Page - 292 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1561 mixing, theEEMD[46],defines the“true”modesas theaverageof thecorrespondingIMFsobtained from an ensemble of the original signal plus different realizations of finite variancewhite noise. But incompletion of decomposition still exists, and thenumber ofmodeswill bedifferent due to thenoise added. Taking these short comes into account,CEEMDANisproposed. Thedetails are describedas follows: letEk(·)betheoperatorwhichproduces thekthmodeobtainedbyEMDandw(i) bearealizationofwhitenoisewithN (0, 1).AndthentheprocessofCEEMDANcanbeexpressedas several stages: 1ststep. For every i = 1,. . . , I decompose each x(i) = x+β0w(i) byEMD, until the firstmode is extractedandcompute d˜1 by: d˜1= 1 I I ∑ i=1 di1= d1 (1) 2ndstep.At thefirst stage (k=1)calculate thefirst residueas r1= x− d˜1. 3rdstep.Obtain the first mode of r1+ β1E1(wi) , i=1,. . . I, by EMD and define the second CEEMDANmodeas: d˜2= 1 I I ∑ i=1 E1(r1+β1E1(w(i))) (2) 4thstep.Fork=2,. . .K calculate thekthresidue: rk= r(k−1)− d˜k (3) 5thstep.Obtain thefirstmodeof rk+βkEk(w(i)) , i=1,. . . , I, byEMDuntil define the (k+1)th CEEMDANmodeas: d˜(k+1) = 1 I I ∑ i=1 E1(rk+βkEk(w(i))) (4) 6thstep.Goto4thstepfor thenextk. Iterate thesteps4 to6until theobtainedresiduecannotbe furtherdecomposedbyEMD,either because it satisfies IMFconditionsorbecause ithas less than three local extremums. Observe that, byconstructionofCEEMDAN,thefinal residuesatisfies: rK= x− K ∑ k=1 d˜k (5) withKbeingthe totalnumberofmodes. Therefore, thesignalof interestxcanbeexpressedas: x= K ∑ k=1 d˜k+rk (6) whichensures thecompletenesspropertyof theproposeddecompositionandthusprovidinganexact reconstructionof theoriginaldata. Thefinalnumberofmodes isdeterminedonlybythedataandthe stoppingcriterion. Thecoefficientsβk= εkstd(rk)allowtheselectionof theSNRateachstage. TheCEEMDANmethodcanaddalimitednumberofself-usewhitenoisesateachstage,whichcan achieve almost zero reconstruction error with fewer average times. Therefore, CEEMDAN can overcomethe“mode-mixing”phenomenonexisting inEMD,andcanalsosolve the incompletenessof EEMDdecompositionandreduce thecomputationalefficiencybyreducingthereconstructionerrorby increasingthenumberof integrations. 292