Multi-objective optimization of a building envelope for thermal performance using genetic algorithms and artificial neural network

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artificial neural network

Didier Gossard, Bérangère Lartigues, Françoise Thellier

To cite this version:

Didier Gossard, Bérangère Lartigues, Françoise Thellier. Multi-objective optimization of a building

envelope for thermal performance using genetic algorithms and artificial neural network. Energy and

Buildings, Elsevier, 2013, 67, pp.253-260. �10.1016/j.enbuild.2013.08.026�. �hal-02175513�

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ContentslistsavailableatScienceDirect

Energy

and

Buildings

jo u r n al h om ep a g e :w w w . e l s e v i e r . c o m / l o c a t e / e n b u i l d

Multi-objective

optimization

of

a

building

envelope

for

thermal

performance

using

genetic

algorithms

and

artificial

neural

network

D. Gossard,

B. Lartigue

∗

_,

_F.

_Thellier

UniversitéToulouseIII–PaulSabatier,LaboratoirePHASE,118,routedeNarbonne,31062Toulousecedex9,France

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received16May2013

Receivedinrevisedform31July2013 Accepted18August2013 Keywords: Multi-objectiveoptimization Buildingenvelope Energyperformance Comfortdegree ANN Geneticalgorithm

a

b

s

t

r

a

c

t

Theobjectiveofthispaperistopresentamethodtooptimizetheequivalentthermophysicalproperties

oftheexternalwalls(thermalconductivitykwallandvolumetricspecificheat(c)wall)ofadwellingin

ordertoimproveitsthermalefficiency.Classicaloptimizationinvolvesseveraldynamicyearlythermal

simulations,whicharecommonlyquitetimeconsuming.Toreducethecomputationalrequirements,

wehaveadoptedamethodologythatcouplesanartificialneuralnetworkandthegeneticalgorithm

NSGA-II.ThisoptimizationtechniquehasbeenappliedtoadwellingfortwoFrenchclimates,Nancy

(continental)andNice(Mediterranean).Wehavechosentocharacterizetheenergyperformanceofthe

dwellingwithtwocriteria,whicharetheoptimizationtargets:theannualenergyconsumptionQTOTand

thesummercomfortdegreeIsum.First,usingadesignofexperiments,wehavequantifiedandanalyzed

theimpactofthevariableskwalland(c)wallontheobjectivesQTOTandIsum,dependingontheclimate.

Then,theoptimalParetofrontsobtainedfromtheoptimizationarepresentedandanalyzed.Theoptimal

solutionsarecomparedtothosefrommono-objectiveoptimizationbyusinganaggregativemethodand

aconstraintprobleminGenOpt.Thecomparisonclearlyshowstheimportanceofperforming

multi-objectiveoptimization.

1. Introduction

Consideringthepresentenergybalance[1],buildingsdesignhas tointegratethermalperformance.Thisnotiontakesintoaccount energysavingsandcomfortoftheoccupants,asthereductionof buildingenergyconsumptioncannotbeachievedattheexpenseof theindoorenvironmentquality(IEQ).Improvingthethermal per-formanceofabuildingcanbedoneintwoways.Thefirstapproach isbasedonatrial-and-errormethod.Asetofvariables correspond-ingtodesignbuildingparameters ischosen;severalvalues are pickedfromthissetandaretested.Thismethodcanrevealsome interestingtrends,butitcannotachieveanoptimaldesignwithout fail.Thesecondapproachensuresamorereliablemethodbyusing optimizationalgorithms–wecancite,forexample,thestudyof Tuhus-Dubrow[2]fortheoptimizationofthebuildingshape. How-ever,sinceinthefieldofbuildingphysics,theobjectivefunctions aregenerallycalculatedoveroneyear,andbecausethe optimiza-tion algorithms require hundredsor thousands of calculations, thetotaloptimizationcomputationdurationcanquicklybecome prohibitive. Toresolve this computation-time problem,another

∗ Correspondingauthor.Tel.:+33561556897;fax:+33561558154. E-mailaddress:[email protected](B.Lartigue).

methodexists.Itconsistsofusingartificialneuralnetworks(ANN) toevaluatecostfunctionsfaster,withoutdegradingtheiraccuracy, bymimickingthebehaviorofexternalsimulationprograms.ANN haveproventheirefficiencyinbuildingphysicsstudies[3–6].Once validated,theANNiscoupledtoamulti-objectivealgorithmtofind theproblem’soptimalsolutions.

