Dynamic Congestion Pricing Research Articles

Optimal control of day-to-day traffic dynamics under user learning with dynamic congestion pricing

Abstract A continuous day-to-day dynamic traffic assignment model is proposed in this study, where several typical day-to-day models are integrated into a more general form. The effects of the past information on the travelers’ routing behavior are characterized by the second-order derivatives of the traffic flow. Thereupon, the day-to-day dynamic takes the form of a second-order non-homogeneous ordinary differential equation with constant coefficients. We prove that the dynamical process is globally asymptotically stable at the user equilibrium with the existence of travelers’ route swapping behavior. Moreover, we prove that there always exists a limited time point, after which the non-negative flows can be guaranteed in the proposed model. The optimal control of the system is realized by utilizing dynamic congestion pricing. Different optimization objectives are investigated, including cost, time, revenue, and combined minimizations. Necessary conditions for optimal congestion prices are analyzed to uncover the bang-bang charging strategy. Numerical examples are provided to illustrate the trajectory of the flow evolution under the optimal control.

MAGT-toll: A multi-agent reinforcement learning approach to dynamic traffic congestion pricing.

Modern urban centers have one of the most critical challenges of congestion. Traditional electronic toll collection systems attempt to mitigate this issue through pre-defined static congestion pricing methods; however, they are inadequate in addressing the dynamic fluctuations in traffic demand. Dynamic congestion pricing has been identified as a promising approach, yet its implementation is hindered by the computational complexity involved in optimizing long-term objectives and the necessity for coordination across the traffic network. To address these challenges, we propose a novel dynamic traffic congestion pricing model utilizing multi-agent reinforcement learning with a transformer architecture. This architecture capitalizes on its encoder-decoder structure to transform the multi-agent reinforcement learning problem into a sequence modeling task. Drawing on insights from research on graph transformers, our model incorporates agent structures and positional encoding to enhance adaptability to traffic flow dynamics and network coordination. We have developed a microsimulation-based environment to implement a discrete toll-rate congestion pricing scheme on actual urban roads. Our extensive experimental results across diverse traffic demand scenarios demonstrate substantial improvements in congestion metrics and reductions in travel time, thereby effectively alleviating traffic congestion.

Parking design and pricing for regular and autonomous vehicles: a morning commute problem

Autonomous vehicles can profoundly change parking behaviour in the future. Instead of searching for parking, the occupants alight at their final destination and send their occupant-free cars to a parking spot. This paper studies the impact of parking in a morning commute problem with autonomous and regular vehicles. We simplify the complex problem with distinct cost functions to a classic bi-class problem that can be analytically solved. To optimize the system, we develop temporal and spatial parking pricing strategies and a new parking supply design scheme, as practical alternatives for the conventional dynamic congestion pricing.

A reinforcement learning-based dynamic congestion pricing method for the morning commute problems

A reinforcement learning-based dynamic congestion pricing method for morning commute problems is proposed. In this method, tolls are iteratively updated day by day based on observable information such as traffic volume. The advantage of the method is two fold. First, the method does not require travelers personal preferences such as value of time. Second, the method does not require manually designed toll update scheme; instead, the method automatically find the most efficient toll by a data-driven manner. Results of numerical experiments show that the proposed method properly decreased traffic congestion in various types of networks. Especially, in the experiment with a single bottleneck model, total waiting time decreases to nearly zero in about 200 days. One of future issues is more stable coordination among the time zones.

Reinforcement Learning and Adaptive Optimal Control of Congestion Pricing

The increasing road traffic congestion has urged researchers to look for solutions to tackle the problem. Many different interventions reduce traffic jams including, optimizing traffic-lights, using video surveillance to monitor road conditions, strategic road network resilience, and congestion pricing. This paper uses a nonlinear model for dynamic congestion pricing, considering manual-toll and automatic toll lanes using wireless communication technologies. The model can adjust the traveling demand and improve traffic flow performance by charging more for entering express lanes. We linearize the model about the equilibrium states and propose a reinforcement learning-based adaptive optimal control approach to learn the optimal control gain of the linearized model. Further, we rigorously show that the developed optimal controller can ensure the stability of the original nonlinear closed-loop system by making its output asymptotically converge to zero. Finally, the proposed approach is validated by numerical simulations.

