Using detection of vehicular presence to estimate shockwave speed and upstream traffics for a signalized intersection

Using detection of vehicular presence to estimate shockwave

speed and upstream traffics for a signalized intersection

Hsun-Jung Cho

, Ming-Te Tseng

, Ming-Chorng Hwang

a

Department of Transportation Technology and Management, National Chiao Tung University, Hsinchu 300, Taiwan

b

Chien-Chen Technology Corporation, No.79, Hsien Cheng 16th Street, Chupei, Hsinchu 302, Taiwan

c

Microelectronics and Information Systems Research Center, National Chiao Tung University, Hsinchu 300, Taiwan

a r t i c l e

i n f o

Keywords: Shockwave speed LWR traffic flow theory Signal control Vehicle detection

a b s t r a c t

Based on the Lighthill–Whitham–Richards (LWR) traffic flow theory, this paper provides alternative methods to compute shockwave speed mainly by using detection data that reflects three states of vehicular presences: vehicles in moving, vehicles stopped, and void of vehicles. As the duration of a state is firmly identified within a cycle, the proposed meth-ods compute shockwave speeds directly by means of Euclidian geometrics on time–space trajectory of shockwaves. This approach is also applicable to congested signal links with a long queue (but a residual queue) beyond detection zone. In addition, given signal timing and the shockwave speeds calculated by the methods, characteristics of arrival traffics, i.e. upstream flow rate and speed, can be predicted before the end of current cycle. It justifies that the methods are capable of whether to extend green phase before next cycle or not and will be a promising tool for real-time operations of signal control. Finally, the predicted shockwave speeds, upstream flow rate, and space mean speed by the proposed method are testified using simulated data from CORSIM. The mean absolute percentage errors of the estimated speeds of forward recovery shockwave and backward forming shockwave are 4.0% and 12.4% respectively. For the predicted flow rate and space mean speed of down-stream arrival traffics, the mean absolute percentage errors are 18% and 4%, respectively. The results demonstrate the effectiveness of the presented approach.

Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction

A traffic signal, like a switch, built at an intersection conducts passing traffics when to go and when to stop alternatively. Such an operation at signalized approaches causes cyclic change of queuing phases, both accumulating and discharging pro-cesses, that begins from stop line and proceeds to upstream. Oversaturation takes place at urban signal links frequently over peak hours. As defined conceptually, it reveals traffic queues being unable to disperse fully during a cycle either due to not enough green time or because of downstream blockage when traffic demand exceeds the capacity of signalized intersection. How to estimate oversaturated queues effectively by real-time data from detectors is an important research issue on solving the congestion problem of city traffics. Most existing studies evaluate signal queues mainly relying on traffic volumes, speed and occupancies observed from surveillance systems no matter shockwave theory or queuing models applied. However, vehicle detectors usually fail to provide such data in peak periods because stopped vehicles occupy detection zone. This

http://dx.doi.org/10.1016/j.amc.2013.12.180

0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

⇑Corresponding author.

E-mail addresses:[email protected](H.-J. Cho),[email protected](M.-T. Tseng),[email protected](M.-C. Hwang).

Contents lists available atScienceDirect

Applied Mathematics and Computation

paper concentrates on the subject of how to determine shockwave speed on congested arterials by using detection of vehic-ular presence but not by that of traffic volumes, speed and occupancies. The discussed shockwave structure also follows the Lighthill–Whitham–Richards (LWR) traffic flow theory[12,18]. As the detection of vehicle presence can be clearly identified, the proposed methods calculate shockwave speeds by employing Euclidian geometrics on the time–space diagram of shock-wave propagations. The following paragraphs are a brief review on shockshock-wave theory based studies and end with remarks on their limitations.

The application of wave behavior to traffic study was first presented independently by Lighthill and Whitham in 1955 and Richards in 1956. Shockwave structure derived from LWR traffic flow theory was widely applied to describe queue dynamics and performance measures for a signalized intersection. These two classic works also provided a preliminary attempt to how the theory might be utilized to describe traffics at road junctions. Michalopoulos et al.[16]formally extended the shockwave theory to the case at signalized links. The approach is macroscopic in nature and considers interrupted traffic as a continuum fluid to demonstrate the shockwave propagation and the evolution of queue periodically downstream of a traffic signal. Stephanopoulos et al.[15]and Michalopoulos et al.[17]formulated the dynamics of formation and dissipation of queues at isolated signalized intersections by solving the conservation equations along the street. The studies also developed a real-time control policy by minimizing total intersection delays subject to queue length constraints. However, these theo-retic models consume full information of arrival traffics that limits their empirical applications to conditions with perfect detection data. We will rearrange shockwave speed equations in Section2with the assumed flow-density curve in a specific style. It forms an introductory material that is insightful for the proposed methods to compute shockwave speeds without such restrictions. For other notable reviews of shockwave-related traffic flow theories, we refer to[20,9,11,14,6,7]. Following the shockwave terminology used in[14], a typical illustration of multiple shockwaves at a signalized approach is show in

