A bi‐level programming model and differential evolution for optimizing offshore wind farm layout

The optimization of offshore wind farms is mainly performed through the deployment of wind farms and submarine cables to maximize power output and minimize cable costs. However, the above results are affected by the wake effect, equipment layout, and the cost of cables. To effectively complete the deployment of wind turbines and submarine cable lines, first, a bi‐level constrained optimization model based on maximum profit and the shortest route of cable is proposed in this paper; then, a differential evolution and improved Prim algorithm (IPADE) are used to optimize the upper‐ and lower‐level objective function, respectively. Moreover, the fitness values are used to divide the population, and a surrogate model is used to evaluate approximate fitness values for the sub‐population with poor performance; the best individual is selected as the offspring individual according to the approximate fitness values. Next, a clustering method is used to divide the position of the wind farm, and a Prim algorithm based on roulette wheel selection is designed to deploy submarine cables of every subwind farm. Finally, the proposed algorithm is compared with five other popular algorithms under the two wind conditions. The simulation experimental results show that algorithm IPADE performs better than other algorithms in terms of the power output, profit, and the length of cables.

optimization problem of micrositing is NP hard; it is difficult to solve the problem using traditional methods. The evolutionary algorithm, as a heuristic and swarm intelligence optimization algorithm, is widely used to solve complex optimization problems. When the evolutionary algorithm is used to solve the problem of wind farm deployment, the codes of grids or coordinates are usually used to represent the location of a wind turbine; for example, the genetic algorithm proposed by Mosetti et al. 3 adopts grid coding, which divides the wind farm area into some small square grids, and binary encoding is used to represent the deployment of a wind turbine. Moreover, the genetic algorithm is used to evolute population and output the optimal deployment. In the same way, binary particle swarm optimization (PSO) 4 and binary differential evolution (DE) 5 are also used to solve the grid-based optimization model for the micrositing problem of wind farms. Obviously, the above evolutionary algorithms have the characteristics of simple coding, easy evolution operation, and fast convergence, but it is difficult to determine the size of the grid. Different from the encoding method of the grid, the coordinate adopts real number encoding, and a twodimensional coordinate is used to represent the location of a wind turbine, which means that the wind turbine can be located anywhere in the wind farm. Obviously, this encoding method is more flexible and realistic. Based on the output maximum wind energy, a single objective constrained optimization model is established by Sun et al., 6 and the coordinate is taken as the decision variable. Moreover, the evolution operation is used to update the wind farm location. Simulation results show that coordinate encoding is better than binary encoding. Some common optimization models can be established on the basis of the preference of the decisionmaker and the basic requirements of the wind farm, including single objective constraint, multi-objective constraint and bi-level constraint, and so forth, where minimization of the cost of kilowatt hour (kWh) or maximization of the total power output of a wind farm can be used as an objective function. For example, Feng et al. 7 adopted minimization of kWh cost as the optimization objective, and a random search algorithm was used to solve the problem. Long et al. 8 performed maximization of the total power output as the optimization objective. In this paper, a two-echelon (grid and coordinate) layout planning model is proposed to determine the optimal wind farm layout. Due to the complexity of models in both echelons, the random key genetic algorithm and the PSO algorithm are separately applied to obtain the optimal solutions in the first and second echelons. Moreover, the convex hull area, the maximized output power, and the minimum spanning tree distance are the objective functions in the study carried out by Shekar et al., 9 and an improved genetic algorithm is proposed to solve the multi-objective problem. Experimental results show that the proposed algorithm performs better than other algorithms in solving three scenarios. Moreno et al. 10 proposed a novel multi-objective algorithm based on the cost of annual energy production, the overall wind farm area, and the wake effect losses. Moreover, the lightning search algorithm is designed to efficiently complete the wind farm layout. Song et al. 11 maximized the annual energy production and minimized the annual production cost as bi-objectives, and improved the multi-objective harmony algorithm to obtain better nondominated solutions. It is worth noting that the evolutionary algorithm has some limitations in solving the above optimization problems, such as slow convergence speed and the algorithm falling into local optimum. Therefore, some studies aim to speed up the convergence of algorithms by designing efficient evolutionary operations. For example, a new encoding mechanism for the locations of wind turbines is designed in the study carried out by Wang et al. 12 based on the characteristics of the wind farm layout. A hybrid optimization strategy is proposed in the study carried out by Yang et al., 13 which is based on the genetic algorithm and the particle swarm algorithm, to effectively reduce the cost of energy. Besides, embedding other strategies in evolutionary algorithms is a method to improve the search ability of algorithms. For example, Bai et al. 14 transformed the position of the wind turbines into a single-player reinforcement learning problem, and used the Monte Carlo tree search to complete the training and output the optimal solution. Long et al. 15 proposed a data-driven evolutionary algorithm to save the computation cost, where the data-driven surrogate model is constructed by the general regression neural network, and the surrogate model is trained and updated by using the data, which are generated by an adaptive DE.
