Dynamic User Equilibrium Research Articles

A cross entropy based multiagent approach for multiclass activity chain modeling and simulation

This paper attempts to model complex destination-chain, departure time and route choices based on activity plan implementation and proposes an arc-based cross entropy method for solving approximately the dynamic user equilibrium in multiagent-based multiclass network context. A multiagent-based dynamic activity chain model is developed, combining travelers’ day-to-day learning process in the presence of both traffic flow and activity supply dynamics. The learning process towards user equilibrium in multiagent systems is based on the framework of Bellman’s principle of optimality, and iteratively solved by the cross entropy method. A numerical example is implemented to illustrate the performance of the proposed method on a multiclass queuing network.

Dynamic traffic assignment approximating the kinematic wave model: System optimum, marginal costs, externalities and tolls

System marginal costs, externalities and optimal congestion tolls for traffic networks are generally derived from system optimising (SO) traffic assignment models and when they are treated as varying over time they are referred to as dynamic. In dynamic system optimum (DSO) models the link flows and travel times or costs are generally modelled using so-called ‘whole link’ models. Here we instead develop an SO model that more closely reflects traffic flow theory and derive the marginal costs and externalities from that. The most widely accepted traffic flow model appears to be the LWR (Lighthill, Whitham and Richards) model and a tractable discrete implementation or approximation to that is provided by the cell transmission model (CTM) or a finite difference approximation (FDA). These handle spillbacks, traffic controls and moving queues in a way that is consistent with the LWR model and hence with the kinematic wave model and fluid flow model. An SO formulation using the CTM is already available, assuming a single destination and a trapezoidal flow-density function. We extend the formulation to allow more general nonlinear flow density functions and derive and interpret system marginal costs and externalities. We show that if tolls computed from the DSO solution are imposed on users then the DSO solution would also satisfy the criteria for a dynamic user equilibrium (DUE). We extend the analysis to allow for physical or behavioural constraints on the link outflow proportions at merges and inflow proportions at diverges. We also extend the model to elastic demands and establish connections between the present DSO model and earlier DSO models.

An alternative definition of dynamic user optimum on signalised road networks

SUMMARYIn the literature on dynamic traffic assignment (DTA), dynamic user optimum (DUO) and dynamic user equilibrium (DUE) are interchangeable and refer to a stable state on a road network in which, under the assumption of perfect information on traffic conditions and the assumption that every traveller chooses the least costly path to travel, no one can reduce his or her travel cost by changing his or her path unilaterally. This paper proposes an alternative definition of DUO on signalised road networks; in such a DUO state, the travel times or costs of used paths between the same origin–destination pair can be different because of the existence of signalised junctions. Usually, on signalised road networks, a DUE solution is sought to approximate such a DUO solution by assuming that the capacity of each approach at a signalised junction is equal to the saturation flow rate of the approach times the corresponding split. By comparing the DUE and DUO solutions of a DTA problem on a small signalised road network, this paper discusses advantages and disadvantages of this approximation and shows that such a DUE solution would not be a good approximation to a DUO solution. Copyright © 2012 John Wiley & Sons, Ltd.

A Temporal Domain Decomposition Algorithmic Scheme for Large-Scale Dynamic Traffic Assignment

ABSTRACT With emergent interest of Simulation-Based Dynamic Traffic Assignment (SBDTA) in the field of transportation network modeling, deployment of SBDTA models for traffic operations and transportation planning have increased significantly in recent years. In parallel, research and development of innovative approaches of the SBDTA model have enhanced the quality of both the assignment component, i.e, improvement of convergence quality of the Dynamic User Equilibrium (DUE) problem, and the traffic simulation element. However, computational requirement remains to be one of the great challenges for DTA implementations on large-scale networks with a long analysis period. This paper presents a temporal decomposition scheme for large spatial- and temporal-scale dynamic traffic assignment, in which the entire analysis period is divided into Epochs. Vehicle assignment is performed sequentially in each Epoch, thus improving the model scalability and confining the peak run-time memory requirement regardless of the total analysis period. A proposed self-turning scheme adaptively searches for the run-time-optimal Epoch setting during iterations regardless of the characteristics of the modeled network. Extensive numerical experiments confirm the promising performance of the proposed algorithmic schemes.

