The Intelligent Driver Model with Stochasticity -- New Insights Into Traffic Flow Oscillations
aa r X i v : . [ phy s i c s . s o c - ph ] A ug ISTTT Procedia 00 (2017) 1–14
The Intelligent Driver Model with Stochasticity – NewInsights Into Tra ffi c Flow Oscillations Martin Treiber a, ∗ , Arne Kesting b,a a Technische Universit¨at Dresden, Institute for Transport & Economics,W¨urzburger Str. 35, 01062 Dresden, Germany b TomTom Development Germany, An den Treptowers 1, 12435 Berlin (Germany)
Abstract
Tra ffi c flow oscillations, including tra ffi c waves, are a common yet incompletely understood feature of congested tra ffi c.Possible mechanisms include tra ffi c flow instabilities, indi ff erence regions or finite human perception thresholds (actionpoints), and external acceleration noise. However, the relative importance of these factors in a given situation remainsunclear. We bring light into this question by adding external noise and action points to the Intelligent Driver Modeland other car-following models thereby obtaining a minimal model containing all three oscillation mechanisms. Weshow analytically that even in the subcritical regime of linearly stable flow (order parameter ǫ < ffi ciently far awayfrom the threshold, the amplitude scales with ( − ǫ ) − . . By means of simulations and comparisons with experimental carplatoons and bicycle tra ffi c, we show that external noise and indi ff erence regions with action points have essentiallyequivalent e ff ects. Furthermore, flow instabilities dominate the oscillations on freeways while external noise or actionpoints prevail at low desired speeds such as vehicular city or bicycle tra ffi c. For bicycle tra ffi c, noise can lead to fullydeveloped waves even for single-file tra ffi c in the subcritical regime.c (cid:13) / or peer review under responsibility of Delft Universityof Technology Keywords: car-following model, tra ffi c oscillations, flow instability, acceleration noise, action points, spectralintensity, fluctuations, correlations, order parameter, bicycle tra ffi c
1. Introduction
Tra ffi c flow oscillations, including stop-and-go tra ffi c, are a common phenomenon in congested vehic-ular tra ffi c [1, 2, 3, 4, 5]. Conventionally, this phenomenon is described in terms of linear or nonlinearstring or flow instabilities [6, 7, 8] which are typically triggered by a local persistent perturbation, e.g., lanechanges near a bottleneck [2]. In another approach, the flow oscillations are traced back to indi ff erence re-gions of the human driver [9, 10] or to finite perception thresholds leading to abrupt acceleration changes atdiscrete “action points” [11, 12]. Related to this are finite attention spans [13, 14]. It has also ben proposedthat the oscillations may be caused by event-oriented changes of the driving style switching, e.g., between ∗ Corresponding author.
Email addresses: [email protected] (Martin Treiber), [email protected] (Arne Kesting)
M. Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 “timid” and “aggressive” [15], or, related to this, by over- and underreactions [16]. Finally, direct externaladditive or multiplicative acceleration noise (e.g., caused by perception errors) is postulated to drive the os-cillations. This line of reasoning is typically modelled by cellular automata (e.g., [17]) which need some sortof stochasticity, anyway, for a proper specification. However, there are also approaches to incorporate accel-eration noise into time-continuous car-following models leading to stochastic di ff erential equations [14, 18].One of the simplest approaches is the “Parsimonious Car-Following Model” (PCF model) [19] which addswhite acceleration noise to the free-acceleration part of Newell’s car-following model with bounded acceler-ation [20] and, as [14], also provides explicit numerical stochastic update rules by integrating the stochasticdi ff erential equation over one time step.While all of the above approaches can explain certain observations, it remains an open question whetherthese approaches are connected with each other, and if so, in which way. Another open problem is to identifythe situations where oscillations are caused predominantly by flow instabilities, by indi ff erence regions, orby noise.In this contribution, we bring light into these questions by proposing a general scheme for addingnoise and indi ff erence regions (in form of action points) to a class of deterministic acceleration-based car-following models. Suitable underlying models include the Intelligent Driver Model (IDM) [21], the FullVelocity Di ff erence Model (FVDM) [22], or Newell’s car-following model [23] with bounded accelera-tions [20]. In this way, we obtain a minimal model containing all three mechanisms which we then analyzeanalytically and numerically. The focus is on the generic instability mechanisms and their relative impor-tance rather than on specific car-following models.The rest of the paper is organized as follows: In the next section, we specify the minimal model. InSection 3, we introduce the order parameter ǫ denoting the relative distance to linear string instability andanalytically derive, as a function of ǫ , the statistical fluctuation properties induced by white noise, includingspectral, modal, and overall intensity of the vehicle gap and speed fluctuations, and the associated spatiotem-poral correlations. In the Sections 4 and 5, we investigate the oscillation mechanisms for high-speed andlow-speed tra ffi c (cars and bicycles, respectively), and compare the results with experimental observations.Finally, Section 6 concludes with a discussion.
