Framework of Research

In order to achieve research objectives, the framework of this research is shown as Figure 1-1.

1. Car-following model development

A simple car-following will be developed. It can:

„ reflect the difference among different drivers,

„ avoid some deficiencies of traditional car-following models, such as drivers have to determine the deceleration capability of their lead vehicle, and

„ reproduce equilibrium and disequilibrium traffic phenomena.

2. Sensitivity analysis

It will discuss how the proposed model output varies with changes in model inputs.

3. Stability analysis

Local stability between two moving cars will be discussed. The discussion is on the stability of a following vehicle when its lead vehicle is in equilibrium state and the following vehicle has no acceleration limit. A method of discussing linearized stability of a dynamical system will be applied.

4. Equilibrium state discussion

A system is either in equilibrium or in disequilibrium. A vehicle is in equilibrium state if its speed and spacing never change as time passes. The microscopic and macroscopic equilibrium states of the proposed car-following model will be discussed. Fundamental diagram based on the microscopic equilibrium state will be also discussed.

5. Disequilibrium state discussion

Some traffic phenomena of disequilibrium state will be discussed, such as closing-in, shying-away, stop-and-go, and traffic hysteresis. Simulation examples will be provided.

6. Relaxation time discussion

Relaxation time refers to the time needed for a system to relax under external stimuli. When a perturbation occurs at an equilibrium system, the system will depart from equilibrium state. This study discusses how much time the system needs to return to the equilibrium state.

7. Parameters discussion and mathematical analysis

This study possesses analytical properties that are logical in representing physical phenomena, and prove that some traffic phenomena still hold under any parameters or some traffic phenomena have various patterns under different parameter values.

Stability Analysis

Parameters Discussion / Mathematical Analysis

Car-Following Model

Microscopic Traffic Flow

Macroscopic Traffic Flow Equilibrium State

Homogeneous Drivers Static Macroscopic Traffic

Flow Model (Fundamental Diagram)

Disequilibrium State

Traffic Hysteresis Traffic Flow

Sensitivity Analysis

Relaxation Time Equilibrium State

Equilibrium Speed Equilibrium Spacing

Disequilibrium State Stop-and-go

Closing-in Shying-away Traffic Hysteresis

Figure 1-1 Framework of research

Chapter 2 Literature Review

The research purpose of this dissertation is to develop a simple car-following model which can analyze traffic properties, represent traffic flow phenomena, save execution time, and have potential for extending to macroscopic models. Thus, traffic flow characteristics are reviewed in section 2.1. Section 2.2 reviews some car-following models. Section 2.3 presents some static macroscopic traffic flow models. Section 2.4 review linearized stability of dynamical systems. Finally, a brief summary and discussion is given in section 2.5.

2.1 Traffic Flow Characteristics

A system is either in equilibrium state or disequilibrium state. Section 2.1.1 reviews traffic stability that discusses whether a following car will reach the equilibrium state or not. Section 2.1.2 reviews some traffic phenomena of disequilibrium states.

2.1.1 Traffic Stability

Car-following models describe both the space-time behavior of vehicles and their interactions individually on a single lane. After car-following for a long time, the speed or spacing of the vehicle might be kept at a particular value (i.e., stable traffic) or changed again and again over time (i.e., unstable traffic). The fundamental diagram (as shown in Fig. 2-1.) of traffic flow indicates that traffic flow is unstable at low speed (i.e., under heavy traffic), and stable at high speed.

Critical Speed Critical

Density Density

Speed Unstable flow Stable flow

Figure 2-1 Relationship among speed and density [Mschane & Roess, 1990].

Traffic stability can be analyzed from the viewpoint of macroscopic traffic flow.

Zhang [1999b] found that various instability criteria can be reduced to a single criterion derived from first order waves traveling faster than slow second order waves in the higher order theories. Nagatani [2000] pointed out when the density is larger than a critical value, the traffic becomes unstable. Yi et al [2003] derived a nonlinear traffic flow stability criterion using a wavefront expansion technique. Jiang and Wu [2003] found that stability depends on the equilibrium speed density relationship, and it is also affected by the sensitivity parameters in the corresponding car-following model.

Some researchers analyzed traffic stability from the viewpoint of microscopic traffic. It is found that unstable traffic is likely to occur under higher reaction time and higher sensitivity response based on stimulus-response models [Herman et al, 1959;

May, 1990; Zhang & Jarret, 1997; Holland, 1998]. This cannot explain why heavy traffic is unstable, unless drivers have different reaction time or different sensitivity response under different spacings.

