Modeling and simulation of the car-truck heterogeneous traffic flow based on a nonlinear car-following model

Applied Mathematics and Computation 273 (2016) 706–717

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Modeling and simulation of the car-truck heterogeneous traffic flow based on a nonlinear car-following model Lan Liu a,b, Liling Zhu a,b, Da Yang a,b,∗ a b

School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610031, China National United Engineering Laboratory of Integrated and Intelligent Transportation, Southwest Jiaotong University, Chengdu 610031, China

a r t i c l e

i n f o

Keywords: Modeling Simulation Car and truck traffic flow Intelligent Driver Model Car-following combinations

a b s t r a c t The traffic flow heterogeneity caused by vehicle type difference has drawn increasing attention recently. This paper uses real data to explore the characteristics of the traffic flow consisting of the four types of car-truck car-following combinations, car-following-car (CC), carfollowing-truck (CT), truck-following-car (TC) and truck-following-truck (TT). To overwhelm the shortcoming that the existing car-following model, Optimal Velocity Model, cannot reflect the complexity of real heterogeneous traffic flow, a new model is proposed based on a nonlinear ordinary differential car-following model, Intelligent Driver Model (IDM). Next Generation Simulation (NGSIM) vehicle trajectory data is applied to calibrate and evaluate the proposed model. Based on the calibrated model, the traffic regime, linear stability, fundamental diagrams, and shock wave characteristics of the car-truck heterogeneous traffic flow are investigated. The results reveal some new findings of the car-truck heterogeneous traffic flow. The mixture of congested and free flow regime occurs when the trucks reach their maximum speeds. Cars and trucks can both stabilize and destabilize the traffic flow, depending on the combination type and the equilibrium velocity. Moreover, the fundamental diagrams of different car-truck combinations converge to several clusters with the same proportion difference between the CC and TT combinations. The speeding-up effect of trucks on shock wave propagation in the car-truck heterogeneous traffic flow is observed in the simulation. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Heterogeneity is a crucial characteristic of real-world traffic flow. Although traffic flow theories and models are usually developed first for the homogeneous traffic flow, most of them can be easily converted into their heterogeneous forms. The difficulty in studying the heterogeneous traffic flow is the lack of fine data to calibrate the heterogeneous traffic flow models. With the development of the new data collection technology in recent years, such as the video-based method and the GPS-based method, traffic information, especially individual vehicle dynamics, can be obtained in detail. Many researchers use new data sources to investigate the characteristics of car-truck heterogeneous traffic flow in the past years. Early researchers focused on investigating the driving behavior differences between cars and trucks. Huddart and Lafont [1], McDonald et al. [2] and Sayer et al. [3] compared the headway differences between the two cases, car-following-car and car-following-truck. However, their studies did not reach conclusive results that which case had the larger headway. Peeta et al. [4,5] analyzed interactions of cars and trucks in multiple lanes. Highway Capacity Manual [6] presented that trucks occupied more space, had poorer operating capabilities ∗

Corresponding author. Tel.: +86 18780065505. E-mail address: [email protected] (D. Yang).

http://dx.doi.org/10.1016/j.amc.2015.10.032 0096-3003/© 2015 Elsevier Inc. All rights reserved.

L. Liu et al. / Applied Mathematics and Computation 273 (2016) 706–717

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and could create larger gaps than cars. However, these studies [1–5] did not mention that the car-following behavior in car-truck flow also depended on the type of the following vehicle. Ye’s study [7] first explored the impact of the following vehicle type (car or truck) on traffic flow. He concluded that the four types of car-truck combination should be taken into account in the study of the car-truck heterogeneous traffic flow, that is, car-following-car (CC), car-following-truck (CT), truck-following-car (TC) and truck-following-truck (TT). Sarvi [8] also studied the driving behavior of the three car-following combinations, car-following-car, truck-following-car, and car-following-truck. Aghabayk et al. [9] further empirically studied variations of the distance headway, time headway, reaction time and car-following threshold among the four types of combination. One major limitation of the early studies on car-truck traffic flow is their lack of modeling of the dynamic car-truck traffic flow. Mason and Woods [10] extented the homogeneous Optimal Velocity (OV) car-following model to a heterogeneous form to describe the interaction between cars and trucks. The derived OV heterogeneous car-following model is as follows,

d2 xn (t ) = λn (Un (xn−1 − xn ) − vn ) dt 2

(1)

where xn (t) and vn (t) respectively denote the location and velocity of the vehicle n at time t, xn− 1 (t) denotes the location of the vehicle n − 1 (the preceding vehicle of n) at time t, λn denotes the sensitivity parameter of the vehicle n, and Un (xn+1 − xn ) denotes the optimal velocity function the vehicle n wishes to take and it is the function of the headway of vehicle n. However, the Optimal Velocity Model is a simple model, which cannot reflect the real characteristics of heterogeneous cartruck traffic flow. Therefore, in this study, based on a more realistic nonlinear ordinary differential car-following model, Intelligent Driver Model (IDM), we develop a new model for the heterogeneous traffic flow. The model has a sub-model for each car-truck following combination. The model parameters for each car-truck car-following combination are calibrated separately using the real car-following data extracted from NGSIM (Next Generation Simulation) data. Based on the calibrated model, we study the four characteristics of the car-truck heterogeneous traffic flow, traffic regime, linear stability, fundamental diagram, and shock wave.

