Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 769251, 9 pages

http://dx.doi.org/10.1155/2015/769251

## Research on Congestion Pricing in Multimode Traffic considering Delay and Emission

^{1}MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing 100044, China^{2}Shandong Provincial Transport Scientific Research Institute, Jinan 250031, China^{3}Center of Cooperative Innovation for Beijing Metropolitan Transportation, Beijing 100044, China^{4}School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China^{5}Changchun Municipal Engineering Design and Research Institute, Changchun 130033, China

Received 18 July 2014; Revised 24 September 2014; Accepted 25 September 2014

Academic Editor: Wuhong Wang

Copyright © 2015 Hongna Dai et al. 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

Rapid development of urbanization and automation has resulted in serious urban traffic congestion and air pollution problems in many Chinese cities recently. As a traffic demand management strategy, congestion pricing is acknowledged to be effective in alleviating the traffic congestion and improving the efficiency of traffic system. This paper proposes an urban traffic congestion pricing model based on the consideration of transportation network efficiency and environment effects. First, the congestion pricing problem under multimode (i.e., car mode and bus mode) urban traffic network condition is investigated. Second, a traffic congestion pricing model based on bilevel programming is formulated for a dual-mode urban transportation network, in which the delay and emission of vehicles are considered. Third, an improved mathematical algorithm combining successive average method with the genetic algorithm is proposed to solve the bilevel programming problem. Finally, a numerical experiment based on a hypothetical network is performed to validate the proposed congestion pricing model and algorithm.

#### 1. Introduction

With the development of urbanization and automation, the supply and demand contradiction of urban traffic has become increasingly prominent as well as traffic jam, which has resulted in a series of problems, such as increasing travel delays and traffic emissions, more frequent traffic accidents, and reducing transportation efficiency. The primary reason of urban traffic congestion is the sharp contradiction between urban transport developing and land use. To ease this problem effectively, thousands miles of urban roads have been built in many Chinese cities recently, but this is not the feasible solution to mitigate congestion. For a few decades, congestion pricing has been considered to be an effective way for traffic demand management and revenue regeneration in many cities worldwide. It can balance the spatiotemporal distribution of travel demand by making travelers readjust their travel modes and routes to avoid congested roads. Thus, the traffic congestion can be alleviated and urban traffic system-wide operation efficiency can be improved.

The original motivation of congestion pricing is to reduce traffic congestion [1–5]. Now these corresponding models can generally be classified into two categories, namely, static and dynamic. Waiters proposed an optimal static congestion pricing model according to marginal cost pricing theory, and this model firstly defined that toll charge on each road section was the difference of marginal social cost and marginal individual cost [6]. Dafermos and Sparrow established a road charging model based on marginal charging theory [7], which was applied in multiclass-user transportation network subsequently [8]. Yang and Huang extended to study marginal cost pricing with the constraint of road traffic capacity [9]. In the field of dynamic congestion pricing, Vickrey built a congestion pricing model, considering the departure time of the travelers at the bottleneck, to make toll pricing equal to queue time cost for system equilibrium [10]. Wie and Tobin developed two types of dynamic congestion pricing models based on the marginal cost pricing theory, and two dynamic charging models were appropriate for a network with stable travel demand and fluctuant travel demand, respectively [11]. Arnott et al. researched the bottleneck charging problem under the condition of random travel capacity and demand for further study [12]. Yang and Huang formulated a time-varying pricing model of a road bottleneck with elastic traffic demand based on optimal control theory [13]. Liu et al. proposed a mathematical programming with equilibrium constraint model for the speed-based toll design [14].

In recent years, given the ever-increasing concern on the sustainability of transportation, travel environments have received much attention and have been comprehensively considered in the travel mode choice. Nowadays, both emissions and other environmental factors are often taken into account in road pricing. It is believed that congestion pricing could lead to emission reduction and urban environment improvement. In this field, Johansson discussed how to apply marginal cost pricing theory to obtain the maximal net social benefit by internalizing marginal emissions and fuel consumption costs [15]. Nagurney et al. carried out a series of pioneering work on market-based policies and proposed a novel charging strategy to keep traffic emission within the limit of an environmental quality standard [16–18]. Yin and Lawphongpanich showed that a traffic flow distribution on a network with minimum emissions can always be induced by a toll charging scheme if link emission functions are increasing [19]. Chen and Yang studied nonnegative link toll schemes and cum rebate schemes for Pareto system optimum of congestion and emissions on a road network using a biobjective optimization approach [20]. Almodóvar et al. proposed a bilevel approach for estimating pollution tax to meet environmental goals [21]. Li et al. designed a road toll model considering congestion and environmental externalities on a congested network with uncertain demand [22]. Yang et al. presented an optimal toll approach for link-based emission pricing [23].

These aforementioned methods of urban road pricing mainly focus on single-mode transportation system, which consider environmental factors such as emission. However, few articles have been devoted to focusing on multimode transportation system, particularly on environment and congestion. Hence, a comprehensive congestion pricing approach, under multimode and environmental condition, should be developed to reduce travel delay and pollution emission, for the efficiency and sustainability of entire urban transportation system.

