- Email: [email protected]
Stability analysis of the mixed traffic flow of cars and trucks using heterogeneous optimal velocity car-following model
Physica A 395 (2014) 371–383
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Stability analysis of the mixed traffic flow of cars and trucks using heterogeneous optimal velocity car-following model Da Yang a,b,∗ , Peter (Jing) Jin c , Yun Pu a , Bin Ran b a
School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China
b
Department of Civil and Environmental Engineering, University of Wisconsin-Madison, Madison, 53706, USA
c
Department of Civil and Environmental Engineering, University of Texas-Austin, Austin, 78701, USA
highlights • • • • •
Stability of the mixture of cars and trucks on traffic flow is studied. A new method is used to derive the stability criterion of the heterogeneous traffic flow. Percentage and stability function of different vehicles are main factors influencing stability. Both car and truck can stabilize and destabilize the mixed traffic flow. Slow trucks can trigger traffic capacity drop.
article
info
Article history: Received 23 May 2012 Received in revised form 22 September 2013 Available online 12 October 2013 Keywords: Traffic flow Heterogeneous car-following model Car and truck Stability
abstract Real-world traffic flow usually contains a mixture of passenger vehicles (PV) and heavy vehicles (HV). In this paper, the four types of car–truck following combinations are considered: the car-following-car case, car-following-truck case, truck-following-car case and truck-following-truck case. The effect of different combinations on the stability of traffic flow is explored by converting the original Bando’s optimal velocity (OV) model to a heterogeneous form. A new linear stability analysis method that can derive the stability criterion of the heterogeneous traffic flow mixed by cars and trucks is introduced. Moreover, the effect of the proportions of the four car–truck following combinations on traffic flow is examined through the trajectory analysis. It concludes that the linear stability of the car–truck mixed traffic flow is determined more by the proportions of the different car–truck following scenarios, rather than the numbers of the cars and trucks. Moreover, cars and trucks can both stabilize and destabilize traffic flow depending on the density of the traffic flow and the parameters of the heterogeneous OV model. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Traffic flow is a complex system with many particles (vehicles) and complex interactions among them. The study on traffic flow can help people understand the travel characteristics of human, which can further help the understanding on the other aspects of human life, such as the diffusion of virus [1–5]. Researchers from both the applied physics and traffic flow theory have been developing models to describe such a system for more than 50 years. According to the reviews of the several researchers (Helbing [6], Schadschneider [7], Chowdhury et al. [8], Kerner [9], Schadschneider et al. [10]) on traffic
∗ Corresponding author at: Department of Civil and Environmental Engineering, University of Wisconsin-Madison, Madison, 53706, USA. Tel.: +1 6083344281. E-mail address: [email protected] (D. Yang). 0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.10.017
372
D. Yang et al. / Physica A 395 (2014) 371–383
flow models, the existing models include the hydrodynamic models, gas kinetic models, Cellular Automata (CA) models, car-following models and coupled-map lattice models. In recent years, the heterogeneous phenomena in traffic flow have drawn a lot of attention among traffic flow researchers [11–17]. The existing studies mainly focus on three types of traffic heterogeneity, driver heterogeneity, vehicle heterogeneity, and traffic environment heterogeneity. Taylor et al. [11] explored driver heterogeneity within a trip using the Dynamic Time Warping method. Ossen and Hoogendoorn [12,13] focused on the impact of car-following styles on traffic flow. Davis [14] examined the effect of the percentage of ACC (Adaptive Cruise Control) vehicles on traffic flow near an on-ramp, and Islam et al. [15] analyzed the characteristics of the traffic flow mixed by motorized and non-motorized vehicles. Representative works on environment heterogeneity are Jin et al.’s formula [16,17] about visual impact of traffic environment on traffic flow using a non-lane-based car-following model. In this study, the primary focus is to explore the heterogeneity of vehicle types, more specifically, the mixture of trucks and cars based on car-following models. The traffic flow considered in this paper is the one-dimensional traffic flow. Sayer et al. [18], Huddart et al. [19] and McDonald et al. [20] compared the headway difference between the car-following-car case and car-following-truck case. Peeta et al. [21,22] introduced a discomfort level to study the influence of trucks on surrounding vehicles. Most recently, Aghabayk et al. [23] pointed out that the effect of car–truck mixture on traffic flow is not only determined by the leading vehicle type, but also influenced by the following vehicle type; based on this idea they further studied the characteristics of the four types of car–truck combination, the car-following-car (CC), car-followingtruck (CT), truck-following-car (TC) and truck-following-truck (TT). According to the paper [23], the four different combinations exhibit different driving behavioral characteristics, especially different reaction times (the reaction time order is CC < CT < TC < TT). However, their studies only presented the statistical results without further investigating the stability of the car–truck mixed traffic flow. Mason et al. [24] concluded that trucks had unstable effect on traffic flow compared to cars and may induce traffic congestion using both linear and nonlinear stability analysis methods; however, the study only considered the influence of the leading vehicle types on traffic flow. Jin et al. [25] drew the contrary conclusion that trucks can better stabilize traffic flow than cars due to their bigger sizes based on a visual angle car-following model, but the study only took into account the size difference of cars and trucks without considering the difference in car–truck combinations. In this paper, we attempt to improve the existing studies by considering the two critical characteristics in car–truck mixed traffic flow, the reaction time difference of the four car-following scenarios presented by Aghabayk et al. [23] and the desire velocity difference of cars and trucks. Compared to other one-dimensional traffic flow models, the car-following model is adopted taking advantage of its ability to describe microscopic driving characteristics of cars and trucks [6]. Meanwhile, due to its strong behavioral background and mathematical convenience, Bando’s Optimal Velocity (OV) car-following model [26] is developed to a heterogeneous form to model the car–truck heterogeneous traffic flow. What needs to be stated here is that lane-changing is not considered in this paper, and instead we just think it as an external perturbation in the stability analysis of traffic flow as suggested by Wilson [27]. The remainder of the paper is organized as the following. In Section 2, a heterogeneous OV model is proposed to describe the car–truck mixed traffic flow. Then, the stability condition of the heterogeneous OV model is derived using the new linear stability method verified numerically in Section 3. Section 4 analyzes the stability impacts of the four types of car–truck combination from two perspectives: one is the individual impact of each type of car–truck combination, and the other is the combined impact with different proportions of the CC, CT, TC and TT combinations. Section 5 concludes the paper. 2. Model description In the homogeneous traffic flow, all vehicles and drivers have the same characteristics. However, it is general that vehicles have different types, and drivers have different behavior patterns in real traffic. Fig. 1 illustrates one example of the heterogeneous traffic flow consisting of cars and trucks. All the four types of car–truck following combinations (CC, CT, TC and TT) are displayed in this example. The OV model assumes that a driver chooses the acceleration that can make the velocity of the vehicle equal to the optimal velocity under prevailing vehicle spacing. The formula of the Bando’s OV model is as following:
2 dxn (t ) d xn (t ) dt 2 = a f (∆xn (t )) − dt max V f (∆xn (t )) = [tanh(∆xn (t ) − ds ) + tanh(ds )]
(1)
2
where, xn (t ) is the position of the vehicle n at time t, a is the sensitivity coefficient, ∆xn (t ) is the headway of the following vehicle n ∆xn (t ) = xn−1 (t ) − xn (t ), f (∆xn (t )) is the optimal velocity function following the formula proposed in Ref. [28], V max is the maximum velocity which depends on the vehicle type, and ds is the safe distance. In this paper, we attempt to improve the existing studies by considering the two critical characteristics in car–truck mixed traffic flow, the reaction time difference of the four car-following scenarios presented by Aghabayk et al. [23] and the desire velocity difference of cars and trucks. In the OV model, the sensitivity coefficient a increases with the decreasing of the reaction time [24], namely, the driver with higher a is more sensitive. In this study, the different sensitivity level among the four types of the car–truck following scenarios can be reflected by different a values, which is similar to the work done by
D. Yang et al. / Physica A 395 (2014) 371–383
373
Fig. 1. A basic scenario of car–truck mixed traffic flow.
Table 1 The relationship between OV model parameters and the four car–truck following scenarios.
Response sensitivity Maximum velocity
CC
CT
TC
TT
High High
medium–High High
Medium–low Low
Low Low
Mason et al. [24]. Meanwhile, the maximum velocity difference between a car and a truck can be analyzed using V max . Table 1 shows the level combinations of response sensitivity and maximal velocity for different car–truck following scenarios. To model the car–truck heterogeneous traffic flow, the homogeneous OV car-following model is reformulated into a heterogeneous OV form (denoted as HEOV) by giving subscripts to the parameters a and V max as following,
2 dxn (t ) d xn ( t ) = a f ( ∆ x ( t )) − n n n dt 2 dt max V n fn (∆xn (t )) = [tanh(∆xn (t ) − ds ) + tanh(ds )]
(2)
2
where, n indicates the nth vehicle that can be either a car or a truck. an and Vnmax have the same definition as in the homogeneous model; however, their values are vehicle specific depending on the characteristics of the different car–truck following combinations. More specifically, an has four alternatives, acc , act , atc , and att ; while the Vnmax only has two alternatives, Vcmax and Vtmax . 3. Linear stability analysis of the heterogeneous OV model 3.1. Uniform flow in the car–truck mixed traffic flow The uniform flow needs to be defined to analyze the stability of the proposed model. In the equilibrium state of the homogeneous traffic flow, all vehicles have zero acceleration, equal velocity, and uniform spacing. However, for the heterogeneous flow, in order to maintain zero acceleration, the distance headway may vary among different types of vehicle. Therefore, at the equilibrium state of the heterogeneous flow, all vehicles have zero acceleration and the same velocity, but cars and trucks can have different distance headways. Such an equilibrium state of HEOV can be described as follows,
dxn (t ) =0 a f ( ∆ x ( t )) − n n n dt max Vn fn (∆xn (t )) = [tanh(∆xn (t ) − ds ) + tanh(ds )] .
(3)
2
The above equations indicate that the relationship of uniform velocity and uniform headway in HEOV model can be defined by the optimal velocity function. Therefore, only two uniform flow functions exist for the four car–truck combinations, as the following.
∗
hc = ds − log
∗
ht = ds − log
Vcmax + Vcmax e2ds
v∗
+ Vcmax + v ∗ e2ds
Vcmax + Vcmax e2ds
v∗
+ Vtmax + v ∗ e2ds
−1
(4)
−1
(5)
where, h∗c is the uniform headway of CC and TC corresponding to the uniform velocity v ∗ , and h∗t is the uniform headway of TC and TT corresponding to the uniform velocity v ∗ . In summary, the uniform flow of the HEOV model satisfies vn = v ∗ and an = 0, and the headway of vehicle n depends on the vehicle types. The distance headways of cars and trucks are given by Eqs. (4) and (5) respectively.
