Page - 27 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1009 ranging frompredictable tochaos, i.e.,withgoodergodicuniformity [40]. Thispaper thusapplies the tent chaoticmapping function to behybridizedwith theCSalgorithm todetermine the three parametersofanSVRmodel. The tentchaoticmappingfunction isshownasEquation(8): xn+1= { 2xn x∈ [0,0.5] 2(1−xn) x∈ (0.5,1] (8) wherexn is the iterativevalueof thevariablex in thenthstep,andn is thenumberof iterationsteps. 2.2.2.CuckooSearch(CS)Algorithm TheCSalgorithm is anovelmeta-heuristic optimizationalgorithm, inspiredbycuckoobirds’ obligate broodparasitic behavior of laying their eggs in thenests of otherhost birds. Meanwhile, byapplyingLévyflightbehaviors, thesearchspeed ismuchfaster thanthatof thenormal random walk. Therefore, cuckoo birds can reduce the number of iterations and thus speed up the local searchefficiency. ForCSalgorithmimplementation,eachegg inanest representsapotential solution. Thecuckoobirdscouldchoose,byLévyflightbehaviors, recently-spawnednests to lay theireggs in thehostnests toensure theireggscouldhatchfirstdueto thenaturalphenomenonthatcuckooeggs usuallyhatchbefore thehostbirds’ eggs. It takes times for thehostbirds todiscover that theeggs in theirnestsdonotbelong to them,basedon theprobability, pa. When these“stranger”eggsare discovered, theyeither throwout thoseeggsorabandonthewholenest tobuildanewnest inanew location. Thecuckoobirdswouldcontinuously layneweggs (solutions), andtheywouldchoose the nest,byLévyflightbehaviors,aroundthecurrentbest solutions. TheCSalgorithmcontains three famous idealized rules [31]: (1) each cuckoo lays one eggat a timeinarandomlyselectedhost; (2)high-qualityeggsandtheirhostnestswouldsurvive to thenext generation; (3) thenumberofavailablehostnests isfixed,andthehostbirddetects the“stranger”egg withaprobability pa∈ [0,1]. In thiscase, thehostbirdcaneither throwawaytheeggorabandonthe nest, andbuildacompletelynewnest. The last rulecanbeapproximatedbyafraction(pa)of then hostnests thatare replacedbynewnests (withnewrandomsolutions). Thevalueof pa isoftenset as0.25 [37]. TheCS algorithm couldmaintain the balance between twokinds of search (randomwalks), the local search and theglobal search, by a switchingparameter, pa. The switchingparameter pa determinesthecuckoobirdstoabandonafractionoftheworstnestsandbuildnewonesfordiscovering new andmore promising regions in the search space. These two randomwalks are defined by Equations (9)and(10), respectively: xt+1i = x t i+αs⊗H(pa−δ)⊗ ( xtj−xtk ) s (9) xt+1i = x t i+αL(s,λ) (10) where xtj and x t k are current positions randomly selected; α is the positive Lévy flight step size scalingfactor; s is thestepsize;H(·) is theHeavy-side function;δ isa randomnumber fromuniform distribution;⊗ represents theentry-wiseproductof twovectors;L(s,λ) is theLévydistributionandis usedtodefinethestepsizeof randomwalk, it isdefinedasEquation(11): L(s,λ)= λΓ(λ)sin(πλ/2) π 1 s1+λ (11) where λ is the standard deviation of step size; the gamma function, Γ(λ), is defined as Γ(λ)= ∫∞ 0 t λ−1e−tdt, and represents an extension of factorial function, if λ is a positive integer, then,Γ(λ)=(λ−1)!. Lévyflightdistributionenablesaseriesofstraight jumpschosenfromanyflight 27