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A cellular automata model for car–truck heterogeneous traffic flow considering the car–truck following combination effect
Physica A 424 (2015) 62–72
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Physica A journal homepage: www.elsevier.com/locate/physa
A cellular automata model for car–truck heterogeneous traffic flow considering the car–truck following combination effect Da Yang a,b , Xiaoping Qiu a,b,∗ , Dan Yu a , Ruoxiao Sun a , Yun Pu a a
School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China
b
National United Engineering Laboratory of Integrated and Intelligent Transportation, Chengdu, 610031, China
highlights • • • • •
A new cellular automata-based car–truck traffic flow model is proposed. The model discriminates differences among the four car–truck following combinations. The effect of car–truck following combinations is limited to a certain density range. Traffic congestion can be reduced up to 6.3% by mixing cars and trucks. When the density is high, trucks will aggravate traffic congestion obviously.
article
info
Article history: Received 22 September 2014 Received in revised form 7 December 2014 Available online 29 December 2014 Keywords: Cellular automata Car–truck traffic flow Following combinations Safe distance
abstract To better understand the characteristics of car–truck heterogeneous traffic flow that is very common on freeway, a cellular automata-based traffic flow model is proposed for single lane traffic in this paper. The proposed model discriminates the four types of car–truck following combination, car-following-car (CC), car-following-truck (CT), truck-followingcar (TC) and truck-following-truck (TT). The four combinations are considered in terms of the safety distance, reaction time and randomization probability. The parameter values in the proposed model are derived from NGSIM data. Simulations are conducted based on the new model and some new conclusions about the characteristics of the car–truck traffic flow are drawn. First, in the density range of (23–36) vehs/km, the fundamental diagram mainly depends on the car–truck following combination, especially, on the proportion of CC combination. In this range, the fundamental diagram curves with the same proportion of CC gather into a cluster, and the flow rate increases with the increment of the proportion of CC for the same traffic density. Second, traffic congestion can be effectively reduced up to 6.3% by increasing the proportion of TC or CT combination. This finding provides a possible way to alleviate traffic congestion on freeway. Third, reducing randomization probability of the four combinations can effectively increase traffic capacity and alleviate traffic congestion. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Car–truck heterogeneous traffic flow is very common on freeway. Studies have shown that although the number of trucks on freeway is relatively smaller than the number of cars, it has a significant influence on traffic flow characteristics [1–8].
∗
Corresponding author at: School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China. E-mail addresses: [email protected] (D. Yang), [email protected] (X. Qiu).
http://dx.doi.org/10.1016/j.physa.2014.12.020 0378-4371/© 2015 Elsevier B.V. All rights reserved.
D. Yang et al. / Physica A 424 (2015) 62–72
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Fig. 1. Schematic of vehicle occupying cells.
Regarding the car–truck traffic flow, early researchers mainly focused on investigating the difference between following a car and following a truck. Huddart and Lafont [3], Sayer et al. [8] and McDonald et al. [9] compared the headway differences between the two cases: car-following-car and car-following-truck, but their studies did not reach the conclusion that which case had larger headway. Peeta et al. [4,5] analyzed the interaction of cars and trucks in multiple lanes. However, the above studies [3–5,8,9] did not recognize that the driving behavior also depends on the lag vehicle type (car or truck). Ye’s study [10] first explores the impact of the lag vehicle type on traffic flow. He concluded that the four types of car–truck following combination should be taken into account in the study of the car–truck heterogeneous traffic flow: car-following-car (CC), car-following-truck (CT), truck-following-car (TC) and truck-following-truck (TT). Sarvi [6] also studied the driving behavior of the three following combinations, CC, TC and CT, but neglecting the TT combination. Sarvi et al. [7] further studied the distance headway, time headway, reaction time and car-following threshold variations among the four types of combination based on NGSIM data. They concluded that the effect of car–truck mixture on traffic flow was not only determined by the leading vehicle type, but also influenced by the lag vehicle type. One major limitation of the early studies on car–truck mixed traffic flow is their lack of modeling of the dynamic flow characteristics. It still needs an effective model focusing on features of the traffic flow consisting of the four following combinations. Mason and Woods [11] developed the homogeneous Optimal Velocity (OV) car-following model into a heterogeneous traffic flow model to describe the interaction between cars and trucks. The authors of this paper also extended the original OV car-following model to its heterogeneous form and further studied the stability of the car–truck heterogeneous traffic flow [12]. However, the in-depth analysis on characteristics of car–truck heterogeneous traffic flow consisting of the four following combinations has not been performed, such as the influence of four types of following combination on fundamental diagram and traffic congestion characteristics. This paper attempts to propose a cellular automata-based car–truck traffic flow model taking into account the car–truck combination effect. As an important tool for research in microscopic traffic simulation, cellular automata models, have the merits of simple rules, fast calculation speed and easy realization, and are favored by many scholars [13–18]. Since 1990s, cellular automata has been widely used in many fields, e.g., epidemic spreading [19–21], personal social behavior [22–24], dynamical systems [25–27], traffic flow [14,28,29], etc. In 1986, Cremer and Ludwig [30] first applied cellular automata in describing the transport system and proposed the first cellular automata traffic flow model. In 1992, Nagel and Schreckenberg [14] improved the No.184 cellular automata model and extended the maximum speed of the vehicle to the case of more than one, and introduced the randomization probability into the model. NaSch model [14] is the most representative cellular automata model for freeway traffic flow. Takayasu and Takayasu [15] and Fukui and Ishibashi [16] further developed and improved NaSch model and greatly enriched the freeway cellular automata-based traffic flow model. In 2003, based on the famous NaSch and improved FI model [16], Moussa and Daoudia [17] studied the impact of truck on traffic flow. In 2010, Jetto and Ez-Zahraouy [31] studied the traffic flow characteristics consisting of vehicles with different lengths and different velocities, and they found that the transition from active phase to absorbing phase depended on the length of the slow vehicles. In 2004, Ez-Zahraouy et al. [32] used numerical simulations with both open and periodic boundaries to explore the traffic flow mixed by different lengths of vehicles. However, the existing cellular automata-based car–truck traffic flow models did not discriminate the differences among the four car–truck following combinations, CC, CT, TC and TT, which will be considered in this paper. The differences between cars and trucks considered in this paper are their occupying cellular numbers, maximum velocities, maximum decelerations etc. The differences among the four types of following combination are the randomization probability, driver’s reaction time and safe distance adopted. The model parameter values are derived from NGSIM data. Based on the proposed cellular automata model, simulations are further conducted to analyze the characteristics of the heterogeneous car–truck traffic flow. 2. Model description This paper uses one-dimensional discrete cellular to simulate the single-lane traffic flow. Every cellular grid has two states at any time t, which are free or occupied by a vehicle. The length of a single cellular is L. Due to the different lengths of cars and trucks, this paper refines the cellular size and defines L equals 0.5 m, as shown in Fig. 1. Following is the proposed cellular automata car–truck traffic flow model. a. Safe distance rule Safe distance is a space that vehicles must maintain to the preceding vehicle so as to avoid rear-end collision when the preceding vehicle brakes emergently. The safe distance is with respect to the braking ability of both the lag and preceding vehicles and the reaction time of the driver of the lag vehicle, as shown in Fig. 2. Thus, due to the different braking abilities of cars and trucks and the different drivers’ reaction times for the four following combinations, the safe distances are different among the four following combinations as well.
