Delegates are invited to meet and discuss with the poster presenters in this topic directly after the session 'How does the wind blow behind wind turbines and in wind farms?' taking place on Tuesday, 11 March 2014 at 16:30-18:00. The meet-the-authors will take place in the poster area.

**George Fitton**ENPC, France

Co-authors:

George Fitton (1)

^{F}

^{P}Ioulia Tchiguirinskaia (1) Daniel Schertzer (1) Shaun Lovejoy (2)

(1) ENPC, Champs-sur-Marne, France (2) McGill University, Montreal, Canada

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## Abstract

**EVIDENCE OF A MULTIFRACTAL ATMOSPHERIC SURFACE-LAYER AND ITS IMPLICATIONS FOR THE ENERGY COMMUNITY****Introduction**

Despite significant advances in wind-energy related technologies, extracting energy (or torque) from the wind remains difficult, intermittent and unpredictable in many respects. The European Union would like to reduce the uncertainties in wind resource assessment and forecasting to below 3% by 2030. If this is managed wind energy will be competitive. We discuss why and how evidence of a strongly anisotropic and multifractal surface-layer may give some insight into why we have for so long underestimated the wildness of the wind we so often subject our wind turbines to.

**Approach**

Quantifying the multifractality of the surface/boundary-layer required an in-depth and long-winded (no pun intended) empirical data analysis. The wind velocity data used in the analyses came from two atmospheric wind measurement campaigns from two very topographically different nest sites: the German Growian experiment in a near homogenous terrain setting and the French Ersa wind park experiment where measurements were taken from within a wind farm on top of a mountain in Corsica. Both sites are subject to convective sea breezes and both exhibit wind shear distributions where even coarse time-scale extremes are much more frequent than those predicted by a Gaussian curve.

The Growian wind turbine experiment was a German Federal Ministry of Research and Technology’s project that took place over the years 1983 to 87. A two-bladed, 3-megawatt, wind turbine was constructed for research purposes in Kaiser-Wilhelm-Koog, near the German coastline of the North Sea. This particular experiment is of great interest to us due to the two 150m measuring masts, positioned 65m West-South-West of the turbine. Sixteen cup anemometers and wind vanes, eight-per-mast, were installed on the ends of 12m booms at 50, 75, 100, 125 and 150m from the ground; covering an effective area of 75-by-100m. This grid-like set-up meant that (limited) space-time measurements were possible.

The Corsica experiment is the result of a wind measurement campaign performed by EDF in the Ersa wind park from the 16th November to the 15th of May. With respect to the measurements, sonic anemometers at 22, 23 and 43m measured three-component wind velocities and temperature at 10Hz.

**Main body of abstract**

Numerical attempts to model the complex and highly non-linear processes found in the surface/boundary-layer typically require a truncation of scales and more often than not complex parameterisations. Due to the symmetries of the governing equations of fluid motion for a high-Reynolds number flow, the statistics of the wind are scaling and multifractal. It is therefore unnecessary to truncate the scales of the process.

Our results confirm that the atmospheric surface-layer is scaling and is multifractal, however, they also show that this scaling and multifractality are not only strongly anisotropic, a feature that already requires the framework of (linear) generalised scale invariance, (GSI), but there is a strong lack of translation invariance that requires non-linear GSI. By analysing the scaling exponent H - corresponding to the measure of the long term memory of time series and estimated on the log-log plot of the spectra - in a rotated frame-of-reference we showed that the horizontal velocity components are highly correlated. This is contrary to popular belief that is, that rotational invariance allows for an isotropic velocity field as is often invoked in industrial applications.

We have developed several new techniques to estimate the multifractality index α, in particular to take care of the extremes present in a sample that easily introduce statistical biases in classical estimation methods, such as the trace- moments (TMs) and double trace moments (DTMs). Let us recall that this index α has several important properties: it measures the multifractality of our field: α = 0 corresponds to a mono-fractal field, whose intermittency is independent of the considered activity of the field, i.e. the extremes are not that much different from the mean. Its maximal value α = 2 corresponds to the misnamed ‘log- normal’ cascade model, whose extremes are much larger than those of a log-normal distribution. It also characterises the generator of the cascade process; more precisely it corresponds to the Lévy stable index of the generator in the framework of universal multifractals (UM).

We believe that it is particularly significant that we constantly estimate this multifractality index α ≈ 1.4. This not only confirms a significant multifractality, but also confirms that, despite the complexity of the surface-layer, a given universality of the generator of the cascade exists. Furthermore, this generator is significantly different from a normal generator, i.e. the log-normal model is not applicable. We found that the estimates of two other fundamental scaling exponents were much less stable and difficult to obtain although their interpretations are much simpler. Indeed, they are only fractal exponents (contrary to α) – the scaling exponent H of the mean field (H = 0 for a strictly scale invariant mean field) and the co-dimension C1 of its support that measures the mean intermittency (C1 = 0 for a homogeneous field). This difficulty, surprising at first glance, is rather typical of non-linear GSI.

We developed several methods to better estimate the parameters – in particular making detailed studies of the sensitivity of parameter estimation methods to instrumental noises: structure functions (SF), TMs and DTMs. This led us to introduce a hybrid method somewhere between the SF and the DTM method; the double structure function (DSF), that rather combines both of the other methods advantages for large statistics. Developments in the structure function led to analyses based on the q-normalised method: f(q) = ζ(q)/q, where ζ(q) is the scaling exponent of the structure function.

**Conclusion**

We highlight several key results:

• The scaling anisotropy of the samples: to avoid shadow effects from masts, we are compelled to deal with samples whose ‘mean’ velocity is near-perpendicular to the masts. The anisotropy of these samples turns out to be beyond a trivial component-wise anisotropy corresponding to pre-factors depending on the direction, i.e., the scaling exponents themselves (in particular H) depend on the direction.

• We derive an analytical expression for the direction dependence of H and plot the corresponding ‘potatoid’ shapes of the isolines of the exponent values. The expression is based on the cross-correlation between orthogonal components of the singularities.

• We show that this scaling anisotropy has important consequences for first-order multifractal phase transitions; it decreases the critical order qD (the analog of the inverse of a critical temperature) at which the transitions occur. This decrease is so significant that these transitions may occur on a unique sample whereas usually they are expected only a very large number of samples since they correspond to a divergence of moments for an infinite number of samples.

• Not taking care of these phase transitions leads to spurious estimates of the multifractal exponent α.

• The lack of translation invariance of the fluctuations along the vertical space axis forces us to deal with non-linear GSI as opposed to simple GSI in order to avoid inconsistencies.

These results show that the complexities of the surface/boundary-layer are far out of the reach of classical engineering methods that involve linear approximations and parameterisations. It is in attempting to parameterise these highly variable processes where we are falling down. In particular we must learn to take advantage of the fundamental properties of the turbulent processes of the atmosphere. This involves using and exploiting the scaling symmetries of the wind and other variables that can be dimensionally described through the wind. We are finding more and more evidence that multifractals are not only the most appropriate framework to deal with these kind of problems but they are the necessary framework.

**Learning objectives**

By very carefully applying a range of statistical methods we have shown that turbulence is far more complex than its face value shows. Nonetheless, it is multifractal and scaling, although only in a non-linear and generalised manner. Going from a multifractal analysis to a multifractal surface-layer model would provide not only very accurate energy predictions but also meaningful nowcasts.

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