Page - 295 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1561 Tomathematicallymodel thesalpchains, thepopulation isfirstdividedto twogroups: leader andfollowers. The leader is thesalpat the frontof thechain,whereas therestof salpsareconsidered as followers.As thenameof thesesalps implies, the leaderguidesswarmandthe followers follow eachother. Similar toother swarm-based techniques, thepositionof salps isdefined inann-dimensional searchspacewheren is thenumberofvariablesofagivenproblem. Therefore, thepositionsofall salpsarestored ina two-dimensionalmatrixcalledx. It is alsoassumedthat there isa foodsource calledF in thesearchspaceas theswarm’s target. Definition1.Toupdate thepositionof the leader the followingequation isproposed: x1j = { Fj+c1 (( ubj− lbj ) c2+ lbj ) c3≥0 Fj−c1 (( ubj− lbj ) c2+ lbj ) c3 <0 (12) where x1j shows thepositionof thefirst salp (leader) in the jthdimension, Fj is thepositionof the foodsource in the jthdimension,ubj indicates theupperboundof jthdimension, lbj indicates the lowerboundof jthdimension, c1, c2, andc3 are randomnumbers. Equation (12) shows that the leaderonlyupdates itspositionwith respect to the foodsource. Definition2.Thecoefficient c1 is themost importantparameter in theSalp swarmalgorithm(SSA)because it balances explorationandexploitation isdefinedas follows: c1=2e−( 4l L ) 2 (13) where l is the current iterationandL is themaximumnumberof iterations. Theparameter c2 and c3 are randomnumbersuniformlygenerated in the intervalof [0, 1]. In fact, theydictate if thenextpositionin jthdimensionshouldbetowardspositive infinityornegative infinity aswellas thestepsize. Definition3.Toupdate thepositionof the followers, the followingequations isutilizeddependingonNewton’s lawofmotion: xij= 1 2 aijt2+v0t (14) where i ≥ 2, xij shows the position of ith follower salp in jth dimension, t is time, v0 is the initial speed, andaij= vij−v0 t wherevij= xij−x0 t , i≥2, j≥1. Because the time inoptimization is iteration, thediscrepancybetween iterations is equal to 1, andconsideringv0 =0, thisequationcanbeexpressedas follows: xij(t) = 1 2 ( xij(t−1)+x i−1 j(t−1) ) (15) where i≥2andxij(t) showthepositionof ith followersalp in jthdimensionat t-th iteration. Accordingto themathematicalemulationexplainedabove, theswarmbehaviorofsalpchains canbesimulatedvividly. Whendealingwithmulti-objectiveproblems, thereare twoissues thatneedtobeadjusted for SSA.First,MOSSAneedtostoremultiplesolutionsas thebest solutions foramulti-objectiveproblem. Second, ineachiteration,SSAupdates thefoodsourcewiththebestsolution,but in themulti-objective problem,singlebest solutionsdoesnotexist. 295