A Validation Study of the Fadhloun-Rakha Car-following Model

Karim Fadhloun

1

, Hesham Rakha

1a

, Amara Loulizi

2b

and Jinghui Wang

1

1

Virginia Tech Transportation Institute, Virginia Tech, 3500 Transportation Research Plaza, Blacksburg VA, U.S.A.

2

LR11ES16 Laboratoire de Matériaux, d'Optimisation et d'Environnement pour la Durabilité,

École Nationale d'Ingénieur de Tunis, Tunis, Tunisia

Keywords: Rakha-Pasumarthy-Adjerid Car-following Model, Car-following Behavior, Vehicle Dynamics.

Abstract: The research presented in this paper investigates and validates the performance of a new car-following model

(the Fadhloun-Rakha (FR) model). The FR model incorporates the key components of the Rakha-Pasumarthy-

Adjerid (RPA) model in that it uses the same steady-state formulation, respects vehicle dynamics, and uses

very similar collision-avoidance strategies to ensure safe following distances between vehicles. The main

contributions of the FR model over the RPA model are the following: (1) it explicitly models the driver throttle

and brake pedal input; (2) it captures driver variability; (3) it allows for shorter than steady-state following

distances when following faster leading vehicles; (4) it offers a much smoother acceleration profiles; and (5)

it explicitly captures driver perception and control inaccuracies and errors. In this paper, a naturalistic driving

dataset is used to validate the FR model. Furthermore, the model performance is compared to that of five

widely used car-following models, namely: the Wiedemann model, the Frietzsche model, the Gipps model,

the RPA model and the Intelligent Driver Model (IDM). A comparative analysis between the different model

outputs is used to determine the performance of each model in terms of its ability to replicate the empirically

observed driver/vehicle behavior. Through quantitative and qualitative evaluations, the proposed FR model

is demonstrated to significantly decrease the modeling error when compared to the five aforementioned

models and to generate trajectories that are highly consistent with empirically observed driver following

behavior.

1 INTRODUCTION

Due to the continuous technological advancement

and proliferation of computational tools both at the

level of hardware and software, traffic engineering is

becoming more and more simulation-oriented.

Relying on computerized traffic simulations for

planning, urbanization and environmental purposes

can be cast as a two-edged activity. On the one hand,

microscopic simulation software allow the user to

evaluate and estimate the outcomes of different

potential scenarios in a fast and cost effective manner

and, most importantly, without inducing any

bottlenecks or disrupting the flow of vehicles in the

real world. On the other hand, it is imperative to not

forget that the results returned by traffic simulators

are directly correlated to the accuracy and precision

of the different models and logics incorporated in

them. Subsequently, it is necessary to ensure that

a

https://orcid.org/0000-0002-5845-2929

b

https://orcid.org/0000-0003-1542-0086

whatever implemented in this type of software, would

constitute good descriptors of real traffic conditions

and empirical behavior.

A main component of microscopic simulation

software is the car-following model. Car-following

models (Chandler, Herman et al. 1958, Gazis,

Herman et al. 1961, Drew 1968, Fritzsche 1994,

Treiber, Hennecke et al. 2000, Jiang, Wu et al. 2001,

Newell 2002, Olstam and Tapani 2004) predict the

temporal and spatial behavior of a following vehicle

when the time-space profile of the leading vehicle is

known. The output of car-following models directly

impact several other factors and measures of

effectiveness (MOE), such as vehicle energy/fuel

consumption and emissions.

This paper describes a research effort that aims to

validate a new innovative acceleration-based car-

following model, which is the Fadhloun-Rakha (FR)

model. The methodology and the procedure that led

180

Fadhloun, K., Rakha, H., Loulizi, A. and Wang, J.

A Validation Study of the Fadhloun-Rakha Car-following Model.

DOI: 10.5220/0009435501800192

In Proceedings of the 6th International Conference on Vehicle Technology and Intelligent Transport Systems (VEHITS 2020), pages 180-192

ISBN: 978-989-758-419-0

Copyright

c

2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

to the functional form of the model was described

extensively in a previous work by Fadhloun and

Rakha (Fadhloun and Rakha 2019). The validation of

the proposed model is conducted by comparing its

performance against the performance of other car-

following models. Gipps (Gipps 1981), Frietzsche

(Fritzsche 1994), Wiedemann (Wiedemann 1974,

Wiedemann 1992), the IDM model (Treiber,

Hennecke et al. 2000) and the RPA model (Rakha

2009) were selected as controls of the proposed

model because of their wide use and their

implementation in some of the most famous traffic

simulators (AIMSUN (Barceló 2001), PARAMICS

(Smith, Duncan et al. 1995), VISSIM (PTV-AG 2012)

and INTEGRATION (Van Aerde and Rakha 2007,

Van Aerde and Rakha 2007). The dataset used in the

validation procedure is extracted from the naturalistic

data of the 100-Car study that was conducted by the

Virginia Tech Transportation Institute (Dingus,

Klauer et al. 2006).

Concerning the layout, this paper is organized as

follows. First, an overview of the Fadhloun-Rakha

(FR) model is provided along with the other state-of-

the-practice car-following models mentioned above.

Subsequently, the dataset used in this study is briefly

described and the analysis related to the calibration

procedure as well as the validation process of the FR

model is presented. Finally, the conclusions of the

paper are drawn and insights into future work are

provided.

2 BACKGROUND

In this section, a brief description of the logic behind

each of the studied models is provided in a

chronological order.

