Layout Optimization Algorithms for the Offshore Wind Farm with Different Densities Using a Full-Field Wake Model

Abstract

To decrease the power deficit of a wind farm caused by wake effects, the layout optimization is a feasible way for the wind farm design stage. A suitable optimization algorithm can significantly improve the quality and efficiency of the optimization process. For exploring the high-performance algorithms under different layout densities, a comparison is conducted by optimizing the layout of a real offshore wind farm with five algorithms, namely two population-based algorithms and three single-point algorithms. Wake effects are considered by a full-field wake model. A penalty function is proposed for the population-based algorithms to handle the constraint violations. Different iterations and constraints of the layout density are applied in the optimization. The random search has the best optimization results in all the cases and the control of the feasibility check is necessary for this algorithm. More iterations can advance the optimization results. The density constraint greatly affects the computational cost of the random search, which is significantly increased under the strict constraint. Except under the strict constraint, the random search has the best performance of optimization efficiency. A combination of the pattern search and random search is recommended when the strict constraint is applied in the layout optimization.

1. Introduction

The climate crisis is a great challenge for human society of today. Reaching net-zero emissions is our hope to tackle this. It requires a total transformation of our energy systems from fossil fuels to renewable energy such as wind. Power losses are an unavoidable problem in most wind farms. Wake effects are one of the most important reasons, which appear in the downstream side of wind turbines leading to velocity deficits. The power losses will reach about 23% of the wind turbine’s power output without wake effects in a crowded offshore wind farm, and the second row of wind turbines will almost lose 70–80% power compared to the first row [1].

Optimization techniques have been widely used in various fields such as manufacturing [2], transportation [3], and energy systems [4] to improve efficiency and reduce costs. The layout optimization of a wind farm provides a feasible way to minimize the power loss at the design stage. It has benefited from the development of analytical wake models and optimization algorithms. This optimization problem has firstly been investigated by Mosetti et al. [5] utilizing the Jensen wake model [6,7] and a genetic algorithm (GA). Due to the flexibility of general-purpose algorithms, such as the GA and particle swarm optimization (PSO), they and their advanced versions have been widely used in the layout optimization problem. Beyer et al. [8] studied this problem using the GA and the PARK code which is based on the Jensen model. Grady et al. [9] and Şişbot et al. [10] advanced the work of Mosetti et al. by the GA. In the earlier studies [9,10], the design variables were discrete including the number and positions of turbines. The turbine was configured in the centre of discrete grids of wind farms in the optimization. The other way is continuous configuration in the optimization, which is based on coordinates [11,12]. Kirchner-Bossi and Porté-Agel [11] applied a GA to optimize the layouts of two real wind farms with a Gaussian wake model. Shin et al. [12] optimized the layout of an offshore wind farm by the PSO and evolutionary algorithm (EA). The variables included the shape and positions of the wind farm in the optimization. Hwang et al. [13] proposed a non-dominated sorting genetic algorithm (NSGA2) for the layout optimization by the Larsen model. In terms of objective functions, most of the studies considered the annual energy production [12], cost of energy [13], or power production [11], while a few studies considered also other aspects. Farajifijani et al. [14] presented an optimization model using the fmincon solver in the MATLAB optimization toolbox. With the analysis of conditional value at risk (CVaR), the production of a wind farm was guaranteed at a certain confidence level. Zhang and Jiang [15] introduced an improved quantum-behaved particle swarm optimization (IQPSO) algorithm for the layout optimization with a bi-level model, which considered the interaction between the wind farm and connected grid. Environmental constraints including an island and a channel were considered in the optimization. A non-genetic evolutionary algorithm (NGEA) was implemented by Gonzalez-Rodriguez et al. [16] to optimize the layout of a real wind farm. The size and shape of the wind farm were also considered as variables in the optimization. In addition to general algorithms, some customized algorithms have been developed targeting at the layout optimization problem. The local search (LS) algorithm was performed by Wagner et al. [17] for the layout optimization. The basic operation of the LS is to move a turbine step by step to determine a position for improving the objective function. The turbine displacement is random but also takes into consideration the closest turbines and previous movements. Feng and Shen [18,19] proposed the random search (RS) algorithm for the layout optimization. The RS algorithm has a similar basic idea to the LS algorithm. Although the former RS [18] determined a new position completely randomly, the new version [19] is improved by considering the previous movements. Liang and Liu [20,21] optimized an offshore floating wind farm composed of self-adaptive platforms using a full-field wake model [22] and the RS algorithm.

