Energy Convers. Mgmt Vol. 30, No. 2, pp. 155-161, 1990 0196-8904/90 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1990 Pergamon Press pie

S T A T I S T I C A L C O M P A R I S O N O F G L O B A L A N D D I F F U S E

S O L A R R A D I A T I O N C O R R E L A T I O N S

G. M. SINGH and S. S. BHATTIt Department of Physics, Guru Nanak Dev University, Amritsar-143 005, India

(Received 5 December 1987; received for publication 28 November 1989)

Abstract--Correlations for estimating monthly average daily global and diffuse radiation from sunshine data and global radiation data are developed using measured values of these parameters at Amritsar. The strength of the correlations has been confirmed from the values of coefficients of correlation and standard errors of estimate obtained thereof. The comparative study of our experimental results with the results obtained from other correlations reveals that, for estimating global radiation, the correlation suggested by Black et aL [Q. JI R. met. Soc. 80, 231 (1954)] is most accurate, whereas for estimation of diffuse radiation, the relationship given by Page [Proc. U.N. Conf. on New Sources of Energy, Vol. 4, p. 378 (1961)] yields the most accurate results.

Global radiation Diffuse radiation Sunshine duration

N O M E N C L A T U R E

a, b, c --- Regression constants /7 = Monthly average daily global radiation incident on horizontal surface (kWh/m z)

/7¢ = Monthly average perfectly clear day radiation incident on horizontal surface (kWh/m 2) /70 --- Monthly average extraterrestrial daily radiation on horizontal surface (kWh/m 2)

n = Number of instrument recorded bright sunshine hours per day N --- Maximum possible duration of sunshine per day I x = Solar constant, 1.353 kW/m z d --- Day of year, e.g. d = 73 for 14 March t$ = Declination (deg) ~b = Latitude (deg) w~ = Sunrise or sunset hour angle (deg) /~ = Monthly average daily diffuse solar radiation on horizontal surface (kWh/m 2)

• ( r =/7//70, monthly average clearness index (dimensionless)

I N T R O D U C T I O N

A knowledge of the solar radiation received at the earth's surface is of fundamental importance in understanding the physical processes that take place in the earth-atmosphere system and in the successful development of projects for the practical utilization of solar energy, particularly in the arid and semi-arid regions of the world, where sunshine is available in abundance and where there is a chronic shortage of conventional sources of power.

The measurement of duration of sunshine is made at most of the research centres, as well as aerodromes, whereas global, diffuse and direct solar radiations are recorded only at some selected centres in India [3] due to the requirement of expensive equipment and careful operation and maintenance. Empirical formulae are, therefore, useful to estimate the solar radiation received in areas where no such measurements are being made. Various climatological parameters, such as humidity, temperature, rainfall, total amount of coverage and, in particular, number of sunshine hours, etc., have been used in developing empirical relations as a substitute for the measurement of solar radiation. The general problem of predicting the amount of solar radiation reaching the earth's surface over a given period of time is a very complicated matter. No general formula is yet available to calculate the solar radiation received at any location and at any time accurately. The design of flat plate collectors depends upon the amount of energy actually available at any locality and the best orientation for optimum efficiency. The average amount of radiation received can be calculated for inclined and vertical surfaces for predicting the performance of focussing solar collectors if the diffuse radiation component can be assessed with reasonable accuracy.

tTo whom all correspondence should be addressed.

155

156 SINGH and BHATTI: GLOBAL AND DIFFUSE SOLAR RADIATION

We have divided our study into two parts. The first part deals with correlations, estimating daily global solar radiation from sunshine duration, whereas correlations concerning diffuse radiation are discussed in the second part.

METHODOLOGY

In the present investigations, solar radiation data has been recorded at Amritsar (31.63°N, 74.87°E, elevation 234 M above sea level) during 1986-1987. This includes the global and diffuse components of solar radiation received on the horizontal surface of the earth in addition to the duration of bright hours of sunshine. The instrumentation used for this study includes the Campbell-Stokes sunshine recorder for observing duration of bright sunshine hours daily, a thermoelectric pyrano- meter, along with a radiation integrator for daily and hourly values of global solar radiation, and a pyranometer with a shading ring for the diffuse component. However, a correction factor is being applied for the diffuse component cut-off due to the shading ring. All these instruments are calibrated from the National Radiation Centre of the Indian Meteorological Department at Pune. According to the criterion [4] laid down by the Commission for Instruments and Methods of Observation of the World Meteorological Organization for rating, the pyranometer used is rated First Class.

