Preliminary results of the fractal classification of daily solar irradiances☆
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 pre-eminent 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). 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 long-term 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 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.
Section snippets
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
Theoretical determination of the fractal dimension
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
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 rise to a daily fractal index . 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 Δτ and breadth ∣E(tn+Δτ)−E(tn)∣. Thus, we calculate the area S needed for the covering process using the following expression which we have defined
Classification of daily solar irradiances
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 sky state without clouds, while a value of close to 1.6 reveals a perturbed sky state with clouds. This is why the daily index is used here as a criterion of the daily irradiance classification. Our
Application to PV systems performance analysis
Due to the costs involved in analytical monitoring in remote areas, very few studies dealing with experimental 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
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 introducing the fractal index based on the Minkowsky–Bouligand method. The obtained estimation
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.
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This paper is dedicated to the Memory of Professor Abdelbaki Maafi.