Local stability of DC microgrids: A perspective of graph laplacians with self-loops

This paper studies the local stability problem of DC microgrids from a network-based perspective that provides new insights. Firstly, the stability problem is transferred to the positive definiteness of the Laplacian matrix of a loopy graph. The graph preserves the information of the original DC network structure, and contains both positive and negative self-loops that are introduced by the differential conductances of net loads in the microgrid. Then, the concept of effective grounded conductance is introduced to measure the impact of self-loops. New criteria in terms of the effective grounded conductance are established to check the positive definiteness of the Laplacian matrix of a loopy graph, and thus can be used to infer the local stability of DC microgrids. It shows that the negative effective grounded conductance is the crucial factor in triggering the instability of DC microgrids. The instability mechanism can be physically interpreted as the loss of passivity of some buses caused by the negative differential conductances of net loads.