Antonella Ferrara

Gian Paolo Incremona

MEMBER, IEEE Eugeniu Birliba

FELLOW, IEEE, GIAN Paolo Incremona
email: gianpaolo.incremona@polimi.it

Paola Goatin

Multi

Keywords: triggered control, optimal control, platoon control, sliding mode control, traffic control

I. INTRODUCTION

T HE problem of traffic congestion and its impact on energy and environmental resources is becoming more and more critical, so that there is a strong commitment of researchers to find solutions oriented to promote a more sustainable mobility [START_REF] Goatin | transition & climate sustainability working groups[END_REF]. Traffic congestion phenomena are in fact the main cause of fuel waste, and of unsafe driving behaviours, which implies a significant cost in terms of resources consumption and direct fatalities [START_REF] Pasquale | Traffic control for freeway networks with sustainability-related objectives: Review and future challenges[END_REF]. As reported in [START_REF] Zhang | Air pollution and health risks due to vehicle traffic[END_REF], road traffic also considerably contributes to air pollution, with the related consequences in terms of health issues and indirect fatalities. As a matter of fact, an efficient control of road traffic systems can be beneficial to ensure regular traffic conditions, optimizing the throughput, and alleviating all the problems previously mentioned.

A. Brief Overview of Traffic Control

In the last decades, traffic control has been a widely investigated research topic, both for its applicative importance, and for its highly challenging theoretical issues. The recent advances in computational power and enabling technologies have fostered the development of control methodologies having as main objective that of mitigating traffic congestion. In this context, classical methods were mainly ramp metering, route guidance or variable speed limits (see [START_REF] Papageorgiou | Review of road traffic control strategies[END_REF]- [START_REF] Ferrara | An overview of traffic control schemes for freeway systems[END_REF]). These methods were based on the regulation of the vehicle flow entering the mainstream from on-ramps via traffic lights, on the suitable routing of vehicles and on the variation of speed limits via variable message signs (see [START_REF] Siri | Freeway traffic control: A survey[END_REF] for a recent detailed overview).

Moreover, traffic control strategies were traditionally based on simple feedback control solutions, as also demonstrated in field implementations [START_REF] Papageorgiou | ALINEA local ramp metering: Summary of field results[END_REF]. For instance, the socalled ALINEA strategy, aimed at keeping the traffic density close to the critical value locally, so as to make the traffic system exploit the maximum local road capacity, was one of the most applied and studied laws in the literature. However, the need to take into account constraints and specific control objectives paved the way to the application of more advanced optimal control strategies (see e.g., [START_REF] Alessandri | Optimal control of freeways via speed signalling and ramp metering1[END_REF]- [START_REF] Gomes | Optimal freeway ramp metering using the asymmetric cell transmission model[END_REF]).

As for the methodologies, the typical choice was model predictive control (MPC), in a standalone fashion or combined with other control strategies, in order to enhance robustness or to reduce the computational burden. In [START_REF] Haddad | Cooperative traffic control of a mixed network with two urban regions and a freeway[END_REF], for instance, MPC is adopted for traffic regulation through perimeter control and ramp metering in urban and freeway environments. In [START_REF] Luo | Real-time route diversion control in a model predictive control framework with multiple objectives: Traffic efficiency, emission reduction and fuel economy[END_REF], instead, the total time spent by vehicles is minimized while reducing fuel consumption. In [START_REF] Liu | Model predictive control for freeway networks based on multi-class traffic flow and emission models[END_REF], a MPC scheme with end-point penalties in the cost function, to account for the behaviour of the traffic system beyond the prediction horizon, is proposed. Another example is given in [START_REF] Van De Weg | Efficient freeway MPC by parameterization of alinea and a speedlimited area[END_REF], where the proposed MPC optimizes the upstream and downstream boundaries of a speed-limited area and the parameters of the ALINEA ramp metering strategy. While these works use as prediction model the so-called METANET model [START_REF] Ferrara | Second-order macroscopic traffic model[END_REF], other contributions rely on the popular cell transmission model (CTM), as for instance [START_REF] Ferrara | An event-triggered model predictive control scheme for freeway systems[END_REF], where a mixed-integer linear finite horizon optimal control problem (FHOCP) is formulated and solved. In alternative, as in [START_REF] Canudas-De Wit | A variable-length cell road traffic model: Application to ring road speed limit optimization[END_REF], a simple but accurate variable-length macroscopic traffic model (VLM) is adopted in [START_REF] De Wit | A variable-length cell transmission model for road traffic systems[END_REF]. In order to circumvent the implementation problems related to the possibly high computational burden of traffic control strategies, efficient solutions are provided by the adoption of event-triggered control logics. In such cases, the optimization is executed only when a specific triggering condition becomes active, see e.g., [START_REF] Ferrara | Event-triggered model predictive schemes for freeway traffic control[END_REF]- [START_REF] Pasquale | Hierarchical centralized/decentralized event-triggered control of multiclass traffic networks[END_REF].

In this framework, other recent solutions adopt decentralized or distributed hierarchical control strategies. Examples of such control architectures are reported for instance in [START_REF] Ghods | An efficient optimization approach to real-time coordinated and integrated freeway traffic control[END_REF], where a game-theoretic approach for the optimal coordination of ramp metering and variable speed limits is presented, and in [START_REF] Frejo | Global versus local MPC algorithms in freeway traffic control with ramp metering and variable speed limits[END_REF] and [START_REF] Ferrara | Freeways as systems of systems: A distributed model predictive control scheme[END_REF], where distributed and decentralized MPCs are designed. Another very recent example is introduced in [START_REF] Ferrara | A hierarchical MPC and sliding mode based two-level control for freeway traffic systems with partial demand information[END_REF], where the robust control of road traffic is pursued by means of a hierarchical MPC and sliding mode based two-level control scheme.

A highly innovative paradigm in traffic control, which has gained much interest in the last few years, and that has the potential to become the most suitable approach for traffic control in the future, is instead based on the exploitation of connected and automated vehicles (CAVs). These are highly automated vehicles that are expected to become increasingly common in vehicular traffic in the coming years. They are assumed to be immersed in the macroscopic freeway traffic, which is also composed of conventional (i.e., low automation) vehicles, and are controlled so as to produce a regularizing effect on the surrounding traffic flow by acting as "moving bottlenecks". For instance, in [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF], a multivariable MPC is proposed to control the length and the speed of a platoon of CAVs, with the aim of reducing the fuel consumption of the overall vehicular traffic. In [START_REF] Chalons | A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem[END_REF], macroscopic models accounting for the trajectory of moving bottlenecks have been proposed, as it is also done, in alternative ways, in [START_REF] Piacentini | VACS equipped vehicles for congestion dissipation in multi-class CTM framework[END_REF]- [START_REF] Piacentini | Highway traffic control with moving bottlenecks of connected and automated vehicles for travel time reduction[END_REF]. In [START_REF] Pasquale | A new micro-macro METANET model for platoon control in freeway traffic networks[END_REF], an original micromacro METANET model for platoon control in freeway traffic networks is formulated and discussed. In [START_REF] Piacentini | Traffic control via moving bottleneck of coordinated vehicles[END_REF], instead, a PDE-ODE model is adopted to describe the dynamics of moving bottlenecks constituted by CAVs, the velocity of which is suitably varied since it is regarded as the control variable. Experimental evidence has demonstrated the effectiveness of CAVs control in dissipating stop-and-go waves deriving from human driving behaviour, with benefits also in terms of emissions, as reported for instance in [START_REF] Stern | Dissipation of stopand-go waves via control of autonomous vehicles: Field experiments[END_REF].

All the previously cited works concerning CAVs are important because they marked the beginning of a totally new line of research in traffic control. Yet, in all the previous papers, the microscopic/mesoscopic dynamics of the platoon of CAVs is neglected in the model formulation. This fact is justifiable considering that the model formulated as a basis to design a control strategy must be, in general, extremely simple. Nevertheless, if it is true that a platoon of CAVs may have an impact on the surrounding traffic contributing to its regulation, it is also true, vice versa, that traffic density in the freeway stretch along which the platoon moves significantly influences the attainment of the optimal velocity and optimal length by the platoon. In other terms, it is not certain that the optimal speed and the optimal length determined at a certain time step by the controller that solves the optimization problem at high level are then achievable, in practice, by the platoon, since this latter must adapt its speed and intervehicular distances to the condition of the traffic in which it moves.

Moreover, it must be considered that there is always a certain degree of uncertainty in any dynamical model, and this is true also for CAVs. More precisely, there may be a mismatch between the individual vehicle expected (i.e., nominal) behaviour and the actual behaviour, so that, if the dynamics of the platoon of CAVs is neglected, the overall traffic model may result in being simple but not sufficiently accurate for control design. This is why, in order to realize a reliable and actually implementable CAV-based traffic control system, the dynamic behaviour of the platoon and that of the CAVs composing it must be explicitly modeled.

Further, just because of the unavoidable inaccuracy in the CAV model, it is advisable that robust control approaches are adopted to control the vehicles forming the platoons. In [START_REF] Zambelli | Robustified distributed model predictive control for coherence and energy efficiency-aware platooning[END_REF], for instance, a hierarchical MPC with sliding mode control (SMC) has been proposed to keep a safe distance between the vehicles of the platoon in spite of the modelling uncertainties affecting the vehicles description. A genuine sliding mode platoon control has been instead proposed in [START_REF] Incremona | Lateral vehicle dynamics control via sliding modes generation[END_REF]. SMC is indeed a powerful control methodology in the case of system models affected by uncertainties [START_REF] Ferrara | Advanced and Optimization Based Sliding Mode Control: Theory and Applications[END_REF], as the CAVs model considered in this paper. This control methodology relies on the design of a variable, called sliding variable, such that when this variable is steered to zero, the control objective is attained. The SMC law is typically a discontinuous signal which allows to make the sliding variable become zero in a finite time, enforcing the so-called "sliding mode". The controlled system in sliding mode results in being robust to matched uncertainties and, by virtue of the choice of the sliding variable, exhibits the desired dynamic behaviour [START_REF] Ferrara | Advanced and Optimization Based Sliding Mode Control: Theory and Applications[END_REF].

Another aspect to mention is that, while many works consider the traffic demand as known, this can determine the generation of erroneous theoretical conclusions since, in real world freeway traffic systems, the traffic demand is not available a priori, and only an estimate can be used in the prediction models upon which optimal control methods rely. Therefore, SMC is a good candidate methodology to enhance the robustness not only of the platooning control layer, but also of the overall traffic control scheme, as seen is [START_REF] Mammar | Coordinated ramp metering via second order sliding mode control[END_REF]- [START_REF] Piacentini | Freeway traffic control via second-order sliding modes generation[END_REF].

