I. Introduction
Grid-forming inverters play a crucial role in future power systems in autonomously regulating grid frequency and voltage. While a grid-forming inverter operates like a voltage source, limiting its current during grid disturbances is critical to prevent potential overcurrent damage. Moreover, grid-forming inverters should sustain transient stability during grid faults, ensuring synchronization while transitioning from one operating state to another. Transient stability is crucial for successful fault ride-through (FRT) and ancillary services during FRT, such as fault current injection and phase jump power provision. These are requirements outlined in recent grid-forming specifications, e.g., the Great Britain, Australian, and European grid codes [1], [2], [3]; see a survey in [4] and [5]. To satisfy these requirements, grid-forming inverters should maintain grid-forming synchronization and provide FRT ancillary services as continuously as possible, even when the current reaches the limit [1], [2], [3], [4], [5]. These requirements involve technical challenges in limiting fault current, maintaining transient stability (synchronization), and providing FRT ancillary services simultaneously [6], [7].