Renewable Energy Vol. 2. No. 4/5. pp. 451-456, 1992 0960-1481/92 $5.00+.00 Printed in Great Britain. Pergamon Press Ltd

DATA BANK

An empirical method for estimation of hourly diffuse fraction of global radiation

M. AL-RIAHI, N. AL-HAMDANI and K. TAHIR Solar Energy Research Centre, Jadiriyah P.O. Box 13026, Baghdad, Iraq

(Received I0 February 1992: accepted 20 February 1992)

Abstract--Modelling of solar energy systems requires computation of both direct and diffuse components of incident solar radiation. Based on data collected between 1984 and 1987 in Fudhaliyah near Baghdad, a correlation is developed to estimate the diffuse fraction of hourly global radiation in terms of clearness index. Results show that this correlation is very comparable to correlations suggested by other studies. Because this correlation gives a significant uncertainty in the estimated hourly diffuse radiation, a new empirical multiparameter correlation is proposed. In this new method, hourly diffuse radiation is derived from hourly clearness index and intensity of sunshine. Comparison of predicted and measured values indicate the method performs well during different sky conditions. The statistical distribution shows that 88.5% of the estimated diffuse radiation lies within a small error (_+0.2 M J/m2). The peak of the error frequency histogram occurs for the class of -0.1 to 0.0 MJ/m 2, and the error distribution is almost symmetrical about zero.

1. INTRODUCTION

Data of global and diffuse solar radiation at a location is of significant use in the design of various utilization devices at that location. Measurements of global radiation on a horizontal surface are now fairly common, but diffuse radia- tion is only measured at a few locations due to the cost of the equipment required, and to the care needed for regular adjustment to allow for the seasonal progression of the path of the sun across the sky. Many approaches have been used to obtain estimates of diffuse radiation for various regions around the world. The first of these approaches is to use theoretical predictions based upon specific models. The more general and universal model has been given by King and Buckius [I]. Recently, this model has been extended by Ideriah [2]. Because such atmospheric models require com- plex calculations, a second approach has been developed to use relationships between the ratios of global to diffuse radiation and global to extraterrestrial radiation. Two of the most widely used correlations in this respect are given by Liu and Jordan [3] and Page [4]. A third approach has been proposed to find correlations between the ratio of diffuse to global radiation and aspects of local climate data, such as the ratio of hours of bright sunshine to the daylength and the total cloud cover. This has been the method adopted by many workers such as Iqbal [5] and Stanhill [6]. The above-mentioned correlations are convenient to derive diffuse radiation on hourly, daily and monthly basis.

Relationships for estimating the beam and diffuse com- ponent of monthly solar radiation have been developed by Liu and Jordan [3], Page [4], Erbs et al. [7], Iqbal [8] and Collares-Pereira and Rabl [9].

To estimate the beam and diffuse components of daily radiation, empirical correlations have been developed by Liu and Jordan [3], Stanhill [6], Erbs et al. [7], Collares-Pereira and Rabl [9], Ruth and Chant [10], Choudhury [I1] and Tuller [12].

Erbs et al. [7], Orgill and Hollands [13], Bruno [14], Boes et al. [15], Bugler [16], and Spencer [17] developed empirical

correlations to calculate the hourly diffuse radiation on a horizontal surface from the global radiation data. Most of these works express the correlation in terms of (Kd)h and (KT)h, whereas Bugler [16] obtained correlation in terms of (Kd)h and (Kc)h,

In a previous study [18], daily correlations between: (i) diffuse fraction of global radiation and clearness index ; (if) diffuse fraction and fractional sunshine duration: and (iii) diffuse fraction and clearness index combined with frac- tional sunshine duration, were developed for Fudhaliyah, Baghdad. In addition, the monthly averages of these cor- relations were also established.

The primary objective of the present study is to develop hourly correlation of the diffuse fraction, including intensity of sunshine explicity as a parameter in addition to the clear- ness index. The advantage of this approach is that it gives more accuracy and best fit in the suggested correlation.

This study is based on hourly data of global radiation, diffuse radiation and intensity of sunshine collected between 1984 and 1987 in Fudhaliyah, Baghdad.

