Optimal distance- and time-dependent area-based pricing with the Network Fundamental Diagram

Abstract

Given the efficiency and equity concerns of a cordon toll, this paper proposes a few alternative distance-dependent area-based pricing models for a large-scale dynamic traffic network. We use the Network Fundamental Diagram (NFD) to monitor the network traffic state over time and consider different trip lengths in the toll calculation. The first model is a distance toll that is linearly related to the distance traveled within the cordon. The second model is an improved joint distance and time toll (JDTT) whereby users are charged jointly in proportion to the distance traveled and time spent within the cordon. The third model is a further improved joint distance and delay toll (JDDT) which replaces the time toll in the JDTT with a delay toll component. To solve the optimal toll level problem, we develop a simulation-based optimization (SBO) framework. Specifically, we propose a simultaneous approach and a sequential approach, respectively, based on the proportional-integral (PI) feedback controller to iteratively adjust the JDTT and JDDT, and use a calibrated large-scale simulation-based dynamic traffic assignment (DTA) model of Melbourne, Australia to evaluate the network performance under different pricing scenarios. While the framework is developed for static pricing, we show that it can be easily extended to solve time-dependent pricing by using multiple PI controllers. Results show that although the distance toll keeps the network from entering the congested regime of the NFD, it naturally drives users into the shortest paths within the cordon resulting in an uneven distribution of congestion. This is reflected by a large clockwise hysteresis loop in the NFD. In contrast, both the JDTT and JDDT reduce the size of the hysteresis loop while achieving the same control objective. We further conduct multiple simulation runs with different random seed numbers to demonstrate the effectiveness of different pricing models against simulation stochasticity. However, we postulate that the feedback control is not applicable with guaranteed convergence if the periphery of the cordon area becomes highly congested or gridlocked.

Surrogate‐based toll optimization in a large‐scale heterogeneously congested network

“…This drop, as was also observed in Gu, Shafiei, et al. (), results from the reduced inflow or demand to the PZ due to the presence of pricing. An extreme and apparently unrealistic scenario is that we implement an exceptionally high toll price whereby no one would enter the PZ.…”

“…The scale effect is a major reason given that the PZ only covers a relatively small area of the entire network (see Figure a). It is therefore no surprise that the performance of the entire network changes very little (see Figure e) when pricing a relatively small subnetwork (Gu, Shafiei, et al., ). The performance of the entire network may even reduce, for example, in the third replication, due to the redistribution of detour vehicles around the PZ that is highly dependent on the network configuration and the structure and magnitude of the demand, and hence being case‐specific.…”

“…See Aboudolas and Geroliminis (); Geroliminis et al. (); Keyvan‐Ekbatani, Kouvelas, Papamichail, and Papageorgiou (); Ramezani, Haddad, and Geroliminis () for NFD‐based perimeter control, and Gu, Shafiei, Liu, and Saberi (); Simoni, Pel, Waraich, and Hoogendoorn (); Zheng, Rérat, and Geroliminis (); Zheng, Waraich, Axhausen, and Geroliminis () for NFD‐based pricing.…”

“…The defect is that the time toll component as part of the JDTT tends to overcharge drivers as a longer link typically requires a larger travel time despite being uncongested. As such, an improved joint distance and delay toll (JDDT) was proposed (Gu, Shafiei, et al., ) and considered in this paper.…”

“…To address this network design problem with an expensive‐to‐evaluate objective function featuring nonconvexity, nonlinearity, and nonclosed form, simulation optimization (SO) or simulation‐based optimization has recently been investigated and advocated (Amaran, Sahinidis, Sharda, & Bury, ; Osorio & Bierlaire, ). When SO is applied to solve the TLP, existing methods can be classified into two broad categories of feedback control (Gu, Shafiei, et al., ; Simoni et al., ; Zheng et al., ; Zheng et al., ) and surrogate‐based optimization (Chen et al., ; Chen, Xiong, He, Zhu, & Zhang, ; Chen, Zhang, He, Xiong, & Zhu, ; Chow & Regan, ; Ekström, Kristoffersson, & Quttineh, ; He, Chen, Xiong, Zhu, & Zhang, ).…”