A security-constrained economic dispatch (SCED) problem is regularly solved by system operators in electric power networks to make day-ahead and realtime dispatch decisions. Preventive SCED is conservative and requires dispatch 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. This thesis analyzes the benefits of allowing recourse actions for simple networks and 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 multiparametric linear programming. This thesis concludes with numerical results on various IEEE test networks.

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Solar hosting capacity analysis (HCA) assesses the ability of a distribution network to host distributed solar generation without seriously violating distribution network constraints. In this paper, we consider risk-sensitive HCA that limits the risk of network constraint violations with a collection of scenarios of solar irradiance and nodal power demands, where risk is modeled via the conditional value at risk (CVaR) measure. First, we consider the question of maximizing aggregate installed solar capacities, subject to risk constraints and solve it as a secondorder cone program (SOCP) with a standard conic relaxation of the feasible set with power flow equations. Second, we design an incremental algorithm to decide whether a configuration of solar installations has acceptable risk of constraint violations, modeled via CVaR. The algorithm circumvents explicit risk computation by incrementally constructing inner and outer polyhedral approximations of the set of acceptable solar installation configurations from prior such tests conducted. Our numerical examples study the impact of risk parameters, the number of scenarios and the scalability of our framework.