Timescale separation via Rayleigh quotient in flexible wind turbines: a singularly perturbed approach

Abstract

The current deep knowledge of the mechanical and structural dynamics of wind turbines has been long considered to optimize their design. From an engineering point of view, this optimized design induces a timescale separation that reduces the complexity of the dynamics by decoupling the main physical mechanisms involved: flexion, torsion, and rotational dynamics. Fortunately, the mature theory of singular perturbations can help engineers to identify the parameters inducing such timescale and thereby reducing the complex dynamics. In this work, a thorough singular perturbations analysis is made, starting from the nonlinear dynamics and considering the first-mode shapes of the blades, rotor and tower. Moreover, the innovative idea of using the Rayleigh quotient has been instrumental to identify the essential physical parameters providing that timescale separation. The reduced-order dynamics arises scaling the singularly perturbed system, recovering most of the well-established heuristic models and also, showing new findings that had not been yet reported in the literature. Among other findings, a contribution of the first bending mode of the blades in the equivalent inertia of the reduced model has been identified, showing that any attempt of identifying its value, without considering the bending mode of the blades, would cause inevitably its overestimation by $$50\%$$ , on average. Moreover, necessary and sufficient conditions and “rules of thumbs” are provided, to check, a priori, if the timescale separation of a specific turbine is satisfied. The approach has been validated by testing it thoroughly on actual wind turbines taken from the literature with the realistic and well-known FAST simulator.