Trial-and-error method for congestion pricing scheme under side-constrained probit-based stochastic user equilibrium conditions

A toll pattern that can restrict link flows on the tolled links to some predetermined thresholds is named as effective toll solution, which can be theoretically obtained by solving a side-constraint traffic assignment problem. Considering the practical implementation, this paper investigates availability of an engineering-oriented trial-and-error method for the effective toll pattern of cordon-based congestion pricing scheme, under side-constrained probit-based stochastic user equilibrium (SUE) conditions. The trial-and-error method merely requires the observed traffic counts on each entry of the cordon. A minimization model for the side-constrained probit-based SUE problem with elastic demand is first proposed and it is shown that the effective toll solution equals to the product of value of time and optimal Lagrangian multipliers with respect to the side constraints. Then, employing the Lagrangian dual formulation of the minimization method, this paper has built a convergent trial-and-error method. The trial-and-error method is finally tested by a numerical example developed from the cordon-based congestion pricing scheme in Singapore.     

Bayesian network-based formulation and analysis for toll road utilization supported by traffic information provision

Congestion pricing has been proposed and investigated as an effective means of optimizing traffic assignment, alleviating congestion, and enhancing traffic operation efficiencies. Meanwhile, advanced traffic information dissemination systems, such as Advanced Traveler Information System (ATIS), have been developed and deployed to provide real-time, accurate, and complete network-wide traffic information to facilitate travelers’ trip plans and routing selections. Recent advances in ATIS technologies, especially telecommunication technology, allow dynamic, personalized, and multimodal traffic information to be disseminated and impact travelers’ choices of departure times, alternative routes, and travel modes in the context of congestion pricing. However, few studies were conducted to determine the impact of traffic information dissemination on toll road utilizations. In this study, the effects of the provisions of traffic information on toll road usage are investigated and analyzed based on a stated preference survey conducted in Texas. A Bayesian Network (BN)-based approach is developed to discover travelers’ opinions and preferences for toll road utilization supported by network-wide traffic information provisions. The probabilistic interdependencies among various attributes, including routing choice, departure time, traffic information dissemination mode, content, coverage, commuter demographic information, and travel patterns, are identified and their impacts on toll road usage are quantified. The results indicate that the BN model performs reasonably well in travelers’ preference classifications for toll road utilization and knowledge extraction. The BN Most Probable Explanation (MPE) measurement, probability inference and variable influence analysis results illustrate travelers using highway advisory radio and internet as their primary mode of receiving traffic information are more likely to comply with routing recommendations and use toll roads. Traffic information regarding congested roads, road hazard warnings, and accident locations is of great interest to travelers, who tend to acquire such information and use toll roads more frequently. Travel time formation for home-based trips can considerably enhance travelers’ preferences for toll road usage. Female travelers tend to seek traffic information and utilize toll roads more frequently. As expected, the information provided at both pre-trip and en-route stages can positively influence travelers’ preferences for toll road usage. The proposed methodology and research findings advance our previous study and provide insight into travelers’ behavioral tendencies concerning toll road utilization in support of traffic information dissemination.     

Enhanced traffic information dissemination to facilitate toll road utilization: a nested logit model of a stated preference survey in Texas

This paper investigates the effects of the provision of traffic information on toll road usage based on a stated preference survey conducted in central Texas. Although many researchers have studied congestion pricing and traffic information dissemination extensively, most of them focused on the effects that these instruments individually produce on transportation system performance. Few studies have been conducted to elaborate on the impacts of traffic information dissemination on toll road utilization. In this study, 716 individuals completed a survey to measure representative public opinions and preferences for toll road usage in support of various traffic information dissemination classified by different modes, contents, and timeliness categories. A nested logit model was developed and estimated to identify the significant attributes of traffic information dissemination, traveler commuting patterns, routing behavior, and demographic characteristics, and analyze their impacts on toll road utilization. The results revealed that the travelers using dynamic message sign systems as their primary mode of receiving traffic information are more likely to choose toll roads. The potential toll road users also indicated their desire to obtain traffic information via internet. Information regarding accident locations, road hazard warnings, and congested roads is frequently sought by travelers. Furthermore, high-quality congested road information dissemination can significantly enhance travelers’ preferences of toll road usage. Specifically the study found that travelers anticipated an average travel time saving of about 11.3 min from better information; this is about 30 % of travelers’ average one-way commuting time. The mean value of the time savings was found to be about $11.82 per hour, close to ½ of the average Austin wage rate. The model specifications and result analyses provide in-depth insights in interpreting travelers’ behavioral tendencies of toll road utilization in support of traffic information. The results are also helpful to shape and develop future transportation toll system and transportation policy.     

