Cascading Failures Analysis Considering Extreme Virus Propagation of Cyber-Physical Systems in Smart Grids

Research Article Cascading Failures Analysis Considering Extreme Virus Propagation of Cyber-Physical Systems in Smart Grids Tao Wang ,1,2 Xiaoguang Wei ,3Tao Huang ,1,4 Jun Wang,1,2 Luis Valencia-Cabrera,5 Zhennan Fan ,1,2 and Mario J. Pérez-Jiménez5 1School of Electrical Engineering and Electronic Information, Xihua University, China 2Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University, Chengdu 610039, China 3School of Electrical Engineering, Southwest Jiaotong University, China 4Department of Energy, Politecnico di Torino, Italy 5Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Seville, Spain Correspondence should be addressed to Xiaoguang Wei; wei [email protected] and Tao Huang; [email protected] Received 3 November 2018; Revised 14 January 2019; Accepted 18 February 2019; Published 13 March 2019 Guest Editor: Riccardo Patriarca Copyright © 2019 Tao Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Communication networks as smart infrastructure systems play an important role in smart girds to monitor, control, and manage the operation of electrical networks. However, due to the interdependencies between communication networks and electrical networks, once communication networks fail (or are attacked), the faults can be easily propagated to electrical networks which even lead to cascading blackout; therefore it is crucial to investigate the impacts of failures of communication networks on the operation of electrical networks. This paper focuses on cascading failures in interdependent systems from the perspective of cyberphysical security. In the interdependent fault propagation model, the complex network-based virus propagation model is used to describe virus infection in the scale-free and small-world topologically structured communication networks. Meanwhile, in the electrical network, dynamic power flow is employed to reproduce the behaviors of the electrical networks after a fault. In addition, two time windows, i.e., the virus infection cycle and the tripping time of overloaded branches, are considered to analyze the fault characteristics of both electrical branches and communication nodes along time under virus propagation. The proposed model is applied to the IEEE 118-bus system and the French grid coupled with different communication network structures. The results show that the scale-free communication network is more vulnerable to virus propagation in smart cyber-physical grids. 1. Introduction The smart grid, as a modern electrical network (EN) infrastructure, can enhance the efficiency, reliability, and security of traditional ENs based on the advancement of cyber-physical systems [1–3]. In a smart grid, the monitoring, control, and management of the EN depend closely on the smart information and communication (cyber) network [4–6], which works such that the EN ensures not only its own secure operation but also reliable operation of the entire communication network. Meanwhile, when the EN fails (especially, through cascading failure), fault cross-propagation between the electrical and communication networks (ECNs), called interdependent network, occurs, which increases the complexity of fault propagation owing to interactions between the ECNs. For example, the Italian blackout of 2003 was triggered by effects of the ECN [7]. Therefore, exploring the propagation mechanism of interactive cascading failures [8] in an interdependent network has been receiving increasing attention. To date, the connection between the different coupling modes between ECNs and the robustness/vulnerability of interdependent networks has been investigated widely [9–12]. Studies have demonstrated that the different types of links between ECNs greatly impact the robustness of the network. For example, [10] reveals the double-network link allocation Hindawi Complexity Volume 2019, Article ID 7428458, 15 pages https://doi.org/10.1155/2019/7428458 2Complexity strategy is superior to single-network link allocation strategy. Therefore, reasonably allocating the interconnecting links betweenECNsisvitalforimprovingtherobustnessof interdependent networks. Accordingly, a few models (e.g., Petri nets) have been introduced to reveal the mechanism of interactions leading to catastrophic blackouts [13]. However, these works have been done merely from the perspective of the structure of the coupling of the ECNs. Meanwhile, the physical and operational characteristics considering the interactions have been focused on as well. In [14], the impact of communication network vulnerability on power system operation was assessed considering both latency and communication interruptions. The data exchange model is introduced for modeling cascading failures in interdependent networks [15]. Reference [16] proposes a simulation platform to analyze the ECN vulnerability by considering the control strategy of power balance. Similarly, other simulations [17, 18] have been proposed to analyze the fault mechanism considering interactions between ECNs. Although these studies have included the interactions between the communication network and EN, their focus is only to study how to set up the simulation platform by considering both the communication network and the EN. Moreover, the operational characteristics are only studied from a steady state point of view. In addition, as communication networks become increasingly smart and as smart grids are increasingly accessed using the Internet owing to advancement of the energy Internet [19, 20], cyber threats (e.g., virus propagation; hacking attacks) leading to interactive cascading failures should be focused on [21,22].Forinstance,the2015blackoutintheUkrainewas a typical coordinated cyber-attack in which malicious code was employed to tamper with data and control the server of the monitoring system [23]. In the face of potential threats, measures to enhance information security of communication networks [24] and a few robust and efficient cyber infrastructures [25] have been proposed. For example, in Qinghai, China, to prevent viral infection of networks, an antivirus system was installed to manage the power dispatching data network. This system successfully detected and neutralized 3384 viruses in 2010-2012 [26]. It is manifest that there is a need to consider the virus propagation in the communication network although it is a low-probability event, as such an event can cause immense harm to the ECN owing to the subtletyofthevirusandthehighspeedatwhichthenetwork is infected by using advanced attack methods. With this background, in our paper, we propose a framework to analyze the performance of the ECN by considering the interactions between two types of propagations: fault propagation in the EN and virus propagation in the communication network. IntheEN,cascadingfailureshavebeenanalyzedfrom theperspectiveoftheoverloadedmechanism[15,27–29]. When a line fails, the power transmitted over the line will be redistributed in the network, and thus, a fault may cause increased flow in other branches and even overload them, leading to the fault propagation. However, those analyses are generally performed by through a steady state fashion. In our study, we improved the dynamic power flow method to redistribute loads and adjust unbalanced power in the network during fault propagation by introducing the primary frequency regulation and the equations of rotors of generators. Notably, we only consider high-voltage transmission networks as the study objects. For the virus propagation, virus spread models [30, 31] with time delay have been developed based on the complex network theory (CNT) from the perspective of the topological structure of the communication network. Generally, the communication network mainly has two topical topological structures: scale-free and small-world networks. This paper focuses on investigating the impacts of two types of networks on electrical networks during virus propagation. In other words, we analyze which type of communication network is more vulnerable from the network-wise perspective, i.e., once viruses are propagated in a communication network, which type of communication network can cause more damage to the electrical network. Meanwhile, we further analyzethatinacommunicationnetwork,thevulnerability of communication nodes with different degree is revealed by investigating the number of fault branches and blackout level of coupled electrical networks. Inaddition,itshouldbenotedthatthemostmodern malware, surely most malware is used in known attacks to power grids and industrial controls, limits its own effectiveness by prematurely destroying/disabling nodes and is not self-replicating. Furthermore, currently in a real-world communication network vertexes will not be homogeneous; thus they will not support most of self-propagating malicious code. However, in order to consider a low-probability but high consequence scenario, in this paper, we assume a random constant spread malicious code (called “virus” in this paper) with the following features to investigate the impact of extreme case of self-propagating virus on the power system from the network-wise perspective: (1) the virus can block the communication between infected vertexes and the control center; (2) the virus can self-propagate among homogeneous vertexes; (3) a few infectious vertexes can be cured with the probability owing to the strengthening of security measures; (4) the differences of security level of each node are not considered. The remainder of this paper is organized as follows. Section 2 describes the interactions