Computing the monthly mean daily diffuse radiation from clearness index and percent possible sunshine

Solar Energ~ Vol. 41, No. 4, pp. 379-385, 1988 Printed in the U.S.A.

0038-092X/88 $3.00 + .00 Copyright ~ 1988 Pergamon Press pie

COMPUTING THE M O N T H L Y M E A N DAILY DIFFUSE RADIATION FROM CLEARNESS INDEX A N D PERCENT POSSIBLE S U N S H I N E K. K. GOPINATHAN Department of Physics, National University of Lesotho, P.O. Roma 180, Lesotho, Africa Abstract--There are many locations where no measured data on diffuse radiation are available and have to be estimated from empirical correlations. Here we develop correlations for predicting monthly mean daily diffuse radiation on a horizontal surface, for locations in India. from the measured data of clearness index and percent possible sunshine. Three models are developed from the experimental data of three stations in India and the applicability of the developed equations are tested by estimating monthly mean daily diffuse radiation for four locations. A correlation connecting monthly mean daily diffuse radiation together with the clearness index and percent possible sunshine is found to be the most accurate one for Indian locations.

I. 1 Type 1 correlations

1. INTRODUCTION One o f the most important requirements in the design of any solar energy c o n v e r s i o n e q u i p m e n t is the information on the intensity of solar radiation and its c o m p o n e n t s at a g i v e n location. There are two m a i n c o m p o n e n t s o f the radiation reaching the ground: the direct radiation and the diffuse radiation. For most solar energy applications, either global-tilt or direct normal solar radiation resource data are needed. H o w e v e r , the diffuse radiation on a horizontal surface too has m a n y applications, particularly in finding global radiation on titled surfaces, illumination design inside a building, and so on. W h e r e a s considerable information exists on global solar radiation on horizontal surfaces, only few d a t a are available for diffuse radiation. One has thus to depend on theoretical estimations of diffuse radiation for such locations where no m e a s u r e d data are available. Several empirical correlations have already been developed to predict diffuse radiation from available meteorological data. There are three types o f correlations for estimating horizontal diffuse radiation. The first type expresses monthly average daily diffuse radiation, Ho/H, as a function of the m o n t h l y average daily clearness index, Kr = H/Ho. Ho, H, and /40 are the m o n t h l y average daily diffuse, global, and extraterrestrial radiation on a horizontal surface. The second type expresses the fraction Ho/H or 1-1o/14o as a function o f the monthly averaged daily values of the bright sunshine hours, S, and the m a x i m u m possible s u n s h i n e hours, So. The ratio S/So is often called the percent possible sunshine. T h e third type of correlation expresses Ho/H as a function of both clearness index and percent possible sunshine. Several correlations have thus been developed for c o m p u t i n g diffuse radiation and the coefficients o f all these correlations are found to be different and site dependent. E x a m ples of some o f the well-accepted correlations that have universal applicability are given below. 379

Liu and Jordan[ 1] developed a statistically based correlation from results obtained from one station in the following form:

Ho/H = 1.39 - 4 . 0 2 7 K r + 5.531K~- 3.108K3;

.3 < Kr < .7

(1)

Page[2] developed correlations between daily total and diffuse radiation for 10 widely spread sites in the 40°N to 40°S latitude and obtained the following relationship:

Ho/H = 1 . 0 0 -

1.13Kr

(2)

Collares-Pereira and Rabl[3] used data obtained from five locations in the United States to derive the following s e a s o n - d e p e n d e n t correlation:

Ho/H = .775 + .347 (rr/180)(Ws - 90 °) - [.505 + .261 (w/180)(Ws - 90°)] x cos [2(Kr - .9)]

(3)

where Ws = the sunset hour angle in degrees Ws = 90 ° during February to April and August to October Ws = 100 ° during M a y to July Ws = 80 ° during N o v e m b e r to January The quantity 2(Kr - 0.9) is in radians.

1.2 Type H correlations Iqbal[4] used data obtained from three locations in C a n a d a to propose the correlation.

Ho/H = .791 - .635 (S/So)

(4)

380

K. K. GOPINATHAN

By eliminating H. he then used the same data to obtain

1to/1-1o = . 163 + .478 (S/So) - .655 (S/So) 2

Ho = ~r (zl,¢) c o s L C o s ~ S i n W s

(5)

Hay[5] considered the relationship between radiation before and after multiple reflections between the earth and the cloud cover. He proposed the following location independent relationship between global (H t) and diffuse (H~) radiation emerging from the atmosphere before striking the ground,

H~o/H ' = .9702 + 1.6688 (Ht/Ho) - 21.303 (HZ/Ho) 2 + 51.2880 (HJ/Ho) 3 - 50.081 (Hl/Ho) ~ + 17.5510 (HI/Ho) 5

(6)

+

2rrWsSin L sin ) 360

(11)

where cos IV, = - t a n L tan

(12)

and (360 x n) z = 1 + .033 c o s 365

(13)

Factor z accounts for the varying earth-sun distance from day to day.

