Page - 26 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1009 yi− f(xi)≤ ε+ξ∗i , −yi− f(xi)≤ ε+ξi, ξ∗i ≥0 ξi≥0 i=1, 2, . . . , N ThesolutionofEquation(4) isoptimizedbyusingLagrangemultipliers,β∗i , andβi, theweight vector,w, inEquation(1) is computedasEquation(5): w∗= N ∑ i=1 (β∗i −βi)ϕ(xi) (5) Eventually, theSVRforecastingfunction iscalculatedasEquation(6): f(x)= N ∑ i=1 (β∗i −βi)K ( xi,xj ) +b (6) whereK ( xi,xj ) is theso-calledkernel function,anditsvaluecouldbecomputedbythe innerproduct ofϕ(xi)andϕ ( xj ) , i.e.,K ( xi,xj ) = ϕ(xi)×ϕ ( xj ) . Theare severalkindsofkernel function, suchas Gaussian function (Equation (7)) andthepolynomialkernel function. Due to its superiorability to mapnonlineardata intohighdimensional space,aGaussianfunction isused in thispaper: K ( xi,xj ) = exp ( −‖xi−xj‖ 2 2σ2 ) (7) Therefore,determiningthe threeparameters,σ,C, and εofanSVRmodelwouldplay thecritical role toachievemoreaccurate forecastingperformances [5,28,29]. Theparameter εdecides thenumber ofsupportvectors. If ε is largeenough, it implies fewsupportvectorswith lowforecastingaccuracy; if εhasavaluethat is toosmall, itwouldincrease theforecastingaccuracybutbetoocomplextoadopt. ParameterC, asmentioned,penalizes the trainingerrors. IfC is largeenough, itwould increase the forecastingaccuracybut suffer frombeingdifficult to adopt; ifChasa too small value, themodel wouldsuffer fromlarge trainingerrors. Parameterσ represents therelationshipsamongdataandthe correlationsamongsupportvectors. Ifσ is largeenough, thecorrelationsamongsupportvectorsare strongandwecanobtainaccurate forecastingresults,but if thevalueofσ is small, thecorrelations amongsupportvectorsareweak,andadoption isdifficult. However, structural methods to determine the SVR parameters are lacking. Hong and his colleagueshavepointedout theadvancedexplorationwaybyhybridizingchaoticmappingfunctions withevolutionaryalgorithmstoovercometheembeddedprematureconvergenceproblem, toselect suitableparameter combination, toachievehighlyaccurate forecastingperformances. Tocontinue thisvaluableexploration, thechaoticcuckoosearchalgorithm, theCCSalgorithm, isproposedtobe hybridizedwithanSVRmodel todetermineanappropriateparametercombination. 2.2. ChaoticCuckooSearch (CCS)Algorithm 2.2.1. TentChaoticMappingFunction The chaoticmapping function is anoptimization technique tomap theoriginaldata series to showsensitivedependenceonthe initial conditionsandinfinitedifferentperiodic responses (chaotic ergodicity), tomaintain thediversityofpopulation in thewholeoptimizationprocedures, toenrich thesearchbehavior,andtoavoidprematureconvergence. Themostpopularchaoticmappingfunction is the logistic function,however,basedontheanalysisonthechaoticcharacteristicsof thedifferent mappingfunctions,a tentchaoticmappingfunction[39]demonstratesarangeofdynamicalbehavior 26