~ Pergamon Renewable Energy, Vol. 13, No. 2, pp. 249 253, 1998

,L3 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain

PII : S0960-1481 (97)00082-7 096(~1481/98 $19.00 + 0.00

D A T A B A N K

The diffuse fraction of hourly solar radiation for Amman]Jordan

MAHER I. HIJAZIN Faculty of Engineering and Architecture, American University of Beirut, Beirut,

Lebanon

(Received 2 September 1997 ; accepted 2 October 1997)

Abstract--This paper presents various correlations of the hourly clearness index Kt with the hourly diffuse fraction Kd on a horizontal surface for Amman/Jordan. The first correlation is based on Orgill and Hollands method, but using weighted averages. The second correlation is based on hourly integrated values of Kt's and K~'s without any grouping. A refinement of the latter correlation is attempted by including a random shock. © 1998 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Hourly integrated values of solar radiation are usually used in simulating solar systems. Measurements of solar radiation in a specific region are usually taken on a horizontal plane, as measurements on various inclinations and orientations are prohibitively expensive.

It is the purpose of this work to make use of hourly integrated values of a one year of data of diffuse and global radiation, and to correlate the diffuse fraction Kd to the clearness index Kt, in the hope that this will enable the extrapolation of the diffuse radiation from a more commonly measured variable such as global radiation. This task will help in better design and evaluation procedures than the existing ones in Amman/Jordan.

2. DATA BASE

The solar radiation data which is used in this paper is taken from a report [1] which contains the only published solar radiation data in Jordan. The measurements have been done on the premises of the Royal Scientific Society (Latitude = 3U' North, Longitude = 36 c East, 980 meters above sea level) in the outskirts of Amman ; the capital. Sixteen integrated hourly values are reported for each day, starting from 4:00 till 20:00 local time. Each value represents the integrated mean value for each hour.

Nothing in the report [1] is mentioned about the methodology or accuracy of the measurements, except that the measurements were carried out by newly calibrated pyranometers.

The measurements of diffuse radiation were carried out using a shadow band pyranometer. No information about the accuracy of the measurements is given and it is thought that the diffuse radiation data were not compensated for the shadow band effect. For this reason, any results which may come out from this work should be cautiously used.

Various simplifying assumptions and inconsistency checks were made, regarding the global and diffuse radiation data. These can be summarised as follows :

(1) The first and last three values of each daily record were truncated, leaving ten daily values. (2) Global and diffuse radiation values were assumed to be representative of the mid hours.

249

250 Data Bank

(3) All values of global and diffuse radiation which were less than 10 W/m 2 were deleted from the record. These amounted to 124 values, representing 3.4% of the total record used.

(4) All values of the diffuse fraction Kd which were greater than one were discarded (instrument inaccuracies)• These amounted to 49 values, representing 1.3% of the total record used.

(5) All values of the K~'s, which were greater than one were reduced to 0.975 (interval width is 0.05). These amounted to 17 values.

A total of 3477 values for global and diffuse radiation were used.

3. CORRELATING THE DIFFUSE FRACTION (Kd) WITH CLEARNESS INDEX (Kt)

The diffuse fraction K a was worked out for each hour of the year. Figure 1 shows the probability density distribution of the Kd's. It is evident that Amman enjoys a great deal of direct radiation.

Since Liu and Jordan [2] correlated the daily and monthly values of the Kd'S to the Kt's, several others investigated other correlations for daily, hourly and instantaneous data. The first to investigate hourly correlation between the two parameters were Orgill and Hollands [3]. They suggested dividing the range of K~ into intervals of width 0.05, and correlating average values of Kd's within each interval with the K,'s.

Orgill and Hollands method was adopted, but correlating the weighted average values of the Kd's with the Kt's. The resulting correlation is shown in Fig. 2, and the frequencies of Kd's values within each interval of the Kt are shown in Fig. 3. The correlation can be represented by the following equation :

Kd=0.744 for 0 .0~<Kt<0.1

Ka = 0.842~).977 Kt for 0.1 ~< Kt ~< 0.8

Kd=0.06 for 0 . 8 < K ~ < 1.0 (1)

The range of Kt greater than 0.8, includes only 2.5% of the data. Following the procedure of Orgill and Hollands [3], a constant value of K~ was chosen. Orgill and Hollands attributed the observed increase in the diffuse fraction to beam radiation being reflected from the clouds and recorded as diffuse radiation when the sun is unobscured. The data frequency in this range is comparatively small to justify fitting a curve to them.

