Cycle-based queue length estimation considering spillover conditions based on low-resolution point detector data

Transportation Research Part C 109 (2019) 1–18

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

Cycle-based queue length estimation considering spillover conditions based on low-resolution point detector data Jiarong Yao, Keshuang Tang

T

⁎

The Key Laboratory of Road and Traffic Engineering Ministry of Education & College of Transportation Engineering, Tongji University, Cao’an Road 4800, Shanghai 201804, China

ARTICLE INFO

ABSTRACT

Keywords: Signalized intersection Queue length Shockwave theory Queue spillover Low-resolution detector data

Queue length is vitally important for signal control optimization and congestion management of urban arterials. Due to the existence of mutual influence of neighboring intersections, cycle-bycycle estimation of queue length under spillover conditions remains a challenging problem given low-resolution detector data. On most urban arterials in China, point detectors are commonly installed at the intersection approaches and the upstream segments, providing volume, occupancy and speed data at a time interval of 1 min, i.e., a double-section low-resolution detection environment. Thus, the objective of this study is to propose a method to estimate the cycle-based queue length at signalized intersections considering spillover in the context of the above detection environment. Considering dense traffic conditions, uniform distribution is used to transform the low-resolution volume data into vehicle arrivals in 1 s. To deal with spillover conditions, mutual effects of neighboring intersections are classified into four different cases, based on the offsets between the neighboring intersections. Detector data at the upstream intersection approach are used to modify the volume data of the downstream intersection when long queue occurs, and the effect of spillover can thus be formulated analytically using the shockwave theory. Finally, the queue length can be calculated by accumulating queue lengths of all offset conditions within a cycle. Evaluation is done through an empirical case of two intersections, and a simulation case of seven intersections. Results show that the accuracy can reach 85.1% and 82.4% respectively. In both cases, spillover cycles can be recognized with stable and reliable performance.

1. Introduction As one of the most fundamental performance measures of signalized intersections, queue length is vitally important for coordinated signal control optimization and allocation of available roadway capacity. Cycle-by-cycle queue length can further reflect the relationship between traffic demand and capacity on a basis of cycle. Therefore, cycle-based queue length estimation plays an important role in signal control optimization and congestion management of urban arterials. However, it is difficult to realize accurate estimation of queue length in case of spillover, due to the existence of mutual influence of neighboring intersections and invalid detection once long queue occurs. In addition, cycle-based queue length estimation often requires high-quality data, e.g., high-resolution loop detector data, floating car data with high pulling frequency and penetration rate, which can provide sufficient information for the synchronization of signal timing and detection intervals. Hence, cycle-based queue length estimation based on

⁎

Corresponding author. E-mail address: [email protected] (K. Tang).

https://doi.org/10.1016/j.trc.2019.10.003 Received 25 July 2017; Received in revised form 5 October 2019; Accepted 8 October 2019 0968-090X/ © 2019 Elsevier Ltd. All rights reserved.

Transportation Research Part C 109 (2019) 1–18

J. Yao and K. Tang

Fixed Point Detector

Fig. 1. Typical Layout of Double-Section Point Detectors at Signalized Arterials in China.

low-resolution detector data remains a challenging research question. Despite the fact that high-quality floating car data as well as data fusion of multiple detectors has shown promising advantages in real-time estimation of queue length at signalized intersections (Cheng et al., 2011; Ban et al., 2011; Cetin, 2012; Tiaprasert et al., 2015; Zhan, et al., 2015), queue length estimation solely based on point detectors is still highly demanded because of the wide implementation of point detectors as well as the unavailability of high-quality data sources for most of researchers and practitioners. Single-section detector with a sampling interval varying from 1 s to 30 s (Mück, 2002, Papageorgiou and Vigos, 2008, Skabardonis and Geroliminis, 2008) is commonly used, while multi-section environment is also studied for improvement of estimation accuracy in empirical application (Anderson et al., 2014). Using such point detector data, various methodologies have been proposed in literature, such as the cumulative traffic input-output model (Webster, 1958, Sharma et al., 2007) and the shockwave model (Lighthill and Whitham, 1955). All the advances in data source and methodology aim to solve the long queue problem under spillover conditions. However, existing solutions of long queue demand supplementary information to obtain judgment rules through statistical analysis (Liu et al., 2009), or historical data for parameter calibration (Mück, 2002), which is site-specific and limits the application to some extent. Whereas, queue length estimation on urban arterials in most Chinese cities mainly relies on low-resolution fixed point detectors with a detection interval of 1-min. Typically, geomagnetic detectors or loop coil detectors are installed at the approach (often 20–30 m away from the stop line) of the intersection connecting to the signal controller for signal timing optimization. While, installed in the middle of links are the microwave detectors or coils for the purpose of traffic monitoring. A common double-section detection environment is shown in Fig. 1. According to the Ministry of Public Security (2013), such double-section detector configuration should be set up in all links of intersections of two arterials whose traffic demands should be both less than 30,000 veh/ day, or intersections of one arterial and one minor road, whose traffic demands should be less than 30,000 veh/day and 12,000 veh/ day, respectively. Taking Shenzhen and Qingdao as examples, more than 50% of the links are installed with double-section detectors in the urban areas, which is the epitome of the widespread detection configuration in most cities in China. Considering the detection condition in China, it is essential to develop a queue length estimation method which can deal with spillover and long queue problems using low-resolution point detector data. However, the deficiency of low-resolution detector data is that the traffic flow parameters is aggregated through 1-min interval, failing to provide timely changes in traffic flow caused by signal phase switching. Besides, long queue problem differs from place to place, dependent upon the location of detectors. Therefore, this study was intended to propose a generalized method to estimate the cycle-based queue length in the context of such a doublesection low-resolution detection environment. In the proposed method, uniform distribution is firstly used to transform the lowresolution volume data of each detection interval (1-min) into individual vehicle arrivals of 1 s. To account for queue spillover conditions, mutual effects of neighboring intersections are classified into four different cases based on the offsets between the neighboring intersections. Detector data at the upstream intersection approach are used to modify the flow rate of the downstream intersection approach if long queue occurs. As a result, the effect of spillover on the queue length of the upstream intersection can thus be formulated analytically using the shockwave theory. Finally, the maximal queue length is calculated through accumulating queue lengths of all offset conditions before the intersection of queuing and dissipating shockwaves within a cycle. For the above-mentioned objectives, the rest of the paper is organized as follows. Firstly, in Section 2, the past related studies on queue length estimation are reviewed to scope and position this study. In Section 3, the queue length estimation method based on low-resolution fixed point detector data is introduced. Then, the evaluation of the method through an empirical case and a simulation case is presented in Section 4. Concluding remarks and future research prospects are finally highlighted in Section 5. 2. Literature review Input-output models (Webster, 1958; May 1975.; Akcelik, 1999; Sharma et al. 2007; Vigos et al., 2008) and shockwave models (Lighthill and Whitham, 1955; Richards, 1956; Stephanopoulos et al., 1979; Liu et al., 2009) have been the main methods of queue length estimation. Among most existing methods based on fixed point sensors, emphasis is always placed on the long queue problem due to either high traffic demand or decreased discharging flow rate caused by spillover of downstream intersections. According to 2

