Modular multilevel converter periodic trajectory determination — A generic Fourier-based collocation method

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This paper presents a generic method for the steady-state periodic trajectory calculation of the modular multilevel converter (MMC) and MMC-based HVDC systems, which is a prerequisite to steady-state performance optimization and harmonic state-space-based small-signal stability analysis, among other applications. In the case of the MMC, the periodic trajectory determination is challenging due to nonlinearity and delays in the differential equations, in particular when control dynamics are taken into account. To this day, most methods rely on lengthy manipulations of waveform representations and cannot generally account for delays and nonlinearities other than products of variables. Hence, there has been missing a more efficient formulation capable of addressing the limitations of existing state-of-the-art methods. This paper fills this gap by presenting a highly flexible Fourier-based collocation method which seamlessly accounts for control dynamics, nonlinearity and delays. Being based in the time-domain, the developed method naturally accounts for nonlinearity in the differential equations and, being real-valued, it is solved efficiently with readily available root-finding Newton-based algorithms. In this paper, the proposed method is also applied to an illustrative simple RLC circuit as well as to a complete arm-averaged model of the MMC; it is validated against simulations and compared with a state-of-the-art shooting method.