Theaimofourstudyistoproposeafastandefficient multi-objective optimization approach to optimize theenvelope of a residential building based on its thermal performance. In this respect,ourarticledetailsthestepsoftheoptimization method-ology.Thevariablescalculatedintheoptimizationaretheeffective thermophysical properties of the external walls (thermal con-ductivity k, and volumetric specific heat (c)). A discussion is performedtoselectthemostappropriateobjectivefunctionsthat definethethermalperformanceofthebuilding.In sucha prob-lem,withmultiplevariablesandnon-linearobjectivefunctions,a parametricstudyisusefultounderstandtheoptimization’s solu-tions.Consequently,wehaveperformedaparametricstudyusing adesignofexperimentsthatquantifiestheimpactofthevariables ontheobjectivefunctions.Then,theresultsoftheoptimization are presented,asa functionof theclimate.Last,the methodol-ogyisdiscussedbycomparingtheresultsofmulti-objectiveand mono-objective optimizations, demonstrating the limits of the latter.

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254 D.Gossardetal./EnergyandBuildings67(2013)253–260

Nomenclature

an principaleffect

ANN artificialneuralnetwork c specificheat(Jkg−1_K−1₎

Isum integrateddiscomfortdegreeinsummer(◦Ch)

Iwin integrateddiscomfortdegreeinwinter(◦Ch)

k thermalconductivity(Wm−1_K−1₎

MSE meansquarederror

NSGA non-sortedgeneticalgorithm p numberofsimulations PMV predictedmeanvote

QTOT annualenergydemand(kWh)

t time(s)

T temperature(◦_C)

Tcom comforttemperature(◦C)

Tin indoortemperature(◦C) To outsidetemperature(◦C) xn parameter Greekletters density(kgm−3₎ 2. Optimizationmethodology

2.1. Objective-functionsandoptimizationvariables

2.1.1. Objective-functions

Toperformanoptimizationproblem,thefirststepistodefine theappropriateindicatorsreflectingthethermalperformanceofa building.Inmoststudies,buildingenergysavingsarecalculatedby consideringannualheatingandcoolingdemands[7,8].

Inourstudy,weconsidertheannualenergyload,QTOT,required

tomaintainthewintertemperaturesetpointto19◦_C_[9]_in_a

free-runningbuilding.

The occupants’ thermalcomfort can beevaluated in several ways.Basedonsteady-stateheattransfertheory,Fanger’smodel

[10]proposedthePredictedMeanVotePMVderivingfromclimatic chamberstudies,asathermalcomfortindex.NicolandHumphreys

[11]consideredtheapplicationoftheadaptiveapproachto ther-malcomfort standards,andpresented therelationshipbetween comforttemperatureTcom andoutdoortemperatureTofor

free-runningbuildings: Tcom=13.8+0.54To. De Dear and Brager [12]

presentedtherevisionstoASHRAEStandard55forthermal com-fortin naturally ventilatedbuildings, includinga new adaptive comfortstandard,whichallowswarmerindoortemperaturesfor naturallyventilatedbuildingsinsummerandinwarmerclimate zones.Zhang et al. [13] proposedthe two parameters Iwin and

Isumthatmeasurethethermalcomfortdegreeinanindoor

envi-ronment.Theyaredefinedasintegrateddiscomfortdegreeforair indoortemperatureinwinterandinsummer,respectively.By def-inition,thethermalcomfortdegreeincreaseswhenIsumandIwin

decreases.

Inourstudy,aswewanttocharacterizesummercomfort,we willusethesummerthermalcomfort index Isum definedonan

entireyearas:

Isum=

Z

8760

0

(Tin−Tcom)dt (1)

whereTinistheindoortemperatureasafunctionoftime;Tcomis

thecomforttemperaturefixedatTcom=28◦C;dtisthetimestep

fixedatdt=1h,8760isthenumberofhoursinayear.

