Voltage stability is a key factor in the reliable operation of distribution grid in the context of dynamic loads and changing power demand. Conventional voltage control methods depend on reactive power control and static adjustments, which make them ineffective in managing the rapid fluctuations caused by the unpredictable and variable nature of new load patterns such as electric vehicles and smart appliances. This paper focuses on a new approach to ensure voltage stability in low-voltage (LV) distribution grids using machine learning based forecasting and mixed integer linear programming (MILP) based optimization models for household load forecasting and optimization. The long short-term memory model is used on historical time series household load data, which accurately forecasts the day-ahead (short-term) household load. Then the MILP based optimization model for the home energy management system uses the forecasted load data to optimize load and ensure voltage stability in the LV distribution grid. In the base case, there are 238 instances out of 960, where voltage violations occur, but in the optimized case, the voltage violations were reduced to 36 instances in one day for 10 houses, leading to improved voltage stability in the LV distribution grid.