# Integrated Spacing Policy Considering Micro- and Macroscopic Characteristics

- 164 Downloads

## Abstract

An appropriate spacing policy improves traffic flow and traffic efficiency while reducing commuting time and energy consumption. In this paper, the integrated spacing policy that combines the benefits of the constant time headway (CTH) and safety distance (SD) spacing policies is proposed in an attempt to improve traffic flow and efficiency. Firstly, the performance of the CTH and SD spacing policies is analyzed from the perspective of the microscopic characteristics of human-vehicle and the macroscopic characteristics of traffic flow. The switching law between CTH and SD spacing policies and the integrated spacing policy are then proposed to increase traffic efficiency according to the traffic conditions, and the critical speed for the proposed integrated spacing policy is derived. Using the proposed switching law, the integrated spacing policy utilizes the safety redundancy difference between the CTH and SD spacing policies in a flexible manner. Simulation tests demonstrate that the proposed integrated spacing policy increases traffic flow and that the traffic flow maintains string stability in a wider range of traffic flow density.

## Keywords

Integrated spacing policy Critical speed Critical traffic flow density String stable Traffic efficiency## Abbreviations

*a*_{bmax}Maximum deceleration

*C*Stability factor

- CTH
Constant time headway

*D*_{CTH}Following space of CTH spacing policy

*D*_{com}Following space of integrated spacing policy

*d*_{min}Minimal following space when vehicle stopped

*D*_{SD}Following space of SD spacing policy

*L*Vehicle length

*q*Traffic flow

*s*Braking distance

- SD
Safety distance

- \({th}\)
Time headway

*V*_{c}Critical vehicle speed

*V*_{r}Relative vehicle speed

*V*_{max}Maximum speed

- \(\rho\)
Traffic density

- \(\rho_{\text{c}}\)
Critical traffic density

- \(\tau\)
Equivalent braking system response time

## 1 Introduction

Congestion is a common problem in urban traffic environments, and the spacing policy applied in vehicles has a considerable influence on traffic flow and efficiency. Many researchers have studied spacing policies from micro- or macroscopic perspectives. The main objects of research are the car-following safety, driver habits, and stability performance of the spacing policy.

- 1.
The critical speed of the integrated spacing policy is derived, and the safety redundancy differences between the CTH and SD spacing policies is used to improve traffic efficiency. When the vehicle speed is lower than the critical speed, the SD spacing policy is applied to shorten the car-following space; when the vehicle speed is greater than the critical speed, the CTH spacing policy is applied to promote traffic efficiency.

- 2.
As the traffic flow density varies from sparse to dense, the proposed integrated policy always maintains a large traffic flow to improve traffic efficiency.

- 3.
Compared with the CTH and SD spacing policies, the proposed integrated policy broadens the range of traffic flow density that ensures string stability.

The remainder of this paper is organized as follows. In Sect. 2, the microscopic characteristics and stability performance of the CTH and SD spacing policies are analyzed. In Sect. 3, a switching law is proposed and the critical speed is derived for an integrated spacing policy that combines the advantages of the CTH and SD spacing policies. In Sect. 4, the characteristics of the integrated spacing policy are compared with those of the CTH and SD spacing policies. Finally, the conclusions to this study are summarized in Sect. 5.

## 2 Microscopic Characteristics and Stability Analysis

### 2.1 CTH and SD Spacing Policies

In Fig. 1, \(\tau_{1}\) is the driver reaction time, \(\tau_{2}\) is the time to eliminate the clearance of braking system, \(\tau_{3}\) is the braking force build time, \(\tau_{4}\) is the braking time to stop the vehicle completely, and \(a_{\text{bmax}}\) is the maximum braking deceleration.

### 2.2 Microscopic Characteristics and Critical Speed

The microscopic characteristics of humans, the ego-vehicle, and the preceding vehicle mainly include car-following safety, human driver car-following habits, and stability performance. Car-following safety refers to the ability to ensure there is no collision with the preceding car under any condition during the car-following process. From the analysis in Sect. 2.1, the CTH and SD spacing policies have different car-following performance.

The SD car-following space in Eq. (8) prevents vehicles from colliding in emergency situations, except when the preceding vehicle reverses or drives backwards. When the following space is no less than the safety distance \(D_{\text{SD}}\), the vehicle is protected from collisions and remains absolutely safe. In other word, the vehicle with SD spacing policy is “absolutely safe.” When the vehicle following space is longer than the safety distance, there exists too much safety redundancy and the vehicle is “over-safe.”