between the ECNs in the coupling relationships and topological structures. In Section 3, the virus propagation models with time delay and information exchange model in the communication network are introduced. The dynamic power flow method and the overload mechanism are established in Section 4. The cascading failures model considering the interactions and the corresponding simulation analysis are described in Sections 5and6,respectively.InSection7,wefurtherdiscussthe contribution of this paper and the external validity of the modeling. Finally, conclusions are given with possible future work in Section 8. Complexity 3 Communication vertexControl center Substation Optical fiber Transmission branch Protection Monitoring Power supply Breaker Inductor Other GatewayWorkstation Process layer Bay layer Control layer Communication network Electrical network Figure 1: Diagram of ECN in smart grids. 2. Interaction between Electrical and Communication Networks in Smart Grid 2.1. Interdependent ECN in Smart Grids ECN Spatial Model. The ECN in a smart grid is shown in Figure 1. The communication network, a hierarchical structure, is composed of optical fibers and synchronous digital hierarchies (SDHs), including control centers and communication vertexes [14, 32]. Generally, the substations (generators and loads) in the electrical network have the corresponding communication vertexes. The coupling between the substations and the communication vertexes is modeled by a smart communication module comprising three layers: process,bay,andcontrollayers.Amongthethreelayers, the bay layer is mainly responsible for accepting commands from the control layer to protect and monitor the electric network and realizing real-time interaction of information between the control and the process layers. The control layer is responsible for sending real-time messages from the electrical network to the control center as well as for accepting commands from the control center through the communication network. ECN Operation Model.WeanalyzetheECNoperationmodel from the perspective of energy flow. In an ECN, there are two types of energy flows: power flow and communication flow, asshowninFigure2.InanEN,powerflowchangeswithtime via buses or lines. Conversely, because the communication vertexes transmit and receive messages at regular intervals, thecommunicationflowistransmittedbasedondiscrete time. Communication Topology between Vertexes and Branches. Because there is a consistent one-to-one match between each communication vertex and each bus node, each transmission branch Bjhas two related communication vertexes Vm1 and Vm2during normal operation. In this paper, only Vm1 4Complexity Power flowCommunication flow Figure 2: Information flows of electrical and communication networks in smart grids. is considered for exchanging information packets with the control center [15]. 2.2. Topological Structures of ECN Electrical and Communication Network as Graphs.Forsimplifying analysis of the topological structures, we abstracted the ECN as graphs. The EN can be considered as a complex network with nodes and links. The buses, including generators, loads, and substations, can be viewed as nodes while transmission lines can be viewed as branches; therefore, the electrical network is represented as the graph G𝐸=(N,B). The adjacent matrix 𝐺𝐸=(𝑎 𝑖𝑟)𝑀𝑁×𝑀𝑁is employed to define G𝐸as follows: 𝑎𝑖𝑟 ={ { { 1∃𝑁 𝑖𝑁𝑟=𝐵 𝑗 0¬∃𝑁 𝑖𝑁𝑟=𝐵 𝑗 (1) where 𝑁𝑖𝑁𝑟=𝐵 𝑗represents that there is a branch 𝐵𝑗between nodes 𝑁𝑖and 𝑁𝑟. Similarly, the optical fibers and SDHs of the communication network can be considered as edges and vertexes, respectively; therefore, the communication network is represented as the graph G𝐶=(V,E). The adjacent matrix 𝐺𝐶= (𝑎󸀠 𝑚V)𝑀𝑉×𝑀𝑉is employed to define G𝐶as follows: 𝑎󸀠 𝑚V={ { { 1∃𝑉 𝑚𝑉V=𝐸 𝑎 0¬∃𝑉 𝑚𝑉V=𝐸 𝑎 (2) where 𝑉𝑚𝑉V=𝐸 𝑎represents that there is an edge 𝐸𝑎between vertices 𝑉𝑚and 𝑉V. Because the buses are coupled with the corresponding communication vertexes by the communication module, which is represented as L={𝐿 𝑏|𝐿 𝑏=𝑁 𝑖𝑉𝑚,𝑖 = 𝑚}, the ECN can be developed as an interdependent graph G= G𝐸∪G𝐶=(N∪V,B∪E∪L). Topological Structure Analysis.Weanalyzethestructuralcharacteristics from the perspective of the CNT. Existing literature indicate that ENs have small-world networks [33–35], which demonstrates that ENs have a relatively small average shortest path but a very large cluster coefficient. Thus, the smallworld electrical networks reveal that if a node (or branch) in the network fails, the adjacent and even nonadjacent nodes (or branches) could fail, leading to cascading failures. Meanwhile, ENs has scale-free characteristics, as determined by analyzing changes in the network structure and function when one or more nodes (or branches) are removed from thenetwork,whichshowsthenetworksarehighlyvulnerable under deliberate