Page - 301 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1561 wheremdenotes thenumberofsamples,andθi isabinary indexwhichmeasureswhether the target value iscoveredbyPIs: θi= { 1 , yti∈ [Lˆi,Uˆi] 0 , yti /∈ [Lˆi,Uˆi] (20) where yti denote the ith target value and Lˆi, Uˆi represent the ith lower bound and the upper bound, respectively. AlargerCPmeansmore targetsarecoveredbytheconstructedPIsandatoosmallCPindicates theunsatisfiedcoveragebehaviors. TohavevalidPIs,CPshouldbe largeror at least equal to the nominal confidence level of PIs. Furthermore, in this paper, CP is also an important factor in the processofparameteroptimizationbythemulti-objectiveoptimizationalgorithm. 4.2.2. PredictionInterval (PI)NormalizedAveragewidthandPINormalizedRoot-Mean-SquareWidth Inresearchstudiesonintervalprediction,moreattention isusuallypaidtoCP.However, if the lowerandupperboundsof thePIsareexpandedfromeither side, anyrequirement fora largerCP canbesatisfied,evenfor100%.However, insomecases,anarrower intervalwidth isnecessary for amoreprecisesupport forelectricpowersupply. Therefore, thewidthbetweenthe lowerandupper boundsshouldbecontrolledso that thePIsaremoreconvincing. In this study, theprediction interval width (PIW) isanother factor in theprocessofparameteroptimization.WithCPandPIW, twoobjects compose thesolutionspacewithinwhichtheParetosolutionset isestimated. Inorder to eliminate the impactofdimension, somerelative indexes shouldbe introduced to improve the comparability ofwidth indicators. Inspired by themean absolute percentage error (MAPE) inpoint forecasting,weemployedPInormalizedaveragewidth(PINAW)andPInormalized root-mean-squarewidth(PINRW)[50]: PINAW= 1 mR m ∑ i=1 (Ui−Li) (21) PINRW= 1 R √ 1 m m ∑ i=1 (Ui−Li)2 (22) whereRequals to themaximumminusminimumof the targetvalues.NormalizationbytherangeR isable to improvecomparabilityofPIsconstructedusingdifferentmethodsandfordifferent types ofdatasets. 4.2.3.AccumulatedWidthDeviation(AWD) Accumulatedwidthdeviation(AWD)isacriterionthatmeasuretherelativedeviationdegree,and it canbeobtainedbythecumulativesumofAWDi [55]. ThecalculationformulaofAWDisexpressed asEquations (23) and (24),whereαdenotes the intervalwidthcoefficient and Ii represents the i-th prediction interval. AWDi= ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ L(α)i −zi U(α)i −L (α) i ,zi<L (α) i 0, zi∈ I(α)i zi−U(α)i U(α)i −L (α) i ,zi>U (α) i (23) AWD(α) = 1 n n ∑ i=1 AWDαi (24) 301