A State-of-the-Art Fractional Order-Driven Differential Evolution for Wind Farm Layout Optimization

Abstract

The wind farm layout optimization problem (WFLOP) aims to maximize wind energy utilization efficiency and mitigate energy losses caused by wake effects by optimizing the spatial layout of wind turbines. Although Genetic Algorithms (GAs) and Particle Swarm Optimization (PSO) have been widely used in WFLOP due to their discrete optimization characteristics, they still have limitations in global exploration capability and optimization depth. Meanwhile, the Differential Evolution algorithm (DE), known for its strong global optimization ability and excellent performance in handling complex nonlinear problems, is well recognized in continuous optimization issues. However, since DE was originally designed for continuous optimization scenarios, it shows insufficient adaptability under the discrete nature of WFLOP, limiting its potential advantages. In this paper, we propose a Fractional-Order Difference-driven DE Optimization Algorithm called FODE. By introducing the memory and non-local properties of fractional-order differences, FODE effectively overcomes the adaptability issues of advanced DE variants in WFLOP’s discreteness while organically applying their global optimization capabilities for complex nonlinear problems to WFLOP to achieve more efficient overall optimization performance. Experimental results show that under 10 complex wind farm conditions, FODE significantly outperforms various current state-of-the-art WFLOP algorithms including GA, PSO, and DE variants in terms of optimization performance, robustness, and applicability. Incorporating more realistic wind speed distribution and wind condition data into modeling and experiments, further enhancing the realism of WFLOP studies presented here, provides a new technical pathway for optimizing wind farm layouts.

1. Introduction

Against the backdrop of the global energy transition, the promotion and application of renewable energy are advancing at an unprecedented pace. Faced with the increasingly depleted resources of traditional fossil fuels and the profound impact of carbon emissions on the climate and environment, countries are actively seeking clean and sustainable energy solutions. Wind energy, as an efficient and environmentally friendly renewable energy source, occupies a significant position in the adjustment of the energy structure [1,2]. With its low-carbon characteristics and wide adaptability, wind power has become a key pillar of future energy systems. Thanks to advances in wind power technology and the support of related policies, the share of wind power in global electricity supply continues to rise, becoming a core component of many countries’ energy strategies [3,4]. However, the further promotion of wind power still faces significant challenges, including the high initial construction cost, the dependence of the power generation process on natural conditions, and energy losses caused by the wake effect between wind turbines [5,6,7,8]. Therefore, the wind farm layout optimization problem (WFLOP) has emerged, providing technical means and theoretical foundations to address these issues. The core objective of WFLOP is to improve wind energy utilization efficiency through scientifically optimizing the layout of wind turbines, thereby mitigating the adverse impact of the wake effect on power generation efficiency. Due to its complexity and multi-objective characteristics, WFLOP has gradually become a hotspot for interdisciplinary research. WFLOP not only involves the integration of fluid dynamics modeling and optimization algorithms but also requires the comprehensive consideration of various factors such as terrain constraints, land use, and maintenance costs. This multi-dimensional and multi-variable optimization demand not only makes WFLOP of significant practical engineering value but also provides a unique application scenario for computational intelligence algorithms [9]. As a critical technological support for the development of clean energy, the research outcomes of WFLOP not only contribute to the economic optimization of wind farms but also offer new perspectives for addressing global energy shortages [10,11].

The core challenge of WFLOP lies in the highly nonlinear and discrete nature of its optimization space. As a key influencing factor in this problem, the wake effect not only causes a significant decrease in wind speed behind the turbines but also triggers complex nonlinear dynamic changes within the wind farm, thereby profoundly impacting the overall power generation efficiency. In research, Jensen’s wake model has been widely used to describe the impact of the wake effect on wind speed and power loss due to its simplicity and applicability [12]. This model uses analytical expressions to effectively characterize the wind speed distribution in the wake region of wind turbines, providing a foundation for wind farm power evaluations. However, the propagation of the wake effect changes significantly depending on the relative positions of the turbines, and this spatial variability introduces strong nonlinear characteristics into WFLOP. At the same time, since the layout positions of wind turbines are often restricted to a predefined discrete set of points, this discreteness further increases the complexity of the optimization process. As a result, WFLOP embodies both nonlinearity and discreteness in its computational model, making it a highly challenging optimization problem. To address this problem, optimization algorithms must achieve a dynamic balance between global exploration and local exploitation while effectively handling the nonlinear complexity introduced by the wake effect. Traditional optimization methods, such as mixed-integer programming and classical heuristic algorithms, have achieved some success in early studies. However, their computational complexity and limited adaptability to discrete characteristics restrict their scope of application. Metaheuristic algorithms, with their flexibility and global search capabilities, have been widely applied in WFLOP [13]. Among them, Genetic Algorithms (GAs) and Particle Swarm Optimization (PSO) have shown advantages in matching the modeling requirements of WFLOP due to their discrete encoding approaches. However, they still face challenges in maintaining population diversity and achieving sufficient global search depth. In contrast, Differential Evolution (DE) algorithms, known for their efficient differential operations and strong local exploitation capabilities, perform exceptionally well in continuous optimization problems. Nevertheless, the continuous nature of DE introduces limitations when directly applied to WFLOP. For instance, DE tends to fall into local optima in discrete solution spaces [14,15].

To address the complex nonlinear and discrete optimization demands in WFLOP, metaheuristic algorithms have become essential tools due to their strong global search capabilities and adaptability. Genetic Algorithms (GAs) and Particle Swarm Optimization (PSO), with their flexible encoding approaches and inherent suitability for discrete problems, have been widely applied in WFLOP. For instance, as one of the most classical metaheuristic algorithms, GA simulates the process of natural selection and genetic evolution and has been extensively used to optimize wind turbine layouts to improve wind energy utilization efficiency [16]. Early studies achieved significant improvements in power generation efficiency through GA-based optimization of turbine layouts [17,18]. Statistical research indicates that the GA is the most widely studied and applied metaheuristic algorithm [19], largely because it can handle both binary and continuous encoding problems and its simple structure makes it suitable for resource-constrained industrial scenarios [20]. In recent years, variants of the GA tailored to WFLOP have been further developed. For example, Ju et al. proposed an Adaptive GA (AGA), which integrates the Jensen wake model and land-use constraints, showing significant improvements in WFLOP performance [12]. By adaptively adjusting genetic parameters, the AGA enhances the algorithm’s search efficiency to some extent. However, the optimization process of the AGA is still affected by the stochastic nature of genetic operations, and its parameter adjustment strategy, while improving adaptability, lacks global control over population diversity. This limitation often leads to premature convergence in large-scale turbine layout optimization. Additionally, Ju et al. developed the Support Vector Regression-Guided GA (SUGGA), which, as a highly efficient and widely applicable WFLOP optimizer, established and open-sourced a related benchmark library, significantly advancing WFLOP research and reproducibility [12]. However, SUGGA inherits the simplicity of the GA’s computational process and relies heavily on pre-generated layout data. This dependency limits its performance when tackling new WFLOP problems, as the absence of high-quality data hinders efficient guidance during the optimization process, thereby reducing its generalizability and stability. PSO has also achieved important results in WFLOP applications. For example, Lei et al. proposed an Adaptive Genetic PSO (AGPSO) method, which significantly improved optimization performance in WFLOP through adaptive replacement strategies [14]. PSO demonstrates high convergence efficiency by leveraging information sharing among individuals and collaborative search within the population, making it particularly suitable for low- and medium-dimensional optimization problems. However, the frequent replacement of inferior solutions in AGPSO, while enhancing local exploitation capabilities, to some extent limits the global exploration depth of the population, potentially leading to local optima. Meanwhile, chaotic PSO variants, which introduce chaotic mechanisms to enhance global search capabilities, have demonstrated state-of-the-art performance in WFLOP with 60 to 80 turbines [15]. Nevertheless, PSO’s heavy reliance on the diversity of search paths globally can result in decreased search efficiency when problem dimensions increase or nonlinearities intensify. Differential Evolution (DE) algorithms, known for their efficient differential operations and strong local exploitation capabilities, have shown significant advantages in continuous optimization problems [21]. However, DE was originally designed for continuous variable optimization, and its direct application to WFLOP often exhibits certain limitations. For instance, DE tends to fall into local optima in discrete optimization scenarios due to insufficient dynamic adjustment of the population. Yu et al. proposed the CLSHADE algorithm by introducing chaotic local search and demonstrated its potential performance improvement in WFLOP [22]. Although CLSHADE enhances local search capabilities and performs well in certain benchmark scenarios, its global search performance under complex wind farm conditions remains insufficient. Additionally, the differential operations of DE are highly dependent on the continuity of the solution space, making it difficult to balance exploration and exploitation in discrete optimization problems like WFLOP. Despite the extensive research into GA, PSO, and DE variants for WFLOP, these methods still face significant challenges in handling complex wind farm conditions and achieving globally optimal solutions. For example, while the GA’s stochastic genetic operations offer flexibility, its global search efficiency is constrained by the maintenance of population diversity. PSO’s collaborative search mechanism may lack sufficient exploration depth when dealing with multi-objective or high-dimensional optimization problems. Meanwhile, DE’s reliance on the continuity of differential operations limits its effectiveness in discrete optimization scenarios like WFLOP. Furthermore, the discrete nature of WFLOP and the nonlinear characteristics of the wake effect add additional difficulty to the optimization process. These challenges highlight the limitations of existing optimizers and underscore the need to design an algorithm that not only retains strong global search capabilities but also better adapts to WFLOP’s discrete and nonlinear features. Such an algorithm would be key to overcoming the inherent difficulties in this problem.

