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Assessment of solar radiation models for the Gulf Arabian countries
Renewable Energy Vol. 2, No. I, pp. 65 71, 1992 Printed in Great Britain.
096~1481/92 $5.00+.00 ,! 1992 Pergamon Press Lid
DATA BANK Assessment of solar radiation models for the Gulf Arabian countries N . A L MAHDI, N . S. A L BAHARNA a n d F . F . ZAKI College of Engineering, University of Bahrain, P.O. Box 32038, Isa Town, Bahrain
(Received 12 December 1990 ; accepted 13 September 1991 ) Abstract--A statistical assessment of the accuracy of 12 solar radiation models for five meteorological stations in the Gulf Arabian states namely Abu Dhabi, Bahrain, Doha, Kuwait and Riyadb, is presented. The first six models are of the regression type in which the measured global radiation is correlated with the sunshine hours and other meteorological parameters. The other six models are based on the calculation of clear sky radiation and the effects of cloud amount and sky transmittance. The models are assessed in terms of the root mean square error and the mean bias error. The obtained results indicate some inconsistency of published data for Abu Dhabi, Bahrain and Doha stations. However, using Kuwait and Riyadh stations as a basis for the evaluation, two models are recommended for the estimation of monthly average daily and hourly global radiation for the Gulf states.
low latitude of the region along with a climate, characterized by a very small amount of either rain or cloudy skies, permits radiation levels to reach 7 kW/m 2 during the summer. The weather conditions are very hot and semi-arid during summer days, becoming warm and humid at night. During winter, daytime conditions fall within the comfort zone, while the night times are cool and humid. Wind speeds generally average about 5 m/s and prevail mainly from the north. Unfortunately, the meteorological measurements available for the region [1-11] are limited to global radiation measurements, while data of diffuse radiation are very rare and refer to limited periods. Moreover, the annual global radiation and sunshine hours data available from the various stations appear to be inconsistent, especially since the climatic conditions are very similar and the stations are located within relatively close proximity from each other, as can be seen from Table 1. Published measurements [1] indicate that the annual global radiation for Abu Dhabi is lower than that of other Gulf states, which conflicts with our expectations since the Abu Dhabi station is located at a latitude angle lower than that of the other stations. The variation of the measured monthly mean daily global radiation and the monthly mean daily bright sunshine hours also exhibit inconsistencies, as can be seen from Figs 1 and 2, respectively. In addition to instrumental problems such as misalignment, malfunctions and dust on radiation sensors, the lack of consistency between the data may be attributed to the relatively short periods of measurements, and differences in the extent and quality of data, as well as the procedures used in the evaluation of spatial and temporal averages.
INTRODUCTION Accurate quantitative data of the variation of solar radiation reaching the earth surface, and the relevant meteorological parameters are essential requirements for conducting a wide range of scientific studies. Typical examples are found in hydrological studies when calculating soil moisture deficits, investigation of biological process, climatology, thermal design of environmental control of buildings and quantitative evaluation of ecophysiological systems for the determination of irrigation water needs and the potential yield of crops. The design and estimation of performance of solar heating, cooling and distillation systems also require detailed knowledge of solar radiation data. The variation of solar radiation over a given area for a specified time period has conventionally been obtained through the integration of continuous records of radiation measurements. The gathering of such data requires properly trained personnel and appropriate instruments that are continuously maintained and regularly calibrated, in addition to a reliable supply of accessories and materials. The most frequently needed data are the long term average daily global radiation and its direct and diffuse components on a horizontal surface. The objective of the present work is the determination of the radiation characteristics for the Gulf Arabian countries. To achieve this, solar radiation data available from five meteorological stations, namely Abu Dhabi, Bahrain, Doha, Kuwait and Riyadh, are used to derive correlations for estimating the mean global radiation. The correlations are made with the aid of a number of commonly used solar radiation models. The first group of models correlate the monthly average daily global radiation only to sunshine hours while others introduce the effect of latitude, altitude, maximum temperature and humidity in addition to the sunshine duration. The second group of models predict the clear sky radiation and the cloud cover effects in order to determine the global radiation. The Arabian Gulf is located just north of the equator. The
GLOBAL RADIATION CORRELATIONS The various available methods for the estimation of global radiation can be categorized into two groups. The first group is totally empirical whereby meteorological data are used in conjunction with regression analysis. The second group involves the prediction of clear sky radiation and the 65
66
Data Bank Table 1. Geographical data and annual sunshine and global radiation for the Gulf stations Annual sunshine Station Abu Dhabi Bahrain Doha Kuwait Riyadh
Latitude ()
Longitude (°)
Altitude (m)
g (h)
g/go
24.43 26.16 25.90 29.22 24.70
+54.45 +50.90 +51.55 +47.97 +46.72
15 2 5 45 594
3465.799 3359.399 3501.198 3204.058 3376.910
0.789 0.765 0.798 0.730 0.769
0.7
~o
24 × 3600 7~
Annual radiation H (MJ/m 2) H / H o 6671.389 7125.488 6895.259 7083.529 7204.539
0.552 0.597 0.576 0.608 0.597
f
Go l c o s (q~) cos (6) cos (~o~) L + ~2nOSssin (~b) sin (6) 1
0.6
(2)
,2 where the instantaneous extra-terrestrial radiation Go for the nth day of the year is:
II 0.~
I F
0.4
I M
I A
I M
Go = G~[1 +0.033 cos (2~n/365.24)].
<> Rtyadh
x Bahroin [] Kuwai¢
(3)
The monthly average maximum possible number of daily sunshine hours is evaluated from the daily values as :
I ] J d Month
I A
I S
I 0
I N
I D
n2
go = [1/(n2-n,)])~ So
(4)
nI
Fig. 1. Monthly/tm//q o for the five Gulf stations.
where So = 2/15~0s
(5)
co~ = cos -j [ - t a n (~b) tan (6)].
(6)
and
1.0
The solar declination angle 6 is given by [16] :
0.91
(7)
sin 6 = 0.3979 sin (7)
,,g
0.8
where )' = 7o +0.007133 sin (7o) +0.032680 cos (Yo)
0.7
-0.000318 sin (270)+0.000145 cos (2~o) 0.6
x Bohrain [] Kuwoit
O Riyadh
(8)
and 7o = 2n(n + 284)/365.24.
