IL NUOV0 CIMENTO VOL. 2 C, N. 2 Marzo-Aprile 1979

Statistical Correlation between Daily and Monthly Averages of Solar-Radiation Data (*).

B. BARTOLI, ~. CATALANOTTI, V. CUOMO, M. I~RA1NCESCA

C. SERIO, V. SILVESTRINI a n d G. TROISE

Istituto di Fisica, Facoltk di Ingegneria de~-Universitk - Napoli, Italia

(ricevuto il 26 Gennaio 1979)

Summary. - - We have analysed the way for a correlation between the monthly average of the fraction of solar radia t ion reaching the Ear th and the s tat is t ical dis tr ibut ion of the days belonging to such a month as a function of the dai ly fraction of solar radia t ion; we have found tha t this correlation exists in I t a ly and tha t the dis tr ibut ion functions are independent of the location and of the per iod of the year.

De/inition o] symbols

F~p: dai ly global radiat ion on horizontal surface (langley/day). Ftb: dai ly global radiat ion on horizontal surface outside the atmosphere (langley/day). m: number of days in a month. n: number of sunshine hours in a day. N: number of sunshine hours outside the atmosphere, i.e. the durat ion of the day. I: n/N, monthly averaged. KT: F~/Fth. X: ~KT/m, monthly average of the KT ratios.

month

1. - Introduct ion .

F o r m o r e p r a c t i c a l uses of so lar e n e r g y t h e i n c i d e n t r a d i a t i o n m u s t b e k n o w n

w i t h g o o d reso lu t ion , b o t h in space a n d t ime .

(*) Work financially supported by C.N.R., Progetto Final izzato Encrgetiea, Sotto- progetto Energia Solare.

222

STATISTICAL CORRELATION BETWEEN DAILY AND MONTHLY AVERAGES ETC. 223

I n I t a l y there are only 31 meteorological stations where solar radia t ion has been sys temat ica l ly measured; it is, however, possible to es t imate the mon th ly average values for other locations b y using well-known correlations between solar radiat ion and meteorological da ta of simpler and more widespread meas- urement . We can here recall, for example, the correlation between solar ra- diation and nu m ber of sunshine hours (1,2), which was found to hold also for I t a l y (3,4):

(1.1) X ~- A ~ B I

(where X is the rat io of the actual value of to ta l solar radiat ion measured on horizontal surface to the value of exmaterres t r ia l solar radiat ion (monthly averages) ; I is the fract ion of sunshine hours averaged over the same period of t ime ; A and B are empirical parameters) or a similar correlation found (5) be tween X and the number of complete ly cloudy days.

I n this paper we show tha t , a t least for a region of l imited geographical ex ten t like I t a ly , the stat ist ical distr ibution of daily radiat ion values depends only on their mon th ly average. This result allows us to foresee the energy yield of solar collectors even in places where no measurement of solar radia- t ion is available and only simpler meteorological quanti t ies are known.

An analysis of meteorological data , addressed to a similar purpose for the U.S.A. and Canada, has been per formed b y L I u and JORDAN (6,7). I t is not clear, however , if their results can be extended to other countries; and the sta- t ist ical significance of their approx imat ion has not been discussed in detail.

2. - Stat ist ical corre la t ion b e t w e e n daily and m o n t h l y data in e a c h locat ion .

For our analysis we have used daily values of global radia t ion on hori- zontal surface Fex~ measured during three years (1971, 1972, 1973) in 17

