# Dynamic Reconfiguration of Active Distribution Network Based on Improved Equilibrium Optimizer
## 4 Optimization Algorithms
### Feasible Solution Determination
#### 4.2.2 Feasible Solutions Determination Based on Basic Ring Matrix  
During the reconfiguration process, T.S.s are normally open, and S.S.s are normally closed. After reconfiguration, some switches change their positions based on the optimization results [12]. Distribution network reconfiguration essentially involves the opening and closing of switches. When all switches are closed, the distribution network topology can be represented as a connected graph, denoted as \(\mathbf{G}\). By specifying that the total number of branches equals the difference between the total number of nodes and the number of root nodes in the distribution network, the occurrence of a large number of infeasible solutions can be avoided, as shown in Equation (27):  
\[\sum_{ij\in\mathbf{H}}h_{ij}=n_{b}-n_{s},h_{ij}\in\{0,1\} \tag{27}\]  
where \(H\) is the set of branches of the above distribution network, \(h_{ij}\) is the open and closed state of the branch, with zero denoting disconnection and one denoting closure, and \(n_{b}\) and \(n_{s}\) denote the total number of nodes and the number of root nodes in the distribution network, respectively. In actual distribution networks, common branches exist between loops, which are shared by multiple loops simultaneously. If two switches are disconnected at the same time on a common branch, it may result in the formation of an island, as illustrated in Figure 3. To address this issue, the following three provisions are established: at most, one disconnection is allowed on the common branch \(L_{mn}\) between any loops \(m\) and \(n\), as shown in Equation (28); at most, one non-public branch can be disconnected, as shown in Equation (29); and if \(n\) branches are connected at node \(j\), then no more than \(n-1\) branches can be disconnected at this node, as shown in Equation (30). \[\sum_{(i,j)\in L_{mn}}1-h_{ij}\leq 1,L_{mn}=L_{m}\cap L_{n} \tag{28}\]  
\[\sum_{(i,j)\in C_{L_{m}}L_{mn}}h_{ij}\geq n_{L_{m}}-1,\forall m\in L,\forall n\in L \tag{29}\]  
\[\sum_{(i,j)\in K_{j}}h_{ij}\geq 1,\forall j\in\{1,2,\ldots,n_{b}\} \tag{30}\]  
where \(L_{m}\) denotes the set of branches in the \(m\)th loop, \(n_{L_{m}}\) denotes the number of branches in the \(m\)th loop, \(L\) denotes the total number of loops, and \(K_{j}\) denotes the set of branches connected to the \(j\)th node. The method described above effectively avoids infeasible solutions in the reconfiguration process. However, due to the inherent randomness in the search process of intelligent optimization algorithms, a large number of inferior solutions remain in the solution space, specifically low-quality feasible solutions with high fitness values. The overall lower quality of solutions leads to an increased reconfiguration time. Therefore, this paper calculates the power moment based on the real-time load of each node, removes duplicate branches  
Figure 3: Situations that do not comply with topological constraints. (**a**) Configuration with multiple isolated nodes; (**b**) Configuration with a single isolated node. through branch verification, and generates an adaptive ordered loop feasible solution matrix, denoted as _JM_. The specific process is as follows:  
Step 1: Find the repeated branch \(S_{i}\) in the basic ring matrix \(\mathbf{H}\), with its repetition number \(Nr\), where \(S_{i}\) appears in the basic ring matrix, and find all positions \(H_{l1 kl}\), \(H_{l2k2}\), \(\ldots\), \(H_{lNrkNr}\). Assuming that the switch corresponding to the repeated branch \(S_{i}\) is disconnected and the other switches are closed, with its end node being \(Ne_{i}\), the backflow path formed by the set of branches and nodes from node \(Ne_{i}\) to the source point of the loop is denoted as \(L_{i}\) (\(i=1\), \(2\), \(\ldots\), \(Nr\)).