Road pricing modeling for hyper-congestion

Recently there has been a resurgence in the interest of road pricing. Most studies adopt the static modeling paradigm, typically using either separable monotone or backward-bending link travel time functions for the analysis. In this study, through the shockwave analysis, we show that separable backward-bending functions are not appropriate for modeling hyper-congestion and hence road pricing. In the absence of queue spillback, link travel time is a monotone increasing function of inflow. However, in the presence of queue spillback, we show that the static paradigm even with a monotone travel time function cannot adequately portray the congestion phenomenon. In some cases, the tolls determined by the static paradigm can be even detrimental, worsening rather than alleviating the congestion problem. In the end, to model congested networks properly, perhaps one has no other choices but to adopt a modeling paradigm that faithfully captures both the temporal as well as the spatial dimensions of traffic queuing.     

Capturing value of reliability through road pricing in congested traffic under uncertainty

Empirical studies showed that travel time reliability, usually measured by travel time variance, is strongly correlated with travel time itself. Travel time is highly volatile when the demand approaches or exceeds the capacity. Travel time variability is associated with the level of congestion, and could represent additional costs for travelers who prefer punctual arrivals. Although many studies propose to use road pricing as a tool to capture the value of travel time (VOT) savings and to induce better road usage patterns, the role of the value of reliability (VOR) in designing road pricing schemes has rarely been studied. By using road pricing as a tool to spread out the peak demand, traffic management agencies could improve the utility of travelers who prefer punctual arrivals under traffic congestion and stochastic network conditions. Therefore, we could capture the value of travel time reliability using road pricing, which is rarely discussed in the literature. To quantify the value of travel time reliability (or reliability improvement), we need to integrate trip scheduling, endogenous traffic congestion, travel time uncertainty, and pricing strategies in one modeling framework. This paper developed such a model to capture the impact of pricing on various costs components that affect travel choices, and the role of travel time reliability in shaping departure patterns, queuing process, and the choice of optimal pricing. The model also shows the benefits of improving travel time reliability in various ways. Findings from this paper could help to expand the scope of road pricing, and to develop more comprehensive travel demand management schemes.     

Time-dependent area-based pricing for multimodal systems with heterogeneous users in an agent-based environment

In this paper, we investigate an area-based pricing scheme for congested multimodal urban networks with the consideration of user heterogeneity. We propose a time-dependent pricing scheme where the tolls are iteratively adjusted through a Proportional–Integral type feedback controller, based on the level of vehicular traffic congestion and traveler’s behavioral adaptation to the cost of pricing. The level of congestion is described at the network level by a Macroscopic Fundamental Diagram, which has been recently applied to develop network-level traffic management strategies. Within this dynamic congestion pricing scheme, we differentiate two groups of users with respect to their value-of-time (which related to income levels). We then integrate incentives, such as improving public transport services or return part of the toll to some users, to motivate mode shift and increase the efficiency of pricing and to attain equitable savings for all users. A case study of a medium size network is carried out using an agent-based simulator. The developed pricing scheme demonstrates high efficiency in congestion reduction. Comparing to pricing schemes that utilize similar control mechanisms in literature which do not treat the adaptivity of users, the proposed pricing scheme shows higher flexibility in toll adjustment and a smooth behavioral stabilization in long-term operation. Significant differences in behavioral responses are found between the two user groups, highlighting the importance of equity treatment in the design of congestion pricing schemes. By integrating incentive programs for public transport using the collected toll revenue, more efficient pricing strategies can be developed where savings in travel time outweigh the cost of pricing, achieving substantial welfare gain.     

