The topological structure of the array cables of the offshore wind farm cluster can be regarded as a directed graph. As a collection of wind farm array cables and AC export cables, the OWFCTS can be represented as follows: O W F C T S = G 1 V , A , E C 1 , G 2 V , A , E C 2 , ⋯ , G i V , A , E C i , (1) G V , A = g 1 V , A , g 2 V , A , ⋯ , g j V , A , (2) V = V 0 , V 1 , V 2 , ⋯ , V k , (3) A = A 01 , A 12 , A 23 , ⋯ , A 01 m , n ∈ 1 , 2 , ⋯ , k , m ≠ n , (4) where EC _i is an AC export cable; G (V, A) is the set of array cables of the wind farm, which consist of several feeders g (V, A), where V represents the vertex on each feeder, V ₀ is the substation at the end of the feeder, and V ₁∼V _k stand for wind turbines; A is the set of the length of array cables; and A _mn is the length of the cable from turbine m to turbine n.

The OWFCTS generally uses several types of cables for economic reasons. The price of cables P is shown in Eq. 5, and C _mn (Eq. 6) is the cost of the cable from turbine m to turbine n. Figure 4 represents the array cables of any feeder in a wind farm, and the topology is represented, as shown in Eq. 7. To express a clear representation of the distribution of the cable cost of the OWFCTS, this paper introduces the degree matrix D to represent the cost of each segment of the cable in the wind farm. The elements in the degree matrix D are related to the weight matrix W, where D _i = W _i1+ W _i2+…+ W _ij . The traces of the matrix tr(D) which is the sum of the absolute values of the diagonal elements of the matrix in Eq. 9 is the total cost of the cables of any feeders of the array cables. As shown in Eq. 10, the total cost of the array cables is the sum of the traces of the matrix of the individual feeder. As shown in Eq. 11, the total cable cost of the OWFCTS is the sum of the cable cost of the array cables of the wind farms and the cost of the AC export cables, where Cost _i (G _i (V, A)) is related to the cable section areas and the length of the array cables. The cable section areas are first selected via the maximum long-term operating current method in Section 2.3 and then checked by the voltage drop. The cable length is obtained by the optimization of the Prim method. Cost _i (EC _i ) is related to the path length of the AC export cables that is optimized by the Informed-RRT* method presented in Section 2.2. P = P 1 , P 2 , ⋯ , P i , (5) C m n = A m n ⋅ P i , (6) g V , C = V = V 1 , V 2 , V 3 , V 4 , V 5 , V 6 , V 7 C = C 10 , C 21 , C 31 , C 43 , C 53 , C 65 , C 75 , (7) W = 0 0 0 0 0 0 0 0 C 10 0 0 0 0 0 0 0 0 C 21 0 0 0 0 0 0 0 C 31 0 0 0 0 0 0 0 0 0 C 43 0 0 0 0 0 0 0 C 53 0 0 0 0 0 0 0 0 0 C 65 0 0 0 0 0 0 0 C 75 0 0 , (8) D = 0 C 10 C 21 C 31 C 43 C 53 C 65 C 75 , (9) Cost ( G V , A = ∑ i = 1 n tr i D i , (10) COST OWFCTS = ∑ i = 1 m Cost i ( G i V , A + ∑ i = 1 m Cost i EC i . (11)

Due to the long length and high cost of array cables, the method of maximum allowable continuous working current is applied to select economic cable section areas, which are then checked by the allowable voltage drop.

