The recent large-scale penetration of renewable energy in power networks has also introduced with it a risk of random variability. This new source of power uncertainty can fluctuate so substantially that the traditional base-point forecast and control scheme may fail to work. To address this challenge, we study the so-called robust AC optimal power flow (AC-OPF) so as to provide robust control solutions that can immunize the power system against the intermittent renewables. In this paper we generalize the continuous-time primal-dual gradient dynamics approach to solve the robust AC-OPF. One advantage of the proposed approach is that it does not require any convexity assumptions for the decision variables during the dynamical evolution. This paper first derives a stability analysis for the primal-dual dynamics associated with a generic robust optimization, and then applies the primal-dual dynamics to the robust AC-OPF problem. Simulation results are also provided to demonstrate the effectiveness of the proposed approach.