System Optimal Dynamic Traffic Assignment Research Articles

Congestion-aware system optimal route choice for shared autonomous vehicles

We study the shared autonomous vehicle (SAV) routing problem while considering congestion. SAVs essentially provide a dial-a-ride service to travelers, but the large number of vehicles involved (tens of thousands of SAVs to replace personal vehicles) results in SAV routing causing significant congestion. We combine the dial-a-ride service constraints with the linear program for system optimal dynamic traffic assignment, resulting in a congestion-aware formulation of the SAV routing problem. Traffic flow is modeled through the link transmission model, an approximate solution to the kinematic wave theory of traffic flow. SAVs interact with travelers at origins and destinations. Due to the large number of vehicles involved, we use a continuous approximation of flow to formulate a linear program. Optimal solutions demonstrate that peak hour demand is likely to have greater waiting and in-vehicle travel times than off-peak demand due to congestion. SAV travel times were only slightly greater than system optimal personal vehicle route choice. In addition, solutions can determine the optimal fleet size to minimize congestion or maximize service.

Emission modeling and pricing on single-destination dynamic traffic networks

This paper proposes an emission pricing model for single-destination dynamic traffic networks. The model contains two sub-models derived from the corresponding two sub-problems: a system optimum dynamic traffic assignment problem and a first-best dynamic emission pricing scheme. For the first problem, it proves that under certain conditions, an optimal solution, if exists, must be a free-flow solution to minimize the generalized system cost including the costs of total travel times and total emissions (or fuel consumption). The optimal first-best emission pricing can then be determined by solving an optimal control problem, using the free-flow dynamic system optimal solution as the input. Numerical results are provided to illustrate the proposed models and the solution methods.

Convexity and robustness of dynamic traffic assignment and freeway network control

We study the use of the System Optimum (SO) Dynamic Traffic Assignment (DTA) problem to design optimal traffic flow controls for freeway networks as modeled by the Cell Transmission Model, using variable speed limit, ramp metering, and routing. We consider two optimal control problems: the DTA problem, where turning ratios are part of the control inputs, and the Freeway Network Control (FNC), where turning ratios are instead assigned exogenous parameters. It is known that relaxation of the supply and demand constraints in the cell-based formulations of the DTA problem results in a linear program. However, solutions to the relaxed problem can be infeasible with respect to traffic dynamics. Previous work has shown that such solutions can be made feasible by proper choice of ramp metering and variable speed limit control for specific traffic networks. We extend this procedure to arbitrary networks and provide insight into the structure and robustness of the proposed optimal controllers. For a network consisting only of ordinary, merge, and diverge junctions, where the cells have linear demand functions and affine supply functions with identical slopes, and the cost is the total traffic volume, we show, using the Pontryagin maximum principle, that variable speed limits are not needed in order to achieve optimality in the FNC problem, and ramp metering is sufficient. We also prove bounds on perturbation of the controlled system trajectory in terms of perturbations in initial traffic volume and exogenous inflows. These bounds, which leverage monotonicity properties of the controlled trajectory, are shown to be in close agreement with numerical simulation results.

Link-based system optimum dynamic traffic assignment problems with environmental objectives

Maintaining air-quality standards has been a priority for transportation planners and policy makers worldwide. However, most existing system optimum dynamic traffic assignment (SO-DTA) models do not accommodate environmental objectives. In this paper, we use the link transmission model (LTM) to develop SO-DTA models that minimize total system emissions (TSE) in single destination networks. We use step functions to approximate cumulative flow curves for individual links, and to decompose link inflow into sub-flows according to time intervals at which they leave the link. The decomposed link inflows are used to estimate link emissions. Dynamic network constraints, non-vehicle holding constraints and link inflow decomposition constraints are considered, and SO-DTA problems with environmental objectives are formulated as mixed integer linear programming (MILP) problems. Any average speed based emission functions can be used for our models. Finally, numerical examples are provided to demonstrate the performance of the proposed models.

Impact of Autonomous Vehicles on Traffic Management: Case of Dynamic Lane Reversal

As the future of autonomous vehicles (AVs) becomes more certain, transport network managers may seek ways to reinvent elements of the traffic network to improve efficiency. One possibility is dynamic lane reversal, in which the network operator makes use of AV communications and behavior to change the direction of flow on a road link at smaller time intervals than would be possible with human drivers. Although there is much research into the mechanical details of AVs, this study motivates the need for future research by focusing on a planning application in which AVs are already present. A novel extension to an established system optimal dynamic traffic assignment model based on the cell transmission model was examined. The model determined the optimal lane configuration at small space–time intervals. Results demonstrate the model on a single link and a grid network and explore the dynamic demand scenarios that are most conducive to increasing system efficiency with dynamic lane reversal.