Existence of self-financing and Pareto-improving congestion pricing: Impact of value of time distribution

This paper considers a static congestion pricing model in which travelers select a mode from either, driving on highway or taking public transit, to minimize a combination of travel time, operating cost and toll. The focus is to examine how travelers’ value of time (VOT), which is continuously distributed in a population, affects the existence of a pricing-refunding scheme that is both self-financing (i.e. requiring no external subsidy) and Pareto-improving (i.e. reducing system travel time while making nobody worse off). A condition that insures the existence of a self-financing and Pareto-improving (SFPI) toll scheme is derived. Our derivation reveals that the toll authority can select a proper SFPI scheme to distribute the benefits from congestion pricing through a credit-based pricing scheme. Under mild assumptions, we prove that an SFPI toll always exists for concave VOT functions, of which the linear function corresponding to the uniform distribution is a special case. Existence conditions are also established for a class of rational functions. These results can be used to analyze more realistic VOT distributions such as log-normal distribution. A useful implication of our analysis is that the existence of an SFPI scheme is not guaranteed for general functional forms. Thus, external subsidies may be required to ensure Pareto-improving, even if policy-makers are willing to return all toll revenues to road users.     

Viability of new road infrastructure with heterogeneous users

This paper explores the importance of heterogeneity in value of time and route choice when assessing the viability of new road infrastructure to alleviate congestion problems. The model incorporates strategic interaction between road operators in a cost-benefit framework and several competitive regimes are considered. It is then employed to establish the financial and socio-economic viability of a congestion pricing demonstration entering Madrid city centre, where road users have to choose between a free but highly congested road and a priced free-flowing road (semi-private regime). A logit estimation is undertaken with information from a questionnaire among road users in the Eastern Madrid area to obtain users’ value of time and of congestion.The tolls obtained generate a traffic reallocation towards the new roadway such that revenues suffice to render the infrastructure socio-economically viable. The private and the low toll regimes generate similar welfare gains that are close to the first best. Yet the former supposes large losses to users. The low toll and the semi-private regimes do not raise such distributional concerns. However, the low toll regime requires a sufficiently high traffic growth rate to make it financially viable; this does not happen for the other competitive regimes.     

A forecast air pollution model applied to a hypothetical urban road pricing scheme: An empirical study in Madrid

Reducing the air pollution from increases in traffic congestion in large cities and their surroundings is an important problem that requires changes in travel behavior. Road pricing is an effective tool for reducing air pollution, as reflected currently urban road pricing outcomes (Singapore, London, Stockholm and Milan). A survey was conducted based on establishing a hypothetical urban road pricing system in Madrid (a random sample size n = 1298). We developed a forecast air pollution model with time series analysis to evaluate the consequences of possible air pollution decreases in Madrid. Results reveal that the hypothetical road pricing for Madrid could have highly significant effects on decreasing air pollution outside of the city and in the inner city during the peak operating time periods of maximum congestion (morning peak hours from 7:00 to 10:00 and evening peak hours from 18:00 to 20:00). Furthermore, this system could have significant positive effects on a shift toward using public transport and non-motorized modes inside the hypothetical toll zone. This reveals that the system has a high capacity to motivate a decrease in air pollution and impose more sustainable behavior for public transport users.     

