## Section: New Results

### Transportation networks and vehicular systems

#### Travel time prediction

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

One of the regular performance metrics for qualifying the level of congestion in traffic networks is the travel time. In [24], we addressed the problem of dynamic travel time (DTT) forecasting within highway traffic networks using speed measurements. Definitions, computational details and properties in the construction of DTT are provided. DTT is dynamically clustered using a K-means algorithm and then information on the level and the trend of the centroid of the clusters is used to devise a predictor computationally simple to be implemented. To take into account the lack of information in the cluster assignment for the new predicted values, a weighted average fusion based on a similarity measurement is proposed to combine the predictions of each model. The algorithm is deployed in a real time application and the performance is evaluated using real traffic data from the South Ring of the Grenoble city in France. We consider in a recent paper submitted to European Control Conference 2018 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 and a piecewise linear fundamental diagram. The estimation problem considers three key principles. First, the principle governing traffic models where flow is maximized in a junction. Second, the error minimization between the measured and reconstructed flows and velocities, and finally the equilibrium state of the network which establishes flow propagation within the network. All principles are integrated and the problem is casted as a constrained quadratic optimization with inequality and 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.

#### Urban traffic control

Participants : C. Canudas de Wit [Contact person] , F. Garin, P. Grandinetti.

The PhD thesis of Pietro Grandinetti deals with optimal or near-optimal operation of traffic lights in an urban area, e.g., a town or a neighborhood. The goal is on-line optimization of traffic lights schedule in real time, so as to take into account variable traffic demands, with the objective of obtaining a better use of the road infrastructure. More precisely, we aim at maximizing total travel distance within the network, together with balancing densities across the network. The complexity of optimization over a large area is addressed both in the formulation of the optimization problem, with a suitable choice of the traffic model, and in a distributed solution, which not only parallelizes computations, but also respects the geometry of the town, i.e., it is suitable for an implementation in a smart infrastructure where each intersection can compute its optimal traffic lights by local computations combined with exchanges of information with neighbor intersections. A modified version of the algorithm uses simplified optimization (purely local, instead of distributed) but takes into account the real constraints in Grenoble downtown traffic lights network, such as priority to public transportation, and imposed minimal and maximal green duration, leading to a fully realistic implementation, tested using Aimsun microscopic simulator.

#### Traffic Regulation Via Controlled Speed Limit

Participants : M. L. Delle Monache [Contact person] , B. Piccoli, F. Rossi.

The work [21] address the speed limit problem on a single road. The control variable is the maximal allowed velocity, which may vary in time but we assume to be of bounded total variation, and we aim at tracking a given target outgoing flow. More precisely, the main goal is to minimize the quadratic difference between the achieved outflow and the given target outflow. Mathematically the problem is very hard, because of the delays in the effect of the control variable (speed limit). In fact, the link entering time, which represents the entering time of the car exiting the road at time $t$, depends on the given inflow and the control policy on the whole time interval. Moreover, the input-output map is defined in terms of the Link Entering Time, thus the achieved outflow at time $t$ depends on the control variable on the whole time interval. After formulating the optimal control problem, we consider needle-like variations for the control policy as used in the classical Pontryagin maximum principle. We are able to derive an analytical expression of the one-sided variation of the cost, corresponding to needle-like variations of the control policy, using

fine properties of functions with bounded variation. In particular the one-sided variations depend on the sign of the control variation and involve integrals w.r.t. the distributional derivative of the solution as a measure. This allows us to prove Lipschitz continuity of the cost functional in the space of a bounded variation function and prove existence of a solution. Afterwards, we define three different techniques to numerically solve this problem and we compare the three approaches on two test cases.

#### Scalar conservation laws with moving flux constraints

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

This problem is motivated by the modeling of a moving bottleneck in traffic flow, which can be caused by a large, slow moving vehicle. A slow moving large vehicle, like a bus or a truck, reduces the road capacity and thus generates a moving bottleneck for the surrounding traffic flow. This situation can be modeled by a PDE–ODE strongly coupled system consisting of a scalar conservation law with moving flux constraint accounting for traffic evolution and an ODE describing the slower vehicle motion. In [18], we introduce a novel approach to solve numerically this problem. The main point here is related to the presence of non-classical shocks in the solutions of the model under consideration. It is well-known that, in this context, standard conservative finite volume methods cannot be applied and fail in producing good numerical results. Glimm’s scheme can be used but it is not strictly conservative. In order to propose a numerical scheme which is conservative on fixed meshes and able to compute non-classical solutions, we propose to adapt a reconstruction strategy approach, which allows to precisely capture moving non-classical discontinuities on fixed meshes still guaranteeing conservation, unlike Glimm’s scheme. An important feature of the proposed method is to be exact for isolated classical and non-classical shocks, which means in particular only one point of numerical diffusion (on each cell the approximate value corresponds to the value of the average of the exact solution). In the general case, shocks are still computed without numerical diffusion and convergence is proved numerically.

