Neural networks based predictive control for thermal comfort and energy savings in public buildings

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ContentslistsavailableatSciVerseScienceDirect

Energy

and

Buildings

jo u rn al h om epa 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

Neural

networks

based

predictive

control

for

thermal

comfort

and

energy

savings

in

public

buildings

P.M.

Ferreira

b,c,∗

_,

_A.E.

_Ruano

a,c

_,

_S.

_Silva

a

_,

_E.Z.E.

_Conceic¸

_ão

a

a_University_of_Algarve,_8005-139_Faro,_Portugal

b_Algarve_Science_and_Technology_Park,_Campus_de_Gambelas,_Pav._A5,_8005-139_Faro,_Portugal c_Centre_for_Intelligent_Systems,_IDMEC-IST,_Av._Rovisco_Pais_1,_1049-001_Lisboa,_Portugal

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received17February2012

Receivedinrevisedform17May2012 Accepted2August2012

Keywords:

HVACpredictivecontrol Predictedmeanvote Neuralnetworks

Multi-objectivegeneticalgorithm Thermalcomfort

Wirelesssensornetworks

a

b

s

t

r

a

c

t

ThepaperaddressestheproblemofcontrollingaHeatingVentilationandAirConditioning(HVAC) sys-temwiththepurposeofachievingadesiredthermalcomfortlevelandenergysavings.Theformulation usesthethermalcomfort,assessedusingthepredictedmeanvote(PMV)index,asarestrictionand min-imisestheenergyspenttocomplywithit.Thisresultsinthemaintenanceofthermalcomfortandon theminimisationofenergy,whichinmostconditionsareconflictinggoalsrequiringanoptimisation methodtofindappropriatesolutionsovertime.Adiscretemodel-basedpredictivecontrolmethodology isapplied,consistingofthreemajorcomponents:thepredictivemodels,implementedbyradialbasis functionneuralnetworksidentifiedbymeansofamulti-objectivegeneticalgorithm;thecostfunction thatwillbeoptimisedtominimiseenergyconsumptionandmaintainthermalcomfort;andthe optimi-sationmethod,adiscretebranchandboundapproach.Eachcomponentwillbedescribed,withspecial emphasisonafastandaccuratecomputationofthePMVindices.Experimentalresultsobtainedwithin differentroomsinabuildingoftheUniversityofAlgarvewillbepresented,bothinsummerandwinter conditions,demonstratingthefeasibilityandperformanceoftheapproach.Energysavingsresultingfrom theapplicationofthemethodareestimatedtobegreaterthan50%.

1. Introduction

InEuropeanUnion(EU)countries,primaryenergyconsumption inbuildingsrepresentsabout40%ofthetotalenergyconsumption and,withvariationsfromcountrytocountry,halfofthisenergyis spentforindoorclimateconditioning[1–3].Itisestimatedthatthe useofefﬁcientenergymanagementsystemsinbuildingscansave upto8%oftheenergyconsumptionintheentireEU[4].Around83% oftheEUdwellingswereconstructedbefore1990andabout50%of thembefore1970[1].Thereforeitisoffundamentalimportanceto controlefﬁcientlytheexistingHVACsystems,inordertodecrease energyusageandincreasecompliancewiththeEuropeanDirective (2010/31/EU)ontheenergyperformanceofbuildings[3].

TheuseofArtiﬁcialNeuralNetworks(ANNs)invarious appli-cationsrelated withenergymanagement in buildingshasbeen increasingsigniﬁcantly over the recent years. Withinthis area, ANNshavebeenmainlyappliedinseveralaspectsofHVACcontrol

∗ Correspondingauthorat:AlgarveScienceandTechnologyPark,Campusde Gam-belas,Pav.A5,8005-139Faro,Portugal.Tel.:+351289800950;fax:+351289800098. E-mailaddresses:pfrazao@ualg.pt(P.M.Ferreira),aruano@ualg.pt(A.E.Ruano), ssilva@ualg.pt(S.Silva),econcei@ualg.pt(E.Z.E.Conceic¸ão).

methodologies [5–11], and in forecasting energy consumption [12–20].

Theauthorshaveadvocated,inpastpublications[21,22],the useofmodelbasedpredictivecontrol(MBPC)withthepurposeof efﬁcientlycontrollingexistingHVACsystemsinlargepublic build-ings.Thepresentpaperisthenaturalfollow-upofthispastwork, wherea remote,wired dataacquisitionsystemwasset-upin a secondaryschoolintheSouthofPortugal.Theperformanceof phys-icalmodels,basedonenergyandmassbalanceintegral-differential equationswereevaluatedagainstblack-box,neuralnetwork mod-elsforthepurposeofmodellingtheinsideairtemperature.Itwas foundthattheroleofthesetwodifferenttypesofmodelsis com-plementary,andnotcompeting.Whilephysicalmodelscanbeused inthephaseoftheprojectofbuildings,andtoassessthe conse-quencesofpossiblebuildingsmodiﬁcations,data-drivenmodels suchasneuralnetworksshouldbeused fortheon-linecontrol ofHVACsystems.SimulationsshowedthepotentialofMBPCfor thecontrolofair-conditionedsystems,indicatingboththebetter regulationandtheenergysavingsobtained.

Thisarticleisorganisedasfollows.Theexperimentalsetupis describedinSection2.AsaPMVformulationisemployed,its com-putationisdiscussedinSection3.ToimplementtheMBPCstrategy, severalneuralnetwork modelsareused.Their designis brieﬂy addressedinSection4.Section5describesthebranch&bound(BB) 0378-7788/$–seefrontmatter©2012ElsevierB.V.Allrightsreserved.

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Fig.1.Overviewoftheexperimentalsetupused.

searchtechniquethatisemployedtoimplementtheMBPC,which inturnisformulatedinSection6.Severalexperimentsarethen presentedinSection7,bothinsummerandwinterconditions.The paperendswithconclusionsandadescriptionoffuturework. 2. Experimentalsetup

HVACcontrolexperimentswereconductedinthreeareasofone buildingusedbytheFacultyofSciences&Technologyofthe Uni-versityofAlgarve,inthesouthofPortugal.Intotal,16locations (labs,ofﬁces,corridors)arenowequippedwithdataacquisition devicesandtheirinternalHVACunitsmaybeindependently con-trolledwithnewalgorithms,andmonitored.Fortheworkthatis goingtobepresented,fourroomswereused,denotedbytheletters A,B,C,andD.RoomsAandBareadjacentwithwallsfacingwest andnorth(onlyA).RoomCsharesthesamecorridorwithAandB andhaswallsexposedtothenorthandeast.Thesethreeroomsare onthesecondﬂoor.Finally,roomDisdirectlybelowroomCwith wallsexposedtothenorthandeast.

A weather station locatedin the campus provides air tem-perature(Tao),airhumidity(Hao),andglobalsolarradiation(Rsg)

measurements.All theelementsinvolved are connected tothe TCP/IPnetwork,enablingaPCstationinthecontrolsystems labo-ratorytomonitorandcontrolanyoftherooms.Fig.1providesan illustrationofthesystemintegration.

2.1. Wirelesssensornetworks

WSNsarerapidlygainingpopularity,andarebeingusedina vastspectrumofapplications.Oneofthemostactiveareabeing buildingautomation[23,24].Eachoftheroomswherethe experi-mentswereconductedhasWSNnodeswithsensorstomonitorthe airtemperature(Tai),airhumidity(Hai),globetemperature(Tg),the

stateofwindowsanddoors(open/closed),andmovementusinga passiveinfra-redactivitymonitor.Tgismeasuredbymeansofa

blackglobethermometer[25]anditsmeasurementisthenusedto determinethemeanradianttemperature,Tmr[26].