Offshore wind farms have attracted much attention because of their high wind speed and due to the fact that do not take up a large part of the land, but effective deployment of wind turbines is very important to avoid high investment. Gonzalez et al. take maximization of the economic profitability as an objective function in a study; 16 the available marine plot is divided into some smaller areas to reduce computational complexity and improved genetic algorithms are used to optimize the optimal layout. The PSO algorithm is used to find the optimal solution and the adaptive adjustment strategy is adopted for the related parameters in the study carried out by Peng et al. 17 A mixed integer linear program is developed in the study carried out by Pérez-Rúa et al., 18 which uses total cable length and total electrical power losses as the objective functions, and the model is embedded in an iterative algorithmic framework to obtain a satisfactory solution. Besides, the Gaussian model is used to evaluate the value of energy output and the genetic algorithm is used to output the optimal solution through successive iterations in the study carried out by Gonzáez et al. 19 However, the above studies did not consider the influence of seabed topography on the total energy output. Therefore, Liu et al. used genetic algorithms to optimize the wind farm layouts according to different seabed topographies. 20 Moreover, unlike traditional wind farm layout optimization, Cazzaro et al. solved the problem of selecting the shape and area of a wind farm from a larger area to maximize profitability, 21 that is, by taking macro, meso, and micro as three different scales and designing a heuristic evolutionary algorithm to choose the best shape of a wind farm for the above three scales. Other studies on offshore wind farms can be found in various studies. 22,23 To reduce the investment cost, the optimization problem of offshore wind farm cable layout has received more attention from researchers. Early relevant studies focused on minimizing cable length and adopted the graph theory 24 or the classical evolutionary algorithm 25 to optimize cable length. However, considering that the location of the wind farm unit is set, cable topology still depends on the location of the substation. Wei et al. designed a hierarchical optimization model according to the deployment characteristics of wind farms, and the optimization model was divided into an offshore substation layer, a wind turbine layer, and a submarine cable layer, and the fuzzy clustering algorithm, the single parent genetic algorithm, and the multiple traveling salesman solution technique were applied for sequential optimization of the above model. 26 To obtain a lower-cost optimization scheme, the cable cost and the associated power loss cost are considered the objective functions. 27 The offshore substation location, cable connection layout, and cable sectional area are optimized simultaneously while ensuring an uncrossed cable connection layout using a line segment intersection detection algorithm. Thomas et al. 28 constructed a bi-objective function to simultaneously optimize cable design and micrositing, and used Jensen's equations to approximate the wake effect. Moreover, they used use linear approximations to solve the cable layout problem as a classical MILP. Pérez-Rúa et al. used an improved genetic algorithm to optimize cable topology and cable models, 29 which can reduce the loss costs of cables, where a greedy algorithm is used to generate initial cable connectivity and a genetic algorithm is used to further improve connectivity.
In fact, the deployment of offshore wind farms requires both micrositing and cable deployment, and the two deployment problems are contradictory to each other, that is, the increase of the distance between the wind motors can reduce the wake effect, but requires the use of more expensive cables. Therefore, the combinatorial optimization problems mentioned above are complex. In contrast to micrositing optimization of wind farms, the investment cost of offshore wind farms is higher, especially the cable cost. It is very important to determine the location of wind turbines and the best line connection method between offshore substations; in this way, the cable cost can be minimized under the premise of the maximum output power. This means that if profit maximization is the objective function, the number and location of wind turbines must be determined first; then, the cable deployment is completed by the placement of wind motors. Finally, the cost of cable deployment is fed back to the objective function to calculate the maximum profit. Obviously, the above optimization process has a hierarchical relationship. Therefore, a new bi-level constrained optimization model is established in this paper according to the deployment requirements of offshore wind farms. The main contributions can be concluded as follows to solve the bi-level optimization model: (1) The DE is used to solve the layout of the upper-level wind farm, and some offspring individuals are generated by DE; the surrogate model is used to approximately evaluate these offspring individuals and select more potential individuals. (2) The clustering algorithm and roulette wheel selection are used to modify the Prim algorithm to avoid intersection of lower-level cables.
The rest of the paper is organized as follows: A bilevel optimization model is established and its characteristics are described in Section 2. The handing techniques and algorithms are introduced in Section 3. The related framework and process of the algorithm are described in Section 4. Section 5 discusses the experimental results based on two wind scenarios and Section 6 presents the conclusions.

