Data bank Comparison of monthly mean hourly sunshine fraction estimation techniques from calculated diffuse radiation values

\ PERGAMON

Renewable Energy 06 "0888# 116Ð123

Data bank

Comparison of monthly mean hourly sunshine fraction estimation techniques from calculated di}use radiation values A[ Solera\\ K[ K[ Gopinathanb\ L[ Robledoc a

Departamento de F(sica e Instalaciones\ Escuela Tecnica Superior de Arquitectura\ UPM\ Avda Juan de Herrera 3\ 17939 Madrid\ Spain b Departament of Physics\ The National University of Lesotho\ Roma\ Lesotho c Departamento de Sistemas Inteligentes Aplicados\ Escuela Universitaria de Informatica\ 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 monthly daily 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 elevation at mid hour a[ Two available methods are used to predict the values of sh from sd and the calculated Id data are compared[ Statistical tests performed for a total of six locations\ including those used to develop the models\ show that the best results are obtained when sh predicted with the method developed by Page is employed in the estimation correlation[ Þ 0888 Elsevier Science Ltd[ All rights reserved[

0[ Introduction Among the available techniques to estimate values of the monthly mean hourly di}use radiation Id\ the most common are the relations connecting values of di}use fraction of global radiation Id:I\ with the mean monthly hourly clearness index Kt  I:I9 or with the mean monthly hourly sunshine fraction sh\ where I and I9 are the 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 9859Ð0370:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved PII] S 9 8 5 9 Ð 0 3 7 0 " 8 7 # 9 9 0 0 5 Ð 4

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the di}use fraction of hourly global radiation and of the relation between monthly average hourly di}use and global radiation on solar elevation a "calculated at mid hour# has also been established ð1Ð3Ł[ Relating ah\ the values can be easily obtained by averaging the hourly sunshine fraction for each hour before or after the true solar noon\ but hourly values of the sunshine fraction are not routinely available from Meteorological O.ces\ which usually only provide the number of sunshine hours per day\ that one can use to compute the monthly mean daily sunshine fraction ad but not sh[ Due to the lack of published values of sh\ at least two investigators have developed equations to estimate sh when values of sd are known ð4Ð5Ł[ Regarding the prediction of monthly mean daily values of di}use radiation\ it has been shown from data for di}erent regions and in agreement with experimental results analysed by Ref[ ð6Ł\ that when both\ clearness index and the monthly mean daily sunshine duration are used together in multiple linear correlations\ the accuracy of the estimated values of the monthly mean daily di}use radiation is better than when they are used separately ð7Ð09Ł[ Taking these results in to account\ in a recent work with 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 sunshine fraction added to the basic equation Id:I  a¦b Kt the accuracy of the estimated data improved to a good extent ð00Ł[ However\ no de_nite conclusion on the improvement of 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 improvement of estimated Id values\ when used in multiple linear correlations[ In the present work both 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 are available 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 locations cover 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[ In Page|s method ð5Ł developed from German data sh  "1[4 tan a#"S:S9rel #

for a ³ 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 sunshine hours 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−sd Ł#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 the solar noon were used to develop the Models in the present work\ giving a total of 114 sets of values[ The following are the Models obtained\ with the corresponding values of 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[9158 Model 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[9159 Model 2 Id :I  0[9830−0[2509 I:I9 ¦9[0018 sin a

"4#

with r  9[8477\ s  9[9131 Model 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[9139 Model 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[91 Model 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 Model 5\ the multiple linear correlation of Id:I with I:I9\ sin a and sh as evaluated by Page|s method performs the best[ Figures 0 and 1 show the relation between the experimental and 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 and estimated values is obtained with Model 5[ This is specially noticeable for high values of Id\ with Model 0 showing a tendency to underpredict the experimental values and a higher dispersion in the predicted values[ Values of Id estimated with Model 5\ and the 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 the Models\ in MJ m−1[

