Cellular automata simulation for mixed manual and automated control traffic

Mathematical and Computer Modelling 51 (2010) 1000–1007

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Cellular automata simulation for mixed manual and automated control traffic Shih-Ching Lo ∗ , Chia-Hung Hsu Chung Hua University, Department of Transportation Technology & Logistics Management, Hsinchu, 300, Taiwan

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Keywords: Cellular automata Traffic flow Advanced vehicle control and safety systems Multi-class user traffic

abstract Complex traffic systems seem to be simulated successfully by cellular automaton (CA) models today. Various models are developed in efforts to understand single-lane traffic, multilane traffic, lane-changing behavior and network traffic flow. In this study, four cellular automata (CA) rules for advanced vehicle control and safety systems (AVCSS) are proposed and simulated. The major difference among the rules is the different settings of the gap, which is defined to be the distance between two successive vehicles. The gap of each rule is given depending upon the speed of vehicles. According to the results, CA rules with AVCSS (the H0 , H1 , H2 and H3 models) lead to a more stable traffic flow than rules without AVCSS. Also, the average flow and speed for CA rules with AVCSS are larger than the average flow and speed for rules without AVCSS. However, the average speeds of the H1 and H2 models fluctuate greatly, which is considered unsafe and unreliable, in a congested regime. The results from the H0 and H3 rules are more stable than the results from the H1 and H2 models. The H3 rule keeps a larger gap between two successive vehicles; therefore, the H3 rule is considered the best design of the four AVCSS CA rules. If a combinative rule is considered, the envelope of the speed–density curves of the four models might provide an optimal design, presenting a larger speed and flow than the H3 model. Therefore, an efficient design of AVCSS might be obtained by CA simulation. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, the traffic demand in metropolitan areas has greatly exceeded the vehicular capacity, which causes problems of increasing pollution and growing frequency of accidents. To solve this problem by constructing additional roadway is undesirable due to political and environment objections or impractical because of the high costs of construction. Therefore, the consistent management of large, distributed human-made transportation systems has become more and more important for planning and prediction of traffic. The purpose of intelligent transportation systems (ITS) is to increase transportation safety and efficiency by integrating human beings, vehicles, roadways and call-centers by encompassing wireless and wire line communications-based information, electronics technologies and transportation systems. ITS is an integrated system of communication, electronic and vehicular technology. Among the ITS applications, advanced vehicle control and safety systems (AVCSS) are developed for traffic safety and efficiency. AVCSS involves several safety critical functions, such as vehicle longitudinal and lateral control, so as to prevent or mitigate hazardous conditions. AVCSS becomes important as a research topic for providing safe mobility and accessibility with the trend of an aging population. Moreover, computer simulations as a means of evaluating, planning, and controlling traffic systems have gained considerable importance. In particular, micro-simulation work has been used to simulate the influence of governmental

∗ Corresponding address: Chung Hua University, Department of Transportation Technology & Logistics Management, No. 707, Sec. 2, WuFu Road, Hsinchu 300, Taiwan. E-mail address: [email protected] (S.-C. Lo). 0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2009.08.042