Modeling and solving continuous-time instantaneous dynamic user equilibria: A differential complementarity systems approach

This paper is the second of a two-part research wherein we undertake a mathematically rigorous investigation of the continuous-time dynamic user equilibrium (DUE) problem using the recently introduced mathematical paradigm of differential complementarity systems (DCSs). Based on the thorough study of continuous-time single-destination point-queue models in the previous part, we first extend this special case to multiple destinations respecting the First-In–First-Out property of travel flows. A DCS with constant time delay is then introduced to formulate the continuous-time model of instantaneous dynamic traffic equilibria (IDUE) with a fixed demand profile. We develop a time decomposition scheme based on link free flow travel times to convert the delay DCS to a series of DCSs without time delays that are solved by a numerical time-stepping method. We provide rigorous numerical treatment of the time-decomposed IDUE model, including solvability of the discrete-time complementarity problems and convergence of the numerical trajectories to a continuous-time solution. We present numerical results to validate the IDUE on a small network and also on the Sioux Falls network.

Continuous-time point-queue models in dynamic network loading

Extending a recent work (Pang et al., in press) pertaining to a simple single-bottleneck model, this paper is the first of a two-part research wherein we undertake a mathematically rigorous study of the continuous-time dynamic user equilibrium (DUE) problem using the recently introduced mathematical paradigm of differential complementarity systems (DCSs) (Pang and Stewart, 2008). The first step in this comprehensive research is to gain a thorough understanding of some continuous-time point-queue models, which will be used as the building block of a computationally tractable model for the continuous-time DUE problem that we will study in detail in the accompanying paper (Ban et al., in press). Starting with the original point-queue model introduced by Vickrey (1969), we summarize some desirable properties that a continuous-time point-queue model should possess, and show that one of these properties—the nonnegativity of the queue lengths—is violated by Vickrey’s original model. As a remedy to this drawback of Vickrey’s model and with the goal of extending it to a continuous-time setting, we introduce two continuous-time point-queue models and show that they satisfy the properties we propose. Discretizations of the continuous-time models are discussed and construction of numerical trajectories is presented; convergence of such trajectories as the time step approaches zero is established; regularity of a solution to the continuous-time problem is clarified, and numerical results are presented.

Bi-level optimization for risk-based regional hurricane evacuation planning

Almost all engineering evacuation models define the objective as minimizing the time required to clear the region or total travel time, thus making an implicit assumption that who will or should evacuate is known. Conservatively evacuating everyone who may be affected may be the best strategy for a given storm, but there is a growing recognition that in some places that strategy is no longer viable and in any case, may not be the best alternative by itself. Here, we introduce a new bi-level optimization that reframes the decision more broadly. The upper level develops an evacuation plan that describes, as a hurricane approaches, who should stay and who should leave and when, so as to minimize both risk and travel time. The lower level is a dynamic user equilibrium (DUE) traffic assignment model. The model includes four novel features: (1) it refocuses the decision on the objectives of minimizing both risk and travel time; (2) it allows direct comparison of more alternatives, including for the first time, sheltering-in-place; (3) it uses a hurricane-scenario-based analysis that explicitly represents the critically important uncertainty in hurricane track, intensity, and speed; and (4) it includes a new DUE algorithm that is efficient enough for full-scale hurricane evacuation applications. The model can be used both to provide an evacuation plan and to evaluate a plan’s performance in terms of risk and travel time, assuming the plan is implemented and a specified hurricane scenario then actually occurs. We demonstrate the model with a full-scale case study for Eastern North Carolina.