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Fig. 2. Schematic of safe distance calculation.
First, we will obtain the safe distance of the four following combinations. According to the theory of calculating safe distance introduced by Gipps [33], we obtain the following equation. xn−1 (t ) +
vn−1 (t )2 2bn−1
− ln−1 ≥ xn (t ) + vn (t ) τn +
vn (t )2 2bn
.
(1)
Rewriting Eq. (1) can acquire the expression of the safe distance that the vehicle n have to main to the vehicle n − 1 in driving. gapsafe,n = xn−1 (t ) − xn (t ) − ln−1 = vn (t )τn +
vn (t )2 2bn
−
vn−1 (t )2 2bn−1
.
(2)
In this equation, gapsafe,n denotes the safe distance of the nth vehicle, xn (t ) denotes the location of the vehicle n at time t , xn−1 (t ) denotes the location of the vehicle n − 1 at time t , ln−1 is the length of vehicle n − 1, bn is the emergency braking deceleration of the vehicle n, τn is the reaction time of the driver of the vehicle n, vn (t ) is the velocity of the vehicle n at time t, and vn−1 (t ) is the velocity of the vehicle n − 1 at time t. For the different types of following combination, τn has four alternatives, τcc , τct , τtc and τtt , while bn only has two alternatives, bc and bt . b. Acceleration rule In the moving, when the distance between the lag vehicle and the preceding vehicle is greater than the required safe distance, that is gapn > gapsafe,n , the vehicle n will accelerate in accordance with the following rule:
vn (t ) → min vn (t ) + an , Vmax,n , gapn .
(3)
c. Uniform velocity rule When the distance between the lag vehicle and the preceding vehicle exactly equals to the required safe distance, that is gapn = gapsafe,n , the vehicle n will maintain the original velocity:
vn (t ) → min (vn (t ) , gapn ) .
(4)
d. Slowing down rule When the distance between the lag vehicle and the preceding vehicle is smaller than the required safe distance, that is gapn < gapsafe,n , the vehicle n will slow down. There are two types of vehicle deceleration rules: if the preceding vehicle is stationary, for security reason, the safe deceleration rules will be used, that is the distance between the lag vehicle and the preceding vehicle is more than 0.5 m; if the preceding vehicle is in a non-stationary state, i.e. vn−1 (t ) ̸= 0, the deterministic deceleration rule will be used and the formulations are as follows. Safe deceleration rule:
vn (t ) → max {min (vn (t ) , gapn − 1) , 0} .
(5)
Deterministic deceleration rule:
vn (t ) → min (vn (t ) , gapn ) .
(6)
e. Randomization probability The randomization probabilities of the four following combinations are different as well. In the car–truck heterogeneous traffic flow, the randomization probability of a following combination is mainly determined by the following vehicle type due to the significant difference of flexibility in driving a car and a truck (the difference between following a truck and following a car is relatively less significant [6,7]). Thereby, driving a car no matter following a car or a truck has bigger randomization probability than driving a truck, as cars are much more flexible than trucks, which produces Rcc , Rct > Rtc , Rtt . Furthermore, according to the above safe distance theory, trucks have longer emergency braking distance than cars, which means that the driver following a truck has more time to deal with an emergency situation and has less pressure in deceleration. Thus, following a truck will have a smaller randomization probability. Therefore, the sequence of the four randomization probability
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Table 1 Used model parameters for the car and the truck. Parameters
Car
Truck
Size (including the standstill distance) Maximum velocity Emergency braking deceleration General acceleration General deceleration
15 cells (7.5 m) 75 cells/s (135 km/h) 7 cells/s2 (3.5 m/s2 ) 6 cells/s2 (3.0 m/s2 ) 5 cells/s2 (2.5 m/s2 )
30 cells (15 m) 45 cells/h (81 km/h) 5 cells/s2 (2.5 m/s2 ) 4 cells/s2 (2.0 m/s2 ) 4 cells/s2 (2.0 m/s2 )
Table 2 Used model parameters for the four following combinations.
Reaction time [7] Randomization probability
CC
CT
TC
TT
1.8 s 0.16
1.9 s 0.14
1.9 s 0.12
2s 0.10
in the paper is, Rcc > Rct > Rtc > Rtt . The driving rule is given by,
vn (t ) → max vn (t ) − b′n , 0 .
(7)
f. Location update The location update of vehicles can be obtained through the velocity update rule. The location update rule is as follows, xn (t ) → xn (t ) + vn (t ) .
(8)
In the above equations, gapn is the gap between the vehicle n and the vehicle n − 1. gapn (t ) = xn−1 (t ) − xn (t ) − ln−1 . an , Vmax,n and b′n are the general acceleration, the maximum velocity, and the general deceleration of the vehicle n respectively. R denotes the randomization probability, and there are four types of R, Rcc , Rct , Rtc , and Rtt respectively. According to the types of vehicle, an has two alternatives, ac and at . Vmax,n has two alternatives, Vmax,c and Vmax,t . b′n has two alternatives, b′c and b′t . 3. Experimental design 3.1. Data introduction The data used in this study refers to the trajectory data collected at southbound direction of the US Highway 101 (Hollywood Freeway) in Los Angeles, California from the NGSIM (Next Generation Simulation) project. The entire segment is approximately 2100 ft in length, with five main lanes throughout the section and one auxiliary lane. The entire data collection time period is 45 min between 7:50 and 8:35 a.m. during the morning peak hours. All the parameters in the proposed model have explicit physical meaning, so we can derive them directly from NGSIM data. The specific parameter values for the car and the truck are shown in Table 1 and the parameter values of the four following combinations are shown in Table 2. 3.2. Simulation setup Based on the proposed model, we will use simulation method to study the characteristics of the car–truck heterogeneous traffic flow. The simulation uses the periodic boundary, and the road length equals to 5 km. In the initial state, assume that N vehicles are on the road with uniform distribution, where the numbers of cars and trucks are Nc and Nt respectively. Thus, the proportion of trucks is Pt = NNt . The initial speed of the vehicles is set as 0 km/h. The total vehicle density on the road is
denoted as ρ . The density of cars is ρc and the density of trucks is ρt . There is ρ = ρc + ρt , in which ρc = The average speed of the vehicle traffic flow at time t is v¯ (t ) and v¯ (t ) =
Nc L
and ρt =
Nt L
.
n=1 vn (t ). Q is the flow rate of the traffic. The and Ptt = NNtt , where Ncc , Nct , Ntc and Ntt are the
1 N
N
proportions of the four combinations are: Pcc = NNcc , Pct = NNct , Ptc = NNtc number of the four combinations in the traffic flow. The simulation step is 1 s. Each simulation realization is realized with more than 10,000 s, and we choose the last 1000 s to present the simulation results. 4. Fundamental diagram analysis 4.1. Influence of proportions of the four car–truck combinations on fundamental diagram
Although the previous studies have analyzed the effect of the different truck proportions in traffic flow on the fundamental diagram, they did not study the effects of the different proportions of the four following combinations on the fundamental diagram. In this paper, we use 17 groups of proportions of the four following combinations to simulate the traffic flow and observe the variation of the fundamental diagram curves.