0.5 J
I F
I M
I A
I M
I d
I d
I A
I S
I 0
I N
I D
Month
Fig. 2. Monthly g/so for the five Gulf stations. inclusion of a suitable model for the estimation of cloud cover effects. Regression models A variety of regression correlations for the estimation of global solar radiation exist in the literature. The most commonly used model is the linear regression equation of Angstrom [12] as developed by Page [13], Prescott [14] and others. The equation relates the monthly average daily global radiation/4 to the monthly average daily number of bright sunshine hours g by the linear function :
1:1//7o = a+b(g/go),
(l)
where a and b are the regression constants. The monthly average daily extra-terrestrial radiation/4o is given by [15] :
(9)
Many researchers have used Page model and presented values of the constants a and b which vary with geographic location and climatic conditions. Rietveld [17] examined several published values of a and b and produced the empirical formula : /4//4o = 0.18+0.62(S/So)
(10)
which can reasonably be applied anywhere in the world. On the other hand Bahel et al. [18] developed the worldwide correlation : FlltTlo = O.16+O.87(glgo)--O.16(g/go)Z +O.34(,~/go) 3.
(ll) The correlation based on bright sunshine and global radiation data of 48 stations around the world, with varied meteorological conditions and a wide distribution of geographical locations. Other attempts made to improve the radiation sunshine correlation introduce geographic parameters, such as latitude angle and altitude ; and meteorological factors, such as
67
Data Bank atmospheric temperature, humidity, optical properties of the cloud cover, ground reflectivity and average air mass. Glover and McCulloch [19] included the latitude effects and produced the equation :
any time between sunrise and sunset is given by [15] : 10 = G~[1 +0.033 cos (360n/365)] sin ~
(22)
where
(12)
sin ~ = sin (~) sin (6)+cos (4~) cos (6) cos (~o). (23)
which can be applied for latitude angles of up to _+6ft. Sayigh [20] modified eq. (I) to include the altitude h of the station and presented the model :
The potential solar radiation is depleted as it passes through the atmosphere owing to reflection, scattering and absorption by dust, gas molecules, ozone and water vapour. The clear sky direct normal solar radiation received al the earth surface can be expressed [30] as :
/t//~o = 0.29 cos (q~)+0.52(~¢/So)
(13)
171/171o = a + b ( S / S o ) + c h
where c is also a constant. Hay [21] developed a generalized procedure, that incorporates the monthly average ground albedo pg, cloudless sky albedo p~ and cloud albedo p~, in the form : /q//lo - [0.1572 + 0.5566(~/S~)]/ {1-pg[p,(~/~)+p~(l-
g/eoj)]}.
(14)
Hay used p~,- 0.25 and p ~ - 0.6. The modified monthly average maximum possible number o f daily sunshine hours gj excludes solar zenith angles less than 5~ because the Campbell Stokes sunshine recorder does not function when the solar latitude is less than 5 . Grag and Grag [22] found that a least square correlation of data from 14 pyranometer stations in India gives the following relationship : H//4o = 0.414+0.4(S/So)-0.0055W~,
(15)
where W~, is the atmospheric water vapour content per unit volume of air which is computed by using: W~,t = R ( 4 . 7 9 2 3 + O . 3 6 4 7 T + O . O O 5 5 T 2 + O . O O O 3 T ~ ' ) .
(16)
Sayigh [20] developed the formula: (17)
171 = N K [ d p ( S - R / 1 5 ) - l/Tm],
where N = 1.7- 0.458n(b/180,
(18)
K = 100[).So + ~b~jcos (qS)],
(19)
and 2 = 0.2/(1 +0.1~b).
(20)
O,,j is a seasonal factor in which i = 1, 2, 3, respectively, for inland stations, coastal stations and hilly stations ; and j = 1, 2 , . . . , 12 represents the months of the year. A similar formula has also been developed by Reddy [23]. More recently, Gopinathan [24] related the global radiation to sunshine duration, maximum temperature, humidity, latitude and altitude in an equation of the form : I t / H o = a + b cos ( q b ) + c h + d ( S / , ~ o ) + e T m
+fR
(21)
where a, b, c, d, e and f are regression constants. Various other global radiation regression models have also been developed by Black [25], Sabbagh et al. [26], Hoyt [27] and many other researchers. Some of the aforementioned regression models have been assessed by Ma and Iqbal [28] and by Sambo [29]. Ma and lqbal concluded that Rietveld's and Page's equations generally yield the best results and recommended their use for estimating monthly average daily global radiation. However, Sambo recommended the use of a Gopinathan-like model.
(24)
lL, c = I , , T w . T ~ T w ~ T ~ T ~
where the terms T are transmission functions for water vapour absorption, aerosol or dust absorption, water vapour scattering, Rayleigh scattering and aerosol or dust scattering. These transmission parameters vary considerably in the magnitude of their effects on the depletion of solar radiation. According to the estimates of Watt [31] the ozone layer, upper-layer aerosol, dry air, water vapour and lower-layer aerosol deplete the solar radiation by 0.5-3%, 1.9 11%, 11 13%, 3.5-14% and 0.1 26%, respectively. Many authors [32 35] have developed models for the determination of various transmission parameters. The models normally require detailed meteorological measurements such as water precipitation, mixing height and visibility. Such measurements are not available, in detail, for the Gulf countries. A convenient model for the estimation of the combined effects of the transmission terms have been developed by Hottle [36]. The method takes into account the zenith angle and latitude for a standard atmosphere and for four climate types. The instantaneous atmospheric transmittance for beam radiation is given by : (25)
r~, = a , , + a l e k~i,=
The constants ao, a~ and k for the standard atmosphere with 23 km visibility are defined as : a,, = r°[0.4237-0.00821(6-h)']
(26)
a i = r ~[0.5055 + 0.00595 (6.5 - h) 2]
(27)
k = r~[0.2711 +0.01858(2.5
h) 2]
rd = 0.2710-0.2939rb.
In the absence of atmosphere, the instantaneous flux density of potential solar radiation on a horizontal surface at
(29)
Where rd is the ratio of diffuse to extra-terrestrial radiation. The total clear sky radiation can then be evaluated using the relation : lc - lv, c + Idc = Io (rb + rd).
(30)
Another convenient method is the A S H R A E model [37] as developed by Powell [38]. In this model the total radiation incident upon a horizontal surface and clear sky conditions is evaluated from : 1,. = A(sin ~ + C ) e Bmr
(31)
where the optical air mass at sea level m and the altitude correction factor r are defined as : M = 35/[1224 sin 2 (~) + I] 0.5
Clear and cloudy sky models
(28)
where h is the altitude in km ; and r o, r~ and rk are correction factors that are applied to allow changes in climate types. The clear sky diffuse transmission can be estimated using the empirical relation developed by Liu and Jordan [42]:
r = [1 -- (h/44.308)] 5 257.