2.1.2. Optimizationvariables

Theoptimizationvariablesarethethermophysicalproperties oftheexternalwalls(kwalland(c)wall)andoftheroof(kroofand

(c)roof).Theirrangesofvariationarerespectively0.05≤k≤1.175

W m−1 _K−1_and₄₀_≤_(c)_≤₂₀₀₀ _kJ _m−3 _K−1_, _which _are

conventionalvaluesforbuildingmaterials.Optimizationvariables areconsideredascontinuousvariables.

Thesetwoobjective-functions(QTOTandIsum)aretime

consum-ing becausetheyrequireone-year simulationsto beevaluated. Moreover,consideringthevariables’ranges,alargesearchspace hastobeexploredextensively.Therefore,weproposea methodol-ogytosimplifytheoptimization.

2.2. Multi-objectiveevolutionaryalgorithmoptimization

Simultaneouslyreducingthebuildingenergyconsumptionand maintainingacomfortableindoorenvironmentaretwoconflicting objectives. Since these two functions are nonlinear, stochastic global multi-objective optimization techniques such as genetic algorithmscanbeusedinordertoobtainoptimaldesigns[3,14]. Geneticalgorithms are gradient-freestochastic searchmethods that mimic natural biological evolution. We used the Non-dominated Sorting Genetic Algorithm II, NSGA-II [15]. First, it initializes a random population of several individuals, then it producesoffspringbyrecombinationandmutation,evaluatesthe individuals,andfinallyselectsthefittestones.

The efficiencyof NSGA-II is due tothe non-dominated-and-crowding sorting and selection. This method ensures both the convergenceofthepopulationanditsspreading.Itisbasedonthe twofollowingparameters:

•Therank(orfitness)valueofanindividual.Inapopulation, non-dominatedindividualshaverank1andbelongtothefirstfront. Individualsthataredominatedonlybysolutionsfromthefirst frontbelongtothesecondfront,andthenhaverank2.Thenotion ofrankenablesthecomparisonofanindividualtothewhole pop-ulation.Attheendoftheoptimizationprocess,onlythePareto frontwithrank1iskept.

•Thecrowdingdistanceofanindividual.Itmeasureshowclosean individualistoitsneighbors.Alargeaveragecrowdingdistance resultsinbetterdiversityinthepopulation.Individualswiththe highestcrowdingdistancesshouldbepreferredforthespreading.

2.3. Artificialneuralnetwork(ANN)

Aswehavealreadywritten,thecalculationoftheobjective func-tionsbasedonone-yearsimulationsisrelativelyslow.Inorderto makeitfaster,weuseamultilayerfeed-forwardartificialneural network(ANN).Itiscomposedoflayersofneurons:weclassically callthelayerthatproducesthenetworkoutput,theoutputlayer, andallotherlayersarecalledhiddenlayers.Itsprincipleisstrongly inspiredfromthebiologicalnervoussystem[16].Neuronsmayuse anydifferentiabletransferfunctiontogeneratetheiroutput.The performanceoftheANNisstronglyinfluencedbytheconnections betweenneurons.

TheANNmustbetrainedtoperformaspecifictaskbyadjusting theweightsbetweenneurons.Theweightsareadjustedby com-parisonbetweenoutputsofANNandtarget-valuesfromsampling datasets,untiltheoutputsmatchthetargets.

2.4. Optimizationframework

Theproposedoptimizationmethodologyisshown schemati-callyinFig.1.

