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Renewable Energy Vol. I. No. 3/4, pp. 473478. 1991 0960-1481/91 $3.00+.00 Printed in Great Britain. Pergamon Press plc DATA BANK Simple hourly global solar radiation prediction models M. S. AUDI and M. A. ALSAAD Department of Mechanical Engineering, University of Jordan, Amman, Jordan (Received 20 September 1990; accepted l0 December 1990) Abstract--Three simple prediction models of hourly global radiation, namely, a normal distribution model (ND), a half-sine wave model (HW), and a polynomial model (PO), are tested using data for a period of five years of the area of Amman, Jordan. The results show that none of these models alone can adequately represent the tested data. Specifically, the results show that PO model represents the data in about 42% of the hours of the year, the ND model about 32%, and the HW model about 34%. Thus a new model which is a combination of the three simple models developed to provide a comprehensive representation of the tested data. This model is given as follows : r=@p(i,h)[ao+a~h+a,h2+a~h3+a4h4]+dp~(i,h)[bo+b . frch ~'~-] F 1 -h2~ where h is the hour of the day, i the month of the year, and ~Pp,q)s, and ~, have 0, 0.5 or 1 values depending on i and h. It is found that this model represents the data in about 74% of the time. INTRODUCTION The current worldwide awareness of the cost of conventional energy and the interest in optimizing the utilization of the conventional and the non-conventional energy resources forced an interest in the minute details of energy resources. In the area of solar energy applications, in addition to the seasonal, monthly, and daily variation, the hour by hour variation of solar energy intensities in any desired locality has become increasingly important. Jordan is one of the small countries of the world which imports most of its energy needs. Its solar energy availability, however, is one of the highest in the world [1, 2]. It is natural, therefore, that modeling research [3, 4] and applications [5, 6] be conducted to maximize the utilization of this source and hope to reduce dependence on imported oil. Thus, hourly or instantaneous, if possible, models are needed for the simulation of solar systems for research and the design of solar application systems for implementation of the research results. Several models for the prediction of the ratio of hourly over daily solar radiation energies have been reviewed and published [7]. These models have varying degrees of appli- cability and complexity. The normal distribution model [8] however, is very simple and the claim of its universality is highly intriguing. In this paper, we tested the normal distribution model on the hourly global solar radiation of Amman, Jordan. We, also, tested the same data using two other simple models : a half-sine model and a polynomial model. The mathematical formulation of these three models are given in the next section, and the analysis and discussion of the results are given in the section that follows it. MATHEMATICAL MODELS The three models considered in this research are the normal distribution model (ND), the half-sine wave model (HW), and the polynomial model (PO). The mathematical form of each of these models, and a newly developed model com- bining the three of them are presented in this section. The ND is given by the following expression : r =(~exp -h2 \cr~/2r~/ 2a-' (1) where rg is the ratio of the hourly global solar radiation averaged for a month for five years of data divided by the daily global radiation computed in the same manner, a is the standard deviation which was computed in two different ways as done by [8]. In one computation, the solar radiation at noon was considered, and in the other the daily sunshine hours were considered, h is the hour of the day ; h = 0 at noon, positive in the afternoon hours, and negative in the morning hours. The HW model is given by the following expression : r~ = bo +bl sm~ + (2) where both b0 and b~ are constants to be determined on the basis of the local data. The PO model is based on the following relationship : rg = ao +alh+aeh2 +a~h'~ +a4 h4 (3) where the coefficients are to be determined by using the local data. The degree of the polynomial was determined by testing the data for higher degrees and the results were not improved by going beyond the fourth degrees. In fact, the contribution of the fourth degree term to the accuracy of the model was insignificant. Finally, a combination of these models of the form given below is presented for a comprehensive representation of the data analysed in this research. 473

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Renewable Energy Vol. I. No. 3/4, pp. 473478. 1991 0960-1481/91 $3.00+.00 Printed in Great Britain. Pergamon Press plc

DATA BANK

Simple hourly global solar radiation prediction models

M. S. AUDI and M. A. ALSAAD Department of Mechanical Engineering, University of Jordan, Amman, Jordan

(Received 20 September 1990; accepted l0 December 1990)

