A Conditional Generative adversarial Network for energy use in multiple buildings using scarce data

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Citation for this paper:

Baasch, G., Rousseau, G., & Evins, R. (2021). A Conditional Generative adversarial

Network for energy use in multiple buildings using scarce data. Energy and AI, 5, 1-14.

https://doi.org/10.1016/j.egyai.2021.100087.

UVicSPACE: Research & Learning Repository

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A Conditional Generative adversarial Network for energy use in multiple buildings

using scarce data

Gaby Baasch, Guillaume Rousseau, Ralph Evins

September 2021

© 2021 Gaby Baasch et al. This is an open access article distributed under the terms of the

Creative Commons Attribution License.

https://creativecommons.org/licenses/by/4.0/

This article was originally published at:

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ContentslistsavailableatScienceDirect

Energy

and

AI

journalhomepage:www.elsevier.com/locate/egyai

Perspective

A

Conditional

Generative

adversarial

Network

for

energy

use

in

multiple

buildings

using

scarce

data

Gaby

Baasch

∗

_,

_Guillaume

_Rousseau

_,

_Ralph

_Evins

Energy and Cities Group, Department of Civil Engineering, University of Victoria, Canada

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i

g

h

l

i

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s

•CreatingmultivariatetimeserieswhereeachbuildingrepresentsavariablesigniﬁcantlyeasesstrictdatarequirementsforbuildingGANSwhilemaintaining per-buildingloadcharacteristics.

•ConditioningonmeanmonthlyoutdoortemperatureimprovesGANperformance.

•TheGANmodelledtheresidentialdatawithhighﬁdelitybutcouldnotcompletelycapturethecomplextemporalbehaviourinthecommercialdata.

•MetricsthataremostcommonlyusedtoevaluateGANsinthebuildingdomaindonotsuﬃcientlycapturetemporalbehaviour.

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t

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Article history:

Received 30 January 2021

Received in revised form 20 April 2021 Accepted 8 May 2021

Available online 15 May 2021 Keywords:

Generative adversarial network Building load proﬁle Machine learning Data scarcity

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Buildingconsumptiondataisintegraltonumerousapplicationsincludingretroﬁtanalysis,SmartGridintegration andoptimization,andloadforecasting.Still,duetotechnicallimitations,privacyconcernsandtheproprietary natureoftheindustry,usabledataisoftenunavailableforresearchanddevelopment.Generativeadversarial networks(GANs)-whichgeneratesyntheticinstancesthatresemblethosefromanoriginaltrainingdataset -havebeenproposedtohelpaddressthisissue.PreviousstudiesuseGANstogeneratebuildingsequencedata,but themodelsarenottypicallydesignedfortimeseriesproblems,theyoftenrequirerelativelylargeamountsofinput data(atleast20,000sequences)anditisunclearwhethertheycorrectlycapturethetemporalbehaviourofthe buildings.InthisworkweimplementaconditionaltemporalGANthataddressestheseissues,andweshowthat itexhibitsstate-of-the-artperformanceonsmalldatasets.22diﬀerentexperimentsthatvaryaccordingtotheir datainputsarebenchmarkedusingJensen-Shannondivergence(JSD)andpredictiveforecastingvalidationerror. Ofthese,thebestperformingisalsoevaluatedusingacuratedsetofmetricsthatextendsthoseofpreviouswork toincludePCA,deep-learningbasedforecastingandmeasurementsoftrendandseasonality.Twocasestudies areincluded:oneforresidentialandoneforcommercialbuildings.ThemodelachievesaJSDof0.012onthe formerdataand0.037onthelatter,usingonly396and156originalloadsequences,respectively.

1. Introduction

Theinformationrevolutioncarriesthepromise ofinnovationand high-impact industry disruption. Indeed, the potential for machine learningandbig-datatoassistbuildingdecarbonizationistremendous. TheSmartGridisoptimizedusingdetailedsupplyanddemand informa-tion[1],high-resolutionmetereddatasupportscity-wideretroﬁt strate-gies[2]andforecastingmodelstrainedonbig-data areusedfor en-ergymanagement[3].Infact,areviewbyHongetal.foundover9579 studiesonmachinelearninginbuildings.However,ofthe153studies thattheyselectedforin-depthanalysisnonehadbeenadoptedbroadly bythebuildingindustry[4].Thisislargelybecause,inpractice,

build-Abbreviations:GAN,GenerativeAdversarialNetwork;JSD,Jensen-ShannonDivergence;KLD,Kullback-LeiblerDivergence;PCA,PrincipleComponentAnalysis; MAE,MeanAbsoluteError;MAPE,MeanAbsolutePercentageError;RMSE,RootMeanSquaredError;DTW,DynamicTimeWarping;MMD,MaximumMean Discrepancy;PRD,PrecisionandRecallforDistributions;SSIM,StructuralSimilarity;TOVO,TrainedonOriginal,ValidatedonOriginal;TGVO,TrainedonGenerated, ValidatedonOriginal.

∗_{Corresponding}_author.

E-mailaddresses:gbaasch@uvic.ca(G.Baasch),guillaumer@uvic.ca(G.Rousseau),revins@uvic.ca(R.Evins).

ingdatacollectionispreventedbytechnical,regulatoryandeconomic challenges.Forexample,smartmetersthatmeasuretemporalelectricity consumptionarebecomingincreasinglypervasive(withover1billion devicesinstalledasof2020[5]),butconcernsoverprivacyprevent util-itycompaniesfromdisclosingthisinformation[6,7].

Thelackofbuildingdataavailabilityhaslead toakeenresearch interestinitsgeneration.Oneapproachtogenerativemodellingisto usedetailedphysicalmodelstosimulatetemporalbuildingbehaviour [6].Intheemergingﬁeldofurbanbuildingenergymodelling,for ex-ample,city-widebuildingconsumptionissimulatedbasedonasmaller subsetbuildingarchetypes[8].Theavailabilityofrepresentative build-ingreferencemodelsmakesthisapproachdesirable, butitissubject tomodellingassumptions,thereferencesmaynotcloselyrepresentthe

https://doi.org/10.1016/j.egyai.2021.100087

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Table1

SummaryoftheGANstudiesreviewedoverthecourseofthiswork.Themetrics col-umncontainsthekeyindicatorsthatwereusedtoevaluatetheresultsforthegiven study.Wedonotguaranteethatthereviewisexhaustive,butwebelievethisisstilla goodrepresentationofexistingwork.StudiesthatusedGANsinbuildingsfor diﬀer-entapplicationsthanloadproﬁlegeneration(suchas[19,20]and[21])areexcluded fromthistable.

