# Random Perturbations of a Three-Machine Power System Network

## Abstract

This paper develops an asymptotic method based on averaging and large deviations to study the transient stability of a noisy three-machine power system network. We study the dynamics of these nonlinear oscillators (swing equations) as random perturbations of two-dimensional periodically driven Hamiltonian systems. The phase space for periodically driven nonlinear oscillators consists of many resonance zones. It is well known that, as the strengths of periodic excitation and damping go to zero, the measure of the set of initial conditions which lead to *capture in a resonance zone* goes to zero. In this paper we study the effect of weak noise on the escape from a resonance zone and obtain the large-deviation rate function for the escape. The primary goal is to show that the behavior of oscillators in the resonance zone can be adequately described by the (slow) evolution of the Hamiltonian, for which simple analytical results can be obtained, and then apply these results to study the transient stability margin of power system with stochastic loads. The classical swing equations of a power system of three interconnected generators with non-zero damping and small noise is considered as a nontrivial example to derive the “exit time” analytically. This work may play an important role in designing and upgrading existing electrical power system networks.

## Keywords

Homoclinic Orbit Large Deviation Principle Periodic Excitation Transient Stability Resonance Zone## Notes

### Acknowledgements

The authors acknowledges the support of the AFOSR under grant number FA9550-16-1-0390 and PSERC.

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