First,abuildingmodelisestablishedinTRNSYS[17],whichisa softwaredesignedforthetransientthermalsimulationofcomplex

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ANN le

arnin

g and

validation

Samp

lin

g d

ata

sets

TRNSYS

+ GenOpt

1/ ANN Learning

2/ Multi-objective optimization process

ANN validated

NSGA II

Pareto f

ront

Objective functions

(Q

TOT

, I

sum

)

Individuals

(k

wall

, (ρc)

wall

)

Stopp

ing

criterion

yes

no

Fig.1.Optimizationframework.

systems.Then,a samplingdatasetofvariables(k,c) is gener-atedinordertocoverthemaximumsearchspace.Itiscomposed ofsufficientdatarandomlydistributed overthesearchspaceby thesoftwareGenOpt[18].GenOptperformsparametricstudiesby spanningamulti-dimensionalgridinthespaceoftheindependent parameters,anditevaluatestheobjectivefunctionsateachgrid point.

Oncethe training and validation are completed,the ANNis ready to perform fast and accurate calculations of objective-functions. Finally, NSGA II is coupled to the ANN in order to achieveoptimization.NSGAIIprovidesinputvaluestoANNand ANNperformsevaluationsofobjectivefunctionsrequiredbythe NSGAII.

3. Applicationoftheoptimizationmethodologytoa dwelling

3.1. Descriptionofthecasestudy

Themethodologypresentedaboveisnowappliedtoadwelling (Fig.2),whosespecificationisdetailedin[19].Itisasinglestorey housewithanetfloorareaof112m2_and_a_ceiling_height_of_2.3_m.

Itisdividedinto6thermalzones,3ofwhichareheated.Table1

liststheassumptionsandspecificationsadoptedforthethermal simulationofthedwelling.

Theinertiaoftheinnerwallsislow,sothattheinertialeffects oftheexternalwallsareemphasizemore.Theexternalwallsare highlightedinredintheblueprint(Fig.2).

Theexternalenvelopeisdividedintoverticalexternalwallsand theroof(Fig.3).Thethermophysicalpropertiesoftheplasterand theinsulationarekeptconstantduringtheoptimizationprocess andtheirvaluesareshowninTable1.Keepingtheinsulation con-stantenablestodistinguishtheinsulationfunctionofthewall,from itsstructuralfunctionandtofocusofthisone.

3.2. Impactofthevariablesontheobjective-functions

Beforeperformingtheoptimization,weneedtoknowthe influ-enceofvariablesonthetwothermalperformancecriteria,Isumand

QTOT,fortwodifferentFrenchclimates,Nice(Mediterranean

cli-mate)andNancy(continentalclimate).Thesetwoclimateswere chosenbecauseeachisimpactedbyone-performancecriteriamore than theother,Isum for Nice andQTOT forNancy.Thisinfluence

isanalyzedwitha designofexperiments.Apresentationofthis methodcanbefoundin [20].Thefourchosenvariablesarethe thermophysicalpropertiesof theexternalwalls andof theroof (respectivelykwalland(c)wall,kroofand(c)roof).Atwo-level

facto-rialexperimentaldesignassignsthelevel(−1)tothelowestvalue ofthevariablesandthelevel(+1)totheirhighestvalue(Table2), andshowstheinteractionsbetweenthesefactors.

This technique introduces new reduced dimensionless vari-ables,notedxn.Afirst-degreelawisadoptedwithrespecttoeach

variable.Forfulltwo-levelfactorialdesigns,itisexpressedas fol-lows: Response=a0+ p

X

n=1 anxn (2)

Thedimensionalcoefficientsanrepresenttheprincipaleffectof

eachreducedvariableontheresponse,whichcanbeIsumandQTOT

here.Thecoefficienta0isthemeanvalueoftheresponse.Inorderto

identifythecoefficientsan,weneedtorunp=24=16simulations,

4beingthenumberofvariables:

an= 1 16 16

X

p=1 xn(p)×response(p) (3)

ThevaluesofthedifferentprincipaleffectsanforNiceandNancy

areshowninTable3.Sincetheimpactofthevariableismeasured bytheabsolutevalueofitsprincipaleffect,weconcludethatthe

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Fig.2. Residentialhouseblueprint(dimensionsinmeters). Table1

Assumptionsandspecificationsadoptedforthethermalsimulationofthedwelling.

