Preliminary results of the fractal classification of daily solar irradiances

Solar Energy 75 (2003) 53–61

Preliminary results of the fractal classification of daily solar irradiances q A. Maafi a,b , S. Harrouni a , * a

Solar Instrumentation and Modeling Group /LINS, Faculty of Electronics and Computers, University of Science and Technology H. Boumediene ( USTHB), P.O. Box 32, El-Alia, 16111 Algiers, Algeria b Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34100 Trieste, Italy Received 18 September 2002; accepted 2 May 2003

Abstract This paper deals with the fractal modeling of daily solar irradiances measured with a sampling time of 10 min for one year at Tahifet and Imehrou located in the desert of Algeria. The aim of this modeling was to estimate the fractal index and then to use it to classify the considered daily irradiances. Therefore, daily fractal and clearness indexes were used to propose a classification model that leads to three typical classes. These classes (corresponding to clear sky, partly cloudy sky and completely cloudy sky) allow us to characterize the daily irradiance profiles of both locations. The results of the classification model were then applied to the performance analysis of an autonomous photovoltaic system installed at Tahifet. Good agreement was observed between the long-term performance indicators and those obtained while studying typical cases.  2003 Published by Elsevier Ltd.

1. Introduction In this paper, we introduce a model for the estimation of the fractal dimension of daily solar irradiance based on the Minkowski–Bouligand dimension. The fractal dimension is an important parameter that measures signal shape irregularity. For solar radiation, the irregularity of shapes describes the fluctuations of the phenomenon resulting from weather conditions. Using fractal dimension as a parameter, we seek to establish a classification model of daily solar irradiances that takes into account the preeminent typical weather conditions. The classification of days based on daily solar radiation properties has been investigated in many studies (see, for example, Fabero et al., 1997; Muselli et al., 2000). However, very few works have been published dealing with the classification of daily solar radiation using fractal analysis (Louche et al., 1991; Maafi and Harrouni, 2000). q

This paper is dedicated to the Memory of Professor Abdelbaki Maafi. *Corresponding author. E-mail address: [email protected] (S. Harrouni). 0038-092X / 03 / $ – see front matter  2003 Published by Elsevier Ltd. doi:10.1016 / S0038-092X(03)00192-0

We therefore examined if fractal analysis can contribute to daily solar irradiance classification in a simple way. Previously, we studied the fractal character of solar radiation indirectly by investigating the long-term persistence or correlation in measured time series of this phenomenon using the Hurst approach, which is commonly called Hurst’s rescaled range (R / S) analysis (Feder, 1988). This technique has been applied to study the longterm behaviour of the energy stock of autonomous solar systems and has been implemented using series of daily global irradiation recorded on a horizontal surface in the meteorological stations of Abidjan (Ivory Coast), Algiers and Tamanrasset (Algeria) and Carpentras (France) (Maafi and Delorme, 1996). In the present work we implemented a model using 10-min solar irradiance measurements which represent an average of instantaneous data taken every 10 s over a period of 10 min for one year at Tahifet and Imehrou located in the Tamanrasset province of Algeria (Maafi, 1997). The classification model was then applied to the performance analysis of solar photovoltaic (PV) systems. The basic idea is to compare the results of the long-term performance analysis of PV systems with those obtained

A. Maafi, S. Harrouni / Solar Energy 75 (2003) 53–61

54

while using typical cases resulting from daily solar irradiance classification. In the case of obtaining good agreement among the results, the classification should lead to data compression, since the long-term PV system analysis may be reduced to a simple study of typical cases. Consequently, the economic costs involved in the performance analysis of PV systems should be significantly reduced. Section 2 is devoted to the details regarding the irradiance data used in this work. Section 3 presents the methodology for the determination of the Minkowski– Bouligand dimension. Section 4 deals with the estimation procedure for measuring the fractal dimension of daily solar irradiance, where we also present how an estimated fractal index may be related to the fractal dimension for Tahifet and Imehrou. A classification algorithm is given in Section 5 and in the same section we explain how the classification criteria were determined. In Section 6, we apply the classification model to the performance analysis of PV systems. In Section 7, we discuss the results obtained for the fractal index, the classification model and the performance analysis.

