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Stochastic analysis and generation of synthetic sequences of daily global solar irradiation: Rabat site (Morocco)
Renewable Eneryy Vol. 2. No. 2, pp. 129-138, 1992 Printed in Great Britain.
096(~1481/92 $5.00+.00 Pergamon Press Ltd
STOCHASTIC ANALYSIS A N D GENERATION OF SYNTHETIC SEQUENCES OF DAILY GLOBAL SOLAR IRRADIATION: RABAT SITE (MOROCCO) H . LOUTFI a n d A. KHTIRA Laboratoire d'Energie Solaire, Facult~ des Sciences, B.P 1014, Rabat, Morocco (Received 22 January 1991 ; accepted 1 October 1991) A b s t r a c ~ A study of the stationary and sequential properties of daily global horizontal solar radiation is presented for Rabat (Morocco). The results from this analysis are essential for the analytic modelling of the long term performance of solar energy systems, and for the generation of synthetic daily solar radiation sequences. The only climatic information required is the monthly average of the clearness index. We show that the experimental data can be described by a linear auto-regressive stochastic model of first or second order, and that the probability density function PDF can be fitted by a simple, easy to calculate functional form. We also show that, by using this function, the clearness index sequences can be transformed into sequences with normal distribution but with the same auto-correlation properties. It becomes, then, easy to construct synthetic sequences of solar irradiation with the same statistical properties as the real data. These synthetic sequences are very useful for performance prediction of solar energy systems.
1. I N T R O D U C T I O N The statistical modelling of global horizontal solar irradiation is very interesting for many practical applications such as the design and performance prediction of solar energy conversion systems. In fact, one of the most challenging problems that faces solar systems engineers is a proper accounting of the random nature of solar irradiation. If the statistical distribution of this irradiation is known, a large quantity of irradiation data can be summarized with only a few parameters. A typical example is the use of the distribution function of daily global solar irradiation to calculate the efficiency of solar collectors [1-3]. In many cases, such as the optimal sizing of thermal and photovoltaic systems with a storage unit, it is also necessary to know the sequential properties of solar irradiation data. In this paper, we perform a statistical analysis of daily global horizontal solar irradiation in Rabat (34' N). We use a simple functional form for the stationary probability density, and test several types of linear auto-regressive stochastic models toward the generation of synthetic sequences that have the same statistical properties as actual data. This procedure allows 1o characterize the insolation properties of a locality with only few parameters such as the clearness index monthly average, and auto-correlation coefficients. Moreover, it becomes possible to generate a "typical" year of data.
2. S E L E C T I O N OF AN A P P R O P R I A T E S T A T I S T I C A L VARIABLE
The first step in a time series analysis of solar irradiation is to choose both the series size and the variable temporal scale. Many previous studies were developed on a yearly [4~6] or seasonal [7, 8] basis. We choose to divide the year into monthly periods for practical reasons [6, 9]. First, most system design is performed on a monthly basis. Second, because we prefer a time scale that is short enough to consider the annual trends in solar radiation as constant. Thus, a further filtering out of annual trends is not necessary. This procedure has been adopted in many other studies [6, 9]. We consider a series of 5 years of daily global horizontal solar irradiation (1983-1987). To obtain a stationary variable (necessary for a stochastic analysis) and as our analysis is performed on a monthly basis, we need to extract from the daily irradiation the monthly average and the effect of latitude. It was pointed out that the effect of latitude can be eliminated by using the clearness index, K, [5, 6, 8]. However, K, itself is not adapted to a stochastic analysis, because the K, monthly average (Kt) varies markedly from year to year for a given month. The aim is to transform K, into a stationary variable through which we can take account of fluctuations in monthly average of K,. We consider then a variable defined as follows : X ( d , m , y ) = K,(d,m,y)/K,(m,y) 129
(1)
H. LOUTFIand A. KHTIRA
130
where K, is the monthly average of K,, (d, m,y) denotes day, month and year respectively. The selection of this variable introduces a small correlation error, especially because R, depends on all Kt values for a given month : division of K, by Rt introduces an error of roughly l/J, where J is the number of days in the month ; this variable was used by Gordon and Reddy [ll]. Monthly statistical properties of K, values are calculated for each individual year, and then averaged over all years. We will refer to this kind of average as grand average.
3. S T A T I O N A R Y P R O B A B I L I T Y D E N S I T Y
FUNCTION For analytic modelling purposes, it is preferable to have a functional form for the probability density PDF, P(X), that has been compared with actual data with good results. Several studies have considered prediction of the PDF of daily global solar irradiation on a horizontal surface [1, 10-14]. However, so far there is no universal expression for the PDF of K, [l 3]. A proposed function must obey to universal laws of probability : P(X) must vanish at sufficiently low and high values of X. Many functional closed-forms have been published for P(X) [1, 6, 13], but their major short-coming is that they assume the maximum value of K, either as a universal constant or as a known variable. However, R,.... is not a universal constant [13]. In this analysis, we adopt a closed-form which has been proposed by Gordon and Reddy [9]. This closed form takes account of the major limitations of previous proposed functions for PDF and has no adjustable parameters. It was pointed out that this function has been compared favorably with actual data from different climates [9]. The expression of this closed-form is as follows :
grated in a closed-form ; this integral (cumulative frequency) is very useful for system design where integral of P(X) is required. To decide of the use of eqn (2) for the (PDF), we compare it with actual data. In practice, we do not suppose that the detailed values of K, and O'2(X) are known as input to eqn (2), but only their average over the years denoted (R,) and (a2(X)) respectively. It is worth noting that even (a2(X)) values are not tabulated, but recently there has been claims to include values of (a2(X)) in data tables [9]. To characterize quantitatively the quality of fit we have performed the chi2 test (X2) [15] for both P(X) and its integral F(x). The latter function is very useful for system design and the generation of synthetic sequences of solar irradiation (section 7 below). The PDF calculations for actual data were done by dividing the series, for each month (series of 5 times number of days in this month), into a certain number of intervals. In each interval we assess the number of occurrences (frequency). The degree of freedom is equal to the number of bins considered minus 3, because to define P(X) we supposed normalization and a knowledge of K, and cI2(X). Results from the chi2 test can be appreciated by the fraction of data sets that are accepted by the test, in other words, that correspond to several signification levels of the test. The fraction of data corresponding to a signification level of 0.01C(0.01)) for both P(X) and F(X), for Rabat are as follows :
~(0.01)
P(X)
F(X)
66.6%
83.3%
Figures 1 and 2 represent a sample comparison of
P(X) and F(X), respectively, with the corresponding P(X) = AX"[1 -- (X/Xm,x)]
(2)
n, A and Xm,x are parameters to be determined, for each month, from the three following relations : normalization of P(X), knowledge of K, (X = 1) and knowledge of the variance of X. Development of these three relations leads to : n = -2.5+0.519+(8/a2(X))] '~2
(3)
)(max = ( n + 3)/(n+ 1)
(4)
A = (n+ 1)(n+2)/X'~+x'.
