Correlations between values of daily horizontal beam and global radiation for Beer Sheva, Israel

Energy Vol. IY, No. 7. pp. 751-764. 1994

Pergamon

Copyright 0

03604442(93)00@23-H

I!994 Elscvier Science Ltd

Printed in Great Britain. All rightsreserved 03W5442/Y4 $7.00 + 0.00

CORRELATIONS BETWEEN VALUES OF DAILY HORIZONTAL BEAM AND GLOBAL RADIATION FOR BEER SHEVA, ISRAEL A.

IANETZt$

and A. I.

KUDISH§~

tDepartment of Geography, Bar-Ban University, Ramat-Gan, 521001 and Israel Meteorological Service, R&D Division, P.O.B. 25, Bet Dagan 50205 and #Solar Energy Laboratory, Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel (Received

18 August 1993)

Abstract-Individual monthly and seasonal correlations have been developed for Beer Sheva, Israel, for calculating daily values of horizontal beam radiation from measured global radiation via the clearness index. The data base comprises normal incidence and global radiation measurements. The dependent variable in the correlations was either the beam fraction or the beam transmittance; the latter is defined as the ratio of the horizontal-beam to the extraterrestrial radiation. Three different correlation formats were tested, based on the linear beam fraction and the exponential and linear beam transmittances. The most recent 12 months of data were reserved to validate the empirical regression correlations and to compare their predictive utilities. The results of our analysis for this site showed that replacing the beam fraction with beam transmittance as the dependent variable did not improve the predictive capability of the correlations significantly. The linear beam-transmittance correlations were somewhat inferior to the usual linear beam fraction format. Also, the performance of the exponential-beam transmittance relative to the linear-beam-fraction correlations was erratic (viz. superior for some months and inferior for others). The predictive capabilities for the seasonal correlations were comparable to those for the individual monthly correlations. The correlations for both the linear-beam-fraction and exponential-beam-transmittance formats varied significantly throughout the year, with values of the average monthly coefficients of variation varying from 20%.

INTRODUCTION

Whereas

global

radiation

is measured

at many

meteorological

stations,

the number

of stations

due to budgetary constrictions with regard to equipment and staff. Beam radiation measurements require both more expensive instrumentation (viz. the additional cost of the tracking device) and daily maintenance for the adjustment of the declination angle of the tracking device, which is especially troublesome in remote automated stations. This problem has given impetus to the development of empirical correlations between beam and global radiation. These correlations usually take the form of the beam or diffuse fraction of the global radiation as a function of some kind of normalized index of the global radiation and may be reported on either an hourly or daily basis. The most common format for reporting such correlations is as the daily diffuse fraction of global radiation as a function of the daily clearness index, K,, i.e. the ratio of global to extraterrestrial radiation, KT = H/H,,. It has been observed that such correlations are generally a function of the location of the meteorological station, viz., they are not universal. The meteorological station of the Solar Energy Laboratory at the Ben-Gurion University of the Negev, in Beer Sheva (31”15’N, 34”45’E, 315 m MSL) measures both normal incidence measuring

beam

radiation

is much

smaller

#This work is based upon research performed in partial University. ( To whom all correspondence should be addressed.

fulfillment

751

of the

requirements

for a Ph.D.

at Bar-llan

752

A.

IANETZ AND A.

I.

