Modeling the Effect of the Platoon Size of CAVs on Mixed Traffic Flow: A Cellular Automaton Method

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Introduction
Autonomous vehicles are increasingly entering our range of vision as more advanced technologies are being used in vehicles, making them smarter and safer.In addition, connected and autonomous vehicles (CAVs) have emerged due to the development of 5G Internet technology.CAVs can gather complicated data, including nearby people and objects, road conditions, and trafc infrastructure, by using advanced sensors.Ten, they interpret the information and deliver it to the driving control system.CAVs can also broadcast data about how their car operates to nearby vehicles.Te Chinese government has devised the "Connected Automated Vehicle Technology Roadmap 2.0," projecting that semiautonomous and conditionally autonomous vehicles will capture an estimated 50% share of new car sales by 2025.Furthermore, highly CAVs will be widely deployed in various regions by 2035 [1,2].Due to the quick growth of connected automated technologies, there will be mixed trafc fows consisting of both CAVs and HDVs coexisting for a long time in the future [3][4][5].Compared with traditional vehicles, through vehicle-to-vehicle (V2V) communication and vehicle-to-infrastructure (V2I) communication, CAVs can achieve real-time information sharing, thereby forming fexible vehicle feets, which signifcantly improve road capacity and alleviate trafc congestion [6,7].Extensive research highlights the benefts of vehicle platooning, including a more compact vehicle arrangement, reduced human driving errors, and enhanced road trafc effciency.Notably, the size of vehicle platoons assumes a pivotal role in efective platoon management [8][9][10].Consequently, a comprehensive investigation into the mixed trafc fow characteristics, considering the platoon size of CAVs, assumes paramount signifcance.
Cellular automaton has garnered signifcant attention in trafc fow research due to its computational efciency and capacity to simulate intricate trafc phenomena by setting simple rules [11][12][13][14][15].In the pursuit of understanding trafc fow dynamics, numerous scholars have employed cellular automaton models to capture the microlevel driving behaviors of vehicles within mixed trafc scenarios.As the intelligent connected technology continues to advance, researchers are expanding their focus beyond individual CAVs and exploring the formation of CAV platoons.However, in real-world scenarios, the size of the CAV platoons is often constrained by the limitations of the intervehicle communication range.Existing investigations have highlighted that once CAV platoons reach their maximum size, additional vehicles are precluded from joining the platoon, thus forming new platoons with more signifcant headways.Consequently, there is a pressing need for further research to model the following behavior between CAV platoons [16][17][18][19].
To address this gap, this study considers the platoon size when developing the cellular automaton model.By referring to references [20,21], based on the characteristics of carfollowing behavior, the car-following modes on the road are classifed into human-driven vehicles (HDV), adaptive cruise control (ACC), and cooperative adaptive cruise control (CACC).Accounting for the limitations of the platoon size, the CACC is further subdivided into intraplatoon and interplatoon car-following modes.Ten, separate cellular automaton models are established for all four car-following modes.Subsequently, numerical simulation experiments are devised to comprehensively analyze the characteristics and sensitivity of mixed trafc fow.Te main contributions of this paper are as follows: (1) CACC is subdivided into intraplatoon and interplatoon following modes to consider the limitation of platoon size.Ten, cellular automaton models are established for the two following modes.Tis will better simulate the car-following behavior of CAV platoons in real road trafc fow.(2) We investigate the efects of the penetration rates of CAVs on mixed trafc fow, including fundamental diagrams, trafc congestion, and reaction time.(3) We evaluate the impact of platoon size on trafc capacity and determine the optimal platoon size for diferent trafc densities.
Te remainder of this paper is organized as follows.Section 2 reviews the relevant work.Section 3 analyzes the driving characteristics of diferent car-following modes and establishes a cellular automaton model.Section 4 designs simulation experiments to study the impact of CAV penetration rate on trafc fow and determines the optimal platoon size under diferent trafc densities.Finally, Section 5 concludes this study and proposes future research directions.

