domain: spi math

text:
Chiara Daini
email: chiara.daini@inria.fr

Maria Laura Delle Monache
email: mldellemonache@berkeley.edu

Paola Goatin
email: paola.goatin@inria.fr

Antonella Ferrara

Maria Laura

Delle Monache

Traffic Control via Fleets of Connected and Automated Vehicles

Keywords: Traffic control, autonomous vehicles, multi-scale traffic flow models, PDE-constraint optimal control

The foreseen deployment of Connected and Automated Vehicles (CAVs) on public roads opens the perspective of reducing the social and environmental impacts of traffic congestion using CAVs as optimal control actuators, operating as moving bottlenecks on the surrounding flow. In this paper, we propose three control strategies, based on different levels of cooperation, to improve density dependent traffic performance indexes, such as fuel consumption.

We rely on a multi-scale approach to model mixed traffic composed of a small fleet of CAVs in the bulk flow. In particular, CAVs are allowed to overtake (if on distinct lanes) or queuing (if on the same lane). Controlling CAVs desired speeds allows to act on the system to minimize the selected cost function. For the proposed control strategies, we apply both global optimization and a Model Predictive Control approach. In particular, we perform numerical tests to investigate how the CAVs number and positions impacts the result, showing that few, optimally chosen vehicles are sufficient to significantly improve the selected performance indexes, even using a decentralized control policy. Simulation results support the attractive perspective of exploiting a very small number of vehicles as endogenous control actuators to regulate traffic flow on road networks, providing a flexible alternative to traditional control methods. Moreover, we compare the impact of the proposed control strategies (decentralized, quasi-decentralized, centralized).

I. INTRODUCTION

T HE recent technological advances in connectivity and automation for the automotive industry are transforming the transportation sector and impacting the related socioeconomical aspects. In particular, Connected and Automated Vehicles (CAVs), which are expected to dominate the vehicle market in the next future, have raised the interest of researchers for their potential impact on traffic flow, with the aim of improving traffic conditions and safety. Several studies have shown that CAVs can be employed to control the overall traffic to mitigate congestion and improve throughput, with a consequent reduction of pollutant emissions. This antonella.ferrara@unipv.it has been proved by model based theoretical results [START_REF] Talebpour | Influence of connected and autonomous vehicles on traffic flow stability and throughput[END_REF]- [START_REF] Liard | A PDE-ODE model for traffic control with autonomous vehicles[END_REF], machine learning approaches [START_REF] Kreidieh | Dissipating stop-and-go waves in closed and open networks via deep reinforcement learning[END_REF], [START_REF] Vinitsky | Lagrangian control through deep-rl: Applications to bottleneck decongestion[END_REF] and real world experiments [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF]. All these investigations show that even a small number of automated vehicles among human-driven vehicles can bring benefits to the whole system, by dissipating stop-and-go waves, improving the throughput and reducing traffic flow emissions and consumption. In this perspective, CAV control can offer a valid, flexible and cheap alternative to more traditional traffic management strategies, such as ramp metering and variable speed limits [START_REF] Papageorgiou | Freeway ramp metering: an overview[END_REF]- [START_REF] Delle Monache | Traffic regulation via controlled speed limit[END_REF], which require specific infrastructures.

For traffic control applications, the idea of using multi-scale models in the form of systems of strongly coupled differential equations, representing CAVs as moving bottlenecks acting as variable speed limits, was introduced in [START_REF] Piacentini | Traffic control via moving bottleneck of coordinated vehicles[END_REF], [START_REF] Ramadan | Traffic flow control and fuel consumption reduction via moving bottlenecks[END_REF] and [START_REF] Čičić | Traffic regulation via individually controlled automated vehicles: a cell transmission model approach[END_REF]. Models consisting of Partial Differential Equations (PDEs), describing the spatio-temporal evolution of bulk traffic, coupled with Ordinary Differential Equations (ODEs) tracking specific vehicles trajectories, were introduced in the literature to describe the impact of some slow moving vehicles, like buses or trucks, on the surrounding traffic flow [START_REF] Claudel | Lax-Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton-Jacobi Equation. Part II: Computational Methods[END_REF]- [START_REF] Delle Monache | Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result[END_REF]. In the targeted control scenario, multi-scale approaches allow to mitigate the curse of dimensionality, which penalizes the control design for microscopic models [START_REF] Delle Monache | Feedback Control Algorithms for the Dissipation of Traffic Waves with Autonomous Vehicles[END_REF].