As a typical discrete optimization problem, WFLOP’s core challenges lie in the discreteness of its solution space and the nonlinear complexity caused by the wake effect. The discreteness restricts solutions to a finite set of points, preventing individuals from smoothly traversing the solution space via continuous paths [23]. Moreover, significant differences in objective values between neighboring solutions exacerbate the difficulty of escaping local optima [24]. In contrast, in continuous optimization problems, the solution space often exhibits infinite differentiability, enabling optimization algorithms to smoothly approach the global optimum by dynamically adjusting step sizes and leveraging gradient information [25]. This continuity allows for more flexible search paths and facilitates efficient exploration and exploitation of the solution space structure [26]. In discrete optimization problems, however, the solution space is composed of discrete point sets, making it difficult for optimization processes to utilize continuous information for gradual adjustments. As a result, search paths are more likely to stagnate in local regions. The nonlinear complexity of WFLOP further exacerbates the difficulty of discrete optimization, primarily due to the wake effect between wind turbines. This effect not only reduces the power generation efficiency of downstream turbines but also imposes higher demands on the search behavior of optimization algorithms due to its dynamic variability. Specifically, when the turbine layout is sparse or wind farm conditions are complex, the superposition of wake effects significantly increases the nonlinearity of the objective function. In continuous optimization problems, nonlinear objective functions can often be managed using local exploitation strategies, which adjust the distribution of the population or parameters to gradually approach the global optimum. However, in WFLOP, the combination of the wake effect’s dynamic nature and the discreteness of the solution space makes it difficult for optimization algorithms to balance global exploration and local exploitation, thereby degrading performance. Current applications of metaheuristic algorithms in WFLOP predominantly focus on those with natural discrete encoding properties (e.g., GA and PSO), while relatively few studies have explored the use of DE, which performs exceptionally well in continuous optimization scenarios. DE, with its efficient differential operations and strong local exploitation capabilities, has demonstrated outstanding performance in continuous optimization problems. For instance, in continuous solution spaces, DE can efficiently evaluate the relationships between multiple solution points through differential operations, thereby quickly locating potential regions of optimal solutions. However, DE’s core design is tailored to continuous optimization problems, making its optimization process difficult to adapt directly to the discrete characteristics of WFLOP. In WFLOP, DE’s differential update strategies often fail due to the discreteness of the solutions, resulting in insufficient population search capabilities and eventual stagnation in local optima. Moreover, DE’s limitations in maintaining population diversity and its global exploration capabilities further hinder its effective application in WFLOP. Thus, to address the challenges of WFLOP’s discreteness and nonlinearity, there is a need for an improved optimization algorithm that can simultaneously balance global exploration and local exploitation. Such an algorithm must not only retain the adaptability of metaheuristic methods to nonlinear problems but also redefine the optimization process for discrete solution spaces to overcome the performance bottlenecks of existing methods in WFLOP.

In continuous optimization problems, the Differential Evolution (DE) algorithm has been widely applied due to its strong exploitation capabilities and excellent local search performance, achieving notable success in single-objective, multi-objective, and large-scale optimization tasks [27,28,29]. Its efficiency has been extensively validated in numerous international competitions, such as the IEEE CEC Numerical Optimization Competition [21,30,31]. As a highly adaptive optimization algorithm, DE explores the solution space efficiently through its differential operations, and its dynamic population adjustment capabilities make it particularly effective in solving complex nonlinear problems. Notably, Linear Population Reduction-based Success-History Adaptation for Differential Evolution (LSHADE) has emerged as a representative and efficient variant of DE, exhibiting outstanding performance in challenging continuous optimization tasks and providing technical support for numerous optimization problems [30]. However, the optimization processes of DE and its variants are primarily designed for continuous optimization scenarios, limiting their direct applicability to the discrete nature of WFLOP. DE’s continuous optimization characteristics [32] often cause it to stagnate in local optima when applied to discrete solution spaces. Furthermore, its limitations in maintaining population diversity and enhancing global exploration capabilities become even more pronounced in WFLOP [14,15,33]. As a result, the application of DE in WFLOP has not been widely explored, and its potential performance advantages remain underutilized. To address DE’s limitations in adapting to WFLOP, this study proposes a Fractional-Order Differential Evolution algorithm (FODE) based on the LSHADE framework. Leveraging the memory and non-local characteristics of fractional calculus, FODE offers a novel approach to optimizing problems that combine discreteness and nonlinearity. In FODE, the dynamic population adjustment capabilities of classical LSHADE are preserved, while the introduction of fractional-order differential operations redefines the update strategy for individuals within the population. Fractional-order differences allow individuals to flexibly adjust their search paths in the solution space while incorporating information from both the current and historical generations to enable broader global exploration. This design effectively mitigates DE’s susceptibility to local optima in WFLOP due to discreteness and significantly enhances population diversity. Additionally, FODE integrates the Jensen wake model to precisely model wind farm wake effects, ensuring that optimization results are consistent with theoretical expectations and practical applications. By incorporating an adaptive fractional-order adjustment strategy, FODE dynamically balances global exploration and local exploitation. This ensures rapid convergence to high-quality solution sets in the early optimization stages while expanding the search range in later stages to escape local optima. Experimental results demonstrate that FODE significantly outperforms six state-of-the-art WFLOP optimizers, including GA, PSO, and DE variants, across 10 complex wind farm conditions. FODE achieves breakthroughs in maximizing wind power generation efficiency and exhibits robustness and stability under varying wind farm conditions. Statistical analyses and visualization results further validate FODE’s effectiveness and applicability in solving WFLOP, proving its capability to provide efficient and reliable solutions for wind farm layout optimization. In summary, FODE, by introducing fractional-order differences, not only retains LSHADE’s efficiency in continuous optimization but also achieves significant performance improvements in the discrete optimization context of WFLOP. This method provides a novel solution to complex nonlinear discrete optimization problems and holds significant potential for enhancing the efficiency and performance of wind farm layout optimization.

(1)

The FODE optimizer proposed in this paper incorporates fractional-order difference mechanisms to enable the effective application of the classical DE variant LSHADE in WFLOP, a problem characterized by both discreteness and nonlinearity. By dynamically balancing global exploration and local exploitation, it addresses the adaptability limitations of DE in WFLOP.

(2)

Based on the latest relevant research, this paper incorporates wind conditions and wind speed distributions that are closer to real-world scenarios into the experiments, aiming to obtain more practical and meaningful test results and conclusions.

(3)

Experimental and statistical analysis results show that the proposed FODE performs excellently in WFLOP, significantly outperforming state-of-the-art WFLOP optimizers (including the latest GA, PSO, and DE variants). It not only maintains high robustness but also achieves a comprehensive improvement in the upper limit of wind power generation efficiency.

(4)

The proposed wind farm optimization model, experimental wind condition data, and the relevant code for the FODE optimization algorithm will be open-sourced to support further research and reproducibility in the WFLOP field.

The structure of this paper is organized as follows: Section 2 elaborates on the design principles of the FODE algorithm and its adaptability to WFLOP, providing a mathematical description and implementation details of the fractional-order difference mechanism. Section 3 presents the experimental performance of FODE under various complex wind farm conditions and validates its significant advantages in terms of performance, robustness, and applicability through comparative analysis with state-of-the-art optimization algorithms. Finally, Section 5 summarizes the main contributions of this paper and offers an outlook on future research directions in WFLOP optimization applications.

