Grid-connected converters may operate under grid fault conditions, such as the short/open-circuit and the transmission line switching, which may be accompanied by a phase angle difference between the grid and converter. This variation will affect the normal operation of the converter, since its phase-locked loop (PLL) is not able to track the grid voltage phase angle immediately. In this paper, an instability phenomenon with unregulated AC line current is identified in a modular multilevel converter based high-voltage direct-current (MMC-HVDC) transmission system when there is a sudden phase angle change on the grid side. A set of analytical constraints is developed for the modular multilevel converter (MMC) in terms of the Lyapunov stability, the modulation index and the maximum power transfer capability. Then, a multi-constraint operating region of the MMC under various phase disturbances is derived. Moreover, the parameter influence of the operating region is analyzed and two stability enhancing methods for the MMC system are proposed. Finally, the boundaries of the operating region and the stability enhancing methods are verified by simulations and laboratory measurements of three-phase grid-connected MMCs.