1.3 Type 111 correlations In addition to the above two types, a third type of correlation expressing Ho/H as a function of both clearness index and the percent possible sunshine together has been recently suggested by Gopinathan[6] for locations in Southern Africa. A correlation of the form

the solar constant so L = the latitude of the station in degrees 8 = declination in degrees w~= sunset hour angle in degrees n = the number of the day of the year

Ho/H = .879 - .575Kr - .323(S/So)

Regional values of the regression coefficients in all the three equations along with the correlations coefficient (r), coefficient of determination (~i), and the standard error estimate (S.E.) are given as follows.

3. RESULTS AND DISCUSSION (7)

has been found suitable for stations in Southern Africa. Gopinathan has included both Kr and S/So in the estimation correlation to improve the accuracy of the estimated data by having two measured variables in the prediction formula. The purpose of the present study is to develop empirical correlations for estimating monthly mean daily diffuse radiation on a horizontal surface, from measured data of global radiation and sunshine durations, for locations in India. Available long-term data on Ho, H, and S, reported by various authors for three locations in India, are used in a regression analysis to obtain the regression coefficients of different types of correlations. One equation belonging to each type of correlation is employed in the analysis.

Ho/H = 1 . 4 0 3 -

1.672Kr

cr = .898 r = .948 S.E. = .049

(14)

Ho/H = .931 - .814(S/So) cr = .906 r = .952 S.E. = .046

(15)

H o / H = I. 194 - .838Kr - .446(S/So) 2. M E T H O D O L O G I E S FOR FINDING

Ho

cr = .947

Available experimental data on Ho, H, and S reported elsewhere by Anna Mani[7] and L r f et al.[8] for Delhi, Madras, and Poona were used in a least squares method to find the regression coefficients of the following correlations. Proper computer programs were written to obtain the coefficients of the following equations

H o / H = ao + anKr

(8)

H o / H = bo + bt(S/So)

(9)

H o / H = Co + clKr + c2(S/So)

(10)

where a, b, and c are constants. The monthly mean daily extraterrestrial radiation H0 are calculated from the following equations.

r = .973 S.E. = .036

(16)

Equation (16) has the highest value of the correlation coefficient. Coefficients of eqn (14) are higher than that of Page's equation. The regression coefficients of eqn (15) are nearly equal to Iqbal's values of eqn (4). The coefficients of eqn (16) can be compared with the coefficients of the correlation developed by Gopinathan for Southern African locations. However, such variations in the coefficients are always expected as these coefficients are site dependent. The applicability of the proposed correlations for predicting 1to is tested by estimating Ho values for the three locations used in the analysis. Calculations

Computing monthly mean daily radiation are also carried out for a new location (Calcutta). Calculated values of (Ho) for Calcutta along with the reported experimental data, Hoe, (Anna Mani[7]) are presented in Table 1. The calculated values of/4o and the reported values of S/So (L6f et a/.[8]) are also included in the table. Monthly mean daily diffuse radiation computed from eqns (14-16) are shown as 1-1ol, 1-1o2, and Ho3, respectively. Ho, Ho, and Hoe values shown in the table are in MJ m -z day -~. The relative percentage error

e=

Hoe

-

14o

(17)

100

X

Hoe

calculated for each month of the year, from all the three models, are also presented in the table. The mean percentage error is shown as MPE in the table.

N/Hoe-MpE= [>:1

x 100) ] / n

(18)

The sign of the errors are neglected in the summation and all the errors are added up while calculating the mean. The accuracy of the estimated data on diffuse radiation is also tested by calculating the mean bias and root mean square errors. The mean bias error (MBE) and the root mean square error (RMSE) are defined as follows.

(19)

381

where H~.c,l is the ith calculated value, Hi ..... is the ith measured value, and n is the total number of observations. The lower the R M S E , the more accurate the estimate is. The positive M B E shows an overestimation whereas a negative M B E shows an underestimation. The performance of the correlations for New Delhi, Poona, and Madras are shown in Figs. 1 to 3, respectively. The results shown in Table 1 for Calcutta are again shown in Fig. 4. Diffuse radiation estimated from eqns (14), (15), and (16) along with the experimental data are shown in these figures. The following observations can be made from a study of the results presented in Table 1 and in Figs. 1 to 4. Equation (14) estimates 14o fairly accurately for some of the months of the year for all the four locations. However, the errors in the estimated data are high (up to 30%) during January, February, October, November. and December for Poona; during April. May, and June for Delhi; in October and November for Madras; and in October, November, December, and January for Calcutta. Again, accurate estimations are possible for most of the months, for all the four locations, from eqn (15). However, the error in the estimated values during October for Poona in March and September for Delhi, in Januar 3' and December for Madras, and in January and April for Calcutta are very high. The accuracy of the estimated data from eqn (16) is found to be high as compared to other equations.

