Abstract：With the rapid development of renewable energy, large-scale power electronic devices are increasingly integrated into power systems. Inverters, serving as key interfaces, rely on control strategies for grid synchronization. However, their inherently low inertia and strong nonlinearity differ significantly from those of traditional synchronous generators, leading to complex dynamic behavior under disturbances. Grid-connected inverter systems are prone to transient synchronous instability, threatening system safety. Existing stability assessment methods struggle to cope with the strong nonlinearity of inverters. Furthermore, the evolution of the basin of attraction induced by bifurcations complicates the evaluation of stability margins. Therefore, this paper proposes a quantitative method to characterize the basin of attraction across different dominant bifurcation modes. By analyzing the boundary conditions of transient synchronous instability, this approach achieves accurate stability assessment with minimal computation cost. Firstly, the codimension-2 Bogdanov-Takens bifurcation and its induced generalized saddle-node, Hopf, and homoclinic bifurcations are analyzed. The parameter space is partitioned into four regions with distinct equilibrium characteristics and manifold structures, revealing that Hopf and homoclinic bifurcations are the primary mechanisms of transient synchronous instability. Secondly, the scaling laws for limit cycles in the Hopf-dominated region are derived from normal form theory. Simultaneously, a stability margin assessment method for the homoclinic-dominated region is proposed based on the tangent space of the saddle manifold. Finally, a method for identifying dominant bifurcation modes is developed. This method combines the derived indicators to identify the instability mechanism, thereby avoiding the computational burden of point-by-point time-domain simulations required to calculate the basin of attraction. The correlation analysis between system parameters and stability margins indicates that as the PLL proportional coefficient increases, homoclinic bifurcation is more likely to become the dominant bifurcation mode. Conversely, as the integral coefficient increases, the system tends to exhibit oscillatory instability. Meanwhile, as the electrical distance and current reference values increase, oscillatory instability is more likely to dominate. In the Hopf-bifurcation-dominated region, the stability boundary is defined by an unstable limit cycle. Upon instability, the system's time-domain waveforms undergo a process of “stable-oscillatory divergence-constant amplitude oscillation”. The main reason is that when the system trajectory exceeds the limit cycle's range, although the limit cycle cannot pull it back to the stable equilibrium point, it is still attracted and circles around it for several turns. Such a phenomenon manifests as oscillatory divergence in time-domain waveforms. Consequently, after fault clearance, various electrical quantities continue to oscillate, exhibiting distinct oscillatory instability characteristics. In the homoclinic bifurcation-dominated region, the stability boundary is constituted by a homoclinic orbit. Both the phase angle and angular frequency diverge. Simultaneously, the three-phase voltage and current exhibit large amplitude oscillations after fault clearance, and the system transitions from a stable state to a non-periodic out-of-step state. The following conclusions can be drawn. (1) The unstable limit cycle constitutes the basin of attraction boundary in the Hopf-dominated region. The proposed scaling law effectively quantifies its transient stability margin. (2) The homoclinic orbit forms the boundary in the homoclinic-dominated region. This paper approximates the geometric distance between the homoclinic orbit and the equilibrium point using the tangent space to the saddle's stable manifold, which serves as an indicator of the stability margin. (3) Based on the manifold distribution patterns, a dominant bifurcation mode identification method is proposed, operating by comparing the limit-cycle scaling law with the geometric distance between the equilibrium point and the homoclinic orbit.