(5)
Equations (2) to (5) define completely the functional form ofeqn (2). P(X) can then be incorporated in all kinds of calculations without resorting to numerical solutions. Moreover, P(X) can be inte-
function of actual data for four months representing the different seasons as similarities between different months in the same season were noticed. Taking account of the relatively small sample of data considered, results obtained are encouraging and comparable to those of a previous study [9]. As we mentioned above, values of (~2(X)) are not available in most solar data tables, it would be desirable if we could estimate this variable from other climatic information such as the clearness index which is on the other hand, the most available variable. 4. C O R R E L A T I O N BETWEEN <~rZ(X)> A N D
Several studies have considered the statistical analysis of K, in Rabat [16]. These analyses led to many
Stochastic analysis and generation of synthetic sequences of daily global solar irradiation --
Function Data
]
January
--
Fun~ion
I
131
April
2 ¸
I
l
X
X
51 l 4
3-
(1.
--
Function
-I,-
Data
July
!
--
Function
•+
Data
October
E
2 I
L__
L I
X
X
Fig. 1. Comparison of theoretical (eqn 2) and data-based PDF, P(X).
kinds of correlation between K, and several other climatic components. In this analysis we have considered a correlation between a meteorological variable and its statistical property such as the variance. Figure 3 shows a plot of (/~,) versus for Rabat. It appears clearly that there is a strong correlation between the two variables considered. The relation of correlation found is expressed as follows : (a2(X)> = 0 . 4 7 - 0.70(/~,>
(6)
with R = 0.98, where R is the correlation coefficient. This kind of correlation is very important because it reduces the n u m b e r of input parameters to eqn (2) for P(X). Moreover, from the only information about (/~,), it allows us to have an idea about the spread or the distribution of global irradiation.
5. CONFIDENCE LIMITS OF
132
H. LOUTFI and A. KHTIRA -- Function
1.0
R h
I
--
I0
Function
I
R ,7 o.s
05
,January
April
I
0
I
x
l.O ¸
x
- Fonction ] f
l
1.0
Function ]
,~
--*- Data
R o.5
0.5
LI-
JuLy 0
October 2
I
x
0
I
x
Fig. 2. Comparison of theoretical and data-based distribution function, F(X).
fluctuations of this value, at a specified probability, can be determined by using the following relation :
I~(#;,) = ( K,) + tca(g,)/~/N
(7)
in which #(K,) is the expected mean, N the number of years considered (5 years in this case), a(K',) the standard deviation of K, and 6, a constant read from the Student t-distribution probability table. For each defined degree of freedom and a specified probability corresponds a constant to. To illustrate this method, we tackle a concrete example : given 5 monthly mean /~, for January, we calculate ( / ( , ) = 0.5423 and a(/(,) = 0.0240. With the specified probability of 0.9 (or with confidence limits of 90%), the Student tdistribution gives, for 7 = ( N - 1), a t,: of 1.53. Using eqn (7), the expected mean of K, for January is situated in the following boundaries with the probability of 0.9 :
I~(R,) = 0.5423 + 0.0180. This procedure is very useful for /£, values estimation, it will be used for generating synthetic sequences of solar irradiation (section 11).
6. GENERATION OF SYNTHETIC SEQUENCES The aim of this section is the generation of synthetic daily global solar irradiation on a horizontal surface. By synthetic sequences, we refer to sequences that are simulated numerically from a certain model that characterizes the statistical behavior of the actual data. The advantage of these synthetic sequences is that one year of such data could be used in simulating a solar energy system functioning. For specific objective of generating synthetic sequences, we develop an appropriate stochastic model for the daily global solar
Stochastic analysis and generation of synthetic sequences of daily global solar irradiation
133
0.14 '
0.12
0.10
A V
0.08
006
0.04
0.02
0 0.5
0.6
0.7
Fig. 3. Grand average of variance, (~r2(X)). vs (/~,).
irradiation process drawing upon the A R M A methodology described by Box and Jenkins [18]. This method allows the user to incorporate the probability distribution for X events while maintaining the sequential character of daily events. To accomplish successfully this task, the stochastic models were constructed not within the X domain, but with an intermediate Gaussian variable obtained with an exact mapping between the distribution function of X, F ( X ) , and of the Gaussian domain. 7. T R A N S F O R M A T I O N OF X
Classical methods in time series analysis [ 15, 17, 18] allow us to analyse the structure of series and give good estimates of related statistical parameters only when dealing with normal stationary process. Therefore, before the application of the A R M A methodology, it is necessary to correct the non Gaussian character of X series by transforming them into a new Gaussian random variable Z series. This new variable Z must have invariant statistics (i.e. same mean and variance as X series for all months). Let us consider T,, this unknown monthly transformation function, thus : Z = T,,(X).
(8)
This transformation should ensure that the marginal probabilities of the two variables are unaltered [19]. So, let G ( Z ) be the normal P D F and U ( Z ) the cumulative frequency of Z given by :
U(Z) =
~: G(t) dt = x- / ,_ 2n
.... exp
-
dt
(9)
P ( X ) (eqn 2) is the P D F of X, and F ( X ) its cumulative frequency. The aim is that the two variables X and Z have the same distribution function, we must have then : F(X) = U(Z).
(10)
Because U ( Z ) cannot be expressed in a closed-form, we cxpressed it by its approximation function defined as follows [20] :
The + sign applies to Z The error introduced by the is less than 0.0038 [20]. The can be expressed as follows Z=
U '(F(X))
> 0 and the - to Z < 0. use of this approximation unknown transformation :
and
T , , = U 'OF.
Equation (10) when solved for Z gives new time series with zero mean, unit standard deviation and normal distribution. The transform (10) preserves the structure of Markov process, i.e. if X lbllows a M a r k o v process, then Z follows also a M a r k o v process and conversely. These properties of transform (10) can be demonstrated mathematically in a general way. However, they are intuitive since this transform maps X into Z and conversely. So far, our analysis
134
H. LOUTFI and A. KHTIRA k
will be concerned with statistical properties of Z since all results obtained for Z hold for X as well.
Z(t) = ~ c~,Z(t-i)+ A(t)
where t represents a day number in the series, k is the order of the model, and A(t) is a white noise sequence. 0~ are the regression parameters to be determined from the statistical properties of the series.
8. STOCHASTIC MODEL CONSTRUCTION Model construction details are discussed in Box and Jenkins [18], the basic activities include model identification, parameters estimation and diagnostic checking of models residuals. The major steps are discussed briefly in turn below.
8.2. Parameters estimation The c o m m o n method for parameters' estimation consists of using auto-correlation coefficients. For an A R M A ( k , 0) model, the regression parameters, 0i, are expressed as a function of R~. Expressions for the two particular cases of A R M A ( 1 , 0 ) and A R M A ( 2 , 0) are given in Appendix A. The values of these parameters and the variance of the white noise (a2(A)), for different months are presented in Table 1 for the two models. We remark that the first order auto-regressive parameter is strongly significant for the A R M A ( 2 , 0). We notice also that the variance a2(A) is seen to be hardly affected by a higher order model. This confirms that the most correct model to be applied is an ARMA(1,0).
8.1. Model ident!fication The identification exercise consists of the determination of both the nature and the order of a stochastic model. The principal identification tools are the auto-correlation and partial auto-correlation coefficients [18]. An auto-correlation coefficient of order k ( & ) represents the correlation between sets of Z(t) and Z(t+k) series, it is given by:
R~ = C(k)/C(O) JI
C(k) = ( 1 / ( J l - 1 ) )
(12)
k
(Z(j)-2)(Z(j+k)-2)
~ j-
I
(13) in which 2 is the Z variable mean over the J l days (J1 = J x 5). The type of stochastic model we selected for daily global solar radiation is based on an empirical fact that the time persistence for this variable is relatively short (1-2 days). From the analysis of the two partial auto-correlation coefficients of interest [l 8] : qS(l, 1) = R,
(14)
q~(2,2) = (R2-R~)/(I-R~).