Equations (5) and (8) demonstrate that the CTH and SD spacing policies are both safe, but have different safety redundancies, as shown in Fig. 2. In this paper, these safety redundancy differences are used to improve traffic efficiency. When the vehicle speed is no more than the critical speed \(v_{\text{c}}\), \(D_{\text{SD}} \le D_{\text{CTH}}\) and the vehicle following the CTH spacing policy is “over-safe.” In this case, the SD spacing policy is applied. When the vehicle speed is greater than the critical speed \(v_{\text{c}}\), \(D_{\text{SD}} > D_{\text{CTH}}\) and the vehicle following the SD spacing policy has too much safety redundancy. In this case, the CTH spacing policy is set. In short, the policy with less safety redundancy is set to improve traffic efficiency while satisfying the given safety requirements.

### 2.3 Stability Analysis

#### 2.3.1 Traffic Flow Stability

*t*, respectively. Hence, the traffic flow \(q\) at point \(x\) and time

*t*is

Let \(\rho_{p} \left( {x,t} \right)\) and \(v_{p} \left( {x,t} \right)\) represent the density disturbance and speed disturbance of the traffic flow, respectively. The “string stability” term refers to the non-amplifying upstream propagation of vehicle speed perturbations through a string of vehicles.

#### 2.3.2 String Stability

- 1.
Maintain large traffic flow over a wide range of traffic flow density and promote traffic efficiency;

- 2.
Maintain traffic flow string stability over a wide range of traffic flow density.

- (1)For the CTH spacing policy, the vehicle density \(\rho\) can be described as$$\rho = \frac{1}{{D_{\text{CTH}} + L}}$$(18)

*L*is the vehicle length (all vehicles are assumed to have the same length), and the traffic flow \(q\) can be described as follows:

- (2)For the SD spacing policy, the vehicle density \(\rho\) can be described as$$\rho = \frac{1}{{D_{\text{SD}} + L}}\, ,$$(22)

Then, traffic flow \(q\) can be described as follows:

Figure 3 also shows that, when the traffic flow speed is no greater than the critical speed \(v_{\text{c}}\) or the traffic flow density is no less than the critical density \(\rho_{\text{c}}\), the SD spacing policy improves traffic efficiency; when the traffic flow speed is greater than the critical speed \(v_{\text{c}}\) or the traffic flow density is less than the critical density \(\rho_{\text{c}}\), the CTH spacing policy shows more benefits to improve traffic efficiency. Comparing the analytical solutions for the critical speed \(v_{\text{c}}\) in Eq. (9) and the critical density \(\rho_{\text{c}}\) in Eq. (26), it is apparent that the critical speed \(v_{\text{c}}\) and critical density \(\rho_{\text{c}}\) are at the same point as shown in Fig. 3. In other words, as a switching index, \(\rho_{\text{c}}\) and \(v_{\text{c}}\) are equivalent for the integrated spacing policy. For simplicity, only the critical speed \(v_{\text{c}}\) is set as the switch indicator, as this also reflects the critical density \(\rho_{\text{c}}\).

Figure 4 shows the relationship between the traffic flow density and traffic flow. Points A, B, D indicate the same quantities as in Fig. 3, although the coordinate system in Fig. 4 is different. When the traffic flow density of CTH and SD is below point B and point A, respectively, the slope of the two curves is just \(v_{\rm{max} }\), and \(q = \rho v_{\rm{max} }\). When the traffic flow density of CTH and SD is greater than point B and point A, the two curves follow Eqs. (18) and (22), respectively, and \(q = \rho v\). In this case, the two curves intersect at \(\left( {\rho_{\text{c}} ,\rho_{\text{c}}v_{\text{c}} } \right)\). Namely, when the traffic flow speed is no greater than the critical speed \(v_{\text{c}}\), the SD spacing policy is applied; when the traffic flow speed is greater than the critical speed \(v_{\text{c}}\), the CTH spacing policy is applied.

Figure 5 shows the traffic stability factor of the CTH and SD spacing policies. This figure indicates that the SD spacing policy can maintain string stability over a wide range of traffic flow density range. The SD spacing policy ensures string stability over a wider traffic flow density range of [0, 62.3] veh./km, whereas the CTH spacing policy ensures stability for [0, 23.8] veh./km.

