Steady-state voltage security region (SVSR) is an important criterion that can intuitively and effectively evaluate the power system's operation security and overall safety margin. However, there is no literature on its application to electric vehicle (EV) proliferation. Since EV introduces dynamism and uncertainties to the grid, most existing indices cannot assess the grid accurately. Given this, we propose a load margin index (LMI) based on SVSR considering uncertainties. This is used to assess the impact of EV proliferation via a bi-layer optimal EV scheduling model. The upper layer of the model solves a dispatching plan for EV aggregators (EVA) by maximizing the LMI and minimizing the deviation between the upper and lower layer power schedules. The lower layer models each EV's charging/discharging strategy by minimizing the EV's operating cost, the EV user's target residual energy, and the deviation between the lower and upper layer schedules. We also employ a real-time scheduling plan to further adjust the day-ahead scheduling due to forecast errors. A case study based on the IEEE-33 node distribution network verifies the practicability and effectiveness of the proposed method.