2. Methodology 2.1. An IDM-based nonlinear car-following model for the car-truck heterogeneous traffic flow Treiber et al. [11] proposed Intelligent Driver Model in 2000 for the homogeneous traffic flow. This model is a widely explored car-following model [12–14], and its formulation is as follows,

⎧   δ  2 2 ⎪ d x ( t ) ( t ) S (v ( t ) ,  v ( t )) v ⎪ n n n n ⎪ − ⎨ dt 2 = a 1 − V xn (t ) − l

⎪ ⎪ · vn (t ) ⎪ 0 1 ⎩S(vn (t ), vn (t )) = s + s vn (t ) + τ vn (t ) − vn (t ) √ V

(2)

2 ab

where a denotes the maximum acceleration, V denotes the desired velocity, δ denotes the acceleration exponent, S () denotes the desired minimum gap, s0 and s1 denote the jam distances, τ denotes the safe time headway, b denotes the desired deceleration, l denotes the leading vehicle length, and vn (t ) = vn−1 (t ) − vn (t ) denotes the velocity difference between the vehicle n and its preceding vehicle n − 1. We develop the homogeneous IDM to its heterogeneous form by giving subscripts to the model parameters. This procedure of introducing heterogeneity into car-following models was also described by Treiber and Kesting [15]. The proposed heterogeneous IDM formulates the four different car-truck car-following combinations as follows,

 ⎧  δn  2 2 ⎪ d x ( t ) ( t ) S (v ( t ) ,  v ( t )) v n n n n n ⎪ ⎪ − ⎨ dt 2 = an 1 − Vn xn (t ) − ln

⎪ vn (t ) vn (t ) · vn (t ) ⎪ ⎪ + τn vn (t ) −  ⎩Sn (vn (t ), vn (t )) = s0n + s1n Vn

(3)

2 an bn

where all the parameters an , δ n , Vn , s0n , s1n , τ n and bn have four alternatives. Taking an as an example, an can be acc , act , atc and att . The leading vehicle length l has two alternatives, lc and lt . In the homogeneous traffic flow, all vehicles at equilibrium states have zero acceleration, the same distance headway, and the same velocity. However, for the heterogeneous traffic flow, the equilibrium state is quite different, in which all vehicles have zero acceleration and the same velocity, but their distance headways vary for different types of vehicle. The equilibrium state can be described using the following equations,

vn = v∗ , v˙ n = 0, and hn = h∗n

(4)

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L. Liu et al. / Applied Mathematics and Computation 273 (2016) 706–717

where v∗ denotes the equilibrium velocity that is the same for all vehicles, h∗n denotes the corresponding equilibrium headway of the vehicle n, and h∗n varies among the different types of vehicle. Substituting (4) into the heterogeneous IDM (3) yields,

⎧ 2  ⎪ ⎨1 − v∗ δ − sˆ(v∗ , 0) = 0 v0 h∗IDM − l ⎪  ⎩ ∗ sˆ(v , 0) = s0 + s1 v∗ /v0 + τ v∗

(5)

Rewriting the above equation yields,

h∗ =

s0 + s1



√

v∗ /v0 + τ v∗

1 − (v∗ /v0 )δ

+l

(6)

∗ and h∗ that correspond to the four types of In the car-truck heterogeneous flow h∗ has the four alternatives, h∗cc , h∗ct , htc tt combination. It should be noticed that the vehicle length l in h∗ only depends on the leading vehicle type, namely, l = lc in h∗cc ∗ , and l = l in h∗ and h∗ . and htc t ct tt Therefore, the fundamental diagram function of the heterogeneous IDM can be derived using Eq. (6). Assume a given heterogeneous traffic flow on a single lane contains N vehicles. The length of the entire traffic flow in the equilibrium state is,

Ltotal =

N 

h∗n

(7)

n=1

In the car-truck traffic flow, Ltotal has the following formula, ∗ ∗ Ltotal = Ncc h∗cc + Nct h∗ct + Ntc htc + Ntt htt

(8)

where Ncc , Nct , Ntc and Ntt are the numbers of the CC, CT, TC and TT combinations in the car-truck heterogeneous traffic flow. Hence, the density of traffic flow can be calculated as,

k=

N ∗ + N h∗ Ncc h∗cc + Nct h∗ct + Ntc htc tt tt

(9)

Rewrite Eq. (9) as the following form,

k=

1 ∗ + P h∗ Pcc h∗cc + Pct h∗ct + Ptc htc tt tt

(10)

where Pcc , Pct , Ptc and Ptt denote the proportions of the CC, CT, TC and TT combinations. Thus, the fundamental diagram of the car-truck heterogeneous traffic flow has the following equation,

q=

v∗ Pcc h∗cc

Pct h∗ct

+

(11)

