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IFAC PapersOnLine 51-9 (2018) 37–42 A A coupled coupled PDE-ODE PDE-ODE model model for for bounded bounded A acceleration coupled PDE-ODE model for bounded in traffic flow acceleration in macroscopic macroscopic traffic flow Aacceleration coupled PDE-ODE model for bounded in macroscopic traffic flow models models acceleration in macroscopic traffic flow models ∗,∗∗ ∗ Nicolas Laurent-Brouty , Guillaume Costeseque ∗,∗∗ ∗, models Nicolas Laurent-Brouty , Guillaume Costeseque , ∗ ∗,∗∗ ∗

Paola ∗,∗∗Goatin ∗ Nicolas ,, Guillaume Costeseque Nicolas Laurent-Brouty Laurent-Brouty Guillaume Costeseque ∗ ,, Paola Goatin ∗ ∗ Paola Goatin ∗,∗∗ Paola Goatin Nicolas Laurent-Brouty , Guillaume Costeseque ∗ , ∗ e Cˆ o te d’Azur, Inria, ∗ Universit´ Universit´e Cˆ otePaola d’Azur,Goatin Inria, ∗CNRS, CNRS, LJAD, LJAD, France. France. ∗ (e-mail: [email protected]) ∗ Universit´ e Cˆ o te d’Azur, Inria, CNRS, LJAD, France. Universit´ e Cˆ o te d’Azur, Inria, CNRS, LJAD, (e-mail: [email protected]) France. ∗∗ des Ponts ParisTech, Champs-sur-Marne, France. ∗∗ Ecole (e-mail: ∗ [email protected]) Ecole (e-mail: des Ponts ParisTech, Champs-sur-Marne, France. Universit´ e Cˆ [email protected]) d’Azur, Inria, CNRS, LJAD, France. ∗∗ ∗∗ Ecole des Ponts ParisTech, Champs-sur-Marne, France. Ecole des Ponts [email protected]) ParisTech, Champs-sur-Marne, France. (e-mail: ∗∗ Abstract: we aa new model Ecole des Ponts ParisTech, Champs-sur-Marne, France. for Abstract: In In this this paper, paper, we propose propose new mathematical mathematical model accounting accounting for the the boundedness boundedness of traffic acceleration at a macroscopic scale. Our model is built on a first order macroscopic PDE Abstract: In this paper, we propose a new mathematical model accounting for the boundedness Abstract: In this paper, propose a new mathematical model for the boundedness of traffic acceleration at we a macroscopic scale. Our model is builtaccounting on a first order macroscopic PDE model coupled with an ODE describing the trajectory of the leader of a platoon accelerating of traffic acceleration at a macroscopic scale. Our model is built on a first order macroscopic PDE of traffic acceleration at a macroscopic scale. Our model is built on a first order macroscopic PDE model coupled an we ODE describing the trajectory model of theaccounting leader of afor platoon accelerating Abstract: In thiswith paper, propose a new mathematical the boundedness at a given constant rate. We use Tracking techniques approximate model with describing the trajectory the leader of aa construct platoon accelerating model coupled with an an ODE describing theFront trajectory ofbuilt the leader ofto platoon accelerating a coupled given constant We use Wave Wave Front Tracking techniques to construct approximate of at traffic acceleration atrate. aODE macroscopic scale. Our model isof on a first order macroscopic PDE solutions to the Initial Value Problem. We present some numerical examples including the at a given constant rate. We use Wave Front Tracking techniques to construct approximate atsolutions a given constant rate. We describing use Wave We Front Tracking approximate to the Initial Value Problem. present some numerical examples including the case case model coupled with an ODE the trajectory oftechniques the leader to of construct a platoon accelerating of signals on an arterial road and we compare the solutionincluding to model with solutions the Initial Value Problem. We present some examples the solutions toconstant thetraffic Initial Value Problem. WeFront present some numerical examples including the case case ofa successive successive traffic signals an Wave arterial roadTracking and we numerical compare the to our our model with at givento rate. Weon use techniques tosolution construct approximate the solution given by the classical LWR equation in order to evaluate the impact of bounded of successive traffic signals on an arterial arterial road and some we compare the solution to our model model with of successive traffic signals an and we solution our with the solution given byValue theon classical LWR equation in compare order to the evaluate theto impact ofthe bounded solutions to the Initial Problem. Weroad present numerical examples including case acceleration. the solution given by the classical LWR equation in order to evaluate the impact of bounded the solution given by the classical LWR equation in order to evaluate the impact of bounded acceleration. of successive traffic signals on an arterial road and we compare the solution to our model with acceleration. acceleration. the given by the classical LWR equation Control) in orderHosting to evaluate the Ltd. impact of bounded © solution 2018, IFAC (International Federation of Automatic by Elsevier All rights reserved. Keywords: Keywords: Bounded Bounded acceleration; acceleration; macroscopic macroscopic traffic traffic flow flow model; model; PDE-ODE PDE-ODE coupling; coupling; wave wave acceleration. front scheme. Keywords: Bounded acceleration; Keywords: Bounded acceleration; macroscopic macroscopic traffic traffic flow flow model; model; PDE-ODE PDE-ODE coupling; coupling; wave wave front tracking tracking scheme. front tracking scheme. front tracking scheme. Keywords: Bounded acceleration; macroscopic traffic flow model; PDE-ODE coupling; wave 1. INTRODUCTION queue front tracking scheme. 1. INTRODUCTION queue dissipation dissipation times times (see (see Anderson Anderson (2015) (2015) and and referreferences therein). 1. INTRODUCTION queue dissipation times (see Anderson (2015) and 1. INTRODUCTION queue referencesdissipation therein). times (see Anderson (2015) and references therein). 1.1 review ences therein). 1. INTRODUCTION queue dissipation times (see Anderson has (2015) referThe of acceleration been previously 1.1 Motivation Motivation and and review of of the the literature literature The problem problem of bounded bounded acceleration has beenand previously 1.1 Motivation and review of the literature ences therein). addressed by Lebacque (2002, 2003); Leclercq (2007) start1.1 Motivation and review of the literature The problem of bounded acceleration has been previously The problembyofLebacque bounded (2002, acceleration been (2007) previously addressed 2003); has Leclercq startmodern societies, traffic flows on road networks reping from the LWR model. In particular, the mathematical addressed by Lebacque (2002, 2003); Leclercq (2007) start1.1In Motivation and review of the literature addressed (2002, Leclercq startIn modern societies, traffic flows on road networks rep- The ingproblem fromby theLebacque model. In2003); particular, the(2007) mathematical ofLWR bounded acceleration has been previously a concern public authorities. In order approach by Lebacque leads to two model in In modern societies, traffic flows on networks ing from the the LWR model. In particular, the phase mathematical Inresent modern societies, trafficfor flows on road road networks repfrom LWR model. In particular, the mathematical resent a major major concern for public authorities. In reporder ing approach Lebacque leads to aaLeclercq two phase model addressed byby Lebacque (2002, 2003); (2007) start-in to design efficient control strategies, macroscopic traffic which the bounded acceleration phase is described by resent a major concern for public authorities. In order approach