Fig. 1with dimensions both in time and space. Vehicle tracks are indicated in thin lines and shockwaves in boldface. As red light on, vehicles are forced to stop, that forms a backward forming shockwave, the line segment AEG for example, mov-ing upstream of the approach. At the end time of red light, vehicles begin to release in a maximal flow rate, or saturation flow rate, generating a backward recovery shockwave, e.g. BG. If the two shockwaves meets as shown at point G inFig. 1, a third shockwave, a forward recovery shockwave, such as GD, is actuated simultaneously. The subsequent queues within a cycle can be obviously observed, e.g. BF and CG. The total stopped delay for through vehicles is assessed by adding up the area of zone AEGB.

There are still many studies making use of shockwave analysis in interrupted traffic flow propagation and queuing prob-lem[2]; in stopped delay measurement[1]; in phase time determination[3,5]; and in forecasting traffic system performance

[8,19]. These studies were used to determine shockwave speeds by means of traffic volume, speed, and density[10,17,1,5]. In particular, a serial field studies have been conducted based on the SMART-SIGNAL system on arterials in the Twin Cities. By exploiting high-resolution ‘‘event-based’’ traffic signal data (including both vehicle-detector actuation events and signal phase change events), the serial studies make considerable progress on estimation of long queues[13], on identification of oversaturated intersections by quantifying temporally and spatially the detrimental effects of oversaturation on signal operations with the oversaturation severity index[22], on developing a stable trapezoidal form of the cycle-based arterial fundamental diagram[21], and to renew interrupted traffic flow analysis by the shockwave profile model[23].

However, the above studies have two types of limitations. The first one, shockwave speeds estimated using conventional traffic data by detectors, e.g. volume, headway, and occupancy, are less accurate when traffics in bumper-to-bumper. It is because detectors usually operate inaccurately under heavily congested circumstance. Upstream arrival rate can not be pre-dicted before next cycle is the second deficiency occurred in previous investigations. In order to make further developments from them, Cho and Tseng[4]delivered a rudimentary idea to detect only backward forming shockwave under oversaturated traffics with a fixed phase time. This paper improves the method in[4]and extends to forecast upstream flow rate by the presence of vehicle that remains accurate even under oversaturated condition. Contributions of this paper mainly relies on: (1) expanding the results of[4]on applicability at signalized intersections in terms of calculating more shockwaves speeds and dynamic signal timing considered, (2) the capability of computing extra green time to discharge traffic queue before the end of a cycle, and (3) the development of a method to forecast upstream flow beyond detection zone.

The following paragraphs of this paper are organized as follows. Model formulations describing detection of shockwaves are presented in Section2. The results of proposed methods are numerically testified in several simulation scenarios on COR-SIM and deeply discussed in Section3. Section4concludes this paper with some remarks.

2. Formulation

The formulation section comprises five parts. Section2.1describes the structure among shockwaves, speeds, and flows. The traffic parameters, stopped duration, moving duration, and empty duration, are defined in Section2.2. In Section2.3, detection of the shockwaves, including (1) backward recovery, (2) ideal forward recovery, (3) ideal backward forming, (4) backward forming, and (5) forward recovery, are introduced. In common with existing literatures, a shockwave with the term forward indicates the propagation direction of it the same as vehicle trajectory and with the term backward indicating that in opposite to vehicle trajectory. The estimation method of upstream speed and flow is discussed in Section2.4. An algo-rithm, which detects shockwaves with traffic parameters, is demonstrated in Section2.5. However, this study still assumes (1) no heavily spillover from downstream resulted in the residual queue beyond detection zone, and (2) no incident occurs. The first assumption pertains to the discussed shockwave structure with reasonable formation and discharging processes. It is pertinent to most cases of oversaturation but only the case with the residual queue over detection zone.

