  Coupled nonlinear oscillators, e.g., Kuramoto models, are commonly used to
analyze electrical power systems. The cage model from statistical mechanics has
also been used to describe the dynamics of synchronously connected generation
stations. Whereas the Kuramoto model is good for describing high inertia grid
systems, the cage one allows both high and low inertia grids to be modelled.
This is illustrated by comparing both the synchronization time and relaxation
towards synchronization of each model by treating their equations of motion in
a common framework rooted in the dynamics of many coupled phase oscillators. A
solution of these equations via matrix continued fractions is implemented
rendering the characteristic relaxation times of a grid-generator system over a
wide range of inertia and damping. Following an abrupt change in the dynamical
system, the power output and both generator and grid frequencies all exhibit
damped oscillations now depending on the (finite) grid inertia. In practical
applications, it appears that for a small inertia system the cage model is
preferable.
