Summary and Discussion

According to the above-mentioned review, some traffic phenomena are summarized as following.

(1) Equilibrium spacing: If a following vehicle reaches and keeps at a particular spacing, the particular spacing is only dependent on final speed. Hence, a driver with higher reaction time has longer equilibrium spacing is not very reasonable, since the equilibrium spacing is not dependent on driver’s reaction time.

(2) Traffic stability: From the viewpoint of microscopic traffic flow, higher reaction time makes unstable traffic likely to occur.

(3) Closing-in and shying-away: Relative speed cannot ensure the acceleration is positive or negative.

(4) Traffic hysteresis: Speed-density relationships for acceleration and deceleration traffic are different.

(5) Driver characteristics: Different drivers have different behaviors. Some drivers are aggressive, some are not. Drivers may keep different velocities or different spacings under the same conditions.

(6) Stable traffic versus unstable traffic: unstable traffic occurs at high density.

Some strengths or deficiencies of car-following models are summarized as following.

(1) Relative speed form: If driver’s acceleration is only dependent on relative speed, the model cannot represent closing-in and shying-away phenomena, and cannot describe asymmetric response.

(2) Considering enough spacing: If a driver takes enough spacing into account, he must consider emergency braking of the lead vehicle, but he cannot have the information about the deceleration capability of his leader. Furthermore, he should consider his reaction time, and it results in a driver with higher reaction time keeps longer spacing.

(3) Driver characteristics: Some traditional car-following models cannot reflect the difference between drivers. Some models employ sensitivity or aggressiveness factor to describe the diver difference. However, these factors cannot be measured directly.

(4) Simple models versus complex models: simple models that have one or few functions, such as safe-distance models and stimulus-response model, cannot describe some traffic phenomena. On the other hand, simple models can extend to macroscopic traffic flow more easily. For example, stimulus-response model can extend to macroscopic models, such as Greenshield’s model, Greenberg’s model, and Edie’s model [May, 1990]. The models that have different rules for different conditions describe the traffic flow better, but their computations are more complicated. It is difficult to develop a macroscopic traffic flow model based on these models. It is also difficult to derive traffic properties from complex models.

Static macroscopic traffic flow models frequently serve as a state equation in dynamic macroscopic traffic flow models, and they are regarded as the equilibrium state of traffic flow. According to aforementioned, static traffic flow models may be developed based on disequilibrium field data, i.e., include acceleration and deceleration traffic. For example, Greenberg proposed his model for high density traffic, but heavy traffic could hardly reach its equilibrium state.

Chapter 3

Car-Following Model

A simple car-following model is developed in this section, and the model should achieve the following objectives. First, the model should describe microscopic car-following phenomena, such as closing-in, shying-away, and traffic hysteresis.

Second, the model should reflect differences among individual drivers. Third, it should avoid certain deficiencies mentioned in Chapter 2, such as drivers having to determine the deceleration capability of their lead vehicle. Finally, the model should minimize the number of rules employed to facilitate its extension to macroscopic traffic flow models.

3.1 Model Assumption

The car-following process is influenced by driver characteristics, external environment, and lead vehicle. If there is no lead vehicle, a vehicle will run at a specific speed (its individual maximum speed) influenced only by driver characteristics and external environment. Hence, an individual maximum speed of a vehicle is influenced by driver characteristics and external environment. In other words, the influence of driver characteristics and external environment on a vehicle is presented in the individual maximum speed of the vehicle. Driving alone, different drivers may run at different speeds on the same road, implying that different drivers (i.e. different driver characteristics) have different individual maximum speeds.

Driver individual maximum speed may vary with external environment, such as freeway, urban street, and sunny versus rainy days. As driver characteristics and external environment are difficult to measure, the proposed model considers individual maximum speed to help reflecting the influence of driver characteristics and external environment. The individual maximum speed of a vehicle can be measured under certain situations. Where no lead vehicle is present, the vehicle speed is the individual maximum speed. Otherwise, if the speed of the following vehicle does not change with lead vehicle speed or spacing, its speed is considered to be its individual maximum speed.

If there is a lead vehicle, and as the spacing decreases, the following vehicle may slow down so that it cannot run at its individual maximum speed. According to the literature, following vehicle speed depends on the speed of the lead vehicle, the speed

of itself, and the spacing between vehicles. Hence, the variables of the proposed model are individual maximum speed, the speed of the lead vehicle, the speed of itself, and the spacing between vehicles.

