Cyber–Physical Active Distribution Networks Robustness Evaluation against Cross-Domain Cascading Failures

: Active distribution networks (ADNs) are a typical cyber–physical system (CPS), which consist of two kinds of interdependent sub-networks: power networks (PNs) and communication networks (CNs). The combination of typical characteristics of the ADN includes (1) a large number of distributed generators contained in the PN, (2) load redistribution in both the PN and CN, and (3) strong interdependence between the PN and CN, which makes ADNs vulnerable to cross-domain cascading failures (CCFs). In this paper, we focus on the robustness analysis of the ADN against the CCF. Rather than via the rate of the clusters with size greater than a predeﬁned threshold, we evaluate the robustness of the ADN using the rate of the clusters containing generators after the CCF. Firstly, a synchronous probabilistic model is derived to calculate the proportions of remaining normal operational nodes after the CCF. With this model, the propagation of the CCF in the ADN can be described as recursive equations. Secondly, we analyze the relationship between the proportions of remaining normal operational nodes after the CCF and the distribution of distributed generators, unintentional random initial failure rate, the interdependence between the sub-networks, network topology, and tolerance parameters. Some results are revealed which include (1) the more distributed generators the PN contains, the higher ADN robustness is, (2) the robustness of the ADN is negatively correlated with the unintentional random initial failure rate, (3) the robustness of the ADN can be improved by increasing the average control fan in of each node in the PN and the average power fan in of each node in the CN, (4) the robustness of the ADN with Erdos–Renyi (ER) network topological structure is greater than that with Barabasi–Albert (BA) network topological structure under the same average node degree, and (5) the robustness of the ADN is greater, when the tolerance parameters increase. Lastly, some simulation experiments are conducted and experimental results also demonstrate that the conclusions above are e ﬀ ective to improve the robustness of the ADN against the CCF.


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-time 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]. 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 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. 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.  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.
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 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, Huang et al. [26] identify effective clusters using a threshold-based method. However, this method suffers from poor identification accuracy. To mitigate this problem, Yu 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. (1) A synchronous probabilistic percolation model of the ADN is proposed, and the CCF simulation algorithm is proposed also based on the model, which can describe the propagation process of the CCF. (2) We propose a method to analyze the relationship between the robustness of the ADN and distribution of generators, unintentional random initial failure, interdependence between the sub-networks, network topology and tolerance parameters. We give five conclusions. The first one is that the distribution of generators improved the robustness of the ADN against the CCF. The second one is that the robustness of the ADN was negatively correlated with the unintentional random failure rate. The third one is that given network topology parameters and the unintentional random initial failure rate, the more the average control fan in of a node in the PN and the more the average power fan in of a node in the CN, the better the robustness of the ADN. The fourth one is that the robustness of the ADN with Erdos-Renyi (ER) network topological structure was greater than that with Barabasi-Albert (BA) network topological structure under the same average node degree. The last one is that the robustness of the ADN was greater, when the tolerance parameters increased. (3) We conducted extensive simulation experiments to verify our five conclusions based on the ER network topological structure and the BA network topological structure.

Problem Definition
In this paper, it was assumed that (1) the CN node will fail if its power supply interruption happens. (2) Information overload will occur in the CN when severely repeated transmission, packet congestion, and the invalid information transmission takes place [27]. We assumed that the number of the nodes in the PN and the CN are NoN I and NoN P respectively. Table 1 shows the semantic description of the symbols used in this paper. In Table 1, the relationship can be deduced that the unintentional random initial failure nodes set is the subset of the nodes set of PN (V f-initial ⊆V P ), the connected clusters set S P (V f-initial , ∞) in the PN after the CCF stop is equal to {G P1 (∞), G P2 (∞), . . . }, the connected clusters set S I (V f-initial , ∞) in the CN after the CCF stop is equal to {G I1 (∞), G I2 (∞), . . . }, the connected clusters set S Pg (V f-initial , ∞) whose elements contain generators in the PN after the CCF stop is equal to {G Pg1 (∞), G Pg2 (∞), . . . }⊆S P (V f-initial , ∞), and the connected clusters set S Ig (V f-initial , ∞) in the CN whose elements are interdependent with the elements of the connected clusters set Table 1. Notation used in this paper.

Symbol Description
G A Active distribution networks (ADNs).

G P
The power networks of the ADN.

G I
The communication networks of the ADN.

V P
The nodes set of the PN.

V I
The nodes set of the CN.

NoN P
The number of the nodes in the PN.

NoN I
The number of the nodes in the CN.
The unintentional random initial failure nodes set of the PN. ∞ The steady-state of the ADN after the CCF.
The connected clusters in the PN of the ADN after the CCF stops.
The connected clusters containing generators in the PN of the ADN after the CCF stops.
The connected clusters in the CN of the ADN after the CCF stops.
The connected clusters in the CN are interdependent with the connected clusters containing generators in the PN of the ADN after the CCF stops.

Num
The number of experiments. N The number of iterative steps in the propagation of the CCF.
The expected proportion of remaining nodes in the PN/CN at step N after the CCF occurs.
The expectation of the quotient between the number of the remaining nodes in the clusters that containing generators and the number of the remaining nodes of the whole PN at step N after the CCF occurs.
The expectation of the quotient between the number of the remaining nodes belonging to the connected clusters in the CN interdependent with the connected clusters containing generators in the PN and the number of the remaining nodes of the whole CN at step N after the CCF occurs.
The expected proportion of normal operational nodes in the PN/CN at step N after the CCF occurs.
The final expected proportion of normal operational nodes in the PN/CN after the CCF stops. Θ P The probability that a single node in the PN fails randomly.
The edge set (excluding the edges connecting the PN and the CN) of a node n in the PN/CN.
The average control fan in of a node in the PN.
The average power fan in of a node in the CN.

B Ctrl
The set of the paths involved in an effective control Econ I .
Load threshold of an edge e in the PN/CN. Run(n)/ ¬ Run(n) A node or an edge n runs normally/ abnormally. α P / α I A tolerance parameter of a node n in the PN/CN which represents the ratio of the maximum capacity to nominal capacity of the node n. The final expected proportion of normal operational nodes was an index reflecting the robustness of the ADN against the CCF. The problem to be solved in this paper was to calculate the relationship between the final expected proportion of normal operational nodes R PF (∞)/R IF (∞) in the PN and CN of the ADN and distribution of generators, unintentional random initial failure, interdependence between the sub-networks, network topology, and load redistribution, when the CCF stops. According to the notations and relationship in Table 1, it can also be seen that the final expected proportion R PF (∞)/R IF (∞) of normal operational nodes is shown in Equation (1).

Active Distribution Networks Modeling
There were five types of nodes in the PN, including power generation nodes (distributed generators) V P1 , substation nodes V P2 , distribution nodes V P3 , load nodes V P4 , and external nodes (the power from transmission grid) V P5 . There were four kinds of nodes in the CN, including sensor nodes V I1 , information relay nodes V I2 , control nodes V I3 , and actuator nodes V I4 . They were mutually disjoint sets. Definition 1. An effective control Econ I in the CN refers to an effective control of the PN nodes by the control nodes of the CN, and it satisfies three conditions.
(1) There is at least one complete simple directed path of length k − 1 existing in the CN, that is path(n I1 , n I4 ) = (n 1 , . . . , n k ). The path has k nodes n 1 , . . . , n k . Where the source node n I1 = n 1 ∈V I1 is the sensor node, the destination node n I4 = n k ∈V I4 is the actuator node, and at least one of the remaining nodes belongs to the V I3 nodes set. A sensor node n I1 ∈V I1 in the CN is applied to detect failure events in the PN, and then transmit the event information to a control node n I3 ∈V I3 through one or more information relay nodes n I2 ∈V I2 . After that, the control node n I3 ∈V I3 generates the response information based on specific algorithms and subsequently the response information is transmitted to an actuator node n I4 ∈V I4 through one or more information relay nodes n I2 to control the physical process. (2) All nodes and edges in the path path(n I1 , n I4 ) run normally.
(3) t delay + t react < t interval . t delay denotes the time interval from the occurrence of a failure event to the time when the response information is generated by a node n I3 . t react denotes the time interval from the time when the response information is generated to the time when the PN has been changed by actuators. t interval denotes the minimum time interval between two adjacent failure events.

