Variable Speed Limit Design to Relieve Traffic Congestion ... - Core

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ScienceDirect Procedia - Social and Behavioral Sciences 138 (2014) 427 – 438

The 9th International Conference on Traffic & Transportation Studies (ICTTS’2014)

Variable Speed Limit Design to Relieve Traffic Congestion Based on Cooperative Vehicle Infrastructure System Rui Sun, Jianming Hu, Xudong Xie*, Zuo Zhang Department of Automation, Tsinghua University, Haidian District, Beijing 100084, P.R. China

Abstract To uniform traffic flow, and to improve traffic mobility and safety, variable speed limit (VSL) is usually implemented on freeways. With predictive models, traffic states evolutions can be predicted, so VSL values can be optimized within model predictive control (MPC) frameworks to keep traffic flow at a high efficiency and suppress propagation of shockwaves, especially during congestion periods. METANET model is a widely used predictive tool, as well as its many extensions. In this paper, a new extension of METANET is proposed based on cooperative vehicle infrastructure system (CVIS), also known as vehicle-infrastructure integration (VII) system. By introducing vehicle models, vehicles’ speed and position on each link are recorded, and prediction is changed to a mesoscopic scale, which describes the traffic entities at a high level of detail, but describes their behaviour and interactions at a lower level of detail (Van Woensel et al. 2007). Compared with previous studies that use macroscopic variables for prediction, the new model suits the situations in CVIS, where communication between vehicles and infrastructures will be adopted to replace variable message signs (VMS). Simulation results have shown the capacity of the proposed model in predicting state evolutions. Additionally, VSL’s ability to improve traffic efficiency and prevent congestion propagation is proved. © 2014 2014 Published by Elsevier Ltd. is an Ltd. open access article under the CC BY-NC-ND license © The Authors. Published byThis Elsevier (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of Beijing Jiaotong University (BJU), Systems Engineering Society of China (SESC). Peer-review under responsibility of Beijing Jiaotong University(BJU), Systems Engineering Society of China (SESC). Keywords: Variable speed limit; Model predictive control; METANET; Cooperative vehicle infrastructure system; Traffic congestion

* Corresponding author. Tel:+86- (0)10-6279-0756; Fax: +86-(0)10-6278-6911. E-mail address: [email protected].

1877-0428 © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Peer-review under responsibility of Beijing Jiaotong University(BJU), Systems Engineering Society of China (SESC). doi:10.1016/j.sbspro.2014.07.221