374
D. Yang et al. / Physica A 395 (2014) 371–383
3.2. Linear stability analysis of HEOV model The stability of the homogeneous OV model has been given by Bando et al. [26] as following: a
f <
2 where, f is the derivative of the optimal velocity function at the uniform distance headway, f = V ′ (h∗ ).
(6)
(7)
This stability condition can be uniformly applied for any car-following pair in homogeneous flow thus holding for the entire traffic flow. However, in the car–truck mixed traffic flow, the car-following pairs may have different characteristics. In this case, satisfying the stability criterion Eq. (6) for all car-following pairs in heterogeneous traffic flow can still guarantee the stability of the entire traffic flow, but it is too strict for the stability of the heterogeneous traffic flow. For the entire traffic flow to be stable, not all car-following pairs need to be stable as long as the impact of unstable pairs can be suppressed by the stable pairs. Therefore, it is necessary to develop a less strict stability criterion than Eq. (6) for the heterogeneous traffic flow. In this paper, we adopt Ward’s method [29] to derive the stability condition of HEOV model. We consider the following scenario: N vehicle are running on a circular route, and N can be a large number [26,29]. Introduce the following two types of perturbation, the headway perturbation and the velocity perturbation into the heterogeneous OV model: hn = h∗n + h˜ n
(8)
vn = v + v˜ n where, h˜ n and v˜ n are respectively the small headway perturbation and velocity perturbation. ∗
(9)
The headway perturbation and the location perturbation have the following relationship: h˜ n = x˜ n−1 − x˜ n .
(10)
The first-order and second-order derivatives of Eq. (10) give the relationship of the headway and velocity perturbations,
˙
h˜ n = v˜ n−1 − v˜ n
(11)
¨ h˜ n = v˙˜ n−1 − v˙˜ n .
(12)
Linearizing HEOV model yields,
v˙˜ n = an fn h˜ n − an v˜ n
(13)
where, fn is the derivative of the optimal velocity function of vehicle n at the uniform headway, fn = Vn′ (h∗n ).
(14)
Differencing Eq. (13) of two adjacent vehicles and substituting in Eqs. (11) and (12) yields the following equation:
¨
˙
h˜ n + an h˜ n + an fn h˜ n = an fn h˜ n−1 .
(15)
To study the spatiotemporal development of the perturbation on a circular road, consider the ansatz: hn = Re (An exp(iθ n + λt ))
(16)
in which hn is the Fourier modes, Re denotes the real part, An is a complex constant (independent of t), and θ = 2π k/N is a discrete wave-number, where it suffices to consider k = 1, 2 . . . [N /2], where [.] denotes the integer part. Substituting Eq. (16) into Eq. (15),
λ2 An + an · λAn + an · fn · An = an · fn · An−1 e−iθ .
(17)
The above system of equations can be rewritten in the following form,
A1
A1
λ2 ... = M ... AN
(18)
AN
where,
−a 1 · λ − a 1 · f 1 a2 · f2 · e−iθ 0 M = .. . 0
−a 2 · λ − a 2 · f 2 a3 · f3 · e−iθ .. .
··· ··· ··· .. .
0
···
0
a1 · f1 · e−iθ 0 0
−aN · λ − aN · fN
.. . .
(19)
D. Yang et al. / Physica A 395 (2014) 371–383
375
Thus, the solutions of Eq. (18) must satisfy
2 λ + a1 · λ + a1 · f 1 −a2 · f2 · e−iθ 0 .. . 0
λ + a2 · λ + a2 · f 2 −a3 · f3 · e−iθ .. .
··· ··· ··· .. .
0
···
0 2
−a1 · f1 · e−iθ
0 0 =0 2 λ + aN · λ + aN · fN
(20)
where | · | denotes the determinant. Rewrite Eq. (20) as N N 2 an · fn = 0. λ + an · λ + an · fn − eiN θ
(21)
n =1
n =1
The solution of Eq. (21) can be written as the following form,
λ(θ ) = λR (θ ) + iλI (θ ). ˜
(22)
λ = iλ1 θ + λ2 θ 2 + O(θ 3 ).
(23)
Since the interest of this study is large values of N, we shall typically consider the continuous range 0 < θ < π . Here, a small (positive) value of θ corresponds to very long wavelength fluctuations, and θ = 0 gives the longest possible wavelength in this discrete setting [26,27,29,30]. Wilson’s study [27] shows that instability occurs at long wavelength, and λ = 0, θ = 0 solves Eq. (21), so the system only becomes unstable when the growth rate λR (θ ) bends upwards at θ = 0 [27]. Substituting the formula (22) into Eq. (21) displays the symmetry λ (−θ) = λ (θ ), so λR (θ ) is an even function, and λI (θ ) is an odd function. In addition, the imaginary part −λI (θ ) is the dispersion relation and the real part λR (θ ) is the growth rate. The more detail explanation see Ref. [30]. To establish when this occurs we introduce the perturbation expansion of λ as the following series form and study the situation at θ = 0.
To obtain the wavelength at the marginal stability, we substitute Eq. (23) into Eq. (21). It follows that
N N2 2 [iλ1 θ + · · ·] + an iλ1 θ + λ2 θ + · · · + an fn − 1 + iN θ − θ + ··· (an fn ) = 0.
N
2
2
2
n =1
(24)
n=1
To solve the complex equation, omitting all θ terms with the order of three or more, two equations need to be hold for both the real and imaginary part. The equation for the imaginary part is as the following
iλ1
an ·
n
(am fm ) − iN
(an fn ) = 0.
(25)
n
m̸=n
Solving Eq. (25) yields, N
(an fn )
n
λ1 =
.
an ·
n
(26)
( am f m )
m̸=n
Similarly, the equation for the real part yields
−λ + an · λ2 2 1
n
−λ
am f m
2 1
ai · aj
i̸=j,j>i
m̸=n
am fm
m̸=i,j
+
N2 2
(an fn ) = 0.
(27)
n
Eq. (27) can be simplified as
λ2
an ·
n
=λ
2 1
am f m
n
m̸=n
+λ
am fm
2 1
i̸=j,j>i
m̸=n
ai · aj
m̸=i,j
am f m
−
N2 2
(an fn ) .
(28)
n
In addition, the further deviation of formula (28) can use the following identity (29) (more detail see Appendix A:)
n
m̸=n
am f m
n
=
an fn
2 am fm
m̸=n
n
an f n
.
(29)
376
D. Yang et al. / Physica A 395 (2014) 371–383
Substituting Eq. (29) into Eq. (28) yields
2 an f n am fm ai · aj am fm + an fn · 2 λ1 n n m̸=i,j m̸=n i̸=j,j>i am fm = λ2 an fn · . 2 an f n n m̸=n 1 n an · am f m − 2 n
(30)
m̸=n
Then another identity is used to simplify (30). (The derivation of the identity can be found in Appendix B):
ai · aj
=
am f m
m̸=i,j
i̸=j,j>i
1 2
n
an f n
n
2 an
am fm
m̸=n
−
n
a2n
2 am f m . m̸=n
(31)
Then a simpler expression of λ2 can be obtained N2
λ2 =
an fn
n
an f n ·
n
am f m
2
n
2 am f m an fn − a2n · . 2 m̸=n 1
(32)
m̸=n
According to the linear stability theory, when λ2 < 0, the system is stable, so the stability condition can be written as
an fn −
n
1 2
a2n
2 · am f m < 0.
(33)
m̸=n
Eq. (33) is the stability criterion of the HEOV model. If Eq. (33) is satisfied, the heterogeneous traffic flow ruled by the HEOV model is stable; otherwise, the traffic flow becomes unstable. When only one type of car-following pair exists, Eq. (33) can be simplified as f < 2a , which is the same with Eq. (6) derived by Bando for the homogeneous traffic flow. In the car–truck mixed traffic flow, two types of stability can be defined, the combination stability and the traffic flow stability. The combination stability is achieved when each car-following pair in car–truck mixed traffic flow can suppress any incoming perturbation in linear stability analysis. The stability criterion of combination stability is given by Eq. (6). The direct extension of Eq. (6) in heterogeneous traffic flow with CC, CT, TC, and TT combinations is to have the following stability functions be less than zero. Fcc = fc − acc /2,
Fct = fc − act /2,
Ftc = ft − atc /2,
Ftt = ft − att /2.
(34)
The traffic flow is stable when the perturbation can be absorbed passing through the entire traffic flow. However, the perturbation may be temporarily amplified when passing through part of the traffic flow. The corresponding stability condition is formula (33). To facilitate further discussion, we define Fl as the stability function of the traffic flow. Then the traffic flow stability condition can be written as follows
2
Fl =
an fn −
n
a2n
/2 ·
am f m
< 0.
(35)
m̸=n
Furthermore, the combination stability is a sufficient condition for Eq. (35). Fig. 2 exhibits the neutral stability curve of one pair of car-following vehicles in the car–truck heterogeneous traffic flow. The curves represent the relationship of the distance headway h and the sensitivity a for various Vmax values. The relationship of the distance headway h and the sensitivity a can be derived from Eq. (6) as following, a = V max − V max · [tanh (h − ds )]2 .
(36)
Since tanh(x) is a symmetric function with respect to 0, Eq. (36) is also a symmetric function with respect to h = ds . Set ds = 4, and then we can observe the curves in Fig. 2 are asymmetrically distributed about headway = 4. The neutral stability curve splits the quadrant into two areas: the stable area and the unstable area. The mixed traffic flow is stable above the neutral stability curve; the traffic flow becomes unstable and a traffic jam may appear in regions under the curve. With increasing of V max , the stable area becomes smaller and smaller. If the sensitivity increases, it will produce more stable car-following traffic flow. 3.3. Numerical verification of the stability condition In this section, the numerical simulations are conducted to verify the derived traffic flow stability criterion. In the simulation, 100 vehicles are placed on a circular route, and vehicle No. 1 follows vehicle No. 100. Initially, assume the first 50
D. Yang et al. / Physica A 395 (2014) 371–383
377
Fig. 2. The neutral stability curves of HEOV model for the different maximum velocities.