TheWSNshaveastartopology,whereeachunitiscollecting informationonceperminuteandsendingittoacentralnodewith

storageanddatabasecapabilities.EachnodeconsistsofoneTmote Skyplatformconnectedtotherequiredsensors.Thisplatformisan IEEE802.15.4standardcompliantdevicethatusestheTinyOS[27] operatingsystem.

IneachroomoneWSNnodeisinstalledinthewall,nearly1.75m abovetheﬂoor,whichmeasuresTaiandHaiusingSHT11sensors

fromSensirion.Approximatelyinthemiddleoftheroom,ataheight ofroughly2mabovetheﬂoor(forsecurityreasons),theblackglobe thermometerisinstalled.Itconsistsofamateblackpaintedsphere witha diameterof125mmwitha Tmote(anditstemperature sensingdevice)atitscentre.Plasticwasusedinsteadofcopper, whichismorecommonlyfoundinthisapplication,becauseitwas easilyavailable,butalsobecauseaccordingto[26]itovercomes anundesirablehightimeconstant thatappearswhen copperis employed.Infra-redsensorsareinstalledatonecorneroftheroom tomonitoractivity.Digitalinformation,bothfromthemagnetic switchesindoorsandwindowsandfromthemovementdetector arewiredtooneoftheexistingWSNnodes.

2.2. HVACsystem

TheHVACsystemusedintheexperimentsis composedof3 independentMitsubishiVariableRefrigerantFlow(VRF)systems, each one withan outdoor air cooled inverter compressor unit (denotedinthesequelasoutdoorunit),locatedonthebuilding roof,connectedtoceilingconcealedductedindoorunits(denoted asinteriorunits).Ineachindependentroomthereisatleastone internalunit,withitsownwallcontroller.Thesystemcanbe cen-trallymanagedbyaPCmanagementstationtowhichalltheunits areconnectedviaaLonWorkscommunicationbus.Thisstationis abletomonitorandcontrolmanyaspectsofalltheHVACsystem, throughtheMitsubishiLMAP02interface[28].Thoseofmajor inter-estfortheexperimentsare:specifyingatemperatureset-pointfor agivenunit,switchingtheinternalunitonoroff,anddisablingthe localcontrollersotheoccupantscannotoperatetheinternalunit whileexperimentsarebeingconducted.

The model of the interior units used in the experiments describedinthispaperisthePEFY-P63VMMandthemodelofthe outdoorunitsisthePUHY-250YMF-C.

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3. Predictedmeanvote

Thehollygrailofenergymanagementinbuildingsisthe min-imisationoftheenergyrequiredtomaintainadesiredminimum comfortlevelfortheoccupants.Althoughtheperceptionof com-fortisrelatedtoseveralenvironmentalfactorssuchaslighting, temperatureandairquality,inthisworkonlythethermal com-fortconditionsaspectisaddressed.AsadiscreteMBPCproblem, describedinSection5,issolvedusingtheBBsearchtechnique, ﬁnd-ingtheoptimalsolutiondependsontheabilityofcomputingmany PMVvalues inasmallamountoftime.AsthePMVformulation involvesiterativecomputationsconsumingvariabletime,itis cru-cialtohaveamethodforfast,possiblyconstantexecutiontime, computationofthePMVindex.

Feed-forwardANNs are direct input-to-output connectionist computingstructurescapableofapproximatingasmoothfunction witharbitraryaccuracyprovidedsufﬁcientneuronsareused.These featuresarethemeanstoachievetherequirementsstatedabove. Thefeed-forwardANNdirectinput-to-outputstructureprovides theconstantexecutiontime,theirabilitytoapproximatenon-linear functionsprovidethecapabilityofapproximatingthePMV func-tion.Theaccuracyoftheapproximationisrelatedtothenumberof neuronsusedintheANNhiddenlayer(s),whichinturnislinearly relatedtotheexecutiontime.Consequentlytheproblemconsistsin ﬁndingtheappropriatetrade-offbetweenthePMVindex approxi-mationaccuracyandtheestimationexecutiontime.

TheapplicationofanANNtoestimatethePMVindexfunction hasbeenstudiedbefore[29–31].Inallcasestheapproachtakenwas notthebestforreal-timecontrolapplicationsalthoughthiswasthe mainmotivation.Inthisworkitisshownthatbyfollowinga sim-plerandmoreappropriateapproach,itispossibletoselectadesired compromisebetweenaccuracyandexecutiontime.More impor-tantlyitisshownthat,whencomparedtotheresultsin[29–31],an increasedaccuracywithshorterexecutiontimemaybeachieved. 3.1. ThePMVindex

TheAmericanSocietyofHeatingRefrigeratingandAir Condi-tioningEngineers(ASHRAE)proposedthethermalsensationscale withthepurposeofquantifyingthethermalsensationofpeople [32].Itusesanintegernumericalcodingtoexpressthequalitative thermal sensation, by relating the integer range [−3,3] to the qualitativewordscold,cool,slightcool,neutral,slightwarm,warm, andhot.

Anindex, designatedPMV, was proposedby Fanger[33] in ordertopredicttheaveragevoteofalargegroupofpersonson thethermalsensationscale.Itdependsonsixfactors:metabolic rate,clothinginsulation,airtemperatureandhumidity,air veloc-ity,andthemeanradianttemperature.Itiscomputedbymeansof aheat-balanceequation[34,35]givenby,

PMV=(0.303e−0.036M+0.028)[(M−W)−3.05×10−3 ×[5733−6.99(M−W)−Pa]−0.42[(M−W)−58.15] −1.7×10−5M(5867−Pa)−0.0014M(34−Tai)−3.96×10−8fcl

×[(Tcl+273)4−(Tr+273)4]−fclhc(Tcl−Tai)], (1)

whereMandWarethemetabolicrateandexternalwork,bothin W/m2_,_P

aisthepartialwatervapourpressureinPascal,andTaiand

Tr aretheairtemperatureandmeanradianttemperature,in◦C.

Thesurfacetemperatureofclothing,Tcl,andtheconvectiveheat

transfercoefﬁcient,hc,aregivenby,

Tcl=35.7−0.028(M−W)−Icl[3.96×10−8fcl×[(Tcl+273)4 −(Tr+273)4]+fclhc(Tcl−Tai)], (2) and hc=

h∗ c ifh∗c>12.1

Va 12.1

Va if h∗c<12.1

Va (h∗c=2.38(Tcl−Tai)1/4),(3)

respectively.Va istheair velocityin m/s andIcl istheclothing

thermalresistanceinm2◦_C/W._These_two_equations_are_solved

iter-ativelyuntilaprescribeddegreeofconvergenceisattainedora maximumnumberofiterationsisreached.Finally,in(1)and(2), fcl,whichistheratioofbodysurfaceareacoveredbyclothestothe

nakedsurfacearea,isdeﬁnedby: fcl=

1.00+1.290Icl ifIcl≤0.078

1.05+0.645Icl ifIcl>0.078

. (4)

Themeanradianttemperature,Tr,isaquantitywhichishardto measure.Theinstrumentmostcommonlyemployedinits deter-minationisablackglobethermometer[25].Denotingtheglobe temperaturebyTg,themeanradianttemperaturemaybe

deter-minedas[26], Tr=

(Tg+273)4+1.10×10 8_V0.6 a

D0.4 (Tg−Tai)

1/4 −273, (5)

whereDand

aretheglobediameterin metersandtheglobe emissivitycoefﬁcient,respectively.

