On Robust Distributed Control of Transportation Networks

With the ever-growing traffic demands, the transportation networks are getting more and more congested. While expanding these networks with more roads is both costly and in many cities not even feasible, the rapid development of new sensing and communication techniques has made it possible to perform control of transportation networks in real-time. With the right usage of such technologies, existing transportation networks' capacities can be utilized better in order to lower the congestion levels. However, the control has to be done robustly, since real-time control and close to maximal utilization also make the networks more fragile and if not, even a small perturbation can have a tremendous impact on the traffic network. In this thesis, a few solutions that lead to better transportation network utilization are presented, designed with said robustness requirements in mind. In the first part of the thesis, a decentralized control strategy for traffic signals is presented. The proposed policy, which we call Generalized Proportional Allocation (GPA), is inspired by the proportional fairness allocation for communication networks. The original proportional fairness controller does not explicitly take the overhead time needed to shift between different activation phases into account. We, therefore, enhance the proportional fairness so that it adapts its cycle length to the current demand. When the demand is higher, one wants longer signal cycles not to waste too much of the time overhead, while for lower demands, the cycle lengths should be shorter, so that the drivers do not have to wait for a long time. Stability for an averaged version of this control strategy is proved together with throughput-optimality of the controller. This means that no other control strategy can handle larger exogenous inflows to the network than the GPA-controller. Since the traffic signal controllers such as the GPA may allocate service to an empty line, due to the fact that several lanes can receive green light simultaneously, a model that handles this issue is proposed. For this model, the well-posedness of the dynamical system is shown when the traffic signal controller is Lipschitz continuous.The GPA controller's performance is also evaluated in a microscopic traffic simulator. In the microsimulations, it is shown how the proposed feedback controller outperforms the standard fixed-time controller for a scenario based on all traffic over the duration of one full day in Luxembourg. The controller's performance is also compared to another decentralized controller for traffic signals, the MaxPressure controller, for an artificial Manhattan-like network. From these simulations, it can be concluded that the GPA performs better than MaxPressure during low demands, but the MaxPressure performs better when the demand is high. The fact that the GPA does not require any information about the network, apart from the current queue lengths, makes it robust to perturbations. In other words, the control strategy does not have to be updated when the demand or topology of the network changes.The second part of the thesis is devoted to routing problems. First, the problem of routing a fleet of vehicles in an optimal way for the whole fleet is considered. The objective is then to achieve a minimum delay in average for the entire fleet. The routing algorithm takes into account the presence of regular drivers that are trying to optimize their own traveling time in the network. Conditions are posted for when such a routing assignment exists, and two algorithms to compute it are shown.At last, a type of dynamic routing policies for multicommodity flows is studied. The routing policies are designed with the objective to avoid congested routes. It has previously been shown that if only one class of vehicles are present, the network is robust to perturbations with these routing policies. A model for multicommodity flows is proposed, and it is shown that the robustness properties for the single-commodity case do not necessarily hold in the multicommodity case

A multi-commodity dynamical model for traffic networks

A dynamical model for traffic networks is proposed and analyzed. In the traffic network, the transportation demands are considered as multi-commodity flows where each commodity has a unique destination. The network is modeled by a multigraph where at each node each commodity splits among the outgoing links in a way such that the drivers are more likely to avoid a road when the density on it increases. It will be shown that if the graph has no cycles, the density of each commodity on each link will converge to a unique limit that does not depend on the initial state. Network resilience, namely structural robustness of the network with respect to perturbations, is also studied. In particular, it is shown that if all commodities have access to all outgoing links, the network can manage perturbations whose magnitude is less than a quantity which plays the natural role of residual capacity of an equilibrium. If instead not all commodities have access to all links, overreaction of the network to perturbations implies that even small perturbations might be amplified and start a cascade. Finally, the idea of back-pressure is employed to provide a simple distributed control strategy. Analogously to the single commodity case, such actual strategy is able to back-propagate the information that congestion is happening ahead, thus allowing the drivers to reroute even if their decision is based on local information only

Mode Stability for Gravitational Instantons of Type D

We study Ricci-flat perturbations of gravitational instantons of Petrov type D. Analogously to the Lorentzian case, the Weyl curvature scalars of extreme spin-weight satisfy a Riemannian version of the separable Teukolsky equation. As a step towards rigidity of the type D Kerr and Taub-bolt families of instantons, we prove mode stability, i.e. that the Teukolsky equation admits no solutions compatible with regularity and asymptotic (local) flatness

A Micro-Simulation Study of the Generalized Proportional Allocation Traffic Signal Control

