Fundamental diagram, a graphical representation of the relation among traffic flow, speed, and density, has been the foundation of traffic flow theory and transportation engineering for many years. For example, the analysis of traffic dynamics relies on input from this fundamental diagram to find when and where congestion builds up and how it dissipates; traffic engineers use a fundamental diagram to determine how well a highway facility serves its users and how to plan for new facilities in case of capacity expansion. Underlying a fundamental diagram is the relation between traffic speed and density which roughly corresponds to drivers’ speed choices under varying car-following distances. First rigorously documented by Greenshields some seventy-five years ago, such a relation has been explored in many follow-up studies, but these attempts are dominantly deterministic in nature, i.e. they model traffic speed as a function of traffic density. Though these functional speed-density models are able to coarsely explain how traffic slows down as more vehicles are crowded on highways, empirical observations show a wide-scattering of traffic speeds around the values predicted by these models. In addition, functional speed-density models lead to deterministic prediction of traffic dynamics, which lack the power to address the uncertainty brought about by random factors in traffic flow. Therefore, it appears more appropriate to view the speed-density relation as a stochastic process, in which a certain density level gives rise not only to an average value of traffic speed but also to its variation because of the randomness of drivers’ speed choices. The objective of this dissertation is to develop such a stochastic speed-density model to better represent empirical observations and provide a basis for a probabilistic prediction of traffic dynamics. It would be ideal if such a model is formulated with both mathematical elegance and empirical accuracy. The mathematical elegance of the model must include the features of: a single equation (single-regime) with physically meaningful parameters and must be easy to implement. The interpretation of empirical accuracy is twofold; on the one hand, the mean of the stochastic speeddensity model should match the average behavior of the empirical equilibrium speeddensity observations statistically. On the other hand, the magnitude of traffic speed variance is controlled by the variance function which is dependent on the response. Ultimately, it is expected that the stochastic speed-density model is able to reproduce the wide-scattering speed-density relation observed at a highway segment after being calibrated by a set of local parameters and, in return, the model can be used to perform probabilistic prediction of traffic dynamics at this location. The emphasis of this dissertation is on the former (i.e. the development, calibration, and validation of the stochastic speed-density model) with a few numerical applications of the model to demonstrate the latter (i.e. probabilistic prediction). Following the seminal Greenshields model, a great variety of deterministic speeddensity models have been proposed to mathematically represent the empirical speeddensity observations which underlie the fundamental diagram. Observed in the existing speed-density models was their deterministic nature striving to balance two competing goals: mathematical elegance and empirical accuracy. As the latest development of such a pursuit, we show that the stochastic speed-density model can be developed through discretizing a random traffic speed process using the Karhunen- Lo`eve expansion. The stochastic speed-density relationship model is largely motivated by the prevalent randomness exhibited in empirical observations that mainly comes from drivers, vehicles, roads, and environmental conditions. In a general setting, the proposed stochastic speed-density model has two components: deterministic and stochastic. For the deterministic component, we propose to use a family of logistic speed density models to track the average trend of empirical observations. In particular, the five-parameter logistic speed-density model arises as a natural candidate due to the following considerations: (1) The shape of the five-parameter logistic speed-density model can be adjusted by its physically meaningful parameters to match the average behavior of empirical observations. Statistically, the average behavior is modeled by the mean of empirical observations. (2) A three-parameter and four-parameter logistic speed-density model can be obtained by reducing the shape or scale parameter in the five-parameter model, but the counter-effect is the loss of empirical accuracy. (3) The five-parameter model yields the best accuracy compared to three-parameter and four-parameter model. The magnitude of the stochastic component is dominated by the variance of traffic speeds indexed by traffic density. The empirical traffic speed variance increases as density increases to around 25 - 30 veh/km, then starts decreasing as traffic density gets larger. It has been verified by empirical evidence that traffic speed variation shows a parabolic shape which makes the proposed variance function in a suitable formula to model its variation. The variance function is dependent on the logistic speed-density relationship with varying model parameters. A detailed analysis of empirical traffic speed variance can be found in Chapter 6. Modeling results show that by taking care of second-order statistics (i.e., variance and correlation) the proposed stochastic speed-density model is suitable for describing the observed phenomenon as well as for matching the empirical data. Following the results, a stochastic fundamental diagram of traffic flow can be established. On the application side, the stochastic speed-density relationship model can potentially be used for real-time on-line prediction and to explain phenomenons in a similar manner. This enables dynamic control and management systems to anticipate problems before they occur rather than simply reacting to existing conditions. Finally, we will summarize our findings and discuss our future research directions.

