Techniques for the precise estimation of hourly values of global, diffuse and direct solar radiation

Solar Energy Vol. 31, No. 6. pp. 577-595,1983 0038-092X/83 $3.00+.00 Printed in Great Britain. Pergamon Press Ltd.

TECHNIQUES FOR THE PRECISE ESTIMATION OF HOURLY VALUES OF GLOBAL, DIFFUSE

AND DIRECT SOLAR RADIATION

A. MANI and S. RANGARAJAN Indian Institute of Tropical Meteorology, Field Research Unit, 239A Rajamahal Villas Extension, Bangalore-560 080,

India

(Received 2 July 1982; accepted 24 November 1982)

Abstract--The method usually used to compute solar radiation, when no measured data are available, is the well-known regression technique relating mean daily totals of global and diffuse solar radiation with the mean duration of sunshine. Using this method and taking into account the first order multiple reflections between the ground and the atmosphere, regression parameters were obtained from the monthly mean values of daily totals of global solar radiation and sunshine at a network of 16 stations in India. Daily values of global and diffuse solar radiation were then computed for 121 stations, where sunshine data are available for periods of 6-28 yr, using interpolated values of the regression parameters. Where no sunshine data were available, global and diffuse solar radiation were computed from cloud observations, using the inverse relationship between sunshine and cloudiness. Further, using the empirical relationship between daily totals and the corresponding hourly values of global and diffuse solar radiation, two sets of curves were prepared valid for the whole country, using which mean hourly values of global and diffuse radiation could be deduced from the corresponding daily totals, with a high degree of accuracy. The paper discusses the validity of the techniques used for computing daily and hourly values of global and diffuse solar radiation from sunshine and cloud amounts at an extended network of 145 stations in India and stresses the fact that such techniques are successful, only if accurate data on both radiation and sunshine are available at a widely distributed network of stations for a minimum period from at least 5 to 6 yr, using carefully calibrated and well-maintained instruments of the required quality. Theoretical models have also been used to compute clear sky noon values of global, diffuse and direct solar radiation from the solar constant, allowing for attenuation by atmospheric constituents such as ozone, water vapour, dust and aerosols. Using a simple model, calculations of global and diffuse solar radiation on clear days were made for 145 stations from values of the solar constant and measured values of ozone, water vapour and atmospheric turbidity. A method of extending the technique to overcast skies and partly clouded skies is discussed. The values of the mean annual transmission factor for global solar radiation under cloud-free conditions using the two methods show excellent agreement and establishes the soundness of the regression technique on one hand and the reliability of the theoreti:al model used for computing clear sky radiation, on the other.

1. INTRODUCTION To derive the detailed solar radiation climatology of a region and to estimate its solar energy potential, exten- sive radiation measurements of high quality at a large number of stations covering all major climate zones of the region are essential. The radiation network density normally accepted as satisfactory is one that provides a station-to-station spatial correlation of data of at least 0.7 (0.9 is easily achieved in homogeneous regions) for corresponding monthly mean values of daily totals of global solar radiation[l]. Where no accurate radiation measurements are made with reliable instruments, data have to be computed, (i) from other meteorological parameters using regression techniques, (ii) from the solar constant, allowing for depletion by absorption and scattering by atmospheric gases, dust and aerosols and clouds, or (iii) from satellite measurements of the solar energy scattered and reflected into space by the earth- atmosphere system.

The first method is empirical, the cloud field being defined by either the total coud amount or the measured number of sunshine hours. The second is theoretical and requires some approximations of the absorbing and scattering properties of the atmosphere. Both these methods were used in the present study to "derive the detailed solar radiation climatology of the Indian subcontinent and are described below.

2. THE REGRESSION TECHNIQUE FOR COMPUTING GLOBAL SOLAR RADIATION

Since there is an obvious relationship between sunshine duration and the amount of solar radiation received at the earth's surface, sunshine duration has been used by many investigators in the past to estimate solar radiation through regression equations. Following the first study of this kind by Kimball [2], .~,ngstriSm [3] expressed the relationship between the mean daily duration of sunshine, n, and the mean daily total of global solar radiation, G, in the form

G = Go[a' + (1 - a')n/N] (1)

where N is the maximum possible duration of sunshine, Go the daily global radiation with a cloud-free atmosphere and a' the mean proportion of radiation received on a completely overcast day. In view of the uncertainties involved in the precise evaluation of Go, Prescott[4] replaced Go by the daily total extra-terrestrial solar radiation on a horizontal surface ETR. The new regression has the form

G/ETR = a + b(n/N) (2)

where a and b are the new regression parameters, established empirically for each location. ETR can be

577

578 A. MANI and

precisely evaluated for any place and for any day or month, from the solar constant and relevant astronomical variables. Under overcast conditions, n/N becomes zero and a thus becomes a measure of the global radiation received at the ground through an overcast sky as a fraction of the ETR. The coefficient b expresses the rate of increase of G/ETR with increase of n/N. For a clear sky, n/N becomes unity and the parameter (a + b) is the fraction of the ETR that reaches the earth's surface. This becomes a measure of the overall mean transmission factor for daily G under clear sky conditions. The magnitude of a naturally depends on the type and thickness of the prevailing clouds and that of b on the transmission characteristics of the cloud-free atmosphere, mainly determined by its total water vapour content and turbidity. Most of the investigations made so far have been based on monthly mean values of n and G. Typical values of a, published in literature, range from 0.14 to 0.54 and those of b, from 0.18 to 0.73. Lower values of a are invariably associated with higher values of b and vice versa. The variability of (a + b) is much less than that of either a or b and it assumes values in the range from 0.65, for a moist and turbid atmosphere, to 0.80, for a dry and dust-free atmosphere.

To compute G for any location with only sunshine data, it will therefore be necessary to have besides ETR

and N, (both of which can be precisely calculated), the most appropriate values of the parameters a and b. These can be assigned by careful interpolation or extrapolation of the values of a and b determined from measured values of n and G at selected stations. The accuracy, with which an estimate of the monthly mean value of daily O can be made, is governed by the coefficient of correlation and the statistically determined standard error of the estimate. Surprisingly, though literature abounds in the listings of a and b for several parts of the world, little published information is available on the magnitudes of the correlation coefficient and the standard error.

A significant and satisfactory improvement to the regression equation was made, when Hay[5] analysed solar radiation data for Canada and found that surface albedo plays an important role in determining the magnitude of G, besides the effects 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 earlier investigators, that the astronomical duration of maximum sunshine embodied by N in eqn (2) does not represent the actual conditions when a Campbell-Stokes sunshine recorder is used. He assumed that the sensitised card used in the Campbell- Stokes sunshine recorder 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 daylength, N' is determined by

arc cos (cos 85- sin ~b sin~ ) N' - cos ~b cos

7.5 (3)

instead of the conventional equation for N in the form

S. RANGARAJAN

N = arc cos (-tan ~ • tan ~) 7.5

(4)

Hay correlated n/N' with G'IETR, where G' is the global solar radiation that first strikes the ground before under- going 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 G and G' is given by

G - G' = G.R.[O.25(n/N') + 0.60 (1 - n/N')] (5)

where R is the surface albedo. The regression equation now has the form

G¢ ETR = a + b(n/N'.). (6)

This equation was used to derive G from n in the present study.

3. THE REGRESSION TECHNIQUES FOR COMPUTATION OF DIFFUSE SOLAR RADIATION

Investigations relating to estimation of daily diffuse radiation by statistical methods have been fewer, mainly because of the lack of measured data on diffuse radiation in most countries. The classical work of Liu and Jordon[6] established that, for average conditions, D can be determined with reasonable accuracy from G and ETR, through a linear regression equation connecting DIG and G/ETR. Applications of this technique with improvements have been carried out among others by Collares-Pereira and Rabl[7] in the U.S.A., Neuwirth[8] in Austria and Gupta et al. [9] in India. The general linear regression equation is of the form

DIG = c + d(G/ETR) (7)

where c and d are regression parameters. The constant d will always be of negative sign. Some investigators have attempted to fit in additional terms in eqn (7), involving higher powers of G/ETR to obtain higher accuracy. In the present study, however, only the linear equation has been used for two reasons. Firstly, the correlation coefficients obtained with the linear equation were found to be satisfactory. Secondly, with the values of D, G and ETR normally occurring over India, one has to operate only within the middle portions of the regression line, where linearity is found to exist. As in the case of deriving G from sunshine, better results were found when G was replaced by G' and D by D', where D' is given by

D' = D-G.R.[O.25(n/N') + 0.60(1 - n/N')]. (8)

The linear regression equation for deriving D' takes the form

D'/G' = c + d(G'IETR). (9)

Precise estimation of hourly values of global, diffuse and direct solar radiation

This is somewhat similar to the one used by Hay, but higher powers of G'/ETR have been omitted for reasons stated already.

