  The recent literature has discussed the use of the relaxed Second Order Cone
Programming (SOCP) to formulate Optimal Power Flow problems (OPF) for radial
power grids. However, if the shunt parameters of the lines, composing the power
grid, are considered the proposed methods do not provide sufficient conditions
that can be verified ex ante for the exactness of the optimal solutions.
Additionally, the same formulations have not correctly accounted for the
ampacity constraint of the lines. Similar to the inclusion of upper
voltage-magnitude limit, the SOCP relaxation faces difficulties when the
ampacity constraints of the lines are binding. In order to overcome these
limitations, we propose a convex formulation of the OPF problem applied to
radial power grids for which the AC-OPF equations, including the transverse
parameters, are considered. We augment the formulation with a new set of more
conservative constraints to limit the lines current together with the nodal
voltage-magnitudes. Sufficient conditions are provided to ensure the
feasibility and optimality of the proposed OPF solution. Furthermore, the
proofs of the exactness of the SOCP relaxation are provided. Using standard
power grids, we show that these conditions are mild and hold for real
distribution networks.
