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Diffuse solar radiation over Shambat, Khartoum North
Rem,wahleEneryy. Vol. 4. No. 2. pp. 227 233. Iq94
~
Copyright~! 1994ElsevierScienceLid Printed in Great Britain. All rightsreserved 0960 1481/94$6.00+0.00
Pergamon
DATA BANK Diffuse solar radiation over Shambat, Khartoum North ABDEEN MUSTAFA OMER National Centre for Research, Energy Research Institute, P.O. Box 4032, Khartoum, Sudan
(Receil:ed 6 June 1993 : accepted 19 Ju O' 1993) A b s t r a c t I T h e r e are many locations where no measured data on diffuse radiation are available and have to be estimated from empirical correlations. Here, using correlations for predicting monthly mean daily diffuse radiation on a horizontal surface for locations in Sudan, from the experimental data of Shambat station, the applicability of the developed equations were tested by estimating monthly mean daily diffuse radiation for the location. Measurements of diffuse radiation on a horizontal surface at Shambat tk~r 10 years were compared with predictions made by several independent methods. In the first type of correlation, formulae were used to correlate diffuse solar irradiance to the clearness index. Regression coefficients are obtained and used for prediction of diffuse solar irradiance. The predicted values were consistent with measured values (10.13%). In the second type of correlation, sunshine duration and minimum air mass were used to derive an empirical correlation for diffuse radiation. The predicted values compared well with measured values (16.31%). In the third type of correlation, a correlation connecting monthly mean daily diffuse radiation together with the clearness index and per cent possible sunshine is found to be 9.66%, the most accurate for the Shambat location.
I. I N T R O D U C T I O N
ments. Therefore, this work is an effort to lind reliable methods for prediction of solar radiation data with minimum possible measurements in Shambat, Khartoum North. The present study is to establish correlations for the diffuse component for the city of Shambat, Khartoum North, where daily global and diffuse solar radiation over Shambat (latitude 15 40'N, longitude 32 32'E, and altitude 380 m), Khartoum North, Sudan, has been measured by Robitzsch and Kipp instruments since 5 May 1957.
One of the most important requirements in the design of any solar energy conversion system is the information oll the intensity of solar radiation and its components at a given location. There are two main components of radiation reaching the ground : direct radiation and diffuse radiation. For any solar energy applications, global solar radiation level data arc needed. Knowledge of the diffuse radiation on a horizontal surface is of value in the design of various solar utilization devices. However, the diffuse radiation on a horizontal surface has many applications, particularly in finding global radiation on tilted surfaces, illumination design inside a building, and so on. Whereas considerable information exists on global solar radiation on horizontal surfaces, experimental data on diffuse radiation are very rare and often restricted to very short periods. Diffuse radiation is still measured at a few locations. Thus few data are available for diffuse radiation. One has thus to depend on the theoretical estimations of diffuse radiation for such locations where no measured data are available. Several empirical correlations have already been developed to predict diffuse radiation from available meteorological data. In developing countries where the energy problem is acute, solar radiation data is still scarce. The solar radiation measurements are very important in order to establish a complete solar map for countries such as Sudan. The knowledge of the available solar radiation and its components is valuable for design and assessment of solar energy conversion systems. The solar equipment needed for these purposes is expensive and requires maintenance as well as frequent calibrations. Moreover, it is almost impossible to scan day-by-day solar radiation data. This is due to inaccurate readings which result from the reader or the instru-
2. THEORY
2.1. Prediction fi~rmulae There are three types of correlations for estimating horizontal diffuse radiation. The first type expresses monthly average daily diffuse radiation, Hd/Hm, as a function of the monthly average daily clearness index, K~ - Hm/H~, lid, H .... and/4o are monthly average daily diffuse, global, and extraterrestrial radiation on a horizontal surface. The second type expresses the fraction Ho/Hm or Ha/H~ as a function of the monthly averaged daily values of the bright sunshine hours, S, and the maximum possible sunshine hours, 5'~. The ratio S/S,~ is often called the per cent possible sunshine. The third type of correlation expresses Ha/Hm as a function of both clearness index and per cent possible sunshine. Several correlations have thus been developed for computing diffuse radiation and the coefficients of all these correlations are found to be different and site dependent. Examples of some well-accepted correlations that have universal applicability are given below. 2.2. Prediction of d(ffuse solar irradiance 2.2.1. Type I correlations. Page [1] developed correlations between daily total and diffuse radiation for 10 widely spread 227
228
Data Bank
sites in the 40°N to 40°S latitude and obtained the following relationship :
n d / n m = 1.00--1.13 • KT,
(1)
where H d = the monthly average of the daily diffuse solar irradiance; Hm = the monthly average of the daily total global irradiance; KT = the ratio of cloudiness index or transmission coefficient
Kv = Hm/H~.
(2)
Liu and Jordan [2] developed a statistically based correlation from results obtained from one station which was developed by Klein [3] :
Hd/Hm = 1 . 3 9 0 - 4 . 0 2 7 . K v + 5 . 5 3 1 *Kv2-3.108*K-~ 0.3 < KT < 0.7.
(3)
Collares-Pereira and Rabl [4], used data obtained from five locations in the United States to derive the following season-dependent correlation :
H~/Hm = 0.775+0.347 * (~/180) * ( W ~ - 9 0 C') - [ 0 . 5 0 5 + 0 . 2 6 1 • (7r/180) • (W~-90°)] • cos [2(Kv-0.9)],
(4)
where Ws = the sunset hour angle in degrees ; His ~ 90 ° during February to April and August to October; His ~ I00 ~' during M a y to July ; Ws ~ 80 ° during November to January ; the quantity 2(KT--0.9) is in radians. 2.2.2. Type H correlations. Iqbal [5] used data obtained from three locations in C a n a d a to propose the correlation,
Hd/Ho = 0 . 7 9 1 - 0 . 6 3 5 • (S/So),
(5)
where : S
= The monthly averaged daily values of the bright sunshine hours. So = The m a x i m u m possible sunshine hours. S/So = The ratio is often called the per cent possible sunshine. Iqbal [5] has developed the following correlation for Montreal, Canada, latitude 45°30'N,
Hd/Ho = 0.163+0.478 • (S/S0)-0.655 • (S/So)".
(6)
The purpose of the present study is to develop empirical correlations for estimating the monthly mean of the daily diffuse radiation on a horizontal surface, from the measured data of global radiation and sunshine durations, for locations in Sudan. Available long-term data on Hd, Hm and S, are used in a regression analysis to obtain the regression coefficients of different types of correlations. One equation belonging to each type of correlation is used in the analysis. The above correlations are used for the prediction of diffuse solar irradiance. The direct beam component lh can then be deduced from the relation :
Hm = lid + lb * sin (Sn),
(9)
where the product lb sin (Sn) is the average horizontal beam component ; S, = the noon altitude of the sun of the 15th of the month. Sn = 9 0 - ( ~ b - 3 ) ,
(10)
where q~ = the latitude of the place in degrees; 6 = the solar declination in degrees. The direct beam component is useful for the various types of solar concentrating systems.
