Page - 142 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 2080 2.1.1.Auto-RegressiveModel The auto-regressive model is actually an auto-regressive model with errors that includes exogenous variables. Regressionmodelswith time series errors describe the behavior of a series byaccountingfor lineareffectsofexogenousvariables.However, theerrorsarenotconsideredwhite noisebuta timeseries. This typeofmodel isdescribed inEquation(1). yt= p ∑ i=1 ϕi·et−i+Xt·θ+εt (1) where, theoutputyt is expressedasa linearcombinationofpreviousknownerrors, et-i, exogenous variablesXtandarandomshock, εt. Thecoefficientsϕiandvectorθarecalculatedfromthetraining databyamaximumlikelihoodmethod. Theparameterpexpresses thenumberof lagsof theerror that are includedin themodel. 2.1.2.NeuralNetwork TheNeuralNetworkmodel uses a non-linear auto-regressive systemwith exogenous input. mathematicallyexpressed inEquation(2): yt= f ( yt−1,, . . . ,yt−ny,ut−1,, . . . ,ut−nu ) (2) where, theoutputvalueyt isanon-linearfunctionofnypreviousoutputsandnu inputs. Thisnon-linear function is, inourcase,a feedforwardneuralnetwork. Furtherdescriptionof thismodelcanbefound in [39]. Figure 1 showsavisualizationof this typeofnetworksworkingonline. Thefigure shows a feedforward neural networkwith 119 exogenous inputs and a feedback of 14 previous values, 10neurons in thehiddenlayerand1output. Therandomnatureof the trainingprocessof theNARXsystemsrequirescertainredundancyto estabilize theoutput. This isachievedbyusinganumberNNinparallel.Also, theabilityof theNNto capturenon-linearbehaviordependsonthesizeof thehiddenlayer. Bothof theseparametersaffect thecomputationalburden imposedon thesystem,which isoneof theconditionsunderwhich the modelsare tested. Figure1.Schematicviewof theNARXsystemasshownonaMatlabMathworksvisualization. 2.2. ParametersandForecastingConditions Theforecastingenginesdescribedabovehavebeentestedwithdifferentconfigurationparameters andexternalconditions todeterminehowtheyadapt todifferentsituations. Externalconditionsare historical loadavailability, temperature locationsavailabilityandresponse timeliness,which is related tocomputationalburden.Configurationparametersare temperature treatment, frequencyof training andnumberofauto-regressive lags. 142