Pa,thewatervapourpressureinPascal,iseasilyrelatedtothe

relativehumidityoftheair,Hai,bymeansofAntoine’sequation

[36]:

Pa=10Haie(16.6536−4030.183/(Tai+235)) (6)

Using(1)–(6),thePMVappearsconceptuallyasafunctionofsix variablesthatcanbemeasuredorestimated:

PMV=f(Tai,Tg,Hai,Va,Icl,M) (7)

3.2. Proposedapproachforreal-timecontrol

PreviousANNmodelsofthePMVindexfoundintheliterature employmulti-layerperceptronnetworks[29,31]orLeast-Squares SupportVectorMachines(LSSVM)[30].

For real-time control applications there are two important featuresthatanyPMVindexapproximationmethodshould efﬁ-cientlybalance: accuracyand computingtime. Thismeansthat the PMV models should simultaneously be as simple and as accurate as possible. Although the motivation of past works [29–31] was their use in real-time control systems and the decreaseinPMVindexcomputingtime,theirinput/output struc-turedidnotoptimiseproperlythatbalance,asitwillbeshown later.

FormostHVACreal-timecontrolapplications,theenvironment iscontrolledinclosedspaceswherealloccupantsareassumedtobe dressedsimilarlyregardingthetypeofclothingtheywear. More-overitislikelythatwithineachtypeofclosedspacetheywillbe performingsimilaractivitieslikeattendingalecture,sitting writ-ingaresearchpaper,orhavingbreakfastatthecafeteria.These twoassumptionsmeanthatforagivenspaceitispossibleto spec-ifythevaluesoftheclothinginsulation,Icl,andthemetabolicrate,

M,thereforethesemayberemovedfromthePMVmodelinput.If itisfurtherassumedthattheairvelocity,Va,varieslittlewithin

thespaceanditsvalueisdeterminedbymeasurements,Vamaybe

consideredconstantandmayalsoberemovedfromthePMVmodel input.

BydeﬁningacontextvectorC={Icl,M,Va}andbyusing(1)–(6),

asetofinput–outputdatapairsmaybegeneratedinordertotrain anANNmodelforthePMVindexinthecontextC.Thisapproach suggeststhatwithinanHVACcontrolsystemliketheoneinFig.1,

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Fig.2.UsingasetofPMVmodelsinanHVACcontrolsystem.

thereareasetofdistinctPMVmodels,PMVi,eachforadistinct

contextCi.Inthisscheme,asupervisorysystematanupper

oper-ationallevelwilldeﬁnethecorrespondencebetweeneachroom controlledbyHVACsystemsandapair{PMVi,Ci}.Theapproach

isillustratedinFig.2.Thematchingbetweentheroomsandthe assumedcontextscanbedoneonthebasisoftheyearseason,of thepurposeoftheroom,andmaybeofanystrongdeviationsofthe outsideweatherfromwhatisexpected.

TheconsequenceofusingmultiplePMVindexmodels,isthatfor aspeciﬁccontextCi,themodelhasincreasedaccuracyandreduced

computingtimewhencomparedtomodelsconsideringmultiple contexts.Theincreaseinaccuracycomesfromthefactthatthere arelessfeaturestolearninthetrainingdata,whereasthedecrease incomputingtimeresultsfromusingfewerinputsandfromthe necessitytoemployfewerneuronsinthehiddenlayersinorderto achieveadesiredaccuracy.Inourcase,theairvelocityinallrooms consideredwasmeasuredbyaBABUCprobeina6×6gridwith aspacingofonemeter,ataheightof1.2metersaboveground, andduringaperiodof3min.Allthemeasurementswereinthe range[0.04,0.16]m/s.Thesmallvaluesofairvelocityarejustiﬁed becausetheairﬂowfromeachindoorunitpassesthroughanair diffuser(DFU36,fromFranceAir).Fromtheseresultsanaverage valueofapproximately0.08m/swasselectedasaconstantvalue tobeusedinthecontextvectorofPMVindexmodels.Regarding thetwo additionalhumanfactorsinthecontextvector,avalue of69.78W/m2_(1.2_Met)1_was_selected_as_the_metabolic_rate_of_a

sedentaryactivity(seeappendixAof[32]), andfortheclothing insulationavalueof0.85clowasused(seeappendixBof[32])in thefollowingexperiences.

3.3. Radialbasisfunctionneuralnetwork

TheRBFANNwasusedinthisworkasafunction approxima-tortothePMVindex(1).TheRBFmodelsaretrained usingthe Levenberg–Marquardt(LM)algorithm[37,38]minimisinga modi-ﬁedtrainingcriterion[39,40].

RBFANNshavetheform,

ˆ y(x,w,C,)= n

i=0 wiϕi(x,ci,i), (8)

1_Metabolic_equivalent_of_task_(Met).₁_Met₌_58.15_W/m2_.

wheretypicallyϕiistheGaussianfunction,

ϕi(x,ci,i)=e−(1/2 2

i)x−ci2_, _ϕ

0=1. (9)

Foraspeciﬁednumberofneurons,n,andforadeterminedsetof inputs,Xt_,_off-line_training_a_RBF_NN_corresponds_to_determining

thevaluesofw,C,andsuchthat(10)isminimised:

˚(Xt,w,C,)=1₂y− ˆy(Xt,w,C,)2_. ₍₁₀₎

Pleasenotethat(10)isnowappliedtoasetoftraininginput pat-terns,Xt_,_and_not_to_a_single_input_pattern,_x._As_the_model_output

isalinearcombinationoftheneuronactivationfunctionsoutput (8),(10)canbegivenas,

˚(Xt,w,C,)=1₂y −(Xt,C,)w2_, ₍₁₁₎

whereomittingthedependenceofϕonCand, (Xt_,_C,₎₌_[_ϕ(x(1))_ϕ(x(2))_·_·_·_ϕ(x(N))]T_.

Bycomputingtheglobaloptimumvalue(w*)ofthelinear param-etersw,withrespecttothenonlinearparameters Cand,asa least-squaressolution,

w∗=+(Xt,C,)y, (12)

where“+”denotesapseudo-inverseoperation,andreplacing(12) in(11),thetrainingcriteriontodeterminethenonlinear parame-tersCandis:

(Xt,C,)=1

2y−(X

t_,_C,₎+_(Xt_,_C,_)y2_. ₍₁₃₎

Theinitialvaluesfortheneuroncentrepositionsarerandomly selectedfromthetrainingdata,andthespreadsoftheneuron acti-vationfunctionsareinitialisedusingthesimplerulein[41,p.299]. ThetrainingprocedureprogressesiterativelyusingtheLM algo-rithmminimising criterion(13),until a terminationcriterion is satisﬁed.Formore detailsaboutthetrainingalgorithm and the trainingcriterionthereadingof[40,42–44]isrecommended. 3.4. Datasetsandmodelstructure

Theinputmatrix fortraining theRBFANNs,deﬁnedas Xt₌ [Tai Tr Hai],wasbuiltusingrandomlygenerateddata.Tai,Trand

HaiarevectorsofNvaluestakenfromtheranges[16,32],[13,35]

and[20,70],respectively.Theywereconstructedasfollows: 1.TaiandHaiwereselectedrandomlyfromauniformdistribution

ofvaluesintherangesspeciﬁed;

2.ForeachvalueTaik inTai,acorrespondingvalueTg kinTgwas

generatedusing, T_{g k}=T_aik+(−3.0,3.0),

where(a,b)isarandomnumberfromtheuniformdistribution intherange[a,b].ImplicitlyitisassumedthatTai−3<Tg<Tai+3;

3.Tr wasobtainedbymeansof(5),consideringTai andTg just

described,Va=0.08,D=0.125,and

=0.95.

