Coordinated Variable Speed Limit, Ramp Metering and Lane Change Control of Highway Traffic

Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress Control The of Toulouse, France,Federation July 9-14, 2017 Available online at www.sciencedirect.com The International International Federation of Automatic Automatic Control The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 5307–5312

Coordinated Variable Speed Limit, Ramp Coordinated Variable Speed Limit, Ramp Coordinated Variable Speed Limit, Ramp Coordinated Variable Speed Limit, Ramp Metering and Lane Change Control of Metering and Lane Change Control of Metering and Lane Change Control of Metering and Lane Change Control of Highway Traffic Highway Traffic Traffic Highway Highway Traffic ∗ ∗

Yihang Zhang ∗ Petros A. Ioannou ∗ ∗ Yihang Zhang ∗∗ Petros A. Ioannou Yihang Yihang Zhang Zhang Petros Petros A. A. Ioannou Ioannou ∗ ∗ ∗ University of Southern California, Los Angeles, CA 90089 USA ∗ University of Southern California, Los Angeles, CA 90089 USA Southern California, ∗ University of (e-mail: {yihangzh,ioannou}@usc.edu). University of Southern California, Los Los Angeles, Angeles, CA CA 90089 90089 USA USA (e-mail: {yihangzh,ioannou}@usc.edu). (e-mail: {yihangzh,ioannou}@usc.edu). (e-mail: {yihangzh,ioannou}@usc.edu). Abstract: Integrated variable speed limit and ramp metering control for highway traffic is Abstract: speed and ramp metering control for highway traffic is Abstract: Integrated variable speed limit and ramp metering control highway traffic is expected to Integrated smooth thevariable traffic flow andlimit improve the mobility of highway networks. Our recent Abstract: Integrated variable speed limit and the ramp metering control for for highway traffic is expected to smooth the traffic flow and improve mobility of highway networks. Our recent expected to smooth the traffic flow and improve the mobility of highway networks. Our recent results showed that variable speed limit control alone cannot eliminate capacity drops in many expected to smooth the traffic flow and improve the mobility of highway networks. Our recent results that variable speed limit control eliminate capacity drops in many results showed that speed limit control alone cannot eliminate capacity in many incidentshowed cases, where one or more lanes are closedalone due cannot to microscopic phenomena such as forced results showed that variable variable speedlanes limitare control alone cannot eliminate capacity drops drops in forced many incident cases, where one or more closed due to microscopic phenomena such as incident cases, where one or more lanes are closed due to microscopic phenomena such as forced lane changes and merging dynamics. Inare thisclosed paper, wetodeveloped a coordinated variable speed incident cases, where one or more lanes due microscopic phenomena such as forced lane changes and merging dynamics. In this paper, we developed a coordinated variable speed lane and In paper, we coordinated variable limit,changes ramp metering and dynamics. lane change control which guaranteesaastability of the trafficspeed flow lane changes and merging merging dynamics. In this this paper, we developed developed coordinated variable speed limit, ramp metering and lane change control which guarantees stability of the traffic flow limit, ramp metering and lane change control which guarantees stability of the traffic flow and improves traffic mobility, safety and the environment. In the proposed control strategy, limit, ramp metering and lanesafety change which guarantees the traffic flow and improves traffic mobility, andcontrol the environment. environment. In the thestability proposedofcontrol control strategy, and safety and the In proposed strategy, whileimproves the lane traffic changemobility, control avoids the capacity drop, a feedback linearization variable speed and improves traffic mobility, safety the andcapacity the environment. In the linearization proposed control strategy, while the lane change control avoids drop, a feedback variable speed while the control avoids capacity drop, aa feedback variable speed limit controller coordinates with the the ramp metering and stabilizes linearization the bottleneck flow at the while the lane lane change change controlwith avoids the capacity drop, feedback linearization variable speed limit controller coordinates the ramp metering and stabilizes the bottleneck flow at the limit controller coordinates with the ramp metering and stabilizes the bottleneck flow at the maximum level. Microscopic Monte-Carlo simulations demonstrate significant improvement on limit controller coordinates with the rampsimulations metering and stabilizes significant the bottleneck flow at the maximum level. Microscopic Monte-Carlo demonstrate improvement on maximum level. Microscopic Monte-Carlo simulations demonstrate significant improvement on traffic mobility, safety and environment by using the proposed controller. maximum level. Microscopic Monte-Carlo simulations demonstrate significant improvement on traffic mobility, and environment by using the proposed controller. traffic mobility, safety safety environment by the controller. traffic safety and andFederation environment by using usingControl) the proposed proposed © 2017,mobility, IFAC (International of Automatic Hosting controller. by Elsevier Ltd. All rights reserved. Keywords: Variable speed limit, ramp metering, lane change, feedback linearization Keywords: Keywords: Variable Variable speed speed limit, limit, ramp ramp metering, metering, lane lane change, change, feedback feedback linearization linearization Keywords: Variable speed limit, ramp metering, lane change, feedback linearization 1. INTRODUCTION model and an optimal control approach that decides the 1. model and an optimal control approach that decides the 1. INTRODUCTION INTRODUCTION model and an control approach that the metering rates of multiple ramps in a coordinated manner 1. INTRODUCTION model and an optimal optimal control approach that decides decides the metering rates of multiple ramps in a coordinated manner metering rates of multiple ramps in a coordinated manner and Kotsialos (2002)). SWARM is a dataHighway congestion is detrimental to traffic mobility, (Papageorgiou metering rates of multiple ramps in a coordinated manner (Papageorgiou and Kotsialos Kotsialos (2002)). SWARM is a a dataHighway is to traffic mobility, and SWARM is based ramp metering strategy(2002)). which uses linear regression Highway congestion is detrimental detrimental to occurs traffic when mobility, safety andcongestion the environment. Congestion the (Papageorgiou (Papageorgiou and Kotsialos (2002)). SWARM is a datadataHighway congestion is detrimental to traffic mobility, based ramp metering strategy which uses linear regression safety and the environment. Congestion occurs when the based ramp metering strategy which uses linear regression of measured data to predict the density (Caltrans District safety and the environment. Congestion occurs when the traffic demand is higher than the capacity of a highbased ramp metering strategy which uses linear regression safety and the environment. Congestion occursofwhen the of measured data to predict the density (Caltrans District traffic demand is the measured data predict the (Caltrans District (2006)). intensive application of RM, it is traffic demand To is higher higher than the capacity capacity of aaa highhigh- 7of way bottleneck. preventthan or relieve highway congestion, of measuredDespite data to tothe predict the density density (Caltrans District traffic demand is higher than the capacity of high7recognized Despite the intensive application of RM, it is way bottleneck. To prevent or relieve highway congestion, 77 (2006)). (2006)). Despite the intensive application of RM, it that ramp metering can only control the vehicle way bottleneck. To prevent or relieve highway congestion, different Intelligent Transportation Systems (ITS) tech(2006)). that Despite themetering intensive application ofthe RM, it is is way bottleneck. To prevent or relieveSystems highway (ITS) congestion, recognized ramp can only control vehicle different Intelligent Transportation techrecognized that ramp metering can only control the vehicle density immediately downstream the on-ramp therefore different Intelligent Transportation Systems (ITS) (ITS) tech- recognized niques, e.g. dynamic Transportation routing, driver information systems, that ramp metering can only control the vehicle different Intelligent Systems techdensity immediately downstream the on-ramp therefore niques, dynamic routing, driver information density immediately downstream on-ramp improves the overall traffic the condition in therefore practice, niques, e.g. dynamic routing, driver information systems, variablee.g. speed limit (VSL), ramp metering (RM) systems, etc., are barely density immediately downstream the on-ramp therefore niques, e.g. dynamic routing, driver information systems, barely improves the overall traffic condition in practice, variable speed limit (VSL), ramp metering (RM) etc., are barely improves the overall traffic condition in practice, especially when the mainstream demand is high (Lu et al. variable speed limit (VSL), ramp metering (RM) etc., are widely studied and (VSL), appliedramp to improve the(RM) efficiency of barely improves themainstream overall traffic condition in (Lu practice, variable speed limit metering etc., are especially when the demand is high et al. widely studied and applied to improve the efficiency of especially when the mainstream demand is high (Lu et al. (2010); Scariza (2003)). The above limitations of RM widely studied and applied to improve the efficiency of existing road networks (Abadi et al. (2016); Butakov and especially when (2003)). the mainstream demand is highof(Lu etmoal. widely studied and applied toetimprove the Butakov efficiencyand of (2010); Scariza The above limitations RM moexisting road networks (Abadi al. (2016); (2010); Scariza (2003)). The above limitations of RM motivates the investigation of combining ramp metering with existing road networks (Abadi et al. (2016); Butakov and Ioannou (2015); Zhang and Ioannou (2016); Papageorgiou (2010); Scariza (2003)). The above limitations of RM moexisting road networks (Abadi et al. (2016); Butakov and tivates the investigation of combining ramp metering with Ioannou (2015); Ioannou (2016); Papageorgiou tivates the investigation of metering control strategies suchramp as VSL. Ioannou (2015);LuZhang Zhang and Ioannou (2016); Papageorgiou et al. (1991); et al.and (2011)). For a general highway mainline tivates thetraffic investigation of combining combining ramp metering with with Ioannou (2015); Zhang and Ioannou (2016); Papageorgiou mainline traffic control strategies such as VSL. et al. (1991); Lu et al. (2011)). For aa the general highway mainline traffic control strategies such as VSL. et al. (1991); Lu et al. (2011)). For general highway segment, the traffic demand consists of mainline flow mainline traffic control strategies such as VSL. et al. (1991); Lu et al. (2011)). For a general highway The coordination of RM and VSL considers network mosegment, the demand consists of the mainline flow segment, the traffic traffic demand consists of be theregulated mainline in flow coordination of RM and VSL considers moand the on-ramp flows, which need to a The segment, the traffic demand consists of the mainline flow The coordination of and VSL network mobility, on-ramp queues and fairness between network the mainline and the on-ramp flows, which need to be regulated in a The coordination of RM RM and VSL considers considers network moand the on-ramp flows, which need to be regulated in a bility, on-ramp queues and fairness between the mainline coordinated manner in order to improve the traffic condiand the on-ramp flows, which need to be regulated in a bility, on-ramp queues and fairness between the mainline and the ramps. The objective is to keep a balanced delay coordinated manner in order to improve the traffic condibility, on-ramp queues and fairness between the mainline coordinated manner in order to improve the traffic condiramps. The objective is to keep aa balanced tion at a bottleneck. Ramp metering (RM) and Variable coordinated manner inRamp ordermetering to improve the and traffic condi- and and the the ramps. The is delay time between vehicles on the mainline and the rampsdelay and tion aa bottleneck. (RM) the ramps. The objective objective is to to keep keep a balanced balanced delay tion atLimit bottleneck. Ramp metering (RM)traffic and Variable Variable time between vehicles on the mainline and the and Speedat (VSL) are intensively studied control and tion at a bottleneck. Ramp metering (RM) and Variable time between vehicles on the mainline and the ramps and avoidbetween queues on the ramps from spillingand back toramps the urban Speed Limit (VSL) are intensively studied traffic control time vehicles on the mainline the ramps and Speed Limit (VSL) are intensively studied traffic control avoid queues on the ramps from spilling back to the urban strategies for on-ramp and mainline traffic respectively. Speed Limit (VSL) are intensively studied traffic control avoid queues on the ramps from spilling back