Pergamon Solar Energy, Vol. 53, No. 6, pp. 505-510, 1994

Copyright © 1994 Elsevier Science Ltd Printed in the USA. All rights reserved

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0038-092X(94)00089-1

HOURLY DIFFUSE FRACTION CORRELATION AT A TROPICAL LOCATION

J. CHANDRASEKARAN and S. KUMAR* School of Energy, Environment and Natural Resources, Madurai Kamaraj University, Madurai 625 021, India

Abstract--Five years of hourly global and diffuse radiation data on a horizontal surface at a tropical location (Madras, India) were used to establish the relationship between the hourly diffuse fraction and hourly clearness index. Two types of correlations--on a yearly and a seasonal (wet and dry months)--have been established. Piecewise and single polynomial curves have been tilted to the data. These correlations have been compared to those developed by Orgill and Hollands (1977), Erbs et al. (1982), and Reindl et al. (1990), which have been obtained from data of temperate locations. The comparison has been performed in terms of standard deviation and relative standard deviation. The results indicate that the proposed cor- relations are better. The best tits were obtained when the seasonal effects were taken into account. It is also shown that the hourly diffuse fraction is larger at a tropical location at higher hourly clearness indices.

1. INTRODUCTION

A knowledge of the available solar radiat ion incident at a location is necessary for the design and assessment of solar energy conversion systems. It is generally mea- sured in terms of sunshine durat ion, or in terms of direct, diffuse, and global radiat ion (on a horizontal surface). But solar energy devices are generally tilted at an angle to the horizontal so tha t they receive max- i m u m radiation. In order to est imate the radiat ion fall- ing on tilted surfaces, it is necessary to know the global and diffuse radiat ion falling on a horizontal surface. From these values the total radiat ion falling on tilted surfaces can be est imated using the isotropic model of kiu and Jordan (1977) .

Measured radiat ion data are analysed to find out: • whether there is any appreciable correlation between

different locations and their c o m m o n reasons, and • the influence of various climatic parameters and their

relative significance. In addit ion, the large n u m b e r of data are condensed and presented in simple usable forms for the various types of users. In this fashion, hourly diffuse fraction correlations are available in two forms: hourly and month ly average hourly values (Duffle and Beckman, 1980; lqbal, 1983).

The hourly diffuse fraction of global solar radiat ion can be estimated by using the model proposed by Orgill and Hollands (1977) based on data of Toronto , Can- ada. They related the fraction of hourly diffuse radia- t ion to hourly global radiat ion as a function of the hourly clearness index as follows:

k d = 1 . 0 - 0 . 2 4 9 k for k < 0 . 3 5

kd = 1 . 5 5 7 - 1.84k for 0 . 3 5 _ < k _ < 0 . 7 5

ka = 0 . 1 7 7 for k > 0 . 7 5 . (1)

* ISES Member. * Present address: Dipartimento di Energetica, Universit~

Degli Studi di Ancona, via Brecce Bianche, 60100 Ancona, Italy.

Later, Erbs et al. (1982) used data from five US locations and gave a similar correlation:

ka = 1 . 0 - 0 . 0 9 k tbr k_<0 .22

kd = 0.9511 -- 0.1604k + 4.388k 2 - 16.638k 3

+ 12.336k 4 for 0.22 < k < 0.80

kd = 0.165 for k > 0.80. (2)

Reindl et al. (1990) used data from five locations in the US and Europe to assess the influence of climatic variables and presented correlations for the est imation of hourly diffuse radiation. Their correlation is of the following form:

kd = 1 . 0 2 0 - 0 . 2 4 8 k for k_<0 .3

k d = 1 . 4 5 - 1.67k for 0.3 < k < 0 . 7 8

k d = 0 . 1 4 7 for k>_0.78. (3)

It should be noted that all these studies were based on data measured at latitudes higher than 28 ° in the nor thern hemisphere and there are no reports of work of a similar nature within the tropics (23.50 ° N to 23.50 ° S) except for a study by Hawlader (1984) who has analysed 6 mon ths of radiat ion data in Singapore and proposed a correlat ion of the form

kd = 0 . 9 1 5 for k_<0 .225

k~ = 1.135 - 0.9422k - 0.3878k 2

for 0.225 < k < 0,775

kd = 0.215 for k >_ 0.775. (4)

Because of the very l imited data used for this corre- lation, this has not been considered in this study.