Motivated by the necessity of tackling the uncertainty in the CAVs dynamics and in the freeway traffic system demand, the goal of this paper is to propose a practically applicable robust optimal control architecture to address the problem of CAVs-based traffic control in realistic scenarios.

B. Contributions

Inspired by [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF] and [START_REF] Zambelli | Robustified distributed model predictive control for coherence and energy efficiency-aware platooning[END_REF], in this paper a novel hierarchical multi-level control scheme is proposed to reduce traffic congestion and fuel consumption in freeway traffic systems. The proposal consists of four key elements: a high-level MPC relying on the nominal model of the freeway traffic with a platoon of electric CAVs; an event-triggered logic to enable the high-level optimization only when it is actually needed; medium-level distributed MPCs, one for each CAV in the platoon, in order to track the reference values provided by the high-level control in an energy efficient way; finally, low-level integral sliding mode controllers, acting locally at the level of any single CAV, to enhance the control robustness.

The novelty of the proposal lies in the combination of three different levels of control, acting on a multi-scale traffic system where there is no a priori knowledge of the macroscopic traffic demand and of the modelling uncertainties and disturbances affecting the CAVs microscopic dynamics. Specifically, this paper aims at providing a control architecture suitable for practical implementation with the following features:

• It is capable of optimizing, via a high level MPC, a performance index related to both traffic congestion and fuel consumption of vehicles in a freeway stretch with a platoon of electric CAVs. The high level MPC is based on the nominal traffic model, but, differently from [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF], it is applied to a traffic system where the traffic demand is not a priori known and where the microscopic/mesoscopic dynamics of the CAVs and of the platoon they form is not neglected. • It is based on an event-triggered control logic, which activates the high level optimizer only when necessary. This contributes to make the proposal applicable in practice, since in spite of the large number of optimization variables involved in the control problem, the number of times the measurements are acquired and transmitted, and the number of times the optimization problem is solved are significantly reduced with respect to a non event-triggered implementation. • It also relies on distributed MPCs, applied to control, in an energy efficient way, an arbitrary number of CAVs during the platoon formation and maintenance. The platoon, differently from [START_REF] Zambelli | Robustified distributed model predictive control for coherence and energy efficiency-aware platooning[END_REF], is included in a traffic scenario, so that the effect of traffic local density on the platoon dynamics itself is correctly taken into account. • It includes fast local SMC actions, applied at the level of the individual CAV to compensate for the uncertainty affecting the vehicle model dynamics.

Note that the overall control scheme for freeway traffic in presence of platoons of CAVs, apart from being hierarchical and multi-level, is a multi-rate architecture, since the highlevel, the medium-level and the low-level control operate with different time rates. As previously mentioned, the highlevel control is asynchronous since it operates driven by the event-triggered control logic. The other two levels are instead time-triggered control loops. As such, they operate in a synchronous way.

C. Organization of the Paper

The paper is organized as follows. In Section II, the model of the freeway traffic with the platoon of CAVs is described, and the control problem is formulated. In Section III, the proposed multi-rate hierarchical multi-level control scheme is introduced and each control level is described in details. Simulations carried out in realistic scenarios are illustrated and discussed in Section IV, while some conclusions are drawn in Section V.

II. MODELLING AND PROBLEM FORMULATION

In this section, a macroscopic first-order model of the considered traffic dynamics with a platoon of CAVs is introduced.

Differently from [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF], a detailed kinematic description of the electric vehicles in the platoon is also introduced and its impact on the other vehicles in the neighbouring traffic is evaluated.

A. Freeway Traffic With the Platoon of CAVs

The model adopted to describe the traffic flow is the one presented in [START_REF] Piacentini | A macroscopic model for platooning in highway traffic[END_REF]. This latter represents an extension of the moving bottleneck model studied in [START_REF] Chalons | A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem[END_REF], where the

VOLUME , v d v u z u z d FIGURE 1
. Schematic representation of the platoon on a freeway with 2 lanes.

moving bottleneck was represented as a punctual capacity restriction, with no physical dimension. The platoon model, unlike the moving bottleneck, allows for the platoon to have a physical dimension that can vary in time. The present model is based on a suitably modified version of the traditional macroscopic Lighthill-Whitham-Richards (LWR) model [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]. The model is given by a partial differential equation (PDE) describing the traffic flow and two ordinary differential equations (ODEs), to track the trajectories of the initial and final points of the platoon, resulting in a strongly coupled PDE-ODE model. The platoon act as a moving bottleneck [START_REF] Piacentini | Traffic control via moving bottleneck of coordinated vehicles[END_REF], thus occupying some lanes and reducing the capacity of the freeway (see, e.g., Figure 1). Therefore, two discontinuities in the flow appear in correspondence of the front-end and back-end-points of the platoon and suitable numerical methods have to be applied to treat the flow discontinuities, as detailed in [START_REF] Piacentini | A macroscopic model for platooning in highway traffic[END_REF].

t k t k+1 z d (k) ρα ρ ∆t j d i d-1 i d i d+1 FIGURE 2
. Flux reconstruction at the downstream end-point of the platoon.

Consider Figure 2 illustrating the flux reconstruction at the downstream end-point of the platoon. Let ρ = ρ(x, t) be the density of cars corresponding to position x at time t, v = v(ρ) be the average speed of the flow of vehicles, while F = F (t, x, ρ) represents the flow function. The downstream and upstream end-points of the platoon are respectively denoted as z d = z d (t) and z u = z u (t), while i d = i d (t) and i u = i u (t) indicate the cells in which z d and z u are located. The part of freeway between the interval [z u , z d ] is then occupied by the platoon that reduces the road capacity. Then, flow discontinuities arise in correspondence of z d and z u .

The equations capturing the freeway traffic dynamics therefore are

∂ t ρ + ∂ x F (t, x, ρ) = 0, (t, x) ∈ R + × R, (1a) ρ(0, x) = ρ 0 (x), x ∈ R, (1b) żu (t) = v u (t, ρ(t, z u (t)+)), t ∈ R + , (1c) z u (0) = z 0 u , (1d) żd (t) = v d (t, ρ(t, z d (t)+)), t ∈ R + , ( 1e
) z d (0) = z 0 d , (1f)
where the flow function F is given by

F (t, x, ρ) := f (ρ) if x ̸ ∈ [z u (t), z d (t)], f α (ρ) := αf (ρ/α) if x ∈ [z u (t), z d (t)].
(2) Above, α ∈ [0, 1] is a parameter indicating the reduction in the capacity due to the presence of the platoon. The reduced flow is then indicated as f α . Considering the linear speeddensity relation

v(ρ) := V 1 - ρ ρ max , (3)
one achieves

f (ρ) := V ρ 1 - ρ ρ max , (4a)
f α (ρ) := V ρ 1 - ρ αρ max , (4b)
where V and ρ max are the maximum value of speed and density, respectively. An illustrative example of the trends of the normal flow f and the of reduced flow f α is depicted in Figure 3. Making reference to (1), equations (1c) and (1e) indicate the trajectories of the front-end and back-end-points of the platoon. Specifically, the speed of the initial point is given by

ρ f ,f α ρ max αρ max ρ α cr ρ cr
v d (t, ρ) := min {V d (t), v(ρ)} , (5)
where

V d (t) ∈ [0, V ]
is the maximal allowed velocity that z d can assume. As indicated in (5), the platoon cannot travel faster than the surrounding flow, then it cannot overcome cars and it has to adapt its speed to the downstream traffic velocity. On the other hand, for the back-end-point, the speed is given by

v u (t, ρ) := max {V u (t), -f α (ρ)/(ρ max -ρ)} , (6)
where

V u (t) ∈ [-V, V ]
is the maximal velocity for z u , while constraint ( 6) is derived from an admissibility condition on the feasibility of the corresponding Riemann problems, see [START_REF] Piacentini | A macroscopic model for platooning in highway traffic[END_REF]. Now, having in mind the design of a MPC in discrete time domain, we introduce the discrete time step k ∈ N + . The length of the platoon is then indicated with L(k) = z d (k) -z u (k), which can vary over time depending on the speed of the platoon end-points, i.e.,

L(k + 1) = L(k) + (v d (k) -v u (k))∆t, (7)
with ∆t being a suitably selected sampling time for the highlevel optimizer designed in the next section.

In our framework, the velocities v d (k) and v u (k) become the control variables, and in such a way the length of the platoon is also controlled. The platoon length depends both on the spacing between its vehicles and on the amount of vehicles joining it. As usual the space-discontinuous conservation law (1a) is solved by means of a discretization, in both time and space. The portion of freeway is therefore divided into N cells, indicated by the index i = 1, . . . , N , and of length ∆x. The choice of ∆t and ∆x must be compliant with the Courant-Friedrichs-Lewy (CFL) condition 2V ∆t ≤ ∆x. The interfaces of a generic cell i are instead indicated as x i-1/2 , x i+1/2 . At each time interval k the dynamics of the density in each cell i is updated as

ρ i (k + 1) = ρ i (k) + ∆t ∆x (F i-1 (k) -F i (k)) , (8)
where F i-1 is the flow between cell i -1 and cell i.

The flow F is obtained by applying the standard supplydemand paradigm [START_REF] Daganzo | The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory[END_REF] in all the cells that are not interested by the presence of the initial and final points of the platoon, while the reduced flow f α is considered for the cells inside the platoon. Specifically, the nominal demand and supply of a cell i are given by

D i (k) := f (ρ i (k)) if ρ i (k) < ρ cr , f max if ρ i (k) ≥ ρ cr , (9a)
S i (k) := f max if ρ i (k) < ρ cr , f (ρ i (k)) if ρ i (k) ≥ ρ cr , ( 9b
)
for i < i u or i > i d , i.e.
, where the platoon is not present. Above, ρ cr = 0.5ρ max denotes the critical density, while

f max = f (ρ max )
is the maximal flux. Similarly, for i u < i < i d , where the platoon is instead present, the demand and supply are computed considering the reduced flux

D α i (k) := f α (ρ i (k)) if ρ i (k) < ρ α cr , f max α if ρ i (k) ≥ ρ α cr , (10a)
S α i (k) := f max α if ρ i (k) < ρ α cr , f α (ρ i (k)) if ρ i (k) ≥ ρ α cr , (10b)
where ρ α cr = 0.5αρ max and f max α = f α (ρ α cr ). Note that, differently from [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF] and in analogy with [START_REF] Ferrara | A hierarchical MPC and sliding mode based two-level control for freeway traffic systems with partial demand information[END_REF], in this work we assume to have only partial information on traffic demand. This is realistic since in freeway traffic systems, the mainstream inflow is not a priori known, and only a nominal value estimate can be usually retrieved from historical data.

