A multi-class decentralised event-triggered control framework for congestion and emission reduction in freeway networks

15th 15th IFAC IFAC Symposium Symposium on on Control Control in in Transportation Transportation Systems Systems 15th IFAC Symposium on Control in Transportation Systems June 6-8, 2018. Savona, Italy 15th IFAC Symposium on Control in in Transportation Transportation Systems Systems June 6-8, 2018. Savona, Italy 15th IFAC Symposium on Control Available online at www.sciencedirect.com June 6-8, 2018. Savona, Italy 15th Symposium Control in Transportation Systems June 6-8, Savona, Italy JuneIFAC 6-8, 2018. 2018. Savona,on Italy June 6-8, 2018. Savona, Italy

ScienceDirect

IFAC PapersOnLine 51-9 (2018) 291–298

A multi-class decentralised event-triggered A multi-class decentralised event-triggered A multi-class decentralised event-triggered control framework for congestion and A multi-class decentralised event-triggered control framework for congestion control framework for congestion and and emission reduction in networks control framework forfreeway congestion and emission reduction in freeway networks emission reduction in freeway networks emission reduction in freeway networks ∗∗ C. Pasquale ∗∗ S. Sacone ∗∗ S. Siri ∗∗ A. Ferrara ∗∗

C. Pasquale ∗∗ S. Sacone ∗∗ S. Siri ∗∗ A. Ferrara ∗∗ C. ∗∗ C. Pasquale Pasquale ∗ S. S. Sacone Sacone ∗ S. S. Siri Siri ∗ A. A. Ferrara Ferrara ∗∗ C. Pasquale ∗ S. Sacone ∗ S. Siri ∗ A. Ferrara ∗∗ C. Pasquale S. Sacone S. Siri A. Ferrara ∗ ∗ Department of Informatics, Bioengineering, Robotics and Systems ∗ ∗ Department of Informatics, Bioengineering, Robotics and Systems ∗ DepartmentEngineering, of Informatics, Informatics, Bioengineering, Robotics and Systems Systems University of Genova, Italy Department of Bioengineering, Robotics and ∗ University of Genova, Italy ∗∗ DepartmentEngineering, ofofInformatics, Bioengineering, Robotics and Systems ∗∗ Engineering, University of Genova, Italy Electrical, Computer and Biomedical Engineering, Engineering, University of Genova, Italy ∗∗ Department of Computer Biomedical ∗∗ University ofand Genova, Italy Engineering, ∗∗ Department DepartmentEngineering, of Electrical, Electrical, Computer and Biomedical Engineering, University of Pavia, Italy Computer and Biomedical Engineering, ∗∗ Department of Electrical, University of Pavia, Italy Department of Electrical, Computer and Biomedical Engineering, University of of Pavia, Pavia, Italy Italy University University of Pavia, Italy Abstract: The present work deals with the problem of regulating traffic in freeway networks Abstract: The work deals the of traffic in freeway networks Abstract: The present present work deals with with the problem problem of regulating regulating traffic in freeway networks via ramp metering with the twofold objective of reducing congestion and traffic emissions. In Abstract: The present work deals with the problem of regulating traffic in freeway networks via ramp metering with the twofold objective of reducing congestion and traffic emissions. In Abstract: Theanswer present work deals with the problem of regulating traffic in freeway networks via ramp metering with the twofold objective of reducing congestion and traffic emissions. In this paper, an to this problem is given by proposing a multi-class decentralised control via ramp metering with the twofold objective of reducing congestion and traffic emissions. In this paper, an answer to this problem is given by proposing a multi-class decentralised control via ramp metering with the twofold objective of reducing congestion and traffic emissions. In this paper, an answer to this problem is given by proposing a multi-class decentralised control scheme that acts according to an event-triggered logic. Local control algorithms receive local this paper, an answer to this problem is given by proposing a multi-class decentralised control scheme that according toproblem an event-triggered logic. Local control algorithms receive local this paper, answer to freeway thisto is given by proposing a multi-class decentralised control scheme thatanacts acts according to an event-triggered logic. Local control algorithms receive local measurements from the network and make a periodic prediction on the local system scheme that acts according an event-triggered logic. Local control algorithms receive local measurements from the freeway and make aa periodic prediction on system scheme that acts according to information, annetwork event-triggered logic. Localalgorithms control algorithms receive local measurements from the of freeway network and make periodic prediction on the the local local system evolution. On the basis this the local communicate suitable measurements from the freeway network and make acontrol periodic prediction on the local system evolution. On the basis of this information, the local control algorithms communicate suitable measurements from the freeway network and make a periodic prediction on the local system evolution. On the basis of this information, the local control algorithms communicate suitable parameters to controllers, which produce control actions able to jointly evolution. On basisfeedback of this information, the local control algorithms communicate suitable the local reduce parameters to the local feedback controllers, which produce control actions able to jointly reduce evolution. On basis of this emissions information, the local control algorithms communicate suitable parameters to the local feedback controllers, which produce control actions able to jointly reduce the congestion and the traffic in a neighbourhood of the controlled on-ramp. Since parameters to the local feedback controllers, which produce control actions able to jointly reduce the congestion and the traffic emissions in a neighbourhood of the controlled on-ramp. Since parameters to the local feedback controllers, which produce control actions able to jointly reduce the congestion congestion and the theare traffic emissions in aatype, neighbourhood of ramp the controlled controlled on-ramp. Since these local controllers of the multi-class the resulting metering control actions the and traffic emissions in neighbourhood of the on-ramp. Since these local controllers are of the multi-class type, the resulting ramp metering control actions the congestion and the traffic emissions in a neighbourhood of the controlled on-ramp. Since thesedistinguished local controllers controllers are of of the theclasses multi-class type, the the resulting ramp metering control actions actions are for different of vehicles. The results obtained by these local are multi-class type, resulting ramp metering control applying such are distinguished for different classes of vehicles. The results obtained by applying such aaa these local controllers are of the multi-class type, the resulting ramp metering control actions are distinguished for different classes of vehicles. The results obtained by applying such decentralised scheme compared in the paper with by using aa centralised are distinguished for are different classes of vehicles. The those resultsachieved obtained by applying such a decentralised scheme compared in the paper with by using are distinguished for are different classes of vehicles. The those resultsachieved obtained by applying such a decentralised scheme are compared in the paper with those achieved by using aa centralised centralised control framework. decentralised scheme are compared in the paper with those achieved by using centralised control decentralised scheme are compared in the paper with those achieved by using a centralised control framework. framework. control framework. © 2018, framework. IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. control 1. INTRODUCTION INTRODUCTION Another relevant issue addressed in this paper regards Another relevant relevant issue issue addressed addressed in in this this paper paper regards regards 1. Another 1. Another relevant issue addressed in this paper regards the reduction of traffic pollution; this topic has also 1. INTRODUCTION INTRODUCTION Another relevant issue addressed in this paper regards the reduction of traffic pollution; this topic has also the reduction of traffic pollution; this topic has also 1. INTRODUCTION Another relevant issue addressed in thisshown paperhas regards the reduction of traffic pollution; this topic also been treated in [9], where it has been that the the reduction of traffic pollution; this topic has also been treated in [9], where it has been shown that the been treated in inof[9], [9], where it has has been been shown that also the the reduction traffic pollution; thisthe topic has been treated where it shown that the The persistent growth of traffic congestion and the resultminimisation of the total time spent and minimisation been treated in [9], where it has been shown that the minimisation of the total time spent and the minimisation The persistent growth of traffic congestion and the resultminimisation of the total time