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actions like road-pricing and building new highways in solving traffic problems. The cellular automaton (CA) is one of these microscopic models. Although the concept of CA was first proposed long ago [1], the CA began to receive wide attention from the traffic and transportation community only after the simple formulation by Nagel and Schreckenberg [2], which is known as the NaSch model. Daganzo [3] presented how under some specific assumptions, a CA is equivalent to the kinematic traffic flow model and the car-following model. Furthermore, the CA has been generalized to slow-to-start phenomena [4–6], traffic with high speed vehicles [7], signalized intersection, multilane multi-class traffic flow [8–10], inhomogeneous mixed traffic flow [11] and large traffic networks. The objective of this paper is to explore traffic CA models, which involve advanced vehicle technology, and evaluate the contribution of ITS technology to traffic flow. CA rules with AVCSS are proposed, simulated and compared. According to previous studies [2,4–7], driving behaviors are identified by maximal speed (vmax ), dawdling probability (p) and different CA rules. To compare the CA rules with/without AVCSS, mixtures of different driving behaviors are simulated and the fundamental diagrams of multi-class user traffic flow are analyzed first. From the results, the whole of the traffic is found to be dominated by vehicles with the slowest speed or the maximal dawdling probability. Secondly, four CA rules, which are designed due to different gap setting, are proposed. Gap setting of AVCSS might influence the stability of flow and speed in traffic flow. From the simulated results, AVCSS can improve the level of service of highway systems significantly. Since the optimal gap can be determined by CA simulation, the optimal design of AVCSS can be obtained. According to the optimal gap, the setting of the warning system, speed control and distance control of automated control of vehicles can be designed so as to optimize traffic flow and improve driving safety. Thus, a traffic CA model can also be applied in designing electronics systems for vehicular applications. The rest of this paper is organized as follows. Section 2 introduces advanced vehicle control and safety systems. Next, the basic rule of CA traffic flow model is introduced in Section 3, and then CA rules with AVCSS are proposed in Section 4. Numerical examples and a discussion are presented in Section 5. Section 6 draws the conclusions. 2. Advanced vehicle control and safety systems AVCSS enhances safety and convenience, reduces emissions and fuel consumption, and increases the traffic capacity of existing roadways by handling and sensing related techniques. In the study of Huang et al. [12] traffic capacity was increased from 2000 vph (vehicles per hour) per lane to 5000 vph per lane when autonomous vehicle systems were included in advanced safety vehicle (ASV) developments. Up to now, worldwide representative smart car researches have included those of Navlab-11 at Carnegie Mellon University (CMU), Ohio State University (OSU) and California Partners for Advanced Transit and Highways project (PATH) in the USA, VaMoRs-P at the University of Bundeswehr Munchen (UBM) in Germany, ARGO at Parma University in Italy, THMR-V at Tsinhua University in China and TAIWAN iTS-1 at National Chiao-Tung University (NCTU) in Taiwan [13]. Generally, AVCSS includes a lane departure warning system, a lane keeping system, an automated adaptive cruise control system and an automated stop-and-go system; each of the systems will be introduced briefly in the following. The lane departure warning system indicates unintentional roadway departures to avoid running off the road and sideswipe collisions because drivers may lose concentration when they are drowsy or talking on a cell phone. The objective of the lane keeping system is to track the centerline of the lane with the steering wheel. The automated adaptive cruise control system is to adjust speed and inter-vehicle distance, with respect to the preceding vehicle. The longitudinal control system depends on forward distance measurements from the laser radar and the current speed of the host vehicle to generate commands for the throttle and brake actuators. Both speed control and distance control are involved in this system design. The automated stop-and-go system handles unknown vehicle models and guards against uncertainties arising from the vehicle body and external environments. If the sensed distance is smaller than the desired stop distance, the stop-and-go controller activates the brakes and releases the throttle pedal; otherwise, the throttle control is activated and the brake pedal is released. The stop-and-go system also performs collision avoidance, for obstacles or pedestrians. Not only the preceding vehicle, but also forward obstacles and pedestrians would also be detected, especially when these targets appear on the driving course of the vehicle. In this study, a single-lane highway is simulated by CA. Therefore, the effects of the lane departure warning system and the lane keeping system are ignored. Only the effects of the automated adaptive cruise control system and the automated stop-and-go system are considered. 3. Cellular automata of traffic flow In CA, a road is represented as a string of cells with equal size, which are either empty or occupied by exactly one vehicle. The size of cells is chosen to be equal to the speed of a vehicle that moves forward one cell during one time step. The vehicle’s speed (v ) is assumed to be a limited number of discrete values ranging from zero to vmax . The process of the NaSch model can be split up into four steps: (Step 1) Acceleration. If the time step is less than the total simulation time, let each vehicle with speed smaller than its maximum speed vmax accelerate to a higher speed, i.e. v = min(vmax , v + 1).