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different detection environments and time granularities of the analysis period, some researchers have made the following contributions. Muck (2002) used a linear regression model to solve long queue problem by formulating the maximal queue length as the function of vehicle counts and fill-up time of detectors. In this way, cycle-based queue length can be estimated using a single vehicle detector with a 1-s sampling interval, which is 30 m upstream of the stop-line, even in case of queue up to five to ten times further than the detector location. Single detection section was also studied by Skabardonis and Geroliminis (2008), Geroliminis and Skabardonis (2010), who used a loop detector (30 s-interval resolution) installed in the middle of the link of urban arterials to estimate queue length through an analytical method based on the shockwave theory. Based on the identification of long queue and spillover using the measurement of occupancy, the cycle-based maximal queue length could be easily obtained through geometric calculation. However, calibration was required to determine the value of jam density and thresholds for judgment of long queue and spillover. Geroliminis and Skabardonis (2011) then proposed a method to capture the variations in vehicle length using data of single loop detector, thus queue spillover can be predicted more accurately removing the effects of vehicle length on the determination of occupancy thresholds for identification of spillover. Then, Liu et al. (2009), Wu et al. (2010), Wu and Liu (2011) applied high-resolution event-based detector (single-section detector installed in the middle of the link) data to identify traffic state changes, so that the estimation of time-dependent queue length and even long queues was realized. The applicability for environment like second-by-second and wired detectors was also justified with additional assumptions. This method was then developed into a part of Shockwave Profile Model (SPM) and was used to calculate the residual queue length quantifying the temporal influence of oversaturation. Anderson et al. (2014) studied the LWR (Lighthill, Whitham and Richards)-based queue estimation method under two detection conditions, i.e., single loop detector (located at the upstream boundary of the link) and double-section detectors (a loop detector at the upstream boundary of the link and a vehicle identification sensor at the stop-line). Without specific limit to data resolution on spatial or temporal dimensions, the averaged maximal queue length was obtained ignoring microscopic lane-specific behavior, and spillover was not considered. With the development of new data sources like probe vehicle, mobile detector data are becoming available for queue length estimation. According to the comprehensive review on traffic sensing and estimation using connected vehicle data (Guo et al., 2019), traditional methods fusing mobile data and fixed detector data can be found in many studies while some researchers also proposed tailored methods of queue length estimation considering the properties of new data sources. Chai et al. (2013) proposed an approach for the real-time vehicle queue length measurement in a video-based traffic monitoring system, which was built on the property of a modified local variance in video frames and a simplified local binary pattern (LBP) algorithm. Based on the analysis of traffic flow's shockwave profile on the approach of signalized intersections, Wu and Yang (2013) put forward a real-time queue length estimation model based on RFID detector data. The model solved the problem of measuring intersection queue length by exploiting the queue delay of individual vehicles instead of counting arrival traffic flow in the signal cycle, considering the variations under different traffic volumes and the relationship between queue length and the capacity of the approach. Ramezani and Geroliminis (2015) extracted critical points from high-resolution vehicle trajectory data to realize cycle-by-cycle queue estimation through fitting critical points of both the queue forming and discharging processes. Their method was also suitable for oversaturated conditions, covering the identification of queue spillover. Li et al. (2017) further developed this cycle-by-cycle queue length estimation method based on the inflection points of trajectories. The most significant improvement was that no signal timing was needed, and the phase switching time could also be estimated with satisfactory reliability even under a low penetration rate. Fusing data from both point detectors and probe vehicles, Wang et al. (2015) applied the shockwave theory to model queue evolution over time and space, and developed analytical formulations for calculating the maximum and residual queue length using probe vehicle trajectory points as well as traffic states measured by point detectors. Based on the studies of queue length estimation using fixed detectors, the group of Jeff Ban (Hao et al., 2014, Hao and Ban, 2015) further extended to mobile detector data and enriched the methodology. A queue length estimation method was proposed using vehicle trajectories obtained from mobile sensors to estimate the missing deceleration or acceleration process of a vehicle, and then the queue profile as well as the maximum queue length of a cycle, considering long queue problem. Moreover, a Bayesian Network-based method for estimating the cycle-by-cycle queue length distribution of a signalized intersection was proposed using sample travel time from mobile traffic sensors located upstream and downstream of the intersection. Qi and Hu (2018) proposed a Bayesian model to estimate the occurrence and duration of channelized section queue spillover using travel time and flow rate data. The queue evolution was realized by a macroscopic model considering first-in-first-out behavior at the upstream section, so that the maximal queue tail could be calculated. Shahrbabaki et al. (2018) used the input flow from a fixed detector at the upstream end of the link and connected vehicle data to deduce the queue tail location considering the error associated with the penetration rate of connected vehicles. The saturated case was studied emphatically by modeling a nonlinear function using the average speed of connected vehicles in the queue to obtain the actual vehicle count in the queue. Zhao et al. (2019) analyzed the statistical distribution of the queuing position of probe vehicles and established the queue length distribution based on the Bayes theory. Assuming the saturated spacing as constant in the analysis period, the penetration rate could be estimated by solving its definition formula where the volume was modeled as the function of the penetration rate. Considering both under-saturated and oversaturate cases, the queue length could be calculated using the estimated penetration rate and the number of probe vehicles in the queue. In summary, among the studies using fixed detectors to estimate queue length, cycle-based estimation is most adopted in analytical shockwave-based methods due to the periodical recurrence of queuing and dissipating shockwaves. Though data environment varies from one study to another, shorter sampling interval is preferred because higher resolution can help capture more details of traffic demand, as well as occurrence of long queue under spillover. As for long queue problem and decrease in discharging rate under spillover conditions, solutions like statistical regression and analytical models are proposed but there still exist problems of parameter 3

