Table of Contents
Application of System Dynamics in Car-following Models
Arif Mehmood, Frank Saccomanno and Bruce Hellinga
Department of Civil Engineering, University of Waterloo
Waterloo, Ontario, Canada N2L 3G1
E-mail: saccoman@ uwaterloo.ca
F or presentation at the 20" Annual Conference of the System Dynamics Society, J uly 2002.
ABSTRACT
Over the past 50 years, many different “car-following" models have been proposed to describe the
driver behaviour in a traffic stream. A number of inherent assumptions about human constraints and
preferences in existing car-following models hamper their validity for use in the design and
evaluation of different ITS (Intelligent Transportation Systems) technologies and/or controls such as
AVCSS (Advanced Vehicle Control and Safety Systems).
In this paper we introduce a new Systems Dynamic (SD) car-following model that addresses many of
the shortcomings of existing car-following models and provides a more relevant platform for
simulating driver behavior in all types of car-following situations subject to changing traffic
conditions. The proposed SD model was developed and validated based on observed vehicle tracking
data. Preliminary results suggest that the proposed model yields speed and spacing profiles for
vehicles in "real time" that compare well with those observed empirically.
Keywords: Car-following, Driver behavior, Systems Dynamics, Microscopic traffic simulation
1.0 INTRODUCTION
Driver behavior involves two main responses: 1) speed and 2) steering. The primary objective of
most car-following models is to predict following vehicle speed and spacing profiles based on lead
vehicle stimuli (speeds) for a set of route/traffic conditions and driver characteristics. These models
typically consider a string of vehicles traveling in a single lane. Lane changes are normally not
considered within the scope of simple car-following algorithms. More complex driver responses
considered within more extensive microscopic traffic simulations combine simple car-following
models with models of other driver responses (i.e. lane changes, routing, etc.) to produce a more
practical topology of driver behaviour in actual traffic situations.
Given the increasing demand for using new technologies and techniques in transportation sector, it is
clear that a detailed understanding of driver behaviour under different transportation conditions is
now becoming highly important. For example, the validity of car-following models appears to be
especially important when evaluating different ITS technologies and/or controls such as, AVCSS or
ACC (Adaptive Cruise Control). These technologies are expected to modify driver behavior in a
complex interactive fashion. From the perspective of car-following, these technologies also seek to
replicate driver behavior through partial control of the accelerator, while removing potential hazards
that may occur through misperception of distance and other driver errors. Working prototypes are
currently being investigated and will likely be available commercially within the next few years
(Touran, 1999). To assess the impact of AVCSS or ACC on safety and traffic flow, it becomes
necessary to utilize the results of car-following models and the insights they provide into how drivers
perceive and react to variable speeds and separation distances in actual traffic situations.
The model introduced in this paper makes use of Systems Dynamics (SD) principles. Systems
Dynamics provides the computational platform for describing and investigating the complex process
that reflect driver behaviour in a traffic stream. The SD platform is characterized by many non-linear
relationships (both heuristic and empirical) with numerous feedback loops. As such, the proposed
SD car-following model introduced in this paper relaxes many of the limiting assumptions of existing
car-following models, rendering the process more relevant for microscopic traffic simulation.
This paper has three basic objectives: 1) review existing car-following models and identify their
behavioural shortcomings, 2) develop an SD car-following model that addresses many of these
shortcomings, and 3) compare the SD model to observed vehicle tracking data and assess its ability
to predict speed and spacing profiles over time.
2.0 REVIEW OF CAR-FOLLOWING MODELS
A comprehensive review of the historical development of car-following models is available in the
literature (Brackstone and McDonald, 1999). In this paper we provide a summary of the important
models, their formulation, and limitations.
Car-following models have been studied extensively since the early 1950s. The earliest work focused
on the principle that vehicle separation is govemed by safety considerations by which distance or
time headway between vehicles is a function of relative vehicle speeds. Pipes (1953) developed a
car-following model that assumes that drivers control their speed to maintain a desired spacing. This
spacing is assumed to be linearly dependent on speed.
