Queue length estimation at signalized intersections based on magnetic sensors by different layout strategies

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Transportation Research Procedia 25C (2017) 1629–1647 www.elsevier.com/locate/procedia

World Conference on Transport Research - WCTR 2016 Shanghai. 10-15 July 2016 World Conference on Transport Research - WCTR 2016 Shanghai. 10-15 July 2016

Queue length estimation at signalized intersections based on Queue length estimation at signalized intersections based on magnetic sensors by different layout strategies magnetic sensors by different layout strategies c b c a Haijian Li a,b a,b, Na Chen c, Lingqiao Qin b, Limin Jia c, Jian Rong a* Haijian Li , Na Chen , Lingqiao Qin , Limin Jia , Jian Rong *

a Beijing Key Laboratory of Traffic Engineering, Beijing University of Technology, Beijing 100124, China a Department of Civil and Environmental Engineering,Beijing University of Wisconsin-Madison, Madison, WI 53706, Beijing Key Laboratory of Traffic Engineering, University of Technology, Beijing 100124, China USA b c Department Civil and of Environmental Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA State Key of Laboratory Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China c State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China b

Abstract Abstract This paper modeled and analyzed the queue length estimation mechanisms by different layout strategies. According to the lane This paper of modeled and analyzed queuelayout length estimation mechanisms by different layout strategies. According to the with lane allocations intersections, several the feasible strategies of magnetic sensors are proposed. Furthermore, a layout strategy allocations of intersections, feasible layout strategies of length. magnetic sensors are three proposed. Furthermore, strategy with one single magnetic sensor several is proposed to estimate the queue Specifically, single-sensor baseda layout estimation methods one single magnetic sensor is proposed to estimate the (TIM), queue length. single-sensor estimation were presented, which are: Tail Interval-based Method PassingSpecifically, Time-based three Method (PTM), andbased Tail interval andmethods Passing were presented, which are: Tail Interval-based Method (TIM), Time-based Method (PTM), and Tail interval and Passing time-based Method (T-PM). The optimal layout strategy andPassing the corresponding algorithm of queue length estimation were time-basedbased Method (T-PM). optimalindicated layout strategy andhad thea better corresponding algorithm length presented on filed data. The results that T-PM performance in termsofofqueue accuracy andestimation robustness.were The presented basedwith on filed data.magnetic The results indicated that T-PM bettereconomical performance in terms accuracy robustness. layout strategy a single sensor was verified to behad thea most strategy to of estimate the and queue length ofThe the layout strategy a single sensor was verified be the most economical to estimate queueislength of and the immediate past with signal cycle. magnetic The experimental results alsotoshowed that the proposedstrategy single-sensor basedthe method simple immediate past Thefor experimental results also and showed that the proposed single-sensor based method is simple and economical, andsignal will becycle. suitable large-scale deployment application. economical, and will be suitable for large-scale deployment and application. © 2017 The Authors. Published by Elsevier B.V. © 2017 The Authors. Published by Elsevier B.V. © 2017 The Authors. Published by Peer-review under responsibility responsibility of Elsevier WORLDB.V. CONFERENCE ON ON TRANSPORT TRANSPORT RESEARCH RESEARCH SOCIETY. SOCIETY. Peer-review under of WORLD CONFERENCE Peer-review under responsibility of WORLD CONFERENCE ON TRANSPORT RESEARCH SOCIETY. Keywords: traffic information; queue length estimation; signal intersection; magnetic sensor; data driven Keywords: traffic information; queue length estimation; signal intersection; magnetic sensor; data driven

1. Introduction 1. Introduction Nowadays, traffic problems such as traffic congestion, traffic pollution, and traffic crashes are becoming Nowadays, problems as traffic congestion, traffic pollution, andtraffic trafficsystems crashes (ITS) are becoming emergent as thetraffic increase of roadsuch transportation. The advancement of intelligent can ease emergent as the increase of road transportation. The advancement of intelligent traffic systems (ITS) can ease

* Corresponding author. Tel.: +86-10-6739-6062; fax: +86-10-6739-6062. * Corresponding Tel.: +86-10-6739-6062; fax: +86-10-6739-6062. E-mail address:author. [email protected] E-mail address: [email protected] 2214-241X © 2017 The Authors. Published by Elsevier B.V. 2214-241X 2017responsibility The Authors.of Published Elsevier B.V. ON TRANSPORT RESEARCH SOCIETY. Peer-review©under WORLDbyCONFERENCE Peer-review under responsibility of WORLD CONFERENCE ON TRANSPORT RESEARCH SOCIETY.

2352-1465 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of WORLD CONFERENCE ON TRANSPORT RESEARCH SOCIETY. 10.1016/j.trpro.2017.05.212