∗ + P h∗ + Ptc htc tt tt

where q denotes the flow rate of the car-truck heterogeneous traffic flow in space. Eq. (11) indicates that the proportions of the four types of combination and the distance headways can determine the fundamental diagram of the car-truck heterogeneous traffic flow. 2.2. Linear stability analysis of the car-truck heterogeneous traffic flow The linear stability analysis of traffic flow investigates the perturbation propagation characteristic in a vehicle platoon by adding a small perturbation on the first vehicle [13,16]. The stability criterion of IDM can be derived following the general stability criterion summarized by Wilson and Ward [13]. The derivation of the detailed stability criteria of IDM requires the partial differentials of the velocity, velocity difference and headway between the leading and following vehicles with respect to the equilibrium state of IDM.

fv = −aδ

v∗ δ−1 V

−

2aS∗

(

h∗

− l)

S , 2 v

f v = −

2aS∗

(h∗ − l )2

−3

Sv , and fh = 2a(S∗ )2 (h∗ − l )

(12)

where S∗ = S(v∗ , 0, h∗ ), Sv , Sv and Sh denote the partial differences of the desired minimum gap function with respect to the vn , vn and xn at (v∗ , 0, h∗ ). Hence, according to the general stability criterion introduced by Wilson and Ward [13], the stability criterion of IDM is as follows,

f2 f v f v + f h − v < 0 2

(13)

Substituting Eq. (12) into Eq. (13) yields the stability criterion of IDM,

2(S∗ )2

(h∗ − l )3

+

 ∗ δ−1   ∗ δ−1 2 2Sv S∗ a 2S∗ v v + − + <0 δ δ v0 2 v0 (h∗ − l )2 (h∗ − l )2 (h∗ − l )2 2aS∗ Sv

(14)

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Treiber and Kesting [15,17] also introduced a general method to derive the stability criterion of IDM. However, these two methods are only applicable for the homogeneous traffic flow. In investigating the stability of the car-truck heterogeneous traffic flow, we need new method. Fortunately, Ward [18] presented a general formula of the linear stability criterion of the heterogeneous traffic flow as follows,



⎡  2 ⎤    ⎣ fn,v · fn,v + fn,h − fn,2 v /2 · f j,h ⎦ < 0

n

(15)

j=n

where fn,v , fn,v and fn,h respectively denote the partial differences of the car-following model adopted by the vehicle n with respect to v, v and x at the equilibrium state where vn = v∗ , vn = 0 and xn = h∗n . However, Eq. (15) cannot directly reflect the proportion information of the different combinations, so we rewrite it as the following form using induction method,

⎡

⎤

Q Q     2NPi ⎥ ⎢  2NP −2 f j,h ⎣NPi fi,v · fi,v + fi,h − fi,2v /2 · ( fi,h ) i ⎦ < 0,

(16)

j=i j=1

i=1

where Q denotes the number of the combinations in a given car-truck heterogeneous traffic flow, N denotes the number of N vehicles in the traffic flow, Ni denotes the number of the combination type i in the platoon, Pi = Ni denotes the proportion of the M combination type i in total, and i=1 Pi = 1. In the car-truck heterogeneous traffic flow, Q = 4 and i has the four alternatives, CC, CT, TC and TT respectively. Eq. (16) can be further simplified as follows, Q 



Pi

fi,v · fi,v + fi,h − fi,2v /2 2 fi,h

i=1



<0

(17)

In Eq. (17), the two parts determine the stability criterion of the car-truck heterogeneous traffic flow, the fraction part fi,v · fi,v + fi,h − fi,2v /2 f2

providing the stability characteristic of the combination i and the proportion part Pi of the combination i. We

i,h

define the fraction part as the Stability Function (SF) of the combination, that is,

SFi ( fi,v , fi,v , fi,h ) =

fi,v · fi,v + fi,h − fi,2v /2

(18)

2 fi,h

where SFi denotes the stability function of the combination type i. In addition, comparing the stability criterion of IDM (Eq. (14)) with Eq. (18), it can be found that SF has the same sign with (14), so SF also can be used to judge the stability of the car-following combination i. The stability effect SFi of combination type i in the car-truck heterogeneous traffic flow can be written as follows,

 SFi =

   2 ⎤ 6 ⎡  ∗  2



∗ ∗ ∗ δ −1 δ −1 ∗ ∗ i − li 2 Si 2ai Si Si,v 2Si,v Si 2Si ai v v + +  4 · ⎣  ∗ 3 +  ∗ 2 δi V 2 − 2 δi V 2 ⎦ 2 ∗ ∗ ∗

h∗i

4ai Si

hi − li

hi − li

i

hi − li

i

hi − li

(19)

Moreover, we define the left part of Eq. (17) as the stability function F of the car-truck heterogeneous traffic flow. Thus, in the car-truck heterogeneous traffic flow, there is,

F = Pcc · SFcc + Pct · SFct + Ptc · SFtc + Ptt · SFtt

(20)

When F < 0, the car-truck heterogeneous traffic flow is stable. Eq. (20) indicates that the proportions of the car and truck are not the only factors of deciding the stability of heterogeneous traffic flow, while the stability function of each car-truck following combination is significant as well. Furthermore, on a ring road, the stability function of the car-truck heterogeneous traffic flow and the proportions of four types of combination have the following relationship,