Abstract--Three simple prediction models of hourly global radiation, namely, a normal distribution model (ND), a half-sine wave model (HW), and a polynomial model (PO), are tested using data for a period of five years of the area of Amman, Jordan. The results show that none of these models alone can adequately represent the tested data. Specifically, the results show that PO model represents the data in about 42% of the hours of the year, the ND model about 32%, and the HW model about 34%. Thus a new model which is a combination of the three simple models developed to provide a comprehensive representation of the tested data. This model is given as follows :

r=@p(i,h)[ao+a~h+a,h2+a~h3+a4h4]+dp~(i,h)[bo+b . frch ~'~-] F 1 -h2~

where h is the hour of the day, i the month of the year, and ~Pp, q)s, and ~ , have 0, 0.5 or 1 values depending on i and h. It is found that this model represents the data in about 74% of the time.

INTRODUCTION

The current worldwide awareness of the cost of conventional energy and the interest in optimizing the utilization of the conventional and the non-conventional energy resources forced an interest in the minute details of energy resources. In the area of solar energy applications, in addition to the seasonal, monthly, and daily variation, the hour by hour variation of solar energy intensities in any desired locality has become increasingly important.

Jordan is one of the small countries of the world which imports most of its energy needs. Its solar energy availability, however, is one of the highest in the world [1, 2]. It is natural, therefore, that modeling research [3, 4] and applications [5, 6] be conducted to maximize the utilization of this source and hope to reduce dependence on imported oil.

Thus, hourly or instantaneous, if possible, models are needed for the simulation of solar systems for research and the design of solar application systems for implementation of the research results.

Several models for the prediction of the ratio of hourly over daily solar radiation energies have been reviewed and published [7]. These models have varying degrees of appli- cability and complexity.

The normal distribution model [8] however, is very simple and the claim of its universality is highly intriguing.

In this paper, we tested the normal distribution model on the hourly global solar radiation of Amman, Jordan. We, also, tested the same data using two other simple models : a half-sine model and a polynomial model.

The mathematical formulation of these three models are given in the next section, and the analysis and discussion of the results are given in the section that follows it.

MATHEMATICAL MODELS

The three models considered in this research are the normal distribution model (ND), the half-sine wave model (HW),

and the polynomial model (PO). The mathematical form of each of these models, and a newly developed model com- bining the three of them are presented in this section.

The ND is given by the following expression :

r = ( ~ e x p -h2 \cr~/2r~/ 2a-' (1)

where rg is the ratio of the hourly global solar radiation averaged for a month for five years of data divided by the daily global radiation computed in the same manner, a is the standard deviation which was computed in two different ways as done by [8]. In one computation, the solar radiation at noon was considered, and in the other the daily sunshine hours were considered, h is the hour of the day ; h = 0 at noon, positive in the afternoon hours, and negative in the morning hours.

The HW model is given by the following expression :

r~ = bo +bl s m ~ + (2)

where both b0 and b~ are constants to be determined on the basis of the local data.

The PO model is based on the following relationship :

rg = ao +alh+aeh2 +a~h'~ +a4 h4 (3)

where the coefficients are to be determined by using the local data. The degree of the polynomial was determined by testing the data for higher degrees and the results were not improved by going beyond the fourth degrees. In fact, the contribution of the fourth degree term to the accuracy of the model was insignificant.

Finally, a combination of these models of the form given below is presented for a comprehensive representation of the data analysed in this research.

473

474

rg = ~ ( i , h)[a 0 +a~h +a2h 2 +a3h 3 +a4 h4]

+~,(i ,h)[bo+b," sm ~]-~ + ~ ) ] {Tzh n~'~

r 1 -h27 +~.(i,h) =--exp-q- -~_2~ (4) L o.x/~- ~ 20 ]

where Op(i, h), ~ ( i , h), and ~,(i, h) have a 0, 0.5 or 1 value depending on i, the month of the year (i for January is 1).

RESULTS AND DISCUSSION

The results of the analysis of the three models and the proposed combination of these models are presented in this section.

Data Bank

0.20

THE NORMAL DISTRIBUTION MODEL (ND)

The simplicity of the ND model, eq. (1), as presented by Jain [8] is intriguing. It contends that the ratio of hourly over daily global solar radiation of any day of the year may be predicted from a knowledge of solar radiation at noon of that day. Further, if such data is unavailable for desired localities, the daily bright sunshine hours may be adequate.