Fig. 3. The stable scenario. (a) The headway against vehicle number and time in a stable scenario (b) the wave peak plot. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
vehicles are cars and the subsequent 50 vehicle are trucks, that is, 49 CC combinations, 49 TT combinations, 1 CT combination and 1 TC combination in the platoon. The parameters for different combinations are set as acc = 2, act = 1.7, atc = 1.4, att = 1.1, ds = 3, Vcmax = 3 and Vtmax = 2.2. The value of Fl is −7.6328 × 10−12 and the values of the stability functions of the four combinations are respectively Fcc = 0.2421, Fct = 0.5421, Ftc = −0.3839 and Ftt = −0.0839. Hence, the traffic flow should be stable, the small perturbation decays and finally converges to 0. The CC and CT combinations can reduce the perturbation, while the TT and TC combinations can amplify the perturbation. The perturbation is added to vehicle No. 1. Run simulations to observe the propagation of the perturbation and check if the simulation results are consistent with the above analytical analysis. The simulation results are illustrated by the relationship among the distance headway, the vehicle number, and the simulation time in Fig. 3(a). In this figure, the colors cyan, red, green and blue respectively denote the combinations CC, TC, TT and TC. It is obvious that the car–truck mixed traffic flow have two uniform headways. The headway of the CC and CT combinations is smaller than the one of the TT and TC combinations. Fig. 3(a) indicates that the initial perturbation decreases over time, and the traffic flow is stable. Furthermore, the temporal profile of the wave peak of the perturbation passing through each vehicle in the circular platoon is also analyzed and is shown in Fig. 3(b). It can be observed that the wave peak increases quickly until it reaches vehicle number 50, which illustrates that the CC combinations amplify the perturbation. When the wave peak passes through the vehicle No. 51 where a TC combination is located, the magnitude of the wave peak drops. This indicates that the TC combination can suppress the perturbation. The further shrinking of the wave peak after the vehicle No. 51 illustrates that TT combination can harmonize the perturbation. Thus, the simulation results are consistent with the analytical analysis. Then an unstable scenario is considered. Keep the other parameters the same, and decrease the sensitivity of TT from 1.1 to 1. Then the value of Fl becomes 1.5834 × 109 , and the value of the function Ftt becomes 0.1381. Based on the stability conditions, both traffic flow and the TT combination will amplify the perturbation. The simulation result of the new parameter set is shown in Fig. 4(a), and the wave peak profile is shown in Fig. 4(b). Different from Fig. 3(a), the perturbation increases with time, and the traffic waves appear. The temporal wave peak profile in Fig. 4(b) ascends rather than descends as Fig. 3(b) after the vehicle No. 51. Such phenomenon illustrates that the TT combination amplifies the perturbation. Therefore,
378
D. Yang et al. / Physica A 395 (2014) 371–383
Fig. 4. The unstable scenario. (a) The headway against vehicle number and time in an unstable scenario (b) the wave peak plot.
Fig. 5. The unstable scenario for N = 1000.
the simulation results of both the stable and unstable scenarios are consistent with the analytical results obtained through traffic flow and combination stability criterion. To illustrate that the stability result does not change when N goes to a large number, we conduct one more simulation. Keep the parameters same with the last simulation (the unstable scenario), and increase the vehicle number to 1000. The headway plot is displayed in Fig. 5. We can find that the traffic flow is still unstable, although the perturbation travel pattern has been changed. 4. Car-truck following combination impact Eq. (33) can be rewritten as follows by using the induction method:
Fl =
M M 2Npj Npi · an fn − 1 a2 (ai fi )2Npi −2 a f ( ) m m n 2 i=1 j ̸= i j=1
(37)
where, N is the total number of vehicles in a platoon. M is the number of the combination types. Ni is the total number of N car-following pairs with the combination type i in the platoon. pi = Ni is the proportion of the car-following pairs with
M
combination type i in total car-following pairs, i=1 pi = 1 and Eq. (37) can be further simplified as the following: Fl =
M 2fi − ai pi
i=1
ai f i 2
M
i =1
Ni = N.
.
(38)
In the car–truck mixed traffic flow, Eq. (38) becomes the following form. Fl = pcc
2fc − acc acc fc2
+ pct
2fc − act act fc2
+ ptc
2ft − atc atc ft2
+ ptt
2ft − att att ft2
.
(39)
D. Yang et al. / Physica A 395 (2014) 371–383
379
Fig. 6. The relationship between the SC s of the four combinations and the uniform velocity.
In Eq. (39), we define each
2fi −ai ai fi2
term as the stability contribution (SC ) function of the corresponding type of car–truck
following combinations. Then, the stability of car–truck mixed traffic flow can be analyzed from two perspectives. One is the SC value of each type of combinations, the other is the proportion of each type of combination. The SC can be considered as the intrinsic characteristic of each type of combination determined by their corresponding OV model parameters and can be used to measure how much change (amplification or reduction) can be expected when perturbation passes through one type of combination. 4.1. Stability impact of individual car–truck following combination Four different SC s can be defined for the four types of car–truck combinations. SCcc =
2fc − acc acc fc2
,
SCct =
2fc − act act fc2
,
SCtc =
2ft − atc atc ft2
,
SCtt =
2ft − att att ft2
.
(40)
Similar to the stability criterion of the homogeneous traffic flow, smaller SC values mean more stable traffic flow. Fig. 6 shows that the characteristics of the four SC functions using the plot of the SC value versus uniform flow velocity. The parameters in HEOV model are set as Vcmax = 3, Vtmax = 2.5, dc = 3, acc = 1.4, act = 1.3, atc = 1.1 and att = 1. The four curves in Fig. 6 have four intersections resulting in five scenarios regarding the relationship among the four SC s. (1) TT>TC>CT>CC. This scenario happens before the first intersection point from the left. In this scenario, cars can better stabilize traffic flow than trucks. The TT combination has the least stabilizing effect, while the CC combination provides the most stabilizing effect. This scenario has a wide range of uniform velocity. When the uniform velocity is small, the four SCs experience a sharp drop around 0.5 with the decreasing of velocity. Such phenomenon indicates that the stability of all four combinations increases as velocity decreases during congestion. (2) TT>CT>TC>CC. This scenario happens between the first intersection and the second intersection from the left. Compared to the scenario (1), the order of CT and TC exchanges. That reveals that a truck following a car is the more stable than a car following a truck in this scenario. However, compare to the scenario (1) this scenario has a very narrow range for the uniform velocity. (3) TT>CT>CC>TC. This scenario occurs between the second intersection and the third intersection from the left. In this scenario, the CC combination is no longer the most stable combination and is less stable than the TC combination. However, following a car still can better stabilize the traffic flow than following a truck. Compared to the scenario (2), the scenario (3) has a wider uniform velocity range. (4) CT>TT>CC>TC. This scenario arises between the third intersection and the fourth intersection from the left, in which the CT combination becomes the most unstable combination and following a car is still more stable than following a truck as the scenario (3). However, compared to the scenario (3), the uniform velocity range is much narrow. (5) CT>CC>TT>TC. This scenario occurs after the last intersection point from the left. In this scenario, the SC values of the TT and TC combinations drop significantly with the increasing of the uniform velocity. The TT and TC combinations can better stabilize traffic flow than the CT and CC combinations. The TC combination becomes the most stable combination. Meanwhile, this scenario has a large velocity range similar to the scenario (1), but driving a truck becomes more stable than driving a car under this high-speed situation.
380
D. Yang et al. / Physica A 395 (2014) 371–383
Fig. 7. (a) The velocity against vehicle number and time in a stable scenario (b) the wave peak plot.
From the above five scenarios, the following conclusion can be drawn. Cars and trucks have different magnitudes of impact on traffic flow stability depending on the uniform flow velocity. Furthermore, all four curves can have segments that are greater than zero for some combinations of sensitivity a and maximal velocity V max , and all four types of combinations can both stabilize and destabilize traffic flow. It should be pointed out that results from Mason et al. [24] and Jin et al. [25] are only some special cases of the mixture of cars and trucks effects on traffic flow among the above five scenarios. The numerical simulation is conducted to demonstrate the different effects of the cars and truck on traffic flow. Similar to those in Section 3.3, 100 vehicles are placed on a circular route. However, the sequence of vehicles is changed to 40 alternating cars and trucks, 30 cars, and then 30 trucks. The parameters in HEOV model are the same as those used for Fig. 6. The uniform velocity is chosen as 2.3. In this case, the stability function of the mixed traffic flow Fl = −0.9649, which reveals that the traffic flow is stable. Calculate the four SC s by using the equations in (39): SC cc = 0.4599, SC ct = 0.5630, SC tc = −2.7360, and SC tt = −2.2276. The four stability values reveal that the CC and CT combinations can amplify the perturbation, but the TC and TT combinations will suppress the perturbation. The simulation results are shown in Fig. 7 with a 3D plot illustrating the velocity versus time and vehicle numbers. It can be observed that the perturbation eventually disappears, and the traffic flow returns to uniform state, which is consistent with the result of the traffic flow stability criterion. The temporal profile of the perturbation wave peak is also shown in Fig. 7(b). The peak fluctuates but descends before reaching vehicle No. 40 (alternating cars and trucks). When the perturbation propagates from the vehicle No. 40 to No. 70 (cars only) the peak value slightly ascends and after vehicle No. 70 (trucks only) the peak value descends. Trucks in the TT combination make the traffic flow stable, but they make the traffic flow unstable in the CT combination. Cars make the traffic flow unstable in the CC combination, but stabilize the traffic flow in the TC combination. Thus, the simulation results support the conclusion that cars and trucks have both stable and unstable effects on traffic flow. 4.2. Proportional impact of different car–truck following scenarios The relationship between the stability function of the car–truck mixed traffic flow and the proportions of the four types of combination on a circular route can be written as the following: Fl = pcc · SCcc + pct · SCct + ptc · SCtc + ptt · SCtt pcc + pct + ptc + ptt = 1 pct = ptc .
(41)
The third equation in (41) comes from the pairing of CT and TC combination on a circular road. Meanwhile, if the SC of one type of combination is less than zero, increasing the proportion of the type of combination makes the traffic flow more stable; otherwise, it makes the traffic flow unstable. According to the latter two equations in (41), any two fixed proportions can determine the other two proportions and the available ranges of pcc , pct , ptc and ptt are [0, 1], [0, 0.5], [0, 0.5] and [0, 1] on a circular road. With these valid ranges, a numerical experiment can be designed. In this experiment, acc = 2, act = 1.7, atc = 1.4, att = 1.1, Vcmax = 3 and Vtmax = 2.5. Two uniform velocities are chosen as the examples here, v ∗ = 0.45 and v ∗ = 2.18. When v ∗ = 0.45, SC cc = −0.3737, SC ct = −0.1461, SC tc = 0.1178, and SC tt = 0.6402. In this case, increasing the proportions of the TC and TT combinations can stabilize traffic flow. However, the increase of the proportions of CT and CC combinations can make the traffic flow unstable. When v ∗ = 2.18, SC cc = 0.1317, SC ct = 0.2807, SC tc = −0.7168, and SC tt = −0.0069. In this case, the increase of the proportions of the CC and CT combinations can stabilize the traffic flow, while increasing the proportions of the TC and TT combinations can make the traffic flow unstable. Fig. 8 is the scatter plots of the stability function for the two uniform velocities v ∗ = 0.45 and v ∗ = 2.18. The blue line is the neutral stability line connecting all the points where Fl = 0. Based on the parameters and the formula (41), the proportion of CT combination is a monotonically decreasing function with respect to the proportion of CC combination for
D. Yang et al. / Physica A 395 (2014) 371–383
(a) v ∗ = 0.45.
