This study proposes a new formulation for the departure time stochastic user equilibrium (DTSUE) problem based on the generalized bathtub model. The generalized bathtub model extends the classic bathtub model to capture various distributions of the trip length by introducing a new state variable: the number of active trips with remaining distances greater than or equal to a threshold. The traffic dynamics are reformulated by four partial differential equations that track the distribution of the remaining trip lengths. In this study, we first reformulate the stochastic user equilibrium problem for the morning commute problem using the state variable of the generalized bathtub model in order to capture the heterogeneity of trip attributes. In particular, the proposed framework is able to address any distribution for desired arrival time and trip length. Second, we show that the problem in continuous form is a fixed-point problem. Then we apply the cell discretization methods to address the trip-based setting. Third, Two numerical schemes are proposed to solve the discrete version of the DTSUE model. Fourth, we propose a criterion based on the Kullback-Leibler divergence to analyze the convergence of the DTSUE solution algorithm. Fifth, the proposed framework is applied to the real demand profile of the large-scale network of Lyon North for the morning peak hour. We perform an analysis of the equilibrium characteristics and convergence. The results show that the model is valid for addressing the morning commute problem.