Module 3 : Microscopic Traffic Flow Modeling
Lecture 14 : Car Following Models
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Numerical Example

Let a leader vehicle is moving with zero acceleration for two seconds from time zero. Then he accelerates by 1 $ m/s^2$ for 2 seconds, then decelerates by 1$ m/s^2$for 2 seconds. The initial speed is 16 m/s and initial location is 28 m from datum. A vehicle is following this vehicle with initial speed 16 m/s, and position zero. Simulate the behavior of the following vehicle using General Motors' Car following model (acceleration, speed and position) for 7.5 seconds. Assume the parameters l=1, m=0 , sensitivity coefficient ( $ \alpha_{l,m}$) = 13, reaction time as 1 second and scan interval as 0.5 seconds.

Solution

The first column shows the time in seconds. Column 2, 3, and 4 shows the acceleration, velocity and distance of the leader vehicle. Column 5,6, and 7 shows the acceleration, velocity and distance of the follower vehicle. Column 8 gives the difference in velocities between the leader and follower vehicle denoted as $ dv$. Column 9 gives the difference in displacement between the leader and follower vehicle denoted as $ dx$. Note that the values are assumed to be the state at the beginning of that time interval. At time t=0, leader vehicle has a velocity of 16 m/s and located at a distance of 28 m from a datum. The follower vehicle is also having the same velocity of 16 m/s and located at the datum. Since the velocity is same for both, $ dv$ = 0. At time t = 0, the leader vehicle is having acceleration zero, and hence has the same speed. The location of the leader vehicle can be found out from equation  as, x = 28+16$ \times$0.5 = 36 m. Similarly, the follower vehicle is not accelerating and is maintaining the same speed. The location of the follower vehicle is, x = 0+16$ \times$0.5 = 8 m. Therefore, $ dx$ = 36-8 =28m. These steps are repeated till t = 1.5 seconds. At time t = 2 seconds, leader vehicle accelerates at the rate of 1 $ m/s^2$ and continues to accelerate for 2 seconds. After that it decelerates for a period of two seconds. At t= 2.5 seconds, velocity of leader vehicle changes to 16.5 m/s. Thus $ dv$ becomes 0.5 m/s at 2.5 seconds. $ dx$ also changes since the position of leader changes. Since the reaction time is 1 second, the follower will react to the leader's change in acceleration at 2.0 seconds only after 3 seconds. Therefore, at t=3.5 seconds, the follower responds to the leaders change in acceleration given by equation i.e., a = $ \frac{13\times0.5}{28.23}$ = 0.23 $ m/s^2$. That is the current acceleration of the follower vehicle depends on $ dv$ and reaction time $ \Delta$ of 1 second. The follower will change the speed at the next time interval. i.e., at time t = 4 seconds. The speed of the follower vehicle at t = 4 seconds is given by equation as v= 16+0.231$ \times$0.5 = 16.12 The location of the follower vehicle at t = 4 seconds is given by equation as x = 56+16$ \times$0.5+ $ \frac{1}{2}\times$0.231 $ \times0.5^2$ = 64.03 These steps are followed for all the cells of the table.
Table 1: Car-following example
$ t$ $ a(t)$ $ v(t)$ $ x(t)$ $ a(t)$ $ v(t)$ $ x(t)$ $ dv$ $ dx$
(1) (2) (3) (4) (5) (6) (7) (8) (9)
$ t$ $ a(t)$ $ v(t)$ $ x(t)$ $ a(t)$ $ v(t)$ $ x(t)$ $ dv$ $ dx$
0.00 0.00 16.00 28.00 0.00 16.00 0.00 0.00 28.00
0.50 0.00 16.00 36.00 0.00 16.00 8.00 0.00 28.00
1.00 0.00 16.00 44.00 0.00 16.00 16.00 0.00 28.00
1.50 0.00 16.00 52.00 0.00 16.00 24.00 0.00 28.00
2.00 1.00 16.00 60.00 0.00 16.00 32.00 0.00 28.00
2.50 1.00 16.50 68.13 0.00 16.00 40.00 0.50 28.13
3.00 1.00 17.00 76.50 0.00 16.00 48.00 1.00 28.50
3.50 1.00 17.50 85.13 0.23 16.00 56.00 1.50 29.13
4.00 -1.00 18.00 94.00 0.46 16.12 64.03 1.88 29.97
4.50 -1.00 17.50 102.88 0.67 16.34 72.14 1.16 30.73
5.00 -1.00 17.00 111.50 0.82 16.68 80.40 0.32 31.10
5.50 -1.00 16.50 119.88 0.49 17.09 88.84 -0.59 31.03
6.00 0.00 16.00 128.00 0.13 17.33 97.45 -1.33 30.55
6.50 0.00 16.00 136.00 -0.25 17.40 106.13 -1.40 29.87
7.00 0.00 16.00 144.00 -0.57 17.28 114.80 -1.28 29.20
7.50 0.00 16.00 152.00 -0.61 16.99 123.36 -0.99 28.64
8.00 0.00 16.00 160.00 -0.57 16.69 131.78 -0.69 28.22
8.50 0.00 16.00 168.00 -0.45 16.40 140.06 -0.40 27.94
9.00 0.00 16.00 176.00 -0.32 16.18 148.20 -0.18 27.80
9.50 0.00 16.00 184.00 -0.19 16.02 156.25 -0.02 27.75
10.00 0.00 16.00 192.00 -0.08 15.93 164.24 0.07 27.76
10.50 0.00 16.00 200.00 -0.01 15.88 172.19 0.12 27.81
11.00 0.00 16.00 208.00 0.03 15.88 180.13 0.12 27.87
11.50 0.00 16.00 216.00 0.05 15.90 188.08 0.10 27.92
12.00 0.00 16.00 224.00 0.06 15.92 196.03 0.08 27.97
12.50 0.00 16.00 232.00 0.05 15.95 204.00 0.05 28.00
13.00 0.00 16.00 240.00 0.04 15.98 211.98 0.02 28.02
13.50 0.00 16.00 248.00 0.02 15.99 219.98 0.01 28.02
14.00 0.00 16.00 256.00 0.01 16.00 227.98 0.00 28.02
14.50 0.00 16.00 264.00 0.00 16.01 235.98 -0.01 28.02
15.00 0.00 16.00 272.00 0.00 16.01 243.98 -0.01 28.02
15.50 0.00 16.00 280.00 0.00 16.01 251.99 -0.01 28.01
16.00 0.00 16.00 288.00 -0.01 16.01 260.00 -0.01 28.00
16.50 0.00 16.00 296.00 0.00 16.01 268.00 -0.01 28.00
17.00 0.00 16.00 304.00 0.00 16.00 276.00 0.00 28.00
17.50 0.00 16.00 312.00 0.00 16.00 284.00 0.00 28.00
18.00 0.00 16.00 320.00 0.00 16.00 292.00 0.00 28.00
18.50 0.00 16.00 328.00 0.00 16.00 300.00 0.00 28.00
19.00 0.00 16.00 336.00 0.00 16.00 308.00 0.00 28.00
19.50 0.00 16.00 344.00 0.00 16.00 316.00 0.00 28.00
20.00 0.00 16.00 352.00 0.00 16.00 324.00 0.00 28.00
20.50 0.00 16.00 360.00 0.00 16.00 332.00 0.00 28.00

Figure 1: Velocity vz Time
\begin{figure} \centerline{\epsfig{file=t83-velocity-time-plot.eps,width=8cm}} \end{figure}
Figure 2: Acceleration vz Time
\begin{figure} \centerline{\epsfig{file=t84-acceleration-time-plot.eps,width=8cm}} \end{figure}