The optimization algorithm is one of the important elements of the layout optimization. Brogna et al. [23] carried out a comparison of optimization results of eight optimization algorithms in a real wind farm in the complex terrain and pointed out that the pattern search (PS), LS and RS have good performance in terms of computational cost and optimization results. Croonenbroeck and Hennecke [24] compared six algorithms with a unified framework with two groups of computation time and numbers of turbines. They derived that the computation time, layout efficiency, and farm profit are three criteria for comparison. The wind farm boundary and minimal distance requirements are common constraints considered in the layout optimization problem [23]. With the increasing size of wind farms [25], hundreds or even thousands of wind turbines are installed in a wind farm. In addition, the concept of large-scale floating platforms has emerged in the offshore wind farm [20,21]. In these wind farms, layout densities may be diverse due to different turbines or platforms installed which have different properties and requirements. It leads to the degree of constraints being different in the layout optimization, which affects the performance of algorithms. However, this issue lacks research in the earlier works, as they focus on a single layout density [23]. Although the high performance of the RS algorithm has been shown in many works [19,20,21,23], the parameters of this customized algorithm need further investigation to obtain a better optimization result. Moreover, as a popular choice in the layout optimization study [23,24], the accuracy of the Jensen model has a gap compared to the reality [22], while the optimization results are strongly affected by the wake model [11]. With these considerations, a comparative study is carried out in this paper with five algorithms to analyse the performance of different algorithms under different layout densities. The layout of a real-life offshore wind farm is optimized with different densities. A full-field wake model with high accuracy and efficiency is utilized to account for the wake effects. A penalty function is proposed to deal with the constraint violations. The parameter of the RS is investigated and its effect on the optimization results and efficiency is pointed out. The high-performance algorithms are identified under different layout densities. A highly efficient method of layout optimization is presented for a high degree of constraint with a hybrid optimizer.

This paper is organized as follows: Section 2 introduces the optimization methodology; Section 3 presents the optimization algorithms for comparison; the optimization results are shown in Section 4; and Section 5 gives the final conclusions.

2. Optimization Methodology

2.1. Wake Model

A full-field wake model, proposed by Liu et al. [22], is applied in the present work to evaluate the wake velocity in the wind farm with an arbitrary layout density. It has a better applicability and performance compared with the widely used Jensen model by integrating the modified near-field and far-field wake models. The modified model is expressed by

$$u_m=\frac{\pi \left(u_0-u_{*}\right)}{2}\cos \left(\frac{\pi }{2}\frac{r}{r_d}+\pi \right)+u_0$$

(1)

where, u_m is the wake velocity after modification; u₀ is the freestream wind velocity; $u_{*}$ is the wake velocity calculated by the model before modification; r is the radial distance from the rotor shaft of the wind turbine to the calculated point; and r_d is the wake radius at the calculated point. For the velocity deficit near the centre of the near field, it is considered as a constant distribution. The length of the near field is taken as 3 rotor diameters. The full-field wake model shows a well accuracy in the comparison [21,22] and is adequate for the layout optimization problem.

The superposition of multiple turbine wakes is modelled by the model of sum of squares, which has a well accuracy confirmed by Tian et al. [26], expressed by

$$\frac{u}{u_0}=1-\sqrt{{\displaystyle \sum _i^n{\left(1-\frac{u_i}{u_0}\right)}^2}}$$

(2)

where, u is the superimposed wake velocity; u_i is the wake velocity of the calculated point influenced by the turbine i alone; and n is the number of turbines impacting the calculated point.

2.2. Optimization Problem

In the present work, the primary aim is to carry out a systematic comparison of several algorithms and assess their performance for the application of wind farm layout optimization. Thus, the goal of the layout optimization is finding a specified layout (X, Y) to maximize the power output P of the wind farm. The problem is expressed as

$$\underset{\mathit{X},\mathit{Y}}{\max } P=f\left(\mathit{X},\mathit{Y}\right)$$

(3)

$$s.t.\{\begin{array}{l}\left(x_i,y_i\right)\in B\\ \sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}\ge d_{\min }\\ i,j\in \left\{1,2,\dots ,n_t\right\}, \forall i\ne j\end{array}$$

(4)

where, x_i, y_i, and x_j, y_j are the positions of wind turbine i and j, respectively; B is the set of available position of wind turbines; d_min is the minimum distance between turbines; and n_t is the number of turbines in the wind farm.

