A d-q frame impedance model of modular multilevel converter (MMC), which is suitable for high-frequency resonance (HFR) analysis under the condition that the number of submodules in MMC reaches hundreds level, is presented in this work. A 10th-order dynamic phasor model for the MMC is linearized and combined with a control subsystem model. A modular modeling approach based on transfer matrix is applied to the overall linearized model to derive the impedance model. The validity and accuracy of the model are verified using a 501-level MMC aggregated model in SIMULINK. Results show that the impedance model has satisfactory accuracy in the frequency ranging from 1 to 3000 Hz. Then, the impedance model is used to analyze an HFR accident in a realistic MMC project. Therefore, a high time delay in controller is proved can provide negative resistance in high frequency anc cause instability.

DC fault protection is a major issue in multiterminal high-voltage DC current transmission systems. Most studies have focused on primary protection, which considers a different theory. However, the poor sensitivity of primary protection under high-resistance fault and maloperation still affect the protection feasibility to a certain extent. Thus, backup protection is definitely required to enhance the stability of an entire system considering the limit in primary protection. This study proposes a backupprotection strategy based on wavelet-packet-energy entropy as mitigation for primary protection. The poleto-ground fault with high resistance is highlighted because detecting the fault transient of PG fault with high resistance is difficult in most primary relay. Fault detection and classification are achieved using the proposed backup-protection method based on theoretical study and simulation verification. The proposed method also presents the expected robustness of fault resistance, fault distance, and different operation conditions.

A reduced-order dynamic phasor model of modular multilevel converter (MMC) in long time scale, which is applied for power system low-frequency oscillation analysis, is investigated. A tenth-order dynamic phasor model is derived in consideration of the internal circulation current dynamic of MMC. The model is verified via electromagnetic transient simulation. Then, an application of the singular perturbation method to MMC model reduction is presented. Separation of eigenvalues is used as a characterization of time scales and small time constant states is eliminated in the full-order model of MMC. MMC high-voltage direct current (HVdc) models with various orders are embedded into the four-machine two-area system to compare the low-frequency oscillation analysis results. The frequency domain analysis and time domain simulations show that the second-order model has satisfactory performance in describing the low-frequency characteristics of MMC. Moreover, the influence of MMC-HVdc transmission power and outer loop controller parameters on low-frequency oscillation is investigated using the second-order model. The effectiveness of reduced-order model in supplementary controller parameter design is also validated by comparing the difference in the response of various-order models to supplementary frequency-based control.

Modular multilevel converters (MMCs) are characterized by the multiharmonic coupling interactions among internal dynamics. In this work, multiple $d$ - $q$ references and multiharmonic frames, which are developed through dynamic phasor and harmonic state space, are used to derive the generalized impedance models of MMC in the $d$ - $q$ and modified sequence domain, respectively. Although some previous studies have been devoted to revealing the equivalence of different impedance models, this article attempts to elaborate the relation of different multifrequency modeling approaches and provide a more comprehensive review of MMC impedance models. A rigorous comparison of modeling procedure, impedance response, and stability analysis result between different MMC models is conducted. It is demonstrated that the MMC impedance models in the two domains can be related by a generic linear transformation matrix. Furthermore, the stability analysis results given by the generalized Nyquist criterion are identical in these two domains. Application recommendations for different impedance models are given as well. The developed impedance models and their effectiveness for stability and resonance analysis are validated by a 501-level aggregated model in MATLAB/SIMULINK.

The periodic characteristic of modular multilevel converter (MMC), which originates from the multi-harmonic couplings among internal dynamics, results in challenges to develop the linear time-invariant (LTI) model of MMC. The linearization of MMC is inherently around time-periodic equilibrium trajectory. In this paper, two different methods to assess small-signal stability of MMC are compared. Both are based on the linear time-periodic (LTP) systems theory, but one is in the time domain (TD) and the other is in the frequency domain (FD). The LTP model of MMC is derived from the linearized arm averaged model. Then, an LTI model for the LTP system is constructed in the FD through the harmonic state-space method. Small-signal stability analysis based on Floquet theory and generalized Nyquist criterion, that accompany TD and FD, are introduced and compared. It is demonstrated that models developed in both domains can effectively capture the small-signal behavior of MMC. Third and higher-order harmonic components of the states have significant effect on the impedance-based stability analysis of MMC around mid-frequency range. Analytical results are validated by electromagnetic transient simulations based on the MMC aggregated model.