IL NUOVO CIMENTO VoL. 4 C, N. 2 Marzo-Aprilc 1981

Autocorrelation of Daily Global Solar Radiation.

B. ]~ARTOLI, B. COLUZZI, V. GUOMO~ ~[. FRANCESCA lt, I I ( ] . G. SERIO

Istituto di Fisica della Facoltd di Ingegneria - Napoli, l[alia

(riccwlto 1'1 Agosto 1980)

S u m m a r y . - - We show that the time series of daily values K r (ratios of global solar radiation on a horizontal surface to extra atmospherical radia- tion) cannot be statistically distinguished from a tst-order autoregres- sive stochastic process. The analysis of I tal ian meteorological data shows that the first autocorrelation coefficient is a random variable which is practically independent of locality; moreover, the distributions of the residues "lround the mean value are also independent of locality.

List of symbols.

Fo,~(t):

V,=~(t): Kz(t) :

?it :

X :

u(t):

~s( i ) :

~ :

W,(i) :

experimental global radiation on a horizontal surface on the t-th day of the month;

extra atmospherical radiation oil a horizontal surface on the t-th day of the month;

F , = ~ ( t ) l ~ , , d t ) ,

number of days per mollth;

monthly average value of K~: X - - ~ KT(t)/m; tB|

Kr(t ) - - X;

first autocorrelation coefficient;

estimation of ~, calculated on the i-th month and relative to the s-th station;

average value of the monthly autocorrelation coefficients (relative to the s-th station) ;

.~s(i) - - . ~ ;

average value of the monthly autocorrelation coefficients, all over Italy.

8 - I I N u o v o U i m e n t o C. 113

114 m BARTOLI, B. COLUZZI~ V. CUOMO, M. FRANCESCA and c. SERIO

Introduction.

The design of solar systems requires a good evaluat ion of their performances and consequently a good knowledge of the available solar radiation.

However , a big computing effort is required in o rde r to calculate the per- formances of the system over a period comparable with its t ime of life ((10-:- --20) y) if historical da ta must be used.

~or this reason it is useful to determine a reference year (RY), t ha t is a short-period sequence of meteorological da ta with the same statistical prop- erties as the long-period sequence, so tha t the system performance calculation can be quickly made (~-a).

In previous papers (~-3) we have shown tha t at least for I t a ly the month ly distributions of the K r ratios (experimental global radiat ion on ,~ horizontal surface, over extra atlnospherical radiation) depend only on X, the mean month ly value of KT, independent ly of the locality and of the season.

Long- term performances of solar lint-plate collectors c a r be calculated with a ve ry good approximat ion (4-6) by using daily data of solar radiation. Con- sequently the result in ref. (~.3) is sufficient to determine, s tart ing from monthly averages, the solar-radiation reference year to be used for solar fiat-plate col- lectors and ge~mrall3/ for Systems without storage.

Performances of systems with heat storage depend also on sequences of radiat ion dat~, so t ha t a more refined RY must be used in this case.

In this paper we show tha t 1) for each mon th the sequences of K~ r~tios can be described as a iirst-order autoregressive stochastic process (sect. 1), 2) the first-order autocorrelat ion coefficient is a random variable indepen- dent of the stat ion (sect. 2), 3) the residues of the series are independent of

the place (sect. 3).

(1) S . A . KLEIN, I ). J. COOPER, T. L. FREEMAN, D. ]~[. BEEKMAN, W. A. BECKMAN and J. A. Du~'FI~: Sol. Energy, 17, 29 (1975). (=) V. Cuo~o, M. FRA~CeSCA and C. SERIO: The re]erence year, in Solar Energy Availa- bility, lecture notes of the Permanent School o/ Solar Energy Processes, SoGeSTA, Urbino, 1978, edited by V. C~NA, cap. IV (Urbino, 1979). (a) B. ]3ARTOI.I, S. CATALANOTTI, V. CUOMO, ~r FRANCESCA, C. SERIO, V. S~I~V]~STRI~I and G. TROlS~: Nuovo Cimento C, 2, 222 (1979). (~) G. A~tBRoso~E: Energy delivered by solar collectors, in Heal Load Calculation and Economics: Optimization o] the Building Envelope and o] Solar Components, lecture notes of the Permanent School o/ Solar Energy I~roeess, SoGeSTA, Urbino, 1978. cap. X,=edited by V. SILW'~STI~I.~I (Napoli, 1979). (5) G. AMBROSONE, A. A~DR~:TTA, F. BLOISI, S. CATALA,NOTTI, V. CUOMO, V. SIL- VASTRI~I and L. VICA~I: Appl. Energy, 7, 119 (1980). (6) A. A~DII~ETTA, G. ]3ARO~'E, L. BRUNINI, V. CUOMO, --~[. FRANCESCA, P..~[ATTARELLI and C. SERIO: Appl. Energy, 7, 93 (1980).