KUDISH

beam radiation and global radiation on a horizontal surface. In a previous publication,’ we reported seasonal and annual correlations between the beam fraction, H,,/H (i.e. the ratio of the daily beam to global radiation on a horizontal surface), as a function of the clearness index, KT_ The analysis was constrained to seasonal and annual correlations, since the individual monthly data bases were at that time relatively narrow. We are presently in possession of sufficiently broad individual monthly data bases as a result of the expansion of our data base during the time elapsed since the previous publication was prepared. These enable us to develop individual monthly correlations. The analysis has been further expanded to include monthly correlations between the daily beam transmittance K,,, defined as the ratio of the horizontal beam to the extraterrestrial radiation (HJH,,), in addition to the beam fraction format as reported previously as functions of KT. There are, a priori, two advantages to utilizing the beam transmittance rather than the beam fraction as the dependent parameter: (i) in this format, the parameters Kb and KT are measures of the fraction of the extraterrestrial radiation incident on a horizontal surface available as either beam or global radiation, respectively. In other words, they represent the fraction of the maximum possible amount of solar radiation that is incident on the Earth’s surface, available as either beam or global radiation. In the beam fraction format, it is possible to observe relatively high values for the beam fraction corresponding to low values of KT (e.g., under partially cloudy conditions). This result can be misleading from an energetic point of view, especially with regard to concentrating solar collector systems which only utilize beam radiation. (ii) Furthermore, K. is the ratio of a measured parameter (beam radiation) to the corresponding value for the extraterrestrial radiation that is calculated from astronomical considerations. In the beam fraction format, the correlation utilizes a ratio of two measured parameters (viz., beam and global radiation on a horizontal surface), each with its intrinsic error of measurement. Thus, one would expect a reduction in the standard deviation of the individual monthly correlations utilizing the beam transmittance relative to those based upon the beam fraction. Liu and Jordan’ were among the first to analyse solar radiation data and report correlations between diffuse fraction and clearness index. The number of publications reporting such correlations, since their pioneering work, are too many to list (the interested reader is referred to Refs. 3-4 for more details). We prefer to report our data analysis in the complementary beam-fraction format, since our meteorological station measures normal incidence beam radiation rather than diffuse radiation. The complementary nature of the two formats is evidenced by the change in sign of the slope exhibited by the correlations, viz. positive slopes in the case of the beam fraction and negative slopes in the case of the diffuse fraction. In addition, if the diffuse radiation is measured directly by means of a pyranometer equipped with a shadow ring, the most prevalent technique, both geometric and anisotropic correction factors must be applied to the individual measurements. The magnitude of the correction may also vary throughout the day.5*h Thus, if the accuracy of the tracking device of the normal incidence pyrheliometer is carefully monitored, one would expect such data to be more accurate and, as a consequence, provide a superior correlation as it obviates the need to apply correction factors. Consequently, the diffuse radiation determined from the difference between the global and horizontal beam radiation will also be more accurate than measuring it directly utilizing a shadow ring. There are much fewer papers reporting correlations between beam and global radiation. Anderson ,’ referring to a manuscript that apparently was never published, suggested the following exponential correlation between beam and global radiation; H,, = crHB;

(1)

she applied it to data for Madingley Wood (about 2 km from Cambridge), England. Becker’ reported two correlations based upon monthly average daily values for Chiva-Chiva,

Daily horizontal beam and global radiation

753

Panama. In this case, the monthly mean daily beam radiation was calculated from the difference between measured global and diffuse (using a shadow ring) radiation. He correlated his data both in a linear logarithmic format as suggested by Anderson (viz., linearization of the exponential format) and the usual format, i.e. beam fraction of global radiation as a function of clearness index. He observed that the former was slightly curvilinear, whereas the latter was linear. The results of his analysis are described by log Hi,__ = - 1.98 + 2.33 log HdVg

(2a)

H,,_lHavg

(2b)

and = -0.14

+ 1.26K,,,,,,

with a coefficient of correlation of 0.948 for both cases. He also observed that the data for a number of months characterized by relatively clear skies were lower than expected (i.e., these values were below those predicted by the average yearly empirical equation) and hypothesized that this was due to increased turbidity. He was unable to validate this explanation, since no turbidity data had been recorded for Chiva-Chiva area. Kalma and Fleming’ normalized Anderson’s correlation by dividing both the monthly average daily beam and global radiation values by the average monthly extraterrestrial radiation, HO,a,,gin an attempt to remove the site dependence observed in the case of the beam (diffuse) fraction correlation format. They were successful in correlating average monthly data from 14 sites (from 28”35’N to 51”24”S, 12 of the sites being in the southern hemisphere) by the single correlation &,a&,,avg