In [19] we study well-posedness of a scalar conservation laws with moving flux constraints. In this work we assume that the constraint trajectory is given and it does not depend on the solution of the PDE. In this setting we then show Lipschitz continuous dependence of bounded variation solutions with respect to the initial data ant the constraint trajectory.

#### 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.

#### 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 [25], 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.

#### Measuring trajectories and fuel consumption in oscillatory traffic: experimental results

Participants : F. Wu, R. Stern, M. Churchill, M. L. Delle Monache [Contact person] , K. Han, B. Piccoli.

In [37] we present data collected through a set of experiments with nine to 10 vehicles driving on a ring road constructed on a closed track. Vehicle trajectory data is extracted via a series of vision processing algorithms (for background subtraction, vehicle identification, and trajectory extraction) from a 360-degree panoramic camera placed at the center of the ring. The resulting trajectory data is smoothed via a two-step algorithm which applies a combination of RLOESS smoothing and regularized differentiation to produce consistent position, velocity, and acceleration data that does not exhibit unrealistic accelerations common in raw trajectory data extracted from video. A subset of the vehicles also record real-time fuel consumption data of the vehicles using OBD-II scanners. The tests include both smooth and oscillatory traffic conditions, which are useful for constructing and calibrating microscopic models, as well as fuel consumption estimates from these models. The results show a an increase in fuel consumption in the experiments in which traffic oscillations are observed as compared to experiments where vehicles maintain a smooth ow. However, this is partially due to the higher average speed at which vehicles travel in the experiments in which oscillatory traffic is observed. The article contains a complete, publicly available dataset including the video data, the extracted trajectories, the smoothed trajectories, and the OBD-II logs from each equipped vehicle. In addition to the dataset, this article also contains a complete source code for each step of the data processing. It is the first of several experiments planned to collect detailed trajectory data and fuel consumption data with smooth and unsteady traffic flow in a controlled experimental environment.

#### Large Scale Traffic Networks and Aggregation

Participants : G. Casadei, V. Bertrand, B. Gouin, C. Canudas de Wit [Contact person] .

Large scale traffic networks are a popular topic nowadays due to the impact traffic has in our everyday life, both economically and health-wise. City management are interested in understanding the evolution of traffic and its patterns over the city in order to take decision on potential changes and to design new and more functional infrastructure. However, monitoring the current state of a large scale traffic network is a demanding task. The heterogeneity of available measures poses several question on how to merge different sources of information coming from private and public sources. Furthermore, sparsity is an intrinsic issues related to large scale systems: independently from the source we choose to rely on, we cannot expect the measurements to be sufficiently dense to cover the full network in detail.

For large scale urban network, managing real-time traffic information from thousands of links simultaneously is an overwhelming task and extracting interesting and meaningful insights from these tangle of data can be even a more challenging aim. In recent years more and more data are becoming available from new sources, such as smart phones, GPS navigators, and their technological penetration nowadays allows to have an impressive amount of real-time traffic information, not requiring the placement of physical sensors over the network and thus reducing incredibly costs due to installation and maintenance: in other words, each user becomes a moving sensor inside the network.

One way to deal with this huge amount of data over a urban traffic network is to look at the graph describing the network with a clusterization approach: this would reduce the number of nodes, thus the computational cost, proportionally to the clusterization rate and potentially would help with sparsity by merging areas in which no data are available with areas with
sufficient penetration of information. In this work we presented an aggregation-based technique to analyze
GPS velocity data from a private source (TomTom) and to calculate multi-origin multi-destination travel
time. The technique we propose allows to perform the aggregation and the necessary computation in
such a way that its application in a real time framework is feasible. The information and results we obtain
are of great interest to understand the macroscopic evolution of the traffic from a large-scale point of view
and to evaluate the average time that users spend in transiting between different areas along the day. In practice, we show that reducing the *complexity* of the network by 95$\%$ thanks to aggregation, we introduce an error in the calculation of the traveling times that in the average is below 25$\%$.

#### Two dimensional models for traffic

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

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 aims to develop such models with new designs especially including the network topology, and validation with simulation.