Bifurcation control of optimal velocity model through anticipated effect and response time-delay feedback methods

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Abstract

In order to further improve the adaptability of the optimal velocity model (OVM) in actual traffic flow, the paper introduces a feedback control with considering both driver’s anticipated time and response time-delay. Through the linear analysis and bifurcation analysis, we obtain stability conditions and the balance point of dual time-delay control OVM. Aiming at restraining the Hopf bifurcation caused by small disturbance, a bifurcated controller is designed to reduce the number of unstable eigenvalues of the characteristic equation and determine the definite stability interval under the combination of anticipation and response time-delay. Then followed by the simulated verification for more vehicles, the simulation results show that the controller can effectively suppress traffic congestion without changing the balance point, and significantly improve traffic efficiency and traffic stability. In addition, utilizing NGSIM data to calibrate the controlled model so as to explore the feasibility and advantages of the model.

Introduction

With the rapid development of the automobile industry and urbanization, traffic accidents and congestions are increasingly becoming obstacles to the urban development, for the reason that numerous of traffic flow models have been proposed so as to solve traffic problems and improve traffic efficiency, such as car-following models (Sun et al. [1]); the cellular automation models (Tian et al. [2]), the gas kinetic models (Nagatani et al. [3]), the hydrodynamic lattice models (Redhu et al. [4]) and the continuum models (Cheng et al. [5]). As one of the most classic microscopic car-following models, Bando et al. [6] proposed a brand-new optimized velocity model (OVM) based on Newell’s model [7], which effectively solves the problem of excessive acceleration in the Newell’s model during parking and starting. Its expression is as follow: ant=αVΔxntvntwhere ant and vnt are the acceleration and velocity of the nth vehicle, Δxnt=xn+1txnt represents the headway of the nth vehicle; α is the driver’s sensitivity parameter to the difference between his own velocity and the optimal velocity; While VΔxnt is optimal velocity function determined by the headway. While optimal velocity function has various forms, the following are two types of typical optimal velocity function. VΔxnt=vmax2tanhΔxntL+tanhL VΔxnt=00ΔxntLvmaxΔxntL31+ΔxntL3Δxnt>Lwhere vmax represents maximum velocity; L is the minimum safe headway.

The OVM model does overcome the shortcomings of previous studies in a certain extent, but it still lacks consideration of driver’s autonomy. In the subsequent research, quite a few scholars take the influence of factors such as driver’s reaction time, multi-lead car following, anticipation time, traffic signal, intelligent transportation system environment, etc. [8], [9], [10], [11], [12] into account to extend the OVM so as to better describe the observed complex phenomena of actual traffic flow. For example, Nagatani [8] advocated that the driver’s response time-delay, especially response delay to emergencies during driving, has an impact on traffic flow that cannot be ignored, for this reason, he took the reaction time into consideration on the basis of OVM to solve those problems. Nakayama et al. [9] proposed the backward looking OV (BL-OV) model which solves the shortcomings of OVM neglecting the impact of follower on the target driver. Helbing et al. [10] advocated that OVM is only suitable for ideal environments with excessive acceleration and unrealistic deceleration, which is inconsistent with the actual traffic environment, and then he derived the generalized force model (GFM). Sasaki et al. [11] proved the influence mechanism of traffic lights on traffic flow through the data analysis and simulation, and established three different traffic light control strategies. By incorporating the relative velocity of the preceding and following cars into classical model, Jiang et al. [12] pioneered the full velocity difference model (FVDM) that better fits the actual traffic environment.

While, numerous of scholars’ studies have proved, in fact, that both the reaction time delay and the anticipated time play an important role in the stability of traffic flow. Therefore, more and more researches [13], [14], [15], [16], [17], [18] focus on the impact of time delay on complex traffic environment such as phase change, metastability, hysteresis, go and stop, etc. After Bando et al. [13] proposed OVM for the first time, he immediately took driver’s response time to information of the vehicle ahead into consideration, and the simulation finally proved his conjecture. Zhang [14] pointed out that driver’s memory effect can also affect the traffic flow and proposed a macro traffic flow model that considers the driver’s memory effect to make up for the defect. In the context of the ITS, Tang et al. [15] proposed the anticipation time-delay optimal velocity model (ATDOVM) so as to study the influence of the anticipation effect caused by the guidance information on the car-following behavior. Gasser et al. [16] and Orosz et al. [17] used the bifurcation theory to explore the bifurcation behaviors of the extended OVM including time delay, theoretical analysis and numerical simulation proved the influence of time delay on the occurrence of traffic flow bifurcation. Zhang et al. [18] combined the linear analysis and bifurcation analysis of the full velocity difference (FVD) model with the driver’s response time delay to explored the influence of each parameter on the two stability results and compared them.

In summary, whether it is a response time-delay or an anticipated time, both they can effectively control traffic performance and improve the stability of controlled traffic flow with an appropriate design. Therefore, the time-delay feedback control system that explores the optimal delay has been widely used in traffic flow research [19], [20], [21]. For instance, Konishi et al. [19] established decentralized delayed-feedback control (DDFC) which can effectively reduce the impact of minor traffic disturbances on system stability. On this basis, aiming at the abnormal traffic flow oscillations in OVM, Jin et al. [20] improved the OVM and designed a new controller composed of time-delayed distance and relative speed. Furthermore, aiming at the actuator hysteresis defect of autonomous vehicles, Wang et al. [21] designed a method based on a model anticipated control framework to obtain the optimal delay for the maximum stability of the system. Nevertheless, how to choose the most appropriate feedback benefit and time delay remains to be overcome, which is also the top priority of current research.

