## 1 Introduction
### Flexibility of Distribution Network with DG
#### 2.2.2 Constraints for the Multi-Objective Optimization Model
\begin{array}{ll}\max&f_{1}=\sum\limits_{s=1}^{S}p_{s}(\sum \limits_{t=1}^{T}\sum\limits_{i\in\Omega_{PV}}P_{i,t,s}^{PV}\Delta t+\sum\limits _{t=1}^{T}\sum\limits_{i\in\Omega_{W}}P_{i,t,s}^{WF}\Delta t)\\ \max&f_{2}=\sum\limits_{s=1}^{S}p_{s}\frac{(U_{i}^{W})^{2}}{(\frac{U_{i,t,s}^{ PV}-U_{i,s}^{PV}-U_{i,s}^{WF}}{\sum\limits_{i\in\Omega_{W}}^{P}}(-U_{i}^{W})^{2})}\\ \max&f_{3}=\sum\limits_{k=1}^{N_{line}}\sum\limits_{\begin{subarray}{c}s=1 -1\\ k\end{subarray}}^{\sum\limits_{t=1}^{T}\sum\limits_{t=1}^{T}P_{t}(l_{\max}-l_{t,s}^{L})}{l_{\max}}\times 100\%\\ \text{s.t.}&P_{ij,t,s}=\sum\limits_{(j,k\in\Omega_{line})}P_{j,t,s}-P_{i,t,s} ^{net}\\ &Q_{ij,t,s}=\sum\limits_{(j,k\in\Omega_{line})}Q_{jk,t,s}-Q_{i,t,s}^{net}\\ &U_{i,\min}\leq U_{i,t,s}\leq U_{i,\max}\ \ (i\in\Omega_{node})\\ &\left|P_{ij,t,s}\right|\leq S_{ij}\ \ (i,j\in\Omega_{node})\\ &(1-\theta_{WF})\sum\limits_{s=1}^{S}p_{s}\sum\limits_{t=1}^{T}\overline{P_{ i,s}^{WF}}\Delta t\leq\sum\limits_{s=1}^{S}p_{s}\sum\limits_{t=1}^{T}P_{t,t,s}^{ WF}\Delta t\ \ (i\in\Omega_{WF})\\ &(1-\theta_{PV})\sum\limits_{s=1}^{S}p_{s}\sum\limits_{t=1}^{T}\overline{P_{ t,s}^{PV}}\Delta t\leq\sum\limits_{s=1}^{S}p_{s}\sum\limits_{t=1}^{T}P_{t,t,s}^{ PV}\Delta t\ \ (i\in\Omega_{PV})\\ &P_{i,t,s}^{WF}\leq\overline{P_{t,s}^{WF}}\ \ (i\in\Omega_{WF})\\ &P_{i,t,s}^{PV}\leq\overline{P_{t,s}^{PV}}\ \ (i\in\Omega_{PV})\\ &0\leq P_{i,t,s}^{PS}\leq\lambda P_{i,t,s}^{load}\ \ (i\in\Omega_{node})\\ &P_{i,t,s}^{PS}>0\ \ (i\in\Omega_{G})\\ \end{array} \right. \tag{13}\]  
Solving Methods for the Multi-Objective Optimization Model Based on the CLPSO Algorithm and Mixed Strategy Nash Equilibrium  
Comprehensive Learning Particle Swarm Optimization Algorithm for Solving the Multi-Objective Optimization Model  
To solve the proposed optimization model, which is a multi-objective decision making problem (MODM) [37], the particle swarm optimization algorithm (PSO) is used to obtain the planning results for DG. The PSO algorithm generates the initial particle swarm at random, and then realizes the balance of global and local optimization ability by setting a reasonable velocity parameter and inertia weight parameter [38; 39]. The fitness function is used to assess the optimality of different particle solutions, and the optimal solution is obtained through multiple iterations. The PSO algorithm generally solves the single objective optimization problem, which determines only one globally optimal solution, along with determining the particle historical optimal solution in the optimization process. However, the optimization model of accommodation capacity of DG is a multi-objective optimization problem of multiple Pareto non-inferior solutions, which forms the Pareto frontier solution set. Therefore, a new method for determining the optimal particle is needed to ensure the diversity of Pareto non-inferior solutions and avoid falling into the local optimum in the multi-objective optimization model of accommodation capacity of DG. The CLPSO algorithm guides particles to learn from the globally optimal position or from the particle's own historical optimal position through the randomly generated binary sequence \(\mathbf{H}\). Meanwhile, it guides particles to learn from other particles with a certain probability, making better use of the historical information of the global population particles to ensure the comprehensiveness of the algorithm's search [40]. In order to maximize the accommodation capability of DG, the method for updating the connection solution \(i\) of DG with the distribution network in the globally optimal location \(\mathbf{g}\) is to randomly select two solutions from the external archives. The binary bidding method is utilized to select the best solution to update the global optimal position. Thus, the temporary global optimization solution can provide the optimal direction for particles in the CLPSO algorithm process for solving the multi-objective optimization model of accommodation capacity of DG.