2.1.2 Disequilibrium Traffic Flow

A following car is in equilibrium state if it keeps its velocity and spacing at some particular value. If a following car is in disequilibrium state, it implies the car is accelerating or decelerating. This section reviews some traffic phenomena of acceleration and deceleration traffic.

A. Closing-in and Shying-away

Sometimes the following vehicle accelerates despite the lead vehicle traveling slower than it is (i.e. closing-in), and sometimes the follower decelerates even its speed is slower than its lead vehicle’s speed (i.e.

shying-away) [Chakroborty & Kikuchi, 1999]. Figure 2-2 shows that in about 20% of the points in the second and fourth quadrants of the car-following process, they are closing-in or shying-away. Since closing-in and shying-away often occur in a car-following process, a car-following model should be able to describe closing-in and shying-away.

-5 0 5 10 -6-10

-4 -2 0 2 4

Relative speed (ft/s)

Accel./Decel. rates of following vehicle (ft/s )2

Figure 2-2 Acceleration/Deceleration rate of the following vehicle at time t versus the relative speed of the lead and following vehicles at time t-1 [Chakroborty & Kikuchi,

1999] (reproduced) B. Hysteresis

Static macroscopic traffic flow models describe the relationship between speed and density. These relationship models frequently serve as a state equation in dynamic macroscopic traffic flow models. However, speed-density relationships of equilibrium state and disequilibrium state are different. In fact, when traffic flow is not in an equilibrium state (i.e.

acceleration or deceleration), the speed-density relationship is not one-to-one.

The acceleration curve differs from the deceleration curve, known as the traffic hysteresis phenomenon. Igarashi [Igarashi et al, 2001] said that hysteresis phenomena associated with discontinuous phase transitions. Figure 2-4 is the speed-density relationship obtained by Treiterer and Meyers [Zhang, 1999a]. It indicates that speed-density curve for transient traffic is not unique [Zhang, 1999a]. Such curves contain two branches for acceleration and deceleration traffic, respectively. The curves form two hysteresis loops. The acceleration curve lies above the deceleration curve under low density whereas the deceleration curve lies above the acceleration curve under high density. Maes’ observation is shown as Figure 2-3, and he observed that the deceleration curve lies above the acceleration curve under light and heavy density.

Zhang [1999a, 2001] proposed a mathematical theory of traffic hysteresis, and the model presented that acceleration and deceleration curves lies on both sides of the equilibrium curve. These two branches meet with the equilibrium curve.

Other researchers also proposed models to reproduce hysteresis.

Heidemann [2001] proposed a queueing theory, Daganzo [2002] proposed a macroscopic behavioral theory, and Zhao and Gao [2005] presented a full velocity and acceleration difference model. Wong and Wong [2002] indicated that multi-class LWR model can reproduce hysteresis.

Density Velocity

Figure 2-3 Trajectory for density vs. velocity obtained by Maes (reproduced)

Density Velocity

Figure 2-4 Trajectory for density vs. velocity obtained by Treiterer and Meyers (reproduced)

2.2 Microscopic Traffic Flow Models

According to the level of detail, traffic flow models can be divided into microscopic, mesoscopic, and macroscopic models [Hoogendoorn & Bovey, 2001].

Microscopic traffic flow includes car-following and lane-changing. This study focuses on car-following. Various car-following models are reviewed and discussed below.

Pipes [1953] proposed a safe-distance model, and applied a very simple rule. The PITT [Wicks & Andrews, 1980] model is also a type of safe-distance model. This model assumes that the vehicle follows its leader by maintaining some spacing. It employed a sensitivity factor to describe different driver behaviors. The stimulus-response model [Chandler, 1958; Gazis, 1959; Herman, 1959; Edie, 1961]

expresses the concept that a driver of a vehicle responds to a given stimulus based on the stimulus and its sensitivity. The psycho-physical spacing model [Widemann, 1974;

Leutzbach, 1986, 1988] divides the car-following process into several behavior zones, each with its own behavioral rules. Benekohal proposed the CARSIM model [Benekohal & Treiterer, 1988], which computes various acceleration rates for different situations and chooses the most suitable one. Fuzzy models [Kikuchi &

Chakroborty, 1992; McDonald et al, 1997] comprise a set of fuzzy inference rules related to specific driving environments. The intelligent driver model [Treiber et al, 2000] possesses only a few intuitive parameters with realistic values; the model reproduces a realistic collective dynamics, and leads to the plausible microscopic acceleration and deceleration behavior of single drivers. Newell [2002] designed a very simple car-following rule for a homogeneous highway in which a vehicle follows the same trajectory as its lead vehicle except for a translation in space and time.

