 Research
 Open Access
Nodal vulnerability to targeted attacks in power grids
 Hale Cetinay^{1}Email authorView ORCID ID profile,
 Karel Devriendt^{1} and
 Piet Van Mieghem^{1}
 Received: 16 February 2018
 Accepted: 27 July 2018
 Published: 23 August 2018
Abstract
Due to the open data policies, nowadays, some countries have their power grid data available online. This may bring a new concern to the power grid operators in terms of malicious threats. In this paper, we assess the vulnerability of power grids to targeted attacks based on network science. By employing two graph models for power grids as simple and weighted graphs, we first calculate the centrality metrics of each node in a power grid. Subsequently, we formulate different nodeattack strategies based on those centrality metrics, and empirically analyse the impact of targeted attacks on the structural and the operational performance of power grids. We demonstrate our methodology in the highvoltage transmission networks of 5 European countries and in commonly used IEEE test power grids.
Keywords
 Power grids
 Complex networks
 Centrality metrics
 Targeted attacks
Introduction
The unavailability of electrical power can severely disrupt daily life and result in substantial economic and social costs (Baldick et al. 2008). This vital importance encourages a robust design and operation of power grids (Alvarado and Oren 2002). Robust power grids are able to anticipate, adapt to and/or rapidly recover from a disruptive event or a failure.
In current practice, flowbased simulations play an essential role in the security analysis of power grids. Given the generation and demand profiles, the steadystate analyses estimate the operation of power grids. Additionally, many countries require that the power grids should withstand the scheduled and unscheduled outages of its most critical lines or other components. In these contingency analyses, the component outages are also simulated to determine whether the power grids can still function properly under the failure and consequent loss of an element. However, as power grids grow in size and get more complex, the number of contingencies increases significantly, increasing the associated computational time. This motivates research towards alternative techniques (Bompard et al. 2009; Gutierrez et al. 2013; Hines et al. 2010a).
Disruptions in networks can be caused by unintentional failures or intentional attacks. Unintentional failures can include manufacturing defects, malfunction in network elements or human error. These kinds of failures can occur randomly throughout the grid and are characterized as random failures (Trajanovski et al. 2013). Intentional attacks or targeted attacks, on the other hand, are not random and are aimed at maximizing damage (Rueda et al. 2017). A major challenge in power grids is to evaluate the vulnerability of a power system to these intentional hazards, starting by quantifying the importance of electrical buses and the impact of the attacks on the network performance.
Power grids are amongst the largest and the most complex manmade systems and, like many other complex networks, it features a specific topology which characterizes its connectivity and influences the dynamics of processes executed on the network. The complex nature of power grids and its underlying structure make it possible to analyse power grids relying on network science (Strogatz 2001; Barabási and Albert 1999). The application of network science on power grids has shown the promising potential to capture the interdependencies between components and to understand the collective emergent behaviour of complex power grids (Crucitti et al. 2004; Chassin and Posse 2005).
Topological investigations of power grids have demonstrated that power grids have several components with significant importance compared to the rest of the network (Koç et al. 2014a). These components are crucial for the grid as their removal can significantly disrupt the operation of the power grids. Identifying these critical components in advance can enable power grid operators to improve system robustness by monitoring and protecting these components continuously. (Bompard et al. 2009; Koç et al. 2014a)
Currently, many studies use a complex networks perspective in analysing power system vulnerabilities (Cuadra et al. 2015; Crucitti et al. 2004; Bompard et al. 2008). A significant part of these studies investigates the relationship between the topology and specific performance metrics in the underlying graph of power grids (Crucitti et al. 2004; Negeri et al. 2015; RosasCasals et al. 2007). Such studies focus on the basic structural properties of a graph (such as nodal degree, clustering coefficient (Hernández and Van Mieghem 2011)), which typically ignore the electrical properties, such as flow allocation according to Kirchhoff’s laws or the impedance values of transmissions lines in the grid. Mainly, two different aspects are important in the operation and consequent robustness of power grids: the topology of the network formed by electrical buses and their interconnections, and the operating conditions such as supply and demand distributions (Bompard et al. 2012; Cetinay et al. 2016). Consequently, these purely topological metrics could result in misleading research results, which may be far from real physical behaviours of power grids (Arianos et al. 2009; Hines et al. 2010b; Koç et al. 2014a).
To include the electrical properties of the grid in the analyses, several studies propose extended metrics (such as effective graph resistance (Koç et al. 2014b), the electrical centrality (Hines and Blumsack 2008) and the netability (Bompard et al. 2012)) by introducing a set of link weights (such as distance or resistance (Qi et al. 2015)) and node properties (such as the electrical demand and supply (Bompard et al. 2012)). Additionally, other studies have used topological and electrical metrics to rank the electrical buses and lines in power grids as a selective contingency analysis (Gorton et al. 2009; Nasiruzzaman et al. 2011).