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(Step 2) Deceleration. If the speed is greater than the distance gap d to the preceding vehicle, the vehicle will decelerate: v = min(v, d). (Step 3) Dawdling. With given slow-down probability p, the speed of a vehicle decreases spontaneously: v = max(v − 1, 0). (Step 4) Propagation. Let each vehicle move forward v cells and let time step increase by 1. Then, repeat the procedure: acceleration, deceleration, dawdling and propagation. Takayasu and Takayasu (the TT model or the T2 model) [4] suggested a slow-to-start rule based on the NaSch model first. The TT model describes that a standing vehicle (i.e., a vehicle with the instantaneous speed v = 0) with exactly one empty cell in front accelerates with probability qt = 1 − pt , while all other vehicles accelerate deterministically. The other steps of the update rule (Step 2–Step 4) of the NaSch model are kept unchanged. The TT model reduces to the NaSch model in the limit pt = 0. Another slow-to-start model is the BJH model [5]. An extra step is introduced to implement the slow-to-start rule. The slow-to-start rule is different from the TT model. According to the BJH model, the vehicles which had to brake due to the next vehicle ahead will move on the next opportunity only with probability 1 − ps . Steps 1, 3, 4 of the BJH model are the same as in the NaSch model. An extra step (Step 1.5) is introduced and Step 2 is modified as follows: (Step 1.5)Slow-to-start. If flag = 1, then let v = 0 with probability ps . (Step 20 ) Blockage. v = min(v, d) and let flag = 1 if v = 0; otherwise let flag = 0. Here, ‘flag’ is a label distinguishing vehicles which have to obey the slow-to-start rule (flag = 1) from those which do not have to (flag = 0). Obviously, for ps = 0 the above rules reduce to the NaSch model. The slow-to-start rule of the TT model is a spatial rule. In contrast, the BJH slow-to-start rule requires memory, i.e., it is a temporal rule depending on the number of trials but not depending on the free space available in front of the vehicle. 4. AVCSS cellular automata models In this section, CA models with AVCSS vehicles in traffic flow are proposed. According to the function of the automated adaptive cruise control system and the automated stop-and-go system, several assumptions are considered. First, vehicles can adjust speed deterministically due to external environments; therefore, the dawdling step (Step 3) of the NaSch model is omitted. Second, vehicles can react with the preceding vehicles immediately, so there is no perception lag and slow-to-start phenomenon. Third, the highway and vehicle are not fully automated. An automated highway system means that vehicles are not only equipped with the four systems mentioned in Section 2, but also possess a vehicle-to-vehicle communication system and are controlled by a central control center. Thus, vehicles may establish a platoon, to obtain optimal aerodynamic characteristics so as to minimize fuel consumption and emission. In the extreme case, the capacity (qm ) of an automated highway system is equal to the product of the jam density (kj ) and designed maximum speed (vmax ), i.e., qm = kj ·vmax . There is no vehicle-to-vehicle communication in this study. In addition, the initial state on the highway is generated randomly. According to the assumptions, four AVCSS CA models based on different gap settings are proposed. The gap is defined to be the number of cells between one vehicle and its preceding vehicle. The four models assume that vehicles pursue maximal speed while they cruise. Since the dawdling step (Step 3) is omitted, the processes of the AVCSS CA models only have three steps. The H0 model assumes that there is no gap that should be kept between two successive vehicles. Actually, the H0 model is equal to the automated highway rule. The H1 model assumes that if v ≥ 3, then gap (d) must not be smaller than 1. The H2 model assumes that if v = 3, then d must not be smaller than 1. In addition, if v = 4, then it must not be smaller than 2. The H3 model assumes that if v = 2, then d must not be smaller than 1. In addition, if v ≥ 3, then d must not be smaller than 2. The propagation step (Step 4) is the same as that of the NaSch model. Step 1 and Step 2 for the four AVCSS CA models are given as follows: (H0 Step 1) Acceleration. if

vn < vmax ,

⇒ vn = vn + 1,

where vn is the speed of the nth vehicle. (H0 Step 2) Deceleration. if

vn > vn−1 + d,

⇒ vn = min(v, vn−1 + d),

where vn−1 is the speed of the (n − 1)th vehicle, i.e., the vehicle preceding the nth vehicle. (H1 Step 1) Acceleration. if

vn < 2, 2 ≤ v < vmax and d ≥ 1, otherwise,

⇒ vn = vn + 1, ⇒ vn = vn + 1, ⇒ vn = vn .

(H1 Step 2) Deceleration. if

d ≥ 1, otherwise,

⇒ vn = min(vn , vn−1 + d), ⇒ vn = min(2, vn , vn−1 + d).

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Fig. 1. (a) Flow–density (q–k0 ), (b) speed–density (u–k0 ) and (c) speed variance–density (σ –k0 ) relationships of the NaSch, TT and BJH models.

