~ Pergamon

DATA BANK

Renewable A)leryy, Vol. 4. No. 1, pp. 95 100, 1994

Copyright i~ 1994 Else;let Science Ltd Printed in Great Britain. All rights reserved

0960 1481 94 $6.0fH 0.00

Estimation of the diffuse solar irradiation from global solar irradiation. Daily and monthly average daily values

J. A . M A R T I N E Z - L O Z A N O , * M . P. UTR1LLAS'~ a n d V. GOMEZ*

* Dipartment de Termodinfimica, Facultat de Fisica, Univcrsitat de Valencia, 46100 Burj.assot, Valencia, Spain

"i" Dipartment de Ciencias Experimentales, Universitat Jaume I, Apdo. 224, 12080 Castellon, Spain

(Rece ived 12 March 1993: accepted 26 Apri l 1993)

Abstract Data sets of total and direct solar irradiation at Valencia (Spain) measured during the years 1990 1991 have been used to analyze diverse methods of estimating the daily values and monthly average daily values of the diffuse fraction of solar radiation from the ratio of the total to the extraterrestrial radiation (clearness index k,). In the case of daily values, all the methods described in the bibliography have led to significant deviations concerning the experimental values, with a M A D (mean absolute deviation) close to 20%. For the monthly average daily values, the methods based on the PDF (probability density function) of daily values of kt provide more accurate results than the deterministic methods based on the correlation of monthly average daily values of k ,

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

In the last 10 years a considerable quantitative increase has occurred in the number of stations that register solar radi- ation in the world. However, from all the characteristics of solar radiation that reach the Earth 's surface, the only one which is registered in a systematic method and for which a series of statistical values are available is the global radiation on a horizontal surface (Iv). These values, which are of undoubted interest when elaborating climatic radiation maps and establishing the available energies, are insufficient in defining the efficiency of the majority of the systems which are used in solar energy. In these cases it is essential to have knowledge of the diffuse (ld) and direct (lb) components of solar radiation.

()wing to the lack of experimental values of the direct and diffuse components of solar radiation, several different methods have been proposed to calculate them. The methods described in the bibliography can be classified into three major groups, respectively based on : (a) atmospheric trans- mission coefficients; (b) surface meteorological obser- vations : and (c) global radiation records. The advantage of the latter is that it is only necessary to determine one of the components (direct or diffuse) to determine the other from the values of global irradiation.

In this paper an analysis of the different relations proposed for the estimation of the daily values and the monthly average daily wdues of diffuse radiation from recorded values of global radiation has been made. The location where the measurements have been performed (Campus of Burjassot, Valencia, Spain) is situated 40 m above the sea level at a latitude of 39.5 N. Obstructions above the horizon are in general less than 4 , except in a small area in the northwest. The measurement devices and the data acquisition system have been explained in a previous paper [1]. The data used in this study have been collected during the period between I January 1990 and 31 December 1991.

The diffuse radiation hourly values have been obtained by

95

means of a subtractive method from the available global radiation on a horizontal surface and direct radiation at normal incidence hourly values. This method can lead to significant errors in certain eases, but in the case of daily values, the considered errors are not higher than 5%. From the experimental values, a database has been established of both daily radiation values and the following dimensionless parameters : kt - [ l / lo , kd -- ld/ loyla/l ' i . The extra-terrestrial radiation on a horizontal surface (10) has been evaluated using a method previously described by the authors [2]. All the daily values have been considered without discarding any of them. Previous averages in discrete intervals of the clearness index (k0 have not been considered. The filter proposed by Gordon and Hochman [3] has been applied to all the data.

2. RELATIONSHIPS BETWEEN DIFFUSE AND GLOBAL IRRADIATION FOR DAILY VALUES

The usual process followed to determine the daily values of diffuse radiation from daily global radiation is derived from that proposed by Liu and Jordan [4]. They established that a close relationship exists between clearness index and diffuse radiation from a detailed statistical analysis. From this assumption the determination of the daily values of I d is possible by studying either the relationship l~t/1T =./'(k,) or the relationship kd --f(k,) .

