Generalized Voltage-Based State-Space Modeling of Modular Multilevel Converters With Constant Equilibrium in Steady State

Abstract

This paper demonstrates that the sum and difference of the upper and lower arm voltages are suitable variables for deriving a generalized state-space model of a modular multilevel converter (MMC) which settles at a constant equilibrium in steady-state operation. The presented modeling approach separates the multiple frequency components appearing within the MMC as a first step of the model derivation, to avoid variables containing multiple frequency components in steady state. On this basis, it is shown that Park transformations at three different frequencies ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$+\omega $ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$-2\omega $ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$+3\omega $ </tex-math></inline-formula> ) can be applied for deriving a model formulation where all state-variables settle at constant values in steady state, corresponding to an equilibrium point of the model. The resulting model accurately captures the internal current and voltage dynamics and the coupling between the different frequency components appearing in the variables of a three-phase MMC. Independently of the control system implementation, the derived equations are valid for accurate representation of the MMC in the applied <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$dqz$ </tex-math></inline-formula> reference frames, and they can be linearized for utilization in eigenvalue-based analysis of small-signal dynamics. Furthermore, the model can be utilized for control system design by multivariable methods requiring any stable equilibrium to be defined by a fixed operating point. Time-domain simulations in comparison to an established average model of the MMC, as well as results from a detailed simulation model of an MMC with 400 submodules per arm, are presented as verification of the validity and accuracy of the developed model.