spent and the minimisation The persistent growth traffic and been treated inthe [9],are where it spent has been shown that the ing harmful effects onof human life have have encouraged encouraged the minimisation of total time and minimisation The harmful persistenteffects growth ofhuman traffic congestion congestion and the the resultresultof traffic emissions generally nonconflicting objectives minimisation of the total time spent and the the minimisation of traffic traffic emissions are generally nonconflicting objectives ing on life the of emissions are generally nonconflicting objectives The persistent growth ofhuman traffic congestion and the control resulting harmful effects on life have encouraged the minimisation of the total time spent and the minimisation study and the implementation of different traffic ing harmful effects on human life have encouraged the of traffic emissions are generally nonconflicting objectives that may be jointly achieved by using multi-class PIof traffic areachieved generallyby nonconflicting objectives that mayemissions be jointly jointly achieved by using multi-class multi-class PIstudy and the theeffects implementation of different traffic control control that may be using PIing harmful on humanalthough lifedifferent havedesigned encouraged the that study and implementation of traffic of traffic emissions are generally nonconflicting objectives may be jointly achieved by using multi-class PIstudy and the implementation of different trafficto control strategies. Freeway systems, meet ALINEA regulators. Although the efficacy of ALINEA and that may be jointly achieved by using multi-class PIALINEA regulators. Although the efficacy of ALINEA and strategies. Freeway systems, although designed to meet ALINEA regulators. Although the efficacy of ALINEA and and study and the implementation of different trafficto strategies. Freeway systems, although designed meet that mayregulators. behas jointly achieved byefficacy using multi-class PIAlthough the of ALINEA strategies. Freeway systems, although designed tocontrol meet the mobility needs considerable traffic volumes, have PI-ALINEA been demonstrated in many real cases ALINEA regulators. Although the efficacy of ALINEA and PI-ALINEA has been demonstrated in many many real cases cases the mobility needs of of considerable traffic volumes, have ALINEA PI-ALINEA has been demonstrated in real strategies. Freeway systems, although designed to meet the mobility needs of considerable traffic volumes, have ALINEA regulators. Although the efficacy of ALINEA and suffered the major damage caused by the increasing dePI-ALINEA has been demonstrated in many real cases the mobility needs of considerable traffic volumes, have [10], they show some limits due to their local nature, PI-ALINEA has been demonstrated in many real cases [10], they they show show some some limits limits due due to to their their local local nature, nature, suffered the major major damage caused by by the volumes, increasinghave de- [10], the mobility needs of considerable traffic suffered the damage caused the increasing dePI-ALINEA has some beenthe demonstrated many cases mand. Indeed, the backwardness of freeway is due limits due to their local suffered the major damage caused by the systems increasing de- [10], since they compute control law only on the basis [10], they show some limits due law to in their local nature, since they they show compute the control law only on real thenature, basis mand. Indeed, the backwardness of freeway systems is due since they compute the control only on the basis suffered the major damage caused by the increasing demand. Indeed, the backwardness of freeway systems is due [10], they show some limits due to their local nature, since they compute the control law only on the basis mand. Indeed, the backwardness of freeway systems is due to the significant structural and economical interventions of measurements immediately downstream the on-ramp. since they compute the control law only on the basis of measurements immediately downstream the on-ramp. to the significant structural and economical interventions of measurements immediately downstream the on-ramp. mand. Indeed, thestructural backwardness of freeway systems is due of to and economical interventions since they as compute the control law onlyPI-ALINEA on the basis measurements immediately downstream the on-ramp. to the the significant significant and economical required to the traffic demand. Since Moreover, it has been proven in is of measurements immediately downstream the on-ramp. Moreover, as it has has been proven in [11], [11], PI-ALINEA is required to satisfy satisfystructural the current current traffic demand.interventions Since these these Moreover, as it been proven in [11], PI-ALINEA is to the significant structural and economical interventions required to satisfy the current traffic demand. Since these of measurements immediately downstream the on-ramp. actions are not always feasible, the adoption of specific Moreover, as it has been proven in [11], PI-ALINEA is required to satisfy the current traffic demand. Since these more effective than ALINEA in stabilising the traffic flow Moreover, as it has been proven in [11], PI-ALINEA is more effective than ALINEA in stabilising the traffic flow actions are not always feasible, the adoption of specific more effective than ALINEA in stabilising the traffic flow required to satisfy the current traffic demand. Since these actions are feasible, the of specific Moreover, as it hasALINEA been proven in [11], is control measures represents a possible possible answer to to improve effective than in the traffic actions measures are not not always always feasible, the adoption adoption of improve specific more process in the presence of bottlenecks that are located far more effective than ALINEA in stabilising stabilising the traffic flow flow process in the the presence of bottlenecks bottlenecks that PI-ALINEA are located located far control represents answer process in presence of that are far actions are performance. not always feasible, the adoption ofthe specific control measures represents aaa possible answer to improve more effective than ALINEA in stabilising the traffic flow process in the presence of bottlenecks that are located far control measures represents possible answer to improve the system Among these, one of most downstream of the merge area. process in theof of area. bottlenecks that are located far downstream ofpresence the merge merge area. the system performance. Among these,answer one of of the the most downstream the control measures represents aispossible improve the performance. Among these, one most process in theof presence of area. bottlenecks that are located far downstream the merge the system system performance. Among these, one oftowhich the most widespread control measures ramp metering, aldownstream of the merge area. widespread control measures is ramp metering, which althe system performance. Among these, one of the most The idea behind the present the widespread control measures is ramp metering, which alThe idea behind the present work goes goes further, further, since since the downstream of the merge area. lows to regulate the traffic volume entering the mainstream widespread control measures is ramp metering, which al- The idea behind the present work work goes further, since the the lows to regulate regulate the traffic traffic volume entering the mainstream mainstream idea behind the present work goes further, since widespread control measures is use ramp metering, which al- The adopted extended multi-class PI-ALINEA controllers comThe idea behind the present work goes further, since the lows to the volume entering the adopted extended multi-class PI-ALINEA controllers comfrom the on-ramps through the of traffic lights. In the lows to regulate the traffic volume entering the mainstream adopted extended multi-class PI-ALINEA controllers comfrom the on-ramps through the use of traffic lights. In the adopted The behindlaw the present work sincecomthe multi-class PI-ALINEA controllers lows to regulate theseveral traffic volume entering thelights. mainstream pute the control not only on the basis of measurements adopted extended multi-class PI-ALINEA controllers comfrom the on-ramps through the of In pute idea the extended control law not only on on thegoes basisfurther, of measurements fromtwo the on-ramps through the use use of traffic traffic lights. In the the pute last decades, works have proposed different the control law not only the basis of measurements last two decades, several works have proposed different adopted extended multi-class PI-ALINEA controllers comthe not on the of from the on-ramps throughworks the use of traffic lights. In the pute downstream the on-ramp, but also making use of of aaa more more pute the control control law not only only onalso the basis basis of measurements measurements last decades, several have proposed different downstream the law on-ramp, but