Apart from cells i d and i u , the flow between two generic cells i -1 and i is obtained as the minimum between the demand of the upstream cell i -1 and the supply of the following cell i, that is

F i (k) := min{D i-1 (k), S i (k)}for i ≤ i u -2, i ≥ i d + 1, min D α i-1 (k), S α i (k) for i ≥ i u + 1, i ≤ i d -2.
In cells i d and i u , where the end-points of the platoon lie, discontinuities in the flow function arise and the solution to the scalar conservation law with moving discontinuities needs a special treatment. In [START_REF] Piacentini | A macroscopic model for platooning in highway traffic[END_REF], the numerical scheme able to precisely capture the discontinuities has been developed by applying a conservative reconstruction strategy. Let us consider the discontinuities moving with speeds v u (k) and v d (k) for the upstream and downstream discontinuities respectively, so that the positions of the end-points of the platoon are updated as

z d (k + 1) = z d (k) + min{V d (k), v(ρ i d +1 (k)}∆t, (11a) z u (k + 1) = z u (k) + V u (k)∆t. ( 11b
)
Now, let us start from the discontinuity in the cell i d (k) where the downstream point z d (k) is located at time step k. A discontinuity in the density is expected to appear in cell i d (k). Then, at each time step k, we take into consideration the Riemann-type initial datum ρ l = ρ i d -1 (k) and ρ r = ρ i d +1 (k), expressing the density discontinuity, and we solve the Riemann problem as detailed in [START_REF] Piacentini | A macroscopic model for platooning in highway traffic[END_REF].

The obtained solutions, denoted as ρα and ρ, correspond to the values of density that are expected to appear in cell i d upstream and downstream the position z d (k) at the following time step. At this point, we replace the value of the density ρ i d (k) with a convex combination of the two values ρα and ρ, by defining d i d as

ρα d i d + ρ (1 -d i d ) = ρ i d (k), i.e., d i d = ρ i d (k) - ρ ρα - ρ .
The reconstructed discontinuity is then located at position xi d = x i d -1/2 + d i d ∆x, and the numerical flux at the interface x i d +1/2 can then be reconstructed as [START_REF] Hegyi | Model predictive control for optimal coordination of ramp metering and variable speed limits[END_REF] with

∆tF i d (k) = min ∆t i d , ∆t f (ρ) + max ∆t -∆t i d , 0 f α (ρ α ),
∆t i d = 1 -d i d min{V d (k), v(ρ i d +1 (k))} ∆x
being the time for the discontinuity to reach the position x i d +1/2 where the interface between the two cells lies (see Figure 2). We then set

F i d-1 (k) = min D α (ρ i d-1 (k)), S α (ρ α (k)) . ( 13
)
At this point, the same approach is used at the back-end interface. By applying the same reconstruction strategy, the flux at the interface between cell i u , i u+1 is also obtained,

VOLUME ,
and, together with the flux at the other discontinuity reconstructed in [START_REF] Hegyi | Model predictive control for optimal coordination of ramp metering and variable speed limits[END_REF], and with the classical flows defined above, all the fluxes are computed and the state can be updated.

B. The Platoon of CAVs

The previous traffic model is now further detailed adding a model of the electric vehicles (EVs) constituting the considered platoon of CAVs. Assume to have n EVs and the distance between two vehicles is denoted as d ι , ι = 2, . . . , n, with ι = 1 being the index of the leader (see Figure 4 for an example). By adopting a second-order kinematic model, the longitudinal position of each vehicle can be described along the 1-dimensional space coordinate p ι as

d 3 d 2 p ι EV 3 EV 2 EV 1
ṗι (t) = v ι (t) m vι (t) = u ι (t) -F lossι (v ι (t)) + δ ι (t), (14)
with m the vehicle mass and F lossι (v ι (t)) a term encompassing the aerodynamic drag and the rolling resistance forces, described as

F lossι (v ι (t)) = 1 2 ϱc a Av 2 ι (t) + c r mg, (15)
where ϱ is the air density, c a the aerodynamic coefficient, A is the equivalent vehicle surface, while c r and g are the rolling resistance coefficient and the gravitational acceleration, respectively. The term δ ι (t) is introduced to describe the (matched) uncertainty possibly affecting the vehicle dynamics, including for instance unmodeled dynamics in the actuators as well as mismatches in the description of the losses captured by [START_REF] Luo | Real-time route diversion control in a model predictive control framework with multiple objectives: Traffic efficiency, emission reduction and fuel economy[END_REF]. Due to the adoption of EVs as reference vehicles, the input force u ι (t) can be seen as the result of two additive components, i.e.,

u ι (t) = u 1ι (t) + u 2ι (t), u ι (t) ∈ [F min , F max ] , ( 16
)
where u 1ι (t) is the electric force component, positive during acceleration and negative during deceleration (regenerative braking), u 2ι (t) ≤ 0 is the mechanical braking force, while F min < 0 and F max > 0 are the force bounds. Being the torques on the wheels the actually controlled signals, the assumption that the total force is obtained as the sum of the torques (namely, T inµι ) on each wheel is made, i.e.,

u ι (t) = µ∈M T inµι (t) R e , ( 17
)
where M is the set of electrically driven wheels (dependent on the power-train configuration), and R e is their effective radius. The relationship [START_REF] Van De Weg | Efficient freeway MPC by parameterization of alinea and a speedlimited area[END_REF] directly translates into the assumption that ωµι (t) ≃ 0, given for instance through the adoption of a fast-enough low-lever slip control scheme (see [START_REF] Regolin | Lateral vehicle dynamics control via sliding modes generation[END_REF]) and a proper torque vectoring strategy. Finally, to include electric energy spare as a secondary objective, a simple electric power flow model is here adopted, that is

P mι (t) = µ∈M T inµι (t)ω µι (t)η -sign(T inµι (t)) µ . (18)
In [START_REF] Ferrara | Second-order macroscopic traffic model[END_REF] the wheels rotational speeds ω µι (t) can be well approximated as

ω µι (t) = g r R eµ v ι (t), (19)
where the gear ratio g r is supposed here constant since the description is referred to EVs. The sign(•) term at the exponent in expression [START_REF] Ferrara | Second-order macroscopic traffic model[END_REF] allows to describe the effects of regenerative braking, in which the power flows "backward" from the wheels to the engine. Finally, the total power entering (or exiting) the battery pack is

P bι (t) = P mι (t)η -sign(uι(t)) b , (20)
with η b being the mechanical-chemical energy conversion efficiency.

C. The Control Problem

We are now in a position to introduce the traffic control problem to solve. It consists in minimizing the traffic congestion and the fuel consumption of the overall traffic flow, by acting on the maximum speed of the downstream and upstream endpoints of the platoon, which plays the role of the actuator of the considered traffic system. Moreover, assuming that each of the vehicles in the platoon is independently controlled and a short-distance Vehicle-to-Vehicle (V2V) communication is feasible, the objective is also that of enabling a consensus behaviour of all the CAVs, considering a safe reciprocal distance, and taking into account energy efficiency. These goals have to be reached in spite of both unavoidable modelling uncertainties on the vehicle dynamics, and in presence of uncertain freeway demand, which affects the traffic flow surrounding the platoon of CAVs.

III. THE PROPOSED THREE-LEVEL CONTROL SCHEME

In this paper the control problem presented in the previous section is solved by proposing the hierarchical three-level control scheme depicted in Figure 5. It consists of four key blocks: the high-level optimizer, the event-triggered block, the medium-level MPCs, and the local low-level SMCs. More precisely, the high-level optimizer generates, at each triggering event, the desired platoon length and the corresponding reference speed values of downstream and upstream end-points, thus providing them to the mediumlevel MPCs. The optimization is indeed determined by the event-triggered block on the basis of a specific condition on the traffic density, which is assumed measurable in correspondence of the cell crossed by the leading vehicle. Then, the distributed MPCs on the vehicles are aimed at generating the control action to achieve a consensus behaviour (that is following the preceding vehicle while keeping a prespecified safe distance), and to minimize the electric energy consumption. In the following, all these blocks will be discussed in detail.

A. High-Level Traffic Control

From the traffic point of view, the aim of the high-level optimizer is to minimize the congestion and the fuel consumption of the overall traffic flow by controlling the platoon. Specifically, V d and V u , i.e., the speeds of the downstream and upstream end-points of the platoon, are the control variables, and this is equivalent to control the speed and the length of the platoon itself by virtue of [START_REF] Siri | Freeway traffic control: A survey[END_REF]. The fuel consumption is computed by means of the model presented in [START_REF] Ramadan | Traffic flow control and fuel consumption reduction via moving bottlenecks[END_REF], already adopted in [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF]. The model is based on the fact that vehicle fuel consumption is related to the average velocity of vehicles, and high consumption is related to very low or very high speed. Following [START_REF] Ramadan | Traffic flow control and fuel consumption reduction via moving bottlenecks[END_REF], a curve obtained by averaging the commercial fuel consumption characteristics of different vehicles is computed and approximated by the sixth order polynomial

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, (21)
with K(v) being expressed in L h -1 and v being the traffic speed expressed in km h -1 . Since in the LWR model the velocity is a function of the density, it is possible to derive the trend of the fuel consumption as a function of the traffic density, specifically here a linear relationship is used. The polynomial (21) can be then re-parametrized in terms of the density ρ, to get the function K(ρ) = K(v(ρ)), that is the consumption rate of one car as a function of the traffic density at the vehicle position. At this point, the Total Fuel Consumption (TFC), denoted by F, is given as function of the density as

F(ρ) = ρK(ρ). (22)
Since we also aim at reducing traffic congestion, define ρ ∈ R N as the desired density of cars for the considered freeway segment.

