Solar Energ), Vol. 50, No. 3, pp. 247-258, 1993 0038--092X/93 $6.00 + .00 Primed in the U.S.A. Copyright © 1993 Pergamon Press Ltd.

CALCULATION OF MONTHLY AVERAGE GLOBAL SOLAR RADIATION ON HORIZONTAL SURFACES USING DAILY

HOURS OF BRIGHT SUNSHINE

N. HALOUANI, C. T. NGUYEN, and D. VO-NGOC School of Engineering, Universit6 de Moncton, Moncton, New Brunswick, Canada E IA 3E9

Abstract--Several statistical models calculating the monthly average global solar radiation on horizontal surfaces using the daily hours of bright sunshine have been extensively evaluated and compared for Canadian regions. Results show that Gari6py's model provides the best overall performance and Hay's model is rated next best, whereas Rietveld's model has been found to be the least accurate. Local performances of models have also been established throughout Canada that show that both Gari6py's and Hay's models perform quite well. Recommended models have been suggested for the studied regions.

1. INTRODUCTION

For the design of any active and passive solar system, the knowledge of solar radiation on tilted surfaces is required. Such data are often not available and are generally calculated from the corresponding values on horizontal surfaces, which in turn may be obtained from measured data. However, the network collecting solar radiation in Canada is still very scarce; there are currently 56 stations measuring the horizontal global solar radiation, and only five among them also collect the diffuse component. The main concern is therefore to estimate solar radiation on horizontal surfaces at a considered site. In this objective, a very large number of models have been presented in the past for the gen- eral classification and description of many models dated before 1984[1]. With regard to the sunshine- based models in particular, many publications have also been found in the literature [ 2-21 ]. Among them at least five reports include a statistical comparison of the performance of several models [ 1,13-16 ]. From results obtained in these reports, the following com- ments seem pertinent: 1. In[ l ] , Gari6py's[21]model has been found to be

the most accurate, which is followed by Hay's[17]model; lqbal's[18]model comes in the third rank. On a particular-month basis, Riet- veld's[ 20] model gives acceptable performance, In [ 13 ], the latter yields the best general performance among the compared models, and Hay's [ 17 ] is rated next best. Rietveld's model has also been qualified as "encouraging" in [ 22 ] for a few Canadian stations. In[14], the Page [ 23 ] model is rated the best when compared to Rietveld's model. In a recent work, Gopinathan [ 15 ] claimed that his model is better than Rietveld's, but he did not compare his model with Page's. More recently, Soler [ 16 ], who has per- formed an extensive comparative study of several monthly specific and monthly nonspecific models over 77 European stations, has found that the Dog- niaux and Lemoine [ 9 ] model and, unexpectedly, the Rietveld[20] model perform far better than Page's[23](recommended in[14])and Gopina-

than's [ 15 ] models. Soler also rated the Dogniaux and Lemoine model the best for use in European territories.

2. The data bases used in the above works [ 1,13-16 ] are very different from one study to another, and the number of considered stations is often limited ( 15 or fewer, except for[16]).

3. The supposedly reliable and performant Garir- py's [ 21 ] model has been tested over the eastern re- gion of Canada[1]and throughout Canada in the present study, but it does not appear to be well known by other researchers. In view of the above comments, results obtained

from these studies [1,13-16 ], although very interesting, are quite difficult to interprete and use, particularly in Canada. It is also important to note that there is, to the authors' knowledge, no extensive statistical com- parison of the models' performance over Canadian territories.

The present work is specifically concerned with the calculation of the monthly mean global solar radiation on horizontal surfaces, using the daily hours of bright sunshine. In this paper, we present results obtained from an extensive comparative study carried out on three Canadian models (Garirpy[21], Hay[17], lqbal [ 18 ] ) and the Rietveld [ 20 ] model. The objective of our work is not only to determine the most accurate model that could be used for a given region in Canada, but also to study the behavior of these models (validity and local, regional, and national performances) throughout the country.