InordertodeterminethecorrespondingoutputPMVindex val-ues, thecontextvector wasdeﬁnedas describedbefore, i.e.,as C={0.85,1.2,0.08}.Then,Yt_was_constructed_by_means_of₍₁₎_,_using

eachtripletXt

kinXtalongwiththevaluesinC.Thisproceduregives

risetothetrainingdataset,Dt₌_{Xt_,_Yt_}.

Usingtheapproachjustdescribed,anadditionaldataset,Dv, waspreparedinordertovalidatethemodelswithunseendata, afterthetrainingstage.Dvhas23,100trainingpairs(N=23,100).

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Fig.3. Input–outputvariablesdistributionwithintherangesofvaluesconsidered. TheplotscorrespondtodatasetDv_.

Fig.3 shows how the input–outputpatterns are distributed withintherangesspeciﬁedfortheinputvariables,consideringthe datainDvforthecontextC.Itmaybeseenonthetopplothow TrdistributesdenselyaroundthevaluesofTaiforthewholerange

considered.

Thesameisvisibleinthemiddleplotfortherelativehumidity andair temperature.Thebottom plots,fromleft toright, illus-tratethescatteringofPMVvaluesaroundtheairtemperatureand relativehumidities,respectively.Thisdatasetallowsgoodmodel evaluationasitefﬁcientlycoversthevariabilityofcombinations thatoccursbetweeninputvariables.

Althoughtheinput–output structure of themodel hasbeen specified,therearestill two designparameters thatneedtobe determined:thenumberofneuronsandthenumberoftraining patterns.Forthefirst,anexhaustivesearchwasconductedover therange[2,32].Forthesecond,asearchwasalsoconductedas describedinthefollowing.Itisknownthatthe“ideal”numberof trainingpatternsis,tosomeextent,relatedtothenumberof param-etersofthemodelbeingfitted[45,46].ConsideringtheRBFin(8)for thePMVindexmodelpresented,eachneuronaccountsfor5 param-eters,thereforethetotalnumberofparametersisgivenbyn×5,n beingthenumberofneuronsemployed.Byspecifyinganumberof patterns(p)permodelparameteritbecomespossibletodetermine thevalueofNforthetrainingdatasetDt_,_as_N₌_n_×₅_×_p._In_this_case

asearchwasmadeforpin{20,40,60,80,100,120}.Foreachof thesevalues,nwasvariedintherange[2,32]asalreadymentioned. ConsideringthattheRBFparametersareinitialisedrandomly,20 trialswereexecutedforeach(n,p)pair.Eachtrialconsistedonthe applicationofthemodiﬁedtrainingcriterionLMalgorithmfor200 iterations.

3.5. ResultsforthePMVmodel

Havingdeterminedthemodelsspannedby(n,p)r|20_r=1,whereris

thetrialnumber,theﬁrstresultsoughtwasadecisiononthe num-beroftrainingpatterns.Thedecisionwasmadeonthebasisofthe

Fig.4.Resultsregardingtheselectionofthebesttrainingsetsize,thegeneralisation capacityofthemodels,andtheaccuracyachieved.

modelsmaximumabsoluteerrorobtainedonthevalidationdata set,Dv.Foreachvalueofp,theaverageofthaterrorwascomputed overallthemodels(forallnandr).ThetopplotofFig.4illustrates theresult,whereitisclearlyseenthat,onaverage,itisbesttotrain themodelswith80trainingpatternsforeachmodelparameter.

Themiddleplotillustratesthebestrelationbetweentheresults obtainedwiththetrainingsetandthoseobtainedwiththe vali-dationset,forp=80.Theplotpresentstheminimumandmean oftheaverageabsoluteerrorobtainedoverthe20trialsforeach numberofneuronsn.Thedifferenceisnegligible,whichallowsto concludethatthemodelsprovideexcellentgeneralisation capa-bility.Tothisrespect,itshouldbenotedthatthevalidationdata sethas23,100points,avaluecomparabletothetrainingsetin [29],andthatwithonly1600trainingpatterns(4neuronscase), near25%ofthenumberusedin[30],amaximumabsoluteerror ofapproximately0.015isobtainedbothintrainingandvalidation datasets.Stillforp=80,asitshowedthebestgeneralisation,the lowerplotinFig.4showsforeachnumberofneurons,the mini-mumvalue,obtainedoverthe20trials,ofthemaximumabsolute errorinthevalidationdataset.Itmaybeconcludedthatusingmore than10neuronsisnotnecessaryasnosigniﬁcantimprovements areobtained.

Themodelwith5neuronsinthelowerplotofFig.4, correspond-ingto26parameters,achievesanaverageandmaximumabsolute errorof0.0025and0.011,respectively.Thesevaluesare compa-rabletotheresultsin[30],0.0022and0.0097(obtainedwith100 testingpoints),althoughtheRBFmodelisincomparablysmaller. Onlytworesultsarepresentedin[29]:aSumoftheSquareofthe Errors(SSE)onthetrainingsetof0.11,andafigureshowinghow themodelwith97parametersbehavedona9hexperimentwithin aroom.Regardingthefirstitisabitlessthanhalfthevalueof0.23 obtainedbythe5neuronmodel.Regardingthefigurewemayonly commentthatthewellvisibleerrorsclearlyshowthatthemodel doesnotgeneraliseproperly,probablyduetoalargetrainingset andalso,possibly,duetoovertraining,consideringthesmallvalue ofSSE.

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Fig.5. PMVindexﬁttingonvalidationdataset,obtainedwitha5neuronRBFmodel. Histogramoferroratthetopplot.

Fig.5showstheﬁttingofthevalidationdatasetbythe5neurons RBFmodel,aswellasthehistogramoftheerrorobtained.

InordertoprovideamorerealisticevaluationofthePMVindex model,itwasappliedtoasetofdataacquiredintheroomdescribed previously.Thedataacquisitiontookplaceduringasystem identi-ﬁcationexperimentwherepseudo-randombinarysequenceswere beingappliedtotheairconditioningset-point,hencetherewas sig-niﬁcantvariabilityintheroomenvironment.Theresultisshown inFig.6.Theestimatesprovidedbythemodelareextremely accu-rate:theaverageandmaximumabsoluteerrorswere0.0014and 0.0075.

Anoteisdueontheimportanttrade-offbetweencomputing timeandestimationaccuracy.Fig.7highlightstherelative perfor-manceofRBFmodelsintermsofcomputingtimeandaccuracy.The lowerplotshowstheratiobetweenthePMVgivenby(1)andthe onecomputedbyanRBFANNwithnneurons.Itmaybeseenthata 13neuronsmodelisabout20timesfasterthan(1)anda5neurons modelisapproximately55timesfaster.Thelimitcaseofinterestis 4neuronscorrespondingtoaspeed-upnear70.

Theupperplotinthesameﬁgureshowstherelativeaccuracy intermsofaverageabsoluteerror.Thelimitingcaseofinterestis, maybe,12neurons,about20timesfaster,withthedoubleofthe smalleraverageerrorachievedwith32neurons.Althoughthisplot maysuggestthatlessthan12neuronsachievesabadperformance, themiddleplotclearsthisimpression.Itshowstherelative per-formanceintermsofthemaximumabsoluteerror.Itmaybeseen thatwithonly5neuronsthebestperformanceisalmostachieved, althoughintermsofaverageerrorthemodelisabitworsethanthe bestone.Usingthesecurvesonemayselectaspeciﬁcmodelwith thedesiredbalancebetweenaccuracyandspeed-up.

