On the impact of PMU placement on observability and cross-validation

Significant investments have been made into deploying phasor measurement units (PMUs) on electric power grids worldwide. PMUs allow the state of the power system - the voltage phasor of system buses and current phasors of all incident transmission lines - to be directly measured. In some cases, it is also possible to infer the voltage and current phasors at neighboring buses and lines. Because PMUs are expensive, it is typically not possible to deploy enough PMUs to observe all phasors in a grid network [3, 6]. In this paper, we prove the NP-Completeness of four problems relating to PMU placements at a subset of system buses to achieve different goals: FullObserve, MaxObserve, FullObserve-XV, and MaxObserve-XV. FullObserve considers the minimum number of PMUs needed to observe all nodes, while MaxObserve considers the maximum number of buses that can be observed with a given number of PMUs. While the first of these two has been considered in the past, our formulation here generalizes the systems being considered. Next, FullObserve-XV and MaxObserve-XV consider these two problems under the constraints that PMUs must be placed “close” to each other so their measurements can be cross-validated. FullObserve-XV considers observing the entire network, while MaxObserve-XV considers maximizing the number of observed buses under this new constraint. Motivated by their high complexity, for each problem we investigate the performance of a suitable greedy approximation algorithm for PMU placement. Through simulations, we compare the performance of these algorithms with the optimal placement of PMUs over several IEEE bus systems as well as over synthetic graphs. In our simulations these algorithms yield results that are close to optimal - for all four placement problems, the greedy algorithms yield, on average, a PMU placement that is within 97% of optimal.