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

Abstract

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 effective to improve the robustness of the ADN against the CCF.

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-time electricity [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 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.

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.

2. 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. Notation used in this paper.

Symbol	Description
GA	Active distribution networks (ADNs).
GP	The power networks of the ADN.
GI	The communication networks of the ADN.
VP	The nodes set of the PN.
VI	The nodes set of the CN.
NoNP	The number of the nodes in the PN.
NoNI	The number of the nodes in the CN.
Vf-initial	The unintentional random initial failure nodes set of the PN.
∞	The steady-state of the ADN after the CCF.
GP1(∞), GP2(∞),… ∈ SP(Vf-initial, ∞)	The connected clusters in the PN of the ADN after the CCF stops.
GPg1(∞), GPg2(∞),… ∈ SPg(Vf-initial, ∞)	The connected clusters containing generators in the PN of the ADN after the CCF stops.
GI1(∞), GI2(∞),… ∈ SI(Vf-initial, ∞)	The connected clusters in the CN of the ADN after the CCF stops.
GIg1(∞), GIg2(∞),… ∈ SIg(Vf-initial, ∞)	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.
RP(N)/RI(N)	The expected proportion of remaining nodes in the PN/CN at step N after the CCF occurs.
fP(RP(N))	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.
fI(RI(N))	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.
RPF(N)/RIF(N)	The expected proportion of normal operational nodes in the PN/CN at step N after the CCF occurs.
RPF(∞)/RIF(∞)	The final expected proportion of normal operational nodes in the PN/CN after the CCF stops.
ΘP	The probability that a single node in the PN fails randomly.
EdgP(n)/EdgI(n)	The edge set (excluding the edges connecting the PN and the CN) of a node n in the PN/CN.
<K2>	The average control fan in of a node in the PN.
<O1>	The average power fan in of a node in the CN.
BCtrl	The set of the paths involved in an effective control EconI.
TPL(e)/TIL(e)	Load threshold of an edge e in the PN/CN.
Run(n)/¬Run(n)	A node or an edge n runs normally/ abnormally.
αP/αI	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.

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 S_Pg(V_f-initial, ∞) in the PN is equal to {G_Ig1(∞), G_Ig2(∞),…}⊆S_I(V_f-initial, ∞).

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).

$$\{\begin{array}{l}{R}_{PF}(\infty )\approx \frac{1}{Num}{\displaystyle \sum _{Num}\frac{{\displaystyle \sum _{s\in S_{Pg}(V_{f\_initial},\infty )}\left|s\right|}}{NO{N}_P}}\\ R_{IF}(\infty )\approx \frac{1}{Num}{\displaystyle \sum _{Num}\frac{{\displaystyle \sum _{s\in S_{Ig}(V_{f\_initial},\infty )}\left|s\right|}}{NO{N}_I}}\end{array}.$$

(1)

3. 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) = (n₁, …, n_k). The path has k nodes n₁, …, n_k. Where the source node n_I1 = n₁ ∈ 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 V_P = V_P1 ∪ V_P2 ∪ V_P3 ∪ V_P4 ∪ V_P5.

(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).

$$Loa{d}_P(n_{Pj},N+1)=\{\begin{array}{l}Loa{d}_P(n_{Pj},N)+\frac{Loa{d}_P(n_{Pi},N)\times W_P(n_{Pj},n_{Pi})}{{\displaystyle \sum _{n_P\in Neig{h}_P(n_{Pi})}{W}_P(n_P,n_{Pi})}},n_{Pi}\in Fai{l}_P\\ Loa{d}_P(n_{Pj},N),n_{Pi}\notin Fai{l}_P\end{array}$$

(2)

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.

Definition 3.

The CN of the ADN is also a sextuple system G_I = (V_I, E_I, Load_I, W_I, Trans_I, Thres_I), where

(1) 

V_I represents the set of nodes in the CN, and V_I = V_I1 ∪ V_I2 ∪ V_I3 ∪ V_I4.

