The linear models

Therstandsimplestofthestatespaemodels,Equation8.3and8.4,ontains

theonvetiveinuenefromtheambienttemperatureandtheonvetive

inu-ene bythetemperature dierenein theairgap. Furthermore theonvetive

inuenedfrom the temperature ofthe wooden board and theradiation is

in-luded. Inthemodelitispresumedthattheinsulationbehindthewoodenboard

isdense. Thisimpliesthat thehosenheatbalanesystemisnotinuenedby

theambientairtemperaturefromtherearside.

dT = k air (T air − T )dt + k delta ∆T dt + k wood (T wood − T )dt + k i Idt + dw

^(8.3)

T m = T + e

^(8.4)

Inordertoperformthefollowinganalysesitisneessarytoinvestigatewhether

theestimation in CTSMisadequate. InTable8.1 and8.2 itanbeseenthat

themodelsbasedonthealulateddeltatemperature,

∆T calc

^,andirradiation,

I calc

^{, and the top} temperature of the module have p-values equal to 0.0000.

P-valueslowerthan0.05meansthattheparameterestimatesanbeonsidered

tobedierentfromzero. Forsomeoftheothermodelsthep-valuesarehigher

than 0.05 assuming a95% ondene interval, whih means that someof the

estimationsoftheparametersouldbeequalto zero. Thisneedsto be

investi-gatedifitishosentofurtherapplysuhmodels. Ifatermisinsigniantitan

beassumedto beequalto 0. Theterman then beremovedfrom themodel,

andafterthatthemodelsneedtobereestimated. Inordertomaintaintheheat

balanesystemitishosentoavoidthiskindofonsiderationsinrelationtothe

linearmodels. Itisalsohosentoinludethederivativesofthepenaltyfuntion

intheanalysis,sinethisvaluegivesanindiationofwhethertheparameteris

well inside thexedintervallimits. All thederivativesofthe penalty funtion

values shown in Table 8.1 are aeptably low, indiating that the parameters

arewelldened.

The orrelations between the parameters are listed in Table 8.2. Prior

esti-mationshaveshownthatorrelationvaluesuptoaround0.96,donotinuene

how adequate the model is. It an therefore be onluded that most of the

valuesaresatifatorying. Themodelbasedonthealulatedvariableshastoo

Table8.1: Summaryofthehighestp-valueandthederivativeofthepenaltyfuntionofthe

linearmodels

16thofAugust

Themodel p-value dPen.fun

T moduleavg T moduletop T moduleavg T moduletop

∆T meas I meas

^{0.0166 0.0551 0.0007 0.0006}

∆T meas I calc

^{0.0099 0.1677 0.0007 0.0006}

∆T calc I meas

^{0.1114 0.2911 0.0006 0.0006}

∆T calc I calc

^{0.0000 0.0000 0.0006 0.0006}

∆T divided

^{0.2991 0.1297 -0.0158 0.0239}

16th-18thofAugust

∆T meas I meas

^{0.0000 0.0000 0.0007 0.0006}

∆T meas I calc

^{0.0765 0.3092 0.0007 0.0006}

∆T calc I meas

^{0.1202 0.0177 0.0007 0.0006}

∆T calc I calc

^{0.7860 0.0000 0.0005 0.0006}

∆T divided

^{0.8200 0.8980 -0.2677 0.0494}

Table 8.2: Summary of the highest orrelations between the parameter estimates of the

linearmodels

16thofAugust

Themodel Correlation

T moduleavg T moduletop

∆T meas I meas

^0.8983(

k air − k delta

^{) -0.8496(}

k wood − k irrad

⁾

∆T meas I calc

^0.9037(

k air − k delta

^{) 0.9394(}

k air − k wood

⁾

∆T calc I meas

^0.9678(

k air − k delta

^{) 0.9678(}

k air − k irrad

⁾

∆T calc I calc

^0.9690(

k air − k delta

^{) 0.9753(}

k air − k wood

⁾

∆T divided

^-0.9878(

k air − k out

^{) -0.9908(}

k air − k out

⁾

16th-18thofAugust

∆T meas I meas

^-0.8666(

k delta − k wood

^{) 0.8527(}

k air − k wood

⁾

∆T meas I calc

^0.8877(

k air − k delta

^{) 0.9189(}

k air − k wood

⁾

∆T calc I meas

^0.9550(

k air − k delta

^{) 0.9615(}

k air − k wood

⁾

∆T calc I calc

^0.9666(

k air − k delta

^{) 0.9752(}

k air − k wood

⁾

∆T divided

^-0.9854(

k air − k out

^{) -0.9922(}

k air − k out

⁾

Inmostasesthe

∆T

^and

k air

^{arehighly orrelated,see Table8.2.}

∆T

^{is the}

dierene between the inlet and the outlet temperatures. The inlet and the

ambienttemperaturesmust beonsidered to beorrelatedin aphysial sense,

sine the inlet air is taken from the ambient air. It is therefore deided to

problem that the orrelation between the inlet and the ambient temperature

seemsto reate. This model represents thesame heat balanesystem, but in

thedesignofthismodel,named

divided

^{inthetable,itisdeidedtodividethe}

∆T

^{-termintoaninletandanoutlet}temperature. Furthermore

T in

^{isset equal}

to

T air

^{. ThisdividedmodelanbeseeninEquation8.5 and8.6.}

dT = k air − in T air dt + k T T dt + k out T out dt + k wood T wood dt + k i Idt + dw

^(8.5)

T m = T + e

^(8.6)

Table8.1and8.2 revealsthatthedividedmodelshavehigherp-values,

deriva-tives of the penalty funtion values and orrelations ompared to the former

model.