by Lebacque leads to a two phase model in resent a major concern for flows public In order approach by LWR Lebacque leads to aphase twothe phase model by in a tomodern design efficient control strategies, macroscopic traffic ing which bounded acceleration is mathematical described a In societies, traffic onauthorities. road networks repfrom the the model. In particular, flow models seem better suited than microscopic models non-strictly hyperbolic system of balance laws, while the to design efficient control strategies, macroscopic traffic which the bounded acceleration phase is described by a toflow design efficient control macroscopic which the by bounded acceleration phase described by in a models seem better suited microscopic models approach non-strictly hyperbolic system laws,model while the resent a major concern forstrategies, publicthan authorities. In traffic order Lebacque leads to of a balance twoisphase in most cases since they are robust, computationally effilatter applies to restrictive cases limited to trianflow models seem better than models non-strictly hyperbolic system of phase balance laws, while thea flow models seem better suited than microscopic microscopic models hyperbolic system of balance while the most cases since theysuited are robust, computationally effi- non-strictly latterthe applies to rather rather restrictive cases limited toby triantoindesign efficient control strategies, macroscopic traffic which bounded acceleration is laws, described cient and allow for near-to-real-time computations. Among gular fundamental diagrams. The above descriptions can in most cases since they are robust, computationally effilatter applies to rather restrictive cases limited to trianin most cases since they are robust, computationally effilatter applies to rather restrictive cases limited to triancient and allow for near-to-real-time computations. Among gular fundamental diagrams. The above descriptions can flow models seem better suited than microscopic models non-strictly hyperbolic system of balance laws, while the those models, the first order LWR model (Lighthill and hardly be extended to general situations and implemented cient and allow for near-to-real-time computations. Among gular fundamental diagrams. The above descriptions can cient andmodels, allowsince forthe near-to-real-time computations. Among gular fundamental diagrams. The above descriptions can those first are order LWR model (Lighthill and latter hardly be extended to restrictive general situations and implemented in most cases they robust, computationally effiapplies to rather cases limited to trianWhitham (1955); Richards has deeply invesrigorously in situations. In paper, we a those models, the first order LWR model (Lighthill and be to general situations and implemented those models, the first order (1956)) LWR computations. model (Lighthill and hardly hardly be extended extended to general The situations and implemented Whitham (1955); Richards (1956)) has been been deeply invesrigorously in field fielddiagrams. situations. In this this paper, we propose propose cient and allow for near-to-real-time Among gular fundamental above descriptions can a tigated. It reads as a scalar conservation law in one space general approach to account for bounded acceleration of Whitham (1955); Richards (1956)) has been deeply invesrigorously in field situations. In this paper, we propose a Whitham (1955); Richards (1956)) has been deeply invesrigorously in field situations. In this paper, we propose a tigated. It reads as a scalar conservation law in one space general to general accountsituations for bounded acceleration of those models, the first order LWR model (Lighthill and hardly be approach extended to and implemented dimension vehicles from aa macroscopic point of view, and we tigated. It reads aa scalar conservation law in space approach to account for bounded of tigated. It (1955); reads as asRichards scalar (1956)) conservation law deeply in one one invesspace general general approach to account In for bounded acceleration ofa dimension vehicles from macroscopic point ofpaper, view,acceleration andpropose we provide provide Whitham has been rigorously in field situations. this we a rigorous constructive algorithm to compute solutions dimension vehicles from a macroscopic point of view, and we provide dimension vehicles from a macroscopic point of view, and we provide a rigorous constructive algorithm to compute solutions ∂ ρ + ∂ q(ρ) = 0, x ∈ R, t > 0, (1) tigated. It reads as a scalar conservation law in one space general approach to account for bounded acceleration of ∂tt ρ + ∂xx q(ρ) = 0, x ∈ R, t > 0, (1) a to our model, which can be applied to aa large class of rigorous constructive algorithm to compute solutions a rigorous constructive algorithm to compute solutions to our model, which can be applied to large class of dimension vehicles from a macroscopic point of view, and we provide ∂ ρ + ∂ q(ρ) = 0, x ∈ R, t > 0, (1) ∂t ρ +ρmax ∂xx q(ρ) = 0, density x ∈ R, t cars > 0, (1)→ tofundamental where ]] is diagrams. our model, which can be applied to a large class of where ρ ρ ∈ ∈ t[0, [0, ρmax is the the density of of cars and and qq :: ρ ρ → to our model, which can be applied to a large class of fundamental diagrams. a rigorous constructive algorithm to compute solutions q(ρ) flow-density ∂t ρρ +the ∂]]x q(ρ) = 0, x fundamental ∈ of R, cars t > 0, where ρ ∈ is the density and qq :: ρ → diagrams. max where ρ ρv(ρ) ∈ [0, [0, ρis is the density of cars anddiagram. ρ (1) → fundamental q(ρ) = = ρv(ρ) ismax the flow-density fundamental diagram. fundamental diagrams. toThe ourLWR model, which can be appliedwith to aunbounded large classaccelof model provides solutions The LWR model provides solutions with unbounded accelq(ρ) = ρv(ρ) is the flow-density fundamental diagram. q(ρ) = ρρv(ρ) isρthe flow-density fundamental diagram. where ∈ [0, ] is the density of cars and q : ρ → However, the LWR model has the drawback to only reprefundamental diagrams. erations only in specific Upward jumps velocmax model has the drawback to only repre- The The LWR model model provides solutions with with unbounded accelHowever, the LWR LWR solutions unbounded accelerations only inprovides specific situations. situations. Upward jumps in in velocq(ρ) =equilibrium ρv(ρ) thestates flow-density fundamental sent fails to some physity may happen at discontinuities of the initial However, the LWR model has drawback to only reprein situations. Upward jumps However, the is LWR modeland hasit the drawback todiagram. only represent equilibrium states and itthe fails to reproduce reproduce some phys- erations erations only in specific specific situations. Upward jumps in velocityLWR mayonly happen at downward downward discontinuities of in thevelocinitial The model provides solutions with unbounded accelical properties of traffic. In particular, if the traffic density density datum or can be generated by boundary condisent equilibrium states and it fails to reproduce some physity may happen at downward discontinuities of the initial sent states andhas it particular, fails reproduce some physicalequilibrium properties of traffic. In if the density erations itydensity may happen atordownward discontinuities of the initial datum cansituations. be generated by jumps boundary condiHowever, the LWR model the to drawback totraffic only repreonly in specific Upward in velocupstream is higher