This paper introduces an adaptive super-twisting sliding mode control (ASTSMC) approach for controlling a dual active bridge (DAB) converter with an extended phase shift (EPS) modulation. The conventional single-phase shift (SPS) modulation-based DAB converter is known to be inefficient. Hence, an optimization algorithm based on the Lagrange multiplier method (LMM) is proposed to minimize both backflow power and inductor current stress simultaneously. Unlike the conventional schemes that use an offline optimization (OFFO) method to derive the phase shift ratios, this paper proposes an online optimization method and an ASTSMC method for generating the inner and outer phase shift ratio respectively. Initially, a generalized average modeling (GAM) for the DAB converter under EPS modulation is derived, and then the proposed ASTSMC is introduced according to this model. The conventional STSMC with constant gains suffers from low performance under disturbances such as load current perturbations, input voltage variations, and output voltage reference variations. Additionally, it requires an overestimated gain under steady-state conditions. To address these issues, a variable gain-based STSMC scheme is proposed to enhance the performance of the converter under all operating conditions. The effectiveness of the proposed method is verified through simulation and experimental results, which are compared with the results of the conventional STSMC method.INDEX TERMS Dual active bridge converter, adaptive super twisting sliding mode control, extended phase shift modulation.