Forbes (1958) assumed that drivers control their speed to maintain a minimum time headway. This
time headway is a linear function of the speed of the lead vehicle. Subsequent models (Table 1) have
incorporated factors such as spacing between vehicles, speed differential, and driver sensitivity into
car-following behavior. Car-following models developed by Chandler et al. (1958), Gazis et al.
(1959, 1961), Edie (1961), Newell (1961), Herman and Rothery (1962), and Bierley (1963) all
assume that following vehicle drivers respond solely to changes in speed and position of the lead
vehicle (essentially the vehicle immediately in front). Fox and Lehman (1967), and Bexelius (1968)
have suggested that instead of considering only the vehicle immediately in front, drivers should also
take into account the speed and position of other “lead vehicles” (at least two downstream). This
suggests a more reasonable perception of driver behaviour where following vehicle drivers take a
longer range view of the traffic conditions downstream in setting their respective speed and spacing
over time. A common feature of most of the car-following models in Table 1 is the assumption that
the following vehicle driver's responses are based on spacing and differential speed between the
following and the lead vehicle(s). The underlying assumption for these models is that the following
vehicle driver can accurately perceive spacing and differential speed between the following and the
lead vehicle(s).
A larger number of studies have focused on calibration of parameters (4, m, and 1) in the GHR model
(the model developed by Gazis, Herman and Rothery, 1962) and it variants. Among these the most
notable examples of are: May and Keller (1967), Heyes and Ashworth (1972), Treiterer and Myers
(1974), Ceder et al. (1976), Aron (1988), Ozaki, (1993). According to (Brackstone and McDonald,
1999) not withstanding considerable work on calibration and validation the general level of
agreement on parameter values has led to its general demise.
Another class of models, called psychophysical or action point models, also exists. These models
have been developed in the basis that drivers perceive relative speed by detecting changes in the
apparent size of the downstream vehicles. The threshold for this perception, which is well known,
determines whether or not a driver can perceive a change in relative speed or spacing. Several
existing microscopic traffic simulation programs including Paramics and Mission incorporated action
point car-following models. The difficulty with these models is the lack of objective calibration of
the individual parameters and thresholds, and consequently of the models as a whole.
Table 1: Selected car-following model algorithms.
Source: Corresponding C ar-following Model
Chandler et al. (1958) a, (t+at) = a W,,(t)-V, 0]
Gazis et al. (1959, 1961) g 1
a, (t+dt) = ao vit) -V, 0]
L2 a(t) — Xz (
Edie (1961) q (t)
ap (t+At) = a@ ict
Newell (1961) a, (t+dt) = G, Rul (t) -X, ®]
Herman and Rothery, a or
(1962) a, (t+At) = a(t) V, (
Xp 0-107 7
Bierley (1963) a, (t+at) = a W,,(t) ial + B K,0-X,0
Fox and Lehman (1967) = 4
ap t-tat) = av, (i) AMalVi2(t) Ve (OT Wali (t) Ve OIE
Xi.(t)-X, (OP [Xu -X; ()]
Bexelius (1968) a,(tt+at) = a Wt)V.0] + B Wit—7,]
Rockwell et al. (1968) a,(t+at) = a W,()-V,0] + B a(t)
Follower Lead2 Lead 1
F Ll, Li
X(t) vel
= Xult)
Where:
ap(t+A t) = Acceleration rate of Following Vehicle driver at time t + At
ara(t) = Acceleration rate of Lead Vehicle 2 driver at time t
Ve(t), Vir(t), Vialt) = Speed of Following, Lead Vehicle 1 and Lead Vehicle 2 at time t
Xp(t), Xii(t), Xui(t) = Position of Following, Lead Vehicle 1 and Lead Vehicle 2 at time t
t = Current simulation time (seconds)
At = Perception-reaction time (seconds) or simulation interval
Ga = Empirical relationship between velocity and headway for
acceleration/deceleration
4a,m,],W1,W2 = Model parameters
There are four basic assumptions inherent in many existing models that tend to restrict their ability to
explain and predict driver behaviour in actual traffic situations:
1. The vast majority of car-following models assume that following vehicle drivers can accurately
perceive relative speed of the lead and following vehicles, absolute speed and/or acceleration of
lead vehicle at any point in time. These assumptions are unrealistic given the rectilinear nature of
vehicles moving in a single lane, and problems of depth perception and differences driver
reactions with factors such as, ageing, impairment, disability, etc (Boer, 1999).
