REPORT

ON

CAR FOLLOWING MODELS AND THEIR EVALUATION

Submitted by

Saleh Ragab Mousa

DEPARTMENT OF CIVIL ENGINEERING

2011

CHAPTER 1

INTRODUCTION

Traffic congestion in and around the urban areas of the world is a major problem. Congestion

during peak hours extends for longer periods each day. Congestion affects mobility, safety, and

air quality, which cause direct economic losses due to delays and accidents, and indirect

economic losses due to environmental impact. In most cases, the capacity of the existing

roadway systems cannot be increased by adding additional lanes due to space, resource, or

environmental constraints. Thus it’s more common to improve the utilization of the existing

systems through better traffic management, operations strategies, and improving the geometric

design of the congested roads and highways.

Traffic operations in the congested sections of roadways is very complex, since different

drivers employ different techniques to travel through such sections while interacting with other

drivers. To understand the occurrence of bottlenecks and to devise solutions for it, a

comprehensive analysis of vehicle to vehicle interactions is essential. This requires the

development of traffic theories –Models – to explain driver behavior at the microscopic level, the

main elements of which are the acceleration and lane changing dimensions.

Drivers' acceleration behavior, when they are in the car following system, has been studied

extensively since the 1950s. In this system, drivers are assumed to follow their leaders. However,

estimation of these models using microscopic data, as the speed of a subject and its leader, gap

length, acceleration applied by the subject, collected from real traffic has not received much

attention. On the other hand, researchers started paying attention to the acceleration behavior in

the free flow system beginning early 1980s. In the free flow system, drivers are not close to their

Leaders and therefore, have the freedom to attain their desired speed. The parameters of the

general acceleration model that figures drivers’ acceleration decision in both The car following

and free flow systems, have not been estimated yet.

The principal focus of research in modeling driver’s lane changing behavior has been on

modeling the gap acceptance behavior at stop controlled T intersections. The gap acceptance

phase is a part of the lane changing process. Researchers started paying attention to the lane

changing model as microscopic traffic simulation emerged as an important tool for studying

traffic behavior and developing and evaluating different traffic control and management

strategies. However, the existing lane changing models are rule based and do not explicitly

capture variability within driver and between drivers. Furthermore, the model parameters have

not been estimated formally.

In this report, we represent a comprehensive framework for modeling drivers' acceleration and

lane changing behavior

CHAPTER 2

LITERATURE REVIEW

(2.1) Introduction

The aim of this chapter is to review the existing car-following models. It introduces the

basic concepts of the models and focuses on efforts made in improving the models. In addition, it

summarizes the significant shortcomings of the existing car following models in emulating

drivers’ car-following behavior.

(2.2) Traffic Simulation Models

There are three types of traffic models: macroscopic, mesoscopic, and microscopic. The

macroscopic traffic models describe traffic flow behavior on a segment-by-segment basis instead

of tracking individual vehicles. As a result, the models produce aggregate traffic stream

parameters such as speed, flow, and density and their corresponding relationships. The models

use the equation of conservation of vehicle flow to describe the relationships and how

disturbances such as shockwave propagate in the traffic stream. Examples of macroscopic

models include Greenshield (1935), Greenberg (1959), Underwood (1961), Edie (1961), and Bell

Curve (Duke et al., 1990). The most prevalent benefit of such models is that they can describe

the spatial and temporal extent of traffic congestion particularly that is caused by non-occurring

incidents such as traffic crashes.

The microscopic traffic models describe the movement and interactions of an individual

vehicle with a leading vehicle. The models track vehicles on at a certain time interval and

produce observations of vehicle longitudinal and lateral positions, speed, and

acceleration/deceleration at each time interval. The models use fundamental rules of motion and

rules of driving behavior such as lane changing and car-following behavior for moving vehicles

in the system. The most important benefit of microscopic models is that they are used for

evaluating traffic operational performance of the existing or future planned highways.

The mesoscopic traffic models describe individual vehicle interacting with other vehicles but

aggregate parameters for all vehicles. In essence, the models combine both Characteristics of

microscopic and macroscopic models. For example, the model can be used to evaluate average

travel time and speed of a certain highway segment using individual vehicles equipped with in-

vehicle real-time travel information systems. The models are most beneficial in evaluating

traveler information systems. Kinetic theory based models are typical examples of mesoscopic

models (Prigogine, 1971).

(2.3) Car-Following Models

This study uses the following definitions and notations in describing the car following

models. Consider two following vehicles traveling from left to right as shown in Figure 1-1.

Vehicle (n -1) is a leading vehicle with length L (n-1) and vehicle (n) is the subject vehicle. The

subscript t denotes the time of observation of vehicle position, velocity, and acceleration.

