Page - 238 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1893 4.2.2. BeyondtheStationaryCase InthecasewhereZ isnotastationaryfunctionalprocess, someadaptations in thepredictor (6) must bemade to account for nonstationarity. InAntoniadis et al, (2012) corrections areproposed andtheirefficiency isstudiedfor twotypesofnon-stationarities: thepresenceofanevolutionof the meanlevelof theapproximationsof theseriesandtheexistenceofclassessegments. Letusnowbe moreprecise. It is convenient to express each curve Zi according to two terms Si(t) andDi(t) describing respectively theapproximationandthesumof thedetails, Zi(t)=∑ k c(i)j0,kφj0,k(t)+∑ j≥j0 ∑ k d(i)j,kψj,k(t) =Si(t)+Di(t). WhenthecurvesZm+1haveverydifferentaveragelevels, thefirstproblemappears. Inthiscase, it isuseful tocentre thecurvesbeforecalculatingthe(centred)prediction,andthenupdatetheforecast in thesecondphase. Then, the forecast for thesegmentn+1 is ̂Zn+1(t)= ̂Sn+1(t)+ ̂Dn+1(t). Since the functionalprocessDn+1(t) is centred,wecanuse thebasicmethodtoobtain itsprediction ̂Dn+1(t)= n−1∑ m=1 wm,nDn+1(t), (7) where theweightswm,n aregivenby(5). Then, to forecastSn+1(t)weuse ̂Sn+1(t)=Sn(t)+ n−1∑ m=1 wm,nΔ(Sn)(t). (8) Tosolve thesecondproblem,weincorporate the informationof thegroups in thepredictionstage byredefiningtheweightswm,n accordingto thebelongingof the functionsmandn to thesamegroup: w˜m,n= wm,n1{gr(m)=gr(n)} ∑nm=1wm,n1{gr(m)=gr(n)} , (9) where 1{gr(m)=gr(n)} is equal to 1 if the groups gr(n) of the n-th segment is equal to the groupof them-thsegmentandzeroelsewhere. If thegroupsareunknown, theycanbedeterminedfroman unsupervisedclassificationmethod. Theweightvector cangivean interesting insight into thepredictionpowercarriedoutby the shapeof thecurves. Figure6represents thecomputedweightsobtainedfor thepredictionofaday duringSpring2007. Whenplotted against time, it is clear that the onlydays found similar to the currentoneare locatedinaremarkablynarrowpositionofeachyear inthepast.Moreover, theweights seemtodecreasewith timegivingmorerelevance to thosedayscloser to thepredictionpast.Acloser lookat theweightvector (notshownhere) reveals thatonlydays inSpringareused. Pleasenote that no informationabout thepositionof theyearwasusedtocompute theweights.Only the information coded in the shapeof the curve isnecessary to locate the loadcurveat its effectiveposition inside theyear. 238