05)
(16)
i--I
9. GENERATION OF SYNTHETIC SEQUENCES OF X
Once we have completely defined the model, we can generate new sequences of Z and then calculate the corresponding values of X by using an inverse transformation of eqn (8). This task can be done by following the different steps in turn below : I. Generation of a white noise sequence. 2. Generation of new sequences of Z : knowing the auto-regressive parameters ~bi and the initial values of Z(t), we generate new series of Z(t) from eqn (12). The values needed to start the synthetic sequences generations are taken arbitrarily from a sequence of normal distribution [19]. 3. To determine new sequences of the variable X, we
And according to our previous study for Rabat [21], the model we have selected is an auto-regressive model of a first or second order [ A R M A ( I , 0 ) or A R M A ( 2 , 0), respectively]. An auto-regressive model of order k [ARMA(k, 0)] is defined by the following equation :
Table 1. Auto-regressive parameters for ARMA(I, 0) and ARMA(2, 0) Months
J
F
M
A
May
ARMA(1,0) ~b~ 0.31 a2(A) 0.90
0.25 0.93
0,40 0,84
O.lO 0.99
0.13 0.98
ARMA(2, 0) ~b~ 0.33 ~b2 0.08 a2(A) 0.89
0.28 -0.09 0.93
0.42 -0.04 0.83
0.08 0.09 0.98
0.14 -0.11 0.97
J 0.12 0.99 0.12 0.003 0.98
Jul
A
S
O
N
D
0.26 0.93
0.20 0.96
0.31 0.90
0.30 0.92
0.24 0.94
0.28 0.92
0.27 -0.05 0.93
0.22 -0.08 0.95
0.32 -0.02 0.90
0.30 -0.02 0.91
0.24 -0.007 0.94
0.25 0.11 0.91
Stochastic analysis and generation of synthetic sequences of daily global solar irradiation
135
Table 2. Comparison between actual statistical properties and the corresponding values as calculated from synthetic sequences generated from three different auto-regressive models January : (R,) = 0.542 Actual parameters
ARMA(1,0)
ARMAG(I,0)
1.009 0.095 0.289 --
1.011 0.093 0.316 --
Actual parameters
ARMA(1,0)
ARMAG(1,0)
ARMAG(2, 0)
1.000 0.047 0.098 0.097
1.001 0.048 0.096
1.001 0.048 0.129
1.007 0.048 0.044 0.122
Actual parameters
ARMA(1, 0)
ARMAG(1,0)
ARMAG(2,0)
1.000 0.013 0.260 0.031
1.003 0.014 0.245
1.003 0.015 0.361 --
0.994 0.014 0.298 0.031
ARMA(1,0)
ARMAG(I,0)
ARMAG(2,0)
1.005 0.032 0.276
1.006 0.034 0.339
0.985 0.033 0.277 0.151
)?
1.000 0.092 0.309 -0.082
a'-(X) ~b(l, 1) 4~(2, 2)
ARMAG(2,0) 0.985 0.095 0.260 -0.002
April: (R,) = 0.606
.g a2(X) ~b(l, 1) ~b(2, 2) July: (R,) = 0.644
)? cr2(X) 4)(I, l) q5(2,2)
October : (R,) = 0.633 Actual parameters )~ a2(X) 4~(1, 1) q~(2, 2)
1.000 0.031 0.295 - 0.023
apply an inverse t r a n s f o r m a t i o n of Tm (eqn 8). The generated X series are o b t a i n e d by iteration from this relation : I
1
X,,+,(j) = AF(Z(j))/(n+ l
(n ~X(j) 2~ Xm.ax// (17)
in which n, A a n d zxVmaxare given by eqns (3)-(5). F o r each m o n t h , we have c o m p a r e d statistics of the five years generated sequences, from three kinds of models, with actual data. T h e models considered are as follows : A R M A ( 1 , 0 ) : first order auto-regressive model w i t h o u t G a u s s i a n mapping. - A R M A G ( 1 , 0 ) : first order auto-regressive model with G a u s s i a n mapping. --ARMAG(2,0): second order auto-regressive model with G a u s s i a n mapping. Results' analysis has two m a j o r objectives : first, to
test the ability of the models to generate synthetic sequences with accurate statistics (mean, variance a n d partial auto-correlation coefficients). Second, to test their ability in predicting accurate stationary P D F . We have performed these c o m p a r i s o n s for each m o n t h : Table 2 illustrates the first objective while Table 3 a n d Fig. 4 illustrate the second for the four m o n t h s representing the different seasons.
Table 3. Accuracy of the three models considered in predicting stationary PDF, expressed as ?~2 values, for synthetically generated daily solar radiation versus actual data. The degree of freedom is given in parenthesis Month January April July October
ARMA(I,0) 48.30 44.70 23.41 32.94
(v = (v = (v = (v =
7) 6) 7) 7)
ARMAG(1,0) 28.60 11.50 17.17 13.09
(v = (v = (v = (v =
7) 6) 7) 7)
ARMAG(2,0) 18.70 14.20 13.16 38.67
(v (v = (v = (v =
8) 6) 9) 9)
H. LOUTFI and A. KHTIRA
136
January]
5 ] -e. ARMA (i,O) ]
II -'-ARMAG
'°11
0
-e-ARMA(I,O) ]
April
/
I
2
0
I
-e-ARMA (1,0) I
2
x
X
5
"~
I
Ii
July
October
ARMA (I,O) I-4-ARMAG (I,0)
--*-ARMAG (, ,0) I
[-~- Data
4.
32"
I
0
I
2
x
0
f
x
Fig. 4. Ability of AR models to generate accurate stationary PDF, P(X).
Table 3 shows that for all A R M A models considered, even with the relatively small samples, the generated mean and variance are satisfactory. However, this is less clear for the auto-correlation coefficients, where 150 values appear to be too small for their estimation. Similar observations were mentioned in other studies [9]. Table 3 and Fig. 4 highlight the inadequacy of an A R M A ( 1 , 0 ) model without Gaussian mapping in predicting an accurate stationary PDF. This inaccuracy is due firstly to the fact that eqn (2) for P(X) is not perfect, secondly, because the synthetic sequence is finite. It introduces then uncertainties when selecting a finite sample from its population to perform the chi2 test. However, results are generally acceptable taking account of the reduced size of the basic data set.
