Abstract：The dynamic characteristic of grid-forming inverter (GFM) is mainly affected by the control strategy, and the interaction with the power grid may cause instability such as oscillation. At the same time, the interactive coupling between different time-scale controllers in GFM makes the stability analysis more complicated. Modal analysis based on the state-space model (MASS) uses the participation factor (PF) to quantify the contribution of each state variable to a particular pattern. However, the number of electrical components in new power systems is increasing explosively, and the difficulty of state-space modeling of the whole system is increasing rapidly. In addition, state-space modeling requires detailed system structure topology and complete control parameters of each electrical component, and inverters usually only have impedance models that describe the characteristics of voltage and current ports, with gray box or black box characteristics. In order to explore the interaction characteristics among all electrical components of the system, the dynamic model of the whole system is constructed by the closed-loop feedback formula of the whole system dynamic matrix. Based on this foundation, the modal analysis based on impedance model (MAI) can evaluate the contribution of each power device to oscillation modes at the device level. However, MAI treats inverters as single, holistic components, which limits its ability to identify dominant system dynamics at the control loop or state variable level. Decomposing different control loops into equivalent circuit components enables the stability analysis of internal inverter dynamics. However, the decomposition of synchronization control loops remains to be explored. This paper proposes an extended modal analysis based on impedance model (EMAI) method to address the current challenges faced by MAI. First, a decomposition method for the GFM impedance model based on the matrix inversion lemma was proposed, dividing GFM dynamics into synchronous dynamics (SD), dominated by the power frequency synchronization loop (PFL), and electromagnetic dynamics (ED), governed by the voltage control loop (VCL). The detailed categorization of dynamics facilitates an in-depth exploration of the complex coupling mechanisms among controllers operating on different time scales. Subsequently, overall impedance participation factors and participation ratios (PR) were introduced to characterize different internal dynamics of GFM, enabling the evaluation of SD and ED contributions at the control loop level. These metrics help identify the dominant system dynamics and trace the root causes of system instability. Finally, an explicit parameter PF was introduced to precisely locate the critical control parameters of identified loops, serving as a metric for optimizing control parameters and enhancing system damping. The analysis yields the following conclusion: as the frequency of oscillation modes decreases, the dominant dynamics within each GFM gradually shift from ED to SD. MAI can provide an overall assessment of GFM participation but fails to capture the dominant dynamics of individual GFMs. EMAI not only identifies interactions between various GFMs and the grid but also evaluates the contributions of ED and SD within GFM through overall impedance participation factors, thereby pinpointing the primary causes affecting system dynamics to specific control loops. Moreover, the results of EMAI and MASS in assessing the participation levels of different GFM dynamics are highly consistent, validating the effectiveness of the EMAI method. Furthermore, the explicit parameter PF provides effective recommendations for improving system damping and enhancing stability. EMAI offers nuanced insights into system stability analysis, enabling the rapid identification of the root causes of system instability.
郑乐, 郑佳杰. 构网型逆变器内部动态的扩展阻抗模态分析[J]. 电工技术学报, 2025, 40(9): 2712-2723.
Zheng Le, Zheng Jiajie. Extended Impedance Modal Analysis of the Internal Dynamics in Grid-Forming Inverters. Transactions of China Electrotechnical Society, 2025, 40(9): 2712-2723.
