Dynamic congestion pricing with day-to-day flow evolution and user heterogeneity

This paper investigates evolutionary implementation of congestion pricing schemes to minimize the system cost and time, measured in monetary and time units, respectively, with the travelers’ day-to-day route adjustment behavior and their heterogeneity. The travelers’ heterogeneity is captured by their value-of-times. First, the multi-class flow dynamical system is proposed to model the travelers’ route adjustment behavior in a tolled transportation network with multiple user classes. Then, the stability condition and properties of equilibrium is examined. We further investigate the trajectory control problem via dynamic congestion pricing scheme to derive the system cost, time optimum, and generally, Pareto optimum in the sense of simultaneous minimization of system cost and time. The trajectory control problem is modeled by a differential–algebraic system with the differential sub-system capturing the flow dynamics and the algebraic one capturing the pricing constraint. The explicit Runge–Kutta method is proposed to calculate the dynamic flow trajectories and anonymous link tolls. The method allows the link tolls to be updated with any predetermined periods and forces the system cost and/or time to approach the optimum levels. Both analytical and numerical examples are adopted to examine the efficiency of the method.     

Optimal tolls for multi-class traffic: Analytical formulations and policy implications

This paper puts together an analytical formulation to compute optimal tolls for multi-class traffic. The formulation is comprised of two major modules. The first one is an optimization component aimed at computing optimal tolls assuming a Stackelberg game in which the toll agency sets the tolls, and the equilibrating traffic plays the role of the followers. The optimization component is supported by a set of cost models that estimate the externalities as a function of a multivariate vector of traffic flows. These models were estimated using Taylor series expansions of the output obtained from traffic simulations of a hypothetical test case. Of importance to the paper is the total travel time function estimated using this approach that expresses total travel time as a multivariate function of the traffic volumes. The formulation presented in the paper is then applied to a variety of scenarios to gain insight into the optimality of current toll policies. The optimal tolls are computed for two different cases: independent tolls, and tolls proportional to passenger car equivalencies (PCE).The numerical results clearly show that setting tolls proportional to PCEs leads to lower values of welfare that are on average 15% lower than when using independent tolls, though, in some cases the total welfare could be up to 33% lower. This is a consequence of two factors. First, the case of independent tolls has more degrees of freedom than the case of tolls proportional to PCEs. Second, tolls proportional to PCEs do not account for externalities other than congestion, which is likely to lead to lower welfare values.The analytical formulations and numerical results indicate that, because the total travel time is a non-linear function of the traffic volumes, the marginal social costs and thus the optimal congestion tolls also depend on the traffic volumes for each vehicle class. As a result of this, for the relatively low volumes of truck traffic observed in real life, the optimal congestion tolls for trucks could indeed be either lower or about the same as for passenger cars. This stand in sharp contrast with what is implied in the use of PCEs, i.e., that the contribution to congestion are constant. This latter assumption leads to optimal truck congestion tolls that are always proportional to the PCE values.The comparison of the toll ratios (truck tolls divided by passenger car tolls) for both observed and optimal conditions suggests that the tolls for small trucks are about the right level, maybe a slightly lower than optimal. However, the analysis of the toll ratio for large trucks seems to indicate a significant overcharge. The estimates show that the average observed toll ratio for large trucks is even higher than the maximum optimal toll ratio found in the numerical experiments. This suggests that the tolls for large trucks are set on the basis of revenue generation principles while the passenger car tolls are being set based on a mild form of welfare maximization. This leads to a suboptimal cross-subsidization of passenger car traffic in detriment of an important sector of the economy.     

Integrating congestion pricing,transit subsidies and mode choice

We model and analyze optimal (welfare maximizing) prices and design of transport services in a bimodal context. Car congestion and transit design are simultaneously introduced and consumers choose based on the full price they perceive. The optimization variables are the congestion toll, the transit fare (and hence the level of subsidies) and transit frequency. We obtain six main results: (i) the optimal car-transit split is generally different from the total cost minimizing one; (ii) optimal congestion and transit price are interdependent and have an optimal frequency attached; (iii) the optimal money price difference together with the optimal frequency yield the optimal modal split; (iv) if this modal split is used in traditional stand-alone formulations – where each mode is priced independently–resulting congestion tolls and transit subsidies and fares are consistent with the optimal money price difference; (v) self-financing of the transport sector is feasible; and (vi) investment in car infrastructure induces an increase in generalized cost for all public transport users.     