THE GLOBAL AND DIFFUSE CORRELATIONS

The essential equations involved in this study are:

= 23.45 sin[360(284 + d)/365] (1)

N = 2ws/15 (2)

cos ws = - tan ~b tan 6 (3)

H0 = 24/~[1 + 0.033 cos(360d/365)](ws sin ~b sin 6 + cos q) cos 6 sin w~)3600. (4) ff

Global radiation correlations

The first relationship generally used was that of Angstrom which expresses monthly average global radiation as a function of the number of bright sunshine hours as:

/7/ H, = a + bOi / TV). (5)

Since calculation o f /7 c is complicated and unreliable, Page [2] and others modified equation (5) to base it on the extraterrestrial irradiation/70 which can be computed from the solar constant by equation (4).

Thus,

H/Ho = a + b(~/N) . (6)

Many researchers presented the values of a and b which vary with geographic location and climatic conditions.

Black et al. [1] obtained the following overall regression equation for the stations they have studied.

H/Ho = (0.23 + 0.48 t///vv). (7)

Glover and McCulloch [5], while attempting to improve upon the insolation-sunshine correlation, have included the latitude effects and presented the following formulation (for ~b < 60 °)

/-7//7 o = 0.29 cos ~b + 0.52 r//~. (8)

Rietveld [6] concluded from his studies that a is related linearly and b hyperbolically to the appropriate mean values of r//N" such that

a = 0.10 + 0.24(r//F0 (9)

b = 0.38 + 0.080V/,~). (10)

SINGH and BHATTI: GLOBAL AND DIFFUSE SOLAR RADIATION 157

Substituting equations (9) and (10) into (6), the following relation is obtained

/7//70 = 0.18 + 0.62(ti/N'). ( l 1)

This formulation was believed to be applicable everywhere in the world. A significant and satisfactory improvement in the regression equations was made when Hay [7]

analysed solar radiation data for Canada and found that the surface albedo plays an important role in determining the magnitude of H, besides the effect caused by multiple reflections between the ground and the cloud base and the clear atmosphere. He also took into account another factor already indicated by the early investigators. According to this, the astronomical duration of maximum sunshine embodied by N in earlier equations does not represent the actual conditions when a Campbell-Stokes sunshine recorder is used. He assumed that the sensitized card gets burnt only when the sun is at an elevation of 5 ° (or more) above the horizon. When this factor is taken into account, the modified duration of day length N' is determined by

cos 7.5 N' = (cos 85 ° - s in q~ sin 6)/(cos ~b cos 6). (12)

Hay correlated n IN" with H'/Ho, where H ' is the global solar radiation that first strikes the ground before undergoing multiple reflections with the clear atmosphere and the cloud base. As a first approximation, justified by the limits to the accuracy of the method, Hay assumed the albedo of the cloud-free atmosphere as 0.25 and that of the cloud base as 0.60. With these values and taking only the first order multiple reflections, the numerical relationship between H and H' is given by

H - H ' = H.R[O.Z5(n/N') + 0,60(1 - n/N')l (13)

where R is the surface albedo. H ' is given by the regression equation

H'/Ho = (a + b n/U'). (14)

For Amritsar R = 0.20, a = 0.30 and b = 0.40 as given by Mani [8]. In the present measurements for Amritsar, linear and quadratic regression fits are obtained which

give/7.

Linear correlation:

Quadratic correlation:

/7//70 = [0.1986 + 0.5355 (n/)q)] (15)

H/Ho = 0.3440 + 0.0918ti/.~ + 0.3343 0I/N') 2. (16)

These values o f /7 are compared with the values calculated on the basis of equations (7)-(14). The results are given in Table 1. The coefficient of correlation and standard error of estimate in the case of our linear correlation were found to be 0.9135 and 0.0171, respectively, whereas these values in the case of our quadratic correlation were 0.9145 and 0.0155 correspondingly.