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traffic problems to some extent. As one of the fundamental components for the ITS, vehicle detection system is to provide basic data demands. Signalized intersections are key nodes of urban road networks. Its capacity and control mechanism have a great impact on the level of service (LOS) and the connectivity of urban road networks. Queue length is the distance from the stop line to the tail of the latest vehicle stopped in a single lane during red light within one signal cycle. Practically, it can be replaced by the number of vehicles stopped in a queue for convenience. Queue length plays a crucial role in signal timing optimization and LOS evaluation. In general, loop detectors, probe vehicles, video cameras, and magnetic sensors are applied to estimate vehicle queue lengths. Stable sensors, such as loop detectors, magnetic sensors, and video cameras, are typically stationed at fixed locations along roadways to record passing vehicles. When loop detectors are used alone, queue lengths can be obtained by cumulative arrivals and departures (input-output models) at an intersection, which is also a common method for all stable sensors and can be used only when the queue does not exceed the locations of these stable sensors (Cai et al., 2014). Video cameras have been used to measure queue length mainly through image processing and pattern recognition. The processing algorithms includes level lines (Negri, 2014), virtual loops (Zheng et al., 2013), and so on. Queue length estimation of magnetic sensors data usually relies on the occupancy measurement at ramp entrance, vehicle counts at the ramp entrance and exit, speed measurements at the ramp entrance, and vehicle identification (Sanchez et al., 2011a). As one kind of mobile sensors, probe vehicles were used to estimate queue length based on stochastic modeling approach (Comert, 2013), or shockwave theory (Cetin, 2012). Apart from the methods using single detection sensors, probe vehicles and loop detectors are combined together to make queue length estimation (Cai et al., 2013; Comert, 2013; Li et al, 2013). Besides, Liu et al. (2009) studied the real-time queue length estimation methods for congested signalized intersections, they used high-resolution “event-based” traffic signal data and Lighthill–Whitham–Richards (LWR) shockwave theory to implement the queue length estimation of the immediate past signal cycle. Stable sensors always have high installation and maintenance costs. Probe vehicles are typically vehicles GPS devices. The majority of probe vehicles are taxi vehicles, which leads to a narrow use for estimation of queue length when traffic flow is low. Personal mobile phones will be more suitable for macroscopic traffic status estimation instead of microscopic queue length estimation for an intersection or a ramp. Consequently, stable sensors have distinct advantages for queue length estimation, which is used particularly for the demands of real-time traffic control and information service, or all-weather all-time detection and estimation. Magnetic sensors can overcome the defects of other stable sensors, such as high installation and maintenance cost, obvious pavement damage, easy influence by weather, deficiency of lane detection and differentiation. Moreover, magnetic sensors have some advantages of low cost, small size, and wireless data transmission, which comply with the ideas of green transportation. Using some detection algorithms, a single magnetic sensor can obtain plentiful traffic information including traffic volume, single vehicle speed and average speed, vehicle stopping and existing states, vehicle headway, vehicle classification, and so on (Li, 2014). Magnetic sensors have been used to obtain traffic information through the quantitative and qualitative analysis of the magnetic field. The traffic information includes vehicle detection (Yang et al., 2015; He et al., 2014), vehicle counting (Taghvaeeyan et al., 2014; Kwong et al., 2010), travel time (Kwong et al., 2010; Sanchez et al., 2011b), vehicle speed estimation (Li et al., 2011; Taghvaeeyan et al., 2014), vehicle classification (Yang et al., 2015; Taghvaeeyan et al., 2014; He et al., 2014; Li et al., 2014), and even the strain state of vehicle structure (Gontarz et al., 2015). Also, magnetic sensors can acquire the possibility of collision detection (Gontarz et al., 2015; Taghvaeeyan et al., 2014). This paper focuses on queue length estimation methods by considering different layout strategies of magnetic sensors and lane features at signalized intersections. Based on the number of exit directions of a one-way entrance at a signalized intersection, the entrance of an intersection can be categorized as three types as shown in Fig. 1: one-direction, two-direction, and three-direction. Straight-left lane and straight-right lane can be treated as straight lane because they share the same control mechanism. Turning mostly shares the same lane with left-turn, so it won’t be considered individually. One-direction type is common in small-scale intersections, signalized expressway or freeway ramps (Fig.1 (a)). Two-direction type is common in T-intersections or intersections with right- or left-turn prohibition (Fig.1 (b)). To estimate queue length for two-direction type intersections can be sorted into three different cases: two-direction controlled by a same signal phase, two-direction controlled separately, and one-direction controlled only. Three-direction type (Fig.1 (c)) is common in ordinary intersections (Each direction has one lane) or



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large intersections (Each direction has more than one lane). The control mechanisms of the three-direction type mainly include four cases: • Case 1: the left-turn and straight lanes controlled by a same signal phase with right-turn lane uncontrolled (Common in two-phase signalized intersections); • Case 2: the left-turn and straight lanes controlled separately with right-turn lane uncontrolled (Common in fourphase signalized intersections); • Case 3: all of the three directions controlled by a same signal phase (Common in ordinary two-phase signalized intersections with right-turn controlled); • Case 4: the right-turn and straight lanes controlled by a same signal phase with right-turn lane controlled separately (Common in four-phase signalized intersections with right-turn controlled).

a

c

b

a1

a2

a3

b1

b2

b3

Note: Arrows present the directions of traffic flow, and the number of lanes with same direction may be more than one, here just give one for showing.

Fig.1. Types of one-way entrance at signalized intersections (a, one-direction; b, two-direction; c, three-direction)

Queue length is the basic data for signal cycle optimization, traffic capacity control and traffic LOS evaluation at signalized intersections. Meanwhile, it is also an important input parameter for traffic signal control system. As the development of ITS in modern cities all over the world, the large-scale traffic information perception (including queue length estimation) in urban road network will be necessary and urgent. With a magnetic sensor deployed in the center of a lane, many traffic parameters (vehicle approach and departure time, traffic volume, vehicle speed, etc.) can be obtained (Li et al., 2011; Li et al., 2014; Li, 2014). When one direction of a one-way entrance has more than one lane, the sensors in the same cross-section are treated as a single cross-section sensor. The work of low-cost traffic information perception methods will be meaningful and worthy. The objective of this study is to estimate queue length by considering both of economy and availability based on magnetic sensors. The queue length perception mechanisms by different layout strategies of magnetic sensors will be modelled and revealed. More attention will be paid to the single sensor-based strategy for its efficient detection and low cost. After a single cycle, the queuing status of the vehicles can be estimated by the velocities and the interval times of these vehicles. Because of one single sensor, the single sensor-based strategy has the advantages of relatively low cost, quick installment and concise wireless network structure, and is convenient to collect the field data for large-scale networks and intersections. The rest of this paper is organized as follows. Section 2 presents several layout strategies of magnetic sensors to estimate queue length at signalized intersections and their corresponding estimation algorithms are also been proposed and discussed. Section 3 provides the models and methods of single sensor-based queue length estimation. A field example to verify the methods of single sensor-based queue length estimation is presented in Section 4. Section 5 concludes the paper with suggestion for future directions of research. 2. Queue length estimation by different layout strategies of magnetic sensors This section presents appropriate sensor layout strategies by considering the practical detection precision demands and the queue length estimation methods. Some methods can realize real-time queue length estimation, vehicle state identification, signal optimization, and LOS evaluation at intersections; some others can realize queue length estimation of the immediate past signal cycle and basic traffic data acquisition.

