Classification and Unification of the Microscopic Deterministic Traffic Models with Identical Drivers
CClassification and Unification of the Microscopic Deterministic Traffic Models
Bo Yang and Christopher Monterola
Complex Systems Group, Institute of High Performance Computing, A*STAR, Singapore, 138632. (Dated: September 6, 2018)We identify a universal mathematical structure in microscopic deterministic traffic models (withidentical drivers), and thus show that all such existing models in the literature, including boththe two-phase and three-phase models, can be understood as special cases of a master model byexpansion around a set of well-defined ground states. This allows any two traffic models to beproperly compared and identified. The three-phase models are characterised by the vanishing ofleading orders of expansion within a certain density range, and as an example the popular intelligentdriver models (IDM) is shown to be equivalent to a generalized optimal velocity (OV) model. Wealso explore the diverse solutions of the generalized OV model that can be important both forunderstanding human driving behaviours and algorithms for autonomous driverless vehicles.
PACS numbers: 05.40.-a, 05.20.-y
It is of great interest, both theoretically and prac-tically, to understand via simple models the emergentbehaviours of the complex systems containing a largenumber of interacting components. Examples like crowddynamics[1, 2], highway traffic systems[3, 4], and themore recent urban traffic flows[5] have attracted physi-cists for decades. Despite the observed complex patternsfrom these systems, some of the essential characteristicsare universally well-defined[6, 7]. In contrast to the tradi-tional many-body physical systems (e.g. involving iden-tical particles like electrons), the crowd or traffic systemslack almost any symmetry at the microscopic level: evenindividual components are different from one another,with intrinsic stochasticity. This poses great challenges infinding the simplest model with enough predictive powerto adequately characterise these complex systems.The lack of symmetries at the microscopic level leadsto a certain arbitrariness in the model construction. Themodelling of the highway traffic system has led to aplethora of traffic models[3, 4, 6, 7]. Analysis of the em-pirical data has led to profound understanding of the traf-fic dynamics[8, 9]; however due to non-linear interactionsbetween drivers, simple interactions can lead to complexspatiotemporal patterns. It is thus difficult to decidewhich detailed driving behaviours or specific functionalforms should be used in the model. This is the primaryconcern behind the controversy between the two-phaseand three-phase traffic theories[10, 12], which leads totwo classes of traffic models. While both the two-phaseand the three-phase traffic models can simulate the scat-tering of the data points on the flow-density plane whenthe traffic is congested, the two classes are distinguishedby the presence or absence of a continuous fundamen-tal diagram in the flow-density plot[7]. Moreover, thelack of a fundamental principle in understanding variousdifferent traffic models makes it difficult to decide if, inaddition to the non-linear interactions between the vehi-cles, factors like stochasticity or diversity of driving be-haviours are also essential to certain observed empirical features.To tackle the dilemma of the apparent(over)abundance of traffic models, we need answersto the following questions: a). How do we properlycharacterise the differences between two traffic models?b). Is there a standard way of extending an existingtraffic model or construction of new traffic models? c).Is there a standard way in selecting the best trafficmodel based on the empirical data? In this Letter, weanswer in details the first two questions for deterministicmicroscopic models with identical drivers, by proposinga general framework of obtaining all such models froma master model. The controlled expansion aroundproperly defined “ground states” of the master modelshows that the two-phase and three-phase traffic modelsare both just special