Keywords

1 Introduction

In June 2019, Argentina and Uruguay suffered a massive power outage due to a failure of the Argentine grid that affected the connected system, and further resulted in the inability of the entire country and several provinces in the neighboring countries to supply electricity. Then the damage caused huge economic loss. It is especially important to study the impact of different degrees of collapse on the structure of critical infrastructure, especially for the power grids [1, 2].

Therefore, it is necessary to find the vulnerable nodes of the entire network under node-based attacks. Vulnerability is a new concept for the assessment of the survivability and reliability of network systems in recent years [3]. It has been applied in power networks, aviation networks, and transportation networks [4, 5].

At present, the most important research direction of grid vulnerability research is to use the network theory to judge the vulnerability of the grid and predict the possibility of cascading failure. Internationally, there has been research about safety load on the grid [6], distributed power sources to reduce the burden on the grid to reduce vulnerability [7], power supply composition [8], grid structures [9], grid operations [10], and the impact of important transmission channels on the vulnerability [11].

In this paper, we consider the electrical power grid of the western United States which has 4941 nodes and 6594 edges [12]. And we use the vulnerability to comprehensively evaluate the structural survivability of the US power grid. We simulate the intentional attack by deleting some percentage of nodes, according to the degree, k-shell value, betweenness centrality and clustering coefficient, apparently. Three metrics (Largest Connected Component, Efficiency and Average Path Length) are introduced to explore the vulnerable nodes of US power grid under different node-based attacks. It should be pointed that the degree, k-shell value, betweenness centrality and clustering coefficient represent the different characteristics of nodes. Then, the vulnerable nodes could be identified under these node-based attacks.

The space of this paper is deployed as follows. In Sect. 2, the definition and calculation methods of vulnerability metrics and node attack reference indicators are introduced. In Sect. 3, taking the US power grid as an example, the nodes are attacked by different indicators and the largest connected component, efficiency and average path length of the entire network after each attack are calculated to obtain simulation results. In Sect. 4 we discuss the simulation results and network vulnerability analysis in this paper [13,14,15].

2 The Measuring Metrics

2.1 Largest Connected Component G

The component with the largest number of nodes in the component in which the nodes are connected to each other in the whole network becomes the largest connected component. The number of nodes in this component is S, and the number of nodes is N, then the largest connected component ratio of this network is [16]:

$$ G = \frac{S}{N} $$
(1)

2.2 Efficiency E

Efficiency measures the information transmission capability between network nodes. The shorter the shortest path \( d_{ij} \), the higher the efficiency, so the efficiency is:

$$ E = \frac{1}{N * (N - 1)}\sum {\frac{1}{{d_{ij} }}} $$
(2)

2.3 Average Path Length L

The average path length is the average of the number of edges that pass from one node to another, so the average path length is defined as [17]:

$$ L = \frac{1}{N}\sum {{\text{d}}_{ij} } . $$
(3)

3 Different Node-Based Attacks

3.1 Betweenness Centrality \( B_{i} \)

It is a measure of the contribution rate of each node to the connectivity of other nodes. It refers to the number of times that all other shortest paths of other nodes pass through this node. The betweenness centrality \( B_{i} \) of node i is defined as:

$$ Bi = \sum\limits_{a \ne b} {\frac{{\sigma_{ab} \left( i \right)}}{{\sigma_{ab} }}} $$
(4)

where \( \sigma_{ab} \) is the number of the shortest paths between the node a and b, and \( \sigma_{ab} (i) \) is the number of shortest paths between node a and b passing through the node \( i \), respectively.

3.2 Aggregation Coefficient Ci

The aggregation coefficient is an important parameter to measure the degree of aggregation of network nodes, reflecting the degree of network small grouping. If there are \( K_{i} \) neighbors in a node, the total number of possible connected edges of these neighbors is \( K_{i} *\left( {K_{i} - 1} \right)/2 \), and the actual number of connected edges is \( E_{i} \), so the aggregation coefficient of each node is:

$$ C{\text{i}} = \frac{{ 2 * {\text{E}}_{\text{i}} }}{{{\text{K}}_{\text{i}} * ( {\text{K}}_{\text{i}} - 1 )}} $$
(5)

3.3 K-shell Value

K-shell is an indicator used to describe the influence of a node [18, 19]. First, the nodes with degrees 1, 2, 3… are deleted one by one, and the degrees of all nodes are recalculated when the nodes with degree 1 are deleted. Then delete the node with degree 1 until there is no more node with degree 1 in the entire network. Finally, the k-shell value of the node deleted by this process is assigned to 1. And so on, until all nodes of the entire network are deleted.

