The wind farm layout optimization problem is notoriously difficult to solve because of the large number of design variables and extreme multimodality of the design space. Because of the multimodality of the space and the often discontinuous models used in wind farm modeling, the wind industry is heavily dependent on gradient-free techniques for wind farm layout optimization. Unfortunately, the computational expense required with these methods scales poorly with increasing numbers of variables. Thus, many companies and researchers have been limited in the size of wind farms they can optimize. To solve these issues, we present the boundary-grid parameterization. This parameterization uses only five variables to define the layout of a wind farm with any number of turbines. For a 100-turbine wind farm, we show that optimizing the five variables of the boundary-grid method produces wind farms that perform just as well as farms where the location of each turbine is optimized individually, which requires 200 design variables. Our presented method facilitates the study and both gradient-free and gradient-based optimization of large wind farms, something that has traditionally been less scalable with increasing numbers of design variables.

In 2018, wind energy produced 6.5 % of the electricity use in the United States.

The complexity and multimodality of wind farm layout design space. Shown is the normalized annual energy production of a 100-turbine wind farm as a function of the location of one turbine; 99 turbines remain fixed, while one is moved throughout the wind farm.

Because of the multimodality of the space and the often discontinuous models used in wind farm modeling, the wind industry is heavily dependent on gradient-free techniques for wind farm layout optimization

The number of function calls required to optimize the multi-dimensional Rosenbrock function versus the number of variables. The computational expense of gradient-free and finite-difference gradients scale poorly with the number of variables.

Despite its difficulty, layout optimization is an essential step in wind farm development in order to maximize power production. Power losses of 10 %–20 % are typical from turbine interactions within a wind farm

The most common current wind farm layout definitions include defining the location of each turbine directly

In this paper we present the boundary-grid (BG) layout parameterization, a new wind farm layout parameterization. This new method solves the challenges that have previously made wind farm layout optimization so difficult. BG parameterization uses only five variables and can produce layouts that perform just as well as or better than the layouts achieved by directly optimizing the location of each wind turbine.
With some of the most advanced wind farm optimization methods that have previously been available, we can directly optimize the location of every turbine in a 100-turbine wind farm in 4–5 h. More common methods take on the order of days or longer. With BG parameterization, we can optimize a 100-turbine wind farm in 3 min. Additionally, this new parameterization dramatically reduces the multimodality of the design space compared to direct layout optimization (compare Figs.

When the locations of wind turbines in a farm are optimized directly, the final layout often follows two general rules. First, a large fraction of turbines are grouped on or near the wind farm boundary. Second, the turbines that are not positioned on the boundary are loosely arranged in rows throughout the farm (Fig.

Example 100-turbine wind farm layouts, and parameterized wind turbine layout definition. Each dot is to scale, representing the wind turbine diameter.

In BG parameterization, the turbines are divided into two groups: the boundary and the inner grid (Fig.

There are some discrete values which are important in our formulation, namely the number of turbines which are placed along the boundary and how many are in the grid, how many rows and columns are in the grid, and how the rows and columns are organized. We present some rules that we have found effective in determining these discrete values for all wind roses, wind farm boundaries, and wake models that we tested. Each individual case may benefit slightly from a more specialized selection of these values but our method works well across all cases tested.

The number of turbines placed on the boundary is determined by the wind farm perimeter and turbine rotor diameter. If the perimeter is large enough, 45 % of the wind turbines are placed on the boundary. In some cases, the wind farm perimeter is small and would result in turbines that are too closely spaced if 45 % were placed around the boundary. In this case, the number of boundary turbines is reduced until the minimum desired turbine spacing in the wind farm is preserved. When defining the number of turbines to be placed along the perimeter, the user must consider the most extreme boundary angles, such that minimum turbine spacing is preserved even at boundary corners. No matter how many turbines are placed around the boundary, they are always spaced equally traversing the perimeter, and all of the remaining turbines are placed in the inner grid. Note that the number of boundary turbines is determined before the number of turbines in the inner grid, to ensure that sufficient spacing is maintained between the boundary turbines.

The number of rows, columns, and their organization in the grid is determined with the following procedure. First,

The process outlined to select the discrete variables used in the parameterization is recommended as a starting point, and when computational resources or time is limited. We tested many different methods of how to determine the discrete values, but found that the method shown above consistently produced wind farm layouts with high energy production. With sufficient resources, some scenarios may benefit from optimizing with a different ratio of boundary turbines or different initializations of the boundary grid. However, the results discussed in this paper were produced with the method given in this section. Because these variables are discrete, they cannot be included as design variables when using a gradient-based optimization method because the function space would be discontinuous. But a gradient-free optimization may benefit from including some of these discrete variables as design variables in the optimizations.

In the testing of the BG wind farm layout parameterization method, we modeled the turbine parameters after the IEA 3.35 MW reference turbine

The thrust coefficient curve for the 3.35 MW turbine used in this paper.

