Monotonicity Properties and Spectral Characterization of Power Redistribution in Cascading Failures

In this work, we apply spectral graph theory methods to study the monotonicity and structural properties of power redistribution in a cascading failure process. We demonstrate that in contrast to the lack of monotonicity in physical domain, there is a rich collection of monotonicity one can explore in the spectral domain, leading to a systematic way to define topological metrics that are monotonic. It is further shown that many useful quantities in cascading failure analysis can be unified into a spectral inner product, which itself is related to graphical properties of the transmission network. Such graphical interpretations precisely capture the Kirchhoff's law expressed in terms of graph structural properties and gauge the impact of a line when it is tripped. We illustrate that our characterization leads to a tree-partition of the network so that failure cascading can be localized.