# input temporal metrics sequences period generator? Data augmentation for forecasting

[3] forecasting: building 1: 280 daily no (2019) MAE, MAPE, correlation building 2: 81

[15] forecasting: MAPE, RMSE, (# unclear) unclear no (2019) DTW

[16] PCA, forecasting: 3 case studies unclear no (2020) MAPE, RMSE, MAE (# unclear)

[17] forecasting: MAPE, MAE, 2 case studies unclear yes (2020) (# unclear) R-GAN

General generation: use case unspeciﬁed (same as this work)

[18] MMD, clustering 36,500 daily no

(2018) forecasting: MAPE (type unclear)

[9] JSD, RMSE, auto-correlation, 33,760 weekly no (2019) mean and standard variance (residential)

[6] KLD of 5 key parameters, 56,957 daily no (2020) mean and standard deviation (commercial)

[10] JSD, RMSE, PRD, SSIM, 20,000 weekly no

(2020) mean and standard variance (residential)

actualbuildings,andthereisanestablishedperformancegapbetween simulatedandrealbuildings[6].Data-drivenmethodsofferadifferent approachthathelpstoovercome theselimitations,buttheytypically eitherrequiredetailedend-userdataortheyfailtomodelthefull diver-sity,accuracyandcomplexityoftheoriginal[9,10].Fortunately, gen-erativeadversarialnetworks(GANs)offerapromisingalternative.

InitiallyintroducedbyGoodfellowetal.in2014,aGANconsistsof twoneuralnetworksthatareincompetitionwithoneanother:a genera-torthatactsasacounterfeiterandadiscriminatorthatactsasadetective [11].Theyhavebeenappliedacrossvariousdomainstoproduce real-istichumanfaces[12],tocomposemusic[13]andtocreatepaintings [14]withalmostuncannyfidelity.InbuildingsGANsmayofferthe solu-tiontotheinformationshortagethatpreventsthewide-spreadadoption ofdata-drivenbuildingdecarbonizationtechniques,whilealsooffering privacyguarantees.

1.1. Literaturereview

Thebuildingsresearchcommunityhasacknowledgedthepotential ofGANsandseveralpapershavebeen published,butat thetimeof writingtherehasnotyetbeenanoverviewofthestateoftheresearch. Table1thereforeprovidesasuccinctsummaryofthekeycontributions ofthepreviousworksthatwerereviewedforthisstudy.Theworksin thetablecanbeclassiﬁedintotwocategories:thosethatfocuson fore-castingandthosethatareuse-caseagnostic.Thisworkfallsintothe lattercategorysothosewillbe thefocushere.Severalresearchgaps wereidentiﬁed.First,eventhoughallofthereviewedworksgenerated time-seriessequences,noneoftheuse-caseagnosticstudiesusedan un-derlyingnetworkarchitecturethatdirectlymodeledtemporal dynam-ics,andnoneevaluatedthetemporalcomponentsofthegenerateddata suchastrendandseasonality.Second,theuse-caseagnosticworksuse largepre-existingdatasetstotraintheirgenerativemodels,buttheseare notalwaysavailableinpractice.1_Finally,_most_works_only_test_their

ap-proachonasingledatasetsoitisunclearhowwelltheywillgeneralize, especiallybetweenresidentialandcommercialbuildings.

1 _This_highlights_a_more_general_circularity_problem_with_GANs:_as_with_other deeplearningtechniques,theirtrainingbeneﬁtsfromlargedata.Forexample, theinitialGANdevelopedbyGoodfellowetal.wastrainedontheMNISTdataset

[22]whichhas60,000trainingexamples.

1.2. Keycontributions

Inthisworkweaddresstheaforementionedissues.Weapplya re-centlydeveloped, timeseriesGAN(TimeGAN)thatachieves state-of-the-artperformancefortemporalgeneration[23],andaddanovel ex-tensionsothatitworksinaconditionalsetting.Wedemonstratethat, basedongenerateloaddistribution,theperformanceofourmodelis competitivewithexistingbuildingGANs,evenusingadatasizethatis only1–2%thesize,onbothresidentialandcommercialbuildings.We augmentthemetricsusedbypreviousstudiestoincludePCAand time-seriesspeciﬁcevaluationtoevaluateourresults,anddeterminethatthe standardmetricsthatarecurrentlyusedforevaluationinexistingworks missimportantshortcomingsofgenerativemodels.

The remainder of this paper is organized as follows: Section2presentsthetheoreticaloverviewofGANs(2.1)andTimeGAN (2.2); Section 3overviews the experimental methodology,including the GAN implementation(3.1), the residentialandcommercial case studies (3.2), the 22 diﬀerent modelling experiments (3.3) and the metrics(3.4);Section4presentstheresults(4.1forall22experiments, 4.2 for the residential case study and 4.3 for the commercial case study);Section5presentsthediscussion.

2. Conditionaltimeseriesgenerativeadversarialnetwork

TimeGAN,developedbyYoonetal.in[23],isalogicalextension oftheoriginalGANarchitecturebyGoodfellowetal.,sothissectionof thepaperwillstartbyoverviewingthelatter(Fig.1).Next,TimeGAN andC-TimeGAN(Fig.2)willbedescribed.Formoreinformationonthe originalTimeGANimplementation,thereaderisreferredto[23].2

2.1. OriginalGAN

A generic GANconsists of two neural networks:a generator (𝑔) andadiscriminator(𝑑),whohavelearnableparameters𝜃𝑔and𝜃𝑑,

re-spectively.𝑔 acceptsrandomnoisevectors𝑍∈,whereisavector

2_The_TimeGAN_formulation_presented_in_[23]_includes_both_static_and tempo-ralinputs,buttheimplementationprovidedbytheauthorsdoesnotyetinclude thestaticcomponent.ThisisoneofthereasonsthatweintroduceC-TimeGAN.

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Fig.1. ThearchitectureoftheGANdescribed in Section 2.1. The input values 𝑦 and ̂𝑦 make the GAN conditional, as described in

Section2.1.1.