The considered control problem is then solved by means of a MPC approach, which generates the high-level control inputs u = [V d , V u ] as solution of a FHOCP. Let N p be the number of time steps of the high-level prediction horizon, so that at a fixed time step k, given the current initial state ρ(k), the optimal control sequences [START_REF]Design of networked freeway traffic controllers based on event-triggered control concepts[END_REF] subject to the model dynamics [START_REF] Goatin | transition & climate sustainability working groups[END_REF], to ( 5), [START_REF] Ferrara | An overview of traffic control schemes for freeway systems[END_REF], and to the following constraints

u o (h) = [V o d (h), V o u (h)], h = k, . . . , k + N p ,

are computed by minimizing the objective function

min u k+N p h=k N i=1 γ F F(ρ i (h))∆x∆t + γ ρ ∥ρ i (h) -ρi ∥ ,
L min ≤ L(h) ≤ L max , ( 24a
) V min d ≤ V d (h) ≤ V, ( 24b
) |V d (h) -V u (h)| ≤ c, (24c)
where L min , L max and V min d are the positive bounds on the platoon length and downstream velocity, while c is a positive bound on the velocity difference between the upstream and downstream velocities. Finally, the scalars γ F and γ ρ are instead positive weights, suitably selected by the designer in order to prioritize the minimization of TFC or the density error.

B. Event-Triggered Control Logic

Because of the high number of optimization variables, and in order to avoid packet losses or delays in the communication network, an event-triggered logic is introduced in the scheme. Let {t j } j∈N + be the sequence of the triggering instants, generated such that

t j+1 := inf{t > t j | T (ρ(t)) ≥ 0 ∨ t = t j + N p ∆t}, ( 25
)
where T is a suitably chosen event function. This is selected by taking into account the density of the cell associated to the leading vehicle in the platoon at the current time instant t. More precisely, letting ρ ℓ (t) be the density of the cell VOLUME , associated to the leader position, the event function is defined as

T (ρ(t)) = ρ ℓ (t) -ε ρ (26)
with ε ρ > 0 being a design threshold parameter. Therefore, whenever a new event occurs, a flag equal to 1 is sent to the high-level controller, thus closing the control loop, in order to enable a new optimization given the updated initial conditions

ρ 0 = ρ(t j ), (27a)
z 0 u = z u (t j ) = p n (t j ) -ℓ CAV 2 , ( 27b
)
z 0 d = z d (t j ) = p 1 (t j ) + ℓ CAV 2 , (27c)
with p 1 and p n being the actual position of the leading vehicle (ι = 1) and of the last one (ι = n), respectively, and ℓ CAV being the vehicle length. If no events occur, the second condition in [START_REF] Ghods | An efficient optimization approach to real-time coordinated and integrated freeway traffic control[END_REF] means that a new event happens whenever the optimal reference sequences, previously generated by solving the FHOCP at t = t j , have been completely used by the medium-level, that is after N p time steps from the last optimization.

C. Medium-Level Platoon Control

In order to design the medium-level controller, introduce the following nominal model for each vehicle ι (i.e., consider δ ι (t) = 0) relying on [START_REF] Haddad | Cooperative traffic control of a mixed network with two urban regions and a freeway[END_REF], that is

     ḋι (t) = vι-1 (t) -vι (t) m vι-1 (t) = u ι-1 (t) -F lossι-1 (v ι-1 (t)) m vι (t) = u ι (t) -F lossι (v ι (t)) = fv (v ι (t), u ι (t)), (28)
where dι (t) is the nominal inter-vehicle distance, defined using nominal positions as dι (t) = pι-1 (t) -pι (t) and available for measurement (supposedly with ideal precision) through the range sensors mounted at the front of the ιth vehicle. Similarly, the velocity v ι (t) can be measured by assumption, and hence v ι-1 (t) can be made available to vehicle ι by vehicle ι-1, by virtue of the V2V communication infrastructure.

Let κ be the medium-level sampling time instant, sampled with medium-level period ∆t (thus, corresponding to the selected medium-level MPC discretization time). Vehicle ι computes an optimal control sequence

u o ι (ℏ) = [u o 1ι (ℏ), u o 2ι (ℏ)], ℏ = κ, .
. . , κ + N p , with N p being the medium-level prediction horizon, by solving a specific FHOCP, detailed below.

Before detailing the expression of the cost function, it is needed to introduce the safety policy given by the so-called Constant Time Headway (CTH) approach, which has been proven to be suitable for the case of locally communicating vehicles [START_REF] Swaroop | A review of constant time headway policy for automatic vehicle following[END_REF]. Given the reference values provided by the high-level MPC, that is the desired platoon length L and the velocities v d for vehicle 1 (i.e., the leader) and v u for vehicle n, formally, the required distance d i (t) at any time instant corresponds to

d CTH ι (t) = d min + K L e L + t CTH v ι (t), (29)
where d min > 0 is a constant minimum (safety) distance, arbitrarily designed prior to the establishment of the platoon. Furthermore, in [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF], differently from [START_REF] Zambelli | Robustified distributed model predictive control for coherence and energy efficiency-aware platooning[END_REF], e L = L -p 1 + p n -ℓ CAV is a term to take into account the reference length provided by the high-level, and K L > 0 is a gain suitably selected by the designer. Hence, the distributed MPC controller of the ιth vehicle solves a FHOCP as

min u ι α dι c dι + α eι c eι + α bι c bι , (30)
where α dι , α eι and α bι are positive weights selected by the designer. As for the coefficient α bι , it should always be selected large enough so that the use of mechanical brake is discouraged unless inevitable, while suitably enabling a regenerative braking. Moreover, one has

c dι = κ+N p ℏ=κ+1 ( dι (ℏ) -dCTH ι (ℏ)) 2 , (31)
while

c eι = κ+N p ℏ=κ+1 P mι (ℏ), (32)
is the net energy outcome from usage and regeneration during the prediction horizon, and

c bι = κ+N p ℏ=κ+1 F 2 bι (ℏ) = κ+N p ℏ=κ+1 u 2 2ι (ℏ) (33)
is the cost associated with the mechanical braking force, under the constraints

F min + ϵ ≤ u 1ι (ℏ) + u 2ι (ℏ) ≤ F max -ϵ, ( 34a
) 0 ≤v ι (ℏ) ≤ v max , (34b)
d min ≤ dι (ℏ) ≤ d max , (34c)
where ϵ > 0 is a constant which will be specified in the following. Finally, following the receding horizon principle, only the first computed input is taken for the subsequent interval ∆t, giving rise to a piece-wise constant input signal. All the remaining optimal control sequence is instead sent to vehicle ι + 1 which predicts the sequences

{v ι (κ + 1), • • • , vι (κ+1+N p )} and { dι (κ+1), • • • , dι (κ+1+N p )}
at time instant κ + 1 applying the received sequence to the stored nominal model of vehicle ι. Note that the physical parameters of vehicle ι-1 are sent to vehicle ι at connection time, i.e., when vehicle ι approaches the last vehicle of the platoon to join it.

D. Low-Level Vehicle Control

As already pointed out, the ιth optimal control input u o ι (ℏ) is obtained relying on the nominal model [START_REF] Ferrara | A hierarchical MPC and sliding mode based two-level control for freeway traffic systems with partial demand information[END_REF]. To overcome the consequent performance and possibly stability issues, a fast local robust controller is introduced to compensate for the matched uncertainty δ ι so that the real system actually behaves as the nominal one (thus, optimally with respect to the distributed MPC strategy).

In particular, a continuous second-order suboptimal sliding mode control approach originated in [START_REF] Bartolini | Chattering avoidance by second-order sliding mode control[END_REF] is locally enforced. This choice is possible by suitably selecting the so-called sliding variable so as to have a system to control of relative degree 1. This has beneficial effects in terms of chattering alleviation since an additional integrator is included, thus making the input actually fed into the plant continuous. Moreover, in order to achieve robustness in front of matched uncertainties from the initial time instant, an integral sliding mode (ISM) technique is adopted in combination with the suboptimal algorithm.

Since the action of the ISM correction must in practice be exerted at discrete time instants, the quantity τ ≪ ∆t is defined as the ISM sampling time. The computation of the corrective control terms is then performed at discrete steps κ, each of length τ , where velocity measurements are acquired, while the total control u ι (κ) has to be applied and held for the subsequent τ seconds.

At this point, before defining the ISM law, for every vehicle ι, we assume that there exists a positive constant

∆ such that |δ ι (t)| ≤ ∆, ∀t ≥ 0. (35)
The local sliding variable is then defined as

σ ι (κ) = v ι (κ) -vι (κ), (36)
where vι (κ) is the nominal velocity of vehicle ι, computed, relying on (28), as

vι (κ) = v ι (κ∆t) + κ∆t+κτ κ∆t fv (v ι (t), u o ι (κ)) dt, (37)
given the most recent previous MPC sampling time instant. Then, the total control input to be used in the κth time step is

u ι (κ) = (u o 1ι (κ) + u ISMι (κ)) + u o 2ι (κ) = ũι (κ) + u ISMι (κ) (38)
while ( 28) is left evolving under the nominal input

ũι (κ) = u o 1ι (κ) + u o 2ι (κ), ∀κ : κ∆t ≤ κτ < (κ + 1)∆t. (39)
The suboptimal second-order sliding mode control law (see [START_REF] Bartolini | Chattering avoidance by second-order sliding mode control[END_REF] for further details) is designed as

u ISMι (κ) = u ISMι (κ∆t) - κ∆t+κτ κ∆t ϵ sign σ ι (t) -σι 2 dt, (40)
with σι being the extremal values of the sliding variable σ ι .

The proof of convergence for the adopted ISM robustification strategy directly follows from [START_REF] Zambelli | Robustified distributed model predictive control for coherence and energy efficiency-aware platooning[END_REF]Prop. 1]. Since during the sliding phase the condition u ISMι (t) ≃ -δ ι (t) holds (see [START_REF] Ferrara | Advanced and Optimization Based Sliding Mode Control: Theory and Applications[END_REF]Chs. 1,[START_REF] Pasquale | Traffic control for freeway networks with sustainability-related objectives: Review and future challenges[END_REF]), exploiting [START_REF] Incremona | Lateral vehicle dynamics control via sliding modes generation[END_REF] and [START_REF] Piacentini | Traffic control via moving bottleneck of coordinated vehicles[END_REF], it directly follows that the constant ϵ in (34) must be such that ϵ ≫ ∆.

IV. CASE STUDY

In this section, the proposed control scheme is assessed in simulation considering realistic scenarios with a stretch of freeway of 50 km divided in N = 250 cells, each of them with length equal to 200 m. Moreover, a platoon of n = 20 CAVs is present in the considered freeway segment.

A. Simulation Settings and Scenarios

Since in this paper a multi-rate control scheme is proposed, the integration sampling time is chosen as τ = 0.01 s, while the medium-level and high-level MPC discretization times are ∆t = 1 s and ∆t = 5 s, respectively. This means that the model used to describe the process (i.e., the actual traffic system) is discretized in time with τ ≪ ∆t < ∆t. The total simulation window is 1 h.