Parameters Value Unit

Building Location Nancy,Nice –

Orientation South –

Totalindoorsurface 211.02 m2

Totalheatedvolume(3zones) 197.80 m3

Windowssurfaces 1.08 m2

WindowsU-value 2.83 Wm−2_K−1

Windowssolarfactor 0.76 – Infiltrationfornon-heatedzones 1 ACH Ventilationforheatedzones 0.6(3insummernights) ACH

Occupation Occupancy 3 Persons

Occupancyscenario 17:00–8:00weekdays 24h/24week-ends

Externalenvelope Indoorplasterthickness 1 cm Indoorplasterk 0.35 Wm−1_K−1

Indoorplasterc 900 kJm−3_K−1

Insulationthickness 8 cm

Insulationk 0.04 Wm−1_K−1

Insulationc 29.4 kJm−3_K−1

Optimalmaterialthickness(withunknownkandc) 20 cm

Fig.3.Compositionof(a)theexternalwalls(b)theroof.

Table2

Lowandhighlevelsforeachparameter.

x1=kwall(Wm−1K−1) x2=kroof(Wm−1K−1) x3=(c)wall(kJm−3K−1) x4=(c)roof(kJm−3K−1)

−1 0.10 0.10 40 40

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Table3

PrincipaleffectsofwallandroofthermophysicalpropertiesforNancyandNice.

a1(kwall) a2(kroof) a3((c)wall) a4((c)roof)

Nancy Isum(◦Ch) −17.23 +0.99 −97.46 −3.57

QTOT(kWhm−2) +4.89 +0.01 −1.89 −0.13

Nice Isum(◦Ch) −302.91 +9.93 −512.60 −26.59

QTOT(kWhm−2) +2.08 +0.01 −1.69 −0.15

volumetricspecificheat(c)isthemostinfluentvariableonIsum

andthethermalconductivityisthemostinfluentvariableonQTOT.

Thesignoftheprincipaleffectsangivesthevariationtrendsof

theresponses.Forbothclimates,Table3showsthata1anda3for

Isumareprecededbytheminussign.Itindicatesthattheincrease

ofkwalland(c)wallleadstoareductionofIsum.Indeed,thesummer

comfortcanbeimprovedbytheenhancementofinertiawhichis relatedto(c)_wall[21,22],andbymoreconductivewallsthatallow todissipatetheheatinsummer.ThereductionofQTOTisobtained

bythedecreaseofkwall,underliningtheimportanceofwall

insula-tion,andbytheincreaseof(c)whichemphasizestheadvantage ofinertia.

TheimpactoftheroofonbothIsumandQTOTisnotofthesame

orderofmagnitudeastheexternalwalls.ForNancy,kwallhasover

500timesmoreinfluenceonQTOT thankroof,and17timesmore

influenceonIsum(Table3).Thisisduetothelowthicknessofroof

materialtooptimize(2.5cm)incomparisontotheexternalwalls’ one(20cm).Thus,fortheremainderofthisstudy,wewillperform theoptimizationwithonlykwalland(c)wall.

3.3. DescriptionoftheANN

3.3.1. ParametersfortheANN

Priorto training theneuralnetwork, all inputvariables and objective-functionsarelinearlyscaledtoa rangeof−1 to+1in ordertoeasethetrainingprocess.Inourwork,theANNis com-posedof4layersofneurons.Thereare15neuronsinthefirstlayer, 11neuronsinthesecondlayer,7neuronsforthethirdlayer,and3 neuronsinthefourthlayer.Aswehaveseenin(2.3),neuronsneed transferfunctionsinordertocomputetheiroutput.Itiscommonto chooseahyperbolictangentsigmoidtransferfunctionforthe hid-denlayers(thefirst,secondandthirdlayers)andalineartransfer functionforoutputlayer(thefourthlayer).

3.3.2. TrainingoftheANN

TheLevenberg–Marquardt backpropagationmethod isused tocomputetheweightvalues oftheANN.Thistrainingmethod updatesthenetworkweightsinthedirectioninwhichthetraining performancefunctiondecreasesmostrapidly.Thetraining perfor-manceisdeterminedbythemeansquarederror(MSE)anditis stoppedwhenMSEreaches1×10−7.