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2.1. Layout strategy A: uniform deployment The layout strategy A deploys sensors with the same distance interval of △l. The distance between the first sensor and the stop line is l0 (Fig. 2). If the total number of the deployed sensors is n, then the length between stop line and the nth sensor is L=l0+(n-1)△l. This paper firstly considers the simplest condition just with one-direction, as shown in Fig. 2(a). The sensors are numbered in the opposite direction of vehicle flow. If the kth sensor detects a stopped vehicle, meanwhile, other sensors with bigger number have not detected stopped vehicles, then the relationship between queue length estimation value lqest and k is shown as Eq. (1).

lqest (k ) = l0 + ( k - 1) Dl

(1)

The maximum absolute error between lqest and the real queue length lq is △l/2, and the relative error is △l/(2lq). If the nth sensor detects a stopped vehicle, then the value of lqest is no less than L. In strategy A, the maximum detectable queue length is L, and the absolute error is △l/2. When the intersection entrance has more than one lane and each lane is deployed with magnetic sensors according to layout strategy A (Fig. 2(b)), the lqest of each lane can be obtained by Eq. (1). The queue length of the intersection entrance is the average lqest of all lanes, and the corresponding absolute error is △l/2 as well. Stop Line

Stop Line

l0

△l

No.1

L

No.2

△l

No.1

No.1

No.2

No.2

Sensor

Single lane (a)

No.n

No.n

Single lane

L








No.n




l0

Sensor

Single lane (b)

Fig. 2. Layout of strategy A (a, single lane; b, multiple lanes)

2.2. Layout strategy B: two-point deployment This layout strategy deploy two sensors with a certain distance in a lane, which is regarded as layout strategy B (Fig. 3). The distance between the 1st sensor and the stop line is l0, and the distance between the 1st and the 2nd sensor is l12. For layout strategy B, lqest can be obtained by Vehicle Matching-based Method (VMM) or Deductive Analysisbased Method (DAM). VMM obtains lqest by matching the signal structures of different vehicles. During a measuring time interval, the vehicles indexed i = 1, 2, …, N cross the 2nd sensor at times t1 < t2 < …
(2)



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Where Tq(j) is the queuing time of vehicle j; △ is the average time interval of free driving of a vehicle from the 2nd sensor to the 1st sensor. Stop line

Stop line

Sensor

l0 No.1 l12

Sensor

l0 No.1

(a)

No.1

...

l12

No.2

Sensor

No.2

No.2

(b)

Fig. 3. Layout strategy B of two-point deployment (a, single lane; b, multiple lanes)

In a continuous traffic flow, if there is no lane-changing or lagging behavior, the vehicles will pass the 2nd sensor and the 1st sensor in order. Then the queuing time of vehicle k passing the 1st sensor is Tq(k)=(Tk-tk-△). If Tq(k)>d, the vehicle k is a valid queued vehicle. Typically, the value d should be 0. Practically, the value d can be more than 0 for increasing the robustness of the method. The number of queued vehicles nq can be obtained by comparing the vehicles passing through the 1st and the 2nd sensor in order during a signal cycle. Meanwhile, the lqest of the signal cycle can be calculated by Eq. (3): n

lqest = å li + ( nq - 1) lint er

(3)

i =1

Where li is the length of the ith vehicle; linter is the average queue distance between queued vehicles. The 1st or the 2nd sensor can be treated as a cross-section sensor when each direction of the intersection entrance has more than one lane. In this case, D1 and D2 record vehicle information of several lanes in the same direction, and VMM can also be used to obtain the number of queued vehicles. When queuing, drivers always automatically reach the queue length equilibrium among lanes in the same direction. Then, based on Eq. (3), lqest can still be obtained by computing the average number of queued vehicle of each lane. DAM is a deductive inference method to obtain lqest. In Fig. 3(a), we denote the start time as t0 when there is no vehicle between the 1st and the 2nd sensor. With the increasing vehicle flow passing from the 2nd to the 1st sensor, if the 1st sensor detects a stopped vehicle at t1, the traffic light turns to red light and the vehicles begin to queue. And the 1st sensor detects the first passing vehicle at t2, which means the traffic light turns to green and the queued vehicles begin to dissipate. Denote the number of vehicles passing the 2nd sensor as n2# and the 1st sensor as n1# during t0 to t2, if the 2nd sensor doesn’t detect a stopped vehicle at t2, then the number of queued vehicles nq during this signal cycle is nq=n2#-n1#, and the corresponding lqest is shown in Eq. (4).

lqest =

n2 # - n1#

å i =1

li + ( n2# - n1# - 1) lint er

(4)

Otherwise, if the 2nd sensor detects a stopped vehicle at t2, lqest is no less than l0+l12, as shown in Eq. (5).

lqest ³ l0 + l12

(5)

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In Fig. 3(b), the 2nd and 1st sensor can also be treated as a cross-section sensor. Assume the numbers of vehicles passing the 2nd and the 1st sensor are n’2# and n’1# during t0 to t2, respectively, if the 2nd sensor doesn’t detect a stopped vehicle at t2, the corresponding lqest is expressed in Eq. (6); otherwise, lqest is shown in Eq. (5). ¢ n2¢ # - n1#

lqest =

å i =1

¢ - n1# ¢ - 1) lint er li + ( n2#

(6)

nlane

Where nlane is the number of lanes in one direction of the intersection entrance. When an intersection entrance is the two-direction type, there will be different cases according to the signal control mechanisms of lanes. If all of the lanes are controlled at the same time, the estimation method is same with the onedirection type. In this section, only two cases which are two-direction controlled separately (Fig. 4(a)) and only onedirection controlled (typically left-turn lane is controlled but right-turn lane not, Fig. 4(b)) will be discussed. For sharing a same signal phase, the straight-left lane or straight-right lane can be treated as a straight lane, hence they won’t be discussed independently in this paper. If the 2nd sensor is deployed within the solid lane line, it can be assumed that no lane changing behavior exists between the 1st and the 2nd sensor. In that case, VMM and DAM can be used to obtain lqest. Otherwise, if lane changing behavior exists between the 1st and the 2nd sensor, the vehicle flows in these lanes don’t obey the queue length equilibrium. Then VMM will be more suitable to estimate lqest. If a vehicle pair Xi→Yj is detected by the sensors in a same lane, the lqest can be estimated by Eqs. (2) and (3); otherwise, an additional time factor of lane changing ta is to be calibrated, Eqs. (2) will be rewritten as Eq. (7). Tq(j)=sj-ti-△-ta