In this paper, we study the problem of determining phase activations for signalized junctions by utilizing feedback, more specifically, by measure the queue-lengths on the incoming lanes to each junction. The controller we are investigating is the Generalized Proportional Allocation (GPA) controller, which has previously been shown to have desired stability and throughput properties in a continuous averaged dynamical model for queueing networks. In this paper, we provide and implement two discretized versions of the GPA controller in the SUMO micro simulator. We also compare the GPA controllers with the MaxPressure controller, a controller that requires more information than the GPA, in an artificial Manhattan-like grid. To show that the GPA controller is easy to implement in a real scenario, we also implement it in a previously published realistic traffic scenario for the city of Luxembourg and compare its performance with the static controller provided with the scenario. The simulations show that the GPA performs better than a static controller for the Luxembourg scenario, and better than the MaxPressure pressure controller in the Manhattan-grid when the demands are low

Generalized Proportional Allocation Policies for Robust Control of Dynamical Flow Networks

We study a robust control problem for dynamical flow networks. In the considered dynamical models, traffic flows along the links of a transportation network --modeled as a capacited multigraph-- and queues up at the nodes, whereby control policies determine which incoming queues at a node are to be allocated service simultaneously, within some predetermined scheduling constraints. We first prove a fundamental performance limitation by showing that for a dynamical flow network to be stabilizable by some control policy it is necessary that the exogenous inflows belong to a certain stability region, that is determined by the network topology, link capacities, and scheduling constraints. Then, we introduce a family of distributed controls, referred to as Generalized Proportional Allocation (GPA) policies, and prove that they stabilize a dynamical transportation network whenever the exogenous inflows belong to such stability region. The proposed GPA control policies are decentralized and fully scalable as they rely on local feedback information only. Differently from previously studied maximally stabilizing control strategies, the GPA control policies do not require any global information about the network topology, the exogenous inflows, or the routing, which makes them robust to demand variations and unpredicted changes in the link capacities or the routing decisions. Moreover, the proposed GPA control policies also take into account the overhead time while switching between services. Our theoretical results find one application in the control of urban traffic networks with signalized intersections, where vehicles have to queue up at junctions and the traffic signal controls determine the green light allocation to the different incoming lanes

Lower secondary school students’ reasoning about compound probability in spinner tasks

In this paper we investigate the different ways in which students in lower secondary school (14–15 year-olds) reason about compound stochastic events (CSE). We ask students during clinical interviews to respond to CSE-tasks in a spinner context, where two linked spinners display equal or different sizes of red and white areas. We seek to enrich our knowledge of how students make sense of CSE by not focusing exclusively on sample-space grounded reasoning. We open up the analysis to how students’ reasoning can reflect aspects of multiplicative reasoning in relation to The Product Law of Probability. Our results show that students have difficulty in applying well-grounded combinatorial reasoning as well as multiplicative reasoning to the tasks, but they do show intuitive reasoning that reflect aspects of The Product Law of Probability. Two ways of reasoning identified in the current study are area-based part-whole reasoning and lowest-chance reasoning.publishedVersionUnit Licence Agreemen

Hierarchical Pricing Game for Balancing the Charging of Ride-Hailing Electric Fleets

Due to the ever-increasing popularity of ride-hailing services and the indisputable shift towards alternative fuel vehicles, the intersection of the ride-hailing market and smart electric mobility provides an opportunity to trade different services to achieve societal optimum. In this work, we present a hierarchical, game-based, control mechanism for balancing the simultaneous charging of multiple ride-hailing fleets. The mechanism takes into account sometimes conflicting interests of the ride-hailing drivers, the ride-hailing company management, and the external agents such as power-providing companies or city governments that will play a significant role in charging management in the future. The upper-level control considers charging price incentives and models the interactions between the external agents and ride-hailing companies as a Reverse Stackelberg game with a single leader and multiple followers. The lower-level control motivates the revenue-maximizing drivers to follow the company operator's requests through surge pricing and models the interactions as a single leader, multiple followers Stackelberg game. We provide a pricing mechanism that ensures the existence of a unique Nash equilibrium of the upper-level game that minimizes the external agent's objective at the same time. We provide theoretical and experimental robustness analysis of the upper-level control with respect to parameters whose values depend on sensitive information that might not be entirely accessible to the external agent. For the lower-level algorithm, we combine the Nash equilibrium of the upper-level game with a quadratic mixed integer optimization problem to find the optimal surge prices. Finally, we illustrate the performance of the control mechanism in a case study based on real taxi data from the city of Shenzhen in China