Formulations and Approximations of the Branch Flow Model for Mesh Power Networks

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.