The small-signal impedance modeling of modular multilevel converter (MMC) is the key for analyzing resonance and stability of MMC-based power electronic systems. MMC is a power converter with a multi-frequency response due to its significant steady-state harmonic components in the arm currents and capacitor voltages. These internal harmonic dynamics may have great influence on the terminal characteristics of the MMC, which, therefore, are essential to be considered in the MMC impedance modeling. In this paper, the harmonic state-space (HSS) modeling approach is first introduced to characterize the multi-harmonic coupling behavior of the MMC. On this basis, the small-signal impedance models of the MMC are then developed based on the proposed HSS model of the MMC, which are able to include all the internal harmonics within MMC, leading to accurate impedance models. Besides, different control schemes for the MMC, such as open-loop control, ac voltage closed-loop control, and circulating current closed-loop control, have also been considered during the modeling process, which further reveal the impact of the MMC internal dynamics and control dynamics on the MMC impedance. Furthermore, an impedance-based stability analysis of the MMC-HVDC connected wind farm has been carried out to show how the HSS based MMC impedance model can be used in practical system analysis. Finally, the proposed

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Abstract-The high voltage dc (HVDC) systems are appearing more and more, and it is becoming a requirement that the HVDC voltage source converters (VSCs) operate both as an inverter and a rectifier without changing the controls to provide the flexibility of having power flows in both directions. It is observed that the HVDC system operates stably when the power flow direction is from the power controlled-converter to the dc voltage controlledconverter and it becomes unstable when the power flow direction has been altered. In order to analyze such instability problem and to design the local control, an impedance-based method is proposed. Identifying the source and the load impedance are prerequisite to apply the impedance-based method. Existing method of determining the source and the load impedance cannot predict the stability when the power flow direction is altered; therefore a method based on the power flow direction has been presented to determine the source and the load impedance. The converter which injects power to the dc system is the current source represented with its Norton equivalent parallel impedance while the other converter impedance is considered as the load impedance. The stability of the system is determined by the ratio of the load impedance to the current source impedance. Once the source and the load impedance are analytically obtained, the impedance-based Generalized Nyquist Stability Criteria is applied to determine the stability. The system stability for the two power flow directions is well predicted from the Nyquist plot of impedance ratio. A two-terminal HVDC system has been developed in MATLAB/Simulink to demonstrate the application of this method and the results are compared with the experimental results.

Statement: This paper is original and has never been submitted to any other journals or conferences. Jing Lyu is the corresponding author for this work.