attacks but robust under random attacks [28].However,faultpropagationmechanismofENsstudied from the perspective of pure topological structure is not comprehensive and should more focus on the physical and operational features. In communication networks, generally, there are two types of topological networks: scale-free networks and smallworld networks [15, 36, 37]. Communication networks with scale-free structures contain a few nodes with high degree, and they can be considered center nodes. Compared to the small-world networks, the distributions of degree of which aremoreuniform,scale-freenetworkshavehighercom- munication efficiency but are more vulnerable to deliberate attacks. From the perspective of pure topological structures, compared to ENs, fault (or virus) propagation in communication networks is largely determined by its topological structure. That is, a fault node (or branch) only causes neighboring nodes to fail. Therefore, we employ the CNT to develop the virus propagation models (VPMs) based on the topological structures. In summary, the EN and communication network in smart grids have two essential differences in terms of the interactions of cascading failures. Features 1. From the perspective of time scales, the power flowistransmittedbasedoncontinuoustime,whilethe communication flow is transmitted based on discrete time. Features 2. From the perspective of topological structures, fault propagation in the communication networks depends more on the network structures. Compared with the communication networks, fault propagation in ENs depends more on physical and operational modes because ENs comply with operational rules, for example, Kirchhoff’s law. 3. Virus Propagation and Information Exchange Models in Communication Networks Before analyzing VPMs, we introduce the following topological concepts: The degree 𝑘𝑉𝑚 of 𝑉𝑚isthenumberofneighboring vertexes connected to 𝑉𝑚, as expressed by 𝑘𝑉𝑚 =𝑀𝑉 ∑ V=1𝑎󸀠 𝑚V(3) Complexity 5 Degree distribution p(k) is the distribution function of the degrees. That is, when a vertex is randomly selected from the network, the probability that its degree is equal to k is p(k). 3.1. Virus Propagation Model Based on CNT. According to Feature 1, virus propagation in the communication network depends on the network structure; therefore, we employ the SI [38–40] and SIR [41, 42] models based on CNT to simulate virus propagation. In the SI model, the vertexes of the communication network are divided into two groups: susceptible set Sand infectious set I.InS,theprobabilitythat a susceptible vertex contracts the virus from the infectious vertexes is 𝛽. Meanwhile, because the virus spends some time in destroying the functions of susceptible vertexes (e.g., tampering with data or instructions), the susceptible vertexes take time to get infected. Therefore, we introduce time delay (virus infection cycle) to develop the SI model as follows: 𝑑𝑆(𝑡) 𝑑𝑡 =−𝛽𝐼(𝑡−𝜏 1)𝑆(𝑡−𝜏1) 𝑑𝐼(𝑡) 𝑑𝑡 =𝛽𝐼(𝑡−𝜏 1)𝑆(𝑡−𝜏1) (4) On the basis of the SI model, the SIR model considers that a few infectious vertexes can be cured with the probability owing to the strengthening of related antivirus measures (e.g., formatting operation). Thus, the infectious vertexes can obtain immunity in a certain virus removal cycle. Therefore, the vertexes add a group called removed set R. The SIR model with time delay is given as 𝑑𝑆(𝑡) 𝑑𝑡 =−𝛽𝐼(𝑡−𝜏 1)𝑆(𝑡−𝜏1) 𝑑𝐼(𝑡) 𝑑𝑡 =𝛽𝐼(𝑡−𝜏 1)𝑆(𝑡−𝜏1)−𝛼𝐼(𝑡−𝜏 2) 𝑑𝑅(𝑡) 𝑑𝑡 =𝛼𝐼(𝑡−𝜏 2) (5) Becauseaninfectiousvertexonlytransmitsthevirus to its neighboring vertexes, the topological structure of the network greatly influences virus propagation. When the communication network is a small-world network, which can be regarded as a uniform network owing to the relatively uniform distribution of degree [37], the degree 𝑘𝑉𝑚 of 𝑉𝑚 is approximately equal to ⟨𝑘⟩, and the SIR model can be presented as follows: 𝑑𝑆(𝑡) 𝑑𝑡 =−𝛽⟨𝑘⟩𝐼(𝑡−𝜏1)𝑆(𝑡−𝜏1) 𝑑𝐼(𝑡) 𝑑𝑡 =𝛽⟨𝑘⟩𝐼(𝑡−𝜏1)𝑆(𝑡−𝜏1)−𝛼𝐼(𝑡−𝜏 2) 𝑑𝑅(𝑡) 𝑑𝑡 =𝛼𝐼(𝑡−𝜏 2) (6) 𝑑𝑆(𝑡) 𝑑𝑡 =−𝛽𝑘𝑆 𝑘(𝑡−𝜏1)∑𝑘𝑘𝑝(𝑘)𝐼𝑘(𝑡−𝜏1) ⟨𝑘⟩ 𝑑𝐼(𝑡) 𝑑𝑡 =𝛽𝑘𝑆 𝑘(𝑡−𝜏1)∑𝑘𝑘𝑝(𝑘)𝐼𝑘(𝑡−𝜏1) ⟨𝑘⟩ −𝛼𝐼(𝑡−𝜏 2) 𝑑𝑅(𝑡) 𝑑𝑡 =𝛼𝐼(𝑡−𝜏 2) (7) When the communication network is a scale-free network, the vertexes have different damage levels from the perspective of virus propagation because the distribution of degree follows a power law. That is, the greater the 𝑘𝑉𝑚 of 𝑉𝑚, the more serious it is for the 𝑉𝑚to spread or contract the virus to more vertexes. The SIR model can be expressed as (7) in terms of the vertex degree [43, 44]. Generally, in the SI and SIR models, virus propagation is faster in scale-free networks owing to the power law distribution. In addition, by comparing SI and SIR models, once virus propagation occurs in the communication network, we can investigate whether the related antivirus measures with time delay can play an important role to prevent the fault from spreading across the