## 1 Introduction
### Flexibility of Distribution Network with DG
#### 2.2.2 Constraints for the Multi-Objective Optimization Model  
Based on the power flow constraints and the guidelines for secure operation of the distribution network, the optimization model should satisfy the following constraints in the process of evaluating the maximum accommodation capacity of DG. It is worth mentioning that the proposed model is a basic model composed of load and DG, and that it can be further improved by integrating models of other active management assets, including distributed energy storage [35; 36]. 1. Power flow constraints  
The distribution network with DG has been developed from a traditional unidirectional power flow network and changed into a bidirectional power flow network. It is necessary to establish active and reactive power flow balance models, considering power flow directions. The power flow constraints are presented as:  
\[P_{ij,t,s}=U_{i,s}\sum\limits_{j=1}^{N}U_{j,t,s}(G_{ij}\cos\theta_{ij,t,s}+B_ {ij}\sin\theta_{ij,t,s})=\sum\limits_{(j,k\in\Omega_{\text{line}})}P_{jk,t,s }-P_{i,t,s}^{net} \tag{4}\]  
\[Q_{ij,t,s}=U_{i,t,s}\sum\limits_{j=1}^{N}U_{j,t,s}(G_{ij}\sin\theta_{ij,t,s}-B _{ij,t}\cos\theta_{ij,t,s})=\sum\limits_{(j,k\in\Omega_{\text{line}})}Q_{jk,t,s}-Q_{i,t,s}^{net} \tag{5}\]  
\[P_{i,t,s}^{net}=P_{i,t,s}^{load}-P_{i,t,s}^{sg}-P_{i,t,s}^{WF}-P_{i,t,s}^{PV}- P_{i,t,s}^{ENS} \tag{6}\]  
\[Q_{i,t,s}^{net}=Q_{i,t,s}^{load}-Q_{i,t,s}^{sg}-Q_{i,t,s}^{WF}-Q_{i,t,s}^{PV}- Q_{i,t,s}^{ENS} \tag{7}\]2. Nodal voltage amplitude constraints  
The nodal voltage amplitude should meet the requirements of secure operation of the distribution network at any time, which is represented as:  
\[U_{i,\min}\leq U_{i,t,s}\leq U_{i,\max}(i\in\Omega_{node}) \tag{8}\]  
3. Power flow constraints  
The power flow of the line should not exceed its maximum capacity, which is represented as:  
\[\left|P_{ij,t,s}\right|\leq S_{ij}(i,j\in\Omega_{node}) \tag{9}\]  
4. DG output constraints  
The actual output of DG is constrained by the total amount of abandonment power of WF and PV, and the maximum power during the daily dispatching cycle, which are represented as:  
\[\left\{ \begin{array}{l}(1-\theta_{WF})\sum\limits_{s=1}^{S}p_{s}\sum \limits_{t=1}^{T}\overline{P_{t,s}^{WF}}\Delta t\leq\sum\limits_{s=1}^{S}p_{s} \sum\limits_{t=1}^{T}P_{t,i,s}^{WF}\Delta t(i\in\Omega_{WF})\\ (1-\theta_{PV})\sum\limits_{s=1}^{S}p_{s}\sum\limits_{t=1}^{T}\overline{P_{t, s}^{PV}}\Delta t\leq\sum\limits_{s=1}^{S}p_{s}\sum\limits_{t=1}^{T}P_{t,i,s}^{ PV}\Delta t(i\in\Omega_{PV})\\ P_{t,i,s}^{WF}\leq\overline{P_{t,i,s}^{WF}}(i\in\Omega_{WF})\\ P_{t,i,s}^{PV}\leq\overline{P_{t,i,s}^{PV}}(i\in\Omega_{PV})\end{array} \right. \tag{10}\]  
5. Loss of load constraints  
The loss of load should satisfy the following constraints:  
\[0\leq P_{t,i,s}^{ENS}\leq\lambda P_{t,i,s}^{load}(i\in\Omega_{node}) \tag{11}\]  
6. Constraints of power from the superior grid  
The transmission power from superior grids should satisfy the following constraints:  
\[P_{i,t,s}^{sg}>0(i\in\Omega_{G}) \tag{12}\]  
In the multi-objective optimization model represented in Equation (13), the decision variables are optimized to achieve the maximum value of each objective function at each dispatching period and under each scenario. The number of the variables reaches \(S\times T\times N_{PV}\times N_{WF}\times N_{sg}\). Considering the secure operation of the distribution network, equality constraints in Equations (4)-(7) and inequality constraints (i.e., bounding constraints) in Equations (8)-(12) should be satisfied when solving the proposed model. \[\left\{