Bibliography

(eng): In this work, we are interested in an optimal power flow problem with fixed voltage magnitudes in distribution networks. This optimization problem is known to be non-convex and thus difficult to solve. A well-known solution methodology consists in reformulating the objective function and the constraints of the original problem in terms of positive semi-definite matrix traces, to which we add a rank constraint. To convexify the problem, we remove this rank constraint. Our main focus is to provide a strong mathematical proof of the exactness of this convex relaxation technique. To this end, we explore the geometry of the feasible set of the problem via its Pareto-front. We prove that the feasible set of the original problem and the feasible set of its convexification share the same Pareto-front. From a numerical point of view, this exactness result allows to reduce the initial problem to a semi-definite program, which can be solved by more efficient algorithms.