Complexity

Volume 2019, Article ID 3531209, 14 pages

https://doi.org/10.1155/2019/3531209

## Identification of Two Vulnerability Features: A New Framework for Electrical Networks Based on the Load Redistribution Mechanism of Complex Networks

^{1}School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, China^{2}Department of Energy, Politecnico di Torino, Torino 10129, Italy^{3}School of Electrical Engineering and Electronic Information, Xihua University, Chengdu 610039, China

Correspondence should be addressed to Shibin Gao; moc.qq@9204574151 and Tao Huang; ti.otilop@gnauh.oat

Received 24 September 2018; Accepted 25 October 2018; Published 16 January 2019

Guest Editor: Seyedmohsen Hosseini

Copyright © 2019 Xiaoguang Wei 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.

#### Abstract

This paper proposes a new framework to analyze two vulnerability features, impactability and susceptibility, in electrical networks under deliberate attacks based on complex network theory: these two features are overlooked but vital in vulnerability analyses. To analyze these features, metrics are proposed based on correlation graphs constructed via critical paths, which replace the original physical network. Moreover, we analyze the relationship between the proposed metrics according to degree from the perspective of load redistribution mechanisms by adjusting parameters associated with the metrics, which can change the load redistribution rules. Finally, IEEE 118- and 300-bus systems and a realistic large-scale French grid are used to validate the effectiveness of the proposed metrics.

#### 1. Introduction

Critical component identification is an important part of security analyses for electrical networks [1–3]. The main idea is to rank the weakness of the equipment in an electrical network via a set of metrics.

As an artificial network, electrical grids have topological similarities to other general networks. They also exhibit several typical features of complex networks, such as small-world properties [4–6]. Therefore, complex network theory (CNT) is a popular method to assess the vulnerability of electrical networks[3, 4, 7–13]. The construction of structural metrics is an important branch of vulnerability evaluations [14] based on CNT. CNT uses the connectivity information abstracted from the network to create indices based on statistics and, sometimes, physical features of the network are added to improve the effectiveness of the indices [10, 11].

However, there are still several problems with this method. Compared to general networks (or systems), an electrical network has its own characteristics that limit the wide application of CNT. First, analyzing the topological structures of electrical networks without considering their operational status does not disclose the real features of the systems [10, 11]. Secondly, in most general networks, when a vertex (or an edge) of a network fails, the direct neighbors are the first to be affected or have the largest impact based on CNT. However, this is not generally true for electrical networks [15]. Moreover, the structural metrics are static indices [12, 13, 16] that only consider the normal operational status of the network. To overcome the above problems, statistical graphs [17, 18] are employed to analyze the vulnerability or cascading failures of electrical networks. For example, [19, 20] proposed a sequential attack graph (SAG) to identify critical nodes while [21] proposed a correlation matrix. In addition, [22] proposed influence graphs to analyze cascading failures. Statistical graphs have also provided promising options for security, because they comprehensively consider the topological, physical, and operational characteristics of a system.

In addition, another problem in which features of vertices (edges)(For clarity, hereinafter the terms “network, branch and node” are used only for electric systems and “graph, edge and vertex” only for complex networks.) in complex network vulnerability detection, especially in electrical networks, should be distinguished, is often overlooked. For example, some vertices can easily spread faults leading to a high probability of a network failure event. Conversely, some vertices are easily affected by propagated faults. Therefore, it is necessary to devise a method to identify these two features of vertices and to better reveal the vulnerabilities of networks.

In summary, our main contributions are as follows.

First, we propose a new framework that employs statistical graphs to represent the useful information for analyzing the network vulnerability from the original physical grid, using CNT, compared to a traditional framework that employs the original topological structure of the gird.

Secondly, inspired by [19–24], we propose a correlation graph (CG) generated via critical paths to analyze the electrical network vulnerability.

Thirdly, using benchmarks, we analyze the topological properties of the CGs based on CNT via the cumulative distributions of the vertex degrees. According to the analysis, the CGs are scale-free graphs, which verifies that electrical networks have scale-free properties under deliberate attacks, as opposed to traditional complex network methods, which verify the properties by correlating the drop in the network demand (or efficiency) with the attacked branches.

Finally, we define two vulnerability features from the perspective of CNT and then map the features onto electrical networks. Further, we employ the scale-free structures of the CGs to construct vulnerability metrics for the first time to differentiate the two features from the perspective of the load redistribution mechanism of CNT. The features of the metrics are explained in detail, including their relationship with the degree.

In addition, note that, even though dynamic models analyzed by real-time simulation platforms[25] are more comprehensive for security analyses in the real world, they require much longer simulation times and result in an immense computational burden, which makes it difficult to analyze a large-scale network. Meanwhile, as a media connecting equipment in the power system, the transmission network has notably fast dynamics/transients, compared to rotating devices. In other words, the transmission network* per se* can usually be considered to be a static component. Therefore, static models from the perspective of the load redistribution are widely employed to analyze the network vulnerability in existing literature [10–24]. Based on above, we focus on understanding the nature of the transmission network using static models by the load redistribution from the entire network.

#### 2. Correlation Graph

We constructed a CG to incorporate both the structural features and the operational status of power systems, using critical paths from the point of view of* load redistribution mechanism *(LRM). The constructed graph considers both the topological structures and the operational features under fault operation of the system. For example, branches of an electrical network can be transformed into vertices in a new graph while edges are formed to reflect the adjacent relationships between branches.

##### 2.1. Vulnerability Assessment: A New Framework

To overcome the limitations of structural vulnerability identification methods by applying CNT to the electrical network vulnerability assessment, we need to consider the following two aspects: (1) the importance of a vertex and (2) the adjacent relationships between vertices. To assess the importance of a vertex, there are many indices (e.g., degree and betweenness) that can be used to qualify it from the perspective of LRM. Comparatively, there are few indices for quantifying the importance of branches because it is difficult to assess edges under LRM. In addition, in most general networks, when a vertex (or an edge) of a network fails, the adjacent relationship between vertices usually imply that the immediate neighbors are the first to be affected or suffer the largest impact based on CNT. However, this is not generally true for electrical networks; sometimes, nonadjacent branches are the first to be affected due to the physical laws of electric circuits and the physical and operational constraints [15]. Therefore only using the information of the structure of an electrical network cannot effectively identify the critical branches.

In summary, it is spatially insufficient to analyze the network vulnerability using only the topological structures of the girds. Therefore, we propose a new framework that employs statistical graphs [19–24] to represent information useful for analyzing the network vulnerability from the original physical grid, using CNT, as shown in Figure 1. In the existing methods, the topological structures are employed to assess the electrical network vulnerability based on the CNT on the original physical networks. Its main idea is to focus on the importance of branches by constructing statistical metrics without the evolvement of the operational feature of the system. However, we construct statistical graphs comprehensively considering topological, physical and operational features of power systems, and further based on the constructed statistical graphs which can reveal not only importance of branches but also adjacent relationships among branches we assess the vulnerability with two features by replacing the original electrical networks.