Literature Review
In 1992, Nagel and Schreckenberg [22] proposed an NS model capable of simulating simple trafc fow phenomena.Many scholars have made improvements on this basis.M. Takayasu and H. Takayasu [23] took the lead in introducing the slowstart rule into the NS model and proposed the TT model.Te results show that the model can not only simulate metastable and hysteresis phenomena but also simulate the phenomenon of exit phase separation under high trafc density.Zeng et al. [24] redefned the object of stochastic slowdown and proposed the cruising driving limit model.Benjamin et al. [25] reformulated the slow start rule, leading to the development of the BJH model.Schreckenberg et al. [26] established a probabilistic relationship between stochastic slowdown and vehicle speed, giving rise to the VDR model.Furthermore, through improving the acceleration rules, several models have been proposed, including the comfortable driving model [27], the Fukui-Ishibashi model [28], the three-phase trafc fow model [29], and the optimal velocity model [30].
With the continuous development of intelligent and connected technology, many scholars have begun to study the mixed trafc fow of autonomous vehicles (AVs) and humandriven vehicles (HDVs).Regarding the trafc fow characteristics, Yang et al. [31] introduced the Gipps safe distance rule based on the NS model and used diferent reaction times to diferentiate between AVs and HDVs.Trough numerical simulation, they found that AVs can improve trafc capacity and alleviate trafc congestion.Seeking to delve deeper into the infuence of vehicle reaction time on road trafc fow, Vranken et al. [32] devised a single-lane cellular automaton model with a fne time step of 0.1 s.Tis model incorporates a fnite braking capacity and successfully eliminates velocity fuctuations when the CAVs penetration rate reaches 100%.Focusing on trafc safety considerations, Ye and Yamamoto [33] employed the frequency of hazardous situations and collision time as evaluative indicators to evaluate trafc fow safety.Tey investigated the impact of CAVs' penetration rates on trafc safety through simulations.Te results demonstrated a notable improvement in trafc safety conditions as the prevalence of automated driving vehicles increased.Gong et al. [34] constructed a cellular automaton model that accounts for limited visibility, aiming to examine the infuence of CAVs on road safety under extreme weather conditions.Tis study incorporated diverse car-following scenarios and systematically explored the efects of visibility levels and CAV penetration rates on trafc safety.Te fndings provide valuable insights into managing future mixed trafc fow in dense fog conditions.Regarding the intersection trafc control, Marzoug et al. [35] adopted an open boundary and proposed an equation for calculating the probability of accidents based on the NS model.Te results showed that the probability of accidents exhibits three 2 Journal of Advanced Transportation diferent patterns as the CAV penetration rate changes.To further investigate the trafc control methods under the nonvehicle network intersections and interactions of the connected vehicles (CVs), Zhao et al. [36] considered vehicleto-vehicle and vehicle-to-signal communication and proposed two cellular automaton models, namely, CE-NS and CI-NS.Factors such as preceding vehicle speed and signal conditions were adequately considered when setting the rules.
Te results indicated that CAVs can efectively reduce the queue length at signalized intersections and improve the intersection capacity.Te abovementioned research demonstrates that CA models can be widely applied to trafc fow studies to reproduce the actual trafc characteristics.Te scholars mentioned above have primarily concentrated on investigating the driving characteristics of individual vehicles within mixed trafc fow, thereby inadvertently neglecting the interactive dynamics among CAVs.With the evolution of vehicular networking technology, CAVs synchronize their driving states with preceding vehicles through real-time communication facilitated by V2V and V2X technologies, culminating in forming platoons with a lead vehicle.Such CAV platoons typically comprise multiple trailing vehicles, thereby maintaining smaller headways within the platoon, which can substantially enhance road capacity [37].Building upon this premise, Wu et al. [38] considered CAV platoons and developed a greedy algorithm-based cellular automaton model for interactions in a connected vehicle environment.Te CAV platoons are regarded as the fundamental control element in this model.To pass through the intersection, the platoons cooperate and communicate with each other.However, the only vehicles involved in the trafc fow in this study are CAVs, and the primary responsibility of the CAV platoons is to regulate them as the fundamental unit when crossing the intersection.Yangsheng et al. [39] further considered degraded car-following in mixed trafc fow.Tey categorize vehicles on the road into distinct car-following modes based on their driving characteristics.Particularly, vehicles coalesce into platoons under the CACC car-following mode.Te study revealed that when the penetration rate reaches 100%, all CAVs on the road travel in platoons, with road capacity being solely infuenced by the critical density.However, it is noteworthy that the study did not account for platoon size limitations, leading to a discrepancy between actual and calculated capacity.
Based on the abovementioned research, it can be found that the existing studies have not sufciently addressed the impact of platoon size on mixed trafc fows.When one platoon reaches its maximum size, the following CAVs will form a new platoon.Few scholars have studied the carfollowing behavior between two platoons.Tis paper establishes a cellular automaton model for interplatoon and intraplatoon car-following and determines the optimal platoon size under diferent trafc densities.