Fig.2. ThearchitectureoftheTimeGAN de-scribedinSection2.1.Theinputvalues𝑦and ̂𝑦maketheGANconditional,asdescribedin

Section2.2.1.

spaceofknowndistributions(e.gGaussiandistributions),andmapsit tolearneddistribution ̂𝑝.Itsgoalistolearnadensity ̂𝑝(𝑋)thatbest approximatestheoriginaldistribution𝑝(𝑋),for𝑋∈.Thegenerated samplesareinputinto𝑑,which alsoacceptssamplesfromatraining dataset{𝑥_𝑛}𝑁_𝑖₌₁.𝑑’sobjectiveistooutputtheprobabilitythatexample 𝑥camefrom𝑝(𝑋),asopposedtô𝑝(𝑋),i.e.todistinguishwhichsamples ccmefromtheoriginaldataandwhichweregenerated.𝑔 and𝑑 playa two-playerminimaxgamewithvaluefunction𝑈:

min

𝜃𝑔

max

𝜃𝑑 𝑈

=𝔼_𝑥∼𝑝[log𝑑(𝑥)]+𝔼𝑧∼̂𝑝[log(1−𝑑(𝑔(𝑧)))] (1)

where𝑑 aimstomaximizetheprobabilityofassigningthecorrectlabel totheoriginalandgeneratedsamples[log𝑑(𝑥)],and𝑔 aimstominimize theprobabilitythat𝑑 correctlyclassiﬁessamples[log(1−𝑑(𝑔(𝑧)))]. 2.1.1. Conditionalextension

AconditionalextensiontotheoriginalGANformulationwas pro-posedin[24].𝑔 and𝑑 areconditionedonsome extrainformation𝑦, for𝑌 ∈ byaddinganadditionalinputlayertotheneuralnetworks, resultinginthefollowingextensiontoEq.1:

min 𝜃𝑔 max 𝜃𝑑 𝑈 =𝔼_𝑥∼𝑝[log𝑑(𝑥|𝑦)]+𝔼𝑧∼̂𝑝[log(1−𝑑(𝑔(𝑧|𝑦)))] (2) 2.2. TimeGAN

TimeGANconsistsofthesameadversarialcomponentandloss()

astheoriginalGANformulation,butwithtwomajorextensions:an

au-toencodingcomponentandasupervisedautoregressive(AR)learning objective.Thesearedescribedbelow.

AutoencodingComponent:Anautoencoderisadimensionality reduc-tiontechniquewhereembedding(𝑒)andrecovery(𝑟)functionslearn stochasticmappings𝑝_𝑒(ℎ|𝑥)and𝑝_𝑟(̃𝑥|ℎ)[25].ℎisaninstanceof𝐻∈, where isthelatentvectorspacecorrespondingto.3_Simply_put,_𝑒

transformsthedataintoitslatentcodeℎ,and𝑟undoesthe transforma-tion.InTimeGAN,𝑒and𝑟areneuralnetworksthataretrainedusingthe recoverylossfunction:

𝑅=𝔼𝑥1∶𝑇∼𝑝 [_∑ 𝑡 ||𝑥𝑡 −𝑟(𝑒(𝑥_𝑡))||2 ] (3) InTimeGAN,𝑔 and𝑑 operateinthelatentspaceasfollow:(1)𝑔 trans-formsrandomnoisevectorsintolatentcodesand𝑒transformstraining samplesintolatentcodes,(2)𝑑 classiﬁeswhetherthelatentcodescome fromthegeneratedororiginaldata,and(3)𝑟transformsdatafromthe latentspacebackintoitsoriginalform.

ARLearningObjective:Generatingtemporaldatacanbe adiﬃcult learningproblemforaGAN,especiallyiftheinputdataiscomprised oflongsequences.Tointroducetemporalrelationshipsdirectlyintothe learningarchitecture,TimeGANusesasupervisedlossfunctionbased

3_In_statistics,_latent_-_or_unobservable_-_variables_are_variables_(typically low-dimensional)thatretainimportantfeaturesoforiginal,multidimensionaldata

[26].ThemotivationbehindtheirinclusioninTimeGANisthattheyretain im-portanttemporaldynamicsthatareotherwiselostinGANtraining.

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Fig.3. ThearchitectureoftheTCN.

onARdecomposition,sothattheGANcanspeciﬁcallylearnthe condi-tionaltemporalprobabilities.4_For_simplicity,_this_supervised_loss_is_not

includedinFig.2,butitisusedtotrainboth𝑔 and𝑒. 2.2.1. Conditionalextension(C-TimeGAN)

InthisworkwesuggestanovelextensiontoTimeGANthatissimilar totheextensiontothegenericGAN, so𝑔 and𝑑 are conditionedon someadditionalinformation𝑦,andadversarial objectiveisthesame asinEq.2.Networks𝑒 and𝑟donot changebetween TimeGANand C-TimeGAN.

3. Methodology

3.1. GANImplementation

TwovariantsofC-TimeGANwereimplemented,asinglechanneland amultichannelmodel.Thesinglechannelmodelacceptsinputsofshape [672x1]i.e.asinglevariatetimeseries.Themultichannelmodelinstead acceptsinputsofshape[672x12],whereeachofthe12variablesisa distinctbuilding. Forsimplicity,thethis sectionwillonlyreport the networkshapesfromthemultichannel672timestepmodels.

𝑔,𝑑, 𝑒and𝑟areallimplementedin Tensorﬂow5 _using_Temporal

ConvolutionalNetworks(TCN)thatsharethesamearchitecture6_._Each

TCNconsistsof 4stacksof residualblocks,witheachresidualblock containing5hiddenlayersandeachlayerusinganincreasingnumber ofdilations.ThisarchitecturecanbeseeninFig.3.

Bothnon-conditionalandconditionaldatainputswereusedinthis work(seeSection3.3fordetails).Botharchitecturesaredescribedin Fig.4.Forthenon-conditionalcase𝑔,𝑑,𝑒and𝑟allfollow thesame structure.Samplesofshape[672x12]arepassedthroughtheinputlayer andintotheTCN.TheTCN’soutputisthenfedtoadenselayerto trans-formitintotheﬁnaloutput.𝑔,𝑒and𝑟allcontainasigmoidactivation functionintheiroutputlayerwhile𝑑 doesnot.

Fortheconditionalmodels,only𝑔,𝑑 aremodiﬁed.Asecondinput layerisaddedtoaccepttheconditionalinput,theconditionalinputis thenpassedintoadenselayertotransformsothatitcanbeconcatenated totheoriginalinput.ThisconcatenatedinputisthenpassedintotheTCN inthesameprocessdescribedbeforehand.

𝑔,𝑑,𝑒and𝑟arealltrainedusingAdamoptimizationandabatch sizeof128.Trainingissplitupinto3mainphases,embeddingtraining, supervisedtrainingandjointtraining.Duringtheembeddingtraining,𝑒 and𝑟aretrainedtogetherusingtherecoverylossfor50,000iterations. Forthesupervisedtraining,𝑔 istrainedexclusivelyusingthesupervised lossfor50,000iterations. Lastly, thejoint trainingphaseconsistsof

4 _An_AR_model_is_one_where_the_probability_of_observing_a_value_at_a_speciﬁc time𝑡isconditionalonvaluesfromtheprevious𝑡−1timesteps.