3. METHODOLOGIES FOR FINDING Hd Radiation data are the best source of information for estimating average incident radiation. It is possible, however, to use empirical relationships to estimate radiation from hours of sunshine or cloudiness, if radiation data at the location in question are not available. Data on average hours of sunshine, or average per cent of possible sunshine hours, are available for many stations in Sudan, measured by a Campbell-Stokes instrument. Available experimental data on Hd, Hm and S reported by A n n a Mani [8] and L r f et al. [9] for Delhi, Madras, and Poona were used in a least squares method to find the regression coefficient of the following correlations. Proper computer programs were written to obtain the coefficients of the following equations
Hd/Hm = a0+al * Kv
(11)
Hd/Hm = b0+b, * (S/So)
(12)
Hd/Hm = c0+c, * Kv+c2 * (S/So),
(13)
Hay [6] considered the relationship between radiation before and after multiple reflections between the Earth and the cloud cover. He proposed the following location independent relationship between global (Hi) and diffuse (Ha) radiation emerging from the atmosphere before striking the ground,
where a0, a~, b0, b~, co, c~ and c2 are regression constants or 'empirical constants'. The monthly mean daily extra-terrestrial radiation H0 on horizontal surface at any time between sunrise and sunset is given by Duffle et al. [10] :
Hd/Hm = 0.9702+ 1.6688 • (H~/Ho)-21.303 • (Hm/Ho) 2
H0 = [2(23 * 3600 * l~/Tr) * (1 + 0.33 * cos ((360 * m)/365))]
+51.288 * (Hm/no)3+ 50.081 • (Hm/no) 4 + 17.5510 • (Hm/no) 5.
• [cos ~b * cos 3 * sin ( W0 + 2 * 7r * WJ360) (7)
2.2.3. Type III correlations. In addition to the above two types, a third type of correlation expressing H~/Hm as a function of both clearness index and the per cent possible sunshine together has been recently suggested by Gopinathan [7] for locations in southern Africa. A correlation of the form :
Hd/H m = 0 . 8 7 9 - 0 . 5 7 5 * (K-r)-0.323 * (S/So)
(8)
has been found suitable for stations in southern Africa. Gopinathan has included both KT and S/So in the estimated correlation to improve the accuracy of the estimated data by having two measured variables in the predicted formula.
• sin q~* sin 3],
(14)
where I~ - the solar constant = 1350 W/m2; 6 = the solar declination in degrees; q ~ - the latitude of the station in degrees ; m = the number of the day of the year (1 - 365). The values of $0 are computed from Cooper's formula [11] So = 2/15 * cos- i ( _ tan ~b* tan 6).
(15)
The sunrise, sunset hour angles W~ : W~=cos I(-tan~b,tanr),
(16)
where W~ = the sunrise/sunset hour angle, in degrees. The value of declination can be found from the equation of Cooper [11] :
229
Data Bank 6 = 23.45 * sin [360 * (284 +m)/365).
(17)
The declination angle is given a positive value for the summer time in the northern hemisphere and a negative value in the winter time. It has a seasonal variation. Equation (16) gives an accuracy of some _+ 1 . 4. T E S T I N G P R O C E D U R E Measurements of diffuse solar irradiance on a horizontal surface have been utilized to verify simple methods of prediction of solar radiation. Accurate measurements of solar radiation data require delicate and fairly expensive instruments. As such instruments are not available at all locations, it is often necessary to compute diffuse radiation from theoretical models. M a n y models have been developed to predict the diffuse sky solar radiation since the original work of Liu and Jordan [2]. A review of several of the older models and some of the newer models is contained in Duffle and Beckman [10]. The diffuse solar radiation models may be hourly, daily or monthly models, and are typically based on the measurement of global solar radiation (total solar radiation on horizontal surfaces). The other parameter typically used in the model is usually based on extra-terrestrial solar radiation on a horizontal surface. Recent models of this type include the Collares-Pereira and Rabl [4]. The statistic analysis of the results can be performed on three intervals of time : (1) daily ; (2) monthly ; and (3) yearly. The relative deviation (d,) between the computed values Hd~, Hd2, H~3 and the observed value Hdm can be established by
d, = (H~ - H~,,),/Hd, ,.
(18)
The relative percentage error (e) is : e = [(Hd ..... Ho,)/'Ho~,,j * 100
(19)
calculated for each month of the year, from all the three models. The mean percentage error (MPE) can be estimated by M P E : [ E ( ( ( H d , , - Hd,.,)/Ho,.,,) * 100)]/n
(20)
where n = the total number of observations ; Hd, = the ith calculated value : Hd = the ith measured value. The sign of the errors can be neglected in the summation and all the errors are added up while calculating the mean. The accuracy of the estimated data on diffuse radiation can also be tested by calculating the mean bias error (MBE) and root mean square error (RMSE). The M B E and R M S E are the fundamental measures of accuracy [12]. MBE represents the systematic error or bias and can be estimated by M BE = [ E ( H d , - Hd~,,)]/n,
(21)
MBE = E(d,)/n.
(22)
Dispersion of results is usually characterized by standard deviation (~), given by a = ( R M S E 2 - M B E 2 ) ~ "~
(25)
The lower the RMSE, the more accurate the estimate is. A positive MBE shows an overestimation while a negative M BE shows an underestimation. While calculating the M P E values, the sign of the errors are neglected and the percentage errors are added up to calculate the mean. 5. R E S U L T S AND D I S C U S S I O N
5.1. Introduction For m a n y solar designs and applications it is necessary to have knowledge of both the direct and diffuse components of the incident solar radiation. Whereas considerable information exists for global solar radiation on horizontal surfaces, only a few data are available for diffuse solar radiation. The purpose of the present work is to test the applicability of correlations for the estimations of diffuse radiation against measured values in one location: Shambat (latitude 15°40'N ; longitude 32' 32'E; altitude 380 m). The correlations for estimating horizontal diffuse radiation belong to three types. The first type expresses the monthly average daily diffuse radiation fraction, Ha/Hm, as a function of the monthly average daily clearness index, Ha/Ho. The second type expresses the fraction Hd/Hm or Hd/Ho as a function of the monthly averaged daily values of bright sunshine hours, S, and m a x i m u m possible sunshine hours, So. The third type of correlation expresses He/H mas a function of both clearness index and per cent possible sunshine. 5.2. Computation of diffuse solar irradiance The accuracy of the estimated data is tested by comparing the measured data available for the Shambat location with models proposed by Page, Liu and Jordan, Collares-Pereira and Rabl, Iqbal and Gopinathan. The performance of the correlations for Shambat are shown in Figs 1 3, respectively, and the results are shown in Tables 1 3. Diffuse radiation estimated from eqs (1, 3 8) along with the experimental are shown in these figures and tables. The following observations can be made from a study of the results presented in Tables 1 3, and Figs 1 3. Equation (1) estimates Hd fairly accurately. However, the errors in the estimated data are low (up to 8.46%). Accurate estimations are possible for most months, from eq. (3). However, the
or
RMSE is a measure of non-systematic error and can be computed from RMSE = {[E(H%-H%,)2]}'2,
(23) J
or
I
I
I
I
F
M
A
M
J
J
Months RMSE = (E(d32/n) t/2.
(24)
Fig. 1.
I
I
I
I
A
S
0
N
D
230
Data Bank 10
! 0_
I F
I M
I A
I M
I J
I J
I A
I S
I O
I N
I D
Months
Fig. 2.
I /
/+I÷
45~+
J
I I
•
• Hdm
t o
-.,~.