Methodology
3.1.Car-Following Mode.Amidst the ongoing advancements in intelligent connected technology, it is common to witness CAVs traveling in platoons within mixed trafc fows.Tese platoons typically comprise a leading vehicle and one or more following vehicles.Figure 1 visually shows the four diferent car-following patterns that can be observed in a mixed trafc fow.
3.1.1.Human-Driven Vehicle Mode.As indicated in Figure 1, when the ego vehicle is an HDV, regardless of whether the preceding vehicle is an HDV or a CAV, the ego vehicle adopts the HDV car-following mode.In the trafc fow, when the driving behavior of the preceding vehicle changes, the ego vehicle needs some time to determine the changes in the preceding vehicle's behavior and take corresponding measures.For the convenience of description, this time is referred to as the reaction time in this paper.For the HDV carfollowing mode, the reaction time depends on individual diferences between the drivers.By considering factors such as the driver's age, personality, and driving behavior, the speed of HDVs undergoes a random deceleration process, which is referred to as the random slowdown process in this paper.

Adaptive Cruise Control Mode.
When the ego vehicle is a CAV, and the preceding vehicle is an HDV, this carfollowing mode is known as the ACC mode.In this mode, the ego vehicle cannot communicate directly with the preceding vehicle.However, it can leverage communication with the roadside infrastructure to acquire real-time information regarding the driving behavior of the preceding vehicle and respond promptly.Te response time in this scenario is contingent upon the processing time of the onboard system for handling the received information and it is shorter compared to the reaction time in the HDV mode.

Cooperative Adaptive Cruise Control Mode.
In the CACC car-following mode, the ego and the preceding vehicles are CAVs.In this mode, the ego vehicle can quickly perceive the driving state changes of the preceding vehicle through real-time communication between vehicles and adjust its driving behavior to maintain consistency with the preceding vehicle, thus forming a platoon.However, CAV platoons usually have a certain size limit (S is the maximum platoon size).When a platoon is full, and the S+1 vehicle is a CAV, it will lead another platoon on the road.At this point, the car-following mode between the S+1 vehicle and the preceding platoon is the intraplatoon car-following mode, while the car-following mode between the remaining vehicles is the interplatoon car-following mode.Tis study refects the main diferences between the two car-following modes in the reaction time and the rules of acceleration and deceleration.

Safe Distance.
As shown in Figure 2, this paper introduces the concept of safe distance and defnes it as the minimum distance required to prevent a collision with the preceding vehicle when the preceding vehicle brakes suddenly.Te safe distance is defned as follows: where d n,safe is the minimum safe distance between the vehicle n and n-1, v n (t) and v n−1 (t) are the velocities of vehicle n and vehicle n-1 at time t, respectively, B is the maximum deceleration of the vehicle, and τ is the reaction time of the vehicle.