4. COMPUTATION OF GLOBAL AND DIFFUSE SOLAR RADIATION

FROM SUNSHINE AND CLOUDINESS AT INDIAN STATIONS

The global solar radiation network in India consists of 16 stations, while there are 121 stations where sunshine duration is measured. In order therefore to obtain a detailed picture of the temporal and spatial distribution of G over India monthly mean values of daily totals of global and diffuse radiation were computed for the 121 sunshine stations. The method adopted was as follows.

4.1 Global solar radiation Daily totals of global radiation and duration of sun-

shine were available on magnetic tapes for 16 stations in India for periods ranging from 8 to 21 yr. The monthly means of G and the corresponding values of n were calculated for each month and for each station, for all years of data availability. Next using eqn (5), G' was calculated for each mean value of G assuming a mean value of 0.20 for the ground albedo at all the stations, measured values of ground albedo for several locations in the country being generally in the range of 0.15-0.25. The extraterrestrial solar radiation on a horizontal surface ETR, was calculated for each station and each month from the formula:

ETR = 10.39.k. (cos ¢ cos ~5 sin to5 + o~, sin ¢ sin 5) (10) (kWhm 2day ')

where 10.39 represents the solar constant (assumed equal

579

to 1.36kWm 2) multiplied by (24/7r), k the correction factor for varying earth-sun distance; ¢¢ the lat., ~ the mean declination for the month and ¢o5 the sunset hour angle in radians. N' was calculated from eqn (3). For each station, the regression parameters a and b of eqn (6) were calculated using all available pairs of n/N' and G'/ETR combining all months of the year. Since clouding over most parts of the country is confined to a well-defined period June to October, regression coefficients were first calculated separately for cloudy and clear seasons. However, because of the tendency of points with G'/ETR and n/N' as co-ordinates to lie in preferred regions of limited dimensions on the correlation charts, sufficiently reliable correlations could not be found by this method and the year has therefore been considered as a whole.

Other statistical parameters, r the correlation coefficient, r 2 the coefficient of determination and S.E. the standard error of the estimation were also computed for each of the stations. These are presented in Table 1.

It will be seen that the magnitude of the parameters a and b varies appreciably from station to station, but follows an organised spatial pattern over the country. Maps were, therefore, drawn separately for a and b showing their spatial distribution and isopleths drawn, taking into consideration the topography as well as the general climatology of the country.

Reliable data on the duration of sunshine are available for 121 stations in the country, for periods ranging from 6 to 28 yr. Mean monthly values of the actual duration of sunshine, n in eqn (5) and eqn (6) were used for the computation of monthly mean G for all the 121 stations. With the aid of the maps showing the spatial distribution

Table 1. Regression analysis between G'/ETR and n/N' 2

5 t a t i q n No. o f months e b r , r 8.E.

Ahmedebmd 202 0.252 0.448 0.960 0.921 0.030

Bhsvnagaz 141 0.259 0.433 0.985 0.971 0.020

Bombay 100 0.283 0.397 0.920 0.847 0.046

Calcutta 248 0.271 0.378 0.897 0.807 0.040

Goe 175 0.270 0.423 0.914 0.835 0.048

Jodhpuz 132 0.313 0.401 0.874 0.763 0.042

Kodeikenel 194 0,273 0.521 0.916 0.839 0.044

Medzas 255 0.242 0.436 0.913 0.834 0.032

Mangaloze 115 0.214 0.458 0.944 3.892 0.041

Nagpu~ 218 0.227 0.459 0.975 0.951 0.024

New Delhi 255 0.256 0.454 0.874 0.763 0.042

¢Pozt Blair 114 0.156 0.472 0.860 0.739 0.062

Pune 254 0.269 0.41? 0.944 0.891 0.037

Shillong 140 0.225 0.463 0.920 0.847 0.044

Trivmndzum 215 0.309 0.408 0.912 0.831 0.028

Vishekhapetnam 210 0.247 0.435 0.968 0.938 0.025

580 A. MANI and

of a and b, regression parameters a and b were assigned for the 121 stations. Monthly mean values of ETR and N' were then calculated for these stations using eqn (10) and (3), respectively and with the aid of eqn (5) and (6), G was computed for stations forming the extended network of sunshine stations and plotted on maps for each of the twelve months and for the year as a whole. Figure l(a and b) illustrate typical maps for the months of May and July which represent the sunniest and cloudiest months of the year for India. The monthly maps can be used to obtain monthly mean values of global radiation at any location in the country, with a reasonable degree of confidence.

4.2 Diffuse solar radiation Following the same procedure, the regression

parameters c and d in eqn (9) were calculated for 14 stations in the country for which diffuse radiation data were available. Table 2 lists these parameters with the coefficients of correlation, the coefficient of determination and the standard error. Monthly maps showing the spatial distribution of c and d over the country were prepared as for global radiation and appropriate values

S . R A N G A R A J A N

of c and d assigned to each of the 121 stations. Using the corresponding values of G' and ETR, the monthly means of daily diffuse radiation were calculated using eqns (8) and (9). Maps showing the detailed distribution of D over the country for the months May and July are illustrated in Figure 2(a, b).

4.3 Reliability of the regression technique Table 1 shows that the correlation coefficients between

n/N' and G'/ETR are quite high, within the range 0.91- 0.98 for I I stations and within 0.86--0.90 for the remaining five stations. For the former category, the standard error of the estimate lies in the range 0.02-0.04. With annual mean values of 5.5 kWh m -2 for G, 5.0 kWh m - 2

for G' and 9.6 kWh m -2 for ETR, the error in the evaluation of G for any individual month from the corresponding mean value of n/N' is within 6 per cent. However, the error in the evaluation of a climatogical mean G from the climatological mean n/N' will be less depending on the number of years considered for averaging. Table 3 gives the calculated values of G for the 16 stations together with the measured values. The agreement is seen to be within 2-5 per cent in most

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Table 2. Regression analysis between D'/G' and G'/ETR

S t a t i o n No. o f months c d r ¢2 S.E.

Ahmadsbsd 134 1.329 -1 ,701 -0 ,960 0.921 0.057

Bhavnsgor 141 1.406 - 1 . 8 0 9 -0 .979 0,959 0°044

Bombsy 44 1.306 -1 .769 -0 .953 0.909 0.063

CoZcutts 147 1.124 -1 .366 -0 .860 0.740 0,076

Goo 142 1.285 -1 .659 -0 .941 0.885 0.072

Jodhpu= 58 1.325 -1 .703 -0 .883 0.761 0.071

Madras 193 1.108 -1 .351 -0 .903 0,816 0.048

Nogpu~ 123 1,312 -1 .761 -0 .975 0o951 0.042

~ew De lh i 237 1.137 -1 .373 -0 .893 0.797 0,059

Post 81ai~ 26 1.354 -1 .801 -0 .935 0.875 0 .067

Pune 234 1.469 -1 .963 -0 .959 0.920 0.065

S h i l l o n g 116 1.211 -1 .651 -0 ,915 0.837 0,050

Tr ivsndrum 157 1.271 -1 .832 -0 .915 0 .837 0.050

Vishakhapatnsm 122 1.229 -1 .587 -0.939 0.881 0.056

582 A. MANI and S. RANGARAJAN

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cases, although differences exceeding 10 per cent are found in rare instances, these presumably arising from errors in the input sunshine data.

A question arises as to the minimum number of years of data from which a and b should be determined. This has a practical relevance because it is useful to know the minimum number of years for which a sunshine recorder and a pyranometer have to be operated together at a new location, for enabling one to compute a and b, which can be used for future computations of G using sunshine data alone or interpolate similar radiation data for nearby sunshine stations. It is natural to expect this minimum to depend on the stability of the mean monthly values from year to year, which in turn will depend on the extent of the interannual variations of sunshine and global solar radiation at any given place. Four stations in India New Delhi, Poona Calcutta and Madras have records of sunshine and global radiation for periods exceeding 20 yr. Regression analysis of monthly mean sunshine vs the cor-

583

responding monthly mean G, carried out for these four stations for periods successively varying from one to 6 yr, showed that a minimum of 5 yr data are needed to obtain fairly stable values of a and b. The analysis also showed that whereas variations in both a and b occur from year to year and from one group of years to another group, the values of (a + b) remain remarkably constant within very small limits.

A study of Table 2 shows that in the case of regression between D'/G' and G'/ETR, the correlations are not generally as good as those between n/N' and G'/ETR. In general, the accuracy in the determination of D through the regression technique is in the range 5-10 per cent. The results of the verification of the computed values with those measured at the 14 diffuse radiation stations are given in Table 4. Errors in the determination of climatological monthly mean D are within 5-10 per cent in every 3 out of 4 cases and within 10-15 per cent for the rest.

