In future modern power systems, reliability and resilience could be an extreme challenge caused by the stability issues of the bidirectional power converters (BPCs). The non-linear dynamics of DC link voltage (DCLV) of BPCs in interaction with the existing linear control schemes may decrease the stability margin and cause operating-point-dependent instability issues. Existing approaches may solve this issue by reducing the DCLV control loop bandwidth, which considerably degrades the system performance. To tackle this issue, first, the root cause of the instability challenge is analytically investigated, and then, a non-linear stabilizer control scheme based on Lyapunov theorem is proposed. Considering the non-linear dynamic of the BPCs and the interaction between dynamics of DC link voltage and AC currents in the proposed stabilizer, it guarantees the stability of the converter in both directions of power flow and the full range of loading conditions. The performance of the proposed scheme is verified through simulation of the system under various operating conditions, considering uncertainties, disturbances, and short-circuit events, and comparing it with that of prevalent controllers.