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1. Introduction As freeways increasingly networked and the number of vehicles grows, traffic congestion now has an increasing negative impact on the mobility and efficiency of freeways. Although construction of new roads can to some extent alleviate the problem, a more promising and cost-efficient solution is to use the existing infrastructure more efficiently through dynamic traffic control methods, such as ramp metering (RM), variable speed limits (VSL), route guidance (RG) and so on. Among them, VSL has shown its effect in reducing accidents as well as improving mobility and efficiency of freeways. However, in most field studies, improvement about travel time and capacity is not as significant as improvement about safety (Hadiuzzaman et al. 2013). This is because VSL are mostly implemented in a reactive manner. To prevent traffic breakdown actively instead of responding to it, model predictive control (MPC) methods are proposed. In this way, traffic planners can predict the formation of active bottlenecks and evaluate traffic improvement under different control values and strategies. Therefore they are able to post optimized speed limit values to prevent breakdown and relieve traffic congestions. In MPC approach, accurate predicting models are desirable. As a macroscopic modeling tool, METANET (Kotsialos et al. 2002) and its many extensions have been widely used in many previous works in order to make precise predictions of traffic flow state variables. Hegyi et al. (2005) proposes a new VSL model and incorporates speed limits within METANET model, revising previous methods in which the effect of speed limit is expressed by downscaling the desired speed-density diagram. The paper also includes the extension that describes the different effects of a positive or negative downstream density gradient on the speed, and extension that includes the modeling of a mainstream origin. Later, Carlson et al. (2010) extends METANET model and describes the effect of displayed VSL values using affine functions. By rendering static speed-density relationship rate-dependent, the model includes link-specific VSL rates into the link model. In addition to extensions on VSL effect, density and flow variables are also considered. Hadiuzzaman et al. (2013) includes the capacity drop concept in FD to model active bottleneck and introduces cell transmission model (CTM) (Daganzo, 1994) to represent density dynamics, replacing previous assumption that transition flow among the links is equivalent to average link flow. All these extensions are aimed for a more precise prediction. However, they are based on macroscopic data, collected by loop detectors in the real world. Due to the advancements in information technology, the integration of vehicles and infrastructure has been receiving significant attention. One particular approach is that of cooperative vehicle infrastructure system (CVIS), also known as vehicle-infrastructure integration (VII), which is recognized to have the potentials in improving traffic safety and mobility to a greater level (Paniati, 2005). With complete network information provided by the probe vehicles, such as vehicle’s real-time speed and position in the corridor, the density of links can be estimated, as well as the mean speed. Thus it is feasible to predict traffic flow states by predicting the position and speed of each vehicle, a microscopic approach, instead of predicting the evolution of density and mean speed of each link, in macroscopic ways. This paper proposes an extension of METANET model based on cooperative vehicle infrastructure system, where speed and density evolutions are estimated using microscopic data. Thus prediction scale is transferred from links to vehicles. As a vehicle will probably move to the downstream links during a time period, its speed is affected by several consecutive links instead of one. So its speed prediction should take into account the mean speed evolution of the downstream links as well as the current link. Travel distance of each vehicle is calculated using its mean speed during a single time step. Then, by recording the group of vehicles in the link, link density and meanspeed, the macroscopic variables, during the predicted time horizon can be estimated. It is expected that through the proposed model, traffic state variables can be precisely modeled and predicted. Additionally, based on the new model, VSL values can be optimized using MPC approach. The rest of this paper is organized as followings. The methodology section presents the VSL control strategy and the proposed traffic flow predictive model. Then in order to optimize VSL values, the MPC framework with the objective to improve total travel time and total flow is introduced. In the simulation section, the studied corridor and calibration steps are presented. Details in simulation design and the final results are also included in this part. Finally, a summary including conclusions and future research scope is given.

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2. Methodology 2.1. Control strategy In this paper, VSL is proposed to increase the flow at bottleneck and prevent congestion propagation. According to the explanation in Hegyi et al. (2005), imposing speed limits to the upstream of congestion area will reduce the outflow of the upstream and the inflow to the congestion area as well. When inflow value of congestion area is lower than its outflow value, congestion will dissolve. So to the upstream of bottleneck, a discharge section with adequate length is necessary, usually 500-700m. At this end, a critical VSL is displayed to control the flow into the discharge section, as shown in Fig. 1. 2.2. Basic METANET model In the basic model, in order to predict the evolution of traffic state, the freeway section is divided into several links (1, 2,…, M) and time is discretized by single time step, denoted as T. Section division is based on the principles that each link has uniform characters, such as no on/off-ramps or no major change in geometry (Kotsialos et al. 2002). Each link is characterized by traffic density Ui (k )(veh/ km/ lane) , traffic flow qi (k )(veh/ h/ lane) , and mean speed vi (k )(km/ h) , where k stands for time t = kT ( k 0,1,..., K , K is the time horizon). Evolution of these macroscopic variables is predicted by the following equations.

qi (k )

Ui (k  1)

Ui ( k ) 

vi (k  1) 

Ui (k )vi (k )

T [Oi 1qi 1 (k )  Oi qi (k )  ri (k )  si (k )] Li Oi

(1) (2)

T vi (k )  {Ve,i [ Ui (k )]  vi (k )}

W

1 TX Ui 1 (k )  Ui (k ) T ] vi (k )[vi 1 (k )  vi (k )]  [ Li W Li Ui ( k )  N

(3)

Fig. 1. VSL control strategy at active bottleneck: (a) weaving section; (b) lane drop.

