1. Introduction
In recent years, Industry 4.0 has emerged as a fundamental concept in the new wave of the industrial revolution. This revolutionary force has been instrumental in driving the global manufacturing industry toward digital transformation and intelligent advancement [
1,
2]. Inspired by the ideals of Industry 4.0, the smart grid has gradually embodied comprehensive perception, intelligent decision making, and autonomous control, ultimately transforming into the cyber–physical power system (CPPS) [
3]. CPPS is characterized by its digitalized, networked and intelligent nature. By seamlessly integrating state-of-the-art information and communication technology, automation control technology, and data analysis technology, the power system is equipped to achieve superior optimization and flexible operation.
However, while the fusion of power systems with communication technology has brought a wealth of benefits, it has also introduced potential threats to power systems [
4,
5,
6]. For example, changes in the network topology of the information system have the propensity to cause delays or even obstructions in the transmission of information, thus disrupting real-time monitoring of the power system [
7]. Moreover, the security vulnerabilities inherent in these information systems, such as malicious hacker infiltration and the insidious spread of computer viruses, are apt to jeopardize the safe and stable operation of these power systems. In addition, disruptions and anomalous conditions within the power system can also have a detrimental impact on the information systems [
8]. Unlike other intricate networks, the cyber–physical power system embodies a higher degree of intricacy, which reduces its resilience in the face of abrupt power fluctuations or on-site failures, further accentuating the vulnerabilities within the network.
The failure of a specific component within the CPPS can lead to cascading failures and, in severe cases, to the total collapse of the system. In 2010, Buldyrev [
9] introduced the concept of interdependent networks in power systems, emphasizing that a failure in one network can cause failures in nodes of other networks that depend on it, which is referred to as cascading failures. Subsequently, many researchers built on this concept and used complex network theory to study the mechanisms of cascading failures in various systems. They proposed cascading failure models such as load capacity models [
10,
11], epidemic models [
12], OPA failure models (ORNL-PSERC-Alaska, OPA), sandpile models [
13], and power flow models [
14] to explain the causes behind cascading failure phenomena. Regarding the common load capacity models, various researchers have proposed different load redistribution strategies. Wang [
10], based on the betweenness centrality of each node, defined initial load and overload functions for each node and then proposed an evaluation method for the importance of network nodes based on these measures. Wang [
15] proposed the strategy of redistributing the load among the nearest neighbors. As shown in the approach, the load of each failed node is distributed to its neighbors. On the other hand, Nguyen [
16] ranked the importance of nodes and lines based on the DC power flow in the power system to reduce the damage caused by cascading failure attacks. Cai [
17] proposed a dependency network model between the power grid and the scheduling data network based on dynamic flow. In recent years, numerous scholars and researchers have extended the concept of cascading failure in single-layer networks to the realms of dual-layer and even multi-layer networks. Artime [
18] and Zhou [
19], respectively, delve into the impact of non-local cascading failures and network inter-similarity on the robustness of multi-layered multiplex networks. Meanwhile, Artime [
20] explores the characteristics of networks under varying circumstances, ranging from the perspective of multi-layered structures and dynamics. These modeling methods are effective, but the purpose of studying a system is to gain better control over it. As the scale of CPPS gradually expands, it is imperative for us to study the controllability during the cascading failure process.
The integration of power systems and information systems has improved the controllability and observability of power systems [
21]. However, it has also made the control of cyber–physical power systems more challenging. For a given initial time
and final time
, if there exists a set of control signals
that allow the network to transition from an initial state
to any desired state
, then the system is said to be fully controllable. In this paper, the term “node failure” indicates that a node has ceased to function, thereby interrupting normal operations following the failure of the node. Liu [
22] first proposed the theory of structural controllability, establishing a research framework for the controllability of complex networks. They also showed that driver nodes tend to avoid high-degree nodes, addressing the issue of controllability in directed networks. However, it revealed limitations when dealing with undirected networks, weighted networks, and certain large-scale networks. Therefore, Yuan [
23] introduced the concept of exact controllability in 2013, solving the controllability determination problem for networks with arbitrary topologies, undirected networks, and networks with weighted edges.Wang [
24] extended the concept of structural controllability by considering the controllability of multi-input multi-output systems. They found that in certain cases, even if a system satisfies structural controllability, it may still be uncontrollable. The above problems are based on single-layer networks, but in reality, many networks are multi-layer. Jiang [
25] investigated the controllability of multi-layer networks with high-dimensional node states and analyzed the structural controllability of interdependent networks with known directed subnetworks. In addition, Miao [
26] investigated the controllability problem of matrix-weighted discrete-time leader–follower multi-agent systems (MASs).
