Two models and a heuristic algorithm to address the wind farm layout optimization problem are presented. The models are linear integer programming formulations where candidate locations of wind turbines are described by binary variables. One formulation considers an approximation of the power curve by means of a stepwise constant function. The other model is based on a power-curve-free model where minimization of a measure closely related to total wind speed deficit is optimized. A special-purpose neighborhood search heuristic wraps these formulations with increasing tractability and effectiveness compared to the full model that is not contained in the heuristic. The heuristic iteratively searches for neighborhoods around the incumbent using a branch-and-cut algorithm. The number of candidate locations and neighborhood sizes are adjusted adaptively. Numerical results on a set of publicly available benchmark problems indicate that a proxy for total wind speed deficit as an objective is a functional approach, since high-quality solutions of the metric of annual energy production are obtained when using the latter function as an substitute objective. Furthermore, the proposed heuristic is able to provide good results compared to a large set of distinctive approaches that consider the turbine positions as continuous variables.

Cost reductions for renewable energy generation are on the top of political agendas, with the objective of supporting the worldwide proliferation of clean energy production systems. Subsidy-free tendering processes are becoming more frequent, as has been the case for offshore wind auctions in Germany since 2017 and in Netherlands since 2018 or in China for onshore wind since 2021

The basic wind farm layout optimization (WFLO) problem aims at deciding the positioning of wind turbines (WTs) within a given project area to maximize the annual energy production (AEP), while respecting a minimum separation distance. The classic problem definition aims at placing a fixed number

The main components when building an optimization workflow for the WFLO problem are the wake models (deficit and superposition), the program formulation, and the associated numerical algorithms. For formulating tractable frameworks, the designer needs to rely on the so-called engineering wake models. These are essentially mathematical representations which can be expressed in terms of analytical equations after significantly simplifying complex physics modeling, while still capturing the underlying nature of the phenomenon under analysis to a good extent. Scientific articles in this field have proposed and validated engineering wake models with a smooth and differentiable velocity deficit shape; two examples are Bastankhah's Gaussian model

Optimization techniques for the WFLO problem formulation can be classified, depending on the choice of variables, into continuous and discrete optimization. In the field of continuous optimization, the location

The utilization of simplified objective functions closely related to more sophisticated AEP models is also an emerging research field for continuous gradient-based optimization. In the recent work of

Discrete optimization models can be formulated for this problem by means of sampling the available project area in the form of

A large number of benefits are implicit in the discrete modeling technique over the continuous counterpart, including, among others, (i) the capacity to include the number of WTs as a variable and to model overall economic metrics as net present value (NPV)

Probably the first work within the context of integer programming for the WFLO problem was the thesis of Fagerfjäll in 2010

Several contributions to the field of discrete optimization for WFLO are proposed in the paper. The first contribution is the proposition of new integer linear formulations which are able to capture the underlying physics of the problem to a good extent. The main obstacles for a MILP representation of WFLO problem are the non-linearity of the power curves and the choice of a wake velocity deficit superposition approach. Currently, the scientific literature has fundamental knowledge gaps. For example, as discussed before, previous works have considered aggregation of power deficits instead of velocities, gaining a simplification of the mathematical formulation in detriment to the physics modeling fidelity. This paper presents new strategies for modeling both facets of the class of MILP problems, one with an explicit power curve and wake superposition modeling and another with a proxy objective function based on total wind speed, thus simplifying the original formulation. In contrast to

The second main contribution is the proposition of a new special-purpose neighborhood search heuristic in order to speed up the generation of high-quality solutions. This heuristic, wrapping both formulations, has a twofold functionality: first to increase tractability and second to redirect the optimization search in terms of a specified objective function with higher fidelity. Similar neighborhood search methods have been proposed in the literature, such as the discrete exploration-based optimization (DEBO)

The rest of the paper is structured as follows. Section

The proposed MILP models and general optimization framework in this paper can be easily applied to many wake deficit models. No particular properties on smoothness or differentiability are required from these models for optimization purposes. Additionally, no specific demands on mathematical structure in connection with the controlling wake diameter and deficit

