Multifractal analysis and simulation of wind energy fluctuations

Abstract : From the governing equations of the velocity field, one can not only expect a (highly) non-Gaussian wind but also one that is scaling. By ‘scaling' we mean a given statistical self-similarity; a turbulent cascade of eddies. Stochastic multifractals (with multiple singularities and co-dimensions) easily reproduce the scaling, heavy-tailed probabilities ubiquitous with the wind and essential to quantify for the wind energy community. The few parameters that define these models can be derived either from theoretical considerations or from statistical data analysis. It is sometimes possible to determine the statistics of the velocity shears with the universal multifractal (UM) parameters: alpha - the index of multifractality (0 ≤ alpha ≤ 2), C1 - the co-dimension of the mean intermittency (C1 ≥ 0) and H - the degree of non-conservation - the linear part of the scaling exponents. The latter of the three parameters is often called the Hurst exponent. We inter-compare the results from the rather standard method of empirical estimation of the UM parameters, the Double Trace Moment (DTM) method, with that of the Double Structure Function (DSF), a newly developed method. We found that flux proxies based on the modulus of the wind velocity shears yield non-scaling statistical moments and therefore spurious multifractal parameter estimates. DSF does not require this proxy approximation thus providing the scaling of the structure-function to an extent. We found no truly stable estimate of alpha using standard methods. This no longer occurs when we locally optimise (by fractionally differentiating) the DTM scaling behaviour. We then obtain very stable estimates of the multifractality index that are furthermore consistent (alpha ≤ 2) with other literature. On the contrary, the two other parameters (C1 and H) become non-linear functions of the order q of the statistical moments. These results suggest that the isotropic UM model cannot be used to reproduce the velocity shears in the atmospheric surface-layer. To investigate the above hypothesis we use a rotated frame of reference to analyse the anisotropy of the horizontal velocity in the atmospheric surface-layer. This enables us to quantify the angular dependency of a Hurst exponent. Despite being anisotropic the Hurst exponent is consistent with other surface-layer literature. For time-scales above a few seconds, both data exhibit a strong, scaling anisotropy that decreases with height. We put forward an analytical expression for the angular variation of the Hurst exponent based on the correlation of the horizontal components. It determines the generation of wind shear extremes, including those in the wake of a turbine. We find that the turbulent wind shears are so extreme that their probability distributions follow a power law. The corresponding exponent (qD) is rather the same in both sites at 50m heights (4 ≤ qD ≤ 5), in spite of very different orographic conditions. We also discuss its consequences when analysing the stability of the atmospheric boundary-layer and propose a new method for its classification. Finally, we analytically demonstrate that anisotropy increases the extremes probability. This finding reveals one of the many possible turbulence mechanisms in the atmospheric surface-layer that may seemingly over-generate wind shear extremes if they are studied in an isotropic UM framework. We theoretically analyse the consequences of this on the UM estimates for the DTM method. The obtained analytical results fully support empirical findings. We then discuss how to take into account all of these considerations when simulating multifractal fields of the wind in the atmospheric boundary-layer. The overall results of this dissertation go beyond wind energy, they open up new perspectives for the theoretical predictions of extremes in the general case of strongly correlated data
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George Fitton. Multifractal analysis and simulation of wind energy fluctuations. Other [q-bio.OT]. Université Paris-Est, 2013. English. ⟨NNT : 2013PEST1110⟩. ⟨tel-00962318⟩

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