(7)

The two-direction type with one-direction controlled is shown in Fig. 4(b). The estimation algorithm is the same with Fig. 4(a) for the controlled direction of lanes. If the lanes of each direction are more than one, the method based on cross-section sensors can also be used. It will be the same condition for the three-direction type, which will not be discussed specially in the following texts. Next, we just discuss the conditions with one lane of each direction for the three-direction type. Stop Line

Stop Line

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l0 No.1

No.1

l12

No.1

No.1

No.2

l12

No.2

No.2

No.2

Controlled lane 1

Controlled lane 2

Controlled lane 1

(a)

Free lane 2 (b)

Fig. 4. Layout strategy B for two-direction types (a, controlled separately; b, one-direction controlled)



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As mentioned in Section 1, the controlled mechanisms of three-direction type mainly have four cases, whose queue lengths can also be estimated using the VMM or DAM. For example, the case 1 can be converted into two-direction type with one-direction controlled; the case 3 can be converted into one-direction type; the case 4 can be converted into two-direction controlled separately. The case 2 will be a little more complicated, as shown in Fig. 5. If no lane changing behavior exists between the 1st and the 2nd sensor, the lqest can be obtained by using DAM directly; otherwise, VMM will be more suitable to estimate lqest with an additional time factor ta to be determined. Stop Line

l0 No.1

No.1

No.1

No.2

No.2

No.2

l12

Sensor

Fig. 5. Layout strategy B for three-direction type with left-turn and straight lane controlled and right-turn lane uncontrolled

2.3. Layout strategy C: three-point deployment For the layout strategy C, the 1st sensor is deployed with a distance of l0 away from the stop line with. The 2nd sensor is deployed at the conjunction point of the solid lane line and dotted lane line with a distance of l12 away from the 1st sensor. The 3rd sensor is deployed at the dotted lane line with a distance of l23 away from the 2nd sensor. The location of the 3rd sensor depends on the maximum value of queue length estimation or the distance of adjacent upstream intersection. The layout strategy C combines the computational simplicity of DAM and the improvement of detection accuracy. Its basic theory of queue length estimation is the same with layout strategy B. The maximum valid value of the queue length estimation is l0+l12+l23. Fig. 6 shows the layout strategy C for two-direction and threedirection type. According to the location of each sensor, there will be no lane changing behavior between the 1st and 2nd sensor, and there are possible lane changing behaviors between the 2nd and 3rd sensor. The estimation rules will be discussed separately as follows: (1) If the 2nd sensor of a certain lane doesn’t detect a stopped vehicle, it shows that the queue length of that lane is less than l0+l12, and the corresponding estimation algorithm of lqest is shown as Eq. (4). (2) If the 2nd sensor of a certain lane detects a stopped vehicle, it shows that the queue length of that lane is between l0+l12 and l0+l12+l23. The estimation method of the number of stopped vehicles between the 2nd and the 3rd sensor is based on the queue length equilibrium. (3) If the 3rd sensor of a certain lane detects a stopped vehicle, the queue length will be no less than l0+l12+l23.

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Stop Line

Stop Line

l0

No.1

No.1

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No.1

No.1

No.1

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l12

l12 No.2

No.2

l23

l23

No.3

No.3

lane 1

lane 2

(a)

lane 1

lane 2 (b)

lane 3

Fig. 6. Layout strategy C (a, two-direction type; b, three-direction type)

2.4. Layout strategy D: widened intersection scenarios Some right-turn lanes can be accommodated through road extension. The layout strategy D is introduced to estimate the queue lengths for widened intersections. According to different control mechanisms of the right-turn lanes, the layout strategies for the two-direction and three-direction type are shown in Fig.7 and Fig. 8, respectively. The 1st and 4th sensor are deployed close to the stop line. The 2nd sensor is deployed close to the upstream location where the number of lanes varies. The 3rd sensor is deployed with a distance of l23 away from the 2nd sensor, and the 5th sensor is deployed at the conjunction point of the solid lane line and the dotted lane line. The layout strategy D satisfies the following conditions: (1) Vehicle flow conservation equation: n2#=n1#+n4#. (2) Lane changing restraint: no lane changing behavior between the 5th and the 4th sensor or the 5th and the 1st sensor. In this section, appropriate algorithms will be discussed to estimate lqest. If the 2nd sensor detects a stopped vehicle and the 3rd sensor doesn’t, it means that the queue length is longer than l0+l12, but less than l0+l12+l23. The number of queued vehicles between the 2nd and the 3rd sensor can be estimated by DAM. If either the 2nd or the 3rd sensor doesn’t detect a stopped vehicle, the queue length is less than l0+l12. Then VMM will be more suitable for some lane changing behaviors may exist between the 1st and the 2nd sensor. If the 3rd sensor detects a stopped vehicle, the queue length exceeds l0+l12+l23 and the queue length of right-turn lane reaches the maximum length of l0+l12.



