\PERGAMON Renewable Energy 06 "0888# 116Ð123

9859Ð0370:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reservedPII] S 9 8 5 9 Ð 0 3 7 0 " 8 7 # 9 9 0 0 5 Ð 4

Data bank

Comparison of monthly mean hourly sunshinefraction estimation techniques fromcalculated di}use radiation valuesA[ Solera\�\ K[ K[ Gopinathanb\ L[ Robledoc

aDepartamento de F(�sica e Instalaciones\ Escuela Te�cnica Superior de Arquitectura\ UPM\Avda Juan de Herrera 3\ 17939 Madrid\ Spain

bDepartament of Physics\ The National University of Lesotho\ Roma\ LesothocDepartamento de Sistemas Inteligentes Aplicados\ Escuela Universitaria de Informa�tica\ UPM\

Ctra de Valencia km 6\ 17920 Madrid\ Spain

Received 1 March 0887^ accepted 09 April 0887

Abstract

Mean monthly hourly values of global I and di}use radiation Id\ along with mean monthlydaily values of the sunshine fraction sd available for four locations in the United Kingdom\are used to develop six models relating Id:I with the monthly mean hourly clearness index Kt\the estimated monthly mean hourly sunshine fraction sh and the monthly mean solar elevationat mid hour a[ Two available methods are used to predict the values of sh from sd and thecalculated Id data are compared[ Statistical tests performed for a total of six locations\ includingthose used to develop the models\ show that the best results are obtained when sh predictedwith the method developed by Page is employed in the estimation correlation[ Þ 0888 ElsevierScience Ltd[ All rights reserved[

0[ Introduction

Among the available techniques to estimate values of the monthly mean hourlydi}use radiation Id\ the most common are the relations connecting values of di}usefraction of global radiation Id:I\ with the mean monthly hourly clearness indexKt � I:I9 or with the mean monthly hourly sunshine fraction sh\ where I and I9 arethe monthly mean hourly global and extraterrestrial radiation ð0Ł[ The dependence of

� Corresponding author[ Fax 99 23 0 225 5443^ e!mail] asolerÝcorbu[aq[upm[es

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the di}use fraction of hourly global radiation and of the relation between monthlyaverage hourly di}use and global radiation on solar elevation a "calculated at midhour# has also been established ð1Ð3Ł[ Relating ah\ the values can be easily obtainedby averaging the hourly sunshine fraction for each hour before or after the true solarnoon\ but hourly values of the sunshine fraction are not routinely available fromMeteorological O.ces\ which usually only provide the number of sunshine hours perday\ that one can use to compute the monthly mean daily sunshine fraction ad butnot sh[ Due to the lack of published values of sh\ at least two investigators havedeveloped equations to estimate sh when values of sd are known ð4Ð5Ł[

Regarding the prediction of monthly mean daily values of di}use radiation\ it hasbeen shown from data for di}erent regions and in agreement with experimental resultsanalysed by Ref[ ð6Ł\ that when both\ clearness index and the monthly mean dailysunshine duration are used together in multiple linear correlations\ the accuracy ofthe estimated values of the monthly mean daily di}use radiation is better than whenthey are used separately ð7Ð09Ł[ Taking these results in to account\ in a recent workwith data for Spanish locations ð00Ł developed a relation connecting Id:I with Kt\ a\and sh as estimated using the method by Ref[ ð5Ł[ With the solar altitude and sunshinefraction added to the basic equation Id:I� a¦b Kt the accuracy of the estimated dataimproved to a good extent ð00Ł[ However\ no de_nite conclusion on the improvementof predicted Id values can be made with the results obtained only for Spain[ Further\it was noticed that at least other researcher had developed a method to estimate sh

from sd ð4Ł\ and both methods should be compared in relation to possible improvementof estimated Id values\ when used in multiple linear correlations[ In the present workboth methods of estimating sh have been tested for locations in the United Kingdom[

1[ Data used

Monthly mean hourly values of Id:I and I\ and the corresponding values of sd areavailable for several stations in the United Kingdom ð01Ð02Ł[ To develop the Models\data for the following locations and periods were used ] Lerwick "0855Ð64#\Aldergrove "0858Ð64#\ Cambridge "0855Ð64# and Jersey "0857Ð64#[ These locationscover a wide latitude range as they are respectively located at ] 59[97>N\ 43[28>N\41[02>N and 38[00>N[