3.3.3. ValidationoftheANN

InordertochecktheaccuracyoftheANNtopredictIsumand

QTOT,25samplesarerandomlyselectedandthecorrespondingIsum

andQTOTfromTRNSYSarecomparedtodataissuedfromtheANN.

ThemaximumdeviationsforIsumandQTOTare1.86%and0.22%for

Nancy,andare0.19%and0.04%forNice.Theselowvaluesconfirm theaccuracyoftheANN.

3.4. Optimizationresultsanddiscussion

3.4.1. Optimizationresults

Many numerical computations performed previously by the presentauthorshaveshownthatconvergencetendstobedifficult whensamplingnumbersarelimited(i.e.thenumberofindividuals inapopulation),orwhentoosmallofacrossoverprobabilityis

Table4

ValuesassociatedtothepointsA,B,C,D,A′_,_B′_,_C′_and_D′_in_Figs.₄_and₅_. kwall (Wm−1_K−1₎ (c)wall (kJm−3_K−1₎ QTOT(kWhm−2) Isum(◦Ch) Nancy A 0.10 1727 32.3 49 B 0.10 1587 37.2 30 C 0.16 1540 48.5 23 D 1.75 2000 56.1 2 Nice A′ _0.27 ₁₈₉₂ _13.9 ₁₈₃₁ B′ _0.31 ₂₀₀₀ _14.1 ₁₇₅₁ C′ _1.03 ₂₀₀₀ _16.9 ₁₅₆₇ D′ _1.75 ₂₀₀₀ _17.2 ₁₅₀₃

chosen.TheNSGA-IIneedssomeparametersbasedontwomain geneticoperators:crossoverandmutation.Onehundred individ-uals,i.e.envelopescharacterizedby(k,c),perpopulationleadto asatisfactoryconvergencecalculation.Themaximumgeneration number, here500, isthestoppingcriterionbecauseinprevious numericaltests,wehaveverifiedthatthesolutionsdidnotchange beyondthisnumber.TheNSGA-IIalgorithmusessimulatedbinary crossoverwithacrossoverprobabilityof90%(90envelopesina populationexchange(k,c)withothers).Themutationprobability issetto25%(25envelopesinapopulationarerandomlychanged). OncewehaveanefficientANNandawell-setNSGA-IIalgorithm, wecanperformtheoptimization.Figs.4and5showParetofronts thatarecomposedofwell-spreadoptimalsolutions.

Fig.4. IsumasafunctionofQTOT(Nancy).

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Fig.6. EvolutionofTinfor4optimalresidentialbuildings(Nancy).

Thefirstpart(AtoBandA′_to_B′₎_corresponds_to_a_steep_fall_of_I sum

forlowervaluesofQTOT.Indeed,themostinsulatedexternalwalls

(lowvaluesofkwall)aretheoptimalsolutionsandsummer

com-fortissignificantlydegradedbecauseheatcannotbesufficiently dissipatedtotheoutside.

Thesecondpart(BtoCandB′_to_C′₎_is_almost_linear._The_slope_of

thispartoftheParetofrontisgreaterforNicethanforNancy.For bothclimates,thedecreaseofIsumismainlyduetotheriseofkwall

(Table4).

Thethirdpart(CtoDandC′_to_D′₎_corresponds_to_a_sharp_fall_of

IsumforhighervaluesofQTOT.

Table4indicatesthattheoptimalmaterialshave(c)wallclose

orequaltothehighestvalueof(c)wall,i.e.2000kJm−3K−1.Table3

providestheexplanation:a3isalwaysnegativeforboth

objective-functionsandclimates,demonstratingthattheincreaseof(c)_wall correspondstoadecreaseofIsumandQTOT.Wecanalsonoticethatin

Nancy,theenergyreductionproblemshouldberesolvedinpriority becauseofsmallvaluesofIsum.InNice,itisthethermalcomfort

problemthatshouldbeaddressedfirstbecauseoflowvaluesof QTOT.