Definition 2.
The PN of the ADN is a sextuple system G p = (V P , E P , H NP , Load P , W P , Trans P , Thres P ), where (1) V P represents the set of nodes in the PN, and (2) E P represents the edge set in the PN, and E P ⊆V P × V P .
(3) Load P : V P ×Int → C represents the load (power) of a node n in the PN at step N after the CCF occurs. Where Int represents a set of positive integer numbers, C represents the set of complex numbers. It is assumed that the load (power) of a failure node is redistributed to its neighbor node following the nearest neighbor rule. When the neighbor node n Pi of a node n Pj in the PN is failed, the original load (power) of this node n Pi is redistributed to the node n Pj , and the load (power) of this node n Pj changes according to the following recursive Equation (2).
Load P (n Pj , N) + Load P (n Pi ,N)×W P (n Pj ,n Pi ) n P ∈Neigh P (n Pi ) W P (n P ,n Pi ) , n Pi ∈ Fail P Load P (n Pj , N), n Pi Fail P where Neigh P : V P →2 VP is a mapping that represents the neighbor nodes set of a node in the PN. Fail P is the set whose elements are the contiguous failed nodes of the node n Pj , that is Fail P = {n Pi | n Pi ∈Neigh P (n Pj ) ∧¬Run(n Pi )}. (4) W P : E P →C represents the edge weight mapping in the PN. (5) Trans P = (V P (N), E P (N)) represents the subgraph generated by the load (power) redistribution after a node or an edge fails in the PN at step N during the CCF. For example, if the subgraph is generated by a node n Pi ∈V P failure, then V P (N) = V P (N − 1) − Over P (Neigh P (n Pi )), E P (N) = E P (N − 1) − Edg P (n Pi ). Where Over P (Neigh P (n Pi )) = {n∈Neigh P (n Pi )|load P (n,N) ≥ T PP (n)} represents the set of overloaded nodes in the set of neighbor nodes at step N. T PP (n) represents the load threshold of a node n in the PN, when the load (power) of a node n in the PN is greater than its threshold, then the node will fail. The process of an edge failure in the PN is similar. (6) Thres P = Thres PJ ∪ Thres PL represents the thresholds set of nodes and edges in the PN. Where Thres PJ and Thres PL represent the thresholds sets of nodes and edges in the PN respectively. If the load flowing through an edge of the PN is greater than its threshold, then the edge will fail. The node situation is similar to the edge situation.
Since the ADN includes the PN and the CN, they are interdependent with each other. Thereinafter, the definition of the CN in the ADN is introduced, and the model of interdependence between the PN and the CN is also given.
(2) E I represents the edge set in the CN, and E I ⊆V I × V I .
(3) Load P : V I × Int→Int represents the load (data packets) of a node n in the CN at step N after the CCF occurs. When the neighbor node n Ii of a node n Ij is failed, the original load (data packets) of this node n Ii is redistributed to the node n Ij , and the load (data packets) of the node n Ij changes according to the following recursive Equation (3). where Neigh I : V I →2 VI is a mapping that represents the neighbor nodes set of a node in the CN. Fail I is the set whose elements are the contiguous failed nodes of the node n Ij , that is Fail I = {n Ii | n Ii ∈ Neigh I (n Ij ) ∧ ¬Run(n Ii )}. (4) W I : E I →Int represents the edge weight mapping in the CN. (5) Trans I = (V I (N), E I (N))represents the subgraph generated by the load (data packets) redistribution after a node or an edge fails in the CN at step N during the CCF. (6) Thres I = Thres IN ∪ Thres IL represents the thresholds set of nodes and edges in the CN.
We modeled the interdependence between the CN and the PN in the following ways. The symbol <> represents the average value of the corresponding variable. In the CN, a sensor node n I1 ∈V I1 collects information of the nodes in the PN, and an actuator node n I4 ∈V I4 executes the commands from a control node n I3 ∈V I3 to control the physical process of the PN. Abstractly, it can be seen that a control node n K1 ∈V I3 in the CN can control K 1 nodes of the PN, and the average control fan out of the node n K1 is <K 1 > = (1/|V I3 |) × ΣK 1 . A node n K2 in the PN is controlled by K 2 control nodes in the CN, and the average control fan in of the node n K2 is <K 2 > = |V I3 | × <K 1 >/NON P . A sensor node n L1 ∈V I1 in the CN can collect L 1 nodes in the PN, and the average information collecting fan in of the node n L1 is <L 1 > = (1/|V I1 |) × ΣL 1 . Conversely, a node n L2 in the PN is perceived by L 2 sensor nodes in the CN and the average information collecting fan out of the node n L2 is <L 2 > = |V I1 | × <L 1 >/NON P . Correspondingly, the average power fan out of a node in the PN is <O 2 > = (1/|V P2 |) × ΣO 2 , and a node n O1 in the CN can be power supplied by O 1 nodes of the PN, then the average power fan in of the node n O1 is <O 1 > = |V P2 | × <O 2 >/NON I . For example, as shown in Figure 2a, |V P1 | = 2, |V P2 | = 3, |V P3 | = 1, |V P4 | = 3, |V P5 | = 1, |V I1 | = 3, |V I2 | = 5, |V I3 | = 4, and |V I4 | = 3. Therefore, <K 1 > = 10/4 = 2.5, <K 2 > = 1, <L 1 > = 10/3, <L 2 > = 1, <O 1 > = 1, and <O 2 > = 15/3 = 5. (In Appendix D, a special case about how to find <O 1 > and <O 2 > is shown) According to the aforementioned Definitions 2 and 3, the definition of the ADN is given below. (2) E A represents the edge set in the ADN, and E A = E P ∪ E I ∪ E PI ∪ E IP . The edge set of the ADN includes the edge set of the PN and the edge set of the CN. In addition, the edge set formed by the interdependence between the nodes of the PN and CN is added. Where E PI = E PI~∪ E PI , it includes the virtual edge set E PI~f ormed by the sensor nodes in the CN perceiving the corresponding nodes in the PN, and it indicates the information gathering relationship between a sensor node n I1 ∈ V I1 in the CN and a node in the PN.
The set E PI also includes the solid edge set E PI of the nodes in the PN supplying power to the nodes in the CN. E IP represents the virtual edge set formed by the actuator nodes in the CN acting on the nodes in the PN. (3) Load P : V A ×Int→C represents the load of a node n in the ADN at step N after the CCF occurs. Where Int represents a set of positive integer numbers, C represents the set of complex numbers. It is assumed that the load of a failure node is only redistributed to its neighbor node of the same network following the nearest neighbor rule. (4) W A represents the edge weight mapping in the ADN, and W A : E A →C. C is a set of complex numbers. (5) Trans A represents a subgraph generated by the load redistribution after a node or an edge fails in the ADN at step N during the CCF, and Trans A = (Trans P , Trans I , E PI-N , E IP-N ). E PI-N represents the interdependence edges set from a node in the PN to a node in the CN. E IP-N represents the interdependence edges set from a node in the CN to a node in the PN. (6) Thres A = Thres P ∪ Thres I ∪ Thres PI ∪ Thres IP represents the threshold set of nodes and edges in the ADN.
Thres PI represents the threshold set of edges in the set E PI , Thres IP represents the threshold set of edges in the set E IP .
In order to analyze the robustness of the ADN against the CCF, we need to show the normal operation conditions of nodes and edges in the PN and CN respectively. The description of the relevant conditions of a normal node or an edge in the PN and CN is shown in Table 2. (1) It is connected to at least one power generation node n PP in the PN, i.e., (2) It has at least one edge between the node n P in the PN and a node n II in the CN, and the node n II is part of at least one path in the set B tr . i.e., The load flowing through this node n P does not exceed its threshold T PP (n P ) at step N.
i.e., (load(n P , N) ≤ T PP (n P )) ∧ (n P ∈ V P ) (1) The load flowing through the edge e P does not exceed its threshold T PL (e P ) at step N. i.e., (load(e P , N) ≤ T PL (e P )) ∧ (e P ∈ E P ) (2) The nodes on both sides of the edge e P are in normal operation. i.e., e P = (n Px ∈ V P , n Py ∈ V P ) → (run(n Px ) ∧ run(n Py )) CN Node n I (1) There is at least one normal operating node n p in the PN that provides power to this node n I . i.e., ∃n P . ((e = (n P ∈ V P , n I ∈ V I )) → run(n P )) (2) The data traffic of this node n I in the CN does not exceed its threshold at step N. i.e., 3) The node n I is part of at least one path in the set B tr . i.e., (1) The load (data packets) flowing through the edge e P does not exceed its threshold T IL (e P ) at step N. i.e., (load(e P , N) ≤ T IL (e P )) ∧ (e P ∈ E I ) (2) The nodes on both sides of edge e P are in normal operation. i.e., The interdependence between the PN and the CN Edge e (1) The nodes on both sides of the edge e∈ (E PI ∪ E IP ) are in normal operation. i.e.,

