## 2 Mathematical Model of DAB Converter  
The topology structure of the DAB converter is shown in Figure 1. The primary bridge (PB) and the secondary bridge (SB) are composed of switch transistors (\(S_{1}-S_{8}\)), anti-parallel diodes (\(D_{1}-D_{8}\)), and capacitors (\(C_{1}-C_{8}\)). An equivalent leakage inductance \(L\) and a transformer connect the primary and secondary sides. In the topology, \(V_{\rm i}\) represents the input voltage, \(V_{\rm o}\) represents the output voltage, \(C_{\rm i}\) represents the input capacitance, \(C_{\rm o}\) represents the output capacitance, \(N\):\(1\) represents the transformer turns ratio, \(V_{\rm p}\) represents the voltage on the primary side, \(V_{\rm s}\) represents the voltage on the secondary side, \(i_{\rm L}\) represents the leakage inductance current, \(i_{1}\) represents the input current, \(i_{2}\) represents the output current, \(i_{\rm o}\) represents the load current, \(d\) represents the phase-shift duty cycle in single-phase modulation, and \(T_{\rm s}\) represents the switching period of the converter.  
The average power transfer from PB to SB in the DAB converter can be expressed as:  
\[P=\frac{NV_{\rm o}V_{\rm i}d(1-|d|)}{2 Lf_{\rm s}} \tag{1}\]  
Assuming \(d>0\) and neglecting power losses in the switches and transformer, the output power must be equal to the input power, \(P_{\rm o}=P_{\rm i}\). According to Equation (1), the output current \(i_{2}\) can be expressed as [22]:  
\[i_{2}=\frac{NV_{\rm i}d(1-d)}{2 Lf_{\rm s}} \tag{2}\]Based on [PERSON]'s voltage law, the dynamic average model of the DAB can be represented as:  
\[C_{\text{o}}\frac{\text{d}V_{\text{o}}}{\text{d}f}=\frac{NV_{\text{d}}(1-d)}{2 \Gamma_{\text{s}}}-\frac{1}{R_{\text{o}}}V_{\text{o}} \tag{3}\]  
where \(R_{\text{o}}\) represents the output load. This dynamic average model exhibits high accuracy, simplicity, strong verifiability, and scalability, making it widely used in the analysis and optimization of DAB control strategies in subsequent circuit simulations and designs. Since all these dynamic variables include DC components and small-signal AC components, these variables can be expressed as:  
\[\left\{\begin{aligned} & i_{2}=\overline{I}_{2}+\overline{i}_{2}\\ & V_{\text{i}}=\overline{V}_{\text{i}}+\overline{v}_{\text{i}}\\ & V_{\text{o}}=\overline{V}_{\text{o}}+\overline{v}_{\text{o}}\\ & d=\overline{D}+\overline{d}\end{aligned}\right. \tag{4}\]  
where \(\overline{I}_{2}\), \(\overline{V}_{\text{i}},\overline{V}_{\text{o}},\overline{D}\) represent the DC components of the output current, input voltage, output voltage, and duty cycle, respectively, while \(\overline{i}_{2}\), \(\overline{v}_{\text{i}}\), \(\overline{v}_{\text{o}},\overline{d}\) represent the AC components of the output current, input voltage, output voltage, and duty cycle, respectively. By substituting the above equations into Equation (3), the small-signal model of the DAB can be derived as [23]:  
\[\overline{i}_{2}=C_{\text{o}}\frac{\text{d}\overline{v}_{\text{o}}}{\text{d} t}+\frac{1}{R_{\text{o}}}\overline{v}_{\text{o}}=\frac{N\overline{V}_{\text{i}} \big{(}1-2\overline{D}\big{)}}{2f_{\text{s}}L}\tilde{d}+\frac{N\overline{D} \big{(}1-\overline{D}\big{)}}{2f_{\text{s}}L}\overline{v}_{\text{i}} \tag{5}\]  
Using Laplace transform, the transfer functions of the DAB system can be obtained as [24]:  
\[\left\{\begin{aligned} & G_{\text{i}2}\text{d}=\frac{N\overline{V}_{ \text{i}}\big{(}1-2\overline{D}\big{)}}{2f_{\text{s}}L}\\ & G_{\text{i}_{2}V_{\text{i}}}=\frac{N\overline{D}\big{(}1- \overline{D}\big{)}}{2f_{\text{s}}L}\\ & G_{\text{V}_{\text{o}}\text{d}}=\frac{R_{\text{o}}}{1+C_{\text{o }}R_{\text{o}}}G_{\text{i}2}\end{aligned}\right. \tag{6}\]