(H2 Step 1) Acceleration. if

vn < 2, vn = 2 and d ≥ 1, 3 ≤ vn < vmax and d ≥ 2, otherwise,

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d≥2 d=1 otherwise,

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(H3 Step 1) Acceleration. if

vn = 0 v = 1 and d ≥ 1, v ≥ 2 and d ≥ 2, otherwise,

⇒ vn ⇒ vn ⇒ vn ⇒ vn

= vn + 1, = vn + 1, = vn + 1, = vn .

(H3 Step 2) Deceleration. if

d≥2 d=1 otherwise,

⇒ vn = min(vn , vn−1 + d), ⇒ vn = min(2, vn , vn−1 + d), ⇒ vn = min(1, vn , vn−1 + d).

The H1 , H2 and H3 models consider different gaps due to different speeds, with respect to safety and system tolerance. The mean and variance of flow and speed are compared and discussed in Section 5. 5. Results and discussion Typically, the length of a cell was taken as 7.5 m, the time step is 1 s, the vmax is 5 (i.e., 135 km/h). In Taiwan, the upper speed limit of National Freeway No. 1 is 100 km/h. Therefore, the length of a cell is considered to be 7 m, and the maximum speed vmax is 4 (i.e., 100.8 km/h). The CA results are obtained from simulation on a chain of 1000 sites, which is 7 km. A periodic boundary condition is assumed so that both the total number of vehicles and the density are conserved at each simulated point. For each initial configuration of vehicles, results are obtained by averaging over 10,000 time steps after the first 10,000 steps, so the long-time limit is approached. This criterion was found to be sufficient to guarantee a steady state being reached. In this study, the density is the dimensionless density, which is denoted by k0 = k/kj , where k is the density and kj is the jam density. In this study, kj = 142 veh/km (vehicles per kilometer) per lane. The simulated k0 varies from 0.02 to 0.95. The NaSch, TT, BJH, H0 , H1 , H2 and H3 models are simulated and the simulated parameters are given in Table 1. Fig. 1 illustrates the flow–density, speed–density and speed variance–density curves of the NaSch4 (0.25), NaSch4 (0.75), NaSch3 (0.25), TT and BJH models. Let kc be the critical density, which is defined to be the density corresponding to the maximum flow. When k0 ≤ kc , traffic flow is under free-flow conditions, whereas when k0 > kc , traffic flow is under congested conditions. Also, when k0 ≤ kc the speed decreases slowly as k0 increases, whereas when k0 > kc , the speed decreases dramatically as k0 increases. kc for NaSch4 (0.75) is the smallest. kc for NaSch3 (0.25) is the largest, but the optimal flow for NaSch3 (0.25) is smaller than the optimal flow for NaSch4 (0.25). Actually, when k0 = kc (vmax = 3), the speed of NaSch4 (0.75) and the speed of NaSch3 (0.25) are almost the same. However, when vmax is larger, the speed is more sensitive to k0 . That is, the speed of

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Fig. 2. (a) Flow–density (q–k0 ), (b) speed–density (u–k0 ) and (c) speed variance–density (σ –k0 ) relationships of the H0 , H1 , H2 and H3 models. Table 1 Simulated scenario of CA rules for a single driving behavior. Model