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(a) density–flow.
(b) density-average speed. Fig. 3. Fundamental diagrams under different proportions of the four types of car–truck following combinations.
The simulation results of the density–flow and density–speed are shown in Fig. 3. In view of Fig. 3(a), we find three key points existing, K1 = 23 vehs/km, K2 = 36 vehs/km and K3 = 66 vehs/km, which separates the feasible range of density into four spectrums: a. When ρ < K1 , both the proportion of the truck and the proportions of the four combinations have little influence on flow rate Q . Q only increases with the increment of the density ρ . b. When K1 ≤ ρ < K2, the proportions of the four combinations have obvious influence on the flow-density curve. The significant characteristic is exhibited: the fundamental diagram curves with the same CC proportion gather into a cluster. In addition, the flow rate Q increases with the increment of CC proportion in the car–truck flow for the same density. This phenomenon has not been observed in the existing studies. c. When K2 ≤ ρ < K3 , there is not obvious regularity for the characteristic of the fundamental diagram regarding the proportions of the truck or the four following combinations. Actually, this spectrum is a transition spectrum. In this range, the cluster effect found in the range K1 ≤ ρ < K2 is gradually weakening, and the traffic flow begins to show the characteristic that the fundamental diagram curves with the same truck (or car) proportion gather into a cluster. d. When ρ ≥ K3 , the fundamental diagram curves with the same truck (or car) proportion have totally gathered into a cluster. The characteristic of the car–truck flow is mainly determined by the truck (or car) proportion, and the effect of the following combinations of cars and trucks on traffic flow is negligible. In addition, the flow rate Q increases with the decrement of the truck proportion in the car–truck flow for the same density. Therefore, the proportions of the four following combination act on the fundamental diagram of car–truck flow mainly in the range of K1 ≤ ρ < K3 , and have regular effect only in the range of K1 ≤ ρ < K2 , while the proportion of trucks act on the fundamental diagram mainly in the range of ρ ≥ K3 . In Fig. 3(b), when Pcc = 1, only cars exist in the traffic flow. In a free flow state, the traffic flow rate and average speed are higher than other situations of Pcc ̸= 1. This phenomenon is owing to that the maximum traveling speed of trucks is significantly lower than the maximum speed of cars. When trucks exist in the free traffic flow, cars’ speeds are subject to trucks’ speeds, making the average speed of traffic flow and the corresponding flow rate decrease. Furthermore, let us analyze the characteristics of car–truck mixed traffic flow in the case that the truck proportion is fixed. We will explain this using the case Pt = 0.5 in Fig. 3. As shown in Fig. 4, when the proportion of the truck in the flow is fixed, the traffic capacity will increase in accordance with the proportion of the CC combination. This result implies that
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Fig. 4. Density–flow diagram for different proportions of the four car–truck combinations for Pt = 0.5.
Fig. 5. Fundamental diagrams under different randomization probabilities.
in a certain density region the traffic capacity can be increased by increasing the proportion of CC when the proportion of trucks is fixed. 4.2. Influence of randomization probability on fundamental diagram In this subsection, the influence of randomization probability of the four combinations on fundamental diagram will be discussed. Given the proportions of the four combinations Pcc = 0.5, Pct = 0.2, Ptc = 0.2 and Ptt = 0.1, vary the randomization probability values of the four combinations and observe the variation of fundamental diagram. First, fix the randomization probability sequence of the four combinations, and then gradually increase the randomization probability values from Rcc = 0.2, Rct = 0.15, Rtc = 0.1 and Rtt = 0.05 with same magnitude for the four combinations. The density–flow curves are shown in Fig. 5(a). We can observe that with the increment of the overall value of the randomization probability, the capacity of traffic flow is gradually reduced, and the corresponding critical density of capacity gradually decreases. This is because as the randomization probability increases, the number of randomly decelerating vehicles increases, which leads to a smaller maximum value of the overall average speed, thereby reducing the capacity of traffic flow.
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Second, fix the randomization probability value of TT combination, and adjust the randomization probability values of the other three combinations, and then observe fundamental diagram variation. The density–flow curves are shown in Fig. 5(b). It can be found that the fundamental diagram curve has no obvious fluctuation, and the maximum capacity of traffic flow decreases as the randomization probability increases as a whole. This result indicates that when the randomization probability sequence of the four combinations is fixed, the traffic flow is mainly affected by the values of randomization probability for the four combinations. Thus, we can obtain the following conclusion: by providing better traffic environment to reduce the randomization probability of cars and trucks, the traffic capacity can be effectively improved. Third, adjust the randomization probability sequence and observe the fundamental diagram variation. The sequence Rcc > Rct > Rtc > Rtt is the most often case in real traffic, but the randomization probability is a random variable and there is the possibility that the sequence is changed in some traffic conditions, especially for the order of Rct and Rtc . The sequence change of the randomization possibility may influence the traffic flow characteristic, so here we will discuss the fundamental diagram characteristics for the all three most possible sequences, Rcc > Rct > Rtc > Rtt , Rcc > Rtc > Rct > Rtt and Rcc > Rct = Rtc > Rtt . Fig. 5(c) illustrates the fundamental diagrams of three groups of randomization probability value. In each group, we present the results for the all orders of Rct and Rtc . The curves in Fig. 5(c) indicate that the overall values of the randomization probabilities of the four combinations have more significant impact on fundamental diagrams comparing to the orders of Rct and Rtc . Fig. 5(d) is a figure to display the effect of the order change of Rct and Rtc on fundamental diagram. The curves exhibit that changing the original order Rct > Rtc will reduce the traffic capacity. In addition, comparing the two scenarios, maintaining the value of Rtc and decreasing the value of Rct (scenario 1) and maintaining the value of Rct and decreasing the value of Rtc (scenario 2), we can find the capacity increases more significant for the scenario 2, which implies that adjusting the value of Rtc has a greater impact on capacity. 5. Congestion analysis 5.1. Influence of the presence of a truck in traffic flow on congestion The presence of trucks in traffic flow has a significant impact on traffic flow characteristics. Using the proposed model, we will explore the congestion fluctuation when a truck is put into the traffic flow for both the low and high traffic density situations. We define a vehicle whose velocity is less than 10 km/h as a congestion vehicle, and then the congestion level of traffic can be defined as the ratio of the congestion vehicles in the entire traffic flow: CR =
n′
(9)
∆T · N where CR, n′ , ∆T and N denote congestion ratio, number of congestion vehicles, simulation time, and total number of vehicles in traffic flow respectively. Fig. 6 displays the temporal–spatial diagram and velocity diagram for the pure car traffic flow and car–truck mixed traffic flow in the low density (ρ = 20 vehs/km). In the temporal–spatial diagram, the red color represents trucks and the black color represents cars. We can observe from Fig. 6(a) that when no truck exists in traffic flow, long traffic congestion areas and shock waves appear. When adding a truck in the traffic flow, the truck blocks the evolvement of shock waves, as shown in Fig. 6(b). Fig. 6(c) and (d) give the reasons for the above phenomena. The two diagrams display the velocity of each vehicle in the traffic flow, and the congestion ratio is put in the upper right corner. The color in the two diagrams indicates the value of the velocity: dark blue means low velocity and shallow green means high velocity. We can observe from Fig. 6(d) that, in the low density region, the speeds of cars are generally higher than the speeds of trucks. The truck already reaches its maximum speed, but its speed is still less than the average speed of cars, thereby limiting the speed of cars and the evolvement of shock waves. The blocking effect of the truck alleviates the congestion about 1% in the low density range. Fig. 7 displays the temporal–spatial diagram and velocity diagram for the pure car traffic flow and car–truck mixed traffic flow in the high density (ρ = 40 vehs/km). Due to the increased traffic density, traffic congestion significantly increases compared to Fig. 6. We can observe clearly from Fig. 7(b) that the blocking effect of the truck on shock wave evolvement has been obviously weakened. Fig. 7(c) and (d) indicate that the cars’ speeds are close to the truck’s speeds; the truck does not alleviate congestion, but in contrary increases the congestion rate by about 8%. In summary, the presence of trucks has two types of effect on traffic flow. When the density is low, the truck can block the evolvement of shock waves and alleviate congestion, while when the density is high, the truck will aggravate traffic congestion obviously. 5.2. Influence of the proportions of the four following combination on traffic congestion Following are the effect analysis of different proportions of the four following combinations at different densities on traffic congestion. When the number of vehicles in the traffic flow is fixed, increasing the proportion of TT (CC) will decrease the proportions of TC and CT. Fig. 8(a) shows the congestion ratio variation when increasing Ptt from 0% to 30% for ρ = 40 vehs/km. This result indicates that decreasing the proportion of TT combination (increasing the proportions of CT and TC) can alleviate traffic congestion, up to 6.3%. The finding provides a possible way to alleviate traffic congestion on freeway.