To model the aforementioned phenomena, the proposed model assumes that repulsion and thrust act on the following vehicle, which then sets an appropriate speed accordingly. Figure 3-1 presents the proposed model. The model assumptions are listed below:

Lead Vehic le Following

Vehi cle

Repulsion Thrust

( Individ ual Maximum Speed)

Figure 3-1 Illustration of the car-following concept (1) Aggressiveness

The proposed model assumes that driver aggression increases with individual maximum speed. Drivers with high individual maximum speed maintain a higher speed or shorter spacing than do drivers with low individual maximum speed under identical conditions, and also have faster acceleration or deceleration.

(2) Velocity Decision

The following vehicle decides its appropriate velocity based on existing thrust and repulsion, with the appropriate velocity equaling thrust minus repulsion.

(3) Thrust

Each vehicle has its own individual maximum speed, which is regarded as the thrust. The individual maximum speed thus becomes the force driving the following vehicle forward. If there is no lead vehicle, the vehicle will run at its individual maximum speed ν_n,d. Individual maximum speed depends on external environment and driver characteristics, which are not determined by car-following process. Individual maximum speed thus is an exogenous variable.

(4) Repulsion

Because the lead vehicle can prevent the following vehicle from running at its individual maximum speed ν_n,d, the lead vehicle is considered to be repelling the follower. Since the following vehicle speed is influenced by the lead vehicle speed

t

Vn_−1, , the follower speed V_n,t, and the spacing H_n,t, the repulsion is related to these factors.

(a) Spacing H_n,t

(i) Given longer spacing H_n,t, the repulsion should be reduced because a driver will maintain higher velocity V_n,t+1 under no changes in the lead vehicle speed V_n−1,t and the following vehicle speed V_n,t.

(ii) The following vehicle speed V_n,t+1 varies with changes in thespacing H_n,t. The variation of H_n,t in V_n,t+1 is regarded as the sensitivity to H_n,t, and

n

ψH , denotes the sensitivity. At a large spacing H_n,t, the following vehicle is not influenced by the lead vehicle, and thus V_n,t+1 is not sensitive to the changes in H_n,t, i.e. the sensitivity ψ_{H ,n} is zero. On the other hand, when the spacing H_n,t is shorter, a driver is more sensitive to the changes in spacing. Hence, the sensitivity ψ_{H ,n} increases with reducing spacing.

(iii) Continued from the preceding assumption (ii), when spacing is very short, a following driver may be very sensitive to or not sensitive to the changes in spacing. Because a driver may perceive that the spacing is too short, running at a very low velocity V_n,t+1 is his unique choice even though the spacing becomes slightly longer. Hence, the sensitivity ψ_{H ,n} may be very large or very small at short H_n,t.

(iv) If the spacing is in some specific car-following distance, the following vehicle is influenced by its leader. Otherwise, if the spacing is out of some specific car-following distance, the following vehicle is not influenced by its leader. The specific car-following distance is defined as critical car-following distance. At identical following vehicle speed V_n,t , the critical car-following distance increases with reducing lead vehicle speed

t

Vn_−1, .

(v) At identical lead vehicle speed V_n−1,t, the critical car-following distance increases with the following vehicle speed.

(vi) At identical following vehicle speed, lower lead vehicle speed V_n−1,t makes drivers be more sensitive to the movement of the lead vehicle. Thus,

drivers are more sensitive to the changes in spacing, i.e. the sensitivity

n

ψH , increases as the lead vehicle speed V_n−1,t decreases.

(vii) Continued from the preceding assumption (vi), if the spacing is very short, running at a very low velocity V_n,t+1 is the only choice for the driver whose lead vehicle speed V_n−1,t is low. Otherwise, if the lead vehicle speed is faster, a driver has more flexibility in choosing his vehicle speed V_n,t+1 after a reaction time. Faster lead vehicle speed V_n−1,t makes drivers have more flexibility in choosing their speed V_n,t+1 at short spacing, so drivers are more sensitive to the movement of their lead vehicles. Hence, the sensitivity ψ_{H ,n} increases with the lead vehicle speed V_n−1,t at short spacing.

(viii) At identical lead vehicle speed V_n−1,t, a driver with higher vehicle speed

t

Vn_, is more sensitive to the movement of the lead vehicle, and thus he is more sensitive to the changes in spacing, i.e. the sensitivity ψ_{H ,n} increases with the following vehicle speed.