2.1 Wiedemann Model

The Wiedemann model (Wiedemann 1974) is a

psycho-physical car-following model that is widely

known in the traffic engineering community due to its

integration in the microscopic multi-modal traffic

simulation software VISSIM (PTV-AG 2012). The

initial formulation of the model (Wiedemann 1974),

proposed in 1974, was calibrated mostly based on

conceptual ideas rather than real traffic data. As a

result, a much-needed recalibration of the model

(Wiedemann 1992) was performed in the early-1990s

using an instrumented vehicle.

The Wiedemann model framework, as

implemented in VISSIM, uses five bounding functions

in the Δ𝑣 Δx domain —AX, ABX, SDX, SDV and

OPDV— to define the thresholds between four traffic

regimes — free driving, closing-in, following and

emergency. Depending on the traffic regime in which

the following vehicle is located, the acceleration is set

equal to a predefined specific rate. The mathematical

expressions of the five regime thresholds are given in

Equations (1-5).

𝐴𝑋𝐿

𝐴

𝑋

𝐴

𝑋

𝑅𝑁𝐷1

(1)

𝐴𝐵𝑋𝐴𝑋

𝐵

𝑋

𝐵

𝑋

𝑅𝑁𝐷1

min

𝑢

,𝑢

(2)

𝑆𝐷𝑋𝐴𝑋

𝐸

𝑋

𝐸

𝑋

𝑁𝑅𝑁𝐷 𝑅𝑁𝐷2

𝐵𝑋

𝐵

𝑋

𝑅𝑁𝐷1

min

𝑢

,𝑢

(3)

𝑆𝐷𝑉

∆𝑥 𝐿

𝐴𝑋

𝐶𝑋

(4)

𝑂𝑃𝐷𝑉𝑆𝐷𝑉

𝑂𝑃𝐷𝑉

𝑂𝑃𝐷𝑉

𝑁𝑅𝑁𝐷

(5)

Where RND1, RND2, RND3, RND4 and NRND are

normally distributed parameters that aim to model the

randomness associated with different driving patterns

and behaviors, L

n-1

is the length of the leading vehicle

in meters, u

n-1

is the leading vehicle speed in (m/s), Δx

is the spacing between the lead and the following

vehicles, and CX is a model parameter that is assumed

to be equal to 40. Finally, the remaining variables,

named using the standard format P

add

or P

mult

, are the

model parameters requiring calibration.

It is noteworthy to mention that the formulations

of Equations (1-5) could be further simplified by

removing the random driver-dependent parameters

for the specific case of this study. In fact, the

randomness inducing parameters are of no use when

calibrating the model against empirical data of a

single driver. With that being said, Equations (1-5)

are modified by applying the generic transformation

of Equation 6 resulting in a significant reduction of

the number of calibration parameters. The resultant

set of equations, defined in Equations (7-11), requires

the calibration of a total of four parameters.

𝑃

𝑃

𝑃

𝑃

(6)

𝐴𝑋𝐿

𝐴

𝑋

(7)

𝐴𝐵𝑋𝐴𝑋 𝐵

𝑋

min

𝑢

,𝑢

(8)

𝑆𝐷𝑋𝐴𝑋 𝐸

𝑋

𝐵

𝑋

min

𝑢

,𝑢

(9)

𝑆𝐷𝑉

∆𝑥 𝐿

𝐴𝑋

40

(10)

𝑂𝑃𝐷𝑉𝑆𝐷𝑉 𝑂𝑃𝐷𝑉

(11)

A Validation Study of the Fadhloun-Rakha Car-following Model

181

2.2 Gipps Model

Gipps model (Gipps 1981), developed in the late-

1970s and implemented in the traffic simulation

software AIMSUN (Soria, Elefteriadou et al. 2014), is

formulated as a system of differential difference

equations. Using a time step

t

that aims to model

the reaction time of drivers, the model computes the

following vehicle speed u

n

at time t+Δt as a function

of its speed and the leading vehicle speed u

n-1

at the

preceding time step t.

As shown in Equation 12, the speed of the

following vehicle is estimated by determining the

minimum of two arguments. The first term governs

the cases characterized by uncongested traffic and

relatively large headways. Under such conditions, the

following vehicle speed increases until the free-flow

speed of the facility u

f

is reached. The model

formulation is also inclusive of a condition that

ensures that u

f

is never exceeded once achieved. The

second argument of the model is attained when

congestion prevails and speeds are constrained by the

behavior of the vehicles ahead of them. Due to the

collision avoidance mechanism it implements, the

congested regime branch is the one responsible for

making the Gipps model collision-free.

𝑢

𝑡∆𝑡

𝑚𝑖𝑛

⎝

⎜

⎜

⎜

⎛

𝑢

𝑡

2.5.𝐴

.∆𝑡1

𝑢

𝑡

𝑢

0.025

𝑢

𝑡

𝑢

𝐷

.∆𝑡

𝐷

.∆𝑡

𝐷

2

∆𝑥 𝐿

∆𝑡.𝑢

𝑡

𝑢

𝑡

𝐷

⎠

⎟

⎟

⎟

⎞

(12)

Where 𝐴

and 𝐷

are the respective desired

maximum acceleration and deceleration of the

following vehicle in m/s

2

, and 𝐷

denote the

maximum deceleration rate of the leading vehicle in

m/s

2

. Those three parameters are the ones requiring

calibration for Gipps model.

