Equivalent relaxations of optimal power flow

Several convex relaxations of the optimal power flow (OPF) problem have recently been developed using both bus injection models and branch flow models. In this paper, we prove relations among three convex relaxations: a semidefinite relaxation that computes a full matrix, a chordal relaxation based on a chordal extension of the network graph, and a second-order cone relaxation that computes the smallest partial matrix. We prove a bijection between the feasible sets of the OPF in the bus injection model and the branch flow model, establishing the equivalence of these two models and their second-order cone relaxations. Our results imply that, for radial networks, all these relaxations are equivalent and one should always solve the second-order cone relaxation. For mesh networks, the semidefinite relaxation is tighter than the second-order cone relaxation but requires a heavier computational effort, and the chordal relaxation strikes a good balance. Simulations are used to illustrate these results.Comment: 12 pages, 7 figure

Convex Relaxation of Optimal Power Flow, Part I: Formulations and Equivalence

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.Comment: Citation: IEEE Transactions on Control of Network Systems, 15(1):15-27, March 2014. This is an extended version with Appendices VIII and IX that provide some mathematical preliminaries and proofs of the main result

Convex Relaxation of Optimal Power Flow, Part II: Exactness

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.Comment: Citation: IEEE Transactions on Control of Network Systems, June 2014. This is an extended version with Appendex VI that proves the main results in this tutoria

A Note on Branch Flow Models with Line Shunts

When the shunt elements in the Π circuit line model are assumed zero, it has been proved that branch flow models are equivalent to bus injection models and that the second-order cone relaxation of optimal power flow problems on a radial network is exact under certain conditions. In this note we propose a branch flow model that includes nonzero line shunts and prove that the equivalence and the exactness of relaxation continue to hold under essentially the same conditions as for zero shunt elements

A Note on Branch Flow Models with Line Shunts

When the shunt elements in the Pi circuit line model are assumed zero, it has been proved that branch flow models are equivalent to bus injection models and that the second-order cone relaxation of optimal power flow problems on a radial network is exact under certain conditions. In this note we propose a branch flow model that includes nonzero line shunts and prove that the equivalence and the exactness of relaxation continue to hold under essentially the same conditions as for zero shunt elements.Comment: 6 pages, 1 figur

Exact Convex Relaxation of Optimal Power Flow in Radial Networks

The optimal power flow (OPF) problem determines power generation/demand that minimize a certain objective such as generation cost or power loss. It is nonconvex. We prove that, for radial networks, after shrinking its feasible set slightly, the global optimum of OPF can be recovered via a second-order cone programming (SOCP) relaxation under a condition that can be checked a priori. The condition holds for the IEEE 13-, 34-, 37-, 123-bus networks and two real-world networks, and has a physical interpretation.Comment: 32 pages, 10 figures, submitted to IEEE Transaction on Automatic Control. arXiv admin note: text overlap with arXiv:1208.407