However, it did not deal with the question of what determines speed. Zhang & Kim [2005] developed a theory for explaining car-following behaviors in multiphase traffic flow. It specifies different functional forms of gap-time for different spacings, and it can reproduce both the so-called capacity drop and traffic hysteresis.

Some microscopic traffic flow models are reviewed in detail as following.

(1). Stimulus-Response Car-Following Model

Stimulus-response models were first derived from Reuschel [1950] and Pipes [1953]. Chandler et al [1958] derived the stimulus-response function:

(

t T

) (

x

( )

t x

( )

t

)

x_n′′₊₁ + =λ _n′₊₁ − _n′ (2.1)

where λ is a constant, T denotes the reaction time, x_n

( )

t denotes the position of the lead vehicle at time t, and ^xn 1+

( )

^t denotes the position of the following vehicle at time t.

A series of stimulus-response models were developed later, these models can be summarized as Table 2.1

Table 2.1 Governing equations of car-following models

Model Governing equation

Chandler et al. [1958] x_n′′

(

t+T

)

=λ

(

x_n′₊₁

( )

t −x_n′

( )

t

)

California Chandler et al. [1958] ^xn′′

(

^t+^T

)

=λ

[

^xn₊₁

( )

^t −^xn

( )

^t +^c−^T₁^xn′

( )

^t

]

Gazis et al. [1959]

( ) ( ( ) ( ) )

( ) ( ) (

x t x t

)

t x t T x

t x

n n

n n

n −

′

−

= ′

′′ +

+ + 1

λ 1

Herman et al. [1959] xn′′

(

t+T

)

=λ₁

[

xn′₊₁−xn′

]

t₋T₁+λ₂

[

xn′₊₂ −xn′

]

t₋T₂

Edie [1961]

( ) ( ( ) ( ) )

( ) ( ) (

x ₁ t¹ x t

)

²

t x t x T x

t x

n n

n n

n

n −

− ′

′

= ′

′′ +

+

λ +

Newell [1961] x_n′′

(

t+T

)

=V

(

x_n+1

( )

t −x_n

( )

t

)

Gazis et al. [1961]

(General Form)

( ) ( ( ) ) ( ( ) ( ) )

( ) ( )

(

n n

)

^m

n n

l n

n x t x t

t x t x t T x

t

x −

− ′

′

= ′

′′ +

+ +

1

λ 1

Bando et al. [1995] ^xn′′

(

^t+^T

)

=^a

[

^V

(

^xn₊₁

( )

^t −^xn

( )

^t

)

−^xn′

( )

^t

]

Some macroscopic traffic flow models can be derived from stimulus-response models. Gazis’s model [1961] is a general form of stimulus-response models. The case m=0, l =2 can be identified with a model developed by Greenshield [1934].

The case for m=0 and l =1 generates a steady-state relation that was developed by Greenberg [1959]. When m=1 and l =2, the stimulus-response model can be lead to Edie’s model. While m=1 and l=3, the model can be lead to Edie’s model.

The deficiencies of stimulus-response are described as following [Chakroborty, 1999].

1. Response to stimuli in car-following is asymmetric, but stimulus-response model is symmetric.

2. It cannot represent closing-in and shying-away phenomena.

3. Stable distance headway of stimulus-response model is dependent on number of factors, such as initial conditions, but the stable distance is actually only dependent

on the final speed.

4. It ignores the acceleration capability of a vehicle.

(2). PITT Model

PITT is a FRESIM model in CORSIM which was developed by FHWA. Its theory is keeping specific space headway [Wicks & Andrews, 1980]:

(

L F

)

F

L k V b k V V

L

H = + × +10+ × × − (2.2)

In Eq. (2.2):

H : space headway(ft)

V_F : the velocity of the following car at end of time step VL : the velocity of the lead car at the end of time step L_L : the length of the lead car

k : the sensitivity of a driver

b : constant, when V_L=V_F ≤ 10, b=0.1,otherwise, b=0

For keeping above-mentioned space headway, the acceleration of the follower is:

( ) ( )

[ ]

(

^{TV kk TT}

)

^{b k V V}

L X A X

i F L i

F i

F L

F + × ×

−

×

×

−

×

×

−

−

−

= −

2 10 2

2

2

(2.3)

In Eq. (2.3):

AF : the acceleration of the following car XL : the position of the lead car

i

X : the position of the follower at the beginning of time step F i

V ：velocity of the follower at the beginning of time step F

T ：time step

Considering the reaction time of following car c, the velocity of follower should be ^{V V A}F

(

^{T c}

)

i F

F = + × − . To avoid traffic accident, PITT designs three constraint functions for different traffic conditions.