In this framework, CAV optimal control strategies have been applied to the speed of single vehicles [START_REF] Čičić | Traffic regulation via individually controlled automated vehicles: a cell transmission model approach[END_REF], [START_REF] Piacentini | Traffic control via moving bottleneck of coordinated vehicles[END_REF], small fleets [START_REF]Stop-and-go wave dissipation using accumulated controlled moving bottlenecks in multi-class CTM framework[END_REF], [START_REF] Čičić | Front tracking transition system model with controlled moving bottlenecks and probabilistic traffic breakdowns[END_REF] or platoons [START_REF]Energy-optimal platoon catch-up in moving bottleneck framework[END_REF], [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF], [START_REF] Čičić | Coordinating vehicle platoons for highway bottleneck decongestion and throughput improvement[END_REF] of CAVs, aiming to improve the energy footprint of traffic by either dissipating stop-and-go waves or reducing traffic congestion. A similar goal is tackled by [START_REF] Wang | Data-driven predictive control for connected and autonomous vehicles in mixed traffic[END_REF] using a different approach based on a data-driven cruise control in a microscopic framework. We remark that, even if some of the above works consider the presence of several CAVs, no one allows overtaking among the controlled moving bottlenecks. To our knowledge, before [START_REF] Daini | Centralized traffic control via small fleets of connected and automated vehicles[END_REF], [START_REF] Goatin | Interacting moving bottlenecks in traffic flow[END_REF], moving bottleneck overtaking had been introduced only in [START_REF] Simoni | A fast simulation algorithm for multiple moving bottlenecks and applications in urban freight traffic management[END_REF], but for different optimization purposes.

A. Contribution

In this work, following [START_REF] Daini | Centralized traffic control via small fleets of connected and automated vehicles[END_REF], [START_REF] Goatin | Interacting moving bottlenecks in traffic flow[END_REF], we develop a general simulation environment modeling the interactions of CAVs and human-driven traffic flow, which enables us to study different CAV-based control strategies with the goal of reducing the overall traffic emissions. To this end, we consider CAVs distributed on different lanes, thus being allowed to queue (if on the same lane) or to overtake (if on distinct lanes). We note that, introducing the ability for the CAVs to interact adds a degree of freedom that can be conveniently used to improve the control performance.

Our main contribution is to propose three levels of cooperative driving, ranging from a centralized control scheme which assumes the complete knowledge of the traffic conditions and the simultaneous control of the CAV fleet trajectories, to a fully decentralized strategy in which each CAV optimizes its speed, depending on traffic conditions but without taking into account the rest of the fleet. At an intermediate level, we consider a realistic compromise assuming that CAVs optimize their trajectories depending on traffic conditions and taking into account the presence of other fleet items within a limited radius. In the numerical tests, we perform both a global optimization on the full time horizon considered, requiring full knowledge of boundary inflow and outflow conditions, and a Model Predictive Control (MPC) approach that could realistically be implemented in practice. Both optimization strategies show that, if CAV positions are suitably chosen, the quasi-decentralized strategy performs as well as the centralized one. Remarkably, for MPC all the proposed control approaches reach the same performances. Moreover, for all strategies, the gain increases with the number of CAVs, but tends to stabilize quickly even for small fleet sizes. This shows that few, optimally placed vehicles suffice to maximize the selected performance index.

B. Outline

The paper is organized as follows. Section II details the modeling framework, providing a mathematical description of the fully coupled PDE-ODE model including bottleneck interactions, and its numerical discretization. Section III formulates the control problem and illustrates the proposed control strategies, while Section IV describes the results of the numerical experiments.

II. MIXED TRAFFIC AUTONOMY MODEL

In this section, we present the multi-scale model describing the interaction of a small fleet of Connected and Automated Vehicles with the surrounding traffic flow. Since we assume the CAV penetration rate is small, they can be tracked one by one because of their specific dynamics and for control purposes. However, it is convenient to model the overall traffic flow from a macroscopic point of view, describing the spatiotemporal evolution of the traffic density. Compared to more detailed microscopic approaches, macroscopic models (also called hydrodynamic models for their similarity to fluid dynamics equations) results in a very reduced computational cost and simpler calibration of (few) model parameters, especially when considering large road networks. Still, they provide good estimates of traffic performance indexes, such as average travel times, amount of congestion, fuel consumption, etc.

The interaction dynamics between slow moving vehicles and the surrounding bulk traffic has been studied by several authors, see e.g. [START_REF] Čičić | Traffic regulation via individually controlled automated vehicles: a cell transmission model approach[END_REF], [START_REF] Claudel | Lax-Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton-Jacobi Equation. Part II: Computational Methods[END_REF], [START_REF] Leclercq | Moving bottlenecks in Lighthill-Whitham-Richards model: A unified theory[END_REF], [START_REF] Lattanzio | Moving bottlenecks in car traffic flow: a PDE-ODE coupled model[END_REF], [START_REF] Daganzo | On the numerical treatment of moving bottlenecks[END_REF], [START_REF] Borsche | Mixed systems: ODEsbalance laws[END_REF]. Here we extend to several, possibly interacting CAVs, the model proposed in [START_REF] Lebacque | Introducing buses into first-order macroscopic traffic flow models[END_REF] and then developed in [START_REF] Delle Monache | Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result[END_REF], [START_REF] Garavello | A multiscale model for traffic regulation via autonomous vehicles[END_REF]. It consists of the classical Lighthill-Whitham-Richards (LWR) [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF], [START_REF] Richards | Shockwaves on the highway[END_REF] first order macroscopic model accounting for the general evolution of the traffic density, coupled with Ordinary Differential Equations (ODEs) describing the CAVs trajectories.