(1) A. ANGSTROM: Astron. J., 50, 121 (1924); Ark. Geo/is., 2, 41 (1956). (2) J'. K. PAGE: The estimation o] monthly mean values o/ daily short-wave radiation on vertical and inclined sur]ace ]rom sunshine records ]or latitude 40 ~ N-40 ~ S, in Pro. ceedings o] the U. N. Con/erenee o] New Sources o] Energy, Vol. 4 (1964), p. 378. (a) V. CuoMo et al. : Solar radiation, its correlation with the atmospheric ]actors in Italy, and the in]Inenr o/orientation, in Proceedings o] the I Course on Solar Energy Conversion, S.C.A.P., International College on Applied Physics, edited by A. N. MANCINI and I. F. QUERCIA (Napoli, 1974), p. 557. (4) GRUPPO ENERGIA SOLARE D~LL'UNIV'ERSIT)k DI NAPOLI: II clima come elemento di progetto nell'edilizia (Napoli, 1974). (5) M. FRANCESCA and C. S~.RIO: Thesis (1978). ~e) B. Y. H. LIU and R. C. JORDAN: Sol. Energy, 4, 1 (1960). (7) B. Y. H. LIU and R. C. JORDAN: Sol. Energy, 7, 53 (1969).

224 B. BARTOLI, S. CATALANOTTI, V. CUOMO, M. FRANCESCA, C. SERIO, ETC.

meteorological s t a t ions of the (~ A e r o n a u t i e a Mil i tare I t a l i a n a ~ (8) l i s ted in ta-

ble I . The global r ad i a t i on F was m e a s u r e d wi th p y r a n o m e t e r s of the Fuesz-

R u b i t z s c h t ype (9).

R a t h e r t h a n to work wi th the global so la r - rad ia t ion va lues F~,~ direct ly,

it is c o n v e n i e n t to deal wi th the quan t i t i e s

(2.1) K r = F , J / ~ h ,

where Fth is the solar r ad i a t i on on ho r i zon ta l surface j u s t outs ide the a tmos-

phere. The KT ra t io is more d i rec t ly connec t ed wi th the c loudy or s u n n y char-

ac ter of the days t h a n F~xp, as is com'irmed also b y the we l l -known corre la t ion

exis t ing be tween KT a n d the sunshil~v index I (1.4).

TABLE I. -- Lis t O/ the meteorological stations whose data are used in our analys is wi th their lat i tude and the code used in tables I I and I I a .

Station Code Latitude

Ancona AN 43 ~ 47'

Bologna BO 44 ~ 32'

Bolzano BZ 46 ~ 28'

Briudisi BR 40 ~ 39'

Cagliari CA 39 ~ 15'

Crotone CR 39 ~ 4'

Genova GE 44 ~ 25'

Messina ME 38 ~ 12'

Milano MI 45 ~ 26'

Napoli NA 40 ~ 51'

Pescara PE 42 ~ 26'

Pisa PI 43 ~ 40'

Trapani TR 37 ~ 55 I

Udine UD 46 ~ 2'

Ustica US 38 ~ 42'

Venezia VE 45 ~ 30'

Vigna di Valle VV 42 ~ 5'

(8) AERONAUTICA MILITARE ITALIANA: Dat i della rete att inometrica. (9) Aeronautiea Militare della Repubblica Italiana, Ispettorato Telecomunicazioni ed Assistenza al Volo, I I I Reparto, Servizio Metcorologico, Roma: Durata del sole e radia- zione globale, Nora Tecnica No. 19 (Roma, 1970).

S T A T I S T I C A L C O R R ~ , L A T I O N B ' ~ - T W ' E ~ N D A I L Y A N D M O N T H L Y A V E R A G E S E T C . 225

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STATISTICAL CORR]~LATION BETWEEN DAILY AND MONTn'LY AVERAGES ETC. 227

Le t us consider now the mon th ly average X of the K r rat ios (see the defi- nit ion of symbols):

(2.2) x = ~ K~/~n .

The first hypothesis we tes ted was tha t , for a given locality, the stat is t ical distr ibution of the K~ ratios depends only on their mon th ly average X and not on the season (number order).

This analysis was done b y grouping together months with similar values of X and apply ing to every group a tes t of statist ical compat ib i l i ty among their K~ distributions. The n u m b e r of elements (i.e. of days) in every group is not large, so t ha t the choice of a good statist ical tes t is not easy. We used the so-called F - t e s t (see the appendix) , for var iance comparison (~o), a test which gives significant results also when applied to small samples f rom non- normal populations.

I n tables I I and I I a we show the results of the F - t e s t applied to the samples obta ined grouping together months with X values falling into one of the intervals specified in the above-ment ioned table ; the confidence level of the tes t is 5~/o.