A discrete-time second-best dynamic road pricing scheme considering the existence of multiple equilibria

When multiple equilibria exist, the desired traffic equilibrium, may not be reached through a day-to-day dynamic adjustment process. Han et al. [“Discrete-Time Day-to-Day Dynamic Congestion Pricing Scheme Considering Multiple Equilibria.” Transportation Research Part B: Methodological 104: 1–16.] proposed a day-to-day dynamic pricing scheme charged on all links to ensure the desired traffic equilibrium can be achieved even when multiple equilibria exist. However, due to many practical constraints, the second-best pricing scheme has more practical interests. This study first discussed the issues of existing congestion pricing schemes for second-best case. Then, this study developed a second-best dynamic road pricing scheme that is implemented on a dynamic subset of the road links in the network. Our rigorous theoretical proof and numerical tests prove that the dynamic pricing scheme can drive the traffic state evolution towards the desired second-best traffic user equilibrium state from any initial traffic state.

Dynamic Congestion Pricing for Ridesourcing Traffic: a Simulation Optimization Approach

Despite the documented benefits of ride-sourcing services, recent studies show that they can slow down traffic in the densest cities significantly. To implement congestion pricing policies upon those vehicles, regulators need to estimate how much their congestion effects are. This paper studies simulation-based approaches to address the two technical challenges arising from the representation of system dynamics and the optimization for congestion price mechanisms. To estimate the traffic conditions, we use a meta-model representation for traffic flow and a numerical method for data interpolation. To reduce the burden of replicating evaluation in stochastic optimization, we use a simulation optimization approach to compute the optimal congestion price. This data-driven approach can potentially be extended to solve large-scale congestion pricing problems with unobservable states.

A cell-based dynamic congestion pricing scheme considering travel distance and time delay

This study introduces the dynamic congestion pricing (DCP) problem with the consideration of the actual travel distance and time delay (i.e. a joint distance and time-delay toll, JDTDT) in a dynamic network, which is more equitable and effective compared with existing tolling scheme. The system dynamics can be reflected in two aspects: (a) travelers' path choice decisions follow the dynamic user equilibrium principle and (b) the joint distance and time-delay toll takes a time-varying pattern. A multiperiod demand scheme is adopted during the entire modeling horizon and a bi-level programming model for the DCP is formulated to obtain the optimal toll value. Numerical results indicate that the percentage reductions of the minimum total system travel time in the dynamic JDTDT scheme are 6.28%, 4.30% and 7.45% compared to that obtained by the static joint JDTDT, the dynamic joint distance and time toll, and the dynamic pure distance toll, respectively.

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

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.

Parking Pricing and Design in the Morning Commute Problem with Regular and Autonomous Vehicles

Autonomous vehicles can profoundly change parking behavior in the future. Instead of searching for parking, autonomous vehicle owners get dropped off at their final destination and send their occupant-free cars to a parking spot. In this paper, we study the impact of parking in a bi-class morning commute problem with autonomous and regular vehicles. We consider a spatial distribution of parking spaces, which allows us to capture the parking location of travelers. In the equilibrium condition, autonomous vehicles leave home later (than regular vehicles), and park farther away from their destination. The regular vehicle travelers, however, leave home sooner to park closer to the destination with a smaller walking distance. The reverse occurs if regular travelers abdicate walking and take a faster mode from their parking space to the city center. To optimize the system, we develop temporal and spatial parking pricing strategies and a new parking supply design scheme, as practical alternatives for the conventional dynamic congestion pricing. The proposed parking pricing strategies incentivize commuters to adjust their travel schedules by charging a parking price that increases with distance from the city center. Hence, those who park closer to their destination have to pay less. This is a counter-intuitive finding at first which arises from a trade-off with the earliness penalty of users who park close to the destination. We show that system optimality is also reached by redistributing the parking supply. Our numerical experiments capture the impact the autonomous vehicle penetration rate on the performance of the system and the proposed parking pricing and design strategies.

