Page - 244 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1893 Table1.Meanaveragerunningtimes (over5replicates) fordifferentsamplesizes (in log). SampleSize Time(InSeconds) 25×103 67 25×104 513 25×105 4420 25×106 43,893 The result of thisfirst clustering task is the loadcurvesof 200 super consumers (SC).Wenow explore howmuch time series contains the super consumers. For this,weplot (in Figure 10) the relative frequencyofeachSCcluster (i.e.,proportionofobservations inthecluster)against itssizerank (in logarithmicscale).With this, the leftmostpointof thecurverepresents the largest cluster,while the followingonesaresortedindecreasingorderofsize. Tocomparebetweensamplesizes fourcurvesare plotted,oneforeachsamplesize.Acommondecreasingtrendof thecurvesappearsproducingseveral relativelysmall clusters. This isnotadesiredbehaviour forafinalclustering task.However,weare in an intermediatestepwhichaimsat reducingthenumberofcurvesn toacertainnumberK′ofsuper customers,hereK′= 200. The isolatedsupercustomersmaymerge together in the followingstep, producingmeaningfulaggregates. 1XPEHU RI FOXVWHU [ A [ A [ A [ A Figure10. Relative frequencyofobservationsbycluster, indecreasingorder, fordifferentsamplesizes, n=103,104,105,106. Inwhat followswe focus on the results for the largest dataset, which is the onewith over 25millions of load curves. The resulting 200 super consumers are used to construct theWER dissimilaritymatrix,whichcontains rich informationabout the clustering structure. Onemayuse for instanceahierarchical clusteringalgorithmtoobtainahierarchyofSC.Agraphical resultof this structure in theobjectofFigure11,whichcorresponds to thedendrogramobtainedbyagglomerative hierarchical clusteringusingtheWardlinkfunction. Then,onemaygetapartitioningof theensemble ofSCbysettingsomethreshold (avalueofheight in thefigure).However,wewillnot followthis idea toconcentrateonthebottom-uppredictiontask. TheWERdissimilaritymatrixencodesrich informationabout thepairwiseclosenessbetweenthe 200superconsumers.Awaytovisualize thismatrix is toobtainamultidimensionalscaling, that is to construct a settingof lowdimensioncoordinates thatbest represent thedissimilaritiesbetween thecurves. Figure12containsthematrixscatterplotof thefirst4dimensionsofsuchasetting. Foreach 244