Page - 325 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1449 Toobtain thesolution, theLagrangefunctioncanbeestablishedasEquation(8). L(w,b,ξi,αi)= 1 2 wTw+ 1 2 γ N ∑ i=1 ξ2i − N ∑ i=1 αi [ wTϕ(xi)+b+ξi−yi ] (8) whereαi is theLagrangemultipliers. Takethederivativesofeachvariable in the functionandmake themequalzero: ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ∂L ∂w=0→w= N ∑ i=1 αiϕ(xi) ∂L ∂b =0→ N ∑ i=1 αi=0 ∂L ∂ξ =0→αi=γξi ∂L ∂α =0→wTϕ(xi)+b+ξi−yi=0 (9) Eliminatewaswellasξi andtransformit into the followingproblem:[ 0 eTn en Ω+γ−1 · I ] × [ b a ] = [ 0 y ] (10) where Ω=ϕT(xi)ϕ(xi) (11) en=[1,1,...,1] T (12) α=[α1,α2,...,αn] (13) y=[y1,y2,...,yn] T (14) Thesolutioncanbeobtainedbasedonthe linearequationsabove: y(x)= N ∑ i=1 αiK(xi,x)+b (15) whereK(xi,x) is thekernel function thatmeetsMercer’s condition. Theradialbasis function (RBF) is employedas thekernel functionhereon thebasis of itswide convergence regionandextensive applicationscope,asshowninEquation(16). K(xi,x)= exp { −‖x−xi‖2/2σ2 } (16) whereσ2 represents thekernelparameter that reflects thecharacteristicof trainingsamplesandhas influenceongeneralizationabilityof the technique. Aswe can see, the performance improvement of LSSVMmodel is greatly dependent on the appropriate settingof the followingparameters: regularizationparameterγandkernelparameter σ2 [33]. 3.3.WolfPackAlgorithm Inconsiderationof theblindnessofmanualselection inLSSVMmodelparameters, theoptimal valueof regularizationparameterγandkernelparameterσ2 ofLSSVMisobtainedthroughthewolf packalgorithm.TheWPAtechnique is inspiredbyresearchonthehuntingbehaviorsofwolves [34]. Accordingto their roles inhunting,wolvescanbedividedinto three types: headwolves, safariwolves andferalwolves,whoworktogether tocomplete the task.Randomwalk, call toactionandsiegeare threemainbehaviorsofwolves,whicharesimulated in theWPAmodel. Thedeterminationof the headwolfandthereplacementof thewolfpack followthecommonrules that the“winner is theking” and“thesurvivalof thefittest”, respectively [35].WPAis illustrated inFigure5. 325