Motivated by the increasing need of alternative studies to the flowbased analyses and the merits of network science on the investigation of power grids, in this paper, we combine both of the aforementioned approaches: First, we present two different graph representations for a power grid: a simple graph and an extended graph representation that takes the electrical properties of power grids into account. Next, we develop a methodology to identify the critical electrical buses (nodes) in power grids, and compare the impact of targeted nodeattacks in detail for European highvoltage transmission networks (Cetinay 2018) and for the publicly available IEEE test power grids (IEEE 2018). Our contributions can be summarized as follows: (i) we consider two different graph models for power grids based on either purely topological information or by including the link weight information and the linearised DC power flow equations; (ii) we employ these two graph models to formulate the standard and the extended centrality metrics of nodes in power grids; (iii) we formulate 8 different attack scenarios according to these centrality metrics and empirically investigate the impact of targeted nodeattacks on the structural and operational performance of power grids.
Power grids and network science
In this section, we provide details about power grids, the steadystate power flow equations and our models for power grids as simple and weighted graphs.
Power grids preliminaries
where θ_{ik}=θ_{i}−θ_{k} and \(y^{(\mathrm {R})}_{ik}=\text {Re} (y_{ik})\) and \(y^{(\mathrm {I})}_{ik}=\text {Im} (y_{ik})\) are the real and the imaginary parts of the element y_{ik} in the bus admittance matrix Y corresponding to the i^{th} row and k^{th} column, respectively.

Supply node: A supply node generates the active power p_{i} and controls the voltage magnitude v_{i} at its node i.

Demand node: At a demand node, it is possible to specify the extracted active p_{i} and the reactive powers q_{i} from the type of the electrical loads that are connected to that node. There are also nodes without a supply or a demand connected, which can be modelled as a demand node with no injected power, i.e., p_{i}=0 and q_{i}=0.
Due to the impedance of transmission elements, there are power losses during the operation in power grids. As the losses are dependent on the system state–the supply and demand dispatches–they cannot be calculated in advance. Therefore, a slack node among the supply nodes is assigned in power grids to compensate for the difference between the total supply and the total demand plus the losses.
DC power flow equations
 1
The difference between the phase angles of neighbouring nodes is small such that sinθ_{ik}≈θ_{ik} and cosθ_{ik}≈1.
 2
The active power losses are negligible, and therefore, the bus admittance matrix can be approximated as Y≈iY^{(I)} where Y^{(I)} is the imaginary part of the admittance matrix Y, calculated neglecting the line resistances.
 3
The variations in the voltage magnitudes v_{i} are small and, can be assumed as v_{i}=1 for all nodes.
Although the DC power flow solution is less accurate than the AC power flow solution, in practice, the differences in highvoltage transmission networks between the phase angles of neighbouring buses and the variations in voltage magnitudes are relatively small, thus the error is assumed to be negligible (Van Hertem et al. 2006).
Graph representations of power grids
This section presents our models for power grids as simple and weighted graphs.
Power grid as a simple graph
Power grid as a weighted graph
where the weighted Laplacian \(\tilde {\mathbf {Q}}\) is a symmetric, positive semidefinite matrix that possesses nonnegative eigenvalues apart from the smallest eigenvalue, which is zero (Van Mieghem 2010).
Equivalence between linear systems, adopted from (Van Mieghem et al. 2017)
Power grids  Phase angle  Power 

Electrical circuit  Voltage  Current 
Hydraulic circuit  Pressure (height of liquid)  Volume flow 
Mechanical system  Force  Displacement velocity 
Thermal system  Temperature  Heat flow 
…  …  … 
Targeted attacks on power grids
The threats for power grids can be classified by using multiple criteria considering the causes of the threat, their consequences or the preventive actions to manage the hazards (Ciapessoni et al. 2017). One example of such threats are targeted attacks on power grids, which involve intentional, criminal actions to destroy the network. In modelling these threats, we assume that the attacks are performed with the knowledge of power grid layout and with the intention to maximally disrupt the network performance while attacking as few nodes (electrical buses) as possible. Throughout this section, we describe how network science can be employed to formulate such attack strategies, where target nodes correspond to most critical or most vulnerable nodes whose removal significantly disrupts the network functioning. We first describe the standard centrality metrics, which are purely based on the underlying topology of power grids, and then we extend these metrics to include the information on the link weights, i.e. the admittances of the transmission lines, and the DC power flow equations in power grids.