to the urban roads. Past efforts to integrate RM with VSL control instrategies for on-ramp and mainline traffic respectively. avoid queues on the ramps from spilling back to the urban strategies forRM on-ramp and mainline mainline traffic flow respectively. Past efforts to integrate RM with VSL control inThe aim offor is to adjust the on-ramp into the roads. strategies on-ramp and traffic respectively. roads. Past efforts to integrate RM with VSL control include the following: Alessandri et al. (1998); Caligaris et al. The aim of RM adjust the on-ramp flow into Past efforts to integrateetRM with VSL control inThe aim in of order RM is isto to to adjust the theoverall on-ramp flowcondition. into the the roads. clude the following: Alessandri al. (1998); Caligaris et al. mainline improve traffic The aim of RM is to adjust the on-ramp flow into the clude the following: Alessandri et al. (1998); Caligaris et (2007) chose the optimal VSL and RM commands based mainline in order to improve the overall traffic condition. the following: Alessandri etand al. RM (1998); Caligarisbased et al. al. mainline in order order toinimprove improve the overall traffic condition. (2007) chose the optimal VSL commands RM is widely usedto United the States and traffic Europecondition. (Horton clude mainline in overall (2007) chose the optimal VSL and RM commands based on a second order model in an open-loop manner. Hegyi RM is widely used in United States and Europe (Horton (2007) chose order the optimal VSL and RM commands based RM is widely used in United States and Europe (Horton on a second model in an open-loop manner. Hegyi et al. (2016) and Caltrans District 7 (2006)). ALINEA, one RM is widelyand used in United States(2006)). and Europe (Horton onal. aa second order in Hegyi et (2005) developed a combined VSL andmanner. RM controller et al. District ALINEA, one second order model model in an an open-loop open-loop manner. Hegyi et al. (2016) (2016) and Caltrans Caltrans District 7 (2006)). ALINEA, one on et al. (2005) developed aa combined VSL and RM controller of the most popular RM strategies, is(2006)). a heuristic local feedet al. (2016) and Caltrans District 77is ALINEA, one et al. (2005) developed combined VSL and RM controller by using model predictive control (MPC) based on the of the most popular RM strategies, a heuristic local feedet al. (2005) developed a combined VSL and RM controller of the most popular RM strategies, is a heuristic local feedusing model predictive control based on the back control method with integralisaction (Papageorgiou of thecontrol most popular RM strategies, a heuristic local feed- by by using model predictive control (MPC) based on the METANET model. Papamichail et(MPC) al. (2008) combined back method with integral action (Papageorgiou by using model predictive control (MPC) based on the back control method with and integral action (Papageorgiou (Papageorgiou model. Papamichail et al. (2008) combined et al. control (1991)).method Smaragdis Papageorgiou (2003) pro- METANET back with integral action METANET model. Papamichail et al. (2008) combined VSL and coordinated RM using an optimal control apet al. (1991)). Smaragdis and Papageorgiou (2003) proMETANET model. Papamichail et al. (2008) combined et al. (1991)). (1991)). Smaragdis and Papageorgiou (2003) propro- VSL and an optimal control apposed FL-ALINEA which and includes feedback downstream et al. Smaragdis Papageorgiou (2003) VSL and coordinated RM using an optimal approach. Lucoordinated et al. (2010)RM usedusing a MPC to generate posed FL-ALINEA which includes feedback downstream VSL and coordinated RM using an approach optimal control control apposed FL-ALINEA which includes feedback downstream proach. Lu et al. (2010) used a MPC approach to generate flow rate instead of occupancy and ALINEA/Q algorithm posedrate FL-ALINEA which includes feedback downstream proach. Lu et al. (2010) used a MPC approach to generate the VSL commands which coordinate with pre-existing flow instead of occupancy and ALINEA/Q algorithm proach. Lu et al. (2010) used a MPC approach to generate flow rate instead of occupancy and ALINEA/Q algorithm commands coordinate with pre-existing whichrate combines queue control with ALINEA. Some model- the flow insteadqueue of occupancy and ALINEA. ALINEA/Q algorithm the VSL commands which coordinate with pre-existing RM VSL controllers. Lu etwhich al. (2011) designed which with modelVSL commands which coordinate withaa MPC-based pre-existing which combines queue control control withdeveloped. ALINEA. Some Some model- the RM controllers. Lu et al. (2011) designed MPC-based based combines RM algorithms are also Coordinated which combines queue control with ALINEA. Some modelcontrollers. Lu et al. (2011) designed a MPC-based RM controller with a linearized first-order model which is based RM algorithms are also developed. Coordinated RM controllers. Lu et al. (2011) designed a MPC-based based RM algorithms algorithms areonalso also developed. Coordinated controller with aa linearized first-order model which is ramp metering is basedare a second orderCoordinated traffic flow RM based RM developed. RM controller with linearized first-order model equipped with a heuristic VSL controller. ramp metering is based on aa second order traffic flow RM controller with a linearized first-order model which which is is ramp metering is based on second order traffic flow equipped with a heuristic VSL controller. ramp metering is based on a second order traffic flow equipped with a heuristic VSL controller.  This paper has been supported by the Department of Transequipped with aofheuristic VSL controller.efforts on combiAlthough most the above mentioned  This has been the of Although most of the on combi portation funded Center forby Sustainable Transportation at This paper paper hasNational been supported supported by the Department Department of TransTransAlthough most of above mentioned efforts on  nation of VSL RMabove claimmentioned significantefforts improvement on This paper hasNational been supported by the Department of TransAlthough mostand of the the above mentioned efforts on combicombiportation Center Sustainable nation of VSL and RM claim significant improvement on University of California, Davis andfor METRANS. portation funded funded National Center for Sustainable Transportation Transportation at at nation of VSL and RM claim significant improvement portation funded National Center for Sustainable Transportation at nation of VSL and RM claim significant improvement on on University University of of California, California, Davis Davis and and METRANS. METRANS.