Encouraged by such inviting simplicity, the model was tested for applicability to Amman, Jordan as the primary step of this research. The generality of the results stated in reference [8] was also tested. Two sets of computations were prepared. The first set of two computations was based on the results of Jain [8], and the second set of two computations also was based on similar data collected in Amman, Jordan.

The main parameter in the ND and in these computations is the standard deviation of the data, a. Once this variable is computed, the hourly distribution prediction formulas were derived on the basis of the following variations :

1. ~ as computed by Jain [8] using noon solar radiation data of Montreal, Canada ;

2. a was computed by Jain .[8] using a derived linear regression model based on a correlation between a total daily bright sunshine hours ;

3. cr was computed as in (1) above except that the data of Amman, Jordan was used ; and

4. a was computed as in (2) above except that the data of Amman, Jordan was used.

For easy reference in this discussion, these computational results will be referred to as J1, J2, AI, and A2, respectively.

Visual inspection of the graphic display of the results of this analysis for the twelve months of the year indicates that some of them demonstrate considerable disparity between the data on one side and the results of the prediction models on the other side. Figure I, for example, is a typical display of the data and results for a winter month (January). It illustrates that disparity : only, five of the 15 hours are reason- ably predicted by the J 1 variation of the ND model. None of these hours include the time of maximum radiation. The other three variations of the model, namely, J2, AI, and A3 predict five different hours of the day. These hours included the peak radiation hours of the day.

Figure 2 presents typical summer month (June) data and computational results using the four variations of the ND model. One can see that both the Jl and AI variations of the model are unreliable, and the other two variations : J2 and A2 have nearly equal reliability, but not adequately rep- resentative of the data.

._=-°" "~o o,5 ]~~\ // .~ 0.10

o ~ o o (an . )

-r 0.05 ......

-" -5 - 3 -I 1 3 5 7

Hour (noon = 0 1

Fig. I. Comparison of normal distribution models (winter).

0.20

o Data (June) AI

. . . . . . . . . A2

:~ 0 .15 o - - - - J 2 :6 o

o.,o ?/ Y~ '-.x,.X

o.II \ \o 0.0 //

- 7 - 5 - 3 - 1 i 3 5 7

Hour (noon=O)

Fig. 2. Comparison of normal distribution models (summer).

By further inspection of Figs 1 and 2 and the graphic display of the rest of the calculations one can conclude that the distribution improves when a is computed using local data. Thus, on focusing on the data collected in Amman, Jordan, the correlation coefficient, R 2, values were computed for both A 1 and A2. These are shown in Table 1. It is obvious

Table 1. R 2 for ND model computations

Jan. Feb. March April May June AI 0.9706 0.9437 0.9354 0.9313 0.7733 0.7175 A2 0.9681 0.9488 0.9663 0.9841 0.9690 0.9667

July Aug. Sept. Oct. Nov. Dec. AI 0.7387 0.8137 0.8784 0.9157 0.9313 0.9783 A2 0.9654 0.9694 0.9750 0.9726 0.9405 0.9764

Data Bank

that for variation AI of the model five of the twelve mon ths of the year the R z value is less than 0.9 while for A2 this value is more than 0.94.

Further, guided by the shape of the distribution of the "~ daily sunshine hours over the year, a quadratic, instead of linear, regression representation o f a in terms of sunshine hours was tested. The results show an insignificant improve- - ment in the value of R 2. This value is 0.959 compared with o 0.958. Considerable increase of the s tandard error ratios of 2 the coefficients occur : 0.9 for the linear term of the quadratic e, representation compared with 0.066 for the linear equation. The s tandard error ratio of the coefficient of the quadratic 5 term is 2.3225. An increase in the value of the s tandard error "-- ratio decreases the significance of the term in the prediction ~ model the coefficient of which is considered. A value of up = to 0.1 is justifiable [9].

Nevertheless, by further inspection of the twelve graphic representations of the data and the computed results of the models we found that for 57 hours of the total of 180 hours of the year, or about 32%, the N D model A2 is a reliable model.

This analysis leads to the following two limitations for the implementation of this model.

1. The N D model improves considerably by comput ing a using the sunshine hours regression formula instead of using the measured noon solar radiation data.

2. The results reported by Jain [8] lack the Claimed uni- versality. In the best of cases the value of a depends on the latitude of the location in question. In general, the current model depicts the effects of clouds and air mass on the availability of solar radiation as a normally dis- tributed error. These parameters are very local ; even if the same latitude is considered for comparison.