Table 3. Comparison of computed and observed values of mean daily global solar radiation

Stat ion JAN FEB MAR APR MkY JUN JUL AUG SEP OCT NOV DEC

£alc . 5.01 5.88 6,24 6.97 7.45 6.59 5.02 4.67 5.52 5t95 5,31 4,82

Ahmedabad Dba. 4.90 5.79 6,73 7.33 7,61 6.39 4.85 4,52 5.55 5.77 5,00 4,59

% D i f f . 2,3 1,6 7,3 4,9 2,1 3,1 3.5 3.2 0,5 3,1 6,2 5,0

Calc. 5.19 5.91 6,49 7.12 7,67 6,20 4.67 4,46 5,28 5.98 5.48 4.86

Bhavnaga¢ Obs. 5.14 5.96 6.87 7.27 7.60 6,Q3 4,62 4.31 5.46 5.91 5.22 4.78

% D i f f . 1.0 0.8 5.5 2.0 0,9 2.7 1.1 3.5 3.3 1.2 5.0 1,7

Caic. 5.12 5.83 6.31 6.81 8.91 5.49 4.40 4.30 5,05 5,66 5.33 4.95

Bombay Dbl. 5.03 5.75 6,45 5.99 7.26 5.18 4.06 3.98 4.88 5.44 5.07 4.79

% D i f f . 1.8 1.3 2,1 2.6 4.8 5.9 8,3 8,0 3,5 4.0 5.0 3.3

Calc. 4.36 5.09 5,58 6,21 6.21 4,87 4,74 4.60 4.59 4.69 4,54 4.17

Calcutta Obs, 4.22 5.03 5,79 6.32 6,53 4.97 4.64 4.47 4,47 4.56 4.37 4.10

% D i f f . 3.3 1.2 3.6 1.7 4.9 2.Q 2.2 2.9 2.7 2.9 3.9 1.7

Calc. 5.62 6.33 6.57 6.90 6.73 4.94 4.29 4.93 5.37 5.67 5.72 5.37

Goa Obs. 5.69 6.38 6.76 6.93 6.74 4.79 3.99 4.76 5.31 5,61 5.61 5,39

% D i f f . 1.2 O.B 2.8 0.3 0,1 3.1 5.0 3.6 1.1 1.1 2.0 0.4

Calc. 4.67 5.61 6.19 7,09 7.52 7.19 6.25 6.05 6.24 5.95 5.10 4.51

Jodhpur DbB. 4.71 5.57 6.55 7.23 7,55 7.07 5,98 5,54 6.10 5.83 4,90 4.43

Dif f . 0.9 0.7 5.5 1.9 0.4 1.7 4.5 9.2 2,3 2.1 4.1 1.8

EaJc. 6.03 6.78 6.96 6.64 6.02 5.04 4.49 4,95 5.13 4.88 4.86 5.19

Kodaikanai Dbs. 6.28 6.71 6.87 6.43 5.84 5.25 4,54 4.70 4.80 4,44 4.74 5.22

%Di f f . 4.0 1.0 1.3 3,3 3.1 4.5 1.1 5.3 6,9 9.9 2.5 0,6

Calc, 5.37 6.29 6.59 6,72 6,32 5.52 5.03 5.28 5.41 5.10 4.56 4.57

Madras Obs. 5.23 6.29 6.89 6.92 6.53 5.82 5.40 5.58 5.71 4.90 4.34 4.30

Dif f , 2.7 0.0 4.4 2.9 3.2 5.2 6.8 5.4 5.3 4.1 4.7 6.2

Unit : kWh.m'2.dey - I


Table 3(Contd)

$tet£on JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

Celt* 5.82 6,37 6,37 6,45 6.02 4.24 3,53 4,33 4,99 5.25

ManDeZoze Obe. 5,33 6.02 6,38 6.42 5,73 3,86 3.21 4.07 4.96 4.93

% D i f f . 9.2 5.8 0.2 0.5 5.1 9.5 10,2 6.3 0,6 5.7

Delc, 5.02 5.84 6 , t 8 6,69 6,89 5,51 4,33 4,30 4.91 5.58

Nagpu~ 8bs. 4.91 5,68 6.31 6,78 6.91 5.64 4,39 4,13 5,11 5.61

D i f f . 2.2 2,8 2,1 1.3 0,3 2.3 1,4 4.1 3,9 0,5

Emit. 4.06 5.07 5.60 6.50 6°61 6,22 5.61 5.53 5.58 5.48

New Delhi Obs, 3.99 5.00 6.14 6.93 7,29 6=54 5,33 5.05 5.50 5.35

% Diff. 1.7 1,4 8.5 6.2 9.3 4.9 5,3 9.5 0.4 2,4

Cel t . 4.94 5.75 5.90 5.79 4.23 3.47 3.48 3.57 3.69 4.34

Port B la i~ Obs. 4.87 5.65 6.07 5.95 4.05 3.34 3,59 3,49 3.91 4.20

% D i f f . 1.4 1.8 2.8 2.7 4.4 3.9 3.1 2.3 5.6 3.3

Calc, 5.32 6,10 6.53 6.91 6.99 5.74 4.57 4,68 5.19 5.60

Pun~ ObB, 5.30 6.17 6.81 7.16 7,30 5.68 4.54 4,58 5,31 5.67

Diff, 0.4 1,1 4,1 3.5 4.3 2,4 0,7 2.2 2,3 1.2

Caic. 3,64 4,75 5.34 5.69 5.56 4,26 4,18 4.14 4.18 4.26

5hi l long Obs. 4.01 5,08 5,63 5,70 5,23 4.11 4,26 4,21 3.87 4.02

Diff. 9.2 6.5 5.2 0.2 6.3 3,6 1,9 1.7 8.0 6.0

Calc. 6.08 6.55 6.69 6.24 5.78 5.00 5.06 5.64 5.85 5,52

Trivandzum Obs, 5.93 6.36 6.65 6.17 5.49 5.25 5.03 5.56 5.94 5.23

Diff . 2,5 3.0 0.0 1.1 5,3 4,8 0,6 I .4 1.5 5.5

Ceic. 5.34 6.11 6.24 6.62 6,63 5.04 4.59 5.12 5.18 5.42

Vishakha- Obe. 5.37 6.08 6,51 6.66 6.68 5.17 4.67 4.95 5.16 5.28 patnem

% Diff . 0.6 0.5 4.1 0,6 0.6 2.5 1.7 3,4 0,4 2.7

5.53 5.36

5.17 5,10

7.0 5.1

5.27

5,13

2.7

4.7t

4.52

4,2

4.39

4,01

9.7

4,81

4.67

3.0

3.95

3,84

2.9

4.64

4.31

7,7

5.35 5.03

5.24 4.93

2.1 2,0

3.78 3,96

3,94 4.04

4.1 2,1

5,33 5.50

4.95 5.13

7.7 7.2

5.22 4.99

5,16 5.07

1.2 1,2

Unit = kNh.m-2,day -1

4.4 Determination of mean hourly values of global and diffuse solar radiation

For solar energy applications, it is also necessary to obtain computed values of both G and D on an hourly basis for the different months of the year. Deriving these from the corresponding daily totals is rendered somewhat easy, because of the existence of certain empirical and semi-empirical relationships between the hourly and the daily values of radiation for the different hours of the day. Following the earliest work of Whillier[10], Liu and Jordan[6] developed a theoretical method for deriving the mean hourly diffuse radiation from the mean daily total. Assuming the atmospheric transmission to be in- dependant of solar altitude, a theoretical expression was deduced by them in the form

~r (cos ~o - cos,os) r, = 24' (sin to~ - ~os cos ~o,) (1 1)

where r~ is the ratio of hourly to daily D, w the hour angle centered at the hour and w~ the sunset hour angle, both in radians. Liu and Jordan[6] found that the experimentally determined ratios were in excellent agreement with the theoretically deduced ones. Collares- Pereira and Rabl[7] later found that, while the theoretical equation is valid for D, a semi-empirical equation which is a function of the latitude will be required in the case of G, because of the increased dependence of the attenuation of direct solar radiation on the solar elevation. In view of the complexity of the problem, a purely empirical approch was made in this study.