= 0.~32(&,I&,..,)z

29,

(3)

but they did not report any indication of the data fit to the correlation. Jeter and Balaras”’ and Balaras et al” developed a correlation between the “beam transmittance of the atmosphere” tb, defined as the ratio of normal incidence beam to the normal incidence extraterrestrial radiation, and the clearness index kT for hourly values. In the former case, normal incidence beam radiation was measured directly; in the latter case, it was calculated from shadow ring diffuse radiation measurements. The rh and kT data were grouped into eight clearness index bands (0.05 < k,>0.85) and the regression model to assure continuity is (rhj - Yoi)= Bi(k, - ~0) + sj,

(4)

where x0; = lower limit of band i, yoi = regression model at x0;, /?; = regression coefficient, Ej = residual error for data (kTj, tbj). Jeter and Balaras found a coefficient of correlation of 0.942 based upon a 5-year database (8112 sets of data) from the Solar Total Energy Project at Shenandoah, GA. Balaras et al utilizing 2 years of data from the National Observatory of Athens, Greece (5292 sets of data), arrived at a coefficient of correlation of 0.917. In neither case were data measured at the site reserved for correlation validation studies. In fact, in none of the above described studies were any validation studies performed on independent sets of data, measured at the respective sites, reported.

MEASUREMENTS

The normal incidence beam radiation is measured using an Eppley Normal Incidence Pyrhehometer, Model NIP, and the global radiation is measured using an Eppley Precision Spectral Pyranometer, Model PSP. This station is part of the national network of meteorological stations and the instrument calibration constants are checked at regular intervals by the Israel Meteorological Service. Each of these radiation-measuring instruments was initially

154

A. IANETZ AND A. I. KUDISH

connected to an electronic integrator and printer but they are presently connected (since July 1988) to a rechargeable battery powered datalogger (Campbell Scientific Instruments). The radiation data analysed in this study were collected during the time interval 1984-1993. Due to mechanical and electrical failures (prior to the addition of the rechargeable battery powered datalogger), the actual amount of data available for analysis was reduced significantly. The rechargeable battery powered datalogger scans the inputs every 10 set and records average values at lo-min intervals on a tape cassette. The data are later processed on a Macintosh computer to determine the average hourly radiation values, which are referred to as solar time. Only those days for which a complete set of beam and global radiation measurements were available are included in the analysis. The validity of the individual hourly values were checked in accordance with WMO recommendations.‘2 Those values which did not comply with the WMO recommendations were considered erroneous and rejected. The measured hourly normal incidence beam-radiation values, 1,,,, were converted to the beam radiation incident on a horizontal surface, I,,, by applying the geometric conversion factor to individual hourly values, viz. 1, = & cos e,,

(5)

where 8, is the average hourly incidence angle. These hourly values were in turn summed to determine the daily beam radiation incident on a horizontal surface, Hb.

RESULTS

The data were analysed and correlations developed on a monthly and seasonal basis. The initial individual monthly data bases, utilized in developing the correlations, varied from a maximum of 191 (January) to a minimum of 79 days (August). A set of 12 monthly and four seasonal correlations (the seasons being defined as in the previous publication,’ cf. Table 5) were developed for each of the following formats: (a) beam fraction as a linear function of clearness indexHJH

(b) beam transmittance

= a + bK,;

as a linear function of clearness indexK,,=c+dKT;

(c) beam transmittance

(6)

as an exponential

(7)

function of clearness indexKb = eKk.

(8)

In addition, a yearly correlation was developed for the linear beam fraction format in order to compare it to that reported previously.’ The A.ltab..ises for January, April, July, and October (mid-month for each season) are shown in Figa. 1-4, in both the beam-fraction and beam-transmittance formats. We observe that the latter are, in general, slightly curvilinear, a preliminary indication that a linear beamtransmittance correlation may not be suitable and thus encouraging us to develop exponential correlations. Nevertheless, linear beam transmittance correlations were developed for comparative purposes, since this slight curvilinearity was not as obvious for all months. The correlations corresponding to Eqs. (6-8) are listed in Tables l-3, respectively. Also reported in Tables l-3 are the number of days of data for which the correlation was developed (any day for which the standardized residual was >2, was rejected and this reduced the individual monthly databases by 2.3-5.6%) and the coefficients of correlation. The last 12 months of data recorded at the site (March 1992-February 1993) were reserved for validation studies. The results of these studies for the case of the beam fraction as a function of the clearness index, which are representative of all three formats, are reported in