University of California, Davis and METRANS. Copyright 5487Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 5487 Copyright © under 2017 IFAC IFAC 5487Control. Peer review responsibility of International Federation of Automatic Copyright © 2017 IFAC 5487 10.1016/j.ifacol.2017.08.982

Proceedings of the 20th IFAC World Congress 5308 Yihang Zhang et al. / IFAC PapersOnLine 50-1 (2017) 5307–5312 Toulouse, France, July 9-14, 2017

Fig. 1. Highway Bottleneck traffic mobility, there are also a number of studies questioning the reported improvements on travel time under different incident scenarios, see Torne Santos et al. (2011); Gao (2012); Kejun et al. (2008); Ioannou et al. (2012). Zhang and Ioannou (2016) discovered that the inconsistency between macroscopic and microscopic simulations is mainly caused by the capacity drop phenomenon. A major reason of capacity drop is that forced lane changes in close proximity to the incident or bottleneck bring down the flow speed in neighbor lanes, which is difficult, if not impossible, for VSL to eliminate. Zhang and Ioannou (2016) combined the lane change (LC) control with VSL. By providing lane change recommendations to upstream vehicles, most of the lane changes are performed away from the incident hence the capacity drop is dramatically reduced. In this paper, we use an analytical method to design a coordinated VSL and RM controller based on a cell transmission macroscopic model with triangular fundamental diagram which together with a lane change controller guarantees stability of the traffic flow and convergence of traffic density to the desired equilibrium point exponentially fast. Considering the fact that RM controllers have been widely deployed in the United States, we assume that the RM control command is determined before the VSL and design the VSL controller to coordinate with the RM and stabilize the traffic flow. The coordinated VSL and RM controller with lane change is evaluated using Monte Carlo microscopic simulations and shows significant improvement in traffic mobility, safety and the environment impact. The rest of the paper is organized as follows: Section 2 presents the effect of lane change recommendation and variable speed limit on the fundamental diagram, based on which the cell transmission model is demonstrated. In Section 3, we first design a VSL controller which coordinates with any types of ramp metering controller and guarantees exponential stability of the closed loop system. Then the RM strategy, ALINEA/Q, is adopted in our coordinated VSL and RM controller to adjust both the mainline density and the ramp queue length. In Section 4, we use microscopic simulation results to show the benefits with real world data in peak hours on the I-710 highway. Section 5 presents the conclusion. 2. SYSTEM MODELING 2.1 Capacity Drop and Lane Change Control Consider the highway segment with 3 lanes shown in Fig. 1. A bottleneck is introduced by an incident that blocks one lane. The speed limit upstream the bottleneck is the free flow speed vf = 65 mi/h. The length of the bottleneck is Lb , which we assume is small enough that the effect of the density within Lb is negligible and will not affect the bottleneck flow. The relationship between the vehicle density in the discharging section ρd , and the