The deficiencies of PITT are described as following [Benekohal, 1988] [Aycin, 1999].

1. It dose not take into account the star-up delay of stopped vehicles.

2. The dual behavior of traffic in congested and non-congested conditions has not been taken into consideration.

3. It is difficult to reflect all traffic condition since only one type of spacing equation is applied for different conditions such as stop-and-go and noncongested traffic.

4. It performs car-following by considering emergency braking of the lead vehicle, but a follower cannot have the information about the deceleration capability of its leader.

5. It considers driver’s reaction time, and it results in a driver with higher reaction time keeps longer spacing.

(3). Psycho-Physical Spacing Models

Stimulus-response models presume that the following driver reacts to very small changes in relative velocity even when the spacing is very large. On the other hand, if the relative velocity is zero, the follower’s response is zero even the spacing is very small or large. Researchers developed psycho-physical spacing models to remedy these unreasonable assumptions. The basis of psycho-physical spacing models is:

1. at large spacings, the follower is not influenced by the relative velocity, and 2. at small spacings, there are some combinations of relative speeds and spacings do not yield a response of the follower, because the relative motion is too small.

This implies that there is a perceptual threshold. Only when thresholds are reached, the following driver can perceive the changes in the relative speed or spacing.

Such perceptual thresholds are shown as Figure 2-5.

Widemann [1974] introduces the Psycho-Physical Spacing Model into microscopic simulator and design the INTAC Model to be Behavioral Threshold Model. Traffic flow is classified into several reaction zones (as shown in Figure 2-6).

The meaning of each threshold is [Fellendorf, 1997]:

A. Standstill spacing (AX): the desired distance between two cars in a standing queue.

B. Minimum safe spacing (BX): the minimum safe spacing when the velocity of follower is close to its lead vehicle.

C. Perceptual velocity difference threshold (SDV): action point where a driver consciously observes that he approaches a slower lead vehicle. SDV increases with speed difference.

D. Maximum car-following spacing (SDX): concerning the difference among different drivers, the range of SDX is about 1.5-2.5 times BX.

E. OPDV: action point where a driver notices that he is slower than the lead vehicle and starts to accelerate again.

Spacing dX

+dV -dV

Perceptual threshold

Zone with reaction

Zone with reaction

Relative Velocity Zone

without reaction Perceptual

threshold

Figure 2-5 Perceptual thresholds of car-following process DX

SDX SDV

CLDV BX

AX OPDV

-DV +DV

MAXDX

Spacing

Relative Velocity decreasing spacing

increasing spacing

unconscious reaction

no reaction perception

threshold

reaction

deceleration collision

Figure 2-6 Behavioral zones of behavioral threshold model

(4). CARSIM

CARSIM (CAR-Following Simulation Model) is developed by Benekohal [1988]. Several acceleration or deceleration rates are computed for different conditions, and the most suitable one is selected for each vehicle in each time interval.

Each condition is described as following.

A1: The following vehicle is moving but has not reached its speed limit or desired speed.

A2: The following vehicle has reached its speed limit or desired speed.

A3: The follower was stopped and has to start from a standing still position A4: The follower’s performance is governed by the car-following algorithm while space headway constraint is satisfied. The acceleration is computed from the following equation

( ) ( )( )

(

^{X V DT A DT}

)

^{L K}

X_L− _F + _F +0.5 4 ² ≥ _L+ , (2.4)

where

X_L : the position of the lead vehicle XF : the position of the following vehicle L_L : the length of the lead vehicle

K : the buffer space between vehicles

DT : the simulation scanning time interval (1 second) A4 : the acceleration or deceleration in the condition

A5: The following vehicle is advanced according to the car-following algorithm with non-collision constraint. The following equation is used to assure that enough spacing is provided.

( ) ( )( )

( )

( )( )

[ ] ( )

( )( )

[ ] ( ) [ ( )( ) ]

( ) ( )

^⎥_⎦^⎤

⎢⎣

⎡ + −

+ +

+

≥

−

− +

+

−

L MX

X F

MX DT A BRT V

DT A V

BRT DT A V

K L DT A DT

V X X

L F

F F

L F

F L

. 2 .