A. Model definition

To describe the traffic flow on a stretch of highway, we consider the LWR Partial Differential Equation (PDE). This is a macroscopic first-order model based on the mass conservation equation:

∂ρ ∂t + ∂f (ρ) ∂x = 0,
with x ∈ R and t ≥ 0. Above, ρ = ρ(t, x) is the traffic density, defined as the number of vehicles per unit length of the road (veh/km), and f (ρ) = ρv(ρ) is the flow, defined as the number of vehicles passing the cross-section at location x per unit time (veh/h). The phenomenological speed-density relation v = v(ρ) provides the mean velocity, which is the arithmetic mean speed of the vehicles passing the cross-section per unit time (km/h), as a function of the traffic density. Several speed laws have been proposed in the literature, see e.g. [START_REF] Greenberg | An analysis of traffic flow[END_REF], [START_REF] Underwood | Speed, volume and density relationships[END_REF]. For simplicity, we consider here the linear speed function proposed by Greenshields [START_REF] Greenshields | A study of traffic capacity[END_REF]

v(ρ) = V 1 - ρ R , (1)
where V denotes the maximal free flow speed and R the maximal (bump-to-bump) density on the road. The corresponding fundamental diagram is the quadratic function

f (ρ) = V ρ 1 - ρ R ,
attaining its maximum at ρ cr = R/2 (see Fig. 1). Nevertheless, our study can be extended to any fundamental diagram f :

[0, R] → R + with f (0) = f (R) = 0, which is concave in the free-flow interval [0, ρ cr ].
To account for the presence of CAVs among traffic on a given road segment, we consider the following fully coupled PDE-ODE system:

∂ ∂t ρ (t, x) + ∂ ∂x f (ρ (t, x)) = 0, (2a)
ẏ (t) = min{u (t), v(ρ(t, y (t)+))}, (2b)
f (ρ (t, y (t))) -ẏ (t)ρ (t, y (t)) ≤ αR 4V (V -ẏ (t)) 2 , ( 2c
)
together with initial conditions

ρ(0, x) = ρ 0 (x), (3a)
y (0) = y 0 , (3b) ρ R v u V v(ρ) f (ρ): whole road f α (ρ) = αf (ρ/α) reduced road ρ αR R f 0 ρu ρu u
f (ρ(t, 0)) = f in (t), (4a)
f (ρ(t, L)) = f out (t). (4b)
In ( 2)-( 4), x ∈ [0, L], L > 0 being the length of the considered road segment, t ∈ [0, T f ], with T f the time horizon and ρ 0 the initial traffic density on the road. The CAV trajectories are denoted by y (t), = 1, . . . , N , with N the size of the fleet. The CAV desired speeds are the controlled time dependent variables u : [0,

T f ] → [0, V ], = 1
, . . . , N . Equation (2b) states that each moving bottleneck moves at the prescribed speed u (t) if the downstream traffic density ρ(t, y (t)+) allows it. Otherwise, it adapts to the local traffic velocity if it is lower than u (t) (Fig. 2, left). Besides, the inequalities (2c) account for the moving capacity constraints exerted by CAVs, where α ∈ ]0, 1[ is the reduction rate of the road capacity due to the presence of a CAV, which can be set to α = (M -1)/M for all vehicles, where M ∈ N is the number of lanes on the road. When active, the constraint induces the formation of a non-classical discontinuity in the traffic density, moving with speed u (t). The upstream density value is denoted ρu (t) and downstream density ρu (t) (see Fig. 2, right).

B. Moving Bottlenecks Interactions

To account for interactions between the CAVs, we consider

y i ( t) = y j ( t), for some i, j = 1, . . . , N , i = j and t ∈ [0, T f ].
For simplicity, in this work we assume that CAVs stay always in the same lane and are not allowed to change lane to overtake other cars. However, they can overtake other CAVs if they are on different lanes. We can therefore distinguish two situations:

• Same lane. If the vehicles are on the same lane, the upstream vehicle travelling with higher speed will adapt to the preceding vehicle and follow it: we get y i (t) = y j (t) and u i (t) = u j (t) for t ≥ t. • Different lanes. If the vehicles are on different lanes, the faster simply overtakes the slower.

Besides, the constraints (2c) continue to act after the interaction, see Fig. 3. We refer to [START_REF] Goatin | Interacting moving bottlenecks in traffic flow[END_REF] for a detailed mathematical description of the different interaction dynamics.

C. Model discretization

The approximation scheme used to compute the solutions of the coupled PDE-ODE model ( 2)-( 4) consists of two components: the finite volume discretization of the PDE (2a) with the constraint (2c) and the numerical approximation of the ODEs (2b).