(32) (33)
The values of apparent direct normal intensity at zero optical
68
Data Bank
air mass A, the atmospheric extinction coefficient B and the diffuse radiation factor C for the 21st of each m o n t h are given in A S H R A E [37]. Expressions for A, B and C for any day o f the year are given by the following equation [34] : A = 1160+ 77.2 cos [2~(n-- 1)/365]
(34)
B = 0.17028-3.45545 1 0 - a n - 1.51758 I 0 - 6 n 2 +1.8162 10 7 n 3 - 1 . 3 1 7 8 4 10 9n4 +3.34233 10-12nS--2.861610-15n 6
(35)
C = 0.05363+0.000511n-2.1351 10 Sn2 +3.70022 1 0 - 7 n 3 - 2 . 2 7 1 3 10-gn 4 +5.7539 10-12nS-5.19886 10-15n 6.
(36)
The clear sky radiation is used for the calculations of radiation under normal sky conditions by the inclusion of a suitable cloud a m o u n t model [40, 41]. The monthly average daily direct radiation component, /to, under normal sky conditions can be expressed as /~D = /1pc(1 -- Ca)
(37)
where /qOC is the clear-sky monthly average daily direct radiation component and Ca is the total cloud amount. The monthly average daily diffuse radiation/ld is given by :
l~d = lq,lc(1-Ca) + tZl~fC~ + lrIp, p~p~.
(38)
The total global r a d i a t i o n / ~ can be calculated using eqs (37) and (38) as : /7 = / 4 ~ [ ( 1 - C a ) + f C ~ ] / ( I - p g p c C , ) ,
(39)
where/f~,/tr)c and/~d~ are calculated by integrating eq. (30) or (31) over the month. The term ? is the monthly mean average daily sky transmittance. The ground albedo pg is assumed to be 0.25 and the atmospheric albedo p~ is given by the relation [42] : Pc = 0.118 +0.511 - f ( S / S o ) ] ,
(40)
where n~ and n,_ are the first and last days of the month, respectively. An alternative method to determine f has been developed during the course of this work on the basis of the work of Jeter and Balaras [44] and Bahel [18]. The measurement of beam transmittance rb, presented by Jeter and Balaras [44], as a function of the clearness index k-r where used to find a correlation between % and kT. The Bahel [18] correlation was then used to derive an expression for the average daily beam transmittance fb as a function of S/go. That resulting expression is : i n = 0.0243 -0.4872(~¢/So) + 2 ( g g o ) 2 - 0.8537(,~/~o) 3. (47) The monthly average daily transmittance is then calculated by using the relation : f =
f(S/So) = ~¢/~o for
0 ~< (~¢/So) ~< 0.63,
(41)
and
f(~/~¢o) = 0.001427 + 1.2932(~/~o) - 0.4519(S/So) 2 for
0.63 < (~¢/~¢o) < 1.
(43)
or alternatively the cloud a m o u n t m a y be approximated by [42]:
Ca = 1 --f(8/8o).
(44)
A cubic correlation has been recently formulated by Rangarajan et al. [43] as :
~¢/So = l--alC~--azC~--a3C 3
(48)
METHODS OF ASSESSMENT The accuracy of the correlations described in the previous section have been assessed using two statistical tests, namely, the root mean square error e ~ and the mean bias error emb~The root mean square error is defined as : ~rms = {['~(/tm//~o -- Flp/lto)2]/N} i/2
(49)
where/qp is the predicted value of the monthly average daily solar radiation and N is the number of observations. The root mean square error test is useful in assessing the shortterm performance of the correlations. The mean bias error which is defined as :
[~(l~m/Flo-I~p/lYlo)]/g
(50)
is helpful in the assessment of the long-term performance. The values of the errors are a measure of the accuracy of the model used. For example a positive value of embo provides information on an under-estimation and a negative value indicates an overestimation.
(42)
Different expressions for the total cloud a m o u n t have been suggested by several authors. H a y [21] and Hoyt [27] proposed the linear relation :
6". = 1 - S / S o ,
n~--nl)
where fd is evaluated by using a similar relation to eq. (29).
gmbe =
where
Z~+?d
(45)
where the coefficients a~, a2 and a 3 are location dependent. The monthly average daily sky transmittance can be evaluated by adding eqs (25) and (29) and finding their time weighted average over each day. The daily averages are then used to find the monthly average daily values. That is :
RESULTS The accuracy of the global radiation regression models given by eqs (1), (10) (12), (14) and (15) have been evaluated using the root mean square and mean bias error tests. The results of the statistical comparison of these correlations are presented in Table 2. The results show that all the models used, except Grag's model, yield reasonably accurate results for both Kuwait and Riyadh stations. This is to be expected as Kuwait and Riyadh measurements are the best taken and documented ones for the G u l f region. Page's equation gives the least e~msand em~ errors for all stations with the m a x i m u m errors occurring for A b u Dhabi and Bahrain. Results for A b u Dhabi exhibit a consistent high e,m~ and negative em~ indicating that all models overestimate the value o f solar radiation or that the measurement values are low. The next best model is Bahel's correlation. Although this model gives relatively higher overall errors than Page's model, nevertheless it gives a clear indication of the quality of the
Data Bank
69
Table 2. ~rmsand ~mb*of regression models Station
Model A b u Dhabi
Bahrain
Doha
Kuwait
Riyadh
Page eq. (I)
£rms embe
0.042 -- 0.037
0.038 + 0.008
0.019 -- 0.009
0.031 + 0.026
0.036 + 0.013
Rietveld eq. (10)
trms ~,oi~
0.123 - 0.120
0.075 - 0.062
0.111 0.092
0.036 - 0.026
0.063 - 0.060
Bahel eq. (I 1)
c,,.... Crabe
0.088 --0.085
0.050 --0.030
0.080 --0.059
0.021 +0.004
0.032 --0.017
Glover eq. (12)
e.... e,,he
0.127 -- 0.125
0.077 -- 0.066
0.108 -- 0.094
0.033 -- 0.026
0.068 -- 0.066
Hay eq. (14)
r ..... e,mbe
0.115 -0.112
0.070 -0.055
0.103 -0.085
0.033 -0.020
0.056 -0.052
Grag eq. (15)
e,..... ~°,h:
0.121 -0.117
0.079 -0.064
0.105 -0.101
0.058 -0.053
0.112 -0.111
available measurements for stations under consideration since it was derived from world wide data. The model yields excellent correlation for Bahrain, Kuwait and Riyadh but over-estimates the solar radiation for the other two stations. Since Kuwait and Riyadh data are most reliable a m o n g the data available we recommend the use of this model for all the regions as it gives the best correlation for these two stations. For the assessment of the clear sky with cloud effects, six models have been introduced using the combination shown in Table 3 for the evaluation o f lc, Ca and f. The results for these models are given in Table 4. Clearly, model number 3 gives the best estimates for both Kuwait and Riyadh and shows reasonable values of errors for the other stations. This model is recommended for estimating the monthly average hourly or daily global radiation. CONCLUSIONS Twelve regression and clear sky with cloud effect solar radiation models have been compared statistically in terms of root mean square error and mean bias error for five stations in the G u l f Arabian countries. The results indicate that the measurements of Kuwait and Riyadh stations are reliable and consistent with available world wide correlation models. The Babel et al. [18] model was found to give the best correlation and is recommended for use in the estimation of monthly average daily global solar radiation. For the
Table 3. Equations used for clear sky with cloud effects models
Model
Equation for Ic
Equation for C,
Equation for f
I
(3 I)
(44)
(46)
2 3 4 5 6
(31 ) (31 ) (30) (30) (30)
(43) (44) (44) (43) (44)
(46) (48) (46) (46) (48)
estimation of the monthly average hourly global radiation the use of a clear sky model based on A S H R A E [37] is recommended with inclusion of the cloud effect by the introduction of cloud a m o u n t and average sky transmittance using eqs (44) and (48).
