  Power grids sustain modern society by supplying electricity and thus their
stability is a crucial factor for our civilization. The dynamic stability of a
power grid is usually quantified by the probability of its nodes' recovery to
phase synchronization of the alternating current it carries, in response to
external perturbation. Intuitively, the stability of nodes in power grids is
supposed to become more robust as the coupling strength between the nodes
increases. However, we find a counterintuitive range of coupling strength
values where the synchronization stability suddenly droops as the coupling
strength increases, on a number of simple graph structures. Since power grids
are designed to fulfill both local and long-range power demands, such simple
graph structures or graphlets for local power transmission are indeed relevant
in reality. We show that the observed nonmonotonic behavior is a consequence of
transitions in multistability, which are related to changes in stability of the
\emph{unsynchronized} states. Therefore, our findings suggest that a
comprehensive understanding of changes in multistability are necessary to
prevent the unexpected catastrophic instability in the building blocks of power
grids.
