The prevention of cascading failures and large-scale power outages of power grids by identifying weak links has become one of the key topics in power systems research. In this paper, a vulnerability radius index is proposed to identify the initial fault, and a fault chain model of cascading failure is developed with probabilistic attributes to identify the set of fault chains that have a significant impact on the safe and stable operation of power grids. On this basis, a method for evaluating the vulnerability of transmission lines based on a multi-criteria decision analysis is proposed, which can quickly identify critical transmission lines in the process of cascading failure. Finally, the proposed model and method for identifying vulnerable lines during the cascading failure process is demonstrated on the IEEE-118 bus system.