(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).

$$Loa{d}_I(n_{Ij},N+1)=\{\begin{array}{l}Loa{d}_I(n_{Ij},N)+\frac{Loa{d}_I(n_{Ii},N)\times W_I(n_{Ij},n_{Ii})}{{\displaystyle \sum _{n_I\in Neig{h}_I(n_{Ii})}{W}_I(n_I,n_{Ii})}},n_{Ii}\begin{array}{c}\in Fai{l}_I\end{array}\\ Loa{d}_I(n_{Ij},N),n_{Ii}\begin{array}{c}\notin Fai{l}_I\end{array}\end{array}$$

(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 K₁ nodes of the PN, and the average control fan out of the node n_K1 is <K₁> = (1/|V_I3|) × ΣK₁. A node n_K2 in the PN is controlled by K₂ control nodes in the CN, and the average control fan in of the node n_K2 is <K₂> = |V_I3| × <K₁>/NON_P. A sensor node n_L1 ∈ V_I1 in the CN can collect L₁ nodes in the PN, and the average information collecting fan in of the node n_L1 is <L₁> = (1/|V_I1|) × ΣL₁. Conversely, a node n_L2 in the PN is perceived by L₂ sensor nodes in the CN and the average information collecting fan out of the node n_L2 is <L₂> = |V_I1| × <L₁>/NON_P. Correspondingly, the average power fan out of a node in the PN is <O₂> = (1/|V_P2|) × ΣO₂, and a node n_O1 in the CN can be power supplied by O₁ nodes of the PN, then the average power fan in of the node n_O1 is <O₁> = |V_P2| × <O₂>/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, <K₁> = 10/4 = 2.5, <K₂> = 1, <L₁> = 10/3, <L₂> = 1, <O₁> = 1, and <O₂> = 15/3 = 5. (In Appendix D, a special case about how to find <O₁> and <O₂> is shown)

According to the aforementioned Definitions 2 and 3, the definition of the ADN is given below.

Definition 4.

ADN is a combination of the PN and the CN, so it is also a sextuple system G_A = (V_A, E_A, Load_A, W_A, Trans_A, Thres_A), where

(1) 

V_A represents the set of nodes in the ADN, and V_A = V_P ∪ V_I.

(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~formed 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. Normal operation conditions.

Domain	Type	Normal Operation Conditions
PN	Node nP	(1) It is connected to at least one power generation node nPP in the PN, i.e., $$\left(\exists n_{PP}\in (V_{P1}\cup V_{P5})\right).\begin{array}{c}\end{array}path(n_P,n_{PP})$$ (2) It has at least one edge between the node nP in the PN and a node nII in the CN, and the node nII is part of at least one path in the set Btr. i.e., $$\exists n_{II}.\begin{array}{c}\end{array}\left((n_P,n_{II})\in E_{PI}\right)\wedge \left(path(n_{II},n_{I4})\in B_{tr}\right)$$ (3) The load flowing through this node nP does not exceed its threshold TPP(nP) at step N. i.e., $$(load(n_P,N)\le T_{PP}(n_P))\wedge (n_P\in V_P)$$
	Edge eP	(1) The load flowing through the edge eP does not exceed its threshold TPL(eP) at step N. i.e., $$(load(e_P,N)\le T_{PL}(e_P))\wedge (e_P\in E_P)$$ (2) The nodes on both sides of the edge eP are in normal operation. i.e., $$\left(e_P=(n_{Px}\in V_P,n_{Py}\in V_P)\right)\to (run(n_{Px})\wedge run(n_{Py}))$$
CN	Node nI	(1) There is at least one normal operating node np in the PN that provides power to this node nI. i.e., $$\exists n_P.\begin{array}{c}\end{array}\left(\left(e=(n_P\in V_P,n_I\in V_I)\right)\to run(n_P)\right)$$ (2) The data traffic of this node nI in the CN does not exceed its threshold at step N. i.e., $$(load(n_I,N),n_I\in V_I)\le T_{IP}(n_I)$$ (3) The node nI is part of at least one path in the set Btr. i.e., $$\exists n_{I4}.\begin{array}{c}\end{array}path(n_I,n_{I4})\in B_{tr}$$
	Edge eP	(1) The load (data packets) flowing through the edge eP does not exceed its threshold TIL(eP) at step N. i.e., $$(load(e_P,N)\le T_{IL}(e_P))\wedge (e_P\in E_I)$$ (2) The nodes on both sides of edge eP are in normal operation. i.e., $$\left(e_P=(n_{Ix}\in V_I,n_{Iy}\in V_I)\right)\to (run(n_{Ix})\wedge run(n_{Iy}))$$
The interdependence between the PN and the CN	Edge e	(1) The nodes on both sides of the edge e ∈ (EPI ∪ EIP) are in normal operation. i.e., $$((e=(n_{PPx}\in V_P,n_{IIx}\in V_I))\vee (e=(n_{IIx}\in V_I,n_{PPx}\in V_P)))\to (run(n_{PPx})\wedge run(n_{IIx}))$$

.

