Robust optimal predictive control for real-time bus regulation strategy with passenger demand uncertainties in urban rapid transit

The bus is inevitably affected by various factors during the trip, which causes that the phenomenon of bus bunching often occurs. This paper proposed a nonlinear optimal control model with passenger demand and disturbance uncertainties for the real-time bus regulation in urban rapid transit. The aim of this model is to assure the stability of bus headway in the line, reduce passengers' waiting time and improve bus service level. Considering the system uncertainties and the real-time control requirement of bus regulation, a robust optimal predictive control algorithm is put forward to generate the real-time optimal bus regulation strategy based on updated delay feedback of buses by a rolling horizon strategy. Simultaneously, by using continuous interval number to describe the uncertain parameters, the robust counterpart of the formulated robust optimization model is obtained using the duality theory, which is further converted to a mixed integer convex quadratic programming problem by a linearization method, which can be easily solved to obtain the robust solution with a good performance for any value of uncertain parameters. Finally, some numerical experiments are conducted to verify the validity of the presented optimization model in improving the stability of the bus headway.