2.3 Frietzsche Model

Frietzsche model (Fritzsche 1994) is a car-following

model that shares the same structure as Wiedemann

model. In this model, six threshold parameters are

used to define five driving regimes. The thresholds

are defined for four gap (

x) values and two

differences in speed (

v) values between the leader

and the follower vehicles. The four gap threshold

parameters, AR, AS, AD, and AB are presented in

Equations (13-16); while the two differences in speed

thresholds, PTP and PTN, are given in Equations (17-

18). We note that the expression of the acceleration

rate a

n

associated with the “closing in” regime is

given in Equation (19-20).

𝐴

𝑅𝑠

𝑇

𝑢

(13)

𝐴𝑆𝑠

𝑇

𝑢

(14)

𝐴𝐷𝑠

𝑇

𝑢

(15)

𝐴𝐵𝐴𝑅

∆𝑢

∆𝑏

(16)

𝑃𝑇𝑃𝐾

∆𝑥 𝑠

𝑓

(17)

𝑃𝑇𝑁𝐾

∆𝑥 𝑠

𝑓

(18)

𝑎

𝑢

𝑢

2𝑑

(19)

𝑑

∆𝑥𝐴𝑅𝑢

.∆𝑡

(20)

Where T

r

, T

s

, T

D

and Δb

m

are calibration parameters

expressed in seconds. For the remainder of this study,

d

max,

f

x

, K

ptp

and K

ptn

are set equal to -6 m/s

2

, 0.5, 0.002

and 0.001.

2.4 The Intelligent Driver Model

The IDM model (Treiber, Hennecke et al. 2000) is a

kinematics-based car-following model that is widely

used for the simulation of freeway traffic. It was

developed in 2000 by Treiber et al. (Treiber,

Hennecke et al. 2000) with the main objective of

modeling the longitudinal motion of vehicles as

realistically as possible under all traffic situations.

The fame of this model is mainly due to its

mathematical stability, which results in stable vehicle

trajectories and smooth acceleration profiles. The

acceleration function of the intelligent driver model

(IDM) car-following model is presented in Equations

(21-22).

𝑎

𝑢

,𝑠

,∆𝑢

𝑎1

𝑢

𝑢

𝑠

∗

𝑢

,∆𝑢

𝑠

(21)

𝑠

∗

𝑢

,∆𝑢

𝑠

𝑢

𝑇

𝑢

∆𝑢

2

√

𝑎.𝑏

(22)

Where s

*

denotes the steady state spacing, a is the

maximum acceleration level, b is the maximum

deceleration level, δ is a calibration parameter and T

is the desired time headway.

2.5 Rakha-Pasumarthy-Adjerid Model

The RPA model (Rakha 2009) is a car-following

model that controls the longitudinal motion of the

vehicles in the INTEGRATION traffic simulation

software (Van Aerde and Rakha 2007, Van Aerde and

Rakha 2007). The model is composed of three main

components: the steady-state, the collision avoidance

VEHITS 2020 - 6th International Conference on Vehicle Technology and Intelligent Transport Systems

182

and the vehicle dynamics models. Having the values

of its three components, the RPA model computes the

speed of the following vehicle as shown in Equation

23.

𝑢

𝑚𝑖𝑛

𝑢

,𝑢

,𝑢

(23)

Here 𝑢

, 𝑢

and 𝑢

are the speeds calculated

using the three modules described previously and

which expressions are given in what follows.

2.5.1 First-order Steady-state Car-following

Model

The RPA model utilizes the Van Aerde nonlinear

functional form to control the steady-state behavior of

traffic. The latter model was proposed by Van Aerde

and Rakha (Van Aerde and Rakha 1995) and is

formulated as presented in Equation 24.

𝑠

𝑐

𝑐

𝑢

𝑢

𝑐

𝑢

(24)

Here 𝑠

is the steady state spacing (in meters)

between the leading and the following vehicles, u

n+1

is the speed of the follower, in (m/s), u

f

is the free-

flow speed expressed in m/s, and 𝑐

1

(m), 𝑐

2

(m

2

/s) and

𝑐

3

(s) are constants used for the Van Aerde steady-

state model that have been shown to be directly

related to the macroscopic parameters defining the

fundamental diagram of the roadway.

Finally, it should be noted that from the

perspective of car-following modeling, the main

objective is to determine how the following vehicle

responds to changes in the behavior of the leading

vehicle. Subsequently, a speed formulation is adopted

for the Van Aerde model, as demonstrated in

Equation 25, which is easily derived from Equation

24 using basic mathematics.

𝑢

𝑐

𝑐

𝑢

𝑠

𝑐

𝑐

𝑢

𝑠

4𝑐

𝑠

𝑢

𝑐

𝑢

𝑐

2𝑐

(25)

2.5.2 Collision Avoidance Model

The expression of the collision avoidance term is

shown in Equation 26 and is directly related to a

simple derivation of the maximum distance that a

vehicle can travel to decelerate from its initial speed

to the speed of the vehicle ahead of it while ensuring

that, in the case of a complete stop, the jam density

spacing between the two vehicles is respected.

𝑢

𝑢

2𝑏𝑠

𝑠

(26)

Here b is the maximum deceleration at which the

vehicles are allowed to decelerate and s

j

is the spacing

at jam density.

2.5.3 Vehicle Dynamics Model

The final component of the RPA model is the vehicle

dynamics model (Rakha, Lucic et al. 2001, Rakha,

Snare et al. 2004) that ensures that the vehicle’s

mechanical capabilities do not limit it from attaining

the speeds that are dictated by the steady-state

component. This model computes the typical

acceleration of the following vehicle as the ratio of

the resultant force to the vehicle mass M (Equation

27). The resultant force is computed as the difference

between the tractive force acting on the following

vehicle F

n+1

(Equation 28) and the sum of the

resistive forces acting on the vehicle which include

the aerodynamics, rolling and grade resistances.