.*
5.3. Statistical analysis The accuracy of estimated data on diffuse radiation are evaluated by calculating MPE, MBE and RMSE values. The estimated values of MPE, MBE and RMSE for Shambat location from all models are included in Tables 1-3. As seen from the tables, there is a remarkable agreement between the experimental and estimated values of diffuse radiation for the location by using; for type I correlations, eq. (1) and eq. (3) ; type I1 correlations, eq. (5) ; and type Ill correlations, eq. (8). The lower the RMSE, the more accurate the estimation is. The positive MBE shows an overestimation whereas a negative MPE shows an underestimation. The RMSE values are very low, by using eqs (1, 3, 5 and 8), indicating good agreement. 5.4. Prediction of Ha by methods Regional values of the regression coefficient in all the three equations along with root squared (r), standard error of Y Est (p), and standard error coefficient (S.E.) are given as follows :
Hd/Hm = 1.03--1.17*KT
--+,~+
r = 0.93
+.d7 I F
I M
I A
I M
# = 0.02
I 1 J J Months
I A
I S
I O
I N
S.E. = 0.10
(26)
D
Hd/Hm = 0.69-0.53 * (S/So) r = 0.88
Fig. 3.
p - 0.02 S.E. = 0.06
(27)
Hd/Hm = 1.00-- 1.06 • Kv --0.05 * (S/So)
error in the estimated values reaches 13.78%. The accuracy of the estimated data from eqs (4 and 7) are found to be high compared to other equations. The accuracy of eq. (5) shows the errors up to 15.95%. In eq. (8) the errors reach 22.95% during December and are low during January to September.
r = 0.93 p = 0.02 S.E. = 0.46,0.21.
(28)
Table 1. Experimental and calculated values of diffuse radiation on a horizontal surface for Shambat, Khartoum North, using type I correlations, MJ/m 2 day Month
Hm
Ho
Ho avg.
Kx
Ha]
Ha2
H~3
Hdm
el (%)
e2 (%)
e3 (%)
Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.
20.30 23.10 24.50 25.90 24.70 23.60 23.00 22.80 22.90 21.90 21.00 20.10
34.10 35.50 36.11 35.10 33.15 31.81 32.23 33.99 35.50 35.50 34.33 33.54
28.94 31.99 35.39 37.63 38.26 38.16 38.07 37.66 36.03 32.92 29.57 27.88
0.70 0.72 0.69 0.69 0.65 0.62 0.60 0.61 0.64 0.67 0.71 0.72
4.21 4.25 5.33 5.76 6.68 7.11 7.30 7.20 6.45 5.44 4.15 3.73
4.34 4.53 5.43 5.83 6.40 6.60 6.68 6.60 6.11 5.34 4.34 3.96
6.04 6.81 7.63 8.28 8.53 8.57 8.52 8.30 7.79 6.99 6.21 5.82
3.91 4.46 5.52 5.50 6.16 7.63 7.75 6.97 6.43 5.90 4.10 3.95
-7.66 4.69 3.37 -4.66 -8.46 6.85 5.83 -3.33 -0.36 7.85 - 1.16 5.69
- 11.11 - 1.46 1.57 -6.00 -3.92 13.45 13.78 5.27 4.96 9.51 -5.75 -0.21
-54.54 -52.71 -38.24 -50.54 -38.48 -12.36 -9.90 - 19.10 -21.20 - 18.49 -51.48 -47.36
Yearly average
22.82
34.38
0.67
5.63
5.51
7.46
5.69
4.99 -0.06 1.14
6.42 -0.18 1.78
MPE MBE RMSE
34.53 0.26 6.49
Data Bank
.¢
t+q tt~ o
1
i I
©
2
+,
1771171-
4 Z
+..i
.g
1_ ,,.+
~ o ~ o o o o o o o o o o
. ,....
r~
g
t
t"-,l ¢"I :~ . . . .
~ .
.
P"+- <'% t,-) i~. ¢"+,I ~ID -. . . . . .
¢',"1
e,
t-.,,~ t-,,£
231
Equation (28) has the highest value of the correlation coefficient. Coefficients of eq. (26) are higher than in Page's equation. The regression coefficients of eq. (27) are nearly equal to lqbal's values ofeq. (5). The coefficients ofeq. (28) can be compared with the coefficients of the correlation developed by Gopinathan for Southern Africa locations. However, such variations in the coefficients are site dependent. The applicability of the proposed correlations predicting Hd is tested by estimating Hd values for Shambat location used in the analysis. Calculated values of Hd lbr Shambat along with reported experimental data, Ham, from Dr Posorski et al. [13] are presented in Table 4. The calculated values of Kr, S/& are also included in the table. Monthly mean daily diffuse radiation computed from eqs (26 28) arc shown as Ha~, H,I,~and Hdm, respectively. Hm, Hd,,, values shown in the table are in M J/m= day. The rcsults shown in Table 4 and Fig. 4, are for the Shambat location. Diffuse radiation estimated from eqs (26 28) along with experimental data are shown in this figurc. The following observations can be made from a study of the results. Equation (26) estimates Hj fairly accurately for most months of the year for the Shambat station. However, the errors in thc estimated data are high (up to 10.13%) during May. Accurate estimations are possible for most months, for the Shambat location, from eq. (27). However, the errors m the estimated values during April, May and October are high. The accuracy of the estimated data from eq. (28) is found to be low compared to other equations. The predicted values show an excellent agreement with the measured data for the Shambat location during all the months of the year. The error in the estimated values lie within about 10% and for most of the m o n t h s it is far less than 10%. A comparative study of MPE, MBE and RMSE from all the three models for Shambat supports the above agreement. The average percentage errors incurred in predicting Hd from eqs (2(>28) for Shambat, are 5.11%, 5.90% and 4.96% respectively As seen from Table 4, the M P E value from eq. (28) is the lowest. The percentage mean bias error from the three equations, for Shambat, are 0.02%, 0.04% and 0.01%. Equation (28), with per cent mean bias error of 0.01%, cancels out the two effects and improves the accuracy of the estimated data. Again, the per cent root mean square errors from the three equations are 1.12%, 1.40% and 1.12%, respectively, showing the lowest value of error for eq. (16). This supports the superiority ofeq. (28) over the others. In general, there is a good agreement between the experimental and estimated data of diffuse radiation from eq. (28). However, the accuracy of estimated data from eqs (26) and (27) is not ~ery high. The correlations (26) and (27) employ only one measured variable in the estimated correlation (either Kv or S/So). There are two measured variables in eq. (28), both Kv and S/So, and together these seem to have improved the accuracy of the estimated values.
+B .,.-.
6. C O N C L U S I O N Solar irradiance data for the Shambat station have been collected, investigated and analysed. Three types of correlations are tested against the experimental results. The following are concluded :
eq ¢.~
o=
>.
(1) In a location where no recorded data on diffuse radiation are available, there are different possible ways of predicting H~. Out of the three correlations
232
Data Bank Table 3. Experimental and calculated values of diffuse radiation on a horizontal surface for Shambat, K h a r t o u m North, using type III correlations, M J / m 2 day
Hm
Ho
KT
S
So
S/&
Hdm
Ha7
Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.