Minimizing CO2e emissions by setting a road toll

The main purpose of this paper is to develop a bi-level pricing model to minimize the CO₂e emissions and the total travel time in a small road network. In the lower level of the model, it is assumed that users of the road network find a dynamic user equilibrium which minimizes the total costs of those in the system. For the higher level of the model, different road toll strategies are applied in order to minimize the CO₂e emissions. The model has been applied to an illustrative example. It shows the effects on traffic flows, revenues, total time and CO₂e emissions for different numbers of servers collecting tolls and different pricing strategies over a morning peak traffic period. The results show that the CO₂e emissions produced can be significantly affected by the number of servers and the type of toll strategy employed. The model is also used to find the best toll strategy when there is a constraint on the revenue that is required to be raised from the toll and how this affects the emissions produced. Further runs compare strategies to minimize the CO₂e emissions with those that minimize total travel time in the road system. In the illustrative example, the results for minimizing CO₂e emissions are shown to be similar to the results obtained from minimizing the total travel time.     

Profit maximization by a private toll road with cars and trucks

This paper examines the profit maximizing behavior of a private firm which operates a toll road competing against a free alternative in presence of cars and trucks. Trucks differ from cars in value of time (VOT), congestion externality, pavement damage, and link travel time function. We find that the firm takes either a car-strategy or a truck-strategy for profit maximization. For a traffic mix with relatively large car volume and small truck volume, the car-strategy results in no trucks using the toll road, while the truck-strategy results in all trucks using the toll road. We derive the equilibrium flow pattern under any combination of car-toll and truck-toll, based on which we identify a profit-maximizing frontier and a strategy-switching frontier in the car-toll and truck-toll two-dimensional space. By geometrically comparing the two frontiers, we establish general conditions under which each strategy will be taken, which suggest that the truck-to-car VOT ratio, the total traffic demand, and the difference in travel distance between the two roads are critical in shaping the firm's strategy.     

Learning marginal-cost pricing via a trial-and-error procedure with day-to-day flow dynamics

This paper investigates the convergence of the trial-and-error procedure to achieve the system optimum by incorporating the day-to-day evolution of traffic flows. The path flows are assumed to follow an ‘excess travel cost dynamics’ and evolve from disequilibrium states to the equilibrium day by day. This implies that the observed link flow pattern during the trial-and-error procedure is in disequilibrium. By making certain assumptions on the flow evolution dynamics, we prove that the trial-and-error procedure is capable of learning the system optimum link tolls without requiring explicit knowledge of the demand functions and flow evolution mechanism. A methodology is developed for updating the toll charges and choosing the inter-trial periods to ensure convergence of the iterative approach towards the system optimum. Numerical examples are given in support of the theoretical findings.     

Modelling rain effects on risk‐taking behaviours of multi‐user classes in road networks with uncertainty

This paper proposes a new travel time reliability‐based traffic assignment model to investigate the rain effects on risk‐taking behaviours of different road users in networks with day‐to‐day demand fluctuations and variations in travel time. A generalized link travel time function is used to capture the rain effects on vehicle travel times and road conditions. This function is further incorporated into daily demand variations to investigate those travel time variations arising from demand uncertainty and rain condition. In view of these rain effects, road users' perception errors on travel times and risk‐taking behaviours on path choices are incorporated in the proposed model with the use of a logit‐based stochastic user equilibrium framework. This new model is formulated as a variational inequality problem in terms of path flows. A numerical example is used to illustrate the application of the proposed model for assessment of the rain effects on road networks with uncertainty.     

Traffic congestion pricing methodologies and technologies

This paper reviews the methods and technologies for congestion pricing of roads. Congestion tolls can be implemented at scales ranging from individual lanes on single links to national road networks. Tolls can be differentiated by time of day, road type and vehicle characteristics, and even set in real time according to current traffic conditions. Conventional toll booths have largely given way to electronic toll collection technologies. The main technology categories are roadside-only systems employing digital photography, tag & beacon systems that use short-range microwave technology, and in-vehicle-only systems based on either satellite or cellular network communications. The best technology choice depends on the application. The rate at which congestion pricing is implemented, and its ultimate scope, will depend on what technology is used and on what other functions and services it can perform.     