1) Numerical scheme for PDE (2a) with constraint (2c): To approximate the constrained LWR model, we use the reconstruction scheme introduced in [START_REF] Chalons | A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem[END_REF]. It relies on any conservative finite volume scheme for hyperbolic PDEs, coupled with a flux reconstruction technique at the constraint locations. Let ∆x and ∆t be the fixed space and time steps satisfying the Courant-Friedrichs-Lewy (CFL) condition [START_REF] Courant | On the partial difference equations of mathematical physics[END_REF]:

V ∆t = 0.9 ∆x, and set x j-1/2 = j∆x, x j = (j + 1/2)∆x for j = 0, . . . , J, with x -1/2 = 0, x J+1/2 = L, and t n = n∆t for n = 0, . . . , T f /∆t . The algorithm consists of the following steps:

1) Approximate the initial data ρ 0 by piece-wise constant functions ρ n 0 = {ρ n 0,j } J j=0 s.t.

ρ n 0,j := 1 ∆x x j+1/2
x j-1/2 ρ 0 (x) dx, for j = 0, . . . , J, and boundary data

f n in = 1 ∆t n+1 t n f in (t) dt, f n out = 1 ∆t n+1 t n f out (t) dt, u n = u (t n ),
for n = 0, . . . , T f /∆t . 2) Locate the cell position C m of the -th CAV at time t n , such that y n ∈ C m , = 1, . . . , N . 3) Compute the Godunov numerical fluxes [START_REF] Godunov | A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics[END_REF] at the cell interfaces

F n j+ 1 2 = F (ρ n j , ρ n j+1
), which are given by the supply-demand formula

F (ρ n j , ρ n j+1 ) = min{D(ρ n j ), S(ρ n j+1 )}, (5)
where

D(ρ) = f (min{ρ, ρ cr }), S(ρ) = f (max{ρ, ρ cr })
(see the Cell Transmission Model [START_REF] Daganzo | The cell transmission model. Part I: A simple dynamic representation of highway traffic[END_REF]).

• Set ∆t n m = 1 -d n m u n , ∆x. • Replace F n m -1 2 and F n m + 1 2 by F n m -1 2 = F (ρ n m -1 , ρu n ) and ∆tF n m + 1 2 = min (∆t n m , ∆t)f (ρ u n ) + max (∆t -∆t n m , 0)f (ρ u n ). 6
) For j = 0, . . . , J, update the density with the conservative formula

ρ n+1 j = ρ n j - ∆t ∆x F n j+ 1 2 -F n j-1 2 , (6)
where the boundary numerical fluxes [START_REF] Čičić | Front tracking transition system model with controlled moving bottlenecks and probabilistic traffic breakdowns[END_REF]. Notice that, when they are located in the same cell, the moving bottlenecks are processed sequentially one after the other, first those that are not active (i.e. satisfy (2c)), followed by the active ones (violating (2c)).

F n -1 2 and F n J+ 1 2 are computed setting D(ρ n -1 ) = f n in and S(ρ n J+1 ) = f n out in (
2) Numerical scheme for the ODEs (2b): To track the CAV trajectories, at each time step, we update the position y n of the -th CAV using an explicit Euler scheme: for = 1, . . . , N ,

• y n+1 = y n + u n ∆t n if the constraint (2c) is not satisfied in step 4) above;

• y n+1 = y n + v(ρ n m )∆t n if (2c) is satisfied.

III. TRAFFIC CONTROL PROBLEM

In this section, we describe the control strategies applied to the traffic system described above. The control goal is to determine the appropriate CAV velocities, on the basis of the traffic flow conditions, in order to minimize a selected performance index (the cost function). To this end, in this paper we rely on a Model Predictive Control (MPC) approach. MPC is a well-established technique to control dynamical systems, subject to constraints on the state and the control variables, in an optimized way. Following a long history of success in the process industries, in recent years MPC is rapidly expanding in several other domains, such as in the automotive and aerospace industries, smart energy grids, and financial engineering. MPC has also been applied to single-scale (i.e. macroscopic) traffic control, as discussed in [START_REF] Siri | Freeway Traffic Control: a Survey[END_REF], [START_REF] Ferrara | Freeway Traffic Modelling and Control[END_REF]. The MPC control algorithm is based on the solution of the optimization problem at each time step of size ∆τ > 0, taking into account static and dynamic constrains. It is usually implemented relying on the so-called Receding Horizon Control (RHC) concept. This means that, at each time step, the optimal control sequence is determined by solving the optimization problem for the pre-specified prediction horizon ∆T > ∆τ ; then, only the first sample of the optimal control sequence is applied to the plant and the procedure is repeated at each time step. In this paper, MPC is applied to realize three control solutions that treat the CAVs immersed in the macroscopic traffic differently. They can be classified as centralized, decentralized and quasi-decentralized controls, respectively. In the next sections, we will describe each control strategy separately.