cases. The general framework alsoallows us to classify existing and new traffic model, andcompare explicitly various types of approximations in awell-defined way.The answer to the third question lies in the fact that inprinciple the master model can be obtained empirically,via a renormalization-like procedure by averaging overunimportant factors influencing the reaction of individualdrivers. The details can be found in[16] and will also bediscussed elsewhere. We will now develop the generalframework by assuming the following form of the mastermodel ˜ a n = ˜ f (cid:16) ˜ h n , ˜ v n , ∆˜ v n (cid:17) (1)Here the assumption is that the bumper-to-bumper dis-tance of the n th vehicle, ˜ h n , its velocity ˜ v n , and the ap-proach velocity ∆˜ v n = ˜ v n +1 − ˜ v n are the important fac-tors affecting the acceleration of the n th vehicle, and allother factors are averaged over. One can choose more(or different) dynamic variables in the function ˜ f , butfor the purpose of illustration we choose this simple yetsufficient case.The natural time and length scale of the traffic systemis given by the two statistically robust empirical quanti- a r X i v : . [ n li n . AO ] O c t a Δ v = , h a) v a Δ v = , h b) v a Δ v = , h c) v a Δ v = , h d) v FIG. 1. Various possible sections of the acceleration (given by˜ f in Eq.(1)) at different h , as a function of v and evaluated at∆ v = 0. The non-analyticity of c) can be well approximatedby d). The specific form of such acceleration can in principlebe obtained from empirical data[16]. ties: ρ j , the density of the vehicles within a wide mov-ing jam, and V max , the maximum velocity (or the speedlimit). Since we will carry out a controlled expansionof Eq.(1) later, all quantities in Eq.(1) are dimensionlessafter a chosen set of scales. While formally the modeland its expansion are independent of the choice of thescales, for practical purposes one can choose either thelength scale ρ − j , the time scale ( ρ j V max ) − , or other setsof the physically relevant scales, depending on differentpurposes and the particular system under study.The properties of the master model can be studiedby plotting ˜ f ( h, v, ∆ v ) as a function of v at fixed h inFig.(1), where we take ∆ v = 0 and focus around zeroaccelerations. The master model has a fundamental di-agram if for any value of h the acceleration decreasesmonotonically with v and crosses the x-axis only once,as shown in Fig.(1a) or Fig.(1d). This corresponds tothe two-phase traffic models with a unique relationshipbetween the flow and density. The difference betweentwo-phase models as reflected by the detailed plots canbe quantified by the proper Taylor expansions as we willshow later.The plots in Fig.(1b) and Fig.(1c) are cases wherefor certain range of ˜ h , there are more than one steadystate (given by the vanishing acceleration) at the samedensity ( ∼ ˜ h − ), but with different average velocities(given by the x-axis intersections). They correspondto the microscopic three-phase models proposed in theliterature[14, 15]. For Fig.(1b), the number of steadystates at the same density equal to the number of inter-sections at the x-axis, though only the ones with negativegradient are stable. The Taylor expansions around differ-ent stable points where ˜ a = 0 leads to different optimal velocity functions within different velocity range. If forcertain range of ˜ h the acceleration is not analytic in ˜ v asshown in Fig.(1c), steady states can occur at the sameaverage traffic density but for any velocities at which theacceleration is zero. Since the plot corresponds to thecase where ∆˜ v = 0, this is exactly the case of speedadaptation mechanism that leads to the scattering of theflow-density plot at the synchronized phase. Both thesetwo cases are commonly used in various forms for theconstruction of three-phase models.The key assumption of the three-phase traffic theory isthat the scattering of the flow-density plot in the “syn-chronized phase” corresponds to a multitude of steady states with non-unique flow-density relations[7]. it ishowever empirically difficult to verify if those states areindeed steady states. The macroscopic quantities likeflow and density are empirically obtained from the aggre-gated raw data[11, 12]. A transient state with qualitativeor quantitative features that evolves very slowly will becaptured by the sensor as the “steady state”. Thus inmost cases we cannot distinguish such long lasting quasi-steady states from the steady states.We thus argue that instead of dealing with a non-analytic function as shown in Fig.