2.1. Relations between shockwaves and the assumed asymmetric fundamental diagram

This subsection gives a preliminary illustration of the discussed shockwave structure and flow-density curve. The nota-tions employed here would be favorable throughout this paper. Since the backward moving shockwaves are much slower than the forward moving shockwaves, this study utilizes an asymmetric fundamental diagram of flow-density curve com-posed of a parabolic function before reaching maximal flow state, Qm, and then following a linear equation starting from

Qmand halting at jam density, Kj. The proposed stream flow diagram is demonstrated inFig. 2. Let flow, density, and speed

under flow state x ( inFig. 2) be denoted as Qx, Kx, and Ux, respectively. The flow state 0 ( inFig. 2) is defined as the state of

jam density Kj; while the flow state 1 (r inFig. 2) is defined as the state which has the maximal flow rate Qmand density Km.

The maximal flow rate Qmalso represents the saturation flow rate of green phase at a signalized intersection. With proposed

flow model, we have Q0= 0 and Kj= aKm, where a is a constant needs to be calibrated. If the real traffic flow can be

repre-sented as the Greenshields’ model, which is symmetric on both non-congested and congested part, the constant a equals 2. Since backward moving shockwaves are much slower than forward moving shockwaves, the constant a should be greater than 2. For example, with the asymmetric Greenberg’s model, the constant a is equal to nature base of logarithms e (2.718). With a flow state x, the speed of shockwaves among different states can be graphically seen inFig. 2; while W01denotes

the shockwave between state 1 and 0, Wx1is the shockwave between state x and 1, and Wx0is the shockwave between state x

and 0. Throughout this article, W represents the shockwave speed.The W01can be calculated as

W01¼ Q0 Q1 K0 K1 ¼ Qm ða  1ÞKm : ð1Þ

The Wx0and Wx1can be calculated as some ratio of the W01by using the following equations.

Wx1¼ Qx Q1 Kx K1 ¼pffiffiffiffiffiffiffiffiffiffiffi1  rQm Km ¼ ð1  aÞpffiffiffiffiffiffiffiffiffiffiffi1  rW01; ð2Þ Wx0¼ Qx Q0 Kx K0¼ rQm ð1  a pffiffiffiffiffiffiffiffiffiffiffi1  rÞKm ¼ rð1  aÞ ð1  a pffiffiffiffiffiffiffiffiffiffiffi1  rÞW01; ð3Þ

where r is the flow ratio between Qxand Qm, Qx= rQm.

With the above equations, we have the relation among Wx0, Wx1and W01, which follows Wx1¼ 1 2 Wx0þ W01 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Wx0 W01  2  4ð1  aÞ2Wx0 W01þ 4ð1  aÞ 2 s 0 @ 1 A: ð4Þ

By using a Taylor series expansion, Wx1can be approximated as

Wx1¼

1

2ðaWx0þ 2ð1  aÞW01Þ: ð5Þ

After deriving Wx0, Wx1and W01, the flow ratio r between Qxand Qmcan be calculated as

r ¼Wx0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ða  1Þ2W201 4ða  1Þ 2 W01Wx0þ W2x0 q þ 2ða  1Þ2W01Wx0 W2x0 2ða  1Þ2W2 01 : ð6Þ

Therefore, the corresponding space mean speed Uxand flow rate Qxcan be calculated as

Ux¼ ð1  aÞð1 þ ffiffiffiffiffiffiffiffiffiffiffi 1  r p ÞW01 ð7Þ Qx¼ Wx0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ða  1Þ2W201 4ða  1Þ 2 W01Wx0þ W2x0 q þ 2ða  1Þ2W01Wx0 W2x0 2ða  1Þ2W2 01 Qm ð8Þ

From Eq.(1)to(8), we found that all the shockwave speeds and the volume and the space mean speed for upstream traffics can be expressed as a function of W01, Wx0, and the parameter a.

2.2. Traffic parameters: stopped duration, moving duration and empty duration

This subsection introduces the definition of traffic parameters including stopped duration, moving duration, and empty duration. A typical time–space diagram of signalized intersection is shown inFig. 3, with the vehicle trajectory is indicated as black lines. The traffic parameter, stopped duration, is defined as the vehicle presence at detection zone for an extensive per-iod (for example, 3 s for cars and 10 s for trucks) of time; while moving duration is defined as the vehicle presence less than that period. The parameter, empty duration, represents no vehicle presence at that period. To improve accuracy, multiple detectors can be used to check whether a vehicle is stopped or not. The stopped duration can also be obtained when the speed equals zero.

