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In this paper a unifying energy-based approach is provided to the modelingand stability analysis of power systems coupled with market dynamics. Weconsider a standard model of the power network with a third-order model for thesynchronous generators involving voltage dynamics. By applying the primal-dualgradient method to a social welfare optimization, a distributed dynamic pricingalgorithm is obtained, which can be naturally formulated in port-Hamiltonianform. By interconnection with the physical model a closed-loop port-Hamiltoniansystem is obtained, whose properties are exploited to prove asymptoticstability to the set of optimal points. This result is extended to the casethat also general nodal power constraints are included into the social welfareproblem. Additionally, the case of line congestion and power transmission costsin acyclic networks is covered. Finally, a dynamic pricing algorithm isproposed that does not require knowledge about the power supply and demand.

A unifying energy-based approach to stability of power grids with market dynamics