MATHEMATICAL MODELS FOR NATURAL GAS …

1 CANADIANAPPLIEDMATHEMATICSQUARTERLYV olume17, Number 4, Winter , ,GEORGEF. is vitalfornaturalgasLocalDistributionCompa nies(LDCs)to forecasttheircustomers'naturalgasde-mand accurately. A signi cant errorona singleverycolddaycancostthecustomersof theLDCmillionsof looksat the nancialimplicationof forecastingnaturalgas,thenatureof naturalgasforecasting,thefactorsthatimpa ctnaturalgasconsumption,anddescribes a survey of mathemati-caltechniquesandpracticesusedt o of thetechniquesusedin thispaper currentlyareimple-mentedin a softwareGasDayTM, which is currentlyusedby 24 LDCsthroughouttheUnitedStates,forecastin gabout20% ,commercial, GasDay' IntroductionA naturalgasLocalDistributionCompany (LDC)

2 Facesmany challengesin thebusinessof supplyinggasto anLDCconsistsof gatestations,compres-sors,gasstorage, assuredeliveryof gasin adequatevolumesat e cient, economical,andsafeoperation,thedailygasd emandedby thecustomersmustbe knownin advancewitharelativelyhighdegreeof accuracy. Similarmodelsareusedto predictaggregatedailydemandof thecustomersof anLDCconsistsof many individualcustomers,each ,knownas heatingload,forheatingwater,drying,cooki ngandbaking,andotherprocesses,knownas baseload,andforelectricpower1 Director, :Neuralnetworks,regression,utility AppliedMathematicsInstitute,University of dependent onweather(mostimportantlytemperature)fac torsthata ,baseloadac-counts forotherfactorsthatarenotweatherdependen t andtendto beconstant, althoughit may changeover timewithgrowthin challengingtoforecastasit requiresforecastsof weatherfactors,andbaseloadis di cultto forecastbecauseit re-quiresknowledgeof dividedinto fourcategories:residential,commercial,in dustrial,andelectricpower generation[1].

3 Inthispaper,we discussjustresiden-tial,commercial, ersigni cantly. Theresidentialcustomerde-mandsaretypical lytemperaturesensitive withincreasingconsumptionon commonlyusedmeasurement fornaturalgasenergycon-sumptionis a decatherm,equalto onemillionBritishthermalunits[1].A typicalgas-heatedWisconsinresidenceconsu mesapproximatelyonedecathermof gasona coldwinterday. Commercialcustomerstendtobe temperaturesensitive be lesstemperaturesensitive , customersaresub-dividedinto two rmcustomerhasa servicecontractwhich anticipatesnoserviceinterruptions.

4 Andaninterruptiblecustomerhasa servicecontractthatallowstheLDCto donotdiscussor modelelectricpowergeneratorsbecauseelect ricpower generatedat oneendof thecountrycanbe transportedto theotherendof forecastingnaturalgasdemandforelectricpo wer generationafundamentallydi erent forecastingproblemoutsidethescope of stateutility commissionstosupplyuninter-ruptedgasserv icetotheir rmcustomersin a cost-e ective mannerduringa peakday, theday onwhich largeportionof naturalgasis consumedforspaceheating,naturalgasconsum ptionin many operationalareasis heavilyweather-dependent [46].

5 Thus,thepeakloadday is likelyto example,a largenaturalgasutility mayhave a heat-dependent loadof approximately10, ,000additionaldecathermsof naturalgasareconsumed(forheatingpurposes )foreachdegreeFahrenheitcolderit naturalgashasvariedin recent yearsfromapproximately$ approximately$ decatherm,whereasthespot market priceof gasona highdemandNATURALGASFORECASTING809day canbe 10timesthecontractprice[17].Thus,in thisexample,$400,000to $1,500,000of additionalcostis introducedforeach degreeFahrenheittheLDCoverestimatedthete mperatureduringextremecoldweatherconditi ons,assumingthegaswas purchasedonthespot marketat 10 timesthecostof market is passeddirectlyto airconditioningloadsandof non-temperaturedependent gasdemand(commonlyreferredto as baseloadgas)duringwarmweatherconditionsi s , many methods have beenusedto predictdailydemand[29, 32,33].

