This subsection introduces the estimation of shockwave from vehicle detector data.
The shockwaves for a signalized intersection is shown in Figure 3.5. 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 (○0 in Figure 3.5) represents a traffic state with maximal density and the speed equals zero. Second, the flow state 1 (○1 in Figure 3.5) represents the maximum flow state (defined as flow equals saturated flow rate). Third, flow state 2 (○2 in Figure 3.5) 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 (○3 in Figure 3.5 ) 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; W20 is defined as ideal backward forming shockwave, W21 is defined as ideal forward recovery shockwave, and W01 represents a backward recovery shockwave. Figure 3.5(b) demonstrates a similar situation as Figure 3.5(a) but with higher arrival rate (as state 3). Among state 0, 1, and 3, we have shockwaves of 1) W30, a backward forming shockwave, 2) W31, forward recovery shockwave, and 3) W01, a backward recovery shockwave. Moreover, it can be
observed in Figure 3.5(b), where state 3 has a higher arrival rate than state 2, the propagation speed of shockwaves W30 would be greater than W20 and the speed of W31would be slower than W21. Figure 3.5(c) and 3.5(d) show the relationships among five shockwaves on the fundamental diagram and the time-space diagram.
Figure 3.5. (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 shockwaves relations in time-space diagram.
Relationships among shockwaves, speeds and flows
This subsection gives a preliminary understanding of relationships among shockwaves, speeds and densities, and the notations would be used throughout this article. Since the backward moving shockwaves are much slower than the forward
moving shockwaves, this study utilizes an asymmetric fundamental diagram that comprise a parabolic non-congested part and a linear congested part. The proposed stream flow diagram is demonstrated in Figure 3.6. Let flow, density, and speed under flow state x (○x in Figure 3.6) be denoted as Qx, Kx, and Ux, respectively. The flow state 0 ( ○0 in Figure 3.6) is defined as the state of jam density Kj; while the flow state 1 (○1 in Figure 3.6) is defined as the state which has the maximal flow rate Qm and density Km. The maximal flow rate Qm also 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 represented 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).
Figure 3.6 The relation between shockwaves
With a flow state x, the speed of shockwaves among different states can be graphically seen in Figure 3.6; while W01 denotes the shockwave between state 1 and 0, Wx1 is the shockwave between state x and 1, and Wx0 is the shockwave between state x and 0. Throughout this article, W represents the shockwave speed.
The W01 can be calculated as
With the above equations, we have the relationship among Wx0, Wx1and W01 , which follows
By using a Taylor series expansion, Wx1 can be approximated as
)
Back ward recovery shockwaves detection
When the signal changes to green, a backward recovery shockwave W01 is formed between stopped vehicles and the vehicles start to move forward. If the vehicle stopped
on the detection zone of vehicle detector starts to move after time; Let T 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 (see Figure 3.7.). Following the concept proposed by May [16], the backward recovery shockwave can be calculated by,
W T
D
01 (3.12)
Where D is the distance from stop line to the location of detector.
Figure 3.7 Backward recovery shockwave detection
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 recovery shockwave, the other is the ideal backward shockwave. The ideal forward recovery 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 boundary between state 2 and 1 in Figure 3.5.
The ideal flow rate, Q2, can be calculated through green split of a signal cycle (g/c) and saturation flow rate (Qm).
m
m rQ
cQ
Q2g (3.13)
The flow ratio r between Q2 and Qm is equal to g/c. To replace x with 2 in Eq. (3.8), the ideal forward recovery shockwave can be calculated as,
0 1 2 1 (1 a) 1 g/cW
W . (3.14)
Similarly, to replace x with 2 in Eq. (3.9), the ideal backward forming shockwave can be calculated as, shockwaves can be calculated by Eq. (3.14) and (3.15).
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
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 relation among backward forming shockwave, moving duration and empty duration is illustrated in Figure 3.8.
As a vehicle i with speed V and length Li passes a detecting zone with length Lz, this will result a moving duration mi equal to (Li + Lz)/V. As no vehicle within the detecting zone, it will result an empty duration ei. Let E be the summation of all empty durations ei and M be the summation of all moving duration mi during a time interval ΔT.
Assume the length of detecting zone be approximately the same as the gap between two stopped vehicles, then the summation of moving duration M can be calculated as the
where Lq denotes the queue length resulting from the vehicles during time interval ΔT, and Δt is the time interval shown in Figure 3.8. According the geometry relationship in Figure 3.8, the summation of all empty duration E can be calculated as
t T M T
E . (3.17)
Moreover, the geometry relationship also leads to the following backward forming shockwave equation.
