## 1 Introduction  
The dual-active-bridge (DAB) converter, a high-frequency DC-DC conversion topology, was first proposed in the 1990s [1]. After decades of development, the DAB converter has been widely adopted in various applications, including DC microgrids [2, 3], new energy vehicles [4, 5], energy storage systems [6, 7], and aerospace systems [8], due to its advantages of bidirectional power flow, galvanic isolation, high power density, soft-switching capability, and multi-voltage adaptability [9, 10]. However, achieving high efficiency across a wide load range remains a key challenge in these applications [11]. The current approaches to improving DAB converter efficiency focus on two aspects: optimizing the current waveform of an inductor using advanced control methods to suppress conduction losses, and facilitating soft-switching to minimize switching losses [12, 13]. Among the various strategies developed for inductor current waveform optimization, current-stress optimization (CSO) [14; 15; 16] and root-mean-square current optimization (RMSO) [17; 18; 19; 20] have been shown to effectively reduce conduction losses. In this context, current stress is characterized by the peak value of the inductor current. While both techniques offer comparable current optimization performance, RMSO suffers from significant limitations. Specifically, analytical solutions often involve square root terms, posing implementation challenges for digital controllers [17; 18]. For numerical solutions, the modulation schemes are typically unsuitable for real-time computation, heavily reliant on control chips, and lack generality and portability [19; 20]. In contrast, CSO demonstrates higher practicality and adaptability in engineering applications. For both Si MOSFETs and SiC MOSFETs, turn-on losses dominate over turn-off losses [18; 21], making ZVS particularly critical. ZVS realization criteria are generally classified into three categories: energy-based (EB ZVS) [22; 23], current-based (CB ZVS) [17; 24], and charge-based (QB ZVS) [25; 26]. While CB ZVS is theoretically less accurate than EB ZVS and QB ZVS, it offers higher engineering feasibility and is better suited for practical applications. From a device perspective, SiC MOSFETs, with significantly lower junction capacitance compared to Si MOSFETs, require lower switch-current thresholds for ZVS, making ZVS more easily achievable. ZVS implementation methods fall into two main categories: topology modifications [27; 28] and control strategy optimizations. Although topology modifications can achieve ZVS, they increase system cost and converter complexity. In contrast, control-based ZVS techniques, which rely solely on algorithmic adjustments without additional hardware changes, provide a more economical and practical solution for soft-switching operation. For conventional DAB converters operating at a fixed switching frequency, modulation strategies are typically categorized by their degrees of freedom: single-degree-of-freedom (single phase-shift modulation, SPS), two-degrees-of-freedom (dual phase-shift modulation, DPS, and extended phase-shift modulation, EPS), and three-degrees-of-freedom (triple phase-shift modulation, TPS). Among these, TPS serves as the fundamental form, with SPS, DPS, and EPS regarded as special cases [29; 30]. SPS modulation suffers from significant limitations, including a narrow ZVS range, high current stress, and large inductor current RMS values, especially under light load conditions or voltage conversion ratio mismatches. These issues lead to reduced efficiency and limited applicability [31; 32]. In contrast, higher-degree modulation strategies such as DPS, EPS, and TPS offer greater flexibility, enabling reduced current stress and extended ZVS operation. To address these challenges, various phase-shift modulation methods have been proposed, including DPS [33; 34; 35; 36; 37] and EPS [38; 39].