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Distribution systems are just partially monitored networks. As a consequence, they need ad hoc estimators to evaluate the quantities of interest. Distribution system state estimation (DSSE) is a digital technique capable of processing information from a distribution network. DSSE uses both real time measurements and all a priori information, namely pseudomeasurements, to estimate the status of the system. Typically, DSSE is based on the weighted least squares (WLS) optimization technique. WLS is computationally attractive and efficient but it could be unfit for the uncertainty evaluation, particularly for Smart Grid purposes. An incorrect accuracy estimation for the DSSE results could create decision risks in network management. In particular, this paper faces the problem of uncertainty evaluation in a traditional WLS approach for DSSE with respect to the possible variations in the network parameter values. The results obtained on both a small rural distribution network and a portion of Italian distribution network are presented and discussed.