In this paper, a novel nonlinear control scheme for the on-board DC micro-grid of a hybrid electric aircraft is proposed to achieve voltage regulation of the low voltage (LV) bus and power sharing among multiple sources. Considering the accurate nonlinear dynamic model of each DC/DC converter in the DC power distribution system, it is mathematically proven that accurate power sharing can be achieved with an inherent overcurrent limitation for each converter separately via the proposed control design using Lyapunov stability theory. The proposed framework is based on the idea of introducing a constant virtual resistance at the input of each converter and a virtual controllable voltage that can be either positive or negative, leading to a bidirectional power flow. Compared to existing control strategies for on-board DC micro-grid systems, the proposed controller guarantees accurate power sharing, tight voltage regulation and an upper limit of each source's current at all times, including during transient phenomena. Simulation results of the LV dynamics of an aircraft on-board DC micro-grid are presented to verify the proposed controller performance in terms of voltage regulation, power sharing and the overcurrent protection capability.