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Integrated Photovoltai Modules

Nynne Friling

KongensLyngby2006

Supervisor: Prof. HenrikMadsen-TehnialUniversityofDenmark,IMM

ExternalSupervisor: Ing.J.J.Bloem-JointResearhCenter,Italy

Dr. M.J.Jiménez -CIEMAT,Spain

IMM-M.S.-2006-80

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Building321,DK-2800KongensLyngby,Denmark

Phone+4545253351,Fax+4545882673

reeptionimm.dtu.dk

www.imm.dtu.dk

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The present thesis dealswith mathematial modelling of the heat transfer in

buildingsandbuildingintegratedphotovoltaimodules. Oneofthepurposesof

this thesisis to extendtheknowledge abouttheperformane ofbuilding inte-

gratedphotovoltaimodules.

Due to thelimitedoilresouresit beomesmoreand moreimportantto fous

onrenewableenergy. Themainpurposeofthemodulesistoprodueeletriity,

andasaspin-otheairbehindthemodulesanbeheatedandusedforheating

thebuilding. Thisistwowaysin whih toproduerenewableenergy.

The point of turn of the mathematial methods applied is stohasti dier-

ential equation, state spae models, maximum likelihood estimation, and the

extended Kalmanlter. Subsequentthe model estimation various analysesof

theresidualsandtestsofthemodelshavebeenarriedoutin ordertounover

thereliabilityandusefulnessofthemodel. Theestimatedmodelsarebothlinear

and non-linearmodels. The non-lineareet tobeinvestigatedis theinfrared

radiationandthewindspeed.

Thethesisemanatesfromtheartile'Estimationofnon-linearontinuoustime

models for the heat exhange dynamis of building integrated photovoltai

modules'[Jiménez et al. 2006℄. The basisof the applied models of the pho-

tovoltaimoduleensuefromtheartile. Introdutorymodelsofhowtodesribe

theheat transferin buildingsareoutlined anddeveloped. Tehnis andmeth-

ods suh asRC-modelling and lumpingare touhed on. After the desription

oftheappliedtheorytherestofthethesisisrelatedtotheanalysisofthepho-

tovoltaimodule. The module temperature is highly negative orrelatedwith

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temperature of themodule in order to determinethe produtionof eletriity

andtoinvestigatehowthetemperatureofthemoduleanbedereased. Intro-

dutorilyanin-depthanalysisofhowtodesribethetemperatureofthemodule

isarriedout. Thisis donebothinthesense ofidentifyingthebestdesribing

variables,andalsoinordertodisoverthebestttingmodels. Bothsingleand

multiplestatespaemodelsareestimated. Thismakesitpossibletoexamineif

itisneessarytosplit upthemodule temperatureintwostates.

The resultsof the thesis

Thebest desriptionofthedata isobtainedby usinganextended single state

modelontainingnon-linearinuenes. Thebest resultsarefoundbyapplying

thetemperatureatthetopof themodule astheoutput variable. Thisnding

is supported by thermal images taken of themodule where the measuredtop

temperature oversmost of the module temperature. The analyses showim-

provedperformanefor applied variables alulatedfrom theeletrial owof

themoduleompared to thesimilarmeasuredvariable. Oneofthemain aims

has been to identify the inuene of the ambient wind speed. It wasbefore-

handexpetedthatalteredversionofthewindspeedwasinueningtheheat

transfer between the air and the module. The analyses have revealed that a

lteredwind speedhavingnearlythe sameutuationsasthemeasured wind

aetsthemodule.

Afteridentifyingthepreferablemodel,datawherevariationsoftheset-uphave

beenmade,areanalyzed. Theresultofthetestingrevealsthatthemodelisable

todisriminatethevariationsfromeahother. Itisidentiedthathigherfored

veloity behind the module and obstalesturning the laminar air ow into a

moreturbulentowraisetheheat transferoeientbetweentheambientair

andthemodule. Thisleadstothedesireddereaseofthemoduletemperature.

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Denneeksamensopgaveomhandlermatematiskmodelleringafvarmeoverførelse

ibygningerogbygningsintegreredesolellemoduler. Etafformålenemeddenne

opgaveer,atopnåenøgetvidenom,hvordanbygningsintegreredesolellemod-

ulerfungerer,oghvadderpåvirkerdem.

Detblivermereogmerevigtigtatfokuserepåvedvarendeenergigrundet,debe-

grænsedeolieressourer. Idenne opgaveeret bygningsintegreretsolellemodul

medentvungetluftstrømbagmoduletanalyseret. Hovedfunktionenforsolelle-

modulet erat produere strøm. Som en sideeekt kanden tvungne luftstrøm

bagmoduletanvendestilopvarmningafbygningen. Pådennemådefremstilles

dertoforskelligeformerforvedvarendeenergi.

Dematematiskemetoder,somanvendesidenneopgave,erstokastiskedieren-

tialligninger,state-spaemodeller,maximumlikelihoodestimationogdetudvid-

ede Kalmanlter. Efter at haveestimeret modellerne gennemføres forskellige

residualanalyser og test af modellerne. Dette gøres for at afdække, om mod-

ellerne erpålideligeog brugbare. Deestimeredemodellerdækkerbådelineære

ogikke-lineæremodeller. Vindoginfrarødstrålingundersøgessomikke-lineære

indydelser.

Denne eksamensopgave udspringer af artiklen: 'Estimation of non-linear on-

tinuous time models for the heat exhange dynamis of building integrated

photovoltaimodules'[Jiménez et al. 2006℄. De anvendtemodellerfor solelle-

modulet stammer fra denne artikel. Indledende er modeller til at undersøge,

hvordanvarmeoverførelsekanbeskrivesskitseret. Teknikkerogmetoder,såsom

RC-modeller og lumpning, er berørt. Efter beskrivelse af den anvendte teori

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turen er negativt korreleret med eektiviteten af modulet. Dette betyder, at

det ervigtigat kunne prædiktere modultemperaturen for at kunne bestemme

produktionafstrøm. Ydermeregiverdettemulighedforatundersøge,hvordan

modultemperaturen kanredueres. Indledningsvis er en dybdegående analyse

tilat kortlæggemodultemperaturengennemført. Dette gøresbådeforat iden-

tieredebedstbeskrivendevariabler,menogsåforat ndedenbedstemodel.

State-spae modellermed bådeen ogeretilstandeeridentieret. Dette gør

detmuligatundersøgenødvendighedenafatopdelebeskrivelsenafmodultem-

peratureniereområder.

