Alan C. Liddell | Searcharxiv

arXiv2016

Hybrid Methods in Solving Alternating-Current Optimal Power Flows

Many steady-state problems in power systems, including rectangular power-voltage formulations of optimal power flows in the alternating-current model (ACOPF), can be cast as polynomial optimisation problems (POP). For a POP, one can derive strong convex relaxations, or rather hierarchies of ever stronger, but ever larger relaxations. We study means of switching from solving the convex relaxation to Newton method working on a non-convex Lagrangian of the POP.

math.OC↗

arXiv2016

Numerically validating the completeness of the real solution set of a system of polynomial equations

Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares programming to test whether a given set is "complete" with respect to the real solution set. Specifically, we test whether the Zariski closure of that given set is indeed equal to the solution set of the real radical of the ideal generated by the given polynomials. Examples with finitely and infinitely many real solutions are provided, along with an example having polynomial inequalities.

math.NA↗