HDV-and ACC-Following Modes.
Te ego and preceding vehicles cannot exchange information in HDV and ACC modes.Only the diference in reaction time must be considered in the modeling.Terefore, this paper models the abovementioned two modes together.
(1) Acceleration In actual trafc, when road conditions are favorable, and visibility is excellent, vehicles frequently accelerate to maximize travel efciency.For the HDV and ACC modes, when the distance between the preceding vehicle and the ego vehicle exceeds the safety distance, the following vehicle will accelerate, according to the following equation: where v n (t + 1) is the velocity of the vehicle n at time t + 1, a n is the acceleration of the vehicle, v max is the maximum velocity of the vehicle, To ensure driving safety, the vehicle will decelerate when the distance between vehicle n and the preceding vehicle n − 1 is less than or equal to the safe distance.Te specifc regulation is as follows: Equation ( 3) indicates that when d n is less than d n,safe , the vehicle will decelerate to d n to ensure secure driving; when d n is equal to d n,safe , it will maintain a constant speed while ensuring safety.As with the acceleration rule, the diference between HDV and ACC modes resides in the response time.(3) Random deceleration Due to the impact of personal factors such as age, gender, and temperament on driving behavior, random deceleration may occur during the driving process.Typically, equation (4) represents the specifc rule that vehicles in motion are generally subject to random slowdown with a certain probability p slow , i.e., where b n is the random deceleration of vehicle n.Specifcally, since CAVs are not afected by the  x n (t) x n-1 (t) x n (t + 1) x n-1 (t + 1) driver's actions, the ACC mode does not account for sporadic (4) Position update After updating the speed, the vehicle's position is updated according to equation ( 5).

Intraplatoon-Following Mode
(1) Velocity update Under the intraplatoon mode, acceleration driving can be initiated if either of the following two conditions is met: (a) when the distance between the ego vehicle and the preceding vehicle exceeds the safety distance, the ego vehicle will accelerate; (b) considering the intraplatoon mode with intervehicle communication, even if the distance between the ego vehicle and the preceding vehicle is less than the safe distance, it is considered to be safe when the speed of the preceding vehicle is greater than the speed of the ego vehicle.
When the safety gap between the ego vehicle and the preceding vehicle is less than or equal to the safety distance, and the preceding vehicle's speed is less than the ego vehicle's speed, the vehicle will decelerate.
To better describe the speed change circumstance, this article introduces the concept of expected speed to characterize these two acceleration rules, as shown in equation (6).
where sgn(X) is the speed judgment function, and the exact value is represented by equation (7).
Using equations ( 6) and ( 7), we can derive the intraplatoon mode speed update rule shown in equation (8).
(2) Position update In the intraplatoon mode, the equation for updating the vehicle's location is given in equation (5).

Interplatoon Following Mode.
When the vehicle's following mode is interplatoon, the ego vehicle can constantly travel in a convoy with the preceding vehicle.All vehicles in the convoy maintain the same pace and driving style.Tis paper simulates it from two perspectives: before the formation of CAV platoons and after the formation of CAV platoons.
(1) Before the formation of CAV platoons Before forming a platoon, the distance between the ego vehicle and the preceding vehicle is greater than the safe distance, indicating good road conditions.Under the CACC mode, the ego vehicle can synchronize with the preceding vehicle's state changes through V2V communication, enabling the ego vehicle to accelerate and form a platoon with the preceding vehicle as quickly as possible.Te specifc acceleration rules are as follows: where When the ego vehicle accelerates, the distance between the ego vehicle and the preceding vehicle equals the safe distance, and the two vehicles will merge into a platoon and proceed together.Once the platoon is established, both vehicles will adopt the same driving behavior, by adhering to the specifc evolution rules.
(3) Position update In the interplatoon mode, the equation for updating the vehicle's location is given in equation (5).

