This tutorial provides an overview of 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 discusses sufficient conditions for the exactness of the convex relaxations. The bus injection model (BIM) and the branch flow model (BFM) are two mathematical models used to describe power networks. The BIM models a power network using a connected undirected graph, where each node represents a bus and each edge represents a transmission or distribution line. The BFM models a power network using a connected directed graph, where each node represents a bus and each edge represents a transmission or distribution line with a specified direction. The two models are equivalent, meaning there is a bijection between their solution sets. OPF is formulated within each model, where the power flow solutions define the feasible set of OPF. The complexity of OPF lies in the nonconvexity of the power flow equations, which gives rise to a nonconvex feasible set. Various characterizations of the feasible set are developed, and convex supersets are designed based on these characterizations. Different choices of convex supersets lead to different convex relaxations, and their relationships are proven. The tutorial also discusses the feasible sets and relaxations for the BIM and BFM. For the BIM, three characterizations of the feasible set are presented, leading to three equivalent problems to OPF. For the BFM, an SOCP relaxation is proposed, which is derived by relaxing the phase angles of voltage and current variables and then relaxing a set of quadratic equalities to inequalities. The tutorial also discusses the tightness of the relaxations and provides recommendations for algorithmic implementation. The tutorial concludes with a discussion of the chordal relaxation, which is a relaxation of the BFM that is particularly useful for radial networks. The choice of a chordal extension for the BFM is important but nontrivial, as it affects the complexity of the relaxation.This tutorial provides an overview of 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 discusses sufficient conditions for the exactness of the convex relaxations. The bus injection model (BIM) and the branch flow model (BFM) are two mathematical models used to describe power networks. The BIM models a power network using a connected undirected graph, where each node represents a bus and each edge represents a transmission or distribution line. The BFM models a power network using a connected directed graph, where each node represents a bus and each edge represents a transmission or distribution line with a specified direction. The two models are equivalent, meaning there is a bijection between their solution sets. OPF is formulated within each model, where the power flow solutions define the feasible set of OPF. The complexity of OPF lies in the nonconvexity of the power flow equations, which gives rise to a nonconvex feasible set. Various characterizations of the feasible set are developed, and convex supersets are designed based on these characterizations. Different choices of convex supersets lead to different convex relaxations, and their relationships are proven. The tutorial also discusses the feasible sets and relaxations for the BIM and BFM. For the BIM, three characterizations of the feasible set are presented, leading to three equivalent problems to OPF. For the BFM, an SOCP relaxation is proposed, which is derived by relaxing the phase angles of voltage and current variables and then relaxing a set of quadratic equalities to inequalities. The tutorial also discusses the tightness of the relaxations and provides recommendations for algorithmic implementation. The tutorial concludes with a discussion of the chordal relaxation, which is a relaxation of the BFM that is particularly useful for radial networks. The choice of a chordal extension for the BFM is important but nontrivial, as it affects the complexity of the relaxation.