2. Methodology

This section provides a detailed explanation of the design and application process of the FODE algorithm. First, the classic Jensen wake model is utilized to perform wind farm modeling for WFLOP, providing a core description of wind conditions for the experiments. Subsequently, the concept of fractional-order differences is introduced, upon which the FODE algorithm is proposed. Through the innovative design of optimization strategies, the algorithm is applied to WFLOP optimization tasks under complex wind conditions.

2.1. Modeling

This study employs the classic Jensen wake model [34,35], which is regarded as one of the most reference-worthy wake models in WFLOP research due to its low computational complexity and widespread application [12,14,15]. The Jensen model accurately models wind speed losses without significantly increasing computational resource requirements, thereby ensuring fairness in the comparative evaluation of optimization algorithms in experiments. Its low computational complexity not only facilitates the development of WFLOP optimization algorithms but also supports the design and statistical validation of experiments based on complex and more realistic wind conditions without compromising fairness in performance comparisons. This enables a comprehensive assessment of the robustness and applicability of the algorithms [36,37,38,39]. All algorithms in this study are tested based on this wake model.

The wake effect is primarily modeled using the principle of momentum conservation, as illustrated in Figure 1. The wake expansion from an upstream wind turbine is determined by the following equation:

$$\pi L_W^2{V}_1+\pi (L_X^2-L_W^2)V_0=\pi L_X^2{V}_2,$$

(1)

where $V_0$ represents the undisturbed initial wind speed, $L_W$ denotes the rotor radius of the upstream turbine, and $V_1$ is the output wind speed after passing through the upstream turbine. $L_X$ is the downstream wake expansion coverage, d represents the distance between the upstream and downstream turbines, and $V_2$ is the actual wind speed experienced by the downstream turbine under the influence of the wake effect. Using these variables, the Jensen model effectively captures the dynamic impact of wake expansion on wind speed, providing a foundational wind farm modeling framework for optimization algorithms. $L_X$ and $V_1$ are determined using geometric relationships (Equation (2)) and Betz’s theory (Equation (3)), respectively [40]:

$$L_X=L_W+X·tan\theta ,$$

(2)

$$V_1=(1-a)V_0,$$

(3)

where the angular range of wake expansion is determined by the dimensionless parameter $\theta$, whose typical value varies depending on the application scenario. In onshore wind farms, $\theta$ is commonly set to 0.075, while a smaller value of 0.04 is recommended for offshore wind farms to reflect the lower surface roughness [35]. The calculation of $\theta$ is related to the hub height h of the turbine and the surface roughness parameter $z_0$ of the wind farm, and its expression is given as:

$$\theta ={\displaystyle \frac{0.5}{ln\left(\frac{h}{z_0}\right)}}.$$

(4)

where, in practical environments, the value of $z_0$ varies over a wide range. For instance, on sandy terrain or similar surface conditions, it typically ranges from 0.2 mm to 0.3 mm. By incorporating the calculation of $\theta$, the impact of wake expansion on downstream wind speed can be more accurately modeled, thereby enhancing the applicability and accuracy of the Jensen wake model across different application scenarios.

Finally, the downstream wind speed $V_2$ can be calculated using the following equation:

$$V_2={\displaystyle \frac{L_W^2(1-a)V_0+({(L_W+X·tan\frac{0.5}{In(\frac{h}{z_0})})}^2-L_W^2)V_0}{{L_W+X·tan(\frac{0.5}{In(\frac{h}{z_0})})}^2}}.$$

(5)

In recent years, many studies have commonly employed unidirectional, bidirectional, or symmetric wind direction models in wind speed modeling, often assuming uniform or near-uniform wind speeds. However, seasonal monsoon variations frequently result in atmospheric instability in the troposphere, making it challenging for simplified assumptions to fully reflect real-world wind farm conditions. To better simulate wind conditions, this study randomly incorporates non-uniform wind speeds from multiple directions (no fewer than four). Compared to the traditional approach of modeling wind speeds using normal distributions, the Weibull distribution has become the mainstream choice due to its superior ability to capture the characteristics of actual wind speed patterns. In this study, a Weibull distribution with a mean wind speed of $13\mathrm{m}/\mathrm{s}$ is defined, and a dataset of 100,000 randomly sampled wind speed instances is generated to construct a more realistic wind farm model (as shown in Figure 2) [41]. Additionally, unlike normal distributions, the Weibull distribution is more flexible in capturing wind speed distribution characteristics and remains robust under regional wind speed fluctuations. This modeling approach not only ensures the fairness of experiments but also effectively validates the robustness and applicability of optimization algorithms under complex wind conditions.

2.2. State-of-the-Art Differential Evolution

As a classic metaheuristic algorithm, DE has been widely applied to various optimization problems since its inception, demonstrating exceptional performance, particularly in solving NP-hard problems where its efficiency and robustness have been extensively validated [42]. Among them, a series of algorithms based on the DE variant LSHADE have nearly dominated multiple optimization challenges in the IEEE CEC evolutionary computation competitions over the past decade [30,31,43,44,45,46].

Under the LSHADE framework, the algorithm completes the search process by maintaining and iteratively updating a set of solutions in the solution space. Let the solution set (population) of generation k be:

$$P_x^{(k)}=\{x_1^{(k)},x_2^{(k)},...,x_i^{(k)},...,x_{N{P}^{(k)}}^{(k)}\}$$

(6)

where $N{P}^{(k)}$ represents the number of individuals contained in the k-th generation, each individual being a candidate solution to the optimization problem. In terms of solution representation, if the problem dimension is D, then the i-th individual $x_i^{(k)}$ consists of D real numbers, as shown below:

$$x_i^{(k)}=\{x_{i1}^{(k)},x_{i2}^{(k)},...,x_{ij}^{(k)},...,x_{iD}^{(k)}\}$$

(7)

In the initial generation (i.e., $k=1$), individuals are initialized by randomly selecting values for each dimension within the given search boundary interval $[x_l,x_h]$ to ensure the diversity of the initial solution set.

$$x_{ij}^{(1)}=x_{lj}+(x_{hj}-x_{lj})·rand(0,1]$$

(8)

where $x_{lj}$ and $x_{hj}$ are the minimum and maximum possible values of the j-th dimension, respectively, and $rand(0,1]$ is a random number in the interval $(0,1]$.

In the mutation step, LSHADE adopts an improved differential mutation strategy. By selecting high-quality solutions, random solutions, and current individuals for combination differences, new mutant candidate individuals ${y^{\prime }}_i^{(k)}$ are generated. Specifically, the formula for the mutation operation is as follows:

$${y^{\prime }}_{ij}^{(k)}=x_{ij}^{(k)}+s_i·(x_{pbes{t}_{i}j}^{(k)}+x_{r_{1}j}^{(k)}-x_{r_{2}j}^{(k)}-x_{ij}^{(k)})$$

(9)

where $x_{pbes{t}_{ij}}^{(k)}$ is a superior solution randomly selected from the top $p·NP$ elite solutions ranked by quality in the $k^{th}$ generation, where p represents the proportion parameter used for selecting elite solutions. It is usually set to a small value (such as 0.1) to ensure that only the best portion of individuals participate in mutation operations. $x_{r_{1}j}^{(k)}$ and $x_{r_{2}j}^{(k)}$ are different individuals selected through random integer indexing in the current generation population. The parameter $s_i$ (scale factor) is used to scale the magnitude of the differential term.

After the mutation is completed, LSHADE uses crossover operations to recombine the genes of mutated individuals with original individuals. The crossover step involves exchanging genes between the mutated solution ${y^{\prime }}_{ij}^{(k)}$ and the current solution $x_{ij}^{(k)}$ according to a certain probability, in order to produce the final trial individual $y_{ij}^{(k)}$:

$$y_{ij}^{(k)}=\left\{\begin{array}{cc}{y^{\prime }}_{ij}^{(k)}\hfill & ,j=j_{rand}\vee rand(0,1)<c_i\hfill \\ x_{ij}^{(k)}\hfill & ,\mathrm{otherwise}\hfill \end{array}\right.$$

(10)

where in the crossover operation, $c_i$ (crossover rate) determines the frequency of introducing genes from mutated individuals. $j_{rand}$ is a randomly selected dimension index within $[1,D]$, ensuring that at least one dimension of the trial individual receives mutated genes.