381
(b) v ∗ = 2.18. Fig. 8. The scatter plots of the stability function and the neutral stability line of the car–truck mixed traffic flow.
v ∗ = 0.45 and a monotonically increasing function for v ∗ = 2.18. In Fig. 8(a), since the stability functions of the CC and CT combinations are negative, the smaller the proportions, the bigger the values of the stability function of traffic flow. In this case, the slope of the neutral stability line is negative. The line splits the quadrant into two parts, the unstable area to the left and the stable area to the right. When the proportion of the CC combination is more than 0.65, the traffic flow is always stable for v ∗ = 0.45 regardless of the other three proportions. In Fig. 8(b), the neutral stability line has the positive slope. It also splits the quadrant into two parts, however, with the left-side area being stable and the right-side area being unstable. When the proportion of the CC combination is more than 0.65 for v ∗ = 2.18, the traffic flow is always unstable, while when the proportion of the CT combination is more than 0.19, the traffic flow is always stable. Eq. (39) indicates that, in heterogeneous traffic flow, the proportions of the different car–truck combinations play more critical roles for traffic flow stability than the proportions of different vehicle types. This conclusion can be verified using numerical simulation. We reuse the same parameters in Fig. 8(b). The proportions of the four combinations are initially set as pcc = 0.38, pct = 0.12, ptc = 0.12, and ptt = 0.38. This is a stable case as shown in the headway evolution plot (see Fig. 9(a)). Then we remove one CT combination from the platoon and add one TT combination and one CC combination. This does not change the numbers of the cars and trucks but change the proportions of four combinations to pcc = 0.39, pct = 0.11, ptc = 0.11 and ptt = 0.39. Fig. 9(b) shows the simulation result. Traffic flow becomes unstable from the stable case in Fig. 9(a). Thus, the comparison of two cases proves the significance of proportions of car–truck following combinations on the stability of traffic flow. Moreover, another simulation is conducted to demonstrate that the positions of different combinations are not critical factor to influence the linear stability of traffic flow. Adopt the same parameters with the first case and move the half of CT and TC combinations to the end of the platoon. The simulation result is shown in Fig. 9(c). It reveals that the traffic flow is still stable and not influenced by the position changes of the vehicles. 5. Conclusion and future work In this paper, the Bando’s OV car-following is extended to the heterogeneous OV model (HEOV) that can describe heterogeneous traffic flow with four different car–truck following combinations, namely, a car following a car (CC), a car following a truck (CT), a truck following a car (TC) and a truck following a truck (TT). The linear stability method is adopted to derive the stability criterion of HEOV model, and then the stability criterion is verified by numerical simulations. The stability contribution of each car–truck combination is analyzed. Five scenarios exist for the relationship of the four combinations. Under different scenarios, the car and truck both have stable and unstable effect on traffic flow. Moreover, the influence of the combination proportions is explored by both analytical analysis and simulation methods. The results also reveal that, in mixed traffic flow, the proportions of car–truck combinations rather than the proportions of vehicle types determine the traffic flow stability more. However, this paper only focuses on one-dimensional traffic flow, but the two-dimensional traffic flow is more realistic, so the stability of two-dimensional traffic flow needs to be explored in future. Appendix A. Derivation of the first identity The first identity can be derived from the following equation
n
an fn
m̸=n
2 am fm = an f n am f m = an f n am f m . n
n
m̸=n
n
n
m̸=n
(A.1)
382
D. Yang et al. / Physica A 395 (2014) 371–383
(a) The stable case when pcc = 0.38, pct = 0.12, ptc = 0.12 and ptt = 0.38.
(b) The unstable case when pcc = 0.39, pct = 0.11, ptc = 0.11 and ptt = 0.39.
(c) The stable case when pcc = 0.39, pct = 0.11, ptc = 0.11 and ptt = 0.39. Fig. 9. The headway plots for the different proportions of the four combinations.
Reorganizing Eq. (A.1) yields,
n
am fm
n
2 am fm
an f n
m̸=n
=
m̸=n
.
an fn
(A.2)
n
Appendix B. Derivation of the second identity The second identity is derived from the following equation
αn
n
βm
·
γn
n
m̸=n
βm
m̸=n
=
αn γn
n
2 βm + αn γn βm · βn . (B.1) i ,j
m̸=n
m̸=i,j
n
If αn = γn , rewriting Eq. (B.1) yields,
n
2
αn
m̸=n
βm
=
n
αn2
m̸=n
2 βm + 2 αi αj βm · βn . i̸=j,j>i
m̸=i,j
n
(B.2)
D. Yang et al. / Physica A 395 (2014) 371–383
383
In the case of this paper, αn and βn can be written as the following.
αn = an βn = an fn .
(B.3) (B.4)
Substituting Eqs. (B.3) and (B.4) into (B.2) leads to
i̸=j,j>i
ai · aj
m̸=i,j
am f m
=
1 2
n
an f n
n
2 an
m̸=n
am fm
−
n
a2n
2 am f m . m̸=n
(B.5)
References [1] D. Balcan, V. Colizza, B. Gonçalves, H. Hu, J.J. Ramasco, A. Vespignani, Multiscale mobility networks and the spatial spreading of infectious diseases, Proceedings of the National Academy of Sciences 106 (2009) 21484–21489. [2] L. Hufnagel, D. Brockmann, T. Geisel, Forecast and control of epidemics in a globalized world, Proceedings of the National Academy of Sciences of the United States of America 101 (2004) 15124–15129. [3] S. Ni, W. Weng, Impact of travel patterns on epidemic dynamics in heterogeneous spatial metapopulation networks, Physical Review E 79 (2009) 016111. [4] L. Wang, X. Li, Y.-Q. Zhang, Y. Zhang, K. Zhang, Evolution of scaling emergence in large-scale spatial epidemic spreading, PloS One 6 (2011) e21197. [5] S. Meloni, A. Arenas, Y. Moreno, Traffic-driven epidemic spreading in finite-size scale-free networks, Proceedings of the National Academy of Sciences 106 (2009) 16897–16902. [6] D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics 73 (2001) 1067–1141. [7] A. Schadschneider, Traffic flow: a statistical physics point of view, Physica A 313 (2002) 153–187. [8] D. Chowdhury, L. Santen, A. Schadschneider, Physics Reports 323 (2000) 199. [9] B.S. Kerner, The physics of traffic: empirical freeway pattern features, engineering applications, and theory, in: The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, Springer-Verlag, Berlin, 2004. [10] A. Schadschneider, D. Chowdhury, K. Nishinari, Stochastic transport in complex systems: from molecules to vehicles, in: Stochastic Transport in Complex Systems: From Molecules to Vehicles, Elsevier Science, Amsterdam, 2010. [11] J. Taylor, X. Zhou, N.M. Rouphail, Method for investigating intradriver heterogeneity using vehicle trajectory data: a dynamic time warping approach, in: N.H. Gartner (Ed.), Transportation Research Board 91st Annual Meeting, Transportation Research Board of the National Academies, Washington, DC, 2012. [12] S. Ossen, S.P. Hoogendoorn, B.G.H. Gorte, Interdriver differences in car-following: a vehicle trajectory-based study, Transportation Research Record: Transportation Research Record 1965 (2006) 121–129. [13] S. Ossen, S.P. Hoogendoorn, Driver heterogeneity in car following and its impact on modeling traffic dynamics, Transportation Research Record: Transportation Research Record 1999 (2007) 95–103. [14] L. Davis, Effect of adaptive cruise control systems on mixed traffic flow near an on-ramp, Physica A 379 (2007) 274–290. [15] M.M. Islam, C.F. Choudhury, A violation behavior model for non-motorized vehicle drivers in 4 heterogeneous traffic streams, in: N.H. Gartner (Ed.), Transportation Research Board 91st Annual Meeting, Transportation Research Board, Washington, DC, 2012. [16] S. Jin, D. Wang, P. Tao, P. Li, Non-lane-based full velocity difference car following model, Physica A. Statistical Mechanics and its Applications 389 (2010) 4654–4662. [17] S. Jin, D.H. Wang, X.R. Yang, Non-lane-based car-following model with visual angle information, in: Transportation Research Record: Transportation Research Record, Transportation Research Board of the National Academies, Washington, DC, 2011, pp. 7–14. [18] J.R. Sayer, M.L. Mefford, R. Huang, The Effect of Lead-Vehicle Size on Driver Following Behavior, 2000. [19] K. Huddart, R. Lafont, Close driving—hazard or necessity? in: XV European Annual Conference on Human Decision Making and Manual Control, TNO, Netherlands, 1990, pp. 205–217. [20] M. McDonald, M. Brackstone, B. Sultan, C. Roach, Close following on the motorway: initial findings of an instrumented vehicle study, VISION IN VEHICLES-VII, 1999. [21] S. Peeta, W. Zhou, P. Zhang, Modeling and mitigation of car–truck interactions on freeways, Transportation Research Record 1899 (2004) 117–126. [22] S. Peeta, P. Zhang, W. Zhou, Behavior-based analysis of freeway car–truck interactions and related mitigation strategies, Transportation Research Part B: Methodological 39 (2005) 417–451. [23] K. Aghabayk, M. Sarvi, W. Young, Y. Wang, Investigating heavy vehicle interactions during the car following process, in: Transportation Research Board 91st Annual Meeting, Transportation Research Board of the National Academies, Washington, DC, 2012. [24] A.D. Mason, A.W. Woods, Car-following model of multispecies systems of road traffic, Physical Review E 55 (1997) 2203–2214. [25] S. Jin, D.H. Wang, Z.Y. Huang, P.F. Tao, Visual angle model for car following theory, Physica A. Statistical Mechanics and its Applications (2011). [26] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Physical Review E 51 (1995) 1035–1042. [27] R.E. Wilson, Mechanisms for spatio-temporal pattern formation in highway traffic models, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366 (2008) 2017–2032. [28] H.X. Ge, X.P. Meng, R.J. Cheng, S.M. Lo, Time-dependent Ginzburg–Landau equation in a car-following model considering the driver’s physical delay, Physica A: Statistical Mechanics and its Applications 390 (2011) 3348–3353. [29] J.A. Ward, Heterogeneity, Lane-Changing and Instability in Traffic: A Mathematical Approach, Ph.D. Thesis. Thesis, 2009. [30] J.A. Ward, R.E. Wilson, Criteria for convective versus absolute string instability in car-following models, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 467 (2011) 2185–2208.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Stability analysis of the mixed traffic flow of cars and trucks using heterogeneous optimal velocity car-following model Da Yang a,b,∗ , Peter (Jing) Jin c , Yun Pu a , Bin Ran b a
School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China
b
Department of Civil and Environmental Engineering, University of Wisconsin-Madison, Madison, 53706, USA
c
Department of Civil and Environmental Engineering, University of Texas-Austin, Austin, 78701, USA
highlights • • • • •
Stability of the mixture of cars and trucks on traffic flow is studied. A new method is used to derive the stability criterion of the heterogeneous traffic flow. Percentage and stability function of different vehicles are main factors influencing stability. Both car and truck can stabilize and destabilize the mixed traffic flow. Slow trucks can trigger traffic capacity drop.