Robustness Analysis of the ADN against the CCF
We analyzed the relationship between the robustness of the ADN and the distribution of generators, unintentional random initial failure, interdependence, network topology, and load redistribution using the percolation-based method. We reached five conclusions and the conditions that prevent the occurrence of the CCF caused by the load redistribution.

Robustness Analysis
The following analysis only considered the case of nodes failure. In this section, it was assumed that the nodes failures started from the PN.
According to the description of R P (N), f P (R P (N)), and R PF (N) in Table 1, the expected proportion of normal operational nodes after the CCF occurs at step N is Furthermore, for the CN, according to the description of R I (N), f I (R I (N)), and R IF (N) in Table 1, then the expected proportion of normal operational nodes after the CCF occur at step N is R IF (N) = R I (N)*f I (R I (N)). (Where the derivation of the specific expressions of the mapping f P : R→R and the mapping f I : R→R are in Appendix B.) It is assumed that the unintentional-random-initial-failure-rate is Θ P in the PN. We can get the analysis process represented by a Venn diagram, as shown in Figure 3 (The detailed analysis process is shown in Appendix C). This figure shows the process of calculating the expected proportions of normal operation nodes in the first three iterations after the CCF occurs.

Distribution of Generators and Robustness of the ADN
The resulting-iteration-equation-sets are concluded in (4a,b), where R I (0) =1 and R IF (0) = 1. The iteration Equation (4a) calculates the value of R P (2N + 1) using the value of R IF (2N), and the iteration Equation (4b) calculates the value of R I (2N) using the value of R PF (2N − 1). As N→∞, the iteration Equation (4a,b) calculate the sequences R I (0), R P (1), R I (2) . . . , R P (∞), R I (∞) and R IF (0), R PF (1), (2) ( (2)) When the CCF stops, the ADN enters the steady state and then satisfies the equations , a set of steady-state nonlinear equations is obtained after the replacement as above, shown in Equation (5). When the CCF stops, the ADN enters the steady state and then satisfies the equations R P (2N − 1) = R P (2N) = R P (2N + 1) = λ and R I (2N − 1) = R I (2N) = R I (2N + 1) = γ. According to Equation (4a,b), a set of steady-state nonlinear equations is obtained after the replacement as above, shown in Equation (5).
At steady state, the final expected proportions R PF (∞) and R IF (∞) of remaining normal operational nodes in the PN and CN are shown in Equation (6).
The Equations (5) and (6) are the analytical model to evaluate the robustness of the ADN against the CCF. The relationship between the final expected proportions R PF (∞) and R IF (∞) of remaining normal operational nodes and the load redistribution is embodied in the mapping f P : R→R and mapping f I : R→R (in Appendix B). It can be seen from the conclusion in literature [26] that in the case of distributed generators existing in the PN, f I (γ) and f P (λ) will become larger. According to Equations (5) and (6), the distribution of generators will increase the final expected proportions R PF (∞) and R IF (∞) of remaining normal operational nodes. Therefore, the first conclusion of our analysis is that the distribution of generators improves the robustness of the ADN against the CCF.

Unintentional Random Initial Failure Rate and Robustness of the ADN
From the Equation (5), Equation (7) can be further obtained.
The solution of Equation (7) can be obtained by the graphic method of two curves as follows in Equation (8).
By solving Equation (8), we can get that it has a trivial solution γ = 0, λ = 0. That is, an intersection point of the two curves is the original point. The meaning of this trivial solution is that for any Θ P , the expected proportions R IF (∞) and R PF (∞) of normal operational nodes in the PN and CN are zero. There is another solution which corresponds to the critical value Θ P_critical of Θ P . When the two curves in Equation (8) are tangent, that is ∂λ/ ∂γ =1. We can get the equation set for solving Θ P_critical as follows in Equation (9).
where these equations are nonlinear and the critical value Θ P_critical of Θ P can be solved by numerical methods. The relationship between the final expected proportions R IF (∞) and R PF (∞) of remaining normal operational nodes and the unintentional random initial failure rate Θ P is obtained by solving the nonlinear Equation (5). From Equation (5), it can be seen that the greater the unintentional random initial failure rate Θ P is, the smaller γ and λ. When the value of the unintentional random initial failure rate Θ P is greater than the critical value Θ P_critical , the solution of Equation (7) is only the trivial solution γ = 0, λ = 0. Therefore, the second conclusion of our analysis is that the robustness of the ADN is negatively correlated with the unintentional random failure rate Θ P . Once the value of Θ P is greater than the critical value Θ P_critical , the whole ADN will collapse totally.

Independence and Robustness of the ADN
The relationship between the final expected proportions R PF (∞) and R IF (∞) of remaining normal operational nodes and the interdependence between two sub-networks is given by the following Theorem 1. Theorem 1. Given network topology parameters and the unintentional random initial failure rate Θ P of a node in the PN, the robustness of a ADN is better when the value of average control fan in <K 2 > of a node in the PN and the value of average power fan in <O 1 > of a node in the CN are greater.
Proof. In the nonlinear Equation (5) (1 − λ × f P (λ)) < 1, when the value of average power fan in <O 1 > of a node in the CN is greater, the value of γ is greater. Correspondingly, given the unintentional random initial failure rate Θ P , in the nonlinear Equation (5) (1 -γ × f I (γ)) < 1, when the value of average control fan in <K 2 > of a node in the PN is greater, the value of λ is greater. In addition, according to the conclusion in literature [23], given network topology, f I (γ) and f P (λ) are nondecreasing functions. When the value of average control fan in <K 2 > of a node in the PN and the value of average power fan in <O 1 > of a node in the CN is greater, the final expected proportions R PF (∞) and R IF (∞) of remaining normal operational nodes in the PN and CN are greater, so the robustness of the ADN is better.
Therefore, the third conclusion of our analysis is that the robustness of the ADN is improved when the value of average control fan in <K 2 > of a node in the PN and the value of average power fan in <O 1 > of a node in the CN are greater.