1) Maximum long-term operating current

As shown in Eq. 12, first, we calculate the maximum allowable continuous working current I _g of the wind turbines. I g = N P e ⋅ 1000 3 U N ⋅ c o s   ϕ , (12) where N is the number of wind turbines in the feeder; P _e is the rated power (MW); U _N is the rated voltage of cables (kV); and cos ⁡ ϕ is the power factor and generally taken at 0.95 in wind farms. I _g should be smaller than the allowable cable operating current I _al . K I a l   ⩾   I g , (13) where K is the comprehensive correction factor for different laying conditions. In the case of cable laying in offshore wind farms, K is given by K = K θ K 1 K 2 , (14) where K ₁ = 1.2 is correlated with soil thermal resistance in offshore wind farm cables; K ₂ is correlated with cable laying spacing; and K _θ is correlated with cable working environment temperature. K θ = θ a l   ‐ θ 0 θ a l   ‐ θ N , (15) where θ _al is the maximum operating temperature of the cable (°C), which varies by the cable type; θ ₀ is the working environment temperature (°C); and θ _N = 15 °C is the reference ambient temperature corresponding to the rated load capacity in the offshore wind farm.

2) Voltage drop checking

According to [2], the cable voltage drop in the offshore wind farm array cables should not exceed 5% in each feeder. The percentage voltage drop during normal operation of the cable can be obtained from the following equation: ∆ U % = 3 I g L r c o s ϕ + x s i n ϕ 1000 * U N * 100 % , (16) where L is the length of cables (km) and r and x are the resistance and reactance per unit length of the line (Ω/km), respectively.

The basic idea of the gradient descent algorithm is that the new substation is randomly generated on the circumference of a circle with the previous substation as the center until the cable cost decreases. Meanwhile, the searching step is dynamic due to the change in the cost to effectively find the optimal location. The algorithm also combines with a stochastic method to jump out of the local optimum. Based on the selected starting point, the gradient descent algorithm is used to optimize the location of the substation. The optimized starting point (x ₀, y ₀) is generally selected as the center of the wind turbine sites, as shown in the following equation: x 0 = ∑ i = 1 N N x i , y 0 = ∑ i = 1 N N y i . (17)

The principle of substation location optimization by the gradient descent algorithm is shown in Figure 5 with the following steps:

Step 1:

Calculating the cable cost at the starting point: Taking (x ₀, y ₀) as the location of the substation, the Informed-RRT* method is used to optimize the AC export cable path, Prim’s algorithm is used to optimize the array cables, and then, the cable cost COST ₀ at the starting point (x ₀, y ₀) is calculated.

Step 2:

first step of optimization: Taking (x ₀, y ₀) as the starting point, l ₀ as the initial step, and random azimuth θ _i as the direction, the random point R _i (x _i , y _i ) is obtained. Then, taking R _i (x _i , y _i ) as the boosting station, the Informed-RRT* method and Prim algorithm are used to optimize the AC export cable paths and the array cables, respectively, and the cost COST ₁ is calculated. If COST ₁ <COST ₀, then the point is valid, and it is the starting point for the next step of random point generation; if COST ₁ >COST ₀, the point is invalid, the azimuth θ _i is re-generated, and the aforementioned steps are repeated.

Step 3:

The second step of optimization: Taking (x ₁, y ₁) as the starting point, random azimuth θ _i as the direction, and l ₁ as the step size, we obtain the random point R _i (x _i , y _i ), where l ₁ = l ₀*k, k > 1. The value of k and the decline in the ΔCOST are related; the larger the ΔCOST, indicating a more substantial decline in the cost, the larger the k. Using the same method in step 2, we obtain the starting point of the next random point generation (x ₂, y ₂).

Step 4:

Gradient descent algorithm optimization: Starting with this (x ₂, y ₂), the optimization process produces two scenarios.

Case 1:

With (x ₂, y ₂) as the starting point and l ₂ = l ₁*k, k > 1 as the step, we obtain the random point R _i (x _i , y _i ), and being able to find the point COST _i <COST ₂ is a good case to obtain (x ₃, y ₃).

Case 2:

If the method of Case 1 is used to search for 10 points and still no (x ₃, y ₃) is found, then the step size is modified to l ₃ ^` = l ₂*(1/k), k > 1, random points are generated, and the aforementioned steps are repeated to find (x ₃, y ₃).