A simplified version of Bastankhah's Gaussian model is considered

The absolute velocity deficit

Let the set

The wake velocity deficit superposition

Suitable power curves are required for computing the AEP. Often, power curves are not perfectly suitable for optimization, due to the usual non-differentiability in several points throughout the function. Generally, a power curve is 0 below cut-in wind speed, 0 above the cut-out wind speed, and constant between the rated wind speed and the cut-out wind speed. In this particular study, between the cut-in and rated wind speeds the curve is assumed to be smooth, convex, and monotonically increasing. The simplified power curve for a generic turbine as a function of wind speed

The AEP is calculated as

The MILP program with explicit modeling of the WT power curve, wake deficit, and wake superposition is introduced in Sect.

The main type of variables

Let the index sets

All relevant distances can be preprocessed for all combinations of points

Continuous state variables

The power curve is approximated with a stepwise function. The cubic part of the power curve is first partitioned into

An interval

Piecewise constant approximation of a wind turbine power curve through sampling with

Let binary state variables

With all the variables of the model – activation variables

This program collects the AEP objective function, the constraints of a generalized version of the WFLO problem, and the variables' domain definition. The objective function in Eq. (

Although the formulation of Sect.

The power-curve-free model introduces a strategy to account for the combination of Eqs. (

Combining Eqs. (

Nonetheless, the presence of variables

The new objective function in Eq. (

Compared to

For addressing large-scale problems, a heuristic wrapping the MILP formulations given in Sect.

The first three lines are the main inputs of the algorithm: the candidate set

The next step is to start the iterative process in line 6. Values for

Neighborhood search heuristic (NSH) algorithm. Optimization: op.

After solver termination, solution pool

For a transparent benchmark of the proposed methods, the open-access case studies from IEA Wind Task 37 in

The results of the statistical correlation between the proxy function given by the argument in Eq. (

The main parameters of the wake model in Sect.

The experiments in Sect.

The selected MILP solver is the commercial branch-and-cut algorithm implemented in IBM ILOG CPLEX Optimization Studio V20.1

The number

Wind rose used in the computational experiments. Taken from an open-access source

Example of generation of WT candidate locations

In Fig.

To validate the approach modeled by the MILP formulation of Eq. (

In all cases Pearson product-moment linear correlation coefficients from

The correlation between the AEP and the total theoretical wind speed is shown in Fig.

Pearson product-moment linear correlation coefficients for all case studies.

Correlation plots for 5000 randomly generated wind turbine layouts for Case I.

The general trends of the correlation plots for Case II are very similar. Correlations between the AEP and the total theoretical wind speed (

The very strong linear relation between the AEP and the total theoretical wind speed (

Likewise, correlations stemming from the proxy to calculate the total wind speed deficit are lower in Case III. This is the case for both with the total theoretical wind speed (

This case has a round shape with a radius of

The results of the NSH computing time in Fig.

The initial layout (point 1), labeled in Fig.

Performance of two different optimization approaches for Case I and comparison with existing best benchmark results. See Table

The next considerable jump happens for

The benefit of the proposed neighborhood search strategy is shown in Fig.

The initial and final solution layouts for this case study are illustrated in Fig.

Finally, Table

The third column of Table

Generated wind farm layouts for benchmark Case I with 16 turbines.

Results for all three benchmark cases from other algorithms (G, gradient-based; GF, gradient-free) obtained while allowing for WT locations to vary continuously. Values reproduced from

The power-curve-based model of Eq. (

This case has a round shape with a radius of

Performance of two different optimization approaches for Case II and comparison with existing best benchmark results. See Table

After a 3 h plateau linked to

The full model (i.e., without implementing the NSH algorithm) initially provides better solutions within the first

For this case, the proposed method reaches the best solution, as shown in the fifth column of Table

Generated wind farm layouts for benchmark Case II with 36 wind turbines.

This case has a round shape with a radius of

Comparing the blue lines of Figs.

Performance of two different optimization approaches for Case III and comparison with existing best benchmark results. See Table

Generated wind farm layouts for benchmark Case III with 64 wind turbines.

After point 13 in Fig.