Relying on the quadratic fundamental diagram expressed by (4a), the maximum speed is V = 140 km h -1 , while, given a segment with 3 lanes, the maximum density is ρ max = 400 veh km -1 . The initial density value is set as ρ(0) = 0.3ρ max 1 N . Furthermore, the initial length of the platoon is set equal to 1.42 km, at z d (0) = 4 km, and the initial vehicle velocity is 70 km h -1 . The maximum nominal flux is chosen equal to f max = 14000 veh h -1 . Note that, as discussed in Section II, the traffic demand is only partially known a priori, that is the real traffic demand is equal to an estimate from historical data plus a random unpredictable disturbance, whose amplitude is a percentage of the nominal demand.

The high-level MPC is designed as in Section III, with weights γ F = 10 and γ ρ = 1000 in order to prioritize the traffic congestion reduction. Moreover, the high-level prediction horizon is chosen as N p = 80. The constraint in [START_REF] Pasquale | Hierarchical centralized/decentralized event-triggered control of multiclass traffic networks[END_REF] are instead such that L min = 0.5 km, L max = 10 km, V min d = 40 km h -1 and c = 30 km h -1 . The event-triggered threshold is selected as ε ρ = 180 veh km -1 .

As for the platoon of CAVs, in analogy with [START_REF] Zambelli | Robustified distributed model predictive control for coherence and energy efficiency-aware platooning[END_REF], in order to simulate the mismatches between the models and the real systems, or possible external forces arising from actuation lags, a random uncertainty is injected in the vehicle dynamics, i.e.,

δ ι (t) = β 1ι + β 2ι sin 1 β3 ι t + β 4ι + r ι (t) , (41)
where r ι (t) ∈ [0, 0.25] is an uniform random variable,

β 1ι ∈ [-150, 450], β 2ι ∈ [0, 450], β 3ι ∈ [1, 11], β 4ι ∈ [0, 5],
so that ∆ = 900 in [START_REF] Piacentini | Traffic control via moving bottleneck of coordinated vehicles[END_REF]. The other vehicle parameters are instead reported in Table 1. The medium-level MPC is de- signed according to [START_REF] Chalons | A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem[END_REF], with prediction horizon N p = 10.

In the CTH policy [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF], the safe distance is selected as d min = 15 m, t CTH = 1 s, while the other constraints are the maximum distance d max = 120 m and the maximum VOLUME , speed v max = 140 km h -1 . The ISM control gain is instead selected as ϵ = 1000, while the input saturations are F min = -6500 N and F max = 6500 N.

The following two control scenarios are therefore simulated and analysed: S1) abrupt narrowing of the freeway segment: in this scenario, the traffic congestion is abruptly affected by a narrowing of the freeway segment at location 50 km (for instance due to a possible accident) for a time interval of 0.6 h. In order to simulate this condition the incoming flow is kept equal to f max , while the outcoming flow is reduced to 0.3f max . Then, after the considered interval, it becomes again equal to f max . S2) continuous increasing of the traffic congestion: in this scenario, the traffic congestion is continuously increased for all the simulation window by considering a reduction of the outcoming flow to 0.5f max at location 50 km.

All the simulations have been executed using the MATLAB tool for NLP CasADi, with the primal-dual interior point solver IPOPT.

B. Results and Discussion

The results obtained by applying the proposed hierarchical three-level control scheme are hereafter illustrated for the scenario S1. The proposal is then applied also to the scenario S2 to further assess its performance in a different traffic condition with respect to the uncontrolled cases. The amplitude of the considered stochastic disturbance with zero mean is realistically considered equal to 5% of the nominal demand.

The evolution of the density with respect to time and space in scenario S1 is illustrated in Figure 6, where the red lines indicate the trajectory of the platoon. Specifically, Figure 6(a) shows the evolution in the uncontrolled case. Figure 6(b) reports instead the density evolution when the proposed hierarchical three-level control scheme is applied. It is evident that congestion is reduced, thus demonstrating the effectiveness of the proposal even in presence of the uncertainty terms affecting the traffic system and the vehicles in the platoon. One can observe that in the uncontrolled case, cars vary their speed from the maximum one to the minimum speed due to the reaching of the bottleneck, thus implying the worst situation in terms of fuel consumption. In the controlled case, the length and speed of the platoon are instead modified so that slowing down the CAVs allows to avoid abrupt brakes in presence of the bottleneck and, as a consequence, this is beneficial also for the surrounding traffic flow, which smooths its velocity and reduces fuel consumption.

Figures 7 and8 show in the detail the behaviour of the actual platoon in the same scenario, under the control of the medium-level distributed MPCs and low-level ISM controllers. More precisely, Figure 7 to harmonize towards the one of the leader. Then, the platoon cruises and a consensus is achieved, in spite of the uncertainties affecting the overall system and with benefits in terms of energy consumption. The corresponding vehicles inputs are illustrated in Figure 7(b). The distances among the vehicles are reported in Figure 8(a). As expected, they are very similar, thus further confirming the effectiveness of the ISM controllers which enhance the robustness features of the overall control scheme. Finally, Figure 8 To conclude, the comparison in terms of TFC and computational burden achieved with the proposal applied in scenarios S1 and S2 with respect to the uncontrolled cases is shown in Table 2. Specifically, three different subcases for scenario S1 are considered depending on the percentage level of demand uncertainty with respect to the nominal value, namely ∆D % ∈ {0, 5, 10}. The TFC in the uncontrolled and controlled cases are indicated as TFC uc and TFC c , respectively. Its percentage variation with respect to the uncontrolled case (namely ∆TFC % ) is also computed together with the number of triggering events (#ET). Notice that, for instance in the case of no demand variation (that is in the nominal case in which the demand is known a priori), the proposed approach provides a 2.9% improvement with respect to the uncontrolled case, which is consistent with the results in [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF], where a 2.63% improvement is achieved in case of a purely nominal condition of the traffic system. This is a success considering that the model used in the present paper is definitely more realistic than that used in [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF], since it includes a lot a dynamical effects previously neglected in other papers dealing with CAVs. Because of this additional realism, we could expect a significant performance degradation. Instead, we achieve practically the same performance obtained in the pure nominal case in [START_REF] Piacentini | Traffic control via platoons of intelligent vehicles for saving fuel consumption in freeway systems[END_REF], of course by designing a more complex controller. Specifically, it is worth highlighting that the proposed approach allows to achieve the 2.3% improvement when a 5% variation of the demand uncertainty is present. In the same uncertain condition, but in case of scenario S2, the percentage improvement is 3.4%. Increasing the uncertainty level to 10%, the improvement percentage in scenario S1 reduces instead to 1.6%. Nevertheless, notice that in all the considered cases the uncertainty affecting the dynamics of the platooning CAVs is always present, which makes the overall traffic system further complex and realistic. Finally, it should be noticed that the number of triggering events enabling the high-level MPC re-optimization is very limited with respect to the time-triggered solution (that would require 720 optimizations in 1 h). For instance in the worst case (∆D % = 10%) only 227 events occurs, which corresponds to a reduction of the 68% of the number of high-level optimizations. Overall, it is important to notice that the improvement of the performance of the proposed scheme is obtained by slightly increasing the complexity of the architecture. Specifically, the computational and implementation cost of adding an ET block and the low-level SMC components is minor.

VOLUME ,

C. Limitations and Extensions

The control approach proposed in this paper will be further improved and extended in future works. In particular, the dynamic description of the phenomena involved will be enriched. For instance, it could be appropriate to include the dynamics of the state of charge of the batteries of the CAVs [START_REF] Pozzi | Balancingaware charging strategy for series-connected lithium-ion cells: A nonlinear model predictive control approach[END_REF], investigate the use of different fuel consumption models (e.g., [START_REF] Typaldos | Minimization of fuel consumption for vehicle trajectories[END_REF], [START_REF] Incremona | Design of a switching nonlinear MPC for emission aware ecodriving[END_REF]), and add the description of the pollutant emissions of the traditional vehicles that make up the macroscopic traffic (see for instance [START_REF] Pasquale | Supervisory multiclass event-triggered control for congestion and emissions reduction in freeways[END_REF] and [START_REF] Bandeira | Potential pollutant emission effects of connected and automated vehicles in a mixed traffic flow context for different road types[END_REF]).

As for the implementation point of view, one could enhance the computational aspects related to the MPC components of the control scheme, by using methods which are particularly suitable for the on-line implementation in traffic systems (e.g., fast MPC [START_REF] Lin | Fast model predictive control for urban road networks via milp[END_REF], plug-and-play MPC [START_REF] Hu | Plug and play distributed model predictive control for heavy duty vehicle platooning and interaction with passenger vehicles[END_REF], or regression-tree based MPC [START_REF] Nai Oleari | A model predictive control scheme for freeway traffic systems based on the classification and regression trees methodology[END_REF]).

Finally, it is certainly worth studying how a multi-level approach of the type proposed in this paper can be combined with the more traditional traffic control methods based on ramp metering, variable speed limits and routing, giving rise to a coordinated traffic control system where CAVs suitably cooperate with more conventional control actuators.

V. CONCLUSIONS

In this paper a hierarchical three-level control scheme has been proposed for a freeway traffic system in presence of a platoon of CAVs. The main objective is to reduce the traffic congestion, while minimizing the fuel consumption of the overall traffic, by using the platoon of CAVs as actuator, which generates a bottleneck moving along the road stretch. The length and the velocity of the moving bottleneck are suitably varied by controlling the electric vehicles constituting the platoon in an energy efficient way, so as to track the reference signals provided by the highlevel optimizer. The presence of uncertainties in the CAVs model is dealt with by introducing low-level sliding mode controllers of integral type. They enhance the robustness of the control scheme, which has anyway to operate under the assumption that the actual traffic demand is not a priori known. Due to the high number of optimization variables, the computational load of a three-level control scheme as the one proposed in this paper might be significant. To overcome this drawback, an event-triggered mechanism is designed to reduce the number of state transmissions and optimizations which need to be executed by the high-level controller. The simulation campaign, carried out in realistic scenarios with a priori unknown traffic demand, has shown the effectiveness of the proposal. It appears to be capable of providing an improvement of traffic congestion, while complying with the fuel consumption reduction objective, with performances comparable with those seen in the literature in case of known traffic demand and perfectly known CAVs dynamics.

FIGURE 3 .

 3 FIGURE 3. Example of normal flow f and reduced flow f α .

FIGURE 4 .