⎧ ⎨F = Pcc · SCcc + Pct · SCct + Ptc · SCtc + Ptt · SCtt Pcc + Pct + Ptc + Ptt = 1

⎩

(21)

Pct = Ptc

3. Experimental design 3.1. Data description The NGSIM trajectory data with high time resolution is adopted to analyze the characteristics of the car-truck traffic flow in this paper. This study uses the detail vehicle trajectory data collected at Hollywood Freeway (U.S.101) and Berkeley Highway (I-80) in California. The entire segment of U.S.101 is approximately 640 m in length, with five main lanes throughout the section

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L. Liu et al. / Applied Mathematics and Computation 273 (2016) 706–717

Fig. 1. Car-following gap of the four types of combination.

and one auxiliary lane. The data collection time is 45 min between 7:50 and 8:35 a.m. I-80 section is 503 m long and has five main lanes and one auxiliary lane. I-80 data was collected from 4:00 to 4:15 p.m. and from 5:00 to 5:30 p.m. The data has the resolution of 0.1 s, which is fine enough to investigate the characteristics of car-following behavior. The car-following data for the four combinations are extracted separately from the NGSIM data. Since the data includes lane change and large vehicle gaps, we use the following criteria to extract the car-following datasets.

(a) Divide car-following platoons into multiple car-following pairs. For example, a three vehicle car-following platoon is divided into two car-following pairs in which the middle vehicle is the follower of the first pair and is the leader of the second pair. (b) Each pair of car-following vehicle is composed and decomposed according to the two space thresholds, the engaging threshold DE = 39.62 m and the disengaging threshold DD = 45.72 m. These values are determined based on the critical density at the experimental sites. Using two thresholds rather than one threshold can avoid unnecessary frequent composing and decomposing of car-following vehicle pairs caused by small fluctuation of space value around a single threshold. (c) If a vehicle pair breaks due to space increment, lane-changing or reaching the end of a segment, it will not be exported as a valid sample.

The final extracted dataset includes 477,712, 25,844, 16,471 and 2051 observations of CC, CT, TC and TT combinations respectively. These observations belong to 1518, 87, 83 and 15 trajectories of CC, CT, TC and TT combinations. Each trajectory contains 100 to 1000 observations. The datasets are analyzed to examine whether they can reflect the microscopic characteristics of the car-truck heterogeneous traffic flow. The three main microscopic characteristics of the car-truck heterogeneous traffic flow are analyzed respectively, the car-following gap of the four combinations, the maximum speeds of the car and truck and the response times of the car and truck. The gaps are categorized based on the velocities of following vehicles with 2 m/s intervals for each combination. The gap variation with respect to the velocity of the following vehicle is exhibited in Fig. 1. It is obvious that different types of combination have different car-following gaps. The CC combination always has the smallest average gap, and the TT combination always has the largest average gap in the entire feasible velocity range. When the velocity is 17 m/s, the CT combination has the same gap with the TC combination. When the velocity is less than 17 m/s, the CT combination tends to keep larger gap than the TC combination, while the TC combination has larger gap than CT combination after the velocity is more than 17 m/s. The statistical results show that the truck and the car has the maximum speed of 18 m/s and 30 m/s respectively. In the CT combination, the average speed of the combination is restricted by the leading vehicle (the truck). The statistical average speed of the CT combination is around 18 m/s. In the TC combination, the average speed of the combination is restricted by the following slow truck, but the leading vehicle type also has impact on the average speed. The statistical result of this case is around 23 m/s. The CC and TT combinations respectively have the maximum speeds of 30 m/s and 16 m/s. The response time refers to the period from the occurrence of a stimulus to the reaction of the following vehicle [9]. Since the response time is affected significantly by the vehicle mechanical system [19], driving a car and driving a truck have obvious different response times. The measuring method of the response time used in this paper is the one introduced by Aghabayk et al. [9]. Finally, the average response time of driving trucks is 2.1 s and the average response time of driving cars is 1.3 s. In summary, the extracted data can reflect the microscopic characteristics of the car-truck heterogeneous traffic flow. Thus, the calibration results will be effective in the subsequent study of the characteristics of the car-truck heterogeneous traffic flow.

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3.2. Simulation setup Simulations are conducted to investigate the characteristics of the car-truck heterogeneous traffic flow. The two classical traffic simulation scenarios, the straight road scenario and the ring road scenario, are used in this paper. (a) The straight and single-lane road scenario. One hundred vehicles are placed on a single-lane, flat and straight road. The velocity of the first vehicle is given. The initial state of the car-truck heterogeneous traffic flow is the equilibrium state in which all the vehicles have the same initial velocity, and the headways of the four combinations follow Eq. (6). The study on traffic regimes in this paper will adopt this simulation scenario. (b) The ring and single-lane road scenario. One hundred vehicles are placed on a single-lane, flat and ring road, and the vehicle No. 1 follows the vehicle No. 100. The initial state of traffic flow is the equilibrium state, and then adds a small perturbation on the first vehicle to observe the perturbation development in the platoon. Numerous studies adopted this simulation method to explore the fundamental diagram [20,21], shock wave [21,22], linear stability [13,23], and other properties [24] of traffic flow. 4. Model calibration and evaluation 4.1. Calibration and evaluation method The car-following model calibration process can be abstracted as a nonlinear optimization problem. The objective function is the difference between the simulation results and real traffic data. The independent variables are the model parameters. The constraints are the physical ranges of the model parameters. The following general equation can express this nonlinear optimization problem,

min Fob jective = f (v, x, a, p) hi ( p) = 0(i = 1, 2, · · · nh )