determination of the fractal dimension. Indeed, it is well known that the greater the number of samples of the studied signal, the better the estimation of its fractal dimension. This is due to the fact that all signal fluctuations are taken into account in the fractal dimension estimation procedure. Due to the lack of data recorded with a smaller time step, we performed this study using our data set. Fig. 1 gives representative histograms of the experimental data used for Tahifet. They have typical shapes, such as: • the J shape, i.e. a concentration of data around high values; • the L shape, i.e. a concentration of data around low values;

2. Experimental data To carry out this work, an experimental data bank was used. The data were obtained from the operation of two 720 Wp photovoltaic power installations equipped with a system for analytical monitoring. These experimental installations were put into operation in 1992 at Tahifet and Imehrou by the National Electricity and Gas Company (SONELGAZ) with the aim of testing and disseminating photovoltaic (PV) programs (Maafi, 2000). The geographical coordinates of these sites are given in Table 1. These PV systems are autonomous using a photovoltaic generator with storage. Their consumption essentially represents the lighting, the refrigeration and ventilation. Our data bank contains many other parameters together with ‘the global irradiance’ measured at the two sites. The irradiance data were recorded for one year on a 108-tilted surface with a time step of 10 min (Maafi, 1997). Such solar radiation data recorded with a small time step in remote areas of the desert of Algeria are not available for other locations, even in the north of the country. The global irradiance was measured using a monocrystalline solar cell and an automatic data acquisition system. The 10-min sampling time leads to daily signals with about 60 samples, which are insufficient for the Table 1 Geographical coordinates of Tahifet and Imehrou Site

Latitude (N)

Longitude (E)

Altitude (m)

Tahifet Imehrou

228539 268009

068009 088509

1400 600

Fig. 1. Representative histograms of daily irradiance data with typical shapes, L shape (a), U shape (b) and J shape (c), for Tahifet.

A. Maafi, S. Harrouni / Solar Energy 75 (2003) 53–61

• the U shape, i.e. practically the same concentration around both low and high values (sometimes resulting in a quasi-uniform distribution of the data). All three distributions can be interpreted as arising from the presence of the most frequent weather conditions in this region that we can classify relating to the clearness index KT described later in three categories, namely clear sky, partially cloudy sky and completely cloudy sky. The same kinds of histograms were obtained for Imehrou.

4. Estimation of the fractal index Since we have only a limited data set for each day (about 60 data), estimation of the fractal dimension gives ˆ rise to a daily fractal index D(d). Indeed, Fig. 2a shows an example of daily solar irradiance signals for which the fractal dimension should be estimated. This estimation technique consists of covering this signal by rectangles of length Dt and breadth uE(t n 1 Dt ) 2 E(tn )u. Thus, we calculate the area S needed for the covering process using the following expression which we have defined for this purpose:

3. Theoretical determination of the fractal dimension

D 5 2 2 l(S)

(1)

where l(S) represents the infinitesimal order of area S(Dt ). This is defined as Ln(S(Dt )) l(S) 5 lim ]]] Dt →0 Ln(Dt )

(2)

where Ln is the neperien logarithm. When replacing l(S) by its value in relation (1), it is found that D 5 lim

Dt →0

F

Ln(S(Dt )) 2 2 ]]] Ln(Dt )

G

(3)

The logarithmic properties allow us to write relation (3) as D 5 lim

Dt →0

H

2 Ln[S(Dt ) /Dt ] ]]]] Ln[1 /Dt ]

J

(4)

Using a least-squares estimation, the fractal dimension of the daily solar irradiance is then deduced from the equation

S D

S D

1 S(Dt ) Ln ]] (D ? Ln ] 1 c, with Dt → 0 2 D t Dt

(5)

where c is a constant. The fractal dimension D is represented by the slope of the Ln–Ln plot of relation (5).