The width of the X intervals mus t be chosen as a compromise between two conflicting requirements . I f this width is too large, the months with too dif- ferent daily radiat ion distr ibutions are pu t together in the same class and the F - t e s t can fail because of spurious reasons, while a too small width can result in samples containing a num ber of days insufficient for a significant stat ist ical test. The width finally chosen (0.05) is also compat ib le with the error on X.

F r o m the results of the F - t e s t as shown in tables I I and I I a we conclude t ha t the distr ibution of the daffy K~ ratios can be considered as being depen- dent only on the month ly average X.

3. - Statistical compatibi l i ty o f radiation data in different localities.

After having seen tha t , for a given station, the dis tr ibut ion of the K r ratios depends only on their average X, we have tes ted if the same hypothesis holds also when da ta coming f rom different s tat ions are p u t together.

The samples are much larger than in the preceding case, so t h a t it is pos- sible now to use a statist ical test ing procedure with more discr iminat ing power than the F- tes t . We have used a test in t roduced b y S M ~ o v (11) (see also the appendix) which compares the empirical distributions wi thout requiring

(10) H. Se~LEFF~: The Analysis o] Variance (New York, N.Y. , 1959). (11) M. FIsz: Probability Theory and Mathematical Statistics, third edition (New York, N.Y., 1963).

228 B . B A R T O L I , S . C A T A L A N O T T I , V . C U O . ~ I O , M . l e R A N C ~ S C A , C . S E R I O , E T C .

TABLE I I I . - Results o] the Smirnov test. In t h e f irst co lumn t h e 9 i n t e r v a l s of i n t e r e s t for t h e X - v a r i a b l e are specif ied; in t h e second t h e co r r e spond ing va lues of t he s t a t i s t i c 2 o are p r e s e n t e d ; ia t h e t h i r d c o l u m n we p r e s e n t t l le va lue z o = 1 _ Q ( ~ ) K - 1 (see t he append ix ) .

X ).0 / )

0 .30- -0 .35 0.30 9 8 %

0 .35- -0 .40 0.91 76O/o

0.40 -:- 0.45 1.19 58~o

0.45 --0 .50 1.03 97~o

0 .50- -0 .55 1.21 84~o

0.55 --0 .60 1.62 15~/o

0.60 "-0.65 1.36 46~/o

0.65 : 0.70 1.01 78~o

0.70 -:-0.75 0.59 9 8 %

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0.2 0.4 0.6 0.8 K r

1.0

Fig. 1. - D i s t r i b u t i o n of K r for t h e s t a t i o n s of t a b l e I a) for X � 9 0.30] ( X � 9 ( - - - - - - ) , X � 9 ( . . . . . ); b) for X � 9 ( x e ] 0 . 4 5 , 0 .50] ( - - - - - - ) , x �9 ] 0 . 5 0 , 0 . 5 5 ] ( . . . . . ).

STATISTICAL CORRELATION BETWEEN DAILY AND MONTHLY AV'ERAG~.S ETC. 229

any hypothes is on the theoretical parent distribution (free distribution test). In table H I we show the results of the Smirnov test applied to the data

from all stations grouped in the same classes previously used for the F-test . All samples appear to belong with high probabil i ty to the same parent popu- lation, so that we can confidently say that throughout I ta ly the distributions of the KT ratios depend only on their m o n t h l y average X, wi thout any signif- icant dependence on the season or place.

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F i g . 2. - As fig. 1 fo r X � 9 ( ( . . . . . ), X �9 ]0.70, 0.75] ( - - - - - - ) .

), X e ] 0 . 6 0 , 0.65] ( . . . ) , X e ] 0 . 6 5 , 0.70]

4. - The empirical distributions.

The empirical distributions of the K~ ratios for all the stations are shown in fig. 1 and 2 for the 10 classes in which the interval of interest of X was divided. In fig. 3 we also present the integrals of these same distributions, a form which

16 - I1 N u a v o C i m v n t o C.

