## 1 Introduction
### Flexibility of Distribution Network with DG
#### 2.2.2 Constraints for the Multi-Objective Optimization Model
Multi-Objective, Non-Cooperative Equilibrium Decision-Making Based on Mixed Strategy Nash Equilibrium for Determining the Optimal Accommodation Capacity of DG  
The Pareto frontier set obtained by the proposed CLPSO algorithm is stored in the external archives. Finally, the optimal compromise solution needs to be selected from the external archives as the ultimate maximum accommodation capacity of DG. The decision regarding the optimal compromise solution is made using multiple attribute decision making (MADM) [41]. Techniques such as simple additive weighting (SAW), analytic hierarchy process (AHP), fuzzy reasoning method [42], and technique for order preference by similarity to ideal solution (TOPSIS) are widely used in solving the MADM problem. SAW, fuzzy reasoning, and AHP methods are greatly affected by subjective factors. Most methods for solving multi-objective optimization problems are based on engineering experience and lack a theoretical basis. TOPSIS is a kind of centralized decision-making method considering both the positive and negative ideal solutions. However, these methods do not consider the distribution trade-off characteristics of the frontier solution set. Therefore, with the help of game theory, the mixed strategy Nash equilibrium is adopted to solve the proposed multi-objective optimization problem in this paper. Three optimization objectives, namely, the maximal daily energy supply of DG, the minimal voltage amplitude deviation, and the maximal line capacity margin, are regarded as non-cooperative decision-making participants. Then, the objective function values in the Pareto frontier solution set can be modeled by the action set of decision-making participants [43]. The optimal compromise solution can be obtained by solving the optimization problem of joint probability distribution in the Pareto frontier action set. The objective function is normalized first to solve the inconsistency problem of multi-objective dimensions. Then, a multi-objective, non-cooperative equilibrium decision-making model based on mixed strategy Nash equilibrium is established [43] to determine the optimal accommodation capacity of DG, which can be represented as:  
\[ \begin{array}{ll}\max&Nash(Y_{1},\ldots,Y_{i},\ldots,Y_{S_{ob}},u_{1},\ldots,u_{i},\ldots,u_{S_{ob}})\\ &=\sum\limits_{i=1}^{S_{ob}}\sum\limits_{j=1}^{S_{ob}}\left((\omega_{i}f_{ij}) (\prod\limits_{i=1}^{S_{ob}}y_{ij})\right)-\sum\limits_{i=1}^{S_{ob}}u_{i}\\ s.t.&\sum\limits_{j=1}^{S_{os}}y_{ij}=1,i=1,2,\ldots,S_{ob}\\ &y_{ij}\geq 0,i=1,2,\ldots,S_{ob},j=1,2,\ldots,S_{es}\\ &\sum\limits_{j=1}^{S_{os}}\omega_{i}f_{ij}y_{ij}\leq u_{i},\;i=1,2,\ldots,S_ {ob}\end{array}  \tag{14}\]  
The overall model of the mixed strategy Nash equilibrium is represented in Equation (14). In the model, the real control variables are the equilibrium solution of objective function \(i\), (i.e., \(\boldsymbol{Y}_{i}=(y_{i1},\ldots,y_{ij},\ldots,y_{iS_{es}})\)) and the upper limit of the expected value of the objection function \(i\) (i.e., \(u_{i}\)). The two control variables reach \(S_{ob}\times S_{es}\) and \(S_{ob}\), respectively. The positive control variables \(y_{ij}\) are constrained by equality constraints concerning the sum of equilibrium values of all frontier solution sets and inequality constraints bounded by control variable \(u_{i}\). The values of the equality and inequality constraints are \(S_{ob}\) and \(S_{ob}+S_{es}\), respectively. The frontier solution with the optimal joint equilibrium represents the joint action with which the decision-making participants can obtain the highest return (i.e., the optimal compromise solution of the connection of DG with the distribution network) [44]. The frontier solution with the optimal joint equilibrium is expressed as:\[\max(\prod_{i=1}^{S_{ab}}y_{i1},\ldots,\prod_{i=1}^{S_{ab}}y_{ij}\ldots,\prod_{i=1} ^{S_{ab}}y_{iS_{\alpha}}) \tag{15}\]  
In the optimal connection solution, the distribution network can always operate securely and has better flexibility considering the uncertain output power of DG.