Using the two available procedures sh has been obtained from sd as follows[ InPage|s method ð5Ł developed from German data

sh � "1[4 tan a#"S:S9rel# for a³ 09>

sh � "0[9−"9[0:tan a##"S:S9rel# for a× 09> "0#

where\ S is the usually available monthly mean daily duration of bright sunshinehours and Sorel is a weighted daylength\ shorter than the astronomical daylength[Values of S9rel are given for di}erent latitudes by Ref[ ð5Ł[

In the method developed by Ref[ ð4Ł with data for Uccle\ in Belgium

sh � "9[4¦0[912ð0−exp "−9[9845 a#Łð0−sdŁ#sd "1#

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The methods given by eqns "0# and "1# have been used in the present work for a×4>[Values of I9 and a were obtained following standard procedures ð03Ł[

2[ Results

Averages of values of global and di}use radiation for hours symmetrical about thesolar noon were used to develop the Models in the present work\ giving a total of 114sets of values[ The following are the Models obtained\ with the corresponding valuesof the coe.cients of correlation and the standard deviations ]Model 0

Id:I� 0[9185−0[9535I:I9 "2#

coe.cient of correlation\ r�9[8376 ^ standard deviation\ s�9[9158Model 1

Id:I� 0[9074−9[8950I:I9−9[0314sh "3#

with sh obtained by Dogniaux|s method\ eqn "1# ^ r�9[8419\ s�9[9159Model 2

Id:I� 0[9830−0[2509 I:I9¦9[0018 sin a "4#

with r�9[8477\ s�9[9131Model 3

Id:I� 0[9711−0[1369I:I9¦9[0916 sin a−9[9672sh "5#

with sh obtained by Dogniaux|s method\ eqn "1# ^ r�9[8486\ s�9[9139Model 4

Id:I� 9[8329−9[4350 I:I9−9[2091sh "6#

with sh obtained by Page|s method\ eqn "0# ^ r�9[8549\ s�9[91Model 5

Id:I� 0[9940−9[7312 I:I9−9[1703sh¦9[9834 sin a "7#

with sh obtained by Page|s method\ eqn "0# ^ r�9[8607\ s�9[9190[Considering the values of r and s\ which respectively increase and decrease when

the Model|s number increases from 0Ð5\ it can be tentatively concluded that Model5\ the multiple linear correlation of Id:I with I:I9\ sin a and sh as evaluated by Page|smethod performs the best[ Figures 0 and 1 show the relation between the experimentaland the predicted values of Id for the four locations from Models 0 and 5 respectively[Comparison of Figs 0 and 1 show that a better agreement between experimental andestimated values is obtained with Model 5[ This is specially noticeable for high valuesof Id\ with Model 0 showing a tendency to underpredict the experimental values anda higher dispersion in the predicted values[ Values of Id estimated with Model 5\ andthe experimental values\ are compared for Aberporth in Fig[ 2"a# and "b# respectively\for 9[4 and 2[4 h from solar noon[

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Fig 0[ Experimental Id values vs Id values estimated with Model 0 for the 3 locations used to develop theModels\ in MJ m−1[

Fig 1[ Experimental Id values vs Id values estimated with Model 5 for the 3 locations used to develop theModels\ in MJ m−1[

To validate the tentative conclusion that Model 5 performs better than the other 4Models\ the )MBE and the )RMSE were obtained with Models 0Ð5 for the stationsused in their development\ and also for two other locations in the United Kingdomwith data available from the same sources ] London "40> 20?N# and Aberporth "41>97?N#[ Thus\ data for the following stations were used for this part of the work\ with

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Fig 2[ Experimental Id values vs Id values in MJ m−1 estimated with Model 5 for Aberporth[ "a# 9[4 h fromthe solar noon ^ "b# 2[4 h from the solar noon[

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Table 0Values of the )MBE for the 5 Models in MJ m−1