| A BI-LEVEL CONSTRAINED OPTIMIZATION MODEL
Different from onshore wind farms, the optimization of offshore wind farms mainly includes the deployment of wind turbines and an electrical system, which includes cable layout and substation location. However, the deployment of an electrical system is affected by the location of wind turbines. Moreover, the profit generated by the wind farm is also affected by the cost of the cable.

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Therefore, it is feasible to establish a bi-level optimization model.

| Upper-level model
The objective of upper-level optimization is to maximize profit, which is based on each wind turbine's power output, the unit price of electricity, the service life of wind turbines, the cost of wind farm investment, the cost of the offshore cable, and the length of the cable required.
(A) The power output Each wind turbine's power output is calculated by using the power curve model and integration technology; the detailed steps are described as follows 8 : Step 1: Probability density function of wind speed. The probability density function of the wind speed can be described by a Weibull distribution where p v c θ k θ ( , ( ), ( )) is the Weibull probability density function, v is the wind speed, and c θ ( ) and k θ ( ) are the scale and shape parameters dependent on the wind direction θ, respectively.
Step 2: Wake effect model The velocity deficit of upstream wind turbine j is affected by upstream wind turbine i, and denoted as VD j i , ; the total velocity deficit of wind turbine i affected by all other turbines is calculated, which can be calculated using the following equation: where α is the axial induction factor, C r is the fixed thrust coefficient, 30 z is the tower height of the wind turbine, z 0 is the roughness of the ground, k is the constant of entrainment, v i is the wind speed at the downstream wind turbine i, and v j is the wind speed at the upstream wind turbine j.
Step 3: Power curve model A power curve function P v can be used to describe the relationship between the power output P and the wind speed v. The logistic function is used in this paper and the values of β and γ are determined according to the actual data.
, o t h e r w i s e .
The wind turbine does not work if the wind speed v is (5); on the contrary, the wind turbine maintains the rated power output P r Step 4: Numerical integration of wind power output The expected output power of the ith wind turbine can be obtained by integrating the product of Equations (1) and (7), where g θ ( ) is the probability density function of θ v , is the wind speed, and c θ ′( ) i and k θ ( ) i are the scale and shape parameters dependent on the wind direction θ of the ith wind turbine, respectively, which can be obtained from the empirical values. The above double integration can be simplified by breaking down p v ( ) and the basic properties of the integral, that is, Obviously, the above integral is difficult to calculate; therefore, the Riemann sum is adopted to calculate the above integral. The integral region (wind speed v) is divided into h equal parts with ξ t to obtain the approximated equation of (9),