3.4.2. Analysisofthedynamicthermalcomportmentinsummer InNancy,optimalvaluesofIsum areclosetooneanotherand

low(Table4):thisisthereasonwhywewillanalyzemorefinely thedynamicthermalbehaviorinsummer.Fig.6showsthetime evolutionoftheindoortemperatureofzone1ofthedwellingwhose externalwallsarecomposedofdifferentoptimalmaterialsA,B,C andD(Table4),during3days(20,21and22July,thewarmestdays oftheyear)forNancy.Isumcanberepresentedgraphicallyasthe

computedsurfacebetweenthetimeevolutioncurveofTinandTcom

whenTinisaboveTcom.Thus,weobservethatthecurvesAandB

arealmostidentical,whichmeansthattheoptimalsolutionsAand Barenearlyequivalentintermsofdynamicthermalbehavior.The amplitudeofthecurveCislowerthanthoseofthecurvesAand B,whichactuallyleadstoalowervalueofIsum.ThecurveDhasits

amplitudemoreflattenedthanthoseofthethreeothercurves;itis evenbelowTcomontheperiodconsidered.

3.4.3. Comparisonbetweendifferentoptimizationmethods

Inordertodiscusstheefficiencyofthemethodologyusedinthis article,wearegoingtocompareourresultswiththoseobtained from othercommonly usedoptimization methodsfound in lit-erature. They consist in coupling two distinct tools: a thermal simulationprogramthatcomputesthermalperformancecriteria andanoptimizationprogramfortheminimizationofcostfunctions thatareevaluatedbythethermalsimulationprogram. Contrary tothemethodologyinvolvingtheANNdescribedabove,thecost functionisevaluatedexactlybythethermalsimulationprogram. GenOpt[18],mentionedabove,isalsoanoptimizationtoolthatcan becoupledwithexternalsoftwaresuchasTRNSYSinourcase.It proposesglobalmulti-dimensionaloptimizationalgorithmsfor lin-earcostfunctions.Inourwork,aparticleswarmoptimization(PSO) algorithmisusedin ordertoperformtheoptimization.Several studiesinvolvingtheuseofGenOptinbuildingenergy optimiza-tionshavebeenperformedandhavedemonstrateditsefficiency

[23–25]. GenOptdealswithmono-objectiveoptimization prob-lems.Thetwomethodswearegoingtocomparearetheaggregative methodandpenaltyfunctionmethod.

TheaggregativemethodcombinesbothobjectivefunctionsQTOT

andIsumintoaweighted-sumf:











minf =1 2

Isum−I_summin Imax

sum−Isummin

+1 2

QTOT−Qmin TOT Qmax TOT −QTOTmin

kwall∈[0.1;1.75] (c)_wall∈[40;2000] (4) where Imax

sum =140◦Ch,Isummin=2◦Ch,QTOTmax=65.4kWhm −2

,Q_TOTmin=50.6kWh m−2forNancy Imax

sum =3155◦Ch,Isummin=1503◦Ch,QTOTmax=20.8kWh m −2

,Qmin

TOT =13.7kWhm −2

forNice

TheaboveminimumandmaximumvaluesofIsumandQTOTare

obtainedfromdifferentcombinationsofrangelimitsofthe vari-ableskwalland(c)wall.ThesecombinationsaregiveninTable5.

Inthepenaltyfunctionmethod,weimposeaconstraintonIsum,

whichisformulatedbyaninequality.Inpractice,apenaltytermis addedtoQTOT:everytimetheconstraintisviolated,alargepositive

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Table5

ParametersIsumandQTOTformono-objectiveoptimizations(Eqs.(4)and(5)).

kwall(Wm−1K−1) (c)wall(kJm−3K−1) Isummax(◦Ch) Iminsum(◦Ch) QTOTmax(kWhm

−2₎ _Qmin TOT(kWhm −2₎ Nancy 0.1 40 140 1.75 2000 2 1.75 40 65.4 0.1 2000 50.6 Nice 0.1 40 3155 1.75 2000 1503 1.75 40 20.8 0.1 2000 13.7 Table6

Comparisonofoptimalresultsbetweenaggregativemethod(Ag.)andpenaltyfunctions(Pe.).

kwall(Wm−1K−1) (c)wall(kJm−3K−1) QTOT(kWhm−2) Isum(◦Ch)

Ag. Pe. Ag. Pe. Ag. Pe. Ag. Pe.