article
info
Article history: Received 23 May 2012 Received in revised form 22 September 2013 Available online 12 October 2013 Keywords: Traffic flow Heterogeneous car-following model Car and truck Stability
abstract Real-world traffic flow usually contains a mixture of passenger vehicles (PV) and heavy vehicles (HV). In this paper, the four types of car–truck following combinations are considered: the car-following-car case, car-following-truck case, truck-following-car case and truck-following-truck case. The effect of different combinations on the stability of traffic flow is explored by converting the original Bando’s optimal velocity (OV) model to a heterogeneous form. A new linear stability analysis method that can derive the stability criterion of the heterogeneous traffic flow mixed by cars and trucks is introduced. Moreover, the effect of the proportions of the four car–truck following combinations on traffic flow is examined through the trajectory analysis. It concludes that the linear stability of the car–truck mixed traffic flow is determined more by the proportions of the different car–truck following scenarios, rather than the numbers of the cars and trucks. Moreover, cars and trucks can both stabilize and destabilize traffic flow depending on the density of the traffic flow and the parameters of the heterogeneous OV model. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Traffic flow is a complex system with many particles (vehicles) and complex interactions among them. The study on traffic flow can help people understand the travel characteristics of human, which can further help the understanding on the other aspects of human life, such as the diffusion of virus [1–5]. Researchers from both the applied physics and traffic flow theory have been developing models to describe such a system for more than 50 years. According to the reviews of the several researchers (Helbing [6], Schadschneider [7], Chowdhury et al. [8], Kerner [9], Schadschneider et al. [10]) on traffic
∗ Corresponding author at: Department of Civil and Environmental Engineering, University of Wisconsin-Madison, Madison, 53706, USA. Tel.: +1 6083344281. E-mail address: [email protected] (D. Yang). 0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.10.017
372
D. Yang et al. / Physica A 395 (2014) 371–383
flow models, the existing models include the hydrodynamic models, gas kinetic models, Cellular Automata (CA) models, car-following models and coupled-map lattice models. In recent years, the heterogeneous phenomena in traffic flow have drawn a lot of attention among traffic flow researchers [11–17]. The existing studies mainly focus on three types of traffic heterogeneity, driver heterogeneity, vehicle heterogeneity, and traffic environment heterogeneity. Taylor et al. [11] explored driver heterogeneity within a trip using the Dynamic Time Warping method. Ossen and Hoogendoorn [12,13] focused on the impact of car-following styles on traffic flow. Davis [14] examined the effect of the percentage of ACC (Adaptive Cruise Control) vehicles on traffic flow near an on-ramp, and Islam et al. [15] analyzed the characteristics of the traffic flow mixed by motorized and non-motorized vehicles. Representative works on environment heterogeneity are Jin et al.’s formula [16,17] about visual impact of traffic environment on traffic flow using a non-lane-based car-following model. In this study, the primary focus is to explore the heterogeneity of vehicle types, more specifically, the mixture of trucks and cars based on car-following models. The traffic flow considered in this paper is the one-dimensional traffic flow. Sayer et al. [18], Huddart et al. [19] and McDonald et al. [20] compared the headway difference between the car-following-car case and car-following-truck case. Peeta et al. [21,22] introduced a discomfort level to study the influence of trucks on surrounding vehicles. Most recently, Aghabayk et al. [23] pointed out that the effect of car–truck mixture on traffic flow is not only determined by the leading vehicle type, but also influenced by the following vehicle type; based on this idea they further studied the characteristics of the four types of car–truck combination, the car-following-car (CC), car-followingtruck (CT), truck-following-car (TC) and truck-following-truck (TT). According to the paper [23], the four different combinations exhibit different driving behavioral characteristics, especially different reaction times (the reaction time order is CC < CT < TC < TT). However, their studies only presented the statistical results without further investigating the stability of the car–truck mixed traffic flow. Mason et al. [24] concluded that trucks had unstable effect on traffic flow compared to cars and may induce traffic congestion using both linear and nonlinear stability analysis methods; however, the study only considered the influence of the leading vehicle types on traffic flow. Jin et al. [25] drew the contrary conclusion that trucks can better stabilize traffic flow than cars due to their bigger sizes based on a visual angle car-following model, but the study only took into account the size difference of cars and trucks without considering the difference in car–truck combinations. In this paper, we attempt to improve the existing studies by considering the two critical characteristics in car–truck mixed traffic flow, the reaction time difference of the four car-following scenarios presented by Aghabayk et al. [23] and the desire velocity difference of cars and trucks. Compared to other one-dimensional traffic flow models, the car-following model is adopted taking advantage of its ability to describe microscopic driving characteristics of cars and trucks [6]. Meanwhile, due to its strong behavioral background and mathematical convenience, Bando’s Optimal Velocity (OV) car-following model [26] is developed to a heterogeneous form to model the car–truck heterogeneous traffic flow. What needs to be stated here is that lane-changing is not considered in this paper, and instead we just think it as an external perturbation in the stability analysis of traffic flow as suggested by Wilson [27]. The remainder of the paper is organized as the following. In Section 2, a heterogeneous OV model is proposed to describe the car–truck mixed traffic flow. Then, the stability condition of the heterogeneous OV model is derived using the new linear stability method verified numerically in Section 3. Section 4 analyzes the stability impacts of the four types of car–truck combination from two perspectives: one is the individual impact of each type of car–truck combination, and the other is the combined impact with different proportions of the CC, CT, TC and TT combinations. Section 5 concludes the paper. 2. Model description In the homogeneous traffic flow, all vehicles and drivers have the same characteristics. However, it is general that vehicles have different types, and drivers have different behavior patterns in real traffic. Fig. 1 illustrates one example of the heterogeneous traffic flow consisting of cars and trucks. All the four types of car–truck following combinations (CC, CT, TC and TT) are displayed in this example. The OV model assumes that a driver chooses the acceleration that can make the velocity of the vehicle equal to the optimal velocity under prevailing vehicle spacing. The formula of the Bando’s OV model is as following:
2 dxn (t ) d xn (t ) dt 2 = a f (∆xn (t )) − dt max V f (∆xn (t )) = [tanh(∆xn (t ) − ds ) + tanh(ds )]
(1)
2
where, xn (t ) is the position of the vehicle n at time t, a is the sensitivity coefficient, ∆xn (t ) is the headway of the following vehicle n ∆xn (t ) = xn−1 (t ) − xn (t ), f (∆xn (t )) is the optimal velocity function following the formula proposed in Ref. [28], V max is the maximum velocity which depends on the vehicle type, and ds is the safe distance. In this paper, we attempt to improve the existing studies by considering the two critical characteristics in car–truck mixed traffic flow, the reaction time difference of the four car-following scenarios presented by Aghabayk et al. [23] and the desire velocity difference of cars and trucks. In the OV model, the sensitivity coefficient a increases with the decreasing of the reaction time [24], namely, the driver with higher a is more sensitive. In this study, the different sensitivity level among the four types of the car–truck following scenarios can be reflected by different a values, which is similar to the work done by
D. Yang et al. / Physica A 395 (2014) 371–383
373
Fig. 1. A basic scenario of car–truck mixed traffic flow.
Table 1 The relationship between OV model parameters and the four car–truck following scenarios.
Response sensitivity Maximum velocity
CC
CT
TC
TT
High High
medium–High High
Medium–low Low
Low Low
Mason et al. [24]. Meanwhile, the maximum velocity difference between a car and a truck can be analyzed using V max . Table 1 shows the level combinations of response sensitivity and maximal velocity for different car–truck following scenarios. To model the car–truck heterogeneous traffic flow, the homogeneous OV car-following model is reformulated into a heterogeneous OV form (denoted as HEOV) by giving subscripts to the parameters a and V max as following,
2 dxn (t ) d xn ( t ) = a f ( ∆ x ( t )) − n n n dt 2 dt max V n fn (∆xn (t )) = [tanh(∆xn (t ) − ds ) + tanh(ds )]
(2)
2
where, n indicates the nth vehicle that can be either a car or a truck. an and Vnmax have the same definition as in the homogeneous model; however, their values are vehicle specific depending on the characteristics of the different car–truck following combinations. More specifically, an has four alternatives, acc , act , atc , and att ; while the Vnmax only has two alternatives, Vcmax and Vtmax . 3. Linear stability analysis of the heterogeneous OV model 3.1. Uniform flow in the car–truck mixed traffic flow The uniform flow needs to be defined to analyze the stability of the proposed model. In the equilibrium state of the homogeneous traffic flow, all vehicles have zero acceleration, equal velocity, and uniform spacing. However, for the heterogeneous flow, in order to maintain zero acceleration, the distance headway may vary among different types of vehicle. Therefore, at the equilibrium state of the heterogeneous flow, all vehicles have zero acceleration and the same velocity, but cars and trucks can have different distance headways. Such an equilibrium state of HEOV can be described as follows,
dxn (t ) =0 a f ( ∆ x ( t )) − n n n dt max Vn fn (∆xn (t )) = [tanh(∆xn (t ) − ds ) + tanh(ds )] .
(3)
2
The above equations indicate that the relationship of uniform velocity and uniform headway in HEOV model can be defined by the optimal velocity function. Therefore, only two uniform flow functions exist for the four car–truck combinations, as the following.
∗
hc = ds − log
∗
ht = ds − log
Vcmax + Vcmax e2ds
v∗
+ Vcmax + v ∗ e2ds
Vcmax + Vcmax e2ds
v∗
+ Vtmax + v ∗ e2ds
−1
(4)
−1
(5)
where, h∗c is the uniform headway of CC and TC corresponding to the uniform velocity v ∗ , and h∗t is the uniform headway of TC and TT corresponding to the uniform velocity v ∗ . In summary, the uniform flow of the HEOV model satisfies vn = v ∗ and an = 0, and the headway of vehicle n depends on the vehicle types. The distance headways of cars and trucks are given by Eqs. (4) and (5) respectively.