Jian Rong et al. / Transportation Research Procedia 25C (2017) 1629–1647 Haijian Li, et al./ Transportation Research Procedia 00 (2017) 000–000

Stop line

Stop line

l0

No.1 No.4

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No.1 No.4

l45 l12

l12

No.5

n1# n4#

n1# n4#

No.2

No.2 l23

n2#

l23

n2#

No.3

No.3

(a)

(b)

Fig. 7. Layout strategy D for the two-direction type (a, right-turn uncontrolled; b, right-turn controlled)

l0

No.1 No.1 No.4

l0

No.1 No.1 No.4

l45 l12

l12

l23

No.2 No.2

No.3 No.3

(a)

l23

No.5

No.2 No.2

No.3 No.3

(b)

Fig. 8. Layout strategy D for the three-direction type (a, right-turn uncontrolled; b, right-turn controlled)

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2.5. Layout strategy E: single sensor deployment The layout strategy E is the simplest and cost-optimal strategy among all the strategies above. The single sensor is deployed close to the stop line with a distance of l0 (Fig. 9). The layout strategy E cannot obtain real-time queue length, but can estimate queue length of the immediate past signal cycle. Accordingly, strategy E can achieve traffic signal optimization and control, basic traffic information acquisition, and LOS evaluation of signalized intersections. Stop line

Stop line

Sensor

l0 No.1

Sensor

l0

Sensor

No.1

No.1

...

Single lane

Single lane

(a)

Single lane

(b)

Fig. 9. Layout strategy E (a, one lane; b, multi-lane)

3. Single sensor-based queue length estimation (layout strategy E) Other layout strategies (A to D) can achieve real-time queue length estimation based on DAM. However, because of using more sensors, these strategies have higher costs and more complicated sensor networks, consume more energy, and require higher precision time synchronization algorithms. Consequently, in order to overcome these shortcomings, the single sensor-based queue length estimation method is significant to be studied. 3.1. Scenario The process of vehicle queuing comply with the car-following models. Based on the car-following behavior, a typical vehicle queuing scenario at a signalized intersection is presented in Fig. 10. To simplify the issue, this paper makes the following assumptions: (1) The queued vehicles will not change lanes or overtake other vehicles any more. When they are passing through the intersection, they will stay in the same lane and departure the queue in turn. (2) Each queued vehicle satisfies the ‘uniform-acceleration-uniform’ pattern. A queued vehicle keeps uniform speed (vstart) at the start. As vstart is quiet low, with the speed from 0 to vstart, the corresponding acceleration time can be neglected. If the distance between the vehicle and the front vehicle is dneed, the vehicle begins to accelerate. When traveling at an upper limit speed vmax, the vehicle begin to keep the uniform speed (vmax) again. If the queued vehicle satisfies the acceleration condition when starting to run, it will accelerate to the upper limit speed vmax directly.



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Stop line

d1 linter

Car 1

Car 1

...

d2

Sensor

lcar

...Car 2

...Car 2

Car n

Car n

dn

Fig. 10. Typical road signal intersection queuing scene

3.2. Notations and parameters The arrival and departure time of each vehicle can be obtained using vehicle detection algorithms (Li et al., 2014; Li, 2014). Then, the time pairs of consecutive vehicles passing through a magnetic sensor in turn can be got (Fig. 11). The time-difference between adjacent front- and back-vehicles and the time-difference between the arrival and departure time of each vehicle are denoted as △ttail and △tpass, respectively.

Fig. 11. Time pairs of vehicles passing through a magnetic sensor in turn

In order to clearly describe the process of queue formation and dissipation, this paper also makes the following notations and parameters: (1) The 1st vehicle in the queue faces to the stop line vertically; the sensor can detect the first vehicle; the average vehicle length is denoted as lcar, and accordingly d1=lcar. (2) The following distance between two adjacent vehicles is denoted as linter. (3) The sum of reaction and action time is denoted as tr. (4) The vehicle acceleration is a and the number of queued vehicles is denoted as N. (5) The uniform speed time of the ith queued vehicle is denoted as tc(i). (6) The time of the vehicle tail passing the stop line is denoted as △ttail(i) and the speed is denoted as v(i). (7) The time of the vehicle passing the sensor is denoted as △tpass(i). (8) The time difference between the tail of the ith and the (i-1)th vehicle passing the stop line is denoted as △ttail(i). (9) The distance between the tail of the ith queued vehicle and the stop line is denoted as di, then di=i×lcar+(i-1)linter.

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Denote the time when the first vehicle in the queue starts to move as 0, then △tpass(1) =

ttail(1)-0 =

2lcar / a .

2lcar / a and △ttail(1) =

3.3. Models (1) △ttail model The △ttail model uses the variation rules of △ttail(i) to obtain the number of queued vehicles. ttail(i) should be determined firstly. ttail(i) is related to the performance of vehicle i-1. The following two cases will be discussed. Case I: meet the condition of acceleration In case I, when starting to move, the 2nd vehicle of the queued vehicles satisfies the acceleration condition. The distance between the 2nd and the 1st vehicle is more than dneed ( atr2 / 2 + lint er ³ dneed ). Then the 2nd vehicle and the following vehicles will begin to accelerate directly. By analyzing queue dissipation, it indicates that when N is small, the speed of no vehicle can reach vmax when the vehicle tail reaches the stop line. However, when N is big enough, there must be a vehicle m (m
Dttail (i) = 2di / a - 2di -1 / a + tr ,

i = 2,3,..., N

(8)

If N is big enough, then △ttail(i) can be obtained according to Eq. (9):

ïì 2di / a - 2di -1 / a + tr , 2 £ i < m Dttail ( i ) = í m£i£ N ïî( lint er + lcar ) / vmax + tr ,

(9)

Case II: not meet the condition of direct acceleration In case II, when the 2nd vehicle starts, the distance between the 2nd and the 1st vehicle is less than dneed, that is nd atr2 / 2 + lint er < dneed . The 2 vehicle and the following vehicles will begin with uniform speed and then accelerate. nd The time that the 2 vehicle begins to accelerated needs to be determined. Similarly, if N is small, the 2nd vehicle and the following vehicles will not satisfy the acceleration condition, which means the queued vehicles keep uniform speed to pass the stop. Accordingly, △ttail(i) can be obtained by Eq. (10). The difference of acceleration time between vehicle i and vehicle i+1 satisfies the relationship in Eq. (11).