Furthermore, calibration of car-following model parameters based on measured data has become the central issue of research in the traffic flow recently. Treiber et al. [22] and Ossen et al. [23] use the headway and trajectory individually or jointly as the measures of performance (MOD) to calibrate the model to obtain the calibrated parameter values. Punzo et al. [24] made a more specific and detailed research on the selection of MOD so as to achieve more accurate calibrated parameter values. And the article prepares to use the methods they used to verify the feasibility and effectiveness of response time and anticipated time feedback control strategies.

While the innovation of the paper is comprehensively considering the anticipated time and response time-delay to determine the parameter range of the two time-delay feedback control, which will provide theoretical and data support for the establishment of intelligent transportation systems, especially, providing references for the research and development of driverless cars. Moreover, the anticipated time and response time-delay feedback control strategy can also help in solving traffic problems and managing traffic systems.

The writing framework of the article is as follows. We established the controlled OVM with the anticipated time and response time-delay in Section 2. Linear stability analysis and bifurcation analysis are presented in Sections 3 The linear analysis of the controlled OVM, 4 The bifurcation analysis of the controlled OVM respectively. In Section 5, the control principles of the anticipation and response time feedback control system are given and the control effect under the combination of multiple coefficients is tested. Then in Section 6, the two anticipated time and response time-delay control strategies involving multiple vehicles are designed in order to reflect the actual traffic environment followed by the parameter calibration of controlled OVM. And finally obtaining conclusions of design of feedback control system in Section 7.

Section snippets

The controlled OVM

Being one of the most classic descriptions of actual traffic flow, there are considerable researches on OVM and the analysis methods has been really mature. However, once facing the unstable conditions such as traffic congestion, OVM has certain limitations. And the anticipated time and response time-delay feedback control system proposed in this paper is mainly to reduce the influence of these unstable factors. dvntdt=αVΔxntvnt+φ1vnt+τ1vnt+φ2vntvntτ2where φ1 and φ2 donate the feedback

The linear analysis of the controlled OVM

In this section, we conduct the prerequisites for the stability of the improved model by linear analysis. For a uniform traffic flow, all vehicles have the same characteristics, including everyone in the unified system is moving at the optimal velocity Vh and the same desire headway h. Obviously, we can obtain: vn0t=Vh,Δxnt=h=LN,xn0t=nh+Vhtwhere L and N denotes the number of total vehicle and the length of road respectively. Giving a small perturbation ynt can obtain: vn0t=Vh+ynt

Linearizing the

The bifurcation analysis of the controlled OVM

In this section, we will use bifurcation analysis to explore the bifurcation behavior of the controlled model so as to study the correspondence between the critical point of Hopf bifurcation and instability. Giving a small derivation can obtain: vn0t=Vh+ηnt,xn0t=nh+Vht+ξntwhere ηnt is derivation of velocity, ξnt is derivation of position.

Inserting Eq. (13) into Eq. (6) and linearizing, we can obtain: dηntdt=βVhξn+1tξnt+ηnt+βφ2ηntηntτ2dξntdt=ηntwhere β=11φ1τ1; abbreviating Vh as V.

The design of anticipated time and response time-delay feedback control

In summary, feedback gain and time-delay control have significant effects on system’s stability, which are mainly manifested in suppressing the occurrence of bifurcation, but unsuitable combinations of feedback gain and delay can also bring adverse effects. This section mainly solves the problem to acquire the optimal feedback control parameter set. In 2016 Xu et al. [25] proposed an improved algorithm aims at two feedback time-delay control strategies. The specific control principle is as

Parameter calibration

In this part, using 474 groups dataset of the 39th vehicle in the NGSIM database, which contains the required data such as velocity, acceleration, and the headway between the front vehicle and the following vehicle, to calibrate the optimal parameter. Since the vehicle trajectory data of the modified dataset includes vehicle data in a non-car-following state, and existing the lane-changing behavior, therefore we use the following as filter conditions in order to meet the car-following behavior

Conclusions

In this paper, aiming at OVM, a feedback control with anticipation and response delay is designed to suppress bifurcation caused by unstable factors so as to suppress traffic flow oscillation or to improve the fitting degree of the model to vehicle trajectory. The critical range of the stable region is determined by the comparison of linear analysis and bifurcation analysis. Then, we proposed the definite integral method, which is the core of anticipated time and response time-delay feedback

CRediT authorship contribution statement

Xueyi Guan: Conceptualization, Methodology, Software. Rongjun Cheng: Numerical simulation, Writing - original draft. Hongxia Ge: Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is supported by the Program of Humanities and Social Science of Education Ministry of China (Grant No. 20YJA630008) and the Ningbo Natural Science Foundation of China (Grant No. 202003N4142) and the Natural Science Foundation of Zhejiang Province, China (Grant No. LY20G010004) and the K.C. Wong Magna Fund in Ningbo University, China .

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