374
D. Yang et al. / Physica A 395 (2014) 371–383
3.2. Linear stability analysis of HEOV model The stability of the homogeneous OV model has been given by Bando et al. [26] as following: a
f <
2 where, f is the derivative of the optimal velocity function at the uniform distance headway, f = V ′ (h∗ ).
(6)
(7)
This stability condition can be uniformly applied for any car-following pair in homogeneous flow thus holding for the entire traffic flow. However, in the car–truck mixed traffic flow, the car-following pairs may have different characteristics. In this case, satisfying the stability criterion Eq. (6) for all car-following pairs in heterogeneous traffic flow can still guarantee the stability of the entire traffic flow, but it is too strict for the stability of the heterogeneous traffic flow. For the entire traffic flow to be stable, not all car-following pairs need to be stable as long as the impact of unstable pairs can be suppressed by the stable pairs. Therefore, it is necessary to develop a less strict stability criterion than Eq. (6) for the heterogeneous traffic flow. In this paper, we adopt Ward’s method [29] to derive the stability condition of HEOV model. We consider the following scenario: N vehicle are running on a circular route, and N can be a large number [26,29]. Introduce the following two types of perturbation, the headway perturbation and the velocity perturbation into the heterogeneous OV model: hn = h∗n + h˜ n
(8)
vn = v + v˜ n where, h˜ n and v˜ n are respectively the small headway perturbation and velocity perturbation. ∗
(9)
The headway perturbation and the location perturbation have the following relationship: h˜ n = x˜ n−1 − x˜ n .
(10)
The first-order and second-order derivatives of Eq. (10) give the relationship of the headway and velocity perturbations,
˙
h˜ n = v˜ n−1 − v˜ n
(11)
¨ h˜ n = v˙˜ n−1 − v˙˜ n .
(12)
Linearizing HEOV model yields,
v˙˜ n = an fn h˜ n − an v˜ n
(13)
where, fn is the derivative of the optimal velocity function of vehicle n at the uniform headway, fn = Vn′ (h∗n ).
(14)
Differencing Eq. (13) of two adjacent vehicles and substituting in Eqs. (11) and (12) yields the following equation:
¨
˙
h˜ n + an h˜ n + an fn h˜ n = an fn h˜ n−1 .
(15)
To study the spatiotemporal development of the perturbation on a circular road, consider the ansatz: hn = Re (An exp(iθ n + λt ))
(16)
in which hn is the Fourier modes, Re denotes the real part, An is a complex constant (independent of t), and θ = 2π k/N is a discrete wave-number, where it suffices to consider k = 1, 2 . . . [N /2], where [.] denotes the integer part. Substituting Eq. (16) into Eq. (15),
λ2 An + an · λAn + an · fn · An = an · fn · An−1 e−iθ .
(17)
The above system of equations can be rewritten in the following form,
A1
A1
λ2 ... = M ... AN
(18)
AN
where,
−a 1 · λ − a 1 · f 1 a2 · f2 · e−iθ 0 M = .. . 0
−a 2 · λ − a 2 · f 2 a3 · f3 · e−iθ .. .
··· ··· ··· .. .
0
···
0
a1 · f1 · e−iθ 0 0
−aN · λ − aN · fN
.. . .
(19)
D. Yang et al. / Physica A 395 (2014) 371–383
375
Thus, the solutions of Eq. (18) must satisfy
2 λ + a1 · λ + a1 · f 1 −a2 · f2 · e−iθ 0 .. . 0
λ + a2 · λ + a2 · f 2 −a3 · f3 · e−iθ .. .
··· ··· ··· .. .
0
···
0 2
−a1 · f1 · e−iθ
0 0 =0 2 λ + aN · λ + aN · fN
(20)
where | · | denotes the determinant. Rewrite Eq. (20) as N N 2 an · fn = 0. λ + an · λ + an · fn − eiN θ
(21)
n =1
n =1
The solution of Eq. (21) can be written as the following form,
λ(θ ) = λR (θ ) + iλI (θ ). ˜
(22)
λ = iλ1 θ + λ2 θ 2 + O(θ 3 ).
(23)
Since the interest of this study is large values of N, we shall typically consider the continuous range 0 < θ < π . Here, a small (positive) value of θ corresponds to very long wavelength fluctuations, and θ = 0 gives the longest possible wavelength in this discrete setting [26,27,29,30]. Wilson’s study [27] shows that instability occurs at long wavelength, and λ = 0, θ = 0 solves Eq. (21), so the system only becomes unstable when the growth rate λR (θ ) bends upwards at θ = 0 [27]. Substituting the formula (22) into Eq. (21) displays the symmetry λ (−θ) = λ (θ ), so λR (θ ) is an even function, and λI (θ ) is an odd function. In addition, the imaginary part −λI (θ ) is the dispersion relation and the real part λR (θ ) is the growth rate. The more detail explanation see Ref. [30]. To establish when this occurs we introduce the perturbation expansion of λ as the following series form and study the situation at θ = 0.
To obtain the wavelength at the marginal stability, we substitute Eq. (23) into Eq. (21). It follows that
N N2 2 [iλ1 θ + · · ·] + an iλ1 θ + λ2 θ + · · · + an fn − 1 + iN θ − θ + ··· (an fn ) = 0.
N
2
2
2
n =1
(24)
n=1
To solve the complex equation, omitting all θ terms with the order of three or more, two equations need to be hold for both the real and imaginary part. The equation for the imaginary part is as the following
iλ1
an ·
n
(am fm ) − iN
(an fn ) = 0.
(25)
n
m̸=n
Solving Eq. (25) yields, N
(an fn )
n
λ1 =
.
an ·
n
(26)
( am f m )
m̸=n
Similarly, the equation for the real part yields
−λ + an · λ2 2 1
n
−λ
am f m
2 1
ai · aj
i̸=j,j>i
m̸=n
am fm
m̸=i,j
+
N2 2
(an fn ) = 0.
(27)
n
Eq. (27) can be simplified as
λ2
an ·
n
=λ
2 1
am f m
n
m̸=n
+λ
am fm
2 1
i̸=j,j>i
m̸=n
ai · aj
m̸=i,j
am f m
−
N2 2
(an fn ) .
(28)
n
In addition, the further deviation of formula (28) can use the following identity (29) (more detail see Appendix A:)
n
m̸=n
am f m
n
=
an fn
2 am fm
m̸=n
n
an f n
.
(29)
376
D. Yang et al. / Physica A 395 (2014) 371–383
Substituting Eq. (29) into Eq. (28) yields
2 an f n am fm ai · aj am fm + an fn · 2 λ1 n n m̸=i,j m̸=n i̸=j,j>i am fm = λ2 an fn · . 2 an f n n m̸=n 1 n an · am f m − 2 n
(30)
m̸=n
Then another identity is used to simplify (30). (The derivation of the identity can be found in Appendix B):
ai · aj
=
am f m
m̸=i,j
i̸=j,j>i
1 2
n
an f n
n
2 an
am fm
m̸=n
−
n
a2n
2 am f m . m̸=n
(31)
Then a simpler expression of λ2 can be obtained N2
λ2 =
an fn
n
an f n ·
n
am f m
2
n
2 am f m an fn − a2n · . 2 m̸=n 1
(32)
m̸=n
According to the linear stability theory, when λ2 < 0, the system is stable, so the stability condition can be written as
an fn −
n
1 2
a2n
2 · am f m < 0.
(33)
m̸=n
Eq. (33) is the stability criterion of the HEOV model. If Eq. (33) is satisfied, the heterogeneous traffic flow ruled by the HEOV model is stable; otherwise, the traffic flow becomes unstable. When only one type of car-following pair exists, Eq. (33) can be simplified as f < 2a , which is the same with Eq. (6) derived by Bando for the homogeneous traffic flow. In the car–truck mixed traffic flow, two types of stability can be defined, the combination stability and the traffic flow stability. The combination stability is achieved when each car-following pair in car–truck mixed traffic flow can suppress any incoming perturbation in linear stability analysis. The stability criterion of combination stability is given by Eq. (6). The direct extension of Eq. (6) in heterogeneous traffic flow with CC, CT, TC, and TT combinations is to have the following stability functions be less than zero. Fcc = fc − acc /2,
Fct = fc − act /2,
Ftc = ft − atc /2,
Ftt = ft − att /2.
(34)
The traffic flow is stable when the perturbation can be absorbed passing through the entire traffic flow. However, the perturbation may be temporarily amplified when passing through part of the traffic flow. The corresponding stability condition is formula (33). To facilitate further discussion, we define Fl as the stability function of the traffic flow. Then the traffic flow stability condition can be written as follows
2
Fl =
an fn −
n
a2n
/2 ·
am f m
< 0.
(35)
m̸=n
Furthermore, the combination stability is a sufficient condition for Eq. (35). Fig. 2 exhibits the neutral stability curve of one pair of car-following vehicles in the car–truck heterogeneous traffic flow. The curves represent the relationship of the distance headway h and the sensitivity a for various Vmax values. The relationship of the distance headway h and the sensitivity a can be derived from Eq. (6) as following, a = V max − V max · [tanh (h − ds )]2 .
(36)
Since tanh(x) is a symmetric function with respect to 0, Eq. (36) is also a symmetric function with respect to h = ds . Set ds = 4, and then we can observe the curves in Fig. 2 are asymmetrically distributed about headway = 4. The neutral stability curve splits the quadrant into two areas: the stable area and the unstable area. The mixed traffic flow is stable above the neutral stability curve; the traffic flow becomes unstable and a traffic jam may appear in regions under the curve. With increasing of V max , the stable area becomes smaller and smaller. If the sensitivity increases, it will produce more stable car-following traffic flow. 3.3. Numerical verification of the stability condition In this section, the numerical simulations are conducted to verify the derived traffic flow stability criterion. In the simulation, 100 vehicles are placed on a circular route, and vehicle No. 1 follows vehicle No. 100. Initially, assume the first 50
D. Yang et al. / Physica A 395 (2014) 371–383
377
Fig. 2. The neutral stability curves of HEOV model for the different maximum velocities.
