A security-constrained economic dispatch (SCED) problem is regularly solved by system operators in electric power networks to make day-ahead and real-time dispatch decisions. Preventive SCED is conservative and requires dis-patch decisions that are secure against any single component failure. Corrective (recourse) actions can significantly reduce operational costs. Even with linear power flow models, corrective SCED poses significant computational challenges owing to an increase in the dimensionality arising from additional recourse decisions and the number of contingencies to guard against. In this presentation, I will describe a novel risk-sensitive formulation for SCED that captures the tradeoff between cost and reliability. I will also demonstrate an algorithm that tackles the computational challenges of solving the problem at scale through a decomposition of the problem via a critical region exploration technique that exploits the problem structure using properties of multi-parametric linear programming.