20.30 23.10 24.50 25.90 24.70 23,60 23.00 22.80 22.90 21.90 21.00 20.10
34.10 35.50 36.11 35.10 33.15 31.81 32.23 35.68 35.50 35.50 34.33 33.54
0.70 0.72 0.69 0.69 0.65 0.62 0.60 0.61 0.64 0.67 0.7l 0.72
10.20 10.50 10.20 10.60 10.10 9.30 8.70 8.90 9.20 10.20 10.60 10.50
11.16 11.49 11.89 12.36 12.73 12.93 12.85 12.53 12.08 11.64 11.25 ll.07
0.91 0.91 0.86 0.86 0.79 0.72 0.68 0.71 0.76 0.88 0.94 0.95
3.91 4.46 5.52 5.50 6.16 7.63 7.75 6.97 6.43 5.90 4.10 3.95
3.68 3.90 5.00 5.34 6.21 6.87 7.19 6.87 6.12 4.66 3.48 3.16
Yearly average
22.82
0.67
9.92
0.83
5.69
5.20
Month
MPE MBE RMSE
e7 (%) 5.88 12.62 9.48 2.98 -0.82 10.02 7.20 1.47 4.87 21.06 15.11 22.95
9.54 - 0.49 2.06
Table 4. Experimental and calculated values of diffuse radiation on a horizontal surface for Shambat, K h a r t o u m North, using three methods, M J / m 2 day Hm
Hdm
KT
S/So
Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.
20.30 23.10 24.50 25.90 24.70 23.60 23.00 22.80 22.90 21.90 21.00 20.10
3.91 4.46 5.52 5.50 6.16 7.63 7.75 6.97 6.43 5.90 4.10 3.95
0.70 0.72 0.69 0.69 0.65 0.62 0.60 0.61 0.64 0.67 0.71 0.72
0.91 0.91 0.86 0.86 0.79 0.72 0.68 0.71 0.76 0.88 0.94 0.95
Yearly average
22.82
5.69
0.67
0.83
Month
Hdm/Hm
Hd8
Hd9
Hd,o
e8 (%)
e9 (%)
el0 (%)
0.19 0.19 0.23 0.21 0.25 0.32 0.34 0.31 0.28 0.27 0.20 0.20
4.25 4.28 5.39 5.82 6.78 7.23 7.43 7.33 6.56 5.51 4.18 3.75
4.18 4.75 5.77 6.09 6.66 7.28 7.61 7.15 6.56 4.94 4.01 3.77
4.23 4.30 5.41 5.83 6.76 7.22 7.44 7.30 6.54 5.44 4.15 3.73
-8.67 4.11 2.34 -5.82 - 10.13 5.22 4.10 -5.22 - 1.99 6.59 -1.97 5.10
-6.83 -6.60 -4.53 -10.81 -8.07 4.53 1.75 -2.53 -2.02 16.31 2.26 4.62
-8.08 3.49 2.02 -5.95 -9.66 5.35 4.05 -4.76 - 1.76 7.80 -1.15 5.48
0.25
5.71
5.73
5.11 0.02 1.12
5.90 0.04 1.40
4.96 0.01 1.12
MPE MBE RMSE
developed for estimating monthly mean daily diffuse radiation on a horizontal surface for the Shambat location, the equation connecting diffuse radiation together with clearness index and per cent possible sunshine is found to be the most accurate one. Equation (28) in the form
Hd/Hm = 1 . 0 0 - 1 . 0 6 * KT--0.05 * (S/So) is recommended for computing diffuse radiation for K h a r t o u m Province. It should be possible to calculate
the monthly mean diffuse radiation with an error of about 10%. (2) It is also concluded that it is advantageous to have two measured variables in the estimated equation of diffuse radiation to improve the accuracy of the estimated data. This is more important while developing correlations for a region with stations having quite different climatological conditions. (3) It is observed that solar energy is a b u n d a n t throughout the year in K h a r t o u m Province.
Data Bank
233 REFERENCES
i 4 2"~"
+Hd8 • Hd9
r~
ol,
J
F
,
l
M
A
l
M
,
,
i
J J A Months
,
S
,
O
i
N
D
Fig. 4.
NOMENCLATURE
[10" al, l
correlation constants (regression constants)
b,),
b, j el~el0
error (%)
Hd monthly average of the daily diffuse solar
Hm H,, lh
KT
D1 S
S,~
S0 W, 6
irradiance on a horizontal surface, MJ/m 2 day monthly average of the daily total global irradiance on a horizontal surface, MJ/m -~day monthly average of the daily extra-terrestrial irradiation, MJ/m 2 day direct beam component, MJ/m 2 day solar constant = [350 W/m 2 ratio of cloudiness or transmission coefficient number of the day of the year (1 January - 1, 31 December = 365) monthly average daily number of hours of bright sunshine, hrs noon altitude of the sun on the 15th of the month, degrees monthly average day length, hrs sunrise/sunset hour angle, degrees latitude of the place, degrees solar declination, degrees.
1. J. K. Page, The estimation of monthly mean values of daily total short wave radiation on vertical and inclined surfaces from sunshine records for latitudes 40°N-40~S. Proc. U.N. Conference on New Sources of Energy, paper 598, 4, 378 390 (1961). 2. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristics distribution of direct, diffuse and total solar radiation. Solar Energy 4(3), 1 19 (1960). 3. S. A. Klein, Calculation of monthly average insolation on tilted surfaces. Solar Energy 19, 325 (1977). 4. M. Collares-Pereria and A. Rabl, The average distribution of solar radiation correlations between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22(2), 155-165 (1979). 5. M. Iqbal, Correlations of average, diffuse and beam radiation with hours of bright sunshine. Solar Energy 23(2), 169 173 (1979). 6. J. E. Hay, A revised method for determining the direct and diffuse components of the total short wave radiation. Atmo,sphere 14(4), 278 287 (1976). 7. K. K. Gopinathan, Empirical correlations for diffuse solar radiation. Solar Energy 40(3) (1988). 8. Anna Mani, Handbook of Solar Radiation Data Jot India. Allied Publishers, New Delhi (1961). 9. G. O. G. L6f, J. A. Duffle and C. O. Smith, World distribution of solar radiation. Solar Energy Laboratory, University of Wisconsin, Report No. 21 (1966). 10. J. A. Duffle and W. A. Beckman, Solar Energy Thermal Processes. Wiley Interscience, New York (1974). 11. P. I. Cooper, The absorption of solar radiation in solar stills. Solar Energy 12(3) (1969). 12. J. A. Davies, M. Abdel-Wahab and D. C. McKay, Estimating solar irradiation on horizontal surfaces. Int. J. Solar Energy 2, 405M24 (1984). 13. Dr. R. Posorski and Rahman, Evaluation of the meteorological data of Soba (Khartoum) with respect to renewable energy applications. The Energy Research Council, The Special Energy Programme, German Sudanese Technical Cooperation, Khartoum, August (I984).
A P P E N D I X : STATISTICAL ANALYSIS OUTPUTS
Regression output Constant Std err. of Y est. r squared No. of observations Degrees of freedom X coefficient(s) Std err. ofcoef.
Regression output 1.03 0.02 0.93 12.00 10.00
- 1.17 0.I0
H j H m = 1.03--l.17*Kr
Constant Std err. of Y est. r squared No. of observations Degrees of freedom X coefficient(s) Std err. ofcoef.
(26)
Regression output 0.69 0.02 0.88 12.00 10.00
- 0.53 0.06
Hd/Hm=0.69-0.53*(S/So)
Constant Std err. of Y est. r squared No. of observations Degrees of freedom X coefficient(s) Std err. ofcoef.
