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This article presents a distributed model-predictive control (MPC) design for real-time voltage control in power systems, including an online method to estimate the bus admittance matrix $\mathbf{Y}$ to let it be time-varying and unknown a priori. The prevalent control designs are either (a) centralized, providing optimal solutions but less scalable and susceptible to single-point failures/attacks, or (b) decentralized or localized, having increased scalability and attack resilience but are suboptimal. The proposed distributed solution offers the attractive features of both methodologies, where neighboring nodes share state information to attain a globally optimal solution. In addition, the presented framework provides a data-driven estimation of Y to circumvent the challenging issue of acquiring accurate knowledge of the line impedance (required to form Y). We first introduce the centralized version of the predictive voltage control problem and then transfer it to a distributed version. The distributed version is solved via the alternating direction method of multipliers (ADMM), using only local measurements and communication leveraging the graph structure of the power system. The proposed framework is resilient to prediction uncertainty, modeling error, and communication link failure, and the in-built redundancy within the proposed framework supports anomaly detection in cyberattacks. We validate the proposed methodology for IEEE-30 bus, IEEE-57 bus transmission systems, and IEEE-123 bus distribution systems.