2. I. Correlation between I j l r and k, Numerous authors have correlated ld/lv and k~ by poly-

nomial expressions of first [5 9], second [8, 10], third [7, 11. 12] and fourth [I 3 16] degree. More complicated expressions [9, 12, 17, 18] do not usually improve the results to a sig- nificant level.

Figure 1 shows the experimental values of l~/lt as a func- tion of k , The configuration of points presents a set of characteristics that stand out significantly : (a) for values of

96 Data Bank

~.2

l~/Iv

. ? , / , : . : • . . • " - ! :':~i : •

K v

0.2 0.4 ~.S 0.8

Fig. 1. Experimental daily values of ld/l~ VS k,, for Valencia (1990 1991).

kt ~< 0.20, the values of ld/1T take practically the unit value. This has been reported before by other authors [7, 8, 13, 15] : (b) there are no values of kt higher than 0.80, which results logically when dealing with average values throughout the day: (c) the points present a considerable dispersal in the interval 0.20 < kt < 0.80.

A simple linear regression between the two variables yields :

ld/'I T = 0.99, kt ~< 0.20, (1)

Ij/1T -- 1.36-- 1.65k, 0.20 < k, < 0.80, (2)

r = 0.88, ~ = 0.11, N = 555,

where r is the correlation coefficient, ~ is the estimation error and N is the number of points used.

When polynomial regression models of a higher degree are used for the interval 0 . 2 0 < k , <0.80, the following expressions are obtained :

l d / IT - - 1.26--1.19k, 0.48k~', r = 0 . 8 8 , ~-=0.11, (3)

la/13 = 0.72+2.73kt 9.12k(+5.97k~,

r = 0 . 8 8 , a - 0 . 1 1 . (4)

ld/l~ -- 2.02-- I 0.05k~ + 34.7k, 2 + 57.3k, 3 + 32.7k~,

r 0.88, o ' - 0 . l l . (5)

Results (2 ) (5 ) show that in practice the simple linear regression is sufficient and the use of polynomial models of a higher degree is not justified.

The deviations introduced during the theoretical esti- mation of ld/lT by the expressions (2) (5) have been estab- lished using both MBD (mean bias deviation) and the M A D (mean absolute deviation). When equations (2) (5) are used in conjunction with the same database which has been used for its determination (dependent test), the values given in Table 1 are obtained. In this table, the MBD values show

Table 1. Calculated deviations t~r the estimation of ld/lT values from the several

polynomial fittings

Polynomial degree MBD(%) M AD(%)

1 0.3 19 2 0.4 19 3 0.7 19 4 1.5 19

that these expressions do not introduce systematic deviations in the estimation. However the associated M A D is very high.

Collares-Pereira and Rabl [13] have studied the depen- dence of the monthly average daily values of ld/lv on the period of the year. From the conclusions of Collares-Pereira and RaN, several authors [7 9, 12, 14, 16] have proposed a seasonal division for the estimation of the daily values of diffuse radiation from the clearness index. To check this hypothesis, linear fits have been made classifying the exper- imental records monthly. The characteristics of the fitting coefficients of the monthly linear regressions are given in Table 2. In this table, the values of the relative MBE and relative M A D have been included also. These relative values have been obtained by comparing the daily estimate values, by means of the monthly expressions, with the experimental l, d l ~ daily values. The results show that the estimation intro- duces fewer errors for the warm dry months (March Sep- tember). The coefficients of the monthly fits (intercept and slope) show a great variability for these several months. We have tried to establish a possible relation between the fitting coefficients and the average monthly values of the solar azi- muth and solar elevation. A relation does not seem to exist between these variables, and it has not been possible to establish an analytic relation between them. This seems to confirm that the annual evolution of daily diffuse radiation in a determined place depends, as refereed by Zangvil and Aviv [19], on combined seasonal and meteorological effects.