Step 4 is repeated; if no superior position appears after 50 optimal steps, OWFCTS optimization is accomplished.

The RRT algorithm is a sampling-based path planning method; however, due to the randomness of its sampling, it resulted in the final generated path that is often only a feasible path rather than an optimized path. The Informed-RRT* algorithm improves on the RRT* method in three ways (Gammell et al., 2014): (1) the Informed-RRT* method first uses the RRT* method to find a feasible path, and second, it takes the starting point and the endpoint as the focus of the ellipse, forming an ellipse that envelopes the first path, and then, the random points generated in subsequent searches are only within the ellipse; (2) the Informed-RRT* method optimizes the path from the parent node to the new node after the new node is found; and (3) if the new node is randomly obtained to be on the side of the node start point of the already generated tree, the new node can be used to optimize the already generated tree and shorten the new node’s length of tree branches within the range. The Informed-RRT* method of searching the path is shown in Figure 6.

The main process of the Informed-RRT* algorithm is as follows:

Step 1:

Taking the location of the substation as the starting point (x _start , y _start ) of the randomized tree.

Step 2:

Generating a random point (x _rand , y _rand ) within the whole wind farm area, the random point has a certain probability of being the location of the converter station (x _end , y _end ).

Step 3:

Calculating the distance between each node in the random tree at this point and the generated random point (x _rand , y _rand ) and finding the nearest node to this random point, denoted as (x _nearest , y _nearest ).

Step 4:

(x _nearest , y _nearest ) is connected to (x _rand , y _rand ), and then, it is determined whether this line segment intersects with an obstacle or a sensitive area; if it does not intersect, then this path is available; otherwise, we go back and re-select the random point (x _rand , y _rand ).

Step 5:

Optimizing the parent node of the new node (x _new , y _new ): We search for the already generated nodes using a distance greater than the step size, and if there exists the point (x, y) such that the path from (x, y) to (x _new , y _new ) is less than the path from (x, y) to (x _nearest , y _nearest ) and then to (x _new , y _new ), and there is no intersection with the obstacle yet, then (x, y) is selected as the parent node of the new node (x _new , y _new ) to optimize the path.

Step 6:

After the formation of a complete path through the aforementioned steps, the starting point and the endpoint serve as the vertices of the ellipse, generating an ellipse that encompasses the initial path, as shown in Figure 6, represented by the green line in the figure that is generated by the ellipse.

Step 7:

continue to repeat steps 2 through 7, with the difference that the random points generated in step 2 are all within the ellipse. Optimization is continued until the number of optimizations reaches the set upper limit.

The obstacle to be avoided in the optimization is the wind turbine safety distance, which is defined as follows as shown by the red circle in Figure 6: Safety _ Dist = 1.5 ⋅ H + 0.5 D , (18) where H is the hub height of the wind turbine and D is the diameter.

By using the optimization method and parameter selection proposed in this paper, an integrated optimization of OWFCTS is carried out. Figure 8 shows the layout of the cables and the location of the substation in the wind farm cluster after optimization. The comparison of cable length and cable cost before and after optimization is shown in Table 3. After optimization, the total length of array cables in the wind farm cluster is reduced by 319.79 m, a decrease of 0.17%; the path length of AC export cables is reduced by 5640.66 m, a decrease of 13.82%; and the total cable cost after optimization is reduced by 3.58%. The significant reduction in AC export cables is the main reason for the decrease in the total cable cost. As shown in Figure 9, in the path optimization of the OWFCTS, the total cost of the cable decreases the fastest when the number of optimization steps is between 0 and 40 because the original location of the substation is not optimized, which leads to a large decrease in cable cost at the beginning of optimization. Between 40 and 180 steps, the gradient of cable cost decrease becomes smaller, and the optimization enters the bottleneck position. Between 180 and 200 steps, there is a small stepdown in cable cost, which is due to the use of random methods to jump out of the local optimum, and a relatively optimal substation location is found.