also making use of more last two two decades, several works have proposed different ramp metering controllers in to travelling downstream the on-ramp, but making use ramp metering controllers in order order to minimise minimise travelling pute the control law not only onalso the state. basis of measurements downstream the on-ramp, but making use of a more last two decades, several works have proposed different thorough knowledge of the system Extended multidownstream the on-ramp, but also making use of a more ramp metering controllers in order to minimise travelling thorough knowledge of the system state. Extended multitimes for the drivers [1, 2], and, more recently, also to ramp metering controllers in order to minimise travelling knowledge of the the system system state. Extended multitimes for the the drivers drivers [1, 2], 2], and, more more recently, also to to thorough downstream the on-ramp, alsoalready making use of amultimore knowledge of state. Extended ramp controllers in order to minimise travelling class PI-ALINEA controllers have been described thorough knowledge of thebut system state. Extended multitimes for [1, and, recently, also class PI-ALINEA PI-ALINEA controllers have already been described reduce pollutant emissions or to to increase safety [3, 4, 5, 6]. times metering for the drivers [1, 2], and, more recently, also to thorough class controllers have already been described reduce pollutant emissions or increase safety [3, 4, 5, 6]. thorough knowledge of the system state. Extended multiclass PI-ALINEA controllers have already been described times for the drivers [1, 2], and, more recently, also to in [12], in which a centralised supervisory event-triggered class PI-ALINEA controllers have already been described reduce pollutant emissions or safety [3, 5, in [12], [12], in in which which aa centralised centralised supervisory supervisory event-triggered event-triggered reduce or to to increase increase [3, 4, 4, 5, 6]. 6]. in class PI-ALINEA have been paper, described Aim ofpollutant this work workemissions is to to develop develop a control controlsafety scheme able to in [12], in which centralised supervisory event-triggered reduce pollutant emissions or to increase safety [3, 4, 5, 6]. control framework is proposed. In the present inin [12], in which aa controllers centralised supervisory event-triggered control framework is proposed. proposed. In already the present present paper, inAim of this is a scheme able to control framework is In the paper, inAim of this work is to develop aa control scheme able to in [12], in which a centralised supervisory event-triggered regulate traffic in a freeway network via ramp metering in control framework is proposed. In the present paper, inAim of this work is to develop control scheme able to stead, a decentralised control scheme is considered, in control framework is proposed. In the present paper, instead, a decentralised control scheme is considered, in regulate traffic in a freeway network via ramp metering in stead, decentralised control scheme scheme is considered, considered, in Aim oftothis work tothe develop a control scheme able to regulate traffic in aaisfreeway network via ramp metering in control isalgorithms proposed. In similarly the present paper, inaaaframework decentralised control is in regulate traffic in freeway network via ramp metering in stead, order jointly reduce total time spent by the drivers which local control act to the superstead, decentralised control scheme is considered, in which local control algorithms act similarly to the superorder to jointly reduce the total time spent by the drivers local control algorithms algorithms act similarly to the the supersuperregulate traffic in a freeway network via ramp in which order to reduce the total time spent by the drivers stead, a decentralised control scheme in local control act similarly to orderthe to jointly jointly reduce the total time bymetering the drivers and fuel caused by the vehicles moving in visor described in [12], but referring to local portion which local control algorithms act similarly to theportion supervisor described described in [12], [12], but referring referring tois aaaconsidered, local portion and the fuel emissions emissions caused by thespent vehicles moving in which visor in but to local order to jointly reduce the total time spent by the drivers and the fuel emissions caused by the vehicles moving in which local control algorithms act similarly to the superthe system. Moreover, in this proposal, a heterogeneous but aa local portion and system. the fuel Moreover, emissions in caused by the vehicles moving in visor of the freeway. In particular, in present work, the visor described in [12], but referring referring to local portion of the thedescribed freeway. in In [12], particular, in the the to present work, the the this proposal, proposal, heterogeneous of freeway. In particular, in the present work, the and theconsisting fuel Moreover, emissions caused by thee.g. vehicles moving in of the in this aaacars heterogeneous visor described in [12], but referring to a local portion the freeway. In particular, in the present work, the the system. system. Moreover, in this proposal, heterogeneous stream of aa mix of vehicles, and trucks, extended multi-class PI-ALINEA controllers compute of the freeway. In particular, in the present work, extended multi-class PI-ALINEA controllers compute the stream consisting of mix of vehicles, e.g. cars and trucks, extended multi-class PI-ALINEA controllers compute the the system. Moreover, in of this proposal, heterogeneous stream consisting ofinto aa mix vehicles, e.g. and trucks, of the freeway. particular, in controllers the present work, multi-class PI-ALINEA compute the stream consisting mix of vehicles, e.g. acars cars trucks, extended is explicitly taken account by of multi-class control actions on the basis of measurements from aa timetimeextended multi-class PI-ALINEA controllers compute the control actions onInthe the basis of of measurements measurements from timeis explicitly takenof into account by means means of a a and multi-class control actions on basis from a stream consisting of a mix of vehicles, e.g. cars and trucks, is explicitly taken into account by means of a multi-class extended multi-class PI-ALINEA controllers compute the macroscopic traffic flow model and by designing specific control actions on the basis of measurements from aa timeis explicitly taken into account by means of a multi-class varying cluster of road sections which is communicated by control actions on the basis of measurements from timevarying cluster of road sections which is communicated by macroscopic traffic flow model and by designing specific varying cluster of road sections which is communicated by is explicitly taken into account by means of aofmulti-class macroscopic traffic flow model and by designing specific control actions on the basis of measurements from a timecontrol actions for each vehicle class. The idea proposing varying cluster of road sections which is communicated by macroscopic traffic flow model and by designing specific the local control algorithm. The local control algorithm varying cluster of road sections which is communicated by the local control algorithm. The local control algorithm control actions for each vehicle class. The idea of proposing the local control algorithm. The local control algorithm macroscopic traffic flow model and by designing specific control for each class. idea proposing varying cluster of an road sections which is control communicated by control algorithm. The local algorithm control actions actions formetering each vehicle vehicle class. The The idea of ofrecent proposing multi-class ramp controllers is and acts according to event-triggered nature, i.e. it changes the local control algorithm. The local control algorithm acts local according to an event-triggered nature, i.e. it it changes multi-class ramp metering controllers is rather rather recent and the acts according to an event-triggered nature, i.e. changes control actions formetering each vehicle class. The idea of proposing multi-class ramp controllers is rather recent and the local control algorithm. The local control algorithm acts according to an event-triggered nature, i.e. it changes multi-class ramp metering controllers is rather recent and has been developed in few research works, such as for has been developed developed in fewcontrollers research works, such as for for acts according to an event-triggered nature, i.e. it changes multi-class ramp metering rathersuch recent has as instance indeveloped [4, 6, 6, 7, 7, 8].in has been beenin in few few research research isworks, works, such as and for acts according to an event-triggered nature, i.e. it changes instance [4, 8]. has been developed in few research works, such as for instance instance in in [4, [4, 6, 6, 7, 7, 8]. 8]. instance [4, 6,IFAC 7, 8].(International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 2405-8963 in © 2018,