2.2. Correlation between k a and k~ Bruno [20] and Al-Hamdami et al. [9] have studied the

relationship between diffuse irradiation and global irradiation from analytic expressions of the type k d - f ( k O . In Fig. 2, the pairs of experimental (ktkd) values registered in Valencia in the considered period are given. Owing to the distribution of experimental points, a fitting Fourier series has been chosen for its adjustment. When applying a series of this type truncated at the first harmonic, a relative M A D ot22% is obtained for the estimation of the theoretical values of kd. The MBD shows that the method underestimates the experimental values of 7%. A consecutive increase in the number of harmonics in the expansion does not improve these results. Although the MBD can be reduced to prac- tically zero, the relative M A D maintains an approximate value of 20% when a series e~pansion with seven harmonics is t, sed for the adjustment. The separation of the values in seasonal periods does not lend itself to lesser errors. It has been proposed by some authors [4, 10, 21, 22] to work with average values corresponding to k, discrete bins, usually at an amplitude of 0.05. In this case the number of "'experimental values" is reduced and the errors that affect the results become considerably lower. Applying this procedure to the points of Fig. 2, the kd estimated values through the Fourier series expansion lruncated at the first harmonic differ from the experimental ones only by 5%. However, these results must be considered carefully, since the distribution of the values of k, within each interval of k, is far from being constant, and there exist intervals which only include eight points (0.05 < k~ < 0.10) and others including more than 150 points (0.60 < k, < 0.65). This is why it has been decided that the correct calculation procedure is that where all the experimental values are taken in a direct adjustment.

The results presented in this paper prove quite clearly that the estimation of daily values of the diffuse radiation from daily values of the global radiation is restricted by an impor- tant lack of accuracy, whether it is carried out by means of l j / l r or if it is carried out by means o fk d. This is due to the

Data Bank

Table 2. Calculated values for the fitting parameters of the linear regression 1,~/IT = A + Bk, and standard deviations (a), for each month

Month Intercept Slope r a N MBD(%) MAD(%)

January 1.48 -1 .86 0.90 0.12 44 17 28 February 1.59 -2 .07 0.90 0.09 32 - 1 21 March 1.43 -1 .63 0.91 0.10 40 0 16 April 1.37 - 1.60 0.88 0.09 46 - 1 18 May 1.48 - 1.76 0.94 0.07 55 - 1 14 June 1.33 - 1.53 0.91 0.08 57 1 15 July 1.36 - 1.60 0.86 0.08 49 - 1 16 August 1.38 - 1.68 0.93 0.07 38 0 12 September 1.30 - 1.56 0.92 0.07 48 0 14 October 1.38 - 1.74 0.90 0.11 52 0 20 November 1.51 2.09 0.86 0.12 51 0 26 December 1.42 - 1.80 0.92 0.11 49 0 18

97

great variability of the atmospheric behaviour. For a given value o f k , the values o f l j / I T vary in a wide range. One can presume k~ on its own does not significantly represent the atmospheric conditions throughout the day. In this sense, Perez et al. [23] have recently shown the linfits of the clearness index when it is used to estimate the hourly values of diffuse radiation.

3. RELATIONSHIPS BETWEEN DIFFUSE AND GLOBAL IRRADIATION FOR MONTHLY

AVERAGE DALLY VALUES

In the bibliography, two methods are described to deter- mine the monthly average daily values of diffuse radiation : (a) to establish direct analytical relationships between the average monthly values of the clearness index, k-t, and of the ratio ~,,'~: and (b) defining the function of distribution, known as "'normal curve kt" , from the daily values. The obtained results with both processes when applied to the registered values in Valencia in the considered period are given in the following.

3. [ ~ Correlation be tween Ia and TT The relationship between both variables is established for

the case of daily values through the clearness index by means of:

[~/[v = f(k-0. (6)

Several authors have established a relationship between these variables by means of a linear correlation [5, 8, 9, 11, 12, 15, 16, 22, 24~31] or a correlation of a higher degree [14~ 26, 32 341.

