  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