Nancy 0.10 0.10 2000 2000 50.6 50.6 13 13 Nice 0.10 1.75 2000 2000 13.7 17.2 2298 1503

numberisaddedtoQTOT.Suchaproceduretakingintoaccountthe

constraintbypenaltyfunctionisalreadyimplementedinGenOpt.











minQTOT

Isum≤I_summin kwall∈[0.1;1.75] (c)wall∈[40;2000]

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Table6summarizestheoptimizationresultsobtainedbothby theaggregativemethodandpenaltyfunction.

The results in Table 6 belong to the third part of Pareto fronts (high QTOT and low Isum) for both climates, except for

theNice optimalvalues fromaggregative methodwhich donot appearintheParetofront. Itisimportanttonotethat aggrega-tivemethodand penaltyfunctionsproducecompletelyopposite results for Nice (aggregative: k_wall=0.10Wm−1_K−1_, _penalty:

kwall=1.75Wm−1K−1).Indeed,Isumgivesthemaindirectiontothe

optimizationprocessusingtheaggregativemethodbecausethe principaleffectsofkwalland(c)wallonIsum(whichareboth

nega-tive)arehigherthanthoseofQTOT(Table3).However,thepenalty

functiondrivestheoptimizationprocesstotakelowvaluesofIsum,

whichimpliesthatkwallissettoitshighestvalue.

To conlude on the comparison between these different optimization methods, mono-objective optimization using the aggregativemethodortheconstraintprobleminGenOptis too sensitivetoprivilegeddirections,especiallyinourcasewherethe objectivefunctionshavedifferentrangesofvariationspeed. More-over,themono-objectiveoptimizationonlyprovidesonesolution whichisrelativelyrestrictive,anddoesnotallowchoosingamong optimalsolutionsasthemulti-optimizationdoes.

4. Conclusion

Thisworkpresentsamethodologyforbuildingenvelope opti-mizationintermsofthermalperformance.Inordertoreducethe computationtimewithoutreducingthecomplexityofthe prob-lem,anartificialneuralnetworkhasbeendeveloped:itsroleisto providefastandaccurateevaluationsofobjectivefunctionswhich areusedbyageneticalgorithm.Theefficiencyofthis methodol-ogyhasbeenprovenbyapplyingittoaresidentialhousefortwo Frenchclimates, Nancy(continental)and Nice (Mediterranean). Twoobjectivefunctionshavebeenconsideredasbeing represen-tativeofenergy performance:theannual energyloadQTOT and

thesummercomfortindexIsum.Thermophysicalpropertiesofthe

externalwalls,kwalland(c)wall,havebeenchosenasoptimization

variables.

Adesignofexperimentshasbeenconductedtoquantifythe impactof thevariables kwall and (c)wallontheobjective

func-tionsQTOTandIsum.TheoptimalsolutionsarepresentedasPareto

fronts for the two climates. These optimalsolutions cover the entirerangeofpossiblesolutions.Theyenabletheselectionofthe thermophysicalpropertiesaccordingtotheconflictingobjective functions.These multi-objectiveoptimization resultshavebeen comparedtothosefrommono-objectiveoptimizationbyusingan aggregativemethodandaconstraintprobleminGenOpt.The com-parisonclearlyshowstheadvantageofperformingmulti-objective optimizationsinceitensuresthattheoptimizationisnottrapped inaprivilegeddirection.

Thisstudyalsohighlightsthemajorinfluenceoftheclimateon optimalenvelopes.Indeed,wehaveshownthattheoptimal solu-tionsarevery differentforvariousclimates.However,standard buildingsolutionsdonotadequatelytakeintoaccountthis param-eterastheyareoftenidenticalforanyclimate.

Acknowledgements

TheauthorsthanktheFrenchEnvironmentandEnergy Man-agement Agency(ADEME)and theTechnicalCenterfor Natural BuildingMaterials(CTMNC)fortheirsupport.

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