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Fig. 6. The temporal–spatial and velocity diagrams for the pure car traffic flow and the traffic flow including a truck in the low density.
Fig. 8(b) displays the maximum congestion alleviation value for the densities varying from 10 vehs/km to 90 vehs/km. The alleviation effect by increasing the proportions of CT and TC is only effective within a certain range. When the density is too low or too high, changing the sequence of vehicles in traffic flow has little impact on congestion ratio, almost negligible. When 40 ≤ ρ ≤ 60 vehs/km, traffic congestion can be effectively reduced by mixing different types of vehicle. 5.3. Influence of randomization probability on traffic congestion This paper selects three groups of randomization probability to analyze and compare the influence of randomization probability on traffic congestion. Fig. 9 is the temporal–spatial diagram under different randomization probabilities. As shown in Fig. 9, when Rcc = 0.16, narrow congestion area appears. With the increment of the randomization probability, the number of narrow congestion area increases. When Rcc = 0.56, wide congestion area appears. By comparison, we can conclude that congestion area will increase with the increment of randomization probability. This is because when the randomization probability value increases, the number of vehicles with random deceleration increases, which enhances mutual interference among vehicles, thereby increasing congestion. Fig. 10 shows the congestion ratio change trend for 5 groups of Rcc . It clearly shows the congestion ratio of traffic flow increases with the increment of randomization probability. The 40% increment of the randomization probability value results in almost 40% increment of the congestion ratio. This confirms the above conclusion that traffic congestion can be eliminated or alleviated by improving the traffic environment and reducing the randomization probability. 6. Conclusions and future work The car–truck heterogeneous traffic flow is common on freeway and trucks have significant influence on traffic flow. Although some studies have been conducted on car–truck heterogeneous traffic flow, the majority of them were focusing on the statistical analysis using field data. The studies on modeling car–truck heterogeneous traffic flow are few and especially did not recognized the four car–truck following combinations, CC, CT, TC and TT. The characteristics of the car–truck heterogeneous traffic flow have not been investigated thoroughly. This paper proposes a cellular automata model for the car–truck heterogeneous traffic flow, which discriminates the differences among the four car–truck following combinations. The length of the cells in the cellular automata model is chosen as 0.5 m. The model parameters are chosen according to NGSIM
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Fig. 7. The temporal–spatial and velocity diagrams for the pure car traffic flow and the traffic flow including a truck in the high density.
Fig. 8. Congestion ratio diagram for different proportions of the four following combinations at different densities.
data. Based on the proposed model, simulations are conducted to investigate the characteristics of the car–truck heterogeneous traffic flow consisting of the four following combinations. Finally, the following new findings are found in this study: (1) The influence of combination effect on the fundamental diagram of car–truck flow is only effective in a limited range of 23 vehs/km ≤ ρ ≤ 66 vehs/km. When 23 vehs/km ≤ ρ ≤ 36 vehs/km, the fundamental diagram curves with the same Pcc (the proportion of CC in the mixed traffic flow) gather into a cluster and the flow rate Q increases with
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(a) Rcc = 0.16, Rct = 0.14, Rtc = 0.12, Rtt = 0.10.
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(b) Rcc = 0.36, Rct = 0.34, Rtc = 0.32, Rtt = 0.30.
(c) Rcc = 0.56, Rct = 0.54, Rtc = 0.52, Rtt = 0.50. Fig. 9. The temporal–spatial diagram under different randomization probabilities.
Fig. 10. Congestion ratio for different randomization probabilities.
the increment of Pcc . When 36 vehs/km ≤ ρ ≤ 66 vehs/km, there is no obvious regularity regarding the combination proportions for the characteristic of the fundamental diagram. When ρ > 66 vehs/km, the influence of combination effect is negligible, and the fundamental diagrams with the same Pc (or Pt ) cluster together. The flow rate Q increases with the decrement of the truck proportion in the car–truck flow for the same density. (2) The presence of trucks has two types of effect on traffic flow. When the density is low, the truck can block the evolvement of shock waves and alleviate congestion, while when the density is high, the truck will aggravate traffic congestion obviously. (3) When 40 vehs/km ≤ ρ ≤ 60 vehs/km, traffic congestion can be effectively reduced by mixing different types of vehicle. This is to say, increasing the proportion of TC or CT combination can alleviate traffic congestion up to 6.3%. The finding provides a possible way to alleviate traffic congestion. (4) The randomization probability of the four combinations has significant influence on the fundamental diagram and congestion of the car–truck heterogeneous traffic flow. Reducing randomization probability by improving the traffic environment can effectively increase traffic capacity and alleviate traffic congestion.