Parameter

NaSch NaSch NaSch BJH TT

vmax vmax vmax vmax vmax

=4 =4 =3 =4 =4

Notation p p p p p

= 0.25 = 0.75 = 0.25 = 0.25 = 0.25

ps = 0.75 pt = 0.75

NaSch4 (0.25) NaSch4 (0.75) NaSch3 (0.25) BJH TT

NaSch4 (0.25) decays faster than the speed of NaSch3 (0.25) and the speed variance of NaSch4 (0.25) is larger than the speed variance of NaSch3 (0.25). From Fig. 1(b) and (c), the facts mentioned above can be observed. According to the figures, the slow-to-start rules reduce the flow and speed, especially in the congested flow. When k0 is smaller than 0.14, which is a dilute traffic condition, the speeds and flows of the NaSch4 (0.25), TT and BJH models are almost the same, because vehicles can move forward freely and not be blocked. The flow of the NaSch4 (0.75) model is much lower than the results for the other models. However, the speed of the NaSch4 (0.75) model is higher than the speed of the NaSch3 (0.25) one in the free-flow regime. The reason can be found in Fig. 1(c), i.e., the speed variance of the NaSch4 (0.75) model is the largest one. The flow decreases with increase of the speed variance. From Fig. 1, we see that larger vmax allows higher speed and larger p implies that more vehicles may decelerate in free flow; i.e., large vmax and p may induce unstable traffic in a free-flow regime. Since larger vmax allows higher speed, it also induces larger flow in a free-flow regime. On the other hand, the same phenomena cannot be observed in intermediate and congested flow, because drivers cannot drive freely when density increases; interaction among vehicles decreases the mean speed and flow. The speed and flow of the TT model is slightly larger than that of the BJH model while k0 < 0.5, because the slow-to-start rule of the TT model depends on the empty space. Both slow-to-start and dawdling cases reflect the reaction of perception lag of the human driver, which reduces the capacity of highways. Therefore, advanced vehicle control and safety systems are worth developing. Fig. 2 illustrates the simulated results for the H0 , H1 , H2 and H3 models. The four models are simulated on the basis of the assumption that all vehicles are AVCSS vehicles. Obviously, the flows and speeds of the four models are much larger than for the NaSch, TT and BJH models. Let kt be the transient density, which is defined to be the boundary of the flow–density relation transient, from one phase to another phase. When k0 < kt , vehicles move forward at vmax and the flow increases with k0 . The H0 model does not have kt . The kt for the H1 and H2 models are the same. In both cases, kt = 0.13. kt for the H3 models is 0.24, which is larger than the kt for the H1 and H2 models. From the four models, some interesting results are observed. First, vehicles with AVCSS can improve the level of service of a highway. The service flow might increase as density increases and be a nondecreasing curve. This is quite different to the case for q–k0 curves for rules without AVCSS. The q–k0 curves increase in the free-flow regime and decrease in the congested regime. Second, a transitional state, kt ≤ k0 ≤ 0.4, is observed. When k0 < kt , the traffic is dilute and vehicles can obtain vmax with arbitrary initial condition. Therefore, the average speed decreases when k0 is larger than kc except for the H0 model. When k0 > 0.4, the speeds of the H1 , H2 and H3 models tend toward another equilibrium. Although the gap settings of the H1 and H2 models are different, the results for the H1 and H2 models are almost the same. The designed gap of the H3 model is larger than those of the others. Surprisingly, the H3 model performs better than the H1 and H2 models in the transition state. In the congested regime (k0 > 0.4), the performance of the H3 model decreases with density increase significantly. Thus, the performance of traffic flow with AVCSS vehicles depends on the design of the AVCSS, especially the setting of the gap. Generally, the flow and speed decrease as the gap increases. Nevertheless, the flow and speed of the H3 models are larger than those of the H1 and H2 models within the transitional state. Therefore, optimal flow and speed of traffic flow can be obtained by designing the gap of AVCSS carefully. Further observations are obtained from the speed variance of the simulated models. Speed variances of all simulated models are illustrated in Figs. 1(c) and 2(c). In the uncongested regime, the speed variances of the models without AVCSS

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Table 2 Simulated scenario and notation for multi-class user traffic flow. Model 1

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Proportion (%)

Notation

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0 25 50 75 100

NS4 (0.75) NS4 (0.75%–75%, 0.25%–25%) NS4 (0.75%–50%, 0.25%–50%) NS4 (0.75%–25%, 0.25%–75%) NS4 (0.25)

100 75 50 25 0

NaSch4 (0.25) NaSch4 (0.25) NaSch4 (0.25) NaSch4 (0.25) NaSch4 (0.25)

0 25 50 75 100

NS3 (0.25) NS3 -75%, NS4 -25% NS3 -50%, NS4 -50% NS3 -25%, NS4 -75% NS4 (0.25)

25 50 75

NS4 -75%, H3 -25% NS4 -50%, H3 -50% NS4 -25%, H3 -75%

Scenario 1 NaSch4 (0.75) NaSch4 (0.75) NaSch4 (0.75) NaSch4 (0.75) NaSch4 (0.75) Scenario 2 NaSch3 (0.25) NaSch3 (0.25) NaSch3 (0.25) NaSch3 (0.25) NaSch3 (0.25) Scenario 3 NaSch4 (0.25) NaSch4 (0.25) NaSch4 (0.25)