From the climatological means of the hourly global radiation for the 16 stations based on several years of data, the ratios hourly/daily global radiation were plotted as a function of the mean sunset hour angle of each month for each of the hours 1[2, 1(1/2), 2(112), 3(1/2), 4(1/2) and 5(1/2) measured from solar noon. One smooth


Table 4. Comparison of computed and observed values of mean daily diffuse solar radiation

585

StetLon JAN FEB MAR APR NAY JUN JUL AUG SEP OCT NOV DEE

Ce1¢. 1,10 1.26 1.99 2.15 2.13 2,89 3,32 3,21 2,72 1,60 1.18 1,00

Ahmedabad Ob=. 1.17 ! ,43 1,74 2,18 2.33 3,16 3.48 3,24 2.53 1.43 1.12 1,05

D i f f , 6.6 11.7 14,6 1.6 8,5 8.8 4,6 0,8 7.4 11,3 4.9 4,8

Calc. 1.10 1.39 1,89 2,09 1,92 3,18 3.47 3.36 2,97 1.73 1.11 1,17

Bhevnegmz Ohm. 1.17 1.38 1.68 2.17 2.23 3,24 3,55 3.37 2.61 1.51 1.15 1.09

D i f f . 5.9 0.9 13.7 4.1 14,0 1.8 2.4 0.4 14.0 t4.3 3.6 7.0

£olc. 1,18 1,35 1,77 1.92 2.09 2.97 3,09 3.03 2,73 1.82 1.24 1.10

Bombay Oba, 1.28 1.47 1,78 2,19 2.51 2,97 3.07 3.17 2,86 1.98 1.19 1.11

D i f f . 7.8 8.1 0,6 12,3 16.8 0.0 0,7 4.1 4,5 8.0 4.2 0.5

C=lc. 1.56 1.81 2.23 2,44 2.69 2,97 2.94 2,84 2,64 2o21 1.67 1.49

CIlcuttn Obs. 1.50 1.74 2,16 2.66 3,09 3,20 3.17 2.92 2.60 2.03 1.54 1,33

D i f f . 4.1 4.0 3.3 8.1 13.3 7.4 7.5 2.7 1.3 8.5 8.1 11.6

Celc. 1.39 1,50 2.01 2.18 2.43 3.12 3,12 3,08 2.84 2.28 1.54 1.41

Goa Obs. 1.15 1,34 1.88 2.32 2.69 3.04 3.03 3.34 2.94 2.11 1.47 1.12

Diff. 20.7 12.t 6.8 6.3 9.7 2.6 2.7 7.6 3.2 7.9 4.6 26.1

Celt . 0.99 1.15 1.80 1.94 2.10 2.57 3.07 2.89 2.14 1.25 0.86 0.83

Jodhpur Obs, 1.14 1.36 1.77 2.34 2.68 3.06 3.39 3.21 1.88 1.22 0.91 0.93

D i f f . 13.5 15,9 1.6 16,9 21.6 16.0 9.4 10.1 14.1 2.5 4.9 10.5

Eelc. 1.87 1.83 2.14 2.32 2,54 2.76 2.82 2,80 2.69 2.50 2.23 2,05

Hadzmm Obs. 1.87 1.73 1.87 2.12 2.57 2.86 3,11 3.01 2.58 2.26 2.07 2.04

% D i f f . 0.~ 6,1 16.0 9.5 1.2 3.5 9.3 7.1 4.4 10.8 7.4 0.4

Ealc. 1,06 1.17 1.80 2,03 2.18 3.00 3.09 3.01 2,73 1.76 1.09 1.00

Nlgpuz Obs. 1.21 1.43 1.76 2.16 2.49 2.99 3.22 3.02 2.53 1.64 1.25 1.07

% D i l l , 13.0 17.9 2,5 5.7 12.7 0,3 4.0 0.4 7.8 7.6 12.7 6.3

Calc. 1,35 1,56 2.16 2.45 2.81 3.14 3.12 2.92 2,45 1.65 1.17 1,20

New Delhi Obs. 1.24 1.47 1.86 2,47 2.92 3.54 3.13 2.73 2.15 1,47 1.14 1.12

D i f f . 9.2 5.8 16.2 1.1 3.7 11.2 0.5 7.1 13.9 12.5 2.8 7.1

Calc, 1.23 1.32 1.83 2.11 2.29 3.21 3.43 3.37 3.02 2.18 1.51 1,28

Pune Obs, 1.11 1.19 1.59 2.07 2.24 3.32 3.47 3.49 2.93 1.73 1.22 1.14

% D i f f . 10.2 10.6 15.1 2.0 2.4 3.3 1.1 3,6 3.2 25.5 23.1 12.5

Calc. 2.09 2.05 2.40 2.85 3.03 2.88 2.87 2.91 2.90 2.78 2.38 2.12

Pozt 81eiz Ob=. 2.00 1.90 2.10 2,00 2.80 2.80 2.96 2.90 3,00 2,60 2.30 1.80

% D i f f , 4.5 7.7 14.4 33.7 8.0 2,3 3.1 0.2 3.3 6.9 3.3 18.0

Cel t , 1.52 1.57 1,99 2.40 2.74 2,91 2.87 2.77 2.53 2.07 1.65 1.13

@hillong Obe. 1.21 1.37 1,87 2.36 2.83 3.14 3.20 2.91 2.49 1.75 1.30 0.93

% D i f f . 25.8 14.2 6.3 1.6 3,1 7.4 10.1 4.9 1.6 18.3 26,4 21.5

Cmlc. 1.79 1.91 2.22 2.64 2.77 2.92 2.93 2.84 2.75 2.69 2.43 2.08

Tx£vendcuR Obe. 1.77 1,91 2.20 2.62 2.83 2.97 3.13 3.13 2.77 2.56 2.26 2.03

% g£ f f . 0.9 0,0 1.2 0.6 2.1 1.8 6.6 9.4 0.5 5.0 7.2 2.4

-2 -I Unit : kWh.m .day


Table 4(Contd)

S t l t l o n JAN FEB MAR APR MAY JUN JUL AUG 5EP OCT NOV DEC

Cslc. 1.30 1.41 2.05 2.25 2.40 3.00 3.00 2.92 2.71 2.18 1.64 1.39

V~=hlkhl~,- Obmo 1.42 1.49 1.86 2 . 1 7 2 .64 2 , 9 7 3 .10 3 .05 2 .55 1 .83 1.55 1 .40 patnam

Di l l , B.O 5.6 9.9 3.6 8.9 1.3 3.1 3.9 6.4 19.1 5.5 0.5

Unit : kWh.m-2.day "I

curve was obtained for each hour and thus a set of curves were obtained for each station. After a careful examination of the curves of all stations, it was found that a single set of mean curves was sufficient to relate the ratios hourly/daily G for all stations in India, with a high degree of accuracy. This simplification comes about because of the fact that the transmission factor of global radiation does not vary appreciably with solar zenith angles upto about 50 °, and also because of the fact that the bulk of the contribution to the daily total of G comes when the solar zenith angle is less than about 50-600 . Figure 3 shows the single set of curves obtained for evaluating hourly mean G from the daily total of G for any station in India.

Similarly another set of curves was drawn for the diffuse radiation which is illustrated in Fig. 4. A random verification with actually measured data showed that the accuracy of determining hourly G is within 2-3 per cent and hourly D within 4--8 per cent.

It follows from the method developed in the preceding sections that given the monthly means of sunshine duration and global and diffuse solar radiation for a minimum period of 5 yr for any station, it is possible not only to derive the daily totals of G and D but also the mean

hourly G and D to an accuracy that would meet the requirements of most applications of solar energy.

5. TECHNIQUE OF COMPUTATION OF GLOBAL, DIFFUSE AND DIRECT RADIATION FROM THE SOLAR CONSTANT

5.1 Choice of u model for computing G and D from the solar constant

A large number of theoretical models and formulae have been proposed by many investigators for calculat- ing global and diffuse solar radiation from the solar constant, allowing for its depletion by absorption and scattering by various atmospheric constituents. While some of the models are too elaborate for large-scale computational work, others do not yield results of the desired accuracy. A recent survey of the literature in this field was made by Bener[ll]. Breslau and Dave's[12] model is the most complete but the simpler model evolved by Hoyt[13] was adopted in this study as the most suited for Indian conditions. It had also given results agreeing to within 1-2 per cent with the Breslau and Dave's model[14]. This model enables calculations of clear sky noon values of IH, D and G to be made for any station, provided appropriate values of the concen- tration of ozone, water vapour and particulate matter in

.I 8 0

. 1 6 C

. 1 4 0 - -

. 1 2 0

;+ >, ~ .~ oo

.080

.060

.o4c

.o2c

o K

/

/

/ I

--~-hr I

I T

2~

3± 2

- I 4~-

80 85 90 95 I00 105

SUNSET HOUR ANGLE IN DEGREES FROM LOCAL NOON

Fig. 3. Plot of ratio hourly/daily G vs sunset hour angle for different hours (L.A.T.) from noon.


.I 8 0

. 1 6 0

.140

. 1 2 0

O 0

> .080

.06C

.04C -

.O2C

0 - 75

I

f

--~hr

j -

24

5 ± 2

i

5 4 _

8 0 85 9 0 95 I00

SUNSET HOUR ANGLE IN DEGREES FROM LOCAL NOON

Fig. 4. Plot of ratio hourly/daily D vs sunset hour angle for different hours (L.A.T.) from noon.