AUTOCORRELATION OF DAILY GLOBAL SOLAR RADIATION 115

Our analys is is based on t h e va lues of solar g lobal r a d i a t i o n which h a v e

been m e a s u r e d in t h e years 1971-1972-1973 in t h e me teo ro log ica l s t a t ions of

A e r o n a u t i c a Mil i ta re I t a l i a n a (see t~blo I).

TABLE I. -- Stations considered in our paper.

Station Lat i tude Station Lati tude

Alghero 40.63 ~ Milano 45.43 ~

Amendola 41.53 ~ Monte Cimone 44.20 ~

Ancona 43.62 ~ Olbia 40.93 ~

Brindisi 40.65 ~ Pantelleria 36.82 ~

Capo Mele 43.95 ~ Pescara 42.43 ~

Capo Palinuro 40.02 ~ Pian Ros'~ 42.43 ~

Crotone 39.07 ~ Pianosa 42.58 ~

Gela 37.08 ~ Pisa 43.67 ~

~fessina 38.20 ~ Roma 41.80 ~

(1)

with

1. - L e t us define

u(t) = K : , ( t ) - X

t = 1 , . . . , m,

m: n u m b e r of days iu t h e m o n t h ;

Kr(t) = F~x,(t)/~'o,t(t) in t h e t - th day of t h e m o n t h ;

X = ~ Kr(t) /m;

Fex~(t): h o r i z o n t a l g loba l solar r a d i a t i o n in t h e t - th d a y of t h e m o n t h ;

F , , t ( t ) : ho r i zon t a l e x t r a a t m o s p h e r i c solar r a d i a t i o n in t h e t - th d a y of

t h e m o n t h .

T h e t i m e series (1) can be descr ibed as a l s t - o r d e r au to regress ive s tochas t i c

procoss

(2) u(t) = ~ u ( t - 1) + ~(t) ,

where

t = l , . . . , m ;

e: r a n d o m va r i ab l e wi th e x p e c t a t i o n v a l u e E ( e ) = 0;

o = 1st au toco r re l~ t ion coofficienk

116 m :BAP, T O L I , B . C O L U Z Z I , V. C U O M O , :~I. FB.ANCESCA and c . S E R I O

For finite t ime series (t < ~ ) the es t imat ion of 0 is given by

(3) ~ -

( l /m) ~ [Ke(t) - - tt ,][gf(t -% I) - - p~]

1 / ( m - - I ) ) ~ [KT(t)- - / t , ] '~ ~ [K~(t q- 1)--tt2] * t-~l t~ l

with

/*, . . . . . . . ~_, KTCt), m - - 1 t- ,

1 m--1 / t . = . ~ KT(t 4- 1 ) .

- - I t = 1

With a very good approx imat ion we h ' tve

p, ---- P2 = X .

Note t h a t one m o n t h ( that is 30 data) is fully sufficient to es t imate ~ (7). I n a previous paper (~-) we have shown t h a t KT(t) values are autoeorrelated

and t h a t ,~ per iodogram analysis suggests t h a t there is a ]s t -order autocor- relation.

For this re'~son we have "~ssumed relat ion (2) to hold and we have tested this hypothesis by checking t h a t the t ime series of the residues

(4) ~(2),..., ~(m)

is a r andom series, i.e. tha.t the autocorrelat ion coefficients are all compat ible with zero.

Of course,

e(i) -- u ( i ) - ~ ( i - 1)

with 0 es t imated b y using (3). To tes t this hypothesis , we have used the r ank tes t (7). The hypothesis of randomness of the series ~(t) is fully satisfied, within "r

colffidence level of 5 %. As a consequence, we conclude t h a t the series u(t) is first-order autoeor-

related. An example of the results of the tes t is given in table I I for the Alghero

station.

(7) M. KF, NDALL and A. STU~a{T: The Advanced Theory o] Statistics (London, 1976).

A U T O C O R R E L A T I O N O F D A I L Y G L O B A L S O L A R R A D I A T I O N l l ?

.,-i

~ .

