Non-lane-discipline-based car-following model incorporating the electronic throttle dynamics under connected environment

This study proposes a new car-following model considering the effects of the electronic throttle dynamics to capture the characteristics of connected autonomous vehicular traffic flow without lane discipline. In particular, the proposed model incorporates the effects of both electronic throttle opening angle and lateral gap into the traffic flow model by assuming that the information on electronic throttle dynamics is shared by surrounding vehicles through vehicle-to-vehicle communications. Stability of the proposed model is analyzed using the perturbation method. Numerical experiments analyze three scenarios: start, stop and evolution processes for the scenarios of lane-discipline-based full velocity difference (FVD) model, non-lane-based full velocity difference car-following (NLBCF) model and non-lane-discipline and throttle-based car-following model, respectively. Results from numerical experiments illustrate that the proposed car-following model has a larger stale region compared with the FVD and NLBCF models. In addition, it also demonstrates that the proposed car-following model can better represent the characteristics of connected and autonomous vehicular traffic flow in terms of the responsiveness, smoothness and stability with respect to the position, velocity, acceleration/deceleration and space headway profiles.


Introduction
Recently, connected and autonomous vehicles (CAVs) have received much attention in the field of transportation as CAV technology utilizes on-board or cloudbased sensors, data and computational capabilities to help a vehicle better process and react to its surrounding environment, and then, the transportation mobility, safety and environmental sustainability will be improved accordingly [1][2][3].Under connected environment, a wide variety of information (e.g., position, velocity) on surrounding vehicles within the range is desirable and shared by vehicles via the vehicle-tovehicle (V2V) communications (i.e., dedicated shortrange communications, DSRC).It will reduce the potential collision risk at the junction if the relevant information (e.g., position, velocity) can be shared in advance.In this context, the opening angle of electronic throttle of the preceding vehicles enables a following vehicle in the same platoon to react autonomously to avoid a collision by adjusting its electronic throttle [3,4].Past studies [4,5] have reported that the vehicle 123 speed is related to the opening angle of the electronic throttle.It suggests that the opening angle of the electronic throttle has impacts on traffic behavior.On the other hand, previous studies [6][7][8][9] illustrated that the lateral gap can improve the steady and dynamic performance of traffic flow in terms of the stability and smoothness.However, to the best of our knowledge, no study on traffic flow models considers effects of the electronic throttle dynamics and lateral gap simultaneously.This motivates the construction of a theoretical model to analyze the effects of both electronic throttle opening angle and lateral gap on traffic flow under connected environment.
Traffic flow models aim to capture and replicate the characteristics of vehicle dynamics in traffic flow.To this end, car-following models are proposed from the microscopic viewpoint [10][11][12].Car-following models are formulated using microscopic variables, including position, velocity and acceleration, to capture local interactions between vehicles in traffic flow.In this research line, various car-following models have been developed [13][14][15][16][17][18][19][20][21][22][23][24][25], such as optimal velocity (OV) model [14], full velocity difference (FVD) model [16], multiple headway velocity and acceleration difference model [20] and so on (see [10][11][12] for a review).The aforementioned car-following models can be categorized into lane-discipline-based models, where the models restrict the limitation to the assumption that vehicles follow the lane discipline and travel in the middle of the lane.However, these models may not be valid in the scenario where lanes may not be clearly demarcated on a road though multiple vehicles can travel in parallel [6][7][8][9][25][26][27].Consequently, unlike the lanediscipline-based models, several non-lane-disciplinebased models have been proposed with focus on the lateral gap of road in recent years.To address this scenario, Jin et al. [25] proposed a non-lane-based full velocity difference car-following (NLBCF) model to analyze the impact of the lateral gap on the car-following behavior.However, the NLBCF model cannot distinguish the right-side or the left-side lateral gap on a road.Consequently, Li et al. [26] proposed two-sided lateral gap full velocity difference under non-lane-disciplinebased environment.In addition, Li et al. [27] further studied the effects of lateral gaps on the energy consumption for electric vehicle flow.Recently, Li et al. [9] studied the traffic flow behavior considering the effects of visual angle without lane discipline.Results from numerical experiments verify the impacts of lat-eral gap on traffic flow with respect to the smoothness and stability.
This study is motivated by the need of modeling in the context of CAV environment that enables the incorporation of the lateral gap and electronic throttle opening angle to characterize the performance of CAV traffic flow in terms of the smoothness and stability associated with the acceleration/deceleration, position, velocity and space headway profiles.In particular, a new car-following model is proposed considering the effects of lateral gap and opening angle of electronic throttle on a road without lane discipline.Stability of the proposed model is analyzed using the perturbation method to obtain the stability condition.Simulation-based numerical experiments are performed under three scenarios: start, stop and evolution processes based on the lane-discipline-based FVD model, non-lane-discipline-based NLBCF model and non-lane-discipline and throttle-based car-following model, respectively.Results from numerical experiments illustrate that the proposed model has a larger stale region compared with the FVD and NLBCF models.Also, it demonstrates that the proposed model can better represent the characteristics of CAV traffic flow in terms of the smoothness and stability with respect to the position, velocity, acceleration/deceleration and space headway profiles.
The contributions of this study are summarized as follows.First, a new car-following model considering the effects of both lateral gap and opening angle of the electronic throttle dynamics is proposed to capture the characteristics of connected autonomous vehicular traffic flow without lane discipline.Second, the stability region of the proposed model is enlarged, compared with the FVD and NLBCF models.Third, the dynamic performance of the proposed model is improved in terms of the smoothness and stability with respect to the position, velocity, acceleration/deceleration and space headway profiles.
The rest of the paper is organized as follows.Section 2 proposes the new car-following model considering the effects of both lateral gap and opening angle of ET.Section 3 performs the stability analysis of the proposed model.Section 4 conducts the numerical experiments and comparisons.The final section concludes this study.
Before the model derivation, we present the illustrations of the acronyms used in the paper.It is given in Table 1.