Morning commute problem considering route choice, user heterogeneity and alternative system optima

This paper extends the bottleneck model to study congestion behavior of morning commute and its implications to transportation economics. The proposed model considers simultaneous route and departure time choices of heterogenous users who are distinguished by their valuation of travel time and punctual arrival. Moreover, two dynamic system optima are considered: one minimizes system cost in the unit of monetary value (i.e., the conventional system optimum, or SO) and the other minimizes system cost in the unit of travel time (i.e., the time-based SO, or TSO). Analytical solutions of no-toll equilibrium, SO and TSO are provided and the welfare effects of the corresponding dynamic congestion pricing options are examined, with and without route choice. The analyses suggest that TSO provides a Pareto-improving solution to the social inequity issue associated with SO. Although a TSO toll is generally discriminatory, anonymous TSO tolls do exist under certain circumstances. Unlike in the case with homogenous users, an SO toll generally alters users’ route choices by tolling the poorer users off the more desirable road, which worsens social inequity. Numerical examples are presented to verify analytical results.     

Distance-dependent congestion pricing for downtown zones

A growing literature exploits macroscopic theories of traffic to model congestion pricing policies in downtown zones. This study introduces trip length heterogeneity into this analysis and proposes a usage-based, time-varying congestion toll that alleviates congestion while prioritizing shorter trips. Unlike conventional trip-based tolls the scheme is intended to align the fees paid by drivers with the actual congestion damage they do, and to increase the toll’s benefits as a result.The scheme is intended to maximize the number of people that finish their trips close to their desired times. The usage-based toll is compared to a traditional, trip-based toll which neglects trip length. It is found that, like trip-based tolls, properly designed usage-based tolls alleviate congestion. But they reduce schedule delay more than trip-based tolls and do so with much smaller user fees. As a result usage-based tolls leave most of those who pay with a large welfare gain. This may increase the tolls’ political acceptability.     

集聚经济视角下轨道交通投资对城市系统均衡的影响

随着城镇化进程加快,城市规模增加所带来的集聚效应和交通拥堵等负外部性之间的矛盾越来越突出,优化城市交通基础设施建设是缓解该矛盾的重要举措。本文同时考虑了集聚经济和交通拥挤两种外部性,构建并求解了一个两区域城市空间结构模型,分三种情况探究了轨道交通的投资引入和道路容量的优化对城市规模和城市空间结构均衡的影响。结果表明,轨道交通的投资引入能够增加城市人口规模和社会福利,使城市结构更加紧凑。但是,随着集聚经济水平的提高,城市规划者应逐步完善公交配套设施,以降低公交拥挤成本,提高公交出行分担率;还应制定相应的福利政策,避免由于个体效用下降而导致人口流出。     

Urban highway congestion

This article examines urban highway congestion pricing in the instance in which it is not possible to levy a congestion toll on a major portion of the urban road system. This case is pertinent because of technical and/or political constraints. The article uses economic theory and numerical examples to show that the optimum second-best toll can vary appreciably from the optimal tolls in a regime in which efficient tolls can be imposed on all routes.     

All minutes are not equal: travel time and the effects of congestion on commute satisfaction in Canadian cities

Despite decades of research, it is unclear under which circumstances travel is most onerous. While studies have found that some individuals derive positive utility from aspects of commuting, others have shown that traffic congestion can entail important time, monetary, and mental stress costs. Moreover, responses to traffic congestion-related stressors differs by individual characteristics. In response, this research captures how exposure to traffic congestion events, the duration of this exposure, and individual trait susceptibility to congestion affect the utility of commuting. Working through the lens of individual satisfaction with the duration of their commute, we show that not every minute of travel is valued the same by car commuters in Canadian cities. Results suggest a complex relationship between travel time, congestion, and individual predisposition to congestion-related stress. While improvements in travel time matter for increasing commute satisfaction, it is reductions in travel in congested conditions that matter most, particularly among those susceptible to congestion-related stressors.     

