Page - 5 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 2226 min 12w Tw+c N ∑ i=1 ( ξi+ξ ∗ i ) s.t. ⎧⎪⎪⎪⎨⎪⎪⎪⎩ yi−wTϕ(xi)−b≤ ε+ξi wT(xi)+b−yi≤ ε+ξ∗i ξi≥0, ξ∗i ≥0 i=1, · · · ,N , (2) wherec is thebalance factor,usuallyset to1,andξi andξ∗i are theerrorof introducing the trainingset, whichcanrepresent theextent towhichthesamplepointexceeds thefittingprecision ε. Equation(2)couldbesolvedaccordingtoquadraticprogrammingprocesses; thesolutionof the weight,w, inEquation(2) is calculatedas inEquation(3) [17]: w∗= N ∑ i=1 (αi−α∗i )ϕ(x), (3) whereαi andα∗i areLagrangemultipliers. TheSVRfunction iseventuallyconstructedas inEquation(4) [17]: y(x)= N ∑ i=1 (αi−α∗i )Ψ(xi,x)+b, (4) whereΨ(xi,x), theso-calledkernel function, is introducedtoreplace thenonlinearmappingfunction, ϕ(·), as showninEquation(5) [15]: Ψ ( xi,xj ) =ϕ(xi) Tϕ ( xj ) . (5) 2.1.2. Principleof theLS-SVRModel TheLS-SVRmodel isanextensionof thestandardSVRmodel. It selects thebinomialoferrorξt as the loss function; thentheoptimizationproblemcanbedescribedas inEquation(6) [20]: min 12w Tw+ 12γ N ∑ i=1 ξ2i s.t.yi=wTϕ(xi)+b+ξi, i=1,2, · · · ,N (6) where thebigger thepositiverealnumberγ is, thesmaller theregressionerrorof themodel is. TheLS-SVRmodeldefines the loss functiondifferent fromthestandardSVRmodel, andchanges its inequality constraint into an equality constraint so thatw can be obtained in the dual space. After obtaining parameters α and b by quadratic programming processes, the LS-SVRmodel is describedas inEquation(7) [20]: y(x)= N ∑ i=1 αiΨ(xi,x)+b. (7) It canbeseen thatanLS-SVRmodel contains twoparameters, the regularizationparameterγ andtheradialbasiskernel function,σ2. TheforecastingperformanceofanLS-SVRmodel is related to theselectionofγandσ2. The roleofγ is tobalance theconfidence rangeandexperience riskof learningmachines. Ifγ is too large, thegoal isonly tominimize theexperiencerisk.Onthecontrary, when thevalueofγ is toosmall, thepenalty for theexperienceerrorwill be small, thus increasing thevalueofexperienceriskσ controls thewidthof theGaussiankernel functionandthedistribution rangeof the trainingdata. The smallerσ is, thegreater the structural risk there is,which leads to overfitting. Therefore, theparameterselectionofanLS-SVRmodelhasalwaysbeenthekeyto improve the forecastingaccuracy. 5