~ Pergamon Energy Convers. Mgrnt Vol. 37, No. 4, pp. 501-504, 1996

Copyright © 1996 Elsevier Science Ltd 0196-8904(95)00018-6 Printed in Great Britain. All rights reserved

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E M P I R I C A L R E L A T I O N S H I P T O E S T I M A T E G L O B A L

RADIATION FROM HOURS OF SUNSHINE

O. P. SINGH, S. K. SRIVASTAVA and A. GAUR Energy Laboratory, Depar tment of Applied Sciences, Institute of Engineering and Technology, Lucknow

226 021, Ut tar Pradesh, India.

(Received 16 May 1994; received.for publication 8 February 1995)

Abst rac t - -A new set of constants for an Angst rom type correlation to estimate global solar radiation has been found. Least-squares regression analysis is performed on solar radiation and sunshine duration data recorded at Lucknow (latitude 26°45'N, longitude 80°53'E), India, to derive the new constants. These constants provide good estimates of global solar radiation with a maximum deviation of 9.67%. Statistical comparison of the present results with the correlations of Bahel Energy 11,985 (1986); and Rietveld Agric. Met. 19, 243 (1978) shows that the new constants yield more accurate estimates. The Bahel et al. correlation of third order provides better results than their first order equation [7] and the equation of Rietveld 12, 131 (1987).

Solar radiation Solar constant Extraterrestrial radiation Bright sunshine hours Regression constants Declination Latitude Sunset hour angle.

NOMENCLATURE

H = Monthly mean daily global radiation incident on horizontal surface (kWh/m 2 day) /40 = Monthly mean daily extraterrestrial global radiation incident on horizontal surface (kWh/m 2

day). S = Monthly mean daily bright sunshine hours (h) So = Monthly mean daily max imum possible sunshine duration (h)

H~ = Solar constant (1353 W / m 2) n = Day number starting from first of January L = Latitude (°) 6 = Declination (°)

W, = Sunset hour angle (°) p = Number of observations

Hm = Measured value of H H c = Calculated value of H

a, b, c = Regression constants

INTRODUCTION

Proper design of solar energy systems requires accurate knowledge about the solar radiation obtainable at particular locations. Since the solar radiation reaching the earth's surface depends on climatic conditions of the place, a study of solar radiation under local climatic conditions is essential. The best radiation information of a place is obtained from experimental measurements of the global and diffuse components of the solar insolation at that place. Due to limited solar radiation measuring stations, it has been necessary to estimate solar radiation by theoretical models. An approach, which has been used for estimation purposes, involves the empirical relationships between global radiation and such meteorological parameters as bright sunshine hours, relative humidity and temperature.

There are various empirical correlations [1-17] to estimate the total solar radiation incident on o

a horizontal surface. The first empirical correlation was proposed by Angstrom [1], which correlates global solar radiation and bright hours of sunshine. The Angstrom correlation was modified by Prescott [2] and Page [3], and these are being used most correctly and widely to estimate global irradiance. Various investigators [4-9] have also used sunshine duration to estimate the solar

501

502 SINGH eta/.: ESTIMATION OF GLOBAL RADIATION

radiation, while some other [10-12] have used declination angle and latitude. Mateer [13] and Bennett [14] combined sunshine duration, declination angle and latitude to develop their empirical models. Reddy [15] and Sayigh [16] derived their equations by using sunshine duration, relative humidity and temperature. Sabbagh et al. [17] related the daily global radiation to sunshine duration, relative humidity, maximum temperature, latitude, altitude and location of the place relative to water surfaces.

The objective of the present study is to present an analysis of the global solar radiation data o

recorded at Lucknow, India, and develop new constants for an Angstrom type correlation to estimate the monthly mean daily total radiation incident on a horizontal surface. Performance of the new constants is checked by comparing the estimated and observed values of solar radiation. The present result has been compared statistically with the correlations of Bahel et al. [7, 8] and Rietveld [9]. The purpose of the present study is also to test the suitability of the Babel et al. [7, 8] and Rietveld's equations [9], to estimate global radiation.