A bi-level distributed approach for optimizing time-dependent congestion pricing in large networks: A simulation-based case study in the Greater Toronto Area

Congestion pricing is one of the most widely contemplated methods to manage traffic congestion by charging fees for the use of roads, more where and when it is congested, and less where and when it is not. This study presents a bi-level distributed approach for optimal time-dependent congestion pricing in large networks. The bi-level procedure involves a theoretical model of dynamic congestion pricing and a distributed optimization algorithm. The bi-level toll optimization module is integrated into a testbed of hybrid departure time choice and dynamic traffic assignment simulation models for the Greater Toronto Area (GTA). The integrated system provides a unified (location- and time-specific) congestion pricing system that determines optimal tolling and evaluates its impact on road traffic congestion and travellers’ behavioural choices, including departure time and route choices. For the system’s large-scale nature and the consequent computational challenges, the optimization algorithm is executed concurrently on a parallel computing cluster. The system is applied to a simulation-based case study of tolling major highways in the GTA while capturing the network-wide regional effects of tolling. The travel demand and drivers’ attributes are extracted from regional household travel survey data that reflect travellers’ heterogeneity. The main results indicate that: (1) optimal variable pricing reflects congestion patterns and induces departure time re-scheduling and rerouting patterns, resulting in improved average travel times and schedule delays, (2) optimal tolls intended to manage traffic demand are significantly lower than those intended to maximize toll revenues, (3) tolled routes have different sensitivities to identical toll changes, (4) the start times of longer trips are more sensitive (elastic) to variable distance-based tolling policies compared to shorter trips, (5) toll payers benefit from tolling even before toll revenues are spent, and (6) the optimal tolling policies determined offer a win–win solution in which travel times are improved while also raising funds to invest in sustainable transportation infrastructure.

Discrete-time day-to-day dynamic congestion pricing scheme considering multiple equilibria

In this study, we focus on the discrete-time day-to-day dynamic congestion pricing scheme which varies the toll on a day-to-day basis and aims to drive the traffic system to a given objective traffic equilibrium state. As is well known, due to the asymmetric nature of the travel cost functions, multiple equilibria exist. In this case, without external force, the traffic system cannot converge to the traffic equilibrium state as desired by traffic management through a day-to-day adjustment process if the initial traffic state does not fall into its attraction domain (Bie and Lo, 2010). Therefore, it is imperative for traffic management to propose a traffic control measure to ensure the desired traffic state can be achieved regardless of the initial traffic state. Previous studies on the day-to-day dynamic congestion pricing, either worked on continues-time day-to-day pricing scheme, or took the form of discrete-time day-to-day pricing scheme but did not guarantee the convergence to the desired objective traffic state for the cases when multiple traffic equilibria exist. Both are undesirable. This study aims to develop a discrete-time day-to-day pricing scheme so as to direct the traffic evolution to reach the desired equilibrium from any initial traffic state when multiple traffic equilibria exist. Based on the very general formulation of day-to-day traffic dynamics model, we present a general formulation of such day-to-day pricing schemes and propose a method to obtain one specific road pricing scheme. Moreover, we present rigorous proofs and numerical tests to verify the proposed pricing scheme.