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However, this paper only considers the single-lane traffic. In future, we will extend the study to multiple-lane traffic and investigate the effect of the lane-changing behavior difference of the four car–truck following combinations on traffic flow characteristics. Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos. 51278429 and 51408509), Soft Science Foundation of Sichuan Province (Grant No. 2014ZR0091), Fundamental Research Funds for the Central Universities (Grant No. 2682014BR027) and National Basic Research Program of China (Grant No. 2012CB725405). References [1] K. Aghabayk, M. Sarvi, W. Young, Attribute selection for modelling driver’s car-following behaviour in heterogeneous congested traffic, Transportmetrica A: Transp. Sci. 10 (2014) 1–12. [2] K. Aghabayk, M. Sarvi, W. Young, Y. 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Aghabayk, Investigating heavy-vehicle interactions during car-following process, in: Transportation Research Board 91st Annual Meeting, 2012. [8] J.R. Sayer, M.L. Mefford, R. Huang, The effect of lead-vehicle size on driver following behavior, in: Ann Arbor, 2000, pp. 48109–42150. [9] M. McDonald, M. Brackstone, B. Sultan, C. Roach, Close following on the motorway: initial findings of an instrumented vehicle study, Vision in Vehiclese VII, 1999. [10] F. Ye, Y. Zhang, Vehicle type-specific headway analysis using freeway traffic data, Transp. Res. Rec.: J. Transp. Res. Board (2009) 222–230. Transportation Research Board of the National Academies, Washington, DC. [11] A.D. Mason, A.W. Woods, Car-following model of multispecies systems of road traffic, Phys. Rev. E 55 (1997) 2203. [12] D. Yang, P.J. Jin, Y. Pu, B. Ran, Stability analysis of the mixed traffic flow of cars and trucks using heterogeneous optimal velocity car-following model, Physica A 395 (2014) 371–383. [13] S. 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Physica A journal homepage: www.elsevier.com/locate/physa
A cellular automata model for car–truck heterogeneous traffic flow considering the car–truck following combination effect Da Yang a,b , Xiaoping Qiu a,b,∗ , Dan Yu a , Ruoxiao Sun a , Yun Pu a a
School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China
b
National United Engineering Laboratory of Integrated and Intelligent Transportation, Chengdu, 610031, China
highlights • • • • •
A new cellular automata-based car–truck traffic flow model is proposed. The model discriminates differences among the four car–truck following combinations. The effect of car–truck following combinations is limited to a certain density range. Traffic congestion can be reduced up to 6.3% by mixing cars and trucks. When the density is high, trucks will aggravate traffic congestion obviously.
article
info
Article history: Received 22 September 2014 Received in revised form 7 December 2014 Available online 29 December 2014 Keywords: Cellular automata Car–truck traffic flow Following combinations Safe distance
abstract To better understand the characteristics of car–truck heterogeneous traffic flow that is very common on freeway, a cellular automata-based traffic flow model is proposed for single lane traffic in this paper. The proposed model discriminates the four types of car–truck following combination, car-following-car (CC), car-following-truck (CT), truck-followingcar (TC) and truck-following-truck (TT). The four combinations are considered in terms of the safety distance, reaction time and randomization probability. The parameter values in the proposed model are derived from NGSIM data. Simulations are conducted based on the new model and some new conclusions about the characteristics of the car–truck traffic flow are drawn. First, in the density range of (23–36) vehs/km, the fundamental diagram mainly depends on the car–truck following combination, especially, on the proportion of CC combination. In this range, the fundamental diagram curves with the same proportion of CC gather into a cluster, and the flow rate increases with the increment of the proportion of CC for the same traffic density. Second, traffic congestion can be effectively reduced up to 6.3% by increasing the proportion of TC or CT combination. This finding provides a possible way to alleviate traffic congestion on freeway. Third, reducing randomization probability of the four combinations can effectively increase traffic capacity and alleviate traffic congestion. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Car–truck heterogeneous traffic flow is very common on freeway. Studies have shown that although the number of trucks on freeway is relatively smaller than the number of cars, it has a significant influence on traffic flow characteristics [1–8].
∗
Corresponding author at: School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, 610031, China. E-mail addresses: [email protected] (D. Yang), [email protected] (X. Qiu).
http://dx.doi.org/10.1016/j.physa.2014.12.020 0378-4371/© 2015 Elsevier B.V. All rights reserved.
D. Yang et al. / Physica A 424 (2015) 62–72
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Fig. 1. Schematic of vehicle occupying cells.
Regarding the car–truck traffic flow, early researchers mainly focused on investigating the difference between following a car and following a truck. Huddart and Lafont [3], Sayer et al. [8] and McDonald et al. [9] compared the headway differences between the two cases: car-following-car and car-following-truck, but their studies did not reach the conclusion that which case had larger headway. Peeta et al. [4,5] analyzed the interaction of cars and trucks in multiple lanes. However, the above studies [3–5,8,9] did not recognize that the driving behavior also depends on the lag vehicle type (car or truck). Ye’s study [10] first explores the impact of the lag vehicle type on traffic flow. He concluded that the four types of car–truck following combination should be taken into account in the study of the car–truck heterogeneous traffic flow: car-following-car (CC), car-following-truck (CT), truck-following-car (TC) and truck-following-truck (TT). Sarvi [6] also studied the driving behavior of the three following combinations, CC, TC and CT, but neglecting the TT combination. Sarvi et al. [7] further studied the distance headway, time headway, reaction time and car-following threshold variations among the four types of combination based on NGSIM data. They concluded that the effect of car–truck mixture on traffic flow was not only determined by the leading vehicle type, but also influenced by the lag vehicle type. One major limitation of the early studies on car–truck mixed traffic flow is their lack of modeling of the dynamic flow characteristics. It still needs an effective model focusing on features of the traffic flow consisting of the four following combinations. Mason and Woods [11] developed the homogeneous Optimal Velocity (OV) car-following model into a heterogeneous traffic flow model to describe the interaction between cars and trucks. The authors of this paper also extended the original OV car-following model to its heterogeneous form and further studied the stability of the car–truck heterogeneous traffic flow [12]. However, the in-depth analysis on characteristics of car–truck heterogeneous traffic flow consisting of the four following combinations has not been performed, such as the influence of four types of following combination on fundamental diagram and traffic congestion characteristics. This paper attempts to propose a cellular automata-based car–truck traffic flow model taking into account the car–truck combination effect. As an important tool for research in microscopic traffic simulation, cellular automata models, have the merits of simple rules, fast calculation speed and easy realization, and are favored by many scholars [13–18]. Since 1990s, cellular automata has been widely used in many fields, e.g., epidemic spreading [19–21], personal social behavior [22–24], dynamical systems [25–27], traffic flow [14,28,29], etc. In 1986, Cremer and Ludwig [30] first applied cellular automata in describing the transport system and proposed the first cellular automata traffic flow model. In 1992, Nagel and Schreckenberg [14] improved the No.184 cellular automata model and extended the maximum speed of the vehicle to the case of more than one, and introduced the randomization probability into the model. NaSch model [14] is the most representative cellular automata model for freeway traffic flow. Takayasu and Takayasu [15] and Fukui and Ishibashi [16] further developed and improved NaSch model and greatly enriched the freeway cellular automata-based traffic flow model. In 2003, based on the famous NaSch and improved FI model [16], Moussa and Daoudia [17] studied the impact of truck on traffic flow. In 2010, Jetto and Ez-Zahraouy [31] studied the traffic flow characteristics consisting of vehicles with different lengths and different velocities, and they found that the transition from active phase to absorbing phase depended on the length of the slow vehicles. In 2004, Ez-Zahraouy et al. [32] used numerical simulations with both open and periodic boundaries to explore the traffic flow mixed by different lengths of vehicles. However, the existing cellular automata-based car–truck traffic flow models did not discriminate the differences among the four car–truck following combinations, CC, CT, TC and TT, which will be considered in this paper. The differences between cars and trucks considered in this paper are their occupying cellular numbers, maximum velocities, maximum decelerations etc. The differences among the four types of following combination are the randomization probability, driver’s reaction time and safe distance adopted. The model parameter values are derived from NGSIM data. Based on the proposed cellular automata model, simulations are further conducted to analyze the characteristics of the heterogeneous car–truck traffic flow. 2. Model description This paper uses one-dimensional discrete cellular to simulate the single-lane traffic flow. Every cellular grid has two states at any time t, which are free or occupied by a vehicle. The length of a single cellular is L. Due to the different lengths of cars and trucks, this paper refines the cellular size and defines L equals 0.5 m, as shown in Fig. 1. Following is the proposed cellular automata car–truck traffic flow model. a. Safe distance rule Safe distance is a space that vehicles must maintain to the preceding vehicle so as to avoid rear-end collision when the preceding vehicle brakes emergently. The safe distance is with respect to the braking ability of both the lag and preceding vehicles and the reaction time of the driver of the lag vehicle, as shown in Fig. 2. Thus, due to the different braking abilities of cars and trucks and the different drivers’ reaction times for the four following combinations, the safe distances are different among the four following combinations as well.