(27)
1.00 0.02 0.93 12.00 9.00 - 1.06 - 0.05 0.46 0.21
HjH,,,= 1.O0-1.06*KT-O.05*(S/So)
(28)
~
Copyright~! 1994ElsevierScienceLid Printed in Great Britain. All rightsreserved 0960 1481/94$6.00+0.00
Pergamon
DATA BANK Diffuse solar radiation over Shambat, Khartoum North ABDEEN MUSTAFA OMER National Centre for Research, Energy Research Institute, P.O. Box 4032, Khartoum, Sudan
(Receil:ed 6 June 1993 : accepted 19 Ju O' 1993) A b s t r a c t I T h e r e are many locations where no measured data on diffuse radiation are available and have to be estimated from empirical correlations. Here, using correlations for predicting monthly mean daily diffuse radiation on a horizontal surface for locations in Sudan, from the experimental data of Shambat station, the applicability of the developed equations were tested by estimating monthly mean daily diffuse radiation for the location. Measurements of diffuse radiation on a horizontal surface at Shambat tk~r 10 years were compared with predictions made by several independent methods. In the first type of correlation, formulae were used to correlate diffuse solar irradiance to the clearness index. Regression coefficients are obtained and used for prediction of diffuse solar irradiance. The predicted values were consistent with measured values (10.13%). In the second type of correlation, sunshine duration and minimum air mass were used to derive an empirical correlation for diffuse radiation. The predicted values compared well with measured values (16.31%). In the third type of correlation, a correlation connecting monthly mean daily diffuse radiation together with the clearness index and per cent possible sunshine is found to be 9.66%, the most accurate for the Shambat location.
I. I N T R O D U C T I O N
ments. Therefore, this work is an effort to lind reliable methods for prediction of solar radiation data with minimum possible measurements in Shambat, Khartoum North. The present study is to establish correlations for the diffuse component for the city of Shambat, Khartoum North, where daily global and diffuse solar radiation over Shambat (latitude 15 40'N, longitude 32 32'E, and altitude 380 m), Khartoum North, Sudan, has been measured by Robitzsch and Kipp instruments since 5 May 1957.
One of the most important requirements in the design of any solar energy conversion system is the information oll the intensity of solar radiation and its components at a given location. There are two main components of radiation reaching the ground : direct radiation and diffuse radiation. For any solar energy applications, global solar radiation level data arc needed. Knowledge of the diffuse radiation on a horizontal surface is of value in the design of various solar utilization devices. However, the diffuse radiation on a horizontal surface has many applications, particularly in finding global radiation on tilted surfaces, illumination design inside a building, and so on. Whereas considerable information exists on global solar radiation on horizontal surfaces, experimental data on diffuse radiation are very rare and often restricted to very short periods. Diffuse radiation is still measured at a few locations. Thus few data are available for diffuse radiation. One has thus to depend on the theoretical estimations of diffuse radiation for such locations where no measured data are available. Several empirical correlations have already been developed to predict diffuse radiation from available meteorological data. In developing countries where the energy problem is acute, solar radiation data is still scarce. The solar radiation measurements are very important in order to establish a complete solar map for countries such as Sudan. The knowledge of the available solar radiation and its components is valuable for design and assessment of solar energy conversion systems. The solar equipment needed for these purposes is expensive and requires maintenance as well as frequent calibrations. Moreover, it is almost impossible to scan day-by-day solar radiation data. This is due to inaccurate readings which result from the reader or the instru-
2. THEORY
2.1. Prediction fi~rmulae There are three types of correlations for estimating horizontal diffuse radiation. The first type expresses monthly average daily diffuse radiation, Hd/Hm, as a function of the monthly average daily clearness index, K~ - Hm/H~, lid, H .... and/4o are monthly average daily diffuse, global, and extraterrestrial radiation on a horizontal surface. The second type expresses the fraction Ho/Hm or Ha/H~ as a function of the monthly averaged daily values of the bright sunshine hours, S, and the maximum possible sunshine hours, 5'~. The ratio S/S,~ is often called the per cent possible sunshine. The third type of correlation expresses Ha/Hm as a function of both clearness index and per cent possible sunshine. Several correlations have thus been developed for computing diffuse radiation and the coefficients of all these correlations are found to be different and site dependent. Examples of some well-accepted correlations that have universal applicability are given below. 2.2. Prediction of d(ffuse solar irradiance 2.2.1. Type I correlations. Page [1] developed correlations between daily total and diffuse radiation for 10 widely spread 227
228
Data Bank
sites in the 40°N to 40°S latitude and obtained the following relationship :
n d / n m = 1.00--1.13 • KT,
(1)
where H d = the monthly average of the daily diffuse solar irradiance; Hm = the monthly average of the daily total global irradiance; KT = the ratio of cloudiness index or transmission coefficient
Kv = Hm/H~.
(2)
Liu and Jordan [2] developed a statistically based correlation from results obtained from one station which was developed by Klein [3] :
Hd/Hm = 1 . 3 9 0 - 4 . 0 2 7 . K v + 5 . 5 3 1 *Kv2-3.108*K-~ 0.3 < KT < 0.7.
(3)
Collares-Pereira and Rabl [4], used data obtained from five locations in the United States to derive the following season-dependent correlation :
H~/Hm = 0.775+0.347 * (~/180) * ( W ~ - 9 0 C') - [ 0 . 5 0 5 + 0 . 2 6 1 • (7r/180) • (W~-90°)] • cos [2(Kv-0.9)],
(4)
where Ws = the sunset hour angle in degrees ; His ~ 90 ° during February to April and August to October; His ~ I00 ~' during M a y to July ; Ws ~ 80 ° during November to January ; the quantity 2(KT--0.9) is in radians. 2.2.2. Type H correlations. Iqbal [5] used data obtained from three locations in C a n a d a to propose the correlation,
Hd/Ho = 0 . 7 9 1 - 0 . 6 3 5 • (S/So),
(5)
where : S
= The monthly averaged daily values of the bright sunshine hours. So = The m a x i m u m possible sunshine hours. S/So = The ratio is often called the per cent possible sunshine. Iqbal [5] has developed the following correlation for Montreal, Canada, latitude 45°30'N,
Hd/Ho = 0.163+0.478 • (S/S0)-0.655 • (S/So)".
(6)
The purpose of the present study is to develop empirical correlations for estimating the monthly mean of the daily diffuse radiation on a horizontal surface, from the measured data of global radiation and sunshine durations, for locations in Sudan. Available long-term data on Hd, Hm and S, are used in a regression analysis to obtain the regression coefficients of different types of correlations. One equation belonging to each type of correlation is used in the analysis. The above correlations are used for the prediction of diffuse solar irradiance. The direct beam component lh can then be deduced from the relation :
Hm = lid + lb * sin (Sn),
(9)
where the product lb sin (Sn) is the average horizontal beam component ; S, = the noon altitude of the sun of the 15th of the month. Sn = 9 0 - ( ~ b - 3 ) ,
(10)
where q~ = the latitude of the place in degrees; 6 = the solar declination in degrees. The direct beam component is useful for the various types of solar concentrating systems.