Network Topology and Robustness of the ADN
The relationship between the final expected proportions R PF (∞) and R IF (∞) of remaining normal operational nodes and the network topology is mainly reflected in the probability of occurrence of the event Evt P3 and the event Evt I3 , then in f P (λ) and f I (γ) (in Appendix B). Evt P3 is an event which is defined to represent the failure of a node n of the PN due to the load redistribution caused by the failure of its neighbor nodes. Evt I3 is an event which represents the failure of a node n in the CN due to the load (data packets) redistribution caused by the failure of its neighbor nodes. Under the same average node degree, f P (λ) and f I (γ) of a BA network is smaller than that of an ER network (see Appendix B analysis). Therefore, the fourth conclusion of our analysis is that the robustness of the ADN with ER network topological structure is greater than that with BA network topological structure under the same average node degree.

Relationship Analysis of Robustness and Tolerance Parameters
With the increase of the value of parameter reflecting the nodes thresholds under random attack strategy, the survivability of the single PN increased gradually against cascading failures, as shown in Figure 4. Therefore it was necessary to analyze the nodes thresholds influence on the robustness of the ADN. These nodes thresholds analyses provided conditions for preventing the CCF.
Assuming that the degree (it equals the sum of fan in and fan out) of a node n Pi in the PN is d P . Considering the power fluctuation of the distributed generators and the initial load of the node n Pi at step 0 is shown in Equation (10), which is modified from the conclusions in the literature [23].
where the load F P (d P ) of the node n Pi in the PN is the function of its degree d P . According to the Central Limit Law, this paper used the conclusions in literature [13] to model the randomness of the output power of distributed generators. So, ξ Pi is a random variable obeying Gaussian distribution.
average node degree, fP(λ) and fI(γ) of a BA network is smaller than that of an ER network (see Appendix B analysis). Therefore, the fourth conclusion of our analysis is that the robustness of the ADN with ER network topological structure is greater than that with BA network topological structure under the same average node degree.

Relationship Analysis of Robustness and Tolerance Parameters
With the increase of the value of parameter reflecting the nodes thresholds under random attack strategy, the survivability of the single PN increased gradually against cascading failures, as shown in Figure 4. Therefore it was necessary to analyze the nodes thresholds influence on the robustness of the ADN. These nodes thresholds analyses provided conditions for preventing the CCF. Assuming that the degree (it equals the sum of fan in and fan out) of a node nPi in the PN is dP. Considering the power fluctuation of the distributed generators and the initial load of the node nPi at step 0 is shown in Equation (10), which is modified from the conclusions in the literature [23]. The expected load of the node n Pi in the PN is shown in Equation (11).
where g represents the expected output power by the distributed generators at the node n Pi , if the node n Pi belongs to the set of generation nodes V P1 , g is not equal to zero, otherwise, it is zero. Prob P (d P ) is the probability that the degree of the node n Pi is d P .
The threshold of a node n Pi in the PN at step N is equal to T PP (n Pi ,N) = α P × F P (d P ). Where α P is a tolerance parameter of the node n Pi and satisfies α P ≥ 1. The expected value of the threshold of the node n Pi in the PN is shown in Equation (12).
Similarly, assuming that the degree of a node n Ii in the CN is d I , then the initial load (data packets) of node n Ii at step 0 after the CCF occur is shown in Equation (13).
The expected load of the node n Ii in the CN is shown in Equation (14).
where Prob I (d I ) is the probability that the degree of the node n Ii is d I .
The threshold of a node n Ii in the CN is equal to T IP (n Ii ,N) = α I × F I (d I ). Where α I is a tolerance parameter of the node n Ii and satisfies α I ≥ 1. The expected value of the threshold of the node n Ii in the CN is shown in Equation (15).
It is assumed that the unintentional random initial failure rate of nodes in the PN and CN are Θ P and Θ I respectively. The expected proportion of the remaining normal operational nodes in the PN is (1 − Θ P ) × f P (1 − Θ P ). When the node fails, the load of the original node will be redistributed to the remaining nodes in the PN, and the total redistributed load in the PN is In order to avoid the subsequent nodes failure after the failure of a node, it is required that the redistributed load should not exceed the margin of the remaining nodes in the PN. After simplification, the condition is ( Proposition 1. When the tolerance parameters increase to infinity, that are α P →∞ and α I →∞, the expected proportions R PF (2N + 1) and R IF (2N) of the remaining normal operational nodes reach their upper limits, which are the results of a pure interdependence model without considering the load redistribution. (The proof of Proposition 1 is given in Appendix E.) Therefore, according to Proposition 1, the fifth conclusion of our analysis is that the robustness of the ADN is greater, when the tolerance parameters increase.
If the redistributed load is equal to its margin of the remaining nodes, then the expected tolerance parameters are shown in Equations (16) and (17).
If the tolerance parameters α P and α I are greater than these critical ones respectively, the CCF caused by the load redistribution will not occur, which improves the robustness of the ADN against the CCF.

Evaluation Robustness
Based on the normal operation conditions of nodes and edges and the nearest neighbor rule of the load redistribution, a CCF simulation algorithm for the ADN is proposed in Algorithm 1, which is called stead-state subgraph generating algorithm. This algorithm accepts the ADN and its initial failure set, and then generates subgraphs of the ADN at steady state after the CCF stop. // Load redistribution for G P and G I 4 for n∈ (V P ∪ V I ) do 5 Load redistribution according to the nearest neighbor rule; // Intra-network failures for G P 6 for u P ∈ V P (t) do 7 for q P ∈ V P1 (t) do 8 if ((load P (u P , t) >T PP (u P )) ∨ ( run(q P ) ∧path(u P , q P ) = Ø) ) then 9 FL P (t) ←FL P (t) ∪ {u P }; // Inter-network failures for G P 10 for v P ∈ V P (t) do 11 for // Inter-network failures for G I 19 for v I ∈ V I (t) do 20 for u P ∈ V P (t) do 21 if ((v I , u P ) ∈ E IP )) ∧ (¬run(u P ) ) then 22 FL I (t) ←FL I (t) ∪ {v I };

23
V I (t) ←V I (t − 1) -FL I (t); Output: Sub-graphs of the ADN at steady state after the CCF stops.
After the steady state subgraphs were generated, we needed to evaluate the robustness of the ADN using Equation (1). The evaluation algorithm is proposed in Algorithm 2, which is used to evaluate the proportion of normal operational nodes in the ADN.

Algorithm 2 Evaluation Algorithm
Input: Sub-graphs of the ADN at steady state after the CCF stops.
2 Count the number of connected clusters in Sub-graphs of the ADN, generate sets S P (V f-initial , ∞) and S I (V f-initial , ∞); 3 Decide generators in sets S P (V f-initial , ∞) and S I (V f-initial , ∞), generate sets S Pg (V f-initial , ∞) and S Ig (V f-initial , ∞); // Evaluate the proportion of normal operational nodes in the ADN 4 for Output: The proportion of normal operational nodes in the ADN.
According to Equation (1), the robustness evaluation algorithm is proposed in Algorithm 3, which is used to evaluate the robustness of the ADN.

