  This paper studies the problem of frequency regulation in power grids, while
maximizing the social welfare. Two price-based controllers are proposed; the
first one an internal-model-based controller and the second one based on a
continuous gradient method for optimization. Both controllers can be
implemented in a fully distributed fashion, with freedom in choosing a
controller communication network. As a result, two real-time dynamic pricing
models described by port- Hamiltonian systems are obtained. By coupling with
the port- Hamiltonian description of the physical network we obtain a
closed-loop port-Hamiltonian system, whose properties are ex- ploited to prove
asymptotic stability of the set of optimal points. Numerical results show the
performance of both controllers in a simple case study.