2. Many existing car-following models assume that following vehicle drivers respond only to the
lead vehicle immediately in front without observing other vehicles downstream. A number of
researchers have observed that in actual traffic situations, drivers take a more extensive view of
traffic conditions ahead (which may include several lead vehicles) in setting the following
vehicle desired speeds and spacing (Fox and Lehman, 1967; Bexelius, 1968, Ozaki, 1993, and
Toruran, 1999).
3. Many existing car-following models, particularly the GRH models, assume a mathematical
expression that is empirically based but fails to explain actual behaviour in a mechanistic fashion
(cause-effect). Best fit expressions fail to clarify or explain, why certain relationships are
specified as they are (Winsum, 1999). These expressions have little, if any, basis on actual
behaviour, and the model parameters have no obvious connection with identifiable driver and
vehicle traits that explains behaviour (Gipps, 1981).
4, Many existing car-following models assume symmetrical driver responses to changing traffic
stimuli involving lead vehicles. To illustrate, we consider two cases, one with a positive relative
speed (i.e. lead vehicle is travelling faster) and the other with a negative relative speed (lead
vehicle slower). For the same magnitude of speed difference, the following vehicle driver in the
first instance will increase his or her speed without incurring higher collision risks. In the latter
instance, the following vehicle driver will need to decelerate to avoid a potential collision, since
both vehicles are moving closer to each other. From a safety perspective, we would expect the
acceleration/deceleration rate in the first case to be less than the acceleration/deceleration rate in
the second case. Many existing car-following models assume the magnitude of
acceleration/deceleration to be the same. This situation is normally outside the scope of existing
car-following models and is explained using separate collision avoidance algorithms (Leutzbach,
1988). When both the lead and following vehicle are traveling at the same speed, many existing
car-following models assume zero following vehicle deceleration/acceleration rates regardless of
the spacing between vehicles. This assumption is clearly unrealistic (Chakroborty and Kikuchi,
1999).
3.0 PROPOSED SD CAR-FOLLOWING MODEL
The car-following situation considered in this paper assumes a string of three vehicles (two lead
vehicles and one following vehicle) traveling along a single lane. It is assumed that all vehicles travel
in the same lane and only adjustments in speed are permitted for all drivers involved. The profile of
the first lead vehicle is determined exogenously based on predominant traffic conditions. The speed
and spacing profiles for the second lead and the following vehicle are determined internally.
One of the basic differences between the proposed model and the existing car-following models is
that in existing car-following models following vehicle drivers consider only one lead vehicle ahead,
while in the proposed model following vehicle drivers consider all vehicles travelling ahead within
their comfort zone. For example, in case of three vehicles situation considered in this paper the
following vehicle driver would perceive information either from both lead vehicles (1% and 2) or
from only second lead vehicle, depending on whether one or both lead vehicles are travelling within
his/her comfort zone. The comfort zone of a driver is defined based on his/her current speed and
perception of crash risk.
Unlike many existing car-following models, the proposed model assumes that in a rectilinear travel
system with variable speeds and conditions, following vehicle drivers do not have the required depth
perception to accurately ascertain spacing, differential speeds, and/or acceleration of lead vehicle at
any point in time. In addition to his own speed and safe comfort zone, the following vehicle driver
can only ascertain his or her spacing to the vehicle immediately in front (the second lead vehicle),
and possibly the spacing between both lead vehicles if they are sufficiently close. We note that in the
proposed model the speeds and/or acceleration of the lead vehicles are not required as inputs in
setting the following vehicle acceleration/deceleration rates and spacing. This assumption differs
from many existing car-following models and can be viewed as being more parsimonious than these
models in estimating the following vehicle speed and position over time.
Underlying assumptions
The proposed SD model differs from existing car-following models in several important aspects:
=
. A simplified acceleration/deceleration rule is used for following drivers that includes only
spacing and rate of change in spacing between the lead and the following vehicle.