NOMENCLATURE constants cloud a m o u n t s solar constant, 1367 W / m 2 instantaneous extraterrestrial radiation, W / m 2 monthly average daily global radiation, J/m 2 monthly average daily clear sky radiation, J/m 2 monthly average daily diffuse radiation, J/m 2 monthly average daily diffuse clear sky radiation, J/nl z monthly average daily direct radiation, J/m 2 monthly average daily direct clear sky radiation, J/m e measured monthly average daily global radiation, J/m 2 monthly average daily extraterrestrial radiation, J/m 2 predicted monthly average daily global radiation, J/m 2 altitude, km clear sky radiation intensity, W / m 2 clear sky direct radiation intensity, W / m 2 clear sky diffuse radiation intensity, W / m 2 extraterrestrial radiation intensity, W / m 2 constant optical air mass at sea level number of observations day number relative humidity altitude correction factor monthly average daily sunshine hours, hour monthly average m a x i m u m possible daily sunshine hours, hour temperature, °C m a x i m u m temperature, "C
a, b, c, d, e, f
(7, G~ Go /] /to /7~ /1de /~D /Tt~ Hm /to /-/p h 1¢ 10¢ ld¢ Io k m N n R r S So T Tm
70
Data Bank Table 4. e~,~ and ~
of clear sky with cloud effects models
Model
T~a Ta~ T~ Twa Tw~ War 6 gmbe
e~s Pa Pc pg Zb fb zd fd f ~b ~o~
Station Abu Dhabi
Bahrain
Doha
Kuwait
Riyadh
1
e~ emb~
0.062 --0.060
0.034 --0.008
0.042 --0.030
0.026 +0.023
0.040 --0.031
2
e,,~ em~
0.087 --0.085
0.050 --0.034
0.058 --0.054
0.020 --0.006
0.061 -0.053
3
er~ ~mbe
0.044 --0.071
0.042 --0.020
0.042 --0.039
0.019 +0.010
0.031 --0.021
4
e~ e~
0.035 --0.030
0.034 +0.022
0.056 --0.001
0.057 0.052
0.039 --0.024
5
er~ ~mbe
0.056 --0.053
0.025 --0.003
0.054 --0.023
0.033 +0.025
0.056 --0.047
6
er,,~ /~mbe
0.043 --0.040
0.028 0.011
0.049 --0.009
0.044 +0.040
0.036 --0.025
transmission function for aerosol absorption transmission function for aerosol scattering transmission function for Rayleigh scattering transmission function for water vapour absorption transmission function for water vapour scattering water vapour content per unit volume azimuth angle, degrees solar declination angle, degrees mean bias error root mean square error cloudiness sky albedo cloud albedo ground albedo instantaneous atmospheric transmittance for beam radiation average daily beam transmittance clear sky diffuse transmisttance average daily diffuse transmittance monthly average daily transmittance latitude angle, degrees sunset hour angle, degrees REFERENCES
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71
Data Bank
22.
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32.
33.
for horizontal and inclined surfaces. Solar Energy 23, 301-307 (1979). H. P. Grag and S. N. Grag, Prediction of global solar radiation from bright sunshine hours and other meteorological parameters, Solar-India 1982, Proc. National Solar Energy Convention, 1.004-1.007. Allied Publishers, New Delhi (1982). S. J. Reddy, An empirical method for the estimation of net radiation intensity. Solar Energy 13, 291 995 (1971). K. K. Gopinathan, A new model for estimating total solar radiation. Solar Wind Technol. 5, 107 109 (1988). J. N. Black, The distribution of solar radiation over the earth's surface. Arch. Met. Geoph. Biokl. 7, 165-189 (1956). J. A. Sabbagh, A. A. M. Sayigh and E. M. A. El Salam, Estimation of the total radiation from meteorological data. Solar Energy 19, 307-311 (1977). D. V. Hoyt, A model for the calculation of solar global insolation. Solar Enerqy 21, 27 35 (1978). C. C. Y. Ma and M. lqbal, Statistical comparison of solar radiation correlations. Solar Energy 33, 143 148 (1984). A. S. Sambo, Empirical models for the correlation of global solar radiation with meteorological data for northern Nigeria. Solar Wind Technol. 3, 89 93 (1986). P. W. Suckling and J. E. Hay, Modeling direct, diffuse and total solar radiation for cloudless days. Atmosphere 14, 298 308 (1976). A. D. Watt, On the Nature and Distribution of Solar Radiation, HCP/T2552-01, U.S. Department of Energy, Washington, D.C., U.S.G.P.O. (1978). 3. E. Sherry and C. G. Justus, A simple hourly clear-sky solar radiation model based on meteorological parameters. Solar Energy 30, 425~431 (1983). M. Santamouris and B. D. Katsoulis, Solar radiation
over the northwest part of Greece. Solar Wind Technol.