Numerical simulation of a continuum model for bi-directional pedestrian flow

An algorithm for an extended reactive dynamic user equilibrium model of pedestrian counterflow as a continuum is developed. It is based on a cell-centered high-resolution finite volume scheme with a fast sweeping method for an Eikonal-type equation on an orthogonal grid. A high-order total variation diminishing Runge–Kutta method is adopted for the time integration of semi-discrete equations. The numerical results demonstrate the rationality of the model and efficiency of the algorithm. Some crowd pedestrian flow phenomena, such as dynamic lane formation in bi-directional flow, are observed which are helpful for a global comprehension of pedestrian dynamics. Also, the model can be utilized with different potential applications.

A Cumulative Perceived Value-Based Dynamic User Equilibrium Model Considering the Travelers’ Risk Evaluation on Arrival Time

This paper presents a cumulative perceived value-based dynamic user equilibrium model by applying the prospect theory to formulate the travelers’ risk evaluation on arrival time. The network uncertainty caused by link exit capacity degradation is incorporated into the analysis. The model which considers departure time and route choices simultaneously is expressed by a variational inequality in a discrete time space. Numerical results show that the travelers’ risk preference indeed has big influence on flow distribution. Our study constitutes a deepening of cognition in developing more realistic dynamic traffic assignment technologies.

Complementarity formulations for the cell transmission model based dynamic user equilibrium with departure time choice, elastic demand and user heterogeneity

In this paper we formulate the dynamic user equilibrium problem with an embedded cell transmission model on a network with a single OD pair, multiple parallel paths, multiple user classes with elastic demand. The formulation is based on ideas from complementarity theory. The travel time is estimated based on two methods which have different transportation applications: (1) maximum travel time and (2) average travel time. These travel time functions result in linear and non-linear complementarity formulations respectively. Solution existence and the properties of the formulations are rigorously analyzed. Extensive computational experiments are conducted to demonstrate the benefits of the proposed formulations on various test networks.

A solution algorithm for a dynamic deterministic user equilibrium assignment model with departure time choice

A route-based combined model of dynamic deterministic route and departure time choice and a solution method for many origin and destination pairs is proposed. The divided linear travel time model is used to calculate the link travel time and to describe the propagation of flow over time. For the calculation of route travel times, the predictive ideal route travel time concept is adopted. Solving the combined model of dynamic deterministic route and departure time choice is shown to be equivalent to solving simultaneously a system of non-linear equations. A Newton-type iterative scheme is proposed to solve this problem. The performance of the proposed solution method is demonstrated using a version of the Sioux Falls network. This shows that the proposed solution method produces good equilibrium solutions with reasonable computational cost.

Dynamic user equilibrium with side constraints for a traffic network: Theoretical development and numerical solution algorithm

This paper investigates a traffic volume control scheme for a dynamic traffic network model which aims to ensure that traffic volumes on specified links do not exceed preferred levels. The problem is formulated as a dynamic user equilibrium problem with side constraints (DUE-SC) in which the side constraints represent the restrictions on the traffic volumes. Travelers choose their departure times and routes to minimize their generalized travel costs, which include early/late arrival penalties. An infinite-dimensional variational inequality (VI) is formulated to model the DUE-SC. Based on this VI formulation, we establish an existence result for the DUE-SC by showing that the VI admits at least one solution. To analyze the necessary condition for the DUE-SC, we restate the VI as an equivalent optimal control problem. The Lagrange multipliers associated with the side constraints as derived from the optimality condition of the DUE-SC provide the traffic volume control scheme. The control scheme can be interpreted as additional travel delays (either tolls or access delays) imposed upon drivers for using the controlled links. This additional delay term derived from the Lagrange multiplier is compared with its counterpart in a static user equilibrium assignment model. If the side constraint is chosen as the storage capacity of a link, the additional delay can be viewed as the effort needed to prevent the link from spillback. Under this circumstance, it is found that the flow is incompressible when the link traffic volume is equal to its storage capacity. An algorithm based on Euler’s discretization scheme and nonlinear programming is proposed to solve the DUE-SC. Numerical examples are presented to illustrate the mechanism of the proposed traffic volume control scheme.

