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

In 2015, wind energy growth accounted for almost half of global
electricity supply growth. In the United States, it accounted for 41 % of
new power capacity, raising the wind energy supply to 4.7 % of the total
electricity generated in 2015 and on target to reach 10 % by 2020

Wind farm optimization is a complex, multidisciplinary, and high-dimensional
problem. The wind farm may contain dozens or even hundreds of wind turbines,
and each turbine may be parametrically described using several design
variables. Furthermore, the wind conditions (wind direction, wind speed, wind
turbulence, etc.) are stochastic (uncertain), and thus we need a statistic to
evaluate the performance of the wind farm. A common statistic is the expected
power or the annual energy production (AEP). Many model simulations are
needed to estimate the statistic

Examples of applications that require many model evaluations. In
deterministic optimization

We see three approaches to improving wind farm optimization capabilities.
Each approach focuses on the different blocks of the OUU problem (Fig.

The first approach increases the fidelity of the model, whereas the second (optimization) and third (uncertainty quantification) approaches seek to reduce the number of model evaluations, as this enables the study of larger and more realistic problems.

Here, we focus on the uncertainty quantification approach (the third
approach), as it has not been considered in detail before. The most recent
and thorough review of the wind farm optimization literature

In this paper, which is meant as a comprehensive introduction to uncertainty
quantification methods applied to wind farm simulations, we describe in
detail the polynomial chaos (PC) method and show that, for the efficient
(small number of model simulations) computation of the AEP, the PC method
based on regression should be used. An additional benefit of the PC method is
that it makes it feasible to consider multiple uncertain variables (e.g.,
wind direction, wind speed, wind turbulence, wake model parameter) that
impact the computation of the AEP. An example in which a wake model parameter is
considered an uncertain variable in addition to the wind speed and wind
direction can be found in

We first discuss the details of computing the power and the AEP of a wind
farm in Sect.

We first describe the aerodynamic wake model we use (Sect.

The Floris (FLow Redirection and Induction in Steady-state)

We use
the name Floris for the model, instead of FLORIS, the name used
in

Schematic of the Floris wake model. The model has three zones with
varying diameters,

We will consider the power of the wind farm to be a function of three classes
of variables: uncertain variables

For the problems considered in this work, Table

The variables used for calculating the power.

The power of the wind farm for a given wind direction and wind speed is equal
to the sum of the power produced by each turbine

The annual energy production (AEP) is an important metric used to describe a
wind farm. The AEP is a statistic. Specifically, it is a mean, as it is a
function of the expected power multiplied by the number of hours in a
year:

The expected power, and hence the AEP, is normally computed as a weighted
average, which amounts to the rectangle rule of integration
(Sect.

Uncertainty quantification (UQ) is the process of (1) characterizing input
uncertainties and then (2) propagating these input uncertainties through a
computational model with the goal of quantifying their effect on the model's
output. There are many sources of uncertainty in the modeling of a problem,
and different classifications of the uncertainties have been
proposed

In this work, we consider aleatory uncertainties that arise from the
variability in the inputs to our model caused by changing environmental
conditions. We describe this input variability as random variables with
associated probability distributions. Thus, the first step of characterizing
the input uncertainties is concerned with finding the probability
distributions that describe the model's inputs. This process is known as
statistical inference, model calibration, and inverse uncertainty
quantification

The goal of uncertainty propagation methods is to compute the statistics that
describe the effect of the uncertain inputs on the model output. There are
several methods to propagate the uncertainties and compute
statistics

As the name implies, this method numerically evaluates the integrals in the
definition of the statistics. The integrals to evaluate the mean (or expected
value) and the variance are

There are many quadrature methods to evaluate integrals

The rectangle rule, or midpoint rule,
is the simplest and most straightforward quadrature method. To approximate
the mean or expected value,

For simplicity, we describe the one-dimensional case,

Polynomial chaos (PC) is the name of an uncertainty quantification (UQ)
method that approximates a function with a polynomial expansion made up of
orthogonal polynomials. This function has

We first describe the polynomial chaos method in Sect.

