Optimal Power Flow for AC–DC Grids: Formulation, Convex Relaxation, Linear Approximation, and Implementation

HVDC is becoming an increasingly important part of the present day transmission systems. Accurate models of active and reactive power control capabilities of HVDC converter stations are required to analyze the operation of power systems consisting of ac and dc grids, including ancillary services and security. Different converter station technologies exist, with varying control characteristics. This paper develops an optimal power flow model for ac and dc grids. A variety of formulations, from non-linear to convexified to linearized, are developed and implemented in an open-source tool. A convex relaxation formulation of a parameterized ac-dc converter model is developed. The hierarchy of common ac optimal power flow formulations is mapped to formulations for converter stations and dc grids. Numerical illustrations for a number of test cases, up to 3120 ac nodes and up to ten dc nodes and converters, are provided.