Unifying Analytical Methods with Numerical Methods for Traffic System Modeling and Control

Shockwaves lead to speed variation and capacity drop, which hamper the stationarity and throughput of traffic network greatly in reality. In order to dominate or suppress shockwaves, there exist two philosophies: the analytical and numerical methods to investigate various traffic management schemes. However, both are studied completely separately in the existing literature. In this paper, we primarily focus on the uniformity and combination of the two philosophies, especially in terms of traffic evolution and shockwave trajectory extraction. Aiming at exploring the uniformity, numerical methods are equipped with gradient boundary detection and polar-parameter coordinate projection to iteratively calculate traffic states and extract shockwave trajectories as line segments. By contrast, analytical methods derive traffic evolution from fundamental diagrams and present shockwave trajectories as vector graphics in the traffic time-space diagram. Furthermore, to rationally measure the accuracy of extracted trajectories, a calibration model is established to decrease angle and distance errors yielded by the two methods. In the uncontrolled and variable speed limit (VSL)-controlled bottleneck scenarios, the simulation results have shown that: 1) both analytical and numerical methods have the capability to precisely describe traffic evolution and the effects of VSL strategies on recovering traffic throughput; 2) shockwave trajectories extracted by the two methods are coincident, with the significant reduction of relative distance errors from 15%-80% to 0.1%-3%; and 3) it is quite promising to take full advantage of both the visual/intuitive nature of analytical methods and the iterative optimization of numerical methods to investigate efficient traffic management strategies.