Continuum traffic models & Dynamic traffic assignment

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Mobility in multimodal transport network

Mobility is fundamental in big cities. People move, travel for infinite reasons. These move-ments or displacements are carried through available means of mobility and transport re-sources. Since there are different modes of transportation in industrialized cities, to move from one location to another location, one regularity take more than one mode of transporta-tion in daily trips. These mobilities imply interactions of solicited modes of transportation. They give rise to traffic incidents, for instance traffic jams, traffic breakdowns, traffic acci-dent and time delays. The variability of traffic is complex. It is more complex when different transport modes operating by separated systems are involved. Such a global transport system is called a multimodal transportation system. The word “multimodal” refers to the different transport modes, transport services, and traffic services available for the global system. Let us give in the next section a clear definition of a multimodal transport network with respect to two major aspects: the physical and the functional points of view.

Continuum traffic models & Dynamic traffic assignment

Industrialized cities and countries mostly cope with traffic congestion and traffic incident. These are one of the major problem on the economic, social and societal front. We need to clearly understand what causes traffic congestion and traffic incidents. We need to know too how traffic operations work in order to be able to properly monitor and control traffic in any congested transport network. Infinite mobile jams often occur in dense networks. During the last fifty years and so far, research in traffic theory encompasses a rich set of mathematical and physical models. The models are often designed for specific networks according to required level-of-details. The models aim at:
• predict and estimate traffic states over different types of networks.
• calculate traffic indicators such as travel times.
• provide alternative paths or routes in case of traffic breakdowns and also when certain roads become inaccessible.
• ensure a greater safety and reduce multiple risks of traffic incidents, etc.
Many new trends of modeling and controlling emerged in the field of transportation net-works modeling. They are used as decision tools for the traffic management, traffic control-ling, traffic supervision, etc. In cases of networks traffic controlling, we note that specific physical/mathematical models are required according to requirements of governmental de-cisions makers, transport operators or transport planners. According to the level-of-details, needs and expected results, right models are appropriate. They are numerically and compu-tationally handleable in specific tools for traffic flow management.
These assets help the decision makers for an optimized management of transport net-works. Several scales of modeling exist. Each scale represents a particular aspect of what is modeling. Depending on desired level-of-details in the representation of traffic states, one uses one approach (one scale of representation) rather than the other. The main scales of traffic modeling range from microscopic, to mesoscopic, macroscopic and bi-dimensional. In our research, we mainly dig on models of surface transport networks. In our review, we do not include microscopic models since we are more interesting on mathematical and physical models that represent the traffic flow of networks. We do not include possible interactions between vehicles and passengers within networks. The thesis concerns mostly macroscopic and bi-dimensional traffic flow models and dynamic traffic assignment deriving from related network loading models. We say mostly since we know that microscopic models are the limit of kinematic and macroscopic models.
To the best of author’s knowledge, there is no available and reliable general multimodal model (or seamlessly integrated models and algorithms) for the traffic controlling that takes into account every available transport mode of a large-scale transportation system. We count several families of traffic flow models that describe vehicular dynamics given a certain traffic infrastructure and, if applicable, given additional routes choice information. The infrastruc-ture comprises the considered road system in terms of topology (its geometry, characteris-tics, right-of-way laws, and traffic signaling available on it), road sections speed limits and capacities, and intersection properties, etc. Continuum traffic flow models are divided into mesoscopic/gas-kinetic traffic flow models and fluid-dynamical models also named macro-scopic traffic flow models.

Macroscopic transport simulators

Computer simulation is more and more popular discipline in the filed of science in general. There exist many strategies to simulate traffic systems which fall under the following three categories:
1. The microscopic simulation including cellular automata, multi-agent simulation, par-ticles system simulation.
2. The macroscopic simulation including statistical dispersion models, freeway traffic mo-dels, generic second order modeling family models.
3. The two-dimensional simulation including macroscopic fundamental diagram based traffic models, two-dimensional traffic models and continuous approach based traffic models.
Computer scientists have come up with theses above models and with strategies of hy-bridization to cope with traffic issues. We are interesting on macroscopic simulators and two-dimensional simulators since they are appropriate tools for large-scale traffic manage-ment. There are plenty of macroscopic traffic simulators such like Transims, Transmodeler, Dynamit, Dynasmart-P, Magister, Matsim [36]. The list is not exhaustive. The choice of sim-ulators we deployed in the rest of the manuscript comes up with the simulators’ accessibility. They are essentially macroscopic simulators. That means that the simulators result from im-plementation of mathematical/physical macroscopic traffic flow models, and/or stochastic multi-agent transport models.
These are appropriate tools for traffic analysis and traffic controlling. They allow to make dynamic traffic assignment and rerouting of network flows in order to achieve dynamic user equilibrium, and other purposes relevant for transportation safety and re-liabilities.
1. TRANSIMS. The Transportation Analysis and Simulation System (TRANSIMS) is an open source transportation modeling and simulation toolbox. It is an integrated set of tools developed to conduct regional transportation system analyses. With the goal of establishing TRANSIMS as an ongoing public resource available to the transportation community, TRANSIMS is made available under the NASA Open Source Agreement Version 1.3 and is supported by this online community.
2. TransModeler. TransModeler is a versatile traffic simulator with many advanced fea-tures including support for key aspects of Intelligent Transportation Systems. Trans-Modeler simulates a wide variety of facility types, including mixed urban and freeway networks, and can be applied to specific geographic areas such as downtowns, highway corridors, or beltways. It integrates traffic simulation models such as:
• model for freeway and urban networks
• model rotaries with driver behavior models that capture the unique interactions between vehicles entering and vehicles inside the rotary.
• model high occupancy vehicle (HOV) lanes, bus lanes and toll facilities to better understand their effects on traffic system dynamics.
• model evacuation plans and scenarios for response to natural disasters, hazardous spills, and other emergencies.
• model work zones to manage traffic during the construction and maintenance projects.
3. DynaMIT. DynaMIT aims to operate an ATIS (Advanced Traveler Information System) to improve travel decisions. Its applications include:
• Generation of unbiased and consistent information to drivers.
• Optimizing the operation of TMCs through the provision of real-time predictions.
• Efficient operation of Variable Message Signs (VMS).
• Real-time incident management and control.
• Off-line evaluation of real-time incident management strategies.
• Evaluation of alternative traffic signals and ramp meters operation strategies.
• Co-ordination of evacuation and rescue operations in real-time emergencies (nat-ural disasters, etc.) that could block highway links.
• Generating historical databases.

