## 7 Conclusions  
A reformulation of the MINLP model that represents the optimal phase-balancing in low-voltage distribution networks has been proposed in this paper. A mixed-integer quadratic programming (MI-QP) model was developed, whose main advantages compared to the exact MINLP model are as follows: (i) the global optimum can be guaranteed by combining a B&B algorithm with the quadratic approximation of the three-phase power flow model. (ii) classical heuristic and metaheuristic approaches can be avoided to solve this large-scale combinatorial optimization problem; (iii) by preserving the convex structure of the optimal power flow problem, this model allows one to deal with the uncertainties caused by renewable generation and variable loads while working with probabilities; (iv) under the same simulating conditions, the solution reached the same global solution. The latter implies that it does not require statistical tests to prove its efficiency, which is not possible with metaheuristics.  
As the numerical results have demonstrated, when the probability scenarios of load consumption and power generation are considered, the proposed MI-QP model's solution is robust as it reaches the best configuration possible, thus minimizing the total power losses. This is not possible if the grid is optimized only for a particular load/generation condition.  
For future works, we propose using local measures in load and generation points and extending our approach to real-time phase-balancing in power distribution networks. We also propose that the proposed MI-QP model be used to locate and size three-phase distributed generators, finding unbalanced distribution networks for power loss minimization, or enlarge the voltage stability margin.  
Figure 6: Voltage in generation and demand nodes: (**a**) Before phase-balancing and (**b**) after phase-balancing.  
**Supplementary Materials:** The codes are available at [[https://la.mathworks.com/matlabcentral/fil](https://la.mathworks.com/matlabcentral/fil)]([https://la.mathworks.com/matlabcentral/fil](https://la.mathworks.com/matlabcentral/fil)) eexchange/87447-mixed-integer-quadratic-for-the-phase-balancing.  
Conceptualization, A.G., W.G.-G. and O.D.M.; Methodology, A.G., W.G.-G. and O.D.M.; Investigation, A.G., W.G.-G. and O.D.M.; Writing--review and editing, A.G., W.G.-G., O.D.M., H.R.C. and L.A.-B. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Laboratorio de Simulacion Hardware-in-the-loop para Sistemas Ciberfisicos (LaSSiC). Codigo: 350202.  
Not applicable.  
Not new data were created or analyzed in this study. Data sharing is not applicable to this article.  
The first author was supported by the Vicerrectoria de Investigacion y Extension, Universidad Tecnologica de Pereira, under the grant 6-21-4, project ODIN (Optimal Distribution Networks); The second author was supported by the Centro de Investigacion y Desarrollo Cientifico de la Universidad Distrital Francisco Jose de Caldas under grant 1643-12-2020 associated with the project: \"Desarrollo de una methodologia de optimizacion para la gestion optima de recursos energeticos distribuidos en redes de distribuicion de energia electrica.\" and in part by the Direccion de Investigaciones de la Universidad Tecnologica de Bolivar under grant PS2020002 associated with the project: \"Ubicacion optima de bancos de capacitores de paso fijo en redes electricas de distribucion para reduccion de costos y perdidas de energia: Aplicacion de metodos exactos y metaheuristics.\"  
The authors declare no conflict of interest.