2. DESCRIPTION OF SELECTED M O D E L S

The first three models considered in this study, namely Garirpy [ 21 ], Hay [ 17 ], and Iqbal [ 18 ], have been selected because they were specifically developed from a Canadian data base. The fourth model, Riet- veld's [ 20 ], has also been included because this model, which was derived from a relatively large data base, has been found to be reliable in previous stud- ies [ 13,16,22 ]. Because detailed descriptions of these

247

248 N. HALOUANI, C. T. NGUYEN, and D. Vo.-NGOC

models have been presented in their original papers, only a brief but exhaustive review of their features is given here.

All of the models are based on the well-known Kimbal l -Angstr rm-Page equation, which relates the monthly average daily value of the global solar radia- tion H to the fraction of possible sunshine hours a as follows [ 23 ]:

where p, pa, and pc are, respectively, the monthly av- erage ground aibedo, cloudless sky albedo, and cloud albedo (pa and Pc are assigned to be 0.25 and 0.6 for Canadian conditions, according to Hay [ 17 ]), and a ' is the modified monthly average daily value of the fraction of possible sunshine hours, defined as

a' =- n / N ' , (6)

with

H / H o = a + ba (1) where N" is the modified day length, which takes into account the sensibility of the Campbell-Stokes sun- shine recorder, given by

,~ =- n / N o , (2)

where Ho is the monthly average daily extraterrestrial radiation; n is the monthly average daily hours of bright sunshine; No is the monthly average of the maximum possible daily hours of bright sunshine, i.e., the day length of the average day of a month; and a and b are empirical coetficients depending on the location under study. Coefficients a and b may vary considerably, as one can expect, from one region to another, and in general, they are to be determined for each climatic region. The character of the regional dependency of any model expressed via eqn ( 1 ) is obvious.

2.1 Gari@y's model

Gari~py [ 21 ] considered six stations located in the province of Quebec. In his model, the variation of the transmission of solar radiation through the atmosphere and clouds is taken into account by introducing to the statistical analysis two additional climatic variables, namely the monthly average ambient temperature (Ta) and the total precipitation (P) at a site under consid- eration. In this model, coefficients a and b of eqn ( 1 ) are given by

a = 0.3791 - 0.0041Ta - 0.0176P (3)

b = 0.4810 + 0.0043Ta + 0.0097P, (4)

where Ta is in degrees Centrigrade and P in centi- meters.

2.2 Hay's model

Hay [ 17 ] developed his model using data collected from several western and northwestern regions of Canada. This model calculates both global and diffuse solar radiation on horizontal surfaces and also proposes an interesting method to compute solar radiation on tilted planes. Hay's model feature resides in the con- sideration of the effects of multiple reflections between the sky and the ground, thus requiring as inputs values of the ground albedo at a site under study. It is inter- esting to note that Hay's model has been tested and used in many countries [ 10-12 ]. Hay's correlation for the calculation of H is given as follows:

H / H o = [0.1572 + 0.5566a'] /

{1 - p [ p a ~ ' + pc(1 - ~ ' ) ] } , ( 5 )

N" = {cos-~[(cos 85 °

- sin • sin tS)/(cos ~ cos 6)] }/7.5 (7)

(~ and 6 are, respectively, the latitude of the site under consideration and the monthly average solar declina- tion).

2.3 Iqbal's model

lqbal [ 18 ] used data collected from three Canadian stations (Montreal, Toronto, and Goose Bay) and proposed three statistical correlations that allow the calculation of the monthly average diffuse and beam solar radiation on the horizontal surface from the local fraction of possible sunshine hours a:

Hb/Ho = -0.176 + 1.45a - 1.12a 2

( ~ r a > 0 ) (8)

Hd/Ho = 0.163 + 0.478a - 0.655a 2, (9)

where Hb and Hd are, respectively, the monthly average daily values of the beam and diffuse solar radiation, from which H will be computed.

