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Power flow studies are typically used to determine the steady state or operating conditions of power systems for specified sets of load and generation values, and is one of the most intensely used tools in power engineering. When the input conditions are uncertain, numerous scenarios need to be analyzed to cover the required range of uncertainty. Under such conditions, reliable solution algorithms that incorporate the effect of data uncertainty into the power flow analysis are required. To address this problem, this paper proposes a new solution methodology based on the use of affine arithmetic, which is an enhanced model for self-validated numerical analysis in which the quantities of interest are represented as affine combinations of certain primitive variables representing the sources of uncertainty in the data or approximations made during the computation. The application of this technique to the power flow problem is explained in detail, and several numerical results are presented and discussed, demonstrating the effectiveness of the proposed methodology, especially in comparison to previously proposed interval arithmetic's techniques.