Dynamic System Optimum Research Articles

Dynamic optimal congestion pricing in multi-region urban networks by application of a Multi-Layer-Neural network

Traffic management by applying congestion pricing is a measure for mitigating congestion in protected city corridors. As a promising tool, pricing improves the level of service in a network and reduces travel delays. However, previous advancements in pricing research that are responsive to the prevailing regional traffic conditions did not consider real-time applications and the effect on users’ route choices. This work uses real-time dynamic pricing’s influence and predicts pricing functions to aim for a system optimal traffic distribution. The framework models a large-scale network where every region is considered homogeneous, allowing for the Macroscopic Fundamental Diagram (MFD) application. We compute Dynamic System Optimum (DSO) and Dynamic Route Choice (DRC) of the macroscopic model by formulating a linear optimization problem and utilizing the Dijkstra algorithm and a Multinomial Logit model (MNL), respectively. The equilibria allow us to find an optimal pricing methodology by training Multi-Layer-Neural (MLN) network models. We test our framework on a case study in Zurich, Switzerland, and showcase that (a) our neural network model learns the complex user behavior and (b) allows predicting optimal pricing functions. Results show a significant performance improvement when operating a transportation network in the DSO and highlight how dynamic pricing influences the user’s route choice behavior towards the system optimal equilibrium.

Multiclass dynamic system optimum solution for mixed traffic of human-driven and automated vehicles considering physical queues

Dynamic traffic assignment (DTA) is an important method in the long term transportation planning and management processes. However, in most existing system optimum dynamic traffic assignment (SO-DTA), no side constraints are used to describe the dynamic link capacities in a network which is shared by multiple vehicle types. Our motivation is based on the possibility for dynamic system optimum (DSO) to have multiple solutions, which differ in where queues are formed and dissipated in the network. To this end, this paper proposes a novel DSO formulation for the multi-class DTA problem containing both human driven and automated vehicles in single origin-destination networks. The proposed method uses the concept of link based approach to develop a multi-class DTA model that equally distributes the total physical queues over the links while considering explicitly the variations in capacity and backward wave speeds due to class proportions. In the model, the DSO is formulated as an optimization problem considering linear vehicle composition constraints representing the dynamics of the link capacities. Numerical examples are set up to provide some insights into the effects of automated vehicles on the queue distribution as well as the total system travel times.

A time-dependent shared autonomous vehicle system design problem

The emergence of the shared autonomous vehicle (SAV) provides new opportunities and challenges for the fashionable car-sharing mode. This study proposes a time-dependent SAV system design problem by jointly optimizing fleet size, parking infrastructure deployment, and daily operation of the system for infrastructure planning in the long run. The dynamic system optimum (DSO) principle in terms of total daily system cost (TDSC) is adopted to formulate the daily operation of the SAV system, i.e., users’ departure time choices and SAVs’ route choices. By incorporating the link transmission model (LTM) as the traffic flow model, the daily operation problem (DOP) of the SAV system is formulated as a linear programming (LP) problem. Further, the time-dependent SAV system design problem is formulated as a mixed integer linear programming (MILP) problem. The LP relaxation of the proposed MILP problem could provide a tight lower bound, and a diving heuristic algorithm is developed to solve the proposed MILP problem. Finally, numerical examples are designed to illustrate the properties of the model and the efficiency of the proposed solution algorithm.

Dynamic System Optimum Analysis of Multi-Region Macroscopic Fundamental Diagram Systems With State-Dependent Time-Varying Delays

This paper investigates the dynamic system optimum (DSO) problem with simultaneous route and departure time assignments for a general traffic network partitioned into multiple regions. Regional traffic congestion is modeled with a well-defined macroscopic fundamental diagram (MFD) mapping the trip completion rate to the vehicular accumulation. To overcome the limitation of inconsistent flow propagation between region boundaries and the corresponding travel time, the state-dependent regional travel time function is explicitly incorporated in the flow propagation of the conventional MFD dynamics. From a systems perspective, the traffic dynamics within a region can be regarded as a dynamic system with an endogenous time-varying delay depending on the system state. Equilibrium condition for the DSO problem is analytically derived through the lens of Pontryagin minimum principle and is compared against the static SO counterpart. The structure of path specific marginal cost is analyzed regarding the path travel cost and early-late penalty function. In contrast to existing analytical methods, the proposed method is applicable for general MFD systems without linearization of the MFD dynamics. Neither approximation of the equilibrium solution nor constant regional delay assumption is required. Numerical examples are conducted to illustrate the characteristics of DSO traffic equilibrium and the corresponding marginal cost together with other dynamic external costs.

Graphical solution for system optimum dynamic traffic assignment with day-based incentive routing strategies

This paper analyzes the dynamic traffic assignment problem on a three-alternative network with day-based incentive routing strategies by using graphical solution method. It is assumed that the cumulative count curve of vehicles is known and that the arrival rate is unimodal. The dynamic system optimum (DSO) allocation lines are first drawn based on calculus of variations. Three possible optimal allocation lines are analyzed. A day-based incentive routing strategy is designed and conditions that when and how to implement the incentive scheme to realize DSO are then derived. Extension to general parallel networks is also given. Examples are presented to demonstrate the effectiveness of the scheme.

