  This paper investigates the theory of resilience and stability in
multi-microgrid networks. We derive new sufficient conditions to guarantee
small-signal stability of multi-microgrids in both lossless and lossy networks.
The new stability certificate for lossy networks only requires local
information, thus leads to a fully distributed control scheme. Moreover, we
study the impact of network topology, interface parameters (virtual inertia and
damping), and local measurements (voltage magnitude and reactive power) on the
stability of the system. The proposed stability certificate suggests the
existence of Braess's Paradox in the stability of multi-microgrids, i.e. adding
more connections between microgrids could worsen the multi-microgrid system
stability as a whole. We also extend the presented analysis to
structure-preserving network models, and provide a stability certificate as a
function of original network parameters, instead of the Kron reduced network
parameters. We provide a detailed numerical study of the proposed certificate,
the distributed control scheme, and a coordinated control approach with line
switching. The simulation shows the effectiveness of the proposed stability
conditions and control schemes in a four-microgrid network, IEEE 33-bus system,
and several large-scale synthetic grids.