587

105

the atmosphere are available. The equations that are relevant in this model are:

In = I,,' cos z (1 - i = , ~' (1 - so)(1 - sa) (12)

D = I,, " COS Z( I -- . ~,~ a, )(O.5 s. + O.75 Sd) (13)

s,, = I - f(m*) "° (14)

s . = 1 - g(/3)m. ( 1 5 )

where z is the solar zenith angle, Io is the solar constant corrected for the varying sun-earth distance,

i =1

is the summation of the absorption caused by water vapour, carbon dioxide, ozone, oxygen and dust in the atmosphere, s, the scattering coefficient for Rayleigh scattering and sd the scattering coefficient for dust. Ea~ includes the separate contributions a~, a2, a3, a4 and a~ arising from absorption by water vapour, carbon dioxide, ozone, oxygen and dust, respectively. The empirical formulae, relating the absorptions a~ to a~ with the quantity of the absorbing constituent in the atmosphere, are after Hoyt[13]. For water vapour:

a, = 0.110 (u, +6.31 x 10-4)°3- 0.0121 (16)

where u, is the pressure-corrected precipitable water in the atmospheric path in g cm 2.

For carbon dioxide

a2 =0.00235 (u2 + 0.0129)°z6- 7.5 × 10 -4 (17)

where uz is the pressure-corrected path length of carbon dioxide in cm at S.T.P.: uz= 126cm for air mass = 1.

For ozone

~3 = 0.045(u3 + 8.34 × 10 4)°3"-3.1 × 10 3 (18)

where u~ is the ozone path length in cm at S.T.P.

For oxygen

a4 = 7.5 x 10-3(m*) °"7~ (19)

where m* is the pressure-corrected air mass.

For dust

~ = (1 - , o , ) g ( / 3 ) " * (20)

where ~o~ is the albedo of single scattering and is taken as 0.95. sa = 1 - f ( m * ) ' * , where f(m*) is a function of the pressure-corrected air mass m* as tabulated by Hoyt. sd = I - g ( / 3 ) " " where /3 is the Angstr6m turbidity coefficient or the aerosol optical depth at l t~m wavelength. The aerosol optical depth as a function of the wavelength is expressed by the relation

To = / 3 / ~ ~ (21)

where ,~ is the wavelength of light in /~m. The wavelength exponent a is assumed to be equal to one

588 A. MAM and S. RANGARAJAN

and with this assumption, the function g(fl) was calculated by Hoyt.

One half of the Rayleigh scattering and three quarters of the dust scattering are assumed to be in the forward direction and thus contributing to the diffuse radiation reaching the ground. The sum of In and D in eqns (12) and (13) will make up the global radiation that first strikes the earth's surface. There is an additional component of diffuse radiation caused by the first order reflection between the earth's surface and the clear atmosphere. The magnitude of this component which may be designated as G J, is given by:

G J,=R(I•+D)(0.5 s'+0.25 s,~)(1-~=,~ a'~) (22)

where s'a and s,~ are evaluated for 1.66 times the zenith air mass (P/Po) to account for the Lambert reflectivity of the surface and the absorption values Y,a'i are evaluated for the modified air mass (m* + 1.66 P/Po). P is the mean surface barometric pressure and Po the standard pressure of 1013.2 mb. The use of a modified air mass is necessary to account for the depletion of solar radiation as it initially passes through the atmosphere and to account for the Lambert reflectivity of the surface.

The global solar radiation on a horizontal surface at noon is thus given by:

G = I , + D + G ,1,. (23)

The magnitude of G ~ is usually about 10 per cent of D but under conditions of high turbidity, it can be of the order of 15 or 20 per cent. Compared with the normal values of clear sky noon G, the magnitude of G + is only about 1-2 per cent

The transmission factor t of the atmosphere for global solar radiation at noon is given by:

t = GlIo" cos z (24)

5.2 Inputs for the computations of clear sky noon values of IH, D and G

The inputs for the model fall into two categories (1) those derivable from astronomical factors and (2) those derivable from meteorological or climatological factors. In the first category the most important is the mean solar zenith angle z at noon for the month. This is simply related to the latitude of the station 4~ and the mean declination of the month 6 according to:

z = ( 6 - ~ ) . (25)

For values of z occurring in low latitudes, the relative air mass m = l/cos z. This multiplied by the factor P/Po gives the absolute air mass m*. The mean monthly values of P for each station and each month were found by averaging the values for a consecutive 5-yr period 1964-67 from the data contained in the Monthly Weather Reports published by the India Meteorological Depart- ment.

In the second category fall the mean values of the concentrations of ozone, water vapour and dust in the atmosphere for each of the locations.

5.2.1. Ozone. Total ozone is measured at a network of 6 stations in India, Srinagar, New Delhi, Varanasi, Mount Abu, Poona and Kodaikanal, using Dobson spectropho- tometers and data are available for periods ranging from 4 to 20 yr. The data of Dum Dum where measurements were made from 1963-72 were also used in the study. The gradient of total ozone over the country is small in all months and it was possible to obtain reliable mean values of ozones concentrations for all the 121 stations by interpolation.

5.2.2. Mean precipitable water. Radiosonde data at 19 stations for the period 1971-75 were used for the calculation of total precipable water in the atmosphere. Since the interannual fluctuations in the mean water vapour content in the atmosphere are small, the average of a 5-yr period was considered to be adequate for arriving at climatological averages needed for solar radiation calculations. From the 5-yr averages of dew point temperature at the ground and at 50mb intervals upto 250 mb, mean values of the total precipitable water in the atmosphere for each month were determined for the 19 radiosonde stations. Because of the occurrence of sharp spatial gradients in the distribution of precipitable water over the country, arising from both climatological factors and variations in the terrain, radiosonde data at 19 stations were found to be quite inadequate for interpolating and assigning values for each of the 121 stations. It was, therefore, decided to make use of the values of surface water vapour mixing ratios from routine meteorological observations at about 100 surface observatories evenly located over the country. Since the variability of the total precipitable water arises mostly from the variability in the first kilometre above the ground, the computation of total precipitable water for the larger network of stations was carried out in two stages. First, the total water vapour from the surface upto the 900 mb level (~- 1 km) was calculated from the surface humidity mixing ratios and appropriate lapse rates of the mixing ratios, deduced from available radiosonde data after suitable interpolation. Next the spatial distribution of precipitable water vapour above the 900mb level was obtained by drawing monthly mean maps for the country an interpolating the values for each of the stations. The sum of the two contributions gives the total water vapour in a vertical column for each of the stations. Further details of the procedure adopted and maps showing the distribution of mean monthly precipitable water vapour over the country in the different months will be published elsewhere. These show that the total water vapour content is highly variable in space and time. Large values exceeding 6gmcm 2 are found over the central and northeastern parts of India during July and August, while low values below 2 gm cm -2 occur over large areas of northwest India from December to February.

5.2.3. Pressure correction for water vapour absorption. The absorption of solar radiation by a given amount of water vapour depends on the prevailing temperature and pressure. The temperature effect can be neglected

Precise estimation of hourly values of global, diffuse and direct solar radiation 589

because of the slow variation of the absolute temperature of the atmosphere with height in the lower levels where most of the water vapour is present. The scaling factor for pressure was taken as (5.3e+ P)/I013.2, where e is the water vapour pressure in a slab of the atmosphere with a mean total pressure P(15). Using data on the water vapour profiles at the different radiosonde stations and the scaling factor given above, the overall correction factor for water vapour absorption to be applied to the total precipitable water was determined for each month and for the different radiosonde stations representing different geographical locations in the country. Its value is found to vary from 0.77 to 0.86.

5.2.4. Atmosphere turbidity. Of the 5 factors involved in the absorption of solar radiation in the atmosphere and embodied in the term

i= l

of eqn (12), the dust absorption function is the most uncertain and difficult to compute. Similarly the value of sa occurring in eqn (15) is also subject to uncertainties due to lack of precise information on atmospheric turbidity. The Angstr6m turbidity coefficient/3 is regularly measured at 13 stations in India using /~ngstrom pyr- heliometers and standard Schott RG2 broad-band pass filters. Using climatological mean monthly values of /3 thus measured over a period of several years, g(/3) was derived and made use of in the computations of IH, D and G for clear sky noon at 13 stations. A comparison with actual measured values of D and G for clear sky noon conditions, however, showed that the computed D values which are sensitive to g(/3) were very much lower than the observed values for the period March-June, which are characterised by high values of turbidity in India. G, on the other hand, is not sensitive to uncertainties in /3 or g(/3). The discrepancy can be attri- buted to the measured/3 values being too low, presumably as a result of the circumsolar radiation from the solar aureole being neglected in the computation of the Angstrom turbidity coefficient.