𝑎

𝐹

0.5𝜌𝐶

𝐶

𝐴

𝑔𝑢

𝑀𝑔𝐶

𝐶

𝑢

𝐶

𝑀𝑔𝐺

𝑀

(27)

𝐹

min3600𝜂

𝛾𝑃

𝑢

,𝑀

𝑔𝜇

(28)

Here η is the driveline efficiency (unitless); 𝑃 is the

vehicle power (kW); 𝑀

is the mass of the vehicle on

the tractive axle (kg);

is the vehicle throttle level

(taken as the percentage of the maximum observed

throttle level that a certain driver uses); 𝑔 is the

gravitational acceleration (9.8067 m/s

2

); 𝜇 is the

coefficient of road adhesion or the coefficient of

friction (unitless); 𝜌 is the air density at sea level and

a temperature of 15°C (1.2256 kg/m

3

); 𝐶

is the

vehicle drag coefficient (unitless), typically 0.30; 𝐶

is the altitude correction factor equal to 1-0.000085h,

where ℎ is the altitude in meters (unitless); 𝐴

is the

vehicle frontal area (m

2

), typically 0.85 multiplied by

the height and width of the vehicle; 𝐶

is a rolling

resistance constant that varies as a function of the

pavement type and condition (unitless); 𝐶

is the

second rolling resistance constant (h/km); 𝐶

is the

third rolling resistance constant (unitless); 𝑚 is the

total vehicle mass (kg); and 𝐺 is the roadway grade

(unitless).

The acceleration computed using the dynamics

model is then used to calculate the maximum feasible

speed 𝑢

using a first Euler approximation.

2.6 Fadhloun-Rakha Model

The Fadhloun-Rakha (FR) model (Fadhloun and

Rakha 2019) is an acceleration-based car-following

model that uses the same steady-state formulation and

respects the same vehicle dynamics as the RPA

model. Additionally, the model uses very similar

collision-avoidance strategies to ensure a safe follow-

A Validation Study of the Fadhloun-Rakha Car-following Model

183

ing distance between vehicles.

The mathematical expression of the FR model,

presented in Equation 29, estimates the acceleration

of the following vehicle as the sum of two terms. The

first term models the vehicle behavior in the

acceleration regime, while the second governs the

deceleration regime.

𝑎

𝐹𝑎

𝐶𝐴

𝑢

,𝑠

,∆𝑢

(29)

In the acceleration regime, the vehicle behavior is

governed by the vehicle dynamics, as demonstrated

in Equation 27 to ensure that vehicle accelerations are

realistic. A reducing multiplier F (Equation 30),

which ranges between 0.0 and 1.0, is then applied to

the vehicle dynamics acceleration. The F factor is a

function that is sensitive to 𝑋

(Equation 31) which

represents the ratio of 𝑢

/𝑠

divided by the ratio

of the steady state speed to the steady state

spacing 𝑢

/𝑠

. It aims to guarantee that two

objectives are met. First, it ensures the convergence

of the vehicles’ behavior towards the Van Aerde

steady state model. Second, it attempts to model

human behavior and the different patterns of driving

by acting as a reduction factor to the vehicle dynamics

model.

𝐹

𝑋

𝑒

1 𝑋

𝑒

(30)

𝑋

𝑠

𝑠

∙

𝑢

𝑢

(31)

Where a, b, and d are model parameters that are

calibrated to a specific driver and model the driver

input to the gas pedal.

The second term in the expression of the FR

model considers vehicle deceleration to avoid a

collision with a slower traveling lead vehicle as

shown in Equations (32-33). As shown, collision

avoidance is ensured by the function CA which

computes the needed deceleration to apply as the ratio

of the square of the kinematics deceleration needed to

decelerate from the current speed to the leading

vehicle speed at a desired deceleration level that is set

by the user.

𝑑

𝑢

𝑢

𝑢

𝑢

4𝑠

𝑠

(32)

𝐶𝐴

𝑢

,𝑠

,∆𝑢

𝑑

𝑑

𝑔𝐺

(33)

Where d

des

is the desired deceleration level.

Finally, to model the effect of the driver error in

estimating the leading vehicle speed and the distance

gap between the two vehicles, two wiener processes

are incorporated in the model formulation at the level

of u

n

and s

n+1

. Additionally, a white noise signal is

added to the model’s expression to capture the

driver’s imperfection while applying the gas pedal.

The compounding effect of those three signals makes

the model output more representative of human

driving behavior.

3 NATURALISTIC DATASET

The data used herein represents a small subset that

was extracted from the naturalistic driving database

generated by the 100-Car study (Dingus, Klauer et al.

2006) that was conducted by the Virginia Tech

Transportation Institute (VTTI) in 2002. In fact,

VTTI initiated a study where 100 cars were

instrumented and driven by a total of 108 drivers

around the District of Columbia (DC) area. The

resulting database from the 100-Car study (Dingus,

Klauer et al. 2006) contained detailed logs of more

than 207,000 completed trips with a total duration of

around 20 million minutes of data.

The naturalistic dataset that was used to validate

the proposed model contains information relating to

1,659 car-following events which spans over a

duration of around 13 hours which is significant for

the task of validation of car-following models. The

car-following data composing the dataset comes from

six different drivers and was collected on a relatively

short segment of the Dulles Airport access road

(approximately an 8-mile long section) in order to

maintain facility homogeneity.

Finally, it is noteworthy to state that both the

characteristics of the different vehicles are known due

to the naturalistic nature of the dataset. This makes

the determination of the different FR and RPA model

variables straightforward and exclusive of bias.

