Wind farm power fluctuations and spatial sampling of turbulent boundary layers

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The fluctuations in power output from wind farms display significantly reduced spectra compared to single wind turbines due to power smoothing and averaging. In order to better understand these spectral features and to relate them to properties of turbulent boundary layers, we perform a wind tunnel experiment in which we measure spatio-temporal characteristics of an experimental surrogate of the power output from a micro wind farm with 100 porous disk models. The experimental results show that the frequency spectrum of the total wind farm power follows a power law with a slope between $-5/3$ and $-2$, and up to lower frequencies than seen for any individual turbine model. In agreement with previous studies in the literature, peaks in the spectrum are observed at frequencies corresponding to the mean flow convection time between consecutive turbines. In the current work we interpret the sum of power extraction from an array of turbines as a discrete spatial filtering of a turbulent boundary layer and derive the associated transfer function. We apply it to an existing model for the wavenumber–frequency spectrum of turbulent boundary layers. This approach allows us to verify the individual roles of Doppler shift and broadening of frequencies on the resulting spatially sampled frequency spectrum. Comparison with the wind tunnel data confirms that the approach captures and explains the main features in the spectrum, indicating the crucial role of the interaction between the spatial sampling and the space–time correlations inherently present in the flow. The frequency spectrum of the aggregated power from a wind farm thus depends on both the spectrum of the incoming turbulence and its modulation by the spatial distribution of turbines in the boundary layer flow.