Teoretical Trafc Capacity Calculation.
Trafc capacity refers to the maximum number of vehicles traversing a given stretch of road in each period under specifed road conditions.Tis paper computes the trafc capacity by using the average headway time.When the CAV penetration rate is p (0 < p < 1), the HDV penetration rate is 1 − p.If the vehicles are independent of each other, then p − p 2 can be used to calculate the proportion of ACC.By referring to [40], it is possible to calculate the probabilities of intraplatoon and interplatoon.When the trafc fow is stable and vehicle velocities are equivalent, the average headway h can be calculated as follows: Journal of Advanced Transportation where τ HDV , τ ACC , τ LC , and FC are the reaction times of HDV, ACC, intraplatoon, and interplatoon, respectively.By substituting equation (11) into the equation for calculating the road capacity C � 3600/h, we can obtain When the CAV penetration rate reaches 100%, vehicles on the road form platoons, with each platoon consisting of one vehicle in the intraplatoon mode and S-1 vehicles in the interplatoon mode.When the trafc fow reaches a stable state, the average headway can be calculated as follows: By substituting equation ( 13) into the equation for calculating the road capacity C � 3600/h, we can obtain In summary, the following equation is used to calculate the theoretical trafc capacity:

Numerical Analysis
4.1.Simulation Environment Setup.Tis paper concentrates on the single-lane highway as the subject of its investigation.Te lane consists of 4,000 cells, each measuring 1 meter in length.Terefore, the length of the road is 4000 meters.Te simulation is conducted with periodic boundary conditions, a one-second time step, and four thousand cumulative steps.Te data collected after stabilized trafc fow (2,000 seconds) will be analyzed as simulation results.Each car has a random speed and is evenly distributed on the road in the initial condition.Te vehicle's maximum speed has been set to 35 m/ s (126 km/h), and its conventional acceleration and random deceleration have been set to 2 m/s 2 and 3 m/s 2 .Te maximum deceleration B is set to 5 m/s 2 , vehicle length is 5 m, and reaction times τ HDV , τ ACC , τ FC , and τ LC are, respectively, set to 2 s, 1.5 s, 1 s, and 0.4 s.Te probability of random stalling P slow is 0.3.To enhance the precision of the results, each simulation group is executed 10 times with diferent random speeds, and the fnal average value is recorded.

Analysis of the Basic Trafc Flow
Diagram. Figure 3 depicts the density-fow-speed fundamental diagram for various CAV penetration rates when S � 6.As shown in Figure 3(a), when the CAV penetration rate remains constant, the average speed remains constant as the density increases and then starts to decrease.Tis indicates that vehicles can maintain high-speed driving in free-fow conditions, and the introduction of CAVs has little impact on trafc fow.However, an increase in the CAV penetration rate alleviates trafc congestion in congested fow conditions.When the CAV penetration rate reaches 100%, it can maintain high speed at a vehicle density of 60 veh/km.As demonstrated in Figure 3(b), the density range of the free-fow phase varies at diferent penetration rates.A higher CAV penetration rate leads to a wider density range in the free-fow stage than a lower CAV penetration rate.Tis can be attributed to the superior performance of CAVs compared to the conventional vehicles.A larger freefow phase indicates a higher capacity.As the CAV penetration rate increases, the road capacity consistently improves.At a CAV penetration rate of 60%, the road capacity is 1.6 times that of purely HDVs.When the CAV penetration rate reaches 80%, the road capacity increases to 2.2 times to that of purely HDVs.When the penetration rate reaches 100%, the road capacity is amplifed to 4.3 times to that of purely HDVs.Tis is because CAVs do not undergo random slowdown processes and can adjust their speed to form platoons with preceding vehicles quickly.Once a platoon is formed, vehicles within the platoon travel at the same speed, thereby signifcantly improving the road space utilization.Hence, CAVs play a crucial role in enhancing the road capacity.
To verify the accuracy of the simulation model, this paper calculates the theoretical trafc capacity by substituting the platoon size S � 6 into equation [12] and compares it to the 6 Journal of Advanced Transportation simulation results shown in Table 1 to determine if the model accurate.Te error between the theoretical and simulation trafc capacity under difering CAV penetration rates is within 9%, and it is only 0.46% when the CAV penetration rate is 100%.Terefore, the model error in this paper is relatively small, and it can accurately simulate the actual road trafc situation.