than downstream, for instance tions, for downstream aa merge or bottleneck ical properties of traffic.and In it particular, if the thelike traffic density or be by condiical properties traffic. In particular, if traffic upstream is of higher than downstream, like for density instance density datum or can be generated generated by boundary boundary conditions, for instance instance downstream merge orofa athe bottleneck sent equilibrium states fails to reproduce some phys- density ity may datum happen at can downward discontinuities initial when a traffic light turns green, then the LWR model (1) whose positions are known. To handle these situations, upstream is higher than downstream, like for instance tions, for instance downstream a merge or a bottleneck upstream is higher than downstream, like for instance when a traffic light turns green, then the LWR model (1) tions, for instance downstream a merge or a bottleneck whose positions are known. To handle these situations, ical properties of traffic. In particular, if the traffic density density datum or can be generated by boundary condisolution displays unbounded traffic acceleration, since we the moving bottlenecks proposed when a light turns green, then the LWR (1) positions are known. these situations, when a traffic traffic light an turns green, then thelike LWR model (1) solution displays an unbounded traffic acceleration, since whose whose positions aredownstream known.of To handle these situations, we adapt adapt the framework framework ofTo moving bottlenecks proposed upstream is higher than downstream, formodel instance tions, for instance ahandle merge or a bottleneck the traffic velocity jumps instantaneously to a higher value. in the engineering literature by Lebacque et al. solution displays an unbounded traffic acceleration, since we adapt the framework of moving bottlenecks proposed solution displays anturns unbounded trafficthe acceleration, the traffic velocity jumps instantaneously to a higher value. weinadapt the framework of moving bottlenecks proposed thepositions engineering literature by Lebacque etsituations, al. (1998). (1998). when a traffic light green, then LWR modelsince (1) whose are known. To handle these This poses a limitation to the of LWR The reader interested by mathematical of the traffic velocity jumps instantaneously to a a higher higher value. the engineering literature by Lebacque et al. (1998). the traffic velocity jumps instantaneously to value. This poses a severe severe limitation to the coupling coupling of the the LWR in in the engineering literature by Lebacque etanalysis al.proposed (1998). The reader interested byofthe the mathematical analysis of the the solution displays an unbounded traffic acceleration, since we adapt the framework moving bottlenecks model (and also second order models like Payne (1971); moving bottleneck problem is referred to Delle Monache This poses a severe limitation to the coupling of the LWR The reader interested by the mathematical analysis of the This poses a severe limitation to the coupling of the LWR model (and also second order models like Payne (1971); The reader interested by the mathematical analysis of the moving bottleneck problem is referred to Delle Monache the traffic velocity jumps instantaneously to a higher value. in the engineering literature by Lebacque et al. (1998). Whitham and Aw and (2000); Zhang (2002)) and Goatin (2014) and references therein. In case, we model (and also second order models like bottleneck is referred to Monache model (anda(1974) also second order models like Payne Payne (1971); Whitham (1974) and Aw andtoRascle Rascle (2000); Zhang (2002)) moving bottleneck problem ismathematical referred to Delle Delle Monache andreader Goatin (2014)problem and references therein. In our our case, This poses severe limitation the coupling of the(1971); LWR moving The interested by the analysis of thewe with consumption and pollution models (see for example assume that all vehicles accelerate at the same constant Whitham (1974) and Aw and Rascle (2000); Zhang (2002)) and Goatin (2014) and references therein. In our case, we Whitham (1974) Aw and Rascle (2000); Zhang with (and consumption and pollution models for(2002)) example moving and Goatin (2014) and references therein. In ourMonache case, we assume that all vehicles accelerate attothe same constant model also and second order models like (see Payne (1971); bottleneck problem is referred Delle Treiber and (2013, Chapter 20) and references rate, and that overtaking is possible. Thus, the first with consumption and pollution models (see for example example assume that(2014) all vehicles vehicles accelerate at the the In same constant with consumption and models (see for Treiber and Kesting Kesting (2013, Chapter 20) and references that all accelerate at same constant rate, and that overtaking is not nottherein. possible. Thus, the we first Whitham (1974) and Awpollution and Rascle (2000); Zhang (2002)) assume and Goatin and references our case, therein), in which the acceleration component plays aa rate, vehicle will accelerate and regulate upstream traffic. Hence Treiber and Kesting (2013, Chapter 20) and references rate, and that overtaking is not possible. Thus, the first Treiber and Kesting (2013, Chapter 20) and references therein), in which the acceleration component plays and that overtaking is not possible. Thus, the first vehicle will accelerate and regulate upstream traffic. Hence with consumption and pollution models (see for example assume that all vehicles accelerate at the same constant crucial role. Also, bounded acceleration of it behave as bottleneck. The passing rate is therein), in acceleration component plays aa vehicle will accelerate and upstream traffic. therein), in which which the acceleration component plays may crucialand role. Also, the bounded acceleration of vehicles vehicles may vehicle will accelerate and regulate regulate upstream traffic. Hence it will will behave as aa moving moving The passing rate Treiber Kesting (2013, Chapter 20) and references rate, and that overtaking is bottleneck. not possible. Thus, theHence first is have an impact on management procedures, such as traffic thus set to zero. crucial role. Also, bounded acceleration of vehicles may it will behave as a moving bottleneck. The passing rate is crucial role. Also,onbounded acceleration of vehicles have an impact management procedures, suchplays as may traffic it thus will behave as a moving bottleneck. The passing is set zero. therein), in which the acceleration component a vehicle willto accelerate and regulate upstream traffic. rate Hence signals optimization, which require a estimation of thus set have an role. impact on management management procedures, such as traffic traffic to zero. zero. have an impact on procedures, as signals optimization, whichacceleration require a precise precise estimation to crucial Also, bounded of such vehicles may of thus it willset behave as a moving bottleneck. The passing rate is signals which a estimation of signals optimization, which require require a precise precise such estimation of thus set to zero. have anoptimization, impact on management procedures, as traffic Copyright © 2018 IFACwhich require a precise estimation of 37 signals optimization, Copyright © 2018 IFAC(International Federation of Automatic Control)37 2405-8963 © 2018, IFAC Hosting by Elsevier Ltd. All rights reserved.