THE HALF-SINE WAVE M O D E L (HW)

In further considering other simple models H W model fitting of the data was tested. The description of this model is described above. Equat ion (2) is the mathematical expression of this model.

Figures 3 and 4 represent the worst and the best fit of this

0 . 2 0 o D a t a ( J a n . )

HW

. . . . . . . . PO

g o ~ - . - . 0 0 . " D. :5 o.15

.h

o 0 . 1 0 - -

0 . 0 5 ?, "t-

- - 5 - 5 - 1 1 5 5 7

H o u r ( n o o n = O )

Fig. 3. Compar ison of half-sine wave and polynomial models with data (winter).

475

0 . 2 0

0 . 1 5

o D a t a ( J u n e )

~ H W

. . . . . PO

. - 'O"" "'O.

0 . 1 0

0 . 0 5

O' I I - - 3 t 5

H o u r ( noon = 0 )

Fig. 4. Comparison of half-sine wave and polynomial models with data (summer).

model to the data, respectively. By inspecting the twelve graphic displays of the data and predictions we found that the best fits occurred for the mon ths May, June, July, and August . Some data points fall on the prediction line in April and September. For a total of 62 hours of the 180 of the year or about 34% of the time, rg could be predicted by this model with acceptable accuracy.

The results of the statistical analysis are given in Table 2. It is shown that the R 2 value is more than 0.9 for eight of the twelve months of the year, namely for March through October. For these months the value of the s tandard error ratio of the coefficient of the sinusoidal term, ez,, is of the desired magni tude: less than 0.1. Thus, from this limited view this model is applicable for reliable prediction uses for these months.

Nevertheless, these overall statistical results do not indi- cate where the agreement and disagreement between data and prediction results in a given mon th takes place. The critical hours of the day for solar energy collection where we are interested in a reasonable agreement between the two data values are the few hours around noon. If this element is taken into consideration, then the months where this model

Table 2. Statistical results of HW model

Month R 2 b0 ez0 b~ ez,

Jan. 0.8364 -0 .0356 0.79 0.1728 0,120 Feb. 0.8559 -0 .0299 0.82 0.1633 0.110

March 0.9160 -0 .0241 0.71 0.1534 0.084 April 0.9374 -0 .0161 0.83 0.1398 0.072 May 0.9791 -0 .0108 0.65 0.1310 0.041 June 0.9837 -0 .0072 0.79 0.1249 0.035 July 0.9772 -0 .0085 0.84 0.1270 0.043 Aug. 0.9674 -0 .0135 0.67 0.1355 0.051 Sept. 0.9506 --0.0196 0.62 0.1457 0.063 Oct. 0.9288 -0 .0235 0.67 0.1524 0.077 Nov. 0.8350 -0 .0338 0.82 0.1699 0.123 Dec. 0.8380 -0 .0361 0.78 0.1738 0.120

476 Data Bank

is applicable are reduced to four, namely, May, June, July, and August. This restriction is further justifiable by observing the error in the constant of the equation as listed in Table 2.

THE POLYNOMIAL M O D E L (PO)

The third model considered in this research is the PO model as stated in eq. (3), above. The results reported in this research are limited to polynomials of the fourth power. Although, higher power polynomials were tested, they were discarded because no significant contribution to the accuracy of the results was noted. Other terms could also be discarded on the basis of the criterion used in the statistical analysis [9].

By visual inspection of the graphic display of the data and the computed results one outstanding feature of this model is its close agreement between data and results during the significant hours of the day, namely, the four to eight hours around noon. Figures 3 and 4 which also show PO pre- dictions for a typical month in winter (January) and a typical month in summer (June), respectively, illustrate this feature.

The statistical analysis confirm the above observation. Table 3 shows the R 2 values; the constants and the coefficients o f the polynomial terms, and the standard error ratios of these coefficients. The fourth term of the polynomial was discarded because the coefficients of that term were zero or very small. The following observations could be stated :

1. the R 2 values of the twelve months are greater than 0.98,

2. the third terms could also be discarded because, as seen from Table 3 the coefficients are very small, and

3. the linear term could be ignored for all months of the year except April, because the standard error ratios of the other months are more than the acceptable limit, namely, 0.I.

Thus the polynomials could be reduced in most cases to the constants and the quadratic terms of the polynomial.