Convexity and Robustness of Dynamic Network Traffic Assignment for Control of Freeway Networks

We study System Optimum Dynamic Traffic Assignment (SO-DTA) for realistic traffic dynamics controlled by variable speed limits, ramp metering, and routing controls. We consider continuous-time cell-based Dynamic Network Loading models that include the Cell Transmission Model with FIFO rule at the diverge junctions as well as non-FIFO diverge rules. We consider SO-DTA formulations in which the total inflow into and the total outflow from the cells are independently constrained by concave supply and demand functions, respectively, thus preserving convexity. We design open-loop controllers that guarantee that the optimal solutions are feasible with respect to realistic traffic dynamics, and we develop this methodology for two variants of the SO-DTA problem, one of which accounts for exogenous turning ratios. Robustness of the system trajectory under the proposed controllers is evaluated by deriving bounds on the deviations induced by perturbations of the initial condition and of the external inflows.

On the System Optimum Dynamic Traffic Assignment and Earliest Arrival Flow Problems

This paper investigates the cell-transmission model (CTM)-based single destination system optimum dynamic traffic assignment (SO-DTA) problem, focusing attention on a case where the cell properties are time-invariant. We show the backward propagation of congestion in CTM does not affect the optimal arrival flow pattern of SO-DTA, if the fundamental diagram is of triangular/trapezoidal shape as in the CTM. We mathematically prove that the set of earliest arrival flows (EAFs) not constrained by the traffic wave propagation equations obtained on the node-arc network without cell division is a subset of the SO-DTA. This finding leads to a new approach to the SO-DTA that solves the EAF. Such an EAF can be obtained by merely applying static flow techniques and turning the static flows into dynamic flows over time. Therefore, SO-DTA can theoretically be solved with a run time at the link level depending polynomially on log T. We use numerical examples to verify the results and report the computational benefits of the proposed method by solving SO-DTA on a real-world network.

Evaluation of Incident Management Impacts Using Stochastic Dynamic Traffic Assignment

In this paper, a dynamic traffic assignment (DTA) formulation with probabilistic capacity constraints is suggested in order to incorporate accident-induced random capacity reductions into evaluation of incident management strategies. For this purpose, a cell transmission model (CTM) based system optimal dynamic traffic assignment (SODTA) formulation is used as the underlying network model. Hypothetical scenarios are devised in which the potential incident management (IM) strategies are assumed to reduce either the average or the variation of the incident duration. For each case, a small scale Monte Carlo simulation is also performed and compared with the analytic results of the stochastic DTA model. It was shown that the stochastic DTA model not only provides the changes in total system travel time within the reliability measure, but it also provides the analytical results which requires significantly less computational burden. The model also incorporates the impacts of rerouting which is not possible with a queuing theory based analysis on a single link. The results also show that rather than reducing the average duration, comparable delay reductions can be achieved by reducing the variance while keeping the average accident duration unchanged. Hence, IM strategies, solely targeting average duration may be deemed not to be successful, yet, they may be an effective policy to reduce delay. Overall, the proposed model provides a computationally efficient network-wide analysis of incident induced delay without ignoring the highly stochastic nature of roadway incidents.

Emission Modeling and Pricing in Dynamic Traffic Networks

This paper proposes an emission pricing model for dynamic traffic networks with single destinations. The model contains two sub-problems: a system optimum dynamic traffic assignment problem and a first-best dynamic emission pricing scheme. It proves that under certain conditions, there exists a free-flow optimal solution to minimize the generalized system cost including total travel times and total emissions. The optimal first-best emission pricing can then be determined by solving an optimal control problem, using the free-flow dynamic system optimal solution as the input. Numerical results are provided to illustrate the proposed model and its solution method.

Discrete-Time System Optimal Dynamic Traffic Assignment (SO-DTA) with Partial Control for Physical Queuing Networks

We consider the System Optimal Dynamic Traffic Assignment (SO-DTA) problem with Partial Control for general networks with physical queuing. Our goal is to optimally control any subset of the networks agents to minimize the total congestion of all agents in the network. We adopt a flow dynamics model that is a Godunov discretization of the Lighthill–Williams–Richards partial differential equation with a triangular flux function and a corresponding multicommodity junction solver. The partial control formulation generalizes the SO-DTA problem to consider cases where only a fraction of the total flow can be controlled, as may arise in the context of certain incentive schemes. This leads to a nonconvex multicommodity optimization problem. We define a multicommodity junction model that only requires full Lagrangian paths for the controllable agents, and aggregate turn ratios for the noncontrollable (selfish) agents. We show how the resulting finite horizon nonlinear optimal control problem can be efficiently solved using the discrete adjoint method, leading to gradient computations that are linear in the size of the state space and the controls. The online appendix is available at https://doi.org/10.1287/trsc.2017.0800 .

