## 3 Principle of the Proposed Eios-TPS
### Modulation Strategy in the Low-Power Range When k > 1  
When the DAB converter operates in buck mode under low-power conditions, mode E is identified as the optimal switching strategy for the DAB converter. Taking mode E as a representative case, the corresponding Lagrange function is established to characterize the interdependence among normalized current stress, normalized transferred power, ZVS constraints, and the duty-cycle limitations inherent to the operating mode, as formulated below: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode** & **Maximum Current Stress When k > 1** & **Maximum Current Stress When k < 1** \\ \hline D & \(D_{1}(-1+k)+2\Phi\) & \(D_{2}(1-k)+2k\Phi\) \\ \hline E & \(D_{1}(-1+k)+2\Phi\) & \(D_{2}(1-k)+2k\Phi\) \\ \hline F & \(D_{1}(-1+k)+2\Phi\) & \(\max\{D_{2}-D_{1}k,D_{1}(-1+k)+2\Phi\}\) \\ \hline G & \(D_{1}k-D_{2}\) & \(\max\{D_{2}(1-k)+2k\Phi,D_{1}k-D_{2}\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum current stress of modes D, E, F, and G. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mode** & \(i_{Ln}(t_{x})\) & \(i_{Ln}(t_{x})\) & \(i_{Ln}(t_{x})\) & \(i_{Ln}(t_{x})\) & \(i_{Ln}(t_{x})\) \\ \hline D & \(D_{1}(1-k)-2\Phi\) & \(2-D_{1}(1+k)-2\Phi\) & \(D_{2}(1+k)+2k(-1+\Phi)\) & \(D_{2}(1-k)+2k\Phi\) & \(D_{1}(-1+k)+2\Phi\) \\ \hline E & \(D_{1}(1-k)-2\Phi\) & \(D_{2}-D_{1}k\) & \(D_{2}-D_{1}k\) & \(D_{2}(1-k)+2k\Phi\) & \(D_{1}(-1+k)+2\Phi\) \\ \hline F & \(D_{1}(1-k)-2\[PERSON]\) & \(D_{2}-D_{1}k\) & \(D_{2}-D_{1}k\) & \(D_{1}(1-k)+2\[PERSON]\) & \(D_{1}(-1+k)+2\[PERSON]\) \\ \hline G & \(D_{2}-D_{1}k\) & \(D_{2}(1-k)+2k\[PERSON]\) & \(D_{2}(-1+k)+2k\[PERSON]\) & \(D_{1}k-D_{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The normalized inductor current values at each time instant for modes D–G. \[ \begin{array}{l}L_{\text{f}}=-D_{1}+D_{1}k+2\[PERSON]+\lambda\left(-\frac{(D_{1}-D_{2}) ^{2}}{2}+2(D_{1}+D_{2})\Phi-2\[PERSON]^{2}-p\right)\\ \qquad+\mu_{1}\left(\frac{D_{1}-D_{2}}{2}-\[PERSON]\right)+\mu_{2}\left(\[PERSON]-\left(1 -\frac{D_{1}+D_{2}}{2}\right)\right)+\mu_{3}\left(\frac{D_{2}-D_{1}}{2}-\[PERSON] \right)+\mu_{4}\left(\[PERSON]-\frac{D_{1}+D_{2}}{2}\right)\end{array}  \tag{2}\]  
where \(\lambda\) denotes the equality constraint multiplier, and the \(\mu_{i}\) values (\(i=1,2,3,4\)) denote the inequality constraint multipliers. By applying the LMM and the KKT conditions, the following results can be obtained:  
\[ \begin{array}{l}\left\{\begin{array}{l}\frac{\partial L_{\text{f}}}{ \partial D_{1}}=0,\frac{\partial L_{\text{f}}}{\partial D_{2}}=0,\frac{ \partial L_{\text{f}}}{\partial\Phi}=0\\ \lambda\
eq 0,\mu_{1},\mu_{2},\mu_{3},\mu_{4}\geq 0\\ -\frac{(D_{1}-D_{2})^{2}}{2}+2(D_{1}+D_{2})\Phi-2\Phi^{2}-p=0\\ \mu_{1}\left(\frac{D_{1}-D_{2}}{2}-\[PERSON]\right)=0,\mu_{2}\left(\[PERSON]-\left(1- \frac{D_{1}+D_{2}}{2}\right)\right)=0,\mu_{3}\left(\frac{D_{2}-D_{1}}{2}-\[PERSON] \right)=0,\mu_{4}\left(\[PERSON]-\frac{D_{1}+D_{2}}{2}\right)=0\\ \frac{D_{1}-D_{2}}{2}-\[PERSON]\leq 0,\[PERSON]-\left(1-\frac{D_{1}+D_{2}}{2}\right)\leq 0, \frac{D_{2}-D_{1}}{2}-\[PERSON]\leq 0,\mu_{4}\left(\[PERSON]-\frac{D_{1}+D_{2}}{2}\right)\leq 0 \\ k>1\end{array}\right.\end{array}  \tag{3}\]  
By solving (3), the optimal duty-cycle combination can be expressed as:  
\[\left\{ \begin{array}{l}D_{1}=\sqrt{\frac{p}{2(k-1)}}\\ D_{2}=k\sqrt{\frac{p}{2(k-1)}}\\ \[PERSON]=\frac{1}{2}\sqrt{\frac{(k-1)p}{2}}\end{array} \right. \tag{4}\]  
As shown in (4), for any given \(p\) and \(k\), there always exists a set of corresponding \(D_{1}\), \(D_{2}\), and \(\Phi\) values. Moreover, the intrinsic relationship among \(D_{1}\), \(D_{2}\), and \(\Phi\) can be expressed as:  
\[\left\{ \begin{array}{l}\Phi=\frac{1}{2}(D_{2}-D_{1})\\ D_{2}=kD_{1}\end{array} \right. \tag{5}\]  
Applying the same procedure to mode F yields the optimal duty-cycle combination, which is consistent with the results in (5).