A fast stochastic solution method for the Blade Element Momentum equations for long‐term load assessment

Abstract

Unsteady power output and long-term loads (extreme and fatigue) drive wind turbine design.However, these loads are difficult to include in optimization loops and are typically only assessed in a post-optimization load analysis or via reduced-order methods. Both alternatives yield suboptimal results. The reason for this difficulty lays in the deterministic approaches to long-term loads assessment. To model the statistics of lifetime loads they require the analysis of many unsteady load cases, generated from many different random seeds-a computationally expensive procedure. In this paper, we present an alternative: a stochastic solution for the unsteady aerodynamic loads based on a projection of the unsteady Blade Element Momentum (BEM) equations onto a stochastic space spanned by chaos exponentials. This approach is similar to the increasingly popular polynomial chaos expansion, but with 2 major differences. First, the BEM equations constitute a random process, varying in time, while previous polynomial chaos expansion methods were concerned with random parameters (ie, random but constant in time or initial values). Second, a new, more efficient basis (the exponential chaos) is used. This new stochastic method enables us to obtain unsteady long-term loads much faster, enabling unsteady loads to become accessible inside wind turbine optimization loops. In this paper we derive the stochastic BEM solution and present the most relevant results showing the accuracy of the new method.

Surrogate models for unsteady aerodynamics using non-intrusive Polynomial Chaos Expansions

“…For future work, we want to test the possibility of truncation of the polynomials in order to make the MCs of the surrogate model more cost-efficient. Also, using non-conventional polynomials, such as what Fluck and Crawford (2018) did, might result in a more efficient polynomial. In this study, we used the collocation method for calculating the PCE coefficients.…”

“…The polynomials do not necessarily need to be selected from the specific family of polynomials as long as they are orthogonal polynomials. For instance, Fluck and Crawford (2018) showed exponentials components work at best for their purposes. As the randomness in this study comes in the form of a uniform distribution for φ j , the surrogate model is based on the Legendre polynomials (Xiu and Karniadakis, 2003).…”

“…However, very few have looked at building a surrogate model of the aerodynamic model of wind turbine. Fluck and Crawford showed an initial attempt to build a surrogate model based on intrusive Polynomial Chaos Expansion (PCE) on simple lifting line and BEM models (Fluck, Manuel and Crawford, Curran, 2016;Fluck and Crawford, 2018). As they quickly faced with curse of dimensionality, they showed it is possible to reduce the number of random variables in Veers' unsteady wind model significantly.…”

“…As they quickly faced with curse of dimensionality, they showed it is possible to reduce the number of random variables in Veers' unsteady wind model significantly. Afterwards, they used this reduced dimension wind model to propagate stochasticity thorough a simple lifting line or BEM (Fluck and Crawford, 2018) model. However, with intrusive PCE it is necessary to change the model fundamentally to be able to incorporate the random variables (Sudret, 2007).…”

Surrogate models for the blade element momentum aerodynamic model using non-intrusive Polynomial Chaos Expansions

“…As they were quickly faced with curse of dimensionality, they showed it is possible to reduce the number of random variables in Veers' unsteady wind model significantly. Afterwards, they used this reduced dimension wind model to propagate stochasticity through a simple lifting line (Fluck and Crawford, 2016b) or BEM (Fluck and Crawford, 2018) model. However, with intrusive PCE it is necessary to change the model implementation fundamentally to incorporate the random variables (Sudret, 2007).…”