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Published February 2012 | public
Journal Article

Zero Duality Gap in Optimal Power Flow Problem


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.

Additional Information

© 2011 IEEE. Manuscript received July 11, 2010; revised November 15, 2010 and February 27, 2011; accepted April 04, 2011. Date of publication August 04, 2011; date of current version January 20, 2012. This work was supported by ONR MURI N00014-08-1-0747 "Scalable, Data-driven, and Provably-correct Analysis of Networks,"ARO MURI W911NF-08-1-0233 "Tools for the Analysis and Design of Complex Multi-Scale Networks," the Army's W911NF-09-D-0001 Institute for Collaborative Biotechnology, and NSF through NetSE grant CNS 0911041 and the Southern California Edison.

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