(B) The cost of wind farm investment
The cost of wind farm investment C I mainly includes construction C 1 and C 2 maintenance, that is, The cost of construction C 1 mainly includes the wind turbine unit cost, the tower, the foundation cost, and so forth. It can be computed using the following equation: where d and P max are the related parameter and the maximum generating power, respectively. N and A represent the number of wind turbines and the occupied area of the wind farm (km 2 ), respectively. c 11 represents the cost of purchase and the installation for every wind turbine and c 12 represents the land expropriation for per unit area; their values are fixed. The cost of maintenance C 2 mainly includes management costs, equipment failure maintenance, and others, and it can be calculated using the following equation: where c c b b b b , , , , , 1 2 1 2 3 4 , and δ are intermediate parameters, and the above constant coefficients can be found in the study carried out by Kiranoudis et al. 31 Y represents the life of the wind farm.
(C) The cost of the offshore cable Considering the high cost of offshore cables, cable cost C L is part of the objective function in this paper, which includes the cost of investment C 3 and construction C 4 , that is, where the cost of investment C 3 can be calculated according to the following equation: where B ij is a Boolean variable, and if a cable is connected between wind motors i and j, the value of B ij is 1; otherwise, the value of B ij is 0. L ij represents the distance between wind motors i and j T , ij is the type of cable connected between i and j C , T ij is the price per unit length of the cable, which is related to cross sections.
The cost of construction C 4 can be calculated using the following equation: where C t is the construction cost for unit length of the cable, which can be set by professional and experienced engineers. 32

(D) Wind farm layout model
The maximization of the profit is taken as the objective function of a wind farm layout, and its upperlevel optimization model is as follows: where the upper-level objective function represents the profit of the wind farm after Y years of operation, W represents the time of year when the generator generates electricity, and P represents the unit price of electricity. The variable is the position of wind turbines. The first two inequality constraints ensure that all wind turbines are deployed in the restricted wind farm area, and the third constraint ensures the minimum distance between any adjacent two wind turbines. SONG | 2779

| The lower-level model
Different from onshore wind farms, offshore wind farms focus on the layout optimization of wind turbines and the electrical system, where the electrical system mainly includes an offshore substation location and cable layout. To simplify the calculation, many studies have focused on the optimization of cable layout under the assumption that the offshore substation is fixed, which means that the optimization of the electrical system minimizes the required cable length. For optimum construction, cable crossover or overlap should be avoided when the cable topology is optimized. Besides, the number of feeders in the offshore substation should be no more than the upper bound value L U . Therefore, the lower-level model is as follows: In summary, the following bi-level constrained optimization model is constructed in this section by combining Equations (17) and (18).
where the upper-level objective function represents the profit from a wind farm operation for 20 years, and the variable x y ( , ) is the position of the wind turbine and the lower-level objective function L represents the shortest line connection length obtained under the condition that the constraint is satisfied after the upper variable x y ( , ) is fixed.

| HANDING TECHNIQUES AND ALGORITHM
To reasonably use an evolutionary algorithm to solve the bi-level optimization model and make the algorithm converge to the optimal solution as soon as possible, DE is used to optimize the upper model and an effective surrogate model is established to replace the upper objective functions for the poorer subpopulation, so as to produce better offspring individuals, and a modified Prim algorithm is used to optimize another model, which is described in the following section.

| Differential evolution
Storn et al. 33 proposed the classical DE for solving complex optimization problems and DE has been widely used to solve a series of optimization problems. It has the characteristics of a simple structure and strong adaptability. DE iteratively searches for the optimal solution by initialization, mutation, crossover, and selection; the detailed process is described below.
Mutation: Individuals are regarded as vectors and mutated individuals are generated through the compound operation between vectors; three common mutation operations are listed as follows: where the indices are randomly selected from x {1, …, NP}, best denotes the best individual in the current population, and F is the mutation factor, which determines the search space of DE.
Crossover: The crossover operation is a recombination operation between the parent individual and the mutant offspring, where the components of the crossover individual are selected from the parent or the mutation individual by the crossover probability CR, that is, where i NP j D = 1, …, , = 1, …, . j rand is an integer randomly chosen in D rand [1, ], (0, 1) is a random number in [0,1], and CR is the crossover probability.
Selection: The selection process of DE is a kind of greedy selection, that is, the individual with a better fitness value is selected between the parent individual and the corresponding crossover individual.
Obviously, generation of offspring individuals by DE is mainly related to mutation and crossover operators; the mutation operation can generate new individuals, and the crossover operator ensures the diversity of the population. However, the probability that the offspring individuals generated by DE are feasible solutions is relatively worse in this paper. Therefore, the fitness values are used to divide the parent population in this section, and differential mutation operations are designed according to the characteristics of the subpopulations. The specific process is described as follows.
Step 1: Parent individuals are divided into upper and lower subpopulations according to power output values, where the upper subpopulation is obtained from individuals with better power output values, and the lower subpopulation is made up of the remaining individuals; the size of the upper and lower subpopulations is half of the population NP.
Step 2: An external archive set is established to store the excellent individuals of history, that is, the best individuals from the previous generation are stored in the external archive.
Step 3: The mutation operator of the upper subpopulation is designed with the external archive, which can enhance the exploitation where x i represents the ith parent individual of the upper subpopulation, x hbest is an individual that is randomly selected from the set of the external archive, and x r1 and x r2 are two random individuals different from x i , which are selected from upper subpopulations, respectively. Obviously, the mutation operator of the upper subpopulation is designed with the external archive, which can enhance the exploitation capability of DE.
Step 4: The crossover operation and boundary constraint processing are performed for each mutation individual. Then, it is determined whether the crossover individual satisfies the constraint conditions. If all constraint conditions are satisfied, the crossover individual is selected to as an offspring individual; otherwise, the parent individual is retained.
Step 5: The selection process of DE is performed to select the next-generation parent individuals of the upper subpopulation.
Step 6: The mutation operator of the lower subpopulation is designed by DE/rand/1 to enhance the exploration capability of DE, where a certain number of mutation individuals are generated for each individual of the lower subpopulation, and the crossover operation and boundary constraint processing are performed for the above mutation individuals to formulate potential offspring.