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

To validate the tentative conclusion that Model 5 performs better than the other 4 Models\ the )MBE and the )RMSE were obtained with Models 0Ð5 for the stations used in their development\ and also for two other locations in the United Kingdom with 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 from the solar noon ^ "b# 2[4 h from the solar noon[

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

Lerwick Aldergrove Cambridge Jersey London Aberporth Mean of absolute values

Model 0

Model 1

Model 2

Model 3

Model 4

Model 5

−2[00 −2[93 −9[98 2[58 6[87 −9[25 2[94

−1[90 −1[77 −9[37 2[22 5[35 −9[92 1[42

−1[97 −0[80 0[22 3[72 09[82 −9[70 2[54

−0[46 −0[82 9[88 1[09 00[02 9[63 2[97

−9[04 −1[26 −9[88 0[70 2[06 0[95 0[48

9[33 −0[37 9[17 9[69 5[98 9[43 0[48

mean values of Id as given in brackets ] Lerwick "9[352 MJ m−1#\ Aldergrove "9[498 MJ m−1#\ Cambridge "9[406 MJ m−1#\ Jersey "9[422 MJ m−1#\ London "9[356 MJ m−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 the absolute values of the )MBE are given for each Model[ Models 4 and 5 show the best performance[ In Table 1 a ranking of Models 0Ð5 is established relating the absolute values of the )MBE[ For each location the best Model is given 5 points\ 4 points to the second best\ and so on\ till the worst Model has 0 point ð04Ł[ Model 5 ranks the best with 20 points followed by Model 4[ In Table 2 the values of the )RMSE are given for the 5 locations\ and the means of 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 1 Points for each location and Model\ assigned from values of the absolute )MBE\ as obtained with the procedure outlined in the text

Lerwick Aldergrove Cambridge Jersey London Aberporth Total points

Model 0

Model 1

Model 2

Model 3

Model 4

Model 5

0 0 5 1 2 4 07

2 1 3 2 3 5 11

1 4 0 0 1 1 02

3 3 1[4 3 0 2 07[4

5 2 1[4 4 5 0 12[4

4 5 4 5 4 3 20

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A[ Soler et al[ : Renewable Ener`y 06 "0888# 116Ð123 Table 2 Values of the )RMSE for the 5 Models in MJ m−1

Lerwick Aldergrove Cambridge Jersey London Aberporth Mean values

Model 0

Model 1

Model 2

Model 3

Model 4

Model 5

5[74 5[35 2[32 4[92 8[55 2[70 4[76

4[36 5[11 2[53 3[72 7[97 2[42 4[29

4[45 3[08 2[27 2[42 09[82 2[77 4[14

3[88 3[15 2[02 2[50 02[75 3[95 4[54

2[69 4[36 3[98 3[94 3[89 2[28 3[16

2[65 2[75 1[81 2[10 6[81 2[35 3[08

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

Lerwick Aldergrove Cambridge Jersey London Aberporth Total points

Model 0

Model 1

Model 2

Model 3

Model 4

Model 5

0 0 2 0 2 2 01

2 1 1 1 3 3 06

1 4 3 4 1 1 19

3 3 4 3 0 0 08

5 2 0 2 5 5 14

4 5 5 5 4 4 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 better than Dogniaux|s\ when sh values are used in multiple linear correlations developed to improve the accuracy of estimated Id values[ This is not surprising\ as Dogniaux|s method was developed using only Uccle|s data\ while Page|s method was developed from data for several German stations[ "1# Among the Models developed and tested\ Model 5 is recommended to estimate Id values for the United Kingdom if values of I are known or can be predicted\ that is ] 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 which suggests the wide applicability of this method in improving the accuracy of predicted Id values[

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

References ð0Ł ð1Ł ð2Ł ð3Ł ð4Ł ð5Ł ð6Ł ð7Ł ð8Ł ð09Ł ð00Ł ð01Ł ð02Ł ð03Ł ð04Ł

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