Model derivation
As shown in Fig. 1, the vehicles of interest (i.e., vehicle n, n + 1, n + 2 in Fig. 1) are installed with on-board units.Consequently, under connected environment, the installed vehicles can receive and use the information related to vehicle states (e.g., position x(t) and velocity v(t) in Fig. 1) from surrounding vehicles through the on-board units via V2V communication (i.e., DSRC).
In particular, if the information on vehicle dynamics (i.e., electronic throttle opening angle θ(t) in Fig. 1) can be shared among the installed vehicles, it will lead to a quick reaction by using the autonomous capability of the vehicle, and therefore, the acceleration of the vehicle of interest will be changed accordingly.Hence, this study focuses on the influence of electronic throttle opening angle on traffic flow without lane discipline under connected environment.
Based on [3][4][5], the dynamics of electronic throttle angle to vehicle velocity can be formulated as follows: where v 0 is the steady-state vehicle velocity for a throttle input θ 0 and θn = θ n − θ 0 as the throttle deviation from θ 0 ; b and c vary with v 0 ; d i is the perturbation that captures the effect of unmodeled dynamics; and a n (t) and v n (t) are the acceleration and velocity of the vehicle n at time t, respectively.On the other hand, vehicles may not travel in the middle of the road or follow any lane discipline.Figure 1 shows the scenario that all vehicles travel on a road without lane discipline.The dashed lines in the figure indicated the center lines associated with the various vehicles and are used in the modeling process.The leading vehicle (i.e., vehicle n + 2 in Fig. 1) is traveling in front of the following vehicle (i.e., vehicle n in Fig. 1).Vehicle (i.e., vehicle n +1 in Fig. 1) is traveling laterally with respect to the following vehicle and constitutes the lateral gap.Denote Lg n,n+1 as the lateral gap between the following vehicle n and the vehicle traveling laterally n + 1 and Lg max as the maximal lateral gap.Then, the ratio p n = Lg n,n+1 /Lg max ∈ [0, 1] is the parameter to measure the effect of lateral gap [9,[25][26][27].
Given the above considerations, we propose a new car-following model incorporating the electronic throttle dynamics to describe the effects of electronic throttle opening angle and lateral gap on CAV traffic flow.The model is labeled as non-lane-discipline and throttlebased (NLDT) model hereafter.The proposed NLDT model is formulated as follows: where x n (t) represents the position of the vehicle n. t ∈ R represents the time (s).x n,n+1 (t) ≡

