Using adjoint optimization and three-dimensional steady-state
Reynolds-averaged Navier–Stokes (RANS) simulations, we present a new
gradient-based approach for optimally siting wind turbines within
utility-scale wind plants. By solving the adjoint equations of the flow
model, the gradients needed for optimization are found at a cost that is
independent of the number of control variables, thereby permitting
optimization of large wind plants with many turbine locations. Moreover,
compared to the common approach of superimposing prescribed wake deficits
onto linearized flow models, the computational efficiency of the adjoint
approach allows the use of higher-fidelity RANS flow models which can capture
nonlinear turbulent flow physics within a wind plant. The steady-state RANS
flow model is implemented in the Python finite-element package

Optimizing wind turbine locations within a wind plant is a uniquely challenging problem that combines turbulent flow control with practical engineering challenges concerning the economical development of renewable energy. The problem is further complicated by the strong nonlinear coupling between turbine locations, power production, atmospheric boundary layer turbulence, and mechanical loads on turbine components. Wind plant optimization techniques used in industry often rely on heuristic guidelines and simplified linear flow models that limit computational costs. However, these approaches neglect important turbulent flow physics and can result in the underperformance of wind plants relative to their pre-construction estimates. The reduced power output and increased uncertainty due to optimization with low-fidelity flow models ultimately increases the levelized cost of energy (LCOE) and associated project risk for investors.

Major impediments to improving the LCOE for utility-scale wind plants are the
difficulty of performing the turbine layout optimization using fluid
dynamical models with sufficient fidelity and performing optimization in
high-dimensional design spaces. In order to improve wind plant optimization,
we present ^{™})

This paper is organized as follows: background on wind plant flow modeling,
layout optimization, and adjoint optimization techniques is provided in
Sect. 2; a description of the methods used in

Utility-scale wind plants in the United States typically involve tens to hundreds of turbines arranged in semi-regular arrays. The layout topology is generally an outcome of optimizing the power production or net capacity factor subject to competing influences from site constraints, the local wind resource, and construction costs. These constraints include the patchwork of viable building areas formed by leases and setbacks due to environmental concerns or physical infrastructure, terrain and soil characteristics like slope or vegetation, turbine manufacturer spacing requirements, and continuity requirements imposed by access roads and electrical connections. This results in a complex design problem, with turbine layouts varying drastically between different geographic locations and exhibiting complex topologies.

The AEP from a wind plant layout has traditionally been, and generally
continues to be, assessed using reduced-order linear flow models. Such models
estimate the relative wind speed across a site based on the linearized
Navier–Stokes equations and treat terrain features as perturbations in
boundary conditions. The underlying governing equations, introduced by

A velocity deficit representing the turbine wake is then superimposed on the
background wind resource at each turbine location. The PARK model developed
by

Recently, higher-fidelity computational fluid dynamics (CFD) models have been
increasingly applied to the study of wind plant performance. Several RANS
models have been developed for simulating atmospheric flows in wind plants
using actuator disk turbine representations, with a particular emphasis on
modified

To date, wind plant layout optimization has been performed primarily using
linear flow models with analytical wake deficits. The relatively cheap
computational cost of such models allows gradient-free methods to be used in
the optimization process. A number of layout optimization studies have
utilized gradient-free approaches to minimize wake losses

Recently, higher-fidelity CFD flow models have been used in a limited range
of wind plant optimization applications. The Technical University of Denmark
has developed TOPFARM

In the present study, we use adjoint techniques to enable gradient-based
optimization of wind turbine locations within a plant, subject to realistic
turbulent flow fields. A steady 3-D RANS flow solver is employed as a
first-principles model that can predict new turbulent flow physics, rather
than prescribing fixed wake behaviors as in linear flow models. The RANS
model provides an accurate model of a neutral atmospheric boundary layer at
moderate computational cost and without requiring calibration using LES. The
RANS model is also amenable to automatic differentiation and gradient-based
optimization, as explained in the next section. This gradient-based approach
permits the use of high-dimensional control spaces, thereby providing
optimized layouts of arbitrary complexity (i.e., optimized layouts are not
restricted to grids or any other regular arrangement). We further demonstrate
layout optimization for the full plant AEP based on a real site wind rose,
going beyond the uniform speed optimization considered previously

The greater expense of CFD flow models such as RANS and LES compared to linearized flow models requires the use of an efficient optimization technique that minimizes the number of flow model evaluations. Gradient-based optimization methods are promising candidates for CFD-driven wind plant optimization since they require orders of magnitude fewer function evaluations than gradient-free techniques. However, finding these gradients can be challenging when using complex flow models or when there are many control variables. Calculating such gradients with a finite-difference approach requires a function evaluation for each control variable, making this approach prohibitively expensive for optimizing utility-scale plants with many turbines.

