## 1 Introduction  
In recent years, many countries strongly support the access of a great many distributed generators to low/medium voltage distribution networks [1; 2], which are called active distribution networks (ADNs). The basic goal of designing an ADN is to provide users with convenient and real-timeelectricity [3]. As shown in Figure 1, the ADN is a large complex network that consists of two interdependent sub-networks: power networks (PNs) and communication networks (CNs) [4]. The PN is composed of the power equipment, such as generators, transmission lines, substations, and loads, etc. The CN is composed of the information equipment, such as sensors, computers, communication lines, data storage, and actuators, etc. All these equipment are interconnected according to a certain topological structure. Some nodes in the PN supply power to the nodes in the CN, meanwhile some nodes in the CN collect information of the nodes in the PN and thus to control the actions of them. In general, the operational process of the ADN includes the physical process and the computational process. The physical process can be described by the continuous change of the PN parameters (e.g., voltage, current), which are caused by power consumption or power supply change. The computational process is the procedure that the CN collects the data from the PN via various sensors and thus to control the PN via various actuators. The PN and CN are deeply interdependent. Due to the intrinsic complexity and the characteristic of deep interdependence between the PN and CN, the ADN is vulnerable to cross-domain cascading failures (CCFs) which propagate across nodes from the PN and CN interchangeably [5, 6]. Specifically, when a node or an edge in the CN/PN fails, some other nodes or edges in the CN/PN may be removed by protective devices, which may incur the CCF sequence and finally result in large-scale blackouts [7, 8]. A simple running example of a CCF in an ADN is shown in Figure 2. Initially, the simplified PN and CN work normally. The nodes in the PN and CN are coupled through the control relationship, the power supply relationship and the information collection relationship denoted by dashed arrow  
Figure 1: A brief architecture of an active distribution network (ADN). lines, shown in Figure 2a. In the first step, a substation node in the PN, as well as the edges linked to this node, fails, shown in Figure 2b. In the second step, two connected clusters in the PN containing generators (surrounded by red dotted closed curves) are formed. The failed edges in the first step result in the failure of corresponding nodes in the CN, then two connected clusters (surrounded by red dotted closed curves) are also formed, shown in Figure 2c. The isolated nodes in the CN fail and the corresponding edges fail as well, shown in Figure 2d. The CCF finally stops when no failure nodes or edges exist in both the sub-networks and the ADN enters the steady state as shown in Figure 2e. There are many factors that may be related with the steady state of the ADN after the CCF, such as initial failure rate, interdependence, the distribution of distributed generators, network structure, and load redistribution and so on. Previous studies are focused on the electrical power system. Various approaches have been proposed to study the relationship between the steady state of the electrical power system and the aforementioned factors. For a single network (i.e., PN or CN), researchers study the relationship mentioned above using static geometric methods [9; 10]. However, the dynamic load redistribution is common in both the PN and CN. Static geometric methods are not applicable to the case of the dynamic load redistribution. Therefore, the dynamic methods considering load redistribution are adopted to analyze the relationship [11; 12; 13; 14]. These methods are all designed for a single network only. They are unsuitable for the electrical power system, which is composed of two coupled networks: PN and CN.