Dynamic User Equilibrium Research Articles

The corridor problem with discrete multiple bottlenecks

This paper presents a transparent approach to the analysis of dynamic user equilibrium and clarifies the properties of a departure-time choice equilibrium of a corridor problem where discrete multiple bottlenecks exist along a freeway. The basis of our approach is the transformation of the formulation of equilibrium conditions in a conventional “Eulerian coordinate system” into one in a “Lagrangian-like coordinate system.” This enables us to evaluate dynamic travel times easily, and to achieve a deep understanding of the mathematical structure of the problem, in particular, about the properties of the demand and supply (queuing) sub-models, relations with dynamic system optimal assignment, and differences between the morning and evening rush problems. Building on these foundations, we establish rigorous results on the existence and uniqueness of equilibria.

Formulation, existence, and computation of boundedly rational dynamic user equilibrium with fixed or endogenous user tolerance

This paper analyzes dynamic user equilibrium (DUE) that incorporates the notion of boundedly rational (BR) user behavior in the selection of departure times and routes. Intrinsically, the boundedly rational dynamic user equilibrium (BR-DUE) model we present assumes that travelers do not always seek the least costly route-and-departure-time choice. Rather, their perception of travel cost is affected by an indifference band describing travelers’ tolerance of the difference between their experienced travel costs and the minimum travel cost. An extension of the BR-DUE problem is the so-called variable tolerance dynamic user equilibrium (VT-BR-DUE) wherein endogenously determined tolerances may depend not only on paths, but also on the established path departure rates.This paper presents a unified approach for modeling both BR-DUE and VT-BR-DUE, which makes significant contributions to the model formulation, analysis of existence, solution characterization, and numerical computation of such problems. The VT-BR-DUE problem, together with the BR-DUE problem as a special case, is formulated as a variational inequality. We provide a very general existence result for VT-BR-DUE and BR-DUE that relies on assumptions weaker than those required for normal DUE models. Moreover, a characterization of the solution set is provided based on rigorous topological analysis. Finally, three computational algorithms with convergence results are proposed based on the VI and DVI formulations. Numerical studies are conducted to assess the proposed algorithms in terms of solution quality, convergence, and computational efficiency.

A bi-level model of dynamic traffic signal control with continuum approximation

This paper proposes a bi-level model for traffic network signal control, which is formulated as a dynamic Stackelberg game and solved as a mathematical program with equilibrium constraints (MPEC). The lower-level problem is a dynamic user equilibrium (DUE) with embedded dynamic network loading (DNL) sub-problem based on the LWR model (Lighthill and Whitham, 1955; Richards, 1956). The upper-level decision variables are (time-varying) signal green splits with the objective of minimizing network-wide travel cost. Unlike most existing literature which mainly use an on-and-off (binary) representation of the signal controls, we employ a continuum signal model recently proposed and analyzed in Han et al. (2014), which aims at describing and predicting the aggregate behavior that exists at signalized intersections without relying on distinct signal phases. Advantages of this continuum signal model include fewer integer variables, less restrictive constraints on the time steps, and higher decision resolution. It simplifies the modeling representation of large-scale urban traffic networks with the benefit of improved computational efficiency in simulation or optimization. We present, for the LWR-based DNL model that explicitly captures vehicle spillback, an in-depth study on the implementation of the continuum signal model, as its approximation accuracy depends on a number of factors and may deteriorate greatly under certain conditions. The proposed MPEC is solved on two test networks with three metaheuristic methods. Parallel computing is employed to significantly accelerate the solution procedure.

Better Convergence for Dynamic Traffic Assignment Methods

An essential feature in many of the dynamic traffic assignment methods that are used to study planning and traffic management problems is to compute the dynamic user equilibrium (DUE). In this paper the focus is on efficiently computing the deterministic and stochastic DUE. Both these types of equilibria are found using an iterative solution procedure. An important step in the solution algorithm for deterministic assignments is the choice for the contraction factor which determines how fast the related quadratic programming problem is solved. For stochastic assignments the step size to determine the starting point for the next iteration is important for convergence speed. As we show, the contraction factor and the step size heavily affect both the convergence efficiency and stability. We discuss various fixed contraction factors, a number of step size adjustments proposed by others, and a newly proposed dynamically adjusted step size. The solution method is evaluated on two transport networks of different scale. The comparative analysis suggests that, particularly for larger networks where equilibrium is harder to obtain, a dynamic adjusted contraction factor for deterministic assignments or a dynamic step size for stochastic assignments is preferable as it consistently converges considerably faster, because it does not suffer from a decreasing convergence rate.