2.4 Rietveld's model

Rietveld[20], using measured data collected from 42 stations located in different countries (nine stations in Yugoslavia, six in Sweden, five in Belgium, nine in the United States, but none in Canada), has proposed a unified correlation to compute the horizontal global solar radiation. Because the data base used in Rietveld's study is quite large, this model is expected to have an extended validity, as shown in previous stud- ies [ 1,13,16,22 ]. It has been attempted in the present study, to determine whether this model could be ap- propriately used in Canada. Rietveld's correlation, which is believed to be applicable anywhere in the world, is given as follows:

H / H o = 0.18 + 0.62~r. (lO)

3. METHODOLOGY OF EVALUATION

A total of 46 stations where global solar radiation data are measured have been selected throughout the country for the present comparative study[24]. The locations of these stations are shown on Fig. 1.

\

) , \

#'1

J" i ' gz~ ! ~,m x - -

d v = z , _ - . . , . ,

i

L \ , ~ ~

\

,,,,,

r"

0

k~

249

250 N. HALOUANI, C. T. NGUYEN, and D. VO-NGOC

Table 1. Used values for the ground albedo

Month 1 2 3 4 5 6 7 8 9 10 ! 1 12

o 0.40 0.50 0.40 0.30 0.20 0.20 0.20 0.20 0.20 0.25 0.30 0.35

At each station, the selected models have been used by supplying appropriate input data that were readily compiled as monthly average daily values available from publications[25,26]by the Canadian Atmo- spheric Environment Service. For the ground albedo 19 in particular, which is to be supplied to Hay's model, it is rather difficult to determine typical values for the sites under study, because reflective properties depend strongly on the local and seasonal conditions of the ground. In the present study, typical values of the al- bedo have been adopted (Table 1 ) for all of the con- sidered stations. These values, although mostly con- servative in the winter, are believed to be realistic for Canadian conditions.

Results obtained from models have then been compared to the corresponding measured values of the global solar radiation at each station [27 ]. On a long- term basis, one can have a total of 552 points of com- parison per model ( 12 points per station). The devia- tions, with respect to measured values, are summarized by the following statistical parameters:

M B E (%) = IO0(1/Hm)(~_~Ei/N) (11)

RMSE (%) = 1 0 0 ( 1 / H m ) ( ~ E~/N) °5, (12)

where Ei is the "calculated-measured" deviation of an individual month, defined as

E i : /]calculated -- H ... . . . . d, i = I, 2 . . . . . N (13)

and N is the total number of observation points; Hm is the arithmetic mean value computed from a set of N measured values of the global solar radiation; and MBE and RMSE are, respectively, mean bias error and root mean square error, both expressed in percentages of rim.

The test of MBE provides information on the long- term performance of studied models. A positive MBE value gives the average amount of overestimation in the calculated values and vice versa. In general, a low MBE is desirable. It should be noted, however, that overestimation of an individual observation will cancel underestimation in a separate observation. On the other hand, the test on RMSE provides information on the short-term performance of models, as it allows a term- by-term comparison of the actual deviation between the calculated value and the measured value. In the present study, the test on the parameter RMSE is con- sidered a predominant criterion, provided that the MBE value remains acceptable (e.g., I MBE I -< 10%).

Furthermore, the distribution of the relative cal- culated-measured deviations has been established on both a regional and a national basis. To better study and compare performance between models, these rel-

ative deviations have also been classified into different intervals of 10%. Three additional variables, namely P~0-, P~o-~o, and P20+, have been introduced. They represent the percentage of the N comparison points for which absolute relative deviations are, respectively, lower than 10% (P~o-), higher than 10% but lower than 20% (P1o-2o), and higher than 20% (P20+). It is im- portant to note that deviations varying between - 10 and + 10% are considered acceptable, whereas those lower than -20% or higher than +20% are not.

