  We derive the branch ampacity constraint associated to power losses for the
convex optimal power flow (OPF) model based on the branch flow formulation. The
branch ampacity constraint derivation is motivated by the physical
interpretation of the transmission line {\Pi}-model and practical engineering
considerations. We rigorously prove and derive: (i) the loop constraint of
voltage phase angle, required to make the branch flow model valid for meshed
power networks, is a relaxation of the original nonconvex alternating current
optimal power flow (o-ACOPF) model; (ii) the necessary conditions to recover a
feasible solution of the o-ACOPF model from the optimal solution of the convex
second-order cone ACOPF (SOC-ACOPF) model; (iii) the expression of the global
optimal solution of the o-ACOPF model providing that the relaxation of the
SOC-ACOPF model is tight; (iv) the (parametric) optimal value function of the
o-ACOPF or SOC-ACOPF model is monotonic with regarding to the power loads if
the objective function is monotonic with regarding to the nodal power
generations; (v) tight solutions of the SOC-ACOPF model always exist when the
power loads are sufficiently large. Numerical experiments using benchmark power
networks to validate our findings and to compare with other convex OPF models,
are given and discussed.