Trafc Congestion Analysis.
To quantitatively describe trafc congestion, this article uses the ratio of congested vehicles to refect trafc congestion [31].Te equation for calculation is as follows: where ∆T is the simulation time step, N is the total number of vehicles, and n is the number of severely congested vehicles, which are those moving slower than 10 km/h on the roadway.Figure 4 depicts the correlation between vehicle density and trafc congestion.Te congestion rate progressively rises with increasing vehicle density.Nonetheless, the density threshold for congestion onset varies across diferent CAV penetration rates, whereby a higher CAV penetration rate indicates a higher threshold.At a CAV penetration rate of 0, congestion initiates at a density of 15 vehicles/km.At a penetration rate of 60%, congestion commences at a density of 25 veh/km.When the penetration rate reaches 100%, congestion emerges at a density of 150 vehicles/km.Meanwhile, at the same density, the higher the CAV penetration rate, the smaller the congestion ratio.
To delve deeper into this phenomenon, this study examines trafc congestion under diferent CAV penetration rates at a density of 100 veh/km, as presented in Table 2. Te results highlight a signifcant reduction in the congestion rate as the CAV penetration rate increases.Specifcally, compared to exclusively HDVs, the congestion rate decreases by 63.71% at a CAV penetration rate of 80% and by 100% at a CAV penetration rate of 100%.Tis outcome is attributed to the diminished infuence of random deceleration in HDVs in high CAV penetration rate scenarios.Moreover, the likelihood of vehicles forming stable platoons on the road increases, thereby contributing to higher average speeds and trafc congestion mitigation.To determine the optimal platoon size at various vehicle densities, this study divides the vehicle densities into three sections.Te relationship between trafc fow and platoon size for these densities is depicted in Figure 6.
In the frst section, a density of 20 vehicles/km is chosen to investigate the impact of the maximum platoon size on trafc fow.Te blue line in Figure 6 indicates that, at this vehicle density, the infuence of the platoon size on trafc fow is little.Tis can be attributed to the low vehicle density, where vehicles are in a state of free fow.Although platoon size afects the average headway, the vehicle speed remains constant, resulting in negligible changes in trafc fow.
Moving to the second section, a density of 60 vehicles/km is selected.Te orange line in Figure 6 reveals that the trafc fow gradually increases and stabilizes as the platoon size increases from 2 to 7 vehicles.Tis suggests that, in this particular scenario, a platoon size of 7 vehicles is optimal.Tis is because when the platoon size is between 2 and 7 vehicles, more platoons are on the road, meaning more vehicles are in the intraplatoon mode.Compared to vehicles in the interplatoon mode, vehicles in the intraplatoon mode require a larger safety distance.Due to the high vehicle density in this scenario and the short distances between platoons, the intraplatoon mode's safety distance requirement cannot be met, leading to deceleration.As the platoon size increases, the number of vehicles in the intraplatoon mode decreases, resulting in fewer decelerations between platoons and an overall increase in trafc fow.When the platoon size reaches or exceeds 7, all vehicles in this scenario achieve a free fow state, causing the trafc fow to remain unchanged despite further increases in platoon size.
In the third section, a density of 100 vehicles/km is employed.Te gray line in Figure 6 demonstrates that, in this scenario, the trafc fow increases with larger platoon sizes.Tis is due to the high vehicle density, which makes it challenging to attain a free-fow state.Te average headway decreases as the platoon size increases, thereby improving the trafc fow.