[1] 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. [2] Hatziargyriou Nikos, Milanović Jovica, Rahmann Claudia, et al.Stability definitions and characteri-zation of dynamic behavior in systems with high penetration of power electronic interfaced techno-logies[J]. IEEE PES Technical Report PES-TR77, 2020. [3] Farrokhabadi M, Cañizares C A, Simpson-Porco J W, et al. Microgrid stability definitions, analysis, and examples[J]. IEEE Transactions on Power Systems, 2020, 35(1): 13-29. [4] 李戎, 李建文, 李永刚, 等. 结合特征根及模态分析法的逆变器多机并网系统谐波扰动响应分析[J]. 电工技术学报, 2024, 39(14): 4519-4534. Li Rong, Li Jianwen, Li Yonggang, et al.Analysis of harmonic disturbance response of multi-inverter grid-connected system combining characteristic root and modal analysis method[J]. Transactions of China Electrotechnical Society, 2024, 39(14): 4519-4534. [5] Jovcic D, Fan Lingling.Analytical modeling of high-power converters: an approach for stability analysis of HVDC, FACTS, and wind power delivery systems[J]. IEEE Power and Energy Magazine, 2025, 23(2): 30-45. [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] Gu Yunjie, Green T C.Power system stability with a high penetration of inverter-based resources[J]. Proceedings of the IEEE, 2023, 111(7): 832-853. [8] 孙东伟, 张旸, 温步瀛, 等. 增强同步稳定的构网型变流器虚拟惯量限流方法[J]. 电气技术, 2025, 26(4): 1-6, 60. Sun Dongwei, Zhang Yang, Wen Buying, et al.Virtual inertia current limiting method for grid-forming converters with enhanced synchronous stability[J]. Electrical Engineering, 2025, 26(4): 1-6, 60. [9] Tse C K, Huang Meng, Zhang Xi, et al.Circuits and systems issues in power electronics penetrated power grid[J]. IEEE Open Journal of Circuits and Systems, 2020, 1: 140-156. [10] Fan Lingling, Miao Zhixin.The cause of sub-cycle overvoltage: capacitive characteristics of solar PVs[J]. Electric Power Systems Research, 2022, 209: 108039. [11] Harnefors L, Bongiorno M, Lundberg S.Input-admittance calculation and shaping for controlled voltage-source converters[J]. IEEE Transactions on Industrial Electronics, 2007, 54(6): 3323-3334. [12] Han Miao, Ma Rui, Zhan Meng.Transient syn-chronous stability analysis and assessment of PLL-based VSC systems by bifurcation theory[J]. IEEE Transactions on Energy Conversion, 2025, 40(2): 1312-1324. [13] Ma Rui, Li Jinxin, Kurths J, et al.Generalized swing equation and transient synchronous stability with PLL-based VSC[J]. IEEE Transactions on Energy Conversion, 2022, 37(2): 1428-1441. [14] Qiu Qi, Ma Rui, Kurths J, et al.Swing equation in power systems: approximate analytical solution and bifurcation curve estimate[J]. Chaos: An Interdis-ciplinary Journal of Nonlinear Science, 2020, 30: 013110. [15] 单永鹏, 刘昕宇, 汪莹, 等. 基于等效同步机的新能源基地同步调相机暂态功角失稳机理和稳定裕度分析[J]. 电工技术学报, 2025, 40(17): 5448-5463. Shan Yongpeng, Liu Xinyu, Wang Ying, et al.Transient power angle instability mechanism and stability margin analysis of synchronous condenser co-located with renewable energy base based on equivalent synchronous generator[J]. Transactions of China Electrotechnical Society, 2025, 40(17): 5448-5463. [16] Zhao Jiantao, Huang Meng, Yan Han, et al.Nonlinear and transient stability analysis of phase-locked loops in grid-connected converters[J]. IEEE Transactions on Power Electronics, 2021, 36(1): 1018-1029. [17] 王鹏飞, 陈家伟, 罗超, 等. 跟网型逆变器同步稳定性分析及增强控制方法[J/OL]. 电工技术学报, 2025, 1-14. Wang Pengfei, Chen Jiawei, Luo Chao, et al.Synchronous stability analysis and enhancement control method for grid-following inverters[J/OL]. Transactions of China Electrotechnical Society, 2025, 1-14. [18] 张宇, 张琛, 蔡旭, 等. 并网变换器的暂态同步稳定性分析: 稳定域估计与镇定控制[J]. 中国电机工程学报, 2022, 42(21): 7871-7883. Zhang Yu, Zhang Chen, Cai Xu, et al.Transient grid-synchronization stability analysis of grid-tied voltage source converters: stability region estimation and stabilization control[J]. Proceedings of the CSEE, 2022, 42(21): 7871-7883. [19] 马美玲, 王杰, 李鹏瀚, 等. 电力系统功角稳定域及边界上的全局流形分析[J]. 中国电机工程学报, 2020, 40(18): 5865-5875. Ma Meiling, Wang Jie, Li Penghan, et al.Study of the global manifolds on the boundary of the rotor angle stability region in power systems[J]. Proceedings of the CSEE, 2020, 40(18): 5865-5875. [20] Hu Qi, Fu Lijun, Ma Fan, et al.Large signal synchronizing instability of PLL-based VSC con-nected to weak AC grid[J]. IEEE Transactions on Power Systems, 2019, 34(4): 3220-3229. [21] Ji Taipeng, Huang Shilou, Wang Tong.Synchroni-zation stability domain for power systems integrated with PMSG based on energy function[C]//2021 IEEE 4th International Electrical and Energy Conference (CIEEC), Wuhan, China, 2021: 1-6. [22] Dai Zhiyong, Li Guangqi, Fan Mingdi, et al.Global stability analysis for synchronous reference frame phase-locked loops[J]. IEEE Transactions on Indu-strial Electronics, 2022, 69(10): 10182-10191. [23] Wu Heng, Wang Xiongfei.Design-oriented transient stability analysis of PLL-synchronized voltage-source converters[J]. IEEE Transactions on Power Electro-nics, 2020, 35(4): 3573-3589. [24] 罗聪, 陈燕东, 谢志为, 等. 