Asashortsum-upitanbeonludedthatthevariablesalulatedonthebasis
of the measurements of the eletrial ows give the best preditions. A very
important ndingis that the measuredtemperatureat the topof the module
gives the best desription to the module temperature. Having in mind that
theperformaneofthemoduledereasewhenthetemperatureraises,itmakes
sense to applythetoptemperaturewhihrepresentsthehighesttemperatures
in themoduleomparedtothetemperatureatthebottomofthemodule. The
limitation oftheeieny ofthemoduleis thelowestperformingell. Due to
the orrelationbetween thetemperatureof the module and theeieny, the
worseaseeientismodelled byhavingthetoptemperatureofthemodelas
the output variable. Furthermore, it is evident that the extended non-linear
modelisdeisivebetterthanthetwoothermodelsattempted.
Comparing the performane of the similar models in this hapter and in the
artile[Jiménezetal.2006℄,theperformaneisimproved. Theprimaryreason
isthehangeofoutputvariablefrom theaveragetothetoptemperature. F
ur-thermoreitdoesalsoinuene thatthealulatedirradianeisapplied instead
ofthemeasured.
The residual analysis reveals that none of the models give a perfet
desrip-tionofthedata. Ageneralonlusionfromtheanalysisoftheresidualsisthat
the residuals of the extendednon-linear model has thebest performane. To
name onlyaoupleof theindiations that theextended non-linearmodel has
thebestperformane: theloweststandarddeviations,signianthangeinsign
test,andthemoststraightlineintheumulatedperiodogramOneofthemajor
diultiesforalltheestimatedmodelistoobtainaeptablepreditionduring
thedayhours. Nonof themodelsgotrid ofthebulgeof highresidualsduring
thedayhours.
Anotherndinginthishapteristhatahievedbyaddinganextrastate,where
theausallteredwindisestimated,thebestresultsareattained. Theanalysis
hasrevealedthatonlyverylittleofthevariationinthewindisremoved. Dueto
thisirumstane,itanbedisussedifitisneessarytohavethisextrastate.
Furthermorealikelihoodratiotestofthethree-daymodelsbasedoneither the
measured wind and the asual lteredwind onrm that theimprovementof
themodelissigniant.
L ratio
isequalto262. Thisvaluehasto beomparedto
χ 2 0.05%
(1)=3.841. Whentakeninto aountthat the wind speed atthe siteis relativelylowitanbeassumed thattheausallteranbeofevengreater
importane atloationswherethewindisfaster andmoreutuating.
The ndings above demonstrate that the extended non-linear model has an
aeptableperformane,butitisstillpossibletoimprovethedesriptionofthe
module temperature,whih willbeattempted inthefollowinghapter.
Multiple State Models - Top
and Bottom Divided Model
Inthe previoushapter it hasbeomelearthat itis diultto obtaina
sat-isfatory modelling of the temperature of the module. In this hapter there
willbeattahedimportanetothefatthatsomeofthemeasuredvariablesare
olletedbothatthetopandatthebottomofthemodule. Themodelsinthis
hapter havetwosystemequations: onefordesription ofthe temperatureat
thetopandoneforthebottomtemperature. Thismakesitpossibletoestimate
the two temperatures separately. This is alled amultiple statemodel. This
approah of analyzing the module in several separate setions has been used
in earlieranalyses. In[Christ 2001℄atrisetionsolution ishosenfor the
pur-poseofsimulation. Themain reasonformakingthisfurther developmentisto
investigateifitispossibletogetabetterpreditionofthemoduletemperature.
9.1 The model
Intheprevioushapteritwasfoundthatthemodelwheretheausallterwas
applied tothewindresultedin thebest predition. Unfortunately,despite
nu-merous attempts, ithas notbeenpossibleto estimatethe ausallter for the
tionsorunsuessfulinitialguessesoftheparametervalues. Theresultsshown
intheprevioushapterrevealedthatintheaseofthesedatathemodelsbased
onthe ausallteredwind andthe measuredwind generated quitesimilar
re-sults. It is thereforedeided to estimate these models basedon themeasured
wind. Thiswillstillindiatefairlywhetherthemultiplestatemodelshave
bet-terorworseperformaneomparedto thesinglestatemodels.
Whentaking the available data into aountand also thewish for generating
modelsofsimilarstrutureto theextendedmodelin theprevioushapter,two
feasiblesolutionsareidentied. Therstmodelisverysimilartotheextended
non-linearmodelintheprevioushapter. Theothermodelisamoreadvaned
one,whereCTSMestimatesthetemperaturehalfwayupthegap. Themodels
willbedesribedinfurhterdetailsin thenexttwosetions.
