Wind Farm Cable Connection Layout Optimization with Several Substations

: Green energy has become a media issue due to climate changes, and consequently, the population has become more aware of pollution. Wind farms are an essential energy production alternative to fossil energy. The incentive to produce wind energy was a government policy some decades ago to decrease carbon emissions. In recent decades, wind farms were formed by a substation and a couple of turbines. Nowadays, wind farms are designed with hundreds of turbines requiring more than one substation. This paper formulates an integer linear programming model to design wind farms’ cable layout with several turbines. The proposed model obtains the optimal solution considering different cable types, infrastructure costs, and energy losses. An additional constraint was considered to limit the number of cables that cross a walkway, i.e., the number of connections between a set of wind turbines and the remaining wind farm. Furthermore, considering a discrete set of possible turbine locations, the model allows identifying those that should be present in the optimal solution, thereby addressing the optimal location of the substation(s) in the wind farm. The paper illustrates solutions and the associated costs of two wind farms, with up to 102 turbines and three substations in the optimal solution, selected among sixteen possible places. The optimal solutions are obtained in a short time.


General Context and Motivation
Climatic changes and the population's awareness of pollution have contributed to policymakers changing their policies on energy production. On the other hand, we have seen an exponential increase in the world population resulting in high electricity consumption through the industrial and domestic load. Governments have cherished clean energies to the detriment of fossil energies that contribute to the planet's pollution. Wind energy has benefited from these policies. Wind energy has been a considerable investment for almost all developed countries in the last two decades, with a strong bet on onshore wind farms. This commitment to strengthening renewable energies with a particular focus on wind power will continue in the coming decades. An example of this strategy is the Renewable Energy Directive, Directive (EU) 2018/2001, (RED II), which established a common framework for the promotion of energy from renewable sources in the EU and set a binding target of 32% for the overall share of energy from renewable sources in the EU's gross final consumption of energy in 2030 [1]. In this context, the design of large wind farms using several substations is becoming increasingly common, bringing more efficient and more profitable exploration models to the new wind farm concessions.
Chen et al. [14] use a fuzzy clustering algorithm to assign the turbines to a substation. The optimization of the cable layout is carried on using the minimum spanning tree algorithm. Considering meta-heuristics approaches and the optimization in two independent phases could lead to suboptimal solutions. In their work, a wind farm with three substations and 50 turbines was presented.
Wang et al. [15,23] use an integrated design method to minimize the total cost considering the substation location, connection topology, and cable cross-sections. The method uses an evolutionary algorithm to determine the substations coordinates, the substation associated with each turbine, and the cable layout. They applied the algorithm in a wind farm with two substations and up to 66 turbines.
Pillai et al. [16] use an approach for a wind farm's cable design layout. To solve the problem, they divide it into two phases. The first phase places the substation using the capacitated clustering algorithm. In the second phase, for each subproblem, they use a mixed integer linear program (MILP) to find out the cable connection layout and cable type to be installed. The MILP takes into account initial solutions provided by some heuristics.
Huang et al. [17] propose a genetic algorithms to optimize the Horn Rev wind farm. They consider the transformers connected to wind turbines, substations costs, and cable costs.
Zuo et al. [18] analyzed the economic and reliability benefits of the collector system topology for offshore wind farms' repowering and expansion. The method was based on cross-substation incorporation and radial topology to link repowered old wind farms and new ones. The multilayer optimization consists of an offshore substation (OS) refinement layer, an offshore wind farm partition layer, and an intra-zone cable connection layer. The collector system connection design was based on the predetermined WT selections and layouts, where the WTs were uniformly distributed by the substations. Moreover, the authors considered the repowered farms as one wind turbine. In each zone, the cable connection was based on the minimum spanning tree. A fuzzy clustering technique was used to determine the substation locations.
Dutta and Overbye [19] proposed a clustering-based algorithm for the cable layout of a large-scale wind power plant. Comparison of the proposed method with the radial feeder cable configuration shows that real power losses in the collector system are lowered and greater reliability is achieved with the proposed design. In this work, three different cable layout configurations have been compared.
Wu and Wang [21] use an ant colony algorithm to reduce wind farms' construction costs and to improve the reliability of the collector system. They use the K-means clustering algorithm to partition the wind turbines into several groups to find the shorter collection lines in a radial configuration for the wind farm. They presented a problem with four substations and regions.
Pérez-Rúa et al. [22] propose a mixed-integer linear programming to optimize the cable system of offshore wind farms considering economic costs and losses power. The cable layout encompasses the interconnection between wind turbines (WTs) and transmission systems to couple Offshore Substations (OSSs) to the Onshore Connection Point. The work considers three case studies with up to 175 turbines and eight possible locations for substations, of which two are intended to be chosen. The computational time ranges from 1.48 h to 23.29 h.

