Abstract
Second-order macroscopic models and most car-following models are able to reproduce traffic waves or other observed instabilities of traffic flow. After an intuitive introduction, we define the relevant instability concepts: Local instability, convective string and flow instability, and absolute string and flow instability. We give general analytic criteria for the occurrence of these instabilities for microscopic and macroscopic models. The formulation is more comprehensive than the various accounts in the specialized literature and can be evaluated for any traffic flow model with a well-defined acceleration function. The stability criteria allow us to characterize the influencing factors of traffic flow instabilities and answer the question of if, and in which way, the driving behavior (or new driver-assistance systems) influence traffic flow stability.
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Notes
- 1.
Generally, delays in a feedback control system favor instabilities. This can be experienced intuitively when taking a shower and controlling the water temperature, particularly, if the response time between the controlling action and the result (a change of the water temperature) is rather long.
- 2.
The only way to generate a traffic breakdown in such models is by simulating a bottleneck and assuming upstream boundary conditions corresponding to an inflow exceeding the bottleneck (footnote 2 Continued) capacity. Then, as soon as the flow at the bottleneck exceeds its capacity, the density immediately upstream of the bottleneck jumps to the congested branch of the fundamental diagram at a flow corresponding to the bottleneck capacity (cf. Sect. 8.5).
- 3.
Notice that, in some models, the speed adaptation time may depend on traffic density getting shorter for increased density. This can more than compensate for the destabilizing effects of traffic density itself, so congested traffic may be unstable for most densities but restabilize for high densities near the maximum.
- 4.
- 5.
At least, if the penetration rate of ACC equipped vehicles is sufficiently small. Otherwise, the influence of ACC-driven vehicles on the string instability becomes relevant as will be discussed in the main text below.
- 6.
Some authors stress that there is a conceptual difference between string instability (relevant for microscopic models), and flow instability (macroscopic models). However, observed differences are merely a consequence of an imperfect equivalence between microscopic and macroscopic models with respect to macroscopic phenomena (notice that microscopic models can describe macroscopic phenomena but not vice versa). The unified instability criteria to be developed in the next sections show that the concepts of string and flow instability are identical in a precisely defined sense: For each microscopic model displaying string instabilities in a subset of the space spanned by the model parameters and the steady-state traffic density, there exists a micro-macro relation to a macroscopic model displaying flow instability for exactly the same subset.
- 7.
We do not give the precise mathematical definitions.
- 8.
The distinction may become relevant for models with non-smooth or even non-continuous acceleration functions. Typically, this is the case when the model formulation involves several distinct traffic regimes (e.g., Gipps’ model or the Wiedemann model). Such models may be Ljapunov but not asymptotically stable.
- 9.
Although this is not correct: The Greek \(\lambda \) is mirrored and not upside down.
- 10.
To make the perturbation more massive, the duration of the perturbation must be increased such that it results in a fully-formed initial jam.
- 11.
This technical term originates from the Latin convehi: to move together.
- 12.
At least, if traffic flow is locally stable which is safe to assume.
- 13.
More generally, the perturbations eventually vanish at any fixed location \(x\) for \(t\rightarrow \infty \).
- 14.
For example, Gipps’ model in its original formulation exhibits a short-wavelength instability with the smallest possible wavelength of two car distances.
- 15.
The partial derivatives of the acceleration function are \(a_v=-1/\tau \), \(a_{v_l}=0\), \(a_s=-v^{\prime }_e(s)a_v\).
- 16.
Including multi-anticipation, i.e., considering several leading (or trailing) vehicles, does not pose any technical problem: Starting the investigation from Eq. (12.19) with Eq. (12.15) and proceeding (footnote 16 Continued) along the following lines is straightforward. We restrict ourselves to a single leader so as to not clutter the presentation.
- 17.
This technical term has to be distinguished from passing in the sense of overtaking which is completely unrelated.
- 18.
Beware of the signs: Assuming positive wave numbers \(k\) and our convention to assign to the first vehicle the lowest vehicle index, the angular frequency will generally be negative. This means, the passing rate is positive: The waves propagate in the direction of increasing vehicle indices, i.e., opposite to the movement of the vehicles.
