  We study tightness properties of a Lagrangian dual (LD) bound for the
nonconvex alternating current optimal power flow (ACOPF) problem. We show an LD
bound that can be computed in a parallel, decentralized manner. Specifically,
the proposed approach partitions the network into a set of subnetworks,
dualizes the coupling constraints (giving the LD function), and maximizes the
LD function with respect to the dual variables of the coupling constraints
(giving the desired LD bound). The dual variables that maximize the LD are
obtained by using a proximal bundle method. We show that the bound is less
tight than the popular semidefinite programming relaxation but as tight as the
second-order cone programming relaxation. We demonstrate our developments using
PGLib-OPF test instances.
