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This paper considers the placement of synchronized Phasor Measurement Units (PMUs) in power systems to improve state estimation results. Since currently the number of PMUs deployed in the power system is far away from being sufficient achieve full system observability, one has to install the PMUs on a carefully selected subset of buses, such that the state estimation error is minimized. In this paper, we formulate the optimal PMU placement problem as an optimal experiment design problem, with a class of well-known optimality criteria, which are studied in the statistics literature. Since it has been shown that the optimal experiment design is NP-hard, we consider efficient suboptimal PMU placement algorithms. In particular, we show that the design criteria are usually related to a key property, submodularity, which often allows a greedy algorithm to achieve good approximation ratio. Specifically, we show that for typical power systems, a greedy PMU placement algorithm achieves at least 63% of the optimal reduction in total variance. The performance of the greedy PMU placement algorithms are further verified by simulation results.