It is worth noting that most of the previous work focused on network topologies with different types of links, which have different internal coupling patterns. In addition, the impact of different redistribution strategies on the controllability of CPPS may vary.
Taking these factors into consideration, a load-based redistribution strategy is proposed, and the controllability of CPPS is investigated under different coupling strategies and attack scenarios. This paper focuses primarily on the controllability of interdependent power systems in the face of cascading failure models, as well as the influence of network topology, coupling patterns, and attack scenarios on said controllability. The main contributions of this research are outlined below:
By combining the network topology and functional characteristics of the system, a load-capacity model is developed to address the research gap in cascading failure of CPPS. The validity of the results is verified through the modeling of realistic networks, enhancing the persuasiveness of our findings.
By considering the real-time node loads and the distribution of power flow and information flow in CPPS, a novel load redistribution strategy is introduced. Compared to other strategies, this strategy can quickly terminate failures and prevent system collapse.
This paper comprehensively analyzes different information network topologies and network parameters, and it investigates the controllability of the system after cascading failures under different coupling strategies and capacity parameters. Guidelines for future smart grid planning are provided.
The remaining sections of this paper are organized as follows.
Section 2 outlines the methodology used in this study.
Section 3 focuses on the fault propagation model of CPPS under cascading failures.
Section 4 presents the simulation analysis and related discussions. Finally,
Section 5 provides a summary of the research conducted in this paper.
4. Case Study and Discussion
The load redistribution model proposed in this paper was validated, and its controllability changes during the process were analyzed using the IEEE standard 39-node system and the Chinese 132-node system. This paper assumes the assurance of complete synchronization between the power system and the information system, and unless otherwise specified, the system capacity parameters and are 0.5, respectively. The horizontal axis in the graph of this section, times of attacks, represents the system’s change in terms of controllability after enduring x instances of node attacks. The simulations in this paper were performed using MATLAB 2020b on a personal computer equipped with an Intel Core i5 2.4 GHz CPU and 16 GB RAM.
In general, the network coupling methods of CPPS can be categorized into a “one-to-one” coupling strategy and “one-to-many” coupling strategy. This paper adopts the “one-to-one” coupling strategy to analyze the cascading failure in CPPS. Meanwhile, the coupling methods between nodes in two networks can be divided into the following four methods:
DDM: High-degree nodes in the power network are connected to high-degree nodes in the information network.
BBM: High-betweenness nodes in the power network are connected to high betweenness nodes in the information network.
DBM: High-degree nodes in the power network are connected to high betweenness nodes in the information network.
BDM: High-betweenness nodes in the power network are connected to high-degree nodes in the information network.
4.1. Initial Network Topology
The power-side electrical network topology established on the basis of the IEEE 39-node system and the Chinese 132-node system is shown in
Figure 5a and
Figure 5b, respectively. The IEEE 39-node system consists of a total of 10 generators, 39 busbars, and 12 transformers. In contrast, the Chinese 132-node system represents a more streamlined provincial network consisting of 25 generators, 101 loads, and 180 transmission lines. The degree distribution for both systems is shown in
Figure 6, and the statistical characteristics of the networks are summarized in
Table 1.
The degree distributions in these two networks show notable differences. In the IEEE 39-node system, a significant proportion of nodes are connected to three or two other nodes, resulting in an intricate and highly clustered pattern. Conversely, in the 132-node system in China, the majority of nodes are connected to only one or two other nodes, resulting in a sparse and straightforward network structure. Referring to
Table 1, there is a notable difference in driver node density and network clustering coefficient between the two networks. A higher minimum driver node density implies a higher number of required driver nodes. The network clustering coefficient [
33] measures the degree to which neighboring nodes of a node are connected and is defined as the ratio of the actual number of connections between a node’s neighbors to the maximum possible number of connections. A larger clustering coefficient indicates more connections between nodes and a denser network. Indeed, these two networks exhibit different characteristics, and using both the IEEE standard 39-node system and the Chinese 132-node system for simulation helps improve the universality of the models.
