## 2 Mathematical Model of Distribution Network  
### The Operation Constraints of Distribution Network  
Figure 1 shows a DES under an IEEE 33 distribution network including two photovoltaic power stations, two wind power generation stations, distributed generation (DG) and a BESS. It can be seen that there are multiple energy units scattered throughout the IEEE 33 distribution network. The distributed photovoltaic and wind-power generation stations provide green energy for the DES, while DG and ESS are used to ensure the energy balance and economic of the DES. In the application of the DES, certain operation constraints should be guaranteed, including the interactions of power, current and voltage among power nodes in the system, which conform to some basic physical laws such as energy conservation and node voltage equations [27]. These con \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Reference**} & \multicolumn{4}{c}{**System Components**} & \multirow{2}{*}{**Grid**} & \multirow{2}{*}{**Number of Objective Functions**} & \multirow{2}{*}{**Methodology**} \\  & & & & & & & & \\ \hline
[5] & ✓ & & & ✓ & & ✓ & 2 & PSO-TLBO \\
[6] & ✓ & ✓ & & ✓ & & ✓ & 2 & ECA + TOPSIS \\
[8] & & & ✓ & & ✓ & ✓ & 3 & MO-NSGA-II \\
[10] & ✓ & & & ✓ & & ✓ & 3 & MC + TOPSIS \\
[11] & & ✓ & ✓ & ✓ & ✓ & 3 & EW + ABC \\
[13] & & & & ✓ & ✓ & ✓ & 11 & Weighted summation \\
[14] & & & & ✓ & ✓ & & 3 & MPC + adaptive weight \\
[16] & & ✓ & ✓ & ✓ & ✓ & ✓ & 2 & Weighted summation \\
[17] & & ✓ & & ✓ & ✓ & ✓ & 2 & B-L optimization model \\
[19] & & ✓ & ✓ & ✓ & ✓ & ✓ & 3 & TSDRO \\
[21] & ✓ & & ✓ & ✓ & ✓ & 3 & Multi-OPF model \\
[26] & & & ✓ & ✓ & & ✓ & 3 & MOMIUS \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relevant research review. straints for operation can be roughly divided into two categories, equality constraints and inequality constraints. Firstly, in the operation process of the DES mentioned above, the equation constraints satisfied are the power-balance equation as shown in Equation (1). At time \(t\), \(P_{\text{DG,}i}(t)\), \(P_{\text{ESS,}i}(t)\) and \(P_{\text{New,}i}(t)\) respectively represent the active power of the DG, ESS and new energy station on the \(i\)-th power node. Additionally, \(Q_{\text{DG,}i}(t)\) and \(Q_{\text{ESS,}i}(t)\) are the reactive power of the DG and ESS. Similarly, the branch current and active and reactive power have been described as \(I_{i,j}(t)\), \(P_{i,j}(t)\) and \(Q_{i,j}(t)\), respectively. \[\left\{ \begin{aligned} & P_{\text{DG,}i}(t)+P_{\text{ESS,}i}(t)+P_{ \text{New,}i}(t)-P_{\text{LOAD,}i}(t)=\sum\limits_{j=1}^{N_{\text{b}}}P_{i,j}( t)-\sum\limits_{j=1}^{N_{\text{b}}}I_{i,j}(t)R_{i,j}\\ & Q_{\text{DG,}i}(t)+Q_{\text{ESS,}i}(t)-Q_{\text{LOAD,}i}(t)= \sum\limits_{j=1}^{N_{\text{b}}}Q_{i,j}(t)-\sum\limits_{j=1}^{N_{\text{b}}}I_{ i,j}(t)X_{i,j}\end{aligned} \right. \tag{1}\]  
where \(P_{\text{LOAD,}i}(t)\) and \(Q_{\text{LOAD,}i}(t)\) are the active and reactive power of the load. \(R_{i,j}\) and \(X_{i,j}\) are the resistance and reactance between the \(j\)-th and the \(i\)-th power node. The number of power nodes is \(N_{\text{b}}\). Then, [PERSON]'s law equation for each node in the power grid is considered as shown in Equation (2). \(U_{i}(t)\) is the voltage of the \(i\)-th power node. \[U_{i}(t)=U_{j}(t)+I_{i,j}(t)\times\left(R_{i,j}^{2}+X_{i,j}^{2}\right)-2\times \left(P_{i,j}(t)\times R_{i,j}-Q_{i,j}(t)\times X_{i,j}\right) \tag{2}\]  
Meanwhile, to balance the energy of the ESS during each typical day, Equation (3) represents the relationship between the energy state of the ESS and the charging and discharging power. At time \(t\), the state of energy (SOE), the charge and discharge active power of the ESS on the \(i\)-th power node are \(C_{\text{SOE,}i}(t)\), \(P_{\text{ESS,}c,i}(t)\) and \(P_{\text{ESS,}d,i}(t)\). In a typical day, the starting and ending SOEs of the ESS are \(C_{\text{SOE,}i}(0)\) and \(C_{\text{SOE,}i}(T)\).