Fig. 3. The stable scenario. (a) The headway against vehicle number and time in a stable scenario (b) the wave peak plot. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
vehicles are cars and the subsequent 50 vehicle are trucks, that is, 49 CC combinations, 49 TT combinations, 1 CT combination and 1 TC combination in the platoon. The parameters for different combinations are set as acc = 2, act = 1.7, atc = 1.4, att = 1.1, ds = 3, Vcmax = 3 and Vtmax = 2.2. The value of Fl is −7.6328 × 10−12 and the values of the stability functions of the four combinations are respectively Fcc = 0.2421, Fct = 0.5421, Ftc = −0.3839 and Ftt = −0.0839. Hence, the traffic flow should be stable, the small perturbation decays and finally converges to 0. The CC and CT combinations can reduce the perturbation, while the TT and TC combinations can amplify the perturbation. The perturbation is added to vehicle No. 1. Run simulations to observe the propagation of the perturbation and check if the simulation results are consistent with the above analytical analysis. The simulation results are illustrated by the relationship among the distance headway, the vehicle number, and the simulation time in Fig. 3(a). In this figure, the colors cyan, red, green and blue respectively denote the combinations CC, TC, TT and TC. It is obvious that the car–truck mixed traffic flow have two uniform headways. The headway of the CC and CT combinations is smaller than the one of the TT and TC combinations. Fig. 3(a) indicates that the initial perturbation decreases over time, and the traffic flow is stable. Furthermore, the temporal profile of the wave peak of the perturbation passing through each vehicle in the circular platoon is also analyzed and is shown in Fig. 3(b). It can be observed that the wave peak increases quickly until it reaches vehicle number 50, which illustrates that the CC combinations amplify the perturbation. When the wave peak passes through the vehicle No. 51 where a TC combination is located, the magnitude of the wave peak drops. This indicates that the TC combination can suppress the perturbation. The further shrinking of the wave peak after the vehicle No. 51 illustrates that TT combination can harmonize the perturbation. Thus, the simulation results are consistent with the analytical analysis. Then an unstable scenario is considered. Keep the other parameters the same, and decrease the sensitivity of TT from 1.1 to 1. Then the value of Fl becomes 1.5834 × 109 , and the value of the function Ftt becomes 0.1381. Based on the stability conditions, both traffic flow and the TT combination will amplify the perturbation. The simulation result of the new parameter set is shown in Fig. 4(a), and the wave peak profile is shown in Fig. 4(b). Different from Fig. 3(a), the perturbation increases with time, and the traffic waves appear. The temporal wave peak profile in Fig. 4(b) ascends rather than descends as Fig. 3(b) after the vehicle No. 51. Such phenomenon illustrates that the TT combination amplifies the perturbation. Therefore,
378
D. Yang et al. / Physica A 395 (2014) 371–383
Fig. 4. The unstable scenario. (a) The headway against vehicle number and time in an unstable scenario (b) the wave peak plot.
Fig. 5. The unstable scenario for N = 1000.
the simulation results of both the stable and unstable scenarios are consistent with the analytical results obtained through traffic flow and combination stability criterion. To illustrate that the stability result does not change when N goes to a large number, we conduct one more simulation. Keep the parameters same with the last simulation (the unstable scenario), and increase the vehicle number to 1000. The headway plot is displayed in Fig. 5. We can find that the traffic flow is still unstable, although the perturbation travel pattern has been changed. 4. Car-truck following combination impact Eq. (33) can be rewritten as follows by using the induction method:
Fl =
M M 2Npj Npi · an fn − 1 a2 (ai fi )2Npi −2 a f ( ) m m n 2 i=1 j ̸= i j=1
(37)
where, N is the total number of vehicles in a platoon. M is the number of the combination types. Ni is the total number of N car-following pairs with the combination type i in the platoon. pi = Ni is the proportion of the car-following pairs with
M
combination type i in total car-following pairs, i=1 pi = 1 and Eq. (37) can be further simplified as the following: Fl =
M 2fi − ai pi
i=1
ai f i 2
M
i =1
Ni = N.
.
(38)
In the car–truck mixed traffic flow, Eq. (38) becomes the following form. Fl = pcc
2fc − acc acc fc2
+ pct
2fc − act act fc2
+ ptc
2ft − atc atc ft2
+ ptt
2ft − att att ft2
.
(39)
D. Yang et al. / Physica A 395 (2014) 371–383
379
Fig. 6. The relationship between the SC s of the four combinations and the uniform velocity.
In Eq. (39), we define each
2fi −ai ai fi2
term as the stability contribution (SC ) function of the corresponding type of car–truck
following combinations. Then, the stability of car–truck mixed traffic flow can be analyzed from two perspectives. One is the SC value of each type of combinations, the other is the proportion of each type of combination. The SC can be considered as the intrinsic characteristic of each type of combination determined by their corresponding OV model parameters and can be used to measure how much change (amplification or reduction) can be expected when perturbation passes through one type of combination. 4.1. Stability impact of individual car–truck following combination Four different SC s can be defined for the four types of car–truck combinations. SCcc =
2fc − acc acc fc2
,
SCct =
2fc − act act fc2
,
SCtc =
2ft − atc atc ft2
,
SCtt =
2ft − att att ft2
.
(40)
Similar to the stability criterion of the homogeneous traffic flow, smaller SC values mean more stable traffic flow. Fig. 6 shows that the characteristics of the four SC functions using the plot of the SC value versus uniform flow velocity. The parameters in HEOV model are set as Vcmax = 3, Vtmax = 2.5, dc = 3, acc = 1.4, act = 1.3, atc = 1.1 and att = 1. The four curves in Fig. 6 have four intersections resulting in five scenarios regarding the relationship among the four SC s. (1) TT>TC>CT>CC. This scenario happens before the first intersection point from the left. In this scenario, cars can better stabilize traffic flow than trucks. The TT combination has the least stabilizing effect, while the CC combination provides the most stabilizing effect. This scenario has a wide range of uniform velocity. When the uniform velocity is small, the four SCs experience a sharp drop around 0.5 with the decreasing of velocity. Such phenomenon indicates that the stability of all four combinations increases as velocity decreases during congestion. (2) TT>CT>TC>CC. This scenario happens between the first intersection and the second intersection from the left. Compared to the scenario (1), the order of CT and TC exchanges. That reveals that a truck following a car is the more stable than a car following a truck in this scenario. However, compare to the scenario (1) this scenario has a very narrow range for the uniform velocity. (3) TT>CT>CC>TC. This scenario occurs between the second intersection and the third intersection from the left. In this scenario, the CC combination is no longer the most stable combination and is less stable than the TC combination. However, following a car still can better stabilize the traffic flow than following a truck. Compared to the scenario (2), the scenario (3) has a wider uniform velocity range. (4) CT>TT>CC>TC. This scenario arises between the third intersection and the fourth intersection from the left, in which the CT combination becomes the most unstable combination and following a car is still more stable than following a truck as the scenario (3). However, compared to the scenario (3), the uniform velocity range is much narrow. (5) CT>CC>TT>TC. This scenario occurs after the last intersection point from the left. In this scenario, the SC values of the TT and TC combinations drop significantly with the increasing of the uniform velocity. The TT and TC combinations can better stabilize traffic flow than the CT and CC combinations. The TC combination becomes the most stable combination. Meanwhile, this scenario has a large velocity range similar to the scenario (1), but driving a truck becomes more stable than driving a car under this high-speed situation.
380
D. Yang et al. / Physica A 395 (2014) 371–383
Fig. 7. (a) The velocity against vehicle number and time in a stable scenario (b) the wave peak plot.
From the above five scenarios, the following conclusion can be drawn. Cars and trucks have different magnitudes of impact on traffic flow stability depending on the uniform flow velocity. Furthermore, all four curves can have segments that are greater than zero for some combinations of sensitivity a and maximal velocity V max , and all four types of combinations can both stabilize and destabilize traffic flow. It should be pointed out that results from Mason et al. [24] and Jin et al. [25] are only some special cases of the mixture of cars and trucks effects on traffic flow among the above five scenarios. The numerical simulation is conducted to demonstrate the different effects of the cars and truck on traffic flow. Similar to those in Section 3.3, 100 vehicles are placed on a circular route. However, the sequence of vehicles is changed to 40 alternating cars and trucks, 30 cars, and then 30 trucks. The parameters in HEOV model are the same as those used for Fig. 6. The uniform velocity is chosen as 2.3. In this case, the stability function of the mixed traffic flow Fl = −0.9649, which reveals that the traffic flow is stable. Calculate the four SC s by using the equations in (39): SC cc = 0.4599, SC ct = 0.5630, SC tc = −2.7360, and SC tt = −2.2276. The four stability values reveal that the CC and CT combinations can amplify the perturbation, but the TC and TT combinations will suppress the perturbation. The simulation results are shown in Fig. 7 with a 3D plot illustrating the velocity versus time and vehicle numbers. It can be observed that the perturbation eventually disappears, and the traffic flow returns to uniform state, which is consistent with the result of the traffic flow stability criterion. The temporal profile of the perturbation wave peak is also shown in Fig. 7(b). The peak fluctuates but descends before reaching vehicle No. 40 (alternating cars and trucks). When the perturbation propagates from the vehicle No. 40 to No. 70 (cars only) the peak value slightly ascends and after vehicle No. 70 (trucks only) the peak value descends. Trucks in the TT combination make the traffic flow stable, but they make the traffic flow unstable in the CT combination. Cars make the traffic flow unstable in the CC combination, but stabilize the traffic flow in the TC combination. Thus, the simulation results support the conclusion that cars and trucks have both stable and unstable effects on traffic flow. 4.2. Proportional impact of different car–truck following scenarios The relationship between the stability function of the car–truck mixed traffic flow and the proportions of the four types of combination on a circular route can be written as the following: Fl = pcc · SCcc + pct · SCct + ptc · SCtc + ptt · SCtt pcc + pct + ptc + ptt = 1 pct = ptc .
(41)
The third equation in (41) comes from the pairing of CT and TC combination on a circular road. Meanwhile, if the SC of one type of combination is less than zero, increasing the proportion of the type of combination makes the traffic flow more stable; otherwise, it makes the traffic flow unstable. According to the latter two equations in (41), any two fixed proportions can determine the other two proportions and the available ranges of pcc , pct , ptc and ptt are [0, 1], [0, 0.5], [0, 0.5] and [0, 1] on a circular road. With these valid ranges, a numerical experiment can be designed. In this experiment, acc = 2, act = 1.7, atc = 1.4, att = 1.1, Vcmax = 3 and Vtmax = 2.5. Two uniform velocities are chosen as the examples here, v ∗ = 0.45 and v ∗ = 2.18. When v ∗ = 0.45, SC cc = −0.3737, SC ct = −0.1461, SC tc = 0.1178, and SC tt = 0.6402. In this case, increasing the proportions of the TC and TT combinations can stabilize traffic flow. However, the increase of the proportions of CT and CC combinations can make the traffic flow unstable. When v ∗ = 2.18, SC cc = 0.1317, SC ct = 0.2807, SC tc = −0.7168, and SC tt = −0.0069. In this case, the increase of the proportions of the CC and CT combinations can stabilize the traffic flow, while increasing the proportions of the TC and TT combinations can make the traffic flow unstable. Fig. 8 is the scatter plots of the stability function for the two uniform velocities v ∗ = 0.45 and v ∗ = 2.18. The blue line is the neutral stability line connecting all the points where Fl = 0. Based on the parameters and the formula (41), the proportion of CT combination is a monotonically decreasing function with respect to the proportion of CC combination for
D. Yang et al. / Physica A 395 (2014) 371–383
(a) v ∗ = 0.45.