75 50 25

H3 H3 H3

vehicles are much larger than those of the models with AVCSS. Among the NaSch, TT and BJH models, we found that the slowto-start and dawdling cases induce variance. Vehicles with AVCSS present stable and homogeneous behavior, relatively. Furthermore, the speed variances of the H1 and H2 models are much larger than the speed variances of the H0 and H3 models in the transition regime. This is because the H1 and H2 rules induce lots of acceleration and deceleration so as to keep the gap in the transition state. Unstable behavior not only increases system risk, but also consumes fuel inefficiently. In the congested regime, vehicles without AVCSS cannot move forward freely and are restricted by the external environment and the driver’s reaction lag; therefore, the speed variance is small. The speed variance of the H3 model is small which might be caused by the designed gap, which eliminates the oscillation of speed and flow. The analytical analysis will be left for further research. Comparing with the results for H1 and H2 models, the results for the H0 and H3 models are relatively stable. Generally, a stable system is much more reliable than an unstable one. Therefore, we suggest that an advanced control vehicle may be designed using the H0 or H3 rules. Since the H3 rule provides a wider gap at high speed, it may be safer than the H0 rule under a non-fully automated highway situation. Next, multi-class user traffic flow based on a mixture of different vmax , p and rules is discussed. The scenario and notation are given in Table 2. The results are shown in Figs. 3–5, where Figs. 3 and 4 show the simulation results for two nonAVCSS mixed CA rules and Fig. 5 shows the simulation results for AVCSS and non-AVCSS mixed rules. The highest speed of the NaSch4 (0.25) and NaSch4 (0.75) mixture models is the same as the highest speed of NaSch4 (0.75). In addition, the highest speed of the NaSch4 (0.25) and NaSch3 (0.25) mixture models is the same as the highest speed of NaSch3 (0.25). On the other hand, the average speed and flow decrease with the number of slow vehicles increasing. Therefore, we can conclude that the slowest behavior dominates the maximum speed of the whole traffic flow in single-lane traffic flow. In scenario 1, the reduction of the average speed and flow is nonlinear; this cannot be estimated by interpolation of the speed and flow of the two mixed models. For example, the maximum flow of NaSch4 (0.25) is 1749 veh/h and that of NaSch4 (0.75) is 616 veh/h. If we estimate the maximum flow of NS4 (0.75%–75%, 0.25%–25%) by interpolation, it will be 616×0.25+1749×0.75 = 1465.75 > 1234 veh/h, which is the simulation result. Furthermore, the speed variance of multiclass user traffic is larger than that of single-user traffic, especially in the transition state (0.18 ≤ k0 ≤ 0.4) in scenario 1. In scenario 2, the maximal flow of multi-class user traffic is higher than that for single-user traffic. From Fig. 4(c), which shows the speed variance–density curves of scenario 3, we can find the reason. The maximal flow occurs at the critical density, kc , which is in the transition state (0.18 ≤ k0 ≤ 0.4). Within the transition state, the speed variance of multi-class user traffic is smaller than that of single-user traffic flow. Therefore, the speed of multi-class user traffic is more stable and a bit larger than that for single-user traffic. Consequently, the flow of multi-class user traffic is larger than that of single-user traffic in the transition state. Several important phenomena are observed: (1) large vmax and p induce large speed variance in both single-user and multi-class user traffic flow; (2) large speed variance reduces flow in both single-user and multi-class user traffic flow; (3) the maximal speed of mixed traffic flow is dominated by the slowest behavior; (4) the average flow and speed of multiclass user traffic are not absolutely smaller than the flow and speed of single-user traffic, which depends on the mixed rules. Finally, mixed manual and automated control traffic flow is simulated and discussed. Since the H3 model is considered, the best rule among the four AVCSS rules, the H3 and NaSch4 (0.25) mixed traffic flow, are simulated and the results are depicted in Fig. 5. If there are more vehicles with the H3 rule, the flow and speed distributions are more similar to the distributions of the H3 model. When k0 < 0.2, the curves are overlapped because all vehicles including AVCSS vehicles and non-AVCSS vehicles can move forward freely. The difference between the curves become significant with k0 increase. Flow and speed increase as the number of H3 vehicles increases. If there is 25% automated control vehicle, the maximal

