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In this work, we consider two-stage quadratic optimization problems under ellipsoidal uncertainty. In the first stage, one needs to decide upon the values of a subset of optimization variables (control variables). In the second stage, the uncertainty is revealed, and the rest of the optimization variables (state variables) are set up as a solution to a known system of possibly nonlinear equations. This type of problem occurs, for instance, in optimization for dynamical systems, such as electric power systems as well as gas and water networks. We propose a convergent iterative algorithm to build a sequence of approximately robustly feasible solutions with an improving objective value. At each iteration, the algorithm optimizes over a subset of the feasible set and uses affine approximations of the second-stage equations while preserving the nonlinearity of other constraints. We implement our approach and demonstrate its performance on Matpower instances of AC optimal power flow. Although this paper focuses on quadratic problems, the approach is suitable for more general setups. 

The admittance matrix encodes the network topology and electrical parameters of a power system in order to relate the current injection and voltage phasors. Since admittance matrices are central to many power engineering analyses, their characteristics are important subjects of theoretical studies. This paper focuses on the key characteristic of invertibility. Previous literature has presented an invertibility condition for admittance matrices. This paper first identifies and fixes a technical issue in the proof of this previously presented invertibility condition. This paper then extends this previous work by deriving new conditions that are applicable to a broader class of systems with lossless branches and transformers with off-nominal tap ratios. 

We propose an approach for verifying that a given feasible point for a polynomial optimization problem is globally optimal. The approach relies on the Lasserre hierarchy and the result of Lasserre regarding the importance of the convexity of the feasible set as opposed to that of the individual constraints. By focusing solely on certifying global optimality and relaxing the Lasserre hierarchy using necessary conditions for positive semidefiniteness based on matrix determinants, the proposed method is implementable as a computationally tractable linear program. We demonstrate this method via application to the optimal power flow problem used to operate electric power systems. 

To compute reliable and low-cost operating points, electric power system operators solve optimization problems that enforce inequality constraints such as limits on line flows, voltage magnitudes, and generator outputs. A common empirical observation regarding these constraints is that only a small fraction of them are binding (satisfied with equality) during operation. Furthermore, the same constraints tend to be binding across time periods. Recent research efforts have developed constraint screening algorithms that formalize this observation and allow for screening across operational conditions that are representative of longer time periods. These algorithms identify redundant constraints, i.e., constraints that can never be violated if other constraints are satisfied, by solving optimization problems for each constraint separately. This paper investigates how the choice of power flow formulation, represented either by the non-convex AC power flow, convex relaxations, or a linear DC approximation, impacts the results and the computational time of the screening method. This allows us to characterize the conservativeness of convex relaxations in constraint screening and assess the efficacy of the DC approximation in this context.