Transportation Research Part C 109 (2019) 1–18

J. Yao and K. Tang 1 min detection data of a single lane Occupancy threshold

Section data

Step1:Data preparation

Turning proportion modification

Flow of each direction Signal timing

Monocyclic flow 1-s flow data of each lane q1

Initial queue

Shockwave speed w1

=0? Yes

No

Initial queue length Lt=0

Step2: Calculation of the maximal queue length

Initial queue length

Queue length during red Lt=w1*R

Downstream entrance detector occupied ? Yes

Flow modification No

Timepoint of dissipation wave intersecting with queuing wave No

The maximal queue length

Downstream spillover ? Yes

Timepoint of spillover

Step3: Modification under spillover

Signal timing

Flow modification

Condition 1

Decide the offset condition the spillover happens

Condition 2

Calculate upstream residual queue and modify the downstream maximal queue length

Condition 3 Condition 4

Queue clear? Yes No

Calculate downstream residual queue length Output

of each cycle

Fig. 2. Major Steps of Queue Length Estimation.

calibration and use of historical field data, while accurate identification of spillover using high-resolution data is still impractical in most places because of data availability. Besides, the influence of spillover over the relationship of queue lengths of neighboring intersections is less mentioned, while considering an isolated intersection is quite common. Therefore, given the typical detection environment in Chinese cities explained in Fig. 1, this paper is intended to propose a cycle-based queue length estimation method using double-section low-resolution detectors through the shockwave theory, especially focusing on spillover conditions accounting for relationship of the shockwaves between neighboring intersections. 3. Queue length estimation method 3.1. Major Steps The core of the method is to calculate the shockwave speed of each lane of a certain direction based on shockwave theory and lowresolution detector data. As shown in Fig. 2, detector data are processed in advance to determine the occupancy threshold used for identification of long queue, and initial queue at the beginning of a cycle. Using the signal timing data, the queuing shockwave can be calculated once it begins to form after the red interval starts. Similarly, the dissipation shockwave begins to form as the green interval begins and the dissipation shockwave speed remains constant. Once long queue occurs, modification is conducted using the flow data of the upstream detector. Under spillover case, the effects on the queue length of the upstream intersection are also quantified based on the offset condition. After reconstructing the shockwaves between consecutive intersections in a cycle, the maximal queue length 4

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and the residual queue length can be calculated. ● Step 1: Data preprocessing In this step, volume and occupancy data are used to calculate the occupancy threshold, turning proportion, initial queue and the flow rate at 1-s interval, which is the input to calculate the queue length in Step 2. Firstly, transform the 1-min detection data into second level assuming uniform arrival. Then judge whether the detector is occupied by a vehicular queue using the occupancy value of point detector. The occupancy threshold is given by Eq. (1) according to Geroliminis and Skabardonis (2011). If the detector is not occupied, then the flow of 1-second level derived from the approach detector is used as input to calculate the accumulating queuing length. Otherwise, the flow data from the upstream detector should be used to modify the 1-second flow input considering the long queue problem.

