| Summary: | Phase balancing is a classical optimization problem in power distribution grids that involve phase swapping of the loads and generators to reduce power loss. The problem is a non-linear integer and, hence, it is usually solved using heuristic algorithms. This paper proposes a mathematical reformulation that transforms the phase-balancing problem in low-voltage distribution networks into a mixed-integer convex quadratic optimization model. To consider both conventional secondary feeders and microgrids, renewable energies and their subsequent stochastic nature are included in the model. The power flow equations are linearized, and the combinatorial part is represented using a Birkhoff polytope <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">B</mi><mn>3</mn></msub></semantics></math></inline-formula> that allows the selection of phase swapping in each node. The numerical experiments on the CIGRE low-voltage test system demonstrate the use of the proposed formulation.
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