Section: New Results

Transportation networks and vehicular systems

Traffic estimation: sensors placement and data fusion

Participants : C. Canudas de Wit [Contact person] , E. Lovisari, A. Kibangou.

Ability to reconstruct the state of a transportation network is of paramount importance. Indeed, such an information is used to forecast traffic evolution, to inform drivers in real-time through navigation systems, to provide statistical information to public authorities to detect in a timely fashion accidents and predict hazardous scenarios, and finally to compute controls and to actuate the network through traffic lights, ramp metering, or adaptive speed limits.

A primary source of information on the state of the network are fixed traffic detectors, namely, devices able to measure density, flow and average speed of vehicles crossing the section of the road where they are placed. We have addressed the Optimal Sensor Placement problem [31] , namely, the problem of finding the best physical location for sensors. This is based on a trade-off of two contrasting objectives: the first, to maximize the performance of state reconstruction; the second, to minimize the total economic cost of the network. To simplify the setting, we consider the related problem of reconstruction in a static setting, by considering as performance metric the error covariance of an estimator of the cumulative flows in the network over a long period of time. Since the resulting trade-off problem remains a combinatorial problem, we relax it using a method that we call Virtual Variance algorithm, based on the idea to associate to each sensor a virtual variance, which is large when the sensor is not needed for good reconstruction of the flow vector. The only input that the algorithm needs is an estimate of the matrix of splitting ratios and the nominal variance of each sensor. Since in real application a pre-existing sensor network is often unavailable, possible alternatives are field surveys with operators visually counting vehicles, as commonly done for calibration of traffic software, or temporary non-invasive equipment such as radar traffic detectors.

In addition to fixed traffic detectors, the spread of wireless devices allows new sensing and communication capabilities. In particular, for the traffic application, any vehicle equipped with a GPS device can act as a probe in the traffic and provide Floating Car Data (FCD). If a non negligible fraction of vehicles acts as probe, the collected data provides an estimate of the evolution of speed in the network. Due to privacy reasons, single vehicles traces are usually not directly used, but rather aggregated as average speed of vehicles in segments of road. Advanced methodologies, such as the one used by INRIX, ensure a very fine spatial partition of the network, with segments as short as 250 meters (see the INRIX official website http://www.inrix.com/xd-traffic ). Compared to fixed sensors, this technology is less precise, but since it exploits existing communication systems it is relatively less expensive and already covers all major traffic networks. In our work [30] , [29] , we propose an algorithm that aims at reconstructing the traffic density by fusing fixed sensors measurements and Floating Car Data. We employ a macroscopic model, partitioning the network in cells and assigning to each cell a density of vehicles. The latter evolves dynamically according to a first order mass-conservation law. Our approach inherits from the CTM the cell-based topology, but we do not directly employ the resulting dynamical model. Instead, inflows and outflows are estimated on the basis of the available flow measurements only, and speed measurements are employed to compute a pseudo-measurement of the density. These quantities are the inputs for the density observer. In addition, we propose a gradient descent method to calibrate the Fundamental Diagram, and we implement the proposed solution using real fixed sensor measurements from the Grenoble Traffic Lab [14] and speed FCD measurements provided by INRIX, one of the most well known traffic solutions companies.

Traffic forecasting

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

Traffic forecasting is one of the most desired tools for traffic management, requested by operators and commuters. In the era of data deluge in which we are, measurements collected by sensors are important sources of information that require analysis, classification and processing in order to detect patterns and behaviours that can be exploited for traffic prediction ([30] , [37] ). The collected information can be classified by clustering algorithms such as K-means; each cluster collects traffic patterns, which in some cases characterize typical regimes such as congestion. Based on clustered data, we have first developed forecasting schemes based on adaptive Kalman filtering [14] . These schemes were designed for specific origin-destination (OD) pairs, assuming availability of meaurements whatever the time instants. Recently, within the PhD thesis in progress of Andres Ladino Lopez, we considered a network-oriented forecasting scheme, where travel time measurements are assumed to be available only for a few sets of OD pairs and sporadically (missing data), but forecasting is to be achieved for all the OD pairs of the network. To reduce the dimensionality of the problem, we actually predicted the travel time for the internal state of the network. In addition, since travel time measurements for all the OD pairs cannot be available all the time, we faced a missing data problem. To overcome this issue, we resorted to a data imputation based on a dictionary learning approach. From the imputed data, a clusterization was achieved, defining different clusters characterized by a centroid containing the mean of the data and a given dispersion around it. The evolution of the centroid can be used as future observation, herein called pseudo-observation, that can feed a Kalman filter. Therefore the prediction problem was solved as a filtering one. However, the main question was, how to associate the current day data to a specific cluster, since we didn't know its future? To solve this issue, we run Kalman filters for each cluster and then made the fusion of the obtained forecasts.

Traffic control

Participants : C. Canudas de Wit [Contact person] , F. Garin, D. Pisarski, P. Grandinetti, E. Lovisari, G. Como [U. Lund] , K. Savla [U. of Southern California] .

The activities of the team on traffic control can be organized in three parts: freeway traffic control, urban control, and analysis and control of monotone flows.

