Abstract-The optimal power flow (OPF) problem is nonconvex and generally hard to solve. In this paper, we propose a semidefinite programming (SDP) optimization, which is the dual of an equivalent form of the OPF problem. A global optimum solution to the OPF problem can be retrieved from a solution of this convex dual problem whenever the duality gap is zero. A necessary and sufficient condition is provided in this paper to guarantee the existence of no duality gap for the OPF problem. This condition is satisfied by the standard IEEE benchmark systems with 14, 30, 57, 118 and 300 buses as well as several randomly generated systems. Since this condition is hard to study, a sufficient zero-duality-gap condition is also derived. This sufficient condition holds for IEEE systems after small resistance (10 −5 per unit) is added to every transformer that originally assumes zero resistance. We investigate this sufficient condition and justify that it holds widely in practice. The main underlying reason for the successful convexification of the OPF problem can be traced back to the modeling of transformers and transmission lines as well as the non-negativity of physical quantities such as resistance and inductance.

No abstract

This paper is concerned with the optimal power flow (OPF) problem. We have recently shown that a convex relaxation based on semidefinite programming (SDP) is able to find a global solution of OPF for IEEE benchmark systems, and moreover this technique is guaranteed to work over acyclic (distribution) networks. The present work studies the potential of the SDP relaxation for OPF over mesh (transmission) networks. First, we consider a simple class of cyclic systems, namely weakly-cyclic networks with cycles of size 3. We show that the success of the SDP relaxation depends on how the line capacities are modeled mathematically. More precisely, the SDP relaxation is proven to succeed if the capacity of each line is modeled in terms of bus voltage difference, as opposed to line active power, apparent power or angle difference. This result elucidates the role of the problem formulation. Our second contribution is to relate the rank of the minimum-rank solution of the SDP relaxation to the network topology. The goal is to understand how the computational complexity of OPF is related to the underlying topology of the power network. To this end, an upper bound is derived on the rank of the SDP solution, which is expected to be small in practice. A penalization method is then applied to the SDP relaxation to enforce the rank of its solution to become 1, leading to a near-optimal solution for OPF with a guaranteed optimality degree. The remarkable performance of this technique is demonstrated on IEEE systems with more than 7000 different cost functions.

This paper is concerned with the securityconstrained optimal power flow (SCOPF) problem, where each contingency corresponds to the outage of an arbitrary number of lines and generators. The problem is studied by means of a convex relaxation, named semidefinite program (SDP). The existence of a rank-1 SDP solution guarantees the recovery of a global solution of SCOPF. We prove that the rank of the SDP solution is upper bounded by the treewidth of the power network plus one, which is perceived to be small in practice. We then propose a decomposition method to reduce the computational complexity of the relaxation. In the case where the relaxation is not exact, we develop a graph-theoretic convex program to identify the problematic lines of the network and incorporate the loss over those lines into the objective as a penalization (regularization) term, leading to a penalized SDP problem. We perform several simulations on large-scale benchmark systems and verify that the global minima are at most 1% away from the feasible solutions obtained from the proposed penalized relaxation.

This paper is concerned with the optimal power flow (OPF) problem. We have recently shown that a convex relaxation based on semidefinite programming (SDP) is able to find a global solution of OPF for IEEE benchmark systems, and moreover this technique is guaranteed to work over acyclic (distribution) networks. The present work studies the potential of the SDP relaxation for OPF over mesh (transmission) networks. First, we consider a simple class of cyclic systems, namely weakly-cyclic networks with cycles of size 3. We show that the success of the SDP relaxation depends on how the line capacities are modeled mathematically. More precisely, the SDP relaxation is proven to succeed if the capacity of each line is modeled in terms of bus voltage difference, as opposed to line active power, apparent power or angle difference. This result elucidates the role of the problem formulation. Our second contribution is to relate the rank of the minimum-rank solution of the SDP relaxation to the network topology. The goal is to understand how the computational complexity of OPF is related to the underlying topology of the power network. To this end, an upper bound is derived on the rank of the SDP solution, which is expected to be small in practice. A penalization method is then applied to the SDP relaxation to enforce the rank of its solution to become 1, leading to a near-optimal solution for OPF with a guaranteed optimality degree. The remarkable performance of this technique is demonstrated on IEEE systems with more than 7000 different cost functions.