Page - 240 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1893 5.1. ClusteringbyFeatureExtraction FromEquation(3),wecansee that theglobalenergyof thecurve isapproximatelydecomposed into energy components associatedwith the smooth approximationof the curve (c20) plus a set of componentsrelatedtoeachdetail level. In[33]thesedetail levelswerecalledtheabsolutecontributions ACj = ‖dj‖22, j = 0,. . . , J−1 of each scale to the global energy of the curve. Notice that the approximationpart isnotofprimary interest since theunderlyingprocessofelectricaldemandmaybe highlynonstationary.With this,wefocusonlyontheshapeof thecurvesandonits frequencycontent inorder tounveil thestructureofsimilar individualconsumers toconstructclusters.Anormalized version of absolute contributions can be considered,which is called relative contributions and is definedasRCj =ACj/∑jACj. After this representationstep, the result isdepictedby theschema inFigure2,wheretheoriginalcurvesarenowembeddedintoamultidimensionalspaceofdimension J. Moreover if relativecontributionsareused, thepointsare in thesimplexofRJ. Let us describe now the clustering stepmoreprecisely. For this any clustering algorithmon multivariatedata canbeused. Since the timecomplexityof this step isnotonlydependanton the sample sizeN but also on the number of variablesP, we choose to detect and remove irrelevant featuresusingavariableselectionalgorithmforunsupervised learning introducedin [34]. Besides the reduction of the computation time, feature selection allows also to gain in interpretability of the clusteringsince ithighlydependsonthedata. Theaimofthisfirstclusteringstepistoproducefirstacoarseclusteringwitharatherlargequantity K′ofaggregatedcustomers, eachof themcalledsuper-customer (SC).Thesynchronedemand—that is tosay, thesumofall individualdemandcurves inaspecificgroup—iscomputedtheninall clusters. Aparallel canbedrawnwith theprimarysituation:wenowobtainK′ coarselyaggregateddemands overP features, thatcanbe interpretedasadiscretenoisysamplingofacurve. Figure8showsthisfirst clusteringroundonthefirst rowof theschema. Figure8.Twostepclustering. 5.2. ClusteringUsingaDissimilarityMeasure The second clustering stage consists in grouping the SC into a small quantity K of (super-)aggregates,andbuildingahierarchyofpartitions—asseenbefore.Weconsider thesamples as functionalobjectsandthusdefineadissimilaritybetweencurves, toobtainadissimilaritymatrix betweentheSC.Theprojectionalternative (workingwithcoefficients)wasdiscardedbecauseof the lossof information. 240