Dynamic Activity-Travel Assignment in Multi-State Supernetworks

The integration of activity-based modeling and dynamic traffic assignment for travel demand analysis has recently attracted ever-increasing attention. However, related studies have limitations either on the integration structure or the number of choice facets being captured. This paper proposes a formulation of dynamic activity-travel assignment (DATA) in the framework of multi-state supernetworks, in which any path through a personalized supernetwork represents a particular activity-travel pattern (ATP) at a high level of spatial and temporal detail. DATA is formulated as a discrete-time dynamic user equilibrium (DUE) problem, which is reformulated as an equivalent variational inequality (VI) problem. A generalized dynamic link disutility function is established with the accommodation of different characteristics of the links in the supernetworks. Flow constraints and non-uniqueness of equilibria are also investigated. In the proposed formulation, the choice of departure time, route, mode, activity sequence, activity and parking location are all unified into one time-dependent ATP choice. As a result, the interdependences among all these choice facets can be readily captured. A solution algorithm based on the route-swapping mechanism is adopted to find the user equilibrium. A numerical example with simulated scenarios is provided to demonstrate the advantages of the proposed approach.

The Influence of Assignment Criteria for the Solution of Dynamic Travel Demand Estimation

This study deals with an investigation of the influence of several assignment criteria, combined with a different set of information about traffic conditions, on the efficacy and stability of the solution of the off-line dynamic demand estimation problem.Motivations of such investigation derive from the need to better reproduce the real users’ behaviors during the demand estimation and benefit from the extensive information, which is provided by traffic monitoring systems that collect advanced traffic data ubiquitously distributed on the network. Several synthetic experiments have been conducted on a test network, obtaining interesting results about the common adoption of dynamic user equilibrium and about the possibility to introduce new types of information during the optimization.

Simulation-based Optimization of Mixed Road Pricing Policies in a large Real-world Network

The joint optimization of various types of road pricing strategies for a large-scale real-world network is challenging. Dynamic network supply models can overcome this shortcoming, and provide more detailed information regarding the system performance. However, the computational burden of simulation is still a big challenge for the optimization. Utilizing a simulation-based dynamic traffic assignment model, this paper proposes a simulation-based optimization method to solve the mixed road pricing problem, which is characterized by expensive-to-evaluate and non-closed-form multi-objectives. The mixed road pricing problem is formulated to satisfy dynamic user equilibrium conditions. It enables to capture the time-varying network performance via dynamic traffic assignment. The simulation-based optimization method is successfully applied to the joint optimization of a variety of toll facilities in a real-world network of the Montgomery County in Maryland, i.e. an HOT lane, express toll lanes, and a toll road.

Developing a Large-Scale Hybrid Simulation Model

Long-term, regional travel demand models are essential tools used by planning organizations for resource management, project scheduling, and impact studies. Developed primarily at the macroscopic level, these tools lack sufficient detail to capture the influence of local geometry, dynamic traffic controls, or advanced transportation demand management strategies. To bridge the gap, a hybrid mesoscopic–microscopic model was developed. The core of the model, surrounding two light rail corridors in Minneapolis–Saint Paul, Minnesota (the Twin Cities), was developed at high resolution for microscopic simulation to capture the interaction between traffic signals, transit systems, and the road network. The remainder of the greater Twin Cities area was implemented according to the regional planning model (RPM) maintained by the Metropolitan Council. Interfacing the AIMSUN-based hybrid model with the CUBE-based RPM, the Twin Cities metro hybrid simulation was used to improve mode choice and traffic assignment iteratively to achieve a dynamic user equilibrium state. Significant lessons were learned about the effort needed to develop and to maintain such a model, with implications for future large-scale regional modeling.

