  The synchronization stability has been analyzed as one of the important
dynamical characteristics of power grids. In this study, we bring the
operational perspective to the synchronization stability analysis by counting
not only full but also partial synchronization between nodes. To do so, we
introduce two distinct measures that estimate the operational resilience of
power-grid nodes: functional stability and functional resistance. We
demonstrate the practical applicability of the measures in a model network
motif and an IEEE test power grid. As a case study of German power grid, we
reveal that the modular structure of a power grid and particular unidirectional
current flow determine the distribution of the operational resilience of
power-grid nodes. Reproducing our finding on clustered benchmark networks, we
validate the modular effect on power grid stability and confirm that our
measures can be the insightful tools to understand the power grids'
synchronization dynamics.
