Page - 294 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1561 upperboundofPIdependingonthemulti-outputneuralnetwork. Thestructureweemployedin this paper is showninFigure1. Theoutputof thenormalLUBEstructure [50] just consistof twoneuronswhichdenote theupper boundand the lowerbound,while theoutputs inour structure ofLUBEconsist of threeneurons. Thefirstoutputcorrespondsto theupperboundof thePI, thesecondoutputdenotes thepredicted value,andthirdoutputapproximates the lowerboundof thePI. In the literature, thePIconstruction techniquesattempt toestimate themeanandvarianceof the targets forconstructionofPIs. Incontrast toexisting techniques, theproposedmethodtries todirectlyapproximateupperandlowerboundsof PIsbasedonthesetof inputs. Therefore, in the trainingprocess, loss functionof thisLUBEmethod basedonneuralnetworkshouldbesetaccordingto thekeycriterionofPIs (CPandPIW). 2.4.Multi-ObjectiveOptimizationAlgorithm The multi-objective optimization algorithm has been widely used to solve multi-objective optimization problem. In this paper, a novel multi-objective optimization algorithm named Multi-ObjectiveSalpSwarmAlgorithm(MOSSA) is introduced. 2.4.1.Multi-ObjectiveOptimizationProblem Inmulti-objectiveoptimization, all of theobjectives areoptimized simultaneously. Themain concern is formulatedas follows: Minimize :F(X)={f1(X), f2(X), . . . , fo(X)} (8) Subject to : gi(X) ≥0, i=1,2, . . . ,m (9) hi(X)=0, i=1,2, . . . ,p (10) lbi≤ xi≤ubi, i=1,2, . . . ,n (11) where o is thenumberof objectives,m is thenumberof inequality constraints, p is thenumberof equalityconstraints,and lbi is the lowerboundof the ithvariable,andubi is theupperboundof the ith variable. Withoneobjectivewe can confidently estimate that a solution is better thananother dependingoncomparingthesinglecriterion,while inamulti-objectiveproblem, there ismore than onecriteriontocomparesolutions. Themaintheorytocompare twosolutionsconsideringmultiple objectives iscalledParetooptimaldominanceasexplainedin [51]. Thereare twomainapproachesforsolvingmulti-objectiveproblems: aprioriandaposteriori [52]. In thepriorimethod, themulti-objectiveproblem is transformed to a single-objectiveproblemby aggregatingtheobjectiveswithasetofweightsdeterminedbyexperts. Themaindefectof thismethod is that theParetooptimal setandthe frontneedtobeconstructedbyre-running thealgorithmand changingtheweights [53].However, theaposteriorimethodkeeps themulti-objective formulation in thesolvingprocess,andtheParetooptimalset canbedeterminedinasingle run.Withoutanyweight tobedefinedbyexperts, thisapproachcanapproximateany typeofParetooptimal front. Because of theadvantagesofaposteriorioptimizationover theaprioriapproach, the focusofourresearch is aimedataposteriorimulti-objectiveoptimization. 2.4.2.Multi-ObjectiveSalpSwarmAlgorithm(MOSSA) As an a posteriori multi-objective optimization, MOSSA [54] is similar to some swarm multi-objective optimization algorithm such as MOPSO [31], MOACO [33] and MOGO [35]. Bysimulatingthebiologicalbehaviorofecological communities, theoptimalsolution isachieved. Salpsbelongto the familyofSalpidaeandhave transparentbarrel-shapedbody. Their tissuesare highlysimilar to jellyfishes. Theyalsomoveverysimilar to jellyfish, inwhichthewater ispumped throughbodyaspropulsiontomoveforward. Indeepoceans, salpsoften formaswarmcalledasalp chain. Themainconcernaboutsalps inMOSSAis their swarmingbehavior. 294