2. The information from more than one vehicle ahead is used for decision-making process of
following vehicle drivers.
3. The proposed model permits changes in perception/reaction time of following vehicle drivers
to account for supplementary lead vehicle stimuli, such as, the status of lead vehicle(s) brake
lights.
4. The concept of a comfort zone for the following vehicle driver is introduced to reflect his/her
desired speed and spacing for different driving conditions.
Figure 1 illustrates the dynamic relationships inherent in the proposed SD car-following model. For
every decision interval, a driver sets a unique "safe comfort zone". This comfort zone reflects speed
and spacing status that the driver considers to be safe over time and changing traffic conditions. Here
we assume that the desired speed is based on the current spacing and rate of change in spacing with
respect to the lead vehicle immediately in front. If the current spacing is shorter than that dictated by
the driver's comfort zone and is decreasing in length, the following vehicle driver will decelerate to
achieve a desired comfort zone or separation distance. Conversely, if the current spacing exceeds that
set by the driver's comfort zone, and the vehicle is travelling at a speed below desired speed, the
following vehicle driver will accelerate.
The proposed model assumes that the level of alertness of a driver affects the perception/reaction
time component of the acceleration/deceleration rate. If a driver is alert, less time is needed to
perceive and react to a given situation. In the proposed SD car-following model, the following
vehicle driver will modify his or her personal perception/reaction time with respect to the status of
the lead vehicle brake lights. In the proposed SD model, we assume that the following vehicle driver
becomes more alert with reduced perception/reaction times when the lead vehicle brake lights are on
and the lead vehicle is within the following vehicle driver comfort zone. The status of the brake
lights can be ascertained internally. Ozaki (1993) suggests that brake light status can be determined
as a function of vehicle deceleration rates, such that: if deceleration rate < - 0.013 times the speed
of the vehicle, then brake lights are assumed to be lit.
As indicated in Figure 1, the following vehicle driver considers both the first and second lead vehicle
position in changing his/her speed. The question is how to balance the stimuli between the first and
second lead vehicles in setting the following vehicle driver response. While both lead vehicles
provide stimuli to the following vehicle driver, the importance that the following vehicle driver
places on one lead as compared to the other depends on the spacing between the following vehicles
and the lead vehicle immediately in front, driver comfort zone for prevailing speed, and the spacing
between the two lead vehicles.
Model Formulation
The proposed car-following model consists of four sectors: 1) the first lead vehicle, 2) the second
lead vehicle (vehicle immediately in front of following vehicle), 3) the following vehicle, and 4) the
spacing sector. The stock flow diagram for the proposed model is given in Figure 2 (a and b). Each
sector performs certain functions to produce speed and spacing profiles for individual vehicles in the
three-vehicle string. Functions in each sector interact with functions in the other sectors through
feedback links. This reflects how the speed and spacing of one vehicle acts to affect the speed and
spacing of another vehicle in the string. The first lead vehicle sector is specified exogenously and
prescribes the lead vehicle target conditions for input into the second lead and following vehicle
sectors. The acceleration/deceleration rate, speed and spacing of the second lead and following
vehicles are determined within the model, subject to rules and assumptions pre-scribed in the
following paragraphs. Road geometry, pavement conditions, and weather conditions are set
exogenously.
The process describing the second lead vehicle sector is similar to that associated with the following
vehicle sector. The only difference is that the following vehicle driver sets his or her spacing and
rate of change in spacing on the basis of spacing between the first and second lead vehicle and
between the second lead vehicle and itself. The second lead vehicle driver on the other hand
considers only its spacing with the first lead vehicle in setting his/her spacing and speed. The
assumption here is that we are dealing with a three vehicle string. This can be extended to include
longer strings, with an appropriate number of lead vehicle sectors.