Fig.6.PMVgivenby(1)(thickblackline)andby5neuronRBFANN(thinwhiteline) duringasystemidentiﬁcationexperiment.Pleasenotethatthelinesarecoincident, hencethewhitelineiswithintheblackline.

Fig.7. RelativeperformanceoftheRBFPMVmodelsregardingcomputingtimeand accuracy.

Whencomparedtothemodelsin[29]or[30],thosehere pre-sentedshowbetterestimationaccuracy,speciﬁcallyonunseendata inrealapplication,provideawidercoverageofthePMVinput vari-ables andof thethermalsensationscale, andachieve speed-up improvements.Inthislastcasethegainissigniﬁcant,beingvery largewhencomparedto[30]andestimatedtobeabout3.5times fasterthanthemodelin[29].

4. Predictivemodels

Allthepredictivemodels,neededin theMBPC,were imple-mented by RBF NNs, and were trained using the procedure described in Section3.3. The topology of the model, i.e., its input–outputstructureandthenumberofneurons,wasidentiﬁed usinga Multi-ObjectiveGeneticAlgorithm (MOGA).Acomplete description of the model structure identiﬁcation procedure is beyondthescopeofthispaper,theinterestedreadershouldconsult [47,43,48,44].

Three auto-regressive predictive models were selected by MOGAsforTao,Hao,andRsg.Theyareusedtoforecasttheoutside

weatherandareemployedwheneveroneofTao,Hao,orRsgis

neces-sarytoobtainroomtemperatureandhumiditymodelpredictions. Inordertoobtainpredictiveairtemperatureandhumidity mod-elsforaspeciﬁcroom,theﬁrststepwasthepreparationofcontrol inputsignalsfortheHVACinternalunit.Forthat,theroomwas con-trolledrandomlybyvaryingthetemperatureset-pointwithinthe range[18,19,...,27]orbyswitchingofftheunitforvaryingtime intervals.ThistaskwasaccomplishedbymeansofPseudoRandom BinarySignals(PRBS),asdescribedin[49,50].Twosetsof predic-tiveairtemperatureandhumiditymodelsweredesigned,onetobe usedinwinterconditions,andtheotherinsummer.Thefollowing subsectionsdescribetheirdesign.

4.1. Summermodels

PRBSsignalswith4416datapatternsweregenerated, corre-sponding to approximately 15 days of data at 5min sampling interval. Different times-of-day werecovered and distinct days (concerningtheoutsideweather)wereconsidered,allduringearly summer.Fig.8showsasamplePRBSsequenceofset-pointsand theresultingroomairtemperatureandrelativehumidity.

Using the MOGA and thetechniques referenced above, pre-dictivemodelsfortheroomairtemperatureandhumiditywere selected.Theﬁrstuses14neurons,theseconduses11.Theinput variablesanddelaysusedbytheselectedmodelsaredetailedin Table1.Asexpected,thereissomedelayfromtheoutsidevariables

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Fig.8.SampleofPRBSsequenceappliedtoHVACsystemandtheresulting temper-aturesandhumidities.Summerconditions.

totheroomclimatemodels,asopposedtotheinsidevariablesand HVACtemperatureset-point.

Withinthemodelidentiﬁcationproceduresthemodelswere evaluatedforlong-termpredictioncapabilitiesusingasubsetof data.Theyweresimulatedforpredictionover anhorizonof 48 steps,whichcorrespondsto4h.WithinthishorizontheRMSerror increasedfromabout0.06◦Cuptoapproximately0.65◦Cforair temperature,and fromnear 0.5% upto about3.0% for relative humidity.Theseresultswereobtainedbyfollowingtheminimum errorpredictionapproach,whichmeansthatmeasuredvalueswere usedeverytimethatfuturevaluesofexogenousvariableswere nec-essaryatthemodelinputs.Consideringa4hpredictionhorizon, theerrorvaluesmaybeconsideredquitesmallandadequatefor thepurposeofinclusioninaMBPCscheme.

4.2. Wintermodels

Inthiscase,2199datapatternsweregenerated,corresponding toapproximately8daysofdataat5minsamplinginterval.Fig.9 showsasampleofthePRBSsequenceoftemperatureset-points andtheresultingroomairtemperatureandrelativehumidity.

UsingtheMOGA,predictivemodelsfortheroomair temper-atureandhumiditywereselected.Theﬁrstuses7 neurons,the seconduses11.Theinputvariablesanddelaysusedbytheselected modelsarepresentedinTable2.

Asforsummermodels,thewinteroneswerealsoevaluatedfor long-termprediction,inthiscasewithinapredictionhorizonof2h (24steps-ahead).WithinthishorizontheRMSerrorincreasedfrom about0.13◦Cuptoapproximately0.26◦Cforairtemperature,and from0.5%uptonear2.5%forrelativehumidity.Aswithsummer

Table1

Input–outputstructureforroomclimatesummermodels.

Variable DelaysinTaimodel DelaysinHaimodel

Tai 0,1,6,7,8,10,11 1,6,8 Hai 0,1,2,7,9 0,1,4,6,7,8,9,10 Tao 2,3,4,5,8 – Hao – 3,5,6,8,11 Rsg 7 6,8,11 Tsp 0,1,4 0,1,2,3,6,9,10,11

Fig.9.12hsampleofPRBSsequenceoftemperatureset-pointsappliedtotheHVAC

systemandtheresultingtemperatures(top,redline)andhumidities(bottom,blue

line).Winterconditions.(Forinterpretationofthereferencestocolourinthisﬁgure

legend,thereaderisreferredtothewebversionofthearticle.)

modelstheminimumerrorapproachwasused,andtheerrorvalues

mayalsobeconsideredquitesmallandamenabletobeemployed

inaMBPCscheme.

5. DiscreteMBPCusingbranchandbound

AsmentionedinSection1,whenthecontrolspaceisdiscrete

ordiscretised it becomespossibleto employsearchtechniques suchastheBBmethodinordertoﬁndanoptimalsequenceof controlactions thatminimises a cost function.BBmethods are structuredsearch techniquescommonly used tosolvecomplex discreteoptimisationandcombinatorialprogrammingproblems bydividingthem intosmallersub-problemsusing atree struc-ture.In a discrete MBPCformulation, the global problemis to ﬁndtheoptimalsequenceofcontrolactionsovertheprediction horizon.Thechoiceofanadequatecontrolactionateveryinstant constitutes thevarious sub-problems tobesolved. Assuming A isavectorofnpossiblecontrolactions,attheinitialstepofthe optimisation,intimeinstantk,theBBmethodcreatestheinitial treenodecorrespondingtothedecisionofwhatactionshouldbe takenatthattimestep.Asncontrolcombinationsareavailable, thecorrespondingnumberof branchesis createdbycomputing thepredicted systemoutput, ˆy(k+1), and for each branchthe costfunction,J(k+1),isevaluated.Inthenextpredictionstep,for k+1,theprocessisrepeatedforthenodescreatedineachbranch resultingfromthepreviousstep,creatingn2 _new_branches._The

wholeprocessisrepeateduntiltimeinstantk+PH−1isreached, where thenumber of createdbranches isnPH_. _The_exponential

natureofthewholeprocessisclearandevenforasmallnumberof controloptionsandnottoolargepredictionhorizons,thenumber of available solutions quickly becomes prohibitively large. The optimal solution is chosen by selecting the control trajectory,

Table2

Input–outputstructureforroomclimatewintermodels.

