A framework for the superposition of wind turbines wake properties

Abstract. Wind turbines extract kinetic energy from the wind flow and convert it to electric power, while leaving downstream complex, non-stationary, and meandering plumes of reduced wind speed and increased turbulence, called wakes. Several analytical models have been proposed in the literature to describe the structure of wake properties, such as wind speed deficit (∆U), turbulence intensity (TI), or turbulence kinetic energy (TKE), all of which have been expressed as a function of the upstream, or undisturbed, value of said properties. While this dependency on undisturbed values is natural and practical for a single wind turbine, it becomes less so in the presence of multiple turbines, to the point of being meaningless in large wind farms, where the distance between front- and last-row turbines may exceed tens of kilometers and therefore the very concept of undisturbed flow is moot for the majority of turbines. As such, superposition methods, which aim at obtaining the final distribution of wake properties in overlapping wakes from multiple wind turbines, struggle to provide consistent results, especially for turbulence properties. In addition, the literature is lacking coherent definitions of the superposition methods, with each study introducing its own naming convention and often slightly different formulations. Here we propose a framework to standardize wake superposition for use with any analytical model of wake properties, including ∆U, TI, or TKE (but not temperature or pressure). The framework is based on the concept of “inflow”, as opposed to undisturbed, values, which are in general affected by upstream turbines and are truly undisturbed only for front-row turbines. Within this framework, we propose that the inflow property should not be a constant, but rather a function of z, taken at a fixed distance upstream of each turbine, for example one diameter. Furthermore, the superposition methods are grouped into two main categories (i.e., simple or normalized summation), with the value of an exponent m controlling the linearity or non-linearity of the summed terms. The performance of the superposition methods within the framework is assessed for three analytical wake models (one for each of the three wake properties ∆U, TI, and TKE) against LES results from independent studies under various wind farm layouts, turbine models, and atmospheric stabilities. General findings are that the value of m controls the magnitude of the resulting wake property, with higher values for smaller m, and that, for the same value of m, the simple summation methods tend to create stronger overlapping wakes than the normalized ones in terms of turbulence properties, but the opposite is found for wind speed deficit. We conclude that the selection of the superposition method depends in part on the bias of the analytical wake model itself.