Contribution and Document Structure
This work proposes an exact methodology to solve the optimization wind farm layout problem with several substations. Additionally, restrictions are used to limit connections on walkways. This paper's main contribution is to efficiently solve the wind farm layout optimization, considering the topology and cable connection simultaneously, with several substations and many wind turbines. The proposed integer linear programming model determines the topology, selecting the connections and the substations that should be in the optimal solution and optimal cable to be used, minimizing energy losses and cable installation costs, all at the same time. The model also determines the optimal set of substations to be in the optimal solution.
The paper is divided into five sections. Section 2 describes the model of the electrical power grid underlying the wind farms. In Section 3, an integer linear programming model to optimize the wind farm layout considering several substations is presented. Section 4 presents and discusses the obtained results. Finally, Section 5 draws the main conclusions.

Electrical Grid Power Flow
The steady-state analysis of the power flow in the wind farm distribution network, where each wind turbine represents a node, assumes a fundamental role in this kind of research. It is essential to know all the network parameters to develop an equivalent model for all the constituent elements. Usually, this collector system is established with voltage levels between 20 kV and 30 kV. Since these networks have a radial structure, it is crucial to calculate the power flow circulating in each branch. Some research papers address exactly the problem of power flow in radial networks [5,[24][25][26]. In addition, research such as [27], dedicated to the problem of reactive energy optimization in radial networks, present contributions to the calculation of the load flow also adapted to the radial feature of these networks.
As referred to in Reference [5], in the study of wind farms collecting system, the short line model must be used. This is due to the fact that the networks that connect the various elements of a wind farm are not longer than hundreds of meters or a few kilometers, where the R/X ratio is high. Considering the short line model, several simplifications can be made in the model, and the cables' shunt admittance can be neglected. Therefore, the network branches can be represented by the model of Figure 1, where active (P n ) and reactive (Q n ) power flows between the buses are calculated by using Equations (1) and (2), respectively. The bus voltage is calculated using Equation (3) [28]. In this model, the turbines are represented as injector points of active and reactive power in the collecting system.
where R n,n+1 and X n,n+1 represents the branch resistance and reactance between buses n and n + 1, respectively. Considering the active and reactive branch losses are given by Equations (5) and (6), respectively.
Therefore, the total power losses, P loss total , can then be obtained by adding all losses from all line sections as given by P loss total = ∑ k∈B P loss(k,k+1) + Q loss(k,k+1) (7) where B is the set of the buses in the network. The resistance R n,n+1 and reactance X n,n+1 values between buses depend on the cable characteristics, namely: resistance R, inductance L, and the maximum current I z that can support. This latter value limits the number of turbines connected through a cable. In fact, the rated current drawn by each turbine, with a rated power P r and a U interconnection grid voltage, is defined by: where the value of cos ϕ is the turbines' power factor. For example, assuming that P r = 2 MW and U = 20 kV, the rated current drawn by each turbine is I r = 57.735 A. The I r value restricts the cable type used in a connection depending on the number of downstream wind turbines.  Figure 2, with two substations, nodes 19 and 20, and eighteen wind turbines, nodes 1 to 18. This layout has four branch lines connected to each substation. The total current reaching a substation from a branch line is the sum of the currents drawn by all turbines connected through this branch. For example, the branch starting in connection (19, 3) supports three wind turbines (including turbine 3), and so, the current crossing this cable is 3 × I r = 173.22 A. The branch starting in connection (20,12) supports five wind turbines (including turbine 12), and so, the current crossing this cable is 5 × I r = 288.68 A. In this work, the available cable types are presented in Table 1. So, in the connection (19, 3) of Figure 2 a cable type less than k = 2 cannot be used, which has I z = 207 A, and in connection (20, 12) a cable type less than k = 5 cannot be used, which has I z = 313 A.
Among all the available cable types, the one with the highest I z value is k = 10 with I z 10 = 585 A. So, in the case that I r = 57.735 A, any branch line cannot have more than I z 10 I r = 585 57.7 = 10 wind turbines, where a denotes the maximum integer not greater than a.