The acceleration/deceleration rate of the following vehicle depends on the driver's perception
reaction time, current and desired speed. The desired speed depends on two factors: 1) current
spacing between the following vehicle and lead vehicle immediately in front, and 2) rate of change in
spacing between the following vehicle and the lead vehicle immediately in front. In the SD model,
the former factor is calibrated based on observed individual vehicle tracking data, while the latter is a
non-linear function of rate of change in spacing between second lead and following vehicles. This
relationship is based on a heuristic understanding of the situation as opposed to empirical results
from observed field data. The boundary limits of this non-linear function are set so as to satisfy the
extreme limits of a driver's perception reaction time as reported by Ozaki (1993). The product of
factors (1) and (2) above yields the desired speed of the following vehicle.
The perception reaction time of the following vehicle driver depends upon his/her level of alertness.
Alertness is defined in terms of driver's perception reaction time as modified by brake light status, as
discussed above. When the value of alertness level is one, the perception reaction time is assumed to
be 2.5 sec (Olson, 1986). The perception reaction time decreases as the vehicles get close to each
other and the brake lights on the lead vehicle(s) are lit (Ozaki,1993).
In Figure 2b, a fourth sector is defined that reflects vehicle spacing (separation distance) profiles,
between the first and second lead vehicles, and between the second lead and following vehicle. The
factors such as pavement conditions, pavement friction, road geometry, and traffic conditions can
affect the distance travelled by a vehicle at a particular speed. For this paper, we have assumed ideal
weather and pavement conditions.
4.0 CALIBRATION AND VALIDATION USING SAVE DATA
The major component of the proposed car-following model (relationship between current spacing
and desired speed) is calibrated based on observed individual vehicle tracking data obtained from the
SAVME (System for Assessment of Vehicle Motion Environment) database (Ervin, 2001). The
University of Michigan Transportation Research Institute (UMTRI) developed this SAV ME database
for the National Highway Traffic Safety Administration. This database provides a complete
microscopic record of trajectories and distance headway observed for individual vehicles in a traffic
stream over a period of time. The SAVME database contains 18 hours of vehicle trajectory data
representing over 30,500 vehicles. All data were collected during daylight hours.
Trajectory data for a random sample of 132 vehicle pairs traveling in the shoulder lane were
extracted from the SAVME database. For each pair of vehicles, the speed of the following vehicle
and the spacing were extracted. For each observed speed, the mean distance headway from all
vehicles observed to travel at this speed was computed. The results are illustrated in Figure 3 as the
desired speed versus mean spacing. To ensure realistic behavior at the boundaries of relationship
10
shown in Figure 3, constraints are incorporated such that the desired speed must be non-negative and
not greater than the maximum assumed speed of 70 ft/sec (77 Km/h). The relationship illustrated in
Figure 3 is consistent with the data obtained from a Newcastle University research team in the United
Kingdom (May, 1990). Like SAVME database, the data collected by a research team at Newcastle
University also tends to demonstrate a fairly aggressive car-following behaviour at short spacing and
less aggressive car-following behaviour at longer spacing, as illustrated by Figure 3.
Observations in SAVME suggest that desired speed for a given spacing differs between drivers. This
is likely due to individual driver differences of age, gender, risk taking propensity, skills, vehicle size
and performance characteristics. Moreover, the situational factors such as time of day, day of week,
road geometry, traffic conditions, weather and road conditions also influence the desired speed of a
driver for a given spacing. As an initial step, we have assumed ideal roadway conditions and
individual driver differences and situational factors are not explicitly considered into the proposed
car-following model in this paper.
5.0 EVALUATION OF PROPOSED SD CAR-FOLLOWING MODEL
The microscopic evaluation of the proposed model is conducted by comparing model estimates of
speed and spacing for the second lead and the following vehicle to those observed in the SAVME
database. The trajectories of first lead vehicles were randomly selected from the SAVME database.
The trajectories of the two vehicles following the selected lead vehicle (second lead and following)
were also extracted from the SAV ME database and were used to compare to the model outputs.
The trajectory of the first lead vehicle, the initial speed and position of the second lead and following
vehicles were provided as inputs to the proposed car-following model. The model was then used to
estimate the behavior of the second lead and following vehicle in response to the known behavior of
the first lead vehicle.
Figure 4 illustrates the observed and model predicted results for the first data set extracted from the
SAVME database. Figure 4a illustrates observed and predicted speed and spacing associated with the
second lead vehicle. Figure 4b illustrates the same for the following vehicle. As indicated by the
results illustrated in Figure 4 (a and b), the speed and spacing profiles predicted by the proposed car-
following model closely follow those in the observed field data.
