  A new method for the stability assessment of inverter-based microgrids is
presented in this paper. Directly determining stability boundaries by searching
the multidimensional space of inverters' droop gains is a computationally
prohibitive task. Instead, we build a certified stability region by utilizing a
generalized Laplacian matrix eigenvalues, which are a measure of proximity to
stability boundary. We establish an upper threshold for the eigenvalues that
determines the stability boundary of the entire system and demonstrate that
this value depends only on the network's R/X ratio but does not depend on the
grid topology. We also provide a conservative upper threshold of the
eigenvalues that are universal for any systems within a reasonable range of R/X
ratios. We then construct approximate certified stability regions representing
convex sets in the multidimensional space of droop gains that could be utilized
for gains optimization. We show how the certified stability region can be
maximized by properly choosing droop gains, and we provide closed-form analytic
expressions for the certified stability regions. The computational complexity
of our method is almost independent of the number of inverters. The proposed
methodology has been tested using IEEE 123 node test system with 10 inverters.
