In this paper, we consider the optimal power flow (OPF) problem which consists in determining the power production at each bus of an electric network by minimizing the production cost. Our contribution is an exact solution algorithm for the OPF problem. It consits in a spatial branch-and-bound algorithm based on a compact quadratically constrained convex relaxation. This compact relaxation is computed by solving the rank relaxation once at the beginning of the algorithm. The key point of this approach is that the lower bound at the root node of the branch-and-bound tree is equal to the rank relaxation value, but is obtained by solving a quadratic convex problem which is much faster than solving a SDP. To construct this compact relaxation, we add only O(n) variables that model the squares of the initial variables, where n is the number of buses in the power system. The relations between the initial and auxiliary variables are therefore non-convex. By relaxing them in our relaxation, we have only O(n) equalities to force by the branch-and-bound algorithm to prove global optimality. Our first experiments show that our new algorithm Compact OPF (COPF) performs better than the methods of the literature we compare it with