Stability analysis of power systems with voltage dynamics based on an invariant probability distribution

To maintain the transient stability, which is crucial for the proper functioning of power systems, the fluctuations in frequency, voltage, and phase difference need to be within a required secure range. In this study, based on the third-order model of synchronous machines, we model stochastic disturbances by the Brownian motion and use the variance in the invariant probability distribution of a Gaussian process corresponding to the power systems to characterize the state fluctuations. A procedure for calculating these variance matrices is formulated, with which the interplay of the fluctuations between the phase angle dynamics and the voltage dynamics is investigated. It is revealed that increasing line capacity helps improve the phase angle stability but may also lead to more serious voltage fluctuations. Based on the invariant probability distribution, an optimization framework is established to increase the probability of the state staying in the required secure range. Simulations show that the optimization framework can significantly extend the mean first hitting time of the state to the boundary of the required secure set, thus effectively enhancing the transient stability.