+

+

4

W”

++

++++, +

+

+%+

+ +

+++++b ++

$++++L + l +

x

tc++*+ +: ++ ++ +++ + + ++* +I,’

+&+d++

++

TRANSMITTANCE

Ip+ ++ + +++ + +*++* ++ + + +

;=+ + ;+ +++ + + +++ + ++ + + + *+t-t

$

t

BEAM

L

0

b,

0

b

0

h,

0

b

00

0

++++ +

b

+

+

+

4

+

$+ +

+

.

t

+

+

+

0

0

I

T

,+

+++++‘+

+

++

+

+

+*++

+

,

en

0

++ +

+++ +%*

+* ++ ++ e+++

+$++Z+t+ + # +

*+

+

+$

+$+ +t$*++++#

+++ *+ ++ ++ 44

+.+$

++t++*

,

b,

0

FRACTION

+ +7 ++ + ++ + + + # ++;+ +$+ + ++ + + ++

++t

+

$+

I

;c,

0

BEAM

A b

A. IANEIZ AND A. I. KUDISH

756 1.0

(a) April

+$f+-+ +:+>#Jf +

0.8

pbt+ + + ++++p ++ + + + .+++:>* *+ ++ +++++ + + + ++ ++ ++ + + + + + +* + + + + + + + .

I

0.0

+

0.2

0.0

CLEARNESS

I

I

0.4

0.6

0.8

INDEX

0.8 (b) April

%$

*+4+

0.6 $

+I++

+k ++++t+ +++ Q 4*+++ +$

+ +

$;+I++?+ +++ + ++*

+ %* ‘+ + ++ ++ + + 0.0

0.2 CLEARNESS

+ 0.4

0.6

0.8

INDEX

Fig. 2. Database for April: (a) beam fraction H,,/Has a function of clearness index K,.; (b) beam transmittance K, as function of clearness index K.,.

coefficient of correlation and the coefficient of variation reported as a percent [%CV]. The latter is defined as the ratio of the RMSE to H ,,_(calc). It gives an indication of the average error to be expected in calculating the daily horizontal beam radiation from the correlation. In addition, the values for the monthly coefficients of variation for the three formats are reported in Figs. 5 and 6 for the individual monthly and seasonal correlations, respectively.

+ +

BEAM

+g* ++ +++

1F$+

*++b9

++ +Y

++ *+ + ++++%+ ++++++ +

TRANSMITTANCE

00

0

+ ++

OD

+

0

al

FRACTION

0

BEAM

-6 b

T +z$+ it

*Hw++ + ++ * ++*:#+++

TRANSMITTANCE

$* + + ++ +*+ + +

++ +

BEAM

1

h,

0

b,

0

b

0

N

0

0

00

0 L

0

+

+

++

BEAM

A

a

VI

0

$2

0

0

G + =++**+++++ + #+ + ++ =+ %

+++ +++++

+

+ + *+ + + + + + +

b

0

FRACTION

A b

Daily horizontal beam and global radiation Table 1. HJH Month

183

May

137

June July Aug. Sept.

88 99 75

Oct. Nov. Dec. Winter Spring Summer Autumn Year

vs K,., linear beam fraction empirical correlations.

No. of days

Jan. Feb. Mar. Apr.

172

144 137

85

Linear correlation

Correlation coefficient

H,IH = -0.263 H,IH = -0.310

H,,/H = HJH H,,IH H,,IH H,,IH H,,IH HJH

= = = = = =

101 114

H,,IH =

96

H,,IH =

431 403 249 285 1293

H,,IH = HJH

=

H,,IH = H,,IH =

H,,IH = H,,IH =

+ 1.475 K, + I.508 K, -0.330+ 1.516 K, -0.498 + 1.735 K, -0.417 + 1.646 K, -0.719 + 2.080 K, -0.757 + 2.172 K, -1.005 + 2.560 K, -0.474 + 1.766 K, -0.359 + 1.566 K, -0.270 + 1.503 K, -0.301+ 1.532 K, -0.280 + 1.483 K, -0.410 + 1.627 K, -0.702 + 2.089 K, -0.247 + 1.421 K, -0.307 + 1.505 K,

