This paper studies the problem of frequency regulation in power grids, whilemaximizing the social welfare. Two price-based controllers are proposed; thefirst one an internal-model-based controller and the second one based on acontinuous gradient method for optimization. Both controllers can beimplemented in a fully distributed fashion, with freedom in choosing acontroller communication network. As a result, two real-time dynamic pricingmodels described by port- Hamiltonian systems are obtained. By coupling withthe port- Hamiltonian description of the physical network we obtain aclosed-loop port-Hamiltonian system, whose properties are ex- ploited to proveasymptotic stability of the set of optimal points. Numerical results show theperformance of both controllers in a simple case study.
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