Equivalent Multiple $dq$-Frame Model of the MMC Using Dynamic Phasor Theory in the $\alpha \beta z$-Frame

This article introduces an equivalent multiple dq-frame model of the modular multilevel converter (MMC) that is derived from a dynamic phasor based small-signal state-space MMC model in the stationary αβz frame. When compared to a model in the stationary ABC frame, the order of the model in the αβz frame can be reduced for balanced operation, during which some voltage, and current harmonics are inherently separated in an αβz representation. The proposed method enables further model order reduction through a generalized transformation towards multiple dq-frames. The development of the model, and the generalized transformation are explained in detail, and the obtained equivalent multiple dq-frame models are verified against a nonlinear averaged model in MATLAB/Simulink. An eigenvalue-based small-signal stability analysis highlights the effect of higher-order harmonics in system-level small-signal stability studies, and two case studies of active harmonic suppression illustrate how the presented model allows an in-depth investigation of the impact of extended control functionalities on the small-signal stability.