Equation (1) is the fundamental relationship between speed, flow and density and Equation (2) illustrates the conservation of vehicles. In Equation (3), W , X , N are global model parameters and desired speed Ve,i [ Ui (k)] is calculated by

Ve,i [ Ui (k )] v free,i exp[

1 Um,i (k ) am ( ) ] am Ucrit , m

(4)

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where am is a model parameter, v free,i represents the free-flow speed of link i , Ucrit , m represents the critical density at which traffic flow reaches its maximum value. 2.3. Proposed model Based on cooperative vehicle-infrastructure system, information about individual vehicles is incorporated. To illustrate this vehicle model, the following notifications are used.

u (k ) vvehicle (k ) xvehicle (k ) ivehicle (k ) xsta ,i

speed limit value at time t = kT speed of vehicle at time t = kT global position of vehicle at time t = kT link index of vehicle at time start point of link i

xend ,i

end point of link i

Li Nvehicle,i (k )

length of link i total count of vehicles on link i at time t = kT

vtotal ,i (k )

total sum of vehicles speed on link i at time t = kT

vi (k )

mean speed of link i at time t = kT

Ui ( k )

density of link i at time t = kT

As vehicle is likely to move to the downstream links during the time step, speed prediction should take evolutions of several consecutive links into account. Equation (3) is modified under the assumption that vehicle only travels through the current link and its next downstream link, which is reasonable given the length of each link and the length of a single time step. Vehicle’s next speed is assumed to change with the evolution of link j ,

vvehicle (k  1) 

vvehicle (k ) 

T

W

[Ve , j [ U j (k )]  v j (k )]

T 1 TX U j 1 (k )  U j (k ) v j (k )[v j 1 (k )  v j (k )]  [ ] Lj W Lj U j (k )  N

(5)

where

j ivehicle (k )  1 , ­ ½ 1 U (k ) ° ° Ve,i [ Ui (k )] min ®u (k ), v free,i exp[ ( m,i ) am ]¾ a U ° ° m crit , m ¯ ¿

(6)

Equation (6) modified the desired speed of each link, which results from the VSL model in Hegyi et al. (2005). It is the minimum value between displayed speed limit and the speed based on the experienced speed-density relation. And with vehicle’s coordinate in the corridor recorded, its position is predicted by

xvehicle (k  1)

1 xvehicle (k )  (vvehicle (k  1)  vvehicle (k ))* T 2

(7)

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if xsta,i  xvehicle (k  1) d xend ,i , it means the vehicle will appear on link i at next time step. Vehicle speed will thus be included into vtotal ,i (k  1) , and vehicle will be counted into Nvehicle,i (k  1) . Macroscopic variables are estimated for evaluation of traffic states by

Ui (k  1)

vi (k  1)

Nvehicle,i (k )

Oi * Li vtotal ,i (k ) N vehicle,i

(8)

(9)

and transition flow is assumed to be the average flow, as it is in Equ(1). Because flow prediction is only used for evaluating traffic states according to these equations, accuracy of the proposed model will not be influenced by the accuracy of flow prediction. But the proposed model requires high accuracy of speed prediction. Besides that, prediction methods based on each vehicle are more suitable for situations in CVIS, where communications are mostly between vehicles and vehicles and infrastructures. Variable message signs (VMS) are likely to be replaced by messages sent to drivers, so that VSL effect should be reflected in the vehicle model. 2.4. Objective Function Objective to maximize the traffic flow at bottleneck is equivalent to minimize total travel time (TTT) based on the MPC approach. However, only minimizing TTT tends to keep link density lower, which conflicts with the purpose to manage traffic flow close to its capacity and to improve freeway efficiency (Hadiuzzaman et al. 2013). Moreover, to maximize total travel distance (TTD) only tends to keep link density at a high level by increasing flow, which makes traffic state vulnerable to congestion and breakdown. To balance these two factors, a weight coefficient is introduced and the following objective function is used.