EN. 3.2. Information Exchange Model in Communication Network. In the smart grid, the communication vertexes send operational data (parameters) associated with branches to thecontrolcenterandreceivecommandsfromthecontrol center step-by-step through the communication network in the form of information packets. At every step, the same information packets can be received and sent only by each communication vertex. Before constructing the information exchange model, three simplifications are made as follows. (1) The communication blocks of vertexes (or edges) are not considered in process of the information transfer when the vertexes work orderly. That is, the capacity of the vertexes is adequate to exchange/handle the information packets. (2) Because the vertexes send and receive information packets at intervals of 0.833 ms [45], the time required for information exchange between the vertexes and control center can be ignored because it is very small compared to the time required for fault propagation in the EN. (3) Because we focus on the interactions between the ECNs, the methods of gathering and dealing with the information (such as the measuring units, transmission channels and protocols, encryption and decryption algorithms, etc.) are not considered. Based on the above simplifications, the information exchange model is constructed based on the structure of the communication network. Communication Rules between Target Vertex and Control Center. The vertexes abide by the rule of first-in-first-out to send out information packets to avoid exchange of the information packets to be in the same edge. At first, the target vertex produces information packets. Then, the information packets are sent to all its neighbor vertexes. If the control center is one of the neighbor vertexes, the information 6Complexity transfer ends; otherwise, all neighbor vertexes, acting as target vertexes, continue to send the information packets to their corresponding neighbor vertexes until the information packets are sent to the control center. In the above process, the information packets are transmitted successfully between the target vertex and control center if a path exists between them in the communication network. 4. Dynamic Power Flow and Overload Mechanism Models in Electrical Networks 4.1. Dynamic Power Flow in Electrical Networks. To redistribute power flow during disturbances, we employ primary frequency regulation [46, 47] and rotor equation [48] to model the dynamic power flow method. System Frequency Characteristics.Weemployprimaryfrequency regulation to adjust the power flow. The characteristics of load and generation frequency are given by (8) and (9), respectively. Δ𝑃𝑋=𝐾 𝑋Δ𝑓 (8) Δ𝑃𝑊=𝐾 𝑊Δ𝑓 (9) 𝐾𝑋and 𝐾𝑊are calculated as follows: 𝐾𝑋=∑𝑁𝑋 𝑞=1 (𝐾𝑅𝑞 ⋅𝑃 𝑅𝑟𝑞) ∑𝑁𝑋 𝑞=1 𝑃𝑅𝑟𝑞 (10) 𝐾𝑊=∑𝑁𝑊 𝑐=1 (𝐾𝑊𝑐 ⋅𝑃 𝑊𝑟𝑐) ∑𝑁𝑊 𝑐=1 𝑃𝑊𝑟𝑐 (11) Unbalanced Power Redistribution.Toredistributetheunbalanced power Pun due to disturbances of the system, we first calculate the change in system frequency by using the primary frequency regulation and the rotor equation. 𝑇𝐽𝑑Δ𝜔 𝑑𝑡 =𝑃 𝑢𝑛 −𝐾 𝐸⋅Δ𝜔(𝑡)(12) In (12), Pun is calculated as follows: 𝑃𝑢𝑛 =𝑀𝑊 ∑ 𝑐=1 𝑃𝑊𝑐 −𝑀𝑋 ∑ 𝑐=1 𝑃𝑋𝑞 (13) When Pun <0, we consider the characteristics of load and the generation frequency to adjust the system frequency: 𝐾𝐸=𝐾 𝑋+𝐾 𝑊(14) When Pun >0, we consider only the generation frequency characteristic: 𝐾𝐸=𝐾 𝑊(15) By using (8)-(15), the changes in every generator and load canbeexpressedasfollows: Δ𝑃𝑋𝑞 =𝐾 𝑋𝑞 ⋅Δ𝜔(𝑡)(16) Δ𝑃𝑊𝑐 =𝐾 𝑊𝑐 ⋅Δ𝜔(𝑡)(17) Then, we employ the P-Q power flow to calculate the power flows of each bus (𝑞=𝑐=𝑖) as follows: 𝑃𝑖=−Δ𝑃 𝑊𝑞 +Δ𝑃 𝑋𝑐 +]𝑖 𝑀𝑁 ∑ 𝑢=1 ]𝑢(𝐺𝑖𝑢 cos 𝜃𝑖𝑢 +𝐵𝑖𝑢 sin 𝜃𝑖𝑢) =(−𝐾 𝑊𝑞 +𝐾 𝑋𝑐)Δ𝜔 +]𝑖 𝑀𝑁 ∑ 𝑢=1 ]𝑢(𝐺𝑖𝑢 cos 𝜃𝑖𝑢 +𝐵𝑖𝑢 sin 𝜃𝑖𝑢) (18) 𝑄𝑖=]𝑖 𝑀𝑁 ∑ 𝑢=1 ]𝑢(𝐺𝑖𝑢 sin 𝜃𝑖𝑢 −𝐵𝑖𝑢 cos 𝜃𝑖𝑢)(19) Dynamic Power Flow Method. When branches fault during fault propagation in the network, the dynamic power flow can be calculated in Algorithm 1. 4.2. Overload Mechanism of Electrical Networks. In this paper, cascading failures in the EN are analyzed from the perspective of the overload mechanism. When one or more lines are cut off, the other lines are overloaded owing to the redistribution of power flow in the EN [27, 28]. When a branch is overloaded, the larger the power flow over the branch, the shorter is the operational time for which the branch is permitted to continue working [15]. As most of other studies [15, 49], in this paper, we assume that during the fault propagation, the control center tries to maintain thesecureoperationoftheENandlowertheloadshedding amount, thus the control strategy includes which branches to trip, how to adjust the generators output, as well as to shed which load of how many percentages, etc. Therefore, for some of the branches, the tripping command has to come from the control center. Of course, for some of the faulted lines,thetrippingsignalshouldbeissuedbyalocalprotection unit. However, to simplify the process, we simply assume that the tripping command comes from the control center. Thus, under thus simplification, when the corresponding communication vertexes send information about overloading to the control center via the communication network, the control center must quickly send trip commands to the target vertexes. We employ the inverse-time overcorrect protection scheme [15, 49] to calculate the overloaded operational time. 