Opgavens resultater

Denbedstebeskrivelseafdatafåsvedatbrugedenmestavaneredeikke-lineære

state-spaemodel meden tilstand. Debedste resultateropnåsvedatanvende

dentemperatur,somermåltitoppenafmodulet. Detteresultatunderstøttesaf

varmebillederaf modulet, hvor denmåltetoptemperaturrepræsentererstørst-

edelenaftemperaturenimodulet. Analyserharogsåvistenøgetgradafbeskriv-

else,nårvariabler,somerestimeretudfradenelektriskestrøm,anvendesfrem

for detilsvarende måltevariabler. Etaf hovedformåleneharværet at identi-

ereudendørsvindensindydelsepåmodulet. Påforhåndvardetforventet, at

enform forltreretvindhastighedville haveindydelsepåmodulet. Analysen

harafsløret,atdenltreredevindhastighed,sompåvirkermodulet,kunafviger

megetlidti variationfradenmåltevindstyrke.

Efter at have identieret den foretruknemodel, erdata, hvorder er varieret

påtest-opsætningen,anvendtoganalyseret. Resultaterneafdette forsøgviser,

atdetermuligtformodellenatskelnemellemvariationerneiopsætningen. Det

eri denneforbindelse fundet, aten øgetventilationshastighedbag moduletog

modstande,somændrerstrømningenfralaminærtilturbulentiluftrummetbag

modulet,øgervarmeoverførelseskoeientenmellemudendørsluftenogmodulet.

Detteførertildenønskedereduktionafmodultemperaturen,sombetyderøget

strømproduktion.

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Thismaster'sthesiswaspreparedatInformatisMathematialModelling(IMM),

Tehnial University of Denmark (DTU) in fullment of the requirementsfor

aquiring the Master degree in Engineering. The thesis represents 40 ECTS

pointsoutoftherequired300ECTSpoint. Theprojethasrun fromSeptem-

ber2005to theend ofAugust2006.

Thethesisdealswithstohastimodellingofaphotovoltaimodule. Thepur-

poseistoidentifythephysialfatorsinueningthetemperatureandeieny

of the module. One of the ornerstones of the report is to pinpoint the non-

linear inuene from the outdoor wind speed. The basis of the thesis is the

artile'Estimationofnon-linearontinuoustimemodelsfortheheatexhange

dynamisofbuilding integratedphotovoltaimodules'.

ThethesisissupervisedbyProfessorHenrikMadsenfromIMM.Externalsuper-

visors are Ing. J. J.Bloem, European CommisionJointResearh Center,and

Dr. M.J.Jiménez, ResearhCentre for Energy,Environmentand Tehnology

in Spain.

Lyngby,30

th

ofAugust2006

Nynne Friling

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Thebakgroundofthisthesisistheartile'Estimationofnon-linearontinuous

timemodelsfortheheatexhangedynamisofbuildingintegratedphotovoltai

modules'byM.Jiminez,H. MadsenandH. Bloem. Themodelstreatedin the

artilearereusedanddevelopedfuther. Alsosomeoftheapplieddataoverlap.

Inthepresentthesismoredataandombinationsofdataareinvestigated.

In order to make this thesis get o properly a few introdutory items have

to belaried.

All the models have been estimated by means of the software CTSM.

[Kristensen&Madsen2003b℄givesashortintrodutiontothemodellingin

CTSM.Inthisguidealsoimagesareinluded. Thismayhelpunderstand

someofthestatementsandexplanationsinthethesis.

Thex-axisoftheplotsovertime hasto beexpound. Thevalueisstated

inminutes. Valuezeroisintimeaordanewithmidnight. Observation

720orrespondsto12a.m. andsoforth.

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I wishtothankmyProfessorHenrikMadsen whoduring thewholeperiod has

helpedand guidedmethroughthepreparationofthis thesis. I would alsolike

to thank Henrik Madsen for giving me the oppertunity of patiipating in the

Dynastee Conferenein Athensin Otober2006andthe stay attheJointRe-

searhCenterin Ispra,Italy.

I would like to thank Ing. J. J. Bloem from the European Commision Joint

Researh Center in Ispra for therewardingweek at theCenter. I appreiated

the visit at thetest site sineit gaveme some hands-on experiene. Further-

moreI wishtothankPhD.MarìaJoséJiménez,CIEMAT,ResearhCentrefor

Energy,EnvironmentandTehnology,inSpain,forbakgroundinformationon

the models. Thankyouboth verymuh forinputsand orretions during the

proessofwritingthisthesis.

DuringtheprojetIhavealsohadmeetingswithresearhersfromRisøNational

Laboratory,DanishTehnologialInstituteand DepartmentofCivil Engineer-

ing at Tehnial Universityof Denmark, whohavetaken the time to help me

intheproess. Theresearhershavehelpfullyprovidedmewithnewknowledge

andartiles.

I would alsoliketo thankmyfellowstudents, Anna, Stig,andSøren,fortheir

help, support,andideasduringthiswork.

Finally I would liketo thank Annalise Dühring Wiberg for herhelpful inputs

onthelinguistisofthethesis.

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Summary i

Resumé iii

Prefae v

Introdution vii

Aknowledgements ix

1 RenewableEnergy 1

1.1 Renewableenergymethodsfordwellingsandmathematialmod-

elling. . . 1

2 The MathematisBehindHeat Dynamisof Buildings 3

2.1 Introdution. . . 3

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2.3 Lumpedparameters . . . 4

2.4 Dierentkindsofheat transfers. . . 5

2.5 RC-models . . . 8

2.6 Modelextentionsanddataimprovements . . . 14

3 Non-linear Heat Transfer Phenomenain Buildings 15 3.1 Detailsofthephenomena . . . 16

4 The MathematialMethodsUsed in the Modelling 19 4.1 Continuous-disretestohasti statespaemodels. . . 20

4.2 Testing ofthemodel . . . 25

4.3 Filteringmethods. . . 31

4.4 Modelvalidation . . . 32

5 Modellingin CTSM 35 5.1 Preparationsforthemodelling . . . 35

5.2 Theatualmodelling. . . 36

5.3 Afterthemodelling. . . 37

6 Photovoltaiin General 39 6.1 Introdution. . . 39

6.2 WhatisPVtehnology? . . . 40

6.3 TheonstrutionofaPVmodule . . . 40

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7 The Applied Variablesand theMeasuring Methods 47

7.1 TheTest RefereneEnvironment . . . 47

7.2 Introdutiontothemeasurements . . . 49

7.3 Outputvariable . . . 49

7.4 Inputvariables . . . 51

8 SingleState Models- Model identiation 55 8.1 Thermalmodels . . . 55

8.2 Theapplieddata . . . 56

8.3 Thedesignof themodels . . . 56

8.4 Whyuseheattransfermodels? . . . 57

8.5 Theproedureforthemodellingand theanalysis . . . 58

8.6 Thelinearmodels . . . 60

8.7 Introdutiontothenon-linearmodels . . . 63

8.8 Simplenon-linearmodel . . . 64

8.9 Extendednon-linearmodel . . . 68

8.10 Analysisoftheparameterestimates . . . 71

8.11 Analysisoftheresiduals . . . 72

8.12 Comparisonofthemodels . . . 73

8.13 Plotofresiduals . . . 74

8.14 ACFandPACF. . . 76

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8.16 Tests oftheresiduals . . . 80