381
(b) v ∗ = 2.18. Fig. 8. The scatter plots of the stability function and the neutral stability line of the car–truck mixed traffic flow.
v ∗ = 0.45 and a monotonically increasing function for v ∗ = 2.18. In Fig. 8(a), since the stability functions of the CC and CT combinations are negative, the smaller the proportions, the bigger the values of the stability function of traffic flow. In this case, the slope of the neutral stability line is negative. The line splits the quadrant into two parts, the unstable area to the left and the stable area to the right. When the proportion of the CC combination is more than 0.65, the traffic flow is always stable for v ∗ = 0.45 regardless of the other three proportions. In Fig. 8(b), the neutral stability line has the positive slope. It also splits the quadrant into two parts, however, with the left-side area being stable and the right-side area being unstable. When the proportion of the CC combination is more than 0.65 for v ∗ = 2.18, the traffic flow is always unstable, while when the proportion of the CT combination is more than 0.19, the traffic flow is always stable. Eq. (39) indicates that, in heterogeneous traffic flow, the proportions of the different car–truck combinations play more critical roles for traffic flow stability than the proportions of different vehicle types. This conclusion can be verified using numerical simulation. We reuse the same parameters in Fig. 8(b). The proportions of the four combinations are initially set as pcc = 0.38, pct = 0.12, ptc = 0.12, and ptt = 0.38. This is a stable case as shown in the headway evolution plot (see Fig. 9(a)). Then we remove one CT combination from the platoon and add one TT combination and one CC combination. This does not change the numbers of the cars and trucks but change the proportions of four combinations to pcc = 0.39, pct = 0.11, ptc = 0.11 and ptt = 0.39. Fig. 9(b) shows the simulation result. Traffic flow becomes unstable from the stable case in Fig. 9(a). Thus, the comparison of two cases proves the significance of proportions of car–truck following combinations on the stability of traffic flow. Moreover, another simulation is conducted to demonstrate that the positions of different combinations are not critical factor to influence the linear stability of traffic flow. Adopt the same parameters with the first case and move the half of CT and TC combinations to the end of the platoon. The simulation result is shown in Fig. 9(c). It reveals that the traffic flow is still stable and not influenced by the position changes of the vehicles. 5. Conclusion and future work In this paper, the Bando’s OV car-following is extended to the heterogeneous OV model (HEOV) that can describe heterogeneous traffic flow with four different car–truck following combinations, namely, a car following a car (CC), a car following a truck (CT), a truck following a car (TC) and a truck following a truck (TT). The linear stability method is adopted to derive the stability criterion of HEOV model, and then the stability criterion is verified by numerical simulations. The stability contribution of each car–truck combination is analyzed. Five scenarios exist for the relationship of the four combinations. Under different scenarios, the car and truck both have stable and unstable effect on traffic flow. Moreover, the influence of the combination proportions is explored by both analytical analysis and simulation methods. The results also reveal that, in mixed traffic flow, the proportions of car–truck combinations rather than the proportions of vehicle types determine the traffic flow stability more. However, this paper only focuses on one-dimensional traffic flow, but the two-dimensional traffic flow is more realistic, so the stability of two-dimensional traffic flow needs to be explored in future. Appendix A. Derivation of the first identity The first identity can be derived from the following equation
n
an fn
m̸=n
2 am fm = an f n am f m = an f n am f m . n
n
m̸=n
n
n
m̸=n
(A.1)
382
D. Yang et al. / Physica A 395 (2014) 371–383
(a) The stable case when pcc = 0.38, pct = 0.12, ptc = 0.12 and ptt = 0.38.
(b) The unstable case when pcc = 0.39, pct = 0.11, ptc = 0.11 and ptt = 0.39.
(c) The stable case when pcc = 0.39, pct = 0.11, ptc = 0.11 and ptt = 0.39. Fig. 9. The headway plots for the different proportions of the four combinations.
Reorganizing Eq. (A.1) yields,
n
am fm
n
2 am fm
an f n
m̸=n
=
m̸=n
.
an fn
(A.2)
n
Appendix B. Derivation of the second identity The second identity is derived from the following equation
αn
n
βm
·
γn
n
m̸=n
βm
m̸=n
=
αn γn
n
2 βm + αn γn βm · βn . (B.1) i ,j
m̸=n
m̸=i,j
n
If αn = γn , rewriting Eq. (B.1) yields,
n
2
αn
m̸=n
βm
=
n
αn2
m̸=n
2 βm + 2 αi αj βm · βn . i̸=j,j>i
m̸=i,j
n
(B.2)
D. Yang et al. / Physica A 395 (2014) 371–383
383
In the case of this paper, αn and βn can be written as the following.
αn = an βn = an fn .
(B.3) (B.4)
Substituting Eqs. (B.3) and (B.4) into (B.2) leads to
i̸=j,j>i
ai · aj
m̸=i,j
am f m
=
1 2
n
an f n
n
2 an
m̸=n
am fm
−
n
a2n
2 am f m . m̸=n
(B.5)
References [1] D. Balcan, V. Colizza, B. Gonçalves, H. Hu, J.J. Ramasco, A. Vespignani, Multiscale mobility networks and the spatial spreading of infectious diseases, Proceedings of the National Academy of Sciences 106 (2009) 21484–21489. [2] L. Hufnagel, D. Brockmann, T. Geisel, Forecast and control of epidemics in a globalized world, Proceedings of the National Academy of Sciences of the United States of America 101 (2004) 15124–15129. [3] S. Ni, W. Weng, Impact of travel patterns on epidemic dynamics in heterogeneous spatial metapopulation networks, Physical Review E 79 (2009) 016111. [4] L. Wang, X. Li, Y.-Q. Zhang, Y. Zhang, K. Zhang, Evolution of scaling emergence in large-scale spatial epidemic spreading, PloS One 6 (2011) e21197. [5] S. Meloni, A. Arenas, Y. Moreno, Traffic-driven epidemic spreading in finite-size scale-free networks, Proceedings of the National Academy of Sciences 106 (2009) 16897–16902. [6] D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics 73 (2001) 1067–1141. [7] A. Schadschneider, Traffic flow: a statistical physics point of view, Physica A 313 (2002) 153–187. [8] D. Chowdhury, L. Santen, A. Schadschneider, Physics Reports 323 (2000) 199. [9] B.S. Kerner, The physics of traffic: empirical freeway pattern features, engineering applications, and theory, in: The Physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory, Springer-Verlag, Berlin, 2004. [10] A. Schadschneider, D. Chowdhury, K. Nishinari, Stochastic transport in complex systems: from molecules to vehicles, in: Stochastic Transport in Complex Systems: From Molecules to Vehicles, Elsevier Science, Amsterdam, 2010. [11] J. Taylor, X. Zhou, N.M. Rouphail, Method for investigating intradriver heterogeneity using vehicle trajectory data: a dynamic time warping approach, in: N.H. Gartner (Ed.), Transportation Research Board 91st Annual Meeting, Transportation Research Board of the National Academies, Washington, DC, 2012. [12] S. Ossen, S.P. Hoogendoorn, B.G.H. Gorte, Interdriver differences in car-following: a vehicle trajectory-based study, Transportation Research Record: Transportation Research Record 1965 (2006) 121–129. [13] S. Ossen, S.P. Hoogendoorn, Driver heterogeneity in car following and its impact on modeling traffic dynamics, Transportation Research Record: Transportation Research Record 1999 (2007) 95–103. [14] L. Davis, Effect of adaptive cruise control systems on mixed traffic flow near an on-ramp, Physica A 379 (2007) 274–290. [15] M.M. Islam, C.F. Choudhury, A violation behavior model for non-motorized vehicle drivers in 4 heterogeneous traffic streams, in: N.H. Gartner (Ed.), Transportation Research Board 91st Annual Meeting, Transportation Research Board, Washington, DC, 2012. [16] S. Jin, D. Wang, P. Tao, P. Li, Non-lane-based full velocity difference car following model, Physica A. Statistical Mechanics and its Applications 389 (2010) 4654–4662. [17] S. Jin, D.H. Wang, X.R. Yang, Non-lane-based car-following model with visual angle information, in: Transportation Research Record: Transportation Research Record, Transportation Research Board of the National Academies, Washington, DC, 2011, pp. 7–14. [18] J.R. Sayer, M.L. Mefford, R. Huang, The Effect of Lead-Vehicle Size on Driver Following Behavior, 2000. [19] K. Huddart, R. Lafont, Close driving—hazard or necessity? in: XV European Annual Conference on Human Decision Making and Manual Control, TNO, Netherlands, 1990, pp. 205–217. [20] M. McDonald, M. Brackstone, B. Sultan, C. Roach, Close following on the motorway: initial findings of an instrumented vehicle study, VISION IN VEHICLES-VII, 1999. [21] S. Peeta, W. Zhou, P. Zhang, Modeling and mitigation of car–truck interactions on freeways, Transportation Research Record 1899 (2004) 117–126. [22] S. Peeta, P. Zhang, W. Zhou, Behavior-based analysis of freeway car–truck interactions and related mitigation strategies, Transportation Research Part B: Methodological 39 (2005) 417–451. [23] K. Aghabayk, M. Sarvi, W. Young, Y. Wang, Investigating heavy vehicle interactions during the car following process, in: Transportation Research Board 91st Annual Meeting, Transportation Research Board of the National Academies, Washington, DC, 2012. [24] A.D. Mason, A.W. Woods, Car-following model of multispecies systems of road traffic, Physical Review E 55 (1997) 2203–2214. [25] S. Jin, D.H. Wang, Z.Y. Huang, P.F. Tao, Visual angle model for car following theory, Physica A. Statistical Mechanics and its Applications (2011). [26] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Physical Review E 51 (1995) 1035–1042. [27] R.E. Wilson, Mechanisms for spatio-temporal pattern formation in highway traffic models, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366 (2008) 2017–2032. [28] H.X. Ge, X.P. Meng, R.J. Cheng, S.M. Lo, Time-dependent Ginzburg–Landau equation in a car-following model considering the driver’s physical delay, Physica A: Statistical Mechanics and its Applications 390 (2011) 3348–3353. [29] J.A. Ward, Heterogeneity, Lane-Changing and Instability in Traffic: A Mathematical Approach, Ph.D. Thesis. Thesis, 2009. [30] J.A. Ward, R.E. Wilson, Criteria for convective versus absolute string instability in car-following models, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 467 (2011) 2185–2208.