Model 0 Model 1 Model 2 Model 3 Model 4 Model 5

Lerwick −2[00 −1[90 −1[97 −0[46 −9[04 9[33Aldergrove −2[93 −1[77 −0[80 −0[82 −1[26 −0[37Cambridge −9[98 −9[37 0[22 9[88 −9[88 9[17Jersey 2[58 2[22 3[72 1[09 0[70 9[69London 6[87 5[35 09[82 00[02 2[06 5[98Aberporth −9[25 −9[92 −9[70 9[63 0[95 9[43Mean of absolute values 2[94 1[42 2[54 2[97 0[48 0[48

mean values of Id as given in brackets ] Lerwick "9[352 MJ m−1#\ Aldergrove "9[498MJ m−1#\ Cambridge "9[406 MJ m−1#\ Jersey "9[422 MJ m−1#\ London "9[356 MJm−1# and Aberporth "9[4910 MJ m−1#[

In Table 0 values of the )MBE are given for the 5 locations\ and the means of theabsolute values of the )MBE are given for each Model[ Models 4 and 5 show thebest performance[

In Table 1 a ranking of Models 0Ð5 is established relating the absolute values ofthe )MBE[ For each location the best Model is given 5 points\ 4 points to the secondbest\ and so on\ till the worst Model has 0 point ð04Ł[ Model 5 ranks the best with 20points followed by Model 4[

In Table 2 the values of the )RMSE are given for the 5 locations\ and the meansof the )RMSE are given for each Model[ Model 5 shows the best performance\followed by Model 4[

In Table 3 the ranking of Models 0Ð5 is established relating the values of the)RMSE[ The rating of Models for each location is established as for the )MBE[Model 5 gets a total of 22 points followed by Model 4 with 14[

Table 1Points for each location and Model\ assigned from values of the absolute )MBE\ as obtained with theprocedure outlined in the text

Model 0 Model 1 Model 2 Model 3 Model 4 Model 5

Lerwick 0 2 1 3 5 4Aldergrove 0 1 4 3 2 5Cambridge 5 3 0 1[4 1[4 4Jersey 1 2 0 3 4 5London 2 3 1 0 5 4Aberporth 4 5 1 2 0 3Total points 07 11 02 07[4 12[4 20

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Table 2Values of the )RMSE for the 5 Models in MJ m−1

Model 0 Model 1 Model 2 Model 3 Model 4 Model 5

Lerwick 5[74 4[36 4[45 3[88 2[69 2[65Aldergrove 5[35 5[11 3[08 3[15 4[36 2[75Cambridge 2[32 2[53 2[27 2[02 3[98 1[81Jersey 4[92 3[72 2[42 2[50 3[94 2[10London 8[55 7[97 09[82 02[75 3[89 6[81Aberporth 2[70 2[42 2[77 3[95 2[28 2[35Mean values 4[76 4[29 4[14 4[54 3[16 3[08

Table 3Points for each location and Model\ assigned from values of the )RMSE\ as obtained with the procedureoutlined in the text

Model 0 Model 1 Model 2 Model 3 Model 4 Model 5

Lerwick 0 2 1 3 5 4Aldergrove 0 1 4 3 2 5Cambridge 2 1 3 4 0 5Jersey 0 1 4 3 2 5London 2 3 1 0 5 4Aberporth 2 3 1 0 5 4Total points 01 06 19 08 14 22

3[ Conclusions

The following conclusions are clearly obtained in the present work ]

"0# For the locations studied\ Page|s method to estimate sh from sd performs betterthan Dogniaux|s\ when sh values are used in multiple linear correlations developedto improve the accuracy of estimated Id values[ This is not surprising\ as Dogniaux|smethod was developed using only Uccle|s data\ while Page|s method was developedfrom data for several German stations[

"1# Among the Models developed and tested\ Model 5 is recommended to estimateId values for the United Kingdom if values of I are known or can be predicted\ thatis ]

Id:I� 0[9940−9[7312 I:I9−9[1703sh¦9[9834 sin a

with sh being estimated by Page|s method[Although Dogniaux|s method was not tested\ an equation similar to Model 5 was

found to give the best estimated Id values for Spanish locations as well ð00Ł whichsuggests the wide applicability of this method in improving the accuracy of predictedId values[

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Acknowledgement

This research was funded by the DGICYT as part of the project PB849926[

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