Copyright © 2018 IFAC 291 Copyright © 2018 291 Copyright © under 2018 IFAC IFAC 291 Control. Peer review responsibility of International Federation of Automatic Copyright © 291 Copyright © 2018 2018 IFAC IFAC 291 10.1016/j.ifacol.2018.07.048 Copyright © 2018 IFAC 291

2018 IFAC CTS 292 6-8, 2018. Savona, Italy June

C. Pasquale et al. / IFAC PapersOnLine 51-9 (2018) 291–298

the control laws of the PI-ALINEA controllers only when suitable triggering conditions are met. The main novelty of the proposed control framework is to combine an event-triggered logic with the concepts of decentralised control. Event-triggered control, with the general objective of limiting the sensor and control computations and/or communications to the cases in which the system needs attention, has already been applied to freeway traffic networks [13, 14]. On the other hand, decentralised or distributed frameworks in the field of freeway traffic control have been proposed in recent papers, e.g. [15, 16, 17]. The paper is organised as follows. In Section 2 the multiclass traffic flow model for freeway networks is presented, while in Section 3 the emission model is described. Section 4 presents the proposed control scheme, and some simulation results are reported in Section 5. Finally, some concluding remarks are given in Section 6. 2. THE MULTI-CLASS TRAFFIC FLOW MODEL The considered traffic flow model is based on METANET [18], in the destination-oriented mode, extended to the multi-class case. Such model is briefly described in this section, interested readers can find further details in [6]. In this model, the freeway network is represented by means of a directed graph, composed of M freeway links, O origin links, and N nodes. Each freeway link m = 1, . . . , M is further divided into Nm sections with length Lm and λm lanes. The set of destinations reachable from link m is denoted as Jm . Analogously, J¯o is the set of destinations reachable from origin link o = 1, . . . , O. For each node n = 1, . . . , N , J¯n is the set of reachable destinations, On is the set of exiting links, and In , I¯n are the set of entering freeway links and entering origin links, respectively. The time horizon is divided into K time steps, with sample time interval T [h], and C classes of vehicles are considered. Let ς c , c = 1, . . . , C, represent a conversion factor of vehicles of class c in Passenger Car Equivalent (PCE). The main variables referring to the freeway links are: ρcm,i,j (k) is the partial traffic density of class c in section i of link m at time instant kT with destination j ∈ Jm [veh of class c/km/lane], ρcm,i (k) is the traffic density of class c in section i of link m at time instant kT [veh of class c/km/lane]; ρm,i (k) is the total traffic density in section c i of link m at time instant kT [PCE/km/lane]; vm,i (k) is the mean traffic speed of class c in section i of link m c at time instant kT [km/h]; qm,i (k) is the traffic volume of class c leaving section i of link m during time interval c [kT, (k + 1)T ) [veh of class c/h]; γm,i,j (k) is the portion of the traffic volume of class c in section i of link m at time instant kT having destination j ∈ Jm (composition rate). The main variables referring to the origin links are: dco,j (k) is the partial origin demand of class c entering origin link o at time instant kT with destination j ∈ J¯o [veh of class c/h]; dco (k) is the origin demand of class c entering origin c link o at time instant kT [veh of class c/h]; lo,j (k) is the partial queue length of class c at origin link o with destination j ∈ J¯o at time instant kT [veh of class c]; loc (k) 292

is the queue length of class c at origin link o at time instant c kT [veh of class c]; γo,j (k) is the portion of the traffic volume of class c leaving origin link o at time instant kT c having destination j ∈ J¯o (composition rate); θo,j (k) is the portion of the demand of class c originating in origin link o at time instant kT having destination j ∈ J¯o ; qoc (k) is the traffic volume of class c leaving origin link o during time interval [kT, (k + 1)T ) [veh of class c/h]; qo (k) is the total traffic volume leaving origin link o during time interval [kT, (k + 1)T ) [PCE/h]. The variables referring to the nodes are: Qcn,j (k) is the flow of class c entering node n during time interval [kT, (k + c (k) 1)T ) with destination j ∈ J¯n [veh of class c/h]; βm,n,j is the splitting rate, i.e. the portion of the traffic volume present in node n at time instant kT which chooses link m to reach destination j ∈ J¯n . The two dynamic equations for the freeway links are  T c ρcm,i,j (k + 1) = ρcm,i,j (k) + γc (k)qm,i−1 (k) Lm λm m,i−1,j  c c − γm,i,j (k)qm,i (k) (1) c vm,i (k

  T c c + 1) = + c V (ρm,i (k)) − vm,i (k) τ   T c c c + vm,i (k) vm,i−1 (k) − vm,i (k) Lm   ν c T ρm,i+1 (k) − ρm,i (k)   (2) − τ c Lm ρm,i (k) + χc c vm,i (k)

and the steady-state speed-density relation and the traffic volume are respectively obtained as  lc  m c   ρm,i (k) f,c · 1− (3) V c (ρm,i (k)) = vm,i ρmax m,i c c (k) = ρcm,i (k)vm,i (k)λm (4) qm,i C c c f,c is the freewhere ρm,i (k) = c=1 ς ρm,i (k) and where vm,i flow speed in section i of link m for class c [km/h], ρmax m,i is the jam density in section i of link m [PCE/km/lane].

The dynamic equation for the origin links is given by   c c c lo,j (k + 1) = lo,j (k) + T dco,j (k) − γo,j (k)qoc (k) (5)

and the traffic flow of class c leaving each origin link o, having m as downstream link, is computed as  lc (k) qoc (k) = min dco (k) + o , qomax,c , T  ρmax m,1 − ρm,1 (k) max,c (6) qo · cr ρmax m,1 − ρm,1 where qomax,c is the maximum flow of class c in origin link o, and ρcr m,1 is the critical density of the first section of link m [PCE/km/lane]. In case the considered origin link o is a controlled onramp, in which q¯oc (k) is the on-ramp flow computed by the controller for class c, q¯oc (k) is added as fourth term in the minimum function in (6) to compute the entering traffic flow.

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

C. Pasquale et al. / IFAC PapersOnLine 51-9 (2018) 291–298

Moreover, the incoming traffic flow in a given node and the traffic flow entering the first section of a link exiting a node are computed, respectively, as � � c c c Qcn,j (k) = (k) (7) (k) + qoc (k)γo,j qµ,N (k)γµ,N µ µ ,j µ∈In

c qm,0 (k) =

�

o∈I¯n

c (k) · Qcn,j (k) βm,n,j

(8)

j∈Jm

3. THE EMISSION MODEL The adopted emission model is VERSIT+ [19], which computes the emission factor E [g/s] of a given vehicle on the basis of the value of its speed v [km/h] and on the combination of its acceleration a [m/s2 ] and speed, through the variable w = a + 0.014v (see [6] for further details). Specifically, the emission factor E [g/s] is given by  u0 if v < 5 and a < 0.5    u + u w + u (w − 1) if v ≤ 50 1 2 + 3 + E= (9) if 50 < v ≤ 80 u + u5 w+ + u6 (w − 1)+    4 if v > 80 u7 + u8 (w − 0.5)+ + u9 (w − 1.5)+ where uh , h = 0, . . . , 9, are specific coefficients of the emission model, and the function (x)+ projects x on the set of nonnegative numbers.