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2d-Anisotropic Continuous Network

Definition 3.1 (Network anisotropy). In traffic theory, a network is said anisotropic when there are many possible interactions and several directions of propagation of the traffic flow at ‘almost’ any location of the network.
For instance, the road network of the city of Paris is anisotropic. It forms a spiderweb, ranking in the type of anisotropic networks. In the United States, cities are new and their networks are rather orthotropic since roads are not gradually constructed as cities grow.
Definition 3.2 (Dense network within a continuum area). We mean by dense network, a network with very high number of secondary roads that are very close to each other. The density of the network in traffic refers to the plurality of road sections of short length and their closeness’s. Definition 3.3 (Anisotropic continuous network: ACN). An ACN is a network whose traffic area can be approached by a continuum media with preferred directions of propagation of the flows.
In the remainder of the chapter, we discuss on the traffic dynamics of a dense surface net-work and the bi-dimensional traffic flow theory. We will see that the dynamic bi-dimensional traffic flow (BTF model for short) introduced in the present chapter is particularly timely re-sponding to the issues of the traffic flow estimation over large and dense networks.
Definition 3.4 (Bi-dimensional traffic flow model). A bi-dimensional traffic flow model is a mathematical traffic model that aggregates a network domain to anisotropic continuous network and formulates the relationships among traffic flow characteristics like density, flow, mean speed per dominant/preferred directions of propagation of the network traffic stream.
Consequence 3.1 (Bi-dimensional modeling approach). Traffic streams at the bi-dimensional level are comparable to fluid streams flowing in bounded and euclidian two space dimensions. Consequence 3.2. The bi-dimensional modeling approach shall fit both homogeneous and he-terogeneous networks. The interest of this approach is more visible in the case of heterogeneous networks.
Lemma 3.1. A continuum approach comes out by the aggregation of the road networks (as it is showed in [93, 74, Wong, Prez and Benitez]) which makes vehicles behave on any 2d-ACN (anisotropic continuous network) like a two-dimensional fluid.
Let us denote by U the traffic area of a surface network. We assume that the surface network is dense and large. U is also called the network domain. It shall be a bounded and open subspace of the Euclidean space R2 since any city has a frontier and therefore its urban road network also. Clearly U R2 and meas(U) < +1, with meas the Lebesgue measure in two space dimensions. Let P(x, y) be a point of the network domain and (x, y) its coor-dinates. Four preferred directions of propagation of the flow for the movement of vehicles at the point P(x, y) are distinguished. Figure 3.1(a) depicts these dominant directions: four outflows and four inflows to refer to the directions of propagation of the traffic.

Waves propagation in 2d elementary cells

Waves should propagate in a multidimensional manner and affect other cell averages besides those adjacent to the interface. Waves are represented by fluctuations they produce. These fluctuations in multidimensional domain are transverse and are splitted into two categories: the up-going transverse fluctuations and the down-going transverse fluctuations. Spatial domain for Riemann problem is thus either half plane or quater plane. Let us zoom in on an internal cell of a network domain. Through each interface of the cell, there are plane waves crossing it. At the corner of the cell, there is interference of such waves. Let assume constants u the traffic velocity in directions 1 and 3 in the one hand, and v the traffic velocity in the direction 2 and 4. In such a case a single wave should propagate in the direction (u, v). There is a triangular portion of the wave originating from the cell C , which should move into cells C , +1 and C 1, 1, rather than cells C 1, or C , (see Fig. 3.4). Between the latter, the first are up-going transverse fluctuations, the second are down going transverse fluctuations.

Table of contents :

1 General introduction 
1.1 Context & Problem formulation
1.2 Research objectives
1.3 Conventions & Notations
1.4 Scientific Publications
1.5 Outline of this Report
2 Background and Related work 
2.1 Introduction
2.2 Mobility in multimodal transport network
2.3 Continuum traffic models & Dynamic traffic assignment
2.4 Macroscopic transport simulators
2.5 Summary
3 Bi-dimensional dynamic traffic flow modeling 
3.1 Introduction
3.2 2d-Anisotropic Continuous Network
3.3 Traffic dynamic within 2d Anisotropic continuous network
3.4 Numerical methods
3.5 Validation
3.6 Conclusion
4 Vehicular multimodal traffic flow modeling 
4.1 Introduction
4.2 Modeling skyTran network
4.3 Towards vehicular multimodality
4.4 Multiclass traffic flow modeling
4.5 Conclusion
5 Multiscale traffic flow simulation 
5.1 Introduction
5.2 Hybrid traffic flow modeling
5.3 Multiscale coupling
5.4 Perspectives
6 RDTA over large transport networks 
6.1 Introduction
6.2 Reactive dynamic assignment
6.3 Numerical experiments
6.4 Conclusion
7 Conclusion et perspectives 
7.1 Summary
7.2 Research relevance
7.3 Open Problems & Future Prospects
A CVXOPT convex optimization python package 
A.1 Resolution of the linear-quadratic optimization problem

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