5 _{https://www.tensorﬂow.org/}

6 _The_TimeGAN_repository_is_available_at_{https://github.com/jsyoon0823/}

TimeGAN

training𝑔,𝑑,𝑒and𝑟usingtheirrespectivelossfunctionsforanother 50,000iterations

3.2. Residentialandcommercialdatasets

Twodatasetswereusedinthiswork:(1)aresidentialsmartmeter datasetand(2)asubsetofeducationandgovernmentbuildingsfrom theBuildingDataGenomeProject[27].Eachconsistsof12buildings withhourlyelectricload.Fortheformer,thestartingindexesarenot alignedbytimestep,andthe(hourly)weatherdatawasobtainedfrom theGovernmentofCanadahistoricalweatherservice,forthe VANCOU-VERHARBOURCSstationbetween2015and20197_._For_latter,_the

start-ingindexesarealignedbytimestamp,andtheweatherwasalready included.OtherfeaturesofthetwodatasetsarelistedinTable2and samplesequencesareplottedinFig.5.

3.2.1. Datapreprocessing

Forbothdatasets,eachbuildingwasindividuallystandardizedtothe range[0,1]usingMin-Maxnormalization

̂𝑥= 𝑥−𝑚𝑖𝑛(𝑥)

𝑚𝑎𝑥(𝑥)−𝑚𝑖𝑛(𝑥) (4)

where𝑥isthevectorofallloadvaluesfromoneofthebuildingsinthe dataset.Forthesinglechannelmodels,thedatasetswereinstead nor-malizedovereverybuilding,therefore𝑥wouldrepresentthevectorof allloadvaluesfromeverybuildinginthiscase.Missingvalueswere re-placedbyvaluessampledfromthatbuilding’sdistribution.Dataisthen brokenupintosmallersequencesof672or168timestepsdepending onthemodelbeingtrained.Tocreatemoredatafortraining,asliding windowwithalagof1timestepwasalsoappliedduringthesequencing step.

3.3. Datainputexperiments

Theresidentialdatasetwasusedtotestcaseforalargearrayof dif-ferentexperimentswhoseparametersaresummarizedinTable3.GANs haveahighcomputationalcostsoonly12CwithLandLMwerealso testedusingthecommercialdata.Manyoftheresidentialexperiments werenotabletogeneratedatathatrepresentedtheoriginal,buttosave thetimeofotherresearcherswebelievethatitisimportanttopublish negativeresults.Therefore,theresultsofallunsuccessfulexperiments willstillbepresented,8_but_only_the_best_performing_will_be_evaluated

inmoredetail.

The clustering(LC)case was includedbecauseallof [9,10], and [6]usedclustering.Itwasdoneusingak-meansmodeltrainedusing thesetofrealloadcurves.Adifferentk-meansmodelwastrainedfor boththe168and672timestepsets.Inordertofindtheoptimal num-berofclusters𝑘,twocommonclusteringmetricsareused.Thefirstis Davie-BoulinIndex(DBI),whichisusedtoquantifytheclusterscatter andseparation.ThesecondistheErrorSumofSquares(SSE),which quantifies thedifferencebetweensamplesin each individualcluster. Forbothsetsofloadcurvesthenumberofclusterstestedrangedfrom2 to18.Theoptimalnumberofclusterswas6and7forthe672andthe 168timestepsetrespectively

3.4. Metrics

Themetricsthatareusedinthisstudywerechosen(1)toprovidea numericalcomparisonwiththeresultsfrompreviousworks,and(2)to describethedistributionandtemporalbehaviouroftherealvs. gener-ateddata.JSDwaschosenasthekeynumericalmetricbecauseithas

7_{https://climate.weather.gc.ca/historical_data/search_historic_data_e.html} 8_For _brevity, _only _the _JSD _and _the _forecaster _errors _described _in

Section3.4.1willbereportedinthemainarticle.RefertoAppendixAforall thedistributionplots.

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Fig.4.Themodelarchitecturesforthe uncon-ditionalandconditionalmodel.Inthe condi-tionalcase,𝑒and𝑟stillusetheunconditional architecture.

Table2

Therelevantfeaturesofthetwodatasetsusedtogeneratestochasticloadproﬁles.Thenames specifywhethertheresidentialorcommercialdatasetwasused,aswellasthelengthofthe inputsequences.168isaweeklysequencesand672is28days.

Name Location Timespan Input Sequences Input Sequences Output Sequences (sliding window) (raw)

res 168 Vancouver 3 years 269,544 1608 1608

res 672 Vancouver 3 years 263,496 396 396

comm 672 London 1 year 97,056 156 156

Fig.5. Examplesequencesfromthatillustratethetypesoffeaturesfoundintheresidentialandcommercialdata.

beenusedincomprehensiveGANstudiesthatcoveramultitudeof dif-ferentmodellingapproaches(seeTable1).Forecastingvalidationerror isalsoappliedtoindicatethefidelityofthegeneratedsequences,and toofferasanitycheckonJSD.Ofthedistributionmetrics,normalized loadhistograms,standardvarianceandmeanloadarewellusedinthis domain,PCAwasimplementedbythecreatorsofTimeGAN,and,tothe bestofourknowledge,wearethefirsttointroducethestrengthofthe seasonalityandtrend.Allofthesemetricsaredescribedinthefollowing sections.

3.4.1. Numericalscores

JensenShannonDivergence(JSD):JSD wasappliedtoquantifythe diﬀerencebetweentheoriginalandgeneratedloaddistributions,using aprocesssimilartothatin[10].BeforetakingtheJSD,theoriginaland generateddatawere(1)logtransformed,(2)normalizedusingEq.4and (3)transformedintotheprobabilitydistributions𝑃𝑜 and𝑃𝑔. For(3),

eachofthedatasetsweredividedinto𝐾 segments(inthiswork𝐾= 100),sothattherangeofthe𝑘𝑡ℎintervalis[𝑘_𝐾−1,_𝐾𝑘]:

𝑃𝑜= [_𝑁 𝑜,1 𝑁 , 𝑁𝑜,2 𝑁 ,…, 𝑁𝑜 𝑁 ] 𝑃𝑔= [_𝑁 𝑔,1 𝑁 , 𝑁𝑔,2 𝑁 ,…, 𝑁𝑔 𝑁 ] (5)

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Fig.6. The twovalidation schemes (TOVO, andTGVO)usedtoevaluatethe time series forecaster.

Table3

Thenomenclatureforthedatainputsforthe ex-periments.

