Solar Energy Vol. 28, No. 4, pp. 307-311, 1982 0038-092X/82/040307-05503.00/0 Printed in Great Britain. Pergamon Press Lid,

DIFFUSE AND BEAM COMPONENTS OF DAILY GLOBAL RADIATION IN GENOVA AND MACERATA

B. BARTOLI, V. CUOMO and U. AMATO Istituto di Fisica della Facolth di Ingegneria, Napoli, Italy

and

G. BARONE and P. MATTARELLI SoGesTA, Urbino, Italy

(Received 1 July 1981; accepted 18 August 1981)

Abstract--In this paper we estimate, for the stations of Genova and Macerata, daily values of direct and diffuse radiation starting from daily values of global radiation. We propose a fit to the experimental points and analyse their statistical distributions.

We have estimated daily values of direct (B) and diffuse (D) solar radiation on horizontal surface starting from global radiation (G), for the stations of Genova and Macerata, using the same approach of Liu and Jordan [1].

The Liu and Jordan fit, with some variations depending on the station, gives satisfactory results all over the world except for very high latitudes [2-6].

We have used the data of daily global and diffuse solar radiation for the years 1964--69 (Macerata), and 1%4-72, except for 1969 (Genova), measured respectively by "Osservatorio Geofisico di Macerata"[7] and "Istituto Geofisico e Geodetico delrUniversith de Genova"[8].

We have analysed the ratios KD = D/G as a function of the ratio Kr = G / H where H is the extraat- mospherical radiation on horizontal surface, finding two satisfactory fitting formulae:

( _ bKr ~ K D = f ( K T ) = a + ( 1 - a ) ' e x p \ 1 - K T / (1)

Ko = g(Kr) = a + B Kr + y Kr2 + S Kr 3. (2)

In Table 1 we report the values of the parameters of the best fits and the corresponding root mean square deviations RMS:

RMS= [ ~ (KD,¢xp..- KD, c.~c..)2/N] 1/z

where N is the number of experimental points; n is the current index of experimental points; KD. exp is the experimental value of KD; and KD.¢.~c is the value calculated using the fit.

In order to evaluate the predictive power of the fit we have divided the interval of variability of K r into I00 equal sub-intervals Kr.i of length 0.01. For each interval Kr.i we have calculated the corresponding mean value /~D.i and the root mean square deviation ~rg.o.~ from the

mean value: / .Ii \

O'/~,D,i

where j is the current index of experimental points belonging to the ith sub-interval; J~ is the number of experimental points in the ith sub-interval.

In Figs. 1 and 2 we show KD as a function of KT in Genova and Macerata with both of the best fits and the Liu and Jordan curve. Both fits (1) and (2) are equivalent as shown also by the comparison of the corresponding RMS.

I GO

8 0

60

4 0

20

L l 0 0.20

Genovo

I 1 k I I L L I 0 .40 0.60 0.80 1.00

KT Fig. 1. Ratio of daily diffuse solar radiation on daily global radiation (/Co) as a function of the ratio of daily global solar radiation on extraterrestrial radiation (Kr) in Genova. The dotted curve is the Liu and Jordan fit; the dash-dotted curve is pol-

inomial fit (2) and the continuous curve is transcendent fit (1).

307

308 B. BARTOLI et al.

2 3

K D = ~ + 8 K T + Y K T + ~ K T

MACERATA GENOVA

.931 . 8 7 9

8 -.513 .324 Y -1.849 -3.724

5 1.613 2 . 6 5 8

P4~S . i 0 9 .I00

(bK~) K D = a + (l-a) exp - - -

1 - K T

a .IS4 .O712

b 1.062 .872

c .861 .947

RSM .i09 .iO1

points taken

into account

2022 2801

~ t 0.50 1.00

° N _o o o I x, ,

0"/~, O X X

o,,o l_x ~oXx~ x~X%~x~ [] l . x .~xx ,~ o ~ x ~ w ~ . . ~

o. ,o I -~9~= o x ; ~ . ~ ,~

o " ' 0'20 ' o',o ' o',o ' o'~o ' ,!oo 0 0 20 0.40 0.60 0.80 I00

KT

Fig. 2. Ratio of daily diffuse solar radiation on daily global radiation (KD) as a function of the ratio of daily global solar radiation on extraterrestrial radiation (Kr) in Macerata. The dotted curve is the Liu and Jordan fit; the dash-dotted curve is polinomial fit (2) and the continuous curve is transcendent fit (l).