A. The cost function

Road transport is one of the main source of air pollution [START_REF] Pasquale | Traffic control for freeway networks with sustainability-related objectives: Review and future challenges[END_REF]. In particular, vehicular traffic consumption and pollutant emissions are strictly related to the congestion phase, when vehicles accelerate and stop repeatedly [START_REF] Pathak | Real world vehicle emissions: Their correlation with driving parameters[END_REF]. Since in this work we refer to a first order macroscopic model, thus neglecting the traffic acceleration component, we can consider consumption models based on the average speed of vehicles. The cost functional will therefore provide local emission factors, describing average fuel consumption or emissions in kg (or liters) per meter, volume or mass of consumed fuel or emitted pollutant per kilometer and per vehicle [START_REF] Treiber | Traffic Flow Dynamics[END_REF]. In particular, we will focus on the consumption model derived in [START_REF] Ramadan | Traffic flow control and fuel consumption reduction via moving bottlenecks[END_REF], which is based on the fuel consumption efficiency data (Liters/km) for four types of vehicles (Ford Explorer, Ford Focus, Honda Civic, and Honda Accord), as functions of the vehicle speed, presented in [START_REF] Berry | The effects of driving style and vehicle performance on the real-world fuel consumption of U.S. light-duty vehicles[END_REF]. Multiplying the fuel consumption efficiency by the vehicle speed yields the fuel consumption rate (Liters/hr). The average fuel consumption rate as a function of the speed derived in [START_REF] Ramadan | Traffic flow control and fuel consumption reduction via moving bottlenecks[END_REF] is given by

K(v) = 5.7 • 10 -12 • v 6 -3.6 • 10 -9 • v 5 + 7.6 • 10 -7 • v 4 -6.1 • 10 -5 • v 3 + 1.9 • 10 -3 • v 2 + 1.6 • 10 -2 • v + 0.99, (7)
see Fig. 4, left. Replacing v in ( 7) by the speed-density law (1), we recover the fuel consumption rate as a function of the traffic density ρ,

K(ρ) = K(v(ρ)), (8)
which represents the fuel consumption rate of one vehicles as a function of the traffic density at the vehicle's position. The total fuel consumption rate F C(ρ) is therefore obtained by setting F C(ρ) = ρK(v(ρ)) and is depicted in Fig. 4, right. Lastly, the total fuel consumption of the road stretch [a, b] in the selected time interval [t 1 , t 2 ] is given by:

T F C(a, b; t 1 , t 2 ) = t2 t1 b a F C(ρ(t, x)) dx dt. (9)
Consider that fuel consumption and greenhouse gas emissions are strictly related: on average, a litre of unleaded petrol produces 2.3 kgs of CO 2 . As such, reducing the fuel consumption associated with the overall traffic means correspondingly reducing pollutant emissions [START_REF]Transportation, air pollution, and climate change[END_REF].

B. Centralized control

Assuming an external controller, a centralized control strategy implemented at time t = t ∈ [0, T f ] on a time horizon ∆T takes as input the vector composed by all the controlled CAV positions y ( t), = 1, . . . , N , the current traffic density ρ( t, x), x ∈ [0, L] and the foreseen inflow f in (t) and outflow f out (t), t ∈ [ t, t + ∆T ], and computes the optimal control u * = (u * 1 , . . . , u * N ) = [ t, t + ∆T ] → R N minimizing the selected cost function (here the total fuel consumption T F C(0, L; t, t + ∆T )). The situation is represented in Fig. 5. In principle, this is the most effective strategy, since the control computation is based on the complete information about the CAVs and the macroscopic traffic state, as well as on the knowledge of the inflow and outflow affecting the traffic system in the considered time horizon.

In an MPC perspective, at the k-th iteration step, we compute the optimal (constant) speed value u * (k) for each CAV, = 1, . . . , N , taking the current density value ρ(t k , •) as the initial datum in (3a) over a fixed time horizon [t k , t k +∆T ]:

u * (k) = arg min T F C(0, L; t k , t k + ∆T ), (10)
subject to (2) and the prescribed constraints

u min ≤ u (k) ≤ u max , = 1, . . . , N. (11)
We observe that the considered cost functional T F C(0, L; t k , t k + ∆T ) is non-linear and non-convex.

To avoid local minima, the constrained optimization problem ( 10)-( 11) have been solved using the MATLAB function fmincon, a gradient-based method that is designed for non-linear constrained problems, initialized with the optimal velocities computed by bayesopt, a Bayesian optimizer which uses a Gaussian process model to minimize the objective function, thus better exploring the admissible control domain. We refer to Algorithm 1 for an overview of the steps. In the present case, any individual CAV can be regarded as a subsystem. As such, at the k-th iteration step, we compute separately the -th CAV optimal (constant) speed value u * (k), as it were the only controlled vehicle on the road, taking as input the selected CAV position y ( t), the current traffic density ρ( t, x), x ∈ [0, L] and the foreseen inflow f in (t) and outflow f out (t), t ∈ [ t, t + ∆T ]. For this reason, this control approach can be seen as fully decoupled. The concept is schematized in Fig. 6 and we refer to Algorithm 2 for an overview of the steps. We notice that, compared to the centralized algorithm, the optimization loop has been inserted in another loop, which splits the different vehicles and processes them separately.

