## 1 Introduction  
With the advantages of a simple structure for active and reactive power decoupling control and easy current limiting ability, control strategies based on the d-q rotating coordinate system and phase-locked loop (PLL) synchronization have been widely used in grid-feeding VSCs [1]. At present, the majority of grid-feeding VSCs still use PLL to achieve synchronization [2]. However, numerous theoretical studies and real-world engineering cases have indicated that grid-feeding converters with PLL are exposed to the risk of transient synchronization instability, particularly under grid faults [3, 4]. Thus, it is essential to investigate the transient stability of grid-connected VSC systems under fault conditions. Current research predominantly focuses on the transient synchronization stability of VSC grid-connected systems under symmetric fault conditions. References [5, 6] point out the mathematical similarity between PLL dynamic models and synchronous machine rotor motion equations, using the equal area criterion to reveal the transient synchronization instability mechanism of VSC grid-connected systems. References [7, 8, 9, 10] and [11, 12, 13] employ the phase trajectory method, inverse trajectory method, and direct method, respectively, to quantitatively analyze the transient synchronization stability of VSC grid-connected systems. Additionally, references [14, 15, 16] analyze the impact of dynamic interactions within the current inner loop on the transient synchronization stability of the system under fault conditions. However, these studies primarily address the transient synchronization stability of VSC grid-connected systems under symmetric grid faults. In practice, asymmetric faults are more common. As reported in [17], PLL synchronization instability during asymmetrical line-to-line and single line-to-ground faults in California in 2016 led to the disconnection of 700 MW in a large photovoltaic power plant. Thus, analyzing the transient synchronization stability of grid-connected VSCs under asymmetric faults is equally important. Existing studies on synchronization stability under asymmetric faults mainly investigate three aspects: the existence of equilibrium points during asymmetric faults, the stability of these equilibrium points, and the system's ability to return to equilibrium after a fault. For the existence of equilibrium points, reference [18; 19; 20] establishes a reduced-order dynamic model suitable for analyzing PLL synchronization stability under asymmetric faults and derives static synchronization stability criteria to ensure the existence of equilibrium points post-fault. Clearly, if the system lacks equilibrium points during a fault, it will inevitably experience transient instability. However, the existence of equilibrium points during a fault does not guarantee transient stability, as it also depends on the system's dynamic processes. Regarding the small-signal stability analysis of equilibrium points, reference [21] constructs a complex frequency domain impedance model of the VSC grid-connected system under asymmetric grid faults, using the generalized Nyquist criterion to analyze small-signal stability during asymmetric faults. Reference [22] develops a high-order state-space analysis model under asymmetric faults and uses eigenvalue analysis to study the impact of positive and negative sequence currents on the small-signal stability of the VSC grid connection. It is important to note that references [21; 22] base their analysis on linearized models, which cannot fully describe the transient synchronization process under large disturbances like asymmetric grid faults, where the post-fault operating point deviates significantly from the pre-fault steady-state operating point. In such cases, the nonlinear characteristics of the VSC grid connection must be considered, and transient stability analysis methods are required.