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The predicted values show an excellent agreement with the measured data for all stations during all the months of the year. The error in the estimated values lie within about 10% and for most of the months it is far less than 10%. A comparative study of M P E , M B E , and R M S E from all the three models for Calcutta supports the a b o v e argument. The average percentage error incurred in predicting Ho from eqns ( 1 4 - 1 6 ) for Calcutta, are 9.8, 9.3, and 5.4, respectively. As seen from the table, the M P E value from eqn (16) is the lowest.

The percent mean bias error from the three equations, for Calcutta, are + 4 . 6 , - 4 . 0 , and + 0 . 8 , respectively. Equation (14) overestimates the radiation, whereas eqn (15) underestimates the data. Equation (16), with a percent m e a n bias error o f 0.8, cancels out the two effects and improves the accuracy o f the estimated data. Again, the percent root m e a n square error from the three equations are 9.1, 9.5, and 6.7, respectively, showing the lowest value of error for eqn (16). This supports the superiority of eqn (16) over the others.

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384

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Fig. 4. Experimental and calculated values of diffuse radiation on a horizontal surface for Calcutta, in M J m - : d a y -I.

In general, there is good a g r e e m e n t between the experimental and estimated data o f diffuse radiation from eqn (16). H o w e v e r , the accuracy of the estimated data from eqns (14) and (15) is not very high. Even though eqns (14) and (15) were also derived from the data o f Poona, Delhi, and Madras, the agreement between the m e a s u r e d and calculated data from these equations for these stations is not good. This may be due to the fact that the climatic conditions of these stations are quite different and a single set of constants are not able to satisfy the results of all the m o n t h s for these stations. Separate constants for each location may thus be necessary w h e n any of these equations are to be employed. H o w e v e r , this is not the case with eqn (16). The correlations (14) and (15) e m p l o y only one m e a s u r e d variable in the estimation correlation (either K r or S/So). There are two m e a s u r e d variables in eqn (16), both Kr and S/So, and together these seems to have i m p r o v e d the accuracy of the estimated values. As an example, the error in the c o m p u t e d values during January for Calcutta, from eqn (14) and (15) are - 1 5 . 4 and 21.1, respectively. The variables in these two equations are Kr and S/So, respectively. W h e n the two variables are put together in eqn (16), the discrepancy is brought d o w n to 4 . 4 % . The accuracy o f the estimated values from eqn (16) for Calcutta, a station that has not been included in the regression analysis, suggests that this correlation in the present form can also be e m p l o y e d for other locations in India.

4. CONCLUSION In a location where no recorded data on diffuse radiation are available, there are different possible

ways of predicting Ho. Out of the three correlations developed, for estimating monthly mean daily diffuse radiation on a horizontal surface for locations in India, the equation connecting diffuse radiation together with clearness index and percent possible sunshine is found to be the most accurate one. Equation (16) in the form

H o / H = 1. 194 - . 8 3 8 K r - .446(S/So) ts r e c o m m e n d e d for c o m p u t i n g diffuse radiation for locations in India. It should be possible to calculate the monthly mean daily diffuse radiation with an accuracy of about 10%. It is also concluded that it is advantageous to have two measured variables in the estimation equation of diffuse radiation to improve the accuracy of the estimated data. This is more important while developing correlations for a region with stations having quite different climatological conditions. REFERENCES

I. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristics distribution of direct, diffuse and total solar radiation. Solar Energy 4(3), 1-19 (1960). 2. J. K. Page, The estimation of monthly mean values of daily total short wave radiation on vertical and inclined surfaces from sunshine records for latitudes 40°N-40°S. Proc. U.N. Conference on New Sources of Energy. Paper S 98, Vol. 4, 378-390 (1961). 3. M. Collares-Pereira and A. Rabl, The average distribution of solar radiation correlations between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22(2) 155-165 (1979). 4. M. Iqbal, Correlations of average, diffuse, and beam radiation with hours of bright sunshine. Solar Energy 23(2), 169-173 (1979).

Computing monthly mean daily radiation 5. J. E. Hay, A revised method for determining the direct and diffuse components of the total short wave radiation. Atmosphere 14(4), 278-287 (1976). 6. K. K. Gopinathan, Empirical correlations for diffuse solar radiation. Solar Energy 40(3), (1988).

385

7. Anna Mani, Handbook of Solar Radiation Data for India. Allied Publishers, New Delhi (1961). 8. G. O. G. Lrf, J. A. Duffle, and C. O. Smith, World distribution of solar radiation, Solar Energy laboratory, University of Wisconsin, Report No. 21 (1966).