Fig. 3. Root mean square deviation of experimental KD values from their correspondent average value as a function of Kr: the interval of variability of KT has been divided into 100 sub- intervals of length 0.01: for each sub-interval we have calculated the correspondent average value of KD and the root mean square

deviation of experimental points respect to it.

In the following considerations we will use the tran- scendental fit only since it has only three parameters.

In Fig. 3 we show ~;,¢.o as a function of KT; Cr2g.D is the variance of experimental points around the mean

value. Such variance shows which is the statistical limit in predicting diffuse radiation starting from global radia- tion.

For each sub-interval we have also calculated the root

Diffuse and beam components of daily global radiation in Genova and Macerata 309

mean square deviation ~g.D.~ of experimental points from the fit:

( K D . =xp. u - Ko.c,lc. U) 2 U2 ~e'~"=( ~'~ (],- 1) ) '

In Fig. 4 we show ~e.O as a function of KT. It can be easily seen by comparison with Fig. 3 that ~z.~ and ~e.D are of the same order of magnitude; this means that the fit approximates experimental points very well; therefore the error we make in calculating diffuse radiation is not due to the fit but to the statistical distribution of experimental points KD.~, around the mean value /(D.,.

Such dispersion depends on physical atmospherical

0.50 --

0 .40

0 3 0

x Moceroto o Genovo

o

0 2 0 - x x x x x,x~( _

x x v xx -Xn vx tJ

0 0. 20 0.40 0.60 0.80 1.00 K,.

Fig. 4. Root mean square deviation of experimental KD values from the transcendent fit (1) as a function of KT.

parameters as torbidity and water vapour; no infor- mation about these parameters can be extracted from the measurement of global radiation only.

Since our purpose is to estimate diffuse radiation D, we have also calculated ~rD, that is the root mean square deviation of calculated diffuse radiation with respect to experimental radiation.

We have divided the interval of experimental global radiation into 100 sub-intervals of equal length. For each

0 5 0 - -

0.40

0 . 3 0

x Mocerol"o o Genovo

o .2or - x x x x

l ~ o I- X x ,o x , x L _ ~ L

" / J 1 I I I I J t ~ I o 0.20 0.40 0.60 0 80 noo

K, Fig, 6. Root mean square deviation of experimental Ks values from the transcendent fit (3) as a function of ET: the interval of variability of Kr has been divided into 100 sub-intervals of length 0.01, for each sub-interval we have calculated the root mean square deviation of experimental values in the sub-interval

from the values calculated using fit (1).

I 0 0 0 0 F

I.- x Mocerol"o

80,00 t o Genovo

F 60.00 L ~3 x

/ ~ o x x o / o°~oo mo x

o o xxX= x~x~

[ ., Nn ' °

f J W i I i I i ~ I i I

0 160.O0 320,00 480.00 6 4 0 0 0 8(30.00 G

Fig. 5. Root mean square deviation of experimental value of diffuse radiation from values calculated using the transcendent fit as a function of daily global radiation: the interval of variability of global radiation (G) has been divided into 100 sub-intervals of equal length; for each sub-interval we have calculated the root mean square deviation of experimental values of diffuse radiation in the sub-interval from the value calculated using transcendent

fit (I).

iOQO0

8 0 0 0

6 0 0 0

40.00

2000

x Mocero'ro o Genovo

~] o ~ x'-~ ~oou - ° x

L~o~o o~p. x~ x o x

o~o ? o

n

I I t l J I I I r I 160.00 320.00 48000 640.00 80000

G

Fig. 7. Root mean square deviation of experimental KB values from the values calculated using fit (3) as a function of experi- mental global solar radiation G The interval of variability of global radiation G has been divided into 100 sub-intervals of equal length; for each sub-interval we have calculated the root mean square deviation of experimental values of beam radiation in the sub-interval from the values calculated using transcendent

fit (3).

310 B. BARTOLI et al.

I 00 ,00

80 .00

60. O0

4 0 0 0

2000

x MoceroTo

m Genovo

x o 13 x x x

×x ××~B ~ . ~ × x x ~ × ~ ' ~ [] ~4~ ~ %

~ x o % ~ x []

Xn X ~ [2 X X X {

x x x ~ % _

i I i I 0 40 .00 80 .00 12000 160.00 200 .00

D

Fig. 8. Root mean square deviation of experimental values of diffuse radiation from values calculated using the transcendent fit (1), as a function of experimental diffuse radiation D: the interval of variability of D has been divided into 100 sub-intervals of equal length; for each experimental value of each sub-interval we have calculated the value of diffuse radiation expected on the basis of fit (1); and for each subinterval we have calculated the root mean square deviation of experimental points from the

calculated values.