2.3. Penalty Function

There are two kinds of constraints in the optimization, i.e., the boundary constraint and distance constraint. The boundary constraint describes the boundaries and obstacles such as the channels or reefs in the wind farm which is formulated by the upper and lower bounds and the linear inequality constraints. The distance constraint limits the distance between turbines in the wind farm to ensure safety, formulated by nonlinear constraints. Due to some algorithms only handling the simple constraints, a penalty function f_penalty is proposed to consider the violations of complex constraints such as the linear inequality constraint and nonlinear constraint, expressed by

$$f_{penalty}=k\cdot {\displaystyle \sum _1^{n_s}\mathrm{\Delta }{d}_i^2}$$

(5)

where, k is the penalty coefficient which is 100 in the present work; n_s is the number of violations; and Δd_i is the exceeding distance in the constraint formula i. By squaring the constraint violations, larger violations are penalized more heavily.

By integrating the penalty function and the original objective function, an augmented objective function is expressed by

$$f^{\prime }=f\left(\mathit{X},\mathit{Y}\right)-f_{penalty}$$

(6)

By adding the penalty function, the problem becomes

$$\underset{\mathit{X},\mathit{Y}}{\max } P=f^{\prime }\left(\mathit{X},\mathit{Y}\right)$$

(7)

$$s.t.: x_i\in \left[x_{\min },x_{\max }\right], y_i\in \left[y_{\min },y_{\max }\right], \forall i\in \left\{1,2,\dots ,n_t\right\}$$

(8)

where x_min, y_min and x_max, y_max are the lower and upper bounds, respectively. The optimization algorithm will attempt to maximize the objective function while also satisfying the linear inequality constraints and nonlinear constraints. This penalty function can ensure that all violation distance will not exceed 1 m and the majority of them less than 0.01 m in the following optimizations.

3. Optimization Algorithm

3.1. Population-Based Algorithm

GA and PSO are both population-based optimization algorithms, which are widely used in the wind farm layout optimization [25,27]. They applied in the present work are provided by the Global Optimization Toolbox of MATLAB. No initial value is required in both optimizers. The GA in the Toolbox supports customized bounds of design variables, linear inequality constraints and nonlinear constraints, and the PSO can only handle customized bounds of design variables. However, the GA cannot provide feasible results satisfying the boundary and distance constraints applied in this study. Therefore, the penalty function is implemented in the GA and PSO. The default parameters used here are provided in

Table 1. Default parameters of GA and PSO.

Algorithm	Population/Swarm Size	MaxIteration	MaxStallIteration
GA	200	16,000	50
PSO	100	32,000	20

.

3.2. Single-Point Algorithm

The single-point optimization algorithms used in this study are the global search (GS), PS and RS. The GS generates start points by a scatter-search mechanism and uses a gradient-based local solver [28]. The solver used here is the interior-point algorithm (default) which is the only gradient-based algorithm used in this study. The PS is a direct search algorithm which means it does not require the gradient [29]. It searches the point by a mesh around the current one to approach the optimal answer. The GS and PS are also provided by the Global Optimization Toolbox of MATLAB. The initial values need to be provided in the optimization at first. The default parameters used here are provided in

Table 2. Default parameters of GS and PS.

Algorithm	MaxFunctionEvaluation	MaxIteration
GS	3000	1000
PS	320,000	16,000

.