(Figure 1-1) Definitions and notations used in car-following model

The following are definitions of the variables resulting from Figure 1-1.

In the car-following mode the leading vehicle influences driving behavior of the subject

vehicle. Therefore, the driver of the subject vehicle reacts to the perceived stimulus resulting

from driving behavior of the leading vehicle. The stimulus could be a speed differences and/or

separation between the two vehicles. Furthermore, the driver of the subject vehicle responds to

the stimulus after a certain time lag (Reaction Time). The driver response time lag -Reaction

time- is the interval of time between occurrence of stimulus and initiation of response.

Car-following models can be broadly divided into following main types:

1. Safe distance car-following models (Gipps model, Krauss model)

2. Stimulus-response car-following models (Chandler model, generalized General Motors

model)

3. Psychophysical car-following models (Leutzbach model)

4. Cell based model (cellular automata model)

5. Optimum velocity model (Bando et al)

6. Trajectory based model (Newell model)

We shall deal and focus in our report only on

o Safe distance car-following models (Gipps model, Krauss model)

o Stimulus-response car-following models (Chandler model)

(2.3.1) Safe Distance Car-following Models

Reuschel (1950) and Pipes (1953) were the early pioneers who developed minimum safe

distance models. They were both independently inspired by the law of vehicle separation

stipulated in the California Vehicle Code, which states that “A good rule for following another

vehicle at a safe distance is to allow yourself the length of a car (about fifteen feet) for every ten

miles per hour you are traveling”. They developed traffic models that emulate such driving

behavior. The models expressed minimum safe distance maintained by a subject vehicle behind a

leading vehicle as a linear function of speed.The models assumed that drivers of vehicles obeyed

this rule at all times and derived

model that emulate such driving behavior. The developed models assumed that drivers

reacted instantaneously to the actions of a leading vehicle. Pipes model has the following

form:

2-1

Where:

b is the prescribed legal distance in feets when vehicles are at standstill and T is time constant

in seconds is a as prescribed by the California Driver Code i.e.

(T = 15 𝑓𝑡

10 𝑚𝑝ℎ= 1.023 seconds)

This results in a minimum safe vehicle separation distance (d) equal to:

2-2

Forbes (1958) developed a car-following model that incorporated a driver reaction time

component. This was based on the fact that there is a time lag between the occurrence of

stimulus and initiation of response (Reaction Time). The model assumed that a driver of a

subject vehicle maintains minimum safe time headway at least equal to the driver reaction time.

This time headway is the summation of the driver reaction time and the time taken to travel a

distance equivalent to the length of a leading vehicle. Forbe’s model is defined as follows:

Gipps (1981) derived a model by setting the limits of performance of driver and vehicle and

used the limits to calculate a safe speed with respect to a leading vehicle. In other words, the

driver should not exceed his/her desired maximum speed and the vehicle should not exceed its

maximum acceleration and deceleration capabilities. Furthermore, the study used additional

safety margin to compensate for driver related errors equal to half of driver reaction time. The

assumption made is that a driver of a subject vehicle maintains a speed which allows the driver

to bring the vehicle to safe stop should a vehicle ahead come to a sudden stop.

Gipps (1981) describes models current to that time to be in the general form of:

which is defined primarily by one vehicle (noted by n) following another (noted by n-1).

Gipps sets limitations on the model through safety considerations and assuming a driver would

estimate his or her speed based on the vehicle in front to be able to come to a full and safe stop if

needed due to ahead sudden stop (1981).

Model Notation

an is the maximum acceleration which the driver of vehicle n wishes to

undertake,

bn is the most severe braking (deceleration) that the driver of vehicle n

wishes to undertake (bn < 0),

sn − 1 is the effective size of vehicle n-1, that is, the physical length plus a

margin into which the following vehicle is not willing to intrude, even when

at rest,

Vn is the speed at which the driver of vehicle n wishes to travel,

xn(t) is the location of the front of vehicle n at time *t,

vn(t) is the speed of vehicle n at time t, and

τ is the apparent reaction time, a constant for all vehicles.[2]

Constraints Leading to Development of Model

Gipps defines the model by a set of limitations. The following vehicle is limited by two

constraints: that it will not exceed its driver’s desired speed and its free acceleration should first

increase with speed as engine torque increases then decrease to zero as the desired speed is

reached.

For safety, the driver of vehicle n (the following vehicle) must ensure that the difference between

point where vehicle n-1 stops ( ) and the effective size of vehicle n-1 (sn − 1) is greater than

the point where vehicle n stops ( ).

