  Motivated by recent interest for multi-agent systems and smart power grid
architectures, we discuss the synchronization problem for the network-reduced
model of a power system with non-trivial transfer conductances. Our key insight
is to exploit the relationship between the power network model and a
first-order model of coupled oscillators. Assuming overdamped generators
(possibly due to local excitation controllers), a singular perturbation
analysis shows the equivalence between the classic swing equations and a
non-uniform Kuramoto model. Here, non-uniform Kuramoto oscillators are
characterized by multiple time constants, non-homogeneous coupling, and
non-uniform phase shifts. Extending methods from transient stability,
synchronization theory, and consensus protocols, we establish sufficient
conditions for synchronization of non-uniform Kuramoto oscillators. These
conditions reduce to and improve upon previously-available tests for the
standard Kuramoto model. Combining our singular perturbation and Kuramoto
analyses, we derive concise and purely algebraic conditions that relate
synchronization and transient stability of a power network to the underlying
system parameters and initial conditions.
