Page - 4 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 2226 prematureproblemofFOA.Eventually,determinemoreappropriateparametersofanLS-SVRmodel. Themajorcontributionsareas follows: (1) QCMisemployedtoempower thesearchabilityofeachfruitflyduringthesearchingprocesses ofQFOA.Thecatchaoticmappingfunction is introducedintoQFOAandimplements thechaotic globalperturbationstrategytohelpafruitflyescapefromthelocaloptimawhenthepopulation’s diversity ispoor. (2) Weproposeanovelhybridoptimizationalgorithm,namelyCQFOA,tobehybridizedwithan LS-SVRmodel,namelytheLS-SVR-CQFOAmodel, toconducttheMELforecasting.Othersimilar alternativehybridalgorithms(hybridizingchaoticmappingfunction,QCM,andevolutionary algorithms) inexistingpapers, suchas theCQPSOalgorithmusedbyHuang[36], theCQTSand CQGAalgorithmsusedbyLeeandLin[37,38], andtheCQBAalgorithmusedbyLietal. [39], areselectedasalternativemodels to test thesuperiorityof theLS-SVR-CQFOAmodel in termsof forecastingaccuracy. (3) The forecasting results illustrate that, in threedatasets, theproposedLS-SVR-CQFOAmodel is superior to other alternativemodels in termsof forecasting accuracy indexes; in addition, itpasses thesignificance testata97.5%confidence level. 1.4. TheOrganizationofThisPaper The rest of this paper is organized as follows. Themodeling details of an LS-SVRmodel, theproposedCQFOA,andtheproposedLS-SVR-CQFOAmodelare introduced inSection2. Section3 presentsanumericalexampleandacomparisonof theproposedLS-SVR-CQFOAmodelwithother alternativemodels. SomeinsightdiscussionsareprovidedinSection4. Finally, theconclusionsare given inSection5. 2.MaterialsandMethods 2.1. LeastSquaresSupportVectorRegression (LS-SVR) TheSVRmodel isanalgorithmbasedonpatternrecognitionofstatistical learningtheory. It is anovelmachine learningapproachproposedbyVapnik in themid-1990s [17]. TheLS-SVRmodel was put forward by Suykens [20]. It is an improvement and an extension of the standard SVR model, which replaces the inequality constraints of an SVRmodel with equality constraint [21]. TheLS-SVRmodelconvertsquadraticprogrammingproblemintolinearprogrammingsolving,reduces thecomputationalcomplexity,andimproves theconvergentspeed. It cansolve the loadforecasting problemsdueto itscharacteristicsofnonlinearity,highdimension,andlocalminima. 2.1.1. Principleof theStandardSVRModel Setadatasetas{(xi,yi)}Ni=1, xi∈Rn is the inputvectorofn-dimensional system,yi∈R is the output (notasingle realvalue,butan-dimensionalvector)of system.Thebasic ideaof theSVRmodel canbesummarizedasfollows:n-dimensional inputsamplesaremappedfromtheoriginalspacetothe high-dimensional featurespaceFbynonlinear transformationϕ(·), andtheoptimal linear regression function isconstructed in this space,asshowninEquation(1) [17]: f(x)=wTϕ(x)+b, (1) where f(x)representstheforecastingvalues; theweight,w, andthecoefficient,b,wouldbedetermined duringtheSVRmodelingprocesses. ThestandardSVRmodel takes the ε insensitive loss functionasanestimationproblemforrisk minimization, thus theoptimizationobjectivecanbeexpressedas inEquation(2) [17]: 4