Introduction
Power grids and network science
Power grids preliminaries
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Supply node: A supply node generates the active power pi and controls the voltage magnitude |vi| at its node i.
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Demand node: At a demand node, it is possible to specify the extracted active pi and the reactive powers qi 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., pi=0 and qi=0.
DC power flow equations
Graph representations of power grids
Power grid as a simple graph
Power grid as a weighted graph
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
Ranking nodes in the simple graph representation of a power grid
Degree centrality
Eigenvector centrality
Betweenness centrality
Closeness centrality
Ranking nodes in the weighted graph representation of a power grid
Weighted degree centrality
Weighted eigenvector centrality
Flow betweenness centrality
Electrical closeness centrality
Identifying the effect of node removals in power grids
Performance metrics
The size of the giant component
The capacity of the giant component
Properties of the networks used in simulations
The effects of targeted node removals in power grids
Main lessons learned from the analyses
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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.
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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.
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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 shortest-path 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.
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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.