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Analyse de séries temporelles de production éolienne: Loi de Taylor et propriétés multifractales
Abstract
Depuis quelques décennies, l'énergie éolienne connaît une croissance considérable. Cependant cette énergie est dépendante de la vitesse du vent variant en intensité sur des échelles de temps qui incluent l'année, la journée à quelques secondes. Il est donc fondamental de bien parvenir à comprendre et décrire le caractère non-linéaire et stochastique de ces fluctuations dans la production électrique issue des éoliennes.
L'objectif de ce travail est de caract\'eriser les fluctuations d'une série temporelle de production éolienne. Dans un premier temps, nous vérifions l'utilisation de la loi de Taylor, relation de puissance entre l'écart-type et la moyenne. Cette relation fut observée en écologie, en finance, dans les sciences du vivant et pour des données de traffic internet. De récents travaux fournissent des hypothèses d'explication quant à l'origine de cette loi \cite{Agata2010,Kendall2011}. L'exposant alpha caractérise le type de dynamique du processus considéré et varie entre 1/2 et 1. Dans notre cas d'étude, l'estimation de l'exposant alpha est proche de 1. Quand alpha=1, les processus considérés sont à invariance d'échelle \cite{eisler2008}.
Pour mettre en évidence les propriétés d'invariance d'échelle de notre série temporelle, nous effectuons une analyse multifractale pour estimer la fonction exposant d' échelle l'aide des moments d'ordre q de l'incrément temporel des données de production éolienne , telle que \cite{Schertzer1997}. La fonction est concave et non-linéaire: plus elle est concave, plus la série analysée sera intermittente. Nous montrons que la série temporelle de la production éolienne considérée est intermittente et possède des propriiétés multifractales. De plus le modèle de cascades aléatoires log-normal se révèle pertinent pour décrire ces fluctuations.
... This scaling relationship has been highlighted for the power output delivered by a wind farm [20]. Here, as an extension of this early study, fluctuation scaling is investigated for the power output delivered by five wind farms and a single turbine. ...
... They also showed that Taylor law is a scaling relationship, compatible with the presence 1/f scaling and multifractal properties, characteristic of a self-similar process. On the other hand, several authors have shown the presence of 1/f scaling [26] and recently multifractal properties for wind energy data [20,[27][28][29]. A way to highlight multifractal properties is the use of a multi-scaling analysis including q th -order central moments versus the mean value, a natural generalization of Taylor law where q = 2, or multifractal analysis [8,27,29]. ...
The Taylor power law (or temporal fluctuation scaling), is a scaling relationship of the form σ ~ (P)λ where !! is the standard deviation and hPi the mean value of a sample of a time series has been observed for power output data sampled at 5 min and 1 s and from five wind farms and a single wind turbine, located at different places. Furthermore, an analogy with the turbulence field is performed, consequently allowing the establishment of a scaling relationship between the turbulent production IP and the mean value (P).
... In (Sachs et al., 2002), a power law behavior is also observed for the spectrum of cloud radiances obtained from ground-based photography: the spectra exponent b = 2.10 for light transmitted through clouds, in comparison to b = 2.26 for star light transmitted through interstellar dust, b = 1.67 for clouds over ocean and b = 1.43 for clouds over land. We recall here that a power law behavior with b close to 5/3, for the spectrum of a cluster of wind turbines has been also observed (Apt, 2007;Calif et al., 2013;Calif and Schmitt, 2012). ...
... In (Sachs et al., 2002), a power law behavior is also observed for the spectrum of cloud radiances obtained from ground-based photography: the spectra exponent b = 2.10 for light transmitted through clouds, in comparison to b = 2.26 for star light transmitted through interstellar dust, b = 1.67 for clouds over ocean and b = 1.43 for clouds over land. We recall here that a power law behavior with b close to 5/3, for the spectrum of a cluster of wind turbines has been also observed (Apt, 2007;Calif et al., 2013;Calif and Schmitt, 2012). ...
A good knowledge of the intermittency of global solar radiation is crucial for selecting the location of a solar power plant and predicting the generation of electricity. This paper presents a multifractal analysis study of 367 daily global solar radiation sequences measured with a sampling rate of 1 Hz over one year at Guadeloupean Archipelago (French West Indies) located at 16°15'N latitude and 60°30'W longitude. The mean power spectrum computed follows a power law behaviour close to the Kolmogorov spectrum. The intermittent and multifractal properties of global solar radiation data are investigated using several methods. Under this basis, a characterization for each day using three multifractal parameters is proposed.
... These studies have not provided a comparison between their data and the Kolmogorov's spectral slope. More recent works [1,74,5] have shown that the spectrum of the output wind power from a wind farm located follows a Kolmogorov spectrum. ...
a b s t r a c t In this paper, we have presented a spectral and a multifractal analysis performed on 412 time series of wind speed data each of duration of 350 s and sampled at 20 Hz. The average spectrum for the wind speed displays a scaling behavior, in the inertial range, over two decades, with b ¼ 1:68 close to the Kolmogorov value À 5/3. A multifractal analysis has been motivated by the presence of scaling invariance in data set. Then we have considered their scaling properties in the framework of fully developed turbulence and multifractal cascades. The results obtained for wind speed confirm that the exponent scaling function z V ðqÞ is nonlinear and concave. This exponent characterizes the scaling functions in the inertial range indicating that the wind speed is intermittent and multifractal. Moreover the theoretical quadratic relation for lognormal multifractals is well fitted. We investigate the consequence for wind energy production: we generate stochastic simulations of a multifractal random walk, and using a power curve derived from experimental data, we generate the associated power time series. We show that, due to the saturation of the power curve for large speed values, when the input time series (turbulent wind speed) is multifractal, the output can be almost monofractal.
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