The rest of the paper is organized as follows: Section 2 constructs a bilevel programming model of dual-mode urban traffic congestion pricing considering delay and emissions and then proposes the algorithm combining the successive average method with the genetic algorithm. Section 3 performs a numerical simulation to test the applicability of the congestion pricing method on system operation and performance. Section 4 concludes the paper with a summary of the general findings.

#### 2. Model Construction and Algorithm Solution

##### 2.1. Model Construction

Firstly, the dual-mode pricing model in the paper assumes that the car and bus network are completely separated, and commuters could only transfer within the bus network. Secondly, the ultra-network theory, discussed by Nagurney and Dong [24], is applied in the urban multimode traffic network according to adding virtual nodes and links [25]. In the following network, traveler’s cost perceptual psychology is described by proportional expansion and absolute expansion and extended to links and sections [26]. Finally, the Logit-SUE model is applied to analyze route choice behavior in the multimode traffic network.

Based on the above analysis, a dual-mode congestion pricing model considering delay and emission is established for car-commuters in this section. It can be represented by the following bilevel programming model.

###### 2.1.1. The Upper-Level Model

The upper-level model of bilevel programming is the minimum sum of total delay and total emission caused by two modes:

In this paper, buses and cars are regarded as heavy vehicles and light vehicles, respectively, and their emissions are calculated by (2). In the calculation, the specific parameter values of carbon dioxide, as the only emission gas, are described by Yao and Song [27]. Therefore,in which,

The optimization goal of the upper-level model is to minimize the summation of total delay and total emission on the multimode network, where is the objective function; is the total delay weight of this function, ; is the monetary cost of unit time (RMB/time); is the commuter flow of link ; is the function of travel time cost on link ; is the monetary cost of unit emission; is the emission function of link ; is the average passengers number by car; is the length of link ; is the frequency of bus line ; is the toll charge of link ; is the upper bound of toll charges on link ; is the set for car routes; is the set for bus paths; , , , and are the parameters of emission function by car; , , and are the parameters of emission function by bus; when , is the free-flow travel time on link ; when , is the average travel time of bus on link ; , are the parameters of BPR function; is the average travel speed on link . Besides, is a 0-1 variable; if bus line goes through path , then ; otherwise .

###### 2.1.2. The Lower-Level Model

In the lower-level model, (5) is the probabilistic loading equation for Logit-SUE model. Equation (6) is the constraint for travel demand equilibrium. Equation (7) is the nonnegative flow constraint. Equation (8) is link flow conservation constraint. Equation (9) reflects commuters’ travel variance of each mode in the relative cost structure and perceived cost:wherewhere is the commuters’ flow for path between OD pair ; is the travel demand between OD pair ; is the parameter for Logit-SUE model; is the total cost on path between OD pair ; is the set of OD pairs; is the valid path set between OD pair , and ; is the incidence matrix of link , if link is on the path , , otherwise ; is the monetary cost of fuel consumption per kilometer; is the travel time perception expansion by car, and assume that ; is the travel time perception expansion on the bus, and we have that ; is the monetary cost per comfortable degree loss by bus; is the function of comfort cost on link ; when , is the traffic capacity on link ; when , is the average traffic capacity of bus line ; is the sum of bus replacement fare and transfer time cost on path between OD pair ; is the replacement fare on path ; is the perceived transfer time cost, which has an inverse relationship with the transferred bus frequency.

Valid path set, as the basic component of Logit-SUE assignment model, has important implications for traffic assignment results. There are a variety of effective path set methods proposed for SUE, such as Dial algorithm [28], full path set [29], and cumulative path set under user-equilibrium condition [30]. For simplicity, the efficient path set of absolute cost constraints is adopted to filter the feasible path. Based on multimode cost function in (9), commuters’ preferences, and travel habits, we can obtain valid paths of each mode:where is the path set between OD pair on car network; is the path set between OD pair on bus network; and are the total monetary cost (RMB) of any link between OD pair by car and bus under free-flow state, respectively; is the total cost (RMB) of path between OD pair under free-flow state.

##### 2.2. The Algorithm

In the paper, the algorithm combining the successive average method (MSA) with the genetic algorithm (GA) is developed to solve the proposed bilevel programming pricing model. It aims to solve the lower Logit-SUE model and then to accurately evaluate the applicability of each chromosome in GA. The detail algorithm solution is shown as follows.

*Step 1. *According to (12), find the valid path set between OD pair .

*Step 2. *Enter the predetermined model parameters and set the initial path flow , .

*Step 3. *Update the link flow and travel cost and calculate and substitute the generalized travel cost of each path in the set based on (8).

*Step 4. *Load network traffic flow with Logit-SUE model and calculate the additional path flow , according to (2).

*Step 5. *Calculate , according to , .

*Step 6. *Examine the astringency, if , then stop; otherwise, set and go to Step 3.

#### 3. Experimental Results

##### 3.1. Experimental Road Network

The hypothetical network for numerical test, as shown in Figure 1, consists of two traffic modes: car and bus (c and b, resp.). The label of each link has a unique two-part name, mode-symbols and serial-codes. It is important to note, however, that the serial codes of bus network are the number combination of line and section, for example, b12 represents the second link on bus line 1.