  We study tightness and scalability properties of a Lagrangian dual (LD) bound
for the nonconvex alternating current optimal power flow (ACOPF) problem. We
show that the LD bound is as tight as that provided by the powerful and popular
semidefinite programming relaxation. However, a key advantage of the proposed
bound is that it can be computed in a parallel, decentralized manner.
Specifically, in the proposed approach we partition the network into a set of
subnetworks, we dualize the coupling constraints (giving the LD function), and
we maximize 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 bundle method and we provide a proof of convergence
for such method. We demonstrate our developments using PGLib test instances.