(1c), we can for mostpractical purposes replace it with an analytic func-tion with an inflection point at zero acceleration (i.e.Fig.(1d)). Indeed the function in Fig.(1c) can be approx-imated by an analytical function to any degree of accu-racy. Such a model still has a fundamental diagram, butfor any state not too far away from the inflection point, itevolves very slowly with very small accelerations. Numer-ical simulation of such analytic models show very similarscattering of the flow-density plot and the spatiotemporalpatterns as the proposed three-phase models within anyphysically reasonable time scale. Therefore fundamen-tally there is no strict boundary between the two-phaseand three-phase models, and we will proceed to show theway to differentiate them is no different from the way ofdifferentiating different two-phase models.We will now only deal with analytic master models.The ground states of the model are defined as the solu-tions with all vehicles equally spaced apart with headway˜ h and travelling at the same time-independent velocity˜ v . In Eq.(1) the distance (or headway) is scaled by ρ − j while the time is scaled by ρ − j V − ; the tilded velocitiesand accelerations are derivatives of the scaled distance bythe scaled time. Thus all quantities with tilde are scaledto be dimensionless. The ground states are indexed bythe average density ˜ h − , and at some densities there maybe more than one ground states (i.e. Fig.(1b)). We canthus expand the master model around a chosen groundstate: ˜ a n = (cid:88) p,q κ p,q (cid:16) ˜ h n (cid:17) (cid:16) ˜ v n − V op (cid:16) ˜ h n (cid:17)(cid:17) p ∆˜ v qn (2) κ p,q (cid:16) ˜ h n (cid:17) = 1 p ! q ! ∂ p + q ˜ f∂ p ˜ v n ∂ q ∆˜ v n (cid:12)(cid:12)(cid:12)(cid:12) ˜ v n = ˜ V op ( ˜ h n ) ∆˜ v n =0 (3)where V op ( h ) satisfies ˜ f ( h, V op ( h ) ,
0) = 0. Clearly anyexisting two-phase models can be expanded this way, andthey can be quantitatively compared by the set of coef-ficients κ p,q . The optimal velocity (OV) model[17] is thespecial case where κ , is a negative constant while allother κ p,q vanishes. The related FVD model has an ad-ditional constant κ , [18], while the asymmetric FVD[19]can be understood as having additional non-linear correc-tions with non-vanishing κ ,q , q > κ p, vanishes for p < p so that the leading terms involv-ing ˜ v n are of higher orders. The larger the value of p ,the more long lasting the transient states are around theinflection points. Thus if one can obtain ˜ f empirically,the assumptions of the three-phase traffic theory can betested microscopically.Seemingly different traffic models can be shown to bequalitatively equivalent under this scheme of controlledexpansion. As an example we look at the IDM modelwhich has the following form: a n = a (cid:32) − (cid:18) v n v (cid:19) δ − (cid:18) h ∗ ( v n , ∆ v n ) h n (cid:19) (cid:33) (4) h ∗ ( v, ∆ v ) = s + s (cid:114) vv + T v + v ∆ v √ ab (5)Here we use dimensionful and physical quantities, withthe standard set of parameters[13] a = 0 . ms − , b =1 . ms − , v = 33 ms − , s = 2 m, s = 0 m, T =1 . s, δ = 4. The ground state is unique for any density;the relationship between h and V op is given by h = ( s + V op T ) (cid:32) − (cid:18) V op v (cid:19) δ (cid:33) − (6)The Taylor expansion around the ground state configu-ration has a finite number of terms: a n = p =4 ,q =2 (cid:88) p =0 ,q =0 λ p,q ( v n − V op ( h n )) p ∆ v qn (7)In general the majority of the approach velocities of thehighway traffic, as well as the deviation of the vehi-cle velocities from the optimal velocities, is on the or-der of 1 ms − , as verified from the numerical simulation FIG. 2. (Color Online.)The coefficients of expansion of theIDM in Eq.