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The paper presents a fully distributed framework for sequential recursive state estimation in inter-connected electrical power systems. Specifically, the setup considered involves a grid partitioned into multiple control areas that communicate over a sparse communication network. In the absence of a global sensor data fusion center (the conventional centralized SCADA) and with sensing model uncertainties, an adaptive distributed state estimation approach, the DAE, is proposed in which the system control areas engage in a collaborative joint (model) learning and (state) estimation procedure through sequential information exchange over the pre-assigned communication network. The proposed distributed estimation methodology is recursive, in that, each system control area refines its state estimate at a given sampling instant by suitably combining its past estimate with the newly collected local measurement(s) and the information obtained from its communication neighbors. Under rather weak assumptions of global observability and connectivity of the control area communication network, the proposed distributed adaptive scheme is shown to yield consistent system state estimates (i.e., estimates that converge to the true system state in the large sample limit), the convergence rate being optimal in the Fisher information sense. As discussed, the proposed approach based on local communication and computation is suitable for real-time implementation as opposed to conventional centralized SCADA based estimation architectures with periodic data gathering and processing, thus being potentially more responsive and adaptive to sensed data generated by advanced non-conventional sensing resources like the PMUs with significantly higher system sampling rates.