~.. -,"~ ~

2 ~

,--i

~ •

~ •

~ •

~J

r

rZ

o0

I

I

~q

•

00

.,$

~q

~q

•

I

b

, = r

r :

•

gq

I

r

.•

r

•

e-4

c q ~'q

1-.- t ~- @,1 gq

I I

r r

I

~,1 r

e.~ r

• •

I

~1 ~ i

gq

• •

r ~1

I

t-.-

r : r :

• •

@,1 ~q

• •

]

gq r

• •

I

p,. I.=. r gq

�9 ~

t-~ t-- r r

, I

I I

r r

• •

I

r ~q

r r

• •

I ]

I

r

~q

r

l i

•

I

I

o0

I

r

~q

c , 1

I I

I :

~1 c q

r ~-q

• •

I

r r

• •

[

e~ o

118 B. B A R T O L I , B . C O L U Z Z I , V. C U O M O , ) I . F R A N C E S C A and C. S E R I O

TABLE I I I . - Value o] Os (.mean value o] ~(i)) /or all statio~zs. We show also <O> (average value of ~, all over Italy) and its variance.

Station 5, Station 0,

Alghero 0.19 Milano 0.31

Amendola 0.24 Monte Cimone 0.38

Ancona 0.26 Olbia 0.21

Brindisi 0.20 Pantelleria 0.24

Capo ~ele 0.22 Pescara 0.32

Capo Palinuro 0.20 Plan Ros~ 0.23

Crotone 0.24 Pianosa 0.16

G ela 0.17 Pisa 0.31

l~essina 0.25 Roma 0.30

0.06

<~> o.25

I n table I I I we show the values of ~, for all stations:

-c 1 A �9 (5) q = ~ ~ q,(,), i = l

where

~,(~): first autocorrelation coefficient of the series Kr(t), calculated in the i- th month and relative to the s-th station;

number of months.

2. - In order to identify the solarr-adiation RY, we need a model which

allows us to estimate the value of ~,(i) which must be a t t r ibuted to the various months of the RY.

To this purpose we consider the t ime series

(6) w,(l) ... w,(n) ,

where

w,(i) = ~ , ( i ) - ~,.

~5, and ~,(i) are estimated, respectively, by using eqs. (5) and (3).

Using the rank test, we have shown tha t the series w,(i) is not autocorrelated. The results are shown in table IV. Moreover, the analysis of the variance of

this series, performed by using the F- tes t (s), shows tha t all the samples w,(i) are compatible in belonging to the same parent distribution as shown in table V.

(s) H. SCItAFF~: The Analysis o/ Variance (New York, N.Y., 1959).

AUTOCORR:ELATION OF DAILY GLOBAL SOL.4.R RADIATION 119

TABLF IV. - Results o] the rank tes~ ]or the series w,(i), which are the residuals o] O,(i) to 5~ /or all slatious, v is the value of the rank tes t s tat is t ics , C[ is the corresponding confidence interval . I f v belongs to CI, the series is r andom (in our case at the 5 % level of significance).

Sta t ion v CI

Alghero - - 0.216 [-- 0.228, 0.228]

Amendo la 0.063 [-- 0.228, 0.228]

Ancona 0.020 [-- 0.244, 0.244]

Brindisi 0.291 [-- 0.236, 0.236]

Capo Mcle - - 0.067 [-- 0.228, 0.228]

Capo 1)alinuro - - 0.069 [-- 0.262, 0.262]

Crotone 0.071 [-- 0.248, 0.248]

Gela 0.099 [-- 0.253, 0.253]

3~essina - - 0.094 [-- 0.236, 0.236]

Milano - - 0.093 [-- 0.244, 0.244]

3[onte Cimone - - 0.064 [-- 0.244, 0.244]

01bin - - 0.036 [-- 0.286, 0.286]

l ' an te l l e r i a 0.098 [-- 0.240, 0.240]

Pescara 0.045 [-- 0.232, 0.232]

P ian Rosh - - 0.030 [-- 0.236, 0.236]

P ianosa 0.042 [-- 0.232, 0.232]

Pisa - - 0.196 [-- 0.286, 0.286]

R o m e 0.029 [-- 0.328, 0.328]

TAm,E V. - Results of the F-test on the samples w,(i). F o is the va lue o f the stat is t ics ~'. Wc present the ob ta ined value of t he s tat is t ics /70 "rod the corresponding cri t ical va lue I~'o.o5, nx, n 2 re la ted to t he 1 t' var iable wi th nx, *t2 degrccs of f reedom at the 5 % level of s ignifcance.

f ro T t n 1 ~12

0.69 1.60 17 126

F o r t h i s r e a s o n w e c a n e s t i m a t e t h e c u m u l a t i v e d i s t r i b u t i o n of w u s i n g

t h e d a t a of a l l s t a t i o n s . T h i s d i s t r i b u t i o n is s h o w n in Jig. 1.