[1] Blaabjerg F, Yang Yongheng, Kim K A, et al.Power electronics technology for large-scale renewable energy generation[J]. Proceedings of the IEEE, 2023, 111(4): 335-355. [2] Kim J K, Kang J, Shim J W, et al.Dynamic performance modeling and analysis of power grids with high levels of stochastic and power electronic interfaced resources[J]. Proceedings of the IEEE, 2023, 111(7): 854-872. [3] 黄萌, 舒思睿, 李锡林, 等. 面向同步稳定性的电力电子并网变流器分析与控制研究综述[J]. 电工技术学报, 2024, 39(19): 5978-5994. Huang Meng, Shu Sirui, Li Xilin, et al.A review of synchronization-stability-oriented analysis and control of power electronic grid-connected converters[J]. Transactions of China Electrotechnical Society, 2024, 39(19): 5978-5994. [4] 刘其辉, 洪晨威, 逄思敏, 等. 基于弹性系数的双馈风电机组控制参数对次同步振荡作用分析及调整方法[J]. 电工技术学报, 2022, 37(14): 3528-3541. Liu Qihui, Hong Chenwei, Pang Simin, et al.Analysis and adjustment method of doubly-fed fan control parameters on subsynchronous oscillation based on impedance elastic sensitivity[J]. Transactions of China Electrotechnical Society, 2022, 37(14): 3528-3541. [5] 邵冰冰, 赵峥, 肖琪, 等. 多直驱风机经柔直并网系统相近次同步振荡模式参与因子的弱鲁棒性分析[J]. 电工技术学报, 2023, 38(3): 754-769. Shao Bingbing, Zhao Zheng, Xiao Qi, et al.Weak robustness analysis of close subsynchronous oscillation modes' participation factors in multiple direct-drive wind turbines with the VSC-HVDC system[J]. Transactions of China Electrotechnical Society, 2023, 38(3): 754-769. [6] Hatziargyriou N, Milanovic J, Rahmann C, et al.Definition and classification of power system stability-revisited & extended[J]. IEEE Transactions on Power Systems, 2021, 36(4): 3271-3281. [7] Kundur P.Power System Stability and Control[M]. New York: McGraw-Hill, Inc, 1994. [8] Kouki M, Marinescu B, Xavier F.Exhaustive modal analysis of large-scale interconnected power systems with high power electronics penetration[J]. IEEE Transactions on Power Systems, 2020, 35(4): 2759-2768. [9] Cao Wenchao, Ma Yiwei, Yang Liu, et al.D-Q impedance based stability analysis and parameter design of three-phase inverter-based AC power systems[J]. IEEE Transactions on Industrial Electronics, 2017, 64(7): 6017-6028. [10] Xu W, Huang Zhenyu, Cui Yu, et al.Harmonic resonance mode analysis[J]. IEEE Transactions on Power Delivery, 2005, 20(2): 1182-1190. [11] Huang Zhenyu, Cui Yu, Xu W.Application of modal sensitivity for power system harmonic resonance analysis[J]. IEEE Transactions on Power Systems, 2007, 22(1): 222-231. [12] Cui Yu, Wang Xiaoyu.Modal frequency sensitivity for power system harmonic resonance analysis[J]. IEEE Transactions on Power Delivery, 2012, 27(2): 1010-1017. [13] Zhan Ying, Xie Xiaorong, Liu Huakun, et al.Frequency-domain modal analysis of the oscillatory stability of power systems with high-penetration renewables[J]. IEEE Transactions on Sustainable Energy, 2019, 10(3): 1534-1543. [14] Li Yang, Shuai Zhikang, Liu Xuan, et al.Stability analysis and location optimization method for multiconverter power systems based on nodal admittance matrix[J]. IEEE Journal of Emerging and Selected Topics in Power Electronics, 2021, 9(1): 529-538. [15] Chou S F, Wang Xiongfei, Blaabjerg F.Frequency-domain modal analysis for power-electronic-based power systems[J]. IEEE Transactions on Power Electronics, 2021, 36(5): 4910-4914. [16] Gu Yunjie, Li Yitong, Zhu Yue, et al.Impedance-based whole-system modeling for a composite grid via embedding of frame dynamics[J]. IEEE Transactions on Power Systems, 2021, 36(1): 336-345. [17] Zhu Yue, Gu Yunjie, Li Yitong, et al.Participation analysis in impedance models: the grey-box approach for power system stability[J]. IEEE Transactions on Power Systems, 2022, 37(1): 343-353. [18] Zhu Yue, Gu Yunjie, Li Yitong, et al.Impedance-based root-cause analysis: comparative study of impedance models and calculation of eigenvalue sensitivity[J]. IEEE Transactions on Power Systems, 2023, 38(2): 1642-1654. [19] Li Yitong, Gu Yunjie, Green T C.Mapping of dynamics between mechanical and electrical ports in SG-IBR composite grids[J]. IEEE Transactions on Power Systems, 2022, 37(5): 3423-3433. [20] Hu Guang, Xiao Yi, Xin Huanhai, et al.An extended impedance model for power electronics converters retaining explicit synchronization dynamics[J]. IEEE Transactions on Power Electronics, 2025, 40(1): 2355-2370. [21] Li Yitong, Gu Yunjie, Zhu Yue, et al.Impedance circuit model of grid-forming inverter: visualizing control algorithms as circuit elements[J]. IEEE Transactions on Power Electronics, 2021, 36(3): 3377-3395. [22] 高磊, 吕敬, 马骏超, 等. 基于电路等效的并网逆变器失稳分析与稳定控制[J]. 电工技术学报, 2024, 39(8): 2325-2341. Gao Lei, Lü Jing, Ma Junchao, et al.Instability analysis and stability control of grid-connected inverter based on impedance circuit equivalent[J]. Transactions of China Electrotechnical Society, 2024, 39(8): 2325-2341. [23] Fan Lingling, Miao Zhixin.Time-domain measurement-based DQ-frame admittance model identification for inverter-based resources[J]. IEEE Transactions on Power Systems, 2021, 36(3): 2211-2221. [24] Li Yitong, Gu Yunjie, Green T C.Revisiting grid-forming and grid-following inverters: a duality theory[J]. IEEE Transactions on Power Systems, 2022, 37(6): 4541-4554.
2025, Vol. 40 