4. 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.

4.1. 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 R_PF(N) = R_P(N) × f_P(R_P(N)).

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.

4.1.1. 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), R_IF(2)…, R_PF(∞), R_IF(∞).

$$\{\begin{array}{l}{R}_P(2N+1)=(1-{\theta }_P)\times \left[1-{\left(1-R_{IF}(2N)\right)}^{\langle K_2\rangle }\right]\\ R_{PF}(2N+1)=R_P(2N+1)\times f_P(R_P(2N+1))\end{array},N=0,1,2,\dots$$

(4a)

$$\{\begin{array}{l}{R}_I(2N)=1-{\left[1-R_{PF}(2N-1)\right]}^{\langle O_1\rangle }\\ R_{IF}(2N)=R_I(2N)\times f_I(R_I(2N))\end{array},N=1,2,\dots$$

(4b)

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).

$$\{\begin{array}{l}\gamma =1-{\left[1-\lambda \times f_P(\lambda )\right]}^{\langle O_1\rangle }\\ \lambda =(1-{\theta }_P)\times \left[1-{\left(1-\gamma \ast f_I(\gamma )\right)}^{\langle K_2\rangle }\right]\end{array}$$

(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).

$$\{\begin{array}{l}{R}_{PF}(\infty )=\underset{N\to \infty }{\lim }{R}_{PF}(N)=\lambda \times f_P(\lambda )\\ R_{IF}(\infty )=\underset{N\to \infty }{\lim }{R}_{IF}(N)=\gamma \times f_I(\gamma )\end{array}$$

(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.

4.1.2. Unintentional Random Initial Failure Rate and Robustness of the ADN

From the Equation (5), Equation (7) can be further obtained.

$$\gamma =1-{\left[1-(1-{\theta }_P)\times \left[1-{\left(1-\gamma \times f_I(\gamma )\right)}^{\langle K_2\rangle }\right]\times f_P((1-{\theta }_P)\times \left[1-{\left(1-\gamma \times f_I(\gamma )\right)}^{\langle K_2\rangle }\right])\right]}^{\langle O_1\rangle }$$

(7)

The solution of Equation (7) can be obtained by the graphic method of two curves as follows in Equation (8).

$$\{\begin{array}{l}\lambda =\gamma \\ \lambda =1-{\left[1-(1-{\theta }_P)\times \left[1-{\left(1-\gamma \times f_I(\gamma )\right)}^{\langle K_2\rangle }\right]\times f_P((1-{\theta }_P)\times \left[1-{\left(1-\gamma \times f_I(\gamma )\right)}^{\langle K_2\rangle }\right])\right]}^{\langle O_1\rangle }\end{array}$$

(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).

$$\{\begin{array}{l}1={\left[1-\lambda \times f_P(\lambda )\right]}^{\langle O_1\rangle -1}\times \left[\frac{\partial \lambda }{\partial \gamma }\times \left(f_P(\lambda )+\lambda \times \frac{\partial f_P(\lambda )}{\partial \gamma }\right)\right]\\ \gamma =1-{\left[1-(1-{\theta }_{P\_critical})\times \left[1-{\left(1-\gamma \times f_I(\gamma )\right)}^{\langle K_2\rangle }\right]\times f_P((1-{\theta }_{P\_critical})\times \left[1-{\left(1-\gamma \times f_I(\gamma )\right)}^{\langle K_2\rangle }\right])\right]}^{\langle O_1\rangle }\end{array}$$

(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.

4.1.3. 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 <K₂> of a node in the PN and the value of average power fan in <O₁> 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 <O₁> 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 <K₂> 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 <K₂> of a node in the PN and the value of average power fan in <O₁> 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 <K₂> of a node in the PN and the value of average power fan in <O₁> of a node in the CN are greater. □

4.1.4. 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.