(4) as the function of the bumper-to-bumper dis-tance h , and λ pq is defined in Eq.(7) and normalised to havethe same dimensions . of Eq.(4). We can thus de-dimensionalize velocities inEq.(7) in this unit, normalizing all the non-vanishing co-efficients of expansion to have the unit of the physicalacceleration, and plot them in Fig.(2) as functions of thedensity. One can clearly see that the dependence of λ pq on traffic density is quite intuitive. The driver is onlysensitive to ∆ v n at intermediate traffic densities. In thecongested traffic the drivers are more alert (large λ ),and to keep a safe bumper-to-bumper distance is moreimportant than to synchronize the velocity.It is thus probably enough from Fig.(2) to just keep λ , λ , λ for the qualitative features of the traffic dy-namics to be preserved. Numerical simulations show evenignoring λ may be good enough. Such a truncated IDMmodel can also qualitatively simulate the approach dy-namics of a single vehicle rather well (see Fig.3), eventhough in this special case the truncated terms are pre-sumably not small due to the large approach velocity andlarge deviation of the vehicle’s velocity from its optimalvelocity. For the resulting truncated model, further tun-ing and optimization of the model are possible by fitting λ and λ using simple functions (such as the step-likefunctions), to improve the quantitative agreements withthe original IDM.Truncation of higher orders and simplification of theexpansion coefficients can be applied to other models likeShamoto’s[20]. We would also like to emphasize that itis not a priori true that the IDM or Shamato’s mod-els are more realistic than their simplified counterparts.The tuning of the expansion coefficients should be basedon the experimental measurement of the master model, 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Time (min) V e l o c i t y ( m / s ) llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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− . − . . Time (min) A cc e l e r a t i on ( m / s / s ) 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Time (min) V e l o c i t y ( m / s ) llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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− − Time (min) A cc e l e r a t i on ( m / s / s ) FIG. 3. Here we simulate the dynamics of a single vehicleaccelerating from zero velocity and approaching an obstaclelocated at 2 . km away from the starting point. The top twoplots are the time evolution of the velocity and acceleration,simulated from the full IDM model. The bottom two plots arethe time evolution of the velocity and acceleration, simulatedfrom the linearlized IDM model by keeping only λ and λ .. and it is possible to achieve better agreement with theempirical observations even with the truncated models.The original OV models[17–19] are the simplest case ofEq.(2) where the coefficients of expansion are constantsand only the term linear in ( v n − V op ( h n )) is kept. Thephase diagram of such models are well-known[22–26]. Tounderstand the more realistic traffic models as shownabove, and for tuning and construction of new trafficmodels, it is also important to understand the emergentcharacteristics of the generalized OV models with generalcoefficients of expansion and higher orders of expansion.This is in general difficult due to the non-linear inter-action, and we will just explore some simple examples,leaving a more detailed discussion elsewhere.We start by generalizing the simplest artificial OVmodel as follows: a n = λ ( h n ) (tanh ( h n ) − v n ) (8)It is useful to plot λ together with the coexistence curve(CC) as well as the neutral stability (NS) line of the orig-inal OV model. The CC at h < h min -branch; naturally the other half iscalled the h max -branch[27]. The linear stability conditionat average density h − is given by λ ( h ) > h .The qualitative features of Eq.(8) can be roughly clas-sified by the way λ intersects with the CC and the NSline. If λ intersect with the h min - and the h max -branch atmost once separately (see Fig.