Let

The polynomial basis

In addition to the orthogonal polynomials, the other component of the
expansion Eq. (

For the case of multiple uncertain variables

In total-order expansion a total polynomial-order bound

In tensor-product expansion a per-dimension polynomial-order bound

An example showing the multidimensional basis polynomials,
Eq. (

Note that for both total-order expansion and tensor-product
expansion the number of terms exhibits an exponential increase with an
increase in the number of uncertain dimensions

The mean and variance of the function of interest

The mean is the zeroth coefficient,

The coefficients of the polynomial chaos expansion Eq. (

To obtain the coefficients of the polynomial chaos expansion,

To obtain the coefficients of the polynomial chaos expansion Eq. (

Each row of the matrix

For overdetermined systems, the most popular method (and the one we use) to
estimate the coefficients is least squares, in which we pick
coefficients

For a given number of samples

Let

For most applications, the design and uncertain variables are independent. For instance, the design variables are the wind turbine locations and the uncertain variables are the wind conditions.

. Now the polynomial chaos expansion – over the uncertain variables – becomesWe want to know the gradients of the statistics with respect to the design
variables, and we proceed to derive them below. For simplicity, we drop the
subscript from the statistics

The gradient of the mean from Eq. (

The gradient of the coefficients can be computed with quadrature or regression, similarly to how the coefficients can be calculated with quadrature (Sect.

We start from the equation for the
coefficients Eq. (

To obtain the gradients of the statistics with respect to each design
variable we need to evaluate the multidimensional integral containing

We start from the linear system
Eq. (

Again, the gradient of the mean Eq. (

Similarly to computing the mean and variance with direct numerical
integration (Sect.

As described, the polynomial chaos method assumes that the uncertain
variables are statistically independent. In wind farm layout optimization,
the uncertain variables of wind speed and wind direction are usually
correlated. There are different approaches to use polynomial chaos for
problems with inputs made up of correlated (dependent) random variables. One
approach is to perform a (linear or nonlinear) variable
transformation

Here, we describe the details of computing AEP in the wind farm. We first
describe and discuss the inputs, which are the wind direction and wind speed
(uncertain variables) (Sect.

We consider the wind direction and the wind speed as uncertain variables with
probability distributions shown in Fig.

The uncertain variable probability distributions. The vertical lines show the cut-in and cut-out speed of a single wind turbine.

We construct the wind direction distribution (Fig.

For the wind speed, instead of linearly interpolating the data, we fit a
Weibull distribution

The fitted Weibull –

The truncated distribution is no longer a full Weibull distribution and needs to be scaled to ensure it is a valid probability density function (it integrates to 1).

The probability distributions are the weight functions of the orthogonal
polynomials used in the polynomial chaos method (Sect.

To showcase the results, we will focus on four representative layouts: Grid,
Amalia, Optimized, and Random (Fig.

Representative wind farm layouts used in the results. Each dot represents a wind turbine to scale – the dot represents the swept area of the rotor.

In reality, the 60 turbines in the Princess Amalia wind farm are the Vestas
V80 model. For each of the layouts in our study, we use the NREL 5 MW
reference turbine

We use an ensemble of 10 AEP results to compute the average AEP error. The
average AEP error allows us to better illustrate the differences between the
different methods used to compute the AEP and to avoid drawing conclusions
from one-off solutions. We found that averaging over 10 AEP results is enough
to illustrate the difference between methods and to smooth out the
convergence of the AEP error
(Fig.

We take as the baseline or true AEP the AEP computed with 200 000 Monte
Carlo samples. We picked 200 000 Monte Carlo samples to ensure that the 99 %
confidence interval for the true AEP was smaller than 1 % of the computed
AEP value for all layouts. We consider an AEP within 1 % of the baseline
AEP to be accurate, and we will use it as a reference for the results. AEP
predictions of real wind farms usually have an error of 10 %–20 %

Here, we provide the details of the methods used to compute the AEP and the abbreviations for the
methods.

rectangle rule (Sect.

polynomial chaos based on quadrature (Sect.

polynomial chaos based on regression (Sect.

Monte Carlo

We first characterize the power output of the wake model for different input
conditions (Sect.

The power production, computed with the Floris wake model, for the four wind
farm layouts (Sect.

Wind turbines are aligned in particular directions. The layouts are grid-like and have symmetry planes.