(a) w/ and w/o LC

(b) w/ and w/o VSL

Fig. 2. Effects of LC and VSL on Fundamental Diagrams the bottleneck flow qb is shown as the black dashed line in Fig.2a. When ρd is lower than the critical value ρd,c , qb increases with ρd . When ρd is greater than ρd,c , the demand of the bottleneck will be greater than its capacity, which introduces a queue in the discharging section. Forced lane changes performed by the vehicles in the queue reduce the speed of the flow in open lanes, therefore the flow rate will drop to be lower than the capacity of the bottleneck. This capacity drop phenomenon makes it difficult for VSL controllers to increase qb at the bottleneck as VSL is only able to regulate the average density ρd in the discharging section, but can do nothing to the lane changes at the vicinity of the bottleneck. Zhang and Ioannou (2016) shows that by providing appropriate lane change recommendations to upstream vehicles thus allow drivers to make lane changes before fully stopping in the queue, the capacity drop can be avoided or significantly reduced. The fundamental diagram becomes continuous at the critical density as shown by the red solid line in Fig. 2a, whose triangular approximation is the blue dotted line in Fig. 2a. Application of the lane change control makes it possible for the VSL to stabilize ρd at ρd,c therefore the maximum possible qb is achieved. The details of the design method of the lane change controller can be found in Zhang and Ioannou (2016). In the fundamental diagram with lane change control, the low ρd part is very close to its triangular approximation, which means that the flow speed is close to vf . However, the flow speed decreases as ρd approaches ρd,c . Zhang and Ioannou (2016) attributes the reduction of speed to modeling error, delay of speed limit following and driver’s caution when passing the incident site. This deviation of speed will not harm the benefit of VSL with respect to traffic mobility when designing the VSL controller based on the triangular fundamental diagram as long as ρd is stabilized at ρd,c . However, if the speed limit upstream the bottleneck is vf , vehicles need to decelerate when approaching the bottleneck, which leads to shock waves that propagate upstream. If we decrease the speed limit upstream the bottleneck to vd , such that 0 < vd < vf , according to Papageorgiou et al. (2008), the critical density in the fundamental diagram will be shifted to higher value and the slope of the under-critical part of the fundamental diagram will be decreased and made closer to a straight line. Our microscopic simulations confirm this statement. The black solid line in Fig. 2b shows the fundamental diagram under a speed limit of 40 mi/h. Compared to the one under 65 mi/h, which is shown as the blue solid line in Fig. 2b, the capacity of the bottleneck is not decreased despite under

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where ρj,d = vd ρd,c /wb + ρd,c . 3. CONTROLLER DESIGN In this section, the coordinated VSL and RM controller is designed. We first design the VSL controller by assuming that the RM control command is given. Then we choose the ramp metering strategy, ALINEA/Q, to manage the ramp flows and the queue lengths on ramps.

Fig. 3. Configuration of the Highway Segment a lower speed limit as the critical density is increased from ρ˜d,c to ρd,c . As we can see in the figure, this fundamental diagram is very close to its triangular approximation, that is, the speed deviation at ρd,c is very small. If we design the coordinated VSL and RM controller based on this fundamental diagram and let the VSL command converge to vd at the equilibrium state, the shockwave upstream the bottleneck will be attenuated. We demonstrate this with microscopic simulations in Section 4. To conclude, under speed limit of vd , the highway bottleneck can be modeled with high accuracy as:  v d ρd , ρd ≤ ρd,c (1) qb = wb (ρj,d − ρd ), ρd > ρd,c where ρj,d = vd ρd,c /wb + ρd,c . 2.2 Cell Transmission Model The highway segment to be controlled by the coordinated VSL and RM controller is shown in Fig. 3. The bottleneck is introduced by a lane closure. The highway segment upstream the bottleneck is divided into N + 1 sections, which are indexed as section 0 through section N . For i = 0, 1, ..., N , ρi , qi , ri , si represent the vehicle density, mainline in-flow rate, on-ramp flow rate and off-ramp flow rate in section i respectively, where ρi , si are measurable, ri are determined by the RM controller, therefore also measurable. For i = 0, 1, ..., N − 1, vi denote the variable speed limit in section i. In section N , the speed limit is a constant denoted by vd . qb denotes the flow rate through the bottleneck. Let Ri = ri − si be the net ramp flow and Li the length of section i, for i = 0, 1, ..., N . According to the flow conservation law, we have 1 (qi − qi+1 + Ri ), for i = 0, 1, ..., N − 1 ρ˙i = Li (2) 1 (qN − qb + RN ) ρ˙N = LN Under the assumption of triangular fundamental diagram, we have q0 = min{d, C0 , w0 (ρj,0 − ρ0 )} (3) qi = min{vi−1 ρi−1 , Ci , wi (ρj,i − ρi )}, i = 1, . . . , N where d is the demand flow of this highway segment. Here we assume d is constant. ρj,i is the jam density of section i, at which qi would be 0. wi is the backward propagating wave speed in section i, Ci the capacity, i.e. the maximum possible flow rate in section i, given by the equation Ci = vi wi ρj,i /(vi + wi ). Section N works as the discharging section in Fig. 3. By applying the lane change controller, capacity drop at the bottleneck can be removed. According to (1), we have  v d ρN , ρN ≤ ρd,c qb = (4) wb (ρj,d − ρN ), ρN > ρd,c