Ranking nodes in the simple graph representation of a power grid
In this section, we review some of the existing topological centrality metrics in order to rank the importance and the centrality of nodes in the underlying simple graph of power grids.
Degree centrality
Eigenvector centrality
Betweenness centrality
In other words, the betweenness centrality of a node i shows the fraction of all shortest paths between any pair (s,t) of nodes, that pass through node i.
Closeness centrality
which is the reciprocal of the sum of the hopcounts of node i to all other nodes. A large closeness centrality value thus corresponds to a “central” node that is wellconnected by a few hops to other nodes.
Ranking nodes in the weighted graph representation of a power grid
While the standard centrality metrics are based on purely topological information, it is possible to extend the definition of these metrics by including the link weight information and the power flow equations in power grids. Different definitions of extended centrality metrics (extended betweenness (Bompard et al. 2010), modified betweenness and closeness centrality (Gutierrez et al. 2013), electical degree (Hines and Blumsack 2008)) exist^{2} and are evaluated by simulations via power flow solvers or by calculating power transfer distribution factors (PTDF) in power grids. Such simulationbased definitions are generally computationally expensive and formulations with the absence of slack node(s) may not fully explain the analogy between the extended centrality definitions and the weighted graph model for power grids.
Extended metrics were also defined before (Ellens et al. 2011; Newman 2005) based on the voltage  current relation in electrical circuits. Since the phase angle  active power relations in (7) and (8) in power grids obey the same linear relation as those in electrical circuits (as described before in Table 1), these metrics can identify central nodes in power grids.
We take here a graph theoretical approach using the slacknodeindependent weighted graph representation for power grids described in the previous section. This weighted graph model facilitates both the analogy between the standard and the extended centrality metrics, and the enhanced linear algebra to formulate the closedform expression of centrality metrics via graphrelated matrices.
Weighted degree centrality
A large value of the weighted degree \(\tilde {d}_{i}\) corresponds to larger values of the admittance directly connected to that node, which indicate that node i is well connected to its neighbours.
Weighted eigenvector centrality
Flow betweenness centrality
where \(\mathcal {B}(i)\) denotes the direct neighbours of node i, and f_{s→t}(i,j) is the magnitude of the power flow through the link between i and j when a unit active power is injected at node s and extracted from node t. In Appendix 2, we show how these flows can be calculated from the weighted graph representation of a power grid. Higher values of the flow betweenness centrality \( \tilde {b}_{i}\) indicate the importance of a node with respect to the electrical power transmission in power grids.
Electrical closeness centrality
Compared to the shortestpath hopcount \(H(\mathcal {P}_{i\rightarrow j})\), the effective resistance Ω_{ij} does not depend only on the shortest path, but also incorporates the information of all possible paths between node i and j, where the contribution of each possible path follows from the linear flow equations. In the case of the unweighted tree networks, the effective resistance Ω_{ij} equals the hopcount \(H(\mathcal {P}_{i\rightarrow j})\) for all nodes. Thus, for treelike power grids with equal admittances, the electrical closeness centrality closely resembles the topological closeness centrality, while for power grids with many loops (i.e. nontreelike) both metrics could differ significantly.
Each of the centrality metrics we present captures a certain aspect of the structural and the operational centrality in the network, such as the strength of a direct connectivity (degree and eigenvector centrality), being a part of many important paths (betweenness centrality) or being close to other nodes (closeness centrality). In recent years, another conceptual definition of centrality has emerged. Based on optimal percolation theory (Morone and Makse 2015), which considers the problem of “finding the smallest set of nodes whose removal fragments the network in small disconnected pieces”, a number of new metrics have been proposed (such as the collective influence (CI) (Morone and Makse 2015; Morone et al. 2016), belief propagation decimation (BPD) (Mugisha and Zhou 2016) and CoreHD (Zdeborová et al. 2016)). Such metrics reflect the importance of a node for the global structural coherence as well as their influence in spreading behaviour. However, to the best of our knowledge, extended metrics based on percolation theory have not been studied yet. Therefore, in this work, we focus on the generally accepted and adopted centrality metrics to the power grids.
Identifying the effect of node removals in power grids
In this section, we empirically compare the effects of the targeted node removals based on the centrality metrics presented in the previous section. To evaluate the change in the network functioning, we use two performance metrics that can quantify both the topological and the operational characteristics of the grid after targeted attacks. We consider the networks from 5 realworld power grids of European countries (Cetinay 2018) and 5 synthetic power grids from the IEEE test case database (IEEE 2018).