Analysis of the Response Time of HDV and Interplatoon.
Human drivers impact the response time of HDV, whereas the system's communication time determines the response time of interplatoon.In the sensitivity analysis, the density is set to 40 vehicles/km, where the τ HDV varies from 1.8 s to 2.4 s with a 0.1 s increment and τ FC varies from 0.3 s to 0.6 s with a 0.05 s increment.Te results of the sensitivity analysis are depicted in Figure 7.    8 Journal of Advanced Transportation Figure 7 illustrates notable disparities in the infuence of τ HDV and FC on average speed at varying CAV penetration rates.Regarding HDVs, an augmented reaction time among diferent CAV penetration rates leads to a substantial decline in the average speed.Equation (1) reveals that an increased human driver reaction time necessitates a larger safety distance.Consequently, under the speed update rule for HDVs, there is a reduction in the number of vehicles opting for acceleration in the subsequent time step, accompanied by a corresponding increase in deceleration choices, resulting in a decreased average speed.Concerning CAVs, under low CAV penetration rates (Figures 7(a)-7(c)), the average speed remains unafected by τ .Tis arises due to the dominance of HDVs at low CAV penetration rates, resulting in a scarcity of vehicles forming the interplatoon mode on the road.Consequently, the impact of τ FC is little.However, under high CAV penetration rates (Figures 7(d) and 7(e)), as τ FC increases, the average speed decreases substantially.

Conclusion
Tis study considers the limitation of the platoon size and categorizes vehicles in a mixed trafc fow into four types: HDV, ACC, interplatoon, and intraplatoon.By employing numerical simulations, the analysis focuses on trafc characteristics, congestion levels, and reaction times across varying CAV penetration rates.Te investigation aims to determine the optimal platoon size under diferent trafc densities, leading to the following key fndings: (1) CAVs make substantial improvements in road capacity.Compared to pure HDVs, a penetration rate of 80% enhances the road capacity by 2.2 times, while a 100% penetration rate results in a 4.3-fold increase.(2) Trafc fow experiences a continuous decrease in average speed as the density rises.However, under the same density, the average speed of the vehicles increases with higher penetration rates.(3) Congestion levels escalate with increased vehicle densities.Nonetheless, higher CAV penetration rates efectively alleviate road congestion at the same vehicle density, with a particularly pronounced reduction observed when the CAV penetration rate surpasses 60%.(4) At a 100% CAV penetration rate, road capacity expands with larger platoon sizes.However, an optimal platoon size exists for a specifc vehicle density.(5) HDV vehicles exhibit a signifcant reduction in average speed with increasing τ HDV under diferent penetration rates.Similarly, interplatoon vehicles experience a notable average speed decrease with growing τ FC at high CAV penetration rates.
Tis study primarily distinguishes between CAVs and HDVs.In future investigations, further subdivision into passenger cars, trucks, and small cars, while considering distinct characteristics among vehicles and driver behavior, would enable the development of a comprehensive mixed trafc fow model to validate the trafc characteristics in real-world scenarios.In addition, it is noteworthy that this study's analysis of optimal platoon size solely accounts for its impact on road capacity.Subsequent research endeavors could explore factors such as platoon stability and controllability.
l is the distance between the vehicle n and the vehicle n-1, x n−1 and x n are the positions of the vehicle n and the vehicle n − 1 at time t, l is the length of the vehicle, and d n,safe is the secure distance between the vehicle n and the vehicle n − 1.According to equation (1), this parameter's value depends on the reaction time, which difers between HDV and ACC modes.Consequently, the diference between HDV and ACC modes is refected in the option of the reaction time utilized in the safe distance calculation.(2)Deceleration

Figure 1 :
Figure 1: Te diagram of the car-following models.

Figure 5 :
Figure 5: Analysis diagram of feet size sensitivity.

Figure 6 :
Figure 6: Plot of trafc versus feet size for diferent vehicle densities.

Table 1 :
Comparison table of theoretical capacity and simulation.