This paper presents a fast and reliable method to evaluate risk of single or multiple commutation failures for multi-infeed HVDC systems following faults occurring at the receiving-end ac system. First, a fast calculation method of multi-infeed interaction factor, describing the interactivity and mutual impact between two inverter AC buses on is proposed. It is then refined to being the AC-DC system voltage interaction factor (ADVIF) to represent the relationship in bus voltages between inverter and AC buses at the receiving-end systems. Then, a critical AC-DC system voltage interaction factor (CADVIF) based on the minimum extinction angle criteria is proposed. By calculating and comparing ADVIFs and CADVIFs, which can be obtained quickly, it is possible to identify AC buses that, if a fault occurs, would cause commutation failures at one or more HVDC systems. The validity and accuracy of the proposed approach are demonstrated by comparing with simulations results using a two-infeed HVDC test system and an actual large power grid.