4 PARAMETER CALIBRATION

OF THE STUDIED MODELS

For each of the studied models, a certain number of

inputs is needed. These inputs can be categorized into

two groups. The first category comprises the inputs

that are the same for the different models, namely the

time-space and the time-speed profiles of the leading

vehicle, the starting location and speed of the

following vehicle as well as the free-flow speed (u

f

)

which was estimated specifically for each car-

following event along with any other variables related

to the roadway. The use of the free-flow speed

distribution shown in Figure 1 instead of a constant

value across all of the events, is justified by the

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184

significant heterogeneity of the driver behavior

during the free driving phase. In fact, drivers do not

necessarily drive at the speed limit of the facility

when there is no vehicle ahead of them.

Besides that, the desired speed of a certain

naturalistic event was set equally across all of the

studied models in order to maintain the homogeneity

of driver behavior and road facility for that specific

event.

Figure 1: Distribution of the free-flow speed for the

naturalistic events.

As a side remark, we note that the jam density k

j

,

the capacity q

c

and the speed-at-capacity u

c

, which are

needed to generate a simulated trajectory in the case

of the formulations of the RPA model and the FR

model, were estimated using the calibration

procedure proposed by Rakha and Arafeh (Rakha and

Arafeh 2010). However, unlike the free-flow speed,

those parameters were calibrated using the bulk data

of each driver given their minor influence on the

resulting model outputs. The estimated values for the

latter driver-specific parameters are presented in

Table 1a along with the needed vehicle-specific

parameter values in Table 1b.

The remaining input variables consist of model-

specific parameters that require to be calibrated

accordingly depending on the researcher’s objectives.

Since this study aims to validate a new car-following

model by comparing its performance to that of other

state-of-the-art models, the different parameters need

to be calibrated such that the resulting simulated

behavior of the following vehicle matches its

observed behavior as closely as possible. The

calibration procedure of the different parameters of

each model was conducted heuristically taking the

speed RMSE as the error objective function. The

choice to optimize each model with regards to the

speed RMSE is judged reasonable given that the

optimization operation was done on an event-by-

event basis. In fact, we opted to calibrate each model

separately for each car-following event rather than for

the dataset as a whole. Even though that exponentially

increased the computation time, a more fair

comparison between the results is made possible as

each model was allowed to propose its best possible

fit for each of the 1659 naturalistic events. Hence, the

different model outputs are incorporative of the effect

of the strength points of each model.

Table 1a: Values of k

j

, q

c

and u

c

for each driver.

Driver

kj

(veh/m)

qc

(veh/s)

uc

(m/s)

Driver_124 0.091 0.865 22.22

Driver_304 0.150 0.833 19.00

Driver_316 0.075 0.464 21.36

Driver_350 0.080 0.529 21.28

Driver_358 0.087 0.447 19.53

Driver_363 0.131 0.906 23.69

Table 1b: Characteristics of the different vehicles.

Vehicle Characteristics

Driver

P

(kW)

M (kg) C

d

A

f

(m

2

)

Driver_124 90 1190 0.36 2.06

Driver_304 90 1090 0.40 2.00

Driver_316 90 1090 0.40 2.00

Driver_350 90 1090 0.40 2.00

Driver_358 145 1375 0.40 2.18

Driver_363 145 1375 0.40 2.18

Finally, given the presence of noise in the

proposed model, the calibration was conducted using

a bi-level procedure. First, the model parameters were

calibrated deterministically without the consideration

of the noise signals. Next, to model the effect of the

noise, the optimized parameters of the first step were

used to run a total of 1000 simulations in order to have

valid model outputs and to determine the 95%

confidence interval of the results.

5 RESULTS AND MODEL

VA L I D AT I O N

Having access to the calibrated parameters, the speed

profiles were obtained for each car-following event of

the naturalistic dataset. The corresponding speed

A Validation Study of the Fadhloun-Rakha Car-following Model

185

outputs ensure a minimal RMSE between a model’s

predictions and the measured data over its whole

timespan. To illustrate the results, the probability

distribution of the speed RMSE of the different

models is plotted in Figure 2. The figure demonstrates

that the FR model performs better overall in terms of

fitting the observed data than the other models. That

is demonstrated by the fact that its RMSE distribution

is higher than those of the other models towards the

lower end of the speed errors (between 0 and 0.5).

Then, as the RMSE keeps getting bigger and bigger,

the tendency is reversed and the RMSE distribution

of the FR model becomes the smallest.

Figure 2: Probability distribution of the speed RMSE for the

different models.

To better quantify statistically the difference in

performance between the proposed model and the

other five models, the rank of the new model was

determined for each event based on the calculated

RMSE value (the resulting mean of the 1000 trials).

The ranking was sorted in an increasing direction of

the RMSE value with the best model being the one

offering the lowest error. Table 2.a shows the results

of this analysis where the rank distribution of the

proposed model is presented. From the table, one can

see that the FR model outperformed the other ones. In

fact, this model offered the best fit to the empirical

data for about half of the considered events (735 out

of 1659 events). Furthermore, the number of events

for which the fit of the proposed model was in either

the first or the second position, represents about two

thirds of the total cases (1126 out of 1659 events).