Table 2. K, vs K.,., lineear beam transmittance Month

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Winter Spring Summer Autumn

0.874 0.831 0.885 0.935

0.924 0.894 0.874

0.951

Linear correlation

Correlation coefficient

186 175 145 141 138 87 100 75 85 103 115 97 447 410 247 288

K, = -0.244 + 1.026 K, K, = -0.286 + 1.lOO K, K, = -0.396 + 1.278 K, K, = -0.559 + 1.508 K, K, = -0.560 + 1.532 K, K, = -0.968 + 2.123 K, K, = -0.985 + 2.171 K, K,=-1.064+2.302K, K, = -0.641+ 1.669 K, K, = -0.557 + 1.519 K, K, = -0.373 + 1.264 K, K, = -0.267 + 1.047 K, K, = -0.275 + 1.078 K, K, = -0.501 + 1.437 K, K, = -0.929 + 2.088 K, K, = -0.488 + 1.371 K,

0.907

No. of days 182

Feb. Mar.

169 143

Apr. May June July Aug. Sept.

139 140 88 98 75 85 102 115 92 416 396 247 290

Nov. Dec. Winter Spring Summer Autumn

0.927 0.905 0.877 0.787 0.898 0.929 0.803

No. of days

Jan.

Oct.

0.907 0.949

empirical correlations.

Table 3. K, vs K.,., exponential beam transmittance Month

759

Exponential correlation K, K, K, K, K,

= = = = =

K; K; K; K:

‘et’ “’ ‘W “““

K = 1,508 K’ (‘I” Kh= t,564,&‘” Kh = 2.040 ,& 65: K; = 1.548 K; “” K = K,h= K, = K, = K, = Kh=

0.958

0.885 0.929 0.906

0.917 0.930 0.938 0.950 0.932

empirical correlations. Correlation coefficient

2. I02 K; x3 1.837 1.677 1.537 1.501

0.940 0.931 0.920 0.913 0.885 0.945

1.397 K’ ‘W 1.428 Kj”” 1.846 K; “W 1.793 K; ‘I” 1.540 K; “W 1.478 K’ “”

K h = I.289 K:,.‘Jc1

0.957 0.975 0.970 0.964 0.941

0.895 0.94x 0.965 0.929 0.95 1

0.919 0.939

0.970 0.977

0.933 0.951

760

A. Table 4. Validation

No. of Month Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

93 93 92 92 92 92 92 92 92 92 92 92

A. I. KUDISH

from monthly

H WWg

H,,.ilvg

(talc)

(meas)

6.462 8.334 9.757 14.210 16.595 19.727 21.019 18.704 14.732 11.4% 6.771 4.308

28 25 26 25 28 30 27 26 29 28 27 27

AND

of H,, (as calculated

(MJlm*)

days

IANETZ

H,/H

vs K.r linear correlations.

RMSE

(MJ/m*) 6.140 7.188 8.017 12.785 14.451 18.192 19.726 18.463 14.281 11.727 6.586 4.099

MBE

(MJ/m’) 0.768 1.600 2.567 i ,996 2.445 1.842 1.690 0.978 1.019 0.744 0.882 0.802

Correlation coefficient

(MJlm*) 0.322 1.147 1.741 1.426 2.143 1.534 1.292 0.240 0.451 -0.252 0.185 0.201

0.988 0.981 0.939 0.976 0.983 0.976 0.863 0.933 0.946 0.911 0.972 0.976

%CV 11.9 19.2 26.3 14.1 14.7 9.3 8.0 5.2 6.9 6.5 13.0 18.6

As mentioned previously, we reserved the last 12 months of measured radiation data (i.e., from March 1992 to February 1993) to perform validation studies on the correlations developed (36 monthly and 12 seasonal). The results are summarized in Tables 4 and 5 and Figs. 5 and 6. We performed a comparative analysis of the results of the validation studies using the same database. In this analysis, the results of the two proposed formats (exponential and linear Kb) are compared to those for the beam fraction H,,/H for monthly and seasonal correlations. The pertinent results of this analysis follow. Monthly correlations Monthly average daily horizontal beam-radiation values. In accordance with the MBE analysis, the monthly correlations tend to overestimate the monthly average daily horizontal beam radiation values with the exception of October, regardless of format. Coefficient-of -correlation values. There is no significant difference between the corresponding average monthly values for the coefficient of correlation of the exponential Kh and linear k&,/H formats. The former yields an average improvement in the monthly correlation Table