Copyright ©under 2018 responsibility IFAC 37 Control. Peer review© of International Federation of Automatic Copyright 2018 IFAC 37 10.1016/j.ifacol.2018.07.007 Copyright © 2018 IFAC 37

2018 IFAC CTS 38 June 6-8, 2018. Savona, Italy

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initial datum ρ0 . Then the Cauchy problem (2) admits a solution in the sense of Definition 1.1.

1.2 The PDE-ODE model We consider the following assumptions:

The rest of the paper is organized as follows: Theorem 1 is proved in Section 2. We then provide some numerical examples in Section 3. Finally, Section 4 gives a wrap-up discussion and details for some future research directions.

(A0) The map q : ρ → q(ρ) is strictly concave, differentiable, satisfies q(0) = q(ρmax ) = 0 and there exists a unique ρcrit ∈ [0, ρmax ] such that q is maximal. (A1) The map v : ρ → v(ρ) is decreasing, Lipschitz continuous and satisfies v(0) = Vmax and v(ρmax ) = 0. (A2) The initial datum ρ0 ∈ BV(R; [0, ρmax ]) is a piecewise constant function with a finite number of jumps (any function in BV(R; [0, ρmax ]) can be approximated by a piecewise constant function, see Bressan (2000)[Lemma 2.2]) .

