Successive failures in power transmission lines can cause cascading failures in the power grid, which may eventually affect large parts of the power grid and even cause the power grid system to go down. Collecting and transmitting primary equipment information and issuing load-shedding action commands in the power grid depend on the power communication network. With the help of the power communication network, we can better observe the situation of the power grid in real time and provide a guarantee for the regular working of the power grid. However, the communication network also has the problem of communication delay causing latency in load-shedding action. On the premise of preserving the key physical properties and operational characteristics of the power grid, this paper uses the IEEE 14 and 30 bus systems as examples to establish a direct current (DC) power flow simulation environment. We establish a communication network model based on the power grid topology and the corresponding communication channels. For the problem of cascading failures occurring in the power grid after transmission line failures, a load-shedding strategy using soft actor-critic (SAC) based on deep reinforcement learning (DRL) was developed to effectively mitigate cascading failures in the power grid while considering the impact of communication delay. The corresponding communication delay is obtained by calculating the shortest communication path using the Dijkstra algorithm. The simulation verifies the feasibility and effectiveness of the SAC algorithm to mitigate cascading failures. The trained network can decide on actions and give commands quickly when a specific initial failure is encountered, reducing the scale of cascading failures.