Performance metrics
In an ideal power grid which is robust to targeted attacks, the removal of nodes should not significantly alter the network functioning. In some cases, removing a node from the power grid can partition the network into several components, which are disconnected from each other. This is undesirable as this partition both adversely affects (i) the structure: as the size (i.e. the number of nodes) of the connected component of the network is decreased, and (ii) the operation: since the disconnected structure disrupts the service and the capacity of the network. In this work, we present two performance metrics in our case studies, the size and the capacity of the giant component, to assess both the topological and the operational performance aspect in the network.
The size of the giant component
The giant component (Molloy and Reed 1998) is the connected component of a graph that contains the largest fraction of the entire graph’s nodes. The size of the giant component in the graph reflects the disruptive effect of node removals on the structure of the network.
where 1_{{x}} is the indicator function: 1_{{x}}=1 if the condition {x} is true, else 1_{{x}}=0, and G^{′} is the current giant component of the initial graph G(N,L).
The capacity of the giant component
Each transmission line in a power grid is associated with a maximum flow carrying capability. For the safe operation of a network, the flows through the network links should be below these capability. If the flow limits are exceeded, the situation is detected by protection relays, the circuit breakers are tripped, and the corresponding element is taken out of service. The possibility and the negative impact of cascading failures in power grids increases when the operating point of a power grid is close to the flow carrying capabilities of its links (Koç et al. 2013; Cetinay et al. 2017). Consequently, a network with a high flow carrying capability is desired.
We calculate the total capacity of the network as the sum of the maximum flow carrying capabilities of links in the largest connected component of the graph. When multiple lines are connecting the same pair of nodes, we consider an equivalent capacity between those nodes. This equivalent capacity represents the maximum power that can be transferred between these nodes such that the resulting power flow through each single line is at most at its capacity. In Appendix 3, we describe how this equivalent capacity is calculated.
Properties of the networks used in simulations
The effects of targeted node removals in power grids
We apply both the standard and the extended centrality metrics as nodeattack strategies in power grids. For each centrality metric, we start the attacks by removing the node (and all its links) with the highest value of the chosen centrality metric. After each node removal, we recalculate the values of the centrality metric, and continue by removing the node with the highest value of the centrality metric in the current giant component of the graph. Note that, during the successive node removals, we do not take the cascading dynamics (such as overloading of links or demand/supply redistribution due to cascading failures (Cetinay et al. 2017)) into account. In other words, we focus on the instant just after the removal of nodes to identify the effects on the structure and the operational performance indicators of the power grid.
where γ(k) is the normalized capacity of the giant component after k successive attacks. The structural \(\bar {\sigma } \) and the operational \(\bar { \gamma } \) performance indicators in (19) and (20) are evaluated on a score between 0 and 1: In an ideal power grid which is robust to targeted attacks, the node removals should have slight effects on the network performance. Thus, a performance indicator close to 1 is desirable by the network operators. On the other hand, a lower performance indicator over successive nodeattacks indicates a powerful (destructive) attackstrategy in which few important nodes of the network are identified and removed, with negative operational and structural consequences.
Main lessons learned from the analyses

The degree centrality (9) only provides information on the local structure around a node. Similarly, the weighted degree centrality (13) reflects local connectivity information. Thus, a node that is connected to many other nodes (with high admittance) is not necessarily a central node for the whole network. Therefore, as illustrated by the targeted attack simulations, the degree and the weighted degree centralities cannot always indicate the important nodes.

The betweenness centrality (11) incorporates information about the global network structure, and in the analyses of the test networks, high betweenness centrality values were found to efficiently indicate the nodes whose removal would significantly disrupt the network performance. While successfully indicating vulnerable nodes, the betweenness centrality (11) is based on the shortest paths only. This means that the betweenness centrality does not discriminate nodes that are positioned “close” to many shortest paths (and would be considered central), and peripheral nodes. This limitation is partly addressed by the flow betweenness centrality (15), in which the flows through the network links are distributed throughout the network according to the Kirchhoff’s laws. In the analyses of the test networks, removing nodes with a high flow betweenness usually resulted in the most destructive effects on the network.

The closeness centrality (12) reflects the average shortest path distance from a node to all other nodes in the network. Higher closeness centrality values thus indicate nodes which can easily reach the other nodes in the network. Similarly, higher values of electrical closeness centrality (16) show a node that is on average close to the other nodes in the network, based on the operationally inspired effective resistance distance instead of the shortestpath distance. In the analyses of targeted attacks, the performance of the closeness and the electrical closeness centrality in identifying the important nodes in the tested power grids are found to be similar.