6, 79-84 (1989). 34. D. T. Brine and M. lqbal, Diffuse and global solar spectral irradiance under cloudless skies. Solar Energy 30, 447-453 (1983). 35. R. E. Bird and R. L. Hulstrom, Terrestrial solar spectral data sets. Solar Energy 30, 563 573 (1983). 36. H. G. Hottle, A simple model for estimating the transmittance of direct solar radiation through clear atmosphere. Solar Eneryy 18, 129-134 (1976). 37. ASHRAE Technical Committee, ASHRAE Handbook Fundamentals. American Society of Heating, Refrigerating and Air Conditioning Engineers, New York (1972). 38. G. L. Powell, The ASHRAE clear sky model--an evaluation. ASHRAE J. 32-34 (1982). 39. S. Barbaro, G. Cannisttraro, C. Giaconia and A. Orioli, The ASHRAE clear sky model: An evaluation in the Mediterranean Zone. Solar Wind Technol. 5, 111 116 (1988). 40. J. L. Monteith, Attenuation of solar radiation: a climatological study. Q. J. Met. Soc. 88, 508 521 (1962). 41. S. M. Abughress, Monthly average daily insolation for horizontal and inclined surfaces. Solar Wind Technol. 2, 119 130 (1985). 42. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4, 1 19 (1960). 43. S. Rangarajan, M. S. Swaminathan and A. Mani, Computation of solar radiation from observations of cloud cover. Solar Energy 32, 553 556 (1984). 44. S. M. Jeter and C. A. Balaras, Development of improved solar radiation models for predicting beam transmittance. Solar Energy 44, 149 156 (1990).
096~1481/92 $5.00+.00 ,! 1992 Pergamon Press Lid
DATA BANK Assessment of solar radiation models for the Gulf Arabian countries N . A L MAHDI, N . S. A L BAHARNA a n d F . F . ZAKI College of Engineering, University of Bahrain, P.O. Box 32038, Isa Town, Bahrain
(Received 12 December 1990 ; accepted 13 September 1991 ) Abstract--A statistical assessment of the accuracy of 12 solar radiation models for five meteorological stations in the Gulf Arabian states namely Abu Dhabi, Bahrain, Doha, Kuwait and Riyadb, is presented. The first six models are of the regression type in which the measured global radiation is correlated with the sunshine hours and other meteorological parameters. The other six models are based on the calculation of clear sky radiation and the effects of cloud amount and sky transmittance. The models are assessed in terms of the root mean square error and the mean bias error. The obtained results indicate some inconsistency of published data for Abu Dhabi, Bahrain and Doha stations. However, using Kuwait and Riyadh stations as a basis for the evaluation, two models are recommended for the estimation of monthly average daily and hourly global radiation for the Gulf states.
low latitude of the region along with a climate, characterized by a very small amount of either rain or cloudy skies, permits radiation levels to reach 7 kW/m 2 during the summer. The weather conditions are very hot and semi-arid during summer days, becoming warm and humid at night. During winter, daytime conditions fall within the comfort zone, while the night times are cool and humid. Wind speeds generally average about 5 m/s and prevail mainly from the north. Unfortunately, the meteorological measurements available for the region [1-11] are limited to global radiation measurements, while data of diffuse radiation are very rare and refer to limited periods. Moreover, the annual global radiation and sunshine hours data available from the various stations appear to be inconsistent, especially since the climatic conditions are very similar and the stations are located within relatively close proximity from each other, as can be seen from Table 1. Published measurements [1] indicate that the annual global radiation for Abu Dhabi is lower than that of other Gulf states, which conflicts with our expectations since the Abu Dhabi station is located at a latitude angle lower than that of the other stations. The variation of the measured monthly mean daily global radiation and the monthly mean daily bright sunshine hours also exhibit inconsistencies, as can be seen from Figs 1 and 2, respectively. In addition to instrumental problems such as misalignment, malfunctions and dust on radiation sensors, the lack of consistency between the data may be attributed to the relatively short periods of measurements, and differences in the extent and quality of data, as well as the procedures used in the evaluation of spatial and temporal averages.
INTRODUCTION Accurate quantitative data of the variation of solar radiation reaching the earth surface, and the relevant meteorological parameters are essential requirements for conducting a wide range of scientific studies. Typical examples are found in hydrological studies when calculating soil moisture deficits, investigation of biological process, climatology, thermal design of environmental control of buildings and quantitative evaluation of ecophysiological systems for the determination of irrigation water needs and the potential yield of crops. The design and estimation of performance of solar heating, cooling and distillation systems also require detailed knowledge of solar radiation data. The variation of solar radiation over a given area for a specified time period has conventionally been obtained through the integration of continuous records of radiation measurements. The gathering of such data requires properly trained personnel and appropriate instruments that are continuously maintained and regularly calibrated, in addition to a reliable supply of accessories and materials. The most frequently needed data are the long term average daily global radiation and its direct and diffuse components on a horizontal surface. The objective of the present work is the determination of the radiation characteristics for the Gulf Arabian countries. To achieve this, solar radiation data available from five meteorological stations, namely Abu Dhabi, Bahrain, Doha, Kuwait and Riyadh, are used to derive correlations for estimating the mean global radiation. The correlations are made with the aid of a number of commonly used solar radiation models. The first group of models correlate the monthly average daily global radiation only to sunshine hours while others introduce the effect of latitude, altitude, maximum temperature and humidity in addition to the sunshine duration. The second group of models predict the clear sky radiation and the cloud cover effects in order to determine the global radiation. The Arabian Gulf is located just north of the equator. The
GLOBAL RADIATION CORRELATIONS The various available methods for the estimation of global radiation can be categorized into two groups. The first group is totally empirical whereby meteorological data are used in conjunction with regression analysis. The second group involves the prediction of clear sky radiation and the 65
66
Data Bank Table 1. Geographical data and annual sunshine and global radiation for the Gulf stations Annual sunshine Station Abu Dhabi Bahrain Doha Kuwait Riyadh
Latitude ()
Longitude (°)
Altitude (m)
g (h)
g/go
24.43 26.16 25.90 29.22 24.70
+54.45 +50.90 +51.55 +47.97 +46.72
15 2 5 45 594
3465.799 3359.399 3501.198 3204.058 3376.910
0.789 0.765 0.798 0.730 0.769
0.7
~o
24 × 3600 7~
Annual radiation H (MJ/m 2) H / H o 6671.389 7125.488 6895.259 7083.529 7204.539
0.552 0.597 0.576 0.608 0.597
f
Go l c o s (q~) cos (6) cos (~o~) L + ~2nOSssin (~b) sin (6) 1
0.6
(2)
,2 where the instantaneous extra-terrestrial radiation Go for the nth day of the year is:
II 0.~
I F
0.4
I M
I A
I M
Go = G~[1 +0.033 cos (2~n/365.24)].
