The installation of smart meters at customer premises provides opportunities for the monitoring of distribution grids. This paper addresses the problem of improving the observability of low-voltage distribution grids using smart metering infrastructure. In particular, this paper deals with the application of state estimation algorithm using smart meter measurements for near-real-time monitoring of low-voltage distribution grids. This application is proposed to use a nonlinear weighted least squares method-based algorithm for estimating the node voltages from minimum number of smart meter measurements. This paper mainly deals with sensitivity analysis of the state estimation algorithm with respect to multiple uncertainties for, e.g., measurements errors, line parameter errors, and pseudo-measurements. Simulation studies are conducted to estimate the accuracy of the DSSE under various operating scenarios of a real-life low-voltage grid, and cost-effective ways to improve the accuracy of the state estimation algorithm are also evaluated. The paper concludes that by using smart meter measurements from few locations, voltage profiles of the low-voltage grid can be estimated with reasonable accuracy in near-real-time.