(4a)), the constant density( h n = const) solution of Eq.(8) can be stable or unstableagainst the development of cluster solutions with mini-mum and maximum bumper-to-bumper distances ¯ h min llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll −4 −2 0 2 4 . . . . . h l CoexistencecurveNeutral stability line l (1) l (2) l (3) a) lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l l l l l l 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−4 −2 0 2 4 . . . . . h l CoexistencecurveNeutralstability line l (1) l (2) b) llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll − − Car index h n c) llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll − . . . Car index h n d) llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll . . . Car index h n e) FIG. 4. (Color Online.)All the insets are plots of headways asa function of the car index. a). λ (1) , λ (2) and λ (3) are examplesof different functions of λ in Eq.(8). For cluster solutions themaximum or minimum space gaps can be uniform as c)., from λ (1) and λ (2) ), or fluctuating as d)., from λ (3) ). b). λ (1) and λ (2) are examples of different functions of λ . The vehicledensity corresponding to the thickened part of λ , if unstable,leads to regular cluster solutions. e). shows incomplete clustersolutions when the vehicle density satisfies λ ( h ) < h ,which is linearly unstable. . and ¯ h max , qualitatively the same as the original OVmodel; both ¯ h min and ¯ h max are independent of the den-sity of the traffic. Clearly only λ ∈ (cid:2) λ (cid:0) ¯ h min (cid:1) , λ (cid:0) ¯ h max (cid:1)(cid:3) contributes to the cluster structure, with the followingrelationship: ¯ h max / min ∈ (cid:2) h max / min (cid:0) λ (cid:0) ¯ h min (cid:1)(cid:1) , h max / min (cid:0) λ (cid:0) ¯ h max (cid:1)(cid:1)(cid:3) (9) If λ (cid:0) ¯ h min (cid:1) < h min ) or λ (cid:0) ¯ h max (cid:1) < h max ), thevehicles in the clusters (or anti-clusters) are also linearlyunstable and their bumper-to-bumper distances will fluc-tuate (see λ (3) in Fig.(4a) and the inset), in contrast tothe original OV model.If λ intersects with the h min -branch ( h max -branch)more than once (Fig.(4b)), the traffic dynamics can be different at different vehicle density. For h within thethickened part of λ (1) and λ (2) , the stability of the h n = const solution and the development of the clustersolution are similar as the cases in Fig.(4a). For otherpart of h , if λ ( h ) < h ), the h n = const solu-tion is linearly unstable but clearly the cluster solutioncannot be fully developed due to the conservation of thevehicle density. Partially developed cluster solutions withstrong oscillations are obtained as shown in the inset ofFig.(4b)[21].Higher orders of ( V op − v n ) generally shift the valuesof ¯ h min and ¯ h max . Here we look at simple cases with FIG. 5. a). Cluster solutions in the unstable phase, whenhigher order terms in ( v − V op ) are just small perturbations(Top left). b). Multiple equilibrium densities at the samevelocity, corresponding to the case in Fig.(1b), when the h n = const solution at each x-axis intersection with negativegradient is stable (Top right). c). The unstable phase whenonly ( v − V op ) term is present in the expansion with constantexpansion coefficient (Bottom left). d). The unstable phasewhen only ( v − V op ) term is present in the expansion withconstant expansion coefficient (Bottom right). constant coefficients of expansion. For situations like theones in Fig.(1b), the stable phase can have multiple equi-librium densities with vehicles all traveling at the samevelocity (see Fig.(5b)). For three-phase models with van-ishing low order terms (at least the linear oder), the NSline is no longer applicable. Since the accelerations arevery small even when the vehicles deviates from theirequilibrium density, it is very difficult for the clusters toform, as shown in Fig.(5c) and Fig.