In order to adopt the VERSIT+ emission model for freeway networks, it is necessary to exploit the macroscopic multi-class traffic flow model in order to compute the average acceleration and the number of vehicles for each link, for each class of vehicles and for every simulation time step. In the freeway links, two types of acceleration are considered: • the segmental acceleration aseg,c m,i (k) referring to vehicles of class c in section i of link m between time step k and time step k + 1 (the number of vehicles subject seg,c to this acceleration is denoted as nm,i (k)); cross,c • the cross-segmental acceleration am,i,i+1 (k) of vehicles of class c moving from section i to section i + 1 of link m between time step k and k + 1 (the number of vehicles involved is indicated with ncross,c m,i,i+1 (k)). In the origin links, four types of acceleration are identified: • the acceleration aa,c o (k) of arriving vehicles, i.e. vehicles of class c arriving at the origin link o at k and waiting in queue at k + 1 (let na,c o (k) indicate the number of arriving vehicles); • the acceleration aw,c o (k) of waiting vehicles, i.e. vehicles of class c moving within the queue of the origin link o between k and k + 1 (let nw,c o (k) indicate the number of waiting vehicles); • the acceleration als,c o (k) of leaving vehicles with stop, i.e. vehicles of class c being in the queue of the origin link o at k and exiting link o at k + 1 (let nls,c o (k) indicate the number of leaving vehicles with stop); • the acceleration alns,c (k) of leaving vehicles without o stop, i.e. vehicles of class c arriving at the origin link o at k and exiting link o at k + 1 without any 293

293

(k) indicate intermediate stop in the queue (let nlns,c o the number of leaving vehicles without stop). Table 1. Variables of the multi-class VERSIT model. VERSIT+ E w v a

seg,c Em,i seg,c wm,i c vm,i seg,c am,i

Multi-class VERSIT+ cross,c Em,i,i+1 cross,c wm,i,i+1 c vm,i across,c m,i,i+1

Eoy,c woy,c voy,c ay,c o

In accordance with the accelerations and the number of vehicles previously defined, the macroscopic multiclass VERSIT+ emission model is applied considering the variables indicated in Table 1, where y ∈ Y = {a, w, ls, lns} and voy,c (k) assumes the following values � on,c vo (k) if y = a or y = lns voy,c (k) = (10) voidl,c (k) if y = w or y = ls where voon,c (k) is the speed of vehicles arriving at the origin link o and voidl,c (k) is the speed of the vehicles moving within the queue of origin link o. 4. THE DECENTRALISED EVENT-TRIGGERED CONTROL FRAMEWORK The proposed control scheme aims at regulating traffic in a freeway network via multi-class ramp metering in order to reduce the total time spent by the drivers and the total fuel emissions. This goal could be achieved by adopting multi-class PI-ALINEA feedback regulators, as shown in [9]. Nevertheless, the local nature of PI-ALINEA, which uses only measurements immediately downstream the controlled on-ramp, makes it not completely effective in case of bottlenecks that are located far from the onramp. To overcome this drawback, the control framework proposed in this work is based on extended multi-class PIALINEA controllers, which compute the control action on the basis of measurements of sections in a neighbourhood of the on-ramp. This neighbourhood, associated with a given on-ramp (i.e. an origin link), is called cluster and is a set of road sections downstream the on-ramp. It is worth noting that the clusters are time-varying. In [12], a supervisory event-triggered control scheme is described, based on extended multi-class PI-ALINEA controllers, in which the clusters are decided by a supervisor according to an event-triggered logic. In that control scheme, the supervisor continuously monitors the system state and some indicators referred to the whole network and periodically makes a prediction of the traffic state evolution in the network: according to this information, it changes the size of the cluster for each controlled on-ramp and, consequently, the control laws of the PI-ALINEA controllers, only when it is necessary. A sketch of the supervisory event-triggered control scheme is reported in Fig. 1, while further details on that scheme can be found in [12]. In the present paper, a decentralised version of such a control scheme is proposed. Again, the clusters are timevarying and decided according to an event-triggered logic, but the definition of the clusters is realised by local control algorithms, instead of by a centralised supervisor. These

2018 IFAC CTS 294 6-8, 2018. Savona, Italy June

C. Pasquale et al. / IFAC PapersOnLine 51-9 (2018) 291–298

 q¯oc (k) = max qomin,c , qoc (k−1)−KPc ·[ρcm,1 (k)−ρcm,1 (k−1)]  c,cl cl c + KR · fo (k)[ˆ (11) ρm,1 − ρo (k)]

SUPERVISOR System state

EXTENDED PI-ALINEA Parameters of the control law

where qomin,c is the minimum traffic volume for class c and origin link o, ρcm,1 (k) and ρcm,1 (k − 1) are the density measurements realised in the first section of link m, i.e. immediately downstream link o, and referred to class c, ρˆm,1 is the total density set-point of the first section of c link m, KPc and KR are gain parameters of the regulator specifically identified for each class c.

EXTENDED PI-ALINEA Parameters of the control law

Fig. 1. The supervisory event-triggered control scheme. algorithms compute new parameters of the control law, if necessary, on the basis of local measurements and periodic predictions of the local system state. More specifically, one local control algorithm is associated with each origin link and receives measurements from a local set of road sections starting from the section immediately close to the on-ramp and lasting until a downstream section located before the next controlled on-ramps. This local set of road sections may include sections belonging to different downstream links. Let Ioloc and Moloc denote the set of sections and the set of links belonging to the local set referred to the local controller associated with origin link o. In this decentralised scheme, the neighbourhood (cluster ) of the generic origin link o, on the measurements of which the control law of link o is computed, is by definition a subset of the local set of road sections associated with link o. Specifically, let Iocl (k) and Mocl (k) denote, respectively, the set of sections and the set of links belonging to the cluster of origin link o at time k. In this sense, it yields Iocl (k) ⊆ Ioloc and Mocl (k) ⊆ Moloc .

System state

Parameters of the control law

LOCAL CONTROL ALGORITHM

ω c,cl(k)ς c σ c (k) foc,cl (k) = C o h,cl o h h h=1 ωo (k)ς σo (k)

(12)

where ωoc,cl (k) ∈ [0, 1] is the weight associated to vehicles of class c in the cluster of origin link o at time step k and σoc (k) is the number of vehicles in that cluster, computed as   σoc (k) = loc (k) + ρcm,i (k)Lm (13) m∈Mocl (k) i∈Iocl (k)

Moreover, in (11) the value of the total density to be compared with the set-point is a weighted sum of the densities in the cluster, i.e.   ρcl αcl (14) o (k) = o,m,i (k)ρm,i (k) m∈Mocl (k) i∈Iocl (k)

where αcl o,m,i (k) ∈ [0, 1] are parameters which properly weigh the measurements in the different sections belonging to the cluster of origin link o at time step k. The time-varying parameters of the control law (11)(14) for origin link o are Iocl (k), Mocl (k), ωoc,cl(k), and αcl o,m,i (k). These parameters are computed by the local control algorithm associated with link o according to an event-triggered policy, described below.

System state

EXTENDED PI-ALINEA

In (11), foc,cl(k) is a weighted ratio, at time step k, of the number of vehicles of class c over all the vehicles, which are present in link o and in the sections belonging to the cluster of link o. Such quantity can be computed as

EXTENDED PI-ALINEA Parameters of the control law

4.2 Local control algorithm

LOCAL CONTROL ALGORITHM

Fig. 2. The decentralised event-triggered control scheme. A sketch of the decentralised event-triggered control scheme is reported in Fig. 2. In the following, the logic of the extended multi-class PI-ALINEA controllers and of the local control algorithms are described in detail.

