The recent literature discusses the use of the relaxed second-order cone programming (SOCP) for formulating optimal power flow problems (OPF) for radial power grids. However, if the shunt parameters of the lines that compose 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. The same formulations also have not correctly accounted for the lines' ampacity constraint. 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 for the OPF in radial power grids, for which the ac-OPF equations, including the transverse parameters, are considered. To limit the lines' current together with the nodal voltage-magnitudes, we augment the formulation with a new set of more conservative constraints. 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.