(5d).One should note that our method can be applied tomore general cases, in particular with more realistic op-timal velocity functions. The cluster solutions, the CCand NS line are universal features for the more compli-cated models. Though step functions of λ are used inmany of the arguments above for its simplicity, λ can bedeformed moderately or smoothened without altering thequalitative features. Since only λ ∈ (cid:2) λ (cid:0) ¯ h min (cid:1) , λ (cid:0) ¯ h max (cid:1)(cid:3) determines the instability of the model, one can alterthe shape of λ outside of that range arbitrarily withoutchanging the cluster structure or the phase diagram. Forexample, the inset of Fig.(4b) will be realized by a mono-tonically decreasing λ intersecting the h max -branch of theCC twice, as is the case for the IDM model.In conclusion, we propose that microscopic traffic mod-els should start with identifying the proper optimal ve-locity function, from which the family of ground statesare defined. Simple traffic models are constructed as spe- cial cases of the expansion around those ground states,with coefficients of expansion in general dependent on thevehicle density. In particular the “synchronized phase”can be formally understood as the result of the mod-els where the leading order expansion about the optimalvelocity vanishes within some density range. The theo-retical framework also proposes standard ways of modelconstruction, and validation by the empirical data. Theunderstanding of various possible solutions to a more gen-eral class of OV models could also be useful for designingthe algorithms of autonomous driverless vehicles for largescale highway transportation.We thank Prof. Dirk Helbing for useful discus-sions. This research was partially supported by Singa-pore A (cid:63) STAR SERC “Complex Systems” Research Pro-gramme grant 1224504056. [1] D. Helbing, L. Buzna, A. Johansson and T. Werner,Transport Sci. , 1 (2005).[2] D. Helbing, I. Farkas and T. Vicsek, Nature , 487(2000).[3] N. Bellomo and C. Dogbe, Soc. Ind. App. Math. , 409(2011).[4] D. Chowdhury, L. Santen and A. Schadschneider, Phys.Rep. , 199 (2000).[5] D. Helbing and A. Mazloumian, Eur. Phys. J. B. , 257(2009).[6] D. Helbing, Rev. Mod. Phys. , 1067 (2001).[7] B.S. Kerner, Introduction to Modern Traffic Flow The-ory and Control: The Long Road to Three-phase TrafficTheory, Springer-Verlag Berlin Heidelberg 2009, and thereferences therein.[8] B.S. Kerner, Phys. Rev. E. , 046138 (2002).[9] B.S. Kerner, Phys. Rev. Lett. , 3797 (1998).[10] B.S. Kerner, Physica A, , 5261(2013).[11] M. Treiber and A. Kesting, Traffic Flow Dynamics,Springer-Verlag Berlin Heidelberg 2013, and the refer-ences therein.[12] M. Treiber, A. Kesting and D. Helbing, Transport. Res.B. , 983 (2010). D. Helbing, M. Treiber, A. Kesting andM. Schonhof, Eur. Phys. J. B. , 583 (2009). One shouldalso note that empirically the spatiotemporal patterns ofthe synchronized phase fluctuates over short time anddistance.[13] M. Treiber, A. Hennecke and D. Helbing, Phys. Rev. E. L31-L43 (2002). B.S. Kerner and S.L. Klenov, J. Phys.A: Math. Gen. , 036130(2003).[16] Bo Yang, Xihua Xu, John Z.F. Pang and ChristopherMonterola, arXiv. 1504.01256. While the general frame-work does not favor one traffic model over the other, theverification of the master model by the experimental dataallows us to justify the approximations employed by themodel.[17] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama, Phys. Rev. E. , 1035 (1995).[18] R. Jiang, Q. Wu and Z. Zhu, Phys. Rev. E. , 017101(2001).[19] H. Gong, H. Liu and B. Wang, Phys. A. , 2595 (2008).[20] D. Shamoto, A. Tomoeda, R. Nishi and K. Nishinari,Phys. Rev. E. , 046105 (2011).[21] If λ ( h ) > h ) but below the coexistence curve,the h n = const solution can be absolutely stable or un-stable depending on the detailed form of λ . It is stableif the oscillations of partially developed cluster solutionare small, since a large enough perturbation is needed todestabilize the h n = const solution[22]. [22] B.S. Kerner and P. Konhauser, Phys. Rev. E. , 54(1994).[23] W. Shi, N-G Chen and Y. Xue, Commun. Theor. Phys. , 1088 (2007).[24] H. Hayakawa and K. Nakanishi, Phys. Rev. E. , 3839(1998).[25] T. Nagatani, Phys. Rev. E. , 3564 (2000).[26] H.X. Ge, S.Q. Dai, L.Y. Dong and Y. Xue, Phys. Rev.E. , 066134 (2004).[27] Both of them depend only on λ and independentindependent