2.3. Detection of shockwaves for signalized intersection

This subsection introduces the estimation of shockwave from vehicle detector data. The shockwaves for a signalized intersection is shown inFig. 4. Gray lines represent the trajectories of individual vehicles, while black lines or black dash lines indicate shockwaves. This study defines four dedicated flow states. First, flow state 0 ( inFig. 4) represents a traffic state with maximal density and the speed equals zero. Second, the flow state 1 (r inFig. 4) represents the maximum flow state (defined as flow equals saturated flow rate). Third, flow state 2 (s inFig. 4) is defined as the ideal traffic flow, which means vehicles arrive within a cycle equals the saturation flow of green phase. Fourth, flow state 3 (t inFig. 4) is defined as the uniformly distributed flow over a cycle, which might be different from cycle to cycle.

There are three shockwaves among states 0, 1, and 2; W20is defined as ideal backward forming shockwave, W21is defined

as ideal forward recovery shockwave, and W01represents a backward recovery shockwave.Fig. 4(b) demonstrates a similar

situation asFig. 4(a) but with higher arrival rate (as state 3). Among state 0, 1, and 3, we have shockwaves of (1) W30, a

back-ward forming shockwave, (2) W31, forward recovery shockwave, and (3) W01, a backward recovery shockwave. Moreover, it

can be observed inFig. 4(b), where state 3 has a higher arrival rate than state 2, the propagation speed of shockwaves W30

would be greater than W20and the speed of W31would be slower than W21.Fig. 4(c) and (d) show the relationships among

five shockwaves on the fundamental diagram and the time–space diagram.

The rest of the subsection would describe the shockwave detection method via traffic parameters including stopped dura-tion, moving duradura-tion, and empty duration.

2.3.1. Back ward recovery shockwaves detection

When the signal changes to green, a backward recovery shockwave W01is formed between stopped vehicles and the

vehi-cles start to move forward. If the vehicle stopped on the detection zone of vehicle detector starts to move after time; LetDT be the time difference between the time that green phase begins and the time that state of vehicle detector changes from stopped duration to moving duration (seeFig. 5). Following the concept proposed by May[14], the backward recovery shock-wave can be calculated by,

W01¼ 

D

D

where D is the distance from stop line to the location of detector.

2.3.2. Ideal forward recovery and backward forming shockwaves calculation

To calculate the shockwaves in an intersection, this study introduces two ideal shockwaves; one is the ideal forward recov-ery shockwave, the other is the ideal backward shockwave. The ideal forward recovrecov-ery shockwave is formed at where the ideal arrival traffic flow that catches the forward moving saturation flow; the shockwave can be graphically shown as the bound-ary between state 2 and 1 inFig. 4.

Fig. 4. (a) Ideal shockwaves for a specified green and red time, (b) comparison between the ideal shockwaves and general shockwaves, (c) five shockwave relations in the proposed model, (d) five shockwave relations in time–space diagram.

The ideal flow rate, Q2, can be calculated through green split of a signal cycle (g/c) and saturation flow rate (Qm).

Q2¼ g=c Qm¼ rQm ð10Þ

The flow ratio r between Q2and Qmis equal to g/c. To replace x with 2 in Eq.(2), the ideal forward recovery shockwave can

be calculated as, W21¼ ð1  aÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  g=c p W01: ð11Þ

Similarly, to replace x with 2 in Eq.(3), the ideal backward forming shockwave can be calculated as,

W20¼

g=cð1  aÞ

ð1  a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1  g=cÞW01: ð12Þ

The equations above show that the ideal shockwaves, W21and W20, can be simplified to a ratio of the backward recovery

shockwave W01. Therefore, with backward recovery shockwave W01, green split and calibrated constant a, the ideal

shock-waves can be calculated by Eqs.(11) and (12).

2.3.3. Backward forming shockwaves detection

This subsection discusses the calculation of a backward forming shockwave. The calculation method can be categorized into two types: (1) the method which utilizes moving and empty duration, and (2) the calculation method which utilizes stopped duration. Parameters of moving duration and empty duration are generated from the detector while there are no stopped vehicles within the detection area; otherwise, the parameter of stopped duration is outputted.