3. METHODOLOGIES FOR FINDING Hd Radiation data are the best source of information for estimating average incident radiation. It is possible, however, to use empirical relationships to estimate radiation from hours of sunshine or cloudiness, if radiation data at the location in question are not available. Data on average hours of sunshine, or average per cent of possible sunshine hours, are available for many stations in Sudan, measured by a Campbell-Stokes instrument. Available experimental data on Hd, Hm and S reported by A n n a Mani [8] and L r f et al. [9] for Delhi, Madras, and Poona were used in a least squares method to find the regression coefficient of the following correlations. Proper computer programs were written to obtain the coefficients of the following equations
Hd/Hm = a0+al * Kv
(11)
Hd/Hm = b0+b, * (S/So)
(12)
Hd/Hm = c0+c, * Kv+c2 * (S/So),
(13)
Hay [6] considered the relationship between radiation before and after multiple reflections between the Earth and the cloud cover. He proposed the following location independent relationship between global (Hi) and diffuse (Ha) radiation emerging from the atmosphere before striking the ground,
where a0, a~, b0, b~, co, c~ and c2 are regression constants or 'empirical constants'. The monthly mean daily extra-terrestrial radiation H0 on horizontal surface at any time between sunrise and sunset is given by Duffle et al. [10] :
Hd/Hm = 0.9702+ 1.6688 • (H~/Ho)-21.303 • (Hm/Ho) 2
H0 = [2(23 * 3600 * l~/Tr) * (1 + 0.33 * cos ((360 * m)/365))]
+51.288 * (Hm/no)3+ 50.081 • (Hm/no) 4 + 17.5510 • (Hm/no) 5.
• [cos ~b * cos 3 * sin ( W0 + 2 * 7r * WJ360) (7)
2.2.3. Type III correlations. In addition to the above two types, a third type of correlation expressing H~/Hm as a function of both clearness index and the per cent possible sunshine together has been recently suggested by Gopinathan [7] for locations in southern Africa. A correlation of the form :
Hd/H m = 0 . 8 7 9 - 0 . 5 7 5 * (K-r)-0.323 * (S/So)
(8)
has been found suitable for stations in southern Africa. Gopinathan has included both KT and S/So in the estimated correlation to improve the accuracy of the estimated data by having two measured variables in the predicted formula.
• sin q~* sin 3],
(14)
where I~ - the solar constant = 1350 W/m2; 6 = the solar declination in degrees; q ~ - the latitude of the station in degrees ; m = the number of the day of the year (1 - 365). The values of $0 are computed from Cooper's formula [11] So = 2/15 * cos- i ( _ tan ~b* tan 6).
(15)
The sunrise, sunset hour angles W~ : W~=cos I(-tan~b,tanr),
(16)
where W~ = the sunrise/sunset hour angle, in degrees. The value of declination can be found from the equation of Cooper [11] :
229
Data Bank 6 = 23.45 * sin [360 * (284 +m)/365).
(17)
The declination angle is given a positive value for the summer time in the northern hemisphere and a negative value in the winter time. It has a seasonal variation. Equation (16) gives an accuracy of some _+ 1 . 4. T E S T I N G P R O C E D U R E Measurements of diffuse solar irradiance on a horizontal surface have been utilized to verify simple methods of prediction of solar radiation. Accurate measurements of solar radiation data require delicate and fairly expensive instruments. As such instruments are not available at all locations, it is often necessary to compute diffuse radiation from theoretical models. M a n y models have been developed to predict the diffuse sky solar radiation since the original work of Liu and Jordan [2]. A review of several of the older models and some of the newer models is contained in Duffle and Beckman [10]. The diffuse solar radiation models may be hourly, daily or monthly models, and are typically based on the measurement of global solar radiation (total solar radiation on horizontal surfaces). The other parameter typically used in the model is usually based on extra-terrestrial solar radiation on a horizontal surface. Recent models of this type include the Collares-Pereira and Rabl [4]. The statistic analysis of the results can be performed on three intervals of time : (1) daily ; (2) monthly ; and (3) yearly. The relative deviation (d,) between the computed values Hd~, Hd2, H~3 and the observed value Hdm can be established by
d, = (H~ - H~,,),/Hd, ,.
(18)
The relative percentage error (e) is : e = [(Hd ..... Ho,)/'Ho~,,j * 100
(19)
calculated for each month of the year, from all the three models. The mean percentage error (MPE) can be estimated by M P E : [ E ( ( ( H d , , - Hd,.,)/Ho,.,,) * 100)]/n
(20)
where n = the total number of observations ; Hd, = the ith calculated value : Hd = the ith measured value. The sign of the errors can be neglected in the summation and all the errors are added up while calculating the mean. The accuracy of the estimated data on diffuse radiation can also be tested by calculating the mean bias error (MBE) and root mean square error (RMSE). The M B E and R M S E are the fundamental measures of accuracy [12]. MBE represents the systematic error or bias and can be estimated by M BE = [ E ( H d , - Hd~,,)]/n,
(21)
MBE = E(d,)/n.
(22)
Dispersion of results is usually characterized by standard deviation (~), given by a = ( R M S E 2 - M B E 2 ) ~ "~
(25)
The lower the RMSE, the more accurate the estimate is. A positive MBE shows an overestimation while a negative M BE shows an underestimation. While calculating the M P E values, the sign of the errors are neglected and the percentage errors are added up to calculate the mean. 5. R E S U L T S AND D I S C U S S I O N
5.1. Introduction For m a n y solar designs and applications it is necessary to have knowledge of both the direct and diffuse components of the incident solar radiation. Whereas considerable information exists for global solar radiation on horizontal surfaces, only a few data are available for diffuse solar radiation. The purpose of the present work is to test the applicability of correlations for the estimations of diffuse radiation against measured values in one location: Shambat (latitude 15°40'N ; longitude 32' 32'E; altitude 380 m). The correlations for estimating horizontal diffuse radiation belong to three types. The first type expresses the monthly average daily diffuse radiation fraction, Ha/Hm, as a function of the monthly average daily clearness index, Ha/Ho. The second type expresses the fraction Hd/Hm or Hd/Ho as a function of the monthly averaged daily values of bright sunshine hours, S, and m a x i m u m possible sunshine hours, So. The third type of correlation expresses He/H mas a function of both clearness index and per cent possible sunshine. 5.2. Computation of diffuse solar irradiance The accuracy of the estimated data is tested by comparing the measured data available for the Shambat location with models proposed by Page, Liu and Jordan, Collares-Pereira and Rabl, Iqbal and Gopinathan. The performance of the correlations for Shambat are shown in Figs 1 3, respectively, and the results are shown in Tables 1 3. Diffuse radiation estimated from eqs (1, 3 8) along with the experimental are shown in these figures and tables. The following observations can be made from a study of the results presented in Tables 1 3, and Figs 1 3. Equation (1) estimates Hd fairly accurately. However, the errors in the estimated data are low (up to 8.46%). Accurate estimations are possible for most months, from eq. (3). However, the
or
RMSE is a measure of non-systematic error and can be computed from RMSE = {[E(H%-H%,)2]}'2,
(23) J
or
I
I
I
I
F
M
A
M
J
J
Months RMSE = (E(d32/n) t/2.
(24)
Fig. 1.
I
I
I
I
A
S
0
N
D
230
Data Bank 10
! 0_
I F
I M
I A
I M
I J
I J
I A
I S
I O
I N
I D
Months
Fig. 2.
I /
/+I÷
45~+
J
I I
•
• Hdm
t o
-.,~.