4. M O D E L P E R F O R M A N C E AND DISCUSSION

4.1 Local performance of models Table 2 presents local performances of the four

models studied at each of 46 stations. These results show clearly that the models' performance, as expressed by MBE and RMSE values, varies drastically from one station to another, even for stations located in the same province. Thus, in British Columbia and for Garirpy's model, for example, the best performance is obtained at Nanaimo Departure Bay (MBE = -3.3%, RMSE = 4.7%) and the worst at Port Hardy station (MBE = - 16.1%, RMSE = 18.5% ), whereas for other stations this model performs quite reasonably. This consider- able variation of performance has also been observed for other models and regions under study.

The situation of the Northern Region merits atten- tion here. For most of the stations considered in this region, the four studied models do not perform satis- factorily, in particular Iqbal's and Rierveld's, which produce exaggeratedly high values of both MBE and RSME parameters. Hay's model also exhibits this gen- eral behavior, although its performance is remarkably good at Whitehorse and Fort Smith stations and quite satisfactory at Eureka and Norman Wells stations. The above rather discouraging performances of these mod- els may be attributed to the severe and particular cli- matic conditions that exist in this region, where the environment is characterized by long solar days and solar nights. In addition, low solar elevation angles exist for most of the daylight periods, which causes, as one can expect, serious difficulties to measure both the sunshine duration and the solar radiation[27]. The poor performance of Hay's model in particular over many stations in the Northern Region may partly be attributed to the values used for the ground albedo, values that are perhaps too conservative to represent the almost year-long snow-covered ground conditions.

Results from Table 2 also show that for the majority of the stations considered in this study, Rietveid's model has been found to be the least accurate in terms of MBE and RMSE, when compared to the other three. Hence, this model seems not to be appropriate for Ca- nadian climatic conditions. It is very interesting to ob-

Calculating monthly average global solar radiation 251

Table 2. Local performances of studied models

Gari6py Hay lqbal Rietveld

MBE RMSE MBE RMSE MBE RMSE MBE RMSE Station (%) (%) (%) (%) (%) (%) (%) (%)

British Columbia

Nanaimo Dep. Bay -3 .3 4.7 -5 .0 9.8 -6 .7 11.1 - 10.1 13.1 Port Hardy A - 16.1 18.5 - 1.7 9.2 2.1 4.8 -7 .3 11.0 Prince George A 6.9 9.0 2.0 4.4 - 1.1 4.8 - 4 . 0 6.6 Sandspit A -6.1 10.8 2.3 7.1 1.5 3.1 -8 .0 10.1 Summerland CDA 8.7 9.2 - 1.6 3.5 -9.1 13.0 -7.1 8.1 Vancouver UBC t -4.1 7.4 4.7 15.5 2.4 8.7 - 5.2 7.6

Northern Region

Whitehorse A 9.3 10.3 -2 .7 5.7 -3 .7 7.9 -9 .4 11.9 Baker Lake 2.5 8.1 - 11.0 19.1 - 12.8 21.1 17.6 25.3 Cambridge Bay A 1.2 10.0 - 14.1 23.9 - 17.0 25.9 28.4 41.8 Coral Harbour A 3.0 10.2 -8 .8 18.3 - 12.7 24.0 33.3 24.1 Eureka 10.7 17.8 3.6 9.8 14.9 30.1 -9 .3 13.1 Fort Smith UA 6.2 7.2 -2 .2 5.4 -8.1 11.2 - 8 . 9 11.2 Frobisher Bay A -3 .2 10.0 - 15.2 25.7 - 15.5 24.4 -21 .6 30.8 lnuvik A 7.13 11.2 -1 .9 13.1 -7 .7 16.2 -11 .3 17.6 Isachsen 3.7 12.7 - 12.2 22.1 - 19.6 29.0 -21 .4 34.8 Mould Bay A 2.3 13.0 - 14.5 24.6 -21 .6 31.8 -21.2 31.9 Norman Wells A 0.8 6.2 7.2 10.8 -9 .9 14.2 -14.3 18.8 Resolute A -4 .6 14.5 19.5 31.4 24.0 35.7 -24 .9 39.1