10. C O N S T R U C T I O N OF A "TYPICAL YEAR"
The question which we would like to answer at this stage is, "if we wish to generate only one year of synthetic data how accurately will this year represent the actual statistics of 'multi-years'?". To answer this question we have generated synthetic sequences, from the A R M A G ( I , 0), l0 separate times for each month, each time with a different set of white noise. To highlight the effect of the generated samples' size on final results, we have generated five separate times a series of 300 values for each month. Statistics properties and comparison of the P D F with actual data, in terms of chi2 for July, are summarized in Table 4. We notice that there is a marked fluctuation in the statistics corresponding to the series of 31 values from one run to another. This is due to the important sen-
Stochastic analysis and generation of synthetic sequences of daily global solar irradiation Table 4. Comparison between the actual parameters and the corresponding values as computed from synthetic data sequences of 31 values long from 10 separate generation runs, and 300 values long from 5 separate generations. "Goodness" of fit in predicting the stationary PDF is measured by the X2 test. The degree of freedom is equal to 3 for the short series and 17 for the long one )? Actual synoptic data Series of 31 values Run number 1 2 3 4 5 6 7 8 9 10 Series of 300 values Run number I 2 3 4 5
cr2(X)
4,
Z2
1
0.013
0.269
1.017 0.984 0.995 1.003 0.997 0.989 1.015 0.996 1.004 0.996
0.013 0.022 0.011 0.016 0.016 0.009 0.011 0.013 0.012 0.012
0.360 0.437 0.222 0.405 0.423 0.081 0.028 0.286 0.417 0.366
1.002 1.004 1.015 0.997 1.004
0.013 0.013 0.013 0.013 0.012
0.350 20.54 0.336 24.15 0.398 28.51 0.392 8.996 0.305 17.58
2.897 7.554 2.867 2.102 7.523 2.089 1.336 4.018 1.314 2.348
sitivity of the small white noise sequence to the choice of initial inputs. However, for all runs results remain acceptable. In order to define a "typical year", we have performed at least 10 separate monthly generation runs and we have chosen then the "best" run for each month. For the generation of long series, we notice that there is no remarkable difference between the different runs. Therefore, five generation runs seem enough to choose the run that gives the best sequence. The best runs for July are underlined in Table 4.
It. GENERATION OF SYNTHETIC SEQUENCES OF DALLY GLOBAL SOLAR RADIATION The principal aim of this analysis is the generation of synthetic sequences of daily global solar radiation (H~) rather than those of X. Hence, having series of X~ (generated value of X) and K, for a certain number of years we can determine easily the corresponding values of //.,t for the considered m o n t h (i.e. H~ = XqR, Hoh where Hob is the solar irradiation value outside the Earth's atmosphere). To generate series longer than the basic series we use eqn (7) to estimate values of R, for the required number of years. The structure of stochastic A R M A models ensures that
137
these generated series are not identical to real data but they are typical for the considered location.
12. CONCLUSIONS A stochastic procedure capable of generating sets of synthetic daily irradiation values has been developed. The only information needed to build these sequences is the monthly average of K, so that our approach can be used also for localities where solar radiation data are not directly measured, they can be estimated fairly accurately from very crude information such as average hours of sunshine which is commonly collected by hundreds of meteorological stations around the world. Specifically, for daily global solar radiation we have : --Segmented the year into months and identified an appropriate variable for statistical analysis on the monthly basis. - - T e s t e d and adopted a functional form for the stationary probability for daily solar radiation which requires the knowledge of the mean and the variance only, and compared it against our real data. --Calculated the mean monthly statistical parameters (i.e. mean, variance), considered correlations among them and suggested a procedure based on confidence limits that unables us to ascertain the uncertainty range of the mean ( K , ) given a limited number of years samples for a given month at a particular location. Examined several types of auto-regressive stochastic models toward the generation of synthetic daily solar radiation sequences that capture all the essential statistical stationary and sequential information of the climate. - - P r o p o s e d how an appropriate synthetic sequence can be selected to construct a typical year of data that can be used for solar energy systems simulation. The type of analysis and the results presented here are aiming to provide the user with both time saving calculational procedures and the required solar radiation input data for optimal sizing softwares for solar systems, and particularly for stand-alone systems.
REFERENCES
1. P. Bendt, M. Collares-Pereira and A. Rabl, The frequency distribution of daily insolation values. Solar Eneryy 27, 1 5 (1981). 2. S. A. Klein, Calculation of flat plate collector utilizability. Solar Energy 21,393 402 (1978).
138
H. LOUTFI and A. KHTIRA
3. J. M. Gordon and M. Hochman, On the random nature of solar radiation. Solar Energy 32, 337-341 (1984). 4. E. Boileau, Use of some simple statistical models in solar meteorology. Solar Energy 30, 333 339 (1983). 5. L. Vergara-Dominguez, R. Garcia-Cromez, A. R. Figuerias-Vidal and J. R. Casar-Carredera, Automatic modelling and simulation of daily global solar radiation series. Solar Energy 35, 483~,89 (1985). 6. U. Amato, A. Andretta, B. Bartoli, B. Coluzzi, V. Cuomo and C. Serio, Stochastic modelling of solar radiation data. IL Nuovo Cimento 8, 248-258 (1985). 7. B.J. Brinkworth, Autocorrelation and stochastic modelling of insolation sequence. Solar Energy 19, 343 347 (1977). 8. C. Mustacchi, Stochastic simulation of hourly global radiation sequences. Solar Energy 23, 47 51 (1979). 9. J. M. Gordon and T. A. Reddy, Time series analysis of daily horizontal solar radiation. Solar Energy 41, 215 226 (1988). 10. J. D. Engels, S. M. Pollock and J. A. Clark, Observation on the statistical nature of terrestrial irradiation. Solar Energy 26, 91 92 (1981). 11. K. T. Hollands and R. J. Huget, A probability density function for the clearness index, with application. Solar Energy 30, 195 209 (1983). 12. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4, 1-19 (1960). 13. G. Y. Saunier, T. A. Reddy and S. Kumar, A monthly probability distribution function of daily global irradiation values appropriate for both tropical and temperate locations. Solar Energy 38, 167 177 (1987). 14. A. A. Sfeir, Solar radiation in Lebanon. Solar Energy 26, 497-502 (1981). 15. M. Kendall and A. Stuart, The Advanced Theory of Statistics. Charles Griffin, London (1976). 16. H. Nfaoui and J. Bahraoui-Buret, Trois anndes de mesures de l'ensoteillement fi Rabat. Proc. de la 24Ome Coq/~rence lnternationale de la Comples, Verona, Italy, pp. 179 187 (1986). 17. M. R. Spiegel, ThOorie et Application de la Statistique. Edition Franqaise (1980). 18. G. E. P. Box and G. M. Jenkins, Time Series Analysis." Forcasting and Control. Holden Day, San Francisco (1979).
19. V. A. Graham, K. G. T. Hollands and T. E. Unny, A time series model for K, with application to global synthetic weather generation. Solar Energy 40, 83 92 (1988). 20. A. A. Sfeir, A stochastic model for predicting solar performance. Solar Energy 25, 149 153 (1980). 21. H. Loutfi, A. Khtira and J. Buret-Bahraoui, Daily global irradiation dynamic study : Modelling with a stochastic model. Proc. of the Ist Worm Renewable Energy Congress, Reading, U.K., 23 28 Sept., Vol. 5, pp. 3132 3136 (1990).
APPENDIX
Auto-regressive models (ARMA(k, 0)) [23] are stochastic models in which the variable of the process is expressed as a finite, linear sum o f previous values plus a random white noise. The basic variable, in this analysis, is the daily global horizontal solar irradiation (H(t)). Let Z(t) be the stochastic variable constructed from H(t), where t denotes the day, an ARMA(k, 0) of order k is expressed as: i
Z(t) = ~ ¢ i Z ( t - i ) + A ( t ) . i-I
The regression parameters ~b~,¢~ . . . . .