Using the General Link Transmission Model in a Dynamic Traffic Assignment to Simulate Congestion on Urban Networks

This article presents two new models of Dynamic User Equilibrium that are particularly suited for ITS applications, where the evolution of vehicle flows and travel times must be simulated on large road networks, possibly in real-time. The key feature of the proposed models is the detail representation of the main congestion phenomena occurring at nodes of urban networks, such as vehicle queues and their spillback, as well as flow conflicts in mergins and diversions. Compared to the simple word of static assignment, where only the congestion along the arc is typically reproduced through a separable relation between vehicle flow and travel time, this type of DTA models are much more complex, as the above relation becomes non-separable, both in time and space.Traffic simulation is here attained through a macroscopic flow model, that extends the theory of kinematic waves to urban networks and non-linear fundamental diagrams: the General Link Transmission Model. The sub-models of the GLTM, namely the Node Intersection Model, the Forward Propagation Model of vehicles and the Backward Propagation Model of spaces, can be combined in two different ways to produce arc travel times starting from turn flows. The first approach is to consider short time intervals of a few seconds and process all nodes for each temporal layer in chronological order. The second approach allows to consider long time intervals of a few minutes and for each sub-model requires to process the whole temporal profile of involved variables. The two resulting DTA models are here analyzed and compared with the aim of identifying their possible use cases.A rigorous mathematical formulation is out of the scope of this paper, as well as a detailed explanation of the solution algorithm.The dynamic equilibrium is anyhow sought through a new method based on Gradient Projection, which is capable to solve both proposed models with any desired precision in a reasonable number of iterations. Its fast convergence is essential to show that the two proposed models for network congestion actually converge at equilibrium to nearly identical solutions in terms of arc flows and travel times, despite their two diametrical approaches wrt the dynamic nature of the problem, as shown in the numerical tests presented here.

The Corridor Problem with Discrete Multiple Bottlenecks

This paper presents a transparent approach to the analysis of dynamic user equilibrium and clarifies the properties of a departure- time choice equilibrium of a corridor problem where discrete multiple bottlenecks exist along a freeway. The basis of our approach is the transformation of the formulation of equilibrium conditions in a conventional “Eulerian coordinate system” into one in a “Lagrangian-like coordinate system.” This enables us to evaluate dynamic travel times easily, and to achieve a deep understanding of the mathematical structure of the problem, in particular, about the properties of the demand and supply (queuing) sub-models, relations with dynamic system optimal assignment, and differences between the morning and evening rush problems. Building on these foundations, we establish rigorous results on the existence and uniqueness of equilibria.

Activity-Based Model with Dynamic Traffic Assignment and Consideration of Heterogeneous User Preferences and Reliability Valuation

To forecast the impact of congestion pricing schemes, it is essential to capture user responses to these schemes and the resulting dynamics of traffic flow on the network. The responses of users must include route, departure time, and mode choices. To capture the effects of these decisions, this paper lays out a framework for the integration of the relevant elements of an activity-based model (ABM) with a dynamic traffic assignment (DTA) model and a simulation framework to support the analysis and evaluation of various pricing schemes. In this paper, a multicriterion dynamic user equilibrium traffic assignment model is used; the model explicitly considers heterogeneous users who seek to minimize travel time, out-of-pocket cost, and travel time reliability in the underlying route choice framework. In addition to the methodological developments, various demand and supply parameters are estimated and calibrated for the selected application network (the Greater Chicago, Illinois, network). This paper showcases the integration of ABM components and a DTA in one coherent modeling framework for the implementation and evaluation of congestion pricing in an actual large-scale network.

A semi-analytical approach for solving the bottleneck model with general user heterogeneity

This paper proposes a novel semi-analytical approach for solving the dynamic user equilibrium (DUE) of a bottleneck model with general heterogeneous users. The proposed approach makes use of the analytical solutions from the bottleneck analysis to create an equivalent assignment problem that admits closed-form commute cost functions. The equivalent problem is a static and asymmetric traffic assignment problem, which can be formulated as a variational inequality problem (VIP). This approach provides a new tool to analyze the properties of the bottleneck model with general heterogeneity, and to design efficient solution methods. In particular, the existence and uniqueness of the DUE solution can be established using the P-property of the Jacobian matrix. Our numerical experiments show that a simple decomposition algorithm is able to quickly solve the equivalent VIP to high precision. The proposed VIP formation is also extended to address simultaneous departure time and route choice in a single O–D origin-destination network with multiple parallel routes.