Prairies

Beaverlodge CDA 3.7 6.3 -2.1 6.6 6.8 10.1 -8.1 11.6 Edmonton Mun. A t 0.4 6.6 - 3.5 5.8 - 10.4 12.8 -9 .2 11.0 Suffield A 4.5 6.9 -2 .5 5.0 - 14.8 19.6 -7 .8 9.6 Bad Lake IHD 102 4.2 5.6 -2 .8 5.3 -13.7 16.6 -8 .2 10.2 Swift Current CDA 1.3 5.0 -6 .2 8.1 - 14.4 17.5 11.5 13.2 Churchill A -0 .4 8.3 - 11.0 8.8 - 12.2 18.3 - 16.9 23.7 The Pas A 5.9 7.6 0.5 7.4 -5 .5 10.8 -5 .5 10.8 Winnipeg Int'l At 2.3 4.8 - 1.2 4.8 -9 .6 12.8 -6 .6 9.3

Ontario

Elora Research STN 0.6 5.0 0.4 4.7 -2 .8 6.8 -4 .9 8.4 Guelph OAC - 1.0 4.2 1.1 5.1 4.6 8.3 -6 .4 9.6 Kapuskasing A - 10.2 14.6 - 10.8 14.2 9.3 12.4 - 15.9 19. l Moosonee -0.1 4.8 -3 .7 8.6 1.9 9.9 -9 .4 13.7 Ottawa NRC t - 1.2 4.2 -0.1 4.5 2.3 5.8 -5 .6 8.9 Toronto t 4.3 4.9 4.4 5.1 0.6 5.9 - 1.1 3.5 Toronto Meteo

Research STN t 4.0 5.7 2.6 5.1 -2 .4 6.3 -2 .8 6.6 Toronto Scarborough t -0 .8 4.8 -0 .9 3.9 -4 .5 8.9 -6 .2 7.6

Qurbec

lnoucdjouac A -7 .5 12.9 -21 .6 29.7. -20 .4 27,5 -26.5 34.1 Montrral Jean

Br~beuf t 0.3 3.8 4.6 5.2 -2 .7 5.5 - 1.1 4.9 Nitchequon -5 .6 8.3 -8 .3 13.4 -6 .9 15.6 - 13.8 18.8 Normandin CDA -5 .6 9.3 -5 .5 9.0 -4 .0 10.5 - 11.0 14.8 Sept-lies UA -2.1 6.0 2.9 4.9 3.2 6.6 -3 .0 6.2

Maritimes

Fredericton CDA t -5 .4 6.5 0.3 3.7 2.4 5.1 -5 .3 7.5 Halifax Citadel t -9.1 10.0 3.5 6.0 5.4 7.3 -2.1 5.3 Kentville CDA - 8 . 4 9.1 -3 .0 4.8 - 1.8 4.2 -8 .3 9.7 Sable Island -22 .0 25.2 - 11.4 17.4 7.8 9.9 - 16.3 19.7 Charlottetown CDA t - 6 . 2 7.1 -2 .3 5.3 -0 .8 5.6 -7 .8 10.3 Goose UA -8 .5 12.4 -7.1 11.9 3.0 10.4 -12 .9 17.0 St John's West CDA - 15.6 17.2 -2 .0 6.2 1.1 7.0 -7 .5 10.3

MBE = mean bias error; RMSE = root mean square error. * Major Canadian cities.

252 N. HALOUAN|, C. T. NGUYEN, and D. Vo-NGoc

Table 3. Comparison of several major Canadian cities

Hay Rietveld

MBE RMSE MBE RMSE Station (%) (%) (%) (%)

Vancouver 4.7 15.5 -5.2 7.6 (-2.1)* (6.1) (-2.0) (5.7)

Edmonton -3.5 5.8 -9.2 11.0 (-1.3) (5.5) (-9.0) (11.4)

Winnipeg - 1.2 4.8 -6.6 9.3 (-2.2) (6.1) (-8.3) (11.4)

Toronto 4.4 5.1 - 1.1 3.5 (5.7) (8.9) (-0.9) (8.3)

Montrral 4.6 5.2 - 1.1 4.9 (5.2) (7.3) (2.0) (6.5)

MBE = mean bias error; RMSE = root mean square error. t Values computed from Ma and Iqbal's [13]results.

serve, however, that this model performs satisfactorily over several major Canadian cities (Table 2), where climatic conditions are expectedly quite different from one city to another. In conjunction with the large data base (including 42 worldwide cities) used by Riet- veld [20], and with similar results obtained in previous studies [ ! 3,16 ], it has been suggested that densely pop- ulated cities could have some common climatic char- acteristics that are well handled by this model [ 24 ].