Optimal Control for Congestion Pricing: Theory, Simulation, and Evaluation

This paper presents a mathematical framework for dynamic congestion pricing. The objective is to calculate an optimal toll using the optimal control theory. The problem consists of tolled lanes or routes and alternate non-tolled lanes or routes. The model is developed using a traffic conservation law, the queuing theory, and fundamental macroscopic relationships. A logit model is used for establishing the relationship between the price and the driver's choice behavior. We design a cost function and then use Hamilton–Jacobi–Bellman equation to derive an optimal control law that uses real-time information to determine an optimal tolling price. Simulations are performed to demonstrate the performance of this optimal control congestion-pricing algorithm.

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.

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.

Value of Dynamic Revenue-Maximizing Congestion Pricing in a Highly Congested Corridor

AbstractThis paper examines the value of utilizing dynamic, revenue-maximizing, congestion pricing on a privately operated tolled route, when the only alternative is a highly congested, public, free-access route. Three methods are specified by which revenue can be maximized—through the selection of a fixed percent of users to set the toll price for, through the selection of a fixed level of service on the toll road, and through the use of a nonlinear optimization model. These methods are tested on a case study involving a highly congested corridor. In simulation, the corridor’s flow rates are observed following the posting of a toll, the toll is updated, and the new flow observed until a subsequent round of revenue maximizing is applied to the new conditions. It was concluded that in a highly congested corridor a dynamic, revenue-maximizing toll can finance the construction of new capacity without degrading social welfare.

Implications of Changing Dynamics for Metered On-Street Parking

The parking industry has seen significant changes over the past decade with the infusion of new technology and smart assets. The introduction of networked meters, virtual payment methods (such as pay-by-cell and credit–debit cards at meters), and technology for real-time detection of space occupancy has resulted in better system uptime, proactive maintenance strategies, multiple payment options, real-time information on parking availability, and better use of spaces through dynamic congestion pricing. The new parking assets and payment options have implications for municipalities and vendors supporting their parking programs. Instead of a significant portion of revenue from coins, virtual transactions account for a predominant share of the parking revenue stream. Focusing on Washington, D.C., as a case study, this paper discusses the economic implications of the changes in the context of overall parking revenue and the cost of different revenue streams for parking. The paper also discusses the impact of these changes on program management (such as maintenance, personnel, and contracting models) and program outcomes (such as customer satisfaction and continued innovation). The paper provides agencies with a framework for taking a holistic look at their parking programs and assessing the impacts of various alternative, cost-effective approaches.

Dynamic Congestion and Tolls with Mobile Source Emission

This paper proposes a dynamic congestion pricing model that takes into account mobile source emissions. We consider a tollable vehicular network where the users selfishly minimize their own travel costs, including travel time, early/late arrival penalties and tolls. On top of that, we assume that part of the network can be tolled by a central authority, whose objective is to minimize both total travel costs of road users and total emission on a network-wide level. The model is formulated as a mathematical programming with equilibrium constraints (MPEC) problem and then reformulated as mathematical programming with complementarity constraints (MPCC). The MPCC is solved using a quadratic penalty- based gradient projection algorithm. A numerical study on a toy network illustrates the effectiveness of the tolling strategy and reveals a Braess-type paradox in the context of traffic-derived emission.

Dynamic congestion pricing with demand uncertainty: A robust optimization approach

In this paper, we consider dynamic congestion pricing in the presence of demand uncertainty. In particular, we apply a robust optimization (RO) approach based on a bi-level cellular particle swarm optimization (BCPSO) to optimal congestion pricing problems when flows correspond to dynamic user equilibrium on the network of interest. Such a formulation is recognized as a second-best pricing problem, and we refer to it as the dynamic optimal toll problem with equilibrium constraints (DOTPEC). We then present numerical experiments in which BCPSO is compared with two alternative robust dynamic solution approaches: bi-level simulated annealing (BSA) and cutting plane-based simulated annealing (CPSA), as well as a nominal dynamic solution and a robust static solution. We show that robust dynamic solutions improve the worst case, average, and stability of total travel cost in comparison with the nominal dynamic and the robust static solutions. The numerical results also show that BCPSO outperforms BSA and CPSA in terms of solution quality and computational efficiency.

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.