Table 1. Monthly average daily values of global solar radiation on horizontal surface

Correlation Black Glover and Rietveld Hay Present Present type et al. McCulloch (linear) (quadratic) Month Equation (7) Equation (8) Equation (11) Equation (13) Equation (15) Equation (16) Observed

January 3.044 3.286 3.279 3.457 3.073 3.063 2.979 February 4.033 4.355 4.406 4.480 4.102 4.104 4.170 March 4.813 5.196 5.221 5.282 4.877 4.865 5.040 April 5.786 6.246 6.299 6.375 5.873 5.867 6.188 May 6.350 6.855 6.913 7.077 6.446 6.438 6.397 June 6.352 6.857 6.876 7.163 6.429 6.409 6.290 July 5.450 5.880 6.735 6.380 5.434 5.470 5.609 August 5.432 5.862 5.806 6.116 5.461 5.450 5.197 September 5.139 5.548 5.587 5.647 5.213 5.204 5.307 October 4.384 4.734 4.806 4.782 4.467 4.480 4.406 November 3.605 3.893 3.957 4.007 3.675 3.690 3.609 December 3.050 3.293 3.335 3.273 3.103 3.108 3.062

158 SINGH and BHATTI: GLOBAL AND DIFFUSE SOLAR RADIATION

Diffuse radiation correlations

The correlations for estimating the diffuse radiation on a horizontal surface contain a relationship between the monthly average diffuse fraction (/3/F/) as a function of the monthly average clearness index ~(T, defined as K'T =/7//70. The only input data for these correlations is monthly average horizontal daily global radiation. Some of these correlations are listed below.

Page [2], using data from 10 widely spread sites between 40°N and 40°S lat, recommended the following linear equation:

/)//7 = 1 .00- 1.13 -KT. (17)

Liu and Jordan [9] showed a graphical relationship between/)/ /7 and /(T which can be expressed as

D/H = 1.39 -- 4.027 ~(T + 5.531 (/('7-) 2 -- 3.108(KT) 3 (18)

where 0.3 < K'T < 0.7. They employed global radiation data from several stations with a substantial range of/(T values

(0.3-0.7). The diffuse radiation data, however, were obtained from only one station, where/(T ranged from 0.4 to 0.5. Furthermore, the diffuse radiation data were not corrected for shadow band effects.

Tuller [10] presented the following correlation after studying Canadian data of four stations

D/H = 0.84 - 0.62 KT" (19)

Iqbal [11] established a slightly different correlation using Canadian data which were corrected for shadow band effects.

D//7 = 0.958 - 0.982 K" T (20)

where 0.3 < gx < 0.6. Modi and Sukhatme[12], after analysing data available from the Indian Meteorological

Department for 12 locations in India, gave the following linear correlation

/)//7 = 1.4112- 1.6956 KrT (21)

where 0.34 ~< J(v ~< 0.73 Collares-Pereira and Rabl [13] used pyranometer and pyrheliometer data for U.S. stations

available from the Aerospace Corp. for a period of approx. 2 yr at each station and obtained the following correlation using a least-squares fit.

~ 0.99 for K'T ~< 0.17

/')//7 = .~ 1.188 -- 2.272 (~'T) + 9.473 (KT) 2 -- 21.856(~'~) 3 + 14.648(~'T) 4 ! [ . f o r 0.17 < gv < 0.8. (22)

From the data obtained at Amritsar, we have obtained linear and quadratic correlations using a least-squares fit technique.

Linear:

Quadratic:

/)//7 = 1.1486- 1.3311 KT (23)

/)//7 = 1.5855- 2.9154(J(x)+ 1.4263(KT) 2. (24)

The coefficient of correlation and standard error of estimate in the case of our linear correlation were found to be -0.8980 and 0.0289, respectively, whereas these values in the case of our quadratic correlation were 0.8984 and 0.0263 correspondingly. The results of various correlations are given in Table 2.

Methods of comparison In this study, two statistical tests, root mean square error (RMSE) and mean bias error (MBE),

are used to evaluate the accuracy of the correlations described above.