4~K,are given by:
- - F o r a first order model (ARMA(I,0)) : ¢~ =R~. For a second order model (ARMA(2, 0)) : ¢, = R , ( 1 - R 2 ) / ( I - R ~ z)
ff)z = (R2-- R~)/(I - R~). Where R~ and Rz are the auto-correlation coefficients (eqns 12, 13) for one and two day lags, respectively. The random noise, A(t), should be normally distributed, should have zero mean and have a specified variance given below : For a first order model: A R M A ( I , 0)
a2(A) = 1--R~. - - F o r a second order model : ARMA(2, 0)
~(A) = 1-¢~-¢~-
2¢,4~/(1-4~2).
096(~1481/92 $5.00+.00 Pergamon Press Ltd
STOCHASTIC ANALYSIS A N D GENERATION OF SYNTHETIC SEQUENCES OF DAILY GLOBAL SOLAR IRRADIATION: RABAT SITE (MOROCCO) H . LOUTFI a n d A. KHTIRA Laboratoire d'Energie Solaire, Facult~ des Sciences, B.P 1014, Rabat, Morocco (Received 22 January 1991 ; accepted 1 October 1991) A b s t r a c ~ A study of the stationary and sequential properties of daily global horizontal solar radiation is presented for Rabat (Morocco). The results from this analysis are essential for the analytic modelling of the long term performance of solar energy systems, and for the generation of synthetic daily solar radiation sequences. The only climatic information required is the monthly average of the clearness index. We show that the experimental data can be described by a linear auto-regressive stochastic model of first or second order, and that the probability density function PDF can be fitted by a simple, easy to calculate functional form. We also show that, by using this function, the clearness index sequences can be transformed into sequences with normal distribution but with the same auto-correlation properties. It becomes, then, easy to construct synthetic sequences of solar irradiation with the same statistical properties as the real data. These synthetic sequences are very useful for performance prediction of solar energy systems.
1. I N T R O D U C T I O N The statistical modelling of global horizontal solar irradiation is very interesting for many practical applications such as the design and performance prediction of solar energy conversion systems. In fact, one of the most challenging problems that faces solar systems engineers is a proper accounting of the random nature of solar irradiation. If the statistical distribution of this irradiation is known, a large quantity of irradiation data can be summarized with only a few parameters. A typical example is the use of the distribution function of daily global solar irradiation to calculate the efficiency of solar collectors [1-3]. In many cases, such as the optimal sizing of thermal and photovoltaic systems with a storage unit, it is also necessary to know the sequential properties of solar irradiation data. In this paper, we perform a statistical analysis of daily global horizontal solar irradiation in Rabat (34' N). We use a simple functional form for the stationary probability density, and test several types of linear auto-regressive stochastic models toward the generation of synthetic sequences that have the same statistical properties as actual data. This procedure allows 1o characterize the insolation properties of a locality with only few parameters such as the clearness index monthly average, and auto-correlation coefficients. Moreover, it becomes possible to generate a "typical" year of data.
2. S E L E C T I O N OF AN A P P R O P R I A T E S T A T I S T I C A L VARIABLE
The first step in a time series analysis of solar irradiation is to choose both the series size and the variable temporal scale. Many previous studies were developed on a yearly [4~6] or seasonal [7, 8] basis. We choose to divide the year into monthly periods for practical reasons [6, 9]. First, most system design is performed on a monthly basis. Second, because we prefer a time scale that is short enough to consider the annual trends in solar radiation as constant. Thus, a further filtering out of annual trends is not necessary. This procedure has been adopted in many other studies [6, 9]. We consider a series of 5 years of daily global horizontal solar irradiation (1983-1987). To obtain a stationary variable (necessary for a stochastic analysis) and as our analysis is performed on a monthly basis, we need to extract from the daily irradiation the monthly average and the effect of latitude. It was pointed out that the effect of latitude can be eliminated by using the clearness index, K, [5, 6, 8]. However, K, itself is not adapted to a stochastic analysis, because the K, monthly average (Kt) varies markedly from year to year for a given month. The aim is to transform K, into a stationary variable through which we can take account of fluctuations in monthly average of K,. We consider then a variable defined as follows : X ( d , m , y ) = K,(d,m,y)/K,(m,y) 129
(1)
H. LOUTFIand A. KHTIRA
130
where K, is the monthly average of K,, (d, m,y) denotes day, month and year respectively. The selection of this variable introduces a small correlation error, especially because R, depends on all Kt values for a given month : division of K, by Rt introduces an error of roughly l/J, where J is the number of days in the month ; this variable was used by Gordon and Reddy [ll]. Monthly statistical properties of K, values are calculated for each individual year, and then averaged over all years. We will refer to this kind of average as grand average.
3. S T A T I O N A R Y P R O B A B I L I T Y D E N S I T Y
FUNCTION For analytic modelling purposes, it is preferable to have a functional form for the probability density PDF, P(X), that has been compared with actual data with good results. Several studies have considered prediction of the PDF of daily global solar irradiation on a horizontal surface [1, 10-14]. However, so far there is no universal expression for the PDF of K, [l 3]. A proposed function must obey to universal laws of probability : P(X) must vanish at sufficiently low and high values of X. Many functional closed-forms have been published for P(X) [1, 6, 13], but their major short-coming is that they assume the maximum value of K, either as a universal constant or as a known variable. However, R,.... is not a universal constant [13]. In this analysis, we adopt a closed-form which has been proposed by Gordon and Reddy [9]. This closed form takes account of the major limitations of previous proposed functions for PDF and has no adjustable parameters. It was pointed out that this function has been compared favorably with actual data from different climates [9]. The expression of this closed-form is as follows :
grated in a closed-form ; this integral (cumulative frequency) is very useful for system design where integral of P(X) is required. To decide of the use of eqn (2) for the (PDF), we compare it with actual data. In practice, we do not suppose that the detailed values of K, and O'2(X) are known as input to eqn (2), but only their average over the years denoted (R,) and (a2(X)) respectively. It is worth noting that even (a2(X)) values are not tabulated, but recently there has been claims to include values of (a2(X)) in data tables [9]. To characterize quantitatively the quality of fit we have performed the chi2 test (X2) [15] for both P(X) and its integral F(x). The latter function is very useful for system design and the generation of synthetic sequences of solar irradiation (section 7 below). The PDF calculations for actual data were done by dividing the series, for each month (series of 5 times number of days in this month), into a certain number of intervals. In each interval we assess the number of occurrences (frequency). The degree of freedom is equal to the number of bins considered minus 3, because to define P(X) we supposed normalization and a knowledge of K, and cI2(X). Results from the chi2 test can be appreciated by the fraction of data sets that are accepted by the test, in other words, that correspond to several signification levels of the test. The fraction of data corresponding to a signification level of 0.01C(0.01)) for both P(X) and F(X), for Rabat are as follows :
~(0.01)
P(X)
F(X)
66.6%
83.3%
Figures 1 and 2 represent a sample comparison of
P(X) and F(X), respectively, with the corresponding P(X) = AX"[1 -- (X/Xm,x)]
(2)
n, A and Xm,x are parameters to be determined, for each month, from the three following relations : normalization of P(X), knowledge of K, (X = 1) and knowledge of the variance of X. Development of these three relations leads to : n = -2.5+0.519+(8/a2(X))] '~2
(3)
)(max = ( n + 3)/(n+ 1)
(4)
A = (n+ 1)(n+2)/X'~+x'.
