Page - 290 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1561 With thedevelopmentofartificial intelligence technology, the intervalpredictionmethodsbased onNNhave been proved to be efficient techniques. According to existing research, the popular techniques for constructing PIs are Bayesian [21], delta [22], bootstrap [23], andmean–variance estimation[24]. In the literature, theBayesian technique[25] isusedfor theconstructionofNN-based PIs. Errorbars areassigned to thepredictedNNvaluesusing theBayesian technique. Even if the theories are effective in the constructionofPIs, the calculationof theHessianmatrixwill result in the increaseofmodelcomplexityandcomputationcost. In [26], thedelta techniquewasappliedto construct PIs for STLF, anda simulated annealing (SA) algorithmwas introduced to improve the performanceofPIs throughtheminimizationofa loss function. In [27], accordingtobootstrap,error output, resampling,andmultilinearregression,wereusedwithSTLFfortheconstructionofconfidence intervalswithNNmodels. In [24],amean–varianceestimation-basedmethodusedNNtoestimate thecharacteristicsof theconditional targetdistribution.AdditiveGaussiannoisewithnon-constant variancewas thekeyassumptionof themethodforPIconstruction. Consideringmostof theexistingresearchstudiesofPIsbyNNmentionedabove, thePIswere usually calculateddependingon thepoint forecasting. TheNNswerefirst trainedbyminimizing an error-based cost function, and the PIs were then constructed depending on the outcomes of trained and tunedNNs. Itmay be questionable to construct PIs in thisway. Furthermore, it is amore reasonableway to output the upper and lower boundsdirectly [28]. Comparedwith the Bayesian,delta,andbootstraptechniques, thisapproachcanoutput thePIswithoutbeingdependent onpointprediction. However, in traditional researchapproaches, thecost functionmainlyaimsat guaranteeingcoverageprobability (CP).However,asatisfactorycoverageprobabilitycanbeachieved easily by assigning sufficiently large and small values to the upper and lower bounds of the PIs. Thus, theprediction intervalwidth(PIW) isanotherkeycharacteristicwhichneeds tobeconsidered fully. These two goals, that is, achieving a higherCP and a lower PIW, should be considered in acomprehensivemannerwhentheNNparametersaredetermined. Therefore, in this study,ahybrid, lowerupperboundestimation (LUBE)basedonmulti-objective optimization is proposed. The requirements for higher CP and lower PIW constitute a typical case of the Pareto optimization problem. In the present study, a significant and valid approach wasused to solve theParetooptimizationproblem is themulti-objectiveoptimization [29]. There are many algorithms in the literature for solving multi-objective optimizations. For the GA, themostwell-regardedmulti-objective algorithm is thenon-dominated sortinggenetic algorithm (NSGA) [30]. Other popular algorithms include themulti-objective particle swarmoptimization (MOPSO)[31,32],multi-objectiveantcolonyoptimization(MOACO)[33],multi-objectivedifferential evolution (MODE) [34], multi-objective grasshopper optimization (MOGO) [35], multi-objective evolution strategy (MOES) [36], multi-objective sine cosine (MOSC) [37], andmulti-objective ant lion[38].All thesealgorithmsareprovedtobeeffective in identifyingnon-dominatedsolutions for multi-objectiveproblems.Accordingto the“nofree lunchtheorem”foroptimization[39,40], there is no algorithmcapable of solving optimization algorithms for all types of problems. This theorem logicallyproves thisandproposesnewalgorithms,or improves thecurrentones. In this study, toachieveabetterperformance inSTLF,oneof thenovel recurrentneuralnetworks, the Elman neural network (ENN) [41], is applied to construct the structure of amodified LUBE. The Elman neural network has already been extensively used in time-series forecasting [42–44]. As a type of recurrent neural network, ENN exhibits superiority on the time delay information becauseof theexistenceof theundertaking layerwhichcanconnecthiddenNNlayersandstore the historical information in the trainingprocess. ThisstructuredesignofNNcommonly leads toabetter performance in time-series forecasting. In traditionalSTLF,mostof themethodsconstruct the trainingsetof themodeldirectlyusingthe originaldata.However,data in thenaturalworldoftenreceivesa lotofnoise interference,whichwill causemore difficulties for desired STLF. Furthermore, improving the signal-to-noise ratio of the trainingdatasetwillhelp theeffective trainingof themodel.Amongst theexistingdenoisingmethods, 290