Twenty samples of three-vehicle strings were extracted from SAV ME database. For each sample the
Toot-mean-squared (RMS) error associated with the prediction of speeds and spacing of second and
11
following vehicle was estimated as given in Table 2. The average RMS error associated with the
prediction of second lead and following vehicle speed for the twenty samples was found to be 3.68
Km/h and 4.7 Km/h respectively. The RMS error associated with the prediction of second lead and
following vehicle spacing was 2.56 m and 2.87 m respectively.
Table 2: RMS error associated with twenty sample applications
Sample [Observed Averag¢d Observed Average] _Root-Mean-Squared-Error
Speed (Km/h) Spacing (m) Speed (Km/h) | Spacing (m)
v2 Vf $2 Sf v2 VE $2 Sf
53.31 | 66.11 | 24.98 | 62.17 | 5.16 | 1.46 1.65 [ 1.24
55.72 | 59.96 | 24.02 | 1857 | 1.06 | 615 | 0.28 | 5.63
55.66 | 59.54 | 30.58 | 29.67 | 3.39 | 433 | 2.34 | 0.89
67.17 | 65.27 | 1866 | 1813 | 0.78 | 3.05 | 0.19 | 2.10
52.20 | 58.87 | 16.00 | 30.85 | 5.23 | 5.07 5.94 | 3.82
55.78 | 52.71 | 21.39 | 25.15 | 2.99 | 6.42 3.23 | 4.01
58.67 | 61.43 | 29.69 | 27.40 | 161 | 1.24 | 0.76 | 0.80
41.99 | 48.70 | 23.18 | 4242 | 551 | 844 | 2.54 | 1.15
44.03 | 43.97 | 16.47 | 20.02 | 2.24 | 2.59 1.40 | 1.13
62.35 | 62.87 | 25.38 | 51.91 3.15 | 3.68 | 0.95 | 2.57
67.95 | 60.04 | 45.81 | 61.42 | 3.24 | 6.30 1.05 | 0.79
52.34 | 52.43 | 51.06 | 16.13 | 5.80 | 7.02 3.90 | 8.45
44.63 | 4447 | 15.15 | 20.82 | 3.61 | 2.77 3.61 | 2.01
47.01 | 48.72 | 20.68 | 19.52 | 3.09 | 5.48 5.66 | 3.48
59.99 | 63.47 | 29.16 | 32.43 | 1.57 | 2.63 1.15 | 1.00
50.88 | 48.51 | 29.18 | 19.36 | 4.09 | 3.26 | 3.82 | 5.62
54.93 | 55.25 | 40.68 | 48.83 | 888 | 9.73 | 3.45 | 0.17
47.49 | 50.80 | 17.40 | 17.21 | 466 | 5.34 | 253 | 3.20
35.45 | 36.67 | 21.22 | 12.00 | 2.94 | 5.04 | 488 | 5.56
20 60.05 | 60.52 | 32.89 | 2037 | 455 | 4.04 | 1.84 | 3.75
Average 53.38 | 55.01 | 26.68 | 29.72 | 3.68 | 4.70 | 2.56 | 2.87
By Re] ee] ele} fa] ale
©] Oo] BLO] APB] cof] GE] S] Of O] MPD] | copy
V2 = Second lead vehicle speed
Vf = Following vehicle speed
S2 = Spacing between first and second lead Vehicle
Sf = Spacing between second lead and following Vehicle
12
A one-way ANOVA was carried out to assess the statistical significance of the RMS error with
respect to the following vehicle speed and spacing. The results of ANOVA are given in Table 3. For
this analysis the variation in observed mean speed of following vehicle was grouped into three
classes ( < 50K m/h, 50 - 60 Km/h, and > 60 Km/h).
Table 3: ANOVA results, RMS error versus following vehicle speed and spacing.
