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Distribution systems require ad hoc estimators, distribution system state estimation (DSSE) techniques, to acquire knowledge about the system status. An incorrect evaluation of the accuracy of the DSSE creates decision risks in network management. The possible variations in the network parameter values and the decays of the metrological characteristics of the measurement system elements are uncertainty sources very often not considered. Considering these possible lacks of accuracy, this paper focuses on the robustness of distributed measurement systems aimed to obtain accurate DSSE results. The problem of the proper assessment of the accuracy of the DSSE results obtained through a weighted least squares (WLS) approach is faced. A method capable of including different uncertainty sources in the uncertainty estimation of the WLS approach is presented. Furthermore, this paper proposes an optimal meter placement algorithm robust with respect to possible malfunctions in measurement system components. The results obtained on a portion of an Italian distribution network, along with their accuracy, are presented and discussed.