The major benefit of wind turbine layout parameterization comes for large wind farms. For farms with just a few turbines, the layout can be optimized directly with a small amount of design variables. In such cases with few design variables, there is little to no benefit gained from intelligently parameterizing the design space. In this study, each wind farm layout that we optimized had 100 wind turbines, to demonstrate the benefits of BG parameterization for large wind farms.

We tested the performance of our parameterization method on wind farms with different average turbine spacing: 4, 6, and 8 rotor diameters shown in Fig.

Wind speed deficits in this paper were predicted from turbine wakes with a modified version of the 2016 Bastankhah Gaussian wake model

To find the effective wind speed across the entire wind turbine to be used in turbine power calculation, we averaged the velocities sampled at several points across the rotor. During optimization, we sampled at four points over the swept area of the rotor, shown in Fig.

The sampling points across the swept rotor area to calculate the effective wind speed at the turbine. Wind speeds are sampled at each point and then averaged.

As the goal of this paper is to demonstrate the performance of our layout parameterization method in wind farm optimization for any scenario, we chose three different wind roses from cities in California, USA: North Island, Ukiah, and Victorville.

The three wind roses and associated average wind speeds used in this study. The wind resources are from

Example Weibull distributions for two different average wind speeds. Each wind direction is associated with an average wind speed (shown in Fig.

In this paper we compare how optimizing with BG wind farm layout parameterization compares to two common currently used parameterization methods. We have optimized wind farms using a simple grid parameterization (referred to as “grid optimization”) and BG parameterization (“BG”) and by directly optimizing the location of each turbine independently (“direct optimization”). Examples of these layouts, along with the baseline layout that was used to compare results in Sect.

Example optimal layouts achieved with each parameterization method. These are 100-turbine layouts, with an average turbine spacing of 4 rotor diameters and the Princess Amalia wind farm boundary. They were optimized with the wind rose from North Island, California.

In each case, the objective function of the optimization was to maximize the annual energy production (AEP) of the wind farm, shown in Eq. (

We used exact-analytic gradients in each optimization. The gradients for each portion of the model were obtained with an automatic differentiation source code transformation tool, Tapenade

Using exact, rather than finite-difference, gradients is important in this study because the computational expense required for optimization problems with increasing design variables scales better with exact gradients (see Fig.

For this paper we have used only a gradient-based optimization method. The purpose of this research is to explore a novel wind turbine layout parameterization and how it compares to other more commonly used layout parameterizations. We do not explore how different optimization methods compare when applied to the wind farm layout problem. As mentioned in the Introduction, the relationship of how optimization method performance scales with increasing numbers of design variables is well documented. Additionally, our past work suggests that the large number of random starts allows for a reasonably thorough search of the design space.

In this section we demonstrate how the optimal wind farms using BG parameterization compared to wind farms that have been optimized directly or with a common grid parameterization. We will discuss the best results, the computation expense required to optimize and the multimodality of the design space with each parameterization method.

The best annual energy production achieved with 100 randomly initialized optimizations. Shown are the best results from the grid turbine parameterization (four design variables), our new boundary-grid parameterization method (five design variables), and by directly optimizing the location of each turbine (200 design variables). Results are shown as a percent increase over a baseline grid layout.

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Results from 100 randomly initialized optimizations for wind farms with varied average turbine spacing and 100 wind turbines. The farm optimized had the Princess Amalia boundary and the wind rose from North Island, California. Shown are results using the grid turbine parameterization, our new boundary-grid parameterization, and direct optimization. The optimal annual energy production distribution achieved for each of the optimization runs, in wind farms with varied turbine spacing of 4, 6, and 8 rotor diameters for panels

In terms of the best achievable wind farms with each parameterization method, our new BG method performs almost identically to optimizing the location of each wind turbine directly. In all cases that we tested, the BG optimizations were able to find solutions that slightly outperformed the direct optimizations, although they were almost identical. With only five design variables, we can create wind farms that perform the same as or better than farms that have been designed with 200 variables. While the grid parameterization is able to achieve good results for some wind farms, it often performs much worse than our parameterization. One additional variable is a small price to pay for significant improvement in optimal wind farm design.

The utility of any wind farm layout parameterization is not only measured by the ability to create high energy-producing wind farms, but by the ability to do so quickly and reliably. Figures

Results from 100 randomly initialize optimizations for wind farms with varied wind roses and 100 wind turbines. The farm optimized had the Princess Amalia boundary, and the average turbine spacing was 4 rotor diameters. Shown are results using the grid turbine parameterization, our new boundary-grid parameterization, and direct optimization. The optimal annual energy production distribution achieved for each of the optimization runs, in wind farms with varied wind roses. Wind rose from

Results from 100 randomly initialize optimizations for wind farms with varied wind farm boundaries and 100 wind turbines. The average turbine spacing was 4 rotor diameters, and the wind rose was from North Island, California. Shown are results using the grid turbine parameterization, our new boundary-grid parameterization, and direct optimization. The optimal annual energy production distribution achieved for each of the optimization runs, in wind farms with varied boundary shapes.