Month

5. Validation

of H,, as calculated

from seasonal

H h.avg

HO.il”g

(talc)

(meas)

H,,/H vs K.,. linear

No. of days

RMSE

(MJ/m’)

(MJ/m’)

NJ/m*)

correlations.

MBE

(MJ/m*)

Correlation coefficient

%CV

Winter Dec. 92 Jan. 93 Feb. 93 Spring Mar. 92 Apr. 92 May 92

27 28 25

4.271 6.328 7.914

4.099 6.140 7.188

0.816 0.727 1.272

0.166 0.188 0.726

0.976 0.989 0.981

19.1 17.4 16.1

26 25 28

9.496 14.623 16.458

8.017 12.785 14.451

2.434 2.337 2.314

1.480 1.838 2.007

0.940 0.974 0.983

25.6 16.0 14. I

June 92 July 92 Aug. 92

30 27 26

20.375 20.948 18.307

18.192 19.726 18.463

2.427 1.636 0.861

2.183 I.221 -0.156

0.976 0.862 0.937

II.9 7.8 4.7

Sept. 92 Oct. 92 Nov. 92

29 28 27

14.768 11.802 6.498

14.281 11.727 6.586

I.136 0.739 0.923

0.486 0.076 -0.088

0.942 0.901 0.972

7.7 6.3 14.2

Daily horizontal beam and global radiation

761

40

MONTHLY

CORRELATIONS

0 l

30

l

.

+ +

0

20

0 +

10

0

MONTH Fig. 5. Comparison of percent coeflicient of variation values for individual monthly correlations for linear beam fraction 0, linear beam transmittance 0, and exponential beam transmittance +

coefficient values of about 1%. The linear K,, format yields the poorest results, viz. an average decrease relative to the linear beam fraction of about 3%. Coefficient-of-variation values. This parameter is probably the most significant one in our analysis, since it gives an indication of the average error to be expected in calculating the daily horizontal beam radiation from the empirical regression correlations as a function of the measured daily global radiation (i.e. KT). The monthly average values for the percent coefficient of variation for the three formats studied are shown in Fig. 5. The monthly average values for the coefficient of variation for the exponential K,, format exceed that for the linear H,,/H format for five months (January, February, March, June, and August), essentially equivalent for July and October and are lower for the remaining five months. In the case of the linear K,, format the monthly average values for the coefficient of variation exceed that for the linear HJH format for 10 months, essentially equal for July and lower for September. Seasonal correlations Monthly average daily horizontal beam radiation values. The seasonal correlations also tend to overestimate the monthly average daily horizontal beam radiation values with the exception of August and November (also January for the linear K,, format) based upon the MBE analysis. Coefficient-of-correlation values. The results of the analysis are similar to those for monthly correlations. The exponential K,, format yields an average monthly improvement in the correlation coefficient values of about l%, whereas the linear Kh format yields the poorest results, i.e. an average decrease relative to the linear beam fraction of about 3%. Coefficient-of-variation values. The monthly average values for the percent coefficient of variation for the three formats studied are shown in Fig. 6. The monthly average coefficient of

A.

