Journal of Advanced Transportation

Volume 2018, Article ID 8616120, 11 pages

https://doi.org/10.1155/2018/8616120

## A Two-Layer Network Dynamic Congestion Pricing Based on Macroscopic Fundamental Diagram

^{1}State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China^{2}Center for UAV Applications and ITS Research, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China

Correspondence should be addressed to Daniel(Jian) Sun; nc.ude.utjs@nusleinad

Received 22 February 2018; Revised 15 May 2018; Accepted 27 May 2018; Published 1 August 2018

Academic Editor: David Z. W. Wang

Copyright © 2018 Bangyang Wei and Daniel(Jian) Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Dynamic congestion pricing has attracted increasing attentions during the recent years. Nevertheless, limited research has been conducted to address the dynamic tolling scheme at the network level, such as to cooperatively manage two alternative networks with heterogeneous properties, e.g., the two-layer network consisting of both expressway and arterial network in the urban areas. Recently, the macroscopic fundamental diagram (MFD) developed by both field experiments and simulation tests illustrates a unimodal low-scatter relationship between the mean flow and density network widely, providing the network traffic state is roughly homogeneous. It reveals traffic flow properties at an aggregated level and sheds light on dynamic traffic management of a large network. This paper proposes a bilevel programming toll model, incorporating MFD to solve the unbalanced flow distribution problem within the two-layer transportation networks. The upper level model aims at minimizing the total travel time, while the lower level focuses on the MFD-based traffic assignment, which extends the link-based traffic assignment to network wide level. Genetic algorithm (GA) and the method of successive average were adopted for solving the proposed model, on which an online experimental platform was established using VISSIM, MATLAB, and Visual Studio software packages. The results of numerical studies demonstrate that the total travel time is decreased by imposing the dynamic toll, while the total travel time savings significantly outweigh the toll paid. Consequently, the proposed dynamic toll scheme is believed to be effective from both traffic and economic points of view.

#### 1. Introduction

Congestion pricing has been regarded as an effective traffic demand management policy that has been applied in many cities, among which Singapore was the first one to impose road congestion pricing at city wide level [1]. The development process has experienced two stages from the initial regional license scheme to the real-time dynamic electronic charging system. In United States, most tolled roads are related to new lanes or lanes that have been opened to High Occupancy Vehicles (HOV). In California, private investors have built the 91st fast lane based on the existing state highway S91, providing drivers the option of using an uncharged lane or a time-varying fee lane. These successful road congestion pricing schemes have aroused wide considerations in exerting more efficient and impartial pricing strategies in large urban networks.

The theory of congestion charging was developed by Walters [2] and Vickrey [3] that the users should pay for both the toll equal to their own travel cost and the additional congestion cost imposed on other users due to the extra travel occupying the public road resource, which is consistent with the concept called “marginal cost” in economics. Sheffi [4] found that the additional cost added in the static models can drive the user equilibrium pattern to achieve system optimum. Ferrari [5] studied the charging problem for urban transport network with elastic demand and link capacity constraints and found that the model has no solution which satisfies the capacity constraints, but additional costs can be imposed on some links so that an equilibrium solution that satisfies the capacity constraints always exists. In terms of multiuser classes, Leurent [6] designed a model that allows for differentiating travelers by means of an attribute called value of time that is continuously distributed over a real interval. Yang and Lam [7] and Yang and Bell [8] studied congestion pricing in the queue network under fixed and elastic demand, respectively, through establishing the bilevel programming model, where the lower level problem describes users’ routes choice behavior under condition of both queuing and congestion, while the upper level problem is to determine road toll to optimize a desired system’s performance. In view of the time-varying traffic demand, the static toll models need to be extended to dynamic toll models. Vickrey [9] applied deterministic queuing theory to firstly propose the dynamic pricing scheme called the bottleneck model that the travelers should pay the toll for eliminating the queuing delay without toll. Yang and Meng [10] combined the application of the space-time expanded network representing the time-varying traffic flow and the conventional network equilibrium modeling technologies. Carey and Watling [11] extended the formulation that used Cell Transmission Model (CTM) for constructing system optimum (SO) formulation to allow more general nonlinear flow-density functions and found that if the tolls computed from the dynamic system optimum (DSO) solution are imposed on the users, the DSO solution would also satisfy the criteria for a dynamic user equilibrium (DUE). Ban and Liu [12] formulated a link-nodes discrete-time dynamic toll model as a bilevel problem, where the upper level is to minimize the total weighted travel time, while the lower level is to capture the users’ route choice behavior. Laval and Castrillón [13] analyzed the time-dependent congestion pricing scheme on two alternative routes that have different bottleneck capacities and surprisingly found that the minimum total system delay can be achieved with many different pricing strategies, which are valuable for traffic management. Although both the static and dynamic settings and cases that have been addressed so far are encouraging, there are some certain deficiencies:** (i)** traffic system is not in steady state at link level validated by simulation evaluation and field experiment, and thus toll calculated based on ideal curve is not optimal [14, 15];** (ii)** existing studies on dynamic congestion pricing have been limited to certain idealized networks, because the model and computation of the link-based dynamic traffic assignment (DTA) are so complex that these research cannot be applied to a large city network practically.