L q¯ O¯ = v uf

(1)

where O¯ is the occupancy threshold. L v (m) is the average effective vehicle length. q¯ (veh/h) is the average flow measured by the detector. uf (km/h) is the free flow speed. Then, aggregate the flow of all the lanes of the approach into section flow, use the flow data of the approach detectors to calculate the turning proportion of the last 5-min to decide the flows of different directions for the present analysis time step. For instance, the proportion of through traffic is calculated based on Eq. (2). With this turning ratio, the flow obtained by the mid-link detectors are redistributed to cover the effect of lane changing, ensuring that the flow data obtained at different positions of the same lane can represent the actual demand of this direction. t

=

q5Tmin q5Tmin

+ q5Rmin + q5Lmin

(2)

where t (%) is the proportion of through traffic.q5Tmin , q5Rmin , and q5Lmin (veh/5min) are the summation of through, right-turn and leftturn flow in the last 5mins, respectively. Lastly, determine if there exists initial queue. Multiply the arrival flow rate of the green and red interval of each signal cycle by respective phase duration to obtain the total number of vehicles passing the detectors during one cycle. If the total number of vehicles exceeds the capacity, then calculate the initial queue as the difference of these two indexes. Note that the initial queue is in essence different from the residual queue which means the vehicles failing to clear after joining the queue during red. According to Highway Capacity Manual (HCM) (National Research Council (U.S.), 2010), the initial queue here refers to the number of the vehicles queuing at the start of the red phase of each cycle, including the residual queue and vehicles arriving at the end of the green interval without enough time to pass the intersection.

QC =

qa

ra +

a

Q0 =

qb

gb

(3)

b

Qc

c, Qc > c 0, Qc c

(4)

Here, QC (veh) is the total number of vehicles passing during one signal cycle. Q0 (veh) is the number of vehicles in the initial queue. qi (i = a , b) is the 1-s flow obtained through uniform distribution of a certain 1-min interval in the green (red) interval of interest. ra is the duration of red phase in the 1-min interval a and gb is the duration of green phase in the 1-min interval b within a certain cycle. a (b) denote the ath (the bth) 1-min interval in the red (green) interval. c (veh/lane/h) is the capacity of a single lane for a cycle. ● Step 2: Calculate the maximal queue length The analytical modeling of maximal queue length and the judgment of whether spillover happens is introduced in this step. Firstly, calculate the shockwave speeds of queuing and dissipation within each cycle based on the shockwave theory and then figure out the maximal queue length by Eqs. (5)–(7). In case of light traffic, the queue length of neighboring intersections can be calculated independently. i

=

Lm =

tm =

qbefore

qafter

kbefore

kafter

2 (tm

2 Gs 2

(i = 1, 2, 3, 4)

(5) (6)

Gs ) 1 Rs

(7)

1

Here, qbefore , qafter (veh/h) are the traffic flow rates of two different states and kbefore, kafter (veh/km) are their respective densities. i is the shockwave speed of two different states. Specifically, as shown in Fig. 3, for the queuing shockwave 1, the two states are the free flow state and the queuing state which can also be regarded as traffic jam state. For the dissipating shockwave 2 , the two states are 5

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Fig. 3. Shockwaves at A Signalized Intersection.

the traffic jam state and the saturation flow state. For the compression shockwave 3, the two states are the saturation flow state and the free flow state. For the secondary queuing shockwave 4 , the two states are the saturation flow state and the jam density state, which may occur when queuing vehicles fail to clear in the green phase and are forced to form a queue again. tm (s) is the time point when the maximal queue length is reached. Lm (m) is the length of the maximal queue. Gs (s) is the start of the green interval. If spillover occurs, the upstream traffic condition is influenced by downstream traffic. The calculation of the queue length should consider the neighboring intersections as a whole and here their relationship of signal phases is analyzed, as shown in Fig. 4. In all cases there are totally four offset conditions, where, Reddown and Greendown are the red interval and the green interval of the downstream intersection respectively, Redup and Greenup are the red interval and the green interval of the upstream intersection respectively.C (s) is the cycle length. ● Condition 1 is the period (i.e., [t1, t2 ]) during which both two intersections are in the red interval. ● Condition 2 is the period (i.e., [t2, t3 ]) during which the upstream intersection is in the green interval while the downstream intersection is in the red interval. ● Condition 3 is the period (i.e., [t3, t4 ]) during which both two intersections are in the green interval. ● Condition 4 is the period (i.e., [t4, t1 + C ]) during which the upstream intersection is in the red interval while the downstream intersection is in the green interval. Once spillover occurs at any condition, the maximal queue length of the downstream intersection is surely the link length, while that of the upstream intersection may be obtained at the intersection of the dissipating shockwave and the queuing shockwave either caused by upstream arrivals or by downstream spillover queue, as given by Eq. (8) and Eq. (9). According to which condition where spillover happens, detailed analysis is introduced in Section 3.2. (8)

Ldown, m = Ltotal Lup, m =

2

(Tm

(9)

Gs )

Here, Ltotal (m) is the link length. Ldown, m (m) is the maximal queue length of the downstream intersection while Lup, m is the maximal queue length of the upstream intersection. Tm is the time point when the maximal queue length of the upstream intersection is reached. Correspondingly Gs denotes the start of green interval of the upstream intersection.

Fig. 4. Relationship of Signal Phases between Neighboring Intersections. 6

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Fig. 5. Shockwave Diagram of under Spillover Condition 1.