The quantitative analysis was taken a step further

as the proposed model was compared face-to-face

with each of the studied models. That would allow for

a better understanding of the new model’s

performance. Figure 3.a and Figure 3.b present the

results of this comparison in terms of the optimized

speed RMSE and the one computed from the resulting

acceleration profiles, respectively. In terms of speed

error, the FR model is demonstrated to significantly

outperform the other models. In fact, its speed RMSE

was smaller than that found using the RPA, Gipps,

Wiedemann, Frietzsche, and the IDM models in

between 65% to around 90% of the events. The

previous stated values do not confer enough

information about the new model performance by

themselves as they do not quantify the percentages by

which the error function was reduced. Consequently,

the bar chart of Figure 3 is complemented by Table

2.b which presents key measures (mean, median and

standard deviation) about the distribution of the

relative percentage decrease in the speed RMSE. For

instance, it is found that for the 90% of the total events

for which the proposed model formulation

outperformed the Wiedemann model, the error

reduction percentage had a median equal to 85%. In

the case of the RPA model, the FR model resulted in

an average decrease of the RMSE that is around 56%

for the 88% of the events for which it was the best.

Figure 3: Comparison of the proposed FR model

formulation performance to the other models: a. Based on

the speed RMSE; b. Based on the acceleration RMSE.

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When considering face-to-face comparisons in

terms of the resulting acceleration data from the

optimized speed profiles, only the Wiedemann and

the IDM model outperformed the FR model as it can

be observed in Figure 3.b. While the IDM model is

known for its excellent fit to acceleration data due to

its smooth expression, the results of the Wiedemann

model seem intriguing at first. In fact, it is found that

the results are justified by the structure of the

Wiedemann model itself as it will be described later.

Table 2a: Rank of the FR model in terms of goodness of fit

as a percentage of the total number of events using the speed

RMSE.

Rank Rank Distribution (%)

1 44.30

2 23.57

3 16.88

4 11.63

5 3.32

6 0.30

Table 2b: Distribution characteristics of the decrease

percentage in the speed RMSE for head-to-head

comparisons.

Best Mean Median Std Dev

RPA

FR

56.3 58.6 23.7

G 45.4 46.9 22.4

W 77.0 85.7 20.9

F 45.6 46.9 22.7

IDM 50.5 53.4 21.5

RPA RPA 26.6 22.7 18.7

G G 43.2 45.6 23.3

W W 43.9 45.4 24.4

F F 35.9 35.8 22.1

IDM IDM 30.3 27.5 20.8

In order to examine the performance of the

different models qualitatively, the resulting simulated

speeds are presented for some sample events. In fact,

Figure 4 plots the variation of the observed and

simulated speed profiles for four different events over

time. For each subplot (Figure 4a through Figure 4d),

the results from the studied models are drawn in order

to compare their predictions with the observed

naturalistic behavior. For example, for the event

presented in Figure 4a, the driver accelerated from

about 23.5 m/s to around 26 m/s, maintained his/her

speed around that value, then re-accelerated to about

27 m/s and tried to maintain that speed until the end

of the event. This behavior was well captured by

most of the studied models, except that at the end of

the event all models predicted a decrease in speed.

This is mainly due to the fact that all the studied

models take into account a minimum safe distance in

order to avoid collision with the leading vehicle.

Given that the collision avoidance logics of the

models judged that the spacing maintained by the

driver is unsafe for such high speeds, a decrease in

speed was predicted to keep a safe distance and to

ensure that the collision avoidance conditions are

met. That opposes the actual driver behavior who

maintained his/her driving speed despite being

unsafely close to the leading vehicle. Looking

roughly into this event, it is the FR model that traces

better the actual driver behavior, followed by the IDM

model, then Gipps, the RPA and Frietzsche models,

and lastly Wiedemann model.

It is worth clarifying at this level the reasons

behind the steep decrease in speed observed in the

output of the Wiedemann model. The observed speed

drop, which occurs 30 seconds after the start of the

event, is due to the nomenclature of Wiedemann

model itself. In fact, similar data cliffs were found to

be present in a noticeable number of other events for

this model. Such behaviors result from the abrupt

change in the acceleration value when transitioning

from one traffic regime to another. Besides the latter

aspect, the crossing of one of the boundaries

delimiting the different regions of the Wiedemann

model was found to result in another disparity in the

model output when compared to most of the other

models (FR, RPA, Gipps, IDM). The concerned

disparity is observed when the following vehicle

remains in the same traffic region for the entire

duration of the car-following event, hence arising the

possibility of having a constant acceleration over the

entire duration of the car-following maneuver. The

previous two drawbacks are also manifested in the

Frietzsche model due to its similar structure, however

their presence is not as prevalent. For instance, one

such case in which the following vehicle remained

within the same traffic regime for Frietzsche model is

shown in Figure 4c. The figure illustrates a scenario

in which the driver was trying to maintain his/her

desired speed of 28.5 m/s with minor fluctuations.

Since the vehicle started and finished its trip within

the “Free Driving” regime, the Frietzsche model

resulted in a constant speed profile for the entire

A Validation Study of the Fadhloun-Rakha Car-following Model

187

event. However, the latter aspects of Frietzsche and

Wiedemann models do not necessarily connote an

inability to propose a fitted speed that matches

empirical data. As a matter of fact, while all the

models captured the empirical behavior of the event

presented in Figure 4b, the Frietzsche model was the

best in terms of tracing the actual speed profile. All

other models slightly over-predicted the maximum

reached speed.

Finally, concerning the event described by Figure

4d, the speed profile suggests that the highway is

heavily congested. The driver decelerated from about

9 m/s to come to an almost complete stop for a few

seconds. This was followed by an oscillatory

behavior due to a succession of accelerations and

decelerations. Despite the repeating oscillations, the

FR model traced almost perfectly the driver behavior

for the entire timespan. The RPA model gave

reasonable predictions for this event as well. Overall,

as a qualitative measure, the different events

presented in the figure are consistent with the

goodness of fit results presented earlier. The Gipps

model along with the FR and RPA model appear to

capture the naturalistic data considerably well.

