Aiming to handle the complex situation that the parameters of the powertrain mounting system （PMS） of battery electric vehicles are both uncertain and correlated， the robust design optimization of PMS based on the vehicle model with 13 degree of freedom （DOF） is investigated. Firstly， based on Monte Carlo sampling and the correlation transformation method， the UTI-Monte Carlo （UMC） method for the uncertainty and correlation analysis of PMS inherent characteristic responses was proposed， where the probabilistic parameters were correlated. Then， an efficient method named the UTI-arbitrary polynomial chaos expansion （UAPCE） method was derived for the uncertainty and correlation analysis of PMS responses by combining the correlation transformation method and arbitrary polynomial chaos expansion. Next， based on the UAPCE method and correlation coefficient weighting method， a robust design optimization method of PMS was developed by considering the uncertainty and correlation of responses. Finally， a numerical example was used to verify the effectiveness of the proposed method. The analysis and optimization results based on the PMS model with 6 DOF and those based on the vehicle model with 13 DOF were compared. The results show that the calculated results using the vehicle model with 13 DOF can better reflect the vibration performance of PMS under the vehicle environment. Using the UMC as a reference method， the UAPCE method has good computational accuracy and efficiency in conducting the uncertainty and correlation analysis of PMS responses. The proposed robust design optimization method can configure the PMS parameters reasonably and improve the robustness of the system.