Distributed Primary Frequency Control through Multi-Terminal HVDC   Transmission Systems
  This paper presents a decentralized controller for sharing primary AC frequency control reserves through a multi-terminal HVDC grid. By using Lyapunov arguments, the proposed controller is shown to stabilize the equilibrium of the closed-loop system consisting of the interconnected AC and HVDC grids, given any positive controller gains. The static control errors resulting from the proportional controller are quantified and bounded by analyzing the equilibrium of the closed-loop system. The proposed controller is applied to a test grid consisting of three asynchronous AC areas interconnected by an HVDC grid, and its effectiveness is validated through simulation. 
Martin Andreasson, Roger Wiget, Dimos V. Dimarogonas, Karl H.   Johansson, G\"oran Andersson
math.OC