The available values from the Valencia experimental data are presented in Fig. 3. The data points show a great dispersal in a reduced interval of~]. It can be seen that while the daily values of k, present a range of variation between 0.05 and 0.80 (see Fig. 1), the values of k] rise to between 0.46 and 0.66. The polynomial approximations give the following results :

[d/[v = 0.17+0.81/~, r = 0.62, a = 0.059, (7)

[j/l~ - -0.86+4.59k't-3.46k- ~, r - 0.64, o- - 0.060, (8)

~ / [ . - - 8.38 + 48.9k-~ - 85.0k- t + 49.6/~,

r = 0.65, o" = 0.060. (9)

An increase in the degree of correlation does not significantly improve the correlation coefficient, nor the estimation error. This coincides with the results obtained by other authors [5, 25]. In any case, the correlation indexes are very poor. The estimations of the theoretical values of 1"~ by means of these expressions have a relative MAD that is higher by 15%, in every case.

0.4

K s

• ~ . . ~ ..~ ; 7 ~ -~ • •

~ • i : ' • ~.. :~ ~. ' i• <?~-. • . . . .

0 0.2 0.,# O.S

Fi~. 2. Pairs of experimental (k t ,kd) values registered in Valencia (1990 1991).

0.5

0.4

~.3

0.2

O . 1

~°

• . J . . .

O 0 . 2 0 . 4 0 . 6

Fig. 3. Experimental monthly average daily values of l~/1, vs k,, for Valencia (199g 1991 ).

98 Data Bank

To improve the results which provide the simple cor- relation, Collares-Pereira and Rabl [13] have developed a multiple correlation of the type [d/[V = f(k't, e),). From data registered in five stations in the U.S.A., the following expression has been proposed :

I d / I T = 0.775 +0.347(e)~-Tz/2)

- [0.505 + 0.261 (~o~-- rt/2)l cos [2(k-~ - 0.9)], (10)

where ~9~ is the sunset hour angle. The expression (10) has been applied to the registered data

in Valencia. The estimated values of [d present a relative MBD of 36% and a relative M A D of 36%. Therefore, we can conclude that the coefficients proposed by Collares- Pereira and RaN have a local character.

3.2. Use o f distribution fimctions The use of distribution functions was introduced by Liu

and Jordan [4] through a cumulative function of distribution F(kO, defined as a fraction of time in which k, is smaller than a determined value k*. Liu and Jordan established that if two series of values of k, have the same average value k~, they generate similar functions of distribution. That is, two different monthly series corresponding to two different localities that have the same k-, are governed by the same function of distribution. This rule implies that there exists a "universal two-variable" function F(kt,k-t) valid for any location in the world and for any month of the year [35].

If F(k*, Et) is the fraction of time during which the clear- ness index is below k*, the fractional time distribution is the integral of the frequency distribution. Liu and Jordan associate the fractional time distribution with the mean value k-,, so that :

£, = k~dF. (11)

The existence ofF(k*, k-t) leads to the definition of a universal generalized probability density function (PDF) represented by P(kt, k-t), so that P(k*, k-,) dk* represents the probability that an event takes the value k* ~< kt ~< k* +dk* . Therefore, the relationship between the distribution function and the probability density function is given as :

= fi'* P(k,,k-,)dk, (12) F(k*,k-,)

and the monthly average value of the clearness index can be expressed as the following :

kiM

~, = [ k , P ( k , L ) d k , . (13) ,Jkln,

Similarly, it is also possible to define :

c k,/M

ffd = [ kaP(kd,ffd)dkd, (14) .Jld,,,

where P(kg, k-e) dk* represents the probability that an event takes the value of k* ~< ka ~< k* +dk~'.