2. Model Specification
We consider general stochastic time-continuous car-following models of the form˙ v n = f ( s n , v n , v l ) + ξ n ( t ) , h ξ n ( t ) i = , h ξ n ( t ) ξ m ( t ′ ) i = Q δ nm δ ( t − t ′ ) . (1)Here, f ( · ) denotes the acceleration function of the underlying car-following model for vehicle n as a functionof the (bumper-to-bumper) gap s n , the speed v n , and the leader’s speed v l . Time delays in the independentvariables representing reaction times such as in Newell’s Car-Following Model [23] are allowed. The whiteacceleration noise ξ n ( t ) is completely uncorrelated in time and between vehicles (the Kronecker symbol δ nm = n = m and zero, otherwise; δ ( t − t ′ ) denotes Dirac’s delta distribution), and has the intensity Q . Model (1) can be seen as a simplistic special case of the Human Driver Model (HDM) [14]. Noticethat, when starting with a deterministic initial state v n (0) at t =
0, integration of the stochastic di ff erentialequation (1) leads, in the limit t →
0, to a Gaussian speed distribution whose expectation and variance aregiven by (see, e.g., [24]) h v n ( t ) i = v n (0) + f t , h ( v n ( t ) − ( v n (0) + f t ) i = Qt (2)where the variance is independent of f . Consequently, Q has the unit m / s . By chosing a deterministic car-following model with the ability for string instability, e.g., the Intelligent Driver Model (IDM) [21] or theFull Velocity Di ff erence Model (FVDM) [22], the model (1) contains the two oscillation-inducing factorsnoise and string instability. We introduce the third factor, indi ff erence regions in the form of action points,by updating the deterministic acceleration to the actual value given by f ( · ) only, if (cid:12)(cid:12)(cid:12) f ( s i ( t ) , v i ( t ) , v l ( t )) − f ( s i ( t ′ ) , v i ( t ′ ) , v l ( t ′ )) (cid:12)(cid:12)(cid:12) > ∆ a , ∆ a ∼ U (0 , ∆ a max ) . (3) . Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 -1.5-1-0.5 0 0.5 1 0 20 40 60 80 100 A cc e l e r a t i on [ m / s ] Time [s]Veh 1Veh 5Veh 10Veh 15Veh 25
Fig. 1. Visualization of the action points (maximum step ∆ a max = . / s ) by a time series of the accelerations of four platoonvehicles corresponding to Fig. 7(e) and (f). Also shown is the leader (Vehicle 1) accelerating slowly and deterministically to the finalspeed of 30 km / h. Here, t ′ is the time of the last change (action point) and ∆ a is a uniformly distributed random number drawnat the last action point whose maximum value must be well below the maximum acceleration capability ofthe deterministic model.Since characterizing action points according to model (3) is a novel proposition in itself, we first showhow the mechanism works by displaying typical instances of the resulting acceleration time series (Fig. 1).As expected, the drivers drive at constant accelerations, most of the time. These episodes with typicalirregular durations between 1 s and 20 s are separated by “action points” where the acceleration is changedby a variable amount whose maximum is given by the parameter ∆ a max . Notice that both the irregulardurations and increments agree with observations [12].The Eqs. (1) (or (2)), (3), and a deterministic car-following model specify the proposed minimal