# Electrical Centrality Measures for Power Grids

## Abstract

This chapter investigates measures of centrality that are applicable to power grids. Centrality measures are used in network science to rank the relative importance of nodes and edges of a graph. Here we define new measures of centrality for power grids that are based on its functionality. More specifically, the coupling of the grid network can be expressed as the algebraic equation *YU* = *I*, where *U* and *I* represent the vectors of complex bus voltage and injected current phasors; and *Y* is the network admittance matrix which is defined not only by the connecting topology but also by the network’s electrical parameters and can be viewed as a complex-weighted Laplacian. We show that the relative importance analysis based on centrality in graph theory can be performed on power grid network with its electrical parameters taken into account. In the chapter the proposed electrical centrality measures are experimented with on the NYISO-2935 system and the IEEE 300-bus system. The centrality distribution is analyzed in order to identify important nodes or branches in the system which are of essential importance in terms of system vulnerability. A number of interesting discoveries are also presented and discussed regarding the importance rank of power grid nodes and branches.

## Keywords

Centrality Measure Power Grid Betweenness Centrality Closeness Centrality Eigenvector Centrality## References

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