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The digital control system of modular multilevel converters (MMCs) normally synchronizes with the switching actions. The switching frequency of an MMC is generally selected as low as possible from an efficiency point of view, which leads to an unavoidable and relatively long time delay in the control system. Such a time delay might deteriorate the MMC performance and even reduce the stability margin by introducing infinite uncertain eigenvalues into the system state functions. Without a proper time-delay model and quantitative system stability analyses, MMCs might suffer from uncertainties and unstable risks during their operations, especially when the switching frequency is low. This paper derives an MMC model considering the time delay in the digital control system. A Padé-based stability analysis method is then proposed, which consists of three steps: information integration and critical eigenvalue extraction, and trajectory visualization and critical angle (CA) calculation, and quantitative assessment and extreme value determination. By the proposed model and method, the critical eigenvalue can be effectively extracted. With the Hopf bifurcation, the time delay in the digital control system induces a pair critical conjugate eigenvalues into MMC systems, which is detected as an instability cause. The CA and the critical impact are defined to quantitatively assess the stability, while selecting the controller parameters and minimum switching frequency. The proposed method for the MMC stability assessment is experimentally verified. [ABSTRACT FROM AUTHOR]