The use of equation (14) can be avoided if we assume the existence of a functional relationship between the daily values of ka and kt, and the average value of ka could be determined from the PDF of k, by means of:

f kLM ffd = ] f d ( k t ) P ( k t , k - t ) d k t . ( 1 5 )

m,

In fact, throughout a day it is possible to express kd by :

ka = I,~/Io = (l j l~)(lT/Io) = ktld/[r. ([6)

On the other hand, in Section 2. l it has been shown that for daily values it is possible to establish relations of the type ld/lr - f ( k O . So we can form an expression :

I kiM

k-d = kd(k0P(k,, E0 dk,. (17) m,

This expression provides the monthly average daily values of ka as a function of the PDF ofk t and the relation Ij/IT -./ '(kO obtained from the daily values.

In the last few years, several analytical expressions of a general character for the PDF of kt have been proposed. They are based on the analysis of sets of daily [36, 37], hourly [35] and instantaneous (1 min) [38] values. Other expressions of a regional character have also been proposed [3942].

In this paper the PDF of Bendt et al. [36] has been chosen, because Bendt's PDF was obtained from daily values of kv, and their validity is not restricted to a region of a determined climate. This leads to the expression :

P(k~, k-t) = Ce v~', (18)

where :

C = ),/(e:% e:~,o). (19)

In equations (18) and (19), 7 is an implicit function of k-, that can be obtained by means of an iterative process, and ktm and k,M are the min imum and maximum values of k, in the considered period.

If we assume that the daily values of diffuse radiation are related to the clearness index by means of the linear relationship Id/Iv -- A + Bkt (see Section 2.1), then the inte- gration of equation ( [ 7), using the PDF of Bendt et al. gives :

I~ a = (A - 2B/7)k-, + B(k~M e rk,M -k~,, e"%,)(e:'*,~ - e rk,,,,) '. (20)

The expression (20) has been used to estimate the values of k-d corresponding to Valencia from two different sets of values of),. In the first case (~) the real value o f k , , for every month has been taken. In the second case (3'2), a constant value of kt, n = 0.05 has been considered for all the months, as proposed by Bendt et al. The A (intercept) values and B (slope) values used are the monthly values given in Table 2, as with other additional values obtained by grouping the daily values in other intervals of the year.

In Table 3, the obtained results from the two sets of 7 values are given ; the deviations of the estimated values con- cerning the experimental values are also indicated. The results slightly improve when the real value of ktm of every month is considered. In this case, the average value of the deviations is 1.3% when all the fittings are considered. For the set of 12 monthly fittings the average value of the devi- ations is 1.4%. I f a constant value equal to 0.05 is considered for k,m, the average value of the deviations is 3.7% when all the fittings are considered, and 2.8% for the set of 12 months.

in order to compare the results provided by this method with those by the deterministic method described in Section 3.1, it is essential to establish the deviations that are intro- duced in the estimation of the diffuse radiation values [a. This leads to additional problems that affect the accuracy of the estimation, since the conversion of k-a to [a and To is not mathematically correct, because k d is defined as ld/L~. The determination of [a from E~ implies a constant [o in the considered period. Nevertheless, to reach a mean assessment which compares the methods based on the direct correlation

Table 3. Calculated

Data Bank

deviations for the estimation of k d values from the PDF of Bendt et al. [36]. Method 1 : real ktm. Method 2 : ktm = 0.05

99

Period Kd.p Kd.,~. ~, Deviation (%) Kdm~,~_ ~2 Deviation (%)