4.1 Extended multi-class PI-ALINEA controllers Referring to a generic origin link o, that is a controlled on-ramp having m as downstream link, according to the extended multi-class PI-ALINEA logic, the flow of class c that should enter at time step k is computed as follows 294

The generic local control algorithm associated with origin link o receives measurements from sections i ∈ Ioloc and links m ∈ Moloc at each time step k = 0, . . . , K − 1. Specifically, it receives local measurements of the traffic c densities ρcm,i (k) and mean traffic speeds vm,i (k), ∀c, loc loc i ∈ Io , m ∈ Mo , as well as the queue lengths loc (k) of origin link o, ∀c. Besides monitoring the single values of these state variables, at each time step k the local control algorithm also computes some local indexes defined for each vehicle class c. Let us denote with ηoc (k) the instantaneous number of vehicles of class c in the local subset of origin link o, computed as follows   ηoc (k) = Lm ρcm,i (k) + loc (k) (15) m∈Moloc i∈Ioloc

and with ξoc (k) the instantaneous emissions of vehicles of class c in the local subset of origin link o, computed as

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

ξoc (k) =





C. Pasquale et al. / IFAC PapersOnLine 51-9 (2018) 291–298

cross,c ncross,c m,i,i+1 (k)Em,i,i+1 (k)

m∈Moloc i∈Ioloc

+





m∈Moloc i∈Ioloc

seg,c nseg,c m,i (k)Em,i (k) +



y,c ny,c o (k)Eo (k)

y∈Y

(16)

Each local control algorithm periodically makes a prediction of the local state evolution. In particular, the prediction is computed at each time step k¯ = nP , where n = 0, 1, 2, . . . and P is an integer representing the number of time steps between one prediction and the next one. The prediction is realised over a given prediction horizon of Kp time steps. On the basis of the local system state measured at time k = k¯ and by using the traffic flow model and the emission model described above, the local control algorithm computes the predicted traffic densities ρ˜cm,i (k) c (k), ∀c, i ∈ Ioloc , and the predicted mean traffic speeds v˜m,i m ∈ Moloc , k = k¯ + 1, . . . , k¯ + Kp , as well as the predicted queue lengths ˜loc (k), ∀c, k = k¯ + 1, . . . , k¯ + Kp . With these predicted state variables, the local control algorithm also computes the predicted instantaneous number of vehicles η˜oc (k) and the predicted instantaneous emissions ξ˜oc (k), ∀c, k = k¯ + 1, . . . , k¯ + Kp . On the basis of the measured/predicted variables and of the computed local indexes, the local control algorithm verifies specific triggering conditions and evaluates whether the present control law of the extended multiclass PI-ALINEA controller must be changed or not. The idea is that changes in the control law are applied if there are relevant variations in the local state and/or in the predicted evolution. If a change is required, the local control algorithm defines a new cluster and the associated parameters, i.e. it properly communicates to the local controller the values of Iocl (k), Mocl (k), ωoc,cl(k), and αcl o,m,i (k). To describe the behaviour of the generic local control algorithm of origin link o, it is useful to distinguish between the time steps in which it makes the prediction and the other time steps. Case k �= k¯ • The local control algorithm of origin link o verifies the following triggering conditions on the measured local state ρm,i (k) ≥ ρ¯m,i (17) ρcm,i (k) ≥ ρ¯cm,i (18) c c (k) ≤ v¯m,i (19) vm,i loc (20) loc (k) ≥ ¯ c m ∈ Moloc , i ∈ Ioloc , where ρ¯m,i , ρ¯cm,i , v¯m,i and ¯loc are suitable thresholds. • The local control algorithm of origin link o verifies the following triggering conditions on the trend of the local indicators ηoc (k − χ + 1), . . . , η c (k) monotonic (21) ξoc (k)(k − χ + 1), . . . , ξ c (k) monotonic (22) • If at least one of conditions (17)-(22) is met, the cluster of origin link o is updated, i.e. the sets Iocl (k) and Mocl (k) are changed. Specifically, the new cluster is composed by the links and sections belonging to Moloc and Ioloc , respectively, from the first section downstream the on-ramp until the one characterized by 295

295

the density with the maximum value. Besides, weights ωoc,cl (k) are computed in order to favour/discourage specific classes of vehicles, while weights αcl o,m,i (k) are fixed depending on the type of congestion. • If none of conditions (17)-(22) is met, no changes are applied to the control law (11)-(14) already defined in origin link o. Case k = k¯ • The local control algorithm of origin link o verifies conditions (17)-(22). • The local control algorithm of origin link o verifies triggering conditions on the predicted local state ρ˜m,i (k) ≥ ρ¯m,i (23) ρ˜cm,i (k) ≥ ρ¯cm,i (24) c c (k) ≤ v¯m,i (25) v˜m,i ˜lc (k) ≥ ¯lc (26) o o m ∈ Moloc , i ∈ Ioloc • The local control algorithm of origin link o verifies the triggering conditions on the trend of the predicted local indicators η˜oc (k − χ + 1), . . . , η˜(k)c monotonic (27) ˜ c monotonic (28) ξ˜oc (k − χ + 1), . . . , ξ(k) • If at least one of conditions (17)-(22) is met, the cluster of origin link o is updated, as described before, and the weights ωoc,cl (k) and αcl o,m,i (k) are properly fixed. • If none of conditions (17)-(22) is met, but one of conditions (23)-(28) is met, then the local control algorithm of origin link o firstly computes the first ¯ ¯ time step κ ∈ {k+1, . . . , k+K p } in which one of these conditions is verified. Then, the cluster of origin link o is updated. Specifically, the cluster is composed by the links and sections belonging to Moloc and Ioloc from the first section downstream the on-ramp until the one characterized by the predicted density, referred to time step κ, which has the maximum value. Besides, weights ωoc,cl(k) are computed according to the possible necessity of favouring/discouraging specific classes of vehicles, while weights αcl o,m,i (k) are set depending on the type of predicted congestion. • If none of conditions (17)-(28) is met, no changes are applied to the control law (11)-(14) already defined in origin link o. 5. SIMULATION RESULTS This section is devoted to the presentation and the analysis of the results obtained from the application of the decentralised event-triggered control scheme proposed in Section 4. In order to prove the effectiveness of such control scheme, the obtained results have been compared with the ones achieved by adopting the supervisory event-triggered control scheme described in [12], with the standard multiclass PI-ALINEA controller and with the uncontrolled scenario. Specifically, in order to carry out that comparison, two performance indicators, i.e. the Total Time Spent (T T S) and the Total Emissions (T E), are considered. In these simulations, a data set taking into account two types of vehicles, namely cars and trucks, has been utilised for the freeway network depicted in Fig. 3. Such network is composed of seven freeway links, denoted with m1 to