D. Quasi-decentralized control

As an intermediate, more viable approach, we consider a partially decentralized control strategy: as in the decentralized control described in Section III-C, we optimize each CAV velocity separately, but considering the system evolution Data: Input initial traffic density ρ 0 , CAV positions y 0 and velocities u (0), inflow f in and outflow f out , time horizon T f , optimization time interval

|y m (t k ) y (t k )| ≤ r (12)
to define the set of -neighbors N . We then perform a centralized optimization as in Section III-B, but considering only the CAVs corresponding to the selected neighbor indexes N , and we keep the optimal velocity u * for the originally selected -th vehicle. In Fig. 7, a scheme of the control action is presented. Vehicle A detects other two CAVs within its sensing radius, hence the optimization of its velocity takes into account their presence. On the other hand, vehicle B is too far from other CAVs, thus acting as a fully decentralized controlled vehicle. The optimization procedure is described in Algorithm 3.

As initial condition, we consider an oscillating density

ρ 0 (x) = 0.3 R (sin(0.2πx) + 1) ,
mimicking the presence of stop-and-go waves (see Fig. 9, bottom), while the boundary conditions are given by

f in (t) = f max if t ≤ 0.5 T f , 0 if t > 0.5 T f , (14)
f out (t) = 0.5 f max ∀t ∈ [0, T f ], (15)
on a time horizon T f = 1 hr, where we have set f max = max ρ∈[0,R] f (ρ) = V R/4 = 14 000 veh/hr the capacity of the road. Condition (15) mimics the presence of a fixed bottleneck reducing by half the capacity at the end of the road, due for example to an accident or construction works, and inducing a backward moving congestion whose effects need to be mitigated.

A. Cost functional sensitivity analysis

To better understand the cost functional considered in [START_REF] Ferrara | Multi-scale model based hierarchical control of freeway traffic via platoons of connected and automated vehicles[END_REF], we analyze its dependency on constant CAV desired speed ū ∈ [START_REF] Claudel | Lax-Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton-Jacobi Equation. Part II: Computational Methods[END_REF]140] and initial position y 0 ∈ [2, 50], considering a single CAV on the road. Fig. 8 represents the TFC as a function of ū and y 0 . The global minimum is close to ū = 55 km/h and y 0 = 5 km, while several other local minima are present. In particular, fixing ū = 55 km/h, we observe that the TFC has an increasing trend with respect to the initial position, with local minima located just downstream the higher density regions (x = 5, 15, 25, 35, 45 km), see Fig. 9. This suggests that vehicles starting farther away from the congestion have a greater impact in reducing the total fuel consumption, subject to local traffic conditions. These local minima are good candidates to be the initial CAV positions in the optimization tests described in Section IV-C.

B. Environment setup

The tests have been realized on fleets of up to N = 10 vehicles, distributed on the road starting from the points of local minima observed in Fig. 9; we take y 0 1 = 5 km and y 0 = y 0 -1 + 10 km, = 2, . . . , 5, then y 6 = 2.5 km and y 0 = y 0 -1 + 10 km, = 7, . . . , 10, see Fig. 9, bottom. We assign CAVs sequentially to the different lanes 1 to 3, see Fig. 10.

For each control strategy described in Section III, we implemented the following two setups: This assumes global knowledge of boundary flow conditions since the initial time t = 0. 2) Model Predictive Control with ∆T = 6 min and ∆τ = 5 min. In the perspective of applying this strategy to real environments, the optimization is performed at 4 th minute of the current system evolution interval, to account for the computational time needed to obtain the next control N -tuple. In this way, we guarantee the availability of the newly computed optimal controls for implementation before the end of the current implementation interval.

In all cases, controls will be constrained to the interval [u min , u max ] = [START_REF] Simoni | A fast simulation algorithm for multiple moving bottlenecks and applications in urban freight traffic management[END_REF]100] km/hr, see [START_REF] Kreidieh | Dissipating stop-and-go waves in closed and open networks via deep reinforcement learning[END_REF]. Moreover, for the quasi-decentralized strategy, we fix the radius r = 11 km in [START_REF] Vinitsky | Lagrangian control through deep-rl: Applications to bottleneck decongestion[END_REF], aiming at including at least the preceding and following CAVs.

For each control, the comparison will be made with the uncontrolled case, where no CAV acts on the traffic flow, see Fig. 11. In this case, the total fuel consumption is T F C = 2.7329 • 10 4 liters.