.*
5.3. Statistical analysis The accuracy of estimated data on diffuse radiation are evaluated by calculating MPE, MBE and RMSE values. The estimated values of MPE, MBE and RMSE for Shambat location from all models are included in Tables 1-3. As seen from the tables, there is a remarkable agreement between the experimental and estimated values of diffuse radiation for the location by using; for type I correlations, eq. (1) and eq. (3) ; type I1 correlations, eq. (5) ; and type Ill correlations, eq. (8). The lower the RMSE, the more accurate the estimation is. The positive MBE shows an overestimation whereas a negative MPE shows an underestimation. The RMSE values are very low, by using eqs (1, 3, 5 and 8), indicating good agreement. 5.4. Prediction of Ha by methods Regional values of the regression coefficient in all the three equations along with root squared (r), standard error of Y Est (p), and standard error coefficient (S.E.) are given as follows :
Hd/Hm = 1.03--1.17*KT
--+,~+
r = 0.93
+.d7 I F
I M
I A
I M
# = 0.02
I 1 J J Months
I A
I S
I O
I N
S.E. = 0.10
(26)
D
Hd/Hm = 0.69-0.53 * (S/So) r = 0.88
Fig. 3.
p - 0.02 S.E. = 0.06
(27)
Hd/Hm = 1.00-- 1.06 • Kv --0.05 * (S/So)
error in the estimated values reaches 13.78%. The accuracy of the estimated data from eqs (4 and 7) are found to be high compared to other equations. The accuracy of eq. (5) shows the errors up to 15.95%. In eq. (8) the errors reach 22.95% during December and are low during January to September.
r = 0.93 p = 0.02 S.E. = 0.46,0.21.
(28)
Table 1. Experimental and calculated values of diffuse radiation on a horizontal surface for Shambat, Khartoum North, using type I correlations, MJ/m 2 day Month
Hm
Ho
Ho avg.
Kx
Ha]
Ha2
H~3
Hdm
el (%)
e2 (%)
e3 (%)
Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.
20.30 23.10 24.50 25.90 24.70 23.60 23.00 22.80 22.90 21.90 21.00 20.10
34.10 35.50 36.11 35.10 33.15 31.81 32.23 33.99 35.50 35.50 34.33 33.54
28.94 31.99 35.39 37.63 38.26 38.16 38.07 37.66 36.03 32.92 29.57 27.88
0.70 0.72 0.69 0.69 0.65 0.62 0.60 0.61 0.64 0.67 0.71 0.72
4.21 4.25 5.33 5.76 6.68 7.11 7.30 7.20 6.45 5.44 4.15 3.73
4.34 4.53 5.43 5.83 6.40 6.60 6.68 6.60 6.11 5.34 4.34 3.96
6.04 6.81 7.63 8.28 8.53 8.57 8.52 8.30 7.79 6.99 6.21 5.82
3.91 4.46 5.52 5.50 6.16 7.63 7.75 6.97 6.43 5.90 4.10 3.95
-7.66 4.69 3.37 -4.66 -8.46 6.85 5.83 -3.33 -0.36 7.85 - 1.16 5.69
- 11.11 - 1.46 1.57 -6.00 -3.92 13.45 13.78 5.27 4.96 9.51 -5.75 -0.21
-54.54 -52.71 -38.24 -50.54 -38.48 -12.36 -9.90 - 19.10 -21.20 - 18.49 -51.48 -47.36
Yearly average
22.82
34.38
0.67
5.63
5.51
7.46
5.69
4.99 -0.06 1.14
6.42 -0.18 1.78
MPE MBE RMSE
34.53 0.26 6.49
Data Bank
.¢
t+q tt~ o
1
i I
©
2
+,
1771171-
4 Z
+..i
.g
1_ ,,.+
~ o ~ o o o o o o o o o o
. ,....
r~
g
t
t"-,l ¢"I :~ . . . .
~ .
.
P"+- <'% t,-) i~. ¢"+,I ~ID -. . . . . .
¢',"1
e,
t-.,,~ t-,,£
231
Equation (28) has the highest value of the correlation coefficient. Coefficients of eq. (26) are higher than in Page's equation. The regression coefficients of eq. (27) are nearly equal to lqbal's values ofeq. (5). The coefficients ofeq. (28) can be compared with the coefficients of the correlation developed by Gopinathan for Southern Africa locations. However, such variations in the coefficients are site dependent. The applicability of the proposed correlations predicting Hd is tested by estimating Hd values for Shambat location used in the analysis. Calculated values of Hd lbr Shambat along with reported experimental data, Ham, from Dr Posorski et al. [13] are presented in Table 4. The calculated values of Kr, S/& are also included in the table. Monthly mean daily diffuse radiation computed from eqs (26 28) arc shown as Ha~, H,I,~and Hdm, respectively. Hm, Hd,,, values shown in the table are in M J/m= day. The rcsults shown in Table 4 and Fig. 4, are for the Shambat location. Diffuse radiation estimated from eqs (26 28) along with experimental data are shown in this figurc. The following observations can be made from a study of the results. Equation (26) estimates Hj fairly accurately for most months of the year for the Shambat station. However, the errors in thc estimated data are high (up to 10.13%) during May. Accurate estimations are possible for most months, for the Shambat location, from eq. (27). However, the errors m the estimated values during April, May and October are high. The accuracy of the estimated data from eq. (28) is found to be low compared to other equations. The predicted values show an excellent agreement with the measured data for the Shambat location during all the months of the year. The error in the estimated values lie within about 10% and for most of the m o n t h s it is far less than 10%. A comparative study of MPE, MBE and RMSE from all the three models for Shambat supports the above agreement. The average percentage errors incurred in predicting Hd from eqs (2(>28) for Shambat, are 5.11%, 5.90% and 4.96% respectively As seen from Table 4, the M P E value from eq. (28) is the lowest. The percentage mean bias error from the three equations, for Shambat, are 0.02%, 0.04% and 0.01%. Equation (28), with per cent mean bias error of 0.01%, cancels out the two effects and improves the accuracy of the estimated data. Again, the per cent root mean square errors from the three equations are 1.12%, 1.40% and 1.12%, respectively, showing the lowest value of error for eq. (16). This supports the superiority ofeq. (28) over the others. In general, there is a good agreement between the experimental and estimated data of diffuse radiation from eq. (28). However, the accuracy of estimated data from eqs (26) and (27) is not ~ery high. The correlations (26) and (27) employ only one measured variable in the estimated correlation (either Kv or S/So). There are two measured variables in eq. (28), both Kv and S/So, and together these seem to have improved the accuracy of the estimated values.
+B .,.-.
6. C O N C L U S I O N Solar irradiance data for the Shambat station have been collected, investigated and analysed. Three types of correlations are tested against the experimental results. The following are concluded :
eq ¢.~
o=
>.
(1) In a location where no recorded data on diffuse radiation are available, there are different possible ways of predicting H~. Out of the three correlations
232
Data Bank Table 3. Experimental and calculated values of diffuse radiation on a horizontal surface for Shambat, K h a r t o u m North, using type III correlations, M J / m 2 day
Hm
Ho
KT
S
So
S/&
Hdm
Ha7
Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.