Channels

single channel 1C 12 channel 12C

Load + Conditional Input load only L load + cluster label LC load + mean outtemp. LM load + time stepped, hourly outtemp. LH load + month label LL load + cluster label + mean outtemp. LCM

where𝑁𝑜,𝑘and𝑁𝑔,𝑘arethenumberofloadvalueswithinthe𝑘𝑡ℎ

inter-valoftheoriginalorthegenerateddata.𝐽𝑆𝐷(𝑃𝑜||𝑃𝑔)wasthen calcu-latedbytakingthesquareoftheJSdistance.9,10

ForecastingValidationError(MAE): Timeseriesforecasting canbe usedtotestwhetherthegenerateddataisagoodrepresentationofthe original.Theoriginalandsyntheticdata-setsarebothbrokenupinto se-quencesof24timestepsinlengthtocreatetrainingandtestingdatafor theforecaster.Thegoaloftheforecasteristotakeoneofthesesequences asinputandoutputapredictionforthe25thtimestepofthatsequence. AsimplesinglelayerTemporalConvolutionalNetwork(TCN)wasused forallforecasters.eachforecasterwastrainedfor2000iterationswitha batchsizeof128andanAdamoptimizer.Toquantifyhowwellthe gen-erateddatarepresentedtheoriginal,twocross-validationprocessesare used,andthevalidationscoresbetweenthemarecompared(seeFig.6):

1. TrainedonOriginal,ValidatedonOriginal(TOVO) 2. TrainedonGenerated,ValidatedonOriginal(TGVO)

AdifferentTOVOforecasterwastrainedforthetwodatasets.ForTGVO, adifferent modelis trainedonevery experimentthatis specifiedin Table3.TheMeanAbsoluteError(MAE)isthevalidationmetric,and alowerscore signifyingbetterperformance.Thegoalistoachievea TGVOscorethatisasclosetotheTOVOscoreaspossible.

9 _{https://scipy.github.io/devdocs/generated/scipy.spatial.distance.}

jensenshannon.html.See[10]forthefullequation.

10 _The_equation_for_JSD_uses_logarithms,_and_the_bounds_on_the_results_change dependingonthebaseofthelogs.Forlogbase2itisboundedby0and1;for logbaseeitisboundedby0andln(2).Inbothcases,ascoreof0meansthatthe distributionswereidentical.Inthisworkweuselogbase2becauseitisintuitive tothinkofarangebetween0and1.

3.4.2. Distributions

Allofthefollowingdistributionswerecalculatedbetweenthe gen-erateddataandtheoriginalsequencesthatexistedbeforeapplyingthe slidingwindow.

1. NormalizedLoadHistograms:Histogramsofthelognormalizedloads areusedtoshow howdistributionsof theoriginal andgenerated datadiﬀer.11

2. PrincipalComponentAnalysis(PCA):PCAisusedtocompressdata inton-dimensional spacewhileretaining asmuchof theoriginal variabilityaspossible[28].Here,thelog normalizeddatainto 2 dimensions.Assuggestedby[23],thisprovidesaqualitative assess-mentofthediversityoftheoriginalvs.thegenerateddata. 3. StandardVarianceandMeanLoad:Themeanandthestandard

vari-ancearecalculatedforeverysequenceintheoriginalandgenerated datasetsandtheyareplottedagainsteachothertocomparetheir re-lationship.

4. StrengthofSeasonalityandTrend:Anytimeseriessequence𝑦𝑡 can

bedecomposed intoits seasonal(𝑆𝑡),trend(𝑇𝑡) andresidual(𝑅𝑡)

components.12 _The_{decomposition}_can_be_written _as_𝑦

𝑡=𝑇𝑡+𝑆𝑡+

𝑅𝑡.Thevariationoftheseasonalityandtrendcanthenbemeasured

relativetothevariationintheresidualsusing 𝐹𝑇 =𝑚𝑎𝑥 ( 0,1− 𝑣𝑎𝑟(𝑅𝑡) 𝑣𝑎𝑟(𝑇_𝑡+𝑅_𝑡) ) 𝐹𝑆=𝑚𝑎𝑥 ( 0,1− 𝑣𝑎𝑟(𝑅𝑡) 𝑣𝑎𝑟(𝑆_𝑡+𝑅_𝑡) ) (6) where𝐹_𝑇 and𝐹_𝑆 measurethestrength ofthetrendandseasonality, respectively.Forboth,therangeofpossiblevaluesliesbetween0and 1,where0isthelowestpossiblestrength[29].

Numerousmethodscanbeappliedtodecomposeatimeseriesintoits components.InthisworkSTLdecompositionisused[30].13_Compared

withothermethods,STLismorerobusttooutliersanditcan handle manydiﬀerenttypesofseasonality[29].Basedondomainknowledge aboutthebehaviourofbuildings,adailyseasonalityisspeciﬁedinthis work.

11_The_log_was_taken_before_{regularization}_so_that_the_histograms_look_more normalandareeasiertointerpret.

12_According_to_[29]_,_a_seasonal_pattern_occurs_when_the_time_series_is_aﬀected byseasonalfactorssuchasthedayoftheweekorthehouroftheday,atrend patternoccurswhenthereisalong-termincreaseordecreaseinthedata,and theresidualcomponentiswhateverisleftover.

13_We _use _the _STL _{implementation} _from _the _statsmodels _Python _library:

https://www.statsmodels.org/stable/examples/notebooks/generated/stl_ decomposition.html

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Table4

JSdivergencesforallexperiments(lowerisbetter).Foreachdatasetthe bestperformingcasehighlightedinbold.RefertoTable3forthe nomen-clature. L LC LM LH LL LCM res 168 1C 0.309 0.424 0.995 0.491 0.323 0.314 12C 0.077 - 0.099 0.145 0.201 - res 672 1C 0.953 0.408 0.342 0.568 0.507 0.361 12C 0.15 - 0.012 0.172 0.133 - comm 672 12C - - 0.037 - - - Table5

Thevalidationerrors(MAE)forthetrainedforecasters.RefertoTable3and

Section3.4.1,ForecastingValidationError,forthenomenclature.Thebest per-formingcases(i.e.,theTGVOscoresthatareclosesttotheTOVOscores,andthe TGVOscorethatarethelowest)foreachtrainingsetarehighlightedinbold.