(ix) Continued from the preceding assumption (viii), if the spacing is very short, running at a very low velocity V_n,t+1 is the only choice for the driver whose previous vehicle speed V_n,t is fast. Otherwise, if his previous vehicle speed

t

Vn_, is slower, a driver has more flexibility in choosing his vehicle speed

1 ,t+

Vn . Higher vehicle speed V_n,t makes drivers have less flexibility in choosing their speed V_n,t+1 at short spacing. Hence, the sensitivity ψ_{H ,n} increases with reducing V_n,t at short spacing.

(b) Lead vehicle speed V_n−1,t

(i) Under identical traffic conditions, drivers maintain higher velocity V_n,t+1 at higher lead vehicle speed V_n−1,t, and the repulsion should be reduced.

(ii) The following vehicle speed V_n,t+1 varies with changes in the lead vehicle speed V_n−1,t. The variation of V_n−1,t in V_n,t+1 is regarded as the sensitivity

to V_n−1,t and ψ_V,n−1 denotes the sensitivity. If V_n−1,t approaches infinity, a driver may not perceive the obstacle created by the lead vehicle. Hence,

1 ,t+

Vn is not sensitive to the changes in V_n−1,t, i.e. the sensitivity ψ_V,n−1 is zero. On the other hand, when V_n−1,t is small, a driver is very sensitive to the changes in V_n−1,t . Hence, the sensitivity ψ_V,n−1decreases as V_n−1,t increases.

(iii) Continued from the preceding assumption (ii), when V_n−1,t is very small, following driver may be very sensitive to the changes in V_n−1,t or not sensitive. Because a driver may perceive that the lead vehicle is too slow, running at a very low velocity is his unique choice even though the lead vehicle runs slightly faster. Hence, the sensitivity ψ_V,n−1 may be very large or very small at low V_n−1,t.

(iv) At identical following vehicle speed V_n,t, lower spacing makes drivers pay more attention to the movement of their lead vehicles, and thus be more sensitive to the changes in lead vehicle speeds V_n−1,t, i.e. the sensitivity

1 ,n−

ψV increases with reducing spacing.

(v) Continued from the preceding assumption (iv), if the lead vehicle speed

t

Vn_−1, is very low, running at a very low velocity V_n,t+1 is the only choice for the driver whose spacing is short. Otherwise, if the spacing is longer, a driver has more flexibility in choosing his vehicle speed V_n,t+1 after a reaction time. Longer spacing makes drivers have more flexibility in choosing their speed V_n,t+1 at low lead vehicle speed V_n−1,t, so drivers are more sensitive to the movement of their lead vehicles. Hence, the sensitivity

1 ,n−

ψV increases with spacing at low lead vehicle speed V_n−1,t.

(vi) At identical spacing, a driver with higher velocity V_n,t pay more attention to the movement of his lead vehicle, and thus he is more sensitive to the changes in the lead vehicle speed V_n−1,t , i.e. the sensitivity ψ_V,n−1 increases with the following vehicle speed V_n,t.

(vii) Continued from the preceding assumption (vi), if the lead vehicle runs too slow, running at a very low velocity V_n,t+1 is the only choice for the driver whose previous vehicle speed V_n,t is high. Otherwise, if his previous vehicle speed V_n,t is slower, a driver has more flexibility in choosing his vehicle speed V_n,t+1. Higher vehicle speed V_n,t makes drivers have less flexibility in choosing their speed V_n,t+1 at short spacing. Hence, the sensitivity ψ_V,n−1 increases with reducing V_n,t at low lead vehicle speed

t

Vn_−1, .

(c) Following vehicle speed V_n,t

(i) A driver may slow down if his speed V_n,t is too fast, and may speed up if his speed V_n,t is too slow. Hence, the repulsion increases with the follower speed V_n,t.

(ii) The following vehicle speed V_n,t+1 varies with changes in the lead vehicle speed V_n,t. The variation of V_n,t in V_n,t+1 is regarded as the sensitivity to

t

Vn_, , and ψ_{V ,n} denotes the sensitivity. When V_n,t is very large or very small, a driver may perceive that his speed is too fast or too slow. Thus, running at a low or high speed is his unique choice, and V_n,t+1 is not very sensitive to the changes in V_n,t.

(iii) When the lead vehicle and the following vehicle speeds are identical, the critical car-following distance increases with vehicles speed V_n,t or different speeds result in identical critical car-following distance.