After generating the trial solutions, LSHADE performs greedy selection between parent and offspring individuals to maintain or improve population quality. The selection strategy is as follows:

$$x_{ij}^{(k+1)}=\left\{\begin{array}{c}{y}_{ij}^{(k)},\mathrm{if}f(y_i^{(k)})\le f(x_i^{(k)})\hfill \\ x_{ij}^{(k)},\mathrm{otherwise}\hfill \end{array}\right.$$

(11)

where $f(·)$ is the evaluation function to determine the fitness of individuals. If the new solution is not inferior in evaluation compared to the original solution, it enters the next-generation population, thereby continuously improving overall quality.

To further enhance the search performance of the algorithm at different stages, LSHADE uses a memory-based adaptive mechanism to dynamically adjust $s_i$ and $c_i$. By sampling from historical values recorded in the archive, parameter updates adopt Cauchy and normal distribution random number generation strategies to improve the rationality of parameter settings generation by generation:

$$s_i^{(k)}=Cauchyrand(L_{1m},0.1)$$

(12)

$$c_i^{(k)}=Normrand(L_{2m},0.1)$$

(13)

where in the above two equations, $Cauchyrand(L_{1m},0.1)$ and $Normrand(L_{2m},0.1)$ represent sampling from a Cauchy distribution and a normal distribution, respectively, with historical means $L_{1m}$ and $L_{2m}$ stored in the archive as their centers, and a standard deviation of 0.1. The parameter m starts with an initial value of 1, increases with each iteration, and resets after exceeding the upper limit M. M is the maximum number of cycles used to control the memory length of historical parameters, ensuring that the algorithm can utilize the latest historical information while avoiding excessive reliance on outdated parameter settings, thus forming a cycle of historical parameter memory.

In each generation, the algorithm weights and aggregates the progress of individuals that successfully improve fitness to update the reference value of parameters. The calculation of the reference value is as follows:

$$l_s^{(k)}={\displaystyle \frac{{\sum }_{h=1}^{|S_s^{(k)}|}{\omega }_h^{(k)}{s_h^{(k)}}^2}{{\sum }_{h=1}^{|S_s^{(k)}|}{\omega }_h^{(k)}{s}_h^{(k)}}}$$

(14)

$$l_c^{(k)}={\displaystyle \frac{{\sum }_{h=1}^{|S_c^{(k)}|}{\omega }_h^{(k)}{c_h^{(k)}}^2}{{\sum }_{h=1}^{|S_c^{(k)}|}{\omega }_h^{(k)}{c}_h^{(k)}}}$$

(15)

The weight ${\omega }_h^{(k)}$ is related to the individual improvement amount and is defined as:

$${\omega }_h^{(k)}={\displaystyle \frac{f_h^{(k)}-f_h^{(k-1)}}{{\sum }_{g=1}^{|S^{(k)}|}{f}_g^{(k)}-f_g^{(k-1)}}}$$

(16)

where $|S_s^{(k)}|$ and $|S_c^{(k)}|$ denote the sample sizes of solutions that successfully improve fitness for the parameters $s_i$ and $c_i$, respectively, in generation k. $l_s^{(k)}$ and $l_c^{(k)}$ are weighted statistical mean reference values used to guide the update of the parameter memory archive, thereby establishing a dynamic adaptation mechanism for parameter control.

In addition, LSHADE adopts a linear population size reduction strategy to balance the intensity of exploration and exploitation at different evolutionary stages. The population size gradually shrinks with the accumulation of evaluation times ${\mathit{nfes}}^{(k)}$:

$$N{P}^{(k+1)}=round(N{P}_{init}-nfe{s}^{(k)}·{\displaystyle \frac{N{P}_{init}-N{P}_{min}}{maxFEs}})$$

(17)

where $maxFEs$ represents the maximum number of fitness evaluations and $N{P}_{init}$ and $N{P}_{min}$ denote the initial and minimum population sizes, respectively. A larger $N{P}_{init}$ helps to explore the search space more comprehensively, but it also increases computational overhead. Therefore, it is necessary to balance and choose an appropriate initial population size based on the specific problem. $N{P}_{min}$ defines the lower limit of population size reduction, ensuring sufficient population diversity in the later stages of optimization for detailed local searches. By setting a reasonable $N{P}_{min}$, excessive reduction in population size can be prevented, thus maintaining the algorithm’s search capability and stability in later stages. Typically, since LSHADE variants focus on exploitation search within a local area during the latter part of the search process, $N{P}_{min}$ is set to provide a number value that ensures non-repetitive elements for differential operators; in LSHADE, this is 4. This dynamic population adjustment strategy helps maintain diversity and encourage global exploration during the early stages, while focusing computational resources on more precise local exploitation of potential optimal regions in the later stages.

In summary, these mathematical processes form the framework of LSHADE, which adaptively regulates parameters, dynamically adjusts the population size, and leverages differential mutation and crossover strategies to continuously improve solution quality during the optimization process. This paper, while maintaining consistent formulaic representation, provides a novel textual explanation and interpretation of the steps and principles of this framework, offering a valuable reference for further research and application of such efficient Differential Evolution variants.

2.3. The Proposed FODE

In the proposed FODE, we introduce a fractional-order difference operator to perform weighted integration of historical information into the differential vectors. This allows the mutation operator to consider not only the differential information from the current generation but also that from several previous generations. As a result, the algorithm is endowed with greater flexibility in “memory” and enhanced non-local exploration capabilities in the solution space.

Let a be the fractional order, $d^{(0)}=x_{pbes{t}_i}^{(k)}-x_i^{(k)}$ be the current generation elite differential vector, and $d^{(1)},d^{(2)},\cdots ,d^{(m)}$ be the historical records of differences at corresponding positions from previous generations (in this study, the same fractional processing is applied to the difference term $(x_{r_1}^{(k)}-x_{r_2}^{(k)})$). Then, for any historical depth m, the fractional differential operator ${\Delta }^a[·]$ on a sequence of differences ${d^{(j)}}_{j=0}^m$ is defined as follows:

$${\Delta }^a\left[d\right]=\sum _{j=0}^m\left[{\displaystyle \frac{a(a-1)(a-2)\cdots (a-(j-1))}{j!}}·d^{(j)}\right]$$

(18)

Among them, when $j=0$, the numerator is only a, i.e., ${\displaystyle \frac{a}{1!}}{d}^{(0)}$; when $j>0$, this term becomes a product with descending terms $(a-1),(a-2),\dots$, allowing this operator to flexibly adjust the weight distribution of historical differences. If the historical difference corresponding to a certain j is unavailable or if the iterative algebra has not yet reached that stage, the corresponding term can be set as a zero vector. Fractional-order differences allow the parameter a to take non-integer values, thereby achieving fine-tuning of historical difference weights. In LSHADE, the fractional-order difference operation ${\Delta }^a\left[d\right]$ is used to integrate multi-generational historical difference information, and the value of parameter a determines the weight distribution of different generational historical differences in the current update. For example, choosing a larger a value will give more weight to newer historical differences, enhancing the algorithm’s response to recent search trends, whereas a smaller a value will increase the weight of earlier differences, helping maintain search diversity and preventing premature convergence to local optima. By adjusting the weights of historical difference information through parameter a, LSHADE’s memory capability is enhanced. A reasonable value for a enables the algorithm to utilize past search experiences more effectively, accumulate valuable information, make wiser decisions in subsequent optimization processes, and improve both search efficiency and solution quality. Overall, the fractional-order difference formula achieves a weighted combination of multiple historical difference terms by introducing a non-integer order a. This method not only retains the differential information of the current generation but also comprehensively considers the differential changes of previous generations, thereby enhancing the algorithm’s responsiveness to historical trends and overall search efficiency. Moreover, fractional-order differences can smoothly adjust the influence strength of each generation’s differences, avoiding potential information loss or excessive reliance on single-generation differences that may occur with traditional integer-order differences.

