On string stability of a mixed and heterogeneous traffic flow: A unifying modelling framework

Highlights

• A classification for string stability is proposed to reconcile approaches from different fields.

• We show equivalence between asymptotic and ${\mathcal{L}}_2$ string stability of car-following models.

• The only available condition for car-following string stability of a heterogeneous flow is proven inaccurate.

• Uncertain transfer functions are proposed to study the string stability of a mixed and heterogenous traffic.

• We shift from the study of the stability of a model, to the study of the stability of a traffic flow.

Abstract

Urged by a close future perspective of a traffic flow made of a mix of human-driven vehicles and connected, automated vehicles (CAVs), research has recently focused at making the most of CAVs capabilities to mitigate the instability of the whole, i.e. mixed, traffic flow. In all works, however, either the two sub-flows are studied under a simplifying but unrealistic assumption of flow homogeneity, or drivers’ and vehicles heterogeneity is not correctly taken into account within each sub-flow. We show here that the only condition developed so far to study a car-following model string stability for a heterogeneous flow, is inaccurate. Therefore, we propose a methodology to model string stability that considers drivers’ and vehicles heterogeneity, which is the essence of a real traffic. Uncertain transfer functions are introduced to map the probability distributions of car-following model parameters into a ${\mathcal{L}}_2$ stability measure of a mixed and heterogeneous traffic. Specifically, they allow us to move from the stability analysis of a car-following model, or of a controller, to the stability analysis of a traffic flow, as interpreted by that model, or controller. Eventually, several other theoretical contributions on stability analysis are given in the paper, aiming at reconciling approaches from different fields. Among these, a mathematical justification of the equivalence between the asymptotic stability of a closed-loop platoon system – which has been studied through the famous “traffic wave ansatz” on a ring-road – and the ${\mathcal{L}}_2$ stability of an open-loop platoon system.

Introduction

Sixty years are passed since the seminal works by researchers at the General Motors (GM) Labs on the stability of a stream of traffic forming a platoon of vehicles which follow one another (Chandler et al., 1958; Herman et al., 1959). These studies – inspired by the ambition of a new executive at the GM Labs, Larry Hafstad, a nuclear physicist by background, who wanted to make the lab a leader in “basic science” (see the historical note by Gazis, 2002) – developed in different streams mainly within the traffic flow theory community, and the control theory community.

In the former, the focus has quite obviously been on traffic flow. Researchers have sought to reproduce the instability of real traffic through appropriate car-following model formulations. Methods from physics, which are based on the solution of a car-following differential equation, were mostly applied in recent years (see the famous “traffic wave ansatz” method; Bando et al., 1995, 1998; Wilson, 2008; Treiber and Kesting, 2013; Monteil et al., 2014; Ngoduy, 2015, 2019; Guo et al., 2018; see also the Appendix).

Conversely, research within the control theory community has aimed at developing robust approaches to design vehicle controllers, being the stability of a platoon of vehicles a basic requirement. Customary methods from linear system dynamics and robust H_∞-based control were largely applied (see e.g. Cosgriff, 1965; Bender and Fenton, 1969; Peppard, 1974; Swaroop and Hedrick, 1996; see a review in Ploeg et al., 2014). Within this community, the stability of human-driven vehicles has been generally overlooked until recent years, most of the studies being focused on platoons entirely made of automated vehicles. Urged by a close future perspective of traffic flows made of a mix of conventional (human-driven) vehicles, and connected, automated vehicles (CAVs), a new branch of research has arisen. It aims at making the most of controlling capabilities of CAVs to mitigate the instability of the whole, i.e. mixed, traffic flow (see e.g. van Arem et al., 2006; Wang et al., 2014; Di Vaio et al., 2019, Jia et al., 2019; Monteil et al., 2019, Xie et al., 2019, Zhou et al., 2020), being the instability of human-driven vehicles back at stake again.

Yet, the way in which human-driven traffic is represented in these studies is far from reality. Consequently, also robustness of results in terms of traffic flow stability is questioned. This is the thesis of this work.