Numerical Simulations
In this section, we validate the above theoretical analysis by the data obtained from the simulation examples. We first describe the simulation experiment settings and then explain the simulation results.

Simulation Experiment Setting
An Erdos-Renyi model and a Barabasi-Albert model were used to construct a random network and a scale-free network respectively.
We designed a simulation experiment to illustrate the robustness of the ADN against the CCF in the following way. All the parameters used below are involved in the numerical simulations of References [6][7][8]15,16] except the proportion of distributed generators.
(1) We constructed a PN and a CN respectively, and the topological structure of the PN and CN was divided into two cases: one was that the topology of the PN and CN was both scale-free, the other one was that the topology of the PN and CN was both random. There were two cases about the values of the tolerance parameters of the nodes in the PN and CN: one was both 1.5 and the other was both 2.0. (2) The interdependence model between the PN and CN could be divided into two situations: one was to use one-to-one interdependence model between the two networks (PN and CN), the second one was to use three-to-three interdependence model to combine the two sub-networks (PN and CN) to form a coupling network (ADN). (3) The unintentional random initial failure mode was to select the failure nodes in the PN randomly according to a uniform distribution.
Generators were randomly distributed with uniform distribution in the PN. The number of nodes in the PN and CN were 500 respectively and the average degree of the nodes in the PN and CN was both 5. The unintentional random failure rate Θ P was from 0.002 to 0.7 with an interval of 0.002, and there were 350 non-repetitive points in total. The proportion of distributed generators was from 0.0 to 0.8 with an interval of 0.1. A total of 74 groups were run, and there were 1000 simulation experiments in each group.

Simulation Results
After finishing the simulation experiments, we used Equation (1) to calculate the robustness indicator (R PF (∞) and R IF (∞)). Our main aim was to find the relations between the distribution of the generators, unintentional random initial failure rate, interdependence, the network topology, tolerance parameters, and the ADN robustness against the CCF. The concrete results were as follows.

Distribution of Distributed Generators and Robustness
We first discuss the relationship between the distribution of distributed generators and the ADN robustness, where the unintentional random initial failure rate Θ P was from 0.002 to 0.7 with interval 0.002 and the value of the tolerance parameters of the nodes is 1.5.
The relationship between the final expected proportions R PF (∞) of the remaining normal operational nodes in the PN and the number of distributed generators (distribution of distributed generators) is shown in Figures 5 and 6. In Figures 5 and 6, each contains four sub-graphs respectively. In the following four sub-graphs, the first one shows 50 different curves and the rest sub-graphs show 101 different curves respectively. Each of the following eight sub-graphs show different curves representing the different unintentional random initial failure rate under the ER and BA topological structures respectively.
Of the nodes in the PN, 17.76% survived for Θ P = 0.002 with one-to-one interdependence under the ER topological structure, when the proportion of distributed generators was 0.2. Of the nodes 45.1% survived for Θ P = 0.002 with one-to-one interdependence under the ER topological structure, when the proportion of distributed generators was 0.4. Of the nodes55.17% survived in the PN for Θ P = 0.002 with one-to-one interdependence under the ER topological structure, when the proportion of distributed generators was 0.8.
As shown in Figures 5a and 6a, R PF (∞) fluctuated with the proportion of distributed generators due to the uncertainty of the CCF, and they were positively correlated as a whole. Therefore, this agreed with our first conclusion. That is, the robustness of the ADN was improved when the number of distributed generators was greater.
In the other sub-graphs, Θ P is too high, which led to the whole network crashing, thus making R PF (∞) almost equal to zero. The relationship between the final expected proportions R IF (∞) of the remaining normal operational nodes in the CN and the number of distributed generators was similar.  Figures 5a and 6a, RPF(∞) fluctuated with the proportion of distributed generators due to the uncertainty of the CCF, and they were positively correlated as a whole. Therefore, this agreed with our first conclusion. That is, the robustness of the ADN was improved when the number of distributed generators was greater.
In the other sub-graphs, ΘP is too high, which led to the whole network crashing, thus making RPF(∞) almost equal to zero. The relationship between the final expected proportions RIF(∞) of the remaining normal operational nodes in the CN and the number of distributed generators was similar.

Θ P and Robustness
We then investigated the relationship between the unintentional random initial failure rate Θ P and the ADN robustness under the ER topological structure, where the proportion of distributed generators was 0.8 and the value of the tolerance parameters of the nodes was 1.5.

ΘP and Robustness
We then investigated the relationship between the unintentional random initial failure rate ΘP and the ADN robustness under the ER topological structure, where the proportion of distributed generators was 0.8 and the value of the tolerance parameters of the nodes was 1.5. As shown in Figure 7, it can be seen that the robustness of the ADN deteriorated when the unintentional random initial failure rate ΘP in the PN was greater. That is, the more the unintentional random failure rate was, the robustness of ADN against the CCF was worse. We also observed that the transition phase of RPF(∞) was very sharp, and this indicated that the PN and CN either collapsed or was whole totally. Therefore, this agreed with our second conclusion. That is, the robustness of the ADN was negatively correlated with the unintentional random failure rate ΘP. Once the value of ΘP was greater than the critical value ΘP_critical, the whole ADN collapsed totally.
The relationship between the final expected proportions RIF(∞) of remaining normal operational nodes in the CN and the unintentional random initial failure rate ΘP is similar.

Independence and Robustness
We then investigated the relationship between the interdependence and the ADN robustness under the ER/BA topological structure, which is shown in Figure 8, where the proportion of distributed generators was 0.2 and the value of the tolerance parameters of the nodes was 1.5. One- As shown in Figure 7, it can be seen that the robustness of the ADN deteriorated when the unintentional random initial failure rate Θ P in the PN was greater. That is, the more the unintentional random failure rate was, the robustness of ADN against the CCF was worse. We also observed that the transition phase of R PF (∞) was very sharp, and this indicated that the PN and CN either collapsed or was whole totally. Therefore, this agreed with our second conclusion. That is, the robustness of the ADN was negatively correlated with the unintentional random failure rate Θ P . Once the value of Θ P was greater than the critical value Θ P_critical , the whole ADN collapsed totally.
The relationship between the final expected proportions R IF (∞) of remaining normal operational nodes in the CN and the unintentional random initial failure rate Θ P is similar.

Independence and Robustness
We then investigated the relationship between the interdependence and the ADN robustness under the ER/BA topological structure, which is shown in Figure 8, where the proportion of distributed generators was 0.2 and the value of the tolerance parameters of the nodes was 1.5. One-to-one interdependence represented <K 2 > = 1, and <O 1 > = 1; three-to-three Interdependence represented <K 2 > = 3, and <O 1 > = 3.
Appl. Sci. 2019, 9, 5021 19 of 31 to-one interdependence represented <K2> = 1, and <O1> = 1; three-to-three Interdependence represented <K2> = 3, and <O1> = 3. Of the nodes in the PN, 26.01% survived for ΘP = 0.022 with one-to-one interdependence under the ER topological structure and 62.49% of nodes in the PN survived for ΘP = 0.022 with three-tothree interdependence under the ER topological structure. Of the nodes in the CN, 26.01% survived for ΘP = 0.022 with one-to-one interdependence under the ER topological structure and 64.13% of nodes in the CN survived for ΘP = 0.022 with three-to-three interdependence under the ER topological structure. Accordingly, 21.02% of nodes in the PN survived for ΘP = 0.004 with one-to-one interdependence under the BA topological structure and 25.6% of nodes in the PN survived for ΘP = 0.004 with three-to-three interdependence under the BA topological structure. Of the nodes in the CN, 21.02% survived for ΘP = 0.004 with one-to-one interdependence under the BA topological structure and 25.6% of nodes in the CN survived for ΘP = 0.004 with three-to-three interdependence under the BA topological structure.
As shown in Figure 8, we observed that RIF(∞) and RPF(∞) of three-to-three interdependence were greater than that of one-to-one interdependence both in the ER topology and in the BA topology structure. This agreed with our third conclusion. That is, the robustness of the ADN was improved when the value of average control fan in <K2> of a node in the PN and the value of average power fan in <O1> of a node in the CN were greater.

