Page - 55 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2019,12, 164 whichuses these thresholdvalues for featureselection. For thispurpose,variouschoicesareavailable suchas linearprogramming,non-linearprogramming,quadraticprogramming,convexoptimization, heuristic optimization, etc. However, the first one is not applicable here because the problem is highlynon-linear. Thenon-linearproblemcanbeconverted intoa linearone;however, theoverall processwouldbecomeverycomplex. Thesecondone isapplicablehereandgivesaccurate results bypayingexecution time’s cost. Similarly, the thirdandfourthonessuffer fromslowconvergence time. It isworthmentioninghere that optimizationdoesnot implyexact reachability tooptimum setof solutions, rather,nearoptimalsolution(s) is(are)obtained. Tosumup,heuristicoptimization techniquesarepreferred in thesesituationsbecause theseprovidenearoptimalsolution(s) inrelatively lessexecutiontime. DEisoneof theheuristicoptimizationtechniquesproposedin[45]anditsenhancedversion is usedfor forecasterrorminimization in [28]. In thispaper,wemodify theEDEalgorithmfor thesake accuracy improvement. Thus, in theupcomingparagraphs,detaileddiscussion ispresented. Accordingto [28], ingeneration t, the jth trialvectory for ith individual isgivenas: y ′t i,j= { uti,j if rnd(j)≤FFN(Uti) xti,j if rnd(j)>FFN(U t i) (5) where, xti,j and u t i,j are the correspondingparent andmutant vectors, respectively. In (5), FFN(.) denotes thefitness function(0<FFN(.)<1)andRand(j)∈ [0,1] isarandomnumbercomplyingto uniformdistribution. BetweenXti andY t i , thecorrespondingoffspringof thenextgenerationX (t+1) i is selectedas follows: yti,j= { y ′t i,j ifMAPE(y ′t i )≤EF(xti) xti,j otherwise (6) where,MAPE(.) is theobjective function. From(5)and(6), it is clear thatoffspringselectiondepends onthe trialvectorwhich in turndependsontherandomnumberandthefitness function. Fromthis discussion,we conclude that the selected offspring is not the fittest. Tomake the fittest one, our approacheliminates thechancesofoffspringselectionunder the influenceof randomnumber, i.e.,we modify (5)as follows: y ′t i,j= ⎧⎪⎨⎪⎩ uti,j if Xti Xtimax <FFN(Uti) xti,j if Xti Xtimax ≥FFN(Uti) (7) From(7), it is clear that the trial vectorno longerdependson the randomnumber instead its dependence innowtotallyonthemutantvectorwhichinturndependsontheparentvector.Offspring selectionby thismethodwill ensure selectionof thefittest ones subject to accuracy improvement. Stepwiseoperationsof theoptimizationmoduleareshowninFigure5b. 4. SimulationResults Forevaluationofourproposedmodel,weconductsimulations. Forsimulations,wehaveused MATLABinstalledonIntel(R)Core(TM) i3-2370MCPU@2.4GHzand2GBRAMwithWindows7. TheproposedMI+ANN+mEDE-basedforecastmodel iscomparedwith twoexistingDALFmodels: MI+ANNforecast [27], andbi-level forecast [28]. For simulationpurpose, traces of real timedata forDAYTOWNandEKPC(the twoUSAgrids)are taken fromPJMelectricitymarket. Thisdata is freelyavailableat [46].WehaveusedJanuary–December2014 loadvalues for trainingtheANN,and January–December2015data for testingtheANN.Followingare thesimulationparameters thatare usedinourexperiments(refertoTable2). Justificationoftheseparameterscanbefoundin[27,28,42,43]. 55