## 1 Introduction  
Active distribution network (ADN) is a typical cyber-physical system (CPS), which consists of a wide variety of distributed generators connected to a power network (PN) and a communication network (CN) [1]. An ADN at every moment of time can be abstracted as a directed graph, where equipment is abstracted as nodes, and power lines or communication connections among equipment are abstracted as edges [1], and the directed graph at different time points may be different due to changes in edges or nodes of an ADN. There are five types of nodes in the PN according to the role of nodes: a power generation node type \(V_{P1}\), a substation node type \(V_{P2}\), a distribution node type \(V_{P3}\), a load node type \(V_{P4}\)and an external node type \(V_{P5}\) (the energy from power transmission lines). Similarly, nodes in the CN are categorized into an information relay node class \(V_{C1}\), a sensor node class \(V_{C2}\), an actuator node class \(V_{C3}\), and a control node class \(V_{C4}\). Nodes in the PN are mutually interdependent with nodes in the CN. Specifically, a substation node \(n_{P2}\in V_{P2}\) in the PN supplies power to nodes in the CN, a sensor node \(n_{C2}\in V_{C2}\) in the CN collects the information from nodes in the PN, and an actuator node \(n_{C3}\in V_{C3}\) drives the behaviors of nodes in the PN. There is a risk of the potential cross-domain cascading failures (CCF) spreading alternatively across the PN/CN due to the interdependence between the PN and CN in an ADN. The propagation process of CCF can be divided into two stages [2]. The first stage is a slowly changing process lasting for several minutes. The second stage is an avalanche-collapse process, which could lead to a large area of power outages and cause large economic losses. Hence, it is essential to prevent the propagation of CCF so as to alleviate the side effect of this event and improve the level of the safe and stable operation of an ADN. A directed graph at every time point could be described as a state of an ADN. These states can be divided into steady states and transient states among ADN operations. A steady state means that an ADN is in stable operation, and a transient state is an intermediate state between steady states. During the propagation process of CCF, an ADN goes through a series of transient states from one steady state to another steady state. If the CCF is not prevented when the CCF is propagating, an ADN must be transferred into the next transient state, and it must stop at a final steady state, and the ADN may collapse. Hence, once the propagation process of CCF is initiated, it is better to use the preventive control to block the propagation process of CCF. The preventive control process against CCF in an ADN is shown in Figure 1, there are five steps in each routine loop operation. They are state perception, failure diagnosis, decision, preventive control policies, and control action. The state perception gathers the real-time data (such as the voltage and current of nodes) of an ADN. The failure diagnosis detects failure symptoms and predicts failures that could be fired through some propagation paths of the CCF. The goal of decision is to select an optimal preventive control policy (PCP) to prevent potential accidents and the CCF propagation in an ADN. The preventive control policies (PCPs) are some strategies that could be adopted for decision step, and they are constructed in advance according to safety and stability requirement specifications of an ADN. The control action is the last routine step, which just carries out a sequence of operations on an ADN according to decisions and affects the state transition process of an ADN. In a process of the preventive control routine, a PCP is very important, it provides roadmaps of operations against CCF and determines the performance of the preventive  
Figure 1: The preventive control process against cross-domain cascading failures (CCF) in an active distribution network (ADN). control routine.