Traffic direction
Fig. 1 Car-following scenario with lateral gap under connected environment are the longitudinal space headway, the velocity difference and the electronic throttle opening angle difference between the leading vehicle n + 1 and the following vehicle n at time t, respectively.
are the longitudinal space headway, the velocity difference and the electronic throttle opening angle difference between the leading vehicle n + 2 and the following vehicle n at time t, respectively.α ∈ R (α = 1/τ > 0), λ ≥ 0 and β ≥ 0 are sensitivity coefficients.
The functions V (•, •), G(•, •) and T (•, •) are defined as follows: where U 1 , U 2 , C 1 and C 2 are constant parameters, l c is the vehicle length and tanh is the hyperbolic tangent function.

Stability analysis
In this section, we use the perturbation method to analyze the stability of the proposed NLDT model and obtain the stability condition.Staring from the stability analysis, the following assumption is given.
Assumption The initial state of the traffic flow is steady, and all vehicles in the traffic travel with the identical space headway and the optimal velocity.
Based on the assumption, the position solution to the steady traffic flow is where ) is the optimal velocity in uniform traffic flow, h is the steady headway and x 0 n (t) is the position of the nth vehicle in steady state.
Add a small disturbance y n (t) to the steady-state solution x 0 n (t), i.e., Note that x n (t) = y n (t)+h,v n = ẏn (t)+V (h, 2h), a n (t) = ÿn (t) and substituting Eq. ( 8) into Eq.( 1), we can obtain that Substituting Eqs. ( 3)-( 10) into Eq.( 2) and linearizing the resulting equation using Taylor expansion, it follows that: Set y n (t) to be in Fourier modes, i.e., y n (t) = A exp(ikn +zt) and substitute it into Eq.( 11) to obtain: Let z = z 1 (ik) + z 2 (ik) 2 + . . .and expand it to the second term of (ik) in Eq. ( 12), it follows that Based on Eq. ( 13), we combine similar items in terms of (ik).For simplicity, hereafter, we denote z 1 (ik) and z 2 (ik) as z 1 and z 2 , respectively.It follows from Eq. ( 13) that According to the long wavelengths theory, when z 1 > 0 and z 2 > 0, the stability condition is given by Remark Based on Eqs. ( 2) and ( 15), if β = 0, then the NLDT model is deduced to the NLBCF model and the stability condition also satisfies the NLBCF model in Fig. 2 The critical curves between sensitivity coefficient and the space headway [25]; if β = 0 and p n = 0, then the NLDT model is deduced to the FVD model in [16].The FVD model in [16] is readily applied under the lane-discipline-based road system.The NLBCF model in [25] is an extension of the FVD model by considering the effect of the one-sided lateral gap under the non-lane-disciplinebased road system.The NLDT model in this study is proposed by considering the effects of both the electronic throttle opening angle and lateral gap under the non-lane-discipline-based road system.Hence, the proposed NLDT model is more generalized than the FVD and NLBCF models.