2. ANALYSIS OF THE PROPOSED MODEL Below, we present the main technical elements for proving the existence of a solution to the PDE-ODE model (2). 2.1 The Riemann problem

We are now ready to introduce our coupled ODE-PDE model: x ∈ R, t > 0, (2a) ∂t ρ + ∂x q(ρ) = 0, 0 ρ(0, x) = ρ (x), x ∈ R, (2b) t > 0, (2c) q (ρ(t, yi (t))) − ρ(t, yi (t))y˙ i (t) ≤ 0, y˙ i (t) = ωi (t, ρ(t, yi (t)+)) t > 0, (2d) (2e) yi (0) = yi0 ,

In this paper, we focus on the Riemann problem for (2), i.e. the problem with initial density given by a piecewise constant initial datum with only one discontinuity: if x < y0 , ρL ρ0 (x) = (3) ρR if x > y0 . Note that for sake of simplicity, since I = 1, we drop the lower-script notation and set y0 := y10 and y := y1 .

for every yi0 , i = 1, . . . , I, such that the initial datum has a downward jump: ρ0 (yi0 −) > ρ0 (yi0 +). In the model above, we have defined ωi (t, ρ) := min At + vi0 , v(ρ)

Solutions with unbounded acceleration only appear in the case ρL > ρR , when a platoon of vehicles has to accelerate from an initial velocity v(ρL ) to a higher one v(ρR ). Indeed, from (A1), if ρL > ρR , then v(ρL ) < v(ρR ). The case ρL ≤ ρR gives the classical solution to the LWR model i.e. a shock wave. Definition 2.1. (Classical solutions to (1), (3)). Consider the simple Riemann problem ∂t ρ + ∂x q(ρ) = 0, x ∈ R, t > 0, ρL if x < y0 , ρ(0, x) = ρR if x > y0 .

where vi0 = v(ρ0 (yi0 −)) stands for the initial speed of the moving bottleneck starting at position yi0 and A > 0 is the constant acceleration rate. yi denotes the trajectory of the i-th moving bottleneck. In the model above, the traffic density evolution is described by the scalar conservation law (2a) and the corresponding initial condition (2b). The equation (2c) accounts for the assumption that no overtaking is possible. It enforces that the flux directly upstream of the bottlenecks created by the leader vehicles is bounded by the flux along their trajectory. The ODE (2d) gives the trajectory of each accelerating leader starting from the position given by the initial condition (2e), i.e. any point of downward jump in the density. We note that in this case the LWR model would provide a solution in which the leader vehicles change their speed instantaneously, i.e. with an infinite acceleration.

The unique weak entropic solution (see Garavello and Piccoli (2006)) to the above Cauchy problem is given by x − y0 ρ(t, x) = R(ρL , ρR )(ξ) with ξ := t q(ρL ) − q(ρR ) ρL if ρL < ρR , ξ < ρL − ρ R or ρ ≥ ρ , ξ < q (ρ L R L ), q(ρL ) − q(ρR ) = ρR if ρL < ρR , ξ > ρL − ρ R or ρ ≥ ρ , ξ > q (ρ L R R ), (q )−1 (ξ) if ρ ≥ ρ , q (ρ ) < ξ < q (ρ ). L R L R

Following Delle Monache and Goatin (2014), solutions to (2) are intended in the weak sense specified below. Definition 1.1. (Definition of solutions to (2)). We call (ρ, y) ∈ C 0 (R+ ; L1 ∩ BV(R, [0, ρmax ])) × W1,1 ((R+ ; R) a solution to (2) if and only if

We are now ready to give a constructive definition of solutions to (2) in the case of a Riemann initial datum (3): Definition 2.2. (Constrained solutions to (2), (3)). For a fixed speed of the moving bottleneck y, ˙ we denote by ρˆ the unique solution to the equation v(ρ) = y. ˙ We define a solution ρ(t, x) = Ry˙ (ρL , ρR )(ξ) to (2), (3) as follows:

(i) ρ is a weak solution of (2a), (2b), i.e. ∀φ ∈ Cc1 (R2 ; R), ∞ T ∞ (ρφt +q(ρ)φx )dxdt+ ρ0 (x)φ(0, x)dx = 0; 0

−∞

−∞

(ii) ∀i ∈ I, yi is a Carath´eodory solution of (2d), (2e), t i.e. yi (t) = yi0 + ωi (s, ρ(s, yi (s)+)) ds;

• If q(R(ρL , ρR )(y)) ˙ − y˙ R(ρL , ρR )(y) ˙ > 0, then the bottleneck is active and ˙ R(ρL , ρˆ)(ξ), if ξ < y, Ry˙ (ρL , ρR )(ξ) = R(0, ρR )(ξ), if ξ ≥ y. ˙

0

(iii) the constraint (2c) is satisfied for almost every t ≥ 0.

• If q(R(ρL , ρR )(y)) ˙ − y˙ R(ρL , ρR )(y) ˙ ≤ 0, then the bottleneck is inactive and Ry˙ (ρL , ρR )(ξ) = R(ρL , ρR )(ξ) for all ξ.