## 3 Integrated Spacing Policy Considering Traffic Flow Characteristics

From the analysis in Sect. 2, it is clear that the CTH and SD policies are based on the microscopic system consisting of humans, the ego-vehicle, and the preceding vehicle. The CTH policy emphasizes human drivers’ habits from the microscopic perspective, whereas the SD policy considers the inter-vehicular kinematics from the macroscopic perspective. Moreover, both policies have advantages and limitations in terms of traffic flow efficiency and traffic flow stability. Generally, the CTH spacing policy ensures larger traffic flow at high speed, whereas the SD spacing policy realizes larger traffic flow at low speed and maintains string stability over a wider traffic flow density range.

Critical speed \(v_{\text{c}}\) for various \({th}\)

\({th}\) (s) | 1.0 | 1.5 | 2 |
---|---|---|---|

\(v_{\text{c}}\) (m/s) | 12 | 19.5 | 27 |

The proposed integrated spacing policy uses the critical speed \(v_{\text{c}}\) as a switch law, and the safety redundancy differences are used to improve traffic efficiency. When the vehicle speed is no greater than the critical speed \(v_{\text{c}}\), the vehicle is in the “over-safe” state and the SD spacing policy helps shorten the inter-vehicle distance and increase traffic flow. When the vehicle speed is above the critical speed \(v_{\text{c}}\), the CTH spacing policy is applied to reduce the excessive safety redundancy of SD and keep the following space within a suitable range for improving the traffic efficiency.

## 4 Simulation Test

Figure 7 shows that the traffic flow density varies with time headway \({th}\). As the traffic flow density reflects the traffic congestion situation, a smaller time headway \({th}\) achieves better traffic efficiency with the same traffic flow speed; Fig. 8 shows similar results. The integrated policy with a smaller time headway \({th}\) results in better traffic flow for the same traffic flow density.

The range of string stability offered by the integrated policy with different time headway \({th}\) is illustrated in Fig. 9. Specifically, taking \({th} = 1.0\;{\text{s}}\) as an example, analysis of the stability factor \(C\) shows that the SD spacing policy ensures traffic flow string stability in the density range [0, 62.3] veh./km (Fig. 5); for the integrated spacing policy, the range is [0, 23.8] veh./km and [52.6, 62.3] veh./km; for the CTH policy, the range is [0, 23.8] veh./km. From these results, it can be concluded that the integrated policy broadens the traffic flow density range that has string stability compared with the CTH policy. Thus, the traffic flow remains string stable at high traffic flow density, which is a significant benefit for low-speed conditions in high-traffic-density urban scenarios.

Figure 10 shows the traffic flow improvement ratio of the integrated policy compared with the CTH and SD spacing policies. When the traffic flow density is lower than \(\rho_{\text{c}}\), the integrated policy uses CTH spacing policy and is much better than the SD spacing policy; when the traffic flow density is no less than \(\rho_{\text{c}}\), the integrated policy uses SD spacing policy and performs significantly better than the CTH spacing policy.

Overall, the proposed integrated spacing policy can be used to improve traffic efficiency and string stability, as it combines the benefits of the CTH and SD spacing policies to reduce the excessive safety redundancy. When the vehicle speed is no more than the critical speed \(v_{\text{c}}\), compared with the CTH spacing policy, the integrated policy shows significant improvements in traffic efficiency and broadens the range of string stability. When the vehicle speed is greater than the critical speed \(v_{\text{c}}\), compared with the SD spacing policy, the traffic efficiency is improved, and the integrated spacing policy avoids the large following spaces that cause driver anxiety, thus making the proposed policy more acceptable to human drivers.

## 5 Conclusions

The integrated policy combined the advantages of the CTH and SD policy is proposed via utilizing the safety redundancy and the switching law is presented with the derived critical speed of CTH and SD spacing policies. Simulation tests show that the proposed spacing policy can improve traffic efficiency and achieve better performance in terms of traffic flow string stability and car-following safety.

Compared with the CTH spacing policy, when traveling at less than the critical speed, the integrated policy produces significant benefits in terms of traffic efficiency and broadens the range of string stability. Compared with the SD spacing policy, when traveling at greater than the critical speed, the traffic efficiency is improved, and the integrated spacing policy avoids the large following spaces that cause driver anxiety, making the proposed method more acceptable to human drivers. Furthermore, the integrated spacing policy can maintain efficient traffic flow and stability over a wide traffic flow density range.

The results presented in this paper can be integrated with other spacing policies, and vehicle tests will be extended to calibrate the effectiveness of the integrated spacing policy.