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Fig. 5. (a) Flow–density (q–k0 ), (b) speed–density (u–k0 ) and (c) speed variance–density (σ –k0 ) relationship for the simulated scenario 3.

flow is 2057 veh/h, which achieves an 18% improvement. If there is 50% automated control vehicle, the maximal flow is 2613 veh/h, which achieves a 50% improvement. If there is 75% automated control vehicle, the maximal flow is 2057 veh/h, which achieves an 86% improvement. The more AVCSS vehicles in the mixed traffic, the more improvement of the flow achieved. According to Fig. 5(c), the speed variance is much smaller than single-user and multi-class user CA rules without AVCSS. Thus, AVCSS vehicles can stabilize the mean speed of traffic efficiently, which improves the safety and level of service significantly, especially in the congested traffic condition. 6. Conclusion In this study, four CA models with AVCSS are proposed and compared with the NaSch, TT and BJH models. From the results, we see that AVCSS improves the flow and speed of highway systems significantly. Even if only some of the vehicles are advanced vehicles, the level of service is improved and the speed variance is reduced. Gap is an important design parameter of AVCSS. A good setting of the gap for AVCSS can stabilize and improve traffic flow. Among the four proposed CA models,

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the H3 model presents an efficient and safe design for AVCSS vehicles in the transition state. However, it performs poorly in a congested regime. According to Fig. 2, the envelope curve of the H1 , H2 and H3 models shows good reliability and efficiency. That is, choose the H3 models as the design while k0 ≤ 0.4 and choose the H1 or H2 models as the design while k0 > 0.4. Hence, the CA technique is not only useful for simulating traffic flow on highway systems but also useful for designing electronic systems for AVCSS. Acknowledgements This work was partially supported by the National Science Council (NSC), Taiwan, under Contract NSC 97-2221-E-216015. Also, the author is grateful to the National Center for High-performance Computing for computer time and facilities. References [1] M. Cremer, J. Ludwig, A fast simulation model for traffic flow on the basis of Boolean operations, Mathematics and Computers in Simulation 28 (1986) 297–303. [2] K. Nagel, M. Schreckenberg, A cellular automaton model for freeway traffic, Journal of Physique I, France 2 (1992) 2221–2229. [3] C.F. Daganzo, In traffic flow, cellular automata = kinematic waves, Transportation Research B 40 (2006) 396–403. [4] M. Takayasu, H. Takayasu, 1/f noise in traffics model, Fractals 1 (1993) 860–866. [5] S.C. Benjamin, N.F. Johnson, P.M. Hui, Cellular automaton models of traffic flow along a highway containing a junction, Journal of Physics A 29 (1996) 3119–3127. [6] A. Schadschneider, M. Schreckenberg, Traffic flow models with ‘slow-to-start’ rules, Annalen der Physik 509 (1997) 541–551. [7] M. Fukui, Y. Ishibashi, Traffic flow in 1D cellular automaton model including cars moving with high speed, Journal of Physical Society Japan 65 (1996) 1868–1870. [8] C.F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transportation Research B 28 (1994) 269–287. [9] K. Nagel, D.E. Wolf, P. Wagner, P.M. Simon, Simplified cellular automaton model for city traffic, Physical Review E 58 (1998) 425–1437. [10] K. Nagel, Particle hopping models and traffic flow theory, Physical Review E 53 (1996) 4655–4672. [11] L.W. Lan, C.W. Chang, Inhomogeneous cellular automata modeling for mixed traffic with vehicles and motorcycles, Journal of Advanced Transportation 39 (2004) 323–349. [12] S. Huang, W. Ren, S.C. Chan, Design and performance evaluation of mixed manual and automated control traffic, IEEE Transactions on Systems, Man and Cybernetics, Part A 30 (2000) 661–673. [13] B.-F. Wu, C.-J. Chen, H.-H. Chiang, H.-Y. Peng, J.-W. Perng, L.-S. Ma, T.-T. Lee, The design of an intelligent real-time autonomous vehicle, TAIWAN iTS-1, Journal of the Chinese Institute of Engineers 30 (2007) 829–842.