Multiple equilibria in a dynamic traffic network

This study provides an example in which the dynamic user equilibrium (DUE) assignment of a congested road network with bottlenecks is non-unique. In previous studies, the uniqueness of DUE assignments with the bottleneck model has been shown in limited cases such as single-origin and single-destination networks. Consequently, it is still an important issue whether or not uniqueness is a general property of DUE assignments. The present study describes a network in which multiple patterns of link travel time are found, thus providing a negative answer to this question. The network has a loopy structure with multiple bottlenecks and multiple origin–destination (OD) pairs. Given a certain demand pattern of departure times for vehicles leaving their origins, a non-convex set of equilibria with a non-unique pattern of link travel times is shown to exist.

Comparison of Methods for Path Flow Reassignment for Dynamic User Equilibrium

Models to describe or predict of time-varying traffic flows and travel times on road networks are usually referred to as dynamic traffic assignment (DTA) models or dynamic user equilibrium (DUE) models. The most common form of algorithms for DUE consists of iterating between two components namely dynamic network loading (DNL) and path inflow reassignment or route choice. The DNL components in these algorithms have been investigated in many papers but in comparison the path inflow reassignment component has been relatively neglected. In view of that, we investigate various methods for path inflow reassignment that have been used in the literature. We compare them numerically by embedding them in a DUE algorithm and applying the algorithm to solve DUE problems for various simple network scenarios. We find that the choice of inflow reassignment method makes a huge difference to the speed of convergence of the algorithms and, in particular, find that ‘travel time responsive’ reassignment methods converge much faster than the other methods. We also investigate how speed of convergence is affected by the extent of congestion on the network, by higher demand or lower capacity. There appears to be much scope for further improving path inflow reassignment methods.

B-Dynamic: An Efficient Algorithm for Dynamic User Equilibrium Assignment in Activity-Travel Networks1

Multi-dimensional choice in dynamic traffic assignment (DTA)—for example, a combined model of activity location, time of participation, duration, and route choice decisions—results in exponentially increasing choice alternatives. Any efficient algorithm for solving the multi-dimensional DTA problem must avoid enumeration of alternatives. In this article an algorithm that does not enumerate paths is presented. The algorithm is a novel extension of Algorithm B (Dial, 2006) to dynamic networks and hence referred to as Algorithm B-Dynamic. The DTA model proposed here uses a point queue model for traffic propagation that reduces computational complexity. The activity participation decision dimensions are incorporated through utility functions, which are a linear function of duration and schedule delay (early or late arrival penalty). Numerical examples are then presented to illustrate both the steps of the algorithm and its capabilities. Overall, the algorithm performed well for up to medium-sized networks. Further, the algorithm scales fairly well with increasing demand levels.

A continuous-time linear complementarity system for dynamic user equilibria in single bottleneck traffic flows

This paper formally introduces a linear complementarity system (LCS) formulation for a continuous-time, multi-user class, dynamic user equilibrium (DUE) model for the determination of trip timing decisions in a simplified single bottleneck model. Existence of a Lipschitz solution trajectory to the model is established by a constructive time-stepping method whose convergence is rigorously analyzed. The solvability of the time-discretized subproblems by Lemke’s algorithm is also proved. Combining linear complementarity with ordinary differential equations and being a new entry to the mathematical programming field, the LCS provides a computational tractable framework for the rigorous treatment of the DUE problem in continuous time; this paper makes a positive contribution in this promising research venue pertaining to the application of differential variational theory to dynamic traffic problems.

Congestion Pricing, Heterogeneous Users, and Travel Time Reliability

With the increasing interest in road pricing strategies to alleviate congestion and improve network performance, this study develops a multi-criterion dynamic user equilibrium (MDUE) traffic assignment model that explicitly considers heterogeneous users who seek to minimize three essential decision attributes: travel time, out-of-pocket cost, and travel time reliability in the underlying path choice framework. The value of time in this study is treated as a continuous random variable distributed across the population of trip makers, and travel time reliability is estimated at a path level. Furthermore, the proposed MDUE model is capable of considering different vehicle classes, namely, low-occupancy vehicles and high-occupancy vehicles; thus, with greater realism in trip makers’ path choice behavior, the MDUE model is applicable for analyzing a variety of road pricing scenarios. To address the practical application aspects for large-scale congested networks, novel implementation techniques are proposed to overcome computational obstacles to real-world application. An application to the New York metropolitan regional network is demonstrated, and a set of numerical experiments is conducted on this network to explore the convergence behavior, solution quality, and required computational time of the MDUE algorithm.