2.3.3.1. Using moving duration and empty duration for general backward forming shockwave detection. Before the traffic queue reaches the detector, the sensor would output moving duration and empty duration. The time–space relations among back-ward forming shockwave, moving duration and empty duration, are illustrated inFig. 6. The speed of the first vehicle and the speed of the nth vehicle are both approximated by the space mean speed of the n passing vehicles observed at detection zone. As vehicle i, i = 1, . . ., n, with speed uiand length Lipasses a detecting zone with length Lz, this will result a moving

duration miequal to (Li+ Lz)/ui. As no vehicle within the detecting zone, it will result an empty duration ei. Let E be the

sum-mation of all empty durations eiand M be the summation of all moving duration miduring a time intervalDT. Assume the

length of detecting zone be approximately the same as the gap between two stopped vehicles, then the summation of mov-ing duration M can be calculated as the followmov-ing equation,

M ¼X n i¼1 mi¼ Xn i¼1 ðLzþ LiÞ ui ffiX n i¼1 ðLzþ LÞ ui ¼Lq us ¼

D

where Lqdenotes the queue length resulting from the vehicles during time intervalDT, L the average vehicle length of the n

vehicles, usthe space mean speed of the n vehicles observed at detection zone, andDt is the time interval shown inFig. 6.

According the geometry relations inFig. 6, the summation of all empty duration E can be calculated as

E ¼

D

D

D

Moreover, the geometry relationship also leads to the following backward forming shockwave equation.

W30¼  Lq

D

D

The backward forming shockwave can be calculated using Eq.(15)with the parameters of vehicle speed (V), moving duration (M), and empty duration (E).

2.3.3.2. Using stopped duration for backward forming shockwave detection. After the queue reaches the detector during red phase, stopped duration would be used for shockwave calculation.Figs. 7 and 8illustrate the changes in the stopped dura-tion for two consecutive signal cycles which have same red phase duradura-tion.Fig. 7demonstrates the case which the propa-gation speed of backward forming shockwave is greater than the ideal backward forming shockwave. It should be noted that since the propagation direction of backward forming shockwave is opposite to vehicle trajectory, the term greater actually means the absolute value of W30is greater than the absolute value of W20. In this case, the stopped duration is increasing for

two consecutive cycles.

On the contrary,Fig. 8shows the case which the propagation speed of backward forming shockwave is less than the ideal backward forming shockwave. The stopped duration in this case is decreasing for two consecutive cycles. Since the two cases resemble each other, the following analyses and equations can be applied to both cases.

The stopped duration of the first cycle is O1A1, while that of the second cycle is O2C2. O1A1is almost equal to O2A2if the

traffic flow changes smoothly. Hence, the stopped duration difference for these two consecutive cycles is A2C2, orDSC. Since

the dashed lines have the same slope as the shockwave W30, AC is equal to A2C2:AC is the sum of AB and BC. AB has the same

length asDSRand BC has the same length asDSG. BC, orDSG, is derived from the flow difference between shockwaves W21

and W31during the green phase G. AB, orDSR, can be calculated from the flow difference between shockwaves W20and W30

during the red phase R:Therefore,

D

D

D

To calculateDSR, we should considerFig. 7(b) and (c). Let a Euclidean space represents the time–space diagram ofFig. 7(c)

and set point A to be (0, 0). In this case, the shockwave propagation speed, W, is acted as slope. By using linear algebra, point E can be obtained from lines AE and DE,

EðXE;YEÞ : AE ! :Y ¼ W20X DE ! :Y ¼ W01ðX þ RÞ 8 < : ð17Þ

Furthermore, point B can be derived from lines AB and BE,

B RW01ðW20 W30Þ W30ðW01 W20Þ ;0   : AB ! :Y ¼ 0 BE ! :Y  YE¼ W30ðX  XEÞ 8 < : ð18Þ Hence

D

Similarly,DSGcan be calculated from the flow difference between shockwave W21and W31during the green phase G. To

cal-culateDSG, we should considerFig. 7(b) and (d). Let a Euclidean space represents the time–space diagram ofFig. 7(d) and set

point I to be (0, 0). The point E can then be obtained from lines IE and JE using linear algebra

E GW01 W01 W21 ; GW01W21 W01 W21   : IE ! :Y ¼ W21X JE ! :Y ¼ W01ðX  GÞ 8 < : ð20Þ

Fig. 7. Case for backward forming shockwave |W30| > |W20| (a) shockwaves in flow-density diagram, (b) shockwave and incremental stopped duration, (c)

incremental stopped duration in red phase, (d) incremental stopped duration in green phase.