The new speed of the driver's vehicle is given by the final equation being Gipps' Model:

where the first argument of the minimization regimes describes an uncongested roadway and

headways are large, and the second argument describes congested conditions where headways

are small and speeds are limited by followed vehicles.

(2.3.2) Stimulus-Response Car-Following Models

Researchers associated with the General Motors (GM) (Chandler et al., 1958) developed

five series of models that described acceleration and deceleration response behavior of a subject

vehicle due driving actions of a leading vehicle. The structure of these models was similar to that

of Reuschel, Pipes, and Forbes. However, the upgraded models assumed that a driver response as

a function of driver sensitivity and stimulus. The GM models define stimulus as the relative

speed between the two following vehicles. Negative relative speed, when the leading vehicle

travels slower than the following vehicle, triggers a deceleration response. On the contrary, a

positive relative speed, when the leading travels faster than the following vehicle, triggers an

acceleration response. The magnitudes of the acceleration/deceleration depend on sensitivity

term which includes speed and vehicle spacing. The models have the following general form:

Chandler et al. (1958) developed the first simple linear model that assumes acceleration

/deceleration response of a subject vehicle is proportional to the relative speed between two

following vehicles as shown below.

Where:

The stimulus term, that is, relative speed, at any time step can be either positive, zero, or

negative resulting in drivers’ response in form of acceleration, no response, or deceleration,

respectively. The model assumes that the driver sensitivity-the amount of respose- is constant

across driver population and/or vehicle types.

The study calibrated parameters of the model using instrumented cars on test track of the

GM. The experiment involved two vehicles with a cable on a pulley connected with a wire

wound around a reel mounted on the front of the leading vehicle. The experiment used eight

drivers who drove the test cars while varying driving conditions of mean speed. The driver of the

subject vehicle followed the leading vehicle while maintaining their desired safety distance. The

correlation analysis between observed and estimated acceleration was used to estimate the

parameters of the model. The values that produced the highest correlation were used as the

estimate of the driver response time lag and sensitivity for a particular driver.

Table 2-1 shows the estimated parameter values obtained from this study.

The results obtained from field experiment showed significant variation in the sensitivity

values. The sensitivity term appeared to depend on the distance between the vehicles, therefore,

suggested a modification of the sensitivity term.

Gazis et al. (1959) addressed this weakness of the model by incorporating spacing between

two vehicles in the sensitivity term. The second model proposed that the sensitivity term should

have two states depending on closeness between two following vehicles (separation). This means

that higher sensitivity value a1 is applicable when the two vehicles are close together and lower

sensitivity value a 2 when the two vehicles are far apart. This suggests that drivers are more

sensitive at shorter following distance and less sensitive at larger following distance. The model

is defined as shown below:

Demonstration on the GM third Model using Excel sheet:

If the reaction time (lag time) is = 1 second

The positions & speed of the leading, first, second, third and fourth

following cars at t=0 are as shown

Time step for calculations also = 1 second

Study period =10 seconds

The speed and position of leading car during study period is as given below

Leading 1 2 3 4

T(sec)

x(m)

speed(m/s)

acceleration x

speed

acceleration x

speed

acceleration x

speed

acceleration x

speed

acceleration

0 50 15 10 30 15 5

20 20 10

10 20 15

-30 25 -6

1 70 25 5

2 97.5 30 3

3 129 33 2

4 163 35 0

5 198 35 -2

6 232 33 -4

7 263 29 -8

8 288 21 -10

9 304 11 -10

10 310 1 -1

(2.4.1) Lane Changing Models

The principal focus of research in modeling drivers' lane changing behavior has

been on modeling the gap acceptance behavior at stop controlled T-intersections.

The gap acceptance phase is a part of the lane changing process.

Gipps (1986) presented a lane changing decision model to be used as

microscopic traffic model. The model was designed to cover various urban driving

situations where traffic signals, obstructions, and the presence of heavy vehicles

(for example, bus, truck, semi-trailer) affect a driver's lane selection decision.

Three major factors were considered in the lane changing decision process:

necessity, desirability, and safety. Different driving conditions were examined

including the ones where a driver may face conflicting goals. However, different

goals were prioritized deterministically, and inconsistency and non-homogeneity in

driver behavior were not modeled.

The terms inconsistency implies that a driver may behave differently under

identical conditions at different times, while the term non- homogeneity implies

that different drivers behave differently under identical conditions. The model

parameters were not estimated formally.