Tab

le 2

. M

onth

ly a

vera

ge d

aily

val

ues

of d

iffu

se s

olar

rad

iati

on o

n ho

rizo

ntal

sur

face

Cor

rela

tion

Pa

ge

Liu

and

T

ulle

r Iq

bal

Mod

i an

d C

olla

res-

Per

eira

P

rese

nt

Pre

sent

ty

pe

Jord

an

Suk

hatm

e an

d R

abl

(lin

ear)

(q

uadr

atic

) M

onth

E

quat

ion

(17)

E

quat

ion

(18)

E

quat

ion

(I 9

) E

quat

ion

(20)

E

quat

ion

(2 l)

E

quat

ion

(22)

E

quat

ion

(23)

E

quat

ion

(24)

O

bser

ved

Janu

ary

1.18

5 !.

025

1.51

6 1.

294

1.51

3 1.

613

1.30

9 !.

302

1.17

3 F

ebru

ary

1.32

4 1.

210

1.93

2 1.

519

1.61

7 1.

693

1.43

8 1.

438

1.12

2 M

arch

1.

629

1.49

3 2.

396

1.87

4 1.

983

2.07

5 1.

767

1.76

9 1.

546

Apr

il

1.87

0 1.

735

2.82

8 2.

175

2.25

3 2.

356

2.02

1 2.

036

1.87

9 M

ay

2.19

6 1.

969

3.07

2 2.

478

2.72

2 2.

864

2.39

8 2.

385

1.82

4 Ju

ne

2.28

1 2.

035

3.1 5

4 2.

563

2.83

9 2.

992

2.49

5 2.

479

2.16

4 Ju

ly

2.42

2 2.

067

2.96

7 2.

605

3.13

2 3.

355

2.68

8 2.

692

2.59

3 A

ugus

t 2.

246

1.91

1 2.

720

2.40

6 2.

913

3.12

4 2.

495

2.50

6 2.

156

Sep

tem

ber

1.73

3 1.

585

2.53

6 1.

991

2.11

5 2.

214

1.88

2 1.

883

1.77

4 O

ctob

er

1.39

3 1.

277

2.04

8 1.

602

1.69

7 1.

775

1.51

1 1.

513

1.41

7 N

ovem

ber

1.14

4 1.

048

1.67

9 1.

315

1.39

4 1.

459

1.24

1 1.

242

1.19

7 D

ecem

ber

1.00

8 0.

914

1.44

5 1.

148

1.23

9 1.

300

1.09

8 1.

095

1.17

6

Z

> O

Tab

le 3

. R

oot

mea

n sq

uare

and

mea

n bi

as e

rror

s of

six

cor

rela

tion

s fo

r es

tim

atin

g m

onth

ly a

vera

ge d

aily

glo

bal

radi

atio

n in

cide

nt o

n a

hori

zont

al s

urfa

ce

Cor

rela

tion

B

lack

et

al.

Glo

ver

and

Rie

tvel

d H

ay

Pre

sent

P

rese

nt

type

M

cCul

loch

(l

inea

r)

(qua

drat

ic)

Equ

atio

n (7

) E

quat

ion

(8)

Equ

atio

n (1

1)

Equ

atio

n (1

3)

Equ

atio

n (1

5)

Equ

atio

n (1

6)

RM

SE

0.

1712

0.

3539

0.

3676

0.

5433

0.

1527

0.

1482

M

BE

-0

.068

0.

3126

0.

3305

0.

4821

-0

.008

4 -0

.013

9

Tab

le 4

. R

oot

mea

n sq

uare

and

mea

n bi

as e

rror

s of

eig

ht c

orre

lati

ons

for

esti

mat

ing

mon

thly

ave

rage

dai

ly d

iffu

se r

adia

tion

inc

iden

t on

a h

oriz

onta

l su

rfac

e

Cor

rela

tion

Pa

ge

Liu

and

T

ulle

r lq

bal

Mod

i an

d C

olla

res-

Per

eira

P

rese

nt

Pre

sent

ty

pe

Jord

an

Suk

hatm

e an

d R

abl

(lin

ear)

(q

uadr

atic

) E

quat

ion

(17)

E

quat

ion

(18)

E

quat

ion

(19)

E

quat

ion

(20)

E

quat

ion

(21)

E

quat

ion

(22)

E

quat

ion

(23)

E

quat

ion

(24)

RM

SE

0.

1502

0.

2186

0.

7459

0.