8.17 Modelvalidation . . . 82

8.18 Disussion . . . 83

9 MultipleState Models- Top and Bottom Divided Model 85 9.1 Themodel. . . 85

9.2 Simplemultiplestatemodel . . . 88

9.3 Advanedmultiple statemodel . . . 92

9.4 Summation . . . 95

10Analysis of the Fored Ventilation in the AirGap 97 10.1 Theset-upandthedata . . . 97

10.2 Theresultsoftheanalysis . . . 100

10.3 Summary . . . 109

11FutureWork 111 12Conlusion 113 A Appendixto theChapter Analysis oftheFored Ventilation in the Air Gap 117 A.1 Thedates ofthedataolletion . . . 117

A.2 Thedierenebetween

T moduletop

and

T air

. . . . . . . . . . . . 118

A.3 Cumulatedperiodograms . . . 119

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Renewable Energy

1.1 Renewable energy methods for dwellings and

mathematial modelling

Todayaroundone-thirdtohalfoftheenergyonsumptionintheindustrialised

ountries is used for lighting and making the thermal ondition of dwellings

omfortable[Prasad&Snow2005℄. Furthermoreitisestimatedthathalfofthe

world'soilresoureshavealready beenused. If nobetterinsulation andnew-

thinking energy prodution methods are taken into onsideration in a wider

sense, the remainingoil resouresare estimated to overthe energy needsfor

30 years. Forthemomentthere is nodiretsubstitute foroil. Besidethe lim-

itation of resourestherenewableenergy prodution analso redue thelevel

ofpollutionandglobalwarmingwhihinertainareasisalargesaleproblem.

Therenewableenergysouresarenotdependingonoilressoures,unlessin the

produtionphase,andareforthisreasonlessontaminating.

Favourable regulations and taxations are a way to enourage the appliation

of renewableenergy methods. These ations antake plae on both interna-

tional andnationalsale. Forthemomentonlyfewationshavebeentaken.

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ergy, solarheating, biofuel, andphotovoltaitehnology, whih is the topi of

thisthesis. Therststeptowardsalargerproportionofrenewableenergyisan

eient furtherdevelopmentof equipmentandmethods. This needstobefol-

lowedupbyontinuously improvementsoftheequipments,thematerials, and

the methods. At present, new thinking photovoltai ells are under develop-

ment. It isexpensive, both in termsof energyand eonomis, to produethe

ellsmadeofsilionwhiharethepreferredonesatthemoment. Thereforenew

materialssuhasorganimaterialarebeingresearhedasasubstitutematerial.

Anotherimportantissueinonnetionwithrenewableenergyistoobtainaep-

tanefromtheonsumers. Itisofgreatimportanetoprovidereliableinforma-

tionontheenergy-performaneaswellastheoverallpreformaneofalternative

energysouresto get ahigher aeptanefromthe onsumerofrenewableen-

ergies. Thisisforinstanetheasewhenaonsumerhastodeideifabuilding

integration of photovoltaimodule will be beneial. Canmathematis be of

any benetin relation to theissue? Toobtainagenerally better understand-

ing oftherenewableenergy methods,mathematial modelling anbe applied.

Mathematial modelling an as an example, help to determine whih materi-

als aremost eient. The optimal operationallevel may also be determined.

Furthermore,themathematialmodellingmaygiveanunderstandingofwhih

externalfators,suhaswindspeed,areinueningtheequipment. Sometimes

itisevenpossibleto revealformerunknowninuenes.

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The Mathematis Behind

Heat Dynamis of Buildings

2.1 Introdution

Thissetionisanintrodutiontothemathematisandthephysiallawsbehind

modellingofheatdynamisofbuildings. Furthermore,theadvantagesofusing

statistial and physial knowledge simultaneously, alled grey-box modelling,

willbedisussed. Inthemodelsusedinthishapteritisassumedthattheheat

transferrelationshipsarelinear.

Sine themathematial desriptionof heat dynamis ofbuildings anbevery

ompliated,methodstooveromethiswillbedisussed. Lumpingisusedasa

waytoderease theomplexityofthemodels. The modelsfoundare so-alled

RC-models. In extension of the lumping method RC-models of four dierent

types of houses are outlined. Some of the desriptions and models are also

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2.2 Modelling method

When searhing for a representative model for a physial phenomenon it is

obvioustousethephysiallaws. Thisapproahtotheproblemisalledwhite-

box modelling or the dedutive approah. At the other end of the sale is

the blak-box modelling where only statistial models are used. This way of

analyzingdataisalsoalled anindutiveapproah. Thisapproahrequiresno

preeding knowledge about the data and its nature. This is for instane the

asewhen neuralnetworks areused formodelling data. In betweenthese two

methods is the grey box modelling. This approah uses the knowledge from

bothphysisand statistis,resultingin bettermodels. Grey-boxmodelling in

relationtostohastimodellingistreatedintheartiles[Kristensenetal.2003a℄

and[Kristensenetal.2003b℄. Byusinggrey-boxmodellingitispossibletotest

ifeahoftheparametersin themodelsapplied is signiant,andtherebyitis

oftenpossibletoreduethenumberofparametersinthenalmodel. Thisalso

impliesthat itbeomes easierto interpretthemodel. Anotherreasonwhythe

grey-box models are preferable is that the nal models may sometimes show

physial relationshipswhih were not known in advane, [Andersen 2001℄ and

[Madsen2001℄.

Previousresearhhasshownthatthededutiveapproahisonlyabletodesribe

thelong-timevariationsoftheheattransfer,whihimpliesthatitisnotpossible

togettheshort-timevariationsandutuationsmodelled[Madsen1985℄.Short-

time variation is when the heat transfer hanges due to fast hanges in the

weather onditions, for instane due to a loudpassing by thesun. Byusing

theindutiveapproahwithtwoormoretimeonstantsitispossibleto model

boththeshort-andthelong-timedynamis,whihraisesthelevelofdesription

themodelsprovides.

