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Power domination is a two-step observation process that is used to monitorpower networks and can be viewed as a combination of domination and zeroforcing. Given a graph $G$, a subset $S\subseteq V(G)$ that can observe allvertices of $G$ using this process is known as a power dominating set of $G$,and the power domination number of $G$, $\gamma_P(G)$, is the minimum number ofvertices in a power dominating set. We introduce a new partition on thevertices of a graph to provide a lower bound for the power domination number.We also consider the power domination number of the Cartesian product of twographs, $G \Box H$, and show certain graphs satisfy a Vizing-like bound withregards to the power domination number. In particular, we prove that for anytwo trees $T_1$ and $T_2$, $\gamma_P(T_1)\gamma_P(T_2) \leq \gamma_P(T_1 \BoxT_2)$.

Graphs which satisfy a Vizing-like bound for power domination of Cartesian products