  The formulations and approximations of the branch flow model for mesh power
networks (Mesh-BranchFlow) are given in this paper. Using different sets of the
power flow equations, six formats of the exact Mesh-BranchFlow model are
listed. These six formats are mathematically equivalent with each other. Linear
approximation and second-order cone programming (SOCP) are then used to derive
the six formats of the convex Mesh-BranchFlow model. The branch ampacity
constraints considering the shunt conductance and capacitance of the
transmission line $\Pi$-model are derived. The key foundation of deriving the
ampacity constraints is the correct interpretation of the physical meaning of
the transmission line $\Pi$-model. An exact linear expression of the ampacity
constraints of the power loss variable is derived. The applications of the
Mesh-BranchFlow model in deriving twelve formats of the exact optimal power
flow (OPF) model and twelve formats of the approximate OPF model are formulated
and analyzed. Using the Julia programming language, the extensive numerical
investigations of all formats of the OPF models show the accuracy and
computational efficiency of the Mesh-BranchFlow model. A penalty function based
approximation gap reduction method is finally proposed and numerically
validated to improve the AC-feasibility of the approximate Mesh-BranchFlow
model.