Finally, results from Table 2 show that, in general, Hay's, Iqbal's, and Rietveld's models exhibit the ten- dency to systematically underestimate global solar ra- diation. This underestimation has been found to be drastically pronounced in the case of Rietveld's model for many of the considered stations, with observed

MBE values as large as 33.3% (Coral Harbour station in the Northern Region). Hay's and Iqbal's models, on the other hand, allow less significant underesti- mation of solar radiation. The same behavior has also been observed for Garirpy's model for stations located in the province of Qurbec and the Maritimes.

4.1.1 Comparison with results from previous stud- ies. The performance results obtained from the present study have been found to be in good agreement with those of Gueymard and colleagues[l], who have ex- tensively studied the performances of Gari~py's, Hay's, and lqbal's models for the region of Quebec. Direct comparison with results published in [ 1 ] was, however, not possible as the data base used in [ 1 ] is not the same as that employed in the present study. With respect to results obtained by Ma and Iqbal [ 13 ] for several major Canadian cities, the agreement is quite acceptable (Table 3), although some differences have been noted that may be attributed to the difference in the choice of ground albedo (for Hay's model) and probably to the new solar constant used in[l 3].

4.2 Regional performance of models The regional performance of the models has also

been established throughout Canada. For example, Figs. 2(a-d) illustrate the overall behavior of Gafi~py's, Hay's, lqbal's, and Rietveld's models for the Ontario region. It is quite interesting to observe that all of these models predict horizontal global solar radiation rea- sonably well. Rietveld's model, on the other hand (Fig. 2(d)) , exhibits clearly the tendency to systematically underestimate global solar radiation. It is also observed that, in general, the models perform relatively well

30

e l

10

r~

Izl

/ f

t J , ~ 1 J J J J I J J J J J

10 20 30

H MEASURED ( MJ / m' ) Fig. 2(a). Performance of Gariepy's model for Ontario region.

Calculating monthly average global solar radiation 253

30

i 2O

=

k l / / ~ ~ l ~ l ÷+ I , i J i 1 , t , , I

0 I0 20 30

H M E A S U R E D ( / m 2 )

Fig. 2(b). Performance of Hay's model for Ontario region.

during the cold season where the levels of solar radia- tion are low (e.g., H _< 10 M J / m 2 ) . For the summer, however, the deviations, with respect to the measured values of solar radiation, become more pronounced.

Table 4 summarizes the regional and overall per- formances of the models studied over Canadian re- gions. With respect to regional behavior, it is clearly observed that the performance of models varies ap-

preciably from one region to another. Thus, for Gar- irpy's model, for example, the best performance has been obtained in Ontario (MBE = -0 .5%, RMSE = 6.7%). For other regions under study, the perfor- mance of this model, in terms of both MBE and RMSE parameters, is generally quite acceptable, with the ex- ception of the Mari t ime region, where the largest cal- culated-measured deviations have been found (MBE

!

i =:I

30

20

10

(c)

. /

0 , , I l I J , i , t * I i I 1

0 10 20 30

H M E A S U R E D ( MJ / m ~ )

Fig. 2(c). Performance of Iqbars model for Ontario region.

254 N. HALOUANI, C. T. NGUYEN, and D. VO-NGOC

30

i ,° ['~ 10

( d )

' , l i , ÷

0 I , l , I I I I l 1

0 I0 20

Y

I I ' l

H M E A S U R E D ( MJ / m 2 )

Fig. 2 (d). Performance of Rietveld's model for Ontario region.