The main flaw of previous studies is that human-driven traffic is substantially considered homogeneous. As a matter of fact, heterogeneity of drivers’ characteristics and behaviours, which is the essence of real traffic, has no place in these studies, neither in the methodology nor in the simulations. At best, only one of the parameters of the car-following model, the one describing the conventional traffic, is varied from a simulation to another (typically, the reaction time). Yet, literature in the traffic flow theory highlighted that models will not be able to properly reproduce real traffic if its heterogeneity is neglected (see e.g. Kim and Mahmassani, 2011, Montanino and Punzo, 2015, Punzo and Montanino, 2020). Needless to say, the model results in terms of stability are not expected to be exempt from this consideration.

The reason for this flaw goes back to the lack of an appropriate methodology in the traffic flow theory literature. Until recently, all studies about traffic stability have been based on the assumption of flow homogeneity. In fact, the only results for a heterogeneous flow were first given in Ward (2009) and later revived and extended by Ngoduy (2013). These results were extensively applied in the traffic flow theory field (e.g. Yang et al., 2014, 2015; Li et al., 2015; Talebpour and Mahmassani, 2016; Liu et al., 2016; Wang et al., 2018; Xie et al., 2019; Yao et al., 2019; Jin et al., 2020). The mathematical conditions for stability provided in Ward (2009) and Ngoduy (2013), however, are not entirely useful, as they only provide a Boolean information about the stability of a heterogeneous platoon (i.e. whether it is stable or not), and not a measure of stability. Therefore, they are hardly suitable for e.g. the design of CAVs control algorithms.

All that said, in this paper we show that the condition by Ward (2009) is neither necessary nor sufficient for platoons of finite length, resulting inappropriately applied in the field literature so far. In addition, we quantify its degree of inaccuracy to characterize string stability of a real-world finite heterogeneous flow.

In conclusion, at the best of the authors’ knowledge, there is no effective methodology to study the stability of a heterogenous string of human-driven vehicles so far. Main objective of this paper is to fill this gap in. To this aim, several main contributions are provided, each one corresponding to a different section.

As in the past sixty years, research in traffic flow stability in the two fields followed different paths – thus resulting in several mathematical developments, inconsistent terminologies and apparently unrelated results – a classification framework is proposed in the section “Review of linear stability: a classification”. Studies were classified according to the type of stability – internal vs. external – and the type of system – single-follower system vs. platoon system. This led us to introduce with good reason a new name for the stability of a single-follower system, replacing the confusing ‘local’ or ‘platoon’ labels for stability adopted in the field of traffic flow theory. The classification has highlighted differences, complementarities and overlaps in previous approaches.

Based on the proposed classification, in the section “Reconciling approaches to string stability”, a mathematical justification has been given of the equivalence between the asymptotic stability of a closed-loop car-following system – which has been studied through the famous “traffic wave ansatz” on a ring-road, in the traffic flow theory field (e.g. Bando et al., 1995) – and the ${\mathcal{L}}_2$ stability of an open-loop platoon system, which has been studied in the control theory field (e.g. Monteil et al., 2019).

In the section “Stability of a heterogeneous platoon system”, a sufficient analytical condition for string stability of a heterogeneous platoon of three vehicles is provided. Based on this exact condition, previous results by Ward (2009) and Ngoduy (2013) have been proved to be inaccurate. A coherent approach, straightforwardly following from the adoption of the ${\mathcal{L}}_2$ system induced norm as a stability measure, is presented and discussed also through simulations. The significance of the resonance frequency of unstable vehicles is highlighted, basing on which an analytical sufficient condition for instability has been provided.

Eventually, in the section “Stability of a heterogeneous traffic flow”, we have stressed that the proposed approach, which enables the study of heterogeneous platoons, has to be augmented with uncertain transfer functions, in order to properly analyse the stability of a (mixed and) heterogeneous flow. These functions allow us to map probability distributions of car-following model parameters into the ${\mathcal{L}}_2$ stability measure of a heterogeneous flow. In other words, they allow us to move from the study of the stability of a (car-following) model, to the study of the stability of a traffic flow (as interpreted by that model).

It is worth noting that the same probabilistic framework for investigating stability can be straightforwardly applied also to deal with the heterogeneity of CAVs control parameters.