Les travaux de la thèse se résument en trois parties : La première partie consiste à étudier l’impact de l’automatisation basse vitesse du trafic en termes de sécurité, de consommation en carburant et des émissions des polluants. Dans un premier temps, une étape de modélisation d’un trafic mixte a été élaborée. Ce dernier se définit par la coexistence de deux types de conduite : le premier type est la conduite manuelle ou régis par l’action du conducteur et le deuxième type est la conduite automatisée. Dans un deuxième temps, les scénarios et les indicateurs de sécurité associés ont été développés. Dans ce cadre, un simulateur de trafic a été développé avec le langage C avec une interface graphique sous OpenGL. Les résultats montrent que l’augmentation de la présence des véhicules automatisés améliore la sécurité du trafic dans le cas de collisions aussi que dans le cas de scénarios sans collision. Par ailleurs, l’impact des véhicules automatisés sur la consommation en carburant et les émissions de polluants n’apparait que lorsqu’il s’agit de long parcours. La deuxième partie de la thèse consiste à améliorer le modèle Intelligent Driver Model (IDM) pour pallier ses inconvénients. Ce modèle est un Adaptive Cruise Control (ACC) destiné à représenter la conduite automatisée dans le trafic. Dans ce contexte, une nouvelle version a été développée. L’idée principale est la prise en compte non seulement de l’état du véhicule de tête mais aussi de celle du véhicule précédent. Ce nouveau modèle montre de meilleures performances au niveau de la stabilisation de la file. La troisième partie se concentre sur la modélisation macroscopique du trafic afin d’étudier son diagramme fondamental. Le nouveau modèle macroscopique développé est basé sur les données microscopiques obtenues à partir de la simulation du modèle Intelligent Driver Model. À partir de ces données, les variables macroscopiques du trafic à savoir la densité et la vitesse de la file ont été identifiées par régression non linéaire. Le nouveau modèle macroscopique vitesse- densité possède trois avantages. Le premier avantage est la connaissance de la vitesse de stabilité et la densité d’une file de véhicule dès les premières secondes de simulation en identifiant les modèles de densité et de vitesse. Le deuxième avantage se manifeste dans la possibilité de trouver des modèles existants dans la littérature en fixant quelques paramètres. Le troisième avantage est la possibilité de sa calibration avec un minimum d’erreur pour différentes situations du trafic (ex : congestionné, libre). / My thesis can be summarized in three parts: The first part consists in studying the impact of the low speed automation on driver safety, traffic capacity, fuel consumption and vehicle emissions. As a first step, a pattern of mixed road traffic is established. This one is defined by the coexistence of two driving styles: the first one is the manual and the second one is the automated. Secondly, the scenarios and the associated indicators have been developed. In this context a traffic simulator was developed with the C programming language using the OpenGl library for the graphical interface. Results show that the increase of automated vehicles presence traffic safety increases in case of accidents scenarios as well as in the case of without accidents. Moreover, the impact of automated vehicles on fuel consumption and pollutant emissions only appears when it comes to long path. The second part of my thesis consists in improving the Intelligent Driver Model (IDM) model, Adaptive Cruise Control (ACC) to represent the automated driving in traffic, to overcome its drawbacks. In this context, a new version is developed. The main idea is to take into account not only the state of the preceding vehicle but also the tail vehicle. This new model shows better performance in queue stabilization in terms of accelerations and velocities. The third part is devoted to the macroscopic traffic modeling in which we study the fundamental diagram. The new developed macroscopic model is based on microscopic data given by the Intelligent Driver Model simulation. Then, the traffic density and velocity were identified by non-linear regression. The new macroscopic velocity-density model has three advantages: Knowledge of the stable velocity and density within the first seconds of simulation by identifying. Ability to find existing models in the literature by setting some of its parameters. Calibration with a minimum of error for different traffic situations (eg congestion, free).