The paper introduces a branch flow model for analyzing and optimizing mesh and radial power networks. The model involves two relaxation steps: angle relaxation and conic relaxation. Angle relaxation eliminates voltage and current angles, while conic relaxation approximates the problem by a conic program. For radial networks, both relaxations are exact if there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact, but angle relaxation may not be, and a condition is provided to determine if a relaxed solution is globally optimal. The paper also proposes using phase shifters to convexify mesh networks, ensuring that any relaxed solution can be mapped to an optimal solution for the convexified network. The first part of the paper focuses on the branch flow model, the relaxation steps, and the conditions for exact relaxation. The second part will discuss convexification and simulation results.The paper introduces a branch flow model for analyzing and optimizing mesh and radial power networks. The model involves two relaxation steps: angle relaxation and conic relaxation. Angle relaxation eliminates voltage and current angles, while conic relaxation approximates the problem by a conic program. For radial networks, both relaxations are exact if there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact, but angle relaxation may not be, and a condition is provided to determine if a relaxed solution is globally optimal. The paper also proposes using phase shifters to convexify mesh networks, ensuring that any relaxed solution can be mapped to an optimal solution for the convexified network. The first part of the paper focuses on the branch flow model, the relaxation steps, and the conditions for exact relaxation. The second part will discuss convexification and simulation results.