## 2 Mathematical Model of Distribution Network
### The Objective Function of Distribution Network  
Section 2.1 shows the operational constraints of the DES, and this section will establish the optimization objective function of the DES. There are many elements in a DES, and a variety of different optimization objectives are established from the perspectives of grid performance and economy, such as grid loss, node voltage deviation, new energy utilization rate, system operating cost and so on. However, the SSO method in this paper is more friendly for the BiOP. Therefore, this paper takes the two optimization objectives of grid loss and system operating cost as examples to carry out the subsequent analysis. It should be noted that these two objective functions are only listed for the purpose of illustrating the methodology of this paper, which does not mean that the SSO method is limited to optimizing the following functions.  
1. To reduce the power line loss of the DES, the first optimization objective function \(J_{1}\) is established, as shown in Expression (12), and is a key index that effects the transmission efficiency of the power grid. \[\text{min}\quad J_{1}=\frac{1}{2}\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N_{ \text{b}}}\sum\limits_{i=1}^{N_{\text{b}}}\left(i_{i,j}^{2}(t)\times\sqrt{R_{ i,j}^{2}+X_{i,j}^{2}}\right)\] (12)
2. To reduce the operating costs of the DES, the second optimization objective function \(J_{2}\) is established including the cost of DG, ESS and the new energy generation station, as shown in Expression (13), where \(S_{\text{DG},i}\), \(S_{\text{ESS},i}\) and \(S_{\text{New},i}\) respectively represent the electricity consumption of the \(i\)-th DG, ESS and new energy during time period \(\Delta t\). \(S_{\text{DG},i}=P_{\text{DG},i}\times\Delta t\), \(S_{\text{ESS},i}=P_{\text{ESS},I}\times\Delta t\) and \(S_{\text{New},i}=P_{\text{New},I}\times\Delta t\). In this paper, \(\Delta t=1\) h. Then, where \(k_{1,I}\), \(k_{2,I}\) and \(k_{3,I}\) respectively represent the cost coefficient of DG [30; 31], \(\forall\)/kWh\({}^{2}\), \(\forall\)/kWh, \(\forall\). Significantly, \(k_{1,I}\) represents the cost coefficient of the square of the active power of DG. Therefore, the unit of coefficient \(k_{1,i}\) is \(\forall\)/kWh\({}^{2}\). The levelized cost coefficient of the ESS and the new energy generation station have been defined as \(k_{4,i}\) and \(k_{5,i}\), \(\forall\)/kWh. \[\text{min}\quad J_{2}=\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N_{\text{b}}} \left(k_{1,i}S_{\text{DG},i}^{2}(t)+k_{2,i}S_{\text{DG},i}(t)+k_{3,i}+k_{4,i} S_{\text{ESS},i}(t)+k_{5,i}S_{\text{New},i}(t)\right)\] (13)  
The following can be seen from the above optimization objectives: (1) The physical dimensions of the two optimization objectives are different, so it is difficult to directly compare and analyze the importance between them, and it is not convenient to transform them into a single optimization objective by adding weights. (2) The two optimization objectives are used to evaluate the two aspects of the operation of the DES and do not have dominance, which means that the two optimization objective functions are equivalent in the distributed energy system. (3) Both objective functions exhibit quadratic form, which implies that the optimization and control of the DES is a multi-objective quadratic planning problem.