The RS is a customized algorithm proposed by Feng and Shen [19] specifically for the wind farm layout optimization. In a single iteration, it comprises three steps: random move, feasibility check, and layout evaluation and update. In the first step, a wind turbine is select randomly in the wind farm to adjust its position in a random direction with a random distance ΔS

$$\mathrm{\Delta }S=k_s\cdot k_{rand}{L}_{\max }$$

(9)

where, k_s is the movement coefficient; k_rand is the random coefficient from 0–1; and L_max is the length of the long edge of the wind farm. The movement direction will not update if the power output of the wind farm were improved in the last iteration. In the second step, the new layout is checked by the constraints. The first step will be repeated if the new layout were infeasible. A limitation is applied here to adjust k_s and constrain the number of feasibility check n_check in a single iteration. k_s is 1 and n_check is unknown in the original algorithm. The following settings were used in the earlier works [20,21]

$$k_s=\{\begin{array}{l}1, n_{check}\le 2n_t\\ 0.5, n_{check}\le 3n_t\\ 0.25, n_{check}\le 4n_t\end{array}$$

(10)

and the check was terminated if n_check were exceed 4n_t. The default iteration and k_s of RS are set as 30,000 and 1, respectively. The values of k_s and n_check are further discussed in the next section.

In the present work, the RS is implemented in MATLAB. All algorithms run parallelly in MATLAB with six cores except the RS.

4. Layout Optimization

4.1. Case Settings

The Horns Rev I offshore wind farm [30] has 80 Vestas V80 2 MW wind turbines and a rated capacity of 160 MW. The wind turbine has an 80 m rotor in diameter D and a hub height of 70 m. Figure 1 shows the original layout of the wind farm. The distribution of wind turbines is a rhomboid shape with 8 rows along the x-axis direction and 10 rows along the y-axis direction (a deviation of approximately 7°). Each row has a 7D space in the axis direction.

The boundaries of the wind farm are formulated as

$$\{\begin{array}{l}3920x_i+481y_i\ge 1885520\\ 3920x_i+481y_i\le 21642320\end{array}$$

(11)

$$\{\begin{array}{l}{x}_i\in \left[0,5521\right]\\ y_i\in \left[0,3920\right]\end{array}$$

(12)

A fitted power curve of Vestas V80 2 MW wind turbine, proposed by Liu et al. [22], is utilized to evaluate the power output of wind turbines. It is expressed as

$$P=-0.73u_0^4+20.71u_0^3-186.38u_0^2+776.75u_0-1198.2$$

(13)

where u₀ is from 4–11 m/s.

An ideal condition, a steady inflow wind, along the positive direction of the x-axis, is considered in the optimization. This is a commonly used condition in the layout optimization to assess the performance of algorithms [19,25]. The necessary parameters are summarized in

Table 3. Parameters of the wind farm and wind turbine.

Parameter	Value
Inflow velocity u0 [m/s]	8
Freestream turbulence intensity I0	0.07
Surface roughness height z0 [m]	0.003
Thrust coefficient Ct	0.806

.

For the comparison, the optimization algorithms are applied with four settings. In the setting 1, the d_min is 4D and the options are customized as few as possible to execute the algorithms by default. This is a baseline setting. In the setting 2, more computational time are used for the algorithms by extending the maximal number of iterations and the size of population or particles. In the settings 3 and 4, similar to the setting 1, default options are used but d_min is 2D and 6D, respectively. The settings are summarized in

Table 4. Parameters of the setting.

Setting	dmin	Iteration and Population/Particle
1	4D	default
2	4D	10 times default
3	2D	default
4	6D	default

.

4.2. Effect of the Limitation on n_check

The key to the RS algorithm is the random movement of the wind turbine positions to maximize the power output of the wind farm by finding the optimal layout. However, the movement must satisfy the constraints which are verified by the feasibility check. The n_check is a control parameter in this step. The RS is applied to optimize the layout of the Horns Rev I wind farm under the setting 1 with different limitations on n_check. The optimizer runs 10 times under each limitation. The results are shown in Figure 2.

With the increasing value of limitations, as shown in Figure 2, the average of optimized power outputs is increased at first and then reduced, while the computational cost keeps increasing. Due to the property of the RS, the optimization results have a fluctuation even under the identical option. The fluctuation of mean P is low and it is probably from the fluctuation of the RS itself. However, the growth of computational cost is definite. The maximum computational time is triple the minimum one and obtains the minimum optimization results. Therefore, the limitation of n_check is necessary. Otherwise, the expensive computational cost may be wasted on invalid movements and an inferior optimization result. The limitation of n_check is set as 4n_t in the following optimization.