The seventh column of Table

Although in most projects today the total capacity for grid connection is already decided in the early planning phases, in the future one can envisage situations where flexibility in optimizing the number of wind turbines in a project would yield benefits.

Even if the power-curve-free model (Sect.

For such an optimization, the power-curve-based mathematical program of Sect.

Keeping that in mind for this case, a linear superposition model for the AEP component in the NPV calculation is considered. In this sense, the original WT power curve as depicted in Fig.

The model of Eq. (

Evolution of the AEP, NPV, and number of WTs for the three simulations. The green lines are results for the optimization program with a fixed number of WTs equal to

When the number of turbines is fixed to

An interesting question is whether there is a larger NPV in between the bounds of the WT number. For the optimization program with a variable number of WTs, the evolution of the WT number in Fig.

This result shows the benefit of having optimization models that support a variable number of WTs and accounting for metrics beyond the AEP. The advantages may become even more pronounced for more complex situations, such as, for instance, if the WT investment costs are dependent on the exact installation area or different WT sizes are considered.

The two models proposed in this paper have many of the characteristics of mixed-integer linear programming models. They require significant computational time and memory and exhibit rather low tractability and scalability for global optimization algorithms.

The power-curve-based model, although requiring more computational resources, manages to provide reasonably good solutions for a small-sized problem, being only 1.18 % lower than its power-curve-free counterpart for the case with 16 WTs and 4.41 % for the case with 36 WTs. This diminishing efficiency is to be expected, given the large number of variables and constraints. The power-curve-free model on the other hand, along with the heuristic, is much faster due to its more compact formulation. This translates into the ability to be highly competitive compared to a large set of benchmark algorithms. In situations where there is an interest in optimizing metrics beyond the AEP, such as the NPV, the power-curve-based model becomes very useful given its intrinsic capacity to support this kind of objective functions.

It should be mentioned that there are limitations for the wake models used compared to recent ones

Notwithstanding the listed shortcomings, it is enthralling that these models, in combination with the neighborhood search heuristic, are able to match and in some cases improve upon the results obtained when considering the turbine positions as continuous variables (see Table

This paper contributes both methodologically and empirically to address the WFLO problem. A neighborhood-search-heuristic-embedding integer programming formulation is proposed. For both presented formulations presented in the paper, the stepwise power curve and power-curve-free model, the heuristic notably improves a single execution of full models when calling a state-of-the-art branch-and-cut solver in terms of solution quality. An improvement of up to 3.42 % in the AEP is achieved by applying the neighborhood search strategy for cases where the WT number is fixed compared to solving the full model.

Another important takeaway is the satisfactory performance of the power-curve-free model, which uses an approximation of the total wind speed deficit, when (implicitly) optimizing for the AEP. This is due to the good correlation between the two measures and the correction capability of the heuristic. For the classic WFLO problem definition, the proposed model is able to considerably improve (from 1 % to around 10 %) the AEP compared to benchmark results by multiple gradient-based and gradient-free algorithms. Even when directly compared to methods implementing a continuous-variable technique, the proposed heuristic provides similar or even better results. These are very promising results that would enable getting high-quality solutions for problem instances where continuous-variable modeling approaches may not be able to run or provide with good incumbents.

Finally, the model with an explicit representation of the power curve embedded within the neighborhood search heuristic is able to propose non-trivial solutions when implementing objective functions beyond the AEP, such as the NPV. For these cases, the trade-off between energy revenues and investment costs is studied. For example, the model suggests that installing a lower number of wind turbines than allowed would result in a better NPV value, with a comparable AEP.

Information about the values of

Information about the values of

Information about the values of

Code and data sets are available upon request.

JAPR: conceptualization, methodology, software, validation, formal analysis, data curation, writing. MS: conceptualization, methodology, investigation, resources, supervision. NAC: validation, writing, supervision, project administration, funding acquisition.

At least one of the (co-)authors is a member of the editorial board of

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the Independent Research Fund Denmark (Danmarks Frie Forskningsfond) (grant no. 1127-00188B).

This paper was edited by Michael Muskulus and reviewed by Erik Quaeghebeur and one anonymous referee.