O Dt uE(t 1 Dt) 2 E(t )u

N 21

S(Dt ) 5 Fractals can model many natural phenomena that give rise to different kinds of signals. The fractal dimension is an important parameter in fractal modeling and contains information about the irregularity of signal shapes. Several algorithms are presented in the literature for calculating the fractal dimension of signals (Barnsley, 1988; Dubuc et al., 1989; Falconer, 1990; Maragos and Sun, 1993). As daily global irradiances are one-dimensional discrete time series, the Minkowsky–Bouligand dimension is used to measure their fluctuations. Recall that the Minkowsky–Bouligand method gives a rough theoretical determination of the fractal dimension. Bouligand (1929) defined the fractal dimension D as

55

n

n

(6)

n 50

Here N is the number of samples of the considered signal, E(t n ) is the global irradiance at time n and uE(t n 1 Dt ) 2 E(t n )u is the irradiance variation related to the interval Dt. Therefore, we use different time scales Dt to cover the curve as shown in Fig. 2b, then we measure the corresponding area S(Dt ). We note that, in the Minkowski–Bouligand method, the process of covering signals is performed with the help of disks. In this model, we approximated the disk by the rectangle as a geometric element. This assumption is more convenient for our estimation algorithm because the area S(Dt ) is easily computed (Fig. 2). From a mathematical point of view, this approximation is justified because the topology induced by the rectangle is the same as that of the disk. Thus, this approximation should not increase the estimation error. In other works, instead of traditional geometric elements, morphological covers of profiles are generated and applied to estimate their fractal dimensions (see, for example, Maragos and Sun, 1993). The present estimation is based on the fitting of the straight line described by relation (5) on Ln–Ln plots of discrete data using a least-squares estimation. The discrete Ln–Ln plots are obtained by calculating the pairs

S S D S DD 1 S(Dt ) Ln ] , Ln ]] Dt Dt 2

for Dt ranging from 1 to Dtmax . The latter is an important parameter for a good estimation of the fractal dimension; it is determined experimentally and must not exceed N / 2. To test the validity of our algorithm, very regular and irregular shapes of daily solar irradiation were chosen independently of the considered signals of the experimental data set and their ‘fractal dimensions’ were estimated. The indexes for regular signals are close to 1, while those corresponding to very irregular patterns converge towards 2. Therefore, our algorithm was used for the estimation of the fractal indexes of all daily solar irradiances measured at Tahifet and Imehrou. Fig. 3 shows the detailed results. The daily ˆ fractal index D(d) range from 1 to 1.6 reveals the existence of solar daily irradiances of regular and irregular shapes.

56

A. Maafi, S. Harrouni / Solar Energy 75 (2003) 53–61

These also allow us to detect the months when daily solar irradiance is characterized by fluctuations. June and December are examples of these months for Tahifet and March and June for Imehrou. With the help of the estimated fractal indexes, the month when daily solar irradiances have regular shapes could be determined. October is this month for both sites.

5. Classification of daily solar irradiances

Fig. 2. Covering irradiance signal using rectangles. (a) Original signal. (b) Covered signal at different time scales Dt.