2.3 Lumped parameters

Whendealingwithheatdynamisboththetimeandthespatialoordinatesare

seenas stohastiallyindependent variables. This implies that it is neessary

to introdue partial dierential equations to desribe the orrelations. Sine

partialdierentialequationsanoftenbeveryompliatedtosolve,thelumped

parameterizationmethodwillbeusedtosimplifythemathematis. Themethod

turnsthepartialdierentialequationintoordinarydierentialequations,whih

aremuheasiertohandleandsolve. Lumpedproessesaredenedas: Lumped

proessesareallproessesinwhihthespatialdependeneofthevariablesunder

onsiderationanbenegleted[Keman 1988℄. Theuseof lumped parameters

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hanginginthespaedimensions. Itisthereforealwaysimportanttoinvestigate

ifthemethodisreasonabletouseinthespeiase. Itiswelldoumentedthat

theuseoflumpingisanaeptablewaytomodelthedynamisofbuildingsifthe

building is notinuenedbynon-linearphenomena [Madsenet al.1994℄. The

useoflumpedparametersismostfrequentlyagoodwaytoanalyzeases,whih

otherwisewouldbeomeveryompliated. Figure2.1showsanexampleofhow

lumpedparameterizationanbeusedwhenmodellingasimplewall. Thiskind

of modelling is alledRC-modelling, beausethesystemis built ofresistanes

and apaitanes similar to analogous eletrial systems. From Figure 2.1 it

an be seen that one of the approximations is that the heat transfer is only

distributed horizontally. Therefore,thisapproahisonlyvalidinthis aseifit

is known that the temperature of thewalls are homogeneouslydistributed. If

temperaturesaremeasuredvertiallyatseveralplaesin thewall,thelumping

anbeworkedoutinlayers. Thisapproahanbeimplementedbyaddingmore

stateequationstotheexistingsystem.

0

1 0 1 0 1 0 1

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11

000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111

T u T i

Figure2.1: Illustrationofalumpedsysteminawall[Madsen1985℄

2.4 Dierent kinds of heat transfers

Therearethree waysinwhihheatan betransferred:

Condution

Convetion

Radiation

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Onedenition of heat transferisthat it isnormally transferredfrom ahigher

temperatureobjetto anobjetwithalowertemperature[hyperphysis2006℄.

Oneofthephysialreasonswhyitisdiulttomakeananalytialdesription

oftheheattransferisthatthethreedierenttransmissionwaysareontributing

totheproesssimultaneously.

TheEquations2.1-2.8,arefoundin [Madsen1985℄.

2.4.1 Condution

Condutionmeans energy transmittedthroughrandom splie betweenatomi

partiles. Inrelation to heat dynamis of buildings this kind of energyis ob-

servedasheattransferthroughthewalls,oors,androof.

Themathematialformulationofondutionisbuiltonanempiriallaw,[Both

&Christiansen2002℄,

dQ

dt = − λA dT

dx

(2.1)

where

dQ

dt

istheheatpertimeunit.

λ

isthethermalondutivityonstantwhih

variesquitealotdependingonwhihmaterialsareusedfortheonstrutionof

thewall. Physialfators,suhastemperature,havealsoimpatonthethermal

ondutivity,but sinethethere arenolargevariationsin thephysialfators

whendealingwithheatingofbuildings,onlytheinueneofthematerialsused

is onsidered.

A

is the ross setion of the ondutive solid, in this ase the

wall,perpendiularto thediretionoftheheaturrent.

dT

dx

isthetemperature gradient.

If the part of the building onsidered does not ontain neither heat soures

norsinks,itispossibletosetupanordinarydiusionequationwhihdesribes

thetimerateof thetemperature

∂T

∂t = − λA cρ ( ∂ 2 T

∂x 2 + ∂ 2 T

∂y 2 + ∂ 2 T

∂z 2 )

(2.2)

where

c

istheheatapaityand

ρ

isthedensityofthematerial.

x

,

y

,and

z

are

thespatialoordinates. Intheaseswherewallsonsistofonlyonematerial,it

ispossibleto makeanapproximationtotheproblembyreduingEquation2.2

tobeonlyone-dimensional:

∂T

∂t = − λA cρ

2 T

∂x 2

(2.3)

Equation2.3beomesevensimplerifthetemperaturegradientisseenason-

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isstatedas

d

.

dT

dt = λA T 2 − T 1

d

(2.4)

T 1

and

T 2

arethetemperaturesattheboundariesofthelayer. TomakeEquation 2.4validfortheentirethiknessofthewallEquation2.5anbeused.

T o

and

T u

denotestheoutsideandtheinside surfaetemperatureofthewallrespetively

dT

dt = U A(T o − T u )

(2.5)

where

U

,thetransmissionoeient,isgivenas

1

U = X d i

λ i

(2.6)

i

denotesthenumberofhomogeneouslayers.Thismakesitpossibletomakethe earlierdesribedlumpeddesription. Asstatedthroughoutthesetion,thewall

mustonsistofhomogeneousparallellayersiftheone-dimensionaldesriptionis

to besuient. Inaseswhereanapproximationforaninhomogeneouswallis

satisfyingtheone-dimensionmethodaboveanbeused. Furthermore,itisalso

possibletoapplythelatterproedurewhenmakingapproximatealulationsof

ordinaryonstrutedwallswhenonditionsarenon-stationary.

2.4.2 Convetion

The seondontributorto heat transferof buildingsis onvetion. Thisis the

kind of heat transfer where a uid is in motion as a result of dierene in

temperature. Thestandardequationforonvetionisseenbelow

dQ

dt = hA∆T

(2.7)

where

h

istheonvetiveheattransferoeientwhihisdependingonfators

suhasompressibility,visosity,temperature,theveloityandtheproleofthe

owinthemedium, andthedistanebetweenthelayers[Both &Christiansen

2002℄. Furthermore,this impliesthat theonvetionoeientvaries in value

depending on whether the wall is horizontal or vertial.

A

denotes the area

whih theheat is passing through.

∆T

isthe temperature dierene between thewallandthemainbodyoftheuid.

2.4.3 Radiation

Thelastofthethreewaysinwhihheattransferantakeplaeisbyradiation,

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bodieswithdierenttemperaturesarein optialontatwitheahother. This

istypiallythesituationwhentheheattransferistransmittedthroughwindows.

The heat is exhanged from for instane the sun to the ooring on the other

sideofthewindow. Inaseswhereoneofthebodiesissurroundedbytheother

Equation,2.8anbyappliedtosolvetheproblem.

dQ

dt = Aǫσ(T s 4 − T 4 )

(2.8)

A

is the surfae area of the surrounded body,

ǫ

is the emissivity, and

σ

is

the Stefan-Boltzmann onstant,

σ = 5.67 · 10 8 W m 2 K 4

. As in the aseof

onvetionradiationisoftendiulttoalulateinpratie. Theheattransfer

byradiationinsidebuildingsisratherlimited,butitisneessarytoinludethe

radiationinthemodelforroomswithwindows. ItanbeseenfromEquation2.8

thatthealulationsfortheradiationareheavilydependentonthetemperature

dierenes.

2.5 RC-models

Takingthe previoussetionsinto onsiderationit isobviousthat theheat dy-

namisdesriptionofentirebuildingsanbeveryompliated. Agoodapproxi-

mationofmodellingtheheatdynamisofbuildingsistouselumpedRC-models.

