## 2 DAB Topology and Modeling Analysis
### TPS Mode Classification and Mode Selection of the DAB Converter  
Figure 3 illustrates the typical voltage and current waveforms under the eight TPS modulation modes. The transmitted power for each mode is derived according to (1), and Table 2 summarizes the constraint conditions and normalized power expressions for each mode. In this work, \(p\) denotes the normalized power transferred through the converter. The transmitted power is normalized against the maximum transmitted power of the SPS modulation, defined as \(P_{n}=nV_{1}V_{2}/8\textit{f}_{s}L\).  
\[\text{P}=\frac{1}{T_{\text{hs}}}\int_{0}^{T_{\text{hs}}}v_{1}i_{L}(t)dt \tag{1}\]  
All eight modulation modes support forward power transfer. However, modes A, B, C, and H do not feature intervals in which \(v_{1}\) and \(v_{2}\) exhibit the same polarity. As a result, direct energy transfer from the primary to the secondary side is not possible. Instead, these modes rely on the inductor to temporarily store and transfer energy, which leads to elevated current stress and increased conduction losses. Therefore, under forward power transmission conditions, only modes D-G are practically meaningful for optimization. This  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multirow{2}{*}{**Mode Constraints**} & **Expression of Normalized** \\  & & **Transmitted Power \(p\)** \\ \hline A & \(1+\frac{D_{1}-D_{2}}{2}\Leftrightarrow\Phi\leq 1+\frac{D_{2}-D_{1}}{2}\) & \(4D_{2}(1-\phi)\) & [0, 0.5] \\ \hline B & \(\min\left\{\frac{D_{1}+D_{2}}{2},1-\frac{D_{1}+D_{2}}{2}\right\}\Leftrightarrow \Phi\leq\min\left\{1+\frac{D_{1}-D_{1}}{2},1+\frac{D_{1}-D_{2}}{2}\right\}\) & \(2D_{1}D_{2}-2\left(\phi+\frac{D_{1}+D_{2}}{2}-1\right)^{2}\) & [0, 0.5] \\ \hline C & \(\frac{D_{1}+D_{2}}{2}\Leftrightarrow\[PERSON]\leq 1-\frac{D_{1}+D_{2}}{2}\) & \(2D_{1}D_{2}\) & [0, 0.5] \\ \hline D & \(1-\frac{D_{1}+D_{2}}{2}\Leftrightarrow\[PERSON]\leq\frac{D_{1}+D_{2}}{2}\) & \(1-(1-D_{1})^{2}-(1-D_{2})^{2}-(1-2\Phi)^{2}\) & [0, 1] \\ \hline E & \(\min\left\{\frac{D_{1}-D_{2}}{2},\frac{D_{2}-D_{1}}{2}\right\}\Leftrightarrow \Phi\leq\min\left\{\frac{D_{1}+D_{2}}{2},1-\frac{D_{1}+D_{2}}{2}\right\}\) & \(-\frac{(D_{1}-D_{2})^{2}}{2}+2(D_{1}+D_{2})\Phi-2\Phi^{2}\) & [0, 0.666] \\ \hline F & \(0\leq\phi\leq\frac{D_{2}-D_{1}}{2}\) & \(4D_{1}\Phi\) & [0, 0.5] \\ \hline G & \(0\leq\phi\leq\frac{D_{1}-D_{2}}{2}\) & \(4D_{2}\Phi\) & [0, 0.5] \\ \hline H & \(1+\frac{D_{1}-D_{2}}{2}\Leftrightarrow\Phi\leq 1+\frac{D_{2}-D_{1}}{2}\) & \(4D_{1}(1-\phi)\) & [0, 0.5] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Constraints, transmitted power expressions, and operating ranges of eight modulation modes.  
Figure 3: Typical voltage and current waveforms under eight TPS modulation modes. (**a**) Mode A. (**b**) Mode B. (**c**) Mode C. (**d**) Mode D. (**e**) Mode E. (**f**) Mode F. (**g**) Mode G. (**h**) Mode H.  
study aims to optimize the operational performance of the DAB converter in both buck and boost modes under forward power transfer. The current stress is characterized by the peak inductor current corresponding to each operating mode. By analyzing the voltage and current waveforms of the converter, the current stresses associated with modes D through G in both buck and boost scenarios are analytically derived and systematically presented in Table 2.  
The current stress is normalized as \(I_{\text{n}}=nV_{2}/4f_{\text{s}}L\), where the actual inductor current is represented by \(i_{L}(t_{x})\), and the normalized current is denoted as \(i_{Ln}(t_{x})\), with \(x=0\), 1, 2, 3, 4, The current stress corresponding to each mode is indicated as \(i_{yy}\), where \(y=\) D, E, F, G. The normalized inductor current values at each time instant for modes D-G are presented in Table 3. Table 4 summarizes the peak current stresses for these four operating modes under both buck and boost conditions.