Let us recall that the calculated index, which roughly approaches the fractal dimension, measures the amount of daily solar irradiance fluctuations which are related to weather conditions and, consequently, to the state of the sky. An estimated index close to unity describes a clear ˆ sky state without clouds, while a value of D(d) close to 1.6 reveals a perturbed sky state with clouds. This is why the ˆ daily index D(d) is used here as a criterion of the daily irradiance classification. Our research reveals that some daily solar irradiances have the same fractal index, but correspond to days with different weather conditions. Indeed, a uniformly cloudy day and a sunny day have regular irradiance shapes and practically the same value for ˆ D(d), but have daily different clearness indexes. This is due to the fact that the amount of global irradiation available is not related to the fractal index. Thus, it is not sufficient as a robust classification criterion. That is why the daily clearness index KT (d) is calculated along with ˆ D(d) as a second criterion in the classification algorithm, ˆ which is based on D(d) and KT (d) thresholds. For this purpose, 10-min global irradiation fractions KT (n) were estimated by calculating the integral of solar irradiance on the 10-min basis for all the year, then all derived 10-min global irradiances were divided by their maximum value. Ten-minute histograms of KT (n) were constructed on a daily basis. These are similar to those of Fig. 1. Taking into account the particular shapes of these histograms, KT 0 5 0.5 appears as the threshold value which discriminates between days with a high insolation and the other days. Previously, the same criterion was used to sort daily clearness indexes into two categories (Maafi, 1991). In order to use this criterion in the classification algorithm, the daily clearness index was calculated for all days of the year. On the other hand, when analyzing all daily solar irradiance shapes and their corresponding fractal dimensions, we can observe three kinds of shapes, namely: • a very regular irradiance shape; • a regular irradiance shape with rare fluctuations; • an irregular irradiance shape with many fluctuations. ˆ The threshold values of D(d) that allow us to discriminate daily solar irradiance shapes according to observations are Dˆ 0 5 1.10 and Dˆ 1 5 1.25. Our heuristic approach that ˆ combines D(d) and KT (d) thresholds yields the following classification:

A. Maafi, S. Harrouni / Solar Energy 75 (2003) 53–61

57

ˆ Fig. 3. Yearly behaviour of the estimated daily fractal index D(d) for Tahifet (a) and Imehrou (b).

Class 1 Histogram J / Daily irradiances corresponding to clear sky: ˆ 1 , D(d) # 1.10 and KT (d) $ 0.5

(7)

Class 2 Histogram U / Daily irradiances corresponding to partly cloudy sky: ˆ 1.10 , D(d) # 1.25 and KT (d) $ 0.5

(8)

58

A. Maafi, S. Harrouni / Solar Energy 75 (2003) 53–61

Fig. 3. (continued)

A. Maafi, S. Harrouni / Solar Energy 75 (2003) 53–61 Table 2 Probability of occurrence of daily solar irradiance shapes of each class Site

Class 1

Class 2

Class 3

Tahifet Imehrou

0.56 0.64

0.23 0.22

0.21 0.14

Class 3 Histogram L / Daily irradiances corresponding to completely cloudy sky: ˆ D(d) . 1.25 or ˆ D(d) # 1.25 and KT (d) , 0.5

5

59

of the probability of occurrence of daily solar irradiances of each class. Fig. 4 gives an example of the daily solar irradiance shapes obtained for each class. Daily irradiances of class 1 have the largest probability of occurrence compared with the irradiances of the two other classes. Daily irradiances of class 2 have practically the same probability of occurrence for the two sites and the probability of occurrence of daily irradiances of class 3 is larger for Tahifet than for Imehrou. These results confirm the pre-eminence of days with clear sky for the two sites. Days with partly cloudy sky and those with completely cloudy sky have practically the same probability of occurrence in Tahifet, but not in Imehrou.

(9)

The classification algorithm was used to sort the daily irradiances into three classes. Table 2 gives the distribution

6. Application to PV systems performance analysis Due to the costs involved in analytical monitoring in remote areas, very few studies dealing with experimental

Fig. 4. Examples of daily solar irradiance shapes corresponding to class 1 (clear sky), class 2 (partly cloudy sky) and class 3 (completely cloudy sky) for Tahifet (a) and Imehrou (b).