4.2. 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].

$$\langle loa{d}_P\rangle ={\displaystyle \sum _{d_P=1}^{\infty }{F}_P(d_P)\times Pro{b}_P(d_P)}+g$$

(10)

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.

The expected load of the node n_Pi in the PN is shown in Equation (11).

$$\langle loa{d}_P\rangle ={\displaystyle \sum _{d_P=1}^{\infty }{F}_P(d_P)\times Pro{b}_P(d_P)}+g$$

(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).

$$\langle Thre{s}_P\rangle =\langle {\alpha }_P\rangle \times {\displaystyle \sum _{d_P=1}^{\infty }{F}_P(d_P)\times Pro{b}_P(d_P)}$$

(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).

$$Loa{d}_I(n_{Ii},0)=F_I(d_I)$$

(13)

The expected load of the node n_Ii in the CN is shown in Equation (14).

$$\langle loa{d}_I\rangle ={\displaystyle \sum _{d_I=1}^{\infty }{F}_I(d_I)\times Pro{b}_I(d_I)}$$

(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).

$$\langle Thre{s}_I\rangle =\langle {\alpha }_I\rangle \times {\displaystyle \sum _{d_I=1}^{\infty }{F}_I(d_I)\times Pro{b}_I(d_P)}$$

(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 NON_P × (1 − (1 − Θ_P) × f_P(1 − Θ_P)) × <load_P>. Correspondingly, the total redistributed load (data packets) in the CN is NON_I × (1 − (1 − Θ_I) × f_I(1 − Θ_I)) × <load_I>.

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 (1 − Θ_P) × f_P(1 − Θ_P) × <Thres_P> ≥ <load_P>. Accordingly, the condition in the CN is (1 − Θ_I) × f_I(1 − Θ_I) × <Thres_I> ≥ <load_I>.

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).

$$\langle {\alpha }_P\rangle =\frac{{\displaystyle \sum _{d_P=1}^{\infty }{F}_P(d_P)\times Pro{b}_P(d_P)}+g}{{\theta }_P\times f_P({\theta }_P)\times {\displaystyle \sum _{d_P=1}^{\infty }{F}_P(d_P)\times Pro{b}_P(d_P)}}=\frac{1+\left(\frac{g}{{\displaystyle \sum _{d_P=1}^{\infty }{F}_P(d_P)\times Pro{b}_P(d_P)}}\right)}{{\theta }_P\times f_P({\theta }_P)}$$

(16)

$$\langle {\alpha }_I\rangle =\frac{{\displaystyle \sum _{d_I=1}^{\infty }{F}_I(d_P)\times Pro{b}_I(d_I)}}{{\theta }_I\times f_I({\theta }_I)\times {\displaystyle \sum _{d_I=1}^{\infty }{F}_I(d_I)\times Pro{b}_I(d_I)}}=\frac{1}{{\theta }_I\times f_I({\theta }_I)}$$

(17)

Furthermore, the expected critical tolerance parameters <α_P-Critical> and <α_I-Critical> can be obtained in Equations (18) and (19).

$$\langle {\alpha }_{P-Critical}\rangle =\frac{1+\left(\frac{g}{{\displaystyle \sum _{d_P=1}^{\infty }{F}_P(d_P)\times Pro{b}_P(d_P)}}\right)}{\lambda \times f_P(\lambda )}$$

(18)

$$\langle {\alpha }_{I-Critical}\rangle =\frac{1}{\gamma \times f_I(\gamma )}$$

(19)

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.

4.3. 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.

Algorithm 1 Stead-state Subgraph Generating Algorithm (taking nodes failures of the PN as an example)

Input: ADN GA = (VA, EA, LoadA, WA, TransA, ThresA). Initial failure set Vf-initial⊆VP.