230 B. BARTOLI, 8. CATALANOTTI~ V. CUOI~O, ~f. FRANCESCA, C. SERIO, ETC.

100

%

80

60

40

20

0 0.2 0.4 0.6

Fig . 3. - I n t e g r a l s of t h e d i s t r i b u t i o n s of fig. 1, 2.

0.8 1.0 K r

is sometimes of more pract ical use. Fo r comparison, we have also shown in fig. 4 exper imenta l integral distr ibution curves of K~ (with X = 0.3, X---- 0.4 X = 0.5, X---- 0.6, X = 0.7), as obta ined b y LIV and JO]{DA~ (e,7), using solar radia t ion da ta measured in the U.S.A. and Canada. I t can be seen t h a t these curves are very similar to the corresponding ones for I t a ly , so t h a t some con- fidence is given to the hypothesis t h a t for any given X and K r the distr ibution is pract ical ly the same for all t e m p e r a t e regions.

5 . - C o n c l u s i o n s .

The results of our analysis show t h a t months with the same value of X have a distr ibution of daily solar radia t ion which is the same, within the stat ist ical errors, everywhere in I ta ly . This result is re levant f rom a pure meteorological point of view; bu t it becomes much more useful in view of applications and solar-energy processes. The performances of solar collectors, whose knowledge is a s tar t ing point for the design of solar systems, depend on the detailed t ime behaviour of the solar-radiat ion intensity. However , b y using the results of

STATISTICAL CORRELATION B]~TW'E~N DAILY AND MONTHLY AVERAGES ETC.

100

%

80

60

40

20

231

0 0.2 0.4. 0.6 0.8 1.0 K r

Fig. 4. - As fig. 3 for the stations of the U.S.A. and Canada ( X = 0.3, X = 0.4, X = 0.5, X = 0.6, X = 0.7); ref. (~.7).

our present analysis, af ter the month ly averages are known, daily distribu- tions can be calculated. Such a thing is impor tan t not only because month ly averages are much more widely available than daily data, bu t also because by using weU-known correlation formulae it is possible to compute the month ly average of solar intensi ty start ing from simple meteorological parameters , like the sunshine index, which are known for m an y localities.

The distributions of daily solar-radiation intensi ty can be also used, when in a given locality month ly averages are known over a long period of time, to generate a t the computer the reference year in the considered site with a Monte Carlo procedure (5).

Long- term performances of solar collectors can then be easily computed by simulating their working conditions over the period of one year.

I f the system for solar-energy conversion includes a heat storage unit, then the heat t ransferred to the load depends not only on the distributions of the K~ ratios, bu t also on their sequences.

The s tudy presented above does not examine this point, as it requires the use of different kinds of statistical techniques. An analysis of this problem is being carryied out to be published in a future paper.

16 * I l Nuovo Cimento C.

232 B. BARTOLI , S. CATALANOTTI, V. CUOMO, M. FRANCESCA, C. 8ERIO, ETC.

_A_PPENDIX

We th ink i t is convenient to define here briefly the s tat is t ical tes ts used in our analysis ; in fact , a l though the F - t e s t is quite s t andard (it is summar ized here only for completeness), the Smirnov tes t for the case of several s~.mples is ra ther new, and perhaps unfami l ia r to the reader .

]) The F-lest (~o). Le t us call s 2 the sample wtr iance of '~ r a n d o m sample (x~, . . . , x . ) of n e lements f rom a popula t ion of v 'u ' iance a2:

s 2 = ~_. (x,- ~ ) ~ / ( , ~ - - 1),

where

Then

ii

E(s2) : ~2 ,

where E(s 2) is the expecta t ion value of the s ta t is t ic s 2. We assume tha t the samples under analysis a.re a r ranged in I sets of the type

(X i l , Xi2, . . . , Xik|) , i - ] , . . . , I ,

and we wan t to t es t the (null) hypothes is H tha t the samples belong to the same pa r en t populat ion.

