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The internal currents and voltages of modular multilevel converters (MMCs) contain multiple frequency components in steady-state operation and remain time-periodic even when transformed into a synchronously rotating reference frame. This prevents a straightforward state-space representation, where a constant equilibrium point is reached and all state variables converge to constant values under steady-state conditions. Such steady-state time-invariant (SSTI) representations are needed for linearization and eigenvalue-based analysis of small-signal stability. This paper presents an energy-based model of an MMC with a modulation strategy where the insertion indices are compensated for the oscillations in the sum arm voltage. The formulation of the model allows for deriving, by the application of Park transformations at three different frequencies, an SSTI representation that accurately captures the internal dynamics of the MMC. This model can be simplified to a reduced-order model that maintains accurate reproduction of the external behavior at the ac- and dc-sides while neglecting some of the internal dynamics. The validity and accuracy of these two SSTI MMC models are verified by time-domain simulations and their utilization for eigenvalue-based analysis of MMC dynamics is demonstrated by examples.

Energy-Based State-Space Representation of Modular Multilevel Converters with a Constant Equilibrium Point in Steady-State Operation