3.1 Design of VSL The goals of designing the VSL controller include: (1) Given any type of RM controller, the VSL controller should be able to coordinate with it and stabilize the density ρN in the discharging section at the critical value ρd,c , in order to keep qb at the highest level. (2) Homogenize the traffic flow upstream the bottleneck in order to improve the traffic safety and bring environmental benefits. Consider the subsystem which includes section 1 through section N. Define the error states ei = ρi − ρd,c , for i = 1, 2, ..., N We have 1 (vi−1 ρi−1 − vi ρi + Ri ) e˙ i = Li for i = 1, 2, ..., N − 1 e˙ N = (5)  vN −1 ρN −1 − vd ρN + RN   , ρN ≤ 0 LN   vN −1 ρN −1 − wb (ρj,b − ρN ) + RN , ρN > 0 LN Let

vi =

−λi Li+1 ei+1 + vd ρd,c −

N

j=i+1

Rj

, ρi for i = 0, 1, ..., N − 2

vN −1 =  −λN −1 LN eN + vd ρN − RN  , ρN ≤ ρd,c  ρN − 1   −λN −1 LN eN + wb (ρj,b − ρN ) − RN , ρN ≤ ρd,c ρN − 1 (6)

Substitute the controller (6) into the open-loop system (5), we have the following closed-loop system: Li+1 λi ei+1 e˙ i = −λi−1 ei + Li for i = 1, 2, ..., N − 2 e˙ N −1 =  LN  (λN −1 − vd )eN , ρN ≤ 0  −λN −2 eN −1 + LN −1 LN   −λN −2 eN −1 + (λN −1 + wb )eN , ρN > 0 LN −1 e˙ N = −λN −1 eN (7) Theorem 1. ei = 0, for i = 1, 2, ..., N is the unique and isolated equilibrium point of the closed-loop system (7) and is guaranteed to be globally exponentially stable. The rate of exponential convergence depends on the control design parameters λi , i = 0, 1, ..., N − 1.

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The proof of Theorem 1 can be found in Zhang and Ioannou (2016). According to Theorem 1, the steady state value of ρi is ρi,ss = ρd,c , i = 1, ..., N . The steady state N value of vi is vi,ss = vd − j=i+1 Ri /ρd,c , i = 1, ..., N − 1. Therefore, by applying the coordinated VSL and RM controller, ρ1 through ρN are stabilized and homogenized. The effect of a ramp flow is compensated by its upstream VSL and does not affect downstream traffic. If Ri = 0, then vi,ss = vd , for i = 1, ..., N − 1. That is the upstream speed limit converges to vd . By adjusting the value of vd , we can guarantee that the shockwave resulted by speed deviation between actual traffic flow and the triangular fundamental diagram is eliminated.

manage the queue on the ramps so that the queues do not spill backwards to the urban road network. We adopt the ALINEA/Q, which modifies the classic ALINEA ramp metering strategy with queue adjustment. The original ALINEA/Q method proposed in Smaragdis and Papageorgiou (2003) includes the downstream occupancy and the queue length in the feedback loop. In this paper, to be consistent with the VSL controller, we use the downstream density instead of occupancy.

Now let us consider the dynamics of ρ0 and v0 . Since q1 converges to vd ρd,c , if the demand d > vd ρd,c , ρ0 will increase. Once ρ0 > ρj,0 − d/w0 , we have 1 (w0 (ρj,0 − ρ0 ) − v0 ρ0 + R0 ) (8) ρ˙ 0 = L0 Substitute (6) into (8), we have N  1 ρ˙ 0 = (w0 (ρj,0 − ρ0 ) − vd ρd,c + Rj ) L0 j=0 N Assume that j=0 Rj is constant, then N j=0 Rj − vd ρd,c ρ0 = ρj,0 + w0 N is a stable equilibrium point. As long as j=0 Rj < vd ρd,c , ρ0 will not exceed the jam density ρj,0 and v0 will not go negative, thus the VSL controller is feasible.

rid (k) = r(k − 1) + βd [(ρd,c − ρi (k))] riq (k) = βq (wir − wi (k)) + di (k − 1)