By counting the hours of the total o f 180 of the years where this model is reliable it is found that this model is acceptable for 75 of these hours or about 42% of the time. What is significant about this is that all of these hours are around noon: the hours of interest in solar energy applications.

0.20

g

"~ 0.1,5

0.10

o 1D

cto5 ." A "". 0 "l- .." / "..

- - 5 - 3 - 1 I 5 7

Hour (noon=O)

Fig. 5. Comparison of the combined and other models with data (winter).

THE COMBINATION M O D E L

As illustrated above, none of the three simple models is adequately representative of the data although the PO model is fairly representative of the hours of the year with the highest solar radiation intensity. The coefficients of the com- bined model, given in eq. (4), are computed for best rep- resentation of the data. The values of these coefficients are given in Tables 4, 5, and 6.

A typical application of the listed values of the ~p, ¢P~, and • , is shown in Figs 5 and 6 for a typical winter month (January) and a typical summer month (June), respectively. Predictions by the other three models are also given in the same figures for comparison. The agreement between the data and predictions by the combined model is very good. About 132 hours of the 180 hours of the year or about 74% of the time could be predicted by this model with excellent accuracy. The significance of the proposed model and others which may be developed by following the same procedure is its application in the simulation of systems. The values of

Table 3. Statistical results of PO model

Month R 2 ao eZo a ~ e z ~ a2 e %

Jan. 0.9901 0.1638 0.0482 0.0032 0.3750 --0.0094 -0 .0426 Feb. 0.9933 0.1547 0.0388 0.0058 0.1552

March 0.9980 0.1450 0.0207 0.0031 0.1613 April 0.9993 0.1345 0.0112 0.0038 0.0526 May 0.9983 0.1268 0.0181 0.0000 - - June 0.9982 0.1228 0.0179 0.0008 0.3750 July 0.9985 0.1245 0.0169 0.0016 0.1875 Aug. 0.9983 0.1303 0.0184 0.0013 0.3077 Sept. 0.9984 0.1381 0.0181 0.0002 2.0000 Oct, 0.9975 0.1440 0.0229 -0 .0016 -0.3750 Nov. 0.9815 0.1603 0.0661 - 0.0052 - 0.3077 Dec. 0,9893 0.1647 0.0498 -- 0.0008 -- 1.5000

--0.0082 -0.0366 - 0.0068 --0.0294 -0 .0055 -0.0182 -- 0.0046 - 0.0217 -0.0041 -0 .0244 -0.0043 -0.0233 -0 .0050 -0 .0200 - 0.0059 - 0.0169 -0.0067 -0.0299 -0.0089 - 0.0674 - 0.0095 0.0526

Data Bank

Table 4. The Cp values o f the combinat ion mode l

477

h\i 1 2 3 4 5 6 7 8 9 10 11 12

- 7 0 0 0 0 0 0 0 0 0 0 0 0 - 6 0 0 0 0 0 0 0 0 0 0 0 0 - 5 0 0 0 0 0 0 0 0 0 0 1 1 - 4 1 1 0 0 0 0 0 0 0 1 0.5 0 - 3 0 1 1 0 0 0.5 0.5 0 0 1 1 1 - 2 1 0 0.5 0 I 0.5 0.5 0 l 1 1 1 - 1 1 0 0 0 1 1 0 0 0 0 1 1

0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 1 1 1 1 1 0 0 2 1 l 0 1 1 1 1 1 1 1 1 1 3 0 1 0 1 0 0 0.5 0 0 1 1 1 4 1 1 0 0 0 0 0 0 0 1 I 1 5 0 0 0 0 0 0 0 0 0 0 0 0 6 0 1 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0

Table 5. The ~ , values o f the combinat ion model

h\i 1 2 3 4 5 6 7 8 9 10 11 12

- 7 0 0 0 0 0 0 0 0 0 0 0 0 - 6 1 1 1 1 1 1 1 1 1 0 1 I - 5 0 0 0 0 0 0 0 0 0 0 0 0 - -4 0 0 0 0 1 0.5 0 0.5 0.5 1 0 0 --3 0 0 0 0 1 0.5 0.5 1 1 1 0 0 --2 0 0 0.5 0 0 0.5 0.5 1 0 1 0 0 -- 1 0 0 0.50 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 I 0 0 3 1 0 1 0 1 1 0.5 1 1 1 0 0 4 0 0 0 1 1 1 1 1 0.5 1 0 0 5 0 0 0 1 0 1 1 0.5 0 0 0 0 6 0 0 1 1 1 1 1 1 0 0 1 1 7 ! 0 0 0 0 0 0 0 0 0 0 0