Network Topology and Robustness
Generally, the robustness of the ADN against the CCF was affected by the different topological structure. In Figure 9, the left side shows the node degree distribution comparison diagram when the PN and CN adopted the ER network topological structure, while the right side shows the degree comparison diagram when the BA network topological structure was adopted. We observed from Figure 9 that the degree distribution of the ER network was more uniform than the BA network. The degree distribution of the BA network followed the power law. The average degree of the ER and BA network was 5 and 5 respectively for comparison. Of the nodes in the PN, 26.01% survived for Θ P = 0.022 with one-to-one interdependence under the ER topological structure and 62.49% of nodes in the PN survived for Θ P = 0.022 with three-to-three interdependence under the ER topological structure. Of the nodes in the CN, 26.01% survived for Θ P = 0.022 with one-to-one interdependence under the ER topological structure and 64.13% of nodes in the CN survived for Θ P = 0.022 with three-to-three interdependence under the ER topological structure. Accordingly, 21.02% of nodes in the PN survived for Θ P = 0.004 with one-to-one interdependence under the BA topological structure and 25.6% of nodes in the PN survived for Θ P = 0.004 with three-to-three interdependence under the BA topological structure. Of the nodes in the CN, 21.02% survived for Θ P = 0.004 with one-to-one interdependence under the BA topological structure and 25.6% of nodes in the CN survived for Θ P = 0.004 with three-to-three interdependence under the BA topological structure.
As shown in Figure 8, we observed that R IF (∞) and R PF (∞) of three-to-three interdependence were greater than that of one-to-one interdependence both in the ER topology and in the BA topology structure. This agreed with our third conclusion. That is, the robustness of the ADN was improved when the value of average control fan in <K 2 > of a node in the PN and the value of average power fan in <O 1 > of a node in the CN were greater.

Network Topology and Robustness
Generally, the robustness of the ADN against the CCF was affected by the different topological structure. In Figure 9, the left side shows the node degree distribution comparison diagram when the PN and CN adopted the ER network topological structure, while the right side shows the degree comparison diagram when the BA network topological structure was adopted. We observed from Figure 9 that the degree distribution of the ER network was more uniform than the BA network. The degree distribution of the BA network followed the power law. The average degree of the ER and BA network was 5 and 5 respectively for comparison. The relationship between the final expected proportions RPF(∞) of remaining normal operational nodes in the PN and network topology is shown in Figures 10 and 11. In Figure 10, the interval of the proportion of distributed generators was from 0.0 to 0.8. Of the nodes, 54.57% survived in the PN when the proportion of distributed generators was 0.6 with unintentional random initial failure rate ΘP = 0.002 under the ER topological structure and 16.87% of nodes survived in the PN when the proportion of distributed generators was 0.6 with unintentional random initial failure rate ΘP = 0.002 under the BA topological structure. Of the nodes, 54.57% survived in the CN when the proportion of distributed generators was 0.6 with unintentional random initial failure rate ΘP = 0.002 under the ER topological structure and 16.87% of nodes survived in the CN when the proportion of distributed generators was 0.6 with unintentional random initial failure rate ΘP = 0.002 under the BA topological structure. In Figure 11, the interval of unintentional random initial failure rate ΘP is from 0.002 to 0.008. Of the nodes, 27. As shown in Figures 10 and 11, the RPF(∞) of the ER network intersected with that of BA network due to the uncertainty of the CCF, and the RPF(∞) and RIF(∞) of the ER network were greater than that of BA network as a whole. Therefore, this agreed with our fourth conclusion. That is, the robustness of the ADN with ER network topological structure was greater than that with BA network topological structure under the same average node degree.

Node Degree of ER Network
Node Degree of BA Network Figure 9. Comparison of node degree distribution in the CN and PN.
The relationship between the final expected proportions R PF (∞) of remaining normal operational nodes in the PN and network topology is shown in Figures 10 and 11. In Figure 10, the interval of the proportion of distributed generators was from 0.0 to 0.8. Of the nodes, 54.57% survived in the PN when the proportion of distributed generators was 0.6 with unintentional random initial failure rate Θ P = 0.002 under the ER topological structure and 16.87% of nodes survived in the PN when the proportion of distributed generators was 0.6 with unintentional random initial failure rate Θ P = 0.002 under the BA topological structure. Of the nodes, 54.57% survived in the CN when the proportion of distributed generators was 0.6 with unintentional random initial failure rate Θ P = 0.002 under the ER topological structure and 16.87% of nodes survived in the CN when the proportion of distributed generators was 0.6 with unintentional random initial failure rate Θ P = 0.002 under the BA topological structure. In Figure 11, the interval of unintentional random initial failure rate Θ P is from 0.002 to 0.008. Of the nodes, 27.27% survived in the PN for Θ P = 0.004 when the proportion of distributed generators was 0.8 under the ER topological structure and 25.6% of nodes survived in the PN for Θ P = 0.004 when the proportion of distributed generators was 0.8 under the BA topological structure. Of the nodes, 27.27% survived in the CN for Θ P = 0.004 when the proportion of distributed generators was 0.8 under the ER topological structure and 25.6% of nodes survived in the CN for Θ P = 0.004 when the proportion of distributed generators was 0.8 under the BA topological structure.

Tolerance Parameters and Robustness
We investigated the relationship between the tolerance parameters and the ADN robustness when the proportion of distributed generators was 0.2.
The relationship between the final expected proportions RPF(∞) and RIF(∞) of remaining normal operational nodes and the tolerance parameters αP and αI is shown in Figure 12

Tolerance Parameters and Robustness
We investigated the relationship between the tolerance parameters and the ADN robustness when the proportion of distributed generators was 0.2.
The relationship between the final expected proportions RPF(∞) and RIF(∞) of remaining normal operational nodes and the tolerance parameters αP and αI is shown in Figure 12 As shown in Figures 10 and 11, the R PF (∞) of the ER network intersected with that of BA network due to the uncertainty of the CCF, and the R PF (∞) and R IF (∞) of the ER network were greater than that of BA network as a whole. Therefore, this agreed with our fourth conclusion. That is, the robustness of the ADN with ER network topological structure was greater than that with BA network topological structure under the same average node degree.

