In this article, we consider a dc microgrid composed of distributed generation units (DGUs) trading energy among each other, where the energy price depends on the total current generated by all the DGUs. We then use a Cournot aggregative game to describe the self-interested interaction among the DGUs, where each DGU aims at minimizing the deviation with respect to the given reference signals and maximizing the revenue from the sale of the generated power. Thus, we design a fully distributed continuous-time equilibrium-seeking algorithm to compute the generalized Nash equilibrium (GNE) of the game. We interconnect the designed decision-making algorithm with the dynamics of the microgrid in a passive way, and, by leveraging passivity theory, we prove the convergence of the closed-loop system trajectory to a feasible operating point that is also a Nash equilibrium of the collective aggregative game. Finally, we present extensive simulation results that validate the proposed distributed optimal control scheme, showing excellent performance.

In this article, we treat extended balancing for continuous-time linear time-invariant systems. We take a dissipativity perspective, thus, resulting in a characterization in terms of linear matrix inequalities. This perspective is useful for determining a priori error bounds. In addition, we address the problem of structure-preserving model reduction of the subclass of port-Hamiltonian systems. We establish sufficient conditions to ensure that the reduced-order model preserves a port-Hamiltonian structure. Moreover, we show that the use of extended Gramians can be exploited to get a small error bound and, possibly, to preserve a physical interpretation for the reduced-order model. We illustrate the results with a large-scale mechanical system example. Furthermore, we show how to interpret a reduced-order model of an electrical circuit again as a lower dimensional electrical circuit.