Cost-Benefit Analysis for PMU Placement in Power Grids

Power domination is a graph-theoretic model for the observance of a power grid using phasor measurement units (PMUs). There are many costs associated with the installation of a PMU, but also costs associated with not observing the entire power grid. In this work, we propose and study a power domination cost function, which balances these two costs. Given a graph $G$, a set of sensor locations $S$, and a parameter $\beta$ (which is the ratio of the cost of a PMU to the cost of non-observance of any given vertex), we define the cost function $$\mathrm{C}(G;S,\beta)=|S|+\beta\cdot (|V(G)|-|\mathrm{Obs}(G;S)|)$$ where $|\mathrm{Obs}(G;S)|$ is the number of vertices observed by sensors placed at $S\subseteq V(G)$ in the power domination process. We explore the values of $k$ for which there is a set $S$ of size $k$ that minimizes this cost function, and explore which values of $\beta$ guarantee that it is optimal to observe the entire power grid to minimize cost. We also introduce notions of marginal cost and marginal observance, providing tools to analyze how many PMUs one should install on a given power grid.