 4 FIGURE 4. Platoon of electric CAVs with a leader and 2 followers.

  (a) shows that during the transient the velocities of the following vehicles tend

FIGURE 6 .

 6 FIGURE 6. Density evolution for scenario S1. (a) case with no control. (b) case with the proposed hierarchical three-level control scheme.

  (b) shows the length of the platoon with respect to the reference value provided by the high-level MPC. Moreover, in the same figure, the flag signals indicating the triggering events, whenever a new high-level optimization is enabled, are reported.

FIGURE 7 .

 7 FIGURE 7. Platoon of CAVs in the scenario S1. (a) velocity profiles, vι(t). (b) input profiles, uι(t).

  platoon length and ET flags

FIGURE 8 .

 8 FIGURE 8. Platoon of CAVs in the scenario S1. (a) distances between vehicles, dι(t). (b) platoon length, its reference L and ET flags.

TABLE 1 .

 1 CAVs parameters.

	m	1900 kg
	ℓCAV	4.69 m
	caA	0.8 m 2
	cr	0.008
	gr	10
	ϱ	1.22 kg m -3
	ηµ	0.64
	ηb	0.95

TABLE 2 . Performance indexes.

 2

	Scenario	∆D %	TFCuc [L]	TFCc [L]	∆TFC %	#ET
		0	33013	32057	2.9	194
	S1	5	33214	32444	2.3	200
		10	33222	32688	1.6	227
	S2	5	34135	33020	3.4	19

VOLUME ,

VOLUME ,

VOLUME ,

ANTONELLA FERRARA (S'86, M'88, SM'03, F'20) received the M.Sc. degree in electronic engineering and the Ph.D. degree in computer science and electronics from the University of Genoa, Italy, in 1987 and 1992, respectively. Since 2005, she has been Full Professor of automatic control at the University of Pavia, Italy. Her main research interests include sliding mode control and other nonlinear control methodologies applied to traffic systems, intelligent vehicles, power networks, and robotics. She is author and co-author of more than 450 publications including more than 150 journal papers,


Jean-Claude Bermond

Myriam Di Ianni

Michele Flammini

Stéphane Pérennes

Deadlock prevention by acyclic orientations

come


Davide Regaldo

Aleksandra Bojar

Sean P Dunfield

Pilar Lopez-Varo

Mathieu Frégnaux

Vincent Dufoulon

Shan-Ting Zhang

José Alvarez

Joseph J Berry

Jean- Baptiste Puel

Philip Schulz

Jean-Paul Kleider

On the equilibrium electrostatic potential and light-induced charge redistribution under illumination in halide perovskite structures

Keywords: Perovskite solar cell, surface photovoltage, heterojunction, drift-diffusion modelling, photoemission spectroscopy, defects

Lead halide perovskites are a class of semiconductor materials which are employed as non-intentionally doped absorbers inserted between two selective carrier transport layers (SCTL), realizing a p-i-n or n-i-p heterojunction. In our study, we have developed and investigated a lateral device, based on methylammonium lead iodide (MAPbI3) in which the p-i-n heterojunction develops in the horizontal direction. Our research suggests that the effective doping level in the MAPbI3 film should be very low, below 10 12 cm -3 . Along the vertical direction, this doping level is not enough to screen the electric field of the buried heterojunction with the SCTL. The perovskite work function is therefore affected by the work function of the SCTL underneath. From driftdiffusion simulations we show that intrinsic perovskite-SCTL structures develop mV range surface photovoltages (SPVs) under continuous illumination.

However, perovskite-SCTL structures can develop SPVs of hundreds of mV, as confirmed by our measurements. We therefore analyzed the compatibility between low doping and low defect densities in the perovskite layer and such high SPV values using numerical modelling. It is shown that these high SPV values could originate from electronic processes due to large band offsets in the buried perovskite-SCTL heterojunctions, or at the SCTL -transparent conductive oxide (TCO) buried heterojunction. However, such electronic processes can hardly explain the long SPV persistence after switching off the illumination.

| INTRODUCTION

Metal halide perovskite (MHP) solar cells (PSC) have experienced a rapid increase in efficiency in the last decade. The quality of the perovskite absorber (e.g. high absorption coefficient, low recombination and high diffusion length in the domain of polycrystalline materials) and of the heterointerfaces (selectivity by carrier-blocking band offsets, low interface recombination, transparency of the selective carrier transport layers (SCTLs)) are considered as key factors for the high power-conversion efficiency (PCE) achieved [START_REF] Park | Perovskite solar cells: an emerging photovoltaic technology[END_REF] [START_REF] Ansari | Frontiers, opportunities, and challenges in perovskite solar cells: A critical review[END_REF].

However, despite the record improvements in power conversion efficiency (PCE) of PSCs, the community still faces an ongoing debate about the nature and concentration of defects in the absorber layer. Several studies report the presence of native defects in MHPs, which are dependent on the MHP deposition process [START_REF] Yin | Unusual defect physics in CH3NH3PbI3 perovskite solar cell absorber[END_REF]. Moreover, upon external stimuli, such as illumination, an oxidizing or reducing environment, or a bias voltage, defects could be created or destroyed [START_REF] Meggiolaro | Iodine chemistry determines the defect tolerance of lead-halide perovskites[END_REF] [5].

Polycrystalline MHP thin films show defects both in bulk and on the grain boundaries [START_REF] Jin | It's a trap! On the nature of localised states and charge trapping in lead halide perovskites[END_REF]. In addition, material stability improvements have not kept pace with the PCE increase, mainly due to the many novel properties of this material, like its ionic organic-inorganic structure. Related to the crystal lattice properties, the perovskite's shallow defect levels have very low formation energies and could induce a nonzero doping level in the material [START_REF] Jin | It's a trap! On the nature of localised states and charge trapping in lead halide perovskites[END_REF] [START_REF] Euvrard | Electrical doping in halide perovskites[END_REF].

Direct measurements of the doping level in perovskites, like capacitance-voltage profiling, still depict a major challenge. Perovskites are generally chemically reactive when brought in contact with a wide variety of metals [START_REF] Zhao | Redox chemistry dominates the degradation and decomposition of metal halide perovskite optoelectronic devices[END_REF]. In addition, capacitance techniques rely on the presence of a defined space charge region, a condition which is difficult to achieve for lowly doped and thin (sub-micron) semiconducting layers [START_REF] Almora | On Mott-Schottky analysis interpretation of capacitance measurements in organometal perovskite solar cells[END_REF]. X-ray and ultraviolet photoemission spectroscopy (XPS, UPS) have been employed to characterize electronic and chemical properties of the perovskite material [START_REF] Béchu | Photoemission spectroscopy: photoemission spectroscopy characterization of halide perovskites[END_REF]. Furthermore, transient surface photovoltage (SPV) measurements in fixed capacitor configuration, Kelvin probe and Kelvin probe force microscopy (KPFM) have been employed to measure the charge extraction capabilities of perovskite -SCTL structures [START_REF] Barnea-Nehoshtan | Surface photovoltage spectroscopy study of organo-lead perovskite solar cells[END_REF] [START_REF] Harwell | Probing the energy levels of perovskite solar cells via Kelvin probe and UV ambient pressure photoemission spectroscopy[END_REF].

In several past studies, a combination of these techniques revealed that the Fermi level position in the bandgap of the perovskite can be changed by interfacing the MHP layer with different substrates [13][14][15] [START_REF] Zohar | What limits the open-circuit voltage of bromide perovskite-based solar cells?[END_REF]. However, to date the exact origin of this effect remains unresolved and could be rooted in subtle chemical interactions between the perovskite film and substrate. This supposed variability, together with the effect of different deposition techniques, produces a very complex system, loosely described by the term "p-i-n heterojunction".

In conventional, vertically stacked n-i-p or p-i-n solar cells, the absorber is sandwiched between the SCTLs. Therefore, only indirect measurements can be employed to study the impact of the SCTL on the electronic properties of the junction. In this work, we used a lateral solar cell layout, similar to the design of interdigitated back contact solar cells (IBC). The two SCTLs were deposited prior to the perovskite layer, thus forming two buried heterojunctions. In this way, the absorber properties can be measured with surface sensitive techniques and the contribution of each buried heterojunction can be separated by making the channel between the two SCTLs large enough to contain micrometric space charge regions (SCRs).

Here, we employed XPS, UPS, and KPFM techniques in order to acquire chemical or electronic information about the materials and their mutual interactions. XPS and UPS techniques allow us to probe the equilibrium electrostatic potential distribution across the channel between the two SCTLs. From KPFM measurements performed both in the dark and under illumination we obtain local SPV values that depend on the charge redistribution and thus on their extraction/injection ability at the perovskite-SCTL heterointerfaces. Experimental data are interpreted using a full 3D finite-element modeling tool in order to build a comprehensive energy diagram analysis and get insights into perovskite material parameters related to defect and doping properties.

|Results and discussion

| XPS scan mode surface potential analysis

We consider the perovskite lateral heterojunction device described in Figure 2a. It is composed of an FTO substrate on which two SCTLs, namely TiOx and NiOx, which act as electron transport layer (ETL) and hole transport layer (HTL), respectively, were deposited and laser scribed in order to create a channel between them (l ch > 50 μm). Then, a MAPbI3 perovskite layer was deposited on top, leaving the surface available for characterization and illumination.

The structures were previously characterized using different techniques [START_REF] Dunfield | Understanding carrier gradients and the role of charge selective contacts in lateral heterojunction all back-contact perovskite solar cells[END_REF]. We focus here on the XPS, UPS, KPFM and SPV techniques and on modelling the experimental data to derive conclusions regarding the doping and defects in the perovskite material as well as the impact of the SCTLs on the energy profiles and carrier distributions in the dark and under illumination.

We employed XPS both in scanning and imaging mode. The first of these two methods allowed us to acquire the signal rapidly, minimizing the possible radiation damage during measurement, with a total measurement time of 30 minutes. In this acquisition mode, we swept the probed area across the channel (illustrated in Figure 2a) and performed local XPS measurements. We measured the variation across the channel of the I 3d 5/2 core level peak energy, BEI, which is representative of the electrostatic potential variation, as shown in Figure 1. In (b) the apparent surface electrostatic potential profile across the channel obtained from XPS experimental data (black dashed line) is compared to the one calculated from modelling using an intrinsic perovskite, under various electron collection diameters (𝑑 coll ). The blue solid line shows the calculated surface potential profile without convolution effect with the probe area (also corresponding to 𝑑 coll =0). The reference for the electrostatic potential (0 V) is taken on top of the NiOx side and the origin of the x axis is taken in the middle of the channel. Curves have been shifted with respect to each other for clarity.

