# Analysis of Controllability in Cyber-Physical Power Systems under a Novel Load-Capacity Model
###### Abstract
Meanwhile, [PERSON] [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 \(t_{0}\) and final time \(t_{f}\), if there exists a set of control signals \(u(t)\) that allow the network to transition from an initial state \(x(t_{0})\) to any desired state \(x(t_{f})\), 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. [PERSON] [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, [PERSON] [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. [PERSON] [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, [PERSON] [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:  
1. 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. 2. 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. 3. 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.