modelcontaining all three oscillation factors. Each factor can be controlled by a single parameter. The noise iscontrolled by the noise intensity Q (typical values are of the order of 0 . / s ), the indi ff erence regionby the maximum acceleration step ∆ a max (of the order of 1 m / s or less), and the string instability by therelevant parameter of the underlying car-following model, e.g., the maximum acceleration a for the IDM.It is convenient to define the relative distance to the linear threshold a c (see Section 3 below) by the orderparameter ǫ = − aa c (4)keeping the other IDM parameters fixed. A homogeneous deterministic steady state is linearly stable for ǫ < ǫ >
3. Noise-Induced Subcritical Oscillations
In order to analytically determine the linear response to the white acceleration noise, we assume ahomogeneous ring road of circumference L with N identical drivers / vehicles such that the global densityis given by ρ = N / L . Furthermore, we switch o ff the action points by setting ∆ a max =
0. We generalizethe standard linear stability analysis (see, e.g., Ref. [8]) to include noise. Starting from a homogeneoussteady state v n = v e and s n = s e ( v e ) lying on the fundamental diagram, we decompose the gaps and speedsinto the steady-state contribution and a small time-dependent perturbation by setting s n ( t ) = s e + y n ( t ), v n ( t ) = v e + u n ( t ). This leads to the linearized equationsd y n d t = u n − − u n , d u n d t = f s y n + f v u n + f l u n − + ξ n ( t ) , (5)where f s = ∂ f ∂ s , f v = ∂ f ∂ v , and f l = ∂ f ∂ v l are the gradients of the acceleration function f ( · ) at the deterministicsteady state. M. Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 S yy ( ω ) [ m s ] ω [1/s]k=0.02k=0.04k=0.06k=0.08k=0.10k=0.12 a=1.50 ( ε =-0.22) 0.001 0.01 0.1 1 10 100-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 S vv ( ω ) [ m / s ] ω [1/s]k=0.02k=0.04k=0.06k=0.08k=0.10k=0.12 a=1.50 ( ε =-0.22) (a) (b) Fig. 2. Steady-state fluctuation spectrum of (a) the speeds, and (b) the gaps for the IDM with the parameters of the main text and whiteacceleration noise of intensity Q = . / s . Shown is the spectrum for several modes of dimensionless wavenumber k (numberof vehicles 2 π/ k per wave). The negative values for ω indicate an upstream propagation of − ω/ k vehicles per second in the framecomoving with the vehicles. The underlying deterministic steady state is characterized by v e =
48 km / h corresponding to a density of36.4 vehicles / km. The time-dependent parts can be further decomposed into N harmonic eigenmodes of wave number k = π m / N , m = − N / + , ..., N / | m | indicating the number of travelling waves on the ring and2 π/ | k | the number of vehicles per wave) by the ansatz y n ( t ) u n ( t ) ! = X k ˆ y k ( t )ˆ u k ( t ) ! e ink (6)where i = √− ff erential equations for the modes k , dˆ y k d t = (cid:16) e − ik − (cid:17) ˆ u k , dˆ u k d t = f s ˆ y k + (cid:16) f v + f l e − ik (cid:17) ˆ u k + ˆ ξ k ( t ) , (7)where the noise source now is given by D ˆ ξ k ( t ) E = , D ˆ ξ k ( t ) ˆ ξ l ( t ′ ) E = QN δ kl δ ( t − t ′ ) . (8)Each pair of linear stochastic di ff erential equation (8) can be written in the general formdd t ~ X k = − L k ) ~ X k + ~ξ, h ~ξ ( t ) i = , h ~ξ ( t ) ~ξ ′ ( t ′ ) i = D δ ( t − t ′ ) (9)where ~ X k = (ˆ y k , ˆ u k ) ′ denotes the state vector of mode k , and L k = − e − ik − f s − ( f v + f l e − ik ) ! , D = QN ! , (10)denote the linear dissipation matrix, and the covariance matrix of the white noise, respectively.The stationary solutions to (9) consist of zero-mean Gaussian fluctuations which are therefore com-pletely specified by the amplitude and correlation of the state variables as well as the temporal autocorrela-tion function. Instead of the autocorrelation function, one can also determine the spectral intensity of the gapand speed oscillations of waves of wavenumber k = π/ L (where L is the wavelength), i.e., the di ff erentialenergy content (amplitude squared) contained in waves of wavenumber k at an angular frequency ω = π/ T . Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 S vv ( ω ) [ m / s ] ω [1/s]k=0.02k=0.04k=0.06k=0.08k=0.10k=0.12 a=1.25 ( ε =-0.01) 0.001 0.01 0.1 1 10-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 S vv ( ω ) [ m / s ] ω [1/s]k=0.02k=0.04k=0.06k=0.08k=0.10k=0.12 a=3.00 ( ε =-1.43) (a) (b) Fig. 3. Steady-state speed fluctuation spectrum for the control parameters ǫ = .
01 (a) and ǫ = .
43 (b). The IDM parameters, noiseintensity, and the deterministic steady state is the same as in Figure 2. (where T is the wave period). Remarkably, there exists an analytic solution for the spectral intensity of thestationary fluctuations called the fluctuation-dissipation theorem [24], S k ( ω ) = ( L k + i ω ) − D (cid:16) L ′ k − i ω (cid:17) − . (11)Here, S k ( ω ) = ˜ ~ X k ( ω ) ˜ ~ X ′ k ( ω ) is the spectral intensity with ˜ ~ X k ( ω ) the Fourier transform of ~ X k ( t ), and ˜ ~ X ′ k ( ω )its transposed complex conjugate. Furthermore, L ′ k is the transposed and complex conjugate of the linearmatrix L k . Inserting (10), we finally obtain the subcritical modal fluctuation spectrum S k ( ω ) = QN | Det( L + i ω ) | − cos k ) i ω (1 − e − ik ) − i ω (1 − e ik ) ω ! (12)where | Det( L + i ω ) | = (cid:16) ω + ω f l sin k − f s (1 − cos k ) (cid:17) + ( f s sin k − ω ( f v + f l cos k )) . The diagonal components are of particular interest. S = S yyk ( ω ) gives the spectral intensity of the gaposcillations contained in waves of wavelength 2 π/ ( ρ | k | ) and period 2 π/ω propagating at a velocity ω/ ( ρ k ) inthe comoving frame of reference. The component S = S uuk ( ω ) gives the corresponding spectral intensityof the speed fluctuations. Since the inverse Fourier transform of the spectral intensity with respect to ω and k gives the spatiotemporal correlation function, the fluctuation spectrums S yyk ( ω ) and S uuk ( ω ) completelydescribe the subcritical linear gap and speed fluctuations of the vehicles.Figure 2 displays S yyk ( ω ) and S uuk ( ω ) for the IDM parameters v =
30 m / s, T = . s = b = . / s and a vehicle length of 5 m at the steady state v e =
48 km / h corresponding to a density ρ = . ./ km. For this state, the linear stability condition 2 v ′ e ( s e ) < f l − f v (cf. [8]) leads to the criticalIDM acceleration parameter a c = .
233 m / s . Consequently, the acceleration a = . / s of these plotscorresponds to the control parameter ǫ = − a / a c = − .