XPS scanning data show that a perceived gradual potential drop is present across the channel [START_REF] Dunfield | Understanding carrier gradients and the role of charge selective contacts in lateral heterojunction all back-contact perovskite solar cells[END_REF] (Figure 2b black dashed line).

In this scanning XPS mode, each measured point is the average value of the collected electrons arriving from a given probe area. Therefore, to have a true comparison with the experiments, the surface electrostatic potential values calculated from modeling have been averaged over a simulated collection distance 𝑑 coll , that we varied between 50 µm and 400 µm. Input material parameters for the modelling are reported in Table S2. In Fig. 2b we compare the corresponding apparent surface potential profiles to the XPS experimental data (dark dashed line) and to the true calculated surface potential profile (blue curve without symbols). On the regions above the HTL and the ETL, far from the channel location, the potential is constant. The linear voltage drop in the channel is due to the perovskite being considered undoped and defect-free. A high-curvature electron accumulation region is visible closer to the TiOx side, being the perovskite over the ETL moderately n-type (𝐸 F -𝐸 V = 1.31 eV). Such a high curvature region is not present on the HTL side due to the perovskite being almost intrinsic above NiOx (Table S1). Convoluting the surface potential with the collection area of the XPS setup shows that, when the area increases, the slope of the surface potential decreases, while the perceived channel length increases. We here emphasize that a good agreement is found between the experimental data and the apparent surface potential profile calculated with a collection diameter of 200 µm, which corresponds to the value in our experimental XPS scan mode. However, the perovskite was assumed to be undoped and we thus need to study how doping and defects affect the surface potential variation and the corresponding apparent variation obtained from the XPS scan with the 200 µm collection diameter.

We thus added a trap level in the bandgap close to the band edge. It was considered as an acceptor (EtA -EV=0.02 eV) or as a donor (EC -Etd = 0.02 eV) in order to induce p or n-type doping, respectively [START_REF] Euvrard | Electrical doping in halide perovskites[END_REF]. Trap parameters are summarized in Table S2. In a first example, we chose low bulk trap densities (10 13 cm -3 ). As shown in Figure S3a-c, these low densities are large enough to create two separate SCRs in the channel, while leaving a nearly flat potential region in the middle.

Due to the EF -EV values at the two extremes, introducing a light effective n-type doping pins the channel potential to a value between the two extremes. However, this shift remains undetectable to the measurement probe due to the large collection area. On the other hand, an effective p-type doping causes a larger shift of the potential towards more negative values, which results in a significant deviation from the experimental gradual potential drop. Therefore, Figure S3 shows that the resolution of the measurement allows to classify the perovskite to be intrinsic or lightly n-type, and to rule out the presence of moderate or high p-type doping.

We simulated the surface potential across the channel for various trap concentrations in the bulk perovskite (Figure S5a) and convoluted the data with the experimental electron collection area (dcoll = 200 µm), as shown in Figure 3. As the trap density increases, the potential drifts away from the linear trend, towards a parabolic-like shape, leading to Fermi level pinning inside the channel.

Values of donor trap density up to 10 15 cm -3 produce no effect on the apparent surface potential derived from XPS due to the convolution of the actual potential with the XPS collection area.

Higher donor trap densities are able to screen the buried heterojunction in the vertical direction, which leads to a constant surface potential across the channel. This is confirmed by the vertical perovskite -NiOx band diagram (Figure S5b), and by an analytical model. Indeed, considering an n-type doped perovskitep-type doped HTL structure, we can employ the full depletion approximation and consider the HTL as a metal contact. In this case, assuming a certain perovskite relative permittivity variability (5 < 𝜀 r < 50), the minimum perovskite doping density necessary to vertically screen the buried heterojunction is 10 15 cm -3 (see the modelling section for a more complete analysis).

On the other hand, we find that acceptor trap densities higher than 10 12 cm -3 are not compatible with the experimental findings, as they would modify the gradual potential drop as shown in Figure 3.

Therefore, XPS measurements in scanning mode allow us to classify the perovskite to be intrinsic or slightly n-type, with effective doping concentration up to 10 15 cm -3 , and to rule out the presence of p-type effective doping with concentrations larger than 10 12 cm -3 . In addition to the XPS scanning mode technique, we used the XPS imaging mode on a 500 x 500 μm 2 region of the LHJ device which overlays the channel area with a spatial resolution up to 5 µm. The XPS imaging mode data show, in accordance with the XPS scanning mode, a surface potential difference of ∼ 500 mV between the two sides of the LHJ. In addition, a linear potential variation can be seen in a 60 μm wide region, that corresponds to the length of the LHJ channel (see SI for the detailed analysis).

Finally, we added donor traps on the surface of the perovskite, in a concentration of 10 9 cm -2 , as shown in Figure S3b. This trap density is able to pin the channel potential in a value between the two extremes, similarly to bulk donor traps in Figure S3a.

While perovskite p-type doping character larger than 10 12 cm -3 had already been excluded by the analysis of the XPS scanning data, the linear voltage drop across a ∼ 60 μm region recorded by high-resolution imaging allows us to further confine the range of possible doping densities.

According to the depletion region considerations, we can also set the upper limit for n-type doping in the perovskite to 10 12 cm -3 , since larger doping values would have laterally screened the HTLperovskite heterojunction.

|KPFM surface photovoltage analysis

To gain more insight into the materials properties and the charge redistribution upon illumination, we used KPFM measurements in the dark and under illumination. From the contact potential difference (CPD) between the sample surface and the tip measured by KPFM in dark equilibrium conditions we can deduce the surface work function of the sample. KPFM under variable bias or illumination measures the shift of the surface potential caused by an external stimulus. This shift depends upon the charge redistribution across the layer stack: it is therefore influenced by the presence of surface or bulk defects, or by electric fields present in surface or buried junctions and by band alignments at heterojunctions [START_REF] Barnea-Nehoshtan | Surface photovoltage spectroscopy study of organo-lead perovskite solar cells[END_REF]. Specifically, by subtracting the CPD measured in the dark to that measured under illumination, one obtains the local surface photovoltage (SPV). Preliminary measurements on the MAPbI3 perovskite LHJ device of Figure 2a were reported elsewhere [START_REF] Dunfield | Understanding carrier gradients and the role of charge selective contacts in lateral heterojunction all back-contact perovskite solar cells[END_REF]. We observed a large positive SPV (≈ 500 mV) on top of TiOx, and a smaller but negative SPV on top of NiOx (≈ -320 mV). Here we want to extend the measurements and analysis to triple cation mixed halide perovskite layers over different transport layers. We thus fabricated perovskite-SCTL structures employing a triple cation mixed halide perovskite and three different SCTLs: TiOx, SnOx and NiOx (see SI). We characterized the bilayers with a KPFM setup, in which we illuminated the samples with a 532 nm laser centered on the KPFM tip. We also measured the surface work functions for all the materials under study employing a highly oriented pyrolytic graphite reference sample. We observed the perovskite work function (WF) dependence on the substrate, in line with literature reports [START_REF] Zohar | What limits the open-circuit voltage of bromide perovskite-based solar cells?[END_REF] (Table 1). KPFM measurements under illumination show that the sign of the surface photovoltage (SPV) is consistent with the nature of the SCTL underneath the perovskite, i.e. negative for the perovskite-HTL and positive for the perovskite-ETL structure (Figure 4), and it is reproducible (see Figure S9). This is comparable to our previous measurements on MAPbI3. The sign of the respective SPVs can be attributed to charge separation at the perovskite-SCTL interface. Due to the accumulation of blocked carriers inside the perovskite, a net charge builds up and the surface potential shifts under illumination [START_REF] Qiu | Progress of surface science studies on ABX3-based metal halide perovskite solar cells[END_REF]. The amplitude of the SPV is on the order of 300 mV for the perovskite on both ETL layers, and 170 mV for the perovskite on NiOx. Regarding the perovskite-TiOx structure, SPVs of 150-200 mV have already been observed, while perovskite-HTL structures showed 2 orders of magnitude lower SPVs. This difference has been ascribed to a higher efficiency of charge extraction at the perovskite-ETL interface, with respect to the perovskite-HTL one [START_REF] Barnea-Nehoshtan | Surface photovoltage spectroscopy study of organo-lead perovskite solar cells[END_REF].

The timescales of the SPV buildup after switch-on and decay after switch-off are signatures of the processes involved in the SPV formation. In our case, the rise of the SPV upon switching on the illumination is at least three orders of magnitude faster than the decay of SPV after turning off the illumination, taking into account the low temporal resolution of the instrument (~4 s). In addition, for both transients after the switch-on and switch-off processes, we observe a fast component followed by a slower one. The instantaneous surface potential changes are summarized in Table 2.

SPV We also observe that the fast component after switch-off is very small and, for the perovskite-TiOx sample, is followed by a slow monoexponential decrease with a characteristic time constant of about 700 s. (Figure 4a).

Due to the lack of analytical solutions to the Poisson and carrier continuity equations for many non-equilibrium and non-steady state conditions, we have employed drift-diffusion modelling to unveil the root causes of the SPV and study its dynamics under continuous monochromatic illumination in our perovskite -SCTL bilayers. For comparison with experimental data, we have chosen the same light intensity employed in the KPFM measurements (~8 W/cm 2 ). Simulation parameters are listed in Table S3.

We start with the ideal case, in which the perovskite is intrinsic and without defects states in the bandgap, and the SCTL is also without defects states in the bandgap but doped (𝑁 dop = 10 17 cm -3 ). In addition, we consider zero band offset between the perovskite and the SCTL for the type of carriers that should be extracted at this interface. At equilibrium, the perovskite being undoped and defect-free, the position of the Fermi level in the perovskite is imposed by the doped SCTL underneath, and only a low built-in voltage of about 100 mV forms across the heterojunction.

Under illumination, electrons and holes are separated at the heterointerface: holes accumulate in the perovskite, while electrons accumulate inside the ETL. The sign of the calculated SPV is therefore positive. The charge redistribution is opposite in the perovskite -HTL structure, where the sign is negative. Noticeably, the amplitude of the SPV in both types of structures is lower than 1 mV (Figure 5).

In conclusion, this simple modelling of the perovskite-ETL and perovskite-HTL structures cannot explain the large amplitudes that are experimentally observed. In addition, the calculated transients are much faster (Figure S8). We therefore studied which other factors could affect rise, decay time and magnitude of the SPV.