(22)

g j ( p) ≥ 0( j = 1, 2, · · · ng ) where Fobjective denotes the objective function with respect to the vehicle state (velocity, position, acceleration et al.), v, x and a respectively denote the velocity, position and acceleration of the vehicle, p denotes the parameter set, hi (p) denotes the ith equality constraint, gj (p) denotes the ith inequality constraint, and nh and ng denote the numbers of the equality and inequality constraints. In this paper, Genetic Algorithm (GA) is employed to solve the nonlinear optimization problem. The objective function can have many choices. In this study, Theil’s U function that has been proved to be effective in car-following model calibration [25,26] will be applied. The formula of Theil’s U function is as follows,



1 M



Fob jective =



M

1 M

m=1



M m=1

yreal m

2

sim yreal m − ym



+

1 M

2

M m=1



ysim m

2

(23)

sim where yreal m denotes the real data, ym denotes the simulation result based on the model, and m = 1, …, M denotes the number of the data sample. In Eq. (23), the variable y adopts vehicle acceleration in this paper. There are two reasons for this. The first one is that IDM is an acceleration model, and using the acceleration as y variable is more direct and convenient than using the velocity or space. The second one is that if using the velocity or space as the calibration variable, we need to calculate it first according to the derived acceleration from the model, which may result in error accumulation. The model evaluation is to test the prediction error of the calibrated car-following model by comparing the simulation results with real data. The used evaluation indexes in this paper are Mean Error (ME), Mean Absolute Error (MAE) and Mean Absolute Relative Error (MARE). They are widely used in the evaluation of the car-following models [26,27] and have the following formulas,

M m=1

ME =

sim (yreal m − ym )

M

M MAE =

m=1

  sim  yreal m − ym M

M MARE =

m=1

  sim  real (yreal m − ym /ym ) M

(24)

(25)

(26)

where m denotes the mth sample, M denotes the total number of samples, yreal m denotes the mth real data sample (y is the denotes the mth simulation data sample. acceleration, velocity or position), and ysim m

712

L. Liu et al. / Applied Mathematics and Computation 273 (2016) 706–717 Table 1 Calibration results of the four combinations. Variables

CC

CT

TC

TT

an (m/s2 ) Vn (m/s) s0n (m) s1n (m) τ n (s) bn (m/s2 )

1.01 27 0.85 0.19 1.2 2.26 4

1.03 19.3 1.35 0.27 1.4 2.12 4

0.78 20.6 1.11 0.12 1.8 1.70 4

0.74 17.7 1.53 0.36 2.0 1.61 4

δn

Table 2 Evaluation results of the four combinations. Indexes Acceleration

Velocity

Location

ME MAE MARE ME MAE MARE ME MAE MARE

Values 0.84 1.17 0.11 −0.25 1.31 0.08 2.44 9.71 0.05

0.28 1.67 0.13 0.15 1.62 0.09 3.00 9.6 0.06

−0.03 1.24 0.12 −0.23 1.2 0.07 −1.93 5.75 0.03

0.13 1.16 0.1 −0.04 0.75 0.05 1.23 5.61 0.04

4.2. Calibration and evaluation results Table 1 displays the calibration results of the heterogeneous IDM. The TT combination has the largest jam spaces (s0n and s1n ), while the CC combination has the smallest jam spaces. The order of the safe time headway τ n is CC < CT < TC < TT. Cars have larger deceleration (bn ) than trucks. The acceleration exponents are the same for all the four types of combination. The values of the error indexes of acceleration, velocity, and location for each combination type are presented in Table 2. The acceleration simulation errors (MARE) for the four combinations are all less than 15%, and the velocity and location simulation errors (MARE) are all less than 10%. The results indicate that the calibration results can reflect the car-following behavior characteristics of the four combinations in the NGSIM vehicle trajectory data, with small errors.