In India, the Meteorological Depar tmen t measures sunshine durat ion, global radiation, and diffuse radia- t ion at many locations. The data have been compiled

505

506 J. CHANDRASEKARAN and S. KUMAR

and presented in the form of tables giving the monthly average values of hourly global and hourly diffuse val- ues (Mani and Rangarajan, 1982). There are many studies on the relationship between monthly average daily diffuse to monthly average daily global, and monthly average hourly diffuse to monthly average hourly global radiation (Mani and Rangarajan, 1983; Modi and Sukhatme, 1983; Muneer and Hawas, 1984). But, there has not been any study on the relationship between the individual hourly diffuse and hourly global radiation in India.

The aim of this article is, first, to analyse individual, hourly global and diffuse solar radiation data on a hor- izontal surface in order to find out whether the ratio of hourly diffuse to hourly global radiation, kd, at a tropical location has a similar trend as in the temperate regions; second, to present a correlation between kd and the clearness index k for Madras in order to easily estimate the hourly diffuse fraction of solar radiation falling on a horizontal surface.

2. DATA BASE

Measured hourly global and diffuse solar radiation data from 1983 to 1987 at Madras (latitude 13.00 ° N, longitude 80.18 ° E) were obtained from the Indian Meteorological Department. Table 1 summarises the data used for the hourly diffuse correlations by Orgill and Hollands (1977), Erbs et al. (1982) , Reindl et al. (1990), and the present study. It can be seen that the present study has used the largest database for a single location.

The quality of the data was ensured by performing data checks as suggested in Reindl et al. (1990) . The following data were excluded: • a day with even one missing hourly data (global or

diffuse), • k d > l • beam radiation exceeding the extraterrestrial radia-

tion, • kd < 0.90 when k < 0.20 (overcast sky), and • kd > 0.80 when k > 0.60 (clear sky). After conducting the data checks and eliminating any spurious data by using the checks discussed above,

more than 13,500 hours of data were available for analysis.

The extraterrestrial, hourly radiation on a horizontal surface (Io) was calculated for each hour of the day for the whole year and the hourly clearness index k = Ig/lo for the particular hour of the day for which the hourly diffuse fraction ko = Id/Ig has been measured was established.

3. ANALYSIS

The analysis was done by calculating the diffuse fraction and the clearness index for all hourly data de- pending on the day and the month. All these values were arranged and plotted in a graph so as to obtain the hourly diffuse fraction as a function of hourly clearness index (Fig. 1).

The curve fitting of the data was performed under two conditions: 1. A piecewise correlation, similar to those proposed

by Orgill and Hollands (1977), Erbs et al. (1982) , and Reindl et al. (1990). This type of fit was carried out under two circumstances: (a) taking the whole year, so as to give one correlation applicable to all months of the year, and (b) Dividing the year in two distinct seasons, dry and wet, as the data belongs to a location in the tropics.

2. A single curve (polynomial) correlation that fits best the data (a) for the whole year, and (b) for each of the two seasons. The curve fits were quantitatively assessed by esti-

mating the standard deviation (SD) and relative stan- dard deviation (RSD) given by

SD = [ ( l / N ) ~ (kd . . . . . - - k d , c a l ) 2 ] 1/2

and

RSD = [ ( I / N ) ~ [(kd ..... -kd,caO/kd.¢al]2] 1/2. (5)

International Mathematical and Statistical Library (IMSL) subroutines were used to fit the data using the method of least squares (Draper and Smith, 1981 ).

Taking all the months into account for the three piecewise correlations mentioned above, curve fits were

Table 1. Comparison of data bases used to estimate the hourly diffuse fraction on horizontal surfaces

Study Location Latitude Longitude Years of data used

Orgill and Hollands (1977) Toronto, Canada 43.80°N 79.55°W 4 Erbs et al. (1982) Fort Hood, US 31.08°N 97.85°W 1.8

Livermore, US 37.70°N 121.70°W 1.3 Raleigh, US 35.87°N 78.78°W 1.1 Maynard, US 42.42°N 71.48°W 2 Albuquerque, US 35.05°N 106.62°W 4

Reindl el al. (1990) Albany, US 42.7°N 73.8°W 4 Cape Canaveral, US 28.4°N 80.6°W 1 Copenhagen, Denmark 55.7°N 12.6°E 1 Hamburg, Germany 53.5°N 10.0°E 1 Valentia, Spain 51.95°N 10.22°W 1