EXPERIMENTAL SET-UP

The global solar radiation and sunshine duration data was recorded at Lucknow (latitude 26°45'N, longitude 80°50'E, altitude 120m above sea level), Uttar Pradesh, India. A precision pyranometer, having a calibration factor 5.50 mV/cal/cm2/min, was used to measure the total solar radiation incident on a horizontal surface. A potentiometric chart recorder was used to record the output of the pyranometer. The potentiometric chart recorder had a range of 0-10 mV; chart-width of 160 mm and normal chart speed of 20 mm/h. Measured data from April 1991 to March 1992 were used for the present study. The daily total values of the radiation were computed from the record obtained from the potentiometric chart recorder.

MATHEMATICAL MODELS TO ESTIMATE TOTAL SOLAR RADIATION ON HORIZONTAL SURFACE

The most convenient and most widely used relationship to estimate monthly mean daily radiation o

on a horizontal surface is that given by Angstrom [1] and developed by Page [3]:

H/Ho = a + b(S/So) . (1)

H0 may be obtained by following Klein [18], i.e.:

H0 = (24/~z)H~[1 + 0.33 cos(360n/365)] x [cos L cos 6 sin Ws + (2re WJ360)sin L sin 6]. (2)

Ws is defined as [19]:

W~ = arc[cos(- tan L tan 6)]. (3)

The solar declination angle (6) is :

6 = 23.45 sin[360(284 + n)/365]. (4)

So can be calculated from Cooper's formula [20], i.e.:

So = (2/15)arc[cos(- tan L tan 6)]. (5)

If the regression constants a and b are known, equation (1) can be used to obtain estimates of global solar radiation for other locations with climatological characteristics similar to those of the locations where the original data were obtained.

Correlations o f Bahel et aL

Based on the radiation and sunshine data from Dhahran, Saudi Arabia, Babel et al. [7] developed the following correlation for estimating the global horizontal radiation

n / H o = 0.175 + 0.552(S/So). (6)

SINGH et al.: ESTIMATION OF GLOBAL RADIATION 503

Using the bright sunshine and global radiation data of 48 locations around the world, the following correlation of ' third order was developed by Bahel et al. [8].

H/Ho = 0.16 + 0.87(S/So) - 0.61(S/So) 2 + 0.34(S/So) 3. (7)

Correlation o f Rietveld

Rietveld (9) examined several published values of a and b and arrived at the following correlation:

H / n o = 0.18 + 0.62(S/So). (8)

Several investigators [21, 22] have found that non-linear relationships such as:

H/Ho = a + b(S /So) + c (S /So) 2 (9)

provide slightly better results. In the present study, equation (9) has been used for estimating the global horizontal solar radiation. The constants a, b and c are determined by least square regression analysis. A good agreement has been obtained between the measured and estimated values of global radiation.

Statistical tests were performed for root-mean-square error (RMSE) and mean bias error (MBE) to compare the accuracy of the present correlation as well as for comparison with the other correlations. RMSE and MBE are defined as:

RMSE = [{Z(H c -- Hm):}/p] °5 (10)

MBE = [{Z(Hc - Hm)}/p]. (1 l)

Low values for both RMSE and MBE are desired.

R E S U L T S AND DISCUSSION

The measured daily global radiation H, together with the bright hours of sunshine S, were expressed as ratios of H/Ho and S/So. A least squares regression analysis was used to fit equation (9) to these pairs for the whole year to obtain the regression constants a, b and c. The fit to the data points provides the following correlation.