The eigenvector centrality (10) can rarely identify the critical nodes, and thus, the targeted attacks based on the eigenvector centrality are generally the worst destructive strategy among the traditional centrality metrics in the tested networks. Similarly, the weighted eigenvector centrality (14) seems not to successfully indicate important nodes.
The analyses of the targeted nodeattacks show that centrality metrics, in particular the (flow) betweenness and (electrical) closeness, are very successful in indicating the critical nodes whose removals sharply decrease the selected performance indicators (the size and the capacity of the giant component) of power grids. Identifying these critical components in advance can enable power grid operators to improve system robustness by monitoring and protecting these components continuously. Additionally, although the effect of targeted attacks are more significant when the centrality information is updated after each node removal, the information based on the initial calculation of the centrality metrics is also fairly successful in finding the important nodes. The degree centrality is a good indicator to fragment the network to decrease the structural and operational performance indicators of power grids (See Appendix 1).
Conclusion
In this paper, we took a network science approach to investigate the vulnerability of power grids to malicious targeted attacks. First, we presented two different graph models for power grids: simple and weighted graphs. Subsequently, using these graph models, we ranked the importance of each node according to the standard and the extended centrality metrics that take into account the electrical properties of the grids such as the admittance of the transmission lines and the flow allocation according to the DC power flow equations. Via case studies in both realworld and test power grids, we show that the power grids are highly vulnerable to targeted attacks: sequentially removing the nodes with the highest centrality is a good strategy to fragment the power grids, and to maximally decrease its operational performance. In almost all power grids in our case study, removing approximately 15% of the nodes according to the flow betweenness centrality destroys the network almost completely. Grid operators can use the proposed methodology to analyse the current vulnerability of their network to targeted attacks and to take necessary measures by protecting the important nodes in their networks.
Appendix 1: Targeted attacks based on initial centrality metrics
Instead of recalculating the centrality metrics after each node removal, we consider here a more simplified attack strategy based on calculating the centrality metrics only once, at the beginning of the attacks. The targeted attacks are then performed sequentially according to these initial values.
Figures 7 and 8 show the changes in the normalized size and the capacity of the giant component in French power grids after the targeted attacks, respectively. Figures 7 and 8 illustrates that even these simplified attack strategies could inflict a significant damage on the network functioning: For instance, removal of 15% of the nodes according to the initial rankings nearly destroys French power grids. Compared to Figs. 3 and 4, in Figs. 7 and 8, we observe that the degree centrality is the most destructive attack strategy when the centrality metrics are based on only the initial calculation of the centrality metrics, i.e. when the noderankings are not updated after the targeted attacks.
Appendix 2: Calculation of flow betweenness centrality in power grids
Appendix 3: Multiple lines connecting the same pair of nodes
Multiple lines connecting the same pair of nodes are represented as a single equivalent link in the graph, see Appendix 3.
A broad review of robustness studies in power grids using network science can be found in (Cuadra et al. 2015).
The properties of a distance function D(i,j) between a pair of nodes i and j are: (a) nonnegativity: D(i,j)≥0, (b) zero distance for identical nodes: D(i,j)=0 if and only if i=j, (c) symmetry: D(i,j)=D(j,i) and (d) the triangle inequality: D(i,k)+D(k,j)≥D(i,j). The effective resistance D(i,j)=Ω_{ij} satisfies all four properties (Klein and Randić 1993).
Van Mieghem et al. (2017) defines the node i^{⋆} which is electrically the best spreader to all other nodes based on the flow equations in electrical circuits. This best spreader node i^{⋆} corresponds here to the node with the highest electrical closeness centrality, i.e. \(i^{\star } = \underset {i \in \mathcal {N}}{\operatorname {argmax~}}\tilde {c}_{i}\).
Another metric that can be used to capture the operational performance of power grids is the Yield which is the ratio of the total demand supplied at the end of an attack with respect to the initial demand of the network (Cetinay et al. 2017). This electrical demand information, which is needed to calculate Yield, is not available in our data sets of realworld power grids.
In power grids, the flow through network links are directed. Different from load transmissions in other types of networks, opposite directed flows through a link can cancel out the total flow through that link. Therefore, to calculate the maximum possible flow through a link, the absolute sum is necessary. This definition under the maximum possible loading condition is also used in the definition of full betweenness centrality by (Martín Hernández et al. 2014).
Declarations
Funding
This work was supported in part by Alliander N.V.
Availability of data and materials
The data set used in this article is available in the cited references (Cetinay 2018; IEEE 2018).
Authors’ contributions
Under the supervision of PVM, HC and KD created the methodology and HC carried out the experiments of the manuscript. HC and KD drafted the manuscript. PVM reviewed and revised the manuscript critically. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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