- 19.
The sign is reversed with respect to Eq. (15.35), i.e., negative. The waves propagate upstream, i.e., in negative \(x\) direction.
- 20.
The “order” symbol \(\mathcal{O}(\cdot )\) defines how fast the symbolized contributions converge to zero. Specifically, if a contribution \(f(k)\) is of the order \(\mathcal{O}(k^\gamma )\), then \(\lim _{k\rightarrow 0} k^{-\gamma }f(k)\) is finite, and \(\lim _{k\rightarrow 0} k^{-\gamma +\varepsilon }f(k)=0\) for any positive real-valued \(\varepsilon \).
- 21.
see: www.traffic-simulation.de.
- 22.
Otherwise, Fourier modes cannot be used and the analysis becomes more complicated.
- 23.
Diffusion terms imply infinite speeds. Furthermore, in the presence of speed gradients, negative speeds cannot be excluded. Moreover, local models are numerically more unstable than gradient-free nonlocal models.
- 24.
The equation for \(\lambda \) is of a similar form as the condition (15.27) for local instability of time-delay differential equations.
- 25.
No instabilities are possible for free traffic since \(v^{\prime }_e(s_e)=0\) in this case.
- 26.
On the other hand, short gaps lead to a higher dynamic capacity, see Sect. 11.3.6.
- 27.
In a statistical interpretation, \(\theta _0\) formally denotes the speed variance in analogy to the corresponding term \(\theta =\alpha (\rho )V^2\) of the GKT model. However, in the Kerner–Konhäuser Model, \(\theta _0\) is usually interpreted as a purely phenomenological anticipation term.
- 28.
Notice that this is another hint that it may take some time until an initial perturbation develops to high-amplitude traffic waves, or a traffic breakdown.
- 29.
This is similar to a group of water waves triggered by a localized perturbation, e.g., by a stone thrown at the water surface.
- 30.
This dispersion has the same unit, order of magnitude (\(100\, {\text{m}^2/\text{s}}\)), and effect, as the diffusion terms of some macroscopic models.
- 31.
There is a large body of literature proposing and investigating solitary nonlinear waves which can be investigated analytically. However, the conditions to derive equations for such waves (e.g., a modified Korteweg–de-Vries equation) are extremely restrictive and nearly never satisfied in real traffic situations.
- 32.
A ring road must not be confused with a roundabout which, in contrast to the former, represents a comparatively complex network node.
- 33.
Strictly speaking, convective instability is only well-defined in an infinite or open system. However, for practical purposes, the circumference of the ring must be sufficiently large such that no vehicle drives around the complete ring during the simulation time.
- 34.
For rare combinations of models and parameters, we obtain a region of absolute instability embedded on both sides by regions of convective downstream and upstream instabilities, respectively.
- 35.
We are aware that, in vehicles with manual transmission, it is hard to drive smoothly at very low speeds where the clutch must be operated even when driving in first gear. While this is considered in sub-microscopic models, it is ignored for the models considered here. In effect, the difficulty to drive very slowly leads to persistent noise at a sub-microscopic level. However, if traffic flow is stable at a microscopic or macroscopic level, these perturbations are not collectively amplified, i.e., traffic data show strong fluctuations but no deterministic signal.
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Further Reading
Further Reading
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Huerre, P., Monkewitz, P.: Local and global instabilities in spatially developing flows. Annual Review of Fluid Mechanics 22 (1990) 473–537
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Treiber, M., Kesting, A.: Evidence of convective instability in congested traffic flow: A systematic empirical and theoretical investigation. Transportation Research Part B: Methodological 45 (2011) 1362–1377
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Wilson, R.: Mechanisms for spatio-temporal pattern formation in highway traffic models. Philosophical Transactions of the Royal Society A 366 (2008) 2017–2032
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Treiber, M., Kesting, A.: Validation of traffic flow models with respect to the spatiotemporal evolution of congested traffic 21 (2012) 31–41
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Treiber, M., Kesting, A. (2013). Stability Analysis. In: Traffic Flow Dynamics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32460-4_15
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DOI: https://doi.org/10.1007/978-3-642-32460-4_15
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