(5)
Equations (2) to (5) define completely the functional form ofeqn (2). P(X) can then be incorporated in all kinds of calculations without resorting to numerical solutions. Moreover, P(X) can be inte-
function of actual data for four months representing the different seasons as similarities between different months in the same season were noticed. Taking account of the relatively small sample of data considered, results obtained are encouraging and comparable to those of a previous study [9]. As we mentioned above, values of (~2(X)) are not available in most solar data tables, it would be desirable if we could estimate this variable from other climatic information such as the clearness index which is on the other hand, the most available variable. 4. C O R R E L A T I O N BETWEEN <~rZ(X)> A N D
Several studies have considered the statistical analysis of K, in Rabat [16]. These analyses led to many
Stochastic analysis and generation of synthetic sequences of daily global solar irradiation --
Function Data
]
January
--
Fun~ion
I
131
April
2 ¸
I
l
X
X
51 l 4
3-
(1.
--
Function
-I,-
Data
July
!
--
Function
•+
Data
October
E
2 I
L__
L I
X
X
Fig. 1. Comparison of theoretical (eqn 2) and data-based PDF, P(X).
kinds of correlation between K, and several other climatic components. In this analysis we have considered a correlation between a meteorological variable and its statistical property such as the variance. Figure 3 shows a plot of (/~,) versus
(6)
with R = 0.98, where R is the correlation coefficient. This kind of correlation is very important because it reduces the n u m b e r of input parameters to eqn (2) for P(X). Moreover, from the only information about (/~,), it allows us to have an idea about the spread or the distribution of global irradiation.
5. CONFIDENCE LIMITS OF
132
H. LOUTFI and A. KHTIRA -- Function
1.0
R h
I
--
I0
Function
I
R ,7 o.s
05
,January
April
I
0
I
x
l.O ¸
x
- Fonction ] f
l
1.0
Function ]
,~
--*- Data
R o.5
0.5
LI-
JuLy 0
October 2
I
x
0
I
x
Fig. 2. Comparison of theoretical and data-based distribution function, F(X).
fluctuations of this value, at a specified probability, can be determined by using the following relation :
I~(#;,) = ( K,) + tca(g,)/~/N
(7)
in which #(K,) is the expected mean, N the number of years considered (5 years in this case), a(K',) the standard deviation of K, and 6, a constant read from the Student t-distribution probability table. For each defined degree of freedom and a specified probability corresponds a constant to. To illustrate this method, we tackle a concrete example : given 5 monthly mean /~, for January, we calculate ( / ( , ) = 0.5423 and a(/(,) = 0.0240. With the specified probability of 0.9 (or with confidence limits of 90%), the Student tdistribution gives, for 7 = ( N - 1), a t,: of 1.53. Using eqn (7), the expected mean of K, for January is situated in the following boundaries with the probability of 0.9 :
I~(R,) = 0.5423 + 0.0180. This procedure is very useful for /£, values estimation, it will be used for generating synthetic sequences of solar irradiation (section 11).
6. GENERATION OF SYNTHETIC SEQUENCES The aim of this section is the generation of synthetic daily global solar irradiation on a horizontal surface. By synthetic sequences, we refer to sequences that are simulated numerically from a certain model that characterizes the statistical behavior of the actual data. The advantage of these synthetic sequences is that one year of such data could be used in simulating a solar energy system functioning. For specific objective of generating synthetic sequences, we develop an appropriate stochastic model for the daily global solar
Stochastic analysis and generation of synthetic sequences of daily global solar irradiation
133
0.14 '
0.12
0.10
A V
0.08
006
0.04
0.02
0 0.5
0.6
0.7
Fig. 3. Grand average of variance, (~r2(X)). vs (/~,).
irradiation process drawing upon the A R M A methodology described by Box and Jenkins [18]. This method allows the user to incorporate the probability distribution for X events while maintaining the sequential character of daily events. To accomplish successfully this task, the stochastic models were constructed not within the X domain, but with an intermediate Gaussian variable obtained with an exact mapping between the distribution function of X, F ( X ) , and of the Gaussian domain. 7. T R A N S F O R M A T I O N OF X
Classical methods in time series analysis [ 15, 17, 18] allow us to analyse the structure of series and give good estimates of related statistical parameters only when dealing with normal stationary process. Therefore, before the application of the A R M A methodology, it is necessary to correct the non Gaussian character of X series by transforming them into a new Gaussian random variable Z series. This new variable Z must have invariant statistics (i.e. same mean and variance as X series for all months). Let us consider T,, this unknown monthly transformation function, thus : Z = T,,(X).
(8)
This transformation should ensure that the marginal probabilities of the two variables are unaltered [19]. So, let G ( Z ) be the normal P D F and U ( Z ) the cumulative frequency of Z given by :
U(Z) =
~: G(t) dt = x- / ,_ 2n
.... exp
-
dt
(9)
P ( X ) (eqn 2) is the P D F of X, and F ( X ) its cumulative frequency. The aim is that the two variables X and Z have the same distribution function, we must have then : F(X) = U(Z).
(10)
Because U ( Z ) cannot be expressed in a closed-form, we cxpressed it by its approximation function defined as follows [20] :
The + sign applies to Z The error introduced by the is less than 0.0038 [20]. The can be expressed as follows Z=
U '(F(X))
> 0 and the - to Z < 0. use of this approximation unknown transformation :
and
T , , = U 'OF.
Equation (10) when solved for Z gives new time series with zero mean, unit standard deviation and normal distribution. The transform (10) preserves the structure of Markov process, i.e. if X lbllows a M a r k o v process, then Z follows also a M a r k o v process and conversely. These properties of transform (10) can be demonstrated mathematically in a general way. However, they are intuitive since this transform maps X into Z and conversely. So far, our analysis
134
H. LOUTFI and A. KHTIRA k
will be concerned with statistical properties of Z since all results obtained for Z hold for X as well.
Z(t) = ~ c~,Z(t-i)+ A(t)
where t represents a day number in the series, k is the order of the model, and A(t) is a white noise sequence. 0~ are the regression parameters to be determined from the statistical properties of the series.
8. STOCHASTIC MODEL CONSTRUCTION Model construction details are discussed in Box and Jenkins [18], the basic activities include model identification, parameters estimation and diagnostic checking of models residuals. The major steps are discussed briefly in turn below.
8.2. Parameters estimation The c o m m o n method for parameters' estimation consists of using auto-correlation coefficients. For an A R M A ( k , 0) model, the regression parameters, 0i, are expressed as a function of R~. Expressions for the two particular cases of A R M A ( 1 , 0 ) and A R M A ( 2 , 0) are given in Appendix A. The values of these parameters and the variance of the white noise (a2(A)), for different months are presented in Table 1 for the two models. We remark that the first order auto-regressive parameter is strongly significant for the A R M A ( 2 , 0). We notice also that the variance a2(A) is seen to be hardly affected by a higher order model. This confirms that the most correct model to be applied is an ARMA(1,0).
8.1. Model ident!fication The identification exercise consists of the determination of both the nature and the order of a stochastic model. The principal identification tools are the auto-correlation and partial auto-correlation coefficients [18]. An auto-correlation coefficient of order k ( & ) represents the correlation between sets of Z(t) and Z(t+k) series, it is given by:
R~ = C(k)/C(O) JI
C(k) = ( 1 / ( J l - 1 ) )
(12)
k
(Z(j)-2)(Z(j+k)-2)
~ j-
I
(13) in which 2 is the Z variable mean over the J l days (J1 = J x 5). The type of stochastic model we selected for daily global solar radiation is based on an empirical fact that the time persistence for this variable is relatively short (1-2 days). From the analysis of the two partial auto-correlation coefficients of interest [l 8] : qS(l, 1) = R,
(14)
q~(2,2) = (R2-R~)/(I-R~).
