# Dynamic Reconfiguration of Active Distribution Network Based on Improved Equilibrium Optimizer
## 2 Load Period Partitioning Method Based on Improved Fuzzy C-Means Clustering
### Improved FCM Algorithm with Adaptive Clustering and Temporal Feature Fusion  
Following the computation of both comprehensive similarity between time periods and the time-weighted similarity matrix, this study employs the Fuzzy C-Means (FCM) algorithm to perform time period segmentation. As a distance-based clustering technique, FCM allows each time segment to belong to multiple clusters simultaneously, assigning higher membership values to the most representative cluster. This characteristic makes FCM particularly suitable for handling ambiguous boundaries and transitional patterns in load profiles, thereby more accurately capturing real-world fluctuations in electricity demand. Nevertheless, the conventional FCM algorithm exhibits limitations in clustering stability and accuracy when applied to time-based load partitioning. To address these issues, this paper introduces the following enhancements to improve its overall performance. (1) Introducing Comprehensive Similarity as an Input Feature  
The traditional FCM algorithm utilizes Euclidean distance to assess differences between samples, directly computing distances based on physical attributes such as load and voltage. However, this approach often fails to effectively capture the underlying correlations between features--particularly in scenarios with mild load variations or significant feature overlap--leading to suboptimal clustering outcomes. To overcome these limitations, this paper introduces a comprehensive similarity metric that integrates both load characteristics and optimal network topology. An inter-period similarity matrix is constructed to replace the conventional distance matrix used in FCM. Additionally, the cluster centers are redefined as weighted combinations of all time periods, enabling the formulation of a fused similarity expression between a given time period and a cluster, as detailed in Equation (8):  
\[S_{ij}^{f}=\sum_{k=1}^{N}\left(u_{kj}\right)^{m}\times S_{ik} \tag{8}\]where \(u_{kj}\) is the membership degree of time period \(k\) to cluster \(j\), and \(m\geq 1\) is the fuzzy index and is usually 2. Thus, replacing the original distance-based measurement mechanism, the updated membership calculation formula is adjusted, as shown in Equation (9):  
\[u_{ij}=\frac{1}{\sum\limits_{c=1}^{C}{(\frac{S_{ij}^{f}}{S_{ic}^{f}})\frac{1}{m- 1}}} \tag{9}\]  
where \(C\) represents the candidate value for the number of clusters, which is used to support the adaptive selection process and ultimately determine the optimal value through contour coefficient adaptation. This method realizes the membership relationship determination based on the principle of \"the greater the similarity, the closer it is\", which is more closely related to the actual physical connotation in the problem of time division in the actual distribution network. (2) Adaptive Determination of Cluster Number \(C\)  
Conventional fuzzy C-means clustering requires the number of clusters \(C\) to be specified in advance. However, an inappropriate choice of \(C\) can lead to issues such as over-fitting or under-fitting. In the context of distribution network time segmentation, the optimal number of clusters may vary under different conditions, and cannot be predetermined accurately, thus complicating the partitioning process. To address this, the silhouette coefficient evaluation method is introduced in this study to automatically assess the clustering performance across different values of \(C\). This approach enhances the flexibility and accuracy of cluster number selection and mitigates potential biases arising from manual parameter tuning. (3) Integration of Temporal Sequence Information  
Traditional fuzzy C-means clustering methods typically rely solely on feature data at individual time points, neglecting the inherent temporal correlations. However, in real-world distribution networks, load profiles often exhibit sequential continuity and periodic patterns. Ignoring such temporal dependencies may compromise the interpretability and effectiveness of the clustering outcome.