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Phasor measurement units (PMUs) are time synchronized sensors for power system state estimation. Despite their promising advantages, large scale deployment of PMUs is still limited. One of the reasons is the high cost of their exact synchronization. In this paper, we consider the use of cheaper and less accurately synchronized PMUs, but compensate for these imperfections via signal processing methods. We introduce a statistical model for power state estimation using these impaired units. We then derive an alternating minimization technique and a parallel Kalman filter for static and dynamic estimation, respectively. Numerical examples demonstrate the improvement in estimation accuracy of our algorithm compared with traditional algorithms when phase mismatch is present. Our results suggest that the phase mismatches can be largely compensated as long as there is a sufficient number of PMUs and the delays are small.