Page - 215 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1900 Y= ∑ m i=1yi m , (4) whereyi (i=1,2, . . . ,m) is thesamplingresult calculatedbyEquation(2).Y is theexpectedvalueof theresults. 2.2. TheLoadForecastModelBasedonSVM SVMisthetechniquefirstproposedbyVapniktosolveclassificationandregressionproblems[27]. SVR is amachine learningmethod based on statistical learning theory. It can effectively solve practicalproblemssuchassmall samplesandnonlinearitiesandhasstronggeneralizationability. It mainly includes two regressionmodels: ε-SVRand υ-SVR.The ε-SVRmodel isused in thispaper. By introducingakernel function, thenonlinearproblemof low-dimensional space is transformedinto alinearprobleminhigh-dimensional featurespaceusingnonlinearmapping.After thetransformation, thedecisionfunction[28] is: f(x)=ωT·ϕ(x)+b (5) InFormula(5),ω isaweightvector,b isa threshold,andϕ(x) isasublinearmappingrelationship fromalow-dimensional space toahigh-dimensional space. TheSVMuses theminimumstructural risk todetermine theparametersωandbandintroduces the insensitive loss functionparameter ε,which translates theprobleminto the followingoptimization problems[28]: min 1 2 ωTω+C∑mi=1(ξi+ξ∗i ), (6) s.t. yi−ωT·φ(xi)−b≤ ε+ξi, (7) ωT·φ(xi)+b−yi≤ ε+ξ∗i , (8) ωT·φ(xi)+b−yi≤ ε+ξ∗i , (9) where (x1,y1), . . . , (xm,ym)areapairof inputandoutputvectors,mis thenumberofsamples,ω is weight factor,b is the thresholdvalue,Ciserrorcost, inputsamplesaremappedtohigherdimensional spacebyusingkernel functionφ,ξi is theupper trainingerrorandtheξ∗i is the lower trainingerror subject toε-insensitive tube |y−(ωT·ϕ(x)+b)|≤ ε. TheSVMincludes twoparameters: Intrinsicparametersof thesupportvectormachine, including the penalty parameter ‘C’, the loss functionparameter ‘ε’; andparameters in the kernel function, suchas thekernelwidth in theGaussiankernel. Thechoicesof theseparametersarevery important. Thepenaltyparameter ‘C’directlyaffects thecomplexityandstabilityof themodel. It canmakethe modela tradeoffbetweencomplexityandtrainingerror. The loss functionparameter ‘ε’ controls the simulationofSVR,whicheffects thenumberofsupportvectorsandthegeneralizationabilityof the model. Thewidthcoefficient ‘γ’ in thekernel function that reflects thecorrelationbetweenthevectors. Themain types of kernel functions include linear kernel functions, polynomial kernel functions, Gaussianradialbasiskernel functions,andsigmoidcolonykernel functions.Amongthese functions, Gaussiankernel functions, suitedtorepresent thecomplexnonlinear relationshipbetween inputand output [28,29], have the advantages of computational efficiency, simplicity, reliability, and ease of adaptation.Gaussiankernel functions [28]areas follows: K ( xi,xj ) = exp ( −γ‖ xi−xj ‖2 ) , γ>0 (10) where theγ is thekernelparameter. When trainingSVMmodels, two freeparametersneed tobe identified,whicharekernelparameterγandregularizationconstantC. Since thekeyparametersof theabovesupportvectormachinemodeldirectlyaffect theaccuracy of themodel, thispaperuses theparticle swarmoptimizationalgorithmtodetermine theoptimal 215