This work studies the DC-DC conversion stage in solid-state transformers (SST). The traditional two- or three-level dual active bridge (DAB) topology faces limitations in microgrid interconnection due to power and voltage limitations. For this reason, the use of multilevel topologies such as active neutral point clamped (ANPC) is a promising alternative. Additionally, the efficiency of the SSTs is a recurring concern, and reducing losses in the DC-DC stage is a subject to be studied. In this context, this work presents a new control technique based on an adaptive model- based predictive control (AMPC) to select the modulation technique of an ANPC-DAB DC-DC converter aimed at reducing losses and increasing efficiency. The single-phase shift (SPS), triangular, and trapezoidal modulation techniques are used according to the converter output power with the aim of maximizing the number of soft-switching points per cycle. The performance of the proposed control technique is demonstrated through real-time simulation and a reduced-scale experimental setup. The findings indicate the effectiveness of the AMPC control technique in mitigating voltage source perturbations. This technique has low output impedance and is robust to converter parameter variations. Prototyping tests revealed that, in steady-state, the AMPC significantly improves converter efficiency without compromising dynamic performance. Despite its advantages, the computational cost of AMPC is not significantly higher than that of traditional model predictive control (MPC), allowing for the allocation of time to other applications.