Recent findings on the traffic dynamics at the network level named macroscopic fundamental diagram (MFD) have aroused wide attentions. Geroliminis and Daganzo [14, 15] used both detector data from microscopic simulation test for the San Francisco Business district in California and a data combination of fixed detectors and floating vehicle probes as sensors from field experiment in downtown Yokohama in Japan to find that MFD relating the number of vehicles in network to space mean flow or outflow presents a unimodal low-scatter relationship, if the road network is in homogenous congestion state. Scholars mainly focus on the research about its nature, influence factors, and application. Leclercq and Geroliminis [16] took advantage of the topology of the route and the variation theory to obtain MFD, in order to overcome the homogeneity rule which points out the congestion state at link level should keep pace. The influencing factors on the shape of MFD have also been investigated gradually. Geroliminis and Sun [17] found the “hysteresis phenomenon” that a clockwise loop exists between the flow and density instead of linear curve and then provided the explanation with density inhomogeneity and lack of adequate data. In terms of the influence of turning flow in intersections on MFD, Geroliminis et al. [18] found that the left traffic flow reduces the largest value of MFD. Because of the increasing recognition about MFD, researchers have started to focus on how to apply MFD to region traffic management. One of the main tasks was to investigate network division, thus obtaining a well-defined MFD. The most classic method was developed by Ji and Geroliminis [19] that divided the entire network according to the congestion feature [20, 21], and then the dynamic division problem was also studied. Keyvan-Ekbatani et al. [22] studied the feedback gate control method using the simulation network with perimeter gate control and obtained satisfying results with lower total travel time. Aboudolas and Geroliminis [23] used multireservoir networks with well-defined MFDs to design the perimeter and boundary flow control schemes that aimed at distributing the accumulation of vehicles in each reservoir as homogeneously as possible. However, deficiencies are also existed in these control schemes:** (i)** the change of signal at the cordon may have an influence on the shape of MFD, but the influence was largely ignored;** (ii)** the common perimeter control needs sufficient road space for stopping and queuing [24, 25], which may not be always allowed. Cordon-based or area-based dynamic congestion pricing at network level has been limited to some simple network due to the complexity of link-based dynamic assignment. Geroliminis and Levinson [26] combined the Vickrey’s bottleneck theory with MFD to propose a cordon-based congestion scheme, which is easier implemented in real road networks because of more convenient data collection and much easier computation. Zheng et al. [27] adopted an agent-based simulation to develop and apply a cordon-based dynamic pricing scheme, in which tolls are controlled by MFD. The above works shed light on how to apply MFD to design the dynamic congestion pricing scheme at network level.

This paper aims to combine the MFD with dynamic traffic assignment theory to design a dynamic pricing scheme in heterogeneous networks. A bilevel optimal equilibrium model combining the MFD theory is proposed, which is consistent with the traffic dynamics at network level and can also timely carry out online data analysis and output the expected toll, due to fewer data collection and lower computation cost than link-based equilibrium model. The upper level problem is to minimize the total travel time. The lower level problem is a network equilibrium model with the MFD. In detail, the upper level model is to obtain the optimal toll for the designed system objective from the manager perspective, while the lower level model is to output the expected flow distribution by solving the dynamic user equilibrium assignment with incorporating MFD. Genetic algorithm and method of successive average were used together to solve the bilevel model. Finally, a numerical test for the two-layer network consisting of the loop expressway network and the linear arterial streets was employed in the simulation environment, which combines the microscopic simulation software VISSIM, mathematic solver software MATLAB, and the development tool Visual Studio as the medium to implement the proposed bilevel toll model.

The remainder of the paper is organized as follows. Section 2 introduces the important features of MFD and explains the reliability for applying MFD to dynamic congestion pricing scheme on the two-layer network. Section 3 describes a bilevel dynamic toll programming model, the solving algorithm, and scheme for model application. In Section 4, the proposed model is tested with the two-layer network in the simulation-feedback environment. Finally, conclusions and recommendations for future work are provided in Section 5.

#### 2. Features of MFD and Research Question

##### 2.1. Features of MFD

Technically, MFD represents traffic characteristics at the network level by aggregating the link flow and density [14, 15]. The weighed space mean density and the weighted space mean flow for a road network with the homogeneous traffic distribution can be expressed as follows: where denotes an individual link in the network links set; is the traffic density of link ; denotes the length of link ; is the number of lanes of link ; is the traffic flow of link ; called the accumulation is the existing number of vehicles within the current network; is the network travel production; and denotes the total length (lane-kilometers) of the network.

The network space mean speed can be expressed as follows:A representative fundamental diagram relating the space mean flow to space mean density is resemble with the Figure 1.