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We consider a class of optimal power flow (OPF) applications where some loadsoffer a modulation service in exchange for an activation fee. Theseapplications can be modeled as multi-period formulations of the OPF withdiscrete variables that define mixed-integer non-convex mathematical programs.We propose two types of relaxations to tackle these problems. One is based on aLagrangian relaxation and the other is based on a network flow relaxation. Bothrelaxations are tested on several benchmarks and, although they provide acomparable dual bound, it appears that the constraints in the solutions derivedfrom the network flow relaxation are significantly less violated.

Relaxations for multi-period optimal power flow problems with discrete decision variables