Mathematical Formulation
This section presents an Integer Linear Programming (ILP) model to obtain the optimal wind farm layout considering the topology and cables connection simultaneously for wind farms with several substations.

Data Sets and Parameters
Consider the node-sets N = {1, . . . , n}, corresponding to wind turbines location and S = {n + 1, . . . , n + s} corresponding to substation locations. The goal is to obtain the wind-farm connection layout, i.e., a spanning forest of a graph, where the root of each spanning trees is a node belonging to set S.
In the layout problem is considered the set of available cable types K = {1, . . . , k} presented in Table 1. The maximum current intensity I z k that each cable type k ∈ K can support bounds the number of downstream wind turbines for this cable by I z k I r . So, the maximum number of wind turbines in any branch line is: Therefore, the number of downstream wind turbines that each connection (i, j) can support, being node i in the substation side, is: Another parameter to consider is the F parameter that represents the maximum number of connections that can be linked to the substation, called feeders.

Costs
The total cost is given by the sum of the costs of active losses, c p , and reactive losses, c q , during the expected wind farm lifetime, and the infrastructure cost, c I , which includes the cable costs and digging cost.
For all pairs of nodes (i, j) ∈ A it is known the distance ij , in meters, between them. By using Equations (5) and (6), the cost of the active and reactive losses, during the wind farm lifetime, in a connection (i, j) supporting t downstream turbines, using a three-phase cable of type k, are, respectively, and where h is the number of hours during the expected wind farm lifetime, c ep is the active energy cost, c eq is the reactive energy cost, and l f is the load factor which reflects the real operating conditions during the wind farm lifetime; ω is the angular frequency. The infrastructure cost to make a cable connection (i, j) using a cable of a type k is where D is the digging cost (EUR by meter) and 3 · C k is the three-phase cable cost (EUR by meter). Following [3], a preprocessing calculus is performed to determine the optimal cable type, k t ij , for a connection (i, j) supporting the current of t downstream wind turbines, and the correspondent cost, is the minimum cost for the connection (i, j) with t downstream turbines.

Decision Variables
The decision variables are: • For all (i, j) ∈ A, binary variables x t ij taking value 1 if the nodes i and j are connected (being node i on the substation side) and supports the current of t downstream wind turbines (including the one located in j); otherwise, it takes value zero. • For all i ∈ S, binary variables w i taking value 1 if substation i is in the solution; otherwise, it takes value zero.

Cable Connection Layout Model
The ILP model to optimize the wind farm layout considering multiple substations, WFLMS, is given by: x t ij ∈ {0, 1}, (i, j) ∈ A, t = 1, . . . , Q(i) The objective function (15) minimizes the total cost layout. Equation (16) guarantees that the network connects n wind turbines. Constraints (17) impose the maximum number of feeders, F, that can enter each substation. Constraints (18) guarantee that each wind turbine j ∈ N has one incoming connection. Constraints (19) are the flow conservation constraints and guarantee that for each wind turbine j ∈ N, if there exists an incoming connection supporting t downstream wind turbines, then the outgoing connections from turbine j must support t − 1 downstream wind turbines. Finally, constraints (20) and (21) are the variable domain constraints.

Bounding Connections
If it is desired to bound the number of connections between a set of turbines, B, to the remaining wind turbines and substations, the following constraint must be included, where L is the maximum number of links between set turbines, B, to the remaining wind turbines and substations.

Substations Selection
The model WFLMS enables one to determine also the set of substations to include in the solution. So, it can be used to determine the best substation locations as long as a discrete set of substation locations is available. Furthermore, if a maximum number, M, of substations to be present in the solution is predefined, only Constraint (23) needs to be included.