Variable DelaysinTaimodel DelaysinHaimodel

Tai 0,6,7,8,11 1,5,8 Hai 0,1,5,6,9,10,11 0,1,3,6,8,10 Tao 1,4 – Hao – 3,6,9,11 Rsg 0,3,4,8,9 6,9,10 Tsp 0,1,3,10,11 0,1,2,5,8,9,10,11

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U(k)=[u(k)u(k+1)···u(k+PH−1)],thatminimisestheestimated

accumulatedcostfromtimeinstantk+1tok+PH:

J1:PH(k)=

k+PH

i=k+1

J(i) (14)

The description above assumes unrestricted branching and

resultsinanenumerativesearchovertheentirespaceofcontrol

solutionsspannedbyAoverthepredictionhorizonPH.Asalready

pointedoutthistypeofsearcheasilybecomescomputationally

pro-hibitiveandinordertoreducethenumberofsolutionsenumerated,

twoapproachesaretaken:theuseofboundstorestrictbranching

andperformingthesearchoveracontrolhorizon,CH<PH.As

for-mulatedin[51],twoboundsareemployed:anupperboundonthe

totalcostfrominstantk+1tok+PH,andalowerboundonthe costfrominstantk+itok+PH.Attimestepiintheoptimisationa branchisfollowedonlyifthecumulativecostfromstep1tostep i−1,J_1:i−1(k),plusthelowerboundonthecostfromitoPH, ˆJi:PH(k),

issmallerthantheupperboundonthetotalcost, ˆJ1:PH(k).Thusthe

branchingruleisgivenby:

J_1:i−1(k)+ ˆJi:PH(k)< ˆJ1:PH(k) (15)

Thisrulemaybefurtherdecomposedbynotingthatitssecondterm onthelefthandsideoftheconditionequalsthecostofusinga controlproﬁleAjatstepi,

J(k+i)|u(k+i−1)=Aj,

plustheestimatedcostfromstepi+1toPH:

J_1:i−1(k)+J(k+i)|u(k+i−1)=Aj+ ˆJi+1:PH(k)< ˆJ1:PH(k) (16)

Whentheruledoesnotholdthebranchisnotfollowedbecauseit doesnotcontainanoptimalsolution,thuspruningallthetreenodes thatwouldbecreatedfromthecurrentnode.Thebounds estima-tionmethodandavailabilityareproblemdependent,althougha basicapproachissuggestedby[51]:ateachinstantk,beforethe optimisationstarts,aﬁrstsearchonthetreeofpossiblesolutionsis donebysuccessivelychoosingthecontrolactiongivingthe small-estvaluesofJ(k+i)PH_i=1,asearchusuallycalled“greedy”.Thetotal costfoundistheinitialestimatedupperbound, ˆJ1:PH(k).Ifatalater

stageintheoptimisationasmallervalueisfound,itreplacesthe previousone.Regardingthelowerbound, ˆJ_i+1:PH(k),ifanadequate estimatemaynotbecomputed,itissuggestedthatitissetto0for allstepsioftheoptimisation.Inthetypicalformulation,branching isonlyperformeduntilthecontrolhorizonisreached,therefore theremainingcostmustbeestimatedforallinstantsfromCH+1 uptoPH,forexamplebyusingthegreedyapproachjustdescribed. ItisworthnotingsomeadvantagesoftheBBmethodoverother non-linearoptimisationtechniqueswhenappliedtoMBPC: • Theoptimalsolutionisalwaysfound.Thisguaranteesthatthe

controllerisoptimalinthediscretespaceofcontrolalternatives. Importantly,noassumptionsneedtobemadeontheformulation ofthecostfunction.

• Themethodimplicitlydealswithconstraintswithoutbeing neg-atively affected. Constraints will most certainly improve the efﬁciencyofboundingbyeliminatingthosealternativesthatlead toconstraintviolation.

• As opposed to other iterative optimisation methods, the algorithmoutcomeisnot negativelyinﬂuencedbya poor ini-tialisation,althoughthetimespenttoﬁndtheoptimummaybe greater.

6. Problemformulation

Inordertomaintainthermalcomfortandsimultaneously min-imisetheenergyspent,theproblemmaybeformulatedasfollows. Thecostofselectingonecontrolaction,Tsp,atinstantiisdeﬁned

as, ˆJ(i)=

⎧

⎨

⎩

1+|Tsp−Tao|, Tsp>0 0, Tsp=0 (17)

whereTsp=0encodestheactionofswitchingofftheHVACunit.The

scalingfactorisusedonlytomakethattermsmallwhen com-paredto1.Inpracticeitshouldbechosenbytakingintoaccountan estimateofthemaximumvalueof|Tsp− Tao|.Thetermitselfreﬂects

thenotionthatthehigherthedifference|Tsp− Tao|,thebiggeristhe

energyrequiredtoachieveTsp.Usingthedeﬁnition(17)theHVAC

controlproblemissimply, minimise U(k) J1:PH(k)=

_k+PH

i=k+1 ˆJ(i)

U(k) subjectto | ˆ(i)|<T , (18)

where ˆ(i)istheestimatedPMVindexresultingfromselectingthe set-pointTspattimeinstanti.TisathresholdvalueforthePMV

indexwhichshouldguaranteeacceptablethermalcomfortforthe occupantsofthespace.TheASHRAEstandard[26]recommendsa valueof0.5whichpredictsthatlessthan10%oftheoccupantswill bedissatisﬁed.

7. ResultsoftheMBPCapproach

Usingthemethodologiesdescribedinprevioussections,a num-berofexperiments,bothinsummerandwinterperiods,havebeen carriedouttotestthefunctionalityandassessthecorrectnessand robustnessofthecontrolsystem.

7.1. Summerexperiments

Theresultsthatwillbepresentedanddiscussedinthissection wereobtainedina classroomequippedwithcomputers,where studentshaveanumberofcoursesondifferentcomputerscience topics(roomDdescribedinSection2).Aftermakinganumberof systematicmeasurementsontheairvelocitywithintheroomfor differentsettingsoftheHVACfanspeed, itwasconcludedthat, excludingthevicinityoftheairducts,thevelocitywasonaverage below0.1ms−1.WhencomputingthePMVindex,avalueof0.65clo wasusedfor theclothinginsulationand avalueof1.0Metwas employedforthemetabolicrate.Thismeansthatthecontext,for thePMVmodelsused,wasC=_{0.65,1.0,0.1_}.RegardingtheMBPC systemparameters,thecontrolhorizon,CH,wassetto5samples (25min)andthepredictionhorizon,PH,to48samples(4h). 7.1.1. Experiment1

Fig.10presentsonesituationofahotsummerdaywherethe roomwasinusewhenthesystemstartedoperation.Theinitial thermalcomfortindex,,isabovethe0.5thresholdandtheHVAC takesalmost2hat18◦C set-pointtobringtheroomto accept-ablethermalcomfortconditions.Beyondthispoint,withtheroom withnoload,thesystemisabletomaintainthedesiredconditions byusinghigherset-pointsandbyswitchingofftheHVACwhen possible,thereforeconsuminglessenergy.

The room air temperature model does not have an input accountingfortheroomusage,consequentlythecontrolalgorithm isnotabletoforeseeandactpre-emptivelytocounteractthestrong impactcausedbyaclassenteringtheroom.

