In this paper, we continue the study of power domination in graphs (see [T. W. Haynes et al., SIAM J. Discrete Math., 15 (2002), pp. 519--529; P. Dorbec et al., SIAM J. Discrete Math., 22 (2008), pp. 554--567; A. Aazami et al., SIAM J. Discrete Math., 23 (2009), pp. 1382--1399]). Power domination in graphs was birthed from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A set of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set following a set of rules (according to Kirschoff laws) for power system monitoring. The minimum cardinality of a power dominating set of a graph is its power domination number. We show that the power domination of a connected cubic graph on $n$ vertices different from $K_{3,3}$ is at most $n/4$ and this bound is tight. More generally, we show that for $k \ge 1$, the $k$-power domination number of a connected $(k+2)$-regular graph on $n$ vertices different from $K_{k+2,k+2}$ is at most $n/(k+3)$, where the $1$-power domination number is the ordinary power domination number. We show that these bounds are tight.

The power system monitoring problem asks for as few as possible measurement devices to be put in an electric power system. The problem has a graph theory model involving power dominating sets in graphs. The power domination number $\gamma_P(G)$ of G is the minimum cardinality of a power dominating set. Dorfling and Henning [Discrete Appl. Math., 154 (2006), pp. 1023–1027] determined the power domination number of the Cartesian product of paths. In this paper the power domination number is determined for all direct products of paths except for the odd component of the direct product of two odd paths. For instance, if n is even and C a connected component of $P_m\times P_n$, where m is odd or $m\geq n$, then $\gamma_P(C)=\left\lceil n/4 \right\rceil$. For the strong product we prove that $\gamma_P(P_n \boxtimes P_m) = \max\{\lceil n/3\rceil, \lceil (n+m-2)/4\rceil\}$, unless $3m-n-6 \equiv 4\pmod 8$. The power domination number is also determined for an arbitrary lexicographic product.