Investigating undesired spatial and temporal boundary effects of congestion charging

Two types of reported problems are related to the existing congestion charging projects that levy traffic only in a certain area within one or a few time periods during the day. One is that travellers depart earlier or later than a charging period to avoid paying full or part of the congestion charging tolls, which creates two undesired demand peaks that are often greater than available capacity. One peak comes just before the start of congestion charging and the other follows the end of it. We term this phenomenon ‘temporal boundary effect’ of congestion charging. The other reported problem is that travellers would rather stay away from a charging zone than pay congestion charging tolls, which causes undesired congestion on those roads or paths on the edge of the charging zone. We call this phenomenon ‘spatial boundary effect’ generated by congestion charging. This research investigates these boundary effects in the context of simultaneous route and departure time choice dynamic user equilibrium (SRD-DUE) network flows with an aim to gain new insights into congestion charging design. Numerical experiments investigating constant and time-varying congestion charging toll profiles are presented in this paper. This investigation shows that congestion charging may not be able to eliminate hypercongestion efficiently if schemes are not well designed, and can unfortunately give rise to undesired boundary effects and that a simply designed congestion charging scheme with small level toll or time-varying toll profiles can reduce the magnitude of boundary effects but may not be able to fully eliminate such undesired effects.

Advances in Dynamic Traffic Assgmnt: TAC

Traditionally traffic assignment problems (TAPs) have been formulated as non-cooperative games (NCGs), and Wardrop’s user equilibrium (UE) of a TAP as Nash equilibrium (NE) of the corresponding NCG. In this study, in contrast, we propose a mapping from finite NCGs (F-NCGs) to asymmetric TAPs (A-TAPs) on a two-way or urban network, in which the travel cost of one link depends on flows on other links. With the help of three route-based formulations of UE, we show that NE of an F-NCG is equivalent to UE of the corresponding A-TAP. Based on the new relationship, we extend Nash’s fixed point formulation of NE to prove the existence of static UE. This study provides a more complete picture of the relationships between NCGs and TAPs. In the future we will be interested in studying Nash’s fixed point formulation for dynamic UE.

Fluid Approximation of Point-queue Model

Point-queue model is widely used in the dynamic user equilibrium (DUE) analysis in discrete-time or continuous-time form. In this paper, a continuous time point queue is proposed. In the former studies, the negativity of the queue length of the original point-queue model is shown and some improvement has been made. Based on the observation that the original point-queue model is actually a queuing model with a server and a buffer with infinite capacity, a fluid approximation (FA) model is proposed to interpret the original point-queue model. Three essential components are a flow balance function, an exit flow function and a time-dependent capacity utilization ratio function, which are all in continuous form. During the analysis, the theoretical proof and numerical study of the non-negativity of queue length are accomplished. With the first-order Taylor expansion, this paper applies the Gronwall's inequality to prove the non-negativity of queue-length. Through numerical testing different specific FA models in the solution scheme, we can show that the negativity of the queue length in the FA model is overcome and some differences between the FA and former studies are discussed. Based on the testing, the capability of our model in approximating the point- queue model is demonstrated.

Submission to the DTA2012 Special Issue: Approximating Time Delays in Solving Continuous-Time Dynamic User Equilibria

We develop an approximation scheme, called “pseudo-derivative (PD)” to solving the dynamic user equilibrium (DUE) problem. The PD approximation can convert time-varying, state dependent delays usually involved in DUE to a constant time delay. We study the properties of the proposed PD and the resulting approximate DUE (ADUE) problem after applying the approximation. Some issues of the ADUE, such as the possible violation of the flow conservation at network nodes are also discussed and resolved. It turns out that the original DUE problem can be solved iteratively with an ADUE solved in each iteration. Numerical results are shown on a small test network and the Sioux Falls network. The results show that the iterative algorithm can converge to some reasonable solution, although a formal convergence proof result is not established in the paper.

Submission to the DTA2012 Special Issue: Convergence of Time Discretization Schemes for Continuous-Time Dynamic Network Loading Models

Dynamic Network Loading (DNL) is an essential component of Dynamic Traffic Assignment (DTA) and Dynamic User Equilibrium (DUE). Most DNL models are formulated in continuous time but solved in discrete time to obtain numerical solutions. This paper discusses the importance of choosing proper discretization schemes to numerically solve continuous-time DNL models and further to obtain convergence and other desirable properties of the discretization schemes. We use the recently developed α point-queue model as an example. We first develop theoretical results to prove the consistency, stability and convergence of the implicit and explicit discretization schemes for solving the α point-queue model. We then conduct numerical experiments to show such results accordingly. We also discuss the implications of the implicit and explicit discretization schemes to the developments of DNL and DTA/DUE solution algorithms.