Thehoieofbuildingmaterialshasagreatinueneontheheatbalaneequa-

tions and thereby the ostrution of the models. Four models with dierent

ombinationsofbuildingmaterialsanbefoundinthesetions2.5.1-2.5.4. The

modelsaresimpliedsinethereisonlyoneroom,onewindowandnovariations

intypeofwallsandoorintheindividualmodels. Inthesetionsonlydierene

betweenlightonstrutionsandheavyonstrutionsismade. Thisisduetothe

fat that the heat transfer is transmitted faster through a light onstrution

thanthroughaheavyonstrution. Previousresearh[Hansen1985℄hasshown

thatitisneessarytoinludeoneormoretimeonstantstodesribeforinstane

asolidwall. Inextensiontothisitisobviousthatitisneessarytousedierent

RC-modelsto desribevarious kindsofbuildings ranging from greenhousesto

solidoldhurhes. Thersttwomodels ontaintwotimeonstantswhilstthe

twolatterexamplesontainthreetimeonstants. Itisobviousthatitispossible

toextendthemodelsmadeandreatenewmodels. Commentsonextentionsand

improvementsaresuggestedinasetionafterthedesribtionofthefourmodels.

Themostimportantsimpliationsforallfourmodelsare:

(25)

2. All surfaes, apart from the window, are onsidered to have the same

temperature

3. Radiationtransferas amehanismfor heattransferbetween

T i

,

T o

, and

T m

isnotonsidered.

4. Theheat apaityof theroomairisnegleted.

Thetemperaturesmentionedin bullet numberthree aboveare:

T i

is theroom

temperature,

T o

isthetemperatureofthewallsurfaes,and

T m

isthetemper-

ature of the heat aumulating layer. To reate the model the heat balanes

fortheroomair,thesurfae,andtheheataumulatinglayersneedtobeiden-

tied. Therefore it is neessarytoestablish individual heat balaneequations

onerning eahof the four models. All the generalpriiples are explained in

relationtotherstmodelonly.

2.5.1 Light walls and solid oor

Theharateristisofthis rstmodelarethat thedominatingheatapaityis

loatedin theoorandinthegroundunder theoorwhilethewallsarether-

mally light. A typial building with these harateristis ould bea building

onstrutedofwoodenwallswith aonrete oor. InFigure2.2 bothamodel

and theanalogouseletrial systemforthebuilding areoutlined.

T u

isapure

temperaturesoure,

r

symbolizestheresistanewhihisresponsiblefortheex- hangeofheat.

C

istheapaitane. The

φ

-valuesarealledpureheatsoures

[Davies2004℄.

00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 11111

00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000

11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111

000 000 000 000 000 000 000 000 000

111 111 111 111 111 111 111 111 111

00000000000000000000000000000 00000000000000000000000000000 00000000000000000000000000000 11111111111111111111111111111 11111111111111111111111111111 11111111111111111111111111111

r i

φs

T u T i

C i

T m

c m r u

φh φh

C i

C m

r u

φs φh

T i r i T m T u

Figure2.2: Illustrationofabuildingwithlightwallsandthedominatingheatapaity

intheoor

(26)

In the following the heat balane equations for the building desribed above

will be outlined. Fromthe analogous eletrial systemshown in Figure 2.2 it

ispossibleto seehowsomeoftheheatbalaneequationsarefound. Thethree

heatbalanesare:

Theroomtemperature

dQ k

dt + X

hA h (T o − T i ) + Gc a (T l − T i ) = 0

(2.9)

Thesurfae

dQ r

dt = X

k 1 A 1 (T o − T u ) + X

k 2 A 2 (T o − T m ) + X

hA h (T o − T i )

(2.10)

Theheataumulatinglayer

c m dT m

dt = X

k 2 A 2 (T o − T m )

(2.11)

The explanations of the notation are:

T u

is the temperature of the outside surfae,

T l

is thetemperature oftheventilationair,

Q k

dt

isthe onvetion part

from persons, light, radiators, et.

Q r

dt

is the heat transferred diretly to the surfaeasradiationfrom thesun, light, radiatorset.

G

denotes thequantity

oftheventilationairand

c a

isthespeiheatoftheair.

T o

,

T i

and

T m

denote

respetivelythetemperatureofthesurfae,theroom,andtheheataumulating

layers.

A

denotesareas. Byinsulating

T m

,

T 0

,and

T i

inthethreeheatbalane

equations, Equations2.9-2.11,itispossibleto formulate adeterministilinear

statespae model in ontinuoustime with twosteadystate equationslikethe

oneshown in Equation2.12. Itanbeseenthat only therst equationhas a

dynamiimpat.

dT m

dt = AT m +

B

U T 0 = C 1 T m + D 1 U T i = C 2 T m + D 2 U

 

(2.12)

U

= (T u T l

Q k

dt Q r

dt ) T

(2.13)

T m

is the state vetor and

U

the input vetor.

U

ontains all the external

fatorswhihhaveinueneon thesystem,see Equation2.13.

A

,

C 1

, and

C 2

are onstants while

B

,

D 1

and

D 2

are vetors onsisting of onstants. The

speiexpressionoftheonstantsanbefoundbyisolating

T m

,

T 0

,and

T i

in

theheatbalaneequations.

Themodelformulatedaboveisnotsuient fordesribingtheheat dynamis

ofabuilding,beauseitisonlyabletodesribeslowhangesequivalenttolong-

(27)

toabuildinganbeinuenedbyshorttimedynamissuhasaloudpassing

by the sun. Adding oneortwotime onstant, amodel where both long-and

short-timedynamisanbemodelledisidentied [Hansen1985℄. Thereforean

extratimeonstantisaddedinthemodelbelow. Whenhavingmorethanone

time onstant it an be of advantage to express Equation2.12 by the use of

matries,whihisdoneinEquation2.14

c m dT m

dt

c i dT i

dt

= 1

r i

1 r i

1 r i − ( r 1

u + r 1

i ) T m

T i

+

0 0 A w P

1

r u 1 A w (1 − p) 

 T u dQ t

dQ dt s

dt

(2.14)

where

c m

and

c i

arethetotalheatapaitiesofthewallsandtheair,respetively.

r i

istheresistaneagainsttheheattransferbetweenthelargeheataumulating

mediumandtheroomair.

r u

istheresistaneoftheheattransfertotheoutdoor

air. The details of the expressions of these resistanes anagain be foundby

solvingtheheat balaneequations. Furthermore,thesystemalso ontainsthe

heat supplies for theradiators,

dQ r

dt

, and thesun,

dQ s

dt

.

A w

isthe area of the

window,

p

indiates theeetivepartofthewindowwhere thesolarradiation

ishavinginuene on

T m

.

