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This paper presents three dynamic linear state-space models of the modular multilevel converter (MMC) which are suitable for small-signal dynamic studies and controller design. The three models differ by the number of states (two, six, and ten) and, therefore, are suitable for different applications based on the required accuracy. The 2nd- and 6th-order models ignore dynamics of the second harmonics and circulating current suppression control (CCSC). The main challenges of dynamic analytical modelling of MMC are the nonlinear multiplication terms in ABC frame equations for modulating oscillating signals. The multiplication nonlinear terms are therefore considered directly in the rotating DQ frame. This requires simultaneous modeling in zero sequence, fundamental frequency DQ and double fundamental frequency DQ2 frames. The proposed linear analytical models are implemented in state space in MATLAB. The validity and accuracy of the models are verified against a detailed 401-level MMC model in PSCAD/EMTDC in both time and frequency domains. The results show very good accuracy for the 10th-order model and decreasing accuracy for the lower order models. The influence of phase-locked loop and CCSC gains on the MMC dynamic is also studied using the proposed 10th model, and the results verify the suitability of the model for this purpose.
Far et al. (Mon,) studied this question.