<> Rtyadh
x Bahroin [] Kuwai¢
(3)
The monthly average maximum possible number of daily sunshine hours is evaluated from the daily values as :
I ] J d Month
I A
I S
I 0
I N
I D
n2
go = [1/(n2-n,)])~ So
(4)
nI
Fig. 1. Monthly/tm//q o for the five Gulf stations.
where So = 2/15~0s
(5)
co~ = cos -j [ - t a n (~b) tan (6)].
(6)
and
1.0
The solar declination angle 6 is given by [16] :
0.91
(7)
sin 6 = 0.3979 sin (7)
,,g
0.8
where )' = 7o +0.007133 sin (7o) +0.032680 cos (Yo)
0.7
-0.000318 sin (270)+0.000145 cos (2~o) 0.6
x Bohrain [] Kuwoit
O Riyadh
(8)
and 7o = 2n(n + 284)/365.24.
0.5 J
I F
I M
I A
I M
I d
I d
I A
I S
I 0
I N
I D
Month
Fig. 2. Monthly g/so for the five Gulf stations. inclusion of a suitable model for the estimation of cloud cover effects. Regression models A variety of regression correlations for the estimation of global solar radiation exist in the literature. The most commonly used model is the linear regression equation of Angstrom [12] as developed by Page [13], Prescott [14] and others. The equation relates the monthly average daily global radiation/4 to the monthly average daily number of bright sunshine hours g by the linear function :
1:1//7o = a+b(g/go),
(l)
where a and b are the regression constants. The monthly average daily extra-terrestrial radiation/4o is given by [15] :
(9)
Many researchers have used Page model and presented values of the constants a and b which vary with geographic location and climatic conditions. Rietveld [17] examined several published values of a and b and produced the empirical formula : /4//4o = 0.18+0.62(S/So)
(10)
which can reasonably be applied anywhere in the world. On the other hand Bahel et al. [18] developed the worldwide correlation : FlltTlo = O.16+O.87(glgo)--O.16(g/go)Z +O.34(,~/go) 3.
(ll) The correlation based on bright sunshine and global radiation data of 48 stations around the world, with varied meteorological conditions and a wide distribution of geographical locations. Other attempts made to improve the radiation sunshine correlation introduce geographic parameters, such as latitude angle and altitude ; and meteorological factors, such as
67
Data Bank atmospheric temperature, humidity, optical properties of the cloud cover, ground reflectivity and average air mass. Glover and McCulloch [19] included the latitude effects and produced the equation :
any time between sunrise and sunset is given by [15] : 10 = G~[1 +0.033 cos (360n/365)] sin ~
(22)
where
(12)
sin ~ = sin (~) sin (6)+cos (4~) cos (6) cos (~o). (23)
which can be applied for latitude angles of up to _+6ft. Sayigh [20] modified eq. (I) to include the altitude h of the station and presented the model :
The potential solar radiation is depleted as it passes through the atmosphere owing to reflection, scattering and absorption by dust, gas molecules, ozone and water vapour. The clear sky direct normal solar radiation received al the earth surface can be expressed [30] as :
/t//~o = 0.29 cos (q~)+0.52(~¢/So)
(13)
171/171o = a + b ( S / S o ) + c h
where c is also a constant. Hay [21] developed a generalized procedure, that incorporates the monthly average ground albedo pg, cloudless sky albedo p~ and cloud albedo p~, in the form : /q//lo - [0.1572 + 0.5566(~/S~)]/ {1-pg[p,(~/~)+p~(l-
g/eoj)]}.
(14)
Hay used p~,- 0.25 and p ~ - 0.6. The modified monthly average maximum possible number o f daily sunshine hours gj excludes solar zenith angles less than 5~ because the Campbell Stokes sunshine recorder does not function when the solar latitude is less than 5 . Grag and Grag [22] found that a least square correlation of data from 14 pyranometer stations in India gives the following relationship : H//4o = 0.414+0.4(S/So)-0.0055W~,
(15)
where W~, is the atmospheric water vapour content per unit volume of air which is computed by using: W~,t = R ( 4 . 7 9 2 3 + O . 3 6 4 7 T + O . O O 5 5 T 2 + O . O O O 3 T ~ ' ) .
(16)
Sayigh [20] developed the formula: (17)
171 = N K [ d p ( S - R / 1 5 ) - l/Tm],
where N = 1.7- 0.458n(b/180,
(18)
K = 100[).So + ~b~jcos (qS)],
(19)
and 2 = 0.2/(1 +0.1~b).
(20)
O,,j is a seasonal factor in which i = 1, 2, 3, respectively, for inland stations, coastal stations and hilly stations ; and j = 1, 2 , . . . , 12 represents the months of the year. A similar formula has also been developed by Reddy [23]. More recently, Gopinathan [24] related the global radiation to sunshine duration, maximum temperature, humidity, latitude and altitude in an equation of the form : I t / H o = a + b cos ( q b ) + c h + d ( S / , ~ o ) + e T m
+fR
(21)
where a, b, c, d, e and f are regression constants. Various other global radiation regression models have also been developed by Black [25], Sabbagh et al. [26], Hoyt [27] and many other researchers. Some of the aforementioned regression models have been assessed by Ma and Iqbal [28] and by Sambo [29]. Ma and lqbal concluded that Rietveld's and Page's equations generally yield the best results and recommended their use for estimating monthly average daily global radiation. However, Sambo recommended the use of a Gopinathan-like model.