TGVO TOVO L LC LM LH LL LCM res 168 1C 0.095 0.072 0.119 0.081 0.154 0.096 0.042 12C 0.051 - 0.056 0.078 0.081 - res 672 1C 0.116 0.101 0.118 0.105 0.079 0.090 12C 0.063 - 0.046 0.059 0.052 - comm 672 12C - - 0.166 - - - 0.041 4. Results

Theresultsforall22experimentsarepresentedintheSection4.1. Thesectionsthatpresenttheresultsfortheresidentialandcommercial casestudies(4.2and4.3)willfocusonlyonthebestperformingmodel (i.e.the12Cmonthlymodelconditionedonmeantemperature). 4.1. Allexperiments

4.1.1. JSDivergence

Table4displaystheJSDsforalloftheconductedexperiments.The highJSDsforRES168_1C_LMandRES672_1C_Lindicatesthatthemodels

werenotabletoconvergeduringtraining.Thesewillnotbeconsidered intherestofthispaper.

Asidefromthenon-convergentcases,theJSDsontheresidentialdata hadawiderangeof[0.012,0.568],andthe12Cmodeloutperformed the1Cmodelforeverydatainputcase.Overall,the1Cbaselines(i.e. L,LC) werenotabletogeneratesequencesthatrepresentedthosein theoriginaldataset.Inmanycases,the168modeloutperformedthe 672model,butthelowestJSDwasfromthelatter.ForRES168,thecase

withnoconditioninghadlowestJSDforboththe1Cand12Ccase.For 12C,conditioningonmonthlylabelalsoproducedarelativelylowJSD. RES672,conditioningonmeanoutdoortemperature(LM)signiﬁcantly

outperformedtheotherexperiments,withaJSDof 0.012,compared withthenextlowestof0.078.

4.1.2. Forecasting

The cross-validation results for TOVO and TGVO deﬁned in Section3.4.1arepresentedinTable5.RecallthatifaGANgenerates high-ﬁdelitysequencesthanthevalidationerror ontheoriginaldata shouldbesimilarregardlessofwhethertheforecasterswastrainedon generatedororiginaldata.Ontheotherhand,iftheGANgenerates se-quencesthatdonotrepresenttheoriginals,thentheforecasterthatis trainedontheformerwillnotperformwellwhenvalidatedonthelatter. ThereisonlyoneTOVOscorefortheresidentialdatabecausethe fore-castersweretrainedwithsequencesoflengthtodo,sowhetherlength oftheinputsequenceswasirrelevant.

ThemodelsthathadthelowestJSDsalsohadthelowestforecasting errorsand,aswiththeJSDsthe12Cmodeloutperformedthe1Cmodel foreverytypeofdatainput.Thisshowsthatthereisconsistencybetween thetwometricsandfurtherimpliesthatthe672 step,multi-channel

modelconditionedonmeanmonthlyoutdoortemperaturewasthebest performing. Thismodelwillthereforebepresented inmoredetailin thefollowingsections.Thedistributionsforalloftheotherresidential experimentsareavailableinAppendixA.14

4.2. Casestudy1:Residentialdata 4.2.1. Residentialdistributions

Fig.7illustratesthat,inadditiontoachievingalowJSDandTGVO error,RES672_12C_LMwasabletomodelotherimportantattributesin

thetimeseriesdata.Thedistributionsalongthexandy-axisinFig.7b showthattheamountofvariabilitycapturedbytheprinciple compo-nentsissimilarbetweentheoriginalandthegenerateddata. Qualita-tively, themostnoticeablediﬀerencewasthat theoriginaldatawas more spreadoutalongcomponent1thanthegenerated data,which hadahigherpeakoflowervalues.Thisindicatesthatthegenerateddata containedlessmorediversitythantheoriginal,whichisnotsurprising. Component1hadameanandstandarddeviationof0.0(±2.8)forthe originaland-0.2(±2.75)forthegenerated;component2hadamean andstandarddeviationof0.0(±0.92)fortheoriginaland-0.1(±0.73) forthegenerated.

Themodel’sabilitytocapturetherelationshipbetweenthestandard variancesandthemeansofthesequences(Fig.7c)wasquite impres-sive,sinceitwasabletomodelclustersofoutliers,butthegenerated sequencestendedtohavelowervariancethantheoriginal.This isa similarobservationtothatwhichwasdiscoveredfromthePCA.

Thedistributionsof the𝐹𝑆 and𝐹𝑇 (Fig.7d)exhibited reasonable

overlapbetweentheoriginalandgenerateddata,thoughlesssothan theothertypesofdistributionsdiscussedabove.For𝐹𝑆,themeanand

standarddeviationwere0.22(±0.12)and0.35(±0.17)fortheoriginal andgenerateddata,respectively. TheGANtendedtomodelstronger seasonalitythanwhatwaspresentintheoriginaldata.𝐹𝑇 hadamean

andstandarddeviationof0.07(±0.09)fortheoriginaland0.15(±0.19) forthegenerateddata;boththemeanandvariabilityin𝐹𝑇 iswas

no-tablehigherforthelatter(byabout10%oftherangeofpossible𝐹𝑇s).

4.2.2. Residentialforecasting

Theforecastingmodelthatwastrainedandvalidatedontheoriginal residentialdata(TOVO)achievedaMAEof0.042.Theforecasterwas trainedonthenormalizeddata,sothetotalrangeofvalueswas[0,1]. Relativetothis,anMAEof0.042islow(4.2%ofthetotalrange).The MAEoftheforecasterthatwastrainedonthegeneratedandvalidated ontheoriginal(TGVO)was0.046.Relativetotherangeofvaluesinthe data,theTGVOvalidationonlyincreasedby0.4%whencomparedto TOVO.Thisshowsthatforecasterthatwastrainedgenerateddatawas usefulforpredictingontheoriginal.

4.2.3. Residentialexamples

Fig.8displaystwogeneratedandtwooriginalsequences.Thereisno inherentrelationshipbetweenthechosenexamples;theywereselected tohighlightsomeoftheinterestingfeaturesintheoriginaldatathatthe generatorwasabletocapture.

4.3. Casestudy2:Commercialdata 4.3.1. Commercialdistributions

Fig.9displaysthedistributionplotsforthecommercialdata. Vi-sualizingthelognormalizedloadhistogramshighlightsthediﬀerences betweentheoriginalandgenerateddatathatcannotbeinferredfrom theJSDalone.Herewecanseethattheoriginaldatahadcertainbins withhighercountsthantheoriginaldata(thehighestcountinthe orig-inalwasalmost1000times higherthanthegenerated),andthatthe generateddatahadashapethatlookedwasbi-modalthantheoriginal.

14_Over_the_course_of_the_research,_these_were_also_evaluated_to_determine_that the672step,12C_LMmodelwasindeedthebestperforming.

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Fig.7. Thedistributionresultsontheresidentialdata.

Fig.8.Examplesequencesfromtheoriginalandgeneratedresidentialdata.