Dttail (i ) = (lint er + lcar ) / vstart + tr ,

i = 2,3,..., N

tc ( i + 1) - tc (i ) = 2 ( dneed - lint er - vstart tr ) / a

(10) (11)

If N is big enough, there will be a vehicle k surely begin to accelerate before its tail reaches the stop line and the speed of vehicle k will not reach vmax when its tail passes the stop line. As the number of queued vehicles continues to increase, there must be a vehicle m whose speed will surely reach vmax before its tail leaves the stop line. The values of k and m can be obtained through computation or simulation based methods. Then △ttail(i) can be obtained according to Eq. (12).



Jian Li, Rong et al. / Transportation Research Procedia (2017) 1629–1647 Haijian et al./ Transportation Research Procedia 00 25C (2017) 000–000

ì( lint er + lcar ) / vstart + tr , 2 £ i < k ï ï Dttail ( i ) = ítc (i ) - tc (i - 1) + ( vtail ( i ) + vtail ( i - 1) ) / 2, k £ i < m ï ï î( lint er + lcar ) / vmax + tc (i + 1) - tc (i ), m £ i £ N

(

1641 13

(12)

)

2 Where vtail ( i ) = vstart + 2a di - vstart ( tc ( i ) - ( i - 1) tr ) , and vtail(i) represents the speed of vehicle i when its tail

leaves the stop line. (2) △tpass model The △tpass model applies the variation rules of △tpass to estimate queue length. Similar to the △ttail model, two cases will be discussed correspondingly. Case I: meet the condition of direct acceleration The condition is similar to the case I of the △ttail model. If N is small, the speed of each queued vehicle doesn’t reach vmax when its tail passes the stop line (v(i)
Dt pass (i) =

(

2di / a - 2 ( di - lcar ) / a

)

(13)

If N is big enough, the speed of a vehicle m will surely reach vmax before its tail leaves the stop line (v(m)=

2a ( dm - lcar ) ³ vmax ). △tpass(i) can be obtained as shown in Eq. (14).

(

)

ìï 2d / a - 2 ( d - l ) / a , 2 £ i < m i i car Dt pass ( i ) = í ïîlcar / vmax , m£i£ N

(14)

Case II: not meet the condition of direct acceleration Similar to case II of the △ttail model, if N is small, the queue vehicles will not satisfy the acceleration condition. Then △tpass(i) can be obtained by Eq. (15). △tpass(i)=lcar/vstart

(15)

If N is big enough, △tpass(i) can be obtained by Eq. (16).

2£i
(

)

(

(16)

)

2 Where vh ( i ) = vstart + 2a di -1 + lint er - vstart ( tc ( i ) - ( i - 1) tr ) , and vh(i) means the speed of vehicle i when its head

reaches the stop line. The calibration of k and m is similar to the △ttail model. During the queue dissipation, the variation of tendency of △tpass(i) is similar to △ttail(i) for each case. It indicates that △ttail(i) and △tpass(i) are not incremental and limited, which are the theoretical basis of the queue length estimation for the immediate past signal cycle. The vehicle waveforms obtained by magnetic sensors reflect the changes of the magnetic field. Actually, the arrival and departure time gained by vehicle detection algorithms always differ from the theoretical ones. The arrival time is earlier and the departure time is later than their theoretical time, which will result in an additional length le when calibrating the actual lengths of vehicles (Li et al., 2010; Li, 2014). Denote △tcpas(i) as the revised value of △tpass(i), then △tcpas(i) is shown in Eq. (17).

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△tcpas(i)=r×△tpass(i)

(17)

Where le is extra length, r is correction factor and r=(lcar+le)/lcar. 3.4. Methods The first queued vehicle for each signal cycle in a consecutive traffic flow should be determined firstly. Two criterions can be used to determine the first queued vehicle for the immediate past signal cycle. (1) △ttail(q)+△tcpas(q-1)≥tred. Where q is the vehicle index in a consecutive traffic flow (q=1,2,…), tred is the time duration of red for the queue lane/lanes. The “=” indicates that when the traffic signal turns red, the vehicle q-1 just passes through the intersection and the vehicle q and its following vehicles start to queue. When the traffic signal turns green, the vehicle q starts at the same time and other following vehicles follow their front vehicles. (2) △tcpas(q)≥△tcpas(1). Where △tcpas(1)=r×△tpass(1), which represents the minimum time duration for a queued vehicle. The condition “△tcpas(q)=△tcpas(1)” indicates the vehicle q happens to stop for queuing and the signal turns green at that moment. If a vehicle q in a consecutive traffic flow meets the two criterions above, it indicates that the vehicle q is the 1st queued vehicle of the immediate past signal cycle. Secondly, the last queued vehicle of the same signal cycle will be identified by the methods proposed in this paper below. Finally, the number of the queued vehicles of each signal cycle can be obtained. • Tail Interval-based Method (TIM) TIM uses the △ttail model to estimate queue length. Typically, when the first queued vehicle is identified, the △ttail(i) of the following vehicles can be obtained in turn during the immediate past green and yellow signal. Comparing the two values of △ttail(i) (one is obtained from the magnetic sensor; the other is calculated by Eqs. (8-12)), if the relative error of these two values is within a set value δ1, then it is indicated that the ith vehicle is in the queue; otherwise not. The algorithm based on TIM is presented as following: Algorithm 1: Vehicle queue length estimation algorithm based on TIM 1: Determining the first queued vehicle based on the two criterions for the immediate past signal cycle, and let k=1, i=1 2: if no following vehicle exists during the immediate past green and yellow signal then go to step 4 else i=i+1; go to step 3 end if 3: if the relative error of these two values of △ttail(i) is within a set value δ1 then k=k+1; return to step 2 else go to step 4 end if 4: lest=dk; for the next signal cycle, return to step 1 For the sake of simplicity, in step 3, Algorithm 1 can just use some values of △ttail(i) rather than all of them to computer relative errors of the queues vehicles. For example, the front j vehicles can use △ttail(2) to determine whether they are queued in order, and from vehicle j+1 to N, the algorithm just use △ttail(j+1). • Passing Time-based Method (PTM) PTM just uses the △tpass model. The algorithm of PTM is similar to Algorithm 1 with a slight difference in step 3. The new criterion of PTM in step 3 will be ‘the relative error of these two values of △tcpas(i) is without a set value δ2’. PTM can also just use some values of △tcpas(i). For example, the front j vehicles use △tcpas(j) to determine whether they are queued in order, and the vehicles from j+1 to N use △tcpas(j+1). • Tail interval and Passing time-based Method (T-PM) T-PM takes both TIM and PTM into account and it satisfies both of the criterions of TIM and PTM. Then the criterion of PTM in step 3 will be ‘the relative error of these two values of △ttail(i) is within a set value δ1 & the relative error of these two values of △tcpas(i) is without a set value δ2’. Denote the numbers of queued vehicles obtained by TIM, PTM and T-PM as nt, np, and ntp, respectively, then we have ntp=min{nt, np}.