5. Simulation 5.1. Traffic regime characteristic In the car-truck heterogeneous traffic flow, cars and trucks have different maximum speeds. When the speeds of cars and trucks are below the maximum speed of trucks, the traffic flow is in the congested regime. When the speeds of cars exceed the maximum speed of trucks, trucks cannot match the speeds of the cars. Under this condition, the congested regime is broken, and the car-truck heterogeneous traffic flow enters into a regime mixed by congested flow and free flow. This phenomenon cannot happen in the homogeneous traffic flow in which all the vehicles have the same maximum speeds. We conduct a simulation to illustrate this phenomenon. Adopt the straight road simulation scenario, and set the initial velocity as 10 m/s. The two traffic flows are considered. The first one only contains cars (pure car flow). The second one contains one truck in the position of vehicle No. 51, and all the other vehicles are cars. Let the first vehicles of the two traffic flows accelerate gradually with the fixed acceleration 1 m/s2 . The velocities of the first vehicles will increase gradually from 0 until they reach their maximum velocities. Fig. 2(a) illustrates the pure car traffic flow, and Fig. 2(b) is the result of the heterogeneous traffic flow. Fig. 2(a) and (b) display that the original traffic flow breaks into two segments on the position of vehicle No.51, because the truck cannot catch the velocity of its preceding vehicle (car). Thus, we can conclude that the maximum velocity difference between cars and trucks may induce the mixture of the free flow and congested flow regimes in the car-truck heterogeneous traffic flow. Furthermore, the cars after the truck (No .51) keep the same velocity with this truck, namely, the velocities of the following cars are restricted by the maximum velocity of the truck. This result also reveals that the trucks will restrict the maximum speeds of its following vehicles and the average velocity of the platoon will be heavily impacted by the positions of the trucks in the platoon when trucks reach their maximum velocities. That is why car drivers tend to choose lane-changing to escape from the velocity restrictions of the slow trucks in the multiple-lane situation.

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Fig. 2. Regime characteristic of the car-truck heterogeneous traffic flow. (a) The pure car traffic flow. (b) The car-truck heterogeneous traffic flow.

Fig. 3. Stability functions of the four combinations with respect to the uniform velocity.

5.2. Linear stability 5.2.1. Individual stability of the four combinations The stability function SFi for each combination type i is used to evaluate its individual stability effect on the car-truck heterogeneous traffic flow. Fig. 3 displays the relationship between SF versus the equilibrium velocity for each combination type. The relative order of SF for each combination type has the following three cases. • CT ≈ CC > TC > TT. This case happens before v∗ = 9 m/s. In this case, the CT combination is the most unstable combination, and the TT combination is the most stable combination. The stability effect of the CC combination is close to the CT combination. The truck as the following vehicle is more stable than the car as the following vehicle. • CC > CT > TC > TT. This case occurs in the middle range of velocity from 9 m/s to 11 m/s, and the CC combination becomes the most unstable case. The truck as the leading vehicle in the combination is more stable than the car as the leading vehicle. The truck as the following vehicle in the combination is more stable than the car as the following vehicle. • CC > TC > CT > TT. This case occurs when the velocity is more than 11 m/s. In this case, the CT combination is more stable than the TC combination. The truck as the leading vehicle is more stable than the car as the leading vehicle. From the above analysis, the following conclusion can be drawn. The stability effects of cars and trucks depend on their leading or following states in car-following and the equilibrium velocity of the car-truck heterogeneous traffic flow. The conclusion is more sophisticated than the previous studies [10,28] in which the truck or car only has one single effect (stabilize or destabilize) on the traffic flow. In the following ring-road simulation, we fill the ring road with 30 vehicles alternating between cars and trucks followed by 10 cars, 30 trucks and 30 cars to demonstrate the different effects of the four combinations on traffic flow. Therefore, the

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Fig. 4. The profile of the shock wave peak.

Fig. 5. Scatter plots of the stability function of the car-truck heterogeneous traffic flow. (a) The case v∗ = 1 m/s. (b) The case v∗ = 6 m/s.

proportions of the CC, CT, TC and TT combinations are 39%, 16%, 16% and 29%. Set the equilibrium velocity as 4 m/s. In this case, the stability function of the car-truck heterogeneous traffic flow F = −0.410, which indicates that the traffic flow is stable. The stability functions of the four combinations are SFcc = 0.28, SFct = 0.43, SFtc = −0.83, and SFtt = −1.57, which means that CC and CT combinations can amplify the perturbation, TC and TT combinations will suppress the perturbation, and car-truck alternating can decrease the perturbation (SFct + SFtc < 0). The shock wave peak (see Fig. 4) fluctuates and descends at first (the car-truck alternating part), ascends after the Vehicle No. 31 (the CC combination part), descends after the Vehicle No. 41 (the TT combination part), and ascends slightly again after the Vehicle No. 71 (the CC combination part). Thus, the simulation results are consistent with the analytical analysis. 5.2.2. Impact of the proportions of the four combinations on stability As shown in Eq. (21), the stability function of the car-truck heterogeneous traffic flow is a weighted average of the individual values SFcc , SFct , SFtc , and SFtt according to the proportions of the car-truck combination types. Take the case v∗ = 6 m/s as an example, and obtain the corresponding SF values of the four combinations from Fig. 3. Fig. 5(a) is the scatter plot of the stability function values of the car-truck heterogeneous traffic in the space of the CC and CT proportions (according to Eq. (21), the proportions of CC and TT can determine the proportions of CT and TC). The dots in Fig. 5(a) are the stability function values, and the line is the neutral stability line connecting all the zero stability function values. The neutral stability line splits the quadrant into two areas, the unstable area above the line and the stable area below the line. (0, 0.22) and (0.5, 0) are two critical points. When the proportion of the CC combination is more than 0.5 or the CT combination is more than 0.22, the traffic flow is always unstable regardless of the proportions of the other three combinations. However, when v∗ = 1 m/s (as shown in Fig. 5(b)), the four SF are all negative, which means all four combinations are stable. Thus, no a neutral stability line exists for this case, since the overall stability function values are always less than zero.