Optimal queue placement in dynamic system optimum solutions for single origin-destination traffic networks

The Dynamic System Optimum (DSO) traffic assignment problem aims to determine a time-dependent routing pattern of travellers in a network such that the given time-dependent origin-destination demands are satisfied and the total travel time is at a minimum, assuming some model for dynamic network loading. The network kinematic wave model is now widely accepted as such a model, given its realism in reproducing phenomena such as transient queues and spillback to upstream links. An attractive solution strategy for DSO based on such a model is to reformulate as a set of side constraints apply a standard solver, and to this end two methods have been previously proposed, one based on the discretisation scheme known as the Cell Transmission Model (CTM), and the other based on the Link Transmission Model (LTM) derived from variational theory. In the present paper we aim to combine the advantages of CTM (in tracking time-dependent congestion formation within a link) with those of LTM (avoiding cell discretisation, providing a more computationally attractive with much fewer constraints). The motivation for our work is the previously-reported possibility for DSO to have multiple solutions, which differ in where queues are formed and dissipated in the network. Our aim is to find DSO solutions that optimally distribute the congestion over links inside the network which essentially eliminate avoidable queue spillbacks. In order to do so, we require more information than the LTM can offer, but wish to avoid the computational burden of CTM for DSO. We thus adopt an extension of the LTM called the Two-regime Transmission Model (TTM), which is consistent with LTM at link entries and exits but which is additionally able to accurately track the spatial and temporal formation of the congestion boundary within a link (which we later show to be a critical element, relative to LTM). We set out the theoretical background necessary for the formulation of the network-level TTM as a set of linear side constraints. Numerical experiments are used to illustrate the application of the method to determine DSO solutions avoiding spillbacks, reduce/eliminate the congestion and to show the distinctive elements of adopting TTM over LTM. Furthermore, in comparison to a fine-level CTM-based DSO method, our formulation is seen to significantly reduce the number of linear constraints while maintaining a reasonable accuracy.

Continuous-time dynamic system optimum for single-destination traffic networks with queue spillbacks

Dynamic system optimum (DSO) is a special case of the general dynamic traffic assignment (DTA). It predicts the optimal traffic states of a network under time-dependent traffic conditions from the perspective of the entire system. An optimal control framework is proposed in this paper for the continuous-time DSO problem for single-destination traffic networks. Departure time choice is part of this DSO model. Double-queue model is applied to capture the impact of downstream congestion and possible queue spillbacks. Feasibility conditions and model properties are discussed. A constructive procedure to compute a free-flow DSO solution is also proposed. A discretization method is described to the continuous-time systems and numerical results on two test networks are shown.

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.

A Mixed Integer Linear Program for the Single Destination System Optimum Dynamic Traffic Assignment Problem with Physical Queue

In order to solve the system optimum dynamic traffic assignment problem, the whole link model with physical queue is used to formulate the single destination system optimum dynamic traffic assignment problem as a mixed linear program. A relationship between the cumulative curves and the wave speed presented by Newell (1993) is used to present a dynamic network model in considering spillback queue. And nonlinear constrains are relaxed into mixed linear constrains; the linear program software is used to solve the system optimum dynamic traffic assignment problem. A numerical example illustrates the simplicity and applicability of the proposed approach. KEYWORDS: Dynamic System Optimum Traffic Assignment; physical queue; mixed integer linear programming

DEVELOPMENT OF A SIMULATION ALGORITHM FOR DSO SOLUTION ON A SIMPLE NETWORK

This study develops a DSO (Dynamic System Optimum) simulation algorithm first, which can reproduce the traffic condition so as to minimize total travel time on condition that time-dependent OD demand is given. Next, the algorithm is verified by applying to a simple network with one OD pair. Recently most of dynamic traffic simulation models reproduce traffic flows dynamically under the principle of DUO (Dynamic User Optimal). On the other hand, DSO solution must provide practical information for the effective traffic control such as ramp metering control and road pricing measure. However, there is no simulation algorithm seeking for the DSO solution Therefore, in this study, DSO simulation algorithm has been developed in accordance with Dynamic Marginal Cost equilibrium principle, which equilibrates DMC of all routes. As a result of the verification, it has been confirmed that the proposed model and algorithm efficiently solved DSO solution.

Dynamic System Optimum Traffic Assignment Models and the Solution Methods

This paper studies the basic constraints of dynamic traffic assignment problem and dynamic system optimum model. First, Merchant and Nemhauser's dynamic system optimum model is analyzed in detail. Second, link exit function and other constraints in dynamic traffic assignment problem are qualified strictly. Third, a continuous dynamic system optimum model and a discrete dynamic system optimum model are developed. The models can be applied to the network with multiple origin and multiple destination. Finally, an algorithm for the discrete model is developed. Test example showed that the algorithm can solve the dynamic optimum efficiently.