To check this possibility, an alternate method of deriving /3 from the actual measurements of the ratio D/IH was evolved. It follows from eqn (12) and (13) that:

D _ (0.5 sa + 0.75 sd) I H ( I - s a ) ( 1 - s d ) '

(26)

From measured values of D and G, the ratio D/IH can be obtained after allowing for the small contribution of G ~ to G. Since s, is known precisely sa can be calculated and from eqn (26), sd obtained which with the aid of Hoyt's table yields /3. In the above technique of determining/3, two assumptions have been made, (1) the forward scatter ratio for dust scattering is 0.75 and (2) the wavelength exponent for dust scattering is unity. Both the assumptions may not have universal validity, yet the relationship between th ratio DIlH and g(fl) is a useful one for practical applications. For, any value of/3 corresponding to the actually prevailing value of a in the

atmosphere can be transformed into a new value of /3 that will correspond to a standard value of 1.0 for a and yield the same transmission g(/3) for a standard value of 0.75 for the forward scatter. A comparison of the values determined by the two methods in India gave excellent agreement for the months having low or moderate turbidity in the atmosphere. The disagreement in the months March-June characterised by high turbidities has already been referred to earlier and accounted for. For purposes of the computation of solar radiation for clear sky noon conditions, the /3 values as obtained by the D/IH ratio method have been used.

6. COMPUTATION OF ln,D AND G FROM THE SOLAR CONSTANT

6.1 Clear sky noon values Having assigned appropriate values of ozone, water

vapour and dust scattering functions for all the 121 stations for the middle of each of the 12 months of the year, computations of I, D, G and G ~ were carried out using the Hoyt model. For 16 of the 121 stations considered in this study, observed values of daily G and hourly G for cloud-free days were available. They were averaged monthwise and tabulated and used to verify the results of the computation. Table 5 gives the computed and the observed values of clear sky noon G together with the percentage differences and Table 6 gives similar statistics for D. The agreement between the computed and the observed G is seen to be within 3 per cent for 7 out of the 16 stations and within 5 per cent for the remaining ones. For diffuse radiation, the agreement is seen to be within 3 per cent for 6 stations and within 5 per cent for the remaining 7 stations. Of the different variables that enter into the computations, the dust scattering function g(/3) is the one measured with the least accuracy and reliability and it is probably this un- certainty which places a limit to the overall accuracy of the computed G and D. Even with this limitation, the accuracy obtained is high and appears to be adequate for all practical applications.

6.2 Clear sky daily totals of G From the clear sky noon values of G at noon, it is

possible to obtain daily totals of G for clear days through the relationship given in Fig. 3. To secure greater accuracy for deriving the daily G, two different sets of curves were drawn using observed data, one for stations located to the south of lat. 20 ° N and the other for the remaining ones. Using the two sets of curves for the empirical transformation of the noon values into the daily values of G, clear sky daily totals of G for all 16 stations were obtained. Table 7 lists the computed and the observed mean daily G and the percentage differences.

7. MEAN ANNUAL TRANSMISSION FACTOR FOR G

The mean annual transmission factor for global solar radiation is a useful parameter in studies of the overall transmission of solar radiation through a cloud-free atmosphere. This is the mean of the monthly mean transmission factors on cloud-free days for the 12 different months of the year, and can be derived by two


Table 5. Comparison of computed and observed values of clear sky noon global solar radiation

S t l t i o n JAN FEB MAR APR MAY JUN JUL AUG 5EP OCT NOV DEC

Celc, 0,754 0,851 0,938 0,983 0,985 0,960 0,950 0,947 0,921 0,858 0.779 0,719

Ahmldmbmd Obl. 0,741 0.839 0.941 0.983 0.970 0.957 - 0.880 0.831 0.760 0.715

% Diff, 1,7 1,4 0,3 0,0 1,5 0,3 - - 4,7 3,2 1,3 0,5

CllC, 0,769 0,866 0,947 0,985 0,983 0,957 0,951 0,953 0,928 0,870 0,790 0,736

Bhavnegar Obs, 0,758 0,847 0,934 0,966 0,962 0,981 - 0,899 0,841 0,778 0,725

Diff, 1,5 2.2 1,4 2,0 2,2 0,6 - 3,2 3,4 1,8 1.5

t a l c . 0.786 0.878 0.951 0.986 0.967 0.945 0.947 0.958 0.939 0.884 0.811 0.755

BOmbly Obe. 0.740 0.814 0.898 0.935 0.933 - 0.838 0.771 0.719

% Diff. 6.2 7.9 5.9 5.5 3.6 - - 5.5 5.2 5.0

Celc, 0,744 0,841 0,924 0,963 0,955 0,934 0,929 0,930 0,900 0,837 0,762 0,715

Calcut ta Obo. 0.740 0.847 0.921 0.948 0.929 0.940 - 0.835 0.770 0.675

% D i f f . 0.5 0.7 0.3 1,6 2.8 0.6 - - 0.2 1.0 8.9

ta lc , 0.838 0.917 0.977 0.991 0.971 0,947 0.945 0.967 0,956 0.909 0.850 0,806

GO# Obl. 0.827 0.900 0.962 0.994 0.993 - 0,897 0.853 0.815

% D i f f . 1.3 1.9 1.5 0,3 2.2 - - 1.3 0.4 1,1

CaI©. 0.708 0.816 0.916 0.977 0.993 0,968 0.954 0.935 0.904 0.837 0.742 0.675

Jodhpu/ Obz, 0.713 0,823 0,905 0,967 0.957 0.930 - 0.900 0.851 0,808 0.741 0.664

D i f f . 0,7 0.9 1.2 1,0 3.8 4.1 3.9 6.2 3.6 0.1 1.3

CliC. 0.858 0e936 0.996 0.999 0.959 0,963 0,935 0.959 0.959 0.922 0.867 0.828

SmdIim Obm. 0.878 0.917 0.953 0,950 0.927 0.906 - 0.909 0.830 0.863

D i f f . 2,3 2,1 4,5 5~1 3.4 6.3 - 1.5 4.3 4.1

Cel t . 0.779 0.867 0.954 0.993 0.998 0.962 0.954 0.956 0.931 0.873 0.803 0.748

Negpuz Ohm. 0.739 0,849 0.893 0.933 0.963 0.932 - 0.903 0.837 0.763 0.720

Diff. 5.4 2.1 6.8 6.4 3.6 3.2 - 3.1 4.3 5.2 3.9

C, lc . 0.678 0.792 0.899 0.970 0.993 0.969 0.945 0.919 0.882 0.806 0.711 0.645

New Delhi Obs. 0.666 0.774 0.884 0,941 0.950 0.929 - 0.889 0.852 0.773 0.701 0,642

% D i f f . 1.8 2.3 1.7 3.1 4.5 4.3 . 3.4 3.5 4.3 1.4 0.5

Calc. 0.815 0.907 0.978 1.016 1.004 0.975 0.972 0.980 0.963 0.907 0.832 0.781

Pun, Ohm. 0.762 0.856 0.928 0.993 0.990 0.975 - 0.902 0.860 0.790 0.753

% Di f f . 6.6 5.7 5.1 2.3 1.4 0.0 - 6.3 5.2 5.1 3.6

Calc. 0.733 0.837 0.935 0.990 0.998 0.977 0.975 0.970 0.931 0.851 0.759 0.702

Shi l iong Obs. 0.742 0.821 0.916 0.992 0.995 - 0.834 0.782 0.687

% D i f f . 1.2 1.0 2.0 0.2 0.4 - 2.0 3.0 2.1

Calc. 0.900 0.961 0.997 0.981 0.949 0.922 0.930 0.964 0.979 0.953 0.903 0.869

Trivamdzum Obs. 0.882 0.933 0.983 0.959 0.968 0.935 - 0.948 0.916 0.902 0.855

D i f f . 2.0 3.0 1.4 2.3 2.0 1.4 - 3.3 4.0 0.1 1.6

cel t . 0.806 0.879 0.955 0.975 0.959 0.941 0.943 0.953 0.935 0.88D 0.820 0.775

Vishakha- Obs. 0.785 0.862 0.893 0.915 0.816 0.940 - 0.930 0.869 0.613 0.759 patnem

Di f f . 2.7 1.9 6.5 6.2 4.5 0.2 - 0.5 1.3 0.9 2.1

-2 Unit : kWh,m


Table 6, Comparison of computed and observed values of clear sky noon diffuse solar radiation

591

5rat ion JAN FEB HA8 APR MAY JUN JUL AUG 5EP OCT NOV DEC

EeZc. 0.121 0.141 0.151 0,192 0.191 0.188 0.182 0.179 0.176 0.132 0,109 0.107

Ahmadebad Obs, 0,121 0.t41 0.154 0,194 0.194 0.192 - 0.167 0.131 0,110 0.106

% D i f f . 0.0 0.0 1.8 1.0 1.5 2,1 5,4 0.8 0.9 0.9

Calc. 0.117 0.137 0,154 0,202 0.208 0,185 0.174 0.163 0.157 0.135 0.110 0.107

Bhevnagar Obs. 0.116 0.131 0.153 0.198 0.206 0.187 0.155 0.132 0.110 0.105

D i f f . 0.9 4.6 0.7 1.5 1.0 1.1 1.3 2.3 0.0 1.9

t a l c . 0.1dO 0,163 0.194 0,211 0.319 0.251 0.210 0.180 0,157 0.140 0.107 0.113