// Initialization

1  t ← 0, FLP(t) ← Vf-initial, FLI(t) ← Ø, VP(t) ← VP, VI(t) ← VI;

2  whileadditional failures are possibledo

3   t ← t+1;   // Load redistribution for GP and GI 4   for n ∈ (VP ∪ VI) do 5    Load redistribution according to the nearest neighbor rule;

// Intra-network failures for GP

6   foruP ∈ VP(t) do 7    for qP ∈ VP1(t) do 8      if ((loadP(uP, t) >TPP(uP)) ∨ ( run(qP) ∧path(uP, qP) = Ø) ) then 9       FLP (t) ← FLP(t) ∪ {uP};

// Inter-network failures for GP

10   forvP ∈ VP(t) do 11     for uI ∈ VI(t) do 12      if ((vP, uI) ∈ EPI)) ∧ (¬run(uI) ) then 13       FLP(t) ← FLP(t) ∪ {vP};

14   VP(t) ← VP(t−1) − FLP(t);

// Intra-network failures for GI

15    for vI ∈ VI(t) do 16     for nI4 ∈ VI4do 17      if (loadI(vI, t) >TIP(vI)) ∨ (path(vI,nI4) ∉Btr) then 18       FLI(t) ← FLI(t) ∪ {vI};

// Inter-network failures for GI

19   forvI ∈ VI(t) do 20     for uP ∈ VP(t) do 21      if ((vI, uP) ∈ EIP)) ∧ (¬run(uP) ) then 22       FLI(t) ← FLI(t) ∪ {vI};

23    VI(t) ← VI(t−1) − FLI(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.

// Initialization

1  RPF(∞) ← 0, RIF(∞) ← 0, NONP ← |VP|, NONI ← |VI|; 2  Count the number of connected clusters in Sub-graphs of the ADN, generate sets SP(Vf-initial, ∞) and SI(Vf-initial, ∞); 3  Decide generators in sets SP(Vf-initial, ∞) and SI(Vf-initial, ∞), generate sets SPg(Vf-initial, ∞) and SIg(Vf-initial, ∞);

// Evaluate the proportion of normal operational nodes in the ADN 4  for sP ∈ SPg(Vf-initial, ∞) do 5   RPF(∞) ← RPF(∞) + | sP |; 6  for sI ∈ SIg(Vf-initial, ∞) do 7   RIF(∞) ← RIF(∞) + | sI |; 8  RPF(∞) ← RPF(∞)/NONP; 9  RIF(∞) ← RIF(∞)/NONI;

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.

Algorithm 3 Robustness Evaluation Algorithm

Input: Number of experiments Num.   ADN GA = (VA, EA, LoadA, WA, TransA, ThresA).    Initial failure set Vf-initial⊆ VP.

// Initialization

1  RPF(∞) ← 0, RIF(∞) ← 0, count ← Num, NONP ← |VP|, NONI ← |VI|;   // Evaluate the robustness of the ADN

2  whilecount ≠ 0 do

3   count--; 4   Subgraphs ← Run the Stead-state Subgraph Generating Algorithm; 5   RPF(∞), RIF(∞) ← Run the Evaluation Algorithm; 6   RPF(∞) ← RPF(∞)/(Num×NONP); 7   RIF(∞) ← RIF(∞)/(Num×NONI);

Output: RPF(∞) and RIF(∞).

5. 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.

5.1. 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.

5.2. 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.

5.2.1. 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 Figure 5 and Figure 6. In Figure 5 and Figure 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 Figure 5a and Figure 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.

5.2.2. 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.

Figure 7 clearly shows the relationship between the final expected proportions R_PF(∞)/R_IF(∞) of the remaining normal operational nodes in the PN/CN and the unintentional random initial failure rate Θ_P. Of the nodes in the PN, 55.17% survived for Θ_P = 0.002 when the proportion of distributed generators was 0.8 under the ER topological structure. Of nodes in the PN, 27.27% survived for Θ_P = 0.004 when the proportion of distributed generators was 0.8 under the ER topological structure. Accordingly, 55.17% of nodes in the CN survived for Θ_P = 0.002 when the proportion of distributed generators was 0.8 under the ER topological structure. Of nodes in the CN, 27.27% survived for Θ_P = 0.004 when the proportion of distributed generators was 0.8 under the ER topological structure.

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.

5.2.3. 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 <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 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 <K₂> of a node in the PN and the value of average power fan in <O₁> of a node in the CN were greater.

5.2.4. 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 R_PF(∞) of remaining normal operational nodes in the PN and network topology is shown in Figure 10 and Figure 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.

As shown in Figure 10 and Figure 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.

5.2.5. 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, 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.

6. 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.

7. 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 <O₁> and <K₂>, 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.