In the present optimization framework, the necessary gradients are obtained
relatively inexpensively by using adjoint optimization techniques. A
comprehensive review of discrete techniques for calculating gradients of
engineering design problems, including the adjoint approach, can be found in

Adjoint systems arise naturally from consideration of dynamical systems whose
behavior can be described by differential equations, and they have a rich
history in fluid dynamics and sensitivity analysis. Reviews of adjoint
techniques in various flow control and optimization applications have been
published by

The adjoint operator is defined by the bilinear identity

The following general framework illustrates the computational advantages of
the adjoint approach. Consider a dynamical system with governing equations
that can be expressed in residual form as

A common approach to solving a PDE-constrained problem is to minimize a
reduced functional

Gradient-based optimization algorithms can approach second-order convergence
to local minima and can minimize the evaluations of

In the adjoint approach,

In the present study, wind plant layout optimization is approached as a PDE-constrained optimization problem using the adjoint theory developed in the previous section. The wind plant power output is maximized with gradient-based optimization techniques, subject to a PDE constraint corresponding to the RANS equations, which are used to predict turbulent flow within the plant.

We seek to maximize steady-state power output from

We stress that the strength of this approach is not the particular form of the RANS closure model, but rather its embedding in an adjoint optimization framework. Different objective functions, turbulence closures, or turbine representations can be easily implemented in this framework and still benefit from the adjoint approach.

Turbines in the simulations are represented as non-rotating actuator disks
using actuator disk theory, as described in standard wind energy texts

Standard actuator disk theory assumes that the rotor disk is uniformly
loaded, but this introduces singularities at the rotor edge. To ensure that
the thrust force and power production are continuously differentiable, as
well as to avoid numerical instabilities, the turbine force and power
production are smoothly distributed across the rotor swept area. This
differentiability with respect to position is crucial for gradient-based
layout optimization, and smoothly distributing rotor forces is common
practice in actuator disk

A continuously differentiable modified exponential distribution is
used to smoothly distribute the turbine forces and power production over the
rotor swept area. The smoothing function is demonstrated for an 80 m rotor
diameter turbine with

The 3-D computational domain used in the simulations has horizontal
dimensions of 2.4 km

A coarse computational mesh of finite elements is generated for the entire
domain and then further refined within a circle that circumscribes the site
boundaries, as shown in Fig.

To confirm that the domain is sufficiently large for the number of turbines,
we examined the blockage ratio. In wind tunnel studies, the blockage ratio is
typically defined as the ratio of the total tunnel cross-sectional area to
the total rotor disk area. A worst-case blockage scenario would involve all
16 turbines impeding the flow with no overlap between rotors. This worst-case
results in a blockage ratio of 5.2 %, which is well below the threshold
of 10 % blockage that requires a correction in wind tunnel studies of
horizontal-axis wind turbines

Plan view of the computational mesh at hub height showing additional
refinement around the wind turbine area (left), schematic of the
computational domain showing the turbine locations used to initialize each
optimization (top right), and side view of the mesh showing refinement below
2 rotor diameters (bottom right). The wind plant site constraint is shown as
a red square in the plan view, and horizontal dimensions are normalized by
the rotor diameter

Simulations are performed for a range of different inflow directions and
inflow speeds, corresponding to both idealized and real-world wind roses and
wind speed distributions. Steady-state solutions are found for each of the

Figure

Begin with an initial layout

Perform flow-field simulations for each of the

Calculate the negative of the wind plant power,

Calculate the objective function by taking a weighted sum

Compute adjoint simulations for the forward simulations, and calculate the gradient of the reduced functional

Use the gradient to perform a gradient-based optimization of the layout to obtain

It should be noted that, because the optimization algorithm is
gradient-based, it finds local rather than global minima in the total
objective function

Schematic of the multilevel optimization process for a wind plant
with

The WindSE flow solver is implemented in a software package called

The description of finite-element problems as variational forms in

The 3-D RANS and continuity equations that form the PDE constraint in Eq. (7) are solved
with a nonlinear Newton solver in a coupled fashion using a mixed finite-element space with piecewise linear elements for both the velocity and
pressure fields. To satisfy the Ladyzhenskaya–Babuška–Brezzi (LBB) (or
inf-sup) compatibility condition

