## 1 Introduction
### Flexibility of Distribution Network with DG
#### 2.2.1 Objective Function of Multi-Objective Optimization Model  
In order to accommodate WF as much as possible, the existing accommodate capacity optimization models generally take the maximization of the installed DG capacity as the objective, which may lead to resource waste and uneconomical operation of distribution networks. In addition, the flexibility requirements of distribution network operation should be taken into consideration in the accommodation capability optimization model, so that the distribution network can flexibly adapt to the randomness and fluctuation of distributed generators and loads, and quickly adjust and control flexible resources, such as interruptible loads. Therefore, three indices, namely, the daily energy supply of DG, the voltage amplitude deviation, and the line capacity margin, are considered in the proposed multi-objective optimization model of the accommodation capability of DG. 1. Maximizing the Daily Energy Supply of DG  
Regarding whole distribution systems, the daily energy supply of DG reflects the actual consumption of distributed energy by loads of distribution networks in dispatching periods throughout a day. When the distribution system load demand is constant over the dispatching period, maximizing the daily energy supply of DG could minimize the energy purchased from superior grids, therefore reducing the demand for power outside the local distribution system. Thus, the power of flexible resources such as distributed sources could be maximized, which would promote renewable energy and improve the economic level of the distribution network operation. The objective function\(f_{1}\) of the energy supply of DG over the daily dispatching period can be represented as:  
\[\max f_{1}=\sum_{s=1}^{S}p_{s}(\sum_{t=1}^{T}\sum_{i\in\Omega_{pv}}P_{i,s}^{ PV}\Delta t+\sum_{t=1}^{T}\sum_{i\in\Omega_{WF}}P_{i,s}^{WF}\Delta t) \tag{1}\]  
2. Minimizing the voltage amplitude deviation  
The voltage fluctuation is an important index for measuring the power quality of distribution networks [33; 34]. The voltage fluctuation phenomenon can be caused by the mass connection of DG in the distribution network, which will cause system insecurity and even partial outages in serious cases. Therefore, it is necessary to consider voltage amplitude deviation in the objective function of the accommodation capability optimization of DG. Figure 1: Methodology diagram flows of the multi-objective optimization model. Note: DG = distributed generation. The objective function \(f_{2}\) of the voltage fluctuation can be defined as:  
\[\max f_{2}=\sum\limits_{s=1}^{S}p_{s}\frac{(U_{i}^{N})^{2}}{\sum\limits_{i\in \Omega_{\text{mode}}}\left[\frac{(P_{i,L,s_{s}}-P_{i,L,s}^{\text{PV}}-P_{i,L,s}^ {\text{W}F})}{l_{L,s}^{2}}-(U_{i}^{N})^{2}\right]} \tag{2}\]  
3. Maximizing the line capacity margin  
With the mass connection of newly increased loads and DGs in the distribution network, the fluctuation and randomness of net loads increases, which can easily lead to partial congestion of the lines. The line capacity margin refers to the ratio of the difference between the maximum transmission capacity and the actual transmission capacity at a certain time to the maximum allowable transmission capacity. This mainly reflects the flexible adaptability to randomness and fluctuation for both DG and load, and the power supply transfer capacity to various disturbances in the distribution network. Therefore, the line capacity margin is taken as a flexibility index to evaluate the accommodation capacity of the DG. The objective function \(f_{3}\) of the line capacity margin can be defined as:  
\[\max f_{3}=\sum\limits_{k=1}^{N_{\text{line}}}F_{\text{LMC},k}=\sum\limits_{ k=1}^{N_{\text{line}}}\frac{\sum\limits_{s=1}^{S}\sum\limits_{t=1}^{T}p_{s}(I_{k, \text{max}}-I_{k,t,s})}{I_{k,\text{max}}}\times 100\% \tag{3}\]  
In Equation (3), \(F_{\text{LMC},k}\geq 0\) indicates that the capacity margin of branch \(k\) is sufficient, and that branch \(k\) can adapt to the fluctuation of loads.