The small-signal impedance modeling of a modular multilevel converter (MMC) is the key for analyzing resonance and stability of MMC-based power electronic systems. The MMC is a power converter with a multifrequency 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 multiharmonic 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 the 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 reveals the impact of the MMC internal dynamics and control dynamics on the MMC impedance. Furthermore, an impedance-based stability analysis of the MMC-high-voltage direct current 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 impedance models are validated by both simulation and experimental measurements.

This paper presents a generalized method for converting multi-input and multi-output (MIMO) dq impedance model of a grid-tied voltage source converter system into its sequence domain single-input and single-output (SISO) equivalents. As a result, two types of SISO impedance models were derived, one of which was derived from relatively strong and dq symmetric grid assumption (reduced SISO model) and the other was based on closed-loop equivalent (accurate SISO model). It was proven that the accurate SISO model has the same marginal stability condition as the MIMO model. Accuracy of these models is assessed with respect to the measured impedances in PSCAD/EMTDC simulations, their effects on stability are analyzed as well. Findings show that the accurate SISO model presents identical stability conclusions as the MIMO model. However, the reduced SISO model may lead to inaccurate results if the system is highly dq asymmetric, e.g., VSC with fast phase-locked loop or an actively controlled grid.

Impedance is an intuitive and effective way for dynamical representation of power electronics devices (e.g. VSCs). One of its strengths towards others is the natural association with circuits. However, impedances of VSCs are locally evaluated via linearization, a process dependent on the angle of the reference frame, thus the reference frame transformation (i.e. rotation) is required before connecting them in circuits for the purpose of network analysis. Although this issue was properly treated in the state-space modeling, a counterpart for the impedance-based analysis, particularly the stability impacts of this rotation have not been thoroughly discussed and worth being clarified. On the other hand, there are fundamental differences in applying the impedance-based stability criteria of a single-VSC system to an interconnected one. Several restrictions as revealed (e.g. sensitivity to partition points of the Nyquist-based analysis), if not properly considered, may lead to inaccurate stability assessments. In this respect, a clarification of three commonly employed impedancebased stability criteria is achieved. At last, the capability of the Nyquist-based analysis in identifying the system’s weak point and in facilitating better network design and planning is presented. All the models and analyses are verified by frequency-scanning and time-domain simulations in PSCAD/EMTDC.

A subsynchronous oscillation (SSO) phenomenon has been observed in a modular multilevel converter-based high-voltage DC (MMC-HVDC) transmission system for wind farm integration in the real world, which is independent of the type of wind turbine generator. This kind of oscillation appears different from those in DFIG-based wind farm with series-compensation line or wind farm integration through two-level VSC-HVDC transmission system, because the internal dynamics of the MMC may have significant impact on the oscillation. By far, however, very few papers have reported it. In this paper, the generation mechanism of the SSO phenomenon in an MMC-HVDC transmission system for wind farm integration is revealed from an impedance point of view. The harmonic state-space (HSS) modeling method is applied to model the multi-frequency behavior of the MMC, based on which, the ac-side small-signal impedance of the MMC is analytically derived according to harmonic linearization theory. As a general rule, the controller parameters of the wind power inverter and the HVDC converter are designed separately, to meet the performance requirements of the single converter under ideal conditions, but this practice does not guarantee the stability of the interconnected system. Therefore, an optimal design method for controller parameters is proposed in this paper in order to guarantee the small-signal stability of the interconnected system from a system point of view. Finally, time-domain simulations validate the effectiveness of the theoretical analysis and the proposed optimal design method.

Grid-synchronization stability (GSS) is an emerging stability issue of grid-tied voltage source converters (VSCs), which can be provoked by severe grid voltage sags. Although a qualitative understanding of the mechanism behind the loss of synchronization has been acquired in recent studies, an analytical method for quantitative assessment of GSS of grid-tied VSCs is still missing. To bridge this gap, a dedicated Lyapunov function is analytically proposed, and its corresponding stability criterion for GSS analysis of grid-tied VSCs is rigorously constructed. Both theoretical analysis and simulation result demonstrate that the proposed method can provide a credible GSS evaluation compared to the previous EAC/EF-based method. Therefore, it can be applied for fast GSS evaluation of the grid-tied VSCs as is exemplified in this letter, as well as an analytical tool for some GSS-related issues, e.g., GSS-oriented parameter design and stabilization control.

Small-signal analysis of power electronic converters and systems is often carried out by impedance-based methods. At the core of these methods lies the impedance modeling, which can either be obtained through analytical calculations, simulations, or measurements. The impedance models can be obtained into two main domains: the dq-domain and the sequence domain. In the dq-domain, the impedance model is a 2×2 matrix, while in the sequence domain, it is composed by the positive and negative sequence impedance. Recently, a third domain called the modified sequence domain was defined as an extension to the sequence domain, but also with clear similarities to the dq-domain. The objective of this paper is to unambiguously relate to each other the impedances in these three domains, and to show how this equivalence translates into their respective stability assessments. It is also proven that the sequence domain impedance has the same marginal stability condition as the dq-domain impedance matrix. The three-phase voltage source converter is used as an example converter in this paper, as its impedance model in all three domains is well established and reported by previous research. The results in this paper show that the modified sequence domain model can be derived from the dq-domain model (and vice versa), and that the stability analysis will be identical in these two domains. It is also shown how the original sequence domain model can be derived from the two other models through a model reduction. However, a small discrepancy between the two Nyquist plots is observed in the presence of components such as phase-locked loop or dc-link control.