Stabilization of structure-preserving power networks with market   dynamics
  This paper studies the problem of maximizing the social welfare while stabilizing both the physical power network as well as the market dynamics. For the physical power grid a third-order structure-preserving model is considered involving both frequency and voltage dynamics. By applying the primal-dual gradient method to the social welfare problem, a distributed dynamic pricing algorithm in port-Hamiltonian form is obtained. After interconnection with the physical system a closed-loop port-Hamiltonian system of differential-algebraic equations is obtained, whose properties are exploited to prove local asymptotic stability of the optimal points. 
Tjerk W. Stegink, Claudio De Persis, Arjan J. van der Schaft
math.DS physics.soc-ph