## Section: New Results

### Transportation networks and vehicular systems

#### Density and flow reconstruction in urban traffic networks

Participants : C. Canudas de Wit [Contact person] , H. Fourati, A. Kibangou, A. Ladino, M. Rodriguez-Vega.

In [56], we consider the problem of joint reconstruction of flow and density in a urban traffic network using heterogeneous sources of information. The traffic network is modeled within the framework of macroscopic traffic models, where we adopt Lighthill-Whitham-Richards model (LWR) conservation equation characterized by a piecewise linear fundamental diagram. The estimation problem considers two key principles. First, the error minimization between the measured and reconstructed flows and densities, and second the equilibrium state of the network which establishes flow propagation within the network. Both principles are integrated together with the traffic model constraints established by the supply/demand paradigm. Finally the problem is cast as a constrained quadratic optimization with equality constraints in order to shrink the feasible region of estimated variables. Some simulation scenarios based on synthetic data for a manhattan grid network are provided in order to validate the performance of the proposed algorithm.

In [62], we addressed the conditions imposed on the number and location of fixed sensors such that all flows in the network can be uniquely reconstructed. We determine the minimum number of sensors needed to solve the problem given partial information of turning ratios, and then we propose a linear time algorithm for their allocation in a network. Using these results in addition to floating car data, we propose a method to reconstruct all traffic density and flow.

#### Discrete-time system optimal dynamic traffic assignment (SO-DTA) with partial control for horizontal queuing networks

Participants : S. Samaranayake, J. Reilly, W. Krichene, M. L. Delle Monache [Contact person] , P. Goatin [Acumes, Inria] , A. Bayen.

Dynamic traffic assignment (DTA) is the process of allocating time-varying origin-destination (OD) based traffic demand to a set of paths on a road network. There are two types of traffic assignment that are generally considered, the user equilibrium or Wardrop equilibrium allocation (UE-DTA), in which users minimize individual travel-time in a selfish manner, and the system optimal allocation (SODTA) where a central authority picks the route for each user and seeks to minimize the aggregate total travel-time over all users. It can be shown that the price of anarchy (PoA), the worst-case ratio of the system delay caused by the selfish behavior over the system optimal solution, may be arbitrarily large even in simple networks. System optimal (SO) traffic assignment on the other hand leads to optimal utilization of the network resources, but is hard to achieve in practice since the overriding objective for individual drivers in a road network is to minimize their own travel-time. It is well known that setting a toll on each road segment corresponding to the marginal delay of the demand moves the user equilibrium towards a SO allocation. In [37], we formulate the system optimal dynamic traffic assignment problem with partial control (SO-DTAPC), using a Godunov discretization of the Lighthill-Williams-Richards (LWR) partial differential equation (PDE) with a triangular flux function. We propose solving the SO-DTA-PC problem with the non-convex traffic dynamics and limited OD data with complete split ratios as a non-linear optimal control problem. This formulation generalizes to multiple sources and multiple destinations. We show that the structure of our dynamical system allows for very efficient computation of the gradient via the discrete adjoint method.

#### Priority-based Riemann solver for traffic flow on networks

Participants : M. L. Delle Monache [Contact person] , P. Goatin [Acumes, Inria] , B. Piccoli.

In [20] we introduce a novel solver for traffic intersection which considers priorities among the incoming roads as the first criterion and maximization of flux as the second. The main idea is that the road with the highest priority will use the maximal flow taking into account also outgoing roads constraints. If some room is left for additional flow then the road with the second highest priority will use the left space and so on. A precise definition of the new Riemann solver, called Priority Riemann Solver, is based on a traffic distribution matrix , a priority vector and requires a recursion method. The general existence theorem for Riemann solvers on junctions can not be applied in the present case.Therefore, we achieve existence via a new set of general properties.

#### Dissipation of stop-and-go waves via control of autonomous vehicles

Participants : R. Stern, S. Cui, M. L. Delle Monache [Contact person] , R. Bhadani, M. Bunting, M. Churchill, N. Hamilton, R. Haulcy, H. Pohlmann, F. Wu, B. Piccoli, B. Seibold, J. Sprinkle, D. B. Work.