, it is periodic. The peaks in the contour lines identify wind directions for which there is a poor performance (low power). The grid-like layouts (top row of Fig.Power contours as a function of wind direction and wind speed for
the Grid, Amalia, Optimized, and Random layouts. The grid-like
layouts

The power response as a function of the wind direction has similar shapes for
different wind speeds until the speed is high enough (larger than the rated
speed of the layout) that the response gets clipped at 300 MW (all 60
turbines are operating at their rated power 5 MW). As a function of the wind
speed, for each wind direction, the power response starts after the cut-in
speed of 3 m s

The power response as a function of the uncertain variables (wind direction and wind speed) is a complicated function, which needs to be integrated weighted by the likelihood of the wind conditions, to obtain the AEP of the wind farm. Next, we show that using polynomial chaos instead of the rectangle rule results in better estimates of the AEP.

We consider the AEP as a function of two uncertain variables: the wind speed
and the wind direction. The AEP is usually considered a function of these two
variables. We compare the convergence of the AEP for the different methods to
compute the AEP: polynomial chaos (based on quadrature and regression), the
rectangle rule, and Monte Carlo
(Fig.

The average AEP error as a function of the number of samples for polynomial chaos (based on regression and quadrature), the rectangle rule, and Monte Carlo. The AEP is a function of two uncertain variables: the wind direction and wind speed. The polynomial chaos based on regression performs best for all layouts.

The superior performance of the polynomial chaos based on regression,
especially for the grid-like layouts (Grid and Amalia), is due to the
following: the polynomial chaos fit based on regression does not chase all
the high-frequency oscillations in the power response
(Fig.

For the
two-dimensional problem at 625 samples (

The average standard deviation of the energy (SD) error as a function of the number of samples for polynomial chaos
(based on regression and quadrature), the rectangle rule, and Monte Carlo. The
SD is a function of two uncertain variables: the wind direction and wind
speed. Note that the PC-R response is biased for the Grid and Amalia layouts.
The PC-R underpredicts the true SD at the expense of computing the mean (AEP)
more accurately (see Fig.

In what follows we will compare the PC-R (the best-performing method) with
the rectangle rule (the method most commonly used in practice) to quantify
the reduction of samples needed to compute the AEP accurately. Also, we will
sometimes use polynomial chaos to refer to polynomial chaos based on
regression. Figure

The average AEP error as a function of the number of samples for
both the rectangle rule and polynomial chaos based on regression (this is the
same as Fig.

In Fig.

The polynomial chaos based on regression is not only better on average, as we
have seen in Fig.

Variability in the convergence of the AEP.
Panels

The objective of the wind farm layout optimization is to maximize the AEP
(Sect.

Optimal wind farm layouts achieved for each method to compute the
AEP. The layouts in the first column show the turbine starting positions for
the optimization along with the boundary constraint (dashed line). The
optimal layouts correspond to those with the maximum AEP from
Table

We solve the optimization problem with the gradient-based sequential
quadratic programming optimizer SNOPT

We solve the optimization-under-uncertainty problem,
Eq. (

The AEP of the optimized layouts. We generate the AEP statistics for each method from a set of 10 different optimizations. For a similar number of samples, both coarse and fine, polynomial chaos based on regression produces better optimal layouts than the rectangle rule. Furthermore, using about one-third of the samples, PC-R optima are comparable with the optima obtained with the rectangle rule that used 625 samples to compute the AEP. The values of the AEP reported are computed with 200 000 Monte Carlo samples.

The optimum layouts obtained with polynomial chaos based on regression are
better than those obtained with the rectangle rule. For a similar number of
samples to compute the AEP, the optimizations with polynomial chaos produce
on average optimal layouts with 0.4 % higher AEP than the optimizations
with the rectangle rule. Also, using about one-third of the samples, the
optima obtained with polynomial chaos are comparable to those obtained with
the rectangle rule. Furthermore, polynomial chaos finds better local optima
than the rectangle rule for all the different starting layouts considered
(Table

The initial and optimized position of the turbines for the layouts
obtained with the PC-R-fine method from Fig.

The optimal layouts with the maximum AEP for each method and starting layout
are shown in Fig.