05)
(16)
i--I
9. GENERATION OF SYNTHETIC SEQUENCES OF X
Once we have completely defined the model, we can generate new sequences of Z and then calculate the corresponding values of X by using an inverse transformation of eqn (8). This task can be done by following the different steps in turn below : I. Generation of a white noise sequence. 2. Generation of new sequences of Z : knowing the auto-regressive parameters ~bi and the initial values of Z(t), we generate new series of Z(t) from eqn (12). The values needed to start the synthetic sequences generations are taken arbitrarily from a sequence of normal distribution [19]. 3. To determine new sequences of the variable X, we
And according to our previous study for Rabat [21], the model we have selected is an auto-regressive model of a first or second order [ A R M A ( I , 0 ) or A R M A ( 2 , 0), respectively]. An auto-regressive model of order k [ARMA(k, 0)] is defined by the following equation :
Table 1. Auto-regressive parameters for ARMA(I, 0) and ARMA(2, 0) Months
J
F
M
A
May
ARMA(1,0) ~b~ 0.31 a2(A) 0.90
0.25 0.93
0,40 0,84
O.lO 0.99
0.13 0.98
ARMA(2, 0) ~b~ 0.33 ~b2 0.08 a2(A) 0.89
0.28 -0.09 0.93
0.42 -0.04 0.83
0.08 0.09 0.98
0.14 -0.11 0.97
J 0.12 0.99 0.12 0.003 0.98
Jul
A
S
O
N
D
0.26 0.93
0.20 0.96
0.31 0.90
0.30 0.92
0.24 0.94
0.28 0.92
0.27 -0.05 0.93
0.22 -0.08 0.95
0.32 -0.02 0.90
0.30 -0.02 0.91
0.24 -0.007 0.94
0.25 0.11 0.91
Stochastic analysis and generation of synthetic sequences of daily global solar irradiation
135
Table 2. Comparison between actual statistical properties and the corresponding values as calculated from synthetic sequences generated from three different auto-regressive models January : (R,) = 0.542 Actual parameters
ARMA(1,0)
ARMAG(I,0)
1.009 0.095 0.289 --
1.011 0.093 0.316 --
Actual parameters
ARMA(1,0)
ARMAG(1,0)
ARMAG(2, 0)
1.000 0.047 0.098 0.097
1.001 0.048 0.096
1.001 0.048 0.129
1.007 0.048 0.044 0.122
Actual parameters
ARMA(1, 0)
ARMAG(1,0)
ARMAG(2,0)
1.000 0.013 0.260 0.031
1.003 0.014 0.245
1.003 0.015 0.361 --
0.994 0.014 0.298 0.031
ARMA(1,0)
ARMAG(I,0)
ARMAG(2,0)
1.005 0.032 0.276
1.006 0.034 0.339
0.985 0.033 0.277 0.151
)?
1.000 0.092 0.309 -0.082
a'-(X) ~b(l, 1) 4~(2, 2)
ARMAG(2,0) 0.985 0.095 0.260 -0.002
April: (R,) = 0.606
.g a2(X) ~b(l, 1) ~b(2, 2) July: (R,) = 0.644
)? cr2(X) 4)(I, l) q5(2,2)
October : (R,) = 0.633 Actual parameters )~ a2(X) 4~(1, 1) q~(2, 2)
1.000 0.031 0.295 - 0.023
apply an inverse t r a n s f o r m a t i o n of Tm (eqn 8). The generated X series are o b t a i n e d by iteration from this relation : I
1
X,,+,(j) = AF(Z(j))/(n+ l
(n ~X(j) 2~ Xm.ax// (17)
in which n, A a n d zxVmaxare given by eqns (3)-(5). F o r each m o n t h , we have c o m p a r e d statistics of the five years generated sequences, from three kinds of models, with actual data. T h e models considered are as follows : A R M A ( 1 , 0 ) : first order auto-regressive model w i t h o u t G a u s s i a n mapping. - A R M A G ( 1 , 0 ) : first order auto-regressive model with G a u s s i a n mapping. --ARMAG(2,0): second order auto-regressive model with G a u s s i a n mapping. Results' analysis has two m a j o r objectives : first, to
test the ability of the models to generate synthetic sequences with accurate statistics (mean, variance a n d partial auto-correlation coefficients). Second, to test their ability in predicting accurate stationary P D F . We have performed these c o m p a r i s o n s for each m o n t h : Table 2 illustrates the first objective while Table 3 a n d Fig. 4 illustrate the second for the four m o n t h s representing the different seasons.
Table 3. Accuracy of the three models considered in predicting stationary PDF, expressed as ?~2 values, for synthetically generated daily solar radiation versus actual data. The degree of freedom is given in parenthesis Month January April July October
ARMA(I,0) 48.30 44.70 23.41 32.94
(v = (v = (v = (v =
7) 6) 7) 7)
ARMAG(1,0) 28.60 11.50 17.17 13.09
(v = (v = (v = (v =
7) 6) 7) 7)
ARMAG(2,0) 18.70 14.20 13.16 38.67
(v (v = (v = (v =
8) 6) 9) 9)
H. LOUTFI and A. KHTIRA
136
January]
5 ] -e. ARMA (i,O) ]
II -'-ARMAG
'°11
0
-e-ARMA(I,O) ]
April
/
I
2
0
I
-e-ARMA (1,0) I
2
x
X
5
"~
I
Ii
July
October
ARMA (I,O) I-4-ARMAG (I,0)
--*-ARMAG (, ,0) I
[-~- Data
4.
32"
I
0
I
2
x
0
f
x
Fig. 4. Ability of AR models to generate accurate stationary PDF, P(X).
Table 3 shows that for all A R M A models considered, even with the relatively small samples, the generated mean and variance are satisfactory. However, this is less clear for the auto-correlation coefficients, where 150 values appear to be too small for their estimation. Similar observations were mentioned in other studies [9]. Table 3 and Fig. 4 highlight the inadequacy of an A R M A ( 1 , 0 ) model without Gaussian mapping in predicting an accurate stationary PDF. This inaccuracy is due firstly to the fact that eqn (2) for P(X) is not perfect, secondly, because the synthetic sequence is finite. It introduces then uncertainties when selecting a finite sample from its population to perform the chi2 test. However, results are generally acceptable taking account of the reduced size of the basic data set.