3.2. Treatment of spillover conditions According to Fig. 2, in Step 3, firstly calculate the number of vehicles joining in the queue and judge if the queue length exceeds the upstream boundary of the link detectors. If queue spillover occurs, choose the flow rate of the nearest upstream detector at the same 1-min interval to replace the flow data of the occupied detector. The time point of spillover is also figured out and the offset condition where spillover occurs should be determined. The maximal queue length of downstream intersection is set as the link length and the shockwave is restored for the upstream intersection. At the end of the green interval, calculate the residual queue lengths of the two intersections, then enter the next cycle. Four offset conditions discussing the queue length estimation under spillover are listed as follows with illustrations given in Figs. 5–8. It is noted that these illustrations focus more on temporal-spatial relationship between the upstream intersection and the downstream intersection under spillover case, thus, the time-varying property of the queuing shockwave, namely the updates of queuing shockwave speed 1 once a minute, are not depicted for simplicity. ● Spillover Condition 1 As shown in Fig. 5, LD1 (m) is the distance between the approach detector and the stop bar. LD2 (m) is the distance between the exit access detector and the stop bar.uf is the free flow speed of the downstream traffic flow before queuing. Reddown and Redup represent the duration of red intervals of the downstream and upstream intersection respectively. vs is the speed of saturated flow. If spillover

Fig. 6. Shockwave Diagram of under Spillover Condition 2. 7

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Fig. 7. Shockwave Diagram of under Spillover Condition 3.

Fig. 8. Shockwave Diagram of under Spillover Condition 4.

occurs downstream, the time point of spillover is given by Eq. (10).

ts = t1 +

Ltotal

Ldown, t

(10)

1

where ts (s) is the time point of queue overflowing upstream. t1 (s) is the outset of Spillover Condition 1. Ltotal (m) is the link length. Ldown, t (m) is the queue length of the downstream intersection at the end of each offset condition so it will change after every offset condition. Thus, the maximal queue length of downstream intersection is set as the link length, as given by Eq. (11). (11)

Ldown, m = Ltotal As for the upstream intersection, the queue length at the end of Spillover Condition 1 is calculated by Eq. (12).

Lup, t =

1

'(t2

(12)

t1)

Here, Lup, t is the queue length of the upstream intersection at the end of each offset condition so it will change after every offset condition. t2 (s) is the outset of the next offset condition. 1 ' (km/h) is the speed of queuing shockwave of the upstream intersection. ● Spillover Condition 2 According to Fig. 6, in Condition 2, the upstream intersection enters the green interval while the downstream intersection is in the 8

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red interval.Greenup represents the duration of green interval of the upstream intersection. As the discharge flow approaches the downstream intersection in saturated flow speed vs , the density of traffic flow passing the mid-link detectors will change correspondingly, which will affect the speed of the queuing shockwave 1 in that 1-min interval. The vehicle queue at the upstream link as shown in the left side of the upstream intersection in Fig. 6 denotes the maximal queue length while the vehicles denoted in dotted box mean the vehicles able to clear before spillover spread upstream. The vehicle queue at the right side of Fig. 6 represents the temporal queue accumulated after spillover spread upstream at the end of Condition 2. If spillover occurs, the time point of spillover is given by Eq. (13).

ts = t2 +

Ltotal

Ldown, t

(13)

1

Upstream intersection thus enters a passive red interval, causing the rest of upstream discharging queue to form a new queue once the discharging shockwave meets the spillover queuing shockwave as shown in Eqs. (14) and (15).

t34 =

Lup, m +

4 ts

+

+

4

3

Lup, re =

4

(t34

3 tup, m

(14) (15)

ts )

Here, t34 (s) is the time point of compression shockwave 3 intersecting with the queuing shockwave 4 . Lup, m (m) is the maximal queue length of the upstream intersection. tup, m (s) is the time point of the formation of the maximal queue length in the upstream intersection. Lup, re is the queue length of the upstream intersection which fails to clear before meeting the spillover queuing shockwave. The maximal queue length is obtained once the discharging shockwave meets the queuing shockwave, as shown by Eqs. (16) and (17).

tup, m =

Lup, t + Gs, up

2

2

1

Lup, m =

2

(tup, m

t2

1

(16) (17)

t2)

where Gs, up (s) is the start of the green interval of the upstream intersection. It is noted that if ts is smaller than tup, m , t34 won’t exist. In this way, the upstream queue length at the end of Condition 2 is given by Eqs. (18) and (19).

Lup, t = Ltotal + Lup, re +

1

'(t3

(18)

t34 )

(19)

Lup, t = Ltotal

Here , t3 (s) is the outset of the next offset condition. As shown in Fig. 6, 1 ' denotes the queuing shockwave speed of the upstream intersection.ufup denotes the free flow speed of the upstream intersection. ● Spillover Condition 3 According to Fig. 7, the upstream intersection can be considered as part of the link for the discharge shockwave spreads to meet the upstream queuing shockwave in Spillover Condition 3. Greenup represents the duration of green interval of the upstream intersection. The vehicle queue at the upstream link denotes the maximal queue length of the upstream intersection where vehicles denoted as dotted boxes represent the decrease of queue length at the end of green, that is to say, the solid boxes represent the residual queue length. Thus, the maximal spillover queue can be given through Eqs. (20) and (21).

tdown, m =

Lup, m =

Ltotal + Lup, re +

2

(tup, m

t3

2

2

1

t34 (20)

1

(21)

t3)

If the spillover condition can’t be eliminated, residual queue will form at the end of the green phase of upstream intersection as shown by Eqs. (22)–(25) and the downstream queue length is set as the link length.