In the actual implementation, we perform fractional processing on the elite differential $(x_{pbes{t}_i}^{(k)}-x_i^{(k)})$ and the random differential $(x_{r_1}^{(k)}-x_{r_2}^{(k)})$ separately:

$${\Delta }^a\left[d_{pbest}\right]=\sum _{j=0}^m\left[{\displaystyle \frac{a(a-1)(a-2)\cdots (a-(j-1))}{j!}}·d_{pbest}^{(j)}\right],d_{pbest}^{(0)}=x_{pbes{t}_i}^{(k)}-x_i^{(k)}$$

(19)

$${\Delta }^a\left[d_r\right]=\sum _{j=0}^m\left[{\displaystyle \frac{a(a-1)(a-2)\cdots (a-(j-1))}{j!}}·d_r^{(j)}\right],d_r^{(0)}=x_{r_1}^{(k)}-x_{r_2}^{(k)}$$

(20)

Substituting the above fractional difference results into the variation formula, we can obtain the fractional-order variation operator of FODE:

$${y^{\prime }}_{ij}^{(k)}=x_{ij}^{(k)}+s_i·\left({\Delta }^a\left[d_{pbest}\right]+{\Delta }^a\left[d_r\right]\right).$$

(21)

By introducing the ${\Delta }^a[·]$ operator, FODE accumulates the use of historical differences in the mutation phase, making the evolutionary trajectory of solutions not only influenced by local differences in the current generation but also comprehensively considering historical progress directions and deviations. This fractional-order memory and non-local characteristic effectively alleviate local optimal traps in discrete optimization problems, expand search coverage, and ensure continuous exploration and efficient development of globally excellent solution regions in discrete WFLOP.

Table 1. FODE parameters after tuning.

Parameter	Value
$N^{init}$	$|77-D|$
$N^{min}$	4
p	0.11
M	5
a	0.8

shows a set of parameters for FODE that perform relatively well in multiple WFLOPs, recommended as the practical application parameters for WFLOP in this paper (note that they can be further adjusted in actual applications to suit different wind conditions and turbine number scenarios). The pseudo-code of FODE is shown in Algorithm 1.

Algorithm 1: Pseudo-code of FODE

3. Results

In this section, the experiments are conducted based on a $12\times 12$ wind farm layout to validate the performance of the proposed FODE algorithm on the wind farm layout optimization problem (WFLOP) under various wind field conditions. Specifically, we designed 10 sets of wind field conditions with varying numbers of wind directions (ranging from a single wind direction to 10 directions) and combined these with layouts of 10, 20, 30, 50, and 80 turbines, resulting in a total of 50 test scenarios. The wind speed conditions are based on average wind speed statistics for Japan and relevant studies, with the wind speed distribution modeled as a Weibull distribution with a mean of 13 m/s. A wind speed sample set was obtained through 100,000 independent draws [41]. In this study, FODE is compared against several state-of-the-art WFLOP optimizers proposed in recent years, including two variants of Genetic Algorithms (GAs), two variants of Particle Swarm Optimization (PSO), one variant of Spherical Evolution (SE), and two variants of Differential Evolution (DE). These optimizers have demonstrated strong performance and competitiveness in WFLOP scenarios in prior research. The comparison between FODE and LSHADE also serves to validate FODE’s effectiveness and improvements from both ablation and application extension perspectives. To ensure the reliability and statistical significance of the experimental results, each optimizer was independently run 51 times under each test condition. All comparison algorithms were implemented using the default parameter settings recommended in their original papers, and the basic parameter configurations for FODE were kept consistent with LSHADE. Additionally, to ensure fairness and uniformity, the maximum number of fitness evaluations was set to 24,000 across all experiments. For the selection of test algorithms, this study focused on the most recent and high-performing WFLOP optimizers, including AGA, SUGGA, LSE, AGPSO, CGPSO, CLSHADE, and LSHADE. These algorithms represent the mainstream methods that have demonstrated excellent performance in WFLOP research since 2019. AGA and SUGGA provided the first open-source codes and baseline studies for WFLOP optimization; subsequent research has improved optimization performance through more generalized and robust strategies. LSE introduced a novel spherical evolution concept and, to some extent, outperformed AGA and SUGGA, but its results still lagged behind AGPSO and CGPSO. Later, chaotic search mechanisms were applied to LSHADE, leading to the successful development of CLSHADE for WFLOP. Although CLSHADE’s stability requires further enhancement, it has occasionally outperformed AGPSO and CGPSO in certain scenarios. Due to LSHADE’s popularity and strong performance in the field of metaheuristic algorithms, it serves as a key benchmark in this study. Overall, given that the available state-of-the-art WFLOP optimizers have already achieved a very high level of performance, this study introduces FODE with the goal of determining whether it can further improve WFLOP optimization outcomes beyond the current peak performance.

3.1. Wind Condition Setting

To more realistically reflect wind farm conditions, this study incorporates wind field setups ranging from a single wind direction to multiple wind directions (up to 10 directions), combined with five different turbine counts, resulting in a total of 50 experimental scenarios. For each wind direction condition, wind speed distributions are generated based on non-uniform sampling, making the experiments more representative of real-world wind conditions characterized by instability and variability. Figure 3 illustrates the wind rose graphs for several typical scenarios, showcasing the wind speed distribution characteristics under different numbers of wind directions. The mean wind speed of $13\mathrm{m}/\mathrm{s}$ is modeled accurately using a Weibull distribution. This wind field setup allows the robustness and applicability of optimization algorithms to be evaluated on multi-dimensional and high-complexity WFLOP problems, without introducing unnecessary computational complexity.

3.2. Comparison Results Between FODE and State-of-the-Art WFLOP Optimizers

Table 2. Experimental results of FODE with 4 recently proposed state-of-the-art WFLOP optimizers. (Note that bold in the table is used to highlight the best mean result among all 8 state-of-the-art WFLOP optimizers.)