(24)
lL, c = I , , T w . T ~ T w ~ T ~ T ~
where the terms T are transmission functions for water vapour absorption, aerosol or dust absorption, water vapour scattering, Rayleigh scattering and aerosol or dust scattering. These transmission parameters vary considerably in the magnitude of their effects on the depletion of solar radiation. According to the estimates of Watt [31] the ozone layer, upper-layer aerosol, dry air, water vapour and lower-layer aerosol deplete the solar radiation by 0.5-3%, 1.9 11%, 11 13%, 3.5-14% and 0.1 26%, respectively. Many authors [32 35] have developed models for the determination of various transmission parameters. The models normally require detailed meteorological measurements such as water precipitation, mixing height and visibility. Such measurements are not available, in detail, for the Gulf countries. A convenient model for the estimation of the combined effects of the transmission terms have been developed by Hottle [36]. The method takes into account the zenith angle and latitude for a standard atmosphere and for four climate types. The instantaneous atmospheric transmittance for beam radiation is given by : (25)
r~, = a , , + a l e k~i,=
The constants ao, a~ and k for the standard atmosphere with 23 km visibility are defined as : a,, = r°[0.4237-0.00821(6-h)']
(26)
a i = r ~[0.5055 + 0.00595 (6.5 - h) 2]
(27)
k = r~[0.2711 +0.01858(2.5
h) 2]
rd = 0.2710-0.2939rb.
In the absence of atmosphere, the instantaneous flux density of potential solar radiation on a horizontal surface at
(29)
Where rd is the ratio of diffuse to extra-terrestrial radiation. The total clear sky radiation can then be evaluated using the relation : lc - lv, c + Idc = Io (rb + rd).
(30)
Another convenient method is the A S H R A E model [37] as developed by Powell [38]. In this model the total radiation incident upon a horizontal surface and clear sky conditions is evaluated from : 1,. = A(sin ~ + C ) e Bmr
(31)
where the optical air mass at sea level m and the altitude correction factor r are defined as : M = 35/[1224 sin 2 (~) + I] 0.5
Clear and cloudy sky models
(28)
where h is the altitude in km ; and r o, r~ and rk are correction factors that are applied to allow changes in climate types. The clear sky diffuse transmission can be estimated using the empirical relation developed by Liu and Jordan [42]:
r = [1 -- (h/44.308)] 5 257.
(32) (33)
The values of apparent direct normal intensity at zero optical
68
Data Bank
air mass A, the atmospheric extinction coefficient B and the diffuse radiation factor C for the 21st of each m o n t h are given in A S H R A E [37]. Expressions for A, B and C for any day o f the year are given by the following equation [34] : A = 1160+ 77.2 cos [2~(n-- 1)/365]
(34)
B = 0.17028-3.45545 1 0 - a n - 1.51758 I 0 - 6 n 2 +1.8162 10 7 n 3 - 1 . 3 1 7 8 4 10 9n4 +3.34233 10-12nS--2.861610-15n 6
(35)
C = 0.05363+0.000511n-2.1351 10 Sn2 +3.70022 1 0 - 7 n 3 - 2 . 2 7 1 3 10-gn 4 +5.7539 10-12nS-5.19886 10-15n 6.
(36)
The clear sky radiation is used for the calculations of radiation under normal sky conditions by the inclusion of a suitable cloud a m o u n t model [40, 41]. The monthly average daily direct radiation component, /to, under normal sky conditions can be expressed as /~D = /1pc(1 -- Ca)
(37)
where /qOC is the clear-sky monthly average daily direct radiation component and Ca is the total cloud amount. The monthly average daily diffuse radiation/ld is given by :
l~d = lq,lc(1-Ca) + tZl~fC~ + lrIp, p~p~.
(38)
The total global r a d i a t i o n / ~ can be calculated using eqs (37) and (38) as : /7 = / 4 ~ [ ( 1 - C a ) + f C ~ ] / ( I - p g p c C , ) ,
(39)
where/f~,/tr)c and/~d~ are calculated by integrating eq. (30) or (31) over the month. The term ? is the monthly mean average daily sky transmittance. The ground albedo pg is assumed to be 0.25 and the atmospheric albedo p~ is given by the relation [42] : Pc = 0.118 +0.511 - f ( S / S o ) ] ,
(40)
where n~ and n,_ are the first and last days of the month, respectively. An alternative method to determine f has been developed during the course of this work on the basis of the work of Jeter and Balaras [44] and Bahel [18]. The measurement of beam transmittance rb, presented by Jeter and Balaras [44], as a function of the clearness index k-r where used to find a correlation between % and kT. The Bahel [18] correlation was then used to derive an expression for the average daily beam transmittance fb as a function of S/go. That resulting expression is : i n = 0.0243 -0.4872(~¢/So) + 2 ( g g o ) 2 - 0.8537(,~/~o) 3. (47) The monthly average daily transmittance is then calculated by using the relation : f =
f(S/So) = ~¢/~o for
0 ~< (~¢/So) ~< 0.63,
(41)
and
f(~/~¢o) = 0.001427 + 1.2932(~/~o) - 0.4519(S/So) 2 for
0.63 < (~¢/~¢o) < 1.
(43)
or alternatively the cloud a m o u n t m a y be approximated by [42]:
Ca = 1 --f(8/8o).
(44)
A cubic correlation has been recently formulated by Rangarajan et al. [43] as :
~¢/So = l--alC~--azC~--a3C 3
(48)
METHODS OF ASSESSMENT The accuracy of the correlations described in the previous section have been assessed using two statistical tests, namely, the root mean square error e ~ and the mean bias error emb~The root mean square error is defined as : ~rms = {['~(/tm//~o -- Flp/lto)2]/N} i/2
(49)
where/qp is the predicted value of the monthly average daily solar radiation and N is the number of observations. The root mean square error test is useful in assessing the shortterm performance of the correlations. The mean bias error which is defined as :
[~(l~m/Flo-I~p/lYlo)]/g
(50)
is helpful in the assessment of the long-term performance. The values of the errors are a measure of the accuracy of the model used. For example a positive value of embo provides information on an under-estimation and a negative value indicates an overestimation.