The main result of this paper is the following. Theorem 1. (Existence of solution). Assume (A0), (A1) and (A2) hold, with only one jump discontinuity in the 38

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2.2 Construction of approximate solutions by Wave Front Tracking (WFT) algorithm

39

denote by ρN the approximate solution constructed via WFT method (see Fig. 1): (i) Using the constrained Riemann solution Rvi0 given by Definition 2.2, solve the constrained Riemann problem at x = y0 . The solution consists in a jump discontinuity between ρL,N and 0, moving at speed vi0 , eventually followed by a shock between 0 and ρR,N , moving at speed vi0 −K = v(ρR,N ), and it is defined for t < t1 . (ii) At t = t1 , the speed of the moving bottleneck is set equal to vi0 −1 . We then solve the constrained Riemann problem with Rvi0 −1 : the solution consists in an approximate rarefaction jump of size 2−N ρmax between ρi0 = ρL,N and ρi0 −1 , moving at speed q(ρi0 ) − q(ρi0 −1 ) , λi0 −1/2 = ρi0 − ρi0 −1 followed by a jump discontinuity between ρi0 −1 and 0, moving at speed vi0 −1 . The solution is defined for t ∈ [t1 , t¯2 [, where t¯2 = min{t2 , tint }, tint being the interaction time between the jump discontinuity y˜N and the shock between 0 and ρR,N originated at t = 0. (iii) We repeat step (2) until t = tint ∈ [t1 , +∞]. (iv) If tint < +∞, the solution to the Riemann problem at t = tint and x = y0 + vi0 −K tint consists in a classical shock between the left state before the interaction ρ¯L and ρR,N . Since there is an interaction, the bottleneck propagates faster than the initial shock, and thus ρ¯L < ρR,N . The solution is determined thanks to Definition 2.1. (v) For t > tint only interactions between classical fronts can occur since we restrict ourselves to a Riemann problem, giving rise to classical solutions.

We construct a sequence of approximate solutions (ρN )N ∈N with the Wave Front Tracking algorithm (see for instance Holden and Risebro (2015)) and we show in the next subsection that they converge to a solution of (2) in the sense of Definition 1.1. Definition of the mesh: Consider a given N ∈ N. We set a mesh of densities as follows MN := 2−N ρmax N ∩ [0, ρmax ] such that for any distinct (ρ1 , ρ2 ) ∈ MN , we have ρ1 − ρ2 ≥ 2−N ρmax . The mesh has exactly 2N + 1 elements. We denote by vj,N the speed associated with any density q(ρj,N ) . We consider ρj,N ∈ MN , say vj,N := v(ρj,N ) = ρj,N an approximation of the initial datum taking values on the grid: ρL,N if x < y0 , 0 ρN (x) = if x ≥ y0 , ρR,N such that ρL,N , ρR,N ∈ MN . We then define the two indices i0 and i0 − K such that ρi0 = ρL,N and ρi0 −K = ρ −ρ ρR,N with K = L,NρmaxR,N 2N . In the remaining, when the index N is not necessary for the computations, we will elude it for sake of clarity. Approximated trajectory of the moving bottleneck: According to (2d), while the accelerating vehicle (say the moving bottleneck) has not reached its maximal speed Vmax or caught up with the last of the downstream vehicles, its trajectory is given by the parabola A y(t) = t2 + vi0 t + yi0 , t > 0. 2 We will approximate this parabola by a piecewise affine trajectory, such that the initial slope is equal to v(ρL,N ) = vi0 ,N , and then increases along the grid, taking values vi0 −1,N , vi0 −2,N and so on. The parabola exists on a time horizon included in [t0 = 0, ti0 ], since the velocity cannot exceed v0 = Vmax . We partition the time-horizon with intervals [tn , tn+1 [ where for any n < i0 , tn is defined for each grid-parameter N such that t0 = 0, tn+1 = tn + ∆tn , − vi0 −n v ∀n ∈ N. (4) ∆tn := i0 −(n+1) , A The interval ∆tn corresponds to the time necessary to accelerate between two consecutive velocities vi0 −n and vi0 −(n+1) at a constant acceleration rate A.

t tint

ρi0 −2 −

ρi0 −1 ρi 0

0−

1

t2

2

v i0

ρ=0 ρi0 −K

vi

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vi0

t1

ρL,N

ρR,N

x

Fig. 1. Representation of the WFT approximate solutions and notation used in the algorithm.

The approximate parabola is then defined for each gridparameter N and t ∈ [tn , tn+1 ] as n−1 y˜N (t) := y0 + vi0 −i ∆ti + vi0 −n (t − tn ).

2.3 Proof of Theorem 1 We first show the convergence of the piecewise linear trajectory y˜N towards the parabola described by the accelerating vehicle. Lemma 2. For all T > 0, ˜ yN − yL∞ ([0,T ]) −→ 0.

i=0

After t = ti0 , the leader vehicle trajectory is a straight line, with slope v0 . For t > ti0 , the curve is defined by y(t) = y(ti0 ) + v0 (t − ti0 ). If necessary, we define ti0 +1 the time at which the curve crosses the shock wave emitted by the initial datum.

N →+∞

Proof. Since the sequence {˜ yN }N ∈N is uniformly bounded on any interval [0, T ] and equicontinous, Ascoli-Arzela theorem guarantees that it is uniformly convergent. To show that the limit is y, let now n ∈ N. For any t ∈ [tn , tn+1 [, we have

Algorithm: Let T > 0 given. Following (4), we can partition the time interval [0, T ] in intervals [tn , tn+1 [. We 39

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y(t) ˙ − y˜˙ N (t) = vi0 − vi0 −n + Atn + A(t − tn ) n−1 vi −i−1 − vi −i 0 0 = vi0 − vi0 −n + A + A(t − tn ) A i=0

and Risebro, 2015), using the Riemann solvers defined in Definition 2.1 for the LWR model and in Definition 2.2 for the LWR model with bounded acceleration. The Wave Front Tracking algorithm is known to be very accurate but computationally cumbersome to implement. It is noteworthy that other schemes could be used to compute a numerical solution to the problem (2). For instance, an adapted finite volume scheme has been proposed in Chalons et al. (2018) for a slightly different model of a classical moving bottleneck which is not accelerating. A numerical method based on a semi-analytical Lax-Hopf formula for Hamilton-Jacobi equations has been also recently presented in Simoni and Claudel (2017) for “classical” fixed and moving bottlenecks.