For driver’s acceptance and safety, some constraints are applied on the VSL controller (6): The commands are (a) discretized in time; (b) rounded up to multiples of 5 mi/h; (c) not allowed to decrease too fast between successive sections and time steps. The constrained VSL controller is as follows: Let vi (k) denote the VSL command computed with (6) at time t = kTc , where Tc is the control step size. v˜i (k) = max{[vi (k)]5 , v¯i (k − 1) − Cv , v¯i−1 (k) − Cv } (9)  vmax , if v˜i (k) > vmax (10) v¯i (k) = vmin , if v˜i (k) < vmin v˜i (k), otherwise for i = 1, 2, . . . , N − 1, k = 0, 1, 2, . . .. In (9), [·]5 is the operator which rounds a real number to its closest multiple of 5. v¯i (k) in (10) is the final constrained VSL command for section i at time t = kTc . 3.2 Design of the RM Controller According to Theorem 1, the VSL controller (6) can stabilize the system and improve the mobility as long as the net ramp flow is lower than the bottleneck capacity. It seems that RM control is unnecessary. However, if no RM is applied and large ramp flows flush into the mainline, the merging of ramp flows will severely disturb the mainline flow. Furthermore, when the net ramp flow is high, the VSL controller (6) will suppress the mainline flow in order to spare the capacity for the ramp flows. That is, without RM control, the ramp flow will always have priority which may harm the fairness between the ramp flows and the mainline flow, or even make the VSL controller infeasible. Furthermore, the RM controller should be able to

For an on-ramp i, two RM rates, rid (k) and riq (k), are decided respectively based on the downstream density and the queue length on the ramp at each time step t = kTc . The final RM rate ri (k) is the maximum of the two. i.e.

ri (k) =

(11)

max{rid (k), riq (k)}

where ρi (k) is the density in the highway section that connects to ramp i, wi (k) is the queue length on ramp i at time step k, di (k − 1) is the demand from ramp i within time step k − 1, wir is the reference queue length of ramp i. rid (k) is an integral feedback controller that regulates ρi (k) to be close to ρd,c , which helps maintain the vehicle density on mainline at the desired equilibrium value. riq (k) adjusts the RM rate in order to prevent the queue length from being too large, i.e. if wi (k) is larger than wir , the RM rate will increase to discharge excessive vehicles in the queue and newly arrived vehicles. Since the final RM rate is the maximum of the two, the ramp flow will get the priority to pass the bottleneck if the ramp queue is large, while the mainline flow will get the priority if the vehicle density on the mainline is high. In this way, the ALINEA/Q strategy maintains the fairness between the ramp flows and the mainline flow and avoids the ramp queues from piling up towards the urban road. 4. NUMERICAL SIMULATIONS In this section , we use the microscopic simulator VISSIM to carry out Monte Carlo simulations to evaluate the performance of the coordinated VSL, RM and lane change control on traffic mobility, safety and the environment. 4.1 Scenario Setup We evaluate the proposed controller on a segment of the I-710 freeway in California, United States (between I-105 and the Long Beach Port). As shown in Fig. 4, the highway segment is divided into 8 sections, the VSL signs are deployed at the beginning of section 0 through 6. An incident blocks the middle lane at the end of section 7 and creates a bottleneck. 4 on-ramps, which are equipped with RM, and 5 off-ramps are connected to the highway segment. The lane change control is deployed at the beginning of section 7. The simulation network is calibrated with real world data from the PeMS system (Caltrans (2015)). The incident occurs at 5 minutes after simulation starts, and lasts for 30 min. The capacity of the highway segment is 6800 veh/h without incident. During the incident, the ideal bottleneck capacity is about 4500 veh/h. We load the network with the real demand at 5pm on Monday, which is a peak hour. The mainline demand

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capacity drop, and increases right away after the incident is removed as the queue in the bottleneck area flushes downstream. When the controller is applied, the flow rate decreases to around 4200 veh/h, which is higher than that in the no control case since the capacity drop is avoided by the lane change control and VSL stabilizes the vehicle densities. The bottleneck flow starts increasing about 10 min after the incident is removed as the high density area is held in section 0 by the VSL controller. The high density wave moves forward from section 0 and the flow rate qb starts increasing once the wave front reach the bottleneck.

Fig. 4. Geometry of Simulation Network

Fig. 5. Bottleneck Flow

(a) Density in section 7

with control,

Fig. 6 shows the curve of ρ7 and ρ0 , which are the vehicle density of the discharging section and the first VSL controlled section, respectively. When there is no control, ρ7 starts increasing immediately as the incident occurs at t = 5 min. In addition the shockwave propagates upstream, which makes ρ0 starts increasing at t = 25 min and reaches 500 veh/mi. The high density in section 0 does not discharge until 15 min after the incident is removed. When the coordinated controller is applied, ρ7 increases slightly and is stabilized at 110 veh/mi. ρ0 increases immediately after the incident since v0 decreases to reduce the flow into downstream sections and is stabilized at around 400 veh/h which is lower than that without control.