CORSIM (FHWA 1998) is a microscopic traffic simulator that uses FREESIM

to simulate freeways and NETSIM to simulate urban streets. In CORSIM, a lane

change is classified as either mandatory (MLC) or discretionary (DLC). A driver

performs an MLC when the driver must leave the current lane and performs a DLC

when the driver perceives the driving conditions in the target lane to be better, but,

a lane change is not required. The necessity or desirability of changing lanes is

determined by computing a risk factor that is acceptable to a driver which is a

function of a driver's position relative to the object that gives rise to the need for a

lane change. A default set of model parameters are provided with the flexibility of

using user provided parameters. The gap acceptance behavior is not modeled in a

systematic manner. Minimum gap lengths for different situations are listed and all

drivers are assumed to have identical gap acceptance behavior.

Yang and Koutsopoulos (1996) developed a rule based lane changing model

that is applicable only for freeways. Their model is implemented in MITSIM. A

lane change is classified as either mandatory (MLC) or discretionary (DLC).

Unlike Gipps (1986), they used a probabilistic framework to model drivers' lane

change behavior when they face conflicting goals. A driver considers a

discretionary lane change only when the speed of the leader is below a desired

speed, and checks neighboring lanes for opportunities to increase speed. Two

parameters, impatience factor and speed indifference factor, were used to

determine whether the current speed is low enough and the speeds of the other

lanes are high enough to consider a DLC.

The lane changing model structure is shown in Figure 3-2. As mentioned

above, except for the completion of the execution of the lane change, the whole

decision process is latent in nature. The latent and observable parts of the process

are represented by ovals and rectangles respectively.

figure (3-2)

The MLC branch in the top level corresponds to the case when a driver decides to respond to

the MLC condition. Explanatory variables that affect such decision include remaining distance to

the point at which lane change must be completed, the number of lanes to cross to reach a lane

connected to the next link, delay (time elapsed since the MLC conditions apply), and whether the

subject vehicle is a heavy vehicle (bus, truck, semi-trailer etc.). Drivers are likely to respond to

the MLC situations earlier if it involves crossing several lanes. A longer delay makes a driver

more anxious and increases the likelihood of responding to the MLC situations. And finally, due

to lower maneuverability and larger gap length requirement of heavy vehicles as compared to

their non-heavy counterparts, they have a higher likelihood of responding to the MLC

conditions.

The MLC branch corresponds to the case where either a driver does not respond to an MLC

condition, or that MLC conditions do not apply. A driver then decides whether to perform a

discretionary lane change (DLC). This comprises of two decisions: whether the driving

conditions are satisfactory, and if not satisfactory, whether any other lane is better than the

current lane. The term driving conditions satisfactory implies that the driver is satisfied with the

driving conditions of the current lane. Important factors affecting the decision whether the

driving conditions are satisfactory include the speed of the driver compared to its desired speed,

presence of heavy vehicles in front and behind the subject, if an adjacent on-ramp merges with

the current lane, whether the subject is tailgated etc. If the driving conditions are not satisfactory,

the driver compares the driving conditions of the current lane with those of the adjacent lanes.

Important factors affecting this decision include the difference between the speed of traffic in

different lanes and the driver's desired speed, the density of traffic in different lanes, the relative

speed with respect to the lag vehicle in the target lane, the presence of heavy vehicles in different

lanes ahead of the subject one. In addition, when a driver considers DLC although a mandatory

lane change is required but the driver is not responding to the MLC conditions, changing lanes

Opposite to the direction as required by the MLC conditions may be less desirable. If a driver

decides not to perform a discretionary lane change (i.e., either the driving Conditions are

satisfactory, or, although the driving conditions are not satisfactory, the current is the lane with

the best driving conditions, the driver continues in the current lane. Otherwise, the driver selects

a lane from the available alternatives and assesses the adjacent gap in the target lane.

The lowest level of ovals in the decision tree shown in Figure 3-2 corresponds to the gap

acceptance process. When trying to perform a DLC, factors that affect drivers' gap acceptance

behavior include the gap length, speed of the subject, speed of the vehicles ahead of and behind

the subject in the target lane, and the type of the subject vehicle (heavy vehicle or not). For

instance, a larger gap is required for merging at a higher travel speed. A heavy vehicle would

require a larger gap length compared to a car due to lower maneuverability and the length of the

heavy vehicle. In addition to the above factors, the gap acceptance process under the MLC

conditions is influenced by factors such as remaining distance to the point at which lane change

must be completed, delay (which captures the impatience factor that would make drivers more

aggressive.

Note that, delay cannot be used as an explanatory variable except for very spe-

cialized situations, for example, merging from an on{ramp. This is because the very

inception of an MLC condition is usually unobserved. The speci_cation of the com-

plete model is presented next.

2.4.2 Model Formulation

CHAPTER 3

CONCLUSION