Page - 401 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies 2018,11, 242 Theoutputof theGRBFNNcanbeexpressedas [43,44] yˆ= n1 ∑ j=1 wjGj(x)= n1 ∑ j=1 wjexp ⎛⎝−‖x−cj‖τj d τj j ⎞⎠ , (25) wheren1 is thenumberofhiddenneurons,τj is theshapeparameterof the jthradialbasis function in thehiddenlayer,and cj anddj are, respectively, thecenterandwidthof the jthradialbasis function. Inorder todetermine theparametersτ,c andd in thehidden layerandtheconnectionweightwj, theaforementionedBPalgorithmcanalsobeemployed. 4.1.3. ExtremeLearningMachine TheELMisalsoa feed-forwardneuralnetworkwithonlyonehiddenlayerasdemonstrated in Figure6.However, theELMandGRBFNNhavedifferentparameter learningalgorithmsanddifferent activationfunctions in thehiddenneurons. In theELM, theactivation functions in thehiddenneuronscanbe thehard-limitingactivation function, theGaussianactivationfunction, theSigmoidal function, theSine function,etc. [36,37]. Inaddition, the learningalgorithmfor theELMis listedbelow: • Randomlyassign inputweightsor theparameters in thehiddenneurons. • Calculate thehiddenlayeroutputmatrixH,where H= ⎛⎜⎝ G1(x (1)) · · · Gn1(x(1)) ... ... ... G1(x(N)) · · · Gn1(x(N)) ⎞⎟⎠ N×n1 . (26) • Calculate theoutputweightsw= [w1,w2, · · · ,wn1]T=H+Y,whereY = [y(1),y(2), · · · ,y(N)]T andH+ is theMoore–Penrosegeneralized inverseof thematrixH. This learningprocessisveryfastandcanleadtoexcellentmodelingperformance.Hence, theELM hasfoundlotsofapplications indifferent researchfields. 4.1.4. SupportVectorRegression The SVR is a variant of SVM. It can yield improved generalization performance through minimizingthegeneralizationerrorbound[45]. Inaddition, thekernel trick isadoptedtorealize the nonlinear transformationof input features. Themodelof theSVRcanbedefinedbythe followingfunction yˆ= f(x,w)=wTϕ(x)+b, (27) wherew=[w1, · · · ,wn],ϕ(x) is thenonlinearmappingfunction. Usingthe trainingsetℵ={(x(l),y(l))}Nl=1,wecandeterminetheparametersw andb, andthen obtain theSVRmodelas yˆ= f(x)= N ∑ l=1 w∗Tϕ(x)+b∗, (28) where ⎧⎪⎪⎪⎨⎪⎪⎪⎩ w∗= N ∑ l=1 (α∗l −αl)ϕ(x(l)), b∗= 1 yl −w∗Tϕ(x(l)), (29) 401