January 0.187 0.198 6.2 5.9 0.193 7.0 3.2 February 0.197 0.204 10.8 3.5 0.201 11.2 2.3 March 0.263 0.263 3.8 0.0 0.255 5.0 3.0 April 0.213 0.215 6.8 0.9 0.208 7.7 2.3 May 0.226 0.229 7.7 1.3 0.224 8. l 0.9 June 0.234 0.231 6.3 1.3 0.230 7.1 1.7 July 0.225 0.226 8.2 0.4 0.220 9.2 2.2 August 0.233 0.232 7.6 0.4 0.230 8.4 1.3 September 0.227 0.225 3.5 0.9 0.215 4.7 5.3 October 0.213 0.215 3.8 0.9 0.206 5.1 3.2 November 0.182 0.182 7.6 0.0 0.177 8.5 2.7 December 0.217 0.220 2.2 1.4 0.204 3.9 6.0 Warm-season 0.226 0.225 5.0 0.4 0.216 5.8 4.4 Cold-season 0.209 0.204 3.4 2.4 0.191 4.5 8.2 Winter 0.202 0.202 3.9 0.0 0.191 5.0 5.4 Summer 0.231 0.225 5.9 2.6 0.221 6.7 4.3 Spring 0.232 0.232 5.5 0.0 0.224 6.2 3.5 Au tumn 0.207 0.204 3.8 1.5 0.193 5.1 7.2

with the methods based on the distribution function, a set of estimated values of/d have been calculated from the Ard values given in Table 3 and the monthly average values of extra- terrestrial radiation. These estimated values have been com- pared with the [d values obtained by means of the subtractive method from the records of direct and global radiation in 1990 and 1991 (dependent test). When the PDF proposed by Bendt et al. is used with a constant value of 0.05 for k,m, the eslimated values are affected by a MBD of 0.3% and a M A D of 8.4%, and by a MBD of 3.4% and a M A D of 8.6% for the years 1990 and 1991, respectively. If the real value of ktm corresponding to every month is used, a M B D of 2.4% and a M A D of 6.8% are obtained for the year 1990, and for 1991 a MBD of 0.6% and a M A D of 8.0% are obtained.

These results confirm that the estimation slightly improves when the real value of ktm corresponding to every month is used. In Fig. 4, the theoretical values of the monthly average daily radiation estimated by means of this method (ktm real)

%./ "

Y~(MJ~2;

2 4 6 8 i~ 12

Fig. 4. Compar ison of the calculated values of the monthly average daily diffuse radiation with the experimental values of the monthly average daily diffuse radiation, for Valencia. The continuous line represents 1990 monthly values. The

dashed line represents 1991 monthly values.

against the experimental values of the monthly average daily radiation corresponding to [990 and 1991 are represented. A simple fitting yields :

(/d)rhe,,,- = --0.73+l.09(fd)Exp, r 2=0 .97 , a ( % ) = 6 . 7 , (21)

(~0Theor = 0.58+0.91([d)e~p, r 2 = 0.94, 0"(%) = 8.3, (22)

for the years 1990 and 1991, respectively. The method based on the distribution function performs a significant improve- ment concerning the one described in the anterior section. Working with daily values considerably increases the amount of experimental data, having a wider variety of values in a more ample interval of values of k~.

4. C O N C L U S I O N S

The estimation of the daily values of la from the relations of the type 1~/1~ =f (k t ) and the relations of the type kd = f ( k 0 is affected by considerable inaccuracy. If the error is evaluated with the MAD, this results in being about 20%. When the daily data are grouped monthly, the estimation errors are reduced in the months of the warm season (March September). These results show that due to atmospheric variability, the clearness index on its own is insufficient to explain the evolution of the daily values of diffuse radiation.

The monthly average daily values of diffuse radiation are poorly correlated with the clearness index. The correlation coefficients for the linear and quadratic fits are not higher than 0.65. In any case, the estimated values of [d by means of methods based on correlation with the clearness index give a M A D that is always higher than 15%.

The probability density function of Bendt el al. provides excellent results in determining k'd. The deviations that are introduced by this PDF can be lessened if the real value of k,m corresponding to every month is used in the estimation. From the PDF provided by Bendt et al. with a minimum real ktm, a set of theoretical values of /d has been made for Valencia. The mean deviation of these theoretical values concerning the experimental values is 6.8% for the year 1990 and 8.0% for the year 1991.

100 Data Bank

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