It is neessaryrstly to introdue the onept of modelling the top and
bot-tomtemperaturesofthemoduleseparately. Ashemeoftheestimationset-up
isshownin Figure9.1.
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Figure9.1: Shemeoftheset-upwiththedottedlineindiatingtheplaeoflumping
Thedotted dividinglinebetweenthetopand thebottomparts anbeviewed
as the lumping point. Sine the mean radiant temperature, the ambient air
temperature and the irradiation are aeting the top and the bottom of the
module equally,these ontributorsareidentialin thetwosystemequationsin
Equation9.1.
∆T
is thetemperature dierenein the airgap. Twosenarios areoneivableinrelationto∆T
. Thesimplestassumptionistoapplythe∆T
estimatedovertheentireheightofthegap. Inthisase
∆T
belowthedividinglinewillbelargeromparedtothe
∆T
ofthetopofthegap.Thismeansthatthemodelisapproximativeonerning
∆T
. Intheseondsenarioitisattemptedto estimatethetemperaturehalfwayuptheairgap. Thetermsestimating the
infrared radiationfromthewooden walltothemodulearedierentin thetwo
statespaeequations. Thisisdonesineatemperatureatboththetopandthe
bottomof thewooden board ismeasured. Thisis anopportunityto eliminate
apossiblesoureoferror,sinethereisuptoa5
o
Cdierenebetweenthetop
andthebottomoftheboard.
dT bottom =
1 − 1
1 + exp( − f )
k rad (T rad 4 − T bottom 4 )dt + . . .
dT top = 1 1 + exp( − f )
k rad (T rad 4 − T top 4 )dt + . . . df = dw 3
k air W windair k (T air − T bottom )dt + k delta ∆T dt + . . . k air W windair k (T air − T top )dt + k delta ∆T dt + . . . k woodbottom (T wood,bottom 4 − T bottom 4 )dt + k irrad I irrad dt
+ dw 1
k woodtop (T wood,top 4 − T top 4 )dt + k irrad I irrad dt
+ dw 2
(9.1)T mbottom = T bottom + e 1
T mtop = T top + e 2
(9.2)Inspired by thethermalimages itis onsidered neessaryto beableto weight
theinueneofthetopandthebottomtemperaturesofthemodulerespetively.
This weightingis denoted
f
,and isestimatedin CTSM.f
and(1 − f )
denotehowmuhinuenethetopandthebottomsystemequationsrespetivelyshall
behavein order to obtainthe best desriptionof themoduletemperature. It
is deidedto apply
f
and(1 − f )
in allthe termsin thestateequations. Thefrations anbeseen asindiations asto where the dividingline needs to be
set. Atthe introduing stages of estimation,
f
wasentered in themodel asaonstantparameter. Whenthinkingabouthowutuatingthedataandthereby
theestimatedmodelsare,
f
isnotsuientlydynamial. Thisanalsobeon-luded from thethermalimagespresentedealier. It wasdeidedto reestimate
themodels, where
f
wasappliedasastate,onlydesribedbyanoiseterm. Inthiswayitispossibletolet
f
varyduringtheentire period.The logisti funtion,
1
1+exp( − f)
, is applied in order to assure thatf
islim-ited tobebetween 0and1. Unfortunatelyit beomesevenmoreintensivefor
CTSMto estimate, whenthelogistifuntion is appliedto themodel. CTSM
above150tolessthanten. CTSMusesiterationtondthesolution. The
num-berofiterationsisoneoftheoutputvaluesin CTSM.Thisleadstoasituation
where the estimated parametersare lose to orequal to theinitial parameter
values. The reason for the low number of iterations an be that CTSM has
foundaloalminimumoftheobjetivefuntioninsteadoftheglobalminimum.
Inorder to ndtheglobal minimumitanbeattempted to hange theinitial
parametervaluesinCTSM.Thishasbeenattemptedbyusingseveraldierent
initial values. It has been possible to raise the number of iterationsfrom less
that 10 to about60. Still someof theestimatesare lose to theintialvalues,
whih leadstolesstrustworthyresults.
A general problem in relation to all the models estimated in this hapter is
that theyareverysensitivetovariationsin data andhanges in thedenition
oftheestimationparameters. Thisimpliesthatoneminorhangeintheset-up
hasdetetableinueneontherestofthemodel. Theestimationofthemodels
isverytimeonsuming,thereforeitisnotpossibletoinvestigateallpossibilities.
Inthesetionsbelow,argumentationforthedeisionswillbegiven.
The data applied in the oming analysesare dating from the 16
th
of August.
The previous analyses revealed only very little dierene between the results
basedon one-dayand three-daydata respetively,whih justies for applying
one-daydataintheanalysisbelow.