2. DATA BASE

Since 1984, the Solar Energy Research Centre has operated an Automatic Weather Observation Station (FA 511 Wilh. Lambrecht GmbH) at Fudhaliyah, 35 km northeast of Baghdad (latitude = 3Y'14'N, longitude = 44°14'E, eleva- tion = 34 m above MSL). A description of the experimental setup, the automated data acquisition and processing system and of the adopted quality control measures is given else- where [19]. To repeat briefly, measurements of hourly global radiation, diffuse radiation and intensity of sunshine were made using Dirmhirn Star-shaped pyranometer 1610, Dirmhirn Star-shaped pyranometer with a shadow band, and Transmitter 1620, respectively. The Transmitter is designed for the photoelectric sensing of the duration of sunshine and for the optical signalization of the states: brightness without sunshine, sunshine and darkness.

451

452 Data Bank

The data were simultaneously recorded on charts and stored on cassette tapes. The data for a 30-month period between August 1984 and August 1987, for which we obtained continuous measurements, were included in this study. Seven months of the data were rejected as a number of interruptions occurred during the measurements due to the multifunctioning of the instruments and lack of spare parts. The data were checked to eliminate values of solar radiation recorded at the very low solar altitudes. Data for which the diffuse ratio was equal or greater than 1.0 were also dropped.

3. DATA PROCESSING

The first step in the data processing involves converting all the hourly measurements of global and diffuse solar radiation into values of (Kd)h and (KT)h. In this conversion, extra- terrestrial radiation on a horizontal surface is needed. The values of the instantaneous extraterrestrial radiation ]o can be calculated from :

1o = ScR(s inqbs in6+cos~pcos fcoso~) . (1)

The solar constant, So, was 1353 W/m 2 [20], and the solar hour angle, ~, was taken to be the value at the mid-point of the hour. Calculations of the solar declination angle, 3, and the eccentricity correction factor, R, were carried out using the Fourier series approximation given by Spencer [21].

Since the measured terrestrial values of solar radiation are hourly integrated values, the calculation of (KT), also requires an hourly integrated value Of/o. Orgill and Hollands [I 3] recommended that the instantaneous value of lo be used instead of its integral value over the time period to represent the hourly extraterrestrial radiation, Io. They found that such use made no significant difference to their results.

Using these results the ratio (Kr)h of hourly global radia- tion on a horizontal surface, L to the extraterrestrial radia- tion on a horizontal surface during the same hour, Io, was calculated. Similarly, the ratio of diffuse to global radiation, Id/ l = (Kv)h, on a horizontal surface was calculated for each hour.

The next step in the data processing was to sort the pairs of hourly (Kr)h and (Kd)h into a one-dimensional histogram, using an interval of 0.05 in (Kr)h and (Kd)h. Combined histo- grams for all collected data were also formed.

lected in Highett, Victoria, Australia (latitude 38~'S). Some differences were apparent, especially for values (KT)h < 0.2. Erbs et al. proposed that their correlation was location independent. However, in an interesting study made by Spencer [17], models due to Orgill and Hollands [13], Bruno [14], Boes et aL [15] and Bugler [16], were compared for five Australian locations (latitude range 20~,5°S) using 3~, years of data. He found that a latitude effect to be present in the correlations which based on Orgill and Hollands procedure to perform best.

Following the procedure discussed in the previous studies, a relationship was developed between ld/! and (KT) h using 30-month data of hourly diffuse and global radiation for Baghdad.

The entire range of (KT)h was divided into three convenient regions, and the resulting correlation is given by

r 0.932 ; for (Kv)h < 0.25

Id/l = ~ 1.293--1.631 (Kr)h ; for 0.25 ~< (Kv)h ~< 0.70 (2) / 10.151; for (KT)h > 0.70.

