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Stability analysis of the modular multilevel converter (MMC) requires accurate models over wide frequency ranges, accounting for both the time-periodic nature of internal dynamics and non-passive behavior resulting from time delays. Frequency-lifting techniques such as harmonic state-space (HSS) can be used to tackle the time-periodicity and obtain linear time-invariant models on which classical stability analysis techniques can be applied. However, exact time delays are generally approximated by rational polynomial functions to study the eigenvalues of ordinary differential equations (ODEs), which leads to inaccuracies at higher frequencies. Such approximations can be avoided by preserving delays in their exact form and studying eigenvalues of delay differential equations (DDEs).To that end, this paper presents the generalization of the original ODE-based HSS framework to DDEs, accounting for both time-delay and time-periodic behaviors of the MMC. The generalized HSS framework benefits from the insights of eigenvalues and participation factors analysis while at the same time guaranteeing the derivation of accurate frequency-response models. Using the developed HSS-DDE framework, AC and DCside admittances of the MMC are obtained and validated against frequency scans. Lastly, the stability of a test system is studied in both time and frequency domains to show the advantages of the developed framework.

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