Page - 400 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies 2018,11, 242 wherewkij is theconnectionweightbetweenthe ithunitofkth layerandthe jthunitof (k+1)th layer, and f(·) is the logistic sigmoidfunction. Inorder toobtain theoptimalparametersof theBPNN,theBackwardPropagation(BP)algorithm isadoptedtominimize the followingcost functionforeachtrainingdatapoint E(t,w)=(yˆ(t)−y(t))2, (23) where yˆ(t) andy(t) are thepredictedandactualvalueswithrespect to the inputx(t). Theupdaterule for theweightwkij canbeexpressedas wkij(t+1)=w k ij(t)−η ∂E(t,w) ∂wkij , (24) whereη is the learningrate,and ∂E(t,w) ∂wkij is thegradientof theparameterwkij, andcanbecalculatedby thebackwardpropagationof theerrors. TheBPalgorithmhas twophases—forwardpropagation andweightupdate. In the forward propagation stage,whenan inputvector is input to theNN, it ispropagated forward through the wholenetworkuntil it reaches theoutput layer. Then, theerrorbetweentheoutputof thenetworkand thedesiredoutput iscomputed. In theweightupdatephase, theerror ispropagatedfromtheoutput layerback throughthewholenetwork,until eachneuronhasanassociatederrorvalue thatcanreflect itscontributionto theoriginaloutput. Theseerrorvaluesare thenusedtocalculate thegradientsof the loss functionthatare fedto theupdaterules torenewtheweights [40–42]. 4.1.2.GeneralizedRadialBasisFunctionNeuralNetwork The radial basis function (RBF)NNis a feed-forwardNNwithonlyonehidden layerwhose structure is demonstrated inFigure 6. TheRBFNNhasGaussian functions as its hiddenneurons. TheGRBFNNis amodifiedRBFNNandadopts thegeneralizedGaussian functions as its hidden neurons [43,44]. [ [ Q[ * [ * [ Q* [ ¦ Ö\ Z Z QZ Figure6.Thetopological structureof the feed-forwardsingle-hidden-layerNN. 400