Road Pricing models with maintenance cost

According to the Federal Highway Administration of the United States, maintenance expenditure takes up more than 25% of road revenue disbursement and this percentage has been increasing gradually. The reason for the increment in maintenance cost is that there lacks incentives for road users to take this cost component into their driving behavior. That is, different classes of vehicles should be levied different levels of congestion tax due to the different degrees of damage on the highway if a road pricing policy is implemented. This paper intends to incorporate this concept into road pricing literature by introducing two types of vehicles. After the analysis of the problem, we find that different types of vehicles should be charged different tolls. The toll includes not only the travel delay cost of one's own vehicle and the other types of vehicles, but also the marginal maintenance cost that is dependent on the traffic flow. A set of numerical examples is provided to demonstrate the theoretical analyses. The result shows that both the welfare and cost coverage rate will increase when the road pricing mechanism takes the maintenance cost factor into account.     

A dynamic cordon pricing scheme combining the Macroscopic Fundamental Diagram and an agent-based traffic model

Pricing is considered an effective management policy to reduce traffic congestion in transportation networks. In this paper we combine a macroscopic model of traffic congestion in urban networks with an agent-based simulator to study congestion pricing schemes. The macroscopic model, which has been tested with real data in previous studies, represents an accurate and robust approach to model the dynamics of congestion. The agent-based simulator can reproduce the complexity of travel behavior in terms of travelers’ choices and heterogeneity. This integrated approach is superior to traditional pricing schemes. On one hand, traffic simulators (including car-following, lane-changing and route choice models) consider travel behavior, i.e. departure time choice, inelastic to the level of congestion. On the other hand, most congestion pricing models utilize supply models insensitive to demand fluctuations and non-stationary conditions. This is not consistent with the physics of traffic and the dynamics of congestion. Furthermore, works that integrate the above features in pricing models are assuming deterministic and homogeneous population characteristics. In this paper, we first demonstrate by case studies in Zurich urban road network, that the output of a agent-based simulator is consistent with the physics of traffic flow dynamics, as defined by a Macroscopic Fundamental Diagram (MFD). We then develop and apply a dynamic cordon-based congestion pricing scheme, in which tolls are controlled by an MFD. And we investigate the effectiveness of the proposed pricing scheme. Results show that by applying such a congestion pricing, (i) the savings of travel time at both aggregated and disaggregated level outweigh the costs of tolling, (ii) the congestion inside the cordon area is eased while no extra congestion is generated in the neighbor area outside the cordon, (iii) tolling has stronger impact on leisure-related activities than on work-related activities, as fewer agents who perform work-related activities changed their time plans. Future work can apply the same methodology to other network-based pricing schemes, such as area-based or distance-traveled-based pricing. Equity issues can be investigated more carefully, if provided with data such as income of agents. Value-of-time-dependent pricing schemes then can also be determined.     

The impact of speed limits on traffic equilibrium and system performance in networks

Speed limits are usually imposed on roads in an attempt to enhance safety and sometimes serve the purpose of reducing fuel consumption and vehicular emissions as well. Most previous studies up to date focus on investigation of the effects of speed limits from a local perspective, while network-wide traffic reallocation effects are overlooked. This paper makes the first attempt to investigate how a link-specific speed limit law reallocates traffic flow in an equilibrium manner at a macroscopic network level. We find that, although the link travel time–flow relationship is altered after a speed limit is imposed, the standard traffic assignment method still applies. With the commonly adopted assumptions, the uniqueness of link travel times at user equilibrium (UE) remains valid, and the UE flows on links with non-binding speed limits are still unique. The UE flows on other links with binding speed limits may not be unique but can be explicitly characterized by a polyhedron or a linear system of equalities and inequalities. Furthermore, taking into account the traffic reallocation effects of speed limits, we compare the capability of speed limits and road pricing for decentralizing desirable network flow patterns. Although from a different perspective for regulating traffic flows with a different mechanism, a speed limit law may play the same role as a toll charge scheme and perform better than some negative (rebate) toll schemes under certain conditions for network flow management.     