J

DTTT TTT  DTTDTTD N p 1 M

T ¦ ¦ Oi Li [DTTT Ui (k  j )  DTTD Ui (k  j )vi (k  j )]

(10)

j 1 i 1

2.5. Constraints Displayed VSL is the value to be optimized. When take driver safety and freeway efficiency into consideration, adopted constraints are as followings. C1: ui (k ) d Vmax C2: ui (k ) t Vmin C3: ui (k )  ui (k  1)µd Vmax,dif where, Vmax is determined by the regulations related to studied freeway . Vmin is proposed to guarantee the freeway efficiency. Vmax,dif constrains the change of speed limits between consecutive time steps in order to make sure that the maximum decrease of speed is in a safety range.

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3. Simulation 3.1. Studied corridor The studied site is part of the fourth ring freeway in Beijing, from Haidian Bridge to Zhixin Bridge. The corridor consists of 4 groups of on/off ramps, with a total length aoubt 6 kilometers, as it is shown in Fig. 2. The section between the second group of on/off ramps consists of 4 lanes, and when congestion occurs at the off-ramp, it will propagate to the upstream quickly, affecting the mobility and efficiency of the corridor. In simulation, division of the corridor is shown in Fig. 3.

Vmax , Vmin , Vmax,dif are set to 80 km/h, 30 km/h, 10 km/h.

Fig. 2. Studied corridor

Fig. 3. Corridor division in simulation

3.2. Calibration Calibration of the model includes two steps. For the first step, link specific parameters, such as critical density and free flow speed, are calibrated under the assumption of a triangle-shaped FD, and the method for this can be found in Dervisoglu et al. (2009). For the second step, global parameters in Equation (5) are calibrated, using Equation (10) as the measurement of accuracy to obtain an optimal parameter vector E [W ,X , N ] . Given the objective function is nonlinear, an algorithm derived from SQP (Boggs et al. 1995) is chosen, and search range is set to Emin [0.001,10,10] and Emax [0.1,60,60] .

f (E )

1 vimeasured (k )  vipredicted (k | E ) 2 [ measured ] ¦¦ (k )  vipredicted (k | E ) i 1 k 1 2 vi N

Np

(11)

To do the calibration, real-world data are replaced by simulation data generated by VISSIM. SQP is implemented using MATLAB function fmincon. Finally, the following link specific parameters are adopted with little variance: V free 76 kmph, Ucrit 30 vpkmpl, E [0.03, 43, 24] . 3.3. MPC design Simulation platform used in this study is VISSIM 4.30 and MPC design is implemented using VS2008. More details concerning model predictive control approach can be found in Camacho and Bordons (2004). Single simulation time step ( T ) is 1 min, and total simulation time is 1 h 40 min, namely 100 simulation time steps. Time horizon in MPC ( N p ) is 5 min, which equals to 5 prediction steps. Vehicle inputs at the origin of on-ramps and the main road are set to about 550 vehicles/h and 4500 vehicles/h as the highest demand. In VISSIM, destination flow is given by proportions in routing decisions, and at the active bottleneck, about 20% vehicles from the mainstream

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move to the off-ramp, reducing the mobility of the weaving section. These origin-destination flows are known for model prediction. For each time step, through component object model (COM) provided by VSL, vehicles are all recorded. And in the C++ program, within the MPC framework, different VSL values, usually 2 or 3 values ( ui , ui r 10 ), are compared to find the optimal one that results the minimum value of objective function ( J ) during the future time horizon N p . Control horizon N c is 1 min, which means the optimized value is only applied during the next single time step period. Afterward, time axis shifts, next VSL value is optimized. Then, a sequence of optimal control values is generated. Fig. 4 shows an example of this sequence. 3.4. Prediction results Collected data are analysed using MATLAB and the comparison between predicted data and collected data of links near the bottleneck are shown in Fig. 5. From the diagrams, it can be concluded that the proposed model prediction capture the main features of congestion formation and traffic state evolution. 3.5. No-VSL control results Simulation results without VSL control are presented in Fig. 6.According to definition in Daganzo (1999), active bottleneck is a location where queues exist on the upstream, and unrestricted traffic flows continue on downstream segment. As it is shown by the speed curves in Fig. 6(a), at time step 10, mean speed of link 4 begins to drop while mean speed of link 5 maintains at a high level, reflecting that a bottleneck forms at link 4. The active bottleneck is caused by vehicles’ lane-changing behavior at weaving sections. At time step 35, mean speed of link 3 begins to drop, which is the result of congestion propagation, density curves in Fig. 6(b) also illustrate this.