Qup, se = Qup, m

S0

Ltotal +

2 t3 3

+

+

3 t4

(22)

2

t34 = Qup, se S0

(23)

Lup, se =

(24)

4

t34

(25)

Ldown, t = Ltotal

Here, Qup, se (veh) is the number of vehicles of residual queue in the upstream intersection. Lup, se (m) is the length of the residual queue of the upstream intersection.Qup, m (veh) is the maximal number of vehicles waiting in the queue in the upstream intersection. S0 (veh/ 9

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Signal Timing Scheme of Intersection II

Signal Timing Scheme of Intersection I Phase 1:57s

Phase 2:10s

Phase 3:29s

Phase 4:48s

Phase 5:7s

Phase 6:24s

Phase 1:32s Phase 2:120s Phase 3:23s

Fig. 9. The Case Study Arterial.

70 Detector Data

Section flow /veh

60

Manual Counting Data

50 40 30 20 10 0

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69

1-min interval

Fig. 10. Flow Comparison between Detector Data and Manual Counting Data at Mid-Block Section Flow of Intersection I.

h) is the saturation rate. t4 (s) is the outset of the next offset condition. ● Spillover Condition 4 In this case, the downstream intersection is in the green interval while the upstream intersection enters the red interval, as shown in Fig. 8. The dotted boxes denote the decrease of queue length in Condition 4 and the solid boxes represent the vehicles failing to clear at the end of green, which is the residual queue of the downstream intersection. If spillover hasn’t been cleared, then the residual queue length should be calculated through Eqs. (26) and (27) before the accumulation of upstream queue length in the next cycle. (26)

t34 = Qup, t S0

10

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Table 1 Error Analysis of Queue Length Estimation at the Study Site. Index

Downstream Intersection (Fuzhou South Road-Xianggang Mid. Rd.)

Upstream Intersection (Fuzhou South Road-Zhangzhou 2nd Rd.)

Average

Lup, re =

Lane 1 Lane 2 Lane 3 Lane 4 Average Cycles with Spillover Unsaturated Cycles Lane 1 Lane 2 Lane 3 Lane 4 Average

Estimation Using Detector data

Estimation Using Manual Counting Data

MAE (m)

MAPE (%)

MAE (m)

MAPE (%)

13.3 13.1 12.0 13.1 12.9 17.4 11.3 15.8 12.0 15.8 27.7 17.8 15.4

9.1 9.0 8.6 12.0 9.7 9.8 9.6 21.2 15.5 19.4 24.0 20.0 14.9

6.6 8.8 11.2 10.3 9.2 10.1 8.9 17.1 14.6 12.6 23.3 16.9 13.1

5.1 6.3 8.5 9.5 7.4 5.7 7.9 23.8 19.8 12.1 21.2 19.2 13.3

(27)

t34

4

If downstream queue fails to clear, the residual queue length should be given by Eqs. (28)–(30).

Qdown, se = Qdown, m

S0

2 t3

+

3 (t1

+C

t4 )

(28)

2

t34 = Qdown, re S0

(29)

Ldown, se =

(30)

t34

4

Here, Qdown, re is the number of vehicles in the residual queue in the downstream intersection.Qdown, m is the number of vehicles in the maximal queue of the downstream intersection. Ldown, se is the length of the residual queue at the downstream intersection. It is noted that for intersections that are coordinated in an arterial, they may share a common cycle length so the offset conditions in every cycle of a TOD (Time-of-day) period are the same, while for consecutive intersections whose cycle lengths are different, the types as well as the number of offset conditions in every cycle are different. The maximal queue and whether spillover occurs are determined by the shockwave reconstruction in sequential offset conditions where the Lup, t and Ldown, t are not only temporal outputs of the last condition but also inputs of the next condition. Through the calculation of the offset conditions in a cycle, the maximal queue lengths of both intersection Ldown, m and Lup, m , as well as their residual queues, if exist, can be obtained. 4. Evaluation The evaluation of the proposed method is done through an empirical case and a simulation case. In the empirical case, the queue lengths of two intersections are estimated under a spillover case. In the simulation case, a simulation model of seven intersections is built through VISSIM and two scenarios are set to test the reliability of the proposed method. For both two cases, two evaluation indexes are chosen, given by Eqs. (31) and (32).

MAE =

1 N

MAPE =

|Y

Y|

(31)

N

1 N

Y N

Y Y

× 100%

(32)

Here, MAE (m) is the mean absolute error of the estimated maximal queue length while MAPE (%) is the mean absolute percentage error. N is the number of cycles in the study period. Y (m) is the estimated value of maximal queue length. Y (m) is the true value of the maximal queue length. The maximal queue length here is defined as the distance between the stop-line and the location of the farthest vehicle which stops to join the queue during a cycle. 4.1. Empirical case As shown in Fig. 9, two intersections of South Fuzhou Road, Qingdao City are selected, with field measurement of link length labeled in the picture and the signal timing data known in advance. The southbound direction, namely from Zhangzhou 2nd Road to Xianggang Mid. Road, is used for evaluation so for simplicity, the intersection denoted as I is called the downstream intersection while the intersection denoted by II is called the upstream intersection. The direction of interest consists of four lanes numbered as 1–4 from fast lane to near-side lane. As shown in Fig. 9, the detectors installed in the mid-link are microwave detectors which offer the flow and occupancy of each lane every 1 min. The detectors installed in the entry of each intersection are fixed loop coil detectors 11

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which offer the 1-min flow of each lane, namely the flow of different directions. For validation, empirical queue length data are collected through video recording the traffic flow of these two intersections and field calibration of reference points. The validation is done using the detector data and signal timing data of the morning peak period (7:50–9:00) on Nov. 29th, 2016. Through practical data, parameters used here are obtained through calibration using flow, speed data obtained from detectors, 2 = 20 km h, 4 = 20 km h, 3 = 30 km h, u f = 54 km h, kj = 150 veh km . As the signal controllers of Fuzhou South Road are coordinated, the intersections in Fig. 9 share the same cycle length C = 175 s , and the offset between two intersections is 49 s. The period of interest covers 24 cycles in total and the proportion of heavy vehicles is 2.0%. To investigate the errors caused by miss detection, the proposed method is applied again using the flow data obtained from the videos through manual counting, which is done by counting the vehicles passing the same locations where detectors are installed. Fig. 10 shows the comparison between detector data and manual counting data of the mid-block section of Intersection I. The average error is 11 vehicles for a whole section, which is 2.75 vehicles for each lane at average. It is noted that the aggregated 1-min section 350

Position of the end of the queue/m

300

170m

250

Zhangzhou 2nd Rd.

200

194m

Fuzhou South Road

150

166m 100

Observed Queue Length Estimated Queue Length Using Detector Data

50

0

Xianggang Mid. Rd.

Estimated Queue Length Using Manual Counting Data Spillover

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Cycle

(a) Lane 1 350

Position of the end of the queue/m

300

170m

250

Zhangzhou 2nd Rd.

200

194m

Fuzhou South Road

150

166m

Xianggang Mid. Rd.

100 Observed Queue Length Estimated Queue Length Using Detector Data

50

Estimated Queue Length Using Manual Counting Data

0

Spillover

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Cycle

(b) Lane 2 Fig. 11. Flow Comparison between Detector Data and Manual Counting Data. 12

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J. Yao and K. Tang

Position of the end of the queue/m 400 350

170m

300 250

194m

Fuzhou South Road

Zhangzhou 2nd Rd.

200 150

166m

Xianggang Mid. Rd.

100

Observed Queue Length Estimated Queue Length Using Detector Data

50 0

Estimated Queue Length Using Manual Counting Data Spillover

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Cycle

(c) Lane 3 Position of the end of the queue/m 400 350

170m

300 250

194m

Fuzhou South Road

Zhangzhou 200 2nd Rd.

Xianggang Mid. Rd.

150 166m

100 Observed Queue Length Estimated Queue Length Using Detector Data

50 0

Estimated Queue Length Using Manual Counting Data Spillover

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Cycle

(d) Lane 4 Fig. 11. (continued)

flow obtained from detectors is slightly smaller than that of manual counter, which may be caused by leak detection and lead to shorter queue length. On the other hand, a time lag is inevitable when using detector data to restore the shockwave for upstream section of the detector. This factor can be eliminated using manual counting data, and here the time lag is calculated for each lane using the average speed from detector and flow data are adjusted before estimation. The estimation results of both using the detector data and the manual counting data are listed in Table 1 and Fig. 11 further presents a good fitting result of the time-varying trends of queue length of all four lanes of the downstream and the upstream intersections. It is obvious that the estimation using manual counting data is closer to the observed value than that using detector data, which implies that miss and false detection does exist and cannot be ignored. Also, the cycles where spillover occurs (the maximal queue length is larger than the link length 166 m) are marked to highlight the estimation performance under spillover. Some conclusions are summarized as follows. ● The average estimation error of the downstream intersections is 9.7% for detector data and 7.4% for manual counting data, while the comparison regarding both intersections is 14.9% and 13.3% respectively, which implies that about 10% improvement can be realized if the effect of leak detection is removed. ● During the study period, there are in total 24 saturated cases and 72 under-saturated cases for four lanes (which means there are 13

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J. Yao and K. Tang

Hanjiang Rd.

Zhujiang Rd.

Hehai Rd.

Taihu Rd.

Longjin Rd.

Longcheng Avenue

Jinxiu Rd.

Fig. 12. The Study Scope of the Simulation Case.

4×24 = 96 observations of queue lengths) of the downstream intersection. For spillover cycles of the downstream intersection, the estimation error of manual counting is 5.7%, even lower than that of detector data, 9.8%. Though the mean absolute error (MAE) of the under-saturated cases is smaller than the saturated cases, the mean absolute percentage error (MAPE) of the saturated cases is smaller, showing a little better performance of the queue length estimation for spillover conditions. ● The average estimation error of the downstream intersection is 9.7%, compared with 20% of the upstream intersection. It implies that the downstream intersection enjoys a better performance than the upstream intersection, because the input data to calculate the queue length of the downstream intersection is more complete. 4.2. Simulation case As the empirical case only considers the queue lengths of two intersections, the simulation case further presents the evaluation performance of the proposed method in seven intersections of an arterial. As shown in Fig. 12, a simulation model is established on the background of seven intersections of Tongjiang Road, Changzhou City, Jiangsu Province. The through lanes of the northbound direction are used for evaluation and the simulation model is calibrated using the signal timing data and License Plate Recognition (LPR) detector data from 14:00 to 16:00 on Jan. 9th, 2019. According to the study configuration in Fig. 1, advance detectors are installed at a distance of 30 m away from the stop-line while the mid-block detectors are installed at a distance of 200 m away from the stop-line. The simulation model is run for 8400 s, with a 1200 s-warmup period at the beginning so the queue lengths of 7200 s are estimated and the evaluation indexes given in Eqs. (31) and (32) are calculated. To show the performance of spillover case, another scenario with a 20%-increase input (hereinafter referred to as Scenario 2, while the scenario with the calibrated input is referred to as Scenario 1) is also tested for comparison. Fig. 13(a)–(g) gives the estimation results of seven intersections under two scenarios. As shown in Fig. 13, the estimation of Scenario 1 is shown on the left while the estimation of Scenario 2 is shown on the right. Legends of all the subgraphs are listed on the top of Fig. 13. For the estimation of each intersection, the MAEs and the MAPEs are given in the upper right of each subgraph. To sum 14