Tolerance Parameters and Robustness
We investigated the relationship between the tolerance parameters and the ADN robustness when the proportion of distributed generators was 0.2.
The relationship between the final expected proportions R PF (∞) and R IF (∞) of remaining normal operational nodes and the tolerance parameters α P and α I is shown in Figure 12. Of the nodes, 1.188% survived in the PN for Θ P = 0.01 with the tolerance parameters α P = α I = 1.5 under the ER topological structure. Of the nodes, 93.37% survived in the PN for Θ P = 0.01 with the tolerance parameters α P = α I = 2.0 under the ER topological structure. Of the nodes, 1.188% survived in CN for Θ P = 0.01 with the tolerance parameters α P = α I = 1.5 under the ER topological structure and 93.37% of nodes survived in the CN for Θ P = 0.01 with the tolerance parameters α P = α I = 2.0 under the ER topological structure. Accordingly, 0% of nodes survived in the PN for Θ P = 0.01 with the tolerance parameters α P = α I = 1.5 under the BA topological structure and 1.386% of nodes survived in the PN for Θ P = 0.01 with the tolerance parameters α P = α I = 2.0 under the BA topological structure. None of the nodes survived in the CN for Θ P = 0.01 with the tolerance parameters α P = α I = 1.5 under the BA topological structure and 1.386% of nodes survived in the CN for Θ P = 0.01 with the tolerance parameters α P = α I = 2.0 under the BA topological structure. As shown in Figure 12, RPF(∞) and RIF(∞) with the tolerance parameter 2.0 were greater than that of the tolerance parameter 1.5. Therefore, this agreed with our fifth conclusion. That is, the robustness of the ADN was positively correlated with the tolerance parameters. The robustness of the ADN was improved when the values of the nodes tolerance parameters αP in the PN and αI in the CN were greater. This also shows that the larger the tolerance parameters, the smaller the impact of load redistribution on the robustness of the ADN.

Related Work
Earlier studies mainly focused on the electrical power system. Research on the cascading failures of the electrical power system has a long history. Over the past two decades, the research on cascading failure propagation modes in the electrical power system has focused on two aspects: one is the single network cascading failure propagation modes [9,11,13,28] and the other is the interdependent cascading failure modes between a PN and a CN [16,22,23]. The theories for analyzing cascading failures of large blackouts include self-organization theory [28], percolation theory [15,17,21], power flow model [14,22,24], and so on.
Research on cascading failures in a single network, including random failure modes, with single or multiple nodes removal, the following problems are specifically solved: (1) The impacts exerted on cascading failures are analyzed when the power overloads lead to nodes or edges failures [11]. (2) Considering the intermittent and random nature of new energy generation, the impacts exerted on the cascading failures are analyzed in a single PN [9,11,13].
Research on the CCF of interdependent networks between a PN and a CN includes: (1) The characteristics of the CCF across the PN and CN are summarized: (a) The accidents caused by the CCF have the characteristics of power law distribution [6], and the probability distribution of largescale blackouts caused by the increasing load or unbalanced power flow changes from the exponential tail to the power law tail.  As shown in Figure 12, R PF (∞) and R IF (∞) with the tolerance parameter 2.0 were greater than that of the tolerance parameter 1.5. Therefore, this agreed with our fifth conclusion. That is, the robustness of the ADN was positively correlated with the tolerance parameters. The robustness of the ADN was improved when the values of the nodes tolerance parameters α P in the PN and α I in the CN were greater. This also shows that the larger the tolerance parameters, the smaller the impact of load redistribution on the robustness of the ADN.

Related Work
Earlier studies mainly focused on the electrical power system. Research on the cascading failures of the electrical power system has a long history. Over the past two decades, the research on cascading failure propagation modes in the electrical power system has focused on two aspects: one is the single network cascading failure propagation modes [9,11,13,28] and the other is the interdependent cascading failure modes between a PN and a CN [16,22,23]. The theories for analyzing cascading failures of large blackouts include self-organization theory [28], percolation theory [15,17,21], power flow model [14,22,24], and so on.
Research on cascading failures in a single network, including random failure modes, with single or multiple nodes removal, the following problems are specifically solved: (1) The impacts exerted on cascading failures are analyzed when the power overloads lead to nodes or edges failures [11].
(2) Considering the intermittent and random nature of new energy generation, the impacts exerted on the cascading failures are analyzed in a single PN [9,11,13].
Research on the CCF of interdependent networks between a PN and a CN includes: (1) The characteristics of the CCF across the PN and CN are summarized: (a) The accidents caused by the CCF have the characteristics of power law distribution [6], and the probability distribution of large-scale blackouts caused by the increasing load or unbalanced power flow changes from the exponential tail to the power law tail. (b) Total demand or pressure leads to the operation of the interdependent network under extreme conditions, which is one of the main factors leading to the CCF. (2) After analyzing the CCF propagation mechanisms of different kinds of networks, the conclusions are as follows: (a) Scale-free networks for a single network are robust to random attacks, but two interdependent scale-free networks are sensitive to random attacks [24]. (b) Even if the small disturbances in one network can lead to failures in another network, many clusters can be isolated from the whole network after the CCF stops [15,17,21,23]. (c) From the security point of view, an interaction model is established for the importance of the coupling relationship between a PN and a supervisory control and data acquisition (SCADA) system. (d) A model has been established to analyze the process of cascading failure propagation between the PN and the CN [8,19,21,25]. Its basic idea is to calculate the power flow redistribution after cascading failures occur by the dynamic power flow. (e) A topological model of a CN and a PN is established by the graph theory, and their transmission characteristics are analyzed. The ability of different topologies to resist the CCF under different coupling conditions is analyzed and compared [20,24].
However, the above current research work does not comprehensively consider the impact of the distribution of distributed generators, network topology, interdependence, and load redistribution on the robustness of the ADN against the CCF.

Conclusions and Future Work
In this paper, we analyzed the relationship between the final expected proportion of remaining normal operational nodes and distribution of generators, unintentional random initial failure, interdependence, network topology, and load redistribution. We gave a CCF simulation algorithm and a robustness evaluation algorithm. The robustness of the ADN against the CCF was simulated based on the ER network topological structure and the BA network topological structure. The model analysis and the simulation analysis showed that given the specific network topology parameters and the unintentional random initial failure rate, the more the average control fan in of a node in the PN and the more the average power fan in of a node in the CN, the better the robustness of the ADN, the distribution of generators improved the robustness of the ADN against the CCF. In addition, the robustness of the ADN was improved when the values of the nodes tolerance parameters were greater. Compared with the traditional power distribution network, because of the physical distribution of generators, the CCF caused by the interdependence between the PN and CN were more difficult to occur.
This work is helpful in understanding the CCF in the ADN. Based on our above analysis on numerical simulation results, if we want to improve the robustness of the ADN against the CCF, we should take the following measures: using the more uniform degree distribution of the ADN network topology, improving the values of <O 1 > and <K 2 >, improving the number of distributed generators, improving the tolerance parameters of the nodes in the ADN and protecting the nodes from the random failures. In practice, the PN and CN of the ADN are non-linear and the topological structure of the network is asymmetric due to natural and artificial factors. Considering the effect of factors of non-linear and asymmetry on the robustness of the ADN is one of our future directions. Accordingly, making preventive control measures for the CCF in the ADN is also one of our future directions.