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Fig. 2. Schematic of safe distance calculation.
First, we will obtain the safe distance of the four following combinations. According to the theory of calculating safe distance introduced by Gipps [33], we obtain the following equation. xn−1 (t ) +
vn−1 (t )2 2bn−1
− ln−1 ≥ xn (t ) + vn (t ) τn +
vn (t )2 2bn
.
(1)
Rewriting Eq. (1) can acquire the expression of the safe distance that the vehicle n have to main to the vehicle n − 1 in driving. gapsafe,n = xn−1 (t ) − xn (t ) − ln−1 = vn (t )τn +
vn (t )2 2bn
−
vn−1 (t )2 2bn−1
.
(2)
In this equation, gapsafe,n denotes the safe distance of the nth vehicle, xn (t ) denotes the location of the vehicle n at time t , xn−1 (t ) denotes the location of the vehicle n − 1 at time t , ln−1 is the length of vehicle n − 1, bn is the emergency braking deceleration of the vehicle n, τn is the reaction time of the driver of the vehicle n, vn (t ) is the velocity of the vehicle n at time t, and vn−1 (t ) is the velocity of the vehicle n − 1 at time t. For the different types of following combination, τn has four alternatives, τcc , τct , τtc and τtt , while bn only has two alternatives, bc and bt . b. Acceleration rule In the moving, when the distance between the lag vehicle and the preceding vehicle is greater than the required safe distance, that is gapn > gapsafe,n , the vehicle n will accelerate in accordance with the following rule:
vn (t ) → min vn (t ) + an , Vmax,n , gapn .
(3)
c. Uniform velocity rule When the distance between the lag vehicle and the preceding vehicle exactly equals to the required safe distance, that is gapn = gapsafe,n , the vehicle n will maintain the original velocity:
vn (t ) → min (vn (t ) , gapn ) .
(4)
d. Slowing down rule When the distance between the lag vehicle and the preceding vehicle is smaller than the required safe distance, that is gapn < gapsafe,n , the vehicle n will slow down. There are two types of vehicle deceleration rules: if the preceding vehicle is stationary, for security reason, the safe deceleration rules will be used, that is the distance between the lag vehicle and the preceding vehicle is more than 0.5 m; if the preceding vehicle is in a non-stationary state, i.e. vn−1 (t ) ̸= 0, the deterministic deceleration rule will be used and the formulations are as follows. Safe deceleration rule:
vn (t ) → max {min (vn (t ) , gapn − 1) , 0} .
(5)
Deterministic deceleration rule:
vn (t ) → min (vn (t ) , gapn ) .
(6)
e. Randomization probability The randomization probabilities of the four following combinations are different as well. In the car–truck heterogeneous traffic flow, the randomization probability of a following combination is mainly determined by the following vehicle type due to the significant difference of flexibility in driving a car and a truck (the difference between following a truck and following a car is relatively less significant [6,7]). Thereby, driving a car no matter following a car or a truck has bigger randomization probability than driving a truck, as cars are much more flexible than trucks, which produces Rcc , Rct > Rtc , Rtt . Furthermore, according to the above safe distance theory, trucks have longer emergency braking distance than cars, which means that the driver following a truck has more time to deal with an emergency situation and has less pressure in deceleration. Thus, following a truck will have a smaller randomization probability. Therefore, the sequence of the four randomization probability
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Table 1 Used model parameters for the car and the truck. Parameters
Car
Truck
Size (including the standstill distance) Maximum velocity Emergency braking deceleration General acceleration General deceleration
15 cells (7.5 m) 75 cells/s (135 km/h) 7 cells/s2 (3.5 m/s2 ) 6 cells/s2 (3.0 m/s2 ) 5 cells/s2 (2.5 m/s2 )
30 cells (15 m) 45 cells/h (81 km/h) 5 cells/s2 (2.5 m/s2 ) 4 cells/s2 (2.0 m/s2 ) 4 cells/s2 (2.0 m/s2 )
Table 2 Used model parameters for the four following combinations.
Reaction time [7] Randomization probability
CC
CT
TC
TT
1.8 s 0.16
1.9 s 0.14
1.9 s 0.12
2s 0.10
in the paper is, Rcc > Rct > Rtc > Rtt . The driving rule is given by,
vn (t ) → max vn (t ) − b′n , 0 .
(7)
f. Location update The location update of vehicles can be obtained through the velocity update rule. The location update rule is as follows, xn (t ) → xn (t ) + vn (t ) .
(8)
In the above equations, gapn is the gap between the vehicle n and the vehicle n − 1. gapn (t ) = xn−1 (t ) − xn (t ) − ln−1 . an , Vmax,n and b′n are the general acceleration, the maximum velocity, and the general deceleration of the vehicle n respectively. R denotes the randomization probability, and there are four types of R, Rcc , Rct , Rtc , and Rtt respectively. According to the types of vehicle, an has two alternatives, ac and at . Vmax,n has two alternatives, Vmax,c and Vmax,t . b′n has two alternatives, b′c and b′t . 3. Experimental design 3.1. Data introduction The data used in this study refers to the trajectory data collected at southbound direction of the US Highway 101 (Hollywood Freeway) in Los Angeles, California from the NGSIM (Next Generation Simulation) project. The entire segment is approximately 2100 ft in length, with five main lanes throughout the section and one auxiliary lane. The entire data collection time period is 45 min between 7:50 and 8:35 a.m. during the morning peak hours. All the parameters in the proposed model have explicit physical meaning, so we can derive them directly from NGSIM data. The specific parameter values for the car and the truck are shown in Table 1 and the parameter values of the four following combinations are shown in Table 2. 3.2. Simulation setup Based on the proposed model, we will use simulation method to study the characteristics of the car–truck heterogeneous traffic flow. The simulation uses the periodic boundary, and the road length equals to 5 km. In the initial state, assume that N vehicles are on the road with uniform distribution, where the numbers of cars and trucks are Nc and Nt respectively. Thus, the proportion of trucks is Pt = NNt . The initial speed of the vehicles is set as 0 km/h. The total vehicle density on the road is
denoted as ρ . The density of cars is ρc and the density of trucks is ρt . There is ρ = ρc + ρt , in which ρc = The average speed of the vehicle traffic flow at time t is v¯ (t ) and v¯ (t ) =
Nc L
and ρt =
Nt L
.
n=1 vn (t ). Q is the flow rate of the traffic. The and Ptt = NNtt , where Ncc , Nct , Ntc and Ntt are the
1 N
N
proportions of the four combinations are: Pcc = NNcc , Pct = NNct , Ptc = NNtc number of the four combinations in the traffic flow. The simulation step is 1 s. Each simulation realization is realized with more than 10,000 s, and we choose the last 1000 s to present the simulation results. 4. Fundamental diagram analysis 4.1. Influence of proportions of the four car–truck combinations on fundamental diagram
Although the previous studies have analyzed the effect of the different truck proportions in traffic flow on the fundamental diagram, they did not study the effects of the different proportions of the four following combinations on the fundamental diagram. In this paper, we use 17 groups of proportions of the four following combinations to simulate the traffic flow and observe the variation of the fundamental diagram curves.
