# Dynamic Reconfiguration of Active Distribution Network Based on Improved Equilibrium Optimizer
## 4 Optimization Algorithms
### The Proposed Improved Equilibrium Optimizer Algorithm
#### 4.3.3 Stochastic Differential Mutation Strategy  
EO balances global and local search capabilities through a unique Equilibrium Pool mechanism and a concentration update rule. However, these optimization methods introduce limitations in applying EO to the distribution network reconfiguration problem. First, during the Equilibrium Pool stage, candidate selection is based solely on fitness values, which may lead to candidates being in similar positions, reducing population diversity. Additionally, EO relies exclusively on the concentration update rule for individual updates, causing all concentration vectors to converge towards candidates with better fitness values during iterations. This accelerates convergence, increasing the risk of falling into local optima. Moreover, EO lacks an effective strategy to escape local optima, limiting its ability to find the global optimum. To address the aforementioned issues, this study proposes the Stochastic Differential Mutation Strategy, inspired by the DE/target-to-best/1 mutation strategy in Differential Evolutionary Algorithms. The Stochastic Differential Mutation Strategy combines stochastic selection with a local random wandering mechanism, using the difference vectors and equilibrium candidates of EO as two perturbation terms to generate mutation recombinants. The strategy is defined as shown in Equation (39):  
\[\mathbf{V}_{ij}=\mathbf{C}_{ij}+F1\Big{(}\mathbf{C}_{eq(j)}-\mathbf{C}_{ij}\Big{)}+F2\Big{(} \mathbf{C}_{\mathbf{r}(j)}-\mathbf{C}_{\mathbf{r}2(j)}\Big{)} \tag{39}\]  
where \(V_{ij}\) denotes the \(j\)th component of the variation vector \(V_{i}\); \(C_{ij}\) denotes the \(j\)th component of the current solution vector \(C_{j}\); \(C_{eq}\) is a randomly selected candidate in the Equilibrium Pool; \(C_{r1}\) and \(C_{r2}\) are two solutions randomly selected in the population; and the scale factors \(F1\) and \(F2\) are positive control parameters used to scale the perturbation term in \([0.5,1]\). The perturbation term multiplied by \(F1\) is an arithmetic recombination operation, and the perturbation term multiplied by \(F2\) is a differential mutation operation. Therefore, what is generated in the stochastic global mutation strategy is basically a mutational recombinant rather than a pure mutation. The concentration of particles in an iteration has a certain probability of undergoing mutation. In the distribution network reconfiguration problem, this probability is described in Equation (40):  
\[P_{m}=P_{\min}+(P_{\max}-P_{\min})\times\frac{t}{Max\_Iter}+P_{c} \tag{40}\]  
where \(P_{m}\) is the mutation rate of an individual at the \(t\)th generation; \(P_{max}\) and \(P_{min}\) are the maximum and minimum mutation rates of an individual, respectively; \(Max\_Iter\) is the maximum number of iterations of the population; and \(P_{c}\) is a constant to be determined, the value of which is dependent on the period in which the distribution network is reconfigured. The stage of the distribution network reconfiguration is judged based on the reduction \(Re\) of the active power loss in the three consecutive optimal reconfiguration results. When \(Re>\alpha\) (\(\alpha\in[3\%,5\%]\)), it is judged to be the pre-reconfiguration stage. Because of the large difference in particle concentration in the pre-iteration period, the Equilibrium Pool mechanism is utilized as much as possible to its own advantage, so \(P_{c}=0.03\). When \(Re\leq\alpha\), it is judged as the later reconstruction stage. The particle concentration difference is smaller during the later stage of iteration; thus, the ability of the algorithm to jump out of the local optimum should be improved, so \(P_{c}=0.1\). This study designs a variation strategy for \(\mathbf{JR}\), a feasible solution judgment model based on adaptive ordered loops. First, real number encoding is used to represent the disconnected circuits \(C_{ij}\) within the concentration particle \(Ci\) in the EO algorithm.