(iv) When the lead vehicle and the following vehicle speeds are identical, a driver with higher vehicle speed perceives the repulsion more.

As an aggressive driver may perceive the obstacle created by the lead vehicle as being of greater significance, it is also assumed that a driver with a higher individual maximum speed will perceive higher repulsion under the same traffic conditions.

(5) Safety

Since some drivers exhibit unsafe behaviors, the proposed model assumes that moving vehicles do not consider safe distance. Drivers only consider the standstill spacing.

3.2 Modeling

When both the lead vehicle and the following vehicle are moving, the following vehicle decides its appropriate velocity based on existing thrust and repulsion, with the appropriate velocity equaling thrust minus repulsion. The repulsion is related to the speed of the lead vehicle, the speed of the follower, and the spacing. Hence, repulsion is a function of V_n−1,t, V_n,t, H_n,t and the vehicle speed can be represented as

(

n t nt nt

)

d n t

n RV V H

V~_{, 1 , 1,}, _,, _,

−

+ =ν − , (3.1)

where R

(

Vn_−1,t,Vn_,t,Hn_,t

)

is the repulsion. A driver with a higher individual maximum speed perceives higher repulsion under identical traffic conditions. Hence the repulsion is expressed as

(

Vn t Vnt Hnt

)

ndr

(

Vn t Vnt Hnt

)

R _−1,, _,, _, =ν _{, −1,}, _,, _, . (3.2)

Therefore, the vehicle speed is

( )

(

n t nt nt

)

d n t

n rV V H

V~_{, 1 ,} 1 _1,, _,, _,

−

+ =ν − , (3.3)

and the range of r

(

Vn_−1,t,Vn_,t,Hn_,t

)

is shown as Eq. (3.4):

(

^{, ,}

)

¹

0<rV_n−1,t V_n,t H_n,t < . (3.4) Let

(

Vn t Vnt Hnt

)

I

(

P

(

Vn t Vnt Hnt

) )

r _−1,, _,, _, = _−1,, _,, _, . (3.5)

The repulsion increases with reducing V_n−1,t or H_n,t, and it also increases with

t

Vn_, . To describe closing-in and shying-away phenomena, the relative speed form is not selected because it cannot decide whether the acceleration of the following vehicle is positive or negative. Therefore, r

(

Vn_−1,t,Vn_,t,Hn_,t

)

is represented as Eq. (3.6):

( ) ( )

( )

γ β

α

⎟⎠

⎜ ⎞

⎝

⎛ −

= ⁻

− L

S H V

H V V V

P ^{nt n}

t n

t n t n t n t n

, ,

, 1 ,

, ,

1 , , , (3.6)

where α,β,γ,L are positive parameters. Since drivers take standstill distance S _n into account, H_n,t −S_n is the gap that a driver perceives. As speed and spacing have different units, the model employs the parameter L to standardize.

Since 0<V_n−1,t ≤ν_n−1,d , 0<V_n,t ≤ν_n−1,d , and S_n ≤H_n,t , the range of

More detail discussions about Eq. (3.11) and assumption (4) are discussed in section 3.3.

If both the lead and following vehicles are running, the follower will choose an appropriate speed, which equals thrust minus repulsion (as shown in Eq.(3.11)).

Sometimes the same condition results in different speeds for different drivers, the difference is indicated by Eq. (3.11).

If the speed of the lead vehicle is zero, the following vehicle decelerates its speed so that it can stop before a collision occurs. The distance that the following vehicle can move before collision is H_n,t −S_n. Hence,

( )

Vnt + ant

(

Hnt −Sn

)

= _, ² 2 _{, ,}

0 , (3.12)

where a_n,t is the acceleration. Thus, the acceleration a_n,t is

Hence, the following vehicle speed at the next time step (i.e., after a reaction time T ) is

Under identical condition, the following vehicle speed should increase with its lead vehicle speed. As Eq. (3.14) implicitly assumes V_{n− t1,} =0, the vehicle speed of Eq. (3.11) should be greater than the one of Eq. (3.14), i.e.

( )

approaches zero, the solution of Eq. (3.11) may approach zero. But the solution of Eq.