J

3O

= -10.7% and RMSE = 13.3%). On the other hand, however, in the nearby region, the province of Qurbec, Garirpy's model obtains a respectably good perfor- mance (remember that this model has been recom- mended in[l]for Qurbec). This behavior, which is somewhat surprising, may be explained by the fact that climatic conditions change notably from the continen- tal type in Qurbec province (also in Ontario and the Prairies) to the coastal type in the Maritimes (milder ambient temperature, higher precipitation). Hay's model also performed very well for most of the con- sidered regions, except for the province of Qurbec and the Northern Region, where high levels of RMSE have been found. The performances of both Iqbal's and Rietveld's models are rather discouraging for many re- gions under consideration. The former has been found to be acceptable only in British Columbia, Ontario, and the Maritimes, whereas the latter model gives rea-

sonable performances only in the British Columbia and Ontario regions. On the other hand, drastically large RMSE values have been found for both models in the Northern Region. It is also important to observe from Table 4 that the performance variation from re- gion to region of these two models is, by far, more significant than that found for Garirpy's and Hay's models, clearly indicating that Iqbal's and Rietveld's models are not flexible enough to follow the large vari- ation in Canadian climatic conditions.

Table 5 shows percentages of the relative calculated- measured deviations of the models, as computed on both regional and national bases. The superior perfor- mances of Garirpy's and Hay's models are again ob- served here, with relative deviations lying almost en- tirely between - 10 and + 10% over the major part of the country. For the former, values of Pro- vary from 95 to 100% for the considered regions, with the excep-

Table 4. Regional and overall performances of models

Performance regions

Garirpy Hay lqbal Rietveld

MBE t RMSE l MBE RMSE MBE RMSE MBE RMSE (%) (~) (%) (%) (%) (%) (%) (%)

British Columbia - 1.8 Northern 3.2 Prairies 2.8 Ontario -0.5 Quebec -4.0 Maritime - 10.7 Canada - 1.2

10.3 -0.6 9.3 -2.9 9.0 -7.0 9.7 10.9 -8.7 18.7 -13.6 23.1 -12.0 26.3 6.4 -3.5 8.2 -11.1 15.5 -9.1 12.7 6.7 -1.0 7.0 -3.4 8.2 -6.4 10.4 8.5 -5.2 14.7 -4.7 14.7 -10.7 18.1

13.3 -3.0 9.0 -0.6 7.3 -8.5 12.4 9.6 -4.0 11.4 -6.7 14.2 -9.0 15.8

t MBE = mean bias error. l RMSE = root mean square error.

Calculating monthly average global solar radiation

Table 5. Percentages of the relative calculated-measured deviations of studied models

255

Garirpy Hay lqbal Rietveld

Pio- Pio-2o P2o+ Pio- PIo-2o /920+ Pro- PI0-20 P2o+ PIo- P1o-20 P2o+ Percentage regions (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%)

British Columbia 95.8 2.8 1.4 97.2 2.8 0.0 100 0.0 0.0 86.1 13.9 0.0 Northern 98.6 0.7 0.7 99.3 0.7 0.0 16.7 83.3 0.0 63.2 36.1 0.7 Prairies 100 0.0 0.0 100 0.0 0.0 25.0 75.0 0.0 93.8 2.1 4.1 Ontario 100 0.0 0.0 100 0.0 0.0 96.9 3.1 0.0 95.8 1.0 3.2 Qurbec 95.0 5.0 0.0 68.3 11.7 20 66.7 18.3 15 0.0 73.3 26.7 Maritime 52.4 47.6 0.0 100 0.0 0.0 100 0.0 0.0 95.2 4.8 0.0 Canada 99.4 0.4 0.2 99.6 0.4 0.0 89.3 10.7 0.0 89.3 10.7 0.0

tion of the Maritimes, where only 52.4% of deviations are lower than 10%, whereas the remaining 47.6% fall outside the acceptable range. The performance of Gar- irpy's model, as previously discussed, has been found to be discouraging in this region. Hay's model also provides remarkable values of P~o-, except for the province of Qurbec, where large percentages of un- acceptable deviations have been observed (P~o-2o = 11.7% and ,°2o+ = 20%). Results from Table 5 also confirm the discouraging performances of both lqbal's and Rietveld's models for many Canadian regions.