A. Maafi, S. Harrouni / Solar Energy 75 (2003) 53–61

60

performance analysis of PV systems operating in Algeria are available (Maafi, 1997; Benghanem and Maafi, 1998). In order to reduce these costs, this paper proposes an approach for analyzing PV systems performance based on the results of classification modeling. The approach allows a typical daily irradiance to be built for each of the three classes defined in the previous section. In order to be able to compare the results of performance assessment obtained for days with typical irradiances and those determined for a long period (by using the daily irradiances composing each class), normalized performance indicators are used. The performance indices are as follows (Blaesser, 1997): • the array yield Ya (d), defined as the daily array energy output per KWp of installed PV array; • the final yield Yf (d), which represents the useful output of the PV system per KWp installed; • the reference yield Yr (d) may be defined as the total daily in-plane irradiation divided by the reference inplane irradiance under standard test conditions (STC); • capture losses LC (d) are defined as the difference between the reference yield and the array yield; • system losses Ls (d) are defined as the difference between the array yield and the final yield; • the performance ratio PR (d) is defined by the ratio Yf (d) /Yr (d). For a period of time of m days, averages of yields and losses are obtained by summing these values over the whole period and dividing by m. All yields are in units of (KWh / d) / KWp. In order to asses the performance of the PV system installed at Tahifet, all these indices were determined for each day of the year (Maafi and Harrouni, 2000). Moreover, the classification model allows us to build a typical daily irradiance for each class. In order to compare the results of long-term performance analysis and those obtained for typical cases, the above parameters and the daily PV generator efficiency h(d) were calculated for typical days and their averages were determined for each class. All these results are summarized in Table 3. They show that the performance indices calculated for each class and those determined for typical cases do not differ significantly. Therefore, days with typical irradiances can be used to

obtain an idea about the long-term performance of PV systems. Therefore, the idea of the proposed method is as follows. For a given location and using time series of solar irradiance measurements, we build typical days using the classification model. The typical days will then be used for the performance analysis of any other PV system installed at this location. This is possible since the classification model is based on the solar irradiances, which do not change for the same site. The performance analysis of these PV systems will then be reduced to three typical cases. To exploit the method presented it would be necessary to divide the regions into zones according to the irradiation received; the classification model would then be applied at each zone to determine the corresponding typical days. We thus realise a typical day cartography which would allow us to study the performance of any PV system.

7. Concluding remarks As already mentioned in the Introduction, to our knowledge there is no published work studying the question of whether or not solar radiation exhibits fractal behaviour. This is probably due to the fact that this kind of study needs a significant amount of solar radiation, data that is not always available. In the present work, we estimated the fractal dimension of the daily solar irradiance by introˆ ducing the fractal index D(d) based on the Minkowsky– Bouligand method. The obtained estimation results have the same order of magnitude as those already obtained for a Mediterranean site in the south of France using a different estimation technique (Louche et al., 1991). We have also shown that it is possible to realize a daily ˆ solar irradiance classification using D(d) along with KT (d). In the classification algorithm, KT (d) allows us to define ˆ only two classes, while parameter D(d) gives rise to a third ˆ class. Thus, the combination of D(d) and KT (d) is important. For example, the shape of the daily solar irradiance of class 3 (Fig. 4) corresponds to a rainy day in

Table 3 Comparison between the mean values of the daily performance indicators calculated for each class and those obtained for typical days (the yields and losses are in units of (KWh / d) / KWp)

Averages of class 1 Averages of class 2 Averages of class 3 Typical day of class 1 Typical day of class 2 Typical day of class 3