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(a) density–flow.
(b) density-average speed. Fig. 3. Fundamental diagrams under different proportions of the four types of car–truck following combinations.
The simulation results of the density–flow and density–speed are shown in Fig. 3. In view of Fig. 3(a), we find three key points existing, K1 = 23 vehs/km, K2 = 36 vehs/km and K3 = 66 vehs/km, which separates the feasible range of density into four spectrums: a. When ρ < K1 , both the proportion of the truck and the proportions of the four combinations have little influence on flow rate Q . Q only increases with the increment of the density ρ . b. When K1 ≤ ρ < K2, the proportions of the four combinations have obvious influence on the flow-density curve. The significant characteristic is exhibited: the fundamental diagram curves with the same CC proportion gather into a cluster. In addition, the flow rate Q increases with the increment of CC proportion in the car–truck flow for the same density. This phenomenon has not been observed in the existing studies. c. When K2 ≤ ρ < K3 , there is not obvious regularity for the characteristic of the fundamental diagram regarding the proportions of the truck or the four following combinations. Actually, this spectrum is a transition spectrum. In this range, the cluster effect found in the range K1 ≤ ρ < K2 is gradually weakening, and the traffic flow begins to show the characteristic that the fundamental diagram curves with the same truck (or car) proportion gather into a cluster. d. When ρ ≥ K3 , the fundamental diagram curves with the same truck (or car) proportion have totally gathered into a cluster. The characteristic of the car–truck flow is mainly determined by the truck (or car) proportion, and the effect of the following combinations of cars and trucks on traffic flow is negligible. In addition, the flow rate Q increases with the decrement of the truck proportion in the car–truck flow for the same density. Therefore, the proportions of the four following combination act on the fundamental diagram of car–truck flow mainly in the range of K1 ≤ ρ < K3 , and have regular effect only in the range of K1 ≤ ρ < K2 , while the proportion of trucks act on the fundamental diagram mainly in the range of ρ ≥ K3 . In Fig. 3(b), when Pcc = 1, only cars exist in the traffic flow. In a free flow state, the traffic flow rate and average speed are higher than other situations of Pcc ̸= 1. This phenomenon is owing to that the maximum traveling speed of trucks is significantly lower than the maximum speed of cars. When trucks exist in the free traffic flow, cars’ speeds are subject to trucks’ speeds, making the average speed of traffic flow and the corresponding flow rate decrease. Furthermore, let us analyze the characteristics of car–truck mixed traffic flow in the case that the truck proportion is fixed. We will explain this using the case Pt = 0.5 in Fig. 3. As shown in Fig. 4, when the proportion of the truck in the flow is fixed, the traffic capacity will increase in accordance with the proportion of the CC combination. This result implies that
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Fig. 4. Density–flow diagram for different proportions of the four car–truck combinations for Pt = 0.5.
Fig. 5. Fundamental diagrams under different randomization probabilities.
in a certain density region the traffic capacity can be increased by increasing the proportion of CC when the proportion of trucks is fixed. 4.2. Influence of randomization probability on fundamental diagram In this subsection, the influence of randomization probability of the four combinations on fundamental diagram will be discussed. Given the proportions of the four combinations Pcc = 0.5, Pct = 0.2, Ptc = 0.2 and Ptt = 0.1, vary the randomization probability values of the four combinations and observe the variation of fundamental diagram. First, fix the randomization probability sequence of the four combinations, and then gradually increase the randomization probability values from Rcc = 0.2, Rct = 0.15, Rtc = 0.1 and Rtt = 0.05 with same magnitude for the four combinations. The density–flow curves are shown in Fig. 5(a). We can observe that with the increment of the overall value of the randomization probability, the capacity of traffic flow is gradually reduced, and the corresponding critical density of capacity gradually decreases. This is because as the randomization probability increases, the number of randomly decelerating vehicles increases, which leads to a smaller maximum value of the overall average speed, thereby reducing the capacity of traffic flow.
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Second, fix the randomization probability value of TT combination, and adjust the randomization probability values of the other three combinations, and then observe fundamental diagram variation. The density–flow curves are shown in Fig. 5(b). It can be found that the fundamental diagram curve has no obvious fluctuation, and the maximum capacity of traffic flow decreases as the randomization probability increases as a whole. This result indicates that when the randomization probability sequence of the four combinations is fixed, the traffic flow is mainly affected by the values of randomization probability for the four combinations. Thus, we can obtain the following conclusion: by providing better traffic environment to reduce the randomization probability of cars and trucks, the traffic capacity can be effectively improved. Third, adjust the randomization probability sequence and observe the fundamental diagram variation. The sequence Rcc > Rct > Rtc > Rtt is the most often case in real traffic, but the randomization probability is a random variable and there is the possibility that the sequence is changed in some traffic conditions, especially for the order of Rct and Rtc . The sequence change of the randomization possibility may influence the traffic flow characteristic, so here we will discuss the fundamental diagram characteristics for the all three most possible sequences, Rcc > Rct > Rtc > Rtt , Rcc > Rtc > Rct > Rtt and Rcc > Rct = Rtc > Rtt . Fig. 5(c) illustrates the fundamental diagrams of three groups of randomization probability value. In each group, we present the results for the all orders of Rct and Rtc . The curves in Fig. 5(c) indicate that the overall values of the randomization probabilities of the four combinations have more significant impact on fundamental diagrams comparing to the orders of Rct and Rtc . Fig. 5(d) is a figure to display the effect of the order change of Rct and Rtc on fundamental diagram. The curves exhibit that changing the original order Rct > Rtc will reduce the traffic capacity. In addition, comparing the two scenarios, maintaining the value of Rtc and decreasing the value of Rct (scenario 1) and maintaining the value of Rct and decreasing the value of Rtc (scenario 2), we can find the capacity increases more significant for the scenario 2, which implies that adjusting the value of Rtc has a greater impact on capacity. 5. Congestion analysis 5.1. Influence of the presence of a truck in traffic flow on congestion The presence of trucks in traffic flow has a significant impact on traffic flow characteristics. Using the proposed model, we will explore the congestion fluctuation when a truck is put into the traffic flow for both the low and high traffic density situations. We define a vehicle whose velocity is less than 10 km/h as a congestion vehicle, and then the congestion level of traffic can be defined as the ratio of the congestion vehicles in the entire traffic flow: CR =
n′
(9)