计及电压动态的构网型变流器多机并联系统暂态建模与稳定域估计[J]. 电工技术学报, 2025, 40(9): 2752-2765. Luo Cong, Chen Yandong, Xie Zhiwei, et al.Transient model and stability region estimation for multiple paralleled grid-forming inverter system[J]. Transactions of China Electrotechnical Society, 2025, 40(9): 2752-2765. [25] Zarif Mansour M, Me S P, Hadavi S, et al.Nonlinear transient stability analysis of phased-locked loop-based grid-following voltage-source converters using Lyapunov's direct method[J]. IEEE Journal of Emerging and Selected Topics in Power Electronics, 2022, 10(3): 2699-2709. [26] Zhang Yu, Zhang Chen, Cai Xu.Large-signal grid-synchronization stability analysis of PLL-based VSCs using Lyapunov's direct method[J]. IEEE Trans-actions on Power Systems, 2022, 37(1): 788-791. [27] Zhang Ziqian, Schuerhuber R, Fickert L, et al.Domain of attraction's estimation for grid connected converters with phase-locked loop[J]. IEEE Trans-actions on Power Systems, 2022, 37(2): 1351-1362. [28] Li Xialin, Wang Zhi, Liu Yixin, et al.The largest estimated domain of attraction and its applications for transient stability analysis of PLL synchronization in weak-grid-connected VSCs[J]. IEEE Transactions on Power Systems, 2023, 38(5): 4107-4121. [29] 薛安成, 沈沉, 梅生伟, 等. 基于稳定域边界理论的暂态稳定指标及其应用[J]. 电力系统自动化, 2006, 30(8): 1-6, 26. Xue Ancheng, Shen Chen, Mei Shengwei, et al.Transient stability index of power systems based on theory of stability region and its applications[J]. Automation of Electric Power Systems, 2006, 30(8): 1-6, 26. [30] Nusse H E, Yorke J A, Kostelich E J.Basins of attraction[M]//Dynamics: Numerical Explorations. New York: Springer US, 1994: 269-314. [31] 周驰, 李颖晖, 郑无计, 等. 电力系统稳定域确定及算法特性研究[J]. 浙江大学学报(工学版), 2019, 53(1): 200-206. Zhou Chi, Li Yinghui, Zheng Wuji, et al.Study on stability region determination and algorithm characteristics of power system[J]. Journal of Zhejiang University (Engineering Science), 2019, 53(1): 200-206. [32] 林玉章, 蔡泽祥. 应用哈密顿-雅可比方程计算电力系统稳定域[J]. 中国电机工程学报, 2007, 27(28): 19-23. Lin Yuzhang, Cai Zexiang.Determination of power system stability region using Hamilton-Jacobi equation[J]. Proceedings of the CSEE, 2007, 27(28): 19-23. [33] Aguirre P, Krauskopf B, Osinga H M.Global invariant manifolds near homoclinic orbits to a real saddle: (non) orientability and flip bifurcation[J]. SIAM Journal on Applied Dynamical Systems, 2013, 12(4): 1803-1846. [34] Handbook of Dynamical Systems[M]//Ale Jan Homburg, Björn Sandstede. Chapter 8: Homoclinic and heteroclinic bifurcations in vector fields. Amsterdam: Elsevier, 2010: 379-524. [35] Yang Jingxi, Tse C K, Huang Meng, et al.Com-parison of homoclinic bifurcations between grid-following and grid-forming converters[J]. IEEE Transactions on Industrial Electronics, 2024, 71(5): 4731-4741. [36] Liu C C, Tse C K, Yang Jingxi.Nonlinear behavior and transient stability of grid-following converters using direct power control under weak grid[J]. IEEE Transactions on Sustainable Energy, 2024, 15(4): 2287-2298. [37] 邢光正, 吴琛, 陈磊, 等. 电压源变换器接入电网的小扰动稳定机理分析[J]. 电力自动化设备, 2020, 40(9): 42-52, 193. Xing Guangzheng, Wu Chen, Chen Lei, et al.Analysis of small disturbance stability mechanism for grid-connected voltage source converter[J]. Electric Power Automation Equipment, 2020, 40(9): 42-52, 193. [38] 罗宇航, 肖凡, 郑宇婷, 等. 基于稳定域的PQ控制并网逆变器稳定性分析方法[J]. 电工技术学报, 2025, 40(22): 7334-7348. Luo Yuhang, Xiao Fan, Zheng Yuting, et al.Stability analysis method of grid-connected inverter with PQ control based on stable operating region[J]. Transactions of China Electrotechnical Society, 2025, 40(22): 7334-7348. [39] Ma Rui, Zhang Yayao, Zhan Meng, et al.Dominant transient equations of grid-following and grid-forming converters by controlling unstable equili-brium point based participation factor analysis[J]. IEEE Transactions on Power Systems, 2023, 39(3): 4818-4834. [40] PSCAD. Grid forming inverter models[EB/OL]. (2025-6-27). https://www.pscad.com/knowledge-base/article/894.
2026, Vol. 41 