In general, the grid and the BG optimal AEP results have a similar spread, with the BG results shifted up higher. Compared to the direct optimizations, the grid and BG optimizations have a larger spread in optimal solutions. This is a consequence of the discrete variables that are initialized at the start of each optimization run. The number of rows and columns, as well as their organization in the grid are determined by the randomly initialized rotation design variable,

With regards to the function calls required to converge, the grid optimizations required about one-third of the function calls to converge compared to the BG optimizations, while the direct optimizations required about an order of magnitude more.
The only exception was the circular wind farm, for which the direct optimizations converged quickly, on the same order as the BG optimizations.
Function calls are an important measure of computational expense, as they are correlated with time and processing power required to optimize. Here it is important to remember that our results were obtained with exact-analytic gradients, meaning that one function call was required to obtain the wind farm AEP as well as the gradients with respect to each of the design variables. The same is true of the constraints: one function call gave both the constraint values and the gradients. Without exact gradients, a finite-difference method would need to be used to calculate the gradients. At every optimization step, finite-difference gradients require one (forward or backward difference) or two (central difference) additional function calls for every design variable to approximate the gradients. Thus, if forward-difference gradients were used rather than exact ones, the grid optimizations would need about 4 times as many function calls to reach a solution, the BG optimization would need about 5 times as many function calls, and the direct optimization would need 200 times as many function calls to converge. This is the best-case scenario, as optimizations with finite-difference gradients often have trouble converging. Compared to gradient-free optimization, the exact analytic gradients are vital. The direct optimization with a gradient-free technique would be near impossible because of the massive required computational expense

One of the major difficulties of the wind farm layout optimization problem is the extreme multimodality of the design space (Fig.

One-dimensional sweeps across the design space of each parameterization method discussed in this paper. These figures show the multimodality of each of the design spaces.

Parameterizing the design space with a grid and with the BG method (Fig.

Notice that the ranges of the design variable sweeps is different for the BG and grid parameterizations compared to the direct sweep. This is because the simpler parameterizations are more limited in the feasible design values. The range through which the design variables can sweep is relatively limited, without violating the minimum spacing or the boundary constraints.

BG parameterization requires few variables, produces wind farm layouts that perform similarly to ones that have been optimized directly with much lower computational expense, and reduces the multimodality of the design space. In addition, there are some innate design characteristics that are useful in wind farm design. First, the layouts produced are regular, aesthetically pleasing patterns. To the untrained eye, BG parameterization looks well designed compared to the seemingly random layouts that are often produced when every turbine location is optimized individually. This can play an important role in the public perception of large-scale wind energy. Second, BG parameterization has clear roads or shipping lanes naturally built into the design. Roads and shipping lanes are requirements in wind farm design that are often neglected in research studies.

Often, there are prohibited areas within a wind farm. This could be for many reasons, such as natural geography, roads or shipping lanes, or a variety of other reasons. Although beyond the scope of this paper and not addressed in the results shown in Sect.

In this paper, we have presented the new boundary-grid wind farm layout parameterization method. This method uses only five design variables, regardless of the number of wind turbines but is capable of producing turbine layouts that perform just as well as or better than layouts where the location of each wind turbine has been optimized directly. We optimized the layout of seven different wind farms with three different parameterization methods: a simple grid, directly optimizing the location of each turbine, and our new boundary-grid parameterization. For each wind farm and parameterization method, we ran 100 optimizations with randomly initialized design variables. In every case, the best layout achieved with the BG parameterization perform slightly better than the best layout achieved with the direct optimizations.

In addition to being able to match the optimal energy production of wind farms that were directly optimized, BG parameterization requires an order of magnitude fewer function calls to reach a solution. This is with exact-analytic gradients, which means if finite-difference gradients or a gradient-free optimization method were used instead, our parameterization method would require at least 2 to 3 orders of magnitude fewer function calls to optimize. BG parameterization also reduces the multimodality of the design space, simplifying the optimization process and making it easier to find a good solution.

The BG layout definition places a portion of the wind turbines around the boundary, spaced equally traversing the wind farm perimeter. The rest of the turbines are placed in a grid inside the farm boundaries. The wind farm layouts created have a regular, aesthetically pleasing pattern, naturally defined roads and shipping lanes, and an easily defined cabling pattern. BG parameterizations solve many of the problems that typically accompany wind farm layout optimization. It is a simple, easily implemented technique that can immediately be applied by researchers and wind farm developers, playing an important role in the continued growth of wind energy.

The code written for this paper is included at

APJS led this research, including designing and testing potential parameterization methods, running the optimizations, and writing the paper. AN helped develop ideas and methodology, provided feedback throughout the entire process, and provided editing for the paper.

The authors declare that they have no conflict of interest.

This paper was edited by Rebecca Barthelmie and reviewed by Sebastian Sanchez Perez-Moreno and Ju Feng.