2.5.2 Solid walls and light oor

In this model the dominating heat apaity is found in the outer walls, while

the oorsare onsideredto belight andwell isolated. Atypialbuildingwith

these harateristisisabrikhousewithwoodenoors. Sineitwasoutlined

inthelattersetionhowtondandisolatethebalaneequations,theequations

fortheheattransferin thisasearefoundto be:

c m dT m

dt

c i dT i

dt

=

− ( r i 1 + r 1 u ) r 1 i

1 r i

− 1 r i

T m

T i

+

1

r u 0 0 A w P 0 1 A w (1 − p)

 T u dQ t

dQ dt s

dt

(2.15)

Ingure2.3isadrawingofthebuildingandtheonordantanalogouseletrial

system. The models above an be identied on the basis of the analogous

(28)

000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111

111111

00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111

00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111

00 00 00 00 00 00 11 11 11 11 11 11

r u

φs C m T m

r i C i

T i T u

φh φh

T u

T i

C m

r u T m

φs φh C i

r i

Figure2.3: Illustrationofabuildinghavingthedominatingheatapaityintheouter

wallsandalight oor

2.5.3 Light walls and double solid oor

Thismodeldesribesthesituationwherethedominatingheatapaityisplaed

in the oor. The walls are onsidered to be light. Compared to the former

modelsthismodelhasthreetimeonstants. Thereasonwhytwotimeonstants

havebeenhosenfor thedesription of theheat transfer in theoorareaare

due toheat sensitivity. Thisthird modelhasprovedto beanexellentwayof

modellingagreenhouse[Nielsen1996℄.Thistypeofmodelhasproventobeable

toestimatereliableandauratephysialparameters[Nielsen1996℄. Itwasalso

found that twotime onstantsare not suient in order to desribethe heat

transferofgreenhouses. Themodelis outlinedin Equation2.16andin Figure

2.4.

  c m 1

dT m1

dt

c m 2

dT m2

dt

c i dT i

dt

  =

 

− ( r 1

i + r 1

j ) r 1

j

1 r i

1 r j

− 1

r j 0

1

r i 0 − ( r 1

i + r 1

u )

 

 T m 1

T m 2

T i

 +

0 0 A w P

0 0 0

1

r u 1 A w (1 − P)

 T u dQ t

dQ dt s

dt

(2.16)

(29)

000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111

00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000

11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111

00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 00000000000000000000000000000000000000 00000000000000000000000000000000000000 11111111111111111111111111111111111111 11111111111111111111111111111111111111

r u

φs T i

C i

T

m2

c m 2

c m 1 r i

T u

φh

T

m

φh

φs

C i

r i

r u T i

r j

T m 2

T m 1

C m 1

C m 2

T u

φh

Figure 2.4: A building having the dominating heat apaity inthe oor while the

outerwallsarelightonstrutions

2.5.4 Double solid walls and light oor

The dominating heat apaities in this ase is inside the building. Two heat

aumulatinglayersforthewallsareinludedinthemodelbeauseofthethik

outerwalls. The model is shown in Equation2.17. Figure2.5 showsasheme

of thebuildingalongwith theanalogouseletrialsystem. Thismodel anfor

instanebeusedwhenmodellingoldhurhessinethikwallsareoftenpresent

in theseonstrutions[Madsenetal.1994℄.

  c m 1

dT m1

dt

c m 2

dT m2

dt

c i dT i

dt

  =

− ( r 1

i + r 1

v ) r 1

v

1 r i

1 r v

− 1 r v 0

1

r i 0 r 1

i

 T m 1

T m 2

T i

 +

0 0 A w P

0 0 0

0 1 A w (1 − P)

 T u dQ t

dQ dt s

dt

(2.17)

000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111

111111

00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111

00000000000000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000 11111111111111111111111111111111111111 11111111111111111111111111111111111111 11111111111111111111111111111111111111

00 00 00 00 00 00 11 11 11 11 11 11

000 000 000 000 000 111 111 111 111 111

T u

r u

φs T i

C m 1

φh r i

T

m1

T

m2

r v

C m 1

C i φh

T u

C m 2

r i r v T m 2

r u

C m C m 1

T i T m

φs φh

Figure2.5: Twodominatingheatapaitylayersintheouterwalls,whiletheindoor

oorisalightonstrution

(30)

2.6 Model extentions and data improvements

Thefour models giveaguideline forhowto reatemodels forheat transferof

simpliedbuildings. ByusingtheRC-modellingitispossibletomakeextentions

whih suitapartiularontext. Theextentionsouldbeotherombinationsof

theheataumulatinglayersormoreompliatedbuildingsforinstanehaving

morerooms. However,it isimportantto keepin mind thatit isonly possible

tomodellinearsenarioswiththeRC-models.

Besidesthedevelopmentofdesribingmodelsimprovementsouldalsobemade

in relation to theinput variables. A ontributorto heat transferwhih isnot

taken into onsideration is the temperature of the soil, both underneath the

buildingand in thesurroundingarea. Onereasonwhythesoiltemperaturein

previousmodelsis nottakenintoonsiderationouldbethat thetemperature

underneaththebuilding isonsidered to beapproximatelyonstant. This will

nothangethetotalheat transferofthebuilding.

It ould be interesting to add the heat transfer aused by the surround soil

bothin senseofthediereneintemperatureandthereetiontothemodels.

Thereetionisofspeiinterestforinstaneinareaswheresnowisommon

[Tehnologies2006℄.

Inrelationtoreetionthisisameasurementwhihisnotavailableinallases.

Thesoftwareusedinthisthesisisabletoestimatetheinueneofunmeasured

variables. This isdonebyaddinganextrastatein thesetofsystemequations

orbyinludingthe parameterin themodel. Later in thethesisthelong-wave

radiationtothesurroundingsisestimated.

(31)

Non-linear Heat Transfer

Phenomena in Buildings

IntheprevioushapterdierentRC-modelshavebeenexamined. Asmentioned

RC-modelsanonlybeusedtomodelheattransferofbuildingsthatareinu-

enedbylinearheattransferphenomenaandinaseswhereapproximationsare

adequate. The aim of this hapter is to identify non-linear heat transferphe-

nomenathatausestheneessityofintroduingnon-linearmodellingmethods.

Verylittleresearhhasbeenmadein theeldofnon-linearheattransfer. This

anbeduetothewidespreaduseofRC-models. Anotherreasonisalowlevelof

knowledgeingeneralaboutnon-linearinuenesonheattransfer. Inthisthesis

thephenomenonisallednon-linearheattransfer,inpriniplethiskindofheat

transferanbedesignatedbyother terms,suhasadvanedorunknownways

ofheattransfer. Thisimpliestheriskthatthephenomenaatuallyinvestigated

werenotdisoveredin theresearh.