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Fig. 6. Fundamental diagrams of the car-truck heterogeneous traffic flow. (a) Fundamental diagrams. (b) Fundamental diagram cluster at Pd = 0.2.

Fig. 7. Simulation of the propagation speed of the shock wave. (a) The pure car traffic flow. (b) The car-truck heterogeneous traffic flow.

5.3. Fundamental diagram We explore the fundamental diagram characteristic of the car-truck heterogeneous traffic flow based on Eq. (11). When examining the fundamental diagram curves for all proportion combinations of the CC, CT, TC and TT, we find that the proportion difference of the CC and TT combinations, Pd = Pcc − Ptt , is a critical factor in determining the shape and location of the overall fundamental diagram. It is found that fundamental diagrams with the similar Pd values are close to each other on the flowdensity plot, as shown in Fig. 6(a). Fig. 6(b) illustrates the cluster for the case of Pd = 0.2. The legends denote the proportions of the four combinations CC, CT, TC, and TT respectively. In Fig. 6(b), the curves have an intersection, and the flow decreases with the increment of Pcc before the intersection and decreases after the intersection. Furthermore, the flow rate and critical density both increase with the increment of Pd , which reveals that the most unstable point of the car-truck heterogeneous traffic flow moves towards the high-flow and high-density area with the increment of Pd . 5.4. Shock waves The propagation velocity of the shock wave is not uniform through the platoon due to the response time and vehicle length differences between cars and trucks. According to Treiber and Kesting [15], the shock wave propagation velocity through a vehicle in the car-following regime is given by the following equation,

cn = −

ln Tn

(27)

where cn denotes the shock wave propagation velocity through the vehicle n, and Tn denotes the response time of vehicle n [15]. In the car-truck heterogeneous traffic flow, the length of trucks is generally more than twice of cars while the response time of truck drivers is less than twice that of car drivers, so |c| increases with the truck percentage. That indicates that the shock wave propagates faster in car-truck flow than in pure car flow. We conduct a simulation to verify this point. Fig. 7 illustrates the shock wave propagation characteristics in both pure car and mixed car-truck flows. In the simulation, the initial velocities for the two cases adopt the same value, which is 15 m/s here. The traffic flow in Fig. 7(a) is the pure car traffic