The mean and standard deviations from PCA (Fig. 9b) were 0.0(±2.71)ontheoriginaland0.07(±1.44)onthegenerateddatafor component1;and0.0(±1.66)and3.08(±1.5)forcomponent2.The dif-ferenceinthemeansoftheoriginalandgenerateddistributionsfor com-ponent2isnotable:thegenerateddataappearstohavemoresamples withhigherdiversity.

Thebehaviourofthecommercialgeneratorintermsofthevariances vs.meanswassimilartothatoftheresidential,inthatclustersof build-ingswithaggregatestatisticswereappropriatelymodelled.Theoverlap ofthedistributionson thexandy-axesiftheFig.9cshow thatthe shapeswerealmostperfectlymodelled.Forthetimeseriescomponents (Fig.9d),however,thecommercialmodeldidnotperformwell. Specif-ically,itwasunabletocapturethestrengthofthetrendintheoriginal data.

4.3.2. Commercialforecasting

FromTable5,theTOVOscorewas0.041buttheTGVOscorewas 0.166.TheTOVOscorewassimilartotheresidentialdata,buttheTGVO

scorewasworsethanalloftheTGVOmodels,includingtheonesthat didconverge.Thisisapoorperformanceprovidesfurtherevidencethat eventhoughtheJSDandmeanandstandarddeviationappeared rea-sonable,thegeneratedcommercialdatalikelylacksﬁdelity,anditis notusefulforforecasting.

4.3.3. Commercialexamples

Asdemonstratedbythesamplesequencesin Fig.5, the commer-cialdataexhibitedadifferenttypeofseasonalitythantheresidential data.Specifically,thecommercialbuildingsoftenhadaweekly season-alitywithhighusageontheweekdaysandlowusageontheweekends. Thisisanexpectedpatternincommercialdata.Fig.10showsthat,even thoughthegeneratorisabletoproduceweekdaysequenceswithhigh fi-delity,itisnotabletomodeltheweeklyseasonality.Thislikelyexplains whythedistributionmetricslookedgoodbuttheforecastingvalidation error waslow.Thiscouldalsohelptoexplainwhytheoriginaldata inthedistributionplotshadsomevalueswithmuchhighercounts:the

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Fig.9. Thedistributionresultsonthecommercialdata.

Fig.10. Examplesequencesfromtheoriginalandgeneratedcommercialdata.

lowerusagevaluesontheweekendsarenotpresentinthegenerated sequences.

4.4. Per-buildingdistributions

WhileFigs.7aand9aplottheregularizedloaddistributionsacross theentiredatasets for theresidentialandcommercialcases, Fig.11 showsthedistributionsforeachbuildingindividually,intheformof boxplots.Theseillustratethatthemodelseachindividualbuilding,even forthecommercialcasewherethebuildingshavelargediﬀerences(note that,forvisibility,theloadswereloggedin11b).

5. Discussion

Inthissectionwewilldiscusstheperformanceofourmodelin rela-tiontootherworksusingmetricsthatarestandardinthisﬁeldofstudy (seeTable1).This approachtocomparisonprovidesvaluableinsight intotherelativeperformanceofourapproach,butitdoesnotaccount fortheuseof adiﬀerentGANarchitectureanddatasetthanprevious studies.Weacknowledgethatamoredirectanalysismustbeconducted totrulyestablishtheperformance.Thisleadstoageneralobservation aboutthisdomain;thereisastrongopportunitytointroduceandenforce

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Fig.11. Per-buildingdistributions.

standardpracticesthatwillhelptoacceleratetheresearch.Futurework shouldcompareandbenchmarkdiﬀerentapproachesmorerobustly, in-cludingdiﬀerentapproachestoconditioning.

5.1. Competitiveperformancewithlessdata

Thisresearchsuccessfullydevelopedaloadsequencegeneratorthat iscompetitivewithexistingworks,butthatcanbeappliedfor signif-icantlysmallerdatasets.Unliketheotheruse-caseagnosticstudiesin Table1wedemonstrateourapproachforbothresidentialand commer-cialdata.Ourbestperformingmodel(i.e.the672step,multi-channel model,conditionedonmeanmonthlytemperature)achievedaJSDof 0.012ontheresidentialcasestudyand0.037onthecommercialcase study, using only396 and 156 originalinput sequences (before ap-plying thesliding window). Usinglog e insteadof log 2, these val-uesare0.008and0.024.Thebestperformingarchitecture(ACGAN) intheworkbyWangetal.,whichisthemostcomprehensivestudyon buildingGANstodate,achievedaJSDof0.0463using20,000weekly sequences[10]. Bothourresidentialandcommercialmodels outper-formed[10]usinglessthat2%ofthedata.Guetal.alsousedACGAN andachievedaJSDof0.0045using33,760sequences(loge).Our res-identialresult(0.008)wasclosetotheirs,usingonly1%ofthedata. Ourcommercialresultwasslightlyworse,butourdatasetwas0.5%the size.15

InadditiontoJSD,[9]and[10]alsoplotthedistributionsofthe realandgeneratedmeansandstandardvariances.Thereisno numer-icalmetricthatsummarizesthese,butbasedon visualinspectionwe concludethatourapproachperformsjustaswellorbetter.Thereader isinvitedtocomparetheseplotsbetweenthepapers.Wedonot com-pareourresultswith[18]and[6]becausetheyuse dailysequences, whichisamucheasiergenerationproblem.16

Itisworthnotingthatthecomputationalresourcesrequiredtorun GANsareveryhigh,sohyperparameterexplorationformany experi-mentswasunfeasible.Itis possiblethat ourresultscouldhave been improvedhadmoreoptimalhyperparametersbeenselected,however,

15 _We_had_to_make_several_assumptions_in_this_comparison,_since_neither_of thepapersreportthelogbaseorKvalueinEq.5.Weassumethatbothpapers useK=100tomatchourwork.For[10]weassumetheyuselog2.Sincewe uselogbase2wearepotentiallygivingtheirresultsanadvantageoverours, becausetheJSDforlogereturnssmallerJSDingeneral.Iftheyusedlogbase 10thecomparisonisfair,butiftheyusedlogetheirJSDswillappearsmaller. For[9]weﬁrstsquare theirresultsbecausetheyreportJSdistance,notJS divergence.Thisresultsinavaluethatis2ordersofmagnitudelowerthan[10], eventhoughtheyareusingthesamemodelarchitectureanddataset.Therefore, weassumethattheyareusingloge.

