This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact. The optimal power flow (OPF) problem is a mathematical program that seeks to minimize a certain function, such as total power loss, generation cost or user disutility, subject to the Kirchhoff's laws as well as capacity, stability and security constraints. OPF is fundamental in power system operations as it underlies many applications such as economic dispatch, unit commitment, state estimation, stability and reliability assessment, volt/var control, demand response, etc. Power flow equations are quadratic and hence OPF can be formulated as a quadratically constrained quadratic program (QCQP). It is generally nonconvex and hence NP-hard. A large number of optimization algorithms and relaxations have been proposed. A popular approximation is a linear program, called DC OPF, obtained through the linearization of the power flow equations. To the best of our knowledge solving OPF through semidefinite relaxation is first proposed in [21] as a second-order cone program (SOCP) for radial (tree) networks and in [22] as a semidefinite program (SDP) for general networks in a bus injection model. It is first proposed in [23], [24] as an SOCP for radial networks in the branch flow model of [25], [26]. Convex relaxation of quadratic programs has been applied to many engineering problems. There is a rich theory and extensive empirical experiences. Compared with other approaches, solving OPF through convex relaxation offers several advantages. First, while DC OPF is useful in a wide variety of applications, it is not applicable in other applications. Second, a solution of DC OPF may not be feasible (may not satisfy the nonlinear power flow equations). In this case, an operator may tighten some constraints in DC OPF and solve again. This may not only reduce efficiency but also relieve on heuristics that are hard to scale to larger systems or faster control in the future. Third, when they converge, most nonlinear algorithms compute a local optimal usually without assurance on the quality of the solution. In contrast, a convex relaxation provides for the first time the ability to check if a solution is globally optimal. If it is not, the solution provides a lower bound on the minimum cost and hence a bound on how far any feasible solution is from optimality. Unlike approximations, if a relaxed problem is infeasible, it is a certificate that the original OPF is infeasible. This two-part tutorial explains the main theoretical results on semidefinite relaxations of OPF developed in the last few years. Part I presents two power flow models that are useful inThis tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact. The optimal power flow (OPF) problem is a mathematical program that seeks to minimize a certain function, such as total power loss, generation cost or user disutility, subject to the Kirchhoff's laws as well as capacity, stability and security constraints. OPF is fundamental in power system operations as it underlies many applications such as economic dispatch, unit commitment, state estimation, stability and reliability assessment, volt/var control, demand response, etc. Power flow equations are quadratic and hence OPF can be formulated as a quadratically constrained quadratic program (QCQP). It is generally nonconvex and hence NP-hard. A large number of optimization algorithms and relaxations have been proposed. A popular approximation is a linear program, called DC OPF, obtained through the linearization of the power flow equations. To the best of our knowledge solving OPF through semidefinite relaxation is first proposed in [21] as a second-order cone program (SOCP) for radial (tree) networks and in [22] as a semidefinite program (SDP) for general networks in a bus injection model. It is first proposed in [23], [24] as an SOCP for radial networks in the branch flow model of [25], [26]. Convex relaxation of quadratic programs has been applied to many engineering problems. There is a rich theory and extensive empirical experiences. Compared with other approaches, solving OPF through convex relaxation offers several advantages. First, while DC OPF is useful in a wide variety of applications, it is not applicable in other applications. Second, a solution of DC OPF may not be feasible (may not satisfy the nonlinear power flow equations). In this case, an operator may tighten some constraints in DC OPF and solve again. This may not only reduce efficiency but also relieve on heuristics that are hard to scale to larger systems or faster control in the future. Third, when they converge, most nonlinear algorithms compute a local optimal usually without assurance on the quality of the solution. In contrast, a convex relaxation provides for the first time the ability to check if a solution is globally optimal. If it is not, the solution provides a lower bound on the minimum cost and hence a bound on how far any feasible solution is from optimality. Unlike approximations, if a relaxed problem is infeasible, it is a certificate that the original OPF is infeasible. This two-part tutorial explains the main theoretical results on semidefinite relaxations of OPF developed in the last few years. Part I presents two power flow models that are useful in