𝑡𝐵𝑗=𝜅 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝐼𝐵𝑗/𝐼𝐵𝑗 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝜎−1 (20) If the data exchange between the target vertex and control center to trip the branch Bjis completed within tBj,thecontrol is successful; otherwise, the control is unsuccessful. 5. Interactive Cascading Failure Model A diagram of cascading failures considering the interactions between the ECN is shown in Figure 3. During the cascading Complexity 7 Input: Electrical network information Output: Power flows of branches, frequency violation Step 1: Unbalanced power: Employ Equation (13) to calculate the unbalanced power 𝑃𝑢𝑛 of the network. Step 2: System frequency characteristic: IF 𝑃𝑢𝑛 <0,employ Equation (14) to calculate 𝐾𝑆and go to the next step; ELSE,IF 𝑃𝑢𝑛 >0, employ Equation (15) to calculate 𝐾𝑆andgotothenextstep;ELSE,gotoStep 4. Step 3: Frequency calculation: Calculate variation Δ𝑓(𝑡)of frequency of the network by Equation (12) solved using improved Euler method (Δt = 0.01 s). Step 4: Generator frequency detection: IF Δ𝑓(𝑡)is beyond limits of generator i, cut off generator i and return to Step 3;ELSE,gotothenextstep. Step 5: Frequency violation: IF Δ𝑓(𝑡)is beyond limits the network, end the algorithm and send warnings of of instability; ELSE,gotothenextstep. Step 6: Power flow calculation: Employ the P-Q power flow to calculate the power flows of every branch. Algorithm 1: Dynamic power flow method. failures in the EN, if branch 𝐵𝑗is overloaded, according to the control strategy, 𝐵𝑗generates fault information packets, and the corresponding communication vertexes then send these packets to the control center via the communication network. Thereafter, the control center sends the commands back to 𝐵𝑗 within 𝑡𝐵𝑗. In addition, when the virus spreads through the communication network, the infectious vertex 𝑉𝑙will lose the function of information exchange and connectivity 𝑎󸀠 𝑚𝑙 with its neighbor vertexes, according to 𝑎󸀠 𝑚𝑙 =0 (m=1,2,...,MV)(21) Meanwhile, the virus will cause branches to trip or lead to an outage directly or indirectly because the infectious vertexes lose the function of information exchange. Accordingly, there are four types of fault branches. Type 1. The branch is forced to trip because the corresponding communications get infected, also called forced outage branches B𝐹𝑂. Type 2.Thebranchistrippedproperlybecausethecontrol center successfully sends commands to the corresponding vertexes based on the received overload information and control strategy with 𝑡𝐵𝑗, also called overload tripping branches B𝑂𝑇. Type 3. The branch is damaged irreparably owing to control failures via the communication networks, leading to overload operational time exceeding tBj, also called irreparable fault branches B𝐼𝐹. Type 4. The branch undergoes forced outage owing to networksplitting,alsocallednetworksplittingbranchesB𝑁𝑆. Dynamic power flow calculation Overload tripping Forced outage Irreparable faults Overload Removed vertexes Infectious vertexes Susceptible vertexes Virus propagation Control center SIR model Normal operation Information packets Commands Electrical network Communication network BFO BOT BIF S I R Figure 3: Cascading failures considering interactions between ECNs. Based on the above analysis, cascading failure considering the interactions between the ECNs is modeled in Algorithm 2. 6. Case Study The proposed model was applied to the IEEE 118-bus system and the French grid [50]. The small-world and scale-free networks were adopted to represent the respective communication networks. The computational work was performed in MATLAB running on a laptop. The laptop (Compaq, v3646TU) was equipped with an IntelCore2DuoCPU [email protected]0 GHz, 2.00 GB RAM, and 64-bit Windows 7 operating system. 