Fig. 8. Case for backward forming shockwave |W30| < |W20| (a) shockwaves in flow-density diagram, (b) shockwave and reductive stopped duration, (c)

Also point F can be derived from lines IF and JF, F GW01 W01 W31 ; GW01W31 W01 W31   : IF ! :Y ¼ W31X JF ! :Y ¼ W01ðX  GÞ 8 < : ð21Þ

And point H can be calculated from lines FH and EH,

HðXH;YHÞ : FH ! :Y ¼ YF EH !:Y  YE¼ W30ðX  XEÞ 8 < : ð22Þ Therefore

D

Replace x by 3 in Eq.(3), we would have the following equation,

W31¼

1

2ðaW30þ 2ð1  aÞW01Þ ð24Þ

Substitute W21and W31in Eq.(23)by Eqs.(12) and (24),

D

GW01ðW01 W30ÞðW30 W20Þ

W30ðW01 W21Þð2W01 W30Þ ð25Þ

Therefore,DSCcan be calculated by

D

D

D

The calculation of shockwaves with parameter of stopped duration, the following procedure can be applied. We have the red phase duration R and green phase duration G given by traffic controller; and the backward recovery shockwave W01

cal-culated by Eq.(9). When there is no spillover, the speed of shockwave W01is nearly constant. The ideal shockwaves, W20and

W21, can be calculated from Eqs.(11) and (12)with given R and G. Therefore, the stopped duration differences,DSC, can

immediately be calculated after detecting a stopped vehicle. After deriving stopped duration difference, from vehicle detec-tion, the backward forming shockwave W30can be calculated by Eq.(26). The calculating procedure can be applied toFig. 8

and having the same result.

If the red phase duration (R) is not fixed for two consecutive cycles, then Eq.(26)must be modified as,

D

D

D

D

D

whereDR is the red phase duration difference of two consecutive cycles.

2.3.3.3. Backward forming shockwave detection under heavy congestion. If the queue has the length more than the vehicle detector installation location plus the length of queue that can be discharged during green phase, it would cause the detector to output the traffic parameter of stopped duration be the same as red phase duration, asFig. 9. In this case, Eqs.(26) and (27)cannot be used to calculate the backward forming shockwave. Additional vehicle detectors can be added to solve this problem; the Eqs.(26) and (27)can be applied to new detectors. If the installation of new detector is not possible, the moving average, as Eq.(28), can be used to predict the backward forming shockwave.

W30ðnÞ ¼ 1 5W30ðn  1Þ þ 1 5W30ðn  2Þ þ 1 5W30ðn  3Þ þ 1 5W30ðn  4Þ þ 1 5W30ðn  5Þ ð28Þ

where W30(n) is current shockwave value, W30(ni) is the i-th previous shockwave value.

2.3.4. Forward recovery shockwave detection

The forward recovery shockwave can easily be calculated using Eq.(24). In general, the backward recovery shockwave W01is calculated first, followed by the backward forming shockwave W30. The forward recovery shockwave, W31, is the last

shockwave to be calculated. Therefore, the shockwaves can be derived right after the detection of stopped vehicle. Compare to existing researches, which the shockwaves can only be derived after the beginning of green time; the proposed method can support real-time application.

2.4. Upstream speed and flow detection

This sub-section focuses on the estimation of upstream speed and flow. The upstream means the state which is not af-fected by the queue; state 3 is a common representative. If all shockwaves of state 3 are derived, the arrival flow rate can be calculated by Eqs.(6) and (8). Replace x with 3 in Eq.(6), the arrival flow ratio r is

r ¼W30 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ða  1Þ2W201 4ða  1Þ 2 W01W30þ W230 q þ 2ða  1Þ2W01W30 W230 2ða  1Þ2W2 01 ð29Þ

The flow is calculated as the following equation,

Q3¼ rQm¼ W30 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ða  1Þ2W2 01 4ða  1Þ 2 W01W30þ W230 q þ 2ða  1Þ2W01W30 W230 2ða  1Þ2_W2 01 Qm ð30Þ

where Qmis the saturation flow rate, which can be investigated in advance. The speed of state 3 can be derived from Eq.(6),

U3¼

ð1  aÞr

1 pffiffiffiffiffiffiffiffiffiffiffi1  rW01 ð31Þ

where U3represents the space mean speed of state 3.

