  Many dynamical systems, including power systems, recover from perturbations
more slowly as they approach critical transitions---a phenomenon known as
critical slowing down. If the system is stochastically forced, autocorrelation
and variance in time-series data from the system often increase before the
transition, potentially providing an early warning of coming danger. In some
cases, these statistical patterns are sufficiently strong, and occur
sufficiently far from the transition, that they can be used to predict the
distance between the current operating state and the critical point. In other
cases CSD comes too late to be a good indicator. In order to better understand
the extent to which CSD can be used as an indicator of proximity to bifurcation
in power systems, this paper derives autocorrelation functions for three small
power system models, using the stochastic differential algebraic equations
(SDAE) associated with each. The analytical results, along with numerical
results from a larger system, show that, although CSD does occur in power
systems, its signs sometimes appear only when the system is very close to
transition. On the other hand, the variance in voltage magnitudes consistently
shows up as a good early warning of voltage collapse. Finally, analytical
results illustrate the importance of nonlinearity to the occurrence of CSD.

Physics and Society