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