The microgrid concept has gained enormous popularity in the power industry due to recent advances in the power electronic converter (PEC) technology and environmental concerns over green-house gas emissions from power generation. Among microgrids, the hybrid AC/DC microgrid concept has been promoted as a viable concept to reduce energy conversion losses. However, hybrid AC/DC microgrids are susceptible to stability issues during high penetration of dynamic loads (e.g. induction machines). The non-linear dynamics of induction machines result in sustained voltage/frequency oscillations following disturbances in the microgrid, which is a major challenge for stable operation of the hybrid AC/DC microgrid. The PEC-based energy storage systems (ESSs) are used as an effective solution for power balancing in the microgrid; hence with the fast response of the PEC, microgrid voltage/frequency could be stabilised rapidly. Thus, a supplementary power oscillation damping (POD) controller is proposed in this paper for the ESS to damp low-frequency oscillations (LFOs) in the hybrid AC/DC microgrid. The effectiveness of the proposed damping controller is verified using non-linear simulations considering different penetration levels of dynamic loads and disturbances in a hybrid AC/DC microgrid. Results indicate that the proposed supplementary POD controller can significantly damp the LFOs in the hybrid AC/DC microgrid.