2.5. Proposed algorithm

This section proposes an algorithm to calculate shockwaves that being discussed in the Section2.3. The proposed algo-rithm is demonstrated asFig. 10(a). First, gather presence data from vehicle detector; second, calculate traffic parameters including empty duration, moving duration, and stopped duration. Third, the calculation of backward recovery shockwave and followed by fourth, the calculation of ideal shockwaves. Fifth, backward forming shockwave is calculated. Last, for-ward recovery shockwave is obtained. The estimation method of backfor-ward forming shockwave is detailed inFig. 10(b). This figure demonstrates the usage of multiple detectors to predict backward forming shockwave; although the figure illustrates the procedure by two detectors, it can be easily extended to multiple detectors. The first step is setting a vehicle detector near the stop line and the other at the upstream. The spacing between detectors should be more than the length of queue that can be discharged during maximal green time. If the first detector do not gives a stopped duration, then the moving duration and empty duration of first detector is utilized in the calculation of backward forming shockwave. Other-wise, stopped duration is taken into consideration. Moreover, if the stopped duration is larger than red time, the next detector should be considered; the above procedure should be repeated again for the next detector. The whole procedure ends at the last detector. If all detectors have the stopped durations as red time, estimation method of Eq.(28)should be used.

3. Simulation results

A CORSIM simulation environment has been established to evaluate the traffic parameters and the proposed shockwave detection methods. An independent intersection with four approaches is created in the CORSIM environment. Two vehicle sensors are located 300 and 730 feet from the stop line on an approach, asFig. 11shows.Fig. 12illustrates the phase time and input traffic flow of that approach. During the simulation, the headway distribution is set to be uniform, and the traffic is

only comprised of passenger cars. No spillover occurs during the simulation period; therefore, all stopped vehicles are caused by red signals. This simulation is a sample intersection near ramp of highway or freeway in rush hours. To demonstrate the capability of the proposed method, the signal timing in this simulation is designed to be dynamically changed. The change of phase time may be resulting from some adaptive control methods; as the traffic flow increases, the green time also increases.

3.1. Traffic parameters

Fig. 13shows the traffic parameters derived from both vehicle detectors during the simulation period. Due to changes in phase duration, some cycle length is shorter or longer than others are. With low traffic flow demands, both detectors output empty duration and moving duration. The stopped duration is only presented when traffic queue reaches the detecting zone.

Fig. 14compares the red phase duration and the stopped duration of both detectors. Notably, the stopped duration of both detectors approaches the red time, indicating that queue length of un-discharged vehicles reaches beyond the second detec-tor. Moreover, since the first detector is installed closer to stop line than the second one, the first detector would always have more cycles that reports stopped duration.

3.2. Results of shockwaves

According to Section2, each method of the backward forming shockwave is valid only for a specific circumstance. For example, Eq.(15)should be used when no stopped duration occurs, otherwise Eqs.(26) and (27)should be used instead.

Fig. 10. (a) The algorithm for five shockwaves detection, (b) the algorithm for backward forming shockwave detection.

Sensor 1 Sensor 2

300 feet 730 feet

If the stopped duration be equal to red phase duration, Eq.(28)can be taken into consideration. The constant a in those equa-tions is calibrated as 2.1. The algorithm proposed in Section2.5indicates the proper usage of each equation.Fig. 15 demon-strates the backward forming shockwave calculation of each method and the final result of the proposed algorithm. In

Fig. 15, 4Si represents the tenth of the stopped duration difference of detector i for two consecutive cycles; while Si, M/ Ei, and Mavg denote the shockwave calculated from stopped duration, moving/empty duration, and moving average, respectively. The calculated backward forming shockwave from the proposed algorithm is denoted as W30. It can be observed

inFig. 15that the speed of backward forming shockwave is negatively related to the stopped duration difference. As the stopped duration difference decrease, the backward forming shockwave speed increase, vice versa. The condition should holds theoretically. However, with stochastic driving behavior, some non-ordinary driving behavior occurs near the vehicle detector on the 17th cycle of simulation. Therefore, the statement would be violated on the 17th cycle.