	FODE		AGA		SUGGA		LSE		AGPSO
	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
WS1tn10	100%	0.000%	100%	0%	100%	0%	100%	0%	100%	0%
WS1tn20	100%	0.000%	99.786%	0.077%	99.762%	0.063%	99.807%	0.083%	99.974%	0.063%
WS1tn30	98.953%	0.035%	98.444%	0.102%	98.494%	0.064%	98.295%	0.122%	98.839%	0.082%
WS1tn50	96.022%	0.129%	94.172%	0.223%	94.650%	0.209%	93.728%	0.246%	95.492%	0.315%
WS1tn80	88.450%	0.842%	84.863%	0.224%	85.494%	0.262%	85.326%	0.271%	87.128%	0.376%
WS2tn10	100%	0.000%	100%	0%	100%	0%	100%	0%	100%	0%
WS2tn20	99.730%	0.047%	99.018%	0.121%	98.851%	0.129%	99.041%	0.142%	99.601%	0.103%
WS2tn30	98.124%	0.137%	97.251%	0.196%	96.842%	0.222%	96.524%	0.216%	97.905%	0.270%
WS2tn50	93.233%	0.153%	91.814%	0.255%	91.446%	0.306%	90.356%	0.234%	92.451%	0.404%
WS2tn80	83.389%	0.709%	81.356%	0.249%	81.365%	0.246%	80.795%	0.125%	82.549%	0.361%
WS3tn10	100%	0.000%	99.926%	0.054%	99.943%	0.045%	100%	0%	100%	0%
WS3tn20	99.102%	0.098%	96.945%	0.300%	96.966%	0.259%	98.080%	0.228%	98.589%	0.256%
WS3tn30	95.915%	0.064%	92.134%	0.456%	92.142%	0.517%	93.497%	0.255%	94.739%	0.540%
WS3tn50	85.917%	0.315%	81.675%	0.455%	81.507%	0.393%	82.582%	0.399%	84.860%	0.424%
WS3tn80	69.503%	0.636%	66.483%	0.241%	66.464%	0.232%	67.030%	0.144%	68.440%	0.369%
WS4tn10	100%	0.000%	99.949%	0.037%	99.966%	0.023%	99.995%	0.011%	99.999%	0.006%
WS4tn20	98.891%	0.096%	98.055%	0.178%	98.170%	0.201%	97.736%	0.163%	98.892%	0.178%
WS4tn30	96.362%	0.120%	95.103%	0.209%	95.185%	0.220%	94.402%	0.210%	96.166%	0.266%
WS4tn50	89.764%	0.197%	88.062%	0.188%	88.160%	0.237%	86.992%	0.373%	88.957%	0.383%
WS4tn80	77.363%	0.427%	75.904%	0.238%	75.908%	0.208%	74.950%	0.288%	76.630%	0.328%
WS5tn10	100%	0.000%	99.827%	0.052%	99.850%	0.047%	99.977%	0.023%	99.990%	0.017%
WS5tn20	98.987%	0.047%	97.319%	0.243%	97.982%	0.174%	98.355%	0.110%	98.723%	0.153%
WS5tn30	96.033%	0.105%	93.221%	0.273%	93.935%	0.228%	94.394%	0.222%	95.387%	0.310%
WS5tn50	87.426%	0.108%	84.997%	0.246%	85.616%	0.206%	85.299%	0.262%	86.814%	0.319%
WS5tn80	75.122%	0.776%	73.351%	0.171%	73.632%	0.150%	73.000%	0.219%	74.251%	0.321%
WS6tn10	99.990%	0.013%	99.483%	0.166%	99.452%	0.128%	99.762%	0.092%	99.877%	0.089%
WS6tn20	96.470%	0.165%	93.807%	0.201%	94.046%	0.342%	94.845%	0.260%	95.947%	0.355%
WS6tn30	90.487%	0.499%	87.088%	0.249%	87.081%	0.267%	87.951%	0.308%	89.710%	0.400%
WS6tn50	77.056%	0.359%	74.949%	0.192%	74.912%	0.152%	75.125%	0.160%	76.646%	0.202%
WS6tn80	61.979%	0.060%	60.674%	0.112%	60.639%	0.101%	60.488%	0.127%	61.574%	0.125%
WS7tn10	100%	0.000%	99.697%	0.089%	99.707%	0.072%	99.903%	0.053%	99.957%	0.050%
WS7tn20	99.055%	0.088%	97.331%	0.216%	97.432%	0.221%	98.004%	0.181%	98.691%	0.314%
WS7tn30	97.160%	0.048%	94.224%	0.310%	94.252%	0.212%	94.964%	0.280%	96.557%	0.426%
WS7tn50	90.701%	0.186%	87.122%	0.328%	87.072%	0.283%	87.483%	0.348%	89.687%	0.418%
WS7tn80	78.719%	0.709%	76.214%	0.182%	76.400%	0.280%	76.016%	0.181%	77.505%	0.317%
WS8tn10	99.850%	0.034%	99.279%	0.088%	99.366%	0.100%	99.583%	0.083%	99.706%	0.079%
WS8tn20	97.450%	0.093%	95.527%	0.185%	95.729%	0.174%	96.356%	0.126%	97.111%	0.178%
WS8tn30	93.196%	0.151%	90.717%	0.270%	90.938%	0.197%	91.509%	0.170%	92.646%	0.269%
WS8tn50	83.729%	0.083%	81.649%	0.152%	81.979%	0.148%	81.990%	0.219%	83.275%	0.243%
WS8tn80	71.160%	0.066%	69.903%	0.124%	69.967%	0.112%	69.705%	0.120%	70.809%	0.146%
WS9tn10	99.973%	0.014%	99.424%	0.086%	99.496%	0.126%	99.784%	0.064%	99.839%	0.086%
WS9tn20	97.865%	0.125%	96.129%	0.212%	96.080%	0.261%	96.639%	0.181%	97.413%	0.270%
WS9tn30	94.279%	0.255%	91.310%	0.177%	91.445%	0.240%	91.917%	0.233%	93.189%	0.382%
WS9tn50	83.876%	0.113%	82.061%	0.126%	82.136%	0.141%	82.284%	0.144%	83.324%	0.197%
WS9tn80	71.486%	0.297%	70.427%	0.128%	70.501%	0.108%	70.119%	0.090%	71.034%	0.159%
WS10tn10	99.971%	0.017%	99.799%	0.058%	99.806%	0.046%	99.662%	0.084%	99.896%	0.066%
WS10tn20	98.566%	0.218%	98.118%	0.223%	98.261%	0.169%	96.578%	0.238%	98.481%	0.400%
WS10tn30	96.071%	0.323%	95.004%	0.262%	95.265%	0.239%	92.564%	0.262%	95.778%	0.711%
WS10tn50	88.250%	1.004%	86.885%	0.255%	87.188%	0.229%	84.804%	0.232%	87.578%	0.497%
WS10tn80	75.932%	0.275%	74.943%	0.179%	75.211%	0.172%	73.755%	0.147%	75.174%	0.236%
W/T/L	-/-/-		48/2/0		48/2/0		47/3/0		43/7/0

and

Table 3. Experimental results of FODE with 3 other recently proposed state-of-the-art WFLOP optimizers. (Note that bold in the table is used to highlight the best mean result among all 8 state-of-the-art WFLOP optimizers.)

	FODE		CGPSO		LSHADE		CLSHADE
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
WS1tn10	100%	0.000%	100%	0%	100%	0%	100%	0%
WS1tn20	100%	0.000%	99.942%	0.079%	99.643%	0.076%	99.939%	0.066%
WS1tn30	98.953%	0.035%	98.820%	0.087%	97.372%	0.280%	98.575%	0.079%
WS1tn50	96.022%	0.129%	95.368%	0.275%	90.627%	0.362%	93.952%	0.322%
WS1tn80	88.450%	0.842%	87.147%	0.308%	82.172%	0.237%	85.102%	0.608%
WS2tn10	100%	0.000%	100%	0%	100%	0%	100%	0%
WS2tn20	99.730%	0.047%	99.520%	0.117%	98.854%	0.115%	99.339%	0.106%
WS2tn30	98.124%	0.137%	97.855%	0.239%	95.436%	0.313%	97.102%	0.267%
WS2tn50	93.233%	0.153%	92.615%	0.382%	87.685%	0.233%	90.483%	0.794%
WS2tn80	83.389%	0.709%	82.637%	0.383%	78.495%	0.211%	80.417%	0.681%
WS3tn10	100%	0.000%	100%	0%	100%	0%	99.998%	0.009%
WS3tn20	99.102%	0.098%	98.547%	0.250%	96.963%	0.342%	97.889%	0.367%
WS3tn30	95.915%	0.064%	94.796%	0.523%	90.636%	0.253%	92.526%	0.670%
WS3tn50	85.917%	0.315%	84.821%	0.488%	78.659%	0.221%	81.163%	0.657%
WS3tn80	69.503%	0.636%	68.467%	0.402%	64.431%	0.137%	66.430%	0.640%
WS4tn10	100%	0.000%	100%	0%	100%	0%	99.997%	0.019%
WS4tn20	98.891%	0.096%	98.811%	0.162%	97.618%	0.151%	98.330%	0.140%
WS4tn30	96.362%	0.120%	96.310%	0.251%	93.341%	0.348%	95.112%	0.313%
WS4tn50	89.764%	0.197%	89.027%	0.318%	83.775%	0.312%	87.337%	0.656%
WS4tn80	77.363%	0.427%	76.592%	0.337%	72.072%	0.319%	74.958%	0.332%
WS5tn10	100%	0.000%	99.979%	0.034%	99.983%	0.029%	99.996%	0.006%
WS5tn20	98.987%	0.047%	98.689%	0.160%	97.368%	0.344%	98.333%	0.228%
WS5tn30	96.033%	0.105%	95.434%	0.308%	91.948%	0.596%	94.217%	0.572%
WS5tn50	87.426%	0.108%	86.685%	0.284%	81.605%	0.331%	84.955%	0.422%
WS5tn80	75.122%	0.776%	74.308%	0.232%	69.863%	0.200%	72.246%	0.773%
WS6tn10	99.990%	0.013%	99.886%	0.081%	99.854%	0.071%	99.914%	0.048%
WS6tn20	96.470%	0.165%	95.931%	0.286%	94.304%	0.242%	95.395%	0.339%
WS6tn30	90.487%	0.499%	89.697%	0.448%	86.093%	0.444%	88.502%	0.436%
WS6tn50	77.056%	0.359%	76.567%	0.213%	73.026%	0.239%	74.762%	0.408%
WS6tn80	61.979%	0.060%	61.592%	0.120%	58.672%	0.167%	60.086%	0.462%
WS7tn10	100%	0.000%	99.971%	0.037%	99.938%	0.027%	99.980%	0.018%
WS7tn20	99.055%	0.088%	98.574%	0.304%	97.246%	0.212%	98.271%	0.147%
WS7tn30	97.160%	0.048%	96.518%	0.455%	92.612%	0.299%	95.179%	0.420%
WS7tn50	90.701%	0.186%	89.651%	0.359%	83.638%	0.259%	87.125%	0.515%
WS7tn80	78.719%	0.709%	77.571%	0.309%	73.326%	0.250%	75.575%	0.609%
WS8tn10	99.850%	0.034%	99.678%	0.083%	99.630%	0.050%	99.720%	0.063%
WS8tn20	97.450%	0.093%	97.010%	0.212%	95.807%	0.336%	96.495%	0.324%
WS8tn30	93.196%	0.151%	92.653%	0.239%	89.898%	0.382%	91.661%	0.363%
WS8tn50	83.729%	0.083%	83.260%	0.187%	79.719%	0.190%	81.915%	0.380%
WS8tn80	71.160%	0.066%	70.834%	0.178%	67.826%	0.167%	69.472%	0.489%
WS9tn10	99.973%	0.014%	99.867%	0.089%	99.818%	0.049%	99.887%	0.046%
WS9tn20	97.865%	0.125%	97.446%	0.277%	95.768%	0.389%	96.739%	0.350%
WS9tn30	94.279%	0.255%	93.306%	0.375%	90.499%	0.328%	92.004%	0.306%
WS9tn50	83.876%	0.113%	83.400%	0.225%	80.840%	0.148%	81.858%	0.235%
WS9tn80	71.486%	0.297%	71.083%	0.137%	68.738%	0.122%	69.729%	0.168%
WS10tn10	99.971%	0.017%	99.884%	0.071%	99.762%	0.060%	99.877%	0.046%
WS10tn20	98.566%	0.218%	98.431%	0.405%	96.337%	0.180%	97.526%	0.282%
WS10tn30	96.071%	0.323%	95.520%	0.673%	91.432%	0.248%	93.573%	0.409%
WS10tn50	88.250%	1.004%	87.433%	0.553%	82.497%	0.289%	85.006%	0.419%
WS10tn80	75.932%	0.275%	75.160%	0.310%	71.640%	0.226%	73.326%	0.496%
W/T/L	-/-/-		44/6/0		46/4/0		46/4/0