2 5 5

, 5

of maximum

5 5 . 0

2 2

2

(2.5)

where:

BRT : brake-reaction time of a driver

VL : velocity of the lead car at the end of time interval MX.F : maximum deceleration rate of the following car MX.L : maximum deceleration rate of the lead car A5 : the acceleration and deceleration in the condition

The strengths of CARSIM are described as following [Benekohal, 1988].

1. The vehicles’ acceleration and deceleration rates were kept with the reasonable values observed in actual traffic conditions, and marginally safe spacings were provided for all vehicles.

2. The delay in response of the follower to the lead vehicle’s deceleration was taken into account. The delay is equal to the reaction time of the driver.

3. The start-up delay of a stopped vehicle was taken into consideration.

4. The dual behavior of traffic in congested and non-congested conditions is taken into account.

5. CARSIM uses varying reaction times for an individual driver and different reaction times for different drivers. The reaction time of a driver in congested traffic is less than the reaction time of light traffic.

6. CARSIM can simulate stop-and-go condition.

The deficiencies of CARSIM are described as following [Aycin, 1999].

1. CARSIM performs car-following by considering emergency braking of the lead vehicle, but a follower cannot have the information about the deceleration capability of its leader.

2. CARSIM considers driver’s reaction time, and it results in a driver with higher reaction time keeps longer spacing.

(5). Fuzzy Models

Kakuchi [Kakuchi, 1992, Chakroborty, 1999] proposed a fuzzy inference car-following model. It consists of 396 rules which are based on the relative speed, the spacing, and the acceleration of the lead vehicle. After defuzzifying, the model outputs the acceleration of the following vehicle.

McDonald [1997] proposed another fuzzy model, and it includes car-following and lane-changing. The model takes into account desired car-following spacing, and the inference rules are based on distance divergence and relative velocity.

Most of traditional car-following models are deterministic, but drivers do not completely follow any deterministic behavior. Fuzzy modelsrepresent an approximate nature behavior. They represent the natural language based “rules-of-thumb” of driving which is believed to be reasonable and uses the compromise of more than one rule of behavior.

The strengths of fuzzy models are described as following.

1. Car-following is an approximate nature behavior, and fuzzy models represent the property.

2. Fuzzy models can describe closing-in and shying-away phenomena.

3. The response of fuzzy models is asymmetric.

4. In fact, stable distance headway is only dependent on final speed, and fuzzy models represent it.

2.3 Static Macroscopic Traffic Flow Models

Macroscopic traffic flow models discuss flow, density, and speed. The relationship between these variables is q=ku, where q denotes flow, k denotes density, and u denotes velocity.

Some researchers discussed the relationship between density and velocity based on filed data or some theory. Greenshield [1934], as one of the early investigators of traffic characteristics, proposed a linear relationship. Greenberg [1959], using a theoretical background, has postulated a logarithmic speed-density model.

Greenberg’s model is useful under high density but not under low density. Underwood [1961] proposed a speed-density model for low density traffic. Later, Pipes [1967] and Munjal [1971] developed a general family of speed-density models of which the linear model is a special case. Drew [1968] proposed a family of models of which Greenberg’s logarithmic model is a special case.

As some aforementioned models are only useful under some traffic condition, some researchers proposed multi-regime models. Edie [1961] described a model that

is a composite of Greenberg and Underwood models, where Greenberg is useful at high density and Underwood is useful at low density. Other research results of the static macroscopic model are summarized in Table 2.2 and Table 2.3 May [1990].

Table 2.2 Table of single-regime models

Single-regime models Equations

Greenshields model (1934) u u k

f k

Greenberg model (1959) u u k

k

Northwestern's model (1967) _{( )}

u =u e_f ^{−1 2k ko 2} Drew model (1968)

( )

Pipes-Munjal model (1967)

⎥⎥ Table 2.3 Table of multi-regime models

Multiregime models Free-flow regime Transitional-flow

regime Congested-flow regime

Edie model

(1961) u =u e_f ^{−k k0}

Two-regime linear model

(1967) u u k

Modified Greenberg

model (1967) constant speed

(

^{k ≤k}2

)

－ u u k

Three-regime linear model

(1967) u u k

ki：specified traffic density ，i=1.2.3.4

Different free-flow speeds result in different speed-flow relationships. Figure 2-7

Different free-flow speeds result in different speed-flow relationships. Figure 2-7

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