= A(t − tn ) ≥ 0. Let us fix T ∈ [t0 , ti0 ] and n ∈ N such that T ∈ [tn , tn+1 [. Since yN is differentiable almost everywhere, we can write: T (y(t) ˙ − y˜˙ N (t))dt y(T ) − y˜N (T ) = =

0 n−1 ti+1 i=0

+ =

A 2

ti

T

tn n−1

(y(t) ˙ − y˜˙ N (t))dt

In the two following numerical illustrations, we consider the speed-density function ρ v(ρ) = Vmax 1 − , ρ ∈ [0, ρmax ]. ρmax This leads to a quadratic flow-density fundamental diagram of Greenshield’s type, with ρcrit = ρmax 2 . Even if this choice is arguable for traffic flow applications, different fundamental diagrams can be implemented exactly in the same way.

(y(t) ˙ − y˜˙ N (t))dt

∆ti 2 +

i=0

A (T − tn )2 ≥ 0 2

We also compute n−1 n−1 1 ∆ti 2 = 2 (vi0 −i−1 − vi0 −i )2 A i=0 i=0 ≤

n−1 Lip(v)2 2 |ρi −i−1 − ρi0 −i | A2 i=0 0

The maximal density corresponds to a bumper-to-bumper situation, so we select ρmax = 200 veh/km. The acceler2 ation rate is fixed to A = 2 m/sec , which is a standard value. Finally, the parameter for our density grid is set to N = 10. These values are common to both examples. Density values in the interval [0, 2−N ρmax [, corresponding to the void, are depicted in white color in the figures.

n−1 L2 −2N = 2 2 A i=0

L2 N (2 + 1)2−2N −→ 0 N →+∞ A2 converges to y(t) pointwise almost ev-

≤ Thus {˜ yN (t)}N ∈N erywhere.

3.1 Case of a Riemann problem In this first example, we consider a Riemann problem (3) initially located at y0 = 400 m, with ρL = 180 veh/km and ρR = 80 veh/km. We set Vmax = 30 m/sec.

We define for each solution ρN constructed via Wave Front Tracking the following Glimm type functional: TV(ρ0N (·)) + 2ρR,N if t = 0 Υ(t) = Υ (ρN (t, ·)) = TV(ρN (t, ·)) if t > 0 It easy to check that the approximate solutions constructed with the WFT algorithm satisfy the following lemma. Lemma 3. The map t → Υ(t) is non-increasing. A standard application of Helly’s theorem provides the convergence of the sequence of approximate solutions {ρN }N to some function ρ ∈ C 0 (R+ ; L1 ∩BV(R, [0, ρmax ])). In addition, the approximate parabola y˜˙ N converges to y. ˙ Following Delle Monache and Goatin (2014), one can prove that the limit functions (ρ, y) provide a solution to (2) in the sense of Definition 1.1. This concludes the proof of Theorem 1. 3. NUMERICAL EXAMPLES In this section, we provide two numerical examples to illustrate the construction of the approximate solutions we described in the previous section. We solve Cauchy problems both for the LWR model (1) and for the LWR model with bounded acceleration (2). To do that, our algorithm is based on the Wave Front Tracking algorithm (Holden

Fig. 2. Solution to our PDE-ODE model (2) for a Riemann initial datum (3) with ρL = 180 veh/km and ρR = 80 veh/km. The classical solution to the LWR model with such an initial condition consists in a rarefaction wave. The solution 40

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41

3.2 Example of successive traffic lights

to the LWR model with bounded acceleration is displayed on Fig. 2, while the difference of densities between the classical LWR model and the bounded acceleration model is shown on Fig. 3. One can easily observe the vacuum created in front of the first accelerating vehicle which is specific to the PDE-ODE model. This vacuum lasts for approximately 15 seconds and covers around 300 meters, which is not negligible. Moreover, it is noteworthy that the differences in terms of density spread over a wider area. Indeed, the bounded acceleration of the first vehicle affects all the following vehicles. From this simple example, one can see that the bounded acceleration has not just a local effect. From Fig. 3, we observe that the traffic is first condensed (green and blue zones) and then it is relaxed (yellow to pink zones) in comparison to the classical solution with gaps up to around 80 veh/km in some regions.

In this numerical study, our aim is twofold. First, we want to show that our algorithm can be easily extended to more general initial conditions than a Riemann problem. Second, we highlight the impact of bounded acceleration for a realistic situation of successive traffic lights. We select Vmax = 15 m/sec to match an arterial road behaviour and we consider three traffic signals located respectively at positions x = 300, 700 and 1000 meters. We assume that initially, there are queues standing at each traffic signal with local Riemann problems such that ρL = ρmax and ρR = 0. The initial datum contains additional Riemann problems upstream of each traffic light, accounting for the cars that did not reach the initial queues at time t = 0. We assume that the traffic lights turn simultaneously to green at time t = 0.