Next, the acceleration profiles derived from the

calibrated speed data were examined. For illustration

purposes, a sample event was chosen to visualize and

compare the simulated acceleration profiles to

empirical data. The different profiles are presented in

Figure 5. For clarity of the figure as the overlap

between the outputs of the studied models is

significant, the results are presented in each sub-

figure (Figure 5a through Figure 5f) along with the

Figure 4: Variation of the simulated speeds over time of four sample events.

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Figure 5: Variation of the simulated acceleration over time of a sample car-following event: a. Wiedemann model; b.

Frietzsche model; c. Gipps model; d. RPA model; e. FR model; f. IDM Model.

observed acceleration of the driver. During this 2-

minute car-following event, the driver had

acceleration and deceleration maneuvers with

maximum values of 1.6 m/s

2

and 2.1 m/s

2

,

A Validation Study of the Fadhloun-Rakha Car-following Model

189

respectively. As shown by Figure 5a, the Wiedemann

model results in a zero constant acceleration mainly

because the modeled vehicle behavior remained

within the boundaries of one of the traffic regimes for

the total event duration.

More importantly, the illustrated constant

acceleration behavior of the Wiedemann model,

which was confirmed across several other car-

following events, gives a plausible explanation of the

extremely low values found when the RMSEs related

to the acceleration data were computed. By avoiding

the oscillatory behavior of the other models and, more

importantly, staying within the maximum

acceleration and deceleration values without

overshooting, a constant acceleration profile would

result in a better fit to the empirical behavior in terms

of the RMSE value. Setting aside the car-following

events with a constant simulated acceleration, the

Wiedemann model resulted in a stepped acceleration

profile similar to the acceleration-time diagram of the

Frietzsche model plotted in Figure 5b. As for Gipps

model, the FR model and the RPA model (Figure 5c,

Figure 5d, and Figure 5e, respectively), they resulted

in acceleration values that closely followed the field

data even though the maximum predicted

deceleration was relatively overestimated. More

precisely, the IDM model traced the actual

acceleration profile the best for this specific event

followed by the FR model formulation. Generally

speaking, the new model was found to be the best in

terms of mimicking the real driver behavior as it

successfully avoided the acceleration fluctuations

produced by the other models that are far in excess of

those observed at the level of the empirical data. Even

more, the significance and contribution of the latter

finding is further amplified given the fact that the FR

model formulation is inclusive of three noise signals.

Those noises attempt to account for the driver’s errors

related to estimating the model input variables — the

distance gap to the leading vehicle along with its

speed — as well as his/her imperfection while

applying the gas pedal. Notwithstanding the fact that

the other models are exclusive of such errors giving

them a statistical edge, their predicted acceleration

profiles were still outperformed by the acceleration

predictions of the FR model except for the IDM

model which provides comparable results.

From a traffic researcher standpoint, acceleration

data can be cast as the most important output of a car-

following model. In fact, acceleration information is

the starting point for the computation of other

measures of effectiveness (MOEs). Two specific

MOEs that are very sensitive to the accuracy of

predicted accelerations and quite important from an

environmental perspective, are fuel consumption and

emissions estimations. With that in mind, it seemed

necessary to examine the behavior of the maximum

acceleration distribution of the bulk dataset given its

major impact on any fuel consumption or emissions

calculation.

Subsequently, the observed and predicted

maximum acceleration of each model were extracted

for each event and plotted as shown by Figure 6. We

note here that the maximum acceleration data is

sorted from the highest value to the lowest for each

model independently of the others. This means that

the event numbered as one, for example, in the figure

is not the same physical event for all the studied

models or that calculated from the measured speed

data. It is just the physical event that resulted in the

highest maximum observed or modeled acceleration.

In other words, the figure does not allow making

event-by-event comparisons between the different

models. The main purpose of the plot is to compare

the empirical maximum acceleration distribution of

the whole dataset to the ones resulting from the

calibration of the different studied models.

As a side note, since 1000 simulations were run

using the logic of the FR model to estimate the mean

and the dispersion of the results, the simulated

maximum acceleration using the new model

formulation is plotted using the mean and the 95%

confidence interval of the data which is shown by the

light bounded area in Figure 6. Qualitatively

speaking, the figure demonstrate the superiority of the

FR model in terms of its ability of replicating the

maximum acceleration behavior of the naturalistic

dataset. In fact, the observed data appears to be

successfully covered by the breadth of the 95%

confidence interval of the model output.

Figure 6: Comparison of the maximum acceleration

behavior of the naturalistic dataset to the outputs of the

different models.

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6 CONCLUSIONS AND FUTURE

WORK

This research effort investigates and validates the

statistical performance of the FR car-following model

using naturalistic driving data from the 100-Car

study. The validated model is an acceleration-based

alternative formulation of the RPA model. In fact, the

two models share the same steady state model, respect

the same vehicle dynamics and use different, but very

similar, collision-avoidance strategies to ensure a safe

following distance between cars.

The considered naturalistic data of six drivers was

used to calibrate the FR model along with five state-

of-the-art car-following models, and a comparative

analysis between the resulting model performances

was conducted. By doing so, this study demonstrates

that the FR model outperforms Gipps, Wiedemann,

Frietzsche, the RPA and the IDM models in terms of

statistically matching the empirical data on an event-

by-event basis.