Notice that, because of the computational costs of the optimization routines, in particular for large fleet sizes, we do not compute time-dependent optimal controls, but we limit our study to optimal velocities which are constant in the optimization horizon ∆T . Further efforts are needed to optimize and possibly parallelize the simulation code and the optimization routines to reduce the overall computational costs of finer optimization procedures.

C. Numerical results

In this section, we present and analyze the results of the performed tests. Fig. 12 reports the trends of the global optimization applied to the centralized, decentralized and quasi-decentralized control strategies. We observe that, for all strategies, the TFC reduction improves for increasing fleet sizes up to 5 vehicles, placed at the optimal positions. Moreover, the quasi-decentralized control is as effective as the centralized one, followed by the fully decentralized. We recall that the centralized control has a global and complete view of the traffic evolution, hence it can compute simultaneously the optimal velocities of all CAVs in the fleet. Instead, the fully decentralized control strategy shows the worst results, because each optimizer/actuator ignores the presence and action of the others CAVs. Placing the vehicles at optimal positions, the quasi-decentralized control reaches the same performance as the centralized, despite having an overview of the traffic flow only locally around each controlled vehicle. We remark that, in this configuration, adding more than five vehicles doesn't diminish the fuel consumption. The curves drawn in Fig. 12 are polynomial interpolations of the computed discrete TFC values, meant to capture the TFC trend as a function of the CAV fleet size. Table I displays the total fuel consumption and its reduction rate with respect to the non-controlled dynamics, for each control strategy applied with N = 1, 5, 10 CAVs. On the other hand, Fig. 13 plots the results of MPC optimization corresponding to the three control strategies. Even if the gain is obviously smaller, the trends are similar to the global optimization output reported in Fig. 12: all the strategies improve when increasing the fleet size up to five vehicles, while adding more vehicles in this case seems to deteriorate the result. We notice that with MPC the performances of the three control strategies are practically the same, indicating that the simpler fully decentralized control can be applied as standard procedure. The TFC values and the reduction rates for N = 1, 5, 10, are reported in Table II In Figure 14, we show two examples of the space-time traffic density evolution and the corresponding optimal CAV trajectories for a fleet of N = 10 vehicles following the quasidecentralized strategy for 1 hour optimization and MPC, respectively. The red area corresponds to the congestion induced by the downstream bottleneck, while orange zones are local slowdowns generated by the presence of CAVs, which act as moving variable speed limits, delaying upstream flow in reaching the congestion. In particular, we can observe both overtaking and queuing among CAVs, before they reach the congested zone where they all run at the same speed as the sourrounding traffic. Automated Vehicles as actuators, corresponding to different levels of cooperation: centralized, decentralized and quasidecentralized. We show that a small number of well placed CAVs suffices to consistently reduce the fuel consumption of the entire flow and hence to decrease the pollutant emission. The proposed multi-scale modeling approach is numerically efficient and general enough to allow for CAV interactions (overtaking and queuing), and the optimization procedures (global and MPC based) can be applied to improve any traffic index dependent on aggregated traffic quantities (density, mean velocity, flow). Our results support the perspective of testing and validating CAV based cooperative traffic control strategies in real scenarios.

Possible improvements could be offered by further investigation on CAV optimal number and placement depending on traffic conditions. Also, second order models could be employed to better capture some traffic characteristics that would allow to consider a wider class of cost functionals, possibly depending on traffic acceleration.

Fig. 1 .

 1 Fig. 1. Greenshields' fundamental diagram.

Fig. 2 .

 2 Fig. 2. Left: -th CAV speed depending on downstream traffic density. Right: reduced flow at -th CAV position.

Fig. 3 .

 3 Fig. 3. Examples of interacting CAV trajectories (bold lines). Left: same lane. Right: different lanes.

4 )

 4 For = 1, . . . , N , if the constraint (2c) is not satisfied by the classical Riemann solution corresponding to ρ n m -1 , ρ n m +1 , then a moving bottleneck is located at xm = x m -1/2 + d n m ∆x ∈ C m with d n m = ρu n -ρ n m ρu n -ρu n and we move to Step 5. If the constraint (2c) is satisfied, we move to Step 6. 5) If 0 ≤ d n m ≤ 1, then

Fig. 4 .

 4 Fig. 4. Plots of the fuel consumption rate function K(v) given by (7) (left) and the corresponding total fuel consumption rate function F C(ρ) = ρK(v(ρ)) (right), given by the speed-density function (1).

Fig. 5 .

 5 Fig. 5. Schematic representation of the centralized control framework.

Fig. 6 .

 6 Fig. 6. Schematic representation of the decentralized control framework, where each CAV is treated separately. Its speed is optimized as if it were the only actuator on the road.

Fig. 7 .

 7 Fig. 7. Schematic representation of the quasi-decentralized control framework.

Fig. 8 .

