  We consider the problem of computing equilibria (steady-states) for
droop-controlled, islanded, AC microgrids that are both economic-optimal and
dynamically stable. This work is motivated by the observation that classical
optimal power flow (OPF) formulations used for economic optimization might
provide equilibria that are not reachable by low-level controllers (i.e.,
closed-loop unstable). This arises because OPF problems only enforce
steady-state conditions and do not capture the dynamics. We explain this
behavior by using a port-Hamiltonian microgrid representation. To overcome the
limitations of OPF, the port-Hamiltonian representation can be exploited to
derive a bilevel OPF formulation that seeks to optimize economics while
enforcing stability. Unfortunately, bilevel optimization with a nonconvex inner
problem is difficult to solve in general. As such, we propose an alternative
approach (that we call probing OPF), which identifies an economic-optimal and
stable equilibrium by probing a neighborhood of equilibria using random
perturbations. The probing OPF is advantageous in that it is formulated as a
standard nonlinear program, in that it is compatible with existing OPF
frameworks, and in that it is applicable to diverse microgrid models.
Experiments with the IEEE 118-bus system reveal that few probing points are
required to enforce stability.