In view of the results shown on Tables 4 and 5, the following recommendations may be suggested: 1. Gari~py's model is the best for use in the Northern

Region, the Prairies, Ontario, and the province of Qurbec.

2. In British Columbia, Hay's model is recommended. 3. lqbal's model should be adopted in the Maritime

regions.

4.3 Overall performance of rnodels Figures 3(a--d) show the global behavior of the

models. With respect to the overall performance (Table 4), Garirpy's model gives by far the best performance, with MBE = -1.2% and RMSE = 9.6%, which is closely followed by Hay's model, with MBE = -4.0% and RMSE = 11.4%. It is interesting to note that when removing the 12 stations in the Northern Region (sta- tions where large calculated-measured deviations have been found, see section 4.1 and Table 2), these two models give nearly identical performances, with MBE = -2.5% and -2.7% and RMSE = 9.3% and 9.1%, respectively. The overall performances of Iqbal's and Rietveld's models are, as one can expect, discouraging. Although they give relatively low MBE values, their RMSE values are quite large ( 14.2 and 15.8%, respec- tively). For these two models, the general tendency to systematically underestimate global solar radiation is clearly observed on Fig. 3 (c) and (d). From the point

:30

2O

O ~ 0

(a )

. • .it 4. 4-

-- *" 4 t ~

" J " i "

•

10 20 30

H M E A S U R E D ( M J / m 2 )

Fig. 3 (a). Overall performance of Gariepy's model.

256

30

20

10

N. HALOUANI, C. T. NGUYEN, and D. VO-NGOC

(b)

t I "

TT 4"

t"

l A J a I I I i I 1 J J i 1

10 20

t i MEASURED ( ~ I / m 2 )

Fig. 3(b). Overall performance of l-lay's model.

I

30

of view of statistics, these models are globally not ac- curate enough in predicting the monthly mean solar radiation in Canada. With regard to the percentages of the relative calculated-measured deviation on a global basis (Table 5 ), both Gari6py's and Hay's mod- els provide equivalent and remarkable performances, with more than 99% of deviations lying between - 1 0 and + 10%.

5 . C O N C L U S I O N

Several statistical models calculating global solar radiation on horizontal surfaces, using the daily hours of bright sunshine, have been extensively evaluated and compared throughout Canada. The results have shown that Gari6py's and Hay's models are globally the most reliable and accurate. They have also been found to

20

0

l ( c )

0 10 20 30

H MEASURED ( MJ / m= ) Fig. 3 (c). Overall performance of lqbal's model.

Calculating monthly average global solar radiation 257

"M

r~

3 0 -

20

10

0

( d ) , ~ ~

¢,

' t"

i i I l I I l t I I

10 20

H M E A S U R E D ( MJ / m 2 )

Fig. 3(d). Overall performance of Rietveld's model.

1

3O

be sutficiently flexible to follow variat ions in climatic condi t ions th roughou t the country. With regard to re- gional performance, Gar i rpy 's model is recommended for use in the Nor the rn Region, the Prairies, Ontario, and the province of Qurbec , whereas Hay's and Iqbal 's models should be adopted, respectively, in the British Co lumbia and the Mar i t ime regions.

Acknowledgments--The authors would like to thank the fac- ulty of Graduate Studies and Research of the Universit6 de Moncton for their financial support, and also Dr. Nicolas Galanis of the Universit6 de Sherbrooke (Qurbec, Canada) for valuable discussions.

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258 N. HALOUANI, C. T. NGUYEN, and D. VO-NGOC

sion coel~cients in the formula relating solar radiation to sunshine, Agric. Meteorol. 19, 243-252 (1978).

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