General Comments
It is also possible to incorporate in model WFLMS the topology of the wind turbines farm, choosing a given number of turbines from a set of possible locations of wind turbines. This is done by enlarging the turbine's set, N, to the set of possible locations of the turbines from which it is intended to choose n turbines. In terms of the optimization model, it is not necessary to change the model, it is only necessary to extend the data files.
To take into account the real landscape and forbidden zones, it is necessary to compute the distances between the nodes in order to contemplate the ground situation.

Results and Discussion of the Case Studies
This section presents and discusses the obtained results using the proposed model to optimize the layout of two wind farms, WF-102-S2 and WF-74-S3, generated according to examples found in [29,30]. The first one has two substations and 102 wind turbines, while the second one has 74 wind turbines and sixteen possible sites for a substation, of which a maximum of three are to be chosen.
In all cases, the ten cable types presented on Table 1 were considered. Additionally, it was considered: h = 24 × 365 × 20 as the number of hours during the expected wind farm lifetime, assuming that it is 20 years; c ep = 102.52 × 10 −6 EUR/Wh as the cost of active energy; c eq = 51.26 × 10 −6 EUR/Wh as the cost of reactive energy; l f = 0.35 as the load factor, which reflects the real operating conditions during the wind farm lifetime, and is the ratio between the generated current and the maximum current that can be generated; ω = 100π rad/s as the angular frequency.
The optimization models were constructed using FICO Xpress Mosel (Xpress Mosel Version 4.8.0), and then they were solved with FICO Xpress Optimizer. Computations were performed on a computer Intel(R) Core(TM) i7-8550U CPU @ 1.80 GHz 1.99 GHz with 8GB RAM and 64 bits.
The results of each wind farm are analyzed in the following sections.

WF-102-S2 Wind Farm
The first case study is the WF-102-S2 wind farm with two substations and 102 wind turbines with P r = 2 MW of rated power, interconnected by a U = 20 kV grid. With these parameters, the rated current drawn by each turbine is I r = 57.735 A, and the maximum number of wind turbines per branch line is Q = 10. This value is due to the fact that the maximum current intensity that the available cables can support is I z = 585, Table 1, and so Q = 585 57.7 = 10, Equation (9). The coordinates of the wind turbines and substations are in Table A1. Two scenarios are considered for this wind farm: the wind farm original, WF-102-S2, and the wind farm WF-102-S2W, which includes a limit in the sidewalk connection.
For the first scenario, WF-102-S2, the obtained cable connection layout is presented in Figure 3, the costs in the optimal solution are presented in Table 2, and information about optimal cable connections is presented in Table 3.
The correspondent model WFLMS for this scenario has 207 constraints and 94,758 variables, and the processing time to obtain the optimal solution was 79 s.  The total cost obtained is EUR 10,475,772.3, where 60.5% is the infrastructure cost, corresponding to EUR 6,332,484.5, 23.0% is the active losses cost, corresponding to EUR 2,410,659.4, and 16.5% is the reactive losses cost, corresponding to EUR 1,732,628.4. The highest amount corresponds to the infrastructure cost, and the smallest part is the reactive losses cost during wind farm lifetime. Table 3 shows information about the cable types being used in the optimal solution. In the optimal solution, only five different types of cables are used: type 3 in 41 connections, type 4 in 23 connections, type 7 in 18 connections, type 8 in 6 connections, and type 10 in 14 connections.
The second scenario, WF-102-S2W, is obtained by limiting to two the connections passing on the walkway, adding constraint (22) with L = 2 in the model WFLMS.
The optimal connection layout for WF-102-S2W is presented in Figure 4 and the costs in the optimal solution are presented in Table 2.
The correspondent model has 207 constraints and 94,758 variables, and the processing time to obtain the optimal solution was 85.3 s. In the optimal solution, only five different types of cables are used: type 3 in 39 connections, type 4 in 22 connections, type 7 in 17 connections, type 8 in 10 connections, and type 10 in 14 connections, Table 3.
The total cost is EUR 10,667,045.8, where: 57.3% is the infrastructure cost, corresponding to EUR 6,112,964.7; 23.7% is the active losses cost, corresponding to EUR 2,527,016.9; and 19.0% is the reactive losses cost, corresponding to EUR 2,027,064.2.
There are two sectors in both optimal wind farm solutions, WF-102-S2 and WF-102-S2w: one has an installed capacity of 100 MW, with 50 wind turbines connected to substation 0 1 and the other one has 104 MW of installed capacity, with 52 turbines connected to substation 0 2 . There are ten branch lines linked to each substation, and the number of wind turbines in each branch line ranges between four and eight turbines. In the optimal solutions, only five different types of cables are used 3, 4, 7, 8, and 10, as shown in Table 3.
Comparing both scenarios' optimal solutions, in Tables 2 and 3, as expected, the total cost corresponding to the optimal solution when limiting the number of connections to a subset of turbines, study case WF-102-S2-W, is higher. This phenomenon is due to the increasing of active and reactive loss costs that are not compensated by the decreasing observed in the infrastructure cost. Note that these changes in costs result from the use of lower types of cables, which are cheaper but have higher energy losses.

