  HVDC is a critically important technology for the large-scale integration of
renewable resources such as offshore wind farms. Currently, only point-to-point
and multi-terminal HVDC connections exist in real-life operation. However, with
the advancement of VSC-based converter technologies, future HVDC systems are
foreseen to develop into meshed HVDC grids. Bipolar HVDC grids can be operated
in an unbalanced way during single pole outages or in form of mixed monopolar
and bipolar grids. However, currently, there are no (optimal) power flow tools
to study the feasibility of such systems. Therefore, we develop an optimal
power flow (OPF) model for hybrid AC-DC grids to capture the DC side unbalances
and allowing to efficiently plan and operate such future grids. In this paper,
we present a multi-conductor OPF model with separate modeling of the positive
pole, negative pole, metallic return conductors, and ground return. The
capabilities of the model are demonstrated on a small test case, including
monopolar tapping over a bipolar DC link. It is demonstrated that the developed
OPF model can capture the loop flows between the different poles in unbalanced
conditions, as opposed to the existing single-wire representations in the
literature. Further, numerical results are presented for multiple test cases
with various system sizes, starting from an 11-bus system to a 3120-bus system
to demonstrate the computational tractability of the chosen model formulation.