22. Remarkably, the spectrum of both the gapand the speed modes shows distinct peaks at approximatively ω/ k = − . − . Since 2 π/ k denotes thenumber of vehicles in a wave, ω/ k can be interpreted as the number of vehicles per time unit a wave passes(“passing rate”). Thus, the fluctuations are highly spatio-temporally correlated propagating backwards at 0.6vehicles per second, even significantly below the linear threshold. Notice that fully developed supercriticalfluctuations triggered by linear instabilities have essentially the same passing rate.Figure 3 confirms this observations for other order parameters. While it is expected that the spectralpeaks become more pronounced very near the threshold ( ǫ = − . ǫ = − . M. Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 M oda l i n t en s i t y S uu , k [ m / s ] k [dimensionless]a=1.50 ( ε =-0.22)a=2.00 ( ε =-0.62)a=3.00 ( ε =-1.43) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0 0.02 0.04 0.06 0.08 0.1 S pe c t r a l i n t en s i t y S uu ( ω ) [ m / s ] ω [1/s] a=1.50 ( ε =-0.22)a=2.00 ( ε =-0.62)a=3.00 ( ε =-1.43) (b)(a) Fig. 4. (a) Modal speed fluctuation intensity for a ring road of length L ring =
10 km for three values of the relative proximity ǫ tothe linear stability threshold. Each symbol denotes a possible mode k = n π/ L ring (negative if propagating backwards). (b) Overallspectral intensity (14). The IDM parameters, noise intensity, and the deterministic steady state are the same as in Figure 2. O v e r a ll i n t en s i t y S uu [ m / s ] ε =1-a/a c a c =1.23 Fig. 5. Overall speed fluctuation intensity (15) as a function of the control parameter ǫ for the system of Figure 4. . Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 P o s i t i on a l ong c i r c u m f e r en c e [ m ] Time [s] S peed [ m / s ] Time [s] Car 1Car 10 ε=−0.01
IDM,
Fig. 6. Simulated IDM trajectories for a ring of 10 km circumference, a = .
25 m / s ( ǫ = − . Figure 4 displays one-dimensional integrations of the modal speed fluctuation spectrum for a ring roadof length L ring =
10 000 m. The panel (a) shows the modal fluctuation intensity S uuk = Z S uuk ( ω ) d ω (13)for some of the allowed modes near k =
0. We observe that the integration eliminates the nontrivial peakswhich is to be expected: After all, in the subcritical regime, the linear relaxation rate decreases with decreas-ing absolute values of the wavenumber thereby increasing, according to the fluctuation-dissipation theorem,the fluctuation intensities. Notice, however, that a distinct anisotropy favoring backwards propagating wavesremain.Panel 4(b) displays the overall spectral intensity S uu ( ω ) = X k S uuk ( ω ) (14)where the sum includes all allowed modes of the closed system. In contrast to plot4(a), distinct spectralpeaks remain indicating resonances of the lowest allowed modes. The distance between the peaks is in-versely proportional to the ring circumference.Finally, Figure 5 displays the overall steady-state fluctuation intensity S uu = X k Z S uuk ( ω ) d ω (15)of the speed of every single vehicle which is equal to the speed variance. Su ffi ciently far away from thethreshold ( ǫ < − . − ǫ ) − corresponding to amplitudes propor-tional to ( − ǫ ) − . . Notice that the variance does not diverge at the linear stability limit which is caused (i) M. Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 by finite-size e ff ects, (ii) by nonlinear saturation. As expected, the speed fluctuation amplitude of a givenvehicle is essentially independent of the system size.In summary, the analytical investigation shows that the fluctuations triggered by white noise in thesubcritical regime “anticipate” the characteristics of tra ffi c waves in the collectively unstable regime. Thisis examplified in the simulated trajectories of Fig. 6 whose amplitude and spatiotemporal correlations agreequantitatively with the analytic theory. Moreover, the amplitude of the subcritical fluctuations increasesstrongly when approaching the linear threshold from below. The simulations in the following sectionsdemonstrate that this can even lead to fully developed waves in the subcritical regime.