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J. Yao and K. Tang

up, the average MAPE of the estimation of all seven intersections under Scenario 1 is 17.7% while that of Scenario 2 is 17.6%, showing the reliability of the proposed method. The time-varying trends of the estimation under both scenarios are consistent with the true values, while the estimation of Scenario 1 is more stable and fits better. Under Scenario 2, spillover occurs at Taihu-Tongjiang Intersection, Zhujiang-Tongjiang Intersection and Hanjiang-Tongjiang Intersection, the estimation of the cycles with spillover is relatively satisfactory, which can be implied from the comparatively small MAPEs of such cycles. Although the MAEs under Scenario 2 are rather larger than those of Scenario 1, the differences of MAPEs are not that obvious due to the larger absolute queue length under higher demand scenario.

Fig. 13. Estimated Queue Lengths in the Simulation Case. 15

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J. Yao and K. Tang

Fig. 13. (continued)

As the proposed method estimates the maximal queue length through a dynamic accumulation of queue length based on the shockwave speeds of different detection intervals within each cycle, Fig. 14 gives an intuitive description of the propagation of shockwaves of seven intersections under Scenario 2, which shows how the maximal queue length is obtained through the proposed method. It is found that, during the period from 1,920 s to 2,340 s, spillover occurs at the Intersection of Hanjiang Rd., Zhujiang Rd. and Taihu Rd. As the end of queue increases to the upstream intersection, the signal phases of the upstream intersections are red, which to some extent prevents the mainline congestion from becoming more severe, while the vehicles from the minor roads will suffer from a “passive red phase” and start to form a queue until the queue length on the mainline Tongjiang Rd. begins to decrease. 5. Conclusions and future works In view of the common detection environment in most small-medium cities in China, a method is proposed to calculate the maximal queue length based on shockwave theory using double-section low-resolution detector data. The proposed method can not only estimate the cycle-based maximal queue length for the under-saturated conditions, but also have the ability of spillover identification and data modification when long queue occurs. Evaluation results show an overall accuracy of 85.1% and 82.3% for the empirical case and the simulation case, respectively, which demonstrates promising potentials of the proposed method in traffic monitoring and congestion management on urban arterials. Major methodological contributions of the presented study are highlighted below. ● Analytical solution and favorable expansibility. The proposed method provides analytical solutions for modelling the queue length estimation problem as a dynamic shockwave reconstruction problem, while considering different offset conditions of two 16

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Time/s

Hanjiang Rd.

534m

1920

1980

2040

2100

2160

2220

2280

2340

500

Zhujiang Rd.

564m 1000

Hehai Rd.

784m

1500

Taihu Rd.

289m

2000

612m

2500

Longcheng Avenue 3000

718m Longjin Rd.

3500

4000

Space/m

Fig. 14. Illustration of Shockwave Propagations along the arterial (Scenario 2).

neighboring intersections. In this way, spillover identification is realized based on the estimated queue length, which can be regarded as a functional relation between the intersection performance index and signal timing data. The proposed method can be further extended to optimize the offset between intersections as well as splits, aiming at queue length minimization or no spillover. ● Flexible applicability. In respect of practical significance, the accuracy of the proposed method is satisfactory under the doublesection low-resolution detection environment. As the proposed method offers a general analytical solution for reproducing shockwaves under such a detection environment, it could also be applied for other types of detectors with similar function and configuration as the loop detectors. In addition, given detector data with more refined granularities, i.e., less than 1 min, it is expected that the proposed method can generate more realistic arrival flow profile and thus reach higher estimation accuracy. Though the method is reliable to some extent, the following limitations and shortages still exist affecting the overall performance and demand future research. ● Further verification works should be done using more empirical data in field to examine the reliability of the method in different demand scenarios. Apart from different demand scenarios, empirical evaluation of different types of low-resolution detectors and intersection geometric designs should be implemented, which not only tests the generality of the method, but also identifies the effects of data quality or roadway physical characteristics like roadside interferences on estimation accuracy. ● The modification mechanism in case of spillover is passable without considering the effect of turn-in vehicles. The improvement in accuracy should be studied if detectors installed in minor roads are available. Acknowledgement This study was supported by the Natural Science Foundation of China (No. 61673302). References Akcelik, R., 1999. A Queue Model for HCM 2000. ARRB Transportation Research Ltd., Vermont South, Australia. Anderson, L.A., Canepa, E.S., Horowitz, R., Claudel, C.G., Bayen, A., 2014. Optimization-based queue estimation on an arterial traffic link with measurement

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