Steady performance
Based on the foregoing theoretical analysis, this section presents the simulation to verify the steady performance of the proposed NLDT model in terms of the stability region.Figure 2 is the critical curves between sensitivity coefficient α and the space headway with respect to different values of (λ, β, p n ).In Fig. 2, the space formed by the sensitivity coefficient and the space headway is divided into two regions (stable and unstable regions) by the critical curve.Specifically, the region above each critical curve is the stable region where the traffic flow is stable, while the region below each critical curve is the unstable region where the density waves emerge.In addition, if λ = 0, β = 0 and p n = 0, the NLDT model is reduced to the OV model in [14] and if λ = 0, β = 0 and p n = 0, the NLDT model is reduced to the FVD model in [16], while if λ = 0, β = 0 and p n = 0, the NLDT model is reduced to the NLBCF model in [25].
To conclude, we can find that: (i) the stability region of the uniform traffic flow will be enlarged with the increase of parameter β in the case of the same value of parameter λ; (ii) the critical curve will be shifted left from the initial state (i.e., λ = 0, p n = 0) with the increase of parameter p n ; and (iii) the steady dynamics of the uniform traffic flow will be improved by incor- And the 11th vehicle is the leading one.In the evolution process, we assume that there are N vehicles on a road with length L under a periodic boundary condition to investigate the performance of traffic flow in terms of smoothness and stability.For comparison, we set the same initial disturbance as in [16,25]: The initial conditions are chosen as , where v 0 is the velocity of traffic flow in the steady state.v i (0) and θ i (0) can be calculated according to Eqs. ( 16), ( 17) and ( 1).The values of parameters used in the numerical  2. Other conditions set the same as in [3,16,[25][26][27].

Start process
The start process is set up as follows.Initially, the traffic signal is red, and all vehicles are waiting behind the signal with the uniform space headway.At time t = 0 s, the signal changes to green and the vehicles start to move.The leading vehicle starts to accelerate until it reaches the optimal velocity.The other vehicles follow the leading vehicle and accelerate until they reach their optimal velocities.Eventually, all vehicles travel at the same optimal velocity.For comparison, the position, velocity and acceleration profiles of FVD, NLBCF and NLDT models are illustrated in Figs. 3, 4 and 5, respectively.
Figure 3c shows that the position profile in the NLDT model can not only start responsively but also last longer than that in FVD model (Fig. 3a) and NLBCF model (Fig. 3b). Figure 4c indicates that the NLDT model is more responsive and smooth than the FVD (Fig. 4a) and NLBCF model (Fig. 4b) in terms of the velocity profile, especially in the initial start stage.Based on the comparison between Fig. 4b, c and Fig. 4a, it implies that the information on opening angle of ET can help vehicles react and accelerate quickly.Consequently, the responsiveness of the start process in the NLDT model is improved.In addition, Fig. 5c demonstrates that the magnitude of acceleration in the NLDT model is much less than that of the FVD model (Fig. 5a)

Stop process
The stop process is set up as follows.Initially, the traffic signal is green and all vehicles are traveling at the same constant velocity.At time t = 0 s, the signal changes to red and vehicles begin to slow down.The leading vehicle begins to decelerate until it reaches a full stop.The other vehicles follow the leading vehicle and decelerate.Finally, all vehicles stop behind the signal.For comparison, the position, velocity and deceleration profiles of FVD model, NLBCF model and NLDT model are illustrated in Figs. 6, 7 and 8, respectively.
Figure 6c shows the position profile in the NLDT model can reach the maximum value in a short time and keep the smooth consensus state, compared with the FVD model (Fig. 6a) and NLBCF model (Fig. 6b).7a).In addition, Fig. 7a, b shows that both the FVD model and NLBCF model have unrealistic negative velocity at the end of the stop process.However, the NLDT model (Fig. 7c) avoids this deficiency by incorporating the effect of opening angle of electronic throttle under connected environment.Figure 8c