h

Ya

Yf

YR

LC

LS

PR

0.081 0.089 0.092 0.080 0.088 0.096

4.27 4.25 3.31 4.24 4.27 3.49

3.08 3.35 2.75 3.25 2.85 2.68

8.82 8.01 6.02 8.82 8.01 6.03

4.55 3.76 2.72 4.58 3.74 2.54

1.20 0.90 1.29 0.99 1.42 0.81

0.35 0.42 0.47 0.37 0.36 0.44

A. Maafi, S. Harrouni / Solar Energy 75 (2003) 53–61

Tahifet. Its fractal index is 1.12 and its related KT (d) is 0.4. ˆ Using D(d), this daily irradiance should be classified in ˆ class 2. However, when using D(d) and KT (d) together it is categorized as class 3. Validation of this classification model was achieved by comparing the result of the classification for each day of the year and the related experimental observation. Good agreement was observed. In addition, the information derived from the estimated fractal indexes is useful for sizing and analysing the performance of PV systems. In the case of Tahifet, for example, the installed PV system produces excess energy in October and storage is required in June and December (Maafi, 1997), which is in good agreement with the obtained results. Performance analysis of the PV systems studied using this classification scheme leads to interesting results. The latter allows us to believe that the proposed method could be applied to other system configurations. Therefore, it is necessary to test the method with these other system configurations in order to generalize it. Therefore, further investigations are required for the practical implementation of the method and its validation for all system configurations in order to establish its generalization.

Acknowledgements The authors are grateful to AS-ICTP for support and to SONELGAZ for providing the irradiance data. They are also very grateful to Professor L. Romanelli for valuable discussions. The second author wishes to thank Professor A. Guessoum for his help.

References Barnsley, M., 1988. Fractals Everywhere. Academic Press, San Diego. Benghanem, M., Maafi, A., 1998. Data acquisition system for photovoltaic systems performance monitoring. IEEE Trans. Instr. Meas. 47 (1), 30.

61

Bouligand, G., 1929. Ensembles impropres et nombre dimensionnel. Bull. Sci. Math. 11 (53), 185. Blaesser, G., 1997. PV system measurements and monitoring: the European experience. Sol. Energy Mater. Sol. Cells 47 (1–4), 167. Dubuc, B., Quiniou, J.F., Roques-Carmes, C., Tricot, C., Zucker, S.W., 1989. Evaluating the fractal dimension of profiles. Phys. Rev. A 39 (3), 1500. Fabero, F., Alonso-Abella, M., Chenlo, F., 1997. In: Ossenbrink, H.A., Helm, P., Ehmann, H. (Eds.), Proceedings of EPSEC’97, the 14th European Photovoltaic Solar Energy Conference, Barcelona, Spain, 30 June–4 July. Stephens & Associates, Bedford, UK, p. 2299. Falconer, K., 1990. Fractal Geometry. Mathematical Foundation and Applications. Wiley, New York. Feder, J., 1988. Fractals. Plenum Press, New York. Louche, A., Notton, G., Poggi, P., Simonot, G., 1991. Classification of direct irradiation days in view of energetic applications. Solar Energy 46 (4), 255. Maafi, A., 2000. A survey on photovoltaic activities in Algeria. Renewable Energy 20 (1), 9. Maafi, A., Harrouni, S., 2000. Measuring the fractal dimension of solar irradiance in view of PV systems performance analysis. In: Sayigh, A.A.M. (Ed.), Proceedings of WREC VI, the 6th World Renewable Energy Congress, Brighton, UK, 5–11 July. Elsevier, Oxford, UK, p. 2032. Maafi, A., 1991. Determination of physical aspects of the first order two-state Markov model for solar meteorology: application to photovoltaic conversion. Doctorat d’Etat, USTHB. Maafi, A., 1997. Experimental evaluation of a storage sizing method for autonomous PV systems. In: Ossenbrink, H.A., Helm, P., Ehmann, H. (Eds.), Proceedings of EPSEC’97, the 14th European Photovoltaic Solar Energy Conference, Barcelona, Spain, 30 June–4 July. Stephens & Associates, Bedford, UK, p. 1517. Maafi, A., Delorme, C., 1996. Long-term energy storage modeling and optimisation for autonomous solar systems. J. Phys. III France 6 (4), 511. Maragos, P., Sun, F.K., 1993. Measuring fractal dimension of signals: morphological covers and iterative optimisation. IEEE Trans. Signal Processing 39 (1), 108. Muselli, M., Poggi, P., Notton, G., Louche, A., 2000. Classification of typical meteorological days from global irradiation records and comparison between two Mediterranean coastal sites in Corsica Island. Energy Convers. Manage. 41 (10), 1043.