P-type doping of the perovskite

A built-in voltage across a buried (hetero)junction is usually considered as a primary source of surface photovoltage formation. Regarding the perovskite -TiOx structure, the presence of effective p-type doping in the perovskite could increase the magnitude of the built-in voltage of the heterojunction, since TiOx has been considered as n-type (with a donor concentration 𝑁 d = 10 17 cm -3 ). Thus, an increase of the SPV under the same illumination conditions is expected.

Results of the transient SPV curves for both switch-on and switch-off are displayed in Figure 6,

showing that the magnitude of the SPV and corresponding decay times increase with the perovskite acceptor doping concentration. Perovskite doping values of 7x10 15 cm -3 and 9x10 15 cm -3 provide SPVs of 0.27 V and 0.42 V, respectively, which are comparable to the experimental value. In addition, KPFM measured the perovskite WF dependence on the SCTL WF. For the perovskite -TiOx structure, 𝑊𝐹 pvk -𝑊𝐹 TiO x ≈ 0.4 eV, which in equilibrium could indicate the presence of 5x10 15 cm -3 acceptor dopants in the perovskite, as shown in Figure S10a. However, this heterostructure would develop only a 120 mV SPV, which is at least 100 mV lower than the one found in the experiment.

In addition, the characteristic decay time of the simulated SPV is considerably lower than the 700 s observed experimentally. We also note that the simulated transient after switch-off exhibits a non-exponential decay, in contrast with the exponential behavior observed experimentally for long times (Figure 4a). Therefore, possible p-type doping of the perovskite is unable to explain the experimental SPV results.

Band offset between perovskite and the transport layer

For selectivity purposes, large valence band offset at the ETL interface is preferable in order to prevent holes, generated in the perovskite, from diffusing into the ETL. At the same time the conduction band offset should be small in order to facilitate electron collection. Thus, the conduction band offset defined as the difference at the interface between the bottom of the conduction band in the perovskite, 𝐸 C,pvk , and that in the TiOx, 𝐸 C,TiO x , 𝛥𝐸 𝐶 = 𝐸 C,pvk -𝐸 C,TiO x , should ideally be close to zero. However, we found that a significantly positive value could be a source of large surface photovoltage, as depicted in The same reasoning would be valid for a perovskite -HTL structure with opposite values of the valence band offset with respect to the ones shown in Figure 7. Conduction band offsets of 0.5 and 0.6 eV provide SPVs of 0.28 and 0.38 V, respectively, which are comparable to the experimental values. However, the characteristic decay time of the SPV related to this process is again considerably lower than experimentally observed, and the simulated transient at switch-off is non-exponential. In addition, we expect that such large values of conduction band offset would limit the efficiency of solar cells. In conclusion, the experimental SPV amplitude and transients cannot be explained solely by the conduction band offset at the perovskite-SCTL heterojunction.

Defects in the SCTL

So far, we have considered the SCTLs to be crystalline and moderately doped with no bandgap defects. However, these materials are generally polycrystalline and, especially for NiOx, TiOx and SnOx, not -intentionally doped. Therefore, it is not foregone that these materials possess a high dopant -defect ratio. In addition, the presence of a surface depletion region in polycrystalline oxides has already been reported [19][20]. If acceptor traps are present inside an ETL, under illumination they could trap photogenerated electrons which are injected from the perovskite thanks to the favorable conduction band alignment. However, in order to capture electrons and change the trap occupation, their energy level must lie above the Fermi level at equilibrium.

We found that the only way to significantly increase the SPV was to introduce such a large acceptor trap density at the surface of the ETL, that the Fermi level gets pinned at the trap energy level (Figure S12a). Also in this case, however, SPV decay times are lower than the ones measured in the experiment (Figure 8).

Contact work function

So far, the only buried heterojunction considered in our structures was the one at the perovskite-ETL or perovskite-HTL interface. The interface between the ETL (or HTL) and the back electrode was considered as neutral in the simulation, meaning that the work function of the ETL (or HTL)

and that of the back electrode were taken as identical. In our experiment, the perovskite -TiOx samples were deposited over fluorinated tin oxide (FTO). Such a transparent conductive oxide (TCO) can be modelled by the addition of a Schottky barrier at the back of the SCTL, by setting the work function at the back contact to a value different from that of the SCTL. Thus, a space charge region can form inside the SCTL due to this WF difference. With KPFM, we have measured the FTO WF to be 4.79 eV (Table 1). While the ITO and NiOx WF were found to be nearly equal experimentally, suggesting the presence of an ohmic contact between the two, FTO and TiOx WF values differ by about 400 meV, suggesting the presence of a buried Schottky contact.

Similarly to the perovskite -TiOx affinity mismatch case discussed above, steady state conditions impose a net zero flux of electrons at the TiOx -FTO interface. If the contact WF is increased beyond the TiOx WF, a potential barrier forms between 𝐸 C,TiO x and 𝐸 F,FTO . In order to overcome this barrier by thermionic emission, electrons accumulate in the FTO and thus a spatial separation of charge is obtained (Figure S12b). Thus, both the SPV magnitude and decay time increase with the contact WF, as shown in Figure 9. However, while the SPV decay times could be consistent with the experiment, the magnitude of the SPV is not (Figure S11).

Figure 1 : 2 (

 12 Figure 1: Sketch of the I 3d 5/2 core level, top of the valence band, EV, and bottom of the conduction band, EC, in presence of a space-dependent electrostatic potential, 𝑞𝜑( 𝑥 ) = 𝐵𝐸 I (𝑥) = 𝐸 F -𝐼 3d 5 2

Figure 2 :

 2 Figure2: sketch of the LHJ device with the representation of the XPS electron collection area in the scanning mode (a). In (b) the apparent surface electrostatic potential profile across the channel obtained from XPS experimental data (black dashed line) is compared to the one calculated from modelling using an intrinsic perovskite, under various electron collection diameters (𝑑 coll ). The blue solid line shows the calculated surface potential profile without convolution effect with the probe area (also corresponding to 𝑑 coll =0). The reference for the electrostatic potential (0 V) is taken on top of the NiOx side and the origin of the x axis is taken in the middle of the channel. Curves have been shifted with respect to each other for clarity.

Figure 3 :

 3 Figure 3: Surface potential of the MAPbI3 perovskite across the LHJ device convoluted with the 200 μm collection diameter of the XPS setup employed, for different trap (shallow donor Ntd or shallow acceptor Nta) concentrations in the perovskite. The dark dashed line represents the experimental data.

Figure 4 :

 4 Figure 4: KPFM measurements performed on perovskite-SCTL structures, under 532 nm laser illumination. CPD measurement of (a) perovskite-TiOx, (b) perovskite-SnOx, and (c) perovskite-NiOx. Arrows indicate switch on and subsequent switch off times of the 532 nm laser illumination.

Figure 5 :

 5 Figure 5: perovskite-ETL (a) and perovskite-HTL (b) band diagrams under illumination; the shift of the surface potential is visible in the insets (positive for PVK-ETL, negative for PVK-HTL).

Figure 6 :

 6 Figure 6: Perovskite -TiOx transient SPV simulations for different values of acceptor dopant concentration in the perovskite; build-up of SPV after illumination is switched on (a), and SPV decay after illumination is switched off (b).

Figure 7 .

 7 Under illumination, electron-hole pairs are continuously generated inside the perovskite layer. In steady state conditions, however, the net flux of electrons across the perovskite -ETL heterojunction should be zero (open circuit conditions). For positive values of the conduction band offset, the electrons reinjected towards the perovskite from the TiOx have to overcome a potential barrier in a thermionic emission process. If the barrier height increases, the net amount of electrons accumulated in the TiOx layer has to increase, in order to keep a net zero flux of electrons in steady state. This leads to the building of a net positive charge in the perovskite, which shifts the surface potential towards more positive values.

Figure 7 :

 7 Figure 7: Perovskite -TiOx transient SPV simulations for different values of conduction band offset. Δ𝐸 C = 𝐸 C,pvk -𝐸 C,TiO x ; build-up of SPV after illumination is switched on (a), and SPV decay after illumination is switched off (b)

Figure 8 :

 8 Figure 8: Perovskite -TiOx transient SPV simulations for different values of acceptor trap density on a 5 nm layer thickness on the surface of TiOx(𝑁 ta ), close to the interface with the perovskite: SPV rise (a) and decay (b) after illumination is switched off.

Figure 9 :

 9 Figure 9: Perovskite -TiOx transient SPV simulations for different values of the back contact work function (𝑊𝐹 FTO ); build-up of SPV after illumination is switched on (a), and SPV decay after illumination is switched off (b).

Table 1 :

 1 KPFM surface work function (WF) measurements for the materials under study (TCO -SCTLperovskite layer stacks).

		TCO WF	SCTL WF (fresh)	SCTL WF (aged)	Perovskite	WF
			(eV)	(eV)	(fresh) (eV)
	FTO/TiO2 UVO	4.79	4.40	4.30	4.73
	ITO/SnO2 UVO	5.02	4.74	4.63	4.66
	ITO/NiOx UVO-A	5.02	5.06	4.87	5.09

Table 2 :

 2 Fast SPV component for the bilayers under study. The first column reports the instantaneous surface potential change when the sample is illuminated, while the second column refers to the instantaneous surface potential change when the illumination is switched off.

	fast buildup (V)	SPV fast decay (V)

|Conclusions

In this study, we combined experiments and modelling on specific perovskite-SCTL structures to discuss the equilibrium electrostatic potential distribution and the redistribution of charges upon illumination. On the one hand, our analysis conducted in equilibrium conditions supports the hypothesis of a low effective doping density, smaller than 10 12 cm -3 , inside the perovskite. On the other hand, from drift-diffusion simulations for the bilayers under study, we showed that nearly intrinsic perovskite -moderately doped SCTL structures provide surface photovoltage values lower than 1 mV, while experimental values exceed hundreds of mV under the same continuous illumination conditions. Regarding the perovskite -TiOx structure, we also excluded the effect of band offsets at the buried heterojunction due to decay time after switching off the illumination.

Considering the presence of a TCO at the bottom of the structure, we studied the influence of the contact work function, which affects the SPV. Also in this case, however, varying uniquely this parameter does not allow to fit both the SPV magnitude and decay times.

We can suggest that the large timescale of the photovoltage decays of the perovskite-ETL samples could indicate the occurrence of reversible chemical reactions at the perovskite/SCTL interface, which were not considered in drift-diffusion simulations. Those reactions could be driven by the presence of a net charge inside the perovskite under illumination, due to the presence of the SCTLs and the TCO.


Jean-Claude Bermond

Johny Bond

David Peleg

Stéphane Perennes

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