Power networks and information systems become more and more interdependent to ensure better supports for the functionality as well as improve the economy. However, power networks also tend to be more vulnerable due to the cascading failures from their interdependent information systems, i.e., the failures in the information systems can cause the failures of the coupled portion in power networks. Therefore, the accurate vulnerability assessment of interdependent power networks is of great importance in the presence of unexpected disruptive events or adversarial attacks targeting on critical network nodes. In this paper, we study the Interdependent Power Network Disruptor (IPND) optimization problem to identify critical nodes in an interdependent power network whose removals maximally destroy its functions due to both malfunction of these nodes and the cascading failures of its interdependent communication network. First, we show the IPND problem is NP-hard to be approximated within the factor of (2-ε) . Despite its intractability, we propose a greedy framework with novel centrality functions based on the networks' interdependencies, to efficiently solve this problem in a timely manner. An extensive experiment not only illustrates the effectiveness of our approach on networks with different topologies and interdependencies, but also highlights some important observations which help to sharpen the robustness of interdependent networks in the future.