| Surrogate model based on self-adaptive sampling
Wang et al. designed a new encoding mechanism for the locations of wind turbines 12 that is based on the characteristics of the wind farm layout. The above encoding means that the whole population represents a layout. Experimental results show that the proposed encoding method outperforms other algorithms to find the optimal layout; therefore, this paper also uses the same coding method in the population of initialization for the upper level. However, because of the high dimensionality of the decision variable and the restriction of the constraint function, it is difficult to produce high-quality feasible solutions using this algorithm. Therefore, how to design effective evolutionary strategies to produce potential individuals is important. The surrogate model is essentially an approximate model that can establish the relationship between the decision variables and the objective functions. For expensive optimization problems, a certain number of sample points are selected in the decision space and the response values of sample points are obtained using the SONG | 2781 simulation method. The surrogate model can be established according to the sample points and their response values. For any input variable, the surrogate model can yield its predicted value. Obviously, the optimization of the original problem is transformed into the optimization of the surrogate model, which can improve the optimization efficiency of the problem. Therefore, the surrogate model of the radial basis function (RBF) is used to approximate the objective function of the upper level in this paper, which can reduce the computation and produce better offspring. However, the accuracy of the surrogate model is linked to the sampling method; therefore, an adaptive strategy is designed in this section to ensure the convergence of the algorithm, which is based on the sparsity and the objective value. The detailed process is described below.
Step 1: The Latin square is used to generate an initial sample S x x x = { , , …, } n 0 1 2 of size n and calculate the corresponding objective functions.
Step 2: RBF is used to construct the surrogate model of the objective function where n is the number of sample points,   x x − i is the Euclidean distance between the point to be measured and the ith sample, w i is the weight coefficient of the sample points, ϕ is the basis function, and a Gaussian model is adopted in this section where c is the smoothing parameter and is set to 3.
Step 3: The Tyson polygonal is used to divide the design space into small subregions, which is based on the samples in the design space, and each subregion contains only one sample point.
Step 4: A large number of discrete points can be evenly produced in the design space; obviously, the larger polygon contains more discrete points. Therefore, the ratio of a polygon can be quantified by using the number of discrete points present in the polygon, that is, where R i represents the radius value of the ith polygon, S i represents the number of discrete points in the ith polygon, and S represents the total number of sizes. The polygon with the largest radius value can be considered as the region with sparse samples. Therefore, more samples should be obtained in this kind of region.
Step 5: The objective function values of the offspring individuals are calculated, and m better indivi- are selected from the offspring individuals using the fitness values. Then, these objective values are normalized as follows: where f min and f max represent the maximum value and the minimum value in set X b , respectively.
Step 6: The polygon with the maximum radius value is selected, and the corresponding central sample point is denoted as x c . Then, the Euclidean distance L x x ( , ) bi c is calculated between the above better individuals X x and x c . Next, these Euclidean distances are normalized as follows: where L min and L max represent the maximum value and the minimum value of the Euclidean distances in set X b , respectively. Step 7: The following weight values are designed using Equations (28) and (29), and the individual corresponding to the smallest weight value is selected as the new sample point.
where the weight value ω is set as follows:

| Improved Prim algorithm
The Prim algorithm is one of the classical algorithms for solving the minimum spanning tree, which is a greedy algorithm based on an undirected graph and a weighted value; it is widely used to solve shortest path optimization problems, including the cabling problem of offshore wind farms. However, the Prim algorithm cannot avoid the problem of cables crossing each other and the limitation of the number of offshore substations. Therefore, according to the micro position of the wind farm feedback from the upper-layer optimization, the wind turbines are divided into s subpopulations by the clustering algorithm of the Gaussian mixture model (GMM), where GMM is a probability model to describe the clustering structure. The specific process is described as follows: (a) Set the number of clusters k and initialize the Gaussian distribution parameters of each cluster.
Since there are no data at the beginning, the value can be assigned randomly; (b) scan all sample data and calculate the conditional probability that each sample conforms to each distribution; (c) calculate the new parameters using conditional probability, and update the distribution model using the new parameters; (d) repeat the above two steps to update the model distribution parameters until the iteration is stopped; (e) calculate the conditional probability of each sample according to the new model parameters, and then obtain the maximum value; and (f) divide each sample into the corresponding cluster according to the maximum value. Moreover, wheel selection is designed in this section to select the candidate nodes, which can avoid the intersection problem between cables for every subpopulations as much as possible.
Step 1: The position of the offshore substation is governed by the wind motors' position of the subpopulation and the scale of the wind farm, that is, initial positions of the offshore substation v i0 are randomly generated between the central position of the wind farm and the central position of wind turbines, i s = 1, 2, …, .
Step 2: The distance between any two nodes of the ith subpopulation is calculated as weight values; weight values and these nodes are combined to construct an undirected graph G V W = ( , ) i i , where V i is a set of nodes and W i is a set of weight values.
Step 3: Calculate the distance between all nodes to be selected from ith subpopulation and the v i1 , and select the node with the shortest distance as the next node, denoted as can be selected from V i using W i , which are shorter distances from the last node in set S i .
Step 5: Rank the above weight values and record these ranks R R R { , , …, } i i iK 1 2 . Then, the selection probability is designed from the above ranks.
Step 6: If the selection probability is determined, a potential node v 12 can be obtained by roulette wheel selection, and v 12 is merged into Step 7: Perform Steps 4-6 and if the size of set S i exceeds 3, then the last three nodes v v v , , ij ij ij −1 + 1 in set S i can be used and construct the vectors ; the crossing between cables is determined by calculating the cross product value of vectors, that is, If  λ 0, vectors     AB BC , and other nodes do not form a circle, which means that there is a high probability that the cable lines do not intersect. Therefore, the node v ij+1 is retained and merged into the set S i ; otherwise, the node v ij+1 is discarded and roulette wheel selection is used to find the next available node; the above operation is continued until a candidate solution is found, and if all λ do not satisfy the condition, then the candidate solution is chosen using the maximum values λ.
As shown in Figure 1, taking the undirected graph with seven nodes as an example, the improved prim algorithm is described as follows:    (20) and (23) are first used to produce a certain number T of potential offspring; then, the fitness of offspring individuals is estimated approximately using the surrogate model (25), and the optimal individual is selected as the offspring individual. Finally, the true fitness value of the optimal individual is calculated. (d) The fitness values are used to select offspring individuals and feed back to the lower level. (e) The weighted values are calculated using Equations (28) and (29), and a new sample is selected based on the weighted values to merge into the data sample set. (f) The improved Prim algorithm is used to complete the cable deployment problem of the lower layer and feed T A B L E 2 Distribution of Wind Scenario I.  back to the upper level. (g) The objective function values of the upper level are calculated and the optimal value is output. The above evolution process is iterated sequentially, until the maximum profit value and the minimum cable length value are output when the stopping criteria are satisfied.