Variable | P-value | Remarks
VE 0.024 Significant
Sf 0.206 Not significant
As indicated in Table 3, the ANOVA suggests that the mean speed of following vehicle (Vf) has a
statistically significant effect on the RMS error of following vehicle speed. The P-value for the
following vehicle spacing (Sf) shows the variation in observed mean speed of following vehicle
speed lacks statistical significance at the 5% level. To further investigate the performance of the
proposed model in predicting the speed of following vehicle, the RMS errr of following vehicle
speed is plotted against observed mean speed of following vehicle (Figure 5). As shown in Figure 5,
the RMS error of following vehicle speed at higher observed mean speed is less than the RMS error
at lower observed mean speed. At this point we cannot speculate on the reason for this relationship.
A regression analysis of predicted and observed speed and spacing of following vehicle was carried
out for the sample application. The results are shown in Figures 6 and 7. Figure 6 shows the plot of
predicted versus observed speed of the following vehicle. Figure 7 shows the plot of predicted versus
observed spacing of the following vehicle. The results indicate significant agreement between the
predicted output from the model and the observed field data. While these results are based on a
limited comparison between the proposed SD car-following estimates and observed SAVME data,
they suggest that the proposed model can closely reflect observed speed and spacing profiles for
selected three-vehicle strings, where following vehicle drivers consider both two lead vehicle stimuli
in setting speeds and spacing over time.
13
6.0 CONCLUSIONS
In this paper we have discussed a number of existing car-following models and have identified
several common shortcomings. We have presented a revised car-following model based on System
Dynamics principles, which attempts to address many of these shortcomings. The proposed model
assumes that drivers adjust their speed based on the current spacing and rate of change in current
spacing to next downstream vehicle. The model also takes into account the driver's desired speed and
distance headway in relation to increased risk of collisions.
The proposed model assumes that drivers are capable of estimating the spacing between their own
vehicle and the next downstream vehicle. The model, unlike many existing car-following models,
does not make unrealistic assumptions about drivers' ability to estimate the speed of downstream
vehicles.
In this paper we have compared the model estimates of speed and spacing profiles for the following
and second lead vehicle to the speed and spacing profiles of observed vehicles. These comparisons
suggest that the proposed car-following model yields realistic results in replicating the behavior of
the following vehicle driver from an observed vehicle tracking database. In the proposed model
drivers seek to maintain the speed and spacing that is consistent with their understanding of the risks
involved for any traffic situation.
ACKNOWLEDGEMENTS
The authors are grateful to the researchers at UMTRI, and in particular Drs. Robert Ervin and Jeff
Walker, for providing the SAVME database.
14
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Desired Speed of
IstLead Vet,
Alertness of
Following Veh Dever, Pa
Perceptoneacton Time PerceplioaReaction Tine Alerness of 2nd
| efTolomme Veh dover ° of dnd Hea Veh driver Lead Veh dover
j - sod of Ist
{ 6. lex Veluce ,
t we Desired Speed of 4 Y a
} AccDec of Following Vel AcelDec of nd pes 7
fee dnd Leed Ve
| Folerng ‘ x Lead Veh,
¢, ¢ ¢ = Speed of.
yf (A Desited Spacing of IstLead
Hesredpatneot Current \I between " Lead Veh
j i Followng Veh,
| Curent Speed of _mer 2nd Lead and Followng Current \ al
Folowng Veh” Veh Furrent Speed of. Ist ded Lead Veh. |
\ cm, ( 4 Qt LeadVeh
\ UY
N / /\ UY
Rate of change in af Rate of change in Current
Spacng between 2nd Lead & Spacing between Ist & 2nd
Following Veh 3 Lead Veh
~<
Figure 1: Dynamics hypothesis of proposed car-following model
16
o 1: 1st Lead Vehicle Ab
PercepReaction time of Ist Lead Vehicle driver
AccDec rate of Ist Lead Veh
Current Speed of Ist Lead Vehic
Desired Speed of Ist Lead Vehicle
og 2: Ind Lead Vehicle AB
Alertness of 2nd Lead Veh driver Current Speed of Ist Lead Vehicle
PercepReaction time of 2nd
Current Speed of 2nd Lead Vehicle of Ist Lead Veh
AccDec rate of 2nd Lead Vehicle
Desired Speed of 2nd Lead Vehicle
Effect of Spacing on Desired Speed of 2nd Lead Veh
Eff of rate of change in Spacing on Desired Speed of 2nd Lead Veh
Current Spacing between Ist & 2nd Lead Vehicle
Desired Spacing of 2nd Lead Vehicle Effect of Spacing on Alertness of 2nd Lead Veh driver
Figure 2a: Stock-flow diagram of 1* and 2™ lead vehicle sector
17
18
o 3: Following Vehicle wT
Current Speed of 2nd Lead Vehicle
AccDee rate of 2nd Lead Vehicle > me
Current Speed of Ist Lead Vehicle
Alerness of Following Vehicle driver \.
