Consensus of cyber-physical power systems based on multi-agent systems with communication constraints

J. Mod. Power Syst. Clean Energy https://doi.org/10.1007/s40565-018-0491-4 Consensus of cyber-physical power systems based on multi-agent systems with communication constraints Chi HUANG1,2 , Chengli FENG3, Jinde CAO2 Abstract The consensus protocol of cyber-physical power systems is proposed based on fractional-order multi-agent systems with communication constraints. It aims to enable each generator to reach a time-varying common rotor angle and rotor speed. Communication constraints including event-triggered sampling and partial information transmission are considered to render the consensus protocol more realistic. The Zeno behavior is excluded during the system sampling process. A sufficient condition is derived to solve the consensus problem. The effectiveness of the proposed consensus protocol is demonstrated by a numerical example. Keywords Cyber-physical power system, Fractional-order system, Multi-agent system, Communication constraint CrossCheck date: 17 August 2018 Received: 21 December 2017 / Accepted: 17 August 2018 Ó The Author(s) 2019 & Chi HUANG Chengli FENG Jinde CAO 1 School of Economic Information Engineering, Southwestern University of Finance and Economics, Chengdu 611130, China 2 School of Mathematics, Southeast University, Nanjing 210096, China 3 College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China 1 Introduction The concept of cyber-physical power systems (CPPSs) is first proposed in [1] as a dedicated case of a cyberphysical system in a power system. It is composed of a large number of computing devices (servers, computers, and embedded computing devices), data acquisition devices (sensors, phasor measurement unit, and embedded data acquisition equipment), and physical devices (large-scale generator set, distributed power supply, and load). These devices are connected through communication and transmission networks. CPPSs combine the advantages of cyber-physical systems and power systems. However, it also experiences many technical and security challenges [2, 3]. Owing to the ability to collect and handle massive information compared with traditional power systems, studies on CPPSs such as traffic networks [4] and aviation electric power systems [5] have been performed. Among these, the stability of CPPSs has been emphasized, because an unstable power supply may lead to voltage collapse and breakdown of the entire power system [6, 7]. Most studies on stability in the existing literature are focused on the supply-demand balance and rotor angle stability. Specifically, the target of the supply-demand balance is to continuously match the production and consumption of electricity across networks [8]. The rotor angle stability [9] enables the CPPS to operate smoothly and efficiently by allowing each generator to reach a time-varying common rotor angle and speed, separately. Both of them can be regarded as a consensus perspective. Thus, many studies use the consensus theory to solve stability problems in CPPSs [10, 11]. A multi-agent system (MAS) model has been extensively studied as an effective method for the consensus problem [12, 13]. It is a cooperative system whereby agents 123 Chi HUANG et al. can share their information to communicate with each other. The MAS has many significant real-world applications, such as sensor networks [14], unmanned air vehicles formations [15, 16], and microgrids [17]. Furthermore, many studies have applied the MAS to a CPPS [18–21]. In [18], a distributed multi-agent-based load shedding algorithm that can yield an efficient load shedding decision based on the discovered global information was proposed. In [20], an innovative approach was proposed to use realtime scheduling techniques for the automation of electric loads in CPPSs. In [21], a distributed control method to enhance power system rotor angle stability based on the second-order MAS consensus was proposed, where each generator can be regarded as an agent to communicate with each other. Nevertheless, most of the studies on CPPSs are based on integer-order MASs. In fact, real-world processes are generally (or most likely) fractional-order systems [22, 23]. Many physical systems exhibit fractional dynamical behavior owing to their special materials and chemical properties [24–27]. Thus, it is highly desirable to study CPPSs with fractional-order MASs. Communication constraints that are crucial factors to the system performance cannot be ignored in practice. Among them, the partial information transmission is significant. That is to say, not all of the information can be transmitted to their neighbors perfectly [28–30]. Energy saving is another important topic. Considering the massive communicating and computing work in CPPSs, sampling would be an efficient method to reduce resources. Traditional periodic sampling techniques would typically waste much energy and consequently shorten the lifespan of the system to a certain degree [31]. To address this problem, event-triggered control has attracted significant attention from research into the consensus of MASs [32–37]. The Zeno behavior must be excluded in the event sampling process; otherwise, continuous communication will be required again [38]. However, among the existing literatures, the proofs absent of the Zeno behavior are not presented (or are naturally avoided), because events can only occur at sampling time instants that are a multiple of some given sampling period [34, 39]. Therefore, it is necessary to propose a more specific proof when the Zeno behavior is excluded. Our current study is of practical significance because it can effectively reduce the gap between the physical and cyber worlds. However, few results have been found in the existing literature. Motivated by the discussions above, we herein consider the consensus of CPPSs based on fractional-order MASs with communication constraints. Our objective is to ensure that each generator achieves a timevarying common rotor angle and rotor speed. The contribution of this paper can be summarized in three aspects: 1) a fractional-order MAS modeling with communication 123 constraints is established for the consensus of the CPPS; 2) based on communication constraints, a distributed event condition is introduced for energy conservation, and consensus criteria are obtained that can be applied to more practical systems; 3) a finite number of broadcasts and updates exist by each agent in any finite time period, thus ensuring that the Zeno behavior does not occur in the CPPS. The effectiveness of the proposed consensus criteria is illustrated with a numerical example. 2 Model description and preliminaries In this section, some basic notions and properties of the algebraic graph theory are introduced. Each agent can be abstracted as a node. Let G ¼ ðV; E; GÞ be a weighted directed network of order N, with the set of nodes (vertices) V ¼ f1; 2; . . .; Ng; the set of directed edges E  V  V; and a weighted adja (...truncated)