Implementation of cordon pricing on urban network with practical approach

The road pricing is regarded as a transport policy to realize the efficient use of urban network. The network analysis with variable demand has been often applied to describe the network level of congestion pricing on the urban network. In the study, the cordon pricing system is analyzed to implement in the urban area with practical approach. It is assumed that the congestion tolls are collected in crossing the cordon lines on the network. Therefore, the scale of cordon zones and the values of congestion tolls would be determined simultaneously to produce the maximum social net benefit. The combinatorial optimization with unit price is formulated. The genetic algorithm (GA) is applied as a practical method to provide the solutions for the combinatorial optimization problem. As the pattern of cordon pricing is determined, the performance of system is estimated to confirm the applicability. It is concluded that the cordon pricing can be applied with the practical approach.     

Network equilibrium for congested multi‐mode networks with elastic demand

This paper proposes an elastic demand network equilibrium model for networks with transit and walking modes. In Hong Kong, the multi‐mode transit system services over 90% of the total journeys and the demand on it is continuously increasing. Transit and walking modes are related to each other as transit passengers have to walk to and from transit stops. In this paper, the multi‐mode elastic‐demand network equilibrium problem is formulated as a variational inequality problem where the combined mode and route choices are modeled in a hierarchical logit structures and the total travel demand for each origin‐destination pair is explicitly given by an elastic demand function. In addition, the capacity constraint for transit vehicles and the effects of bi‐directional flows on walkways are considered in the proposed model. All these congestion effects are taken into account for modeling the travel choices. A solution algorithm is developed to solve the multi‐mode elastic‐demand network equilibrium model. It is based on a Block Gauss‐Seidel decomposition approach coupled with the method of successive averages. A numerical example is used to illustrate the application of the proposed model and solution algorithm.     

Pareto-improving ramp metering strategies for reducing congestion in the morning commute

This paper presents an alternative approach to internalize congestion externality during the morning commute. We consider a linear freeway with multiple on-ramps and a downstream bottleneck and commuters accessing the freeway via different on-ramps try to arrive at work on time. Rather than charging congestion tolls as widely suggested by economists, we show that the old-fashioned engineering approach – ramp metering – can be a powerful tool to affect travelers’ departure time choice and thereby alter the congestion externality distribution among travelers. With carefully designed time-dependent metering plans, travelers from different origins can be channelized and will access the freeway bottleneck in different time periods, resulting in less total cost for the system compared to the no-metering case. The metering strategies are Pareto-improving, with travelers from the on-ramp with the highest priority having the smallest individual costs and travelers from the on-ramp with the lowest priority having their costs equal to those in the no-metering scenario. By changing the priority order of the ramps periodically, the benefit of the Pareto-improving metering strategies can be distributed evenly among all travelers. Numerical experiments show that the total user cost can be reduced by up to 40% with the proposed metering strategies. This study offers researchers and policy makers a different angle of looking at congestion externality, and the results provide an overview of the potential long term benefits that dynamic ramp metering strategies can achieve.     

A bi‐level programming approach — Optimal transit fare under line capacity constraints

The fare of a transit line is one of the important decision variables for transit network design. It has been advocated as an efficient means of coordinating the transit passenger flows and of alleviating congestion in the transit network. This paper shows how transit fare can be optimized so as to balance the passenger flow on the transit network and to reduce the overload delays of passengers at transit stops. A bi‐level programming method is developed to optimize the transit fare under line capacity constraints. The upper‐level problem seeks to minimize the total network travel time, while the lower‐level problem is a stochastic user equilibrium transit assignment model with line capacity constraints. A heuristic solution algorithm based on sensitivity analysis is proposed. Numerical example is used to illustrate the application of the proposed model and solution algorithm.     

System optimum and pricing for the day-long commute with distributed demand,autos and transit

