# Dynamic Reconfiguration of Active Distribution Network Based on Improved Equilibrium Optimizer
## 2 Load Period Partitioning Method Based on Improved Fuzzy C-Means Clustering
### Comprehensive Similarity Calculation Based on Load Characteristics and Optimal Network Structure  
The higher the similarity of loads and the optimal network configurations across different time periods, the greater the likelihood that the same network topology can be applied after time periods are merged [30; 31; 32]. Therefore, this study jointly considers both load similarity and optimal network structure similarity to evaluate the degree of resemblance between any two time periods. By ensuring that periods with similar load patterns or identical optimal structures are assigned higher similarity scores, a clustering process based on comprehensive similarity is employed to determine the final time division scheme. In this study, distribution network reconfiguration is performed using next-day load forecasts. The day is initially segmented evenly into \(T\) time intervals (\(T\) = 24 in this case). It is assumed that the load at each node remains constant within a single time period. The load characteristic vectors for time periods \(a\) and \(b\) are defined by Equations (1) and (2), respectively. \[X_{a}=(x_{a1},x_{a2},\ldots,x_{an}) \tag{1}\]  
\[X_{b}=(x_{b1},x_{b2},\ldots,x_{bn}) \tag{2}\]  
where \(n\) is the number of load characteristics, \(x_{ak}\) represents the kth load characteristic of the time period \(a\), \(x_{hk}\) represents the \(k\)th load characteristic of the time period \(b\), and the Euclidean distance \(d(X_{a},X_{b})\) between time periods \(a\) and \(b\) is shown in Equation (3):  
\[d(X_{a},X_{b})=\sqrt{\sum_{k=1}^{n}{(x_{ak}-x_{hk})}^{2}} \tag{3}\]  
In order to facilitate the representation of load similarity, the Euclidean distance is converted into a similarity index, in Equation (4):  
\[S_{ab}^{l}=\frac{1}{1+d(X_{a},X_{b})} \tag{4}\]  
where \(S_{ab}^{l}\) represents the load similarity between time periods \(a\) and \(b\). The smaller the Euclidean distance \(d\), the closer \(S_{ab}^{l}\) is to one, indicating that the load characteristics are more similar. Conversely, the closer \(d\) is to zero, the greater the difference in load characteristics. To further quantify the similarity between optimal network structures across different time periods, this study introduces a structural similarity index based on the degree of overlap among disconnected switches. Specifically, for each initial time interval, static reconfiguration of the distribution network is performed to determine the optimal topology under the prevailing load conditions. The sets of switches that must be disconnected are recorded as \(Q_{i}\) and \(Q_{j}\), corresponding to time periods \(t_{i}\) and \(t_{j}\), respectively. Given the potential variation in the number of switches required to be disconnected across different time periods, directly evaluating similarity using the ratio of the intersection size to that of a single set may introduce bias. To address this issue, this study adopts the Jaccard coefficient, widely used in set similarity analysis, as the metric for defining the similarity of optimal network structures, as presented in Equation (5). \[S_{ab}^{N}=\frac{\left|Q_{i}\cap Q_{j}\right|}{\left|Q_{i}\cup Q_{j}\right|} \tag{5}\]  
where \(\left|Q_{i}\cap Q_{j}\right|\) represents the intersection number of switch numbers that need to be disconnected in two time periods, \(\left|Q_{i}\cup Q_{j}\right|\) represents the number of its union sets. This indicator can adaptively change between 0 and 1, and when the switches that need to be disconnected in two time periods are exactly the same, \(S_{ab}^{N}=1\); if there is no complete overlap, then \(S_{ab}^{N}=0\). This definition not only eliminates the bias caused by an unequal number of disconnections but also has good normalization characteristics, making it easy to integrate with load similarity to construct a comprehensive similarity index in the future. To calculate the comprehensive similarity, a weighted fusion method is used based on the comprehensive influence of load characteristics and network structure, as illustrated in Equation (6).