WF-74-S3 Wind Farm
The second wind farm, WF-74-S3, is formed by 74 turbines with P r = 2 MW of rated power, interconnected by a U = 20 kV grid. With these parameters, the rated current drawn by each turbine is I r = 57.  Table A2.
The goal is to optimize the cable layout, choosing at most three of the available substations. To solve this problem, the model WFLMS including constraint (23) with M = 3 is considered.
The optimal connection layout is presented in Figure 5, the costs in the optimal solution are shown in Table 4, and information about optimal cable connections is presented in Table 5.
The model has 166 constraints and 60,474 variables, and the processing time to obtain the optimal solution was 158.4 s.  There are seven branches connected to substation O 1 , in which the number of turbines varies from one to five. Substation O 8 is linked to eight branch lines; it also has a number of turbines ranging between one and five. There are nine branch lines linked to the substation O 14 , in which the number of turbines varies between one and six. In the optimal solutions, only five different types of cables are used: type 3 in 30 connections, type 4 in 21 connections, type 7 in 13 connections, type 8 in 6 connections, and type 10 in 4 connections, as shown in Table 5.
The total cost is EUR 2,908,787.57, where 64.9% is the infrastructure cost, corresponding to EUR 1,887,148.4,22.7% is the active losses cost, corresponding to EUR 660,641.2, and 12.4% is the reactive losses cost, corresponding to EUR 360,997.9, Table 4.
Once again, the highest amount corresponds to the infrastructure cost and the lowest amount is the reactive losses cost during wind farm lifetime.
Comparing this approach with approaches in the literature, Reference [22], the presented model is much faster at finding the optimal solution.

Conclusions
This work presents an ILP model to solve a cable connection layout considering wind farms with several substations. The model was applied to two wind farms with up to 102 turbines. In the wind farm with 102 turbines, there are two substations, and two variants of this case study are considered: one in which the original model WFLMS is considered and another in which a new model is considered where, to meet the solution installed on the ground, the number of connections between a set of turbines and the rest of the wind farm is limited. An additional constraint was considered in the latter case study. The proposed model was able to determine the optimal solution in both variants in a very short time, around 80 s.
In a wind farm with 74 turbines, sixteen possible locations for substations are considered, and the goal is to optimize the layout of the wind farm with at most three substations. To contemplate this new issue, it is only necessary to include a new constraint (23), to the initially proposed model, WFLMS. The optimal solution for this case study was obtained in only 158.4 s.
To contemplate this new issue, a new constraint (23), to the initially proposed model, WFLMS, was included. The optimal solution for this case study was obtained in only 158.4 s.
The results are promising. The optimal solutions are obtained in a short time and, if desired, the number of possible locations for the substations can be increased, bringing the discretized problem closer to the problem of choosing the best locations globally across the park.
The proposed model can also consider forbidden zones and can be adapted to optimize also the topology of the wind turbines farm, choosing a given number of turbines from a set of possible locations of wind turbines. Furthermore, compared with the literature, the presented model is fast at finding the optimal solution.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Wind Farm Coordinates
This section presents the wind farms' coordinates WF-102-S2 and WF-74-S3 in Tables A1 and A2. Coordinates for wind farms WF-102-S2 and WF-74-S3 are expressed in WGS84 and Cartesian Coordinates, respectively. Substations and wind turbines are labeled on column "No". Columns "Latitude" and "Longitude" show the correspondent coordinates.