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4. Example The queue length estimation methods based on layout strategy E were analyzed by using the field data from a single sensor. According to the deterministic queuing theory (Dai et al., 2008), the queue length can be replaced by the number of queued vehicles. The field experiment results were compared with real queue length of each signal cycle recorded by a video camera. The process of queue length estimation is shown in Fig. 12. Video data processing

Video data

Parameter calibration Sensor output

True queue length nv

TIM

Estimation value nt

PTM

Estimation value np

T-PM

Estimation value ntp

Evaluation

Fig.12. Experiment process of queue length estimation

The field experiment was carried out at Jiaoda East Road in Beijing, on June 30, 2009, as shown in Fig. 13. The road is a two-lane bidirectional street with a traffic signal for pedestrians. A single magnetic sensor was deployed near the stop line of the intersection.

Fig. 13. Example of layout strategy E at Jiaoda East Road in Beijing

4.1. Data This paper utilizes the field traffic data obtained by the single magnetic sensor and the vehicle detection algorithms proposed in Li et al. (2010) to obtain the arrival and departure time as shown in Fig. 11. The values of △ttail(i) and △tcpas(i) can be obtained as well.

4.2. Parameter calibration

The 25-minute sensor data and the corresponding video data of a consecutive traffic flow from 17:26 to 17:51 on June 30, 2009 were analyzed in this study. The signal cycle was 82 seconds, including red tred=24s, yellow tyellow=2s, and green tgreen=56s. The parameter calibration was carried out based on previous research conclusions, related standards, and specifications. The average vehicle length of the traffic flow is set as lcar=4.50m. The additional length

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Jian Rong et al. / Transportation Research Procedia 25C (2017) 1629–1647 Haijian Li, et al./ Transportation Research Procedia 00 (2017) 000–000

is set as le=1.80m (Li et al., 2010). Accordingly, the correction factor is r=(lcar+le)/lcar=1.40. The acceleration of the queued vehicles in different signal cycles is approximated by the average value, a=1.89m/s2. vstart is the speed of nearly stopping vehicle (Zhuang et al., 2013), vstart=5km/h; vmax is set to the maximum speed of the vehicles passing straight lanes in an intersection (Shi, 2011), vmax=25km/h; tr is set to the average reaction time of male and female drivers (Wang, 2008), tr=1.50s; linter is the minimum safe distance when vehicles are running in slow speeds (Liu, 2012; Liu, 2009), linter=2m; when the 2nd queued vehicle begins to accelerate, the average distance between the 1st and the 2nd queued vehicle is set as dneed=10.88m. 4.3. Results According to the given parameters above, we have △ttail(1)=△tpass(1)=2.180s and △tcpas(1)=1.40×△tpass(1)=3.052s. By analyzing the magnetic sensor data, there are 17 queues in total, which is consistent with the actual situation. By computing, both of the △ttail(i) and △tpass models satisfy the case II for this intersection, and k=2. The theoretical values of △ttail(i) and △tcpas(i) respectively calculated by the △ttail and △tpass models are presented in Table 1. Table 1. Values of △ttail(i) and △tcpas(i) based on the △ttail model and △tpass model of the example Vehicle index △ttail(i) (s) 1 2.180 2 3.546 3 3.488 4 3.417 5 and more 3.417

△tcpas(i) (s) 3.052 1.352 1.024 0.909 0.907

Set δ1=δ2=20%, the number of queued vehicles obtained by the methods proposed in this paper and the video data for each signal cycle are shown in Table 2. Table 2. Queued vehicles of the field experiment with different methods (unit: Vehicle) Signal cycle nv(Video) nt(TIM) 1 9 11 2 3 3 3 1 1 4 6 6 5 6 6 6 8 8 7 7 8 8 5 5 9 2 2 10 5 5 11 5 5 12 5 9 13 1 5 14 8 8 15 6 7 16 10 10 17 5 5 Total 92 104

np(PTM) 9 3 1 7 7 15 7 8 4 5 5 5 2 8 7 15 5 113

ntp(T-PM) 9 3 1 6 6 8 7 5 2 5 5 5 2 8 7 10 5 94

According to Table 2, the average relative errors of queue length based on TIM, PTM, and T-PM are εTIM=13.04%, εPTM=22.83%, and εT-PM=2.17%, respectively. Based on Table 2, Fig. 14 shows the results of queue lengths and errors with different methods. It can be concluded that TIM and PTM are always a little overestimated and T-PM fits well with the real data.