16 _Daily_sequences_are_easier_to_generate,_but_they_are_not_as_useful.

theoriginalTimeGANpaper[23]suggeststhatitisfairlystable,sowe donotexpectthistosigniﬁcantlyimpacttheresults.

5.2. Beneﬁtofmultichannelandconditionalformulations

Fromtheanalysisabove,wecanconcludethatourmodelling ap-proachwasabletoreducedatarequirementswithoutdepreciationin performance.Inthissectionweaimtodescribethekeycontributionsof themodellingapproachthatallowedforthisresult.

Thepoorperformanceofthe1Cmodels(Table4)showsthatour ap-proachdidnotacheivelowJSDbecauseofsmalldata,butratherinspite ofit.Theprimaryinsightthatledtotheimprovedperformancewasthe creationofamultivariatetimeseries,inwhicheachvariablerepresents asinglebuilding.Tables4and5showthat,ontheresidentialdata,the 12Cinputdatashapesignificantlyimprovedmodelperformancefor ev-eryinputcase.Intuitively,treatingasetofbuildingsinthiswayhelps theGANtofindtemporalassociationsbetweenthebuildingsbyidentify thedifferencebetweendefinitepatternsandrandomness.Additionally, themulti-channelapproachcircumventstheneedforclustering,which dependsontheuserinterventiontodeterminetheoptimalamountof clusters,andwhichmayhavedifferentdegreesofsuccessondifferent datasets.17

Anothernovelinsightfromthisworkistheconditioningonmean outdoortemperature.Intheresidentialmonthly,multichannelmodel, thisconditioningresultedinalowerJSDandvalidationerrorthanall othercases.Noneoftheotherconditioningcasesoutperformedtheload onlybaseline,whichshowsthattheperformancegainsfromusingthe weatherarenoteworthy,andnotnecessarilyeasytoreplicate. Interest-ingly,however,thebuildingsintheoriginalresidentialdatahadstrong monthlycorrelationswithoutdoortemperaturethatwerenotretained inthegenerateddata,despitetheconditioning.Futureworkshould con-siderinvestigatingwhythesecorrelationswerelost.

5.3. Issueswithtemporalcorrelations

Forthecommercialdata,theJSDandmeanandstandarddeviation plotswerecompetitivewithotherworks,butthetrendwasnot mod-eled,theforecastingvalidationerrorwashigh,andthedistributionsof theprinciplecomponentshadnotablediﬀerencesbetweentherealand generateddata.Fig.10providesanexplanationforthisbehaviour:the dailysequencesinthedatawerecapturedwithhighﬁdelity,butthe weeklypatternswerenot.WetestedthisfurtherbytakingthePearson correlationofthedailymeanloadsbetweenallthebuildingsinthe orig-inaldataset.Insomecases,forinstancebuildingsthathadhighweekday andlowweekendusage,therewasastrongcorrelationthatwasnot re-tainedinthegenerateddata.ThisshowsthattheGANisabletocapture

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dailybutnotweeklyseasonality,whichshouldbeaddressedinfuture work.Further,thisprovides strongevidencethatthestandardsetof metricsusedforevaluationisnotsuﬃcient.

5.4. Generatingdataformultiplebuildings

Fig.11showthattherelativemagnitudesandspreadoftheloads weremaintainedforthebuildings,evenforthecommercialdatawhere thereisalargediﬀerenceinscale.Theabilitytoretainthese distribu-tionsispromisingforpracticalapplicationssuchasstorageandenergy systemdesign.

5.5. Limitationsandfuturework

Alimitationofourapproachisthatitgeneratessetsofsequences, butthereisnowaytoknowwhattimesthesequencesarefor,andsince weusedarollingwindowtocreatemoredataforthetrainingprocess, wedonotknowthestarttimeofanygeneratedsample.Thisalsomeant thatwecouldnotimplementRMSEasametric,sincethedailypeaks inthegenerateddatahappenedatrandomtimestheaggregated them-selvesout.Anattempttoovercomethisissuewastoconditiononhourly temperature(LH)andtousethatasaproxyfortime,buttheresultsdid notexhibitstrongperformance.Futureworkshouldexploreotherways toovercomethisshortcoming.

Importantly,themetricsusedtoevaluatebuildingloadGANsrequire morediscussion.Thisisparticularlyapparentbasedontheevaluation ofthebehaviouroncommercialdata.InthispaperweintroducePCA andseasonalandtrendstrength.Ouranalysisofthesewasstill qualita-tiveandshouldbequantified,buttheresultsshowedthattheyprovide fundamentallyvaluableinformationthatisotherwisemissed[10].also usedPRDandSSIMasscores,buttheseareimage-specificsotheycannot beappliedhere[18].suggesttheuseofMMD,which,likeJSD, quan-tifiesthedifferencebetweentwoprobabilitydistributions[6].usethe KLDofthe5keyparametersforanalyzingelectricloadshapesuggested by[31].Thesewerenotappliedinthisworkbecausetheymeasuredaily

loadproﬁlesanditislessclearhowtheyshouldbeappliedforsequences withweekly ormonthlyseasonality.Boththese andMMD shouldbe exploredfurtherin futurework;astudydedicatedtotheanalysisof diﬀerentmetricsisrequired.

6. Conclusions

GANsarereceivingincreasingattentionforgeneratingbuildingload sequences,buttheyoftendependonlargeamountsofdataarenot al-waysavailableandthetemporalnature ofthegenerateddata isnot quantitativelyevaluated.Thisstudydevelopedanapproachthatuses substantially smaller datasets thanthose of previous works and ex-pandedthemetricsusedforanalysis.Itwasfoundthatamulti-channel TimeGAN,conditionedonmeanmonthlyoutdoortemperature gener-atesloadsequencesthatareapproximately1monthinlengthwithhigh ﬁdelityontheresidentialcasestudy,butthatthecommercialcasehad issuescapturingweeklyseasonality.Inbothcases,thenumericalresults arecompetitivewithotherGANsinthedomain,evenusingonly1–2%of thedata.Usingamultivariatetimeserieswhereeachbuildingrepresents aneuralnetworkinputchannelandconditioningonoutdoor tempera-turearetwonovelinsightsthatleadtostronggenerativeperformance ondata.Thesearekeyinsightsthatprovideimperativeinformation to-wardsreducingdatascarcityinthebuildingsdomain.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingﬁnancial interestsorpersonalrelationshipsthatcouldhaveappearedtoinﬂuence theworkreportedinthispaper.

Acknowledgements

ThisprojectwasfundedbyaCanariegrant.ComputeCanada pro-videdthecloudresourcesusedtoruntrainthenetworks.Theleadauthor wasfundedviaanNSERCBritishColumbiaGraduateScholarship.

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