Le t us assume tha t each set contains a t least two elements . We divide the i - th set into J~ subsets of kij e lements (i = 1 , . . . , I ; j - - 1 , . . . , JD using

Ji ( ) * the var iance of a r a n d o m selection procedure note t ha t k~ = ~ k , ; be s~

t=1

the j - th subset of the i - th set (i -- :l, ..., I ; j - 1, ..., JD. The null hypothesis H is then tes ted b y ver i fy ing if

0"1 : 0"2 - - - - 0"1 ,

where a~ --E(s~j) (note t ha t E(s~j) is not dependen t on the index j, for the r andom procedure of selection).

For what concerns the test , the 2, a, s are es t imated as

where

JI JI kt XitIr ) \ ) 5 , 4 / J , - 2 _ . = - x . ) / ( ~ . - - 1 ? J=l j = l ~k--1

kIJ ~ . = y~ x . , J k . .

STATISTICAL CORRELATION BETWEEN DAILY AND MONTHLY AV'ERAG~S ETC. 2 ~

I t can be demons t ra ted t ha t the s ta t is t ic

where

~, , (~,- ~)~ Me __ ~ l

I - ~ Y. ~_~, , , (y , , - ~,)~ '

v, = ~ ( J , - - 1 ) , v, - - ~ v , , ,

J

has the probabi l i ty dis t r ibut ion of a var iable E with I - - 1, v degrees of f reedom. Hence we reject the null hypothesis H at the ~ confidence level if and only if

M 0 > Ma, I--1, ~,.

where M~.:_~,,,. is the critical value such t h a t

co

f ' ~ r I--I, v e = O~ �9

The M-test (which holds also for small-size samples) tes ts in principle only for normal distr ibutions, a l though it has been shown (~o) tha t i t has good signif- icance also for nonnormal distr ibutions.

2) The Smirnov tes t (~) . Let ( x ~ , . . . , x . ) be a r a n d o m sample of size n ordered as

X I < X 2 < . . . < X . .

Then the real funct ion S/(x) defined as

Sn(X) =

O~ X ~ Xl~

k i n , x~ < x < x,+~ ,

where k = 1, ..., n - - 1 and x e ] - - o o , co[, is called the empir ical dis t r ibut ion function. Assume we have K ( K ~ 2 ) independent r a n d o m samples of size n , , . . . ,n~ f rom K populat ions.

We wan t to tes t the hypothesis H t h a t the K populat ions have the same (continuous) distr ibution function.

The empir ical dis t r ibut ion funct ion of the sample f rom the i - th popula t ion is denoted by Sln,(x); the s ta t is t ic used for the tes t is

(A.1) 3~ = m a x A , ( n l , ..., n~) ,

9 .34 B . BARTOLI , S. CATALANOTTI, V. CUOMO, M. FRANCESCA, C. SERIO, ETC.

where

and

, = V , ' V o,+. , t ~ ' i+1

k ' h D,(~r n,+l) - - m ~ x S,+~ . . . . . ( x ) - (1~iv,) ~ n~S~.,(x).

I f nl , . . . ,nE are large, the dis t r ibut ion funct ion of the s ta t is t ic 20 is ap- p rox ima te ly {Q(2)} ~-1, where

QCX) = ~ (-- 1)" exp [-- 2n~2~],

n ~ r

0 ,

~ > 0 ,

A < 0 .

The Smirnov tes t of H consists in re ject ing H a t the level of significance if and only if

> 1 - - Q ( 2 o ) ~ - z ,

where 2o is the value of the s ta t i s t ic (A.1). The Smirnov tes t is a f ree-dis t r ibut ion tes t , bu t i t has been p roved only for

large-size samples.

�9 RIASSUNTO

Sono stati analizzati i valori quotidiani del rapporto ~ra radiazione totale su superficie orizzontale e radiazione extraatmosferica. Si ~ trovato che ~ali distribuzioni dipendono solo dal valor medio mensile dei rapporti quotidiani e non dalla localits n6 dal numero d'ordine del mese. Tale risultato permette di ricostruire anni di riferimento che hanno lo stesso valor medio e la stessa varianza del lungo periodo.

PC3IOMe He noJIy~leso.