Solid-state transformers (SSTs) are increasingly adopted for microgrids due to their bidirectional power flow capability, availability of medium-voltage dc and low-voltage dc (LVdc) links, and ancillary services. However, linear and nonlinear loads at the low-voltage ac (LVac) side introduce ripple into the LVdc link. These ripples cause capacitor stress, efficiency degradation, and deterioration of power quality. The conventional ripple mitigation techniques, such as oversized dc-link capacitors, passive/active filters, etc. increase the cost and volume of system, and limits the dynamic performance. Another conventional approaches include direct compensation that impose large current stress on the dc-dc converter. To address these challenges, this paper proposes a model predictive control (MPC) strategy for the dual active bridge (DAB) converter to actively suppress LVdc ripple in SST-based microgrids. The proposed method leverages a moving discretized control set MPC (MDCS-MPC) framework that explicitly models both second-harmonic and higher-order ripple dynamics, adaptively adjusting the DAB phase-shift angle for effective ripple mitigation. Simulation results demonstrate reduced LVdc voltage ripple, and improved voltage regulation under nonlinear load conditions.