A Comparison of Dynamic User Optimal States with Zero, Fixed and Variable Tolerances

The notion of dynamic user equilibrium (DUE) and dynamic user optimum (DUO) were still treated interchangeably in the dynamic traffic assignment (DTA) literature until tolerance-based DUO was recently proposed as alternative definitions of DUO. The latter distinguishes DUO from DUE, and the tolerances can be prescribed a priori (i.e. DUO with fixed tolerance) or endogenously (i.e. DUO with variable tolerance). These tolerances reflect the differences in travel costs between used paths of an origin–destination (OD) pair. Since DUE requires zero difference in travel costs between used paths of an OD pair, it can be regarded as DUO with zero tolerance. This paper first compares the three alternative definitions of DUO and discusses their relationships. Secondly, the existence conditions of these DUO solutions are discussed. A key conclusion is that DUO with variable tolerance always exists even when the other two types of DUO solutions may not. The second conclusion is that, when the tolerances are equal to zero, the solution must be a DUE solution. Furthermore, DUE is a special case of DUO with variable tolerance; since the tolerance in the concept of DUO with variable tolerance is an endogenous variable, if a DUE solution exists, an efficient method for solving for DUO with variable tolerance should directly find such a DUE solution. All these are illustrated by means of three examples. Finally, it is acknowledged that the analysis of DUO with variable tolerance in the same rigor as that of DUE is a challenging problem and worthy of further research.

Solving Network Design Problem with Dynamic Network Loading Profiles Using Modified Reinforcement Learning Method

This study aims to solve dynamic user Equilibrium Network Design Problem (ENDP) with dynamic network loading profiles using modified Reinforcement Learning (RL) approach. The bi-level programming technique is used to solve the problem. At the lower level of the problem, the dynamic User Equilibrium (UE) link flows are obtained by simulation based Dynamic Traffic Assignment (DTA) model with DynusT and signal timings are obtained at the upper level by modified RL method. The system Performance Index (PI) is defined as the sum of a weighted linear combination of delay and number of stops per unit time for all traffic streams, which is evaluated by the traffic model of TRANSYT-7F. Q-learning, a model-free approach, is one of the RL methods. The modified RL method is actually based on Q-learning. By integrating the modified RL method, traffic assignment and traffic control, the modified REinforcement Learning TRANSYT-7F DynusT (RELTRAD) model is proposed to solve the dynamic ENDP. The objective function of the proposed RELTRAD is total network PI. The model is tested on the medium sized Allsop and Charlesworth's network. Two scenarios, related to various dynamic network loading profiles, are proposed for numerical application. Encouraging results are obtained. Results showed that the RELTRAD model effectively optimizes the signal timings and values of the network PI. The RELTRAD model improves to the network PI from the initial value to the final value as 65% and 67% for loading profile 1 and 2, respectively.

Park-and-Ride Network Equilibrium with Heterogeneous Commuters and Parking Space Constraint

A dynamic user equilibrium (DUE) model of heterogeneous commuters’ travel choice behaviors including departure time and path and travel mode choices in a schematic park-and-ride network is developed. Commuters have three transportation modes to choose from in a peak period: private car, rail transit, and a combination of private car and rail transit (i.e., an intermodal transportation mode). Two types of DUE patterns are derived with and without the parking space constraint. In this study, an optimal park-and-ride parking fee scheme is proposed: commuters’ travel choice behaviors are characterized by the developed trimodal multiclass DUE model. The optimal park-and-ride parking fee scheme is formulated as a bilevel program in which the upper-level problem is to find the optimal parking fee for improving network performance and the lower-level problem is to evaluate network performance in equilibrium. Because the DUE solution may not be unique, the authors aim to improve the network performance in the worst and best cases. Through numerical tests, it is demonstrated that the parking space constraint has a significant influence on the departure and route choices of commuters and should be considered as an important factor in park-and-ride commuting pattern analysis and parking fee management in order to avoid a misleading decision.