2018 IFAC CTS 296 6-8, 2018. Savona, Italy June

m1 o1

m2 o2

m3 m7

C. Pasquale et al. / IFAC PapersOnLine 51-9 (2018) 291–298

m4 o3

m5 o4

m6 o5

Fig. 3. Layout of the case study freeway network. m7, each one with sections composed by three lanes and 800 [m] long, and five origin links denoted with o1 to o5. Among these origin links, the on-ramps o2, o3 and o5 are controlled and equipped with dedicated traffic lights for each class of vehicles. Moreover, in order to test the efficacy of the control strategies in the prevention of congestion due to distant bottleneck, a temporary reduction of capacity has been simulated in sections i = 4, 5 (red dashed line in Fig. 3). The simulations have been carried out by considering, for both classes of vehicles, a constant traffic demand for the uncontrolled origin links, o1 and o4, and trapezoidal demand profiles for the controlled ones. A sample time T of 10 [s] and a total time horizon of 3 hours (K = 1080) are considered for the simulation tests. As for the temporary bottleneck, this appears after half an hour from the beginning of the simulation and lasts for about an hour. Moreover, the following traffic flow model parameters are f,1 f,2 chosen: vm,i = 120 [km/h], vm,i = 90 [km/h], ρmax m,i = cr 200 [PCE/km/lane], ρm,i = 46.6 [PCE/km/lane], ∀m, ∀i, qomax,1 = 1800 [veh/h], qomax,2 = 450 [veh/h] for all controlled origin links, qomax,1 = 6000 [veh/h], qomax,2 = 1500 [veh/h] for o1 and o4 ; finally, the conversion factors are ς 1 = 1 and ς 2 = 4. Since one of the aims of this work also concerns the containment of polluting emissions, the VERSIT+ emission model has been adopted to compute the CO2 emissions produced by the freeway traffic. In this case, the VERSIT+ model parameters are set as follows: for all origin links, voon,c (k) is considered constant and set equal to 30 [km/h] for the two classes and voidl,c (k), considered constant as well, is set equal to 5 [km/h] for both vehicle classes. As for the the control schemes here considered, the gain parameters of the multi-class feedback regulators are 1 2 set equal to: KP1 =102.1, KP2 =49.8, KR =33.5, KR =5.7. The density set-point ρˆm is chosen equal to the critical density for all the controlled origin links, whereas the threshold values used by the decentralised and the supervisory control schemes have been set as follows: ρ¯m,i =47 [PCE/km/lane], ρ¯1m,i =50 [veh/km/lane], ρ¯2m,i =15 1 2 =50 [km/h] , v¯m,i =45 [km/h], ∀m, ∀i [veh/km/lane], v¯m,i lo2 =25 [veh], ∀o. and ¯lo1 =100 [veh], ¯ Figs. 4–6 report some results obtained by simulating the considered traffic scenario in the uncontrolled case, with a standard multi-class PI-ALINEA controller, with the multi-class supervisory event-triggered control scheme and with the proposed multi-class decentralised eventtriggered control scheme. In particular, Fig. 4 and Fig. 5 show the profiles of the traffic density and CO2 emissions in the four cases, while Fig. 6 reports the traffic density in the road section of the temporary bottleneck (i.e. section 4 of link m4) and in the section downstream the on-ramp o5 (section 1 of link m6), again in the four cases. 296

Let us first of all describe the traffic scenario without control. In Fig. 4, it is possible to observe that in the no-control case the congestion appears at the beginning of freeway link m6 and then propagates backward in m5; moreover, the temporary bottleneck causes a significant traffic jam in the freeway link m4. A similar trend is illustrated in Fig. 5 in which, in the no-control case, it is possible to observe the high concentrations of CO2 emissions in links m1-m6 and m5. Therefore, in absence of control, T T S is equal to 4764 [PCE·h], whereas T E is 46104 [kg]. No control

PI-ALINEA 350

On-ramp o5

300 250

Origin link o4

350

On-ramp o5

300 250

Origin link o4

200

200

150

On-ramp o3 Exiting link m7 On-ramp o2 Origin link o1

100 50

0

0.5

1

1.5

2

2.5

3

0

150

On-ramp o3 Exiting link m7 On-ramp o2 Origin link o1

100 50

0

0.5

1

Time [h]

1.5

2

2.5

3

Supervisory control scheme

Decentralized control scheme 350

On-ramp o5

300 250

Origin link o4

350

On-ramp o5

300 250

Origin link o4

200

200

150

On-ramp o3 Exiting link m7 On-ramp o2 origin link o1

100 50

0

0.5

1

1.5

0

Time [h]

2

2.5

3

0

150

On-ramp o3 Exiting link m7 On-ramp o2 Origin link o1

100 50

0

0.5

1

Time [h]

1.5

2

2.5

3

0

Time [h]

Fig. 4. Mainstream density [PCE/km] in the four considered cases.

No control

PI-ALINEA

5

On-ramp o5

5

On-ramp o5 4

Origin link o4

4

Origin link o4 3

3

2

On-ramp o3 Exiting link m7 On-ramp o2 Origin link o1

1

0

0.5

1

1.5

2

2.5

3

0

2

On-ramp o3 Exiting link m7 On-ramp o2 Origin link o1

1

0

0.5

1

Time [h]

1.5

2

2.5

3

0

Time [h]

Supervisory control scheme

Decentralized control scheme

5

On-ramp o5

5

On-ramp o5 4

Origin link o4

4

Origin link o4 3

3

2

On-ramp o3 Exiting link m7 On-ramp o2 Origin link o1

1

0

0.5

1

1.5

Time [h]

2

2.5

3

0

2

On-ramp o3 Exiting link m7 On-ramp o2 Origin link o1

1

0

0.5

1

1.5

2

2.5

3

0

Time [h]

Fig. 5. CO2 emissions [kg] in the four considered cases. By observing Fig. 4 and Fig. 5 for the controlled cases, it is possible to state that the three control schemes act with different level of effectiveness, but all of them manage to minimise the congestion all over the network and to globally reduce the concentration of CO2 emissions. In case the standard multi-class PI-ALINEA controllers are applied, the ramp metering control is activated only at the on-ramp o5, producing some benefits in the traffic network. Nevertheless, in this case, the traffic jam caused by the bottleneck in link m4 is not solved. As also shown in Fig. 6, that control measure is not able to reduce the traffic

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

C. Pasquale et al. / IFAC PapersOnLine 51-9 (2018) 291–298

200 150 100

Density [PCE/km]

250

250 200 150 100

350

300

300

250 200 150 100

50

50

0

0

0

0.5

1

1.5

2

2.5

3

0

0.5

1

2.5

3

150 100 50

Density [PCE/km]

Density [PCE/km]

200

250 200 150 100 50

0

0.5

1

1.5

0

0.5

1

2

2.5

3

Time [h]

0

1.5

2

2.5

0.5

1

1.5

100

0

0.5

1

2

2.5

3

Time [h]

2

2.5

3

Decentralized scheme - link m6 sec. 1

350

350

300

300

250 200 150 100

0

1.5

Time [h]

50 0

150

0

3

Supervisory scheme - link m6 sec. 1

300

250

200

Time [h]

PI-ALINEA - link m6 sec. 1

350

300

0

2

Time [h]

No control - link m6 sec. 1

350

1.5

250

50

Density [PCE/km]

0

Decentralized scheme - link m4 sec. 4

350

50

Time [h]

Density [PCE/km]