Fig. 3. Density difference between the solution to the LWR model and the solution to the PDE-ODE model.

Fig. 5. Solution to our PDE-ODE model for the series of traffic lights.

We can also compare the solutions of both models by plotting the queues generated along the time. We define a queue length as the distance between extremal points for which the density is higher or equal to 34 ρmax . With our choice of a linear speed-density function, we know that this density is above the critical density ρcrit = ρmax 2 . The result is given on Fig.4. While the queue length is also non-increasing during the considered time period for the LWR model, there is an increase for the PDE-ODE model that corresponds to the very beginning of the bounded acceleration. Interestingly, this gap of more or less 45 meters is conserved for the whole simulated period.

Fig. 6. Solution to the LWR model for the series of traffic lights. The solution to the LWR model with bounded acceleration is displayed on Fig. 5 while the solution to the classical LWR model appears on Fig. 6. The reader can note the formation of rarefactions (dissipation of queues)

Fig. 4. Queue lengths obtained for the LWR model and the PDE-ODE model. 41

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Further work will aim at extending the analytical proof to more general initial data. This model could also be used to analyze the impact of bounded acceleration on congestion in the case of different, potentially desynchronized, traffic light cycles.

and shockwaves as well as the interactions between all these waves when vehicles catch downstream traffic. The difference of densities due to the bounded acceleration shown on Fig. 7 reveals high discrepancies along interacting shockwaves and rarefaction fans (in pink). We recover the same phenomenon than in the previous example with condensed (in green) and relaxed (in yellow) traffic due to the bounded acceleration.

We might also investigate the opportunity of using finite volume schemes, based on the approach of Chalons et al. (2018), to model several moving bottlenecks and compare the numerical results to the previous ones. Finally, we could extend this framework to take into account the deceleration of traffic as well. REFERENCES Anderson, L. A., 2015. Data-Driven Methods for Improved Estimation and Control of an Urban Arterial Traffic Network. University of California, Berkeley. Aw, A., Rascle, M., 2000. Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60 (3), 916–938. Bressan, A., 2000. Hyperbolic systems of conservation laws. Vol. 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford. Chalons, C., Delle Monache, M. L., Goatin, P., 2018. A conservative scheme for non-classical solutions to a strongly coupled pde-ode problem. Interfaces and Free Boundaries 19 (4), 553–570. Delle Monache, M. L., Goatin, P., 2014. Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result. J. Differential Equations 257 (11), 4015–4029. Garavello, M., Piccoli, B., 2006. Traffic flow on networks. Vol. 1 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO, conservation laws models. Holden, H., Risebro, N. H., 2015. Front tracking for hyperbolic conservation laws, 2nd Edition. Vol. 152 of Applied Mathematical Sciences. Springer, Heidelberg. Lebacque, J., 2002. A two phase extension of the LWR Model based on the boundedness of traffic acceleration. In: Transportation and Traffic Theory in the 21st Century. Emerald Group Publishing Limited, pp. 697–718. Lebacque, J., Lesort, J., Giorgi, F., 1998. Introducing buses into firstorder macroscopic traffic flow models. Transportation Research Record: Journal of the Transportation Research Board (1644), 70–79. Lebacque, J.-P., 2003. Two-phase bounded-acceleration traffic flow model: analytical solutions and applications. Transportation Research Record: Journal of the Transportation Research Board (1852), 220–230. Leclercq, L., 2007. Bounded acceleration close to fixed and moving bottlenecks. Transportation Research Part B: Methodological 41 (3), 309–319. Lighthill, M. J., Whitham, G. B., 1955. On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A. 229, 317–345. Payne, H. J., 1971. Models of freeway traffic and control. Mathematical models of public systems. Richards, P. I., 1956. Shock waves on the highway. Operations Res. 4, 42–51. Simoni, M. D., Claudel, C. G., 2017. A fast semi-analytic algorithm for computing solutions associated with multiple moving or fixed bottlenecks: Application to joint scheduling and signal timing. arXiv preprint. Treiber, M., Kesting, A., 2013. Traffic flow dynamics. Springer-Verlag Berlin Heidelberg. Whitham, G. B., 1974. Linear and nonlinear waves. Vol. 42. John Wiley & Sons. Zhang, H. M., 2002. A non-equilibrium traffic model devoid of gaslike behavior. Transportation Research Part B: Methodological 36 (3), 275–290.

Fig. 7. Density difference between the solution to the LWR model and the solution to the PDE-ODE model for the series of traffic lights. The study of the cumulative queues carried over on Fig. 8 with the same threshold ρ ≥ 34 ρmax exhibits a significant difference with and without bounded acceleration. For instance, the model with bounded acceleration captures up to 50 additional meters of queues over a 1000 meters long road, which is significant if one thinks to responsive traffic signals based on an estimation of queue lengths.

Fig. 8. Queue lengths obtained for the LWR model and the PDE-ODE model. 4. DISCUSSION AND FUTURE WORK This paper proposes a new traffic flow model based on a coupled PDE-ODE approach. This model is designed in order to realistically reproduce vehicles acceleration at the macroscopic scale. We compare it to the seminal LWR model on some numerical cases and we observe a significant increase of queue lengths which should be closer to reality due to the fact that cars have finite acceleration. This intuition should nonetheless be verified by confronting the PDE-ODE model to traffic data in future research. 42