T h i s r e s u l t a l l o w s us t o e s t i m a t e ea s i l y t h e a u t o c o r r e l a t i o n coe f f i c i en t ~,(i)

f o r a n y m o n t h a n d l o c a h t y if t h e m e a n v a l u e 5, is k n o w n .

I n f a c t ,

(7) ~,(i) - - g, § o~,

w h e r e o) is a r a n d o m n u m b e r w i t h t h e d i s t r i b u t i o n s h o w n in fig. 1.

19"0 B . BAICTOLI , B . C O L U Z Z I , V. C U O M O , ) I . F R A N C E S C A and c . S E R I O

i i i -~ 0

Fig. 1. - Cumulative distribution of ~he residues.

:Note t h a t ~ does not depend much on locality (table I I I ) , so t ha t we in- t roduce a negligible error if we use (~) instead of ~ , where

28

3. - For the autoregressive model (2) f rom the assumpt ions

(9) a) E(ur -= 0 ,

b) E(e) = 0

(where E indicates the expecta t ion value of the variable), i t follows tha t

(1o) B(~.) = ( 1 - q,,)]~(u,,).

As a consequence of (10) the distr ibution funct ion of the residues e(t) is com- pletely de termined b y the distr ibution funct ion of u(t) and vice versa when

is fixed. Moreover, if hypothesis (9a) is t rue, in our model (2) hypothesis (Pb) follows

as a consequence. I n previous papers (2,3), i t has been shown t h a t the distr ibution funct ion

of the K~ ratios depends in I t a l y only on their mean month ly value X and not on the season nor on the place. This result wi th relat ion (10) implies t h a t the dis tr ibut ion funct ion of the residues depends only on ~ and X.

W e h a v e divided the range of ~ and X, respectively, into 5 and 8 intervals of equal width. The distr ibutions ~re not normal , the hypothesis of some authors (9) to consider Gaussian the distr ibution of residues does not appear to be valid. I n fig. 2a)-h) we give, as an example, the distributions of e for

in the in terva l ]0.2, 0.3].

(9) B.J . BRI~KWORTtt: SOL Energy, 19, 343 (1977).

AUTOCORRELATIOI~ ~ OF DAILY GLOBAL SOLAR RADIATIOL~- 121

30

10

0

20

10

6C

5C

4C

3C

20

10

0

Xe]0.35~0.40] X~]0.40,0.45]

X.-] 0.4 5,0.50]

"ZE]0.60,0.65]

0

re] 0.50,0.55]

"X~]0.65,0.70]

-1

~e]0.55,0.60/

'Xe]0.70,0.75]

-I

Fig. 2. - Distributions of for the various ranges of X attd for O belonging to the interval ]0.2, 0.3]. a) Xe]0 .35 , 0.40], b) Xe]0 .40 , 0.45], c) Xe]0 .45 , 0.50], d) X e ]0.50, 0.55], e) X � 9 ]0.55, 0.60], 1) X � 9 ]0.60, 0.65], g) X �9 ]0.65, 0.70], h) X �9 ]0.70, 0.75].

4. - Conclusions.

The sequences of daily ratios K~ cannot be distinguished statistically fronl the process described by eq. (5).

This result allows one to fit the long-period behaviour of daily solar-ra- diation data by using only two parameters: o and X. Once o and X are known,

a Monte Carlo method can be used to build the reference year (RY). For each month the daily values of the R Y obtained in this way have the

same mean value, the same variance and the same autocorrelation coeiiicient

as the corresponding values over the long period. This result can be used for a quick simulation of the performances of solar

systems both with and without a storage unit.

122 B. BAItTOLI, B. COLUZZI, V. C U 0 ) I 0 , M. FRANCESCA and C. SERIO

�9 R I A S S U N T 0

Si d i m o s t r a t h e la sorie t e m p o r a l e dei va lo r i g io rna l i e r i K z ( r appor to t r a la r ad iaz ione solare g lobale su superf ieie or izzort ta le e la r ad iaz ione ex~raa tmosfer iea) n o n p u b essere s t a t i s f i eamen~e dis~inta d a u n proeesso s toeas t ieo autorega 'essivo di p r i m ' o r d i n e . L ' a n a l i s i d i da~i meteoro logie i i t a l i a n i m o s t r a ehe il p r i m o eoeffieiente d i au~oeorrela- z iono b u n a v a r i a b i l e easuale ehe b p r a f i e a m e n t e i n d i p e n d e n t e da l la loealit 'A; inol t re , anehe le d is~r ibuzioni dei res idu i i n t o r n o al va lo re medio sono i n d i p e n d e n t i da l l a loeal i th.

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