Theresearh hasshown thatradiation andmoistureare twoof themostwell-

desribed and best known ausesof non-linearitiesin relationto heat transfer

in buildings. Another less desribed phenomenon, whih an be assumed to

inuene the heat transfer in buildings with non-linear eets is the outdoor

wind-load.

(32)

Over time dierent non-linear terms seem to be important in the modelling,

due to the hange in hoie of onstrutions and designs of buildings. The

dominating building materials in the past deades have been briks and on-

retewhiharebothvulnerableofmoisture. Oneofthenewtrendsofbuilding

envelopes is the façades primarily made of glass. These types of façades are

expeted to bemoreinuenedby thewind onditionsand thenon-linearities

in the ase of radiation. Furthermore the double envelopes and shading an

ontributetothenon-linearities.

3.1 Details of the phenomena

Thefollowingsetionsgivesomeexplanationsandreasonswhythenon-linearities

ourandin someaseswaystoexpressthephenomenamathematially.

3.1.1 Moisture

Theinterationbetweenheattransferandmoistureinbuildingsisanareawhere

alotofresearhhasbeendone. Inspiteoftheeortdenitionoftheproblemis

stilllaking[Xiaoshu2002℄. [Liesen&Pedersen 1999℄mentionmoistureasthe

most non-linear apaitanedepending primarily on vapourdensity and tem-

perature.

Itispossibleforthemoisturetoenterthebuildingenvelopesinemostbuiding

envelopes are made of porous materials suh as briks [Xiaoshu 2002℄. When

the moisture enters the briks several ompliated proesses will start. It is

diulttodesribetheboundaryonditionswhenmoisturehasperolatedinto

thebriks. Alldegreesofmoistureenompassedfromdrytosaturatedinuene

the material properties suh as thermal ondutivity, density and heat trans-

fer. In[Liesen&Pedersen 1999℄itis notedthat espeially theheattransferis

inuened. This fatoftenleadsto theneessityof simplifyingthe desription

byapproximations. Whenspeimensaretakenitisverydiulttodetermine

experimentallyanyjointonlusions,sinethereisnopatternforthespreadof

themoisture. Theseirumstanesleadtothenesseityofnon-linearmodelling.

Moistureintheouterwallsis notonlyinuenedbytheweatheronditions,it

isknownthat wallsin newbuildingshaveahigherlevelofwaterontentthan

oldwalls. Furthermore,themoistureisnotspreaduniformlyinthewalls[Häupl

etal.1997℄. Thisisalsoanindiationthat theapplied heattransferoeient

variesovertime. Thisisanimportantaspetinrelationtotimeseriesanalysis.

The followingartiles treat the topi of moisture in buildings in anon-linear

(33)

1999℄and[Xiaoshu2002℄.

3.1.2 Wind speed

The inuene ofwindloadhas animpaton theheat transfer. Intheartile,

[Jiménezetal.2006℄,onwhihthisthesisisbased,itisshownthattheambient

windspeedaetstheheattransferfromthephotovoltaimoduletotheambient

air. In theartile the windspeedis raisedto an unknown power,

W k

, whih

makestheinuenenon-linear. Inrelationtothematerialsaetedbythewind

in anon-linear way, it an only beasertained forertain for glass, sine the

photovoltaimodulemainlyonsistsofglass. Thisareaofresearhisinteresting

inrelationtomanynewly-builtbuildingshavingfaçadesdominatedbyglass. In

[Troelsgaard1981℄ it is stated that theambient overall oeient of the heat

transfer,

α u

, is afuntion of the wind speed raised to the power of 0.78.

α u

doesonlyhaveinueneinrelationtotheheattransferthroughglass. Empirial

resultshaveproventhatthealulationof

α u

hastobedividedintwoseparate

equations,whiharefoundinEquation3.1,[Troelsgaard1981℄.

α u =

10.8 + 3.9W W ≤ 5m/s

5.0 + 7.15W 0.78 W ≥ 5m/s

(3.1)

Comparing the statement of the nessiity of dividing the alulation of the

overall oeient of the heat transferto the nding in [Jiménez et al. 2006℄,

there isadivergene. Frombakgroundknowledgeabouttheonditionsof the

wind at the test site it is known that the measured winds are below 5 m/s,

and still the non-linearinuene of thewind is tested to be signiant. This

underlinesthestrengthofestimationmethodswhereitispossibletodetermine

if it is neessary to apply the inuene of the wind as non-linear or if it is

suienttoassumelinearity.

Noartilesoranalyseshavebeenidentiedorhavedesribedthisbehaviourfor

forinstanebriksorotherporousmaterials.

3.1.3 Radiation and the solar angle of inidene

Theinfrared radiation,f. Equation2.8, isalsoanon-linearinuenewhihis

neededtobetakenintoonsiderationespeially,duetothetemperatureraised

to thefourthpower. Inthelateranalysesofthisthesisitis shownthatseveral

infraredradiationontributorsaresigniant.

The relationship betweenthe heat transfer and theangle of inidene is non-

linear[Melgaard1994℄.Theresearhhasshownthatnotonlytheheattransfer

(34)

onthisnon-linearrelationship. Ithasbeenshownthat buildingswithmehan-

iallyventilateddouble envelopefaçadesannot bemodelledbyRC-modelling

sinetheshadingandtherebytheinideneangleoftheradiationarenon-linear

[Manzet al.2004℄.

(35)

The Mathematial Methods

Used in the Modelling

In this hapter the applied mathematial methods will be outlined. Therst

part desribes theory onerning estimation of models while the seond part

desribesmethodsofanalysisofoutputdataofthemodel.

The strengthofthe mathematismathods applied isthat itis possibleto test

the models and estimate not measuredvariables. A very importantaspet of

thisapproahisthattheestimationofthemodelsinorporatepriorknowlegde,

whihinreasesthequalityofthemodels.

4.0.4 Estimation methods

The models applied in the thesis are stohasti dierential models, whih are

dierentialequationswhereoneormoreofthethermsareastohastiproess.

To estimate parameters in stohasti dierential models, two essential issues

havetobeonsideredinadvane. Primarilyifthereareanypriorknowledgethe

parametersinthemodelorifalltheparametershavetobeestimated. Seondly

it isalsoneessarytodeideifthe modelis goingto ontainnon-linearterms.

These twopointshaveimpatonwhihmathematial methodsare tobeused

(36)

4.1 Continuous-disrete stohasti state spaemod-

els

Continuous-disretestohasti statespae modelsare builtupby twokindsof

equations: asetofsystemequations,(4.1),andasetofmeasurementequations,

(4.2). Thesystemequationsarestohatidierentialequationsandaredened

in ontinuous time whereas the measurement equations are in disrete time.

