On the power and control of a misaligned rotor – beyond the cosine law

Abstract. We present a new model to estimate the performance of a wind turbine operating in misaligned conditions. The model is based on the classic momentum and lifting-line theories, considering a misaligned rotor as a lifting wing of finite span, and accounts for the combined effects of both yaw and uptilt angles. Improving on the classical empirical cosine law in widespread use, the new model reveals the dependency of power not only on the misalignment angle, but also on some rotor design parameters and – crucially – on the way a rotor is governed when it is yawed out of the wind. We show how the model can be readily integrated with arbitrary control laws below, above, and around the rated wind speed. Additionally, the model also shows that a sheared inflow is responsible for the observed lack of symmetry for positive and negative misalignment angles. Notwithstanding its simplicity and insignificant computational cost, the new proposed approach is in excellent agreement with large eddy simulations (LESs) and wind tunnel experiments. Building on the new model, we derive the optimal control strategy for maximizing power on a misaligned rotor. Additionally, we maximize the total power of a cluster of two turbines by wake steering, improving on the solution based on the cosine law.