The day-long system optimum (SO) commute for an urban area served by auto and transit is modeled as an auto bottleneck with a capacitated transit bypass. A public agency manages the system’s capacities optimally. Commuters are identical except for the times at which they wish to complete their morning trips and start their evening trips, which are given by an arbitrary joint distribution. They value unpunctuality – their lateness or earliness relative to their wish times – with a common penalty function. They must use the same mode for both trips. Commuters are assigned personalized mode and travel start times that collectively minimize society’s generalized cost for the whole day. This includes unpunctuality penalties, queuing delays, travel times and out-of-pocket costs for users, as well as travel supply costs and externalities for society.It is shown that in a SO solution there can be no queuing and that the set of SO solutions forms a convex set. Furthermore, if the schedule penalty that users suffer due to unpunctuality is separable into morning and evening components, then the set of commuters traveling by the same mode arrive at work and depart from work in the order of their wishes. These orders are in general different in the morning and the evening. It is also shown that there always is a SO solution in which users are at all times, and on both modes, either punctual or flowing at capacity. These problem properties are used to identify search methods, both, for SO solutions and for time-dependent tolls and transit fares that preserve the solutions as Nash equilibriums. In every case studied, these prices exist. They must peak concurrently for the two modes in both periods.In special cases involving only one mode, only one period or concentrated demand the solution to the complete problem decomposes by period conditional on the number of transit users, and this facilitates the solution. In these cases the day-long SO cost is the sum of the SO costs for the two peaks considered separately. However, this is not true in general – the solution obtained by combining the two single-period solutions can be infeasible. When this happens, the optimum day-long cost will exceed the sum of the single-period costs. The discrepancy is about 40% of the total schedule penalty for an example representing a large city. Thus, to develop realistic policies the day-long problem must be addressed head on. An approximate method that yields closed form formulas for the case with uniformly distributed wishes is presented.     

Dynamic Stackelberg equilibrium congestion pricing

This paper considers the problem of dynamic congestion pricing that determines optimal time-varying tolls for a pre-specified subset of arcs with bottleneck on a congested general traffic network. A two-person nonzero-sum dynamic Stackelberg game model is formulated with the assumption that the underlying information structure is open loop. Characteristics of the Stackelberg equilibrium solution are analyzed. The Hooke–Jeeves algorithm that obviates an evaluation of the gradient vector of the objective function is presented with a numerical example. The paper concludes with its future extensions.     

Effect of advanced traveler information systems and road pricing in a network with non-recurrent congestion

The effect of the application of advanced transport information system (ATIS) and road pricing is studied in a transportation system under non-recurrent congestion. A stochastic network deterministic user equilibrium model (SNDUE) with elastic demand is formulated and used to evaluate the welfare and private impacts of different market penetrations of ATIS, together with road pricing for a simple network. Both marginal first-best road pricing and a second-best fixed road pricing are considered. The incentives of private users to use ATIS are analyzed and the characteristics of optimum tolls as a function of ATIS market penetration are shown. We conclude that ATIS is an efficient and necessary tool to reduce the effects of non-recurrent incidents in a transportation network, especially when non-recurrent congestion causes a significant deterioration of operational conditions of the network. If the impact of non-recurrent incidents on free flow costs is small or is reduced only to congestion effects, the use of road pricing would be more efficient. Social benefits obtained when jointly implementing ATIS and road pricing are practically the same whether first-best or second-best road pricing is used. Considering the private costs perceived by the network users, and the benefits experienced by equipped users, the maximum level of market penetration achieved could be limited because private benefits disappear after certain market penetration is obtained.     

Marginal costs of freeway traffic congestion with on-road pollution exposure externality

The health cost of on-road air pollution exposure is a component of traffic marginal costs that has not previously been assessed. The main objective of this paper is to introduce on-road pollution exposure as an externality of traffic, particularly important during traffic congestion when on-road pollution exposure is highest. Marginal private and external cost equations are developed that include on-road pollution exposure in addition to time, fuel, and pollution emissions components. The marginal external cost of on-road exposure includes terms for the marginal vehicle’s emissions, the increased emissions from all vehicles caused by additional congestion from the marginal vehicle, and the additional exposure duration for all travelers caused by additional congestion from the marginal vehicle. A sensitivity analysis shows that on-road pollution exposure can be a large portion (18%) of marginal social costs of traffic flow near freeway capacity, ranging from 4% to 38% with different exposure parameters. In an optimal pricing scenario, excluding the on-road exposure externality can lead to 6% residual welfare loss because of sub-optimal tolls. While regional pollution generates greater costs in uncongested conditions, on-road exposure comes to dominate health costs on congested freeways because of increased duration and intensity of exposure. The estimated marginal cost and benefit curves indicate a theoretical preference for price controls to address the externality problem. The inclusion of on-road exposure costs reduces the magnitudes of projects required to cover implementation costs for intelligent transportation system (ITS) improvements; the net benefits of road-pricing ITS systems are increased more than the net benefits of ITS traffic flow improvements. When considering distinct vehicle classes, inclusion of on-road exposure costs greatly increases heavy-duty vehicle marginal costs because of their higher emissions rates and greater roadway capacity utilization. Lastly, there are large uncertainties associated with the parameters utilized in the estimation of health outcomes that are a function of travel pollution intensity and duration. More research is needed to develop on-road exposure modeling tools that link repeated short-duration exposure and health outcomes.     