3045

0.

5045

0.

6270

0.

2548

0.

2528

M

BE

0.

034

0.14

6 0.

6893

0.

2457

0.

4497

0.

5666

0.

1935

0.

1932

t"*

Z

,-tl

,-11

t~

Oo O 7. ,..]

~D

160 SINGH and BHATTI: GLOBAL AND DIFFUSE SOLAR RADIATION

R o o t m e a n square error

The root mean square error is defined as

where/7,., c is the ith calculated value, Ri, M is the ith measured value, and n is the total number of observations. This test provides information on the short-term performance of the correlations by allowing a term-by-term comparison of the actual deviation between the calculated value and the measured value. However, a few large errors in the sum can produce a significant increase in RMSE.

M e a n bias error

The mean bias error is defined as

MBE = [~.(R~. c - /7 i , M)l /n. (26)

This test provides information on the long-term performance and a low MBE is ideal. A positive value gives the average amount of overestimation and vice versa. One drawback of this test is that overestimation of an individual observation will cancel under estimation in a separate observation.

C O N C L U S I O N S

The performance of six global radiation correlations were evaluated using the RMSE and MBE tests. The results are presented in Table 3. Among the previous correlations available with us, Black et al . ' s [1] equation yields the best results in terms of RMSE and MBE. Only sunshine data are needed as input. Its use is recommended for estimation of global radiation on a horizontal surface. Glover and McCulloch's correlation [5] results in a better estimation as compared to Rietveld and Hay's correlations [7]. In this study, we observe that Hay's equation gives overestimated values. The correlations developed during the present study give excellent results with minimum RMSE and MBE values. From the MBE values of both the linear and quadratic correlations, it is suggested the linear equation is used for further estimation of global radiation.

The performance of the eight diffuse radiation correlations were evaluated following the same procedure. The results are shown in Table 4. It is observed that Page's correlation [2] gives the best results. The low MBE value is particularly remarkable. Its performance coincides with its stated applicability at the locations wi th in_ 40 ° lat. Moreover, it is a very simple equation. Page's correlation is recommended for estimating monthly average daily diffuse radiation on a horizontal surface.

Liu and Jordan's correlation [9] comes next in the order of performance. It has the least RMSE and MBE values. The correlations of Iqbal, Modi and Sukhatme, Collares-Pereira and Rabl and Tuller give results in decreasing order of accuracy. Collares-Pereira and Rabl and Tuller's correlations give maximum over-estimated values. Positive values of MBE in all these cases point out general overestimation (except in the case of Liu and Jordan).

Acknowledgements--Financial support of the Tata Energy Research Institute, New Delhi is gratefully acknowledged. The authors would like to thank Dilbag Singh and B. K. Gupta of the Computer Centre, G.N.D.U. for their help in preparing programmes at various stages.

R E F E R E N C E S

1. J. N. Black, C. W. Bonython and J. A. Prescott, Q. Jl R. met. Soc. 80, 231 (1954). 2. J. K. Page, Proc. U.N. Conf. on New Sources o f Energy, Vol. 4, p. 378 (1961). 3. A. Mani (Ed.), Handbook o f Solar Radiation Data for India, 1980. Allied Publishers, New Delhi (1981). 4. K. L. Coulson, Solar and Terrestrial Radiation Methods and Measurements. Academic Press, New York (1975). 5. J. Glover and J. S. G. McCulloch, Q. Jl R. met. Soc. 84, 172 (1958). 6. M. R. Reitveld, Agric. Met. 18, 243 0978). 7. J. E. Hay, Sol. Energy 23, 301 (1979).

SINGH and BHATFI: GLOBAL AND DIFFUSE SOLAR RADIATION 161

8. A. Mani and S. Rangarajan, Solar Radiation over India. Allied Publishers, New Delhi (1982). 9. B. Y. H. Liu and R. C. Jordan, Sol. Energy 4, 1 (1960).

10. S. E. Tuller, Sol. Energy 18, 259 (1976). 11. M. Iqbal, Sol. Energy 22, 81 (1979). 12. V. Modi and S. P. Sukhatm¢, Sol. Energy 22, 407 (1979). 13. M. Collares-Pereira and A. Rabl, Sol. Energy 22, 155 (1979).