This reetsthefat that themeasurementsare olleteddisretely, whilethe

desriptionofthedataismodelledtobestatedinontinuoustime. Thefuntion

f (x t , u t , t, θ)

isalledthedrifttermand

σ(u t , t, θ)

isthediusionterm. Ifthe

diusion term is not inluded in the model it will redue into a state spae

modelbasedonordinarydierentialequations. Inthisaseitisnotpossibleto

identifytheunertaintyofthesystemequations. Stohasti statespaemodels

enablethepossibilityto model physial systemswhererandom utuationsin

thestatesarepresent.

dx t = f (x t , u t , t, θ)dt + σ(u t , t, θ)dω t

(4.1)

y k = h(x k , u k , t k , θ) + e k

(4.2)

Where

x t ∈ X ⊂ R n

is a vetorof state variables,

u t ∈ U ⊂ R m

is avetorof

input variables,

t ∈ R

is the time variable,

θ ∈ Θ ⊂ R p

is avetorof parame-

ters,

y k ∈ Y ⊂ R l

is avetorof output variables.

f ( · ) ∈ R n

,

σ( · ) ∈ R n × n

and

h( · ) ∈ R l

are possiblenon-linear funtions;

{ ω t }

isan

n

-dimensional standard Wiener proess, also known as arandom walk, and

{ e k }

is an

l

-dimensional white noise proess with

e k ∈ N (0, S(u k , t k , θ))

.

σ( · )

is the gain of the in-

rements of the Wiener proess. It is assumed that

dω t

and

e k

are mutually

unorrelated.

The stohasti term,

dω t

, in Equation 4.1 hanges the equation from being a

determinististatespaemodeltoastohastistatespaemodel. Theapplied

measurementvaluesmayhaveadeviationfrom thetruevalues. Thisdeviation

will then be ontained in the noiseterm. Another reason forintroduing the

noisetermisthatthemodelanbedeientduetovariablesnotonsideredin

themodelhavinganinueneonthesystem. Thisanbeausedbythefollow-

ingirumstanes: either duetoalakofknowledgeaboutthevariables'eet

onthesystem,orbeausemeasurementsofthevariablearenotavailable. The

measurement error,

e k

, ontainserrorsin theoutput signalswhih are aused

bymeasurementnoise[Madsen2001℄and[Andersen2001℄.

Itisobviousthattheaimistohavevaluesofboth

dω t

and

e k

aslowaspossible

(37)

4.1.1 Parameter estimation

The purposeof theparameterestimation isto estimatethe optimalset of pa-

rameters. Thisisobtainedbymaximizingthelikelihoodfuntionofthemodel.

In this thesistwomethods of parameterestimation are desribed, namely the

maximumlikelihoodestimationandthemaximumaposterioriestimation. Max-

imum likelihoodestimation is appliedwhen there isno priorknowledgeabout

the parameters whereas maximum aposterioriestimation is perferable if any

prior knowledge on the parameters is available. Inthis thesis onlythe maxi-

mum likelihood estimation will beapplied. The mathematis behind the two

methods arequitesimilarsinethemaximumlikelihoodestimationisaspeial

aseofthemaximumaposterioriestimation. It isthereforedeidedtooutline

thetheoryofthemaximumaposterioriestimation.

ConsideringavetorY ontaining

S

stohastiallyindependentontinuousse- quenes,Equation4.3.

Y = [ Y N 1 1 , Y N 2 2 , . . . , Y N i i , . . . , Y N S S ]

(4.3)

Eahofthe

Y

'sisdened asin Equation4.4.

Y N i i = [y N i i , . . . , y i k , . . . , y i 1 , y 0 i ]

(4.4)

Furthermore

p(θ)

needstobeintroduedasthepriorprobabilityfuntionofthe parameters,

θ

.

Thepointestimatesinthestatespaemodelanbefoundasthevetorof pa-

rameters,

θ

,thatmaximizetheposteriorprobabilitydensityfuntion,Equation 4.5:

p(θ | Y) ∝ Y S i=1

p( Y N i i | θ)

!

p(θ)

(4.5)

Theexpressioninthelargebraketsisthelikelihoodfuntion.

WhenapplyingBayesrule,

P (A ∩ B) = P (A | B)P (B)

,andtheprinipleofprod-

utsofonditionalprobabilitydensityfuntions,

p( Y N i i | θ)

,inEquation4.5an

bewrittenas:

p( Y N i i | θ) ∝

N i

Y

k=1

p(y k i |Y k i − 1 , θ)

!

p(y 0 i | θ)

(4.6)

It is neessaryto use theonditional densities to form the likelihood funtion

sine the residuals of the ordinary dierential equation part of the stohasti

dierentialequationsareorrelated.

(38)

densityfuntion:

p(θ | Y) ∝ Y S

i=1 N i

Y

k=1

p(y k i |Y k i − 1 , θ)

! p(y 0 i | θ)

! p(θ)

(4.7)

Thediusion term,

dω t

, in Equation 4.2is assumed to theindependent ofall thestatevariables,

x t

, andfurthermoreit isdrivenby aWienerproess. The inrementofaWienerproessisGaussiandistributed,whihmakesitpossibleto

approximatetheonditionalprobabilitydensityfuntionbyGaussiandensities.

This assumptionreates theopportunityto applythe Kalmanltertehnique

for estimating the mean and ovarianeof the onditional probability density

funtion inorder tobeableto alulatethelikelihoodfuntion. TheGaussian

distributionisompletelydesribedbyitsmeanandovariane. Thetehnique

willbedesribedindetailinSetion4.1.2. Thestohastidierentialequations

are driven byaWienerproess. Theharateristisof theWienerproessare

thattheinrementsareGaussianItiswell-knownthat thedensityfuntionfor

themultivariatenormaldistributionisgivenas:

f (x) = exp( − 1 2 (x − µ) T bsΣ 1 (x − µ)) p | 2πΣ |

(4.8)

ˆ

y i k | k 1

istheondititionalmean,

R i k | k 1

representstheondititionalovariane and

ǫ i k

is theresidual. Equation4.9-4.11

ˆ

y k i | k 1 = E { y k i |Y k i − 1 , θ }

(4.9)

R i k | k 1 = V { y i k |Y k i − 1 , θ }

(4.10)

ǫ i k = y i k − y ˆ i k | k 1

(4.11)

TheparametersareassumedtobeGaussiandistributed,whihleadsto:

µ θ = E { θ }

(4.12)

Σ θ = V { θ }

(4.13)

ǫ θ = θ − µ θ

(4.14)

Whentaking these assumptions into aount theposterior probability density

funtion anbeformulatedas:

p(θ | Y) ∝

 

 Y S i=1

 

N i

Y

k=1

exp

1 2i k ) T (R i k | k 1 ) 1 ǫ i k r

det

|2πR i k | k 1 )

 

× p(y i 0 | θ)

!

× exp − 2 1 ǫ T θ Σ θ 1 ǫ θ

p det (Σ θ ) √

p

(4.15)

Figure

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