Appendix B. Derivation of Mapping f P and f I
Firstly, it is assumed the degree of the node n Pi is d P , and the number of neighbor nodes set Neigh P (n Pi ) of a node n Pi in the PN of the ADN is D P1 = d P − O 2 . In order to calculate the conditional probabilities Prob P (Evt P3 /|Neigh P (n Pi )|), then the probability needed is shown in Equation (A1).
is the expected load redistributed to the node n Pi after the failure of a neighbor node n Pj connected to the node n Pi , and then there is Equation (A2).
n P ∈Neigh P (n Pi ) W P (n P ,n Pi ) ×Prob P (d P ) The probability of selecting d P nodes from the neighbors set Neigh P (n Pi ) of the node n Pi in the PN is shown in Equation (A3).
Therefore, under the condition of the number of neighbor nodes of a node n Pi is D P1 = d P − O 2 , the probability of an event Evt P3 occurrence is shown in Equation (A5).
After simplification, it is obtained in Equation (A6).
The following analysis is simplified by referring to the idea of percolation theory. A node n Pi in the PN does not belong to the connected clusters with distributed generators belonging to one of the following four situations: (1) This node n Pi has been removed due to random failures, including power fluctuations (intermittent or random) of distributed generators, etc. (2) This node n Pi exists but does not belong to the connected clusters containing generators. It belongs to a small connected component containing no generators. (3) This node n Pi is removed due to overload. (4) This node n Pi is removed due to the failure of the node in the CN associated with the node n Pi .
In Case (1), the probability is Θ P . In Case (2), the probability is shown in Equation (A7).
The probability in the third case is shown in Equation (A8).
In the fourth case, the probability is considered in Section 4.1. In summary, the probability that a node n Pi with the node failure probability Θ P belongs to the connected clusters containing generators is shown in Equation (A9).
After simplification, it is obtained in Equation (A10).
Thus, the expected probability that any node belongs to the connected clusters containing generators is shown in Equation (A11).
Therefore, considering the power fluctuations (intermittent or random) of distributed generators, it is defined f P (1 − Θ P ) as the expected probability of the connected clusters containing generators divided by 1 − Θ P , then it is as shown in Equation (A12).
The above formula is further equal to Equation (A13).
In addition, Prob P (Evt P2 /Evt P1 ) is the probability that there is a particular neighbor node n P of a node n Pi , and this node n P is not connected to the connected clusters containing generators in subsequent neighbor nodes except this node n Pi . Thus, the conclusion is drawn as shown in Equation (A14).
Secondly, the number of neighbor nodes of a node n Ii in the CN of the ADN is D I1 = d I − O 1 . In order to calculate the conditional probability Prob I (Evt I3 /|Neigh I (n Ii )|), we need to know the conditional probability Prob I (Evt I3 /D I2 = d I ) which is equal to Equation (A15).  The probability of selecting a node from the neighbor nodes set Neigh I (n Ii ) of the node n Ii in the CN is shown in Equation (A17). where Therefore, under the condition that the number of neighbor nodes set is D I1 = d I − O 1 , the probability of an event Evt I3 occurring is shown in Equation (A19).
After further simplifications, the results are obtained as shown in Equation (A20).
Furthermore, it is assumed that NON I × Θ I nodes fail in the CN. A node n Ii does not belong to the connected clusters depending on the connected clusters containing generators in the PN due to the following four situations: (1) This node n Ii has been removed due to random failures, etc.
(2) This node n Ii exists but does not belong to the connected clusters depending on the connected clusters containing generators in the PN. (3) This node n Ii has been removed due to the excessive data traffic flow. (4) This node n Ii fails due to the failure of its depending node in the PN.
The probability in Case (1) is Θ I . In Case (2), the probability is shown in Equation (A21).
In Case (3), the probability is shown in Equation (A22).
In summary, the probability that a node n Ii with a degree d I belongs to the connected clusters depending on the connected clusters containing generators in the PN under the node failure probability Θ I is shown in Equation (A23).
After further simplifications, the results are obtained as shown in Equation (A24).
Thus, the expected probability that any node in the CN belongs to the connected clusters depending on the connected clusters containing generators in the PN is shown in Equation (A25).
In addition, Prob I (Evt I2 /Evt I1 ) is the probability that there is a particular neighbor node n I of a node n Ii , and this node n I is not connected to the connected clusters depending on the connected clusters containing generators in the PN in subsequent neighbor nodes except this node n Ii . Thus, the conclusion is drawn as shown in Equation (A28).

Appendix C. Analysis of Cross-Domain Cascading Failures
The first iteration N=1 of the CCF is shown in Equation (A29).
The number of nodes removed in the PN will further lead to failures of the corresponding nodes in the CN according to the power supply dependence relationship, and its number is shown in Equation (A30).
The expected proportion of the failure of a power supply edge between the PN and CN is shown in Equation (A31).
Thus, the expected proportion of the node failure in the CN is shown in Equation (A32).
When the number of nodes in the CN and the number of nodes in the PN are both large, the above equations hold.
The nodes in the CN with the expected proportion of 1 − R IF (2) fail. Among these failed nodes in the CN, the number of nodes belonging to the control nodes set V I3 is shown in Equation (A34).
where Prob I (V I3 ) = |V I3 |/NON I . The number of failure control edges which are owned by these failure nodes in the CN is shown in Equation (A35).
The total number of control edges is shown in Equation (A36).
Thus, the expected proportion of failure control edges is shown in Equation (A37).
Then the expected failure proportion of a node in the PN due to failure control edges is shown in Equation (A38). [ Then the corresponding expected proportion of a normal operational node in the PN due to failure control edges is shown in Equation (A39).
According to the assumption in literature [26], then the third iteration N=3 of the CCF is shown in Equation (A40).
Appendix D. A Special Case of Evaluation <O 1 > and <O 2 > For a node n Pi belonging to the substation node set V P2 in the PN of the ADN, it is assumed that the number of nodes in the CN that the node n Pi can supply power is proportional to the degree of this node n Pi [23]. The specific relationship is shown in Equation (A41).
where µ P is constant. The expected number of nodes in the CN is supplied by a node n Pi in the PN is shown in Equation (A42).
Prob P (d P )×µ P d P = µ P d P , where Prob P (d P ) represents the probability of a node n Pi with a degree d P belonging to the substation node set V P2 . <d P > represents the average node degree in the substation node set V P2 .
The probability that only one node n Ij in the CN will receive power from a node n Pi with a degree d P belonging to the substation node set V P2 in the PN is shown in Equation (A43).
Then, the probability that o nodes in the CN will receive power from a node n Pi with a degree d P belonging to the substation node set V P2 in the PN is shown in Equation (A44).
where o is less than or equal to d P . Furthermore, the probability that an arbitrary node n Pi belonging to the substation node set V P2 in the PN supplies power to o nodes in the CN can be obtained as shown in Equation (A45).
Finally, the expected number of nodes in the CN is power supplied by any one node in the substation node set V P2 in the PN is shown in Equation (A46).
The expected number of nodes belonging to the substation node set V P2 in the PN which supplies power to any one node in the CN is derived in a similar way. The probability that any one communication node is powered by o nodes belonging to the substation node set V P2 in the PN is shown in Equation (A47). (A47) Thus, the expected number of nodes belonging to the substation node set V P2 in the PN which supplies power to any one node in the CN is shown in Equation (A48).

Appendix E. A Proof of Proposition 1
Proof: According to Equation (A6), if the tolerance parameter α P of the node in the power network is bigger, then the probability Prob P (Evt P3 /D P1 = d P − O 2 ) is smaller. Correspondingly, according to Equations (A9) and (A14), it can be seen that the probability Prob P (Evt P2 /Evt P1 ) is smaller as well. Then according to Equation (A13), f P (1 − Θ P ) becomes bigger as the tolerance α P of the node in the power network is bigger. As the tolerance α P →∞, Prob P (Evt P3 /D P2 = d P )→0, Prob P (Evt P3 /D P1 = d P − O 2 )→0, and then the Equation (A49) can be obtained.
According to Equation (A49) and Equation (6), if R P (2N + 1) 0, R PF (2N + 1) 0, R I (2N) 0, and R IF (2N) 0, then R P (2N + 1), R PF (2N + 1), R I (2N), and R IF (2N) will be bigger. Therefore the expected proportions R PF (2N + 1) of the remaining normal operational nodes reach their upper limits, which are the results of a pure interdependence model without considering the load redistribution. The similar proof process applies to the expected proportions R IF (2N) of the remaining normal operational nodes in the communication network.