Direct integration of non-axisymmetric Gaussian wind-turbine wake including yaw and wind-veer effects

The performance of a wind farm is significantly influenced by turbine-wake interactions. These interactions are typically quantified for each turbine by evaluating its rotor-averaged wind speed, which is impacted by upstream wakes, using numerical methods that involve discrete points across the rotor disk. Although various point distributions exist in the literature, we introduce an analytical expression for integrating a Gaussian wake over a circular disk, which accounts for wake stretching and shearing resulting from upstream turbine yaw and wind veer. This expression is versatile, accommodating any lateral offset and hub-height difference between the wake source (upstream turbines) and the target turbine. Validation against numerical evaluations of the rotor-averaged deficit at various downstream locations from the wake source demonstrates excellent agreement. Furthermore, the analytical expression is shown to be compatible with multiple wake superposition models. The presented solution is differentiable, providing a foundation for deriving mathematical expressions for the gradients of a turbine's power generation concerning its location within a farm and/or the operational conditions of upstream turbines. This capability is particularly advantageous for optimization-based applications