H/Ho = 0.7191 - 1.0673(S/S0) + 1.1448(S/So) 2. (12)

To test the accuracy of these constants, a comparison of the estimated and measured values of monthly mean daily global radiation was made. The new constants provide good estimates of the global radiation with maximum and minimum deviations of 9.67 and 0.06%, respectively. The estimated values are quite accurate to about 5%, and the relative percentage error rarely exceeds 5%. The agreement between experimental and estimated values is remarkable. The low value of RMSE (0.26) shows that equation (12) may be used to obtain accurate estimates of the global radiation. The results of the present analysis are summarised in Table 1 along with the RMSE and MBE values. Table 1 shows that equation (12) provides more accurate results for the winter season (November, December and January).

The present results are also compared with the theoretical estimates of the Bahel et al. [7, 8] and Rietveld [9] models. It shows that the new constants provide more accurate estimates than the Bahel et al. [7, 8] and Rietveld [9] equations. A maximum deviation of 9.7% is observed for the new constants, while it is 15.9, 11.9 and 15.7% for Bahel et al. [7, 8] and Rietveld [9] models, respectively. The RMSE and MBE values for all the correlations are presented in Table 1. A comparative study of the results shows that the Bahel et al. models [7, 8] yield better results than the Rietveld model [9]. The third order Bahel et al. model [8], is better than their first order equation [7]. The present correlation, when applied for Dhahran (26°32'N, 50°13'E), Saudi Arabia [7], provides satisfactory results, and good agreement has been observed between measured and estimated values with a maximum deviation of 7% (results are not shown).

504 SINGH et al.: ESTIMATION OF GLOBAL RADIATION

Table 1. Observed and computed values of monthly mean daily global radiation (kWh/m 2 day) with percentage difference and RMSE and MBE values

Monthly mean daily global radiation Percentage difference

Equation Equation Month Observed (6) (7) (8) (9) (6) (7) (8) (9)

Apr. 1991 7.071 6.724 6.969 7.382 6.746 4.91 1.44 4.40 4.59 May 1991 6.562 6.924 7.177 7.595 6,790 5.52 9.37 15.74 3.47 Jun. 1991 5.747 5.409 5.762 5.890 5,374 5.88 0.26 2.49 6.49 Jul. 1991 5.169 4.734 5.125 5.135 5.197 8.40 0.85 0.66 0.55 Aug. 1991 4.633 3.893 4.319 4.199 5.082 15.97 6.78 9.37 9.69 Sep. 199l 4.607 4.355 4.668 4.735 4.463 5.47 1.32 2.78 3.13 Oct. 1991 5.195 5.192 5.380 5.699 5.182 0.08 3.56 9.70 0.25 Nov. 1991 4.087 5.145 4.577 4.546 4.041 1.42 11.99 11.23 1.13 Dec. 1991 3.098 3.259 3.419 3.561 3.100 5.20 10.36 14.94 0.06 Jan. 1992 3.500 3.640 3.798 3.984 3.460 4.00 8.51 13.83 1.14 Feb. 1992 4.899 4.795 4.970 5.260 4.706 2.12 1.45 7.37 3.94 Mar. 1992 6.073 5.673 5.880 6.222 5.559 6.59 3.18 2.45 8.46

Annual 5.053 4.895 5.170 5.351 4.975 3.13 2.32 5.90 1.54 RMSE - - 0.3274 0.2861 0.4525 0.2622 . . . . MBE - - -0.1415 0.1153 0.2973 -0.0784 . . . .

C O N C L U S I O N

New regression constants give accurate estimates of global radiation incident on a horizontal surface. The estimated values are quite accurate to about 5% and the relative error rarely exceeds 5%. These constants provide more accurate results for the winter season. The present constants yield better results than the Bahel et al. [7, 8] and Rietveld [9] correlations. The Bahel et al. and Rietveld correlations also provide satisfactory results with the Bahel et al. models giving some better results. The Bahel et al. model [8], which is of third order, yields better results than their linear model [7].

Acknowledgement-4)ne of the authors (OPS) is thankful to the Council of Scientific and Industrial Research, New Delhi, for providing financial assistance as Research Associateship.

R E F E R E N C E S

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