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 analyse 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 10 dc nodes and converters, are provided.

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In recent years, the use of HVdc technology has increased and VSC technology has enabled the realization of HVdc grids. This calls for the development of new tools to solve the transmission network expansion planning (TNEP) model. A detailed representation of the dc grid TNEP problem is highly nonlinear and more complex than the traditional ac grid expansion problem due to extra constraints and additional decision variables from the converter station model. The present day industrial solvers have difficulties to tackle the resultant MINLP problem. Therefore, the linearized 'DC' approximation is often used in practice, which may not produce sufficiently accurate answers. In this paper, different relaxations and approximations of the dc grid TNEP problem are presented. The performance of each formulation is evaluated using eight test cases. Although there is no clear formulation that shows the best performance, LPAC approximation and SOC relaxations seem to provide better alternatives to 'DC' approximations. The developed formulations do not guarantee feasibility and a second corrective stage is required to obtain feasible solutions. Index Terms-Transmission planning, HVdc grid, Convex relaxations, Linear approximations, Nonlinear TNEP problem NOMENCLATURE Entities, indices and sets i, j ∈ I ac nodes l ∈ L ac branches lij ∈ T ac ⊆ L × I × I ac topology e, f ∈ E Candidate dc nodes d ∈ D Candidate dc branches def ∈ T dc ⊆ D × E × E Candidate dc topology c ∈ C Candidate converters cie ∈ T cv ⊆ C × I × E Candidates converter topology g ∈ G Generators mi ∈ T ac ac load me ∈ T dc dc load Parameters C d Cost of dc line d Cc Cost of converter c N d Number of poles of dc line (link) d r d Resistance of line d tc Transformation ratio in converter station c ac, bc, cc Coefficients of polynomial converter power loss

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