Existence and Stability of Equilibrium of DC Micro-Grid Under Master-Slave Control

In this paper, we analyze the existence and stability of equilibrium of dc micro-grids under the master-slave control (where some distributed generations (DGs) are under droop control with dual-loop (Droop-DGs) and some DGs are under MPPT control (MPPT-DGs)). Firstly, the power-flow equation of the dc micro-grids under master-slave control with CPLs is obtained. Then, we transform the solvability of the power-flow equation into the existence of a fixed point for a contraction mapping. Based on Banach's fixed point theorem, a sufficient condition to guarantee the existence of the power-flow solution in dc micro-grids is derived. The condition derived in this paper is not only useful for master-slave control but also for droop control. Besides, to calculate the power-flow solution, an iterative algorithm with exponential convergence rate is proposed. Secondly, we use a singular perturbation model to predict the qualitative behavior of the system near the equilibrium point. By analyzing eigenvalues of the boundary layer system and reduced-order system Jacobian matrix, the robust stable analytic conditions of the system are obtained. The effect of the sampling delay on the system stability is analyzed, the robust stability condition is obtained by using linear matrix inequality. The simulation results verify the correctness of the proposed conditions. The obtained conditions provide a reference for establishing a reliable dc micro-grid.