| Comparison algorithm and parameter settings
In this section, five different evolutionary algorithms, a DE with a two-stage optimization mechanism 34 (TSDE), a DE variant using feedback information to adjust the control parameters 35 Tables 2 and 3.
This section describes the parameters involved in the investment cost, the specification of wind turbines, the wind farm scale, and the unit price of cables. Detailed parameter values are listed in Tables 4 and 5. The specification GE1.5-77 is adopted as the model of wind turbines and the detailed parameters are shown in Table 4. Related parameters of investment cost, including installation and maintenance costs for wind turbines and offshore cables, are shown in Table 5. Based on experience, the service life Y of the wind farm is set to 20 years.
Besides, the maximum number of fitness evaluations (MaxFEs) is taken to the stopping standard, and the MaxFEs is set as 30,000. For each compared algorithm, 20 independent runs are executed on the wind scenario with a specified number of wind turbines, and the maximum output power, the mean, and the standard deviation of the profit and the shortest path are calculated. In all the tables of this section, "+" and " − " denote that the performance of other competitor is better than and worse than IPADE, respectively. Moreover, the Friedman test and Wilcoxon's rank sum test, as two no-parameter hypothesis tests, are used to test the statistical significance between IPADE and each compared algorithm, where the significance level is set as 0.05.
T A B L E 4 Model specifications of wind turbines.

Explanation Parameter Value
Rated power output of wind turbine (kW)

P r 1500
Minimum distance D min R 5 The thrust coefficient C r 0.8 The rotor radium (m) R 40 The hub height (m) z 80 Wind direction The scale parameters of the c θ ( ) β 6.0268 The shape parameters of the c θ ( ) γ 0.0007 T A B L E 5 Related parameters of investment cost.

Explanation Parameter Value
The maximum generating power (kW) The life of the wind farm (year) Y 20 The parameter of maintenance cost d 1.74 × 10 −3 The cost of purchase and the installation c 11 1680 The land expropriation for per unit area The parameter of maintenance cost c 21 0.82 The parameter of maintenance cost c 22  Intermediate parameter δ 4 The unit price of cable (¥/km) C t 1680 The unit price of electricity ( ∕ ¥ kWh) For a fair comparison of the algorithms, the size of the population for each comparison algorithm is set to NP = 100, and the parameter settings of the four compared algorithms are described as follows: The parameter pools of mutation and crossover probability for algorithm TSDE are set to F = [1.0, 1.0, 0.8] and CR = [0.1, 0.9, 0.2], respectively. The parameters of PSO include four, where the learning factor of individual is set to 2, the learning factor of social experience is set to 2, the inertia weight is set to 2, and the maximum flying speed of a particle is set to 0.8. The relevant parameters of other comparison algorithms are taken from the original paper.
The parameter settings of IPADE are listed in Table 6, including the size of potential individuals, the factor of mutation F , the probability of crossover CR, and so forth.

| Experimental results of Wind Scenario I
For different wind turbines and scales, Table 7 shows the best performance in terms of the maximum power output when the six comparison algorithms satisfy the stopping criteria. Tables 8 and 9 show the mean values (Mean) and standard values (Std Dev) in terms of the maximum profit and the minimum cable length, where the optimal result is highlighted in bold.
Obviously, the proportion performing optimally on comparing with algorithm IPADE is more than half for the maximum power output. Table 8 summarizes the experimental results for the objective function of the upper level, which shows the Mean and Std Dev of the profit, and the above values are obtained by the six compared algorithms over 20 independent runs. As shown in Table 8, IPADE performs better than the five competitors on N = 40, N = 70, N = 90, and N = 100. The best performance of algorithm IPADE is attributed to the effective mutation algorithm, which enhances the search ability of the algorithm. Meanwhile, the surrogate model can generate potential excellent individuals with a high probability. Besides, IPADE achieves better performance than TSDE, JADEsort, PSO, CSO and RBLSO on 5, 7, 7, 4, and 5 test situations, respectively; the proportion is more than 50%. It is worth noting that the standard deviation of algorithm IPADE is the smallest compared with other algorithms, except for N = 50, N = 80, and N = 100, which means that algorithm IPADE has stronger robustness than others. Table 9 summarizes the experimental results for the objective function of the lower level, which shows the Mean and Std Dev of the cable length. Obviously, the mean values of IPADE increase with increasing number of wind turbines. Moreover, algorithm IPADE has the smallest Std Dev when compared with the other five algorithms, which implies that IPADE has better attributes for achieving a stable solution.
Based on the average of profit and cable length, Friedman's test was carried out using the SPSS platform. Table 10 summarizes the statistical test results. As shown T A B L E 6 Relevant parameters of IPADE.