Present study Madras, India 13.00°N 80.18°E 5

Hourly diffuse fraction correlation 507

"0

0.9

0.8

0.7

0.6 X

X 0.5 X

x

0.4

0.3

0.2

0.1

0 I 0 0.2 0.4 0.6 0.8

lg/Io Fig. 1. The scatter plot of diffuse fraction kd versus clearness index k for Madras.

performed by using second-, third-, and fourth-order, Dividing the year into two seasons, the best fit for

and l inear polynomials . The best fit is given by dry mon ths was

kd = 1.0086 - 0.178k for k_< 0.24

kd = 0.9686 + 0.1325k + 1.4183k 2 - 10.1860k 3

+ 8 . 3 7 3 3 k 4 for 0 . 2 4 < k < 0 . 8

kd = 0.197 for k > 0.8. (6)

Considering the fact that tropical locations have two dist inct seasons - -d ry and wet - -cor re la t ions similar to the above have been obtained. For dry mon ths ( D e c e m b e r - M a y ) :

kd = 1.0039 - 0.192k for k_< 0.22

kd = 0.9486 + 0.3882k - 0.0276k 2 - 7.9667k 3

+ 7.3965k 4 for 0.22 < k < 0.79

k~ = 0.191 for k > 0.79. (7)

For wet mon ths ( J u n e - N o v e m b e r ) :

k ~ = 1 . 0 1 1 7 - 0 . 1 6 6 0 k for k < 0 . 2 5

kd = 0.9802 + 2 .1700k: - 11.4291k 3

- 9 . 0 0 7 4 k 4 for 0.25 < k _ < 0 . 8

k~ = 0.207 for k > 0.8. (8)

Calculat ions were made to see whether it is possible to provide a single curve correlat ion which best fits the data. Taking the complete year data, the best fit was a four th-order polynomial

ko = 1.0060 - 0.3170k + 3.1241k 2 - 12.7616k 3

+ 9.7166k 4 for k_< 0.8

ko = 0.198 for k > 0.8. (9)

ko = 1.0000 - 0.2326k + 2.3633k 2 - 11.6129k 3

+ 9 . 3 1 1 7 k 4 for k_<0 .79

k 0 = 0 . 1 9 2 for k > 0 . 7 9 , (10)

and for wet m o n t h s

kd = 1.0114 -- 0.3714k + 3.5846k 2 - 13.5703k 3

+ 10.1262k 4 for k < 0.8

k d = 0 . 2 0 8 for k > 0 . 8 . (11)

Figure 2 shows the curves obta ined by the earlier correlations (Orgill and Hollands, 1977; Erbs et al., 1982; Reindl et aL, 1990). Figure 2 also shows the piecewise correlat ion developed taking the whole year.

In Fig. 3, the following curves have been drawn (a)

The single and piecewise correlat ion for the two sea- sons, and (b) The single and piecewise correlat ion tak- ing the whole year. It can be seen that the piecewise and single curve correlat ions are very similar. There is a very small difference between the two, and the dif- ference occurs only when k is less than 0.2.

To find out whether the temperate- locat ion-based correlat ions can be used for Madras, the SD and RSD

were calculated for the correlat ions proposed by Orgill and Hollands (1977) , Erbs et al. (1982), and Reindl et a/. (1990). The SD and RSD values for these along with those obtained for Madras-data-based correlations are presented in Table 2. The results indicate tha t the s tandard deviat ions obta ined for the proposed corre- lat ion are substantial ly lower than those of the tem- perate-region-based correlations, while the relative s tandard deviat ions are comparable .

A compar i son of the SD and RSD values of the

508 J. CHANDRASEKARAN and S. KUMAR

Iml

Iml

1.0

0 "8 ~

0 . 6

0 - 4

0-2

0 -0

o

E R ~ E ~ p 0~__ P P

O ~ ~ - - c - - - - - f f - - E R-

O--Orgill E~ Hollands 0977) E - E~bs, et al. (1982) R-- Reindl, et at. (1990) p--Present Study (1994)

I I I I I I o .o 0 .2 o . ~ 0.6 0.8 ,.o

ig/io Fig. 2. The diffuse fraction kd ( = 1Jlg) as a function of hourly clearness index k ( = IJ lo) , showing the

Orgill and Hollands (1977), Erbs et al. (1982), Reindl et al. (1990), and the proposed piecewise correlation.