To account for the uncertain topology of the information network, the simulation will be randomly run for 10,000 iterations. This approach aims to minimize the random errors stemming from the inherent randomness of the network.
4.2. The Controllability of CPPS in Different Network Types
The underlying topology of the information network is unknown, so all cascading failure processes in this paper are based on the failure of power nodes. In typical scenarios, the use of networks of equal scale, such as scale-free (BA) networks or small-world (WS) networks, is considered. Therefore, the impact of different attack strategies on the controllability of the system within these two network models is investigated.
4.2.1. Case A: IEEE 39-Node System
Under the three attack strategies, the information network is constructed into a BA network topology of the same scale, which is coupled with the IEEE 39-node system to form the CPPS. The controllability after cascading failure is shown in
Figure 7, where (a), (b), and (c) represent the controllability changes of the network under random attack, degree attack, and betweenness attack, respectively. As shown in the figure, after the network is subjected to random attacks, the density of driver nodes gradually increases, while the controllability of the network gradually decreases. After the eighth attack, the network is on the verge of collapse under all four coupling modes. In contrast, after four degree attacks and three betweenness attacks, all four networks suddenly collapse. Prior to this, even though the network was under attack, the network still retained some isolated islands due to the redistribution of loads to neighboring nodes. However, the loads of certain nodes had reached the threshold, and after another attack, the network immediately collapsed completely.
After we constructed the information network into a WS network of the same scale and coupled it with the IEEE39 node system to form the CPPS, then three attack strategies of random, degree and betweenness are used to attack the IEEE 39-node system. The controllability of the network after the node load redistribution process is shown in
Figure 8, and like the CPPS coupled with the information network built by the BA network, the CPPS drive node density under random attacks gradually increases and the network controllability gradually decreases. After the number of attacks reaches the 8th time, the network almost collapses under the four coupling methods. When the network was subjected to multiple attacks, the CPPS did not collapse most of the network after the third attack as before. Instead, it still survived most of the nodes and suddenly collapsed after the fourth attack. It is worth noting that after the third betweenness attack, the driver node density of the CPPS has been close to 1, which means that most of the CPPS was decomposed into islands at this time, and it can no longer meet the power demand or communicate with other nodes.
4.2.2. Case B: Chinese 132-Node System
Similarly, the Chinese 132-node system is taken as an example, and the BA network and WS network of the same size are coupled with the Chinese 132-node system to form a CPPS. The controllability of the network with different coupling strategies under three different attack strategies is shown in
Figure 9 and
Figure 10, respectively.
Figure 9 shows the controllability changes of a CPPS constructed with a BA network as its information network topology under different attack strategies. (a), (b), and (c) represent the controllability variation curves of the system under random, degree, and betweenness attack strategies, respectively. When the system is subjected to random attacks, the density of driver nodes gradually increases, while the controllability of the system decreases. The system is already in a state of collapse when the number of attacks reaches about 20. Meanwhile, under degree attacks, the controllability of the CPPS constructed with
DDM and
DBM strategies differs significantly from that of the CPPS constructed with
BBM and
BDM strategies when the number of attacks reaches 4. In comparison, the changes in the controllability of the CPPS under
BBM and
BDM strategies are relatively small when faced with multiple-node attacks, allowing for greater opportunities for system adaptation. Finally, under betweenness attacks, the system suddenly collapses when subjected to the fourth attack, and before that, the CPPS under all four coupling strategies was in a partially dysfunctional state.
Similarly,
Figure 10 shows the changes in the controllability of the CPPS using the WS network as the information network when it is subjected to three attack strategies.
It is noteworthy that in any situation, deliberate attacks on networks do not significantly reduce the controllability of the network compared to random attacks. Deliberate attacks often target “hub” nodes that are connected to many other nodes. As a result, when a hub node fails, its neighboring nodes take over the load, preventing any further propagation of failures. In addition, the BDM coupling approach better withstands node failures or attacks, as the controllability degradation of the CPPS formed by BDM coupling is the slowest compared to the other three methods in all cases.