First,we have studied optimal balancing of vehicle density in the freeway traffic. The optimization is performed in a distributed manner by utilizing the controllability properties of the freeway network represented by the Cell Transmission Model. By using these properties, we identify the subsystems to be controlled by local ramp meters. The optimization problem is then formulated as a non-cooperative Nash game that is solved by decomposing it into a set of two-players hierarchical and competitive games. The process of optimization employs the communication channels matching the switching structure of system interconnectivity. By defining the internal model for the boundary flows, local optimal control problems are efficiently solved by utilizing the method of Linear Quadratic Regulator. The developed control strategy is tested via numerical simulations in two scenarios for uniformly congested and transient traffic. This work is described in the paper [21] .

Second, we have considered optimal or near-optimal operation of traffic lights in an urban area. The goal is on-line optimization of traffic light 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, while also ensuring good servicing of demands of incoming cars in the network from other areas. One way to address the complexity of the resulting optimization problem is to use a simplified averaged model for the traffic variables, and to optimize only the duty-cyles of traffic lights, i.e., the fractions of green time. This, together with a one-step optimization horizon, allows us to turn the problem into a simple linear program [27] . Another approach is to include as optimization variables both duty-cycles and phases of the traffic lights. We show how to turn the resulting problem into a mixed-integer linear program (MILP). Then, to overcome its complexity, we propose a sub-optimal distributed solution, while the global MILP can be used off-line for performance comparison [28] .

Third, stability and throughput properties of monotone dynamical flow networks are studied in [15] . Vehicular density on the cells of the networks evolves according to laws that deterministically split the traffic flow at each intersection as a function of the density of other cells around the intersection. By exploiting the theory of monotone operators it is proven that under certain condition the system achieves an equilibrium that maximizes the throughput of the network, namely, if the inflow is smaller than the network capacity, then asymptotically the total outflow matches the total inflow, otherwise the total outflow matches the network capacity. In [25] a different traffic model is employed which uses demand and supply functions to relate density and flows of the network. The Social Optimum Dynamic Traffic Assignment, which is an optimal control problem with cost corresponding to the total travel time of vehicles in the network, is solved making use of ramp metering and speed limits. The optimization is shown to be a convex optimization problem, making its numerical solution feasible by employing well known tools.

Energy-aware control of communicating vehicles

Participants : C. Canudas de Wit [Contact person] , G. de Nunzio.

The research in this domain focuses mainly on efficient traffic energy consumption and has been carried out at two levels. Strategies for both the vehicles-side and the infrastructure-side eco-management have been proposed or extended. As for the vehicle-side control of communicating vehicles, assuming I2V communication, and therefore full knowledge of the traffic lights timings, the goal is to analyze the driving horizon and compute an energy-efficient speed advisory for the driver. As in previous works, stops at a red traffic light are to be avoided. The novelty of our approach is summarized as follows. Given a set of green traffic light phases, there exist different driving profiles to reach a given destination at a given final time in compliance with traffic lights constraints (i.e. always catching the green light) and city speed limits. The presented strategy is capable of an a priori identification of the most energy-efficient velocity trajectory, by approximating the available paths and their energy cost with an oriented weighted graph. The computational complexity of the graph creation has been reduced in this work from exponential [26] to polynomial, thanks to the introduction of the line graph. The computation time has been consequently significantly reduced. Only after this preliminary stage of path selection, a formal optimization problem is solved in order to calculate the optimal arrival times at each intersection, by explicitly minimizing the energy consumption of the vehicle. This approach qualifies as a pre-trip eco-driving ADAS, since the speed advisory is provided to the driver at the beginning of the driver horizon. However, given the very little computation time required by the algorithm, it may be employed online thus enabling in-trip assistance features. This allows to respond dynamically to traffic perturbations and/or deviations from the speed advisory, and to increase the robustness and the applicability of the strategy in a realistic environment. Simulations in a microscopic traffic simulator demonstrate that the proposed strategy is able to deal online with perturbations coming from traffic and to reduce the overall energy consumption without affecting travel time [16] .

At a lower level, the eco-driving from the vehicle perspective has been also addressed in a comprehensive analysis of the optimal driving strategy for different types of powertrains [22] .

As for the infrastructure-side eco-management, this year's research focused on extending the results published in [26] . The two-way arterial bandwidth maximization problem is addressed with a particular focus on the benefits induced by the speed advisory, and on reducing energy consumption. The problem with internal offsets constraints presents difficulties that make necessary the formulation of the problem as an MILP. The first contribution of our work lies in the addition of terms representing traffic energy consumption and network travel time to the objective function of the two-way arterial bandwidth maximization. The segment speeds, as additional control action, allow to reach higher theoretical bandwidths but might induce driving discomfort and higher energy consumption if the variability of the recommended speeds is too high. Furthermore, optimal solutions with low speeds and high travel time are to be avoided, in trade-off with the energy consumption. The second contribution is given by the extensive evaluation of the benefits of bandwidth maximization via a microscopic traffic simulator. Bandwidth is a theoretical quantity and a correlation with known traffic performance metrics needs to be established in order to justify its use. The combined control of offsets and speed advisory is shown to have a large impact on energy consumption without affecting the travel time. Lastly, an analysis of the traffic performance at different levels of traffic demands has been conducted, testing both under-saturated traffic conditions with the existence of a green wave, and saturated conditions. The goal of this analysis is to identify the best operation conditions of the presented approach, assess the performance degradation with traffic load, and, most importantly, propose a demand-dependent optimization. Several strategies were compared to the presented one in order to assess its performance. This work has been submitted for review to the IEEE Transactions on Control Systems Technology.

Finally, a detailed description of the proposed strategies and the achieved results in the domain of the energy-aware traffic management in urban networks can be found in the dissertation [11] .