Dynamic system optimal model for multi-OD traffic networks with an advanced spatial queuing model

This paper provides an approach to solve the system optimal dynamic traffic assignment problem for networks with multiple O–D pairs. The path-based cell transmission model is embedded as the underlying dynamic network loading procedure to propagate traffic. We propose a novel method to fully capture the effect of flow perturbation on total system cost and accurately compute path marginal cost for each path. This path marginal cost pattern is used in the projection algorithm to equilibrate the departure rate pattern and solve the system optimal dynamic traffic assignment. We observe that the results from projection algorithm are more reliable than those from method of successive average algorithm (MSA). Several numerical experiments are tested to illustrate the benefits of the proposed model.

Method of generating strategic guidance information for driving evacuation flows to approach safety-based system optimal dynamic flows: Case study of a large stadium

In previous evacuation flow planning, a system optimal dynamic traffic assignment (SO-DTA) did not consider the exogenous costs caused by potential traffic accidents. A traffic accident, which might occur as a result of traffic congestion, will impact an evacuation process because of accident-related delays experienced by the downstream vehicles. This paper establishes a safety-based SO-DTA linear programming model in which the generalized system cost incorporates both the travel time and the accident-related delay. The goal is to minimize the generalized system cost under the cell transmission setup. Furthermore, the authors provide strategic guidance information that considers both the objective of the decision maker and the route choice behavior of the evacuees. Mathematically, the authors propose an unconstrained non-linear programming model aimed at minimizing the gap between the safety-based flows and the stochastic real-world evacuation flows, to provide strategic travel time information to be published on variable message signs (VMS). In the case study, the authors found that the safety-based SO-DTA model can reduce congestion and improve the evacuation efficiency; the stochastic real-world evacuation flows, guided by strategic information, can approach the safety-based flows.

System optimal dynamic traffic assignment: Properties and solution procedures in the case of a many-to-one network

Thanks to its high dimensionality and a usually non-convex constraint set, system optimal dynamic traffic assignment remains one of the most challenging problems in transportation research. This paper identifies two fundamental properties of the problem and uses them to design an efficient solution procedure. We first show that the non-convexity of the problem can be circumvented by first solving a relaxed problem and then applying a traffic holding elimination procedure to obtain the solution(s) of the original problem. To efficiently solve the relaxed problem, we explore the relationship between the relaxed problems based on different traffic flow models (PQ, SQ, CTM) and a minimal cost flow (MCF) problem for a special space-expansion network. It is shown that all the four problem formulations produce the same minimal system cost and share one common solution which does not involve inside queues in the network. Efficient solution algorithms such as the network simplex method can be applied to solve the MCF problem and identify such an optimal traffic pattern. Numerical examples are also presented to demonstrate the efficiency of the proposed solution procedure.

Modeling demand management strategies for evacuations

Evacuations are massive operations that create heavy travel demand on road networks some of which are experiencing major congestions even with regular traffic demand. Congestion in traffic networks during evacuations, can be eased either by supply or demand management actions. This study focuses on modeling demand management strategies of optimal departure time, optimal destination choice and optimal zone evacuation scheduling (also known as staggered evacuation) under a given fixed evacuation time assumption. The analytical models are developed for a system optimal dynamic traffic assignment problem, so that their characteristics can be studied to produce insights to be used for large-scale solution algorithms. While the first two strategies were represented in a linear programming (LP) model, evacuation zone scheduling problem inevitable included integers and resulted in a mixed integer LP (MILP) one. The dual of the LP produced an optimal assignment principle, and the nature of the MILP formulations revealed clues about more efficient heuristics. The discussed properties of the models are also supported via numerical results from a hypothetical network example.

A cell based dynamic system optimum model with non-holding back flows

This paper proposes a non-holding back linear programming (NHBLP) model with an embedded cell transmission model (CTM), to account for the system optimum dynamic traffic assignment.