In the range of (Kx)h between 0.25 and 0.70, various degrees of fit were tried for the regressed curve. A straight line fit, with a correlation coefficient r = 0.935, was found to provide the best fit. For values of (Kv)h < 0.25, it seems that a constant value of (ld/l) is a reasonable approximation, this is also suggested by Spencer [17]. The data in this range of (Kv), represent extremely overcast days with about 90% of the radiation being diffuse. Most of the radiation being diffuse due to overcast days, the measured value of global radiation tend to be small and hence affected by the instru- ment sensitivity and accuracy. The data in this range of (Kv), represent 14% of the total data. The values in the range of (Kv)h > 0.70 usually represent relatively clear sky conditions with some clouds, but without any shading of the sun. A constant value of (Kv)h has been assumed for (Kr), > 0.70 due to the unpredictable nature of cloud reflection and limited number of data points. Similar results have been reported by Orgill and Hollands [13] and Erbs et al. [7]. A frequency histogram of (Kd), and (Kv)h data for Baghdad is shown in Fig. 1.

A comparison of our hourly correlation for Baghdad with that of Orgill and Hollands [13] for Toronto, Erbs et al. [7]

4. CORRELATIONS BETWEEN HOURLY DIFFUSE AND GLOBAL RADIATION

Correlating the diffuse fract ion with (Kr)h The idea of developing a relationship between diffuse and

global horizontal radiation is due to pioneering work of Liu and Jordan [3]. Their method is based on 10 years of data from Blue Hill, MA, U.S.A. (latitude 42°13'N), and is ap- plicable to daily rather than hourly values. Following the approach of Liu and Jordan for correlating Kd and KT, Orgill and Hollands [13] developed a correlation between hourly diffuse fraction (Id/l) and hourly clearness index (I/Io). The study of Orgill and Hollands was based upon 4 years of data from Toronto, Canada (latitude 43°48'N). Erbs et al. [7] followed the procedure of Orgill and Hollands to develop a correlation for U.S. locations. They used 65 months of data for four locations (Fort Hoot, TX; Maynard, MA; Raleigh, NC; and Livermore, CA) with a latitude range of 31-42°N. A single correlation, for the four locations showed a close agreement with the Toronto fit [13]. They also checked the applicability of their correlation using 3 years of data col-

1 a _-'~""'~- (K-r)h " " ' " (Kd lh 16 ~Z

. 1 " [ - - >. 14 ~ o ¢,,- i r ~

® 12 i ,,-,z '

10 -a

0

I I t I I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(KT)h, (Kdlh

Fig. 1. The distribution of hourly (KT) h and (Ko)h for a 30 month period for Baghdad.

Data Bank 453

Present study (Baghdad) . . . . . Orgill and Hollands (Canada) [13] - - . ~ Erbs et al. (U.S.A.) [7] . . . . . . . . Spencer (Australia) [17]

1.0 .~. ~ - ~ , - ~ ~

. . . . . . . . . . %. , , , , , .

\..), o.o

°.o ~ ~ 0.5

0.4 "'°, ~

0.3 ~"" .~A 0.2 N ' , ~ - : . " . . . .

0.1

0 0'.1 012 01s 014 015 016 0'.8 019 1.b (KT)h

Fig. 2. Comparison of the present hourly diffuse fraction correlation and some previous studies.

for U.S.A., and Spencer [17] for Melbourn is shown in Fig. 2. This figure shows a typical plot of (loll) vs (KT)h along with the average data of (Kx)h. It is clear that for (KT)h < 0.25 and (K-0h > 0.70, the Baghdad correlation lies approxi- mately between the correlations obtained by those of other regions. In the range of (KT)h between 0.25 and 0.70, Baghdad correlation lies below all other correlations, but deviates slightly from the Spencer correlation [17]. This might be due to the latitude dependence of the Spencer correlation. Generally speaking, no single correlation exists which may be applicable for all regions.

Table 1 summarized the hourly correlations of the studies discussed in this section.

Correlatin 9 the diffuse .fraction with (KT)n and (Fs)h As mentioned earlier, most parameterization schemes for