I00 .00

80.00

60.00

40.00

x Mocerol"o o Genovo

o x x~ xo ~ o

o x _ o x(~ o _~ o°°~ox xx~o × Xx × x L~oxx. x~ ~x o@ x

o x~.~_xx oxj ~ Oo i x'{~30 O{~r x xv ~On []

X O7~inl X 0 - X X ~ -- X)~X x X~ o

20.00 X y o x

I L 0 140.00

I i I J I A I 2 8 0 0 0 420 .00 560 .00 700 .00

8

Fig. 9. Root mean square deviation of experimental values of beam radiation from values calculated using the transcendent fit (1), as a function of experimental beam radiation B; the interval of variability of B has been divided into 100 sub-intervals of equal length; for each experimental value of each sub-interval we have calculated the value of beam radiation expected on the basis of fit (1); and for each sub-interval we have calculated the root mean square deviation of experimental points from the

calculated values.

sub-interval we have calculated:

M

OrD't = \ (ML - l) ]

%

I O0

080

0 . 6 0

0.40

020

x Moceto1"o

o Genovo

[] x o x x

x x~ x~_ x× X ~ .~13Xf~l r'r~13{:l~X 0

x o x ~ o x

x x x ~x x x ~

{~oD ODD0 O []

L I I I I I I I I I 0 4OO0 SO00 ~20.00 160.00 200.00

O

Fig. 10. Relative root mean square deviation of experimental values of diffuse radiation from calculated diffuse radiation (i.e. ratio of root mean square deviation on experimental diffuse

radiation), as a function of experimental diffuse radiation D.

%

8

1.00

1.80

x Mocero1"o

o Genovo

%

o.6o

0 . 4 0 - - x x

~x x x c] o

i I ~ i J I 1 ] t J 0 ~40.00 280.00 ,*20.00 560.00 700.00

B Fig. 11. Relative root mean square deviation of experimental values of beam radiation from calculated beam radiation (i.e. ratio of root mean square deviation on experimental beam radia-

tion), as a function of experimental beam radiation B.

where Dexp.L,, is the mth value of experimental diffuse radiation lying in the Lth sub-interval of global radiation; Dca~c.L,. is the calculated value, corresponding to D=~p.L,.; ML is the number of experimental points lying in the Lth sub-interval of global radiation; and m is the current index of experimental points.

In Fig. 5 we show fro as a function of Gexp. Using X 2 test we have shown that the experimental

values of Ko of Genova and Macerata are not com- patible--neither at a confidence level of 99 per cent nor at 95 per cent to be fitted by the same curve.

Of course the behaviour of ~D VS G depends on the choice of the physical quantities involved in the fit. For such a reason we have investigated if better results can

Diffuse and beam components of daily global radiation in Genova and Macerata 311

be obtained relating Kn = DIH to Kr ; and we have seen that no improvement is obtained by this new fit.

Of course B can be calculated from global radiation when D has been evaluated. In Figs. 6 and 7 we show ~:e.B vs KT and trB vs G; where ~:e.s is the equivalent of ~:e.o for beam radiation and tr8 is the equivalent of trt> In Fig. 8 we show the root mean square deviation or,, of experimental values of diffuse radiation from values calculated using fit (1) as a function of experimental diffuse radiation D. In Fig. 9 we show the root mean square deviation tra of experimental values of beam radiation from values calculated using fit (1) as a function of experimental beam radiation B. Since the absolute values of tro and orb are quite the same, the percentage deviation o d D is consequently comparable with o'dB only in overcast days; in clear sky conditions, as shown in Figs. 10 and I1 we have odD much higher than trdB.

REFERENCES

L B. Y. H. Jordan and R. C. Jordan, The interrdationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4(3), l (1%0).

Z N. K. O. Choudury, Solar radiation at New Delhi. Solar Energy 7(2), 44 0963).

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4. D. W. Ruth and R. E. Chant, The relationship of diffuse radiation to total radiation in Canada. Solar Energy 18(2), 153 (1976).

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6. M. Collares-Pereira and A. Rabl, The average distribution of solar radiation--correlation between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22(2), 155 (1979).

7. Murri et al. Dati delrOsservatorio Geofisico di Macerata. 8. Bossolasco et al., Dati delrIstituto Geofisico e Geodetico

dell'Universita di Genova.