∆T · N where CR, n′ , ∆T and N denote congestion ratio, number of congestion vehicles, simulation time, and total number of vehicles in traffic flow respectively. Fig. 6 displays the temporal–spatial diagram and velocity diagram for the pure car traffic flow and car–truck mixed traffic flow in the low density (ρ = 20 vehs/km). In the temporal–spatial diagram, the red color represents trucks and the black color represents cars. We can observe from Fig. 6(a) that when no truck exists in traffic flow, long traffic congestion areas and shock waves appear. When adding a truck in the traffic flow, the truck blocks the evolvement of shock waves, as shown in Fig. 6(b). Fig. 6(c) and (d) give the reasons for the above phenomena. The two diagrams display the velocity of each vehicle in the traffic flow, and the congestion ratio is put in the upper right corner. The color in the two diagrams indicates the value of the velocity: dark blue means low velocity and shallow green means high velocity. We can observe from Fig. 6(d) that, in the low density region, the speeds of cars are generally higher than the speeds of trucks. The truck already reaches its maximum speed, but its speed is still less than the average speed of cars, thereby limiting the speed of cars and the evolvement of shock waves. The blocking effect of the truck alleviates the congestion about 1% in the low density range. Fig. 7 displays the temporal–spatial diagram and velocity diagram for the pure car traffic flow and car–truck mixed traffic flow in the high density (ρ = 40 vehs/km). Due to the increased traffic density, traffic congestion significantly increases compared to Fig. 6. We can observe clearly from Fig. 7(b) that the blocking effect of the truck on shock wave evolvement has been obviously weakened. Fig. 7(c) and (d) indicate that the cars’ speeds are close to the truck’s speeds; the truck does not alleviate congestion, but in contrary increases the congestion rate by about 8%. In summary, the presence of trucks has two types of effect on traffic flow. When the density is low, the truck can block the evolvement of shock waves and alleviate congestion, while when the density is high, the truck will aggravate traffic congestion obviously. 5.2. Influence of the proportions of the four following combination on traffic congestion Following are the effect analysis of different proportions of the four following combinations at different densities on traffic congestion. When the number of vehicles in the traffic flow is fixed, increasing the proportion of TT (CC) will decrease the proportions of TC and CT. Fig. 8(a) shows the congestion ratio variation when increasing Ptt from 0% to 30% for ρ = 40 vehs/km. This result indicates that decreasing the proportion of TT combination (increasing the proportions of CT and TC) can alleviate traffic congestion, up to 6.3%. The finding provides a possible way to alleviate traffic congestion on freeway.
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Fig. 6. The temporal–spatial and velocity diagrams for the pure car traffic flow and the traffic flow including a truck in the low density.
Fig. 8(b) displays the maximum congestion alleviation value for the densities varying from 10 vehs/km to 90 vehs/km. The alleviation effect by increasing the proportions of CT and TC is only effective within a certain range. When the density is too low or too high, changing the sequence of vehicles in traffic flow has little impact on congestion ratio, almost negligible. When 40 ≤ ρ ≤ 60 vehs/km, traffic congestion can be effectively reduced by mixing different types of vehicle. 5.3. Influence of randomization probability on traffic congestion This paper selects three groups of randomization probability to analyze and compare the influence of randomization probability on traffic congestion. Fig. 9 is the temporal–spatial diagram under different randomization probabilities. As shown in Fig. 9, when Rcc = 0.16, narrow congestion area appears. With the increment of the randomization probability, the number of narrow congestion area increases. When Rcc = 0.56, wide congestion area appears. By comparison, we can conclude that congestion area will increase with the increment of randomization probability. This is because when the randomization probability value increases, the number of vehicles with random deceleration increases, which enhances mutual interference among vehicles, thereby increasing congestion. Fig. 10 shows the congestion ratio change trend for 5 groups of Rcc . It clearly shows the congestion ratio of traffic flow increases with the increment of randomization probability. The 40% increment of the randomization probability value results in almost 40% increment of the congestion ratio. This confirms the above conclusion that traffic congestion can be eliminated or alleviated by improving the traffic environment and reducing the randomization probability. 6. Conclusions and future work The car–truck heterogeneous traffic flow is common on freeway and trucks have significant influence on traffic flow. Although some studies have been conducted on car–truck heterogeneous traffic flow, the majority of them were focusing on the statistical analysis using field data. The studies on modeling car–truck heterogeneous traffic flow are few and especially did not recognized the four car–truck following combinations, CC, CT, TC and TT. The characteristics of the car–truck heterogeneous traffic flow have not been investigated thoroughly. This paper proposes a cellular automata model for the car–truck heterogeneous traffic flow, which discriminates the differences among the four car–truck following combinations. The length of the cells in the cellular automata model is chosen as 0.5 m. The model parameters are chosen according to NGSIM
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Fig. 7. The temporal–spatial and velocity diagrams for the pure car traffic flow and the traffic flow including a truck in the high density.
Fig. 8. Congestion ratio diagram for different proportions of the four following combinations at different densities.
data. Based on the proposed model, simulations are conducted to investigate the characteristics of the car–truck heterogeneous traffic flow consisting of the four following combinations. Finally, the following new findings are found in this study: (1) The influence of combination effect on the fundamental diagram of car–truck flow is only effective in a limited range of 23 vehs/km ≤ ρ ≤ 66 vehs/km. When 23 vehs/km ≤ ρ ≤ 36 vehs/km, the fundamental diagram curves with the same Pcc (the proportion of CC in the mixed traffic flow) gather into a cluster and the flow rate Q increases with
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(a) Rcc = 0.16, Rct = 0.14, Rtc = 0.12, Rtt = 0.10.
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(b) Rcc = 0.36, Rct = 0.34, Rtc = 0.32, Rtt = 0.30.
(c) Rcc = 0.56, Rct = 0.54, Rtc = 0.52, Rtt = 0.50. Fig. 9. The temporal–spatial diagram under different randomization probabilities.
Fig. 10. Congestion ratio for different randomization probabilities.
the increment of Pcc . When 36 vehs/km ≤ ρ ≤ 66 vehs/km, there is no obvious regularity regarding the combination proportions for the characteristic of the fundamental diagram. When ρ > 66 vehs/km, the influence of combination effect is negligible, and the fundamental diagrams with the same Pc (or Pt ) cluster together. The flow rate Q increases with the decrement of the truck proportion in the car–truck flow for the same density. (2) The presence of trucks has two types of effect on traffic flow. When the density is low, the truck can block the evolvement of shock waves and alleviate congestion, while when the density is high, the truck will aggravate traffic congestion obviously. (3) When 40 vehs/km ≤ ρ ≤ 60 vehs/km, traffic congestion can be effectively reduced by mixing different types of vehicle. This is to say, increasing the proportion of TC or CT combination can alleviate traffic congestion up to 6.3%. The finding provides a possible way to alleviate traffic congestion. (4) The randomization probability of the four combinations has significant influence on the fundamental diagram and congestion of the car–truck heterogeneous traffic flow. Reducing randomization probability by improving the traffic environment can effectively increase traffic capacity and alleviate traffic congestion.
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