A Simulation-Based Approximation Algorithm for Dynamic Marginal Cost Pricing

This work offers a simulation-based approximation algorithm for dynamic marginal cost pricing (MCP) congestion pricing that is a direct extension of its static counterpart. The algorithm approximates the time-dependent marginal costs, and is incorporated into the inner approximation dynamic user equilibrium algorithm to evaluate the results of dynamic MCP, which are then compared to static assignment results with MCP from previous study. The status quo and dynamic MCP-on-freeways scenarios are simulated (and then compared) on the Dallas-Fort Worth 35,732-link network. Due to computational requirements for such a large-scale dynamic traffic assignment application, the dynamic MCP scenario is simulated without feedback, and only route choices are permitted to vary. When prices are imposed on freeway users, some minor system benefits are observed, including a delay in the onset of congestion. Dynamic prices vary substantially over the three-hour period of analysis, reflecting changes in freeway congestion. Reasons for any inconsistencies between dynamic and static results are discussed, along with important enhancements to future implementation.

Equilibrium Results for Dynamic Congestion Games

Consider the following game. Given a network with a continuum of users at some origins, suppose users wish to reach specific destinations but they are not indifferent to the cost to reach them. They may have multiple possible routes but their choices modify the travel costs on the network. Hence, each user faces the following problem: Given a pattern of travel costs for the different possible routes that reach the destination, find a path of minimal cost. This kind of game belongs to the class of congestion games. In the traditional static approach, travel times are assumed constant during the period of the game. In this paper, we consider the so-called dynamic case where the time-varying nature of traffic conditions is explicitly taken into account. In transportation science, the question of whether there is an equilibrium and how to compute it for such a model is referred to as the dynamic user equilibrium problem. Until now, there was no general model for this problem. Our paper attempts to resolve this issue. We define a new class of games, dynamic congestion games, which capture this time-dependency aspect. Moreover, we prove that under some natural assumptions there is a Nash equilibrium. When we apply this result to the dynamic user equilibrium problem, we get most of the previous known results.

Approximate network loading and dual-time-scale dynamic user equilibrium

In this paper we present a dual-time-scale formulation of dynamic user equilibrium (DUE) with demand evolution. Our formulation belongs to the problem class that Pang and Stewart (2008) refer to as differential variational inequalities. It combines the within-day time scale for which route and departure time choices fluctuate in continuous time with the day-to-day time scale for which demand evolves in discrete time steps. Our formulation is consistent with the often told story that drivers adjust their travel demands at the end of every day based on their congestion experience during one or more previous days. We show that analysis of the within-day assignment model is tremendously simplified by expressing dynamic user equilibrium as a differential variational inequality. We also show there is a class of day-to-day demand growth models that allow the dual-time-scale formulation to be decomposed by time-stepping to yield a sequence of continuous time, single-day, dynamic user equilibrium problems. To solve the single-day DUE problems arising during time-stepping, it is necessary to repeatedly solve a dynamic network loading problem. We observe that the network loading phase of DUE computation generally constitutes a differential algebraic equation (DAE) system, and we show that the DAE system for network loading based on the link delay model (LDM) of Friesz et al. (1993) may be approximated by a system of ordinary differential equations (ODEs). That system of ODEs, as we demonstrate, may be efficiently solved using traditional numerical methods for such problems. To compute an actual dynamic user equilibrium, we introduce a continuous time fixed-point algorithm and prove its convergence for effective path delay operators that allow a limited type of nonmonotone path delay. We show that our DUE algorithm is compatible with network loading based on the LDM and the cell transmission model (CTM) due to Daganzo (1995). We provide a numerical example based on the much studied Sioux Falls network.