20.30 23.10 24.50 25.90 24.70 23,60 23.00 22.80 22.90 21.90 21.00 20.10
34.10 35.50 36.11 35.10 33.15 31.81 32.23 35.68 35.50 35.50 34.33 33.54
0.70 0.72 0.69 0.69 0.65 0.62 0.60 0.61 0.64 0.67 0.7l 0.72
10.20 10.50 10.20 10.60 10.10 9.30 8.70 8.90 9.20 10.20 10.60 10.50
11.16 11.49 11.89 12.36 12.73 12.93 12.85 12.53 12.08 11.64 11.25 ll.07
0.91 0.91 0.86 0.86 0.79 0.72 0.68 0.71 0.76 0.88 0.94 0.95
3.91 4.46 5.52 5.50 6.16 7.63 7.75 6.97 6.43 5.90 4.10 3.95
3.68 3.90 5.00 5.34 6.21 6.87 7.19 6.87 6.12 4.66 3.48 3.16
Yearly average
22.82
0.67
9.92
0.83
5.69
5.20
Month
MPE MBE RMSE
e7 (%) 5.88 12.62 9.48 2.98 -0.82 10.02 7.20 1.47 4.87 21.06 15.11 22.95
9.54 - 0.49 2.06
Table 4. Experimental and calculated values of diffuse radiation on a horizontal surface for Shambat, K h a r t o u m North, using three methods, M J / m 2 day Hm
Hdm
KT
S/So
Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.
20.30 23.10 24.50 25.90 24.70 23.60 23.00 22.80 22.90 21.90 21.00 20.10
3.91 4.46 5.52 5.50 6.16 7.63 7.75 6.97 6.43 5.90 4.10 3.95
0.70 0.72 0.69 0.69 0.65 0.62 0.60 0.61 0.64 0.67 0.71 0.72
0.91 0.91 0.86 0.86 0.79 0.72 0.68 0.71 0.76 0.88 0.94 0.95
Yearly average
22.82
5.69
0.67
0.83
Month
Hdm/Hm
Hd8
Hd9
Hd,o
e8 (%)
e9 (%)
el0 (%)
0.19 0.19 0.23 0.21 0.25 0.32 0.34 0.31 0.28 0.27 0.20 0.20
4.25 4.28 5.39 5.82 6.78 7.23 7.43 7.33 6.56 5.51 4.18 3.75
4.18 4.75 5.77 6.09 6.66 7.28 7.61 7.15 6.56 4.94 4.01 3.77
4.23 4.30 5.41 5.83 6.76 7.22 7.44 7.30 6.54 5.44 4.15 3.73
-8.67 4.11 2.34 -5.82 - 10.13 5.22 4.10 -5.22 - 1.99 6.59 -1.97 5.10
-6.83 -6.60 -4.53 -10.81 -8.07 4.53 1.75 -2.53 -2.02 16.31 2.26 4.62
-8.08 3.49 2.02 -5.95 -9.66 5.35 4.05 -4.76 - 1.76 7.80 -1.15 5.48
0.25
5.71
5.73
5.11 0.02 1.12
5.90 0.04 1.40
4.96 0.01 1.12
MPE MBE RMSE
developed for estimating monthly mean daily diffuse radiation on a horizontal surface for the Shambat location, the equation connecting diffuse radiation together with clearness index and per cent possible sunshine is found to be the most accurate one. Equation (28) in the form
Hd/Hm = 1 . 0 0 - 1 . 0 6 * KT--0.05 * (S/So) is recommended for computing diffuse radiation for K h a r t o u m Province. It should be possible to calculate
the monthly mean diffuse radiation with an error of about 10%. (2) It is also concluded that it is advantageous to have two measured variables in the estimated equation of diffuse radiation to improve the accuracy of the estimated data. This is more important while developing correlations for a region with stations having quite different climatological conditions. (3) It is observed that solar energy is a b u n d a n t throughout the year in K h a r t o u m Province.
Data Bank
233 REFERENCES
i 4 2"~"
+Hd8 • Hd9
r~
ol,
J
F
,
l
M
A
l
M
,
,
i
J J A Months
,
S
,
O
i
N
D
Fig. 4.
NOMENCLATURE
[10" al, l
correlation constants (regression constants)
b,),
b, j el~el0
error (%)
Hd monthly average of the daily diffuse solar
Hm H,, lh
KT
D1 S
S,~
S0 W, 6
irradiance on a horizontal surface, MJ/m 2 day monthly average of the daily total global irradiance on a horizontal surface, MJ/m -~day monthly average of the daily extra-terrestrial irradiation, MJ/m 2 day direct beam component, MJ/m 2 day solar constant = [350 W/m 2 ratio of cloudiness or transmission coefficient number of the day of the year (1 January - 1, 31 December = 365) monthly average daily number of hours of bright sunshine, hrs noon altitude of the sun on the 15th of the month, degrees monthly average day length, hrs sunrise/sunset hour angle, degrees latitude of the place, degrees solar declination, degrees.
1. J. K. Page, The estimation of monthly mean values of daily total short wave radiation on vertical and inclined surfaces from sunshine records for latitudes 40°N-40~S. Proc. U.N. Conference on New Sources of Energy, paper 598, 4, 378 390 (1961). 2. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristics distribution of direct, diffuse and total solar radiation. Solar Energy 4(3), 1 19 (1960). 3. S. A. Klein, Calculation of monthly average insolation on tilted surfaces. Solar Energy 19, 325 (1977). 4. M. Collares-Pereria and A. Rabl, The average distribution of solar radiation correlations between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22(2), 155-165 (1979). 5. M. Iqbal, Correlations of average, diffuse and beam radiation with hours of bright sunshine. Solar Energy 23(2), 169 173 (1979). 6. J. E. Hay, A revised method for determining the direct and diffuse components of the total short wave radiation. Atmo,sphere 14(4), 278 287 (1976). 7. K. K. Gopinathan, Empirical correlations for diffuse solar radiation. Solar Energy 40(3) (1988). 8. Anna Mani, Handbook of Solar Radiation Data Jot India. Allied Publishers, New Delhi (1961). 9. G. O. G. L6f, J. A. Duffle and C. O. Smith, World distribution of solar radiation. Solar Energy Laboratory, University of Wisconsin, Report No. 21 (1966). 10. J. A. Duffle and W. A. Beckman, Solar Energy Thermal Processes. Wiley Interscience, New York (1974). 11. P. I. Cooper, The absorption of solar radiation in solar stills. Solar Energy 12(3) (1969). 12. J. A. Davies, M. Abdel-Wahab and D. C. McKay, Estimating solar irradiation on horizontal surfaces. Int. J. Solar Energy 2, 405M24 (1984). 13. Dr. R. Posorski and Rahman, Evaluation of the meteorological data of Soba (Khartoum) with respect to renewable energy applications. The Energy Research Council, The Special Energy Programme, German Sudanese Technical Cooperation, Khartoum, August (I984).
A P P E N D I X : STATISTICAL ANALYSIS OUTPUTS
Regression output Constant Std err. of Y est. r squared No. of observations Degrees of freedom X coefficient(s) Std err. ofcoef.
Regression output 1.03 0.02 0.93 12.00 10.00
- 1.17 0.I0
H j H m = 1.03--l.17*Kr
Constant Std err. of Y est. r squared No. of observations Degrees of freedom X coefficient(s) Std err. ofcoef.
(26)
Regression output 0.69 0.02 0.88 12.00 10.00
- 0.53 0.06
Hd/Hm=0.69-0.53*(S/So)
Constant Std err. of Y est. r squared No. of observations Degrees of freedom X coefficient(s) Std err. ofcoef.
(27)
1.00 0.02 0.93 12.00 9.00 - 1.06 - 0.05 0.46 0.21
HjH,,,= 1.O0-1.06*KT-O.05*(S/So)
(28)