4.3. Comparison of Algorithms on Setting 1

The original layout shown in Figure 1 is optimized by five algorithms, i.e., GA, PSO, GS, PS, and RS. Each algorithm was run 10 times. Figure 3 shows the optimization results and computational time. The power output is normalized by the output P₀ of the original layout. In addition, the computational time is normalized by the mean computational time t_GA of GA.

With the application of the penalty function, both GA and PSO have an acceptable optimization result. PS and RS have the best optimization results and the result of RS is a little better. The result of RS is about 7% more than that of GA. GS obtains the worst result among all algorithms. The two population-based algorithms have a high computational cost. GA has the highest cost and fluctuation of time. As a gradient-based algorithm, the cost of GS is better than the two former algorithms, but it is not the best one. RS has the shortest computational time which is even about half that of PS and only needs 5% that of GA.

The results show that the PS and RS have a well performance, especially the RS. The RS has the best performance in terms of the optimization result and computational time. The GS has a bad performance under the setting 1.

The best optimized layout with the most power output is shown in Figure 4, which is from RS. The optimized layout with the most power output of GS is also shown in Figure 4 for a comparison. As shown in Figure 4a, the wind turbines are split into two parts as far as possible in the wind direction. Compared with the original layout, the optimized one shown in Figure 4a has more space for the wake recovery. In Figure 4b, the optimization by GS is far worse than that of RS. It is still much room for improvement.

The Jensen model is also used in the optimization. Optimization results and computational time are shown in Figure 5, where P₀ is estimated by the Jensen model and t_GA is the time of GA with the Jensen model. The best optimized layout is shown in Figure 6. The results show that different wake models have little impact on the relative characteristics of algorithms. RS achieves the best result and efficiency. GA requires the longest time. GS results in the lowest power output. However, wake models differ in their accuracy and efficiency. These performance differences significantly affect the layout optimization results and efficiency. This has also been concluded by Kirchner-Bossi and Porté-Agel [11]. The optimized power output decreases 2–16% and the computational time increases 20–100% compared to those of RS, but the cost of PSO decreases 36%. The optimized layouts show clear differences between different models.

4.4. Expansion of Computational Time

Under the setting 2, more computational time is allocated to the algorithms. Each algorithm has been run 3 times. The comparison of optimization results and computational time is shown in Figure 7.

All algorithms have an improved optimization result after expansion. GS has the greatest improvement, which is about 30% compared to the result under the setting 1. Except GS, the improvement of other algorithms is about 1–3%. On the contrary, the increase in computational cost is significant. All single-point algorithms have a nearly linear growth of computational time.

The results imply that all algorithms can show an acceptable performance in the face of a normal layout optimization problem with the default settings except GS. The GS need more iterations than the default. Under the setting 2, the performance of GA is the worst one which has the lowest optimized power output and the highest computational cost. RS still has the best performance in terms of both optimization results and computational time.

The comparison of optimized layouts is shown in Figure 8. The shown layout is the best one of each algorithm under each setting. Compared with the layout under setting 1, the layout of RS under setting 2 is further split. More turbines are concentrated on the split parts and there are more room for wake recovery. Although more computational time is allocated to GA, the optimized layout is still a nearly scattered configuration which has a great difference compared to the result of RS. It demonstrates that GA is trapped in local optima. The layout of GS under setting 2 shows a trend to split but it is still worse than the layout of RS. Considering the computational cost, GS has a low efficiency in the layout optimization. Low computational efficiency and getting trapped in local optima are the reasons of the differences shown in Figure 7 between different algorithms.

4.5. Different Layout Densities

Due to the different structures of wind turbines installed in the wind farm, the layout may be used different densities to be configured in the optimization to satisfy the different properties and requirements of these structures. Different densities mean different constraints of distance between turbines, i.e., d_min. A greater d_min means that there is less space to adjust the position of wind turbines in the wind farm. Therefore, the constraint is more stringent in this situation. Each algorithm has been run 10 times in the optimization. The optimization results and computational time under setting 1, 3, and 4 are shown in Figure 9 with different d_min. As shown in Figure 9a, the difference of optimization results between GA and PSO gradually decreases with the reduction in d_min. When d_min decreases to 2D under the setting 3, the optimized power output of GA comes from behind compared to that of PSO and the performance of PSO becomes more unstable in terms of optimization results with more deviations in the multiple rounds of execution. The optimization results of RS are still the best which are about 7–14% more than those of GA and a little better, about 1%, than those of PS. Figure 9b demonstrates that computational time of GA, PSO, and RS is more sensitive to the setting of d_min, especially RS. Under the strict condition, i.e., the setting 4, the computational cost of RS is increased more than 4 times compared to that under the setting 1, which is nearly 2.5 times the cost of PS. Therefore, the PS is a better choice in this situation.