(42)
Different expressions for the total cloud a m o u n t have been suggested by several authors. H a y [21] and Hoyt [27] proposed the linear relation :
6". = 1 - S / S o ,
n~--nl)
where fd is evaluated by using a similar relation to eq. (29).
gmbe =
where
Z~+?d
(45)
where the coefficients a~, a2 and a 3 are location dependent. The monthly average daily sky transmittance can be evaluated by adding eqs (25) and (29) and finding their time weighted average over each day. The daily averages are then used to find the monthly average daily values. That is :
RESULTS The accuracy of the global radiation regression models given by eqs (1), (10) (12), (14) and (15) have been evaluated using the root mean square and mean bias error tests. The results of the statistical comparison of these correlations are presented in Table 2. The results show that all the models used, except Grag's model, yield reasonably accurate results for both Kuwait and Riyadh stations. This is to be expected as Kuwait and Riyadh measurements are the best taken and documented ones for the G u l f region. Page's equation gives the least e~msand em~ errors for all stations with the m a x i m u m errors occurring for A b u Dhabi and Bahrain. Results for A b u Dhabi exhibit a consistent high e,m~ and negative em~ indicating that all models overestimate the value o f solar radiation or that the measurement values are low. The next best model is Bahel's correlation. Although this model gives relatively higher overall errors than Page's model, nevertheless it gives a clear indication of the quality of the
Data Bank
69
Table 2. ~rmsand ~mb*of regression models Station
Model A b u Dhabi
Bahrain
Doha
Kuwait
Riyadh
Page eq. (I)
£rms embe
0.042 -- 0.037
0.038 + 0.008
0.019 -- 0.009
0.031 + 0.026
0.036 + 0.013
Rietveld eq. (10)
trms ~,oi~
0.123 - 0.120
0.075 - 0.062
0.111 0.092
0.036 - 0.026
0.063 - 0.060
Bahel eq. (I 1)
c,,.... Crabe
0.088 --0.085
0.050 --0.030
0.080 --0.059
0.021 +0.004
0.032 --0.017
Glover eq. (12)
e.... e,,he
0.127 -- 0.125
0.077 -- 0.066
0.108 -- 0.094
0.033 -- 0.026
0.068 -- 0.066
Hay eq. (14)
r ..... e,mbe
0.115 -0.112
0.070 -0.055
0.103 -0.085
0.033 -0.020
0.056 -0.052
Grag eq. (15)
e,..... ~°,h:
0.121 -0.117
0.079 -0.064
0.105 -0.101
0.058 -0.053
0.112 -0.111
available measurements for stations under consideration since it was derived from world wide data. The model yields excellent correlation for Bahrain, Kuwait and Riyadh but over-estimates the solar radiation for the other two stations. Since Kuwait and Riyadh data are most reliable a m o n g the data available we recommend the use of this model for all the regions as it gives the best correlation for these two stations. For the assessment of the clear sky with cloud effects, six models have been introduced using the combination shown in Table 3 for the evaluation o f lc, Ca and f. The results for these models are given in Table 4. Clearly, model number 3 gives the best estimates for both Kuwait and Riyadh and shows reasonable values of errors for the other stations. This model is recommended for estimating the monthly average hourly or daily global radiation. CONCLUSIONS Twelve regression and clear sky with cloud effect solar radiation models have been compared statistically in terms of root mean square error and mean bias error for five stations in the G u l f Arabian countries. The results indicate that the measurements of Kuwait and Riyadh stations are reliable and consistent with available world wide correlation models. The Babel et al. [18] model was found to give the best correlation and is recommended for use in the estimation of monthly average daily global solar radiation. For the
Table 3. Equations used for clear sky with cloud effects models
Model
Equation for Ic
Equation for C,
Equation for f
I
(3 I)
(44)
(46)
2 3 4 5 6
(31 ) (31 ) (30) (30) (30)
(43) (44) (44) (43) (44)
(46) (48) (46) (46) (48)
estimation of the monthly average hourly global radiation the use of a clear sky model based on A S H R A E [37] is recommended with inclusion of the cloud effect by the introduction of cloud a m o u n t and average sky transmittance using eqs (44) and (48).
NOMENCLATURE constants cloud a m o u n t s solar constant, 1367 W / m 2 instantaneous extraterrestrial radiation, W / m 2 monthly average daily global radiation, J/m 2 monthly average daily clear sky radiation, J/m 2 monthly average daily diffuse radiation, J/m 2 monthly average daily diffuse clear sky radiation, J/nl z monthly average daily direct radiation, J/m 2 monthly average daily direct clear sky radiation, J/m e measured monthly average daily global radiation, J/m 2 monthly average daily extraterrestrial radiation, J/m 2 predicted monthly average daily global radiation, J/m 2 altitude, km clear sky radiation intensity, W / m 2 clear sky direct radiation intensity, W / m 2 clear sky diffuse radiation intensity, W / m 2 extraterrestrial radiation intensity, W / m 2 constant optical air mass at sea level number of observations day number relative humidity altitude correction factor monthly average daily sunshine hours, hour monthly average m a x i m u m possible daily sunshine hours, hour temperature, °C m a x i m u m temperature, "C
a, b, c, d, e, f
(7, G~ Go /] /to /7~ /1de /~D /Tt~ Hm /to /-/p h 1¢ 10¢ ld¢ Io k m N n R r S So T Tm
70
Data Bank Table 4. e~,~ and ~
of clear sky with cloud effects models
Model
T~a Ta~ T~ Twa Tw~ War 6 gmbe
e~s Pa Pc pg Zb fb zd fd f ~b ~o~
Station Abu Dhabi
Bahrain
Doha
Kuwait
Riyadh
1
e~ emb~
0.062 --0.060
0.034 --0.008
0.042 --0.030
0.026 +0.023
0.040 --0.031
2
e,,~ em~
0.087 --0.085
0.050 --0.034
0.058 --0.054
0.020 --0.006
0.061 -0.053
3
er~ ~mbe
0.044 --0.071
0.042 --0.020
0.042 --0.039
0.019 +0.010
0.031 --0.021
4
e~ e~
0.035 --0.030
0.034 +0.022
0.056 --0.001
0.057 0.052
0.039 --0.024
5
er~ ~mbe
0.056 --0.053
0.025 --0.003
0.054 --0.023
0.033 +0.025
0.056 --0.047
6
er,,~ /~mbe
0.043 --0.040
0.028 0.011
0.049 --0.009
0.044 +0.040
0.036 --0.025
transmission function for aerosol absorption transmission function for aerosol scattering transmission function for Rayleigh scattering transmission function for water vapour absorption transmission function for water vapour scattering water vapour content per unit volume azimuth angle, degrees solar declination angle, degrees mean bias error root mean square error cloudiness sky albedo cloud albedo ground albedo instantaneous atmospheric transmittance for beam radiation average daily beam transmittance clear sky diffuse transmisttance average daily diffuse transmittance monthly average daily transmittance latitude angle, degrees sunset hour angle, degrees REFERENCES
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