Fig. 4. Sequence of speed limit values

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Fig. 5. Comparison between collected data and predicted data (density, speed)

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Fig. 6. No-VSL control results: (a) speed evolution near the bottleneck; (b) density evolution near the bottleneck; (c) density profile of each link

3.6. VSL control results VSL is displayed upstream of the identified bottleneck, approximately 600 m. In the objective function, if DTTD is set to 1, then DTTT is the ratio between TTD and TTT, which is equal to speed. In the simulation, DTTT is set to 80. Fig. 7 shows the results of VSL control. Traffic state improvement is obviously observed. Through implementation of the proposed VSL, mean speed of link 4 is kept above 20 km/h and mean speed of link 3 is kept at the free flow level. Also, density of link 4 is reduced, and density of link 3 drops below its critical value. Although speed drop and high density is not eliminated completely, shockwaves have been successfully suppressed, and congestion propagation has been significantly prevented. Flow comparison between no-VSL and VSL cases is shown in Fig. 8. During comparatively low traffic demand, flow may be reduced by implementation of VSL and this can be explained by previous studies (Papageorgiou et al. 2008; Lu et al. 2010). During congestion periods, flow at the bottleneck has been improved, though not very significantly. This is because that, the congestion after a breakdown usually has an outflow that is 5%- 10% lower than the capacity, also known as capacity drop (Cassidy 1999). Here, flow is calculated using Equation (1).

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Fig. 7. VSL control results: (a) speed evolution near the bottleneck; (b) density evolution near the bottleneck; (c) density profile of each link

Fig. 8. Flow comparison between no-VSL and VSL cases

Results above prove the effect of implemented VSL in improving traffic mobility when traffic demand is high. Fig. 9 shows the optimal speed limit values over the simulation time.

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Fig. 9. Optimal VSL values over the simulation time

To generally compare the performance during congestion period, the averaged values over several simulation runs are shown in Table 1 (TF is sum of all the links and each link flow is averaged over congestion periods). Congestion begins from simulation time step 10 to simulation time step 90. Average results show that TTT has been decreased by 22.1% through the implementation of VSL, and TTD has been improved by 2.0%. Moreover, TF has been increased by 2.3%, which indicates a capacity improvement for the studied corridor. Table 1. Table captions should always be positioned above the tables. Average values

Without VSL

With VSL

Improvement

TTT (veh-h )

750

584

-22.1%

TTD(veh-km )

28,872

29,462

2.0%

TF(vph)

25,627

26,227

2.3%

4. Summary In this paper, VSL design is combined with MPC approach to increase traffic flow at bottleneck and prevent congestion propagation. Based on the cooperative vehicle infrastructure system, a new extension of the classic METANET model is proposed. By introducing the vehicle model, a more microscopic way is presented to predict traffic state variables: density dynamics prediction is replaced by position prediction, and in terms of speed prediction, unlike macroscopic models, the proposed model takes into account the movement of vehicles during a time step. Although some assumptions and equations are empirical, the introduction of the vehicle model is instructive and suits the situations in CVIS, where messages are given to drivers instead of in the format of VMS. The simulation results have shown that the new model captures the main features of congestion formation and evolution. Also, they prove that designed VSL could prevent congestion propagation and suppress the shockwaves to a great extent. Compared with the previous models, the proposed model simplifies the prediction of density. On one hand, this may result in better accuracy. But on the other, it requires more accurate prediction of speed, and thus the ways to estimate speed evolution using macroscopic data need to be revised, maybe based on other microscopic predictive models. This is open for future researches. Acknowledgements This work was supported by National Basic Research Program of China (973 Project) 2012CB725405, Hi-Tech Research and Development Program of China (863 Project) 2011AA110405, National Natural Science Foundation China 61273238, National Science and Technology Support Program 2013BAG03B01 and the Henry Fok Foundation 122010.

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