Polynomial chaos and multi-fidelity approximations to efficiently compute the annual energy production in wind farm layout optimization [electronic resource]
Keywords: Aeronautics and Astronautics.

This thesis presents techniques to enable high-fidelity uncertainty quantification and high-fidelity optimization under uncertainty. The techniques developed herein are applied to maximize the Annual Energy Production (AEP) of a wind farm by optimizing the position of the wind turbines. The AEP is the expected power produced by the wind farm over a period of one year, and the wind conditions (e.g., wind direction and wind speed) for the year are described with empirically-determined probability distributions. To compute the AEP of the wind farm, a wake model is used to simulate the power for various sets of input conditions (e.g., wind direction and wind speed). We use polynomial chaos (PC), an uncertainty quantification method, to construct a polynomial approximation of the power from these sets of simulations or samples. We explore both regression and quadrature approaches to compute the PC coefficients. PC based on regression is significantly more efficient than the rectangle rule (the method currently used in practice to compute the expected power): PC based on regression achieves the same accuracy as the rectangle rule with only one-tenth of the required simulations, and, for the same number of samples, its estimates are five times more accurate. We propose a multi-fidelity method built on top of polynomial chaos to further improve the efficiency of computing the AEP. There exists multiple wake models of varying fidelity and cost to compute the power (and hence the AEP). Here, we choose the Floris and Jensen models as our high- and low-fidelity models, respectively. Both models are engineering models that can be evaluated in less than 1 second. The multi-fidelity method creates an approximation to the high-fidelity model and its statistics (such as the AEP) and uses a polynomial chaos expansion that is the combination of a polynomial expansion from a low-fidelity model and a polynomial expansion of a correction function. The correction function is constructed from the differences between the high-fidelity and low-fidelity simulation results. The multi-fidelity method can estimate the high-fidelity AEP to the same accuracy with only one-half to one-fifth of the high-fidelity model evaluations, depending on the layout of the wind farm. Combining the reduction in the number of simulations obtained from using PC and the multi-fidelity method, we have reduced by more than an order of magnitude the number of simulations required to accurately compute the AEP, thus enabling the use of more expensive higher fidelity models in wind farm optimization. Once we can compute the AEP efficiently, we consider the optimization under uncertainty problem of maximizing the AEP of a wind farm by changing its layout subject to geometric constraints— wind turbines must stay within a given area and with a minimum separation between them. We extend polynomial chaos to obtain the gradient of the statistics (AEP) from the gradients of the power at the simulation samples. With the gradient of the AEP, we can make use of a gradient-based optimizer to efficiently maximize the AEP. The optimization problem has many local maxima that are nearly equivalent. To compare the optimizations between methods (polynomial chaos, rectangle rule), we perform a large suite of optimizations with different initial turbine locations and with different samples and numbers of samples to compute the AEP. The optimizations with PC based on regression result in optimized layouts that produce the same AEP as the optimized layouts found with the rectangle rule but using only one-third of the samples. Furthermore, for the same number of samples, the AEP of the optimal layouts found with PC is 1 % higher than the AEP of the layouts found with the rectangle rule. A 1 % increase in the AEP for a modern large wind farm can increase its annual revenue by