PercepReaction time of Following Veh wot
AccDec rate of Ist Lead Veh
CY ar of ot Spacing on Aleriness of Following Veh driver x
\ \
\ AceDeo rate bf Following Vehicle \
\ Current Speed of Following Vehicle 5. ne]
: ee
\ % wo
\ Nos ae Eff of rate of change in Spacing on Desired Speed of Foll Veh
2 { |
Eff of Spacing on Desired Speed of Foll Veh
\ Desired Spacing of Following Vehicle i
x RS.
\ Si Current Spacing between Ind Lead & Following Veh
Effect of Spacing on Aleriness of Following Vehicle driver
Desired Speed of Following Veh!
\
a
Current Spacing between 2nd Lead & Following Vehicle
EJe) 4: Spacing A6
Eff of rate of change in Spacing on Desired Speed of Foll Veh Eff of rate of change in Spacing on Desired Speed of 2nd Lead Veh
fm Spacing between 2nd Lead é Following Veh
oO
Rate of Change in Dist Following Vehicle Rate of change in DjA of 2nd Lead Vehicle
vurrent Spacing between Ist & 2nd Lead ve
13)
Rate of change in Dist of Ist Lead Vehicle
Current Speed of 2nd Lead Vehicle eg
Current Speed of Ist Lead Vehicle
Current Soeed of Following Vehicle
Current Spacing between 2nd Lead & Following Vehicle
Figure 2b: Stock-flow diagram of following vehicle and spacing sector
Desired speed (ft/sec)
80
70
« Mean of observed points
— Relationship used in model w
%
32
vv .
0 20 40 60 80 100 120
Spacing (ft)
Figure 3: Calibrated relationship between spacing and desired speed
19
20
70 70
60 60
mae ee
50 F 50
dao H 40
330 q 30
720 a 20
Observed-V2 Observed VF
10 10
0 = = = ~ Model output -V2 i = = = = Model output-VE
0 1 2 3 4 2 3 4
Observed time(sec) Observed time (sec)
140 120
120 100
=—-..
100 = 80 _
= 80 SS &
Fy = 60
% 60 =
a G
40 40
4% Observed-S2 20 Observed- Sf
A = = = = Model output-$2 0 - = = Model output-Sf
0 1 2 3 5 0 1 2 3 4 5
Observed Time(sec) Observed T ime (sec)
(a) second Lead V ehicle
(b) Following Vehicle
Figure 4: Comparison of Predicted and Observed vehicle speeds and spacing
(Data Set 1)
12
é 10 ,
°
8
3
.
7 : ‘
P
' °
g . *
Bo 4 “4
3 °
E ° °
oe °
Z 2
.
0
30 35 40 45 50 55 60 65
Figure 5: RMS error Vs Observed mean speed of following vehicle (Km/h)
Observed mean speed of following vehicle (Km/h)
70
21
22
y =0.9114x +4.8842
R’ =0.7692
0 r r
0 20 40 60 80
Observed Vf (Km/h)
Figure 6: Predicated Vs Observed speed of following vehicle for twenty samples (n = 1055)
70
y =0.883x + 3.9668
60 R?=0.9181
50 |
w C=
So fo
L L
Predicted Sf (m
re
oo 8S
Ll Ll
30 40 50 60 70
Observed Sf (m)
o
be
=)
iS)
S
Figure 7: Predicated Vs Observed spacing of following vehicle for twenty samples (n = 1055)
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