Polynomial chaos to efficiently compute the annual energy production in wind farm layout optimization
Authors: Padrón Andrés Santiago , Thomas Jared , Stanley Andrew P. J. , Alonso Juan J. , Ning Andrew 
Published: May 2019

Abstract. In this paper, we develop computationally efficient techniques to calculate
statistics used in wind farm optimization with the goal of enabling the use
of higher-fidelity models and larger wind farm optimization problems. We
apply these techniques to maximize the annual energy production (AEP) of a
wind farm by optimizing the position of the individual wind turbines. The AEP
(a statistic) is the expected power produced by the wind farm over a period
of 1 year subject to uncertainties in the wind conditions (wind direction
and wind speed) that are described with empirically determined probability
distributions. To compute the AEP of the wind farm, we use a wake model to
simulate the power at different input conditions composed of wind direction
and wind speed pairs. We use polynomial chaos (PC), an uncertainty
quantification method, to construct a polynomial approximation of the power
over the entire stochastic space and to efficiently (using as few simulations
as possible) compute the expected power (AEP). 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 most
commonly used to compute the expected power). With PC based on regression, we
have reduced on average by a factor of 5 the number of simulations
required to accurately compute the AEP when compared to the rectangle rule
for the different wind farm layouts considered. In the wind farm layout
optimization problem, each optimization step requires an AEP computation.
Thus, the ability to compute the AEP accurately with fewer simulations is
beneficial as it reduces the cost to perform an optimization, which enables
the use of more computationally expensive higher-fidelity models or the
consideration of larger or multiple wind farm optimization problems. We
perform a large suite of gradient-based optimizations to compare the optimal
layouts obtained when computing the AEP with polynomial chaos based on
regression and the rectangle rule. We consider three different starting
layouts (Grid, Amalia, Random) and find that the optimization has many local
optima and is sensitive to the starting layout of the turbines. We observe
that starting from a good layout (Grid, Amalia) will, in general, find better
optima than starting from a bad layout (Random) independent of the method
used to compute the AEP. For both PC based on regression and the rectangle
rule, we consider both a coarse (∼225) and a fine
(∼625) number of simulations to compute the AEP. We find that for
roughly one-third of the computational cost, the optimizations with the
coarse PC based on regression result in optimized layouts that
produce comparable AEP to the optimized layouts found with the fine
rectangle rule. Furthermore, for the same computational cost, for the
different cases considered, polynomial chaos finds optimal layouts with
0.4 % higher AEP on average than those found with the rectangle rule.