(3.14) may not approach zero. Let inequality (3.15) hold, and then

( )

_ν ^α greater than the RHS of (3.16), a following vehicle with a moving leading car will choose a higher speed than it with a stopped leading car under identical spacing and identical following vehicle speed. Hence, the premise of Eq. (3.11) is

threshold following vehicle may regard its leader as a stopped vehicle. Thus the following vehicle starts to slow down and Eq. (3.14) is employed.

threshold

Vn_−1, depends on the spacing H_n,t, the follower speed V_n,t, and the individual maximum speed ν_n,d . Under identical spacing and follower speed, different drivers have different V_{n−1,threshold}, and V_{n−1,threshold} varies with ν_n,d, that is Eq. (3.18) indicates that a driver with higher individual maximum speed has lower V_{n−1,threshold}. It implies that conservative drivers regard a fast lead vehicle as a stopped vehicle under identical spacing and follower speed, and start to slow down.

While an aggressive driver only regards a very slow lead vehicle as a stopped vehicle.

This conforms to the model assumption (1) that aggressive drivers keep higher velocity under identical traffic condition.

If the lead vehicle is moving and the following vehicle is stopped, the follower will not start to move immediately. The follower usually remains stopped, and only moves once the spacing is greater than a specific spacing Z (i.e. the start spacing). _n The follower then moves at the next time step, with its acceleration equaling its desired start acceleration (as shown in Eq. (3.21)). Finally, if the following vehicle stops and the spacing is less than the start spacing Z , the follower remains stopped _n at the next time step (as shown in Eq. (3.22)).

Aside from the repulsion and thrust, the speed of the following vehicle also depends on its capability. Vehicle acceleration should be between the maximum and minimum acceleration of that vehicle. Therefore, the proposed model should be modified as shown in Eq. (3.23), where a_n,max denotes the maximum acceleration of the follower, a_n,min represents the minimum acceleration (i.e. maximum deceleration) of the follower, and T is the length of the time interval.

min

3.3 Sensitivity Analysis

This section discusses how the proposed model output varies with changes in model inputs. The total increment of V^~_n,t+1 is denotes the sensitivity. Thus

( )

Eq. (3.25) indicates that ψ_H,n ≥0. Since driver maintains higher velocity V_n,t+1 with higher spacing, ψ_{H ,n} is greater than zero. But a driver has his maximum speed

d

νn, and V_n,t+1 ≤ν_n,d , a driver cannot always increase his speed with spacing. Thus,

,_n =0

ψH occurs at large spacing. On the other hand, when the spacing approaches infinity, the following vehicle is not influenced by the lead vehicle. Hence, V_n,t+1 is not sensitive to the changes in H_n,t, and then ψ_H,n =0. When spacing approaches values. Eq. (3.26) conforms to assumption (4.a.ii).

Shorter spacing makes drivers pay more attention to the movement of their lead vehicles, and thus the sensitivity ψ_{H ,n} varies with spacing. The variation of H_n,t in

n decides whether

t Both (a) and (b) diagram of Figs 3-2 and 3-3 indicate that when the spacing is not

very short, a driver is more sensitive to the changes in spacing. Hence, the sensitivity

n

ψH , increases with reducing spacing.

(a)γ ≤1

Spacing (Hn,t)

Speed (Vn,t+1)

(b)γ >1

Spacing (Hn,t)

Speed (Vn,t+1)

Figure 3-2 Examples of the relationship between V_n,t+1 and H_n,t under no changes in V_n,t and V_n−1,t

≤1 γ

Spacing (Hn,t)

>1 γ

Spacing (Hn,t)

Figure 3-3 Examples of the relationship between ψ_{H ,n} and H_n,t under no changes in V_n,t and V_n−1,t

Parameter γ of (a) and (b) diagram in Figs 3-2 and 3-3 are different. They reflect the model assumption (4.a.iii). When spacing is very short, a driver may perceive that the spacing is too short, running at a very low velocity V_n,t+1 is his unique choice even though the spacing becomes slightly longer. Hence, the sensitivity

n

ψH , may be very large or very small at short H_n,t

At identical following vehicle speed V_n,t, drivers pay different attention to the lead vehicles with different speeds V_n−1,t. The sensitivity ψ_{H ,n} varies with lead vehicle speed V_n−1,t:

( )

at long spacing. This conforms to the model assumption (4.a.vi) and (4.a.vii). Drivers pay closer attention to the lead vehicle with lower speed V_n−1,t than to the lead vehicle with higher speed V_n−1,t at long spacing. Thus, lower lead vehicle speed

t

Vn_−1, makes drivers be more sensitive to the movement of the lead vehicle. On the

Vn_−1, makes drivers be more sensitive to the movement of the lead vehicle. On the

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