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Sensors called phasor measurement units (PMUs) are used to monitor theelectric power network. The power domination problem seeks to minimize thenumber of PMUs needed to monitor the network. We extend the power dominationproblem and consider the minimum number of sensors and appropriate placement toensure monitoring when $k$ sensors are allowed to fail with multiple sensorsallowed to be placed in one location. That is, what is the minimum multiset ofthe vertices, $S$, such that for every $F\subseteq S$ with $|F|=k$, $S\setminusF$ is a power dominating set. Such a set of PMUs is called a$k$-PMU-defect-robust power domination set. This paper generalizes the workdone by Pai, Chang and Wang in 2010 on fault-tolerant power domination, whichdid not allow for multiple sensors to be placed at the same vertex. We providegeneral bounds and determine the $k$-PMU-defect-robust power domination numberof some graph families.

An Introduction to PMU-Defect-Robust Power Domination: Bounds, Bipartites, and Block Graphs