Table 6. The *n values o f the combinat ion mode l

h\i 1 2 3 4 5 6 7 8 9 l0 11 12

--7 1 1 1 1 0 0 0 1 1 1 1 1 --6 0 0 0 0 0 0 0 0 0 0 0 0 --5 0 0 1 1 I I 1 1 1 1 0 0 - - 4 0 0 1 1 0 0.5 1 0.5 0.5 0 0.5 1 --3 1 0 0 1 0 0 0 0 0 0 0 0 --2 0 1 0 1 0 0 0 0 0 0 0 0 --1 0 0 0.5 1 0 0 l l 1 0.5 0 0

0 0 0 0 1 1 1 1 1 0 0 1 l I 0 0 0 0 0 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0.5 0 0 0 5 0 1 1 0 I 0 0 0.5 0 1 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 1 0 0 0 0 0 1 1 1 1

478 Data Bank

0 . 2 0

o D a t a ( J u n e )

. . . . . . . . . . . . . H W

. . . . . N D

~ - - P O o 0 . 1 5

"1o ~ Combined

o 0 , I 0

o~

> ,

o " o

>, 0.05 _J

o " r

O, I I 7 -3 1 5

Hour ( no0n=O)

Fig. 6. Comparison of the combined and other models with data (summer).

around noon and the two early morning and two late afternoon hours of the day.

4. The best results obtained using the HW model are for the four months of the year from May through August. Yet, these results indicate discrepancy at and around the noon hours.

5. The PO model is most representable of the data at and around the noon hours for about four to eight hours of the day.

6. The newly developed combined model is the most reasonable model to use in simulation application. It could be made to represent any set of data. Most of the deviations of the values predicted by this model occurred at the early and late hours of the day.

The data used in this research covers only five years col- lected in the capital city area of Jordan. In contrast, the data used by Jain [8] covers ten or eleven years. Nevertheless, the statements we made concerning the generalizations made by Jains [8] are supported by the graphics display of the results and the data, and we believe that they will stand future tests when more data becomes available.

@p, ¢~, and On could be calculated for any limited or extended set of data for any desired locality.

CONCLUSION

In comparing the three simple models and the developed combination model for the prediction of the monthly average of the hourly solar radiation in Amman, Jordan, the fol- lowing are the basic results.

I. None of the three models is adequately representative of the data used in this research. The PO model, however, represents 75 hours of the 180 hours of the year com- pared with 57 hours representable by the ND model, and 62 hours representable by the HW model.

2. In computing the tr for the ND model the use of the measurements at noon only must be discarded because of the error incurred as a result of this approach. Instead, the computation of tx using the regression for- mula which gives this parameter in terms of sunshine hours must be used.

3. The best results obtained by ND model, limited as indi- cated in item 2 above, are for the two to three hours

R E F E R E N C E S

I. H. E1-Mulki, Solar and wind energy potential in Jordan. Solar Energy Applications, The Royal Scientific Society, pp. 95-104 0987).

2. M. A. Alsaad, Solar radiation map for Jordan. Solar and Wind Technology 7, 267-275 (1990).

3. M. S. Audi, Estimation of solar energy flux in Jordan. Dirasat 6(2), 121-134 (1979).

4. M.A. Alsaad, Improved correlations for predicting global radiation for different locations in Jordan. Int. J. Solar Energy 8, 97-107 (1990).

5. M. Alsaad, S. Habali, M. Hijazi and N. Rabadi, An inexpensive and reliable solar water heater for Jordan. Dirasat XII(l), I 11-128 (1985).

6. M. A. Alsaad, A sub-atmospheric solar distillation unit. Int. J. Solar Energy 5, 129-141 (1987).

7. M. A. Alsaad and M. S. Audi, Solar radiation charac- teristics in Jordan, Proceedings Fourth Arab International Solar Energy Conference, Amman, Jordan (1990).

8. P. C. Jain, Estimation of monthly global and diffuse irradiation. Solar and Wind Technology 5, 7-14 (1988).

9. M. S. Audi, Simulation models for solar and other weather parameters. Submitted for publication.