the derivation of hourly diffuse radiation from hourly global radiation involve the clearness index, (Kv)h. The parameter KT is an indicator of the relative clearness of the atmosphere. In general, when the atmosphere is clearer, a smaller fraction of radiation is scattered. Many studies of this type were performed by a number of investigators starting with the classic paper of Liu and Jordan [3]. The inclusion of the effect of hours of bright sunshine or "cloudiness" in reducing direct and increasing diffuse radiation, has also been accomplished in various ways and examined by numerous researchers for many parts of the world [22 26]. Erbs et aL [7] used a month of data from Albany, NY, to develop a correlation which demonstrates the dependence of the hourly diffuse fraction on cloud cover. They stated that a corre- lation, which includes the hourly per cent possible sunshine as a parameter, could significantly reduce the standard devia- tion of the hourly diffuse fraction from the correlation. However, after a research in the available literature, no paper was found to study the effect of the two-parameters of (KT)h and the intensity of sunshine, (F0h, on the derivation of hourly diffuse fraction, ld/l. The diffuse fraction of global radiation is probably better described by considering the variation in hourly intensity of sunshine. This intensity (in per cent) is proportional to the total shining of the sky, and follows a skew distribution with zero per cent as the lower boundary (for standard overcast sky) and a value of 100% as the upper boundary (for standard clear sky).

Using the combination of the two parameters of (KT)~ and (F~)h multi-parameter linear regression correlation was developed for the hourly diffuse fraction of global radiation, Id/l. This correlation is based on 30-month data for Baghdad. The first advantage of this approach is that the derived correlation will have greater precision in fitting the data (correlation coefficient r = 0.991), and it covers all of the (Kr)h ranges. The second advantage is that the deduced correlation appears to be seasonally independent. The resulting relationship is represented by the following equation

Id/l = 0.991~0.454(KT)h--0.567(F~)h. (3)

Table 1. Hourly diffuse-clearness index correlation

Investigators Ref. Correlation equation

Erbs et al. [7] la/1 = 1.0-0.09 (KT)h; (KT)h ~< 0.22 = 0.951 lq).1604 (KT)h+4.388 (KT)~

--16.638 (KT)~,+ 12.336 (KT)~, ; 0.22 < (Kr)h ~< 0.80

-- 0.165; (KT)h > 0.80

ld/1 = l.lL0.249 (KT)h; (KT)h < 0.35 = 1.557 1.84 (Kx)h; 0.35 ~< (KT)h <~ 0.75 =0.177; (KT)h>0.75

Ij/l = 0.94+0.0118q~-- (1.185 + 0.0135~b)(Kv)h ; 0.35 < (Kv) h < 0 .75 ~b is the latitude, between 20-45"S

Id/1 = 0.932: (Kr)h < 0.25 = 1.293 1.631 (Kv)h; 0.25 ~< (KT)h ~< 0.70 = 0.151 ; (Kx)h > 0.70

OrgiU and Hollands [13]

Spencer [17]

Present study

454 Data Bank

Table 2. Comparison between measured diffuse radiation (Diff.)m and calculated diffuse radiation (Diff.)~ by eq. (3). Units in Wh/m z

Time (h)

Date 08 09 10 11 12 13 14 15 16 17 18

26 December (Id/l) 0.89 0.43 0.29 0.24 0.31 0.50 0.57 0.49 0.92 0.93 - - 1985 (KT)h 0.I1 0.34 0.49 0.59 0.60 0.54 0.51 0.44 0.20 0.12 - -

(F~)h% 6 78 82 83 74 42 36 47 3 3 - - (Diff.)r. 25 69 92 106 143 210 213 137 86 27 - - (Diff.)~ 26 65 94 109 139 212 210 147 82 27 - -

11 March (/d/1) 0.85 0.74 0.55 0.77 0.48 0.60 0,45 0.44 0.47 0.64 0.86 1987 (KT)h 0.26 0.39 0.53 0.34 0.63 0.54 0.60 0.45 0.51 0.33 0.08

(Fs)h% I0 24 35 8 37 24 46 59 58 47 4 (Diff.).~ 115 223 279 278 335 353 285 230 185 114 18 (Diff.)c 116 204 277 284 347 359 291 236 169 101 20

6 June (~d/l) 0.23 0.18 0.18 0.17 0.17 0.16 0.20 0.23 0.25 0.31 0.36 1987 (KT)h 0.63 0.68 0.70 0.71 0.73 0.72 0.66 0.62 0.57 0.48 0.35

(F~)h% 87 85 87 85 87 85 86 84 85 86 86 (Diff.)r. 109 122 140 149 157 152 167 157 142 118 70 (Diff.)¢ 100 132 144 170 160 168 168 167 143 111 66