The layouts with the best optimization results of the setting 1, 3, and 4 are shown in Figure 10, which are all obtained by RS. With the reduction in d_min, the wind turbines are gradually concentrated on the upstream and downstream edges of the wind farm. The space for wake recovery becomes larger and larger. It is the reason why the optimized power output is increasing as the decrease in d_min.

The comparison above demonstrates that the performance of RS is wonderful and very competent in the most cases except the strict conditions. The optimization method of RS is to adjust the position of wind turbines one by one. Therefore, RS can explore the search space more uniformly and widely to avoid getting trapped in local optima or missing some promising regions. Moreover, due to its specific purpose for handling the layout optimization problem, RS requires no complex parameter to tune compared to other general algorithms. These characteristics make RS efficient and easy to implement than other algorithms. However, this does not mean that RS is always superior to other algorithms. A strict condition probably becomes a big obstacle to find a feasible position of the wind turbine in the movement step for RS. A method has been proposed in the earlier work [20], as shown in Equation (10), to make the movement step easier by reducing the distance of the movement step when the wind farm is full of wind turbines. On the other hand, Feng and Shen [19] introduced that the RS can be used to improve the results of other algorithms. In the following optimization, the RS is applied to refine the results of PS under the setting 4 and the results are shown in Figure 11.

RS-1 is the RS under the default options, but Equation (10) is applied. RS-2 is the RS under the default options, but the iteration is 10,000 to refine the result of PS. RS-3 is similar to RS-2, but the iteration is twice. Compared to the optimization results of RS and PS, the results of the hybrid optimizer improve the power output about 1–2% and save the cost of computational time by up to 30% at most. It is a better method than the application of Equation (10) under a strict constraint.

5. Conclusions

In the present work, the layout of an offshore wind farm is optimized by five algorithms. A full-field wake model is applied to evaluate wake effects in the wind farm. The performance of algorithms is compared in terms of optimization results and computational time. A penalty function is proposed for GA and PSO to strengthen their capabilities to handle the complex constraints. The options of RS are investigated. Different computational time and constraints of layout density are used for the comparison of algorithms. Main conclusions are summarized as follows:

(1)

The RS has the best performance in terms of optimization results and computational cost in the most cases, which is up to 14% more than the results of GA but save 95% of time, except the strict constraint. The PS is the second one with about 1% gap of results and a double cost compared to RS, but it can save 60% time than RS under the strict constraint.

(2)

The combination of PS and RS is a good method to improve the performance of both algorithms under the strict constraint. This hybrid method has a 1–2% improvement in the optimization results and saves 30% computational cost at most;

(3)

The application of the penalty function is a feasible method to strengthen the ability of algorithms to deal with complex constraints. With the penalty function, all algorithms can solve the layout optimization problem under the default options, but GS needs more iterations to obtain acceptable results, which has a 30% improvement with 10 times more iterations than the default;

(4)

In the RS, the limitation of n_check can improve the efficiency of RS. Invalid movements waste the cost of computational time. The discrepancy can reach up to three times. Due to the property of the fluctuation, the limitation depends on the specific problem.

Reducing the invalid movements can probably improve the efficiency of RS. It is the reason that the computational cost of RS is increased significantly under the strict constraints. Due to environmental limitations, except for the layout density, the possible irregularity of the wind farm boundaries is also a realistic scenario which may affect the performance of the optimization algorithm. Moreover, vertical axis wind turbines have fast recovery of wakes compared to the horizontal axis wind turbines [31]. The layout optimization of these turbines probably benefits from the present method due to their high layout density. Finally, the leading edge erosion is a critical problem for wind turbines and has a negative effect on the energy production [32,33]. The consideration of this effect is meaningful in the layout optimization. These aspects deserve further investigation in future works.