## 1 Introduction
### Modelling of Connection Topology and Cable Cross-Section  
In this study, the commonly used tree topology is adopted in the medium-voltage power network. The topology is modelled as a directed graph \(G\left(\mathbf{V},\mathbf{E}\right)\), where \(\mathbf{V}\) are nodes representing the wind turbines or substations in the topology; \(\mathbf{E}\) is the edges connecting the nodes. An essential characteristic of the tree topology is that if there is no loop in the topology, a connection is sent from each node, which inevitably forms a standard tree topology. Based on this, in the constructed topology model, any wind turbine node \(V_{i}\) of the wind farm can randomly generate a pointing node \(V_{j}\) to connect \(\mathbf{E}_{ij}\). When all of the wind turbine nodes have completed the connection generation, a directed graph \(G\left(\mathbf{V},\mathbf{E}\right)\) is automatically formed. The adjacency matrix \(\mathbf{A}\) can be used to describe the relationship between the wind turbines and the substations, as shown in Equation (1). \[\mathbf{A}=\left(a_{i,j}\right)_{n\times n}a_{i,j}=\left\{ \begin{array}{ll}1,& \left(V_{i},V_{j}\right)\in\mathbf{E}\\ 0,&\text{otherwise}\end{array} \right. \tag{1}\]  
where \(n\) is the number of nodes in \(G\), which is the sum of the number of substations \(N_{OS}\) and the number of wind turbines \(N_{WT}\) in the wind farm. As shown in Figure 1, the adjacency matrix \(\mathbf{A}\) is an asymmetric matrix, forming four sub-regions, of which Area I represents the connection relationship between substations. Area II describes the connection relationship of the substations to the wind turbines. Since the substations were directly connected to the land access point, having no directional connection to the wind turbines or other substations, all the elements in this area were zero. Area III describes the connection relationship among wind turbines. Area IV describes the connection of the wind turbines to the substations. In the adjacency matrix \(\mathbf{A}\), any one of the elements \(a_{i,j}\) represents that the current generated by the node \(i\) in the topology flows through the node \(j\) to the substation. For a clearer explanation, a simple topology consisting of eight wind turbines is taken as an example to illustrate the characteristics of the proposed model. As shown in Figure 2a,b, the example topology and the formed adjacency matrix are shown, respectively, where WT represents the wind turbine. The adjacency matrix \(\mathbf{A}\) is a \(9\times 9\) matrix, describing the connections and current flow between one substation and eight wind turbines. The first column represents the connection of the wind turbine to the substation, and its elements in rows 2 and 5 are all 1, indicating that two wind turbines (WT1 and WT4) are connected to the substation. The element in the 3 rd row of the second column is 1, indicating that a wind turbine (WT2) is connected to the wind turbine Similarly, the elements in the 6 nd and 8 th rows of the fifth column are all 1, indicating that two wind turbines (WT5 with WT7) are connected to the wind turbine It can be concluded that the adjacency matrix clearly shows the connection relationship between the wind turbines and the substations in the topology, which is more conducive to the calculation and analysis of the power parameters. Since the proposed connection topology was randomly generated, there can be loops in the topology, which is strictly prohibited in the tree topology. The identification and counting of loops in the topology adopts a very mature union-find set algorithm in the graph theory [34; 35]. \[A=\left[ \begin{array}{cccccccc}a_{1,1}&\cdots&a_{1,N_{OS}}&a_{1,N_{OS}+1}&\cdots& a_{1,n}\\ \vdots&\ddots&1&\vdots&\vdots&\vdots&\ddots&\vdots\\ a_{N_{OS},1}&\cdots&a_{N_{OS},N_{OS}}&a_{N_{OS},N_{OS}+1}&\cdots&a_{N_{OS},n}\\ a_{N_{OS}+1,1}&\cdots&a_{N_{OS}+1,N_{OS}}&a_{N_{OS}+1,N_{OS}+1}&\cdots&a_{N_{OS}+1, n}\\ \vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ a_{n,1}&\cdots&a_{n,N_{OS}}&a_{n,N_{OS}+1}&\cdots&a_{n,n}\\ \end{array}