6 Gascontrollershave usedmethods such as lookingat usepatternsonsimilarhistoricaldays andscatterplotsof areappliedsuccessfullyonlyby expertswithyearsof experienceat gaspricescametheneedto of thelargerLDCshave theability tostoreorwithdrawal gasto cover theirforecasterror,themajority of LDCsdonothave storagecapabilities, LDC'shavedevelopedmathematicalformulasto predictgasdemandwithvaryingdegreesof usinghistoricaldemanddataandotherhistori caldataandinformation,such as weathercondi-tionsandday of Mathematicalmodelsto forecastdailydemandThemostcommonmathemat icalmodelingtechniquesusedto forecastdailyde-mandaremultiplelinearreg ressionandarti ypresents thesetwo methodsusedby GasDayTM, a fore-castingsoftwareapplicationlicensedt o 24 LDCsin present thefactorsusedby GasDayTMthata ectdailynaturalgasdemandanddataquality, respectively.

7 Thesesectionsfocusonthethingswe consideredwhenbuildingGasDayTM. Section5 presents ananalysisof theperformanceof GasDay' (MLR)[15, 18] is oneof themostcommonlyusedmethods forpredictionmod-els,andit hasbeenappliedto utility forecasting[19]. SupposeforNdays (1 k N), we have customerdemandSiandMindependentfactors,x k;j, for1 k Nand1 j Mwe thinkmay a ectSi. The810S. bSk= 0+mXj=1 jxk;j;whereeach jis a parameterthatspeci eshow theoutputis relatedto thej limited,however,by theassumptionof alinearrelationshipbetweentheinputfactor sandtheoutput(gasdemandin thiscase).

8 For thedailydemandmodel, 0may represent baseload, 1may represent theuseper heatingdegreeday factor,andxi;1mayrepresent HeatingDegreeDays, aneight dayforecastinghorizon,witha separateMLRforeach cialneuralnetworksArti cialNeuralNetworks(ANN)[36, 37,41] aremathematicalmodelswhich canapproximateany (non-linear)continuousfunctionarbitraril ywell [23,24]. TheANNacquiresknowledgethrougha trainingprocess[42]. Modelersof gasconsump-tionhave beenattractedto ANN'sbecauseof thiscapability of mappingunknownnonlinearrelationshipsbetw eeninputsandtheoutput[27].

9 Inparticular,thenonlinearpropertiesof theANNallow thedirectinputof temperature,windspeed,andpriorday temperaturesinto theANNnodeswithoutaccountingforinteracti onsandthenonlinearresponseoftheseimpacts [8, 9, 10]. Inaddition,thetrainingprocessbuildsaninp ut-outputrelationshipthatinterpolateswel l to a situationthatmaynotexactlymatch ,whileanANNis quitegood at interpolatinga solutionthatwasnotpresentedduringtrainin g,it is notasgood at extrapolatingoutsidethedomainof example,in thegasestimationproblem,thismeansthatif theANNmodelwas nottrainedwithhistoricaldatafromdays of extremeweather,themodelmay notperformwellonsuch asarti cialneuralnetworksor multiplelinearregressioncanreduceerrorsa risingfromfaulty assumptions,bias,or mistakes indata[21]

10 BatesandGranger[6] suggestthatcombiningseveralfore-caststog ethertendsto decreaseforecastingerrorbecausethecombin edforecasthasequalto or frequentlylessvariancethaneach of thecompo-nent forecasts,andDickinson[14] providesa mathematicalproof of [2] surveyed research oncombiningforecastsover thelast40 NATURALGASFORECASTING811years,concluding thatto obtainthebestcombinedforecastaccuracythe followingguidelinesshouldbe considered. usedi erent component forecastmethods, useat least ve component forecastswhenpossible, useequalweights unlessyou have strongevidenceto supportunequalweightingof forecasts, usetrimmedmeans, usedi erent data,and usethetrack recordanddomainknowledgeto anidealcaseforcombiningforecasts,be-caus etheforecasteris notalwayscertainwhich ,linearregressionextrapolatesbet-terthan ANNs,buttheANNsoftenperformbetterondays similartoonesin component MODELS (MLRandANN)