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Fig. 14. Comparisons of queue lengths and errors with different methods

4.4. Analysis Based on the results in subsection 4.3, T-PM has a better performance comparing to TIM and PTM. In addition, there are two major unavoidable interferences that PTM have to deal with: (1) the mixed vehicle types (namely the existence of large transit bus in intersections) require the adoption of vehicle classification system to improve the precision of lcar; (2) the traffic violations of pedestrians impact on the normal running vehicles when the pedestrian signal is red. Thereby, TIM and T-PM methods have a better robustness under mixed traffic conditions in terms of estimation errors and can be used practically. In addition, for layout strategies, layout strategy A is applicable for all types of intersection and lane control mechanisms, including special shaped intersections and lanes. The error of strategy A is related to △l. Other layout strategies (B to E) are easy to be influenced by road environment and less real time performance. But these layout strategies having the advantages of less devices and lower cost are suitable for large-scale deployment. When the demand estimation precision is required a little high, these layout strategies can be consider with a higher priority. For estimation methods, VMM is more complicated in computation and not real-time in on-line system. However, it can be applied broadly in a variety of situations if the demand for sensor deployment is not high. DAM is a simple and real-time estimation method. The results of DAM can be corrected by lane-changing maneuvers, the time that the following vehicles start to queue, vehicle length, and so on. For estimation errors, DAM will have accumulation errors. In order to conquer the shortcoming, zero flow update can be used in practice. For instance, during the green signal period, if no vehicle is detected by the downstream crosssection sensor and the adjacent upstream cross-section sensor within a period time, then the zero flow can be updated for the two sensors. VMM and the single sensor-based method are based on the consecutive traffic flow of the immediate past signal cycle, therefor there is no accumulation error, which means the estimation errors of a certain signal cycle have no impact on the estimation results of the following signal cycles. Both of VMM and the single sensor-based method have a better robustness. 5. Conclusions This paper mainly focuses on the methods of queue length estimation according to different sensor layout strategies. It gives the details of the concrete layout strategies and the estimation algorithms. The features and applicability of each layout strategy are also analyzed. The major conclusions are summarized as following: • The real-time queue length estimation at signalized intersections can be achieved by strategy A which needs more extra magnetic sensors. Accordingly, because of higher cost and more complicated communication networks, strategy A is not suitable for large-scale deployment. Computation models of layout strategy E are concise. After parameter calibration for an intersection, △ttail(i) and △tcpas(i) can be obtained off-line easily. Then the queue

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length for the immediate past signal cycle can be easily determined on-line by simple comparisons and discriminations instead of complicated machine-learning or data processing algorithms. • This paper proposes DAM and VMM to estimate queue length, and analyzes the applicability, error rules, and computational complexity. When a kind of deployment strategy is settled at a realistic intersection, some instructions to select suitable methods to estimate queue length can be referred. • Compared to the traditional queue length estimation methods based on the data for video cameras, loop inductive detectors, multiple magnetic sensors, and probe vehicles, the layout strategy E presented in this paper only needs one magnetic sensor in each lane. It has a simple sensor network, low cost, and is suitable for large-scale deployment and application. The dissipation analysis of queued vehicles indicates the methods of TIM, PTM, and T-PM are effective. In addition, notable advantages are found for T-PM in terms of accuracy and robustness. Indeed, there are several limitations in this study. The field experiments of layout strategies A to D have not been carried out and will be necessarily verified using field data. The empirical study on strategy E just implemented in a single-lane scenario is very limited and simple. Future research may focus on field experiments for all layout strategies and some multi-lane scenarios for layout strategy E will be paid attention as well. Acknowledgements This work was supported by the Distinguished Young Scholar in Beijing Award, the China Postdoctoral Science Foundation (2015M570912), and the Open Fund for a Key-Key Discipline of Zhejiang University of Technology (2015001). The support from the International Postdoctoral Exchange Fellowship Program (2015037) is also gratefully acknowledged. References Cai, Q., Wang, Z., Guo, X., Wu, B., 2013. New Calculating Method for HCM 2000 Queue Length Estimation Procedures with the Application of Floating Car Data. Procedia-Social and Behavioral Sciences 96, 2201-2210. Cai, Q., Wang, Z., Zheng, L., Wu, B., Wang, Y., 2014. Shock Wave Approach for Estimating Queue Length at Signalized Intersections by Fusing Data from Point and Mobile Sensors. Transportation Research Record: Journal of the Transportation Research Board 2422, 79-87. Cetin, M., 2012. Estimating Queue Dynamics at Signalized Intersections from Probe Vehicle Data: Methodology Based on Kinematic Wave Model. Transportation Research Record: Journal of the Transportation Research Board 2315, 164-172. Comert, G., 2013. Effect of stop line detection in queue length estimation at traffic signals from probe vehicles data. European Journal of Operational Research, 226(1), 67-76. Comert, G., 2013. Simple analytical models for estimating the queue lengths from probe vehicles at traffic signals. Transportation Research Part B: Methodological 55, 59-74. Dai, L. L., Jiang, G. Y., Pei, Y. L., 2008. Prediction of queue length at saturate signalized intersection. Journal of Jilin University 38(6), 1287-1290. (in Chinese) Gontarz, S., Szulim, P., Seńko, J., Dybała, J., 2015. Use of magnetic monitoring of vehicles for proactive strategy development. Transportation Research Part C: Emerging Technologies 52, 102-115. He, Y., Du, Y., Sun, L., 2012. Vehicle classification method based on single-point magnetic sensor. Procedia-Social and Behavioral Sciences 43, 618-627. He, Z., Zhu, H., Yu, F., 2014, April. A vehicle detection algorithm based on wireless magnetic sensor networks. 4th IEEE International Conference on Information Science and Technology (ICIST). Shenzhen, China, 727-730. Kwong, K., Kavaler, R., Rajagopal, R., Varaiya, P., 2010. Real-time measurement of link vehicle count and travel time in a road network. IEEE Transactions on Intelligent Transportation Systems 11(4), 814-825. Li, H., Dong, H., Jia, L., Ren, M., 2014. Vehicle classification with single multi-functional magnetic sensor and optimal MNSbased CART. Measurement 55, 142-152. Li, H., Dong, H., Jia, L., Xu, D., Qin, Y., 2011, October. Some practical vehicle speed estimation methods by a single traffic magnetic sensor. 14th International IEEE Conference on Intelligent Transportation Systems (ITSC). Washington, DC, 1566-1573. Li, H., 2014. Multi-parameter Sensing and Sensor Network Optimization for Road Traffic Information Acquisition. Beijing: School of Traffic and Transportation, Beijing Jiaotong University. (in Chinese) Li, J.Q., Zhou, K., Shladover, S., Skabardonis, A., 2013. Estimating queue length under connected vehicle technology: Using probe vehicle, loop detector, and fused data. Transportation Research Record: Journal of the Transportation Research Board 2356, 1722.



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