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Fig.10.HVACcontrolforthermalcomfortinsummerconditions.Roomisinuse whensystemstarts.Bothplots:shadedareashowsroomactivitymonitor sig-nal.Upperplot:measured(reddash-dot)andone-step-aheadpredicted(reddot) humidity,areshownaboveshadedarea;same(inblue)belowshadedareaforair temperature;dashandsolidlinesshowoutsideairtemperatureandACset-point. (Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereaderis referredtothewebversionofthearticle.)

7.1.2. Experiment2

Fig.11presentsasecondsituation,onanevenslightlyhotter day,whereafteraﬁrsttimespanofroomoccupancyendingwith acceptablethermal conditions,thesystemwasabletomaintain thoseconditionsonasecondclasswithlesseffortthanthatofthe ﬁrstcase.Thiswouldbetheexpectedbehaviourifthesystemcould predicttheroomoccupancy.

7.1.3. Experiment3

Onwhatconcernstheroomairtemperatureandrelative humid-itymodels,itmaybeseenthatthepredictionsarequiteaccurate whichresultsona goodforecast ofthePMV indexand correct operationof thesystemthroughtime. Thisisconﬁrmedbythe resultspresentedinFig.12,whichgivegoodindicationsregarding robustness.Thesystemoperatedforabout48hmaintaininggood thermalconditionsandshowingexcellentroomclimatemodelling, alsoaccountingpositivelyontheoutsideweathermodels.Ifthey wouldnotprovidesufﬁcientlyaccuratepredictionstheroom cli-matemodelsaccuracywouldsuffer.

7.2. Winterexperiments

Theaimofthefollowingexperimentswasnotonlytotestthe functionalityandassessthecorrectnessandrobustnessofthe con-trolsystem,inwinterconditions,butalsotoassessanypossible energysavingsthatitmaygenerate.Inordertofulﬁlthesecond goal,3adjacentsimilarroomswereused.Theobjectiveistorun theMBPCalgorithminoneoftheroomsandcomparetheresults tothoseobtainedbytheHVACbuilt-incontrolsystemintheother rooms.Pleaserecalltheroomsdescriptionthatwasgivenin Sec-tion2.RoomAandCwillemploytheconventionalcontrolsystem, andMBPCwillbeappliedinRoomB.Aftermakinganumberof

Fig.11.HVACcontrolforthermalcomfortinsummerconditions.Pre-emptive con-trolimprovesperformance.Thesamestructure,linestyleandcolourcodingofFig.10 wasused.(Forinterpretationofthereferencestocolourinthisﬁgurelegend,the readerisreferredtothewebversionofthearticle.)

systematicmeasurementsontheairvelocitywithinthe3rooms fordifferentsettingsoftheHVACfanspeed,itwasconcludedthat, excludingthevicinityoftheairducts,thevelocitywasonaverage closeto0.08ms−1.Avalueof1.0clowasusedfortheclothing insu-lationandavalueof1.0Metforthemetabolicrate.APMVmodel withcontextC={1.0,1.0,0.08}wasthereforeemployed.Regarding theMBPCsystemparameters,thecontrolhorizon,CH,wasagain setto5samples(25min)andthepredictionhorizonto48samples (4h).

7.2.1. Experiment4

Fig.13showstheresultsofaﬁrstsmallexperimentcarriedout totesttheexperimentalset-up.

InroomB(solidredline)theset-pointisdeterminedbythe MBPCsystem.InroomsA(blackdash-dotline)andC(bluedash line),ﬁxedset-pointsof23and22◦Cwereemployed,respectively. Thesevaluesarecommonlyusedaspre-programmedset-pointsfor theroomswithintheHVACmanagementsystem.Theoccupants frequentlysetevenlargervalues.Theexperimenttookplaceatthe endoftheday,duringwhichtheaverageoutsideairtemperature was8.8◦C.

Itmaybeseenthatinitiallyalltheroomsareoutsidethecomfort zoneasthePMVindexisbelow−0.5.Afterabout20minallthe roomswerewithinthecomfortzone,inthecaseofBbyactionof theMBPCsystemwhichshowsgoodresponsivenesstothevicinity ofthePMVindexthresholdof−0.5.Wheneverthesystemforesees thatthethresholdwillbereacheditdeterminesoneofthepossible set-pointsandappliesittothesystem.Thisisanindicationofthe goodpredictiveabilityofthemodelsandoftheirusefulnessfor theBBsolutionsearch.Itis alsoobviousthat theHVAC system temperature regulation is very poor,as in rooms A and C the temperaturewasalwaysfarawayfromthetargetvalue.Theresult isthatthesystemwaspermanentlyheatingforapproximately2h and20min.InroomBtheMBPCwasmakinggooddecisionsasit

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Fig.12.HVACcontrolforthermalcomfortinsummerconditions.About48hofoperation.Thesamestructure,linestyleandcolourcodingofFig.11wasused.(For interpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthearticle.)

wasmaintainingthecomfortindexabovethe−0.5limitwithout switchingontheACpermanently.Duringthe2:20hthattheAC washeatinginroomsAandC,thetotalactivetimeinroomBwas 55min,whichcorrespondstoareductionofabout60%.

7.2.2. Experiment5

A second experiment was conducted in order to verify the resultsobtainedintheﬁrstshortexperiment.It startedearlyin theday,around6:00a.m.andlasteduntil12:30.Duringthisperiod theoutsideairtemperatureincreasedfrom7.5to16.5◦C,withan averageof11.7.Thesettingsandset-pointswerethesameasfor theﬁrstexperiment.TheresultsareillustratedinFig.14usingthe samelinestyleandcolourcodingasinFig.13.

Duringtheﬁrst4hofthisexperimentthebehaviourwassimilar totheoneobservedintheﬁrstexperiment.Atthatpointthesystem inroomCstartedtoswitchonandofftheACsystemasifithad

reachedtheset-point,although,asmeasuredbytheWSN,itdid not.Surprisingly, asitseemedtohavereacheditsset-point,up totheendoftheexperimenttheACsystemkeptincreasingthe temperatureinthatroom.InroomA,thesameproblemsasbefore wereobserved,astheACsystemiscontinuouslyactive.

TheMBPCsysteminroomBmanagedverywelltomaintain thethermalcomfortandhaverelativelylargeperiodsofinactivity. WhencomparedtoroomsAandCthereductioninheatingtime wasapproximately41%and45%,respectively.Theresultsachieved reinforcetheideathattheexistingHVACsystemprovides inade-quatetemperatureregulationandhasquitedifferentperformance amongtherooms.Becauseofthisanadditionalexperimentwas carriedoutonlyinroomB.

InordertobeabletocomparetheMBPCinroomBwiththe standardcontrolalgorithminthesameroom,rightafterthesecond experimenttheMBPCwassubstitutedbyaﬁxedset-pointof22◦C.

Fig.13. Winterconditionscontrolexperimentinthreeadjacentrooms:A(dash-dotblackline);B(solidredline);andC(dashblueline).Leftplotsare,fromtoptobottom, theroomairtemperatureandthePMVcomfortindex.Thetop-rightplotshowstheevolutionofthetemperatureset-pointoftheAC,andthebottom-rightplotshowsthe state,onoroff,oftheAC.(Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthearticle.)

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Fig.14. Asecondexperimenttoestimaterelativeenergysavingsinwinter condi-tions.ThesamecolourandlinetypecodingasinFig.13,wasusedforroomsA,B, andC.(Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereader isreferredtothewebversionofthearticle.)