Fig. 16compares the shockwave calculating result derived from detector with the one that directly measured from COR-SIM. InFig. 16, the ideal forward recovery/backward forming shockwave is denoted as W21/W20; while W31/W30denotes the

calculated forward recovery/backward forming shockwave. The directly measured ones are denoted as W⁄

31 and W⁄30,

respectively. InFig. 16, the calculated shockwaves are similar to the directly measured ones, which represent a significant

(a)

(b)

Fig. 12. (a) the phase times of the intersection, (b) the input flow of the link.

(a)

(b)

Fig. 13. (a) The stopped duration, moving duration and empty duration of detector 1, (b) the stopped duration, moving duration and empty duration of detector 2.

result. In the simulation, the intersection has a fixed saturation flow rate, without the disturbance from downstream spill-over; therefore, W01would maintain a stable value (21 ft/s herein).

The mean absolute percentage error (MAPE) of W30is 12.4% and the mean absolute error (MAE) is 0.42 ft/s. While the

MAPE and MAE of W31is 4% and 0.69 ft/s, respectively. The bias mainly comes from the vehicle arrival pattern; as the vehicle

Fig. 14. Relation between the stopped duration and red phase time.

Fig. 15. Results of the backward forming shockwave detection algorithm.

comes uniformly asFig. 17(a), the detected stopped duration and corresponding shockwave detection would be unbiased. If vehicles arrive as platoons as shown inFig. 17(b) or (c), the inaccurate stopped duration would result biased W30.Fig. 17(b)

demonstrates that, as the queue results from the first platoon does not reach detector and there exists a major gap between the first and the second platoon, the stopped duration would be underestimated. The underestimated stopped duration would result in a slower shockwave speed (indicated as dash line).Fig. 17(c) gives the counter example of overestimated stopped duration and the corresponding faster shockwave speed. During the simulation, it is observed that with more con-gested traffic; the less probability of major gap would happen.

3.3. Results of flow and speed

InFig. 18, the estimated upstream flow is compared with the simulation input. To give a better understanding, these flow rates have been transformed into flow ratio (r). The flow ratio (r) is then compared with the flow ratio derived from

Red time Green time

Detector

Red time Green timeRed time

(a)

(b)

(c)

Fig. 17. The comparison of different arrival pattern and its corresponding bias in shockwave estimation, (a) uniform arrival pattern and its corresponding shockwave, (b) arrival pattern that gives an underestimated shockwave speed, and (c) arrival pattern that gives an overestimated shockwave speed.

Fig. 18. The predicted traffic flow of state 3.

simulation (r⁄_{). The comparison of estimated upstream speed (U) and detected speed (U}⁄_{) is illustrated inFig. 19. The}

esti-mated upstream speed (U) is space mean speed but the detected speed is time mean speed; therefore, the detected speed is transformed to space mean speed with the method proposed by Drake, Schofer, and May (1967). The MAPE and MAE of flow ration are 18% and 0.03, respectively. While those of space mean speed are 4% and 1.79 ft/s, respectively. These results dem-onstrate that the proposed algorithm is capable of estimating flow and speed at upstream area.

4. Conclusions

In this paper, we proposed an innovative approach to estimate the upstream traffic information under oversaturated sit-uation using shockwave analysis. A key methodological contribution of the approach is that it estimates shockwaves by com-bining traffic parameters, dynamic signal timing and traffic flow models. By utilizing parameters of stopped duration, moving duration, and empty duration, we are able to calculate shockwaves including (1) forward recovery, (2) ideal back-ward forming, (3) ideal forback-ward recovery, (4) backback-ward forming, and (5) forback-ward recovery shockwave.

To the best of authors’ knowledge, this is the first paper that utilizes real time shockwave by stopped duration to estimate upstream traffic flow and speed far beyond detection zones of vehicle detectors. With the shockwaves, upstream traffic flow and speed information can be estimated accordingly. These models are evaluated by traffic simulation and demonstrate a significant result. The proposed model has some pre-conditions for traffic flow state. These assumptions can be solved by combining linear regression and the information derived from multi-zone sensors to capture the variation of shockwaves.

Acknowledgements

The authors would like to thank the Ministry of Education of the Republic of China, Taiwan (MOE ATU Program), the Na-tional Science Council of the Republic of China, Taiwan (Contract No. NSC-95-2221-E-009-346, NSC-95-2221-E-009-347 and NSC-95-2752-E-009-010-PAE), and the Ministry of Transportation and Communications of the Republic of China, Taiwan (Contract No. MOTC-STAO-101-02) for financially supporting this research.

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