Kinematic wave models of sag and tunnel bottlenecks

Sags and tunnels are critical traffic bottlenecks, as they can cause capacity reduction, capacity drop, and extreme low acceleration rates when vehicles accelerate away from the upstream queue. In this paper, we present a behavioral kinematic wave model to explain the three bottleneck effects of sags and tunnels. Assuming increasing time gaps, we derive location-dependent triangular fundamental diagrams to explain the capacity reduction effect; with a bounded acceleration constraint on the stationary states inside the capacity reduction zone, we demonstrate the occurrence of capacity drop and derive a formula to calculate the dropped capacity from the fundamental diagram, road geometry, and acceleration process; from the structure of continuous standing waves we verify the low acceleration rate out of the upstream queue. We also present a simplified phenomenological model of capacity drop at sag/tunnel bottlenecks and two Cell Transmission Models for numerical simulations. With four stationary trajectories at the Kobotoke tunnel in Japan, we calibrate and validate the behavioral model and find that the theoretical predictions match the observations very well. This study can help to develop better design and control strategies to improve the performance of a sag or tunnel bottleneck.