The grey Bass model serves as a valuable tool in forecasting time series portraying inverted U-shaped characteristics. However, the existing grey Bass model still has some issues in inaccuracies for the Cusum operator and the rationale behind selecting the initial value for parameter estimation. To improve the accuracy and applicability of the grey Bass model, the generalized grey Bass model is developed, incorporating the physics-preserving Cusum operator with second-order accuracy. Initially, the relationship between the generalized grey Bass and the traditional grey Bass models is elucidated via parameter transformation. Subsequently, opting for the first observation as the initial value is proven to be not only accurate but also a highly efficient strategy. Additionally, extensive simulations are conducted to compare both models in terms of robustness against discretization errors, measurement noise, and sample size. Finally, the generalized grey Bass model is applied to two real-world cases and evaluated against competitive models regarding accuracy and stability. Results demonstrate that the generalized grey Bass model exhibits superior precision and exceptional modeling performance.