In the first part of this two-part paper we show that the branch-flow convexification of the OPF problem is not exact and that the ADMM-based decomposition of the OPF fails to converge in specific scenarios. Therefore, there is a need to develop algorithms for the solution of the non-approximated OPF problem that remains inherently non-convex. To overcome the limitations of recent approaches for the solution of the OPF problem, we propose in this paper, a specific algorithm for the solution of a non-approximated, non-convex AC OPF problem in radial distribution systems. It is based on the method of multipliers, as well as on a primal decomposition of the OPF problem. We provide a centralized version, as well as a distributed asynchronous version of the algorithm. We show that the centralized OPF algorithm converges to a local minimum of the global OPF problem and that the distributed version of the algorithm converges to the same solution as the centralized one. Here, in this second part of the two-part paper, we provide the formulation of the proposed algorithm and we evaluate its performance by using both small-scale electrical networks, as well as a modified IEEE 13-node test feeder.