Flow past a single turbine obtained using the 3-D RANS flow solver. The
top panel shows the velocity at hub height and the bottom two panels show
profiles of the velocity deficit in horizontal and vertical planes passing
through the center of the wind turbine rotor. Velocity deficits are relative
to the respective profiles 3 RD upstream of the turbine and the velocities
are normalized by the incoming hub-height velocity. Axes are in units of
rotor diameter

The gradients obtained from

The forward and adjoint problems are parallelized with MPI and can be run on
a desktop or in a high performance computing environment. The discrete
adjoint calculation is automatically parallelized by

Hub-height velocity fields from the 3-D RANS solver used in

Optimal layouts (top row) and flow fields (bottom row) for five test
cases with an increasing number of evenly weighted inflow directions. The
wind roses in the upper right corner of each layout plot show the inflow
directions used in the optimization. The blue triangles show the optimized
turbine locations and the red square indicates the site boundaries. The
velocities are normalized by the hub-height inflow velocity, which is

In the following, we present results for several different layout
optimization cases. First, we provide results for standard test cases of flow
past a single turbine and flow through a very deep wind plant in order to
demonstrate that the RANS flow solver accurately captures wind turbine wakes,
thereby providing confidence that subsequent layout optimizations are
performed according to the correct flow physics. Second, we optimize a
16-turbine wind plant using wind roses with evenly weighted wind directions
and a constant wind speed of 8 m s

Relative efficiency of power production from the test cases compared to 16 turbines with no wake effects. Results from a “naïve” case with two parallel turbine rows and no layout optimization are also included to demonstrate the improvements achieved by the layout optimization.

As a test of the qualitative performance of the RANS flow solver,

The results presented in this section are only intended to demonstrate the qualitative agreement of the RANS solver with prior high-fidelity studies and analytical wake theory. This demonstrates that the present flow solver captures a reasonable level of fidelity to introduce the adjoint optimization framework. A detailed turbulence model verification and validation is beyond the scope of this study as we are instead focused on the integration of the model into a flexible and automated adjoint optimization framework. A comprehensive study on the implementation of more sophisticated turbulence models within the WindSE framework is left for future research.

In order to demonstrate many of the characteristics of optimal layouts and
flow fields found using

Figure

The flow fields shown in Fig.

Spanwise velocities produced by the optimized layout found in the
five-inflow-direction case shown in Fig.

Optimal layout (left) and flow field (right) for the boomerang test
case with the wind rose binned into 36 inflow directions and a uniform
hub-height velocity of 8 m s

Optimal layout (left) and flow field (right) for the NWTC M2
8 m s

Flow curvature also affects the propagation and interaction of the turbine
wakes. Wakes near the edge of the plant are slightly deflected away from the
plant center and reflect the overall spreading of the flow streamlines. This
curvature can be observed near the edges of the plant in the cases shown in
Fig.

It is emphasized that the results in Fig.

We compared the optimized layout for two inflow directions shown in
Fig.

Optimal layout (top left) and flow fields resulting from a full AEP optimization using data from the M2 mast at NWTC. Differences in both wind speed and wind direction are accounted for in the optimization, and the wind rose used is shown in the top right corner of the layout plot. In the top left panel, the site boundary is shown by a solid red line and turbine locations are denoted by blue triangles. Flow fields are shown for inflow winds from the north (top left), east (top right), south (bottom right), and west (bottom left).

In the previous section, each inflow direction was given an equal weight
(i.e.,

The first wind rose considered has two prominent wind directions spaced

The second wind rose considered is given by the directional distribution of
8 m s

As a final demonstration of the power and flexibility of

As shown in Fig.

It should be noted that despite the predominant high-speed winds from the
west-northwest, the turbines in the north–south rows shown in
Fig.

The results presented in this paper show that the nonlinear flow effects
leading to wake curvature and local speedups are significant when optimizing
over a few prominent wind directions. We find consistent rotational symmetry
in the optimal layouts with evenly weighted inflow directions, suggesting
that evenly weighted wind roses may be useful diagnostic tests for wind plant
optimization. As the wind rose is refined into more bin directions, the
optimizer is able to take advantage of prominent wind directions and increase
energy production. As the number of wind direction and inflow speed
combinations increases,

A number of different future studies utilizing

Simulations were performed using the WindSE code that will be made available through NREL's
WISDEM software tool

The authors declare that they have no conflict of interest.

This work was supported by award UGA-0-41026-70 through the Alliance Partner University Program in partnership with the National Renewable Energy Laboratory. Edited by: J. Peinke Reviewed by: three anonymous referees