Modeling the Car-Truck Interaction in a System-Optimal Dynamic Traffic Assignment Model

Several transportation problems, such as implementation of truck-only lanes, require understanding the interaction of heterogeneous dynamic traffic flows in order to provide accurate solutions. System-optimal dynamic traffic assignment can be modeled using a network loading procedure based on the cell transmission model, that is, the hydrodynamic wave model, and solved by linear programming. However, this framework cannot handle the asymmetric integration between the flow of trucks and cars. This article presents a novel formulation for network loading in system-optimal dynamic traffic assignment considering car–truck interactions. By using an embedded cell transmission model, this formulation incorporates a set of assumptions related to the kinematic characteristics of the flow of cars, trucks, and their interactions that can be solved using linear programming. We present numerical results supporting our modeling assumptions. Likewise, the observed emergent behavior captures the car–truck interactions accurately and indicates that minimum system-optimal travel time is obtained by encouraging cars to use highways with shorter distances.

Linear Programming Formulation for Strategic Dynamic Traffic Assignment

This work introduces a novel formulation of system optimal dynamic traffic assignment that captures strategic route choice in users under demand uncertainty. We define strategic route choice to be that users choose a path prior to knowing the true travel demand which will be experienced (therefore users consider the full set of possible demand scenarios). The problem is formulated based on previous work by Ziliaskopoulos (Transp Sci 34(1):37–49, 2000). The resulting novel formulation requires substantial enhancement to account for path-based flows and scenario-based stochastic demands. Further, a numerical demonstration is presented on a network with different demand loading profiles. Finally, model complexity, implications on scalability and future research directions are discussed.

A partial differential equation formulation of Vickrey’s bottleneck model, part II: Numerical analysis and computation

The Vickrey model, originally introduced in Vickrey (1969), is one of the most widely used link-based models in the current literature in dynamic traffic assignment (DTA). One popular formulation of this model is an ordinary differential equation (ODE) that is discontinuous with respect to its state variable. As explained in Ban et al. (2011) and Han et al. (2013), such an irregularity induces difficulties in both continuous-time analysis and discrete-time computation. In Han et al. (2013), the authors proposed a reformulation of the Vickrey model as a partial differential equation (PDE) and derived a closed-form solution to the aforementioned ODE. This reformulation enables us to rigorously prove analytical properties of the Vickrey model and related DTA models.In this paper, we present the second of a two-part exploration regarding the PDE formulation of the Vickrey model. As proposed by Han et al. (2013), we continue research on the generalized Vickrey model (GVM) in a discrete-time framework and in the context of DTA by presenting a highly computable solution methodology. Our new computational scheme for the GVM is based on the closed-form solution mentioned above. Unlike finite-difference discretization schemes which could yield non-physical solutions (Ban et al., 2011), the proposed numerical scheme guarantees non-negativity of the queue size and the exit flow as well as first-in-first-out (FIFO). Numerical errors and convergence of the computed solutions are investigated in full mathematical rigor. As an application of the GVM, a class of network system optimal dynamic traffic assignment (SO-DTA) problems is analyzed. We show existence of a continuous-time optimal solution and propose a discrete-time mixed integer linear program (MILP) as an approximation to the original SO-DTA. We also provide convergence results for the proposed MILP approximation.

On the holding-back problem in the cell transmission based dynamic traffic assignment models

Traffic holding-back is considered an undesirable issue in dynamic traffic assignment since the vehicles are artificially held back on links in spite of the availability of downstream capacity. Holding-back occurs naturally in some system optimal dynamic traffic assignment models. In this paper, we focus on the holding back issue in the cell transmission based models and review the current methods of solving holding-back to understand their advantages and drawbacks. Then, we propose improvements to overcome the drawbacks of the current models. Finally, we propose a novel model which completely eliminates holding-back in a system optimal dynamic traffic assignment for traffic networks with multiple O–D pairs. Rigorous numerical results illustrate the benefits of using the proposed models.

System-optimal dynamic traffic assignment with and without queue spillback: Its path-based formulation and solution via approximate path marginal cost

The knowledge of path marginal cost (PMC) is central to system-optimal dynamic traffic assignment (SO-DTA) problems. In this paper, we propose a method to approximate PMC in general networks when traffic dynamics are modeled by either the point-queue or the kinematic wave traffic flow model. This study examines in detail the flow interactions between downstream and upstream bottleneck links, and shows that the changes in cumulative flows on all the network links caused by an arbitrary flow perturbation can be computed. This offers a way to approximate PMC, which is incorporated in the solution of the least marginal cost problem, a central component of the path-based SO-DTA problem. The approximation scheme allows us to solve path-based SO-DTA problems for general networks with and without queue spillback and/or departure time choices. Numerical examples are provided to demonstrate the effectiveness of the proposed method, and the results show that the SO state produces considerably lower total network cost, shorter congestion duration, and smaller travel delay on bottleneck links than those of produced by the user-optimal state, particularly when the departure time choice is considered.