4 September (ld/l) 0.23 0.18 0.15 0.13 0.14 0.14 0.16 0.18 0.21 0.29 0.42 1985 (KT)h 0.48 0.57 0.63 0.67 0.70 0.70 0.66 0.62 0.56 0.44 0.29

(Fs)h% 99 98 97 94 97 96 96 96 97 90 94 (Diff.)r. 72 92 99 106 116 121 125 117 106 85 48 (Diff.)c 67 92 101 119 102 1 I0 116 113 96 83 38

Table 2 displays the comparison between measured and calculated hourly diffuse radiation by eq. (3). The days were selected arbitrarily from different seasons to verify the sea- sonal independence of the correlation. Figure 3 shows a plot of the results of Table 2. From this figure, it is found that the computed hourly diffuse radiation agreed very well with the measured value. Two statistical tests, the Root Mean" Square Error (RMSE) and Mean Bias Error (MBE), were used to evaluate the accuracy of the predicted diffuse radi- ation. The accuracy of the derived correlation is well checked by computing the error distribution. The best method of

estimating diffuse radiation should yield the largest number of estimates with small errors (e.g. less than 0.2 MJ/m 2 and less than 0.1 M J/m2), and the error distribution should be symmetrical about zero [17].

Figure 4 shows the frequency histogram for the error involved in the prediction of diffuse radiation. The peak occurs in the class - 0.1 to 0.0 MJ/m 2, and the distribution is almost symmetric. The percentage of estimates within - 0 . 2 MJ/m 2 ( - 55.55 W/m s) is 88.5%, while the percentage within - 0.3 MJ/m 2 ( - 83.33 W/m s) is 96.99%. This is an indication that the proposed correlation can perform quite well.

~-~ 360

E 320

-~ 280

v 240 = .o_ 200

~5 160 .m == 12o

80

4O

6

RMSE = 0.037 M B E = -0.002

e ~ e measured o o calculated

f\ , . r \

I I I I I I 8 10 12 14 16 18

RMSE = 0.004 M B E = -0.011

11 Mar. 1987 I I I I I I 8 10 12 14 16 18

RMSE = 0 .072 R M S E = 0 .098 M B E = 0 .023 M B E = -0.053

o aoSe,~

8 j ~ o ' " " " .~ , .

6 June 1987 I I I I f 1 8 10 12 14 16 18

o % o

4 Sept. 1985 I I I I I I I 8 10 12 14 16 18 20

Time (hour)

Fig. 3. Comparison of the recorded diffuse radiation and predicted diffuse radiation for Baghdad.

Data Bank 455

1800 m

1600 O

1400 1200 .-{

" o 800

~ 600

Z 200 I ~

Error = lid, computed - Id, measured), (MJ / m 2)

Fig. 4. Error frequency histogram for prediction of hourly diffuse radiation.

5. CONCLUDING REMARKS

A correlation for the diffuse fraction of hourly global radiation has been developed in terms of clearness index. This correlation is based on 30 months of data for Baghdad. Comparison of the correlation obtained for Baghdad with the work performed for other regions, showed that no single correlation is applicable to all regions; each has its own characteristics. The error rising from this correlation may be as large as 25%, leading to significant differences between predicted and measured data. New empirical multi-par- ameter regression correlation is then proposed to derive the hourly diffuse component of global radiation, in terms of clearness index, (Kv)h, and intensity of sunshine, (F0h. This new correlation is found to be performed best when judged on criterion of absolute error.

F, 1

/o

L

K~ MBE

RMSE R

& 6

(D

Suffix h

NOMENCLATURE

intensity of sunshine, per cent hourly global radiation on a horizontal surface, Wh/m 2 estimate hourly "clear sky" global radiation on a horizontal surface, Wh/m 2 hourly diffuse radiation on a horizontal surface, Wh/m 2 hourly extraterrestrial radiation on a horizontal surface, Wh/m 2 instantaneous extraterrestrial radiation, W/m 2 ratio of hourly global to an estimate of hourly "clear sky" radiation ratio of diffuse to global radiation ratio of global to extraterrestrial radiation Mean Bias Error Root Mean Square Error correction factor for variation in the earth's radius vector solar constant, 1353 W/m 2 solar declination angle, rad latitude, degrees solar hour angle, rad

hourly.

REFERENCES

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456 Data Bank

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