Table8.3: Averagesandstandarddeviationsoftheresidualsforthedierentmodels

alulatedonbasisofdatafromthe16thofAugustand16th-18thofAugust

16thofAugust

T moduleavg T moduletop

Themodel Average Std.dev Average Std.dev

∆T meas I meas 2.445 · 10 ^{− 4} 7.162 · 10 ^{− 2} 2.500 · 10 ^{− 3} 7.171 · 10 ^{− 2}

∆T meas I calc − 8.251 · 10 ^{− 3} 4.645 · 10 ^{− 2} − 6.860 · 10 ^{− 3} 4.211 · 10 ^{− 2}

∆T calc I meas 3.070 · 10 ^{− 3} 7.052 · 10 ^{− 2} 6.260 · 10 ^{− 3} 6.976 · 10 ^{− 2}

∆T calc I calc − 6.979 · 10 ^{− 3} 4.631 · 10 ^{− 2} − 5.039 · 10 ^{− 3} 4.136 · 10 ^{− 2}

∆T divid − 2.876 · 10 ^{− 5} 4.400 · 10 ^{− 2} − 2.265 · 10 ^{− 5} 4.064 · 10 ^{− 2}

16th-18thofAugust

T moduleavg T moduletop

Themodel Average Std.dev Average Std.dev

∆T meas I meas 3.328 · 10 ^{− 3} 7.014 · 10 ^{− 2} 5.868 · 10 ^{− 3} 7.014 · 10 ^{− 2}

∆T meas I calc − 7.609 · 10 ^{− 3} 4.800 · 10 ^{− 2} − 6.159 · 10 ^{− 3} 4.136. · 10 ^{− 2}

∆T calc I meas 5.336 · 10 ^{− 3} 6.787 · 10 ^{− 2} 7.839 · 10 ^{− 3} 6.740 · 10 ^{− 2}

∆T calc I calc − 2.337 · 10 ^{− 4} 4.617 · 10 ^{− 2} − 5.603 · 10 ^{− 3} 4.059 · 10 ^{− 2}

∆T divid − 3.462 · 10 ^{− 4} 4.643 · 10 ^{− 2} − 1.567 · 10 ^{− 4} 4.047 · 10 ^{− 2}

In order to determine whih of the models desribes the data the best, the

residualshavetobeanalyzed. Theaveragesandstandarddeviationsofresiduals

forthelinearmodelsarepresentedinTable8.3. Byomparingtheaveragesand

thestandarddeviationsoftheresidualsthedierentmodelsthefollowingtrends

areidentied:

•

^{Inallasesthestandarddeviationislowerwhenthealulatedirradiane,}

I calc

^{,isappliedinsteadofthemeasuredirradiane}

•

^Thealulated temperature dierene,

∆T calc

^{givesthe lowest standard}

deviationsforallmodels

•

^{Inthemajorityof theasestheuse ofthe} temperature measuredat the topof themodule resultsin lowerstandarddeviations,whihindiates a

betterttingmodel.

•

^{Thedivided model hasthe lowest standarddeviations for the three-day}

period,butnotfortheone-dayestimations. SinealltheCTSMestimated

valuesarehigherforthismoreompliatedmodel,itisdeidednottotake

thismodelinto aount.

•

^{Thereisasmallredutioninthestandarddeviationvalueswhenthe}

mod-elsarebasedinthree-daydataomparedtotheone-daydata.

When takingall the fats aboveinto aount,the models basedon the

alu-latedvariablesshowthrough. Onthebasisoftheseobservationsthenon-linear

modelsinnextsetionwillonlybeestimatedapplyingthealulatedvariablesof

thetemperaturediereneintheairgapandtheirradiation. Fromaphysiist's

point of viewit doesmakesense that thealulatedvariables generate better

results. Thealulatedvariablesarebasedondataolletedinside themodule

viaknowledgeabout theeletrial ows. This way ofestimating thevariables

minimizestheamountofnoiseinthevariablesandtherebytheestimatedmodels.

In general for the results of the linear models it an be onluded that the

orrelationsof theparameterestimatesneedto bereduedin orderto ndan

adequatemodel. Thereforeanaturalprogressis toextendthelinearmodelto

anon-linearmodel.

In document S hai de (Sider 76-79)