8Complexity Input: Electrical network information and parameters, communication network information and parameters, 𝜂,I, 𝛽,𝛼,𝜏1 Output: 𝜗,B𝐹𝑂,B𝑂𝑇,B𝐼𝐹,B𝑁𝑆 Step 1: Initialization:t=0s,𝜗=0,B𝐹𝑂 =0,B𝑂𝑇 =0, B𝐼𝐹 =0,B𝑁𝑆 =0,B𝑂=0,S=0,R=0,R󸀠=0and I󸀠=0. Step 2: WHILE t<𝜂 Electrical network: Step 3: Forced branch outage: IF the corresponding vertexes coupled with the branch 𝐵𝑗(j= 1,2,...,MB)are in I, add the branch 𝐵𝑗to B𝐹𝑂 and trip it. Step 4: Overloaded branch tripping: IF the overload operation time of 𝐵𝑥(𝐵𝑥∈B𝑂)(x= 1,2,...,MO)in tBx =t, and the fault packages between the corresponding vertexes and control center are exchanged successfully, add 𝐵𝑥to B𝑂𝑇,anddelete𝐵𝑥from B𝑂;ELSE,IF tBx >t, add 𝐵𝑥to B𝐼𝐹,anddelete𝐵𝑥from B𝑂 Step 5: Networksplitting:Detectandsplittheelectrical network. IF there exists the forced outage branch 𝐵𝑧 (z=1,2,...,MNS)due to the splitting, add 𝐵𝑧to B𝑁𝑆. Step 6: Network operational status:Calculate the power flow over 𝐵𝑗(j= 1,2,...,MB).IF 𝐵𝑗is overloaded, calculate tBj of 𝐵𝑗by Equation (19), and add 𝐵𝑗to B𝑂. Communication network: Step 8: Infectious vertexes detection: IF the infection time of the candidate vertex 𝑉𝑘(k= 1,2,...,MI󸀠)in I󸀠is equal to t, add 𝑉𝑘to Iand delete it from I󸀠. Step 10: Removal of vertex detection: IF the removal time of the candidate vertex 𝑉𝜔(w= 1,2,...,MR󸀠)in R󸀠is equal to t, add 𝑉𝑤to R,anddeleteitfromR󸀠. Step 7: Virus propagation: The susceptible vertex 𝑉𝑔(𝑔 = 1,2,...,MS)contracts the virus with probability 𝛽 according to Equation (6). IF 𝑉𝑔gets infected, delete 𝑉𝑔 from S,add𝑉𝑔to I󸀠, and label its infectious time t +𝜏 1. Step 9: Vertex immunization: The infected vertex 𝑉𝑙(l= 1,2,...,MI)is immunized with the probability 𝛼according to Equation (6). IF 𝑉𝑙obtains immunity, delete 𝑉𝑙from I,add 𝑉𝑙to R󸀠, and label its immunity time t +𝜏 2. Step11: t=t+Δt; END WHILE. Algorithm 2: Cascading failure model considering interactions between ECNs (SIR model as an example). 6.1. IEEE 118-Bus System. We randomly chose the communication vertexes as the initial infectious vertexes and then performed 1000 cascading events to investigate load shedding andnumberofinstancesofthefourtypesoffaultbranches (MIF,M OT,M FO,andM NS) based on the different VPMs and topological structures of the communication network with the parameters Δt=0.01s,𝜏1=𝜏 2=5𝑠,𝛼 = 𝛽 = 0.3,𝜅=7 and 𝜎 = 1.5. The averaged results are shown in Figure 4 and listed in Table 1. Figure 4 shows that the load shedding changes with the passage of time based on the different topological structures of the communication networks and the SI model. Owing to space limits, the SIR-model-based load shedding is not given herein. In Figure 4, the propagation time of interactive cascading failures depends on the virus propagation time 𝜏1, and the load shedding is the maximum when the propagation time is approximately 10s. Compared with the small-world communication network, the propagation time of interactive cascading failures is longer in the scale-free communication network. Moreover, coupling with the scale-free communication network, the fault branches of the EN result in irreparable faults (MIF = 3.415) with higher probability than that (MIF = 0.245) in the case of coupling with the small-world communication network, as summarized in Table 1. Therefore, the coupling of EN with the scale-free communication network is affected more severely as the propagation time increases because the connectivity between vertexes often depends on a few hub vertexes (i.e., high-degree vertexes), and the exchange of information packets becomes difficult, leading to network paralysis once a few hub vertexes are infected. A comparison of the SI and the SIR models shows that different VPMs have very small impacts on fault propagation Complexity 9 Table 1: Average numbers of four types of fault branches. VPMs Structures MFO MIF MOT MNS SI SW 148.164 0.245 14.589 9.1554 SF 142.977 3.415 14.643 10.1554 SIR SW 148.222 0.139 14.658 9.336 SF 143.104 1.371 16.588 10.064 Small-world Scale-free 0 0.1 0.2 0.3 The percentage of load shedding 2 4 6 8 10 12 14 160 Time (second) Figure 4: Load shedding of IEEE 118-bus system over time based on SI model. in the EN, which indicates that once fault propagation occurs in the EN, the related antivirus measures with time delay can barely prevent the fault from spreading across the EN. However, the number of fault branches with irreparable faults can be reduced, especially in the case of coupling with the scale-free communication network. This is because the immune vertexes treated with the antivirus recover their function of data exchange, and a few overloaded branches can receive trip commands from the control center in a timely manner, thus avoiding irreparable faults. Furthermore, we analyze the interactive cascading failures by selecting different initial infectious vertexes. We used the SI model as an example. Because the degree distributions of the small-world communication network are known, we take the scale-free network as the basis to select the highdegree (vertexes 115 and 116) and small-degree (vertexes 4 and 8) vertexes as the initial infectious vertexes. Figures 5 and6showthetotalandreal-timeloadsheddingchanges with the passage of time for different virus propagation times 𝜏1= 2 s, 5 s, and 8 s. The initial vertexes have small impacts on fault propagation in the EN owing to the known degree