Explanation
Parameter Value The size of the population NP 100 The size of the better individuals m 10 The factor of mutation F 0.5 The probability of crossover CR 0.5 The sample point size of the Tyson polygon S 300 The size of the initial sample S 0 50 The size of individuals using a surrogate model The size of the potential offspring T 30 in the first column of Table 10, IPADE has the best ranking in terms of average profit and it has a significant advantage over CSO, which ranks second. Similarly, the second column of Table 10 shows that IPADE has the best ranking in terms of the average cable length. Table 11 shows the results of the compared algorithms using multiple-problem Wilcoxon's test. The confidence level is set to 0.05. The first three columns of Table 11 show the test results based on the mean value of profits, while the last three columns show the test results based on the mean value of cable length. It is obvious that all R + values are greater than R − , which further implies that IPADE outperforms the other algorithms. In particular, the p value in bold in Table 11 indicates that IPADE and the other algorithms show obvious differences. Tables 12-16 show the related experimental results of Wind Scenario II, including the maximum power output, the maximum profit, and the minimum length cable. The wind distribution of scenario II is relatively simple and the wind speed is higher than that in Wind Scenario I; therefore, the maximum power output and profit of each algorithm in Wind Scenario II are higher than that in Wind Scenario I. The main reason for this is that wind distribution focuses on a small scale, and so the wake effect can be avoided more easily.

| Experimental results of Wind Scenario II
The experimental results in Table 12 show that the output power of IPADE is optimal when the number of wind turbines is 60, 80, and 100, respectively.
The experimental results in Table 13 show that in terms of profit maximization, the best performance of algorithm IPADE accounts for nearly half. In addition, the last row of Table 13 numerically shows how IPADE outperforms each of the compared algorithms. Obviously, algorithms PSO and CSO only account for 1/7 in terms of performance when compared with IPADE. Besides, based on the standard deviation values, the robustness of the IPADE is obviously better than the compared algorithms on N = 40, N = 60, N = 70, and N = 100.
The experimental results in Table 14 show that in terms of cable length minimization, the cable length of IPADE under N = 70, N = 90, and 100 is the smallest. Moreover, although IPADE is worse than TSDE in the cases of N = 40, N = 60, and N = 80, the standard deviation values of IPADE are relatively smaller than TSDE. It is worth mentioning that the average cable length of six compared algorithms is closer under different numbers of wind turbines, but the standard deviation value of IPADE is smaller than the other algorithms in most cases. This implies that IPADE has better stability and convergence. Table 15 shows that the algorithm IPADE has the best ranking in terms of profit and cable length. Table 16 shows the statistical test results based on the multiple-problem Wilcoxon test. The results in the table show that the value of R + for algorithm IPADE is significantly greater than R − for each comparison algorithm, which means that the performance of algorithm IPADE is better than the other algorithms.

| CONCLUSION
To comprehensively consider the deployment optimization of offshore wind farms, a bi-level constrained optimization model is established in this paper, which is constructed using the maximum profit and the shortest cable length as the upper-level and lowerlevel objective functions, respectively. In addition, an evolutionary algorithm (IPADE) based on DE and a data-driven surrogate model is proposed to optimize the upper-level objective function, where the datadriven surrogate model was utilized to approximate the objective functions of offspring individuals and the optimal individual was selected as the offspring individual by the approximate objective value; this operation can improve the search capability of DE. Moreover, an improved Prim algorithm based on roulette wheel selection is designed to solve the problem of cable deployment in the lower level; this process can avoid crossing between cables. The performance of IPADE is verified by simulation experiments comparing the five latest algorithms. The results show that IPADE has better convergence. However, IPADE uses a surrogate model to search for potential solutions, which can generate higher computing costs. In addition, the model in this paper is only applicable to rectangular regions. Therefore, we look forward to solving the deployment of offshore wind turbines in more general situations in future work.