While the RMSE, used herein, is a good indicator

to evaluate a car-following model from a statistical

perspective, it is not generally enough to confirm that

it would be the best with regards to every aspect of

traffic engineering. In fact, the only endpoint that can

be deducted from this study is that the FR model is

the most flexible when compared to the other ones in

terms of its ability to generate a speed profile for the

following vehicle that emulates empirical data such

that the resulting error is at its minimum. Whether the

FR model formulation would offer the best fit when

considering other indicators, such as fuel

consumption or emissions rates, is a completely

separate problem that needs to be investigated before

conclusions can be made.

ACKNOWLEDGMENTS

The authors acknowledge the financial support

provided by the University Mobility and Equity

Center (UMEC) and the Department of Energy

through the Office of Energy Efficiency and

Renewable Energy (EERE), Vehicle Technologies

Office, Energy Efficient Mobility Systems Program

under award number DE-EE0008209.

REFERENCES

Barceló, J. (2001). GETRAM/AIMSUN: A software

environment for microscopic traffic analysis. Proc. of

the Workshop on Next Generation Models for Traffic

Analysis, Monitoring and Management.

Chandler, R. E., R. Herman and E. W. Montroll (1958).

"Traffic dynamics: studies in car following."

Operations research 6(2): 165-184.

Dingus, T. A., S. G. Klauer, V. L. Neale, A. Petersen, S. E.

Lee, J. Sudweeks, M. Perez, J. Hankey, D. Ramsey and

S. Gupta (2006). The 100-car naturalistic driving study,

Phase II-results of the 100-car field experiment.

Drew, D. R. (1968). Traffic flow theory and control.

Fadhloun, K. and H. Rakha (2019). "A novel vehicle

dynamics and human behavior car-following model:

Model development and preliminary testing."

International Journal of Transportation Science and

Technology, ISSN: 2046-0430.

Fritzsche, H.-T. (1994). "A model for traffic simulation."

Traffic Engineering+ Control 35(5): 317-321.

Gazis, D. C., R. Herman and R. W. Rothery (1961).

"Nonlinear follow-the-leader models of traffic flow."

Operations research 9(4): 545-567.

Gipps, P. G. (1981). "A behavioural car-following model

for computer simulation." Transportation Research

Part B: Methodological 15(2): 105-111.

Jiang, R., Q. Wu and Z. Zhu (2001). "Full velocity

difference model for a car-following theory." Physical

Review E 64(1): 017101.

Newell, G. F. (2002). "A simplified car-following theory: a

lower order model." Transportation Research Part B:

Methodological 36(3): 195-205.

Olstam, J. J. and A. Tapani (2004). Comparison of Car-

following models.

PTV-AG (2012). VISSIM 5.40–01 User Manual.

Karlsruhe, Germany.

Rakha, H. and M. Arafeh (2010). "Calibrating Steady-State

Traffic Stream and Car-Following Models Using Loop

Detector Data." Transportation Science 44(2): 151-168.

Rakha, H., I. Lucic, S. Demarchi, J. Setti and M. Aerde

(2001). "Vehicle Dynamics Model for Predicting

Maximum Truck Acceleration Levels." Journal of

Transportation Engineering 127(5): 418-425.

Rakha, H., Pasumarthy, P., and Adjerid, S. (2009). "A

Simplified Behavioral Vehicle Longitudinal Motion

Model." Transportation Letters: The International

Journal of Transportation Research, Vol. 1(2), pp. 95-

110.

Rakha, H., M. Snare and F. Dion (2004). "Vehicle

Dynamics Model for Estimating Maximum Light-Duty

Vehicle Acceleration Levels." Transportation

Research Record: Journal of the Transportation

Research Board 1883(-1): 40-49.

Smith, M., G. Duncan and S. Druitt (1995). "PARAMICS:

microscopic traffic simulation for congestion

management."

Soria, I., L. Elefteriadou and A. Kondyli (2014).

"Assessment of car-following models by driver type

and under different traffic, weather conditions using

data from an instrumented vehicle." Simulation

Modelling Practice and Theory 40: 208-220.

Treiber, M., A. Hennecke and D. Helbing (2000).

"Congested traffic states in empirical observations and

A Validation Study of the Fadhloun-Rakha Car-following Model

191

microscopic simulations." Physical review E 62(2):

1805.

Van Aerde, M. and H. Rakha (1995). Multivariate

calibration of single regime speed-flow-density

relationships [road traffic management]. Vehicle

Navigation and Information Systems Conference, 1995.

Proceedings. In conjunction with the Pacific Rim

TransTech Conference. 6th International VNIS. 'A Ride

into the Future'.

Van Aerde, M. and H. Rakha (2007). "INTEGRATION ©

Release 2.30 for Windows: User's Guide - Volume II:

Advanced Model Features, " M. Van Aerde & Assoc.,

Ltd." (Blacksburg).

Van Aerde, M. and H. Rakha (2007). "INTEGRATION ©

Release 2.30 for Windows: User's Guide – Volume I:

Fundamental Model Features, " M. Van Aerde &

Assoc., Ltd." (Blacksburg).

Wiedemann, R. (1974). Simulation des

Strassenverkehrsflusses. Karlsruhe, Germany,

Schriftenreihe des Instituts für Verkehrswesen der

Universität Karlsruhe.

Wiedemann, R., Reiter, U. (1992). Microscopic traffic

simulation: The simulation system MISSION,

background and actual state. CEC Project ICARUS

(V1052), Final Report. Brussels. 2: 1-53 in Appendix

A.

VEHITS 2020 - 6th International Conference on Vehicle Technology and Intelligent Transport Systems

192