Convex Relaxations of Security Constrained AC Optimal Power Flow Under Uncertainty

System operators have to ensure an N-1 secure operation, while dealing with higher degrees of uncertainty. This paper proposes a semidefinite relaxation of the chance and security constrained optimal power flow (SCOPF). Our main contributions are the introduction of systematic methods to obtain zero relaxation gap, providing a tractable chance constrained SCOPF formulation, and addressing scalability. We introduce a systematic procedure to obtain zero relaxation gap using a penalty term on power losses. To achieve tractability of the joint chance constraint, a piecewise affine approximation, and a combination of randomized and robust optimization is used. To address scalability, we propose an iterative solution algorithm to identify binding constraints, and we apply a chordal decomposition of the semidefinite constraints. We demonstrate the performance of our approach on IEEE 24 and IEEE 118 bus system using realistic day-ahead forecast data and obtain tight near-global optimality guarantees.