After comparing the two methods of constructing information networks, it was found that under the same circumstances, the CPPS composed of a WS network can exhibit stronger controllability when subjected to node attacks or failures. This finding has important implications for future designs of information systems and holds certain guiding significance. Small-world networks possess short average paths and high aggregation, which are highly advantageous in a CPPS, as they enable global connectivity and information transfer while maintaining local communication efficiency. Therefore, in the subsequent research, the information network will be constructed as a small-world network of the same size as the power network. Additionally, the power network and the information network will be coupled in a one-to-one manner based on BDM, thereby enhancing its adaptability to node attacks or failures.
4.3. CPPS Controllability under Different Redistribution Strategies
Network attacks can be divided into static and dynamic attacks. Static attacks refer to the attack sequence that has been determined when the system is established, while dynamic attacks dynamically adjust the attack sequence based on the real-time status and load of the system. Different from static attacks [
34], this paper adopts dynamic attack strategies to make the attacks more targeted, and the network that survives this situation has higher stability. This section compares the load redistribution strategy introduced in this paper with the load redistribution strategies in other studies in the literature [
35,
36], and it analyzes the impact of different load redistribution strategies on the network controllability under different attack strategies.
Strategy I [
35]: Distributing the failed load evenly among the neighboring nodes of the faulty node, based on the average number of neighboring nodes for each fault node.
where
indicates the number of neighbor nodes of node
i, and
is the set of neighbor nodes of node
i.
Strategy II [
36]: Distributing the failed load evenly among the neighboring nodes of the faulty node based on the average number of neighboring nodes for each fault node.
where
indicates the degree of node
n.
For degree attacks and betweenness attacks, the impact of various load redistribution strategies on system controllability was investigated in both scenarios.
4.3.1. Case A: IEEE 39-Node System
Using the IEEE 39-node system as an example, a comparison was made between the reallocation strategy proposed in this paper and strategy I and strategy II. The simulation results are shown in
Figure 11. Under a degree attack on the network, the driver node density of strategies I and II increased significantly after the first attack, reaching about 0.8. However, before the network of the redistribution strategy in this paper was attacked for the third time, the driver node density of the network increased significantly. The density is kept below 0.3. At this point, the controllability of the network is much greater than that of the network under other redistribution strategies. It is worth noting that under betweenness attacks, after two betweenness attacks, the redistribution strategy of this paper still maintains the driver node density of the network below 0.1, which means that most nodes in the network are still in a normal state, while the other two redistribution strategies drive the node density to increase to about 0.8 after one betweenness attack, indicating that most nodes in the network are in an isolated state at this time.
4.3.2. Case B: Chinese 132-Node System
In the simulation of the Chinese 132-node system, under the degree attack, the simulation results are shown in
Figure 12, which is similar to Case A. However, due to the increase in network scale, the driver node density of the network of the other two strategies is reduced after being attacked by two nodes. It reaches 0.8, while the redistribution strategy in this paper drives the node density to remain below 0.4 before being attacked by eight nodes, and the growth rate is much smaller than the other two strategies. Under betweenness attacks, the driver node density of the network of the three redistribution strategies is similar to that of case A, but the number of node attacks required to collapse the network is one more than that of case A, which is the reason why the network scale is larger than that.
4.4. Effect of Various Parameters on the Controllability of CPPS Cascading Failures
Under high-degree attacks, an analysis was conducted to assess the influence of various capacity parameters on the controllability of the network cascading failure process. The simulation results are shown in
Figure 13.
Figure 13a,b show the two networks under different capacity parameters, respectively, indicating the controllable performance of the cascading failure process. In
Figure 13a,b, when
and
are both 1, the driver node density of the network increases the slowest. As
and
decrease, the driver node density of the network increases under the degree attack. The slope of the density curve also increases gradually, meaning that for the same number of node attacks, the driver node density increases as
and
decrease. It is worth noting that when
and
decrease by 0.2 each, the increment of the network’s driver node density is not the same, which means that the node capacity of the power network has a greater impact on the network’s controllability than the node capacity of the information network.