The variability of renewable energy offers significant challenges to the power system security with a large penetration of renewables. The paper models the wind farm penetration as a Gaussian excitation in which the stochastic differential equations (SDEs) are considered to characterize wind energy uncertainties in nonlinear power systems. The SDE-based power system model is first reduced to the averaged Ito SDEs by the stochastic averaging method. Then, a backward Kolmogorov equation for the conditional reliability function and the generalized Pontryagin equations governing the conditional moments of first passage time are established. Finally, numerical results are provided given the designated boundary and initial conditions. The first passage time of both single-machine infinite-bus power system and 3-machine 9-bus system under Gaussian excitation are studied. The analytical results are verified by using a Monte Carlo simulation.