Toll pricing framework under logit‐based stochastic user equilibrium constraints

This paper addresses the toll pricing framework for the first‐best pricing with logit‐based stochastic user equilibrium (SUE) constraints. The first‐best pricing is usually known as marginal‐cost toll, which can be obtained by solving a traffic assignment problem based on the marginal cost functions. The marginal‐cost toll, however, has rarely been implemented in practice, because it requires every specific link on the network to be charged. Thus, it is necessary to search for a substitute of the marginal cost pricing scheme, which can reduce the toll locations but still minimize the total travel time. The toll pricing framework is the set of all the substitute toll patterns of the marginal cost pricing. Assuming the users' route choice behavior following the logit‐based SUE principle, this paper has first derived a mathematical expression for the toll pricing framework. Then, by proposing an origin‐based variational inequality model for the logit‐based SUE problem, another toll pricing framework is built, which avoids path enumeration/storage. Finally, the numerical test shows that many alternative pricing patterns can inherently reduce the charging locations and total toll collected, while achieving the same equilibrium link flow pattern. Copyright © 2013 John Wiley & Sons, Ltd.     

Time-dependent congestion pricing system for large networks: Integrating departure time choice,dynamic traffic assignment and regional travel surveys in the Greater Toronto Area

Congestion pricing is one of the widely contemplated methods to manage traffic congestion. The purpose of congestion pricing is to manage traffic demand generation and supply allocation by charging fees (i.e., tolling) for the use of certain roads in order to distribute traffic demand more evenly over time and space. This study presents a framework for large-scale variable congestion pricing policy determination and evaluation. The proposed framework integrates departure time choice and route choice models within a regional dynamic traffic assignment (DTA) simulation environment. The framework addresses the impact of tolling on: (1) road traffic congestion (supply side), and (2) travelers’ choice dimensions including departure time and route choices (demand side). The framework is applied to a simulation-based case study of tolling a major freeway in Toronto while capturing the regional effects across the Greater Toronto Area (GTA). The models are developed and calibrated using regional household travel survey data that reflect the heterogeneity of travelers’ attributes. The DTA model is calibrated using actual traffic counts from the Ontario Ministry of Transportation and the City of Toronto. The case study examined two tolling scenarios: flat and variable tolling. The results indicate that: (1) more benefits are attained from variable pricing, that mirrors temporal congestion patterns, due to departure time rescheduling as opposed to predominantly re-routing only in the case of flat tolling, (2) widespread spatial and temporal re-distributions of traffic demand are observed across the regional network in response to tolling a significant, yet relatively short, expressway serving Downtown Toronto, and (3) flat tolling causes major and counterproductive rerouting patterns during peak hours, which was observed to block access to the tolled facility itself.     

Nonlinear pricing in linear cities with elastic demands

Nonlinear road pricing charges each traveler based on his/her trip’s corresponding particular attribute level. In order to help authorities in designing road pricing systems at a strategic level, this paper attempts to address two fundamental questions: (i) what is the value of pricing’s nonlinearity for mitigating traffic congestion? (ii) if a nonlinear toll function is implemented, should it be convex, concave or other shape? Specifically, we consider distance-based pricing in linear cities. For linear monocentric cities with heterogeneous travelers, we show that the system optimal distance-based pricing indeed exhibits nonlinearity. It is proved that: (i) the cost-based system optimal toll function is monotonically increasing and concave with respect to the traveled distance; (ii) the time-based system optimal toll function always exists and is unique. If the initial proportion of each traveler group is invariant along a corridor and the demand function is of exponential type, then the time-based system optimal toll function enables the travelers, living further away from a city center, to face a lower toll level per unit distance. For a linear polycentric city, we demonstrate: (i) there always exists the system optimal differentiated (in terms of city centers) toll functions; (ii) it is highly possible that the system optimal non-differentiated toll function does not exist. Hence, we further propose an optimal toll design model, prove the Lipschitz continuity of its objective and adopt a global-optimization algorithm to solve it.