## Notes

### Acknowledgements

Special thanks are due to the National Natural Science Foundation of China [51675217,61790564], the Young Elite Scientists Sponsorship Program by CAST [2016QNRC001], the China Automobile Industry Innovation and Development Joint Fund [U1564213], and the Opening Founding of State Key Laboratory of Automotive Simulation and Control [20161114] for supporting authors’ research.

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that there is no conflict of interest.

## References

- 1.Ioannou, P.A., Chien, C.C.: Autonomous intelligent cruise control. IEEE Trans. Veh. Technol.
**42**(4), 657–672 (1993)CrossRefGoogle Scholar - 2.Yanakiev, D., Kanellakopoulos, I.: Variable time headway for string stability of automated heavy-duty vehicles. In: Proceedings of the 34th IEEE Conference on Decision and Control (1995)Google Scholar
- 3.Yanakiev, D., Kanellakopoulos, I.: Nonlinear spacing policies for automated heavy-duty vehicles. IEEE Trans. Veh. Technol.
**47**(4), 1365–1377 (1998)CrossRefGoogle Scholar - 4.Yanakiev, D., Kanellakopoulos, I.: Speed tracking and vehicle follower control design for heavy-duty vehicles. Veh. Syst. Dyn.
**25**(4), 251–276 (2007)CrossRefGoogle Scholar - 5.Han, D., Yi, K.: A driver-adaptive range policy for adaptive cruise control. Proc. Inst. Mech. Eng. Part D J. Autom. Eng.
**220**(3), 321–334 (2006)CrossRefGoogle Scholar - 6.Treiber, M., Hennecke, A., Helbing, D.: Congested traffic states in empirical observations and microscopic simulations. Phys. Rev. E: Stat. Phys. Plasmas Fluids
**62**(2 Pt A), 1805–1824 (2000)CrossRefzbMATHGoogle Scholar - 7.Kesting, A., Treiber, M.: Calibrating car-following models using trajectory data: methodological study. Transp. Res. Rec.
**2088**(2088), 148–156 (2008)CrossRefGoogle Scholar - 8.Swaroop, D., Hedrick, J.K., Chien, C.C., et al.: A comparision of spacing and headway control laws for automatically controlled vehicles. Veh. Syst. Dyn.
**23**(8), 597–625 (1994)CrossRefGoogle Scholar - 9.Swaroop, D.: String stability of interconnected systems: an application to platooning in automated highway system. University of California, Berkeley (1995)CrossRefGoogle Scholar
- 10.Swaroop, D., Hedrick, J.K.: Constant spacing strategies for platooning in automated highway systems. J. Dyn. Syst. Meas. Control
**121**(3), 462–470 (1999)CrossRefGoogle Scholar - 11.Shrivastava, A., Li, P.Y.: Traffic flow stability induced by constant time headway policy for adaptive cruise control (ACC) Vehicles. Transp. Res. Part C
**10**(4), 275–301 (2002)CrossRefGoogle Scholar - 12.Yi, J., Horowitz, R.: Macroscopic traffic flow propagation stability for adaptive cruise controlled vehicles. Transp. Res. Part C Emerg. Technol.
**14**(2), 81–95 (2006)CrossRefGoogle Scholar - 13.Zhou, J., Peng, H.: Range policy of adaptive cruise control vehicles for improved flow stability and string stability. IEEE Trans. Intell. Transp. Syst.
**6**(2), 229–237 (2005)CrossRefGoogle Scholar - 14.Santhanakrishnan, K., Rajamani, R.: On spacing policies for highway vehicle automation. IEEE Trans. Intell. Transp. Syst.
**4**(4), 198–204 (2003)CrossRefGoogle Scholar - 15.Xiao, L.Y., Gao, F.: Practical string stability of platoon of adaptive cruise control vehicles. IEEE Trans. Intell. Transp. Syst.
**12**(4), 1184–1194 (2011)CrossRefGoogle Scholar - 16.Rödönyi, G.: An adaptive spacing policy guaranteeing string stability in multi-BrandAd HocPlatoons. IEEE Trans. Intell. Transp. Syst.
**19**(6), 1902–1912 (2018)CrossRefGoogle Scholar - 17.Darbha, S., Rajagopal, K.R.: Intelligent cruise control systems and traffic flow stability. Transp. Res. Part C Emerg. Technol.
**7**(6), 329–352 (1999)CrossRefGoogle Scholar

## Copyright information

**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.