present the statistical comparison results of FODE and other state-of-the-art WFLOP optimization algorithms under different wind direction and turbine quantity conditions. In the tables, “WSXtnY” represents a specific WFLOP instance with X wind directions and Y turbines, while “mean” and “std” denote the mean and standard deviation of the results from 51 independent experiments, respectively. The “W/T/L” statistics (Win/Tie/Lose) derived from the Wilcoxon rank-sum test quantify the significant differences in optimization quality between FODE and the comparison algorithms. Note that in order to comprehensively assess the performance differences between FODE and other optimization algorithms, we have chosen to use the Wilcoxon rank-sum test, a non-parametric statistical method. This test not only considers the mean and standard deviation of multiple independent optimization results but also covers the median and other important characteristics of data distribution. The Wilcoxon rank-sum test is a reliable non-parametric statistical method widely adopted and recognized in the fields of evolutionary computation and metaheuristic algorithms, capable of effectively quantifying performance differences between algorithms. Therefore, through the “W/T/L” statistical results derived from the Wilcoxon rank-sum test, it can be fully demonstrated whether there are significant performance differences among different algorithms when solving the same problem. This form of data statistical testing and tabular presentation is consistent with other authoritative studies within the fields of metaheuristic algorithms and evolutionary algorithms.

In terms of average power generation efficiency (mean values), FODE achieves optimal or near-optimal performance in the vast majority of test conditions. In many lower-complexity scenarios (e.g., WFLOP with 10 turbines), FODE consistently achieves or approaches 100% energy conversion efficiency, performing on par with or better than other optimizers. More importantly, in larger-scale and higher-difficulty scenarios (e.g., WFLOP with 80 turbines), FODE continues to achieve the highest mean values. This not only demonstrates its exceptional performance in simpler problems but also highlights its strong robustness and global search capability under complex layout conditions.

A closer examination of the W/T/L statistics further verifies this advantage. For instance, compared to recently proposed high-performance WFLOP optimizers such as AGA, SUGGA, and LSE, FODE demonstrates clear wins (W) or at least ties (T) in nearly all problems, with very few cases where it performs worse (L). Similarly, for AGPSO and CGPSO, which are among the most efficient PSO variants in recent literature, FODE consistently delivers higher-quality solutions and more stable optimization results in most scenarios. Notably, FODE significantly improves the efficiency and adaptability of LSHADE in the WFLOP context. As shown in the tables, FODE outperforms LSHADE in over 90% of the test scenarios, demonstrating that the introduction of the fractional-order difference strategy and associated improvements have successfully transferred and enhanced LSHADE’s strong performance in continuous optimization to the discrete optimization environment of WFLOP.

In summary, FODE demonstrates outstanding performance across scenarios with varying numbers of wind directions and turbines. Whether in simpler, low-dimensional problems or in complex, high-dimensional layout challenges with multiple wind directions, FODE consistently achieves optimal or near-optimal solutions over a wide range of conditions. This fully validates the effectiveness of the proposed strategies in not only significantly enhancing LSHADE’s performance in WFLOP but also providing a more efficient and robust optimization solution for wind farm layout problems compared to other state-of-the-art WFLOP optimizers.

Figure 4 and Figure 5 illustrate the convergence processes and statistical boxplots of various optimizers under multi-wind direction conditions. From the convergence curves in Figure 4, it is evident that FODE significantly enhances the upper bound of solution quality, regardless of whether the number of turbines is 10, 20, 30, 50, or 80. In the lower-complexity 10-turbine problem, FODE quickly approaches 100% power generation efficiency within a few dozen iterations, nearly matching the upper limit of the best existing algorithms. As the problem scale increases (e.g., in 50- and 80-turbine scenarios), FODE’s performance advantage becomes increasingly pronounced, with its final convergence values consistently surpassing those of competing algorithms, showcasing its exceptional adaptability to complex discrete solution spaces. Compared to the suboptimal convergence performance of LSHADE, FODE leverages fractional-order difference strategies to reclaim the performance advantages DE variants possess in complex continuous optimization problems and successfully applies them to the discrete optimization domain of WFLOP. This is particularly evident in the challenging 80-turbine scenarios, where FODE achieves at least a five-percentage-point improvement in power generation efficiency over LSHADE, highlighting the significant performance gap. In the boxplots shown in Figure 5, FODE’s optimization results similarly exhibit high stability and low variance. In the 10-turbine problem, FODE’s solutions are highly concentrated near the optimal value, with no noticeable outliers or significant fluctuations. This indicates that FODE is more likely to reliably achieve high-quality solutions in individual runs, making it more practical compared to algorithms that only approach high-quality solutions in a few runs. In the more complex 20-, 30-, 50-, and 80-turbine scenarios, FODE’s boxplot results remain stable and consistently outperform other DE variants, including CLSHADE, as well as all other state-of-the-art WFLOP optimizers. This phenomenon suggests that, compared to optimizers that focus primarily on local search strategies, FODE’s emphasis on global exploration and integration of historical information is better suited to the structural characteristics of discrete problems, effectively reducing the risk of being trapped in local optima. Similar trends are observed across scenarios with varying wind directions and turbine counts (related results can be found in the figures in Appendix A: Figure A1, Figure A2, Figure A3 and Figure A4). In conclusion, the exceptional performance of FODE on WFLOP arises not only from extending DE’s inherent strengths in continuous optimization but also from the non-local search and historical memory capabilities introduced by the fractional-order difference strategy. This enables FODE to consistently achieve high-quality solutions in complex, high-dimensional discrete optimization problems.

Figure 6, Figure 7 and Figure 8, respectively, illustrate the optimal turbine layouts obtained by FODE and other WFLOP optimizers under varying conditions of wind directions and turbine counts. These layout diagrams intuitively reveal the differences among algorithms in exploring the solution space, balancing wake interference between turbines, and enhancing overall power generation efficiency. They also provide clear evidence for analyzing the relationship between solution distribution patterns and final performance.

In Figure 6, under the low-complexity scenario of five wind directions and 10 turbines, most algorithms achieve 100% energy conversion efficiency. However, what is more noteworthy is FODE’s unique characteristic: in multiple independent optimization runs, FODE not only consistently discovers the optimal layout but also maintains this optimal state in nearly all runs. This level of stability and consistency highlights FODE’s ability to not only identify the optimal solution for simple problems but also repeatedly ensure convergence to the optimal layout without deviation. In contrast, while other algorithms may occasionally achieve the same target, they fall slightly behind in terms of repeatability and reliability. Additionally, FODE demonstrates diverse forms of optimal layouts, reflecting its extensive search capability in the solution space. It avoids reliance on a single layout pattern and is capable of switching between different solutions without compromising performance. This flexibility further underscores FODE’s robustness and adaptability, even in relatively straightforward WFLOP scenarios.