Bombay Obs. 0.135 0.154 0,185 0,209 0.293 - - - 0,136 0.104 0.109

D i f f . 3,7 5.8 4.9 1.0 8.9 - - - 2.9 2.9 3.7

Calc. 0.165 0.171 0.203 0,246 0,311 0.289 0.246 0,208 0,194 0.172 0.148 0.180

Eaicuttm Obs, 0.169 0.176 0,203 0.243 0.287 - - 0,171 0,154 0.188

D i f f . 2.4 2.9 0.0 1.2 8.4 - - 0.6 3.9 4.3

Ca~c. 0,124 0.145 0.184 0.186 0.183 0.170 0.153 0.139 0.132 0.119 0.105 0.102

Gol Obs. 0.122 0.144 0.184 0.195 0.193 - - - 0,119 0.106 0,102

Diff. 1,6 0.7 0.0 4,6 5.2 - - - 0.0 0,9 0.0

t a l c . 0,118 0,136 0.163 0.176 0.239 0.237 0.211 0.197 0.195 0.125 0.096 0.098

Jodhpuz Obs. 0.122 0.145 0.161 0.182 0.211 0.236 - 0.191 0.125 0.098 0,103

% D i f f . 3.3 4.8 1.2 3.3 0.9 0.4 - 2.1 0.0 2.1 5.8

CaZc. 0.118 0.113 0.161 0.162 0.218 0,220 0,209 0.200 0,195 0,130 0.130 0.096

Madzae Obs. 0.118 0.115 0.156 0.175 0,211 0.204 - 0.178 0.130 0.125 0.099

D i f f . 0.0 1.8 3.2 3.9 3.2 7.3 - 8.7 0.0 3,9 3.1

CmlC. 0,125 0,142 0.176 0.197 0,199 0.216 0.210 0.186 0,164 0.131 0.119 0.107

Negpux Obl, 0.120 , 0.137 0.166 0,202 0.194 0.212 0.159 0,125 0.114 0,105

D i f f . 4,2 3.6 6,0 2,5 2,6 1,9 3.1 4.8 4.4 1.9

CmZc. 0.114 0.139 0,148 0.197 0,200 0.216 0.209 0,208 0.190 0.152 0.133 0.114

NI~ Delhi Obl, 0,116 0,140 0.148 0.195 0,193 0.209 0.224 0.187 0,150 0.133 0.114

% D i f f . 1,7 0.7 0.0 1,0 3,6 3.3 7.1 1.6 1.3 0.0 0.0

C l l c . 0.116 0,128 0,154 0.174 0.180 0.177 0.175 0.175 0.176 0.108 0.101 0.096

Pune Oba, 0.111 0.124 0.149 0,171 0.178 0.176 0.167 0.106 0.095 0.095

D i l l , 4,3 3.2 3.3 1,2 1.1 0,6 5.1 1.9 5.9 1.1

C l io . 0.061 0,100 0.131 0.169 0.197 0.167 0.142 0.110 0.065 0,060 0.061 0.055

Sh£1long Ob=. 0,055 0.101 0.134 0.180 0.211 - - - 0.055 0.058

D i f f . 9.8 1,0 2.2 6,5 7.1 - - - 9.8 5.4

Calc, 0,129 0.137 0.163 0.215 0.167 0.200 0.186 0.173 0.152 0.151 0.122 0.124

Txivandxum Oba. 0,128 0.133 0.163 0,209 0.173 0.196 - 0.151 0.153 0.120 0.122

D i f f . 0.9 3.0 0,0 2,9 3.5 2.0 - 0.7 1.3 1.7 1.5

t a l c . 0.135 0.137 0,187 0.220 0.253 0.168 0.160 0.154 0.147 0.142 0.111 0.122

V i | h l k h l - Obl, 0,136 0.137 0.179 0.211 0.239 0.171 - - 0.143 0.111 0,121 potnlm

D i f f . 0.7 0.0 4.3 4.1 5.5 1.7 - - 0.7 0.0 0.8

SE Vol. 31, No. ~ - E

U n i t : kWh.m -2


Table 7. Comparison of computed and observed values of daily global solar radiation on cloudless days

Station JAb FEB riA R APR FAY JUN JUL AUG 5EP OCT NOV DEC

Calc . 5 .043 5 .930 6 .822 7.446 7.695 7.619 7.480 7.285 6 .847 6 .085 5 .300 4.761

Ahmadabad Obs, 5 .099 5.9~0 7 .000 7.690 7.80£ 7 .797 - 6 .537 5 .990 5 .349 4 .837

~ D i f f . 1.0 1.0 2.5 3 .2 1.5 2.3 - 4 .7 1.6 0 .9 1.6

Ca /c . 5 .178 6.055 6 .912 7 .462 7 .650 7.595 7 .488 7 .302 6 .874 6 .170 5 .583 4.906

Bhavnaga= Obs. 5 .313 6.109 7 .048 7 .583 7 .773 7.805 - 6 .787 6 .193 5.491 5 .0n4

,~ ~ i f f . 2.5 0 .9 1.9 1.6 1.6 2 .7 - 1.3 0 .4 1 .7 1.9

Ea lc . 5.365 6.161 6 .923 7.441 7 ,457 7.383 7 .370 7 .313 6 .963 6 .314 5 .593 5.125

5ombay Obs. 5 .182 5 .895 6 .644 7.179 7 .322 - 6 .085 5 .432 4.933

~ D i f f . 3.5 4.5 4.2 3.6 2.0 - 3.8 3.0 3.9

Calc . 4.993 5 .867 6 .720 7.323 7 .468 7 .383 7 .314 7 .692 6 .699 5 .978 5 .184 4.735

C a l c u t t a Obs. 4 .980 5.919 6 .787 7.203 7 .310 7 .517 - 5.990 5 .254 4 .529

~ i f f . 0 .3 0 .9 1.0 1 .7 2 .2 1.8 - 0 .2 1 .3 4.5

Ca lc . 5 ,860 6.621 7.264 7.536 7 .440 7 .284 7.269 7.302 7 .188 6 .635 6 .007 5.559

Goa Obs. 5.~31 6 .585 7 ,167 7.595 7 .857 - 6 .632 6 .180 5 .812

~ ~ i f f . 1.2 0.6 1.3 O.B 5 .3 - 0 .0 2 .8 4 .3

Ca lc . 4 .657 5 .627 6 .637 7 .458 7.819 7 ,744 7.601 7 .256 6.721 5 .894 4.963 4 .369

Obs. 4 .849 5 .907 6.775 7 .678 7.809 7.655 7.036 6.371 5 .883 5 .063 4.635

D i f f . 3.9 4 .7 2.0 2.9 0.1 1.2 3.1 5 ,5 0 .2 2 .0 5 . 7

Calc . 7 .319 7.856 8 .338 8 .423 8 .193 - - 7 .510 7 .106

K o d a i k a n a i Lbs. 7.441 7 .952 8 .427 8.415 8 .284 - - 7 .559 7 .238

~ i f f . 1.6 1.2 1,1 0.1 1.1 - - 0 .7 1 .8

Ca lc . 6 .064 6 ,807 7.4C5 7.600 7.320 7.4O7 7.165 7 .293 7.183 6 .754 6.193 5~810

Ous. 6 ,489 6.81C 7 .167 7 .357 7.298 7,156 6 .822 6 .763 6 .002 6 .204

# z f f . 6.5 F.O 3.3 3 .3 0 .3 3.5 5 ,3 0.1 3 ,2 6 .3

Ca lc . 5 . 264 6 .063 6 .938 7 .522 7.766 7 .634 7 .200 7 .300 6 .896 6 .213 5.481 5 ,003

Nagpur Obs. 5 .122 5 .883 6 .844 7.215 7.761 7.595 6 .834 6 .050 5 .480 4.968

~'~ n i f f . 2 .8 3 .0 1 .4 4.3 0 .0 0 .5 0 .9 2 .7 0 .0 0 .7

Ca lc . 4 .402 5.424 6 .514 7 .404 7.881 7.752 7 .560 7.179 6 .582 5.676 4.708 4 .108

New D e l h i Obs. 4 .457 5 .456 6 ,596 7.476 7.809 7 .738 6 ,989 6 .454 5 .539 4 .790 4 .243

% D i f f . 1.2 0 .6 1.2 1.0 0 .9 0 .2 2 .7 2 .0 2.5 1 .7 3.2

Ca lc . 5 .563 6 .478 7 .244 7 .726 7 .693 7 .500 7.506 7.509 7.241 6.596 5 .778 5 ,248

Rune Obs. 5 .349 6 .169 6 .894 7.619 7.8B0 7.952 - 6 .727 6 .252 5 .598 5 .206

% D i f f . 4 .0 5 .0 5.1 1.4 2,4 5 .7 - 7 .6 5 .5 3 .2 0 .8

t a l c . 4 .838 5 .792 6 ,824 7 .528 7.889 7 .816 7 .746 7.519 6 .922 5 .930 5 .114 4 .573