E MV E
L t T
W Lq q
30 (3.18)
The backward forming shockwave can be calculated using Eq. (3.18) with the parameters of vehicle speed (V), moving duration (M), and empty duration (E).
Fig. 3.8. Relation among backward forming shockwave, moving duration and empty duration.
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. Figure 3.9 and 3.10 illustrate the changes in the stopped duration for two consecutive signal cycles which have same red phase duration.
Figure 3.9 demonstrates the case which the propagation 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 W30 is greater than the absolute value of W20. In this case, the stopped duration is increasing for two consecutive cycles.
On the contrary, Figure 3.10 shows 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.
Figure 3.9. Backward forming shockwave for |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.
The stopped duration of the first cycle is O1A1, while that of the second cycle is
2 2C
O . O1A1 is almost equal to O2A2if the traffic flow changes smoothly. Hence, the stopped duration difference for these two consecutive cycles is A2C2, or ΔSC. Since the dashed lines have the same slope as the shockwave W30, AC is equal toA2C2. AC is the sum of AB and BC. AB has the same length as ΔSR and BC has the same length as ΔSG. BC, or ΔSG, is derived from the flow difference between shockwaves W21 and W31during the green phaseG. AB, or ΔSR, can be calculated from the flow difference between shockwaves W20 and W30 during the red phaseR.
Figure 3.10. Case for backward forming shockwave |W30 |< |W20 | (a) Shockwaves in flow-density diagram. (b) Shockwave and reductive stopped duration. (c) Reductive stopped duration in red phase. (d) Reductive stopped duration in green phase.
Therefore, Euclidean space represents the time-space diagram of Figure 3.9(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,
Furthermore, point B can be derived from lines AB and BE,
Similarly, ΔSG can be calculated from the flow difference between shockwave W21
and W31 during the green phaseG. To calculate ΔSG, we should consider Figure 3.9(b) and Figure 3.9(d). Let a Euclidean space represents the time-space diagram of Figure
Also point F can be derived from lines IF and JF,
And point H can be calculated from lines FH and EH,
Replace x by 3 in equation (3.11), we would have the following equation,
)
Substitute W21 and W31 in equation (3.26) by equation (3.14) and (3.27),
)
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 calculated by Eq.
(3.12). When there is no spillover, the speed of shockwave W01 is nearly constant. The ideal shockwaves, W20 and W21, can be calculated from Eq. (3.14) and (3.15) with given R and G. Therefore, the stopped duration differences, ΔSC, can immediately be calculated after detecting a stopped vehicle. After deriving stopped duration difference, from vehicle detection, the backward forming shockwave W30 can be calculated by Eq.
(3.29).The calculating procedure can be applied to Figure 3.10 and having the same result.
If the red phase duration (R) is not fixed for two consecutive cycles, then Eq. (3.29) must be modified as,
where ΔR is the red phase duration difference of two consecutive cycles.
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, as Figure 3.11. In this case, Eq. (3.29) and (3.30) cannot be used to calculate the backward forming shockwave. Additional vehicle detectors can be added to solve this problem; the Eq. (3.29) and (3.30) can be applied to new detectors. If the installation of new detector is not possible, the moving average, as Eq. (3.31), can be used to predict the backward forming shockwave.
) 5 5 (
) 1 4 5 (
) 1 3 5 (
) 1 2 5 (
) 1 1 5 (
) 1
( 30 30 30 30 30
30 n W n W n W n W n W n
W (3.31)
where W30(n) is current shockwave value, W30(n-i) is the i-th previous shockwave value.
Figure 3.11 A vehicle‟s stopped duration is equal to the red phase time.
Forward recovery shockwave detection
The forward recovery shockwave can easily be calculated using Eq. (3.27). The backward recovery shockwave W01 is calculated first, followed by the backward forming shockwave W30. The forward recovery shockwave, W31, is the last shockwave to be calculated. The sequence of shockwave estimation determines the model‟s prediction capability. Liu and Wu [29, 17, 30] could not provide any predictions because their models had computed the forward recovery shockwave before the backward forming shockwave. In this study, three shockwaves can be derived right after the detection of stopped vehicle and much traffic information can be predicted. Compare to existing researches, which the shockwaves can only be derived after the beginning of green time; the proposed method can predict shockwaves earlier and supports an adaptive traffic control model more efficiency.
Shockwaves detection algorithm
This subsection proposes an algorithm to calculate shockwaves that being
discussed in the previous subsections. The proposed algorithm is demonstrated as figure 3.12(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, forward recovery shockwave is obtained. The estimation method of backward forming shockwave is detailed in Figure 3.12(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. Otherwise, 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. (3.31) should be used.