## 3 Principle of the Proposed Eios-TPS
### Modulation Strategy in the Low-Power Range When \(k<1\)  
Under boost-mode conditions in the operation of the DAB converter, modes E and G are selected as the optimal modulation strategies for low-power operation. These two  
Figure 8: The optimized operating path in the medium and high-power region for \(k>1\). (**a**) A three-dimensional visualization of the optimized path. (**b**) A two-dimensional projection of the optimized path. modes share the same optimal duty-cycle combination, as shown in (11). Based on (11), the intrinsic relationship among \(D_{1}\), \(D_{2}\), and \(\Phi\) can be further derived, as expressed in (12). According to (12), the optimized trajectory lies on the interface between modes E and G. The corresponding waveforms of the optimized solution are shown in Figure 9a, where two switches achieve ZVS and the remaining six switches operate under ZCS conditions. \[\left\{ \begin{array}{l}D_{1}=\sqrt{\frac{p}{2k(1-k)}}\\ D_{2}=\sqrt{\frac{kp}{2(1-k)}}\\ \Phi=\frac{1}{2}\sqrt{\frac{p(1-k)}{2k}}\\ \end{array} \right. \tag{11}\]  
\[\left\{ \begin{array}{l}D_{2}=kD_{1}\\ \Phi=\frac{D_{1}-D_{2}}{2}\\ \end{array} \right. \tag{12}\]  
As illustrated in Figure 9b, by adopting an optimization strategy analogous to that employed under low-power buck-mode operation, the ZVS failure can be effectively mitigated. This is achieved by appropriately tuning the control parameters \(D_{1}\), \(D_{2}\), and \(\Phi\) while imposing the equality constraint \(i_{L}(t_{3})+i_{L}(t_{4})=0\), thereby ensuring soft-switching for all bridge switches under the boost mode. From this condition, the intrinsic relationship among \(D_{1}\), \(D_{2}\), and \(\Phi\) can be derived as:  
\[D_{2}(-2+k)+k(D_{1}+2\Phi)=0 \tag{13}\]  
Combining (13) with the boundary conditions for mode G, \(k>D_{2}\) is derived. For \(i_{L}(t_{4})>0\), by introducing the auxiliary factor \(\lambda\) (0 < \(\lambda\) < 1) and enforcing 1 - \(D_{1}=\lambda(1\) - \(D_{2}/k)\), the normalized inductor current can be expressed as:  
\[i_{Ln}(t_{4})=k(1-\lambda)(1-\frac{D_{2}}{k}) \tag{14}\]  
As shown in Figure 10, the ZVS region of mode F is defined by the planes \(i_{L}(t_{3})=0\) and \(i_{L}(t_{4})=0\), as well as the coordinate planes. Different values of \(\lambda\) correspond to distinct optimized trajectories, all located on the green surface. The green surface lies entirely within the ZVS region of mode F.  
Figure 9: (a) Operating waveforms of mode E and mode G under minimum current-stress conditions. (b) Further optimized waveforms under low-power operation for \(k<1\). Figure 11 illustrates the relationships between the normalized transmitted power \(p\), the normalized current stress \(i_{\text{pn}}\), and the normalized inductor current \(i_{\text{Ln}}(t_{2})\) for different values of \(\lambda\) (0.4, 0.6, 0.8, and 0.9). As shown in Figure 11, for different \(\lambda\) values, the maximum transmitted power of mode G remains constant at [0, \(2(k-1)/k^{2}\)]. As shown in Figure 11a, as \(\lambda\) increases, the current stress decreases. Conversely, as depicted in Figure 7b, for the same transmitted power, the inductor current \(i_{\text{Ln}}(t_{2})\) decreases with increasing \(\lambda\). To ensure sufficient discharge of the parasitic capacitance in SiC MOSFETs, \(i_{\text{L}}(t_{2})\) must maintain an appropriate value, and thus \(\lambda\) should be slightly lower. The specific value of \(\lambda\) should be determined by a comprehensive consideration of the DAB converter design parameters, balancing ZVS realization and current-stress reduction. In this study, \(\lambda\) = 0.8 is selected as the optimal choice. Figure 11: The effects of \(\lambda\) on the normalized current stress \(i_{\text{pn}}\) and the normalized inductor current \(i_{\text{Ln}}(t_{2})\) for mode G. (**a**) The relationship between \(i_{\text{pn}}\) and normalized transmitted power \(p\). (**b**) The relationship between \(i_{\text{Ln}}(t_{2})\) and normalized transmitted power \(p\). Figure 10: The optimized operating path in the low-power region for \(k\) < 1.