This paper proposes a branch flow model for analyzing and optimizing mesh and radial power networks. The model leads to a new approach for solving optimal power flow (OPF) through two relaxation steps: angle relaxation and conic relaxation. For radial networks, both relaxations are exact when there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact, but the angle relaxation may not be. The paper also introduces a method to convexify mesh networks using phase shifters, allowing efficient optimization of the convexified network. The branch flow model focuses on branch currents and powers, unlike the traditional bus injection model that focuses on nodal variables. The model is used to formulate OPF as a conic program (SOCP) after two relaxations. The first relaxation eliminates voltage and current angles, resulting in the Baran-Wu model. The second relaxation converts the quadratic equality constraint into an inequality, yielding a convex problem. For radial networks, the conic relaxation is exact when there are no upper bounds on loads. For mesh networks, the conic relaxation is also exact, but the angle relaxation may not be. The paper provides a way to determine if a relaxed solution is globally optimal by checking the angle recovery condition. This condition ensures that the implied angle differences around each cycle sum to zero modulo $2\pi$. The paper also discusses the equivalence of the branch flow model and the bus injection model, showing that they describe the same physical system. The branch flow model allows for more efficient optimization of mesh networks using phase shifters, which can be placed outside a spanning tree of the network. The paper concludes with a detailed analysis of the branch flow model, the two relaxations, and the conditions for exact relaxation. It also presents simulation results and a proposed algorithm for solving OPF.This paper proposes a branch flow model for analyzing and optimizing mesh and radial power networks. The model leads to a new approach for solving optimal power flow (OPF) through two relaxation steps: angle relaxation and conic relaxation. For radial networks, both relaxations are exact when there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact, but the angle relaxation may not be. The paper also introduces a method to convexify mesh networks using phase shifters, allowing efficient optimization of the convexified network. The branch flow model focuses on branch currents and powers, unlike the traditional bus injection model that focuses on nodal variables. The model is used to formulate OPF as a conic program (SOCP) after two relaxations. The first relaxation eliminates voltage and current angles, resulting in the Baran-Wu model. The second relaxation converts the quadratic equality constraint into an inequality, yielding a convex problem. For radial networks, the conic relaxation is exact when there are no upper bounds on loads. For mesh networks, the conic relaxation is also exact, but the angle relaxation may not be. The paper provides a way to determine if a relaxed solution is globally optimal by checking the angle recovery condition. This condition ensures that the implied angle differences around each cycle sum to zero modulo $2\pi$. The paper also discusses the equivalence of the branch flow model and the bus injection model, showing that they describe the same physical system. The branch flow model allows for more efficient optimization of mesh networks using phase shifters, which can be placed outside a spanning tree of the network. The paper concludes with a detailed analysis of the branch flow model, the two relaxations, and the conditions for exact relaxation. It also presents simulation results and a proposed algorithm for solving OPF.