10. C O N S T R U C T I O N OF A "TYPICAL YEAR"
The question which we would like to answer at this stage is, "if we wish to generate only one year of synthetic data how accurately will this year represent the actual statistics of 'multi-years'?". To answer this question we have generated synthetic sequences, from the A R M A G ( I , 0), l0 separate times for each month, each time with a different set of white noise. To highlight the effect of the generated samples' size on final results, we have generated five separate times a series of 300 values for each month. Statistics properties and comparison of the P D F with actual data, in terms of chi2 for July, are summarized in Table 4. We notice that there is a marked fluctuation in the statistics corresponding to the series of 31 values from one run to another. This is due to the important sen-
Stochastic analysis and generation of synthetic sequences of daily global solar irradiation Table 4. Comparison between the actual parameters and the corresponding values as computed from synthetic data sequences of 31 values long from 10 separate generation runs, and 300 values long from 5 separate generations. "Goodness" of fit in predicting the stationary PDF is measured by the X2 test. The degree of freedom is equal to 3 for the short series and 17 for the long one )? Actual synoptic data Series of 31 values Run number 1 2 3 4 5 6 7 8 9 10 Series of 300 values Run number I 2 3 4 5
cr2(X)
4,
Z2
1
0.013
0.269
1.017 0.984 0.995 1.003 0.997 0.989 1.015 0.996 1.004 0.996
0.013 0.022 0.011 0.016 0.016 0.009 0.011 0.013 0.012 0.012
0.360 0.437 0.222 0.405 0.423 0.081 0.028 0.286 0.417 0.366
1.002 1.004 1.015 0.997 1.004
0.013 0.013 0.013 0.013 0.012
0.350 20.54 0.336 24.15 0.398 28.51 0.392 8.996 0.305 17.58
2.897 7.554 2.867 2.102 7.523 2.089 1.336 4.018 1.314 2.348
sitivity of the small white noise sequence to the choice of initial inputs. However, for all runs results remain acceptable. In order to define a "typical year", we have performed at least 10 separate monthly generation runs and we have chosen then the "best" run for each month. For the generation of long series, we notice that there is no remarkable difference between the different runs. Therefore, five generation runs seem enough to choose the run that gives the best sequence. The best runs for July are underlined in Table 4.
It. GENERATION OF SYNTHETIC SEQUENCES OF DALLY GLOBAL SOLAR RADIATION The principal aim of this analysis is the generation of synthetic sequences of daily global solar radiation (H~) rather than those of X. Hence, having series of X~ (generated value of X) and K, for a certain number of years we can determine easily the corresponding values of //.,t for the considered m o n t h (i.e. H~ = XqR, Hoh where Hob is the solar irradiation value outside the Earth's atmosphere). To generate series longer than the basic series we use eqn (7) to estimate values of R, for the required number of years. The structure of stochastic A R M A models ensures that
137
these generated series are not identical to real data but they are typical for the considered location.
12. CONCLUSIONS A stochastic procedure capable of generating sets of synthetic daily irradiation values has been developed. The only information needed to build these sequences is the monthly average of K, so that our approach can be used also for localities where solar radiation data are not directly measured, they can be estimated fairly accurately from very crude information such as average hours of sunshine which is commonly collected by hundreds of meteorological stations around the world. Specifically, for daily global solar radiation we have : --Segmented the year into months and identified an appropriate variable for statistical analysis on the monthly basis. - - T e s t e d and adopted a functional form for the stationary probability for daily solar radiation which requires the knowledge of the mean and the variance only, and compared it against our real data. --Calculated the mean monthly statistical parameters (i.e. mean, variance), considered correlations among them and suggested a procedure based on confidence limits that unables us to ascertain the uncertainty range of the mean ( K , ) given a limited number of years samples for a given month at a particular location. Examined several types of auto-regressive stochastic models toward the generation of synthetic daily solar radiation sequences that capture all the essential statistical stationary and sequential information of the climate. - - P r o p o s e d how an appropriate synthetic sequence can be selected to construct a typical year of data that can be used for solar energy systems simulation. The type of analysis and the results presented here are aiming to provide the user with both time saving calculational procedures and the required solar radiation input data for optimal sizing softwares for solar systems, and particularly for stand-alone systems.
REFERENCES
1. P. Bendt, M. Collares-Pereira and A. Rabl, The frequency distribution of daily insolation values. Solar Eneryy 27, 1 5 (1981). 2. S. A. Klein, Calculation of flat plate collector utilizability. Solar Energy 21,393 402 (1978).
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H. LOUTFI and A. KHTIRA
3. J. M. Gordon and M. Hochman, On the random nature of solar radiation. Solar Energy 32, 337-341 (1984). 4. E. Boileau, Use of some simple statistical models in solar meteorology. Solar Energy 30, 333 339 (1983). 5. L. Vergara-Dominguez, R. Garcia-Cromez, A. R. Figuerias-Vidal and J. R. Casar-Carredera, Automatic modelling and simulation of daily global solar radiation series. Solar Energy 35, 483~,89 (1985). 6. U. Amato, A. Andretta, B. Bartoli, B. Coluzzi, V. Cuomo and C. Serio, Stochastic modelling of solar radiation data. IL Nuovo Cimento 8, 248-258 (1985). 7. B.J. Brinkworth, Autocorrelation and stochastic modelling of insolation sequence. Solar Energy 19, 343 347 (1977). 8. C. Mustacchi, Stochastic simulation of hourly global radiation sequences. Solar Energy 23, 47 51 (1979). 9. J. M. Gordon and T. A. Reddy, Time series analysis of daily horizontal solar radiation. Solar Energy 41, 215 226 (1988). 10. J. D. Engels, S. M. Pollock and J. A. Clark, Observation on the statistical nature of terrestrial irradiation. Solar Energy 26, 91 92 (1981). 11. K. T. Hollands and R. J. Huget, A probability density function for the clearness index, with application. Solar Energy 30, 195 209 (1983). 12. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4, 1-19 (1960). 13. G. Y. Saunier, T. A. Reddy and S. Kumar, A monthly probability distribution function of daily global irradiation values appropriate for both tropical and temperate locations. Solar Energy 38, 167 177 (1987). 14. A. A. Sfeir, Solar radiation in Lebanon. Solar Energy 26, 497-502 (1981). 15. M. Kendall and A. Stuart, The Advanced Theory of Statistics. Charles Griffin, London (1976). 16. H. Nfaoui and J. Bahraoui-Buret, Trois anndes de mesures de l'ensoteillement fi Rabat. Proc. de la 24Ome Coq/~rence lnternationale de la Comples, Verona, Italy, pp. 179 187 (1986). 17. M. R. Spiegel, ThOorie et Application de la Statistique. Edition Franqaise (1980). 18. G. E. P. Box and G. M. Jenkins, Time Series Analysis." Forcasting and Control. Holden Day, San Francisco (1979).
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APPENDIX
Auto-regressive models (ARMA(k, 0)) [23] are stochastic models in which the variable of the process is expressed as a finite, linear sum o f previous values plus a random white noise. The basic variable, in this analysis, is the daily global horizontal solar irradiation (H(t)). Let Z(t) be the stochastic variable constructed from H(t), where t denotes the day, an ARMA(k, 0) of order k is expressed as: i
Z(t) = ~ ¢ i Z ( t - i ) + A ( t ) . i-I
The regression parameters ~b~,¢~ . . . . .
4~K,are given by:
- - F o r a first order model (ARMA(I,0)) : ¢~ =R~. For a second order model (ARMA(2, 0)) : ¢, = R , ( 1 - R 2 ) / ( I - R ~ z)
ff)z = (R2-- R~)/(I - R~). Where R~ and Rz are the auto-correlation coefficients (eqns 12, 13) for one and two day lags, respectively. The random noise, A(t), should be normally distributed, should have zero mean and have a specified variance given below : For a first order model: A R M A ( I , 0)
a2(A) = 1--R~. - - F o r a second order model : ARMA(2, 0)
~(A) = 1-¢~-¢~-
2¢,4~/(1-4~2).