  In this paper we propose distributed dynamic controllers for sharing both
frequency containment and restoration reserves of asynchronous AC systems
connected through a multi-terminal HVDC (MTDC) grid. The communication
structure of the controller is distributed in the sense that only local and
neighboring state information is needed, rather than the complete state. We
derive sufficient stability conditions, which guarantee that the AC frequencies
converge to the nominal frequency. Simultaneously, a global quadratic power
generation cost function is minimized. The proposed controller also regulates
the voltages of the MTDC grid, asymptotically minimizing a quadratic cost
function of the deviations from the nominal DC voltages. The results are valid
for distributed cable models of the HVDC grid (e.g. $\pi$-links), as well as AC
systems of arbitrary number of synchronous machines, each modeled by the swing
equation. We also propose a decentralized, communication-free version of the
controller. The proposed controllers are tested on a high-order dynamic model
of a power system consisting of asynchronous AC grids, modelled as IEEE 14 bus
networks, connected through a six-terminal HVDC grid. The performance of the
controller is successfully evaluated through simulation.
