## 1 Introduction
In the electrical power system, the power consumers only get their energy from the grid. However, in the ADN, the power consumers not only consume power from the grid, but also supply energy dynamically to the grid using distributed generators. For coupled networks, researchers analyze the CCF using static geometric methods in the early stage [15; 16; 17; 18; 19; 20]. Then, the dynamic methods considering the load redistribution [21; 22; 23; 24; 25] are explored. These static and dynamic methods are based on the giant cluster assumption [15]. Although this  
Figure 2: Simple running example of a cross-domain cascading failure (CCF) in an ADN. assumption can simplify the analysis and alleviate the computational load, it ignores the role of clusters containing power generation nodes of the PN. It assumes that only when a node belongs to the giant cluster, this node is considered to be in a normal state. However, we think the giant cluster assumption dose not accord with the actual situation. As shown in Figure 2c-e, there are two connected clusters (Cis) containing generators in the PN, and there are also two connected clusters in the CN interdependent with the Cis. In fact, if a power generation node is in one of the clusters in the PN (denoted as Ci), meanwhile the state of nodes in the CN that are interdependent with the cluster Ci is running normally, these nodes in the cluster (Ci) in the PN will be considered as the functional nodes. In order to overcome the problem of these methods based on the giant cluster assumption, [PERSON] et al. [26] identify effective clusters using a threshold-based method. However, this method suffers from poor identification accuracy. To mitigate this problem, [PERSON] et al. [27] use a method based on the cluster containing power supplies and assume that a node of the CN is in the normal state if it is powered by the nodes of the PN. However, in reality, even if the node of the CN is power supplied normally, it still will not be considered to be functional if it does not provide the correct control policy to improve the operation of the PN. The previous research results are mainly obtained in the electrical power system environment. Therefore, these research results could not be directly applied to describe the steady-state behavior of the CCF in the ADN environment. Consequently, it is necessary to comprehensively study the relationship between the steady state behavior of the CCF of the ADN and the distribution of generators, unintentional random initial failure (e.g., the terminal voltage of the node is zero or the current flowing through this node is zero), interdependence, network structure, and load redistribution. In order to describe the relationship more accurately, it is necessary to classify the nodes, and adopt more realistic assumptions (denoted as Ai) that clusters in the PN contain power supplies and the CN clusters are interdependent with them. If the nodes belong to these clusters, the nodes are judged to be in the normal state, so as to overcome the limitation of the giant cluster assumption. In this paper we studied the robustness of the ADN against the CCF. We needed a model to analyze the relationship between the proportions of remaining normal operational nodes after the CCF and the distribution of generators, unintentional random initial failure, the interdependence between the sub-networks, network topology, and load redistribution. However, several technical challenges existed when the work of evaluating the robustness of the ADN after the CCF was completed. Firstly, how to describe the propagation process of the CCF. Secondly, how to characterize the influence of the distribution of generators, unintentional random initial failure, the interdependence between the sub-networks, network topology, and load redistribution on the robustness of the ADN based on Ai. Thirdly, how to simulate the propagation process of the CCF. The main contributions of this paper are as follows.