distribution.However,incaseofthecouplingoftheENwith the scale-free communication network, the initial vertexes greatly impact fault propagation. Compared to the smallworld communication network, when the initial vertexes are high-degree vertexes in the scale-free communication network, the propagation time is obviously shorter, which demonstrates the hub can rapidly spread the virus, leading torapidcollapseoftheEN.Bycontrast,thepropagation time is longer when the initial vertexes are the low-degree vertexes, and when the load shedding peaks, as shown in Figure 6, the interactive cascading failures continue to spread, which indicates virus propagation times are longer than fault propagation times. That is, when fault propagation has stopped, virus propagation continues. 6.2. French Grid. ArealFrenchgridwith1951nodesand2956 branches was employed to simulate the interactive model. Owing to computational complexity, we only choose the high-degree vertexes as initial infectious vertexes considering the topological structures of the communication network with the parameters Δt=0.01s,𝜏1=𝜏 2=2s, 𝛼= 𝛽 = 0.3,𝜅=7,and𝜎 = 1.5.Figure7showsthat the total and real-time load shedding of system changes with the passage of time. Compared to the small-world communication network, the propagation times are longer in the case of EN coupled with scale-free communication network, but the load shedding peaks at approximately 8.3 s under both topological structures. Furthermore, we investigated the numbers of the three types of fault branches (MIF,M OT,andM FO) at different moments, as shown in Figures 8(a) and 8(b). Between 6 s and 7 s, the fault propagation is at its height, which indicates thatthenumbersofinfectiousvertexesandforcedoutage branches B𝐹𝑂 are the highest, leading to rapid collapse of the EN and surging load loss. In addition, in the fault propagation process in the EN, the fault branches with irreparable faults arenotfoundwhentheENiscoupledwiththesmall-world communication network. By contrast, there are many fault branches of this type at different moments in the case of EN coupled with the scale-free communication network. Therefore, when the communication network is scale-free, it is more vulnerable which cannot effectively resist the virus propagationleadingtothemoreseveredamagetotheEN. The conclusions obtained from these two cases are summed up in Table 2. In practice, when ENs are faced with hacker attacks, because the attackers find it relatively difficult to obtain complete information about the communication networks, such as topological structures, their attacks are random to some extent, which means scale-free communication networks are more appropriate for the ENs [10, 30, 31]. However, when ENs are faced with the threat of a cyber virus, the virus must be cleared promptly. Once the virus spreads, regardless of whether the initial infectious vertexes are selected randomly or deliberately, the infection will lead to aseveredamageinthescale-freecommunicationnetworks. Therefore, due to the propagation features of the scalefree communication networks, the simulation results show that software engineers should strengthen more the software Hindawi www.hindawi.com Volume 2018 Mathematics Journal of Hindawi www.hindawi.com Volume 2018 Mathematical Problems in Engineering Applied Mathematics Journal of Hindawi www.hindawi.com Volume 2018 Probability and Statistics Hindawi www.hindawi.com Volume 2018 Journal of Hindawi www.hindawi.com Volume 2018 Mathematical Physics Advances in Complex Analysis Journal of Hindawi www.hindawi.com Volume 2018 Optimization Journal of Hindawi www.hindawi.com Volume 2018 Hindawi www.hindawi.com Volume 2018 Engineering Mathematics International Journal of Hindawi www.hindawi.com Volume 2018 Operations Research Advances in Journal of Hindawi www.hindawi.com Volume 2018 Function Spaces Abstract and Applied Analysis Hindawi www.hindawi.com Volume 2018 International Journal of Mathematics and Mathematical Sciences Hindawi www.hindawi.com Volume 2018 Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi www.hindawi.com The Scientific World Journal Volume 2018 Hindawi www.hindawi.com Volume 2018 Volume 2018 Numerical Analysis Numerical Analysis Numerical Analysis Numerical Analysis Numerical Analysis Numerical Analysis Numerical Analysis Numerical Analysis Numerical Analysis Numerical Analysis Numerical Analysis Numerical Analysis Advances in Advances in Discrete Dynamics in Nature and Society Hindawi www.hindawi.com Volume 2018 Hindawi www.hindawi.com Differential Equations International Journal of Volume 2018 Hindawi www.hindawi.com Volume 2018 Decision Sciences Advances in Hindawi www.hindawi.com Volume 2018 Analysis International Journal of Hindawi www.hindawi.com Volume 2018 Stochastic Analysis International Journal of Submit your manuscripts at www.hindawi.com