4 Simulation Result Analysis

In many cases, the collapse of the national grid is caused by the failure of a very small number of nodes, which causes the collapse of the entire grid. Therefore, it is meaningful to study the impact of the click on the entire network structure. The largest connected component describes the connectivity of the entire network. Efficiency measures the ability of information transmission between nodes, that is, the speed at which a node crashes while affecting another node. The average path length refers to the number of paths between two nodes, which reflects the speed of impact on one node when one node crashes. Therefore, all three can reflect the vulnerability of the entire network structure.

Figures 1, 2, and 3 respectively scale the degree \( K_{i} \), k-shell value, Betweenness Centrality-based \( B_{i} \), and aggregation coefficient Ci from large to small, and then attack the nodes proportionally. The largest connected component, efficiency, and average path length are used as performance indicators, and study the impact of the US grid structure on the vulnerability of the network structure. Since the whole network is described in this paper, the degree, the Betweenness Centrality-based, the k-shell value, and the aggregation coefficient of each node are summed, and the ratio of the node to the total number of nodes is used as a measure of the value of the entire network [20, 21].

Fig. 1.
figure 1

Using the largest connected component as a measure of structural vulnerability.

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Fig. 2.
figure 2

Using efficiency as a measure of structural vulnerability.

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Fig. 3.
figure 3

Using the average distance L as a measure of structural vulnerability.

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4.1 The Simulation Results with the Reference Indicators of the Attack Nodes

According to the order of the aggregation coefficient, the proportional destruction node has the least impact on the network structure vulnerability, and the aggregation coefficient is never zero. Namely, the attack coefficient is the reference index, and the attack node is less destructive to the whole network than other methods.

According to the degree and the number of Betweenness Centrality-based, the impact of the crushed nodes on the vulnerability of the entire network structure is basically the same. When they defeat 20%–40% of the nodes, the whole network is basically completely disintegrated.

When the crushed nodes are sorted by k-shell value, we can find the vulnerability by the largest connected component and efficiency. We can find that they are the trend of high-speed decline. When the quarter node is defeated, it gradually reaches a state of gentle decline until the whole network disintegration.

In these three figures, we can also find that when the nodes are defeated according to the size of the aggregation coefficient, the trend of the three performance indicators is not monotonously decreasing. Because the nodes will gradually decrease as the nodes collapse. Therefore, all possible connected edges between neighbors will decrease, resulting in an increase in the aggregation coefficient [22, 23].

4.2 Simulation Results Based on Network Structure Vulnerability Measures

Taking the largest connected component and efficiency as the structural vulnerability measure, the network structure is disintegrated more quickly in the node deletion process. For weaker degrees of power and Betweenness Centrality-based, the G and E values of the entire network are almost zero when 20% of the nodes are deleted.

The average path length is used as a measure of structural vulnerability. It does not change monotonously with the previous two. Because with the deletion of nodes, the path length of some nodes reaching other nodes has increased, but the total number of nodes has changed little, so there will be an undulating state [24, 25].

5 The Conclusions

Through the study of the vulnerability of the US power grid structure, it is found that different attack methods have different destructive effects on the entire network structure. Among them, the size of k-shell and the aggregation coefficient are used as the basis for judging the importance of the node. Attacking the node in these two ways has the least damage to the whole network, and the whole network shows stronger invulnerability. The largest connected component and efficiency are used to measure the vulnerability of the network structure. When the important node is attacked by 20%, the whole network is basically in a discrete state. Therefore, deleting the nodes in these two ways makes the entire network reach the disintegration state faster [26].

Regardless of the attack method, the more important the node, that is, the node with the larger index value, the more damage to the network structure. Therefore, the protection of these nodes is extremely important. By protecting these nodes, the network is improved. The robustness of the structure enables the entire power network to operate normally, reducing the collapse of the entire network system due to the destruction of individual nodes and the resulting economic loss and inconvenience to the national life.

In the future, with limited resources, we will discuss how to protect the key nodes under different attacks.