The Amalia layout is an optimal layout for a different
optimization problem: the designers of the Princess Amalia wind farm likely
considered different optimization constraints and objectives, used a different
wake model in the analysis, and, as mentioned in Sect.

Convergence history of the optimization for the different starting
layouts. The convergence history is for the layouts in
Fig.

To properly compare the results obtained by the different methods used to
compute the AEP in the optimization, we should compute the AEP of the optimal
layout with a method that was not used in the optimization. The values of the
AEP reported in Table

The improvement in AEP of the optimized layouts from
Fig.

Most optimizations required on the order of 100 AEP computations. The
optimization history for the layouts in Fig.

A single wind farm layout optimization requires tens to hundreds of thousands
of model evaluations. During the design phase of a new wind farm, designers
need to perform many optimizations. The designers may explore scenarios
with different turbine types, different sites, larger farms with a different
number of turbines, and possibly even systems of wind farms. Also, the
presence of local optima would require many optimizations with different
restarts to find the best layout. And, furthermore, there is a desire to
increase the fidelity of the models used to simulate the wind farm, which
will increase the time and computational cost of the
optimizations

If instead of using the Floris wake model in the optimization, we had used a high-fidelity fully resolved large eddy simulation to model the wind farm, a back-of-the-envelope calculation shows that the optimization would require the total annual energy production (AEP) of the wind farm to power the computers used in the simulation.

. Thus, to facilitate the design phase of new wind farms and to incorporate the use of higher-fidelity models in the design phase requires significant improvements to reduce the simulation requirements (computational cost) of performing the wind farm problem of optimization under uncertainty (OUU) (Fig.We proposed the use of polynomial chaos (PC) to improve the uncertainty quantification step of the OUU problem. We computed AEP convergence plots and showed that polynomial chaos based on regression (PC-R) could compute the AEP accurately using on average one-fifth of the simulations required using the rectangle rule, the method currently used in practice in the wind industry. PC-R computes the AEP accurately (error less than 1 %), usually with only 200 simulations or fewer depending on the wind farm layout – a significant improvement over the current industry practice of using more than 1000 model evaluations to compute the AEP.

The layout of the wind farm influences the convergence of the AEP because the
layout has a significant effect on the power output of the farm as the wind
conditions vary (Fig.

In addition to computing the AEP efficiently, a benefit of using polynomial
chaos is that it can also compute the gradient of the AEP efficiently. The
use of gradients is essential to enable large-scale wind farm optimization.
We described how to compute the gradients of the AEP in
Sect.

We performed multiple gradient-based wind farm layout optimizations to
compare the optimization results obtained with the different methods to
compute the AEP: polynomial chaos based on regression and the rectangle rule.
The goal of the optimization was to maximize the annual energy production
(AEP) of the wind farm. The layout optimization is an OUU problem (Fig.

An ensemble of optimizations should be used to properly compare the optimal layouts obtained using the different methods to compute the AEP. Also, for proper comparison, the AEP of the optimized layouts should be evaluated in the same way and with a method different from the one used in the optimization. To be confident about the differences between the optimal layouts, we evaluated the AEP of each optimized layout using the same 200 000 Monte Carlo samples. We found that the benefits of being able to efficiently compute the AEP with PC-R translate to being able to find better optima than those obtained when computing the AEP with the rectangle rule when using the same number of simulations. For the cases considered, on average, with PC-R we find optima that are 0.4 % better than those found with the rectangle rule. This is a significant improvement, as a 1 % increase in the AEP for a modern large wind farm can increase its annual revenue by millions of dollars. Also, using about one-third of the simulations, coarse PC-R finds comparable optima to those found by using the fine rectangle rule for the optimization cases considered.

An interesting extension of this work that could further improve the AEP
would be to allow yaw-based control in the optimization problem to take
advantage of wake deflection.

Data are available at

ASP performed the investigations and wrote the article. All authors reviewed and edited the manuscript. JT assisted with the wake model. JT and APJS assisted with setting up the optimization problems. AN and JJA provided direction and supervision for the entire project.

The authors declare that they have no conflict of interest.

Funding for this research was provided by the National Science Foundation under a collaborative research grant (nos. 1539384 and 1539388).

This paper was edited by Johan Meyers and reviewed by three anonymous referees.