Traffic waves are phenomena that emerge when the vehicular density exceeds a critical threshold. Considering the presence of increasingly automated vehicles in the traffic stream, a number of research activities have focused on the influence of automated vehicles on the bulk traffic flow. In [38], we demonstrate experimentally that intelligent control of an autonomous vehicle is able to dampen stop-and-go waves that can arise even in the absence of geometric or lane changing triggers. Precisely, our experiments on a circular track with more than 20 vehicles show that traffic waves emerge consistently, and that they can be dampened by controlling the velocity of a single vehicle in the flow. We compare metrics for velocity, braking events, and fuel economy across experiments. These experimental findings suggest a paradigm shift in traffic management: flow control will be possible via a few mobile actuators (less than $5\%$) long before a majority of vehicles have autonomous capabilities.

#### Cooperative adaptive cruise control over unreliable networks

Participants : F. Acciani, P. Frasca [Contact person] , G. Heijenk, E. Semsar-Kazerooni, A. Stoorvogel.

Cooperative Adaptive Cruise Control (CACC) is a promising technique to increase highway throughput, safety and comfort for vehicles. Enabled by wireless communication, CACC allows a platoon of vehicles to achieve better performance than Adaptive Cruise Control; however, since wireless is employed, problems related to communication unreliability arise. In [45], we design a digital controller to achieve platoon stability, enhanced by an observer to increase robustness against packet losses. Our results confirms the interest of using an observer in combination with a local and cooperative digital controller.

#### Heterogeneity in synchronization: an adaptive control approach, with applications to vehicle platooning

Participants : S. Baldi, P. Frasca [Contact person] .

Heterogeneity is a substantial obstacle to achieve synchronisation of interconnected systems (that is, in control ) In order to overcome heterogeneity, advanced control techniques are needed, such as the use of “internal models’’ or of adaptive techniques. In a series of papers motivated by multi-vehicle platooning and coordinated autonomous driving, we have explored the application of adaptive control techniques. Our results cover both the cases of state-feedback [15] and of output-feedback [16], under the assumption that the topology of the interconnections has no circuits. Further investigation has shown that restrictive assumption can be relaxed (at least for state-feedback on some specific topologies) [47]. This understanding paves the road to use these techniques not only to stabilise heterogeneous platoons, but also to manage their merging or splitting operations [48].

#### Modeling traffic on roundabout

Participants : M. L. Delle Monache [Contact person] , A. Rat, S. Hammond, B. Piccoli.

In [50] we introduce a Riemann solver for traffic flow on a roundabout with two lanes. The roundabout is modeled as a sequence of junctions. The Riemann solver provides a solution at junctions by taking into consideration traffic distribution, priorities, and the maximization of through flux. We prove existence and uniqueness of the solution of the Riemann problem and show some results numerically. This work stems from the fact that there is a general notion among transportation professionals that having a longer additional lane length at a double-lane roundabout entry yields better performances. In [55], we investigate this notion using Lighthill-Whitham-Richards Model. Using Lighthill- Whitham-Richards model, a double-lane roundabout with additional lane design at the entry is analyzed. The additional lane lengths are varied at the entry in order to study the effect of different additional lane lengths on roundabout performance. The results obtained with the PDE model were then compared with similar lane length variations in VISSIM.

#### Two dimensional models for traffic

Participants : S. Mollier, M. L. Delle Monache, C. Canudas de Wit [Contact person] , B. Seibold.

The work deals with the problem of modeling traffic flow in urban area, e. g. a town. More precisely, the goal is to design a two-dimensional macroscopic traffic flow model suitable to model large network as the one of a city. Macroscopic traffic models are inspired from fluid dynamic. They represent vehicles on the road by a density and describe their evolution with partial differential equations. Usually, these models are one dimensional models and, for instance, give a good representation of the evolution of traffic states in highway. The extension of these 1D models to a network is possible thanks to models of junction but can be tedious according to the number of parameters to fit. In the last few years, the idea of models based on a two dimensional conservation laws arose in order to represent traffic flow in large and dense networks. This study starts with a simple model [33] for homogeneous network and where a preferred direction of traffic exists. Our aim is to extend gradually this model by adding complexity. As this approach is uncommon, we investigate a way to compare the results of this model with microsimulation in [73] using Aimsun. Then, in the literature, the network is mainly assumed to be homogeneous. However, in a large-scale scenario, it is unlikely that the traffic network characteristics–such as speed limit, number of lanes, or the network geometry–remain constant throughout the network. Therefore, we introduce a first extension [59] where the fundamental diagram is space-dependent and varies with respect to the area considered. Finally, we have studied more recently a possible way to relax the assumption of a preferred direction of flow by considering several layers of density such that each layer describe a different direction of flow. In this case, the model becomes a system of conservation and is hyperbolic-elliptic which imply special caution in the choice of the numerical method.