no control

(b) Density in section 0

Fig. 7 demonstrates the contour plot of vehicle densities with respect to time and space with different values of vd . When vd = 40 mi/h, high density is held in section 0 during the incident, while downstream sections are highly homogenized. ρ2 is higher than ρd,c at the beginning of the incident as the ramp flows r11 and r12 flush in but then discharged under control. The density in section 6 is slightly higher than ρd,c as vehicles receive the lane change recommendations and make lane changes thus slightly disturbs upstream flow. When vd = 65 mi/h, as explained in Section 2.1, a shockwave propagates upstream. After the incident is removed, the vehicles in section 0 flush downstream and meet with the shockwave, which leads to a high density area in section 2. However in this case, the discharging section is still well protected. As the shockwave propagates upstream, vehicle densities converge to ρd,c gradually from downstream section to upstream section. This is because we use the cascade structure of VSL controller in Fig. 3, which attenuates the shockwave section by section. Thus the controller is robust to parameter selection.

Fig. 6. Vehicle Densities w/ and w/o Control with control, no control

(a) vd = 40 mi/h

(b) vd = 65 mi/h

Fig. 7. Density Contours

(a) Queue length on r11

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(b) Queue length on r4

Fig. 8. Queue Length w/ and w/o Control VSL + RM, RM only is 4500 veh/h, the on-ramp demand from upstream to downstream are 400 veh/h, 500 veh/h, 300 veh/h, 300 veh/h respectively. 4.2 Simulation Results Fig. 5 shows the bottleneck flow with and without the coordinated VSL, RM and lane change control. When there is no control, the flow rate decreases immediately to around 3000 veh/h due to the lane blockage and

Fig. 8 shows the queue length on ramp r11 and r3 , with RM control alone and with the coordinated controller. With RM control alone, the queues pile up fast as the densities in mainline increase. Due to the queue adjustment mechanism of ALINEA/Q, the queue lengths are maintained around the reference value. With the coordinated controller, the queue lengths increase in the transient process when the incident begins and the mainline density is being adjusted to the desired level and then discharge fast. After the incident is removed, large flow flushes downstream, the RM controller decrease the rate to give priority to the mainline, therefore the queue lengths increase. We use the following metrics to evaluate the performance of the coordinated controller. To evaluate traffic mobility, we use: (a) average travel time T¯t ; for traffic safety, use (b) average number of stops s¯ and (c) average number of

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Table 1. Evaluation Results Control Type T¯t (min) s¯ c¯ CO2 (g/veh/mi) Fuel (g/veh/mi)

No Control 15 23 5.1 585 187

RM + VSL 11 4 4.6 538 172

Improvement 27% 82% 10% 8% 8%

lane changes c¯; for the environment, we use (d) average emission of CO2 and (e) average fuel consumption. The detailed definition of the above metric can be found in Zhang and Ioannou (2016). Table 1 shows the evaluation results. The improvement in traffic mobility, safety and the environment is significant. The average travel time is reduced by about 27% as the bottleneck throughput is increased. For traffic safety, the number of stops dramatically decreased by 81% as the lane change control prevented vehicles from stopping at the bottleneck and waiting for lane changes. The 10% reduction in number of lane changes is contributed by both homogenization of mainline flow and the regulated merging behavior of ramp flows. For the environment metrics, the reductions of CO2 emission and energy consumption are usually proportional to each other, which are both around 8% in this case. 5. CONCLUSION In this paper, we developed and evaluated a coordinated variable speed limit, ramp metering and lane change controller for highway traffic. In the proposed method, the lane change control prevents the capacity drop at the bottleneck. A nonlinear VSL controller is designed to coordinate with ALINEA/Q RM strategy in order to achieve maximum flow rate. The ALINEA/Q RM strategy considers both reducing the merging disturbance of the ramp flow to the mainline flow and avoiding the ramp queues from spilling back to the urban road. Microscopic simulations show that the system behaves as predicted by theory with the proposed controller. Traffic mobility, safety and the environment are significantly improved. REFERENCES Abadi, A., Ioannou, P.A., and Dessouky, M.M. (2016). Multimodal dynamic freight load balancing. IEEE Transactions on Intelligent Transportation Systems, 17(2), 356–366. Alessandri, A., Di Febbraro, A., Ferrara, A., and Punta, E. (1998). Optimal control of freeways via speed signalling and ramp metering. Control Engineering Practice, 6(6), 771–780. Butakov, V. and Ioannou, P. (2015). Personalized driver/vehicle lane change models for adas. IEEE Transactions on Vehicular Technology, 64(10), 4422–4431. Caligaris, C., Sacone, S., and Siri, S. (2007). Optimal ramp metering and variable speed signs for multiclass freeway traffic. In 2007 European Control Conference, 1780–1785. IEEE. Caltrans (2015). Caltrans performance measurement system (PeMS). URL http://pems.dot.ca.gov/. Caltrans District 7 (2006). Ramp Metering Annual Report. Technical report, Los Angeles and Ventura Counties.

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