Supervisory scheme - link m4 sec. 4

300

Density [PCE/km]

300

Density [PCE/km]

PI-ALINEA - link m4 sec. 4

350

Density [PCE/km]

No control - link m4 sec. 4

350

297

250 200 150 100 50

0

0.5

1

1.5

Time [h]

2

2.5

3

0

0

0.5

1

1.5

2

2.5

3

Time [h]

Fig. 6. Traffic density in section 4 of link m4 and in section 1 of link m6, in the four considered cases: dashed blue line for cars, dotted black line for trucks [PCE], solid red line for cars plus trucks [PCE]. density in correspondence of the gridlock. The multiclass PI-ALINEA globally reduces T T S to 4263 [PCE·h] (10.51% reduction compared with the uncontrolled case), and decreases T E to 41288 [kg] (10.45% reduction with respect to the uncontrolled case). Better results are obtained by applying the event-triggered control schemes, both in the centralised and in the decentralised cases. Indeed, it is possible to observe that both these control schemes are able to dissolve the tailback due to the reduction of capacity in m4, by activating ramp metering on the on-ramp o3 and o5. More specifically, by observing Figs. 4–6, it can be seen that the adoption of the supervisory event-triggered control scheme produces slightly better results compared with the decentralised one. Numerically, by applying the supervisory control scheme, T T S is reduced to 4096 [PCE·h] (14.03% reduction compared to the uncontrolled case) and T E is 39267 [kg] (14.83% reduction). Very similar results are achieved by adopting the proposed decentralised scheme, that implies a decrease of T T S to 4175 [PCE·h] (12.36 % reduction) and yields a decrement of T E to 39987 [kg] (13.27 % reduction). Although the decentralised scheme performance is slightly lower than the one obtained with the supervisory scheme, the adoption of the former can entail significant advantages with respect to the use of a centralised framework. For instance, it is worth noting that, in case of malfunctioning of the data detection and transmission systems, the decisions taken by the centralised supervisor on the basis of missing or wrong data should compromise the overall controller functionality, while similar failures in a decentralised scheme could have limited repercussions as they only affect local control actions. 6. CONCLUSION In this paper, a multi-class decentralised event-triggered control scheme for freeway networks has been proposed. It consists in providing a local control algorithm for each controlled on-ramp that checks suitable triggering conditions on the basis of real-time local measurements and 297

periodic predictions of the local freeway state evolution. If these triggering conditions are met, the local control algorithm transmits the updated parameters of the control laws to the extended multi-class PI-ALINEA controllers, otherwise these controllers continue to apply the current control law. The time-varying parameters updated and communicated by the local control algorithm are related to the clusters of road sections associated with each on-ramp, which are time-varying as well and whose measurements are used to compute the corresponding control laws. The effectiveness of the proposed decentralised control scheme has been analysed through simulation tests, which have been discussed in detail in the paper. REFERENCES [1] M. Papageorgiou, and A. Kotsialos. Nonlinear optimal control applied to coordinated ramp metering. IEEE Transactions on Intelligent Transportation Systems, 12: 920–933, 2004. [2] S. Sacone, and S. Siri. A control scheme for freeway traffic systems based on hybrid automata. Discrete Event Dynamic Systems: Theory and Applications, 22: 3–25, 2012. [3] S.K. Zegeye, B. De Schutter, J. Hellendoorn, E.A. Breunesse, and A. Hegyi. Integrated macroscopic traffic flow, emission, and fuel consumption model for control purposes. Transportation Research C, 31: 158– 171, 2013. [4] C. Pasquale, I. Papamichail, C. Roncoli, S. Sacone, S. Siri, and M. Papageorgiou. Two-class freeway traffic regulation to reduce congestion and emissions via nonlinear optimal control. Transportation Research C, 55: 85–99, 2015. [5] C. Pasquale, D. Anghinolfi, S. Sacone, S. Siri, M. Papageorgiou. A two-class traffic control scheme for reducing congestion and improving safety in freeway systems. In Proc. 19th IEEE Intelligent Transportation Systems Conference, 1767–1772, 2016. [6] C. Pasquale, S. Sacone, S. Siri, B. De Schutter. A multi-class model-based control scheme for reducing congestion and emissions in freeway networks by com-

2018 IFAC CTS 298 June 6-8, 2018. Savona, Italy

[7]

[8]

[9]

[10] [11]

[12]

C. Pasquale et al. / IFAC PapersOnLine 51-9 (2018) 291–298

bining ramp metering and route guidance. Transportation Research C, 80: 384–408, 2017. T. Schreiter, H. van Lint, and S. Hoogendoorn. Multiclass ramp metering: concepts and initial results. In Proc. of the 14th IEEE Conference on Intelligent Transportation Systems, 885–889, 2011. S. Liu, H. Hellendoorn, and B. De Schutter. Model predictive control for freeway networks based on multi-class traffic flow and emission models. IEEE Transactions on Intelligent Transportation Systems, 18: 306–320, 2017. C. Pasquale, S. Sacone, and S. Siri. Ramp metering control for two vehicle classes to reduce traffic emissions in freeway systems. In Proc. European Control Conference, 2588–2593, 2014. M. Papageorgiou, and A. Kotsialos. Freeway ramp metering: an overview. IEEE Transactions on Intelligent Transportation Systems, 3: 271–281, 2002. Y. Wang, E.B. Kosmatopoulos, M. Papageorgiou, I. Papamichail. Local ramp metering in the presence of a distant downstream bottleneck: theoretical analysis and simulation study. IEEE Transactions on Intelligent Transportation Systems, 15: 2024–2039, 2014. C. Pasquale, S. Sacone, S. Siri, A. Ferrara. Supervisory multi-class event-triggered control for congestion and emissions reduction in freeways. In 20th IEEE Intelligent Transportation Systems Conference, 1535– 1540, 2017.

298

[13] A. Ferrara, S. Sacone, and S. Siri. Event-triggered model predictive schemes for freeway traffic control. Transportation Research C, 58: 554–567, 2015. [14] A. Ferrara, S. Sacone, and S. Siri. Design of networked freeway traffic controllers based on event-triggered control concepts. International Journal of Robust and Nonlinear Control, 26: 1162–1183, 2016. [15] A. Ferrara, A. Nai Oleari, S. Sacone, S. Siri. Freeways as systems of systems: a distributed model predictive control scheme. IEEE Systems Journal, 9: 312–323, 2015. [16] D. Pisarski, C. Canudas-de-Wit. Nash game-based distributed control design for balancing traffic density over freeway networks. IEEE Transactions on Control of Network Systems, 3: 149–161, 2016. [17] J. Reilly, A. M. Bayen. Distributed Optimization for Shared State Systems: Applications to Decentralized Freeway Control via Subnetwork Splitting. IEEE Transactions on Intelligent Transportation Systems, 16: 3465–3472, 2015. [18] M. Papageorgiou, J.-M. Blosseville, and H. HadjSalem. Modelling and real-time control of traffic flow on the southern part of Boulevard P´eriph´erique in Paris: Part I: modelling. Transportation Research A, 24: 345–359, 1990. [19] N. E. Ligterink, R. De Lange, and E. Schoen. Refined vehicle and driving-behavior dependencies in the VERSIT+ emission model. In Proc. ETAPP Symposium, 177–186, 2009.