Evolution process
In this section, we study the performance of perturbation on the NLDT model.For comparison, we use the Runge-Kutta algorithm for simulation with the time step t = 0.01 s.Uniform random noise with the maximum amplitude 10 −3 is added to the vehicle position according to Eq. ( 2) with each time step t.The initial conditions are shown in Eqs. ( 16) and (17).
Figures 9 and 10, respectively, show the variations of space headway and velocity profiles of traffic flow at t = 130 s. Figure 9a, b shows that the space headway  9c shows that the space headway based on the NLDT model of traffic flow is more stable around the constant headway h both upstream and downstream.In addition, Fig. 10 shows that the velocity profile of upstream traffic flow oscillates around the constant velocity v 0 , and the downstream traffic flow moves with constant velocity v 0 .However, the magnitude of the oscillation in the NLDT model (Fig. 10c) is least, followed by the NLBCF model (Fig. 10b), and then the FVD model (Fig. 10a).Figures 9 and 10   eral gap and opening angle of electronic throttle in this study.Theoretical analysis proves that the proposed model is more generalized than the FVD and NLBCF models.Compared with the FVD and NLBCF models, the steady performance in terms of the stability region of the proposed NLDT model is best through the stability analysis using the perturbation method.In addition, numerical experiments are conducted to illustrate the effects of lateral gap and opening angle of electronic throttle on the responsiveness and smoothness with respect to the position, velocity, acceleration/deceleration and space headway profiles in the start, stop and evolution processes.
Simulation results show that the steady and dynamic performance of the proposed NLDT model is better than those of the FVD and NLBCF models in terms of the stability and responsiveness as well as smoothness, respectively.The findings of this study provide insights that are beneficial in understanding the mechanism of CAV traffic flow evolution by incorporating the vehicle dynamics and road geometry characteristics under connected environment.Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Fig. 3
Fig. 3 The position profile of traffic flow in start process.a FVD model, b NLBCF model and c NLDT model

Fig. 4
Fig. 4 The velocity profile of traffic flow in start process.a FVD model, b NLBCF model and c NLDT model

Fig. 5
Fig. 5 The acceleration profile of traffic flow in start process.a FVD model, b NLBCF model and c NLDT model

Fig. 6 Fig. 7
Fig. 6 The position profile of traffic flow in stop process.a FVD model, b NLBCF model and c NLDT model Figure7b, c shows that the effect of lateral gap in the NLBCF and NLDT models can improve the responsiveness, compared with the FVD model (Fig.7a).In addition, Fig.7a, bshows that both the FVD model and NLBCF model have unrealistic negative velocity at the end of the stop process.However, the NLDT model (Fig.7c) avoids this deficiency by incorporating the effect of opening angle of electronic throttle under connected environment.Figure8cdemonstrates that the magnitude of deceleration in the NLDT model is the least (about − 3 m/s 2 ), followed by NLBCF model (Fig.8b, about − 4.5 m/s 2 ), and then FVD model (Fig.8a, about − 5 m/s 2 ), and the deceleration profile of the proposed model is smoothest.It implies that the introduction of electronic throttle opening angle and lateral gap can reduce the magnitude of deceleration.

Fig. 8
Fig. 8 The deceleration profile of traffic flow in stop process.FVD model, b NLBCF model and c NLDT model indicate that both the smoothness and stability of the NLDT model are improved by considering the effects of both opening angle of electronic throttle and lateral gap.Based on the theoretical analyses and numerical experiments, the comparisons between the FVD model, NLBCF model and NLDT model demonstrate that the steady performance in terms of the stable region and the dynamic performance in terms of the responsiveness and smoothness of the NLDT model are improved by considering the effects of both lateral gap and opening angle of ET, compared with the FVD and NLBCF models.

Fig. 9
Fig. 9 The space headway profile of traffic flow in process.a FVD model, b NLBCF model and c NLDT model

Fig. 10
Fig. 10 The velocity profile of traffic flow in evolution process.a FVD model, b NLBCF model and c NLDT model

Table 1
List of acronyms

Table 2
Values of parameters in the CF models