IANETZ

AND

A. I. KUDISH

40

SEASONAL

CORRELATIONS

30

20

10

b

0

MONTH Fig. 6. Comparison of percent coefficient of variation values for seasonal correlations for linear beam fraction 0, linear beam transmittance 0, and exponential beam transmittance +

values for the exponential Kb format exceed that for the linear H,/H format for February and June, are essentially equivalent for July, August, and November and are lower for the remaining eight months. In the case of the linear K,, format the calculated coefficients of variation exceed that for the linear H,,IH format for 9 months are essentially equivalent for the summer season. Based upon the above, it is apparent that the linear K,, format offers no improvement over the linear H,,/H format, in fact its performance is judged to be inferior. It is also questionable, whether the exponential Kh format has any added merit relative to the linear H,,/H format. In general, it is observed that the percent coefficient of variation varies significantly throughout the year. The predictive capability of the monthly correlations for all three formats are best from June to October (5.2 < %CV< 9.7). The linear beam fraction correlations exhibit a %CV between 11.9 and 19.2% for the remaining months with exception of February (%CV = 26.3). The exponential beam transmittance correlations exhibit a %CV between 12.1 and 16.4% for the remaining months with exception of February and March for which it is 25.2 and 27.3%, respectively. The linear beam transmittance correlations exhibit a %CV in excess of 20% for the remaining months with exception of May (%CV = 17.1). The trends are quite similar for the seasonal correlations, cf. Figs. 5 and 6. The enhanced predictive ability of the correlations, irrespective of format and type for June through October, may be a result of the relatively high frequency of clear days and also the relatively high average daily clearness index values (KT > 0.6) observed during this time interval,i3 i.e., of the prevailing climatic conditions in Beer Sheva. In order to determine the minimum number of equations necessary to calculate the daily horizontal beam radiation throughout the year adequately for this site, we compare relative predictive abilities for the monthly and seasonal correlations. We observe that based upon a variation

Daily horizontal beam and global radiation

163

comparison of the validation studies there does not appear to be any distinct advantage to using 12 individual monthly as opposed to four seasonal correlations in the case of Beer Sheva. It is of interest to compare the linear beam-fraction format seasonal correlations reported previously for Beer Sheva’ with those reported in Table 1. They are observed to be very similar. The data base utilized in the present analysis is approximately 40% greater than that used previously (the individual seasonal data bases have been expanded by a minimum of -22% for spring to a maximum of ~74% for autumn). This is an indication of the relative stability of the climatic conditions in Beer Sheva. The yearly correlation is observed to be almost unchanged.

CONCLUSIONS

Empirical regression correlations, both individual monthly and seasonal, have been developed for calculating daily horizontal beam radiation from measured daily global radiation via the clearness index, KT. Three different corelation formats were tested: H,.,/H = a + bK,, K,, = c + dKT and K,, = eK&. It should be noted that the conclusions are based upon the data collected at a single site, Beer Sheva, Israel, and should not be generalized to sites with significantly different climatic conditions. The results of our analysis of the data for this site did not substantiate the hypothesis that replacing the beam fraction, Zf,/H, with the beam transmittance, K,, as the dependent variable would improve the predictive capability of the correlations. The linear beam transmittance correlations were observed to be somewhat inferior to the usual linear beam fraction format on the basis of the individual monthly validation studies. Also, the performance of the exponential beam transmittance correlations relative to the linear beam fraction correlations was erratic (viz., superior for some months and inferior for others). Based upon the data for Beer Sheva, Israel, we conclude that there is no clear advantage for preferring either of the formats (exponential beam and linear beam fractions). The predictive capability of both correlation formats varied significantly throughout the year with the values for the average monthly coefficient of variation varying between months from 20%. In addition, the performance of the seasonal correlations were quite similar to the individual monthly correlations. Thus, for this site there is no need to use individual monthly correlations to calculate the average daily horizontal beam radiation, viz., seasonal correlations are adequate. In all likelihood, the only way to further improve the predictive capability of such correlations is to expand them to include other independent parameters that affect the composition of the global radiation (e.g., turbidity, humidity, etc.). The major drawback being that such additional parameters are not always available at the site. If accurate values of beam radiation are essential the only rigorous solution may be direct measurements with a pyrheliometer. Acknowledgements-We

are indebted to E. Berman for partial support of this research by funding the purchase of the Normal Incidence Pyrheliometer system. We wish to thank M. Sonis of Bar-Ilan University and G. Stanhill of the Volcani Institute for remarks and suggestions during the preparation of this manuscript (thesis advisers to A.I.). We also wish to thank A. Manes and I. Seter of the Israel Meteorological Service for their encouragement of this joint research project.

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