5h i l l ong Obs. 4 .968 5 .836 6 .680 7.749 8 .094 - - 5 .955 5 .467 4 .517

Di f f . 2.6 0 .7 2,1 2,9 2,5 - 0°4 6.5 1.2

Calc . 6 .505 7 .040 7 .413 7.403 7 .216 7 .036 7 .072 7 .275 7 .333 7 .033 6 . 5 6 7 6 .237

Trivendzum Obs. 6 .513 6 .953 7.441 7 .274 7 .452 7 .333 - 7 .155 6 .739 6 .703 6 .240

% D i f f . 0.1 1.3 0 .4 1 .8 3.2 4 .0 - 2.5 4 .4 2 .0 0 .0

Jodhpur

iqadras


Table 7(Contd)

593

S t a t i o n JA N F£~ ;IAR ApR ~ AY JUN JUL AUG 5EP OCT ~'JOV DEC

Calc. 5.559 6.352 6.797 7.149 7.214 7.397 - - 7.180 6.509 5.819 5.288

V£shakha- Obs. 5.539 6.311 6.597 6.977 7.215 7.631 - - 7.262 6.415 5.836 5.337 patnlm

D i f f . 0,4 0.7 3.0 2.5 0.0 3.1 - - 1.1 1.4 0.3 0.9

different methods. Firstly as pointed out in Section 2 the sum of the two regression parameters a and b in eqn (2) is a measure of the mean annual transmission factor and this can be obtained from the statistically determined values of a and b for the different stations. Secondly, as described in Section 2, it is the ratio of the mean daily global radiation under cloud-free conditions to the mean extraterrestrial solar radiation. The annual mean transmission factors obtained by the two methods are tabulated for the 16 stations in Table 8 and it will be seen that the agreement is within 5 per cent in most of the cases. This lends support to the correctness of the statistically derived relationships on the one hand and the computed values of the global solar radiation by the Hoyt model for clear sky conditions on the other. The agreement also shows a method of independantly checking the assumptions made in the two different methods for average monthly conditions.

8. EXTENSION OF THE COMPUTATIONS FOR OVERCAST SKY CON-

DITIONS

Attempts have been made to compute solar radiation by similar techniques under cloudy or overcast ~,on- ditions. Hoyt[3] considered six different types of the clouds and proposed different transmission and absorption characteristics for each one of them. Such computations are possible only if the mean height of bases and tops of the prevailing clouds are known with some certainty. Unfortunately these crucial parameters are seldom known with the required reliability. In the present study, therefore, an empirical method was adopted to compute solar radiation parameters under average overcast conditions. Values of hourly mean and daily mean global radiation measured on overcast days for each of the 16 radiation stations based on all available data were available upto 1975. Using these and the corresponding values for clear sky conditions, mean

=. ,

S t a t i o n

Table 8. Mean annual transmission factor for G compared with (a + b) . , . , , . . . . .

P e r c e n t a g e G(c ) /ETR a + b

d i f f ,

Ahmadabad 0.707 0.739 4.5

Bhavnagar 0.709 0.731 3.1

Bombay 0.700 0.720 2.9

C a l c u t t a 0 .696 0 . 6 8 7 1 .3

Goa 0 .706 0 .733 3 .8

Jodhpuz 0 .709 0 . 7 5 4 6 .3

K o d a i k a n a l 0 . 8 1 7 0 . 8 4 4 3 ,3

Madras 0 .710 0 .716 0 . 8

Nagpur 0 . 7 0 7 0 ,725 2 .5

New Delhi 0 . 7 0 8 0 .749 5 .8

Pune 0 . 7 2 0 0 . 7 2 4 0 .5

Shillong 0.721 0 .731 1 , 4

Tz i vandzum 0 . 7 1 0 0 . 7 6 0 7 .0

V i s h a k h a p a t n a m 0 . 6 9 5 0 .719 3 .0


Table 9. Ratio of mean daily global solar radiation on overcast days to that on cloudless days.

S t s t i o n JAN FEB HAR APR HAY JUN JUL AUG SEP OCT NOV DEC

AhBadabad 0e35 0 . 2 0 - - 0 . 3 5 0032 - 0 . 2 2 0 , 2 7 0048 0 . 1 8

Bhavnlga¢ . . . . 0 . 3 0 0 . 2 8 0 . 3 6 0 . 2 3 -

B ~ b e y 0 . 4 0 . . . . . 0 . 3 6 - -

C = l c u C t e 0 .21 - 0026 0 . 2 7 - 0 . 2 8 0 . 3 0 0 . 2 8 0 , 1 7

Goa . . . . 0 , 2 8 0 . 3 0 - -

J o d h p u r - 0 . 3 2 l I I 0 . 3 3 0 . 3 8 - - 0 . 1 8

K o d e i k a n a l 0 . 2 7 - - - 0 . 3 4 - 0 , 2 5 0 , 2 6

M a d c a l . . . . 0 .21 0 . 3 9 0 , 3 9 0 . 2 4 0 , 2 2 0 . 2 2

Nagpuz 0 . 3 9 0 . 3 9 - - - 0 . 2 8 0 . 3 5 0 . 3 4 0 . 4 0 0 . 2 9

NBw 9 e Z h i 0 . 3 3 0 . 2 6 0 . 3 2 0031 0 . 3 3 0 . 2 7 0 . 2 5 0 . 3 2 0 , 2 5 0 . 3 4

P o r t B l a £ ¢ 0 . 2 4 0 . 3 6 0 . 1 4 - - - 0 . 2 2

Punm - 0 . 2 7 0 . 4 5 - - 0 .41 0 . 3 2 - 0 . 3 7

5 h i l Z o n g 0 . 2 4 0 . 2 3 0 . 2 7 - 0 . 2 3 0 . 2 6 0 . 2 5 0.41

T~£vandrula 0047 0 . 3 2 0 . 4 2 0 ,25 0 . 2 7 0 . 3 5 - 0 . 4 0 0 , 2 9 0 ,31 0 . 3 3 -

V i s h s k h a p e t n s m 0 , 2 4 0 , 1 9 0 , 2 7 - - 0 . 2 4 0 ,19 0 ,31 0 . 3 3

ratios for overcast sky to clear sky daily global radiation were obtained.These ratios which represent climatolo- gically mean values are tabulated in Table 9 and are seen to lie within the limits 0.15-0.35 for monthly means and within a still smaller range of 0.25-0.30 for the annual means. The variation in the ratios among the different stations represent variations caused by the characteristics of the cloud types found in the different regions of the country. They can be used to compute the average global radiation under overcast sky conditions from that for clear sky conditions. For any degree of clouding intermediate between clear sky and overcast sky, the global radiation can be estimated by the method of proportions. For example, if the extent of clouds is 4 octas, the estimated global radiation will be the mean of the clear sky value and the overcast sky value.

9. CONCLUSIONS 1. If reliable records of global and diffuse solar radia-

tion and sunshine are available at a widely distributed network of stations in a region for a minimum period of at least 5 or 6 yr, it is possible using regression techniques to derive with a high degree of correlation, the regression parameters a, b, c and d and in turn compute global and diffuse radiation at an extended network of stations, where sunshine measurements are available. It is also possible to derive, using other empirical relationships, mean hourly values of global and diffuse radiation from

their daily totals. The accuracy possible is 5 per cent for global and 5-10 per cent for diffuse solar radiation for daily totals and 2-3 per cent for hourly of global and 4--8 per cent for hourly values of diffuse solar radiation.

2. Where no sunshine data are available, solar radiation at the ground on clear days can be derived from the solar constant, using theoretical models and allowing for depletion by absorption and scattering by atmospheric gases, dust and aerosols. The accuracy of computing the noon values of the parameters on clear days is about 2-3 per cent and that of the mean daily totals about~3--4 per cent. From observed data covering several years it is also possible to compute the mean global radiation on overcast days.

3. The mean annual transmission factor for global solar radiation derived using the two different methods shows excellent mutual agreement establishing the soundness of the regression technique on the one hand and the correctness of the model calculations on the other.

Acknowledgements--The above work was carried out as part of a Project financed by the Commission for Additional Sources of Energy, (CASE), Department of Science and Technology, Government of India at the Field Unit of the Indian Institute of Tropical Meterology, Bangalore. The authors gratefully ac- knowledge the financial and administrative support extended by CASE and the Indian Institute of Tropical Meteorology, Pune and the facilities provided by the Raman Research Institute and

Precise estimation of hourly values of global, diffuse and direct solar radiation 595

the Raman Research Institute Trust to carry out the work at Bangalore. They are also grateful to the Director-General of Meteorology, for the co-operation extended by him in this joint project and for the supply of solar radiation and radiosonde data recorded at the national network of stations in India. They are indebted to all the members of the Solar Radiation Data Hand- book Project Group for their invaluable help and assistance in the course of the Project.

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Techniques for the precise estimation of hourly values of global, diffuse and direct solar radiation