Network user equilibrium of battery electric vehicles considering flow-dependent electricity consumption

This paper addresses the equilibrium traffic assignment problem involving battery electric vehicles (BEVs) with flow-dependent electricity consumption. Due to the limited driving range and the costly/time-consuming recharging process required by current BEVs, as well as the scarce availability of battery charging/swapping stations, BEV drivers usually experience fear that their batteries may run out of power en route. Therefore, when choosing routes, BEV drivers not only try to minimize their travel costs, but also have to consider the feasibility of their routes. Moreover, considering the potential impact of traffic congestion on the electricity consumption of BEVs, the feasibility of routes may be determined endogenously rather than exogenously. A set of user equilibrium (UE) conditions from the literature is first presented to describe the route choice behaviors of BEV drivers considering flow-dependent electricity consumption. The UE conditions are then formulated as a nonlinear complementarity model. The model is further formulated as a variational inequality (VI) model and is solved using an iterative solution procedure. Numerical examples are provided to demonstrate the proposed models and solution algorithms. Discussions of how to evaluate and improve the system performance with non-unique link flow distribution are offered. A robust congestion pricing model is formulated to obtain a pricing scheme that minimizes the system travel cost under the worst-case tolled flow distribution. Finally, a further extension of the mathematical formulation for the UE conditions is provided.     

Estimating commuter mode choice: A discrete choice analysis of the impact of road pricing and parking charges

Automobile use leads to external costs associated with emissions, congestion, noise and other impacts. One option for minimizing these costs is to introduce road pricing and parking charges to reduce demand for single occupant vehicle (SOV) use, while providing improvements to alternatives to encourage mode switching. However, the impact of these policies on urban mode choice is uncertain, and results reported from regions where charging has been introduced may not be transferable. In particular, revealed preference data associated with cost recovery tolls on single facilities may not provide a clear picture of driver response to tolls for demand management. To estimate commuter mode choice behaviour in response to such policies, 548 commuters from a Greater Vancouver suburb who presently drive alone to work completed an individually customized discrete choice experiment (DCE) in which they chose between driving alone, carpooling or taking a hypothetical express bus service when choices varied in terms of time and cost attributes. Attribute coefficients identified with the DCE were used in a predictive model to estimate commuter response to various policy oriented combinations of charges and incentives. Model results suggest that increases in drive alone costs will bring about greater reductions in SOV demand than increases in SOV travel time or improvements in the times and costs of alternatives beyond a base level of service. The methods described here provide an effective and efficient way for policy makers to develop an initial assessment of driver reactions to the introduction of pricing policies in their particular regions.     

The tax treatment of company cars, commuting and optimal congestion taxes

In Europe, the preferential tax treatment of company cars implies that many employees receive a company car as part of their compensation package. In this paper, we consider a model in which wages and the decision whether or not to provide a company car are the result of direct negotiation between employer and employee. Using this framework, we theoretically and numerically study first- and second-best optimal tax policies on labour and transport markets, focusing on the role of the tax treatment of company cars. We obtain the following results. First, higher labour taxes and a more favourable tax treatment of company cars raise the fraction employees that receives a company car; congestion and congestion tolls reduce it. Second, in countries that provide large implicit subsidies to company cars, eliminating the preferential tax treatment of company cars may be an imperfect but quite effective substitute for currently unavailable congestion tolls. The numerical illustration, calibrated using Belgian data, suggests that it yields about half the welfare gain attainable through optimal congestion taxes. Third, the favourable tax treatment of company cars justifies large public transport subsidies; the numerical results are consistent with zero public transport fares. Finally, we find that earlier models that ignored the preferential tax treatment of company cars may have substantially underestimated optimal congestion tolls in Europe. The numerical illustration suggests that about one third of the optimal congestion toll we obtain can be attributed to the current tax treatment of company cars.     

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.