# Dynamic Reconfiguration of Active Distribution Network Based on Improved Equilibrium Optimizer
## 4 Optimization Algorithms
### Equilibrium Optimizer  
EO is inspired by the mass balance equation in physics. In the EO algorithm's workflow, each particle, along with its concentration, represents an independent individual, with the particles analogous to potential solutions of the optimization problem. Guided by the equilibrium candidate solution, the particles iteratively update their concentrations, converging toward an equilibrium state through the concentration update rule. The parameter design and optimization search process of the algorithm are as follows:  
* Initialization and Function Evaluation  
EO constructs the initial concentration of the population based on random initialization in the search space and initiates the optimization process. The initial concentration is defined in Equation (17). \[\mathbf{C}_{i}^{initial}=\mathbf{C}_{min}+rand(\mathbf{C}_{max}-\mathbf{C}_{ min})\;i=1,2,\ldots,n \tag{17}\]  
where \(\mathbf{C}_{i}^{initial}\) denotes the initial concentration vector of the \(i\)th particle, \(C_{max}\) and \(C_{min}\) are the upper and lower limit vectors of the optimization variables in the search space, and _rand_ is a random vector between [0, 1], \(n\) is the population size. * Constructing the Equilibrium Pool and Candidates  
The equilibrium state represents the algorithm's final converged solution, which corresponds to the desired global optimum. EO searches for unknown equilibrium states via the Equilibrium Pool. At the start of the iterative process, in the absence of any prior knowledge regarding the equilibrium state, EO selects five particles along with their corresponding concentrations as candidate solutions to guide the search for other particles. The Equilibrium Pool, composed of these candidates, is defined as shown in Equation (18):  
\[\mathbf{C}_{eq,pool}=\left\{\mathbf{C}_{eq(1)},\mathbf{C}_{eq(2)},\mathbf{C}_ {eq(3)},\mathbf{C}_{eq(4)},\mathbf{C}_{eq(ave)}\right\} \tag{18}\]  
where \(\mathbf{C}_{eq(1)}\), \(\mathbf{C}_{eq(2)}\), \(\mathbf{C}_{eq(3)}\), and \(\mathbf{C}_{eq(4)}\) are the four solutions with the best fitness values in the current iteration, and the concentration of the fifth candidate \(\mathbf{C}_{eq(ave)}\) is the average of the first four. The probability of these five Candidates being selected is all 0.2. * Computing Exponential Term (\(\mathbf{F}\))  
An accurate definition of the exponential term \(\mathbf{F}\) will assist EO in having a reasonable balance between exploration and exploitation. \[\mathbf{F}=a_{1}\mathrm{sign}(\mathbf{r}-0.5)\Big{[}e^{-\lambda t-1}\Big{]} \tag{19}\]  
where \(a_{1}\) is a constant value that controls exploration ability by amplifying the concentration change; \(\lambda\) and \(r\) are the random vectors in [0, 1] with the same dimension as the optimization space dimension; and \(\mathrm{sign}(r-0.5)\) indicates the control of the direction of exploration and exploitation. The time parameter \(t\) is an iterative function that decreases as the number of iterations increases and is updated as shown in Equation (20):  
\[t=\left(1-\frac{Iter}{Max\_Iter}\right)^{\left(a_{2}\frac{Iter}{Max\_Iter} \right)} \tag{20}\]  
where \(a_{2}\) is a constant value that controls exploitation ability, _Iter_ is the current number of iterations, and _Max_Iter_ is the defined maximum number of iterations. * Computing Generation Rate (\(\mathbf{G}\))  
The Generation Rate \(\mathbf{G}\) is an important parameter to provide the exact solution by improving the exploitation process, which helps EO to accelerate the convergence to the optimal solution in later iterations. The value of \(\mathbf{G}\) is shown as a first-order exponential decay process, defined as shown in Equations (21)-(23):  
\[\mathbf{G}=\mathbf{G}_{0}e^{-\Lambda(t-t_{0})}=\mathbf{G}_{0}\mathbf{F} \tag{21}\]  
\[\mathbf{G}_{0}=\mathbf{G}\mathbf{CP}\big{(}\mathbf{C}_{eq}-\mathbf{\lambda}\mathbf{C}\big{)} \tag{22}\]  
\[\mathbf{GCP}=\left\{ \begin{array}{ll}0.5r_{1}&r_{2}\geq GP\\ 0&r_{2}<GP\end{array} \right.