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This paper introduces a novel multistage state estimation architecture aimed at including synchronized phasor measurements into power system state estimation. The proposed architecture keeps unchanged the internal structure of existing SCADA-based estimators, so that phasor measurements are separately processed by a distinct estimation module. An additional stage makes use of principles from estimation fusion theory in order to combine SCADA- and PMU-based estimates. The proposed multistage scheme improves the quality of the estimates provided by the SCADA-based estimator and, under certain observability and correlation conditions, provides the same optimal results given by a hybrid simultaneous estimator (that is, an estimator that simultaneously processes both SCADA and PMU measurements). The separate processing of conventional and phasor measurements circumvents the technical challenge of conciliating, within the same estimation structure, data obtained from different measuring channels and gathered at very distinct sampling rates. In addition, the execution times tend to be significantly less than those required by hybrid simultaneous schemes. The paper describes in detail the multistage structure of the proposed state estimator and also addresses the benefits brought about by phasor measurement processing to state estimation accuracy. Results of several case studies conducted on the IEEE 57-bus, 118-bus, and 300-bus benchmark systems are used to illustrate the features of the proposed strategy.