Thisset-pointresultsinaPMVindexvaluenotfarawayfrom−0.5, thethresholdthatwasgoverningthebehaviouroftheMBPC.As theoutsidetemperaturewasstillrising,thefixedset-pointcontrol taskwouldnotbeharderwhencomparedtothepreviouscouple ofhoursinthesecondexperimentwheretheoutsidetemperatures weresmaller,thereforetoagoodextentthelast2hofthesecond experimentarecomparabletothefirstcoupleofhoursofthefixed set-pointexperience,withoutfavouringtheMBPC.

Theresultsfromthecomparisonarevisible inFig.15.Asin previousexperimentstheMBPCkeepsgoodtrackofthethermal comfortrestrictionwhichisonlymarginallyviolatedbecauseof theinevitableerrorofpredictivemodels.Thesystemonlyspends energywhennecessarytomaintaintheminimumlevelof com-fortspeciﬁed.Regardingthestandardtemperatureregulation(in dash-dotblackline),itmaybeseenthatregardlessof attaining theset-pointand of switchingonand off sometimes,it keeps increasingtheroomtemperatureunnecessarily,therefore wast-ingenergy.InthiscomparisontheMBPCactivatedtheACsystem approximately37%lessthantheHVACcontrol.

7.2.3. Experiment6

Onwhatconcernstheroomairtemperature,relativehumidity andPMVindex,itmaybeseeninFig.16thattheone-step-ahead predictionsareaccurate, resultingongoodPMVindexforecasts andoncorrectoperationofthesystemthroughtime,whichgive goodindicationsregardingrobustness.Thisﬁgurewasobtained bylettingthesystemrunforapproximately11h,wheretheunit washeatingonly15%ofthisperiod.Duringthistimethesystem operatedcorrectlyandcontinuously,alwaysmaintainingdesired thermalconditionsandshowingexcellentroomclimatemodelling, alsoaccountingpositivelyontheoutsideweathermodels.

Itshouldbenotedthatfromsample40uptosample95,the roomwasbeingusedforlectures.Thisrepresentsathermalload thatactuallycontributestomaintainthecomfortduringwinter. Onsummerconditions,asitwasseen,thesystemisrequiredto react,therefore,asinsummermodels,aninputmustbeaddedto themodelstoaccountfortheroomoccupancyschedule.

Table3

Energy(inkWh)assignedtoeachindoorunit.

Experiment A B C1 C2

4 5.3 2.5 5.3 4.4

5 13.6 6.9 14.6 11.8

6 24.8 5.6 24.8 16.3

7.3. Energysavings

Aswithinthe16locationsmonitoredthereare14interiorair

conditioningunits,it wasnoteconomically possibletoactually

measuretheelectricenergyspentineveryindoorunit.Asa

com-promisesolution,powertransducerswereinstalledinthethree

outdoorunits,thereforemakingtheelectricenergyvalues

avail-ablefortheseunitsthroughtheLMAP02interface.Aprocedure,

suggestedbyMitsubishi,wasthenfollowedtogiveaﬁrst,crude

approximationtotheelectricenergyconsumptionofeachindoor

unit[52].

DenotingbyEo(T)theelectricalenergyconsumedbytheoutdoor

unit,duringaperiodT,byToniandTthithetime(inhours)thatthe

fanisoperatingoftheithindoorunitandthattheunitiscooling orheating,respectively,thentheenergyEi(T)(inWh),assignedto

theithindoorunit,outofNunits,isgivenby: Ei(T )=Eo(T )

Tthi

iTthi+

220Toni+32_N. (19)

Theconstantsin(19)havebeencomputedforthemodelsinuse, following[52].

Inexperiments4–6,theindoorunitsofroomsA,BandCwere connectedtothesameoutdoorunit.RoomAandBhaveasingle indoorunitwhileroomBhad2.RoomsAandCwereunderconstant temperaturecontrol,whilstinroomBtheproposedMBPC method-ologywascontrollingtheindoorunit.Theroomswereadjacent, locatedintheleftcornerofthebuilding(pleaseseeFig.1).Thearea ofroomAisequaltotheareaofroomBandhalfofroomC(hence thetwoindoorunits).Thetotalelectricenergy(inkWh)assignedto eachunit,foreachexperiment,andusingtheproceduredescribed above,isgiveninTable3.

Table4showsthesavings,inpercentage,achievedinthe elec-tricenergyspentbyunitB,comparedwithalltheotherunits,for eachexperiment.Asitcanbeseen,thesavingsrangefrom41%, achievedinthe5thexperimentandtakingthe2ndinteriorunitof roomCasthecomparisonunit,and77%,inexperiment6,forthe interiorunitinroomAandthe1stinteriorunitofroomC.These valuesofenergysavings,ifononehandwereexpectedbecausethe systemhastheabilitytousefutureinformationandbecauseituses moreanddiverseinformation,ontheotherhandtheyalsooccur duetotheinadequateoperationofthestandardcontrolintheHVAC system.Thismayoccurforvariousreasons:malfunctions, inade-quatesensorlocations,badqualityequipment,excessivelysimple controlalgorithms,orinadequateset-pointsbeingset.Whatever thereason is,theresultsobtainedshowthatbyhavinga MBPC commandingtheHVACsystem,importantsavingsinenergy,inthe orderof50%,aretobeexpected.

Table4

Savings(in%)ofBcomparedwithA,C1andC2.

Experiment A C1 C2

4 52 52 43

5 49 52 41

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Fig.15.ComparisonofMBPC(solidredline)withﬁxedset-pointHVACsystemcontrol(dash-dotblackline)inroomB.Winterconditions.Thesamecolourandlinetype

codingasinFig.14,wasusedforroomB.(Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthearticle.)

Fig.16.PredictedPMV,airtemperatureandrelativehumidity,for11hinwinterconditions.Dotsaremodelpredictedvalues.Activitymonitorsignalshowninshadedarea ofmiddleplot.

8. Conclusionsandfuturework

Amodelbasedpredictivecontrolmethodologyusingthebranch andboundmethodwasformulatedandappliedtocontrol exist-ingHVACsystemsinbuildings.Totheextentthemodelsaccuracy allow,theformulationguaranteesthatanoptimalcontrol trajec-toryiscomputedinordertomaintainadesiredlevelofthermal comfortandtominimisetheenergyspentindoingso.Thesystem isconceptuallysimple andintegrateseasilywithexistingHVAC systems.Thefeasibilityandrobustnesshavebeendemonstrated experimentally.Accordingtotheresults,importantenergysavings

areachievedbyhavingthemodelbasedpredictivecontroller deter-miningtheoperationofHVACsystems.Thesesavingsareprobably above50%.

Asthemodelbasedpredictivecontrolneedsseveralcalculation ofthepredictedmeanvalueindex,itscomputation,usingasetof radialbasisfunctionneuralnetworkmodels,hasbeeninvestigated. Themodelsshowgoodestimationaccuracyoverwiderangesof theinputvariablesandprovidegoodcoverageofthethermal sen-sationscale.Whencomparedtopreviousworks,thedesignofthe modelswasplannedindetailwiththepurposeofusingthemin real-timecontrolapplications,thegeneralisationofthemodelswas

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testedthoroughly,andaprocedurewasshowninordertoselecta modelonthebasisofadesiredcompromisebetweenspeed-upand estimation.

Futureworkwillfocusontheimprovementofthecontrol sys-tembymakingitawareoftheroomsoccupancyschedule,andon estimatingmoreaccuratelythepotentialenergysavings.

Acknowledgements

The authors wish to acknowledge the support of the Portuguese National Science Foundation (project grant PTDC/ENR/73345/2006), the European Commission for the grant PERG-GA-2008-239451 and the University of Algarve for theCeratonia2008Award.ThisworkwasalsosupportedbyFCT, throughIDMEC,underLAETA.

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