 8 Fig. 8. TFC dependence on the speed ū ∈ [20, 140] km/h of a single CAV starting at y 0 ∈ [2, 50] km.

1 )

 1 Global optimization on the whole time interval [0, T f ]. Both the optimization horizon ∆T and the simulation interval ∆τ are set equal to T f = 1 hr. In this way, only one (global) optimization is performed, to compute the (constant) optimal control values u * = (u * 1 , . . . , u * N ).

Fig. 9 .

 9 Fig. 9. Top: TFC dependence on the initial position y 0 ∈ [2, 50] of a single CAV with desired speed ū = 55 km/h. Local minima are located at x = 5, 15, 25, 35, 45 km. Bottom: Initial traffic density ρ 0 and CAV initial positions (dashed lines indicate the location of TFC local minima).

Fig. 10 .

 10 Fig. 10. CAVs' distribution on the road at time t = 0.

Fig. 11 .

 11 Fig. 11. Spatio-temporal evolution of traffic density in the uncontrolled case.

Fig. 12 .

 12 Fig. 12. TFC values resulting form 1 hour global optimization, depending on the number of controlled vehicles: comparison between the different control strategies proposed in Section III.

Fig. 13 .

 13 Fig. 13. TFC values resulting form MPC optimization, depending on the number of controlled vehicles: comparison between the different control strategies proposed in Section III.

Fig. 14 .

 14 Fig. 14. Traffic density and CAVs trajectories for the quasi-decentralized strategy with a fleet of 10 CAVs and radius r = 11 km: 1 hour (top) and MPC (bottom) optimizations.

  Input initial traffic density ρ 0 , CAV positions y 0 and velocities u (0), inflow f in and outflow f out , time horizon T f , optimization time interval ∆T , simulation time interval ∆τ Result: CAV optimal velocities u * (k) while t k ≤ T f do for t ∈ [t k , t k + ∆T ] do

	• Solve	(10)	-	(2)	with	function
	bayesopt	for	optimal	solution
	u * bayes (k) = (u * 1,bayes (k) . . . , u * N,bayes (k)) • Taking as initial point u * bayes (k), solve (10)
	-(2) with function fmincon for optimal
	solution u * (k) = (u * 1 (k), . . . , u * N (k))
	end
	for t ∈ [t k , t k + ∆τ ] do
	Apply (2) with the optimal solution u ←
	u

C. Decentralized control

Decentralized control strategies aim at controlling a complex multivariable system by decomposing it into simpler Data: * (k), = 0, . . . , N end t k ← t k + ∆τ and update the traffic variables ρ(t k , •) and y (t k ), = 0, . . . , N end Algorithm 1: Centralized control algorithm subsystems. The latter are controlled by relying only on local information.

  ∆T , simulation time interval ∆τ Result: CAV optimal velocities u * (k) while t k ≤ T f do for = 1, . . . , N do Take as inputs ρ k (t k , •) and y (t k ) for t ∈ [t k , t k + ∆T ] do

	• Solve (10) -(2) with function bayesopt
	for optimal solution u * bayes (k) • Taking as initial point u * bayes (k), solve (10)
	-(2) with function fmincon for optimal
	solution u * (k)
	end
	end
	for t ∈ [t k , t k + ∆τ ] do
	Apply (2) with the optimal solution u ←
	u

* (k), = 0, . . . , N end t k ← t k + ∆τ and update the traffic variables ρ(t k , •) and y (t k ), = 0, . . . , N end Algorithm 2: Decentralized control algorithm knowing the positions and velocities of the other CAVs in a given radius r, acting as moving bottlenecks. More precisely, for any l = 1, . . . , N , let us consider all m ∈ {1, . . . , N }\{ } such that

TABLE I 1

 I HOUR OPTIMIZATION TFC VALUES AND GAINS FOR DIFFERENT CAV

FLEET SIZES, FOR THE THREE CONTROL STRATEGIES.

  .

		TFC with MPC TFC reduction
		[liters/h]	%
	Uncontrolled	2.7329 •10 4	0
	1 CAV -centralized	2.6933 •10 4	1.44
	1 CAV -decentralized	2.6934 •10 4	1.44
	1 CAV -quasi-decentralized	2.6978 •10 4	1.28
	5 CAVs -centralized	2.6285 •10 4	3.82
	5 CAVs -decentralized	2.6314 •10 4	3.71
	5 CAVs -quasi-decentralized	2.6320 •10 4	3.69
	10 CAVs -centralized	2.6431 •10 4	3.29
	10 CAVs -decentralized	2.6507 •10 4	3.01
	10 CAVs -quasi-decentralized 2.6346 •10 4	3.30

TABLE II MPC

 II TFC VALUES AND GAINS FOR DIFFERENT CAV FLEET SIZES, FOR THE THREE CONTROL STRATEGIES.

This work was partially supported by ERASMUS+/KA1 "NORTH SOUTH TRAINEESHIP" academic year 2020/