As problem complexity increases, the three-wind-direction, 50-turbine scenario in Figure 7 further tests the global search and adaptability of the algorithms. In this case, FODE not only achieves a high energy conversion rate but also demonstrates a more compact, almost grid-like structure in its layout pattern. This high regularity minimizes mutual interference between turbines and facilitates practical benefits such as ease of installation, maintenance, and efficient land use. While algorithms such as AGPSO and CGPSO also achieve relatively high efficiency, FODE’s optimal solutions are more consistent across multiple runs, with fewer instances of significant variation. This consistency indicates that FODE is capable of effectively handling uncertainty and complexity in higher-dimensional and more challenging scenarios, consistently identifying and exploiting near-optimal regions of the solution space. The structured and efficient layouts further highlight FODE’s ability to balance performance and practical feasibility in complex WFLOP problems.

In the highly complex scenario of 10 wind directions and 80 turbines described in Figure 8, the advantages of FODE become even more pronounced. Despite the vast and intricate solution space, FODE consistently delivers highly regular, nearly ideal array-like turbine layouts, effectively mitigating the impact of wake effects on power generation efficiency. Compared to other optimizers, FODE excels not only in power generation efficiency (as shown in Table 2 and Table 3) but also in producing more precise and orderly layouts. Such structured and efficient layouts are particularly beneficial in real-world engineering, as they can further reduce construction costs and operational complexity. This highlights FODE’s exceptional adaptability and superiority in tackling high-dimensional, multi-directional WFLOP problems, reinforcing its practical value for large-scale wind farm layout optimization.

Overall, these visualization results align with the quantitative analysis based on tables, convergence curves, and boxplots presented earlier. Whether it is the frequent attainment of perfect solutions in simple scenarios or the formation of highly regularized layouts under medium- to high-difficulty conditions, FODE consistently demonstrates a deep mastery of the discrete solution space and precise exploration of globally optimal regions. This not only provides further compelling evidence of its superiority in WFLOP but also offers highly efficient and practical layout references for real-world wind farm construction and planning.

4. Discussion

In order to enhance readers’ understanding of FODE as much as possible, we use a case study of a method called fuzzy self-tuning Differential Evolution for optimal product line design (hereinafter referred to as fuzzyDE) [47] to further discuss perspectives on parameter adaptation. FuzzyDE is a recently proposed fully adaptive variant of DE.

First, the version of DE on which fuzzyDE is based, as well as its comparison versions, are all based on a basic DE variant, containing only two fundamental parameters (F and CR). In FODE, F and CR have adopted adaptive strategies based on historical success memory used in more advanced DE variants from benchmark studies, thereby ensuring stable and robust optimization capabilities as a foundation.

Second, the presence of parameter p in FODE is due to the use of the current-to-pbest operator. Parameter p, set to 0.11, has demonstrated stable high performance in several recently published adaptive studies. Thus, further adaptation does not necessarily enhance the stability of DE variants. However, in the fuzzyDE paper, the current-to-pbest operator, which performed the best in the benchmark test set, was not used. Therefore, there is no issue regarding whether other parameters require adaptive strategy design.

Third, some parameters in DE series optimizers cannot be completely adapted, such as population size. FODE employs an adaptive mechanism of linearly varying population size, which has been proven in several recent studies to significantly and stably enhance optimization performance. However, this mechanism requires setting maximum and minimum population size parameters to be effective. Similarly, the advanced success memory mechanism for F and CR adopted in FODE introduces another parameter, M. Nevertheless, in multiple studies based on benchmark test sets, setting M to 5 has shown significant, stable, and efficient performance. Therefore, the four DE parameters in FODE are all highly referenced from recent reliable research conclusions.

Fourth, as a typical evolutionary computation method, the computational process of DE variants is difficult to derive or mathematically interpret and is better regarded as a black box. Therefore, apart from the selection of fixed parameters, whether adaptivity can bring performance improvement aligns with the no-free-lunch theorem. In other words, no adaptive method or adaptive theory currently guarantees performance improvement for algorithms with the addition of a certain adaptive strategy across problems with different characteristics. In this sense, the adaptive strategies of fuzzyDE are likely better suited to the optimal product line design problems it targets. At least based on the experimental results in the paper, this is the only conclusion we can draw.

Fifth, the purpose of this study is to maximize the superior performance exhibited by advanced DE variants on standard test sets compared to other peer metaheuristic algorithms in discrete WFLOP. Hence, all DE adaptive strategies and related parameters are drawn from the latest research findings, ensuring an excellent performance foundation when solving WFLOP [31,44]. The discussion of parameters can serve as practical guidance for application or as a next-step research direction to further unlock FODE’s performance potential.

5. Conclusions

The proposed FODE demonstrates exceptional performance in WFLOP across various realistic wind conditions, significantly outperforming state-of-the-art WFLOP optimizers. In problems involving multiple wind directions and large-scale turbine layouts, FODE not only achieves higher energy conversion rates but also delivers highly regular and practical turbine arrangements. This characteristic holds great significance for real-world wind farm construction, operation, and land resource utilization: more uniform and orderly turbine distributions reduce the impact of wake effects on overall power generation efficiency while providing better-planned and more spacious areas for agricultural production and other economic activities surrounding the wind farm. At the algorithmic level, FODE integrates the fractional-order difference strategy into the LSHADE framework, enabling it to maintain strong global search and local exploitation capabilities in discrete optimization scenarios. By incorporating historical differential information into the mutation operation, FODE effectively overcomes the challenge faced by traditional DE variants in discrete problems like WFLOP, where they are prone to getting trapped in local optima. It also balances exploration and exploitation in the high-dimensional solution space under multi-directional and multi-turbine conditions. Compared to other optimizers specifically designed for WFLOP, FODE exhibits significant improvements in parameter adaptivity, convergence speed, and solution quality. This highlights that the potential of DE variants in WFLOP optimization remains largely untapped. From the experimental results of this paper, FODE performs best in high dimensions compared to other WFLOP optimizers. However, since WFLOP cannot bypass energy conservation and the necessary consumption based on physical rules during energy conversion, it is currently difficult to determine whether FODE approaches the physical limits of the tested problems. As the performance potential of DE has not yet been fully developed, we believe that DE variants as WFLOP optimizers still have room for further improvement in energy conversion efficiency.

Future research directions can further expand the contributions of this work on multiple levels. First, the fractional-order difference strategy could be combined with other DE variants or evolutionary algorithms to verify its generality and scalability across different frameworks and problem types. Specifically, some metaheuristic algorithm variants such as DE, PSO, and GA that have been proven to have superior theoretical performance over the past 20 years should be considered a priority. It is important to note that, according to some recent review research, some recently proposed metaheuristic optimization algorithms do not possess or are far from possessing the performance of classic metaheuristic algorithms proposed even 10 or 20 years ago [48]. Therefore, we recommend prioritizing algorithms that perform well in generality and benchmark sets when developing and solving WFLOP or any other complex optimization problems and making specialized improvements based on them for different issues. According to the conclusions of this study, we believe it is imperative to advance further research and application of DE variants in WFLOP. Second, to test the robustness and adaptability of FODE, future studies could apply it to more complex and dynamic wind farm models. For instance, factors such as three-dimensional meteorological conditions, seasonal wind direction variations, and land–sea terrain features could be considered to make optimization results closer to real-world engineering scenarios. Although these applications are currently challenging to implement due to open-source limitations, we hope to collaborate with industry professionals in the future to explore such possibilities. In addition, considering multi-objective scenarios could add more practical significance to wind farm layout optimization. For instance, optimizing both energy generation and land-use efficiency could yield solutions that are more applicable to diverse project requirements. To foster the development of the WFLOP research community, making the code and data used in this study publicly available could encourage more researchers and developers to participate in improving and applying optimization algorithms. Open-source practices can help establish more credible and reproducible benchmarking platforms, facilitating the rapid validation and dissemination of new ideas and strategies. This, in turn, will accelerate innovation and progress in WFLOP optimization algorithms. In summary, the proposal of FODE not only surpasses existing WFLOP optimizers but also provides new insights and directions for fully unlocking the potential of DE variants in discrete optimization problems. With broader and deeper research and collaboration, WFLOP optimization technologies are expected to reach new heights, contributing further to the efficient planning of renewable energy utilization and distributed power generation systems.