinthepeakloadforecast.Theerrorforthepeakloadiscalculatedbythederivationoftheload-temperatureequation.
Sincetheerrorinpeakloadisproportionaltotheerrorintemperature,itcanbemodeledusingatriangularmembershipfunction.
Afuzzyexpertsystemcanbedevelopedusingthemethodappliedforthestatisticalmodel.Amoreaccuratefuzzyexpertsystemcanbeobtainedbydividingtheregionintointervals.Eachintervalhasitsownmembershipfunction.Theintervalsforthetemperatureforecastingerrorsaredefinedasfollows.
Temperaturesmuchlowerthantheforecastedvalue(ML)
Temperatureslowerthantheforecastedvalue(L)
Temperaturesclosertotheforecastedvalue(C)
Temperatureshigherthantheforecastedvalue(H)
Temperaturesmuchhigherthantheforecastedvalue(MH)
Thevaluesfor‘d’are–4,–2,0,1and2forML,L,C,HandMHrespectively.
Themembershipfunctionsaredeterminedusingtrialanderrortechnique.Atriangularmembershipfunctionwiththefollowingco-ordinatesisselected:
Thesevaluesarethensubstitutedinthegeneralequationandthemembershipfunctionforthepeakloadduetoerrorintemperatureforecastingisobtainedas:
Thesemembershipfunctionscanberepresentedgraphicallyasinfigure10.
Figure10.MembershipfunctionsforF2(L2)
ModelUncertainty:
Themodeluncertaintyiscoupledwiththeuncertaintyinforecast-temperature.ThisuncertaintyleadstoathirdtermL3givenby:
L3=L1+L2
Themembershipfunctionforthisnewtermisgivenby:
Thenewmembershipfunctionisshowninthefigure11below:
Figure11.Membershipfunctionswithmodelinguncertaintyincluded
Thecombinedmembershipfunctionswillbeatrianglewiththefollowing
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