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flow, and the traffic flow in Fig. 7(b) is the car-truck heterogeneous traffic mixed with 70 cars and 30 trucks. Given the same perturbation to the two types of traffic flow at the same place. From Fig. 7, it is observed that the shock wave takes around 200 s to propagate to the last vehicle in the pure car traffic flow; while in car-truck heterogeneous flow, it takes only around 100 s. The simulations illustrate that the car-truck heterogeneous traffic flow can speed up the shock wave. 6. Conclusion The traffic flow mixed with cars and trucks is the typical heterogeneous traffic flow in real traffic. This paper focuses on the analysis of the impacts of the four types of combination, car-following-car (CC), car-following-truck (CT), truck-followingcar (TC) and truck-following-truck (TT), on traffic flow characteristics using heterogeneous car-following models. The existing studies on heterogeneous traffic flow adopt the Optimal Velocity Model. However, due to the complexity of traffic, the Optimal Velocity Model cannot reflect the real characteristics of car-truck heterogeneous traffic flow. Therefore, we extend a nonlinear car-following model, Intelligent Driver Model (IDM), to its heterogeneous form and use NGSIM data to calibrate and evaluate the proposed model. The evaluation results show that the calibrated car-following models can reflect the car-following dynamics of the four combinations in the car-truck heterogeneous traffic flow. Based on the calibrated car-following model, we explore the four characteristics of the car-truck heterogeneous traffic flow by simulations, traffic regime characteristic, linear stability, fundamental diagrams and shock wave. We derive the linear stability criterion of the car-truck heterogeneous traffic flow and validate it using the ring-road simulations. Two critical factors, the individual stability function and the proportions of the four combinations, are two factors determining the stability of the car-truck heterogeneous traffic flow. In addition, cars and trucks can both stabilize and destabilize the car-truck heterogeneous traffic flow, and their effects depend on their roles (the leading vehicle or following vehicle) in car-following and the equilibrium velocity of the car-truck heterogeneous traffic flow. We also investigate the impact of the proportions of the four combination types on traffic flow. The results show that the proportions of the four combinations may not have significant influence on the stability of traffic flow with low equilibrium velocity (e.g. on an extremely congested road); however, in most cases, the neutral stability lines can be found for the car-truck heterogeneous traffic flow. Fundamental diagrams of the car-truck heterogeneous traffic flow are determined by the distance headways and the proportions of the four combinations. The simulations further reveal that the fundamental diagrams with the same proportion difference between the CC and TT combinations cluster together. The flow rate and the critical density increase with the increment of the proportion difference between the CC and TT. Finally, the shock wave analysis indicates that trucks can speed up the propagation of the shock wave. Acknowledgment This work was supported by National Science Foundation of China (grant nos. 51278429 and 51408509), Soft Science Foundation of Science and Technology Department of Chengdu (grant no. 2014RK0000056ZF), and National Basic Research Program of China (grant no. 2012CB725405). References [1] K. Huddart, R. Lafont, Close driving-hazard or necessity? in: Proceedings of 18th PTRC Summer Annual Meeting, 1990. [2] M. McDonald, M. Brackstone, B. Sultan, C. Roach, Close following on the motorway: initial findings of an instrumented vehicle study, Vision Veh. 7 (1999) 381–389. [3] J.R. Sayer, M.L. Mefford, R. Huang, The effect of lead-vehicle size on driver following behavior: is ignorance truly bliss? in: Proceedings of 2nd International Driving Symposium on Human Factors in Driver Assessment, Training and Vehicle Design, vol. 1001, 2003, pp. 48109–48150. [4] S. Peeta, P. Zhang, W. Zhou, Behavior-based analysis of freeway car-truck interactions and related mitigation strategies, Transp. Res. Part B: Method. 39 (2005) 417–451. [5] S. Peeta, W. Zhou, P. Zhang, Modeling and mitigation of car-truck interactions on freeways, Transp. Res. Rec. 1899 (2004) 117–126. [6] T.R. Board, Highway Capacity Manual, Transportation Research Board of the National Academy of Sciences, Washington, DC, 2010. [7] F. Ye, Y. Zhang, Vehicle type-specific headway analysis using freeway traffic data, Transp. Res. Rec. 2124 (2009) 222–230. [8] M. Sarvi, Heavy commercial vehicles-following behavior and interactions with different vehicle classes, J. Adv. Transp. 47 (2013) 572–580. [9] K. Aghabayk, M. Sarvi, W. Young, Y. Wang, Investigating heavy vehicle interactions during the car following process, in: Proceedings of Transportation Research Board 91st Annual Meeting, 2012. [10] A.D. Mason, A.W. Woods, Car-following model of multispecies systems of road traffic, Phys. Rev. E 55 (1997) 2203. [11] M. Treiber, A. Hennecke, D. Helbing, Congested traffic states in empirical observations and microscopic simulations, Phys. Rev. E 62 (2000) 1805–1824. [12] M. Treiber, A. Kesting, D. Helbing, Delays, inaccuracies and anticipation in microscopic traffic models, Physica A 360 (2006) 71–88. [13] R.E. Wilson, J.A. Ward, Car-following models: fifty years of linear stability analysis–a mathematical perspective, Transp. Plann. Technol. 34 (2011) 3–18. [14] S.H. Hamdar, H.S. Mahmassani, From existing accident-free car-following models to colliding vehicles: exploration and assessment, Transp. Res. Rec. 2088 (2008) 45–56. [15] M. Treiber, A. Kesting, Traffic Flow Dynamics, Springer-Verlag, Berlin, Heidelberg, 2013. [16] R. Herman, E.W. Montroll, R.B. Potts, R.W. Rothery, Traffic dynamics: analysis of stability in car following, Oper. Res. 7 (1959) 86–106. [17] M. Treiber, A. Kesting, Verkehrsdynamik und-simulation: Daten, Modelle und Anwendungen der Verkehrsflussdynamik, Springer-Verlag, Berlin, 2010. [18] J.A. Ward, Heterogeneity, Lane-Changing and Instability in Traffic: A Mathematical Approach, University of Bristol, 2009. [19] R.W. Rothery, Traffic Flow Theory, Transportation Research Board, Washington, DC, 1998. [20] B.S. Kerner, P. Konhäuser, Structure and parameters of clusters in traffic flow, Phys. Rev. E 50 (1994) 54–83. [21] J. Wang, R. Liu, F. Montgomery, Car-following model for motorway traffic, Transp. Res. Rec. 1934 (2005) 33–42. [22] T. Nagatani, Density waves in traffic flow, Phys. Rev. E 61 (2000) 3564–3570. [23] R.E. Wilson, An analysis of Gipps’s car-following model of highway traffic, IMA J. Appl. Math. 66 (2001) 509–537.

L. Liu et al. / Applied Mathematics and Computation 273 (2016) 706–717

717

[24] R.G. Hoogendoorn, S. Hoogendoorn, K.A. Brookhuis, W. Daamen, Anticipation and hysteresis: parameter value changes and model performance in simple and multianticipative car-following models, in: Proceedings of Transportation Research Board 90th Annual Meeting, 2011. [25] E. Brockfeld, R.D. Kühne, P. Wagner, Calibration and validation of microscopic models of traffic flow, Transp. Res. Rec. 1934 (2005) 179–187. [26] V. Punzo, F. Simonelli, Analysis and comparison of microscopic traffic flow models with real traffic microscopic data, Transp. Res. Rec. 1934 (2005) 53–63. [27] S.H. Hamdar, M. Treiber, H.S. Mahmassani, Calibration of a stochastic car-following model using trajectory data: exploration and model properties, in: Proceedings of Transportation Research Board 88th Annual Meeting, 2009. [28] S. Jin, D.H. Wang, Z.Y. Huang, P.F. Tao, Visual angle model for car-following theory, Physica A 390 (2011) 1931–1940.