The range 0.0 ~< Kt < 0.1 includes only less than 1% of the points in the data set. The observed drop in the diffuse fraction, can be probably attributed to heavily overcast sky while at the same time, the sun is only partially or lightly obscured by clouds. Because of the low frequency of diffuse fraction

0.:30

0.~

• ~ 0.20

O.10

2 0.0 0.I 0.2 0.3 0.4 0.S 0.6 0.7 0.8 0.cJ

Kd

Fig. I. The probability density distribution of the diffuse fraction Kd.

t.0

K d

0.4

1.0

0.8 ¸

0.6

0.2

0.0

1000

I~d= 0.744 0.0- K t *0. I

I~d = 0.842 _ 0.977 K t 0.1-<K t <0.8

Kd= 0.06 0.8_ < K t -~ 1.0

[ ] B

i i i i i = i i

0.0 0.1 0.2 0.3 0.4. 0.5 0.6 0.7 0.8

K t

Fig. 2. Weighted hourly K[s vs Kt 's , and recommended correlation.

[ ] O

i

0.9 1.0

in this region, and in addition to low values of global and diffuse radiation intensities, which consequently are affected by instrument sensitivities and measurement inaccuracies ; a constant value for the diffuse fraction is recommended.

96.5% of the points in the data set lie within the range 0.1 ~< Kt ~< 0.8. The correlation coefficient of eq (l) for this range is 0.996, which means that more than 99% of the grouped data can be explained by eqn (1).

Although the previous method of correlation gives a satisfactory result, and has been adopted and recommended [3 5] to be used in simulation programmes, it is thought, that grouping the Kd's in intervals of Kt's and correlating average values of K[s, leads to losing some important information about the hourly values of/i'd. The next step in the analysis was, to correlate the hourly diffuse

800

• o 600

O

o

~u 400

200

, , l l . . . 0 0.0

Data Bank 251

,,,illl oh o2 o13 o4 o.'5 o6 o17 o.'8

K t

Fig. 3. Frequency distribution of the diffuse fraction Kd.

019 1.0

252 Data Bank

0.4-

1.0

0 . 8 .

0.6.

0.2.

0 , 0

0.0 0.9 1.0

I i i ~ | i i i |

0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8

K t

Fig. 4. Hourly K[s vs Kt's, showing the standard deviations.

fraction to the Kt's, without applying any grouping or averaging. A linear correlation was fitted to the data, for the range 0.1 ~< K~ ~< 0.8. This correlation turned out to be:

Kd = 0.847~).985 K, (2)

The mean bias error ; which is a measure of the average deviation, turned out to be 4 x 10 -5, which is so small, indicating that a linear correlation is sufficient, if a correlation between hourly values of Kd and Kt is accepted. The standard deviation ~ turned out to be 0.104, where 77% and 92% of the data points lie within ~ and 2~r, respectively. Where one standard deviation in the Ka corresponds to 41 W/m 2, in the diffuse radiation. The standard deviation was further investigated by working out the standard deviation over various arbitrary intervals 0.05 of Kt, these, are shown in Fig. 4. It is clear that the standard deviation increases as the sky becomes cloudier, the reason for this maybe is due to the high variability of cloud type and thickness.

Figure 5 shows, the error distribution between the measured Kd and the correlated one. In general,

0.5

0.4-

o2! 0.1

0.0

43.6

. , ,i I,,,,.. i i i i i i i i i i i •

-0.5 -0.,4 -0.3 -0.2 -0,1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Error in K d

Fig. 5. Probability distribution of the errors in K[s.

Data Bank 253

the large size of cr ; indicates that there maybe a considerable error (roughly plus or minus 2a) in estimating the diffuse fraction.

To try to reduce the standard deviation a, a correlation which included a random shock was attempted. This procedure, included the previous correlation eqn (2) plus a random shock R, i.e.

Kd = 0.847~).985 Kt + R (3)

Where R is normally distributed, and has a standard deviation equals to 0.104. This correlation has reduced the a by 12%.

4. CONCLUSION

The correlation developed between the grouped hourly diffuse fraction and clearness index (eqn 1), is found to be significantly different from the relationships developed previously by Orgill and Hollands [3], Erbs et al. [4], and Spencer [5]. While the uncertainty in the estimated diffuse fraction for a particular hour is significant, the correlation predicts the long-term average diffuse fraction accurately. The inclusion of eqn (2) in simulation programmes for PV pumping system would be accepted, as the error would be small. But for concentrating systems, it would give unacceptable errors.

REFERENCES

1. Solar Energy Research Centre, Solar Radiation Measurements, Royal Scientific Society, 1985.

2. Liu, B. Y. H. and Jordan, R. C., The interrelationship and characteristic distr ibution of direct diffuse and total radiation, Solar Energy, 1960, 4.

3. Orgill, J. F. and Hollands, K. G. T., Correla t ion equat ion for hourly diffuse radiat ion on a horizontal surface, Solar Energy, 1977, 19, 357-359.

4. Erbs, D. G., Klein, S. A. and Duffle, J. A., Est imation of the diffuse radiat ion fraction for hourly, daily and month ly average global radiation, Solar Energy, 1982, 28, 293- 302.

5. Spencer, J. W., A compar i son of methods for estimating hourly diffuse solar radiat ion f rom global solar radiation, Solar Energy, 1982, 29, 19-32.