A Study on the Strong Duality of Second-Order Conic Relaxation of AC Optimal Power Flow in Radial Networks

For the popular second-order conic program (SOCP) formulation of AC optimal power flow (OPF) in a radial network, this paper first shows that it does not have the strong duality property in general. Then, through a series of restrictive reformulations, we derive a set of closed-form sufficient conditions on network parameters that ensure its strong duality. Numerical studies on IEEE 33-bus, 69-bus test networks and two real-world distribution systems confirm that non-negligible duality gaps do exist in this SOCP formulation, and also demonstrate the validity of the proposed sufficient conditions on closing the duality gap. Our results provide an analytical tool to ensure the strong duality of the SOCP power flow formulation and to support algorithm developments for its complex extensions.

Automated summary

This paper investigates the popular second-order conic program (SOCP) formulation of AC optimal power flow (OPF) in radial networks, revealing its lack of strong duality in general and proposing closed-form sufficient conditions to ensure strong duality through restrictive reformulations. Numerical studies on test networks and real-world distribution systems confirm the presence of non-negligible duality gaps in this SOCP formulation, validating the proposed sufficient conditions for closing the duality gap.