,-P

I '0

0"8

0 '6

0"4

0.2

0

A - Atl Monlhs W-- Wet Months

D - Dry Months

o . o o .~ 0.4 o-~ 0-8 ,-o

ig/io Fig. 3. The diffuse fraction kd (= IdlI s) as a function of hourly clearness index k (= Igllo), showing the

piecewise and single curve correlations taking wet and dry months separately and the whole year.

Hourly diffuse fraction correlation 509

Table 2. Comparison of standard deviation and relative standard deviation for Madras using

different correlations

Correlation based on SD RSD (%)

/lg values vary from a min imum of 0.283 in March to a maximum of 0.550 in July. Moreover, during the rainy months ( June -November ) , the m i n i m u m / ~ d /

/lg is 0.432.

Orgill and Hollands (1977) 0.142 31.5 Erbs et al. (1982) 0.140 32.8 Reindl et al. (1990) 0.137 32.2 Present Study

Three piecewise Yearly 0.131 29.2 Dry 0.123 28.4 Wet 0.130 27.0

Single Yearly 0.131 29.4 Dry 0.124 28.6 Wet 0.130 27.4

piecewise correlation and the single curve correlation shows that the piecewise correlation is slightly (0.2%) better than the single curve correlation. Similarly, for dry and wet months the piecewise correlation is slightly better than the single curve correlation. Moreover, the SD and RSD values of the seasonwise correlation are slightly smaller than those obtained for the whole year correlation.

A comparison of the earlier correlations with the present study indicates that the diffuse fraction kd is larger at higher clearness indices (k > 0.75 ) at a tropical location as shown below: Orgill and Hollands (1977)

for k > 0.75, kd = 0.177

E r b s ~ a L (1982)

for k > 0 . 8 0 , k ~ = 0 . 1 6 5

Reindl et al. (1990)

4. CONCLUSION

This study was initiated to ascertain the validity of hourly diffuse fraction correlations in tropical locations. Correlations have been proposed based on long term data at Madras which clearly indicate that the diffuse fraction is higher in the tropics than in temperate re- gions, with a substantially significant diffuse fraction during the rainy season when the hourly clearness index is high. These correlations are based on yearly basis and also on a seasonal (dry and wet month) basis. Single curve and piecewise correlations have been pro- posed. Also, it is seen that the SD and the RSD values of the single curve correlation were very similar to the piecewise correlation.

NOMENCLATURE

k k~

k,j.cal kd,meas

N SD

RSD

/)~ Monthly average daily diffuse radiation on a hori- zontal surface (MJ/m 2)

I1 k Monthly average daily global radiation on a horizontal surface (MJ/m 2)

ld Hourly diffuse radiation on a horizontal surface (M J/ m 2 )

I s Hourly global radiation on a horizontal surface ( M J / m 2 )

1o Hourly extraterrestrial radiation on a horizontal sur- face ( MJ/m 2) Hourly clearness index (lg/lo) Hourly diffuse fraction (ld/lg) Hourly diffuse fraction calculated from correlation Hourly diffuse fraction from measured data Number of data Standard deviation as given by eqn ( 5 ) Relative standard deviation as given by eqn ( 5 )

for k > 0 . 7 5 , k d = 0 . 1 4 7

Whereas, for Madras, the best fit shows consistently higher values ofkd ( = 0.197) for higher hourly clearness indices. Even in Hawlader's (1984) model, for higher ks, the diffuse fraction is larger (= 0.215 ).

Therefore, it can be concluded that even though the trend of the hourly diffuse fraction at a tropical location is similar to that of a temperate location, cor- relations obtained from data of temperate locations should not be used for tropical locations. Further stud- ies with more data from tropical locations will be needed to propose a generalised correlation for the tropics.

Moreover, ke is much larger (0.207) during the rainy season due to the dense cloud cover during this period. Such higher values of hourly diffuse fraction are not found at temperate locations and, therefore, the cor- relations based on such locations will err when pre- dicting the hourly diffuse fractions for tropical location. This is also substantiated by the average daily diffuse fraction computed by Veeran and Kumar ( 1993 ): /~d/

Acknowledgments--The authors are grateful to the lndian Meteorological Department, Pune for having provided the data for analysis. One of the authors (SK) is thankful to the International Centre for Theoretical Physics Programme for Training and Research in Italian Laboratories, Trieste, Italy for its help in completing this work.

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510 J. CHANDRASEKARAN and S. KUMAR

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