4. Vehicular Tra ffi c: Platoon Experiments In this section, we simulate platoon car-following experiments and test which of the three possibleoscillation mechanisms, namely instability, noise, or action points, or which combinations thereof, allows forreproducing the empirical findings of a concave increase of the speed fluctuation amplitude as a function ofthe platoon vehicle number [25]). Moreover, by simulating these mechanisms with three underlying models(the IDM, the FVDM and the PCF model), we test to which extent the results are universal, i.e., independentof the specific car-following model. In the IDM and FVDM simulations, the white acceleration noise isintegrated according to (2) resulting in fluctuations that are asymptotically independent of the simulationupdate time step [14]. The PCF model is simulated according to [19], i.e., realisations of the analyticaldistributions of the displacements are added to the locations in each time step which is set equal to itstime-gap parameter τ = T = v lead0 =
30 km / h corresponding to the experiment. The followers have a significantly higher desiredspeed and initially stand in a queue behind the leader. The IDM parameters of the followers that are notdirectly related to the oscillation mechanisms are set always to v =
108 km / h, T = . s = b = / s (notice that the vehicle length is irrelevant in platoon simulations). The three oscillationmechanisms are controlled by the relative IDM acceleration ǫ = − a / a c (where a c = .
25 m / s for theabove parameters and a leading speed of 30 km / h), Q (noise intensity), and ∆ a max (maximum accelerationchange at an action point). In the simulations of each of the three mechanisms, the respective controlparameter ǫ , Q , and ∆ a max has been calibrated to minimize the SSE between the observed and simulatedspeed standard deviations of all the vehicles (right column) over 10 simulation runs with independent seedswhile the other two control parameters have been set to zero.We find that the instability mechanism alone (Panels (a) and (b)) leads to oscillations that do not agreequalitatively with the platoon experiments. Notably, the increase of the fluctuation amplitude along theplatoon vehicles is convex instead of concave. Furthermore, the calibrated IDM acceleration parameter a = . / s , or ǫ = .
6, corresponds to an unrealistically unstable regime which leads to unrealistic resultsin simulation runs with longer platoons. Finally, the result depends sensitively on the noise intensity of theleading vehicle with no sensible results (no fluctuations) obtained for zero noise.Panels 7(c) and (d) display the e ff ect of external noise for a marginal stability ǫ = ∆ a max =
0. In agreement with observations, we obtain a concave growth of the fluctuation amplitudealong the platoon vehicles which, for Q = .
32 m / s , nearly quantitatively agrees with the experiment.Notice that the instability mechanism plays a role as well: The best results are obtained near marginalstability. Panels 7(e) and (f) give the results of simulations with active action points ( ∆ a max = . / s )and deactivated noise ( Q = ǫ =
0) give the best results.Compared to the simulations with pure acceleration noise, the fit quality is even better. For most realisations, . Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 S peed s t anda r d de v i a t i on [ m / s ] Vehicle index IDMData 0 0.5 1 1.5 2 0 5 10 15 20 25 S peed s t anda r d de v i a t i on [ m / s ] Vehicle index IDMData 0 0.5 1 1.5 2 0 5 10 15 20 25 S peed s t anda r d de v i a t i on [ m / s ] Vehicle index IDMData 0 10 20 30 40 50 60 70 0 50 100 150 200 250 300 350 400 S peed [ k m / h ] Time [s] Veh 1Veh 5Veh 10Veh 15Veh 25 0 10 20 30 40 50 60 0 50 100 150 200 250 300 350 400 S peed [ k m / h ] Time [s] Veh 1Veh 5Veh 10Veh 15Veh 25 0 10 20 30 40 50 60 70 0 50 100 150 200 250 300 350 400 S peed [ k m / h ] Time [s] Veh 1Veh 5Veh 10Veh 15Veh 25 v =30 m/sv =30 m/sv =30 m/s (b)(d)(f)(a)(c)(e) Fig. 7. Simulated speed time series (left column) and speed standard deviations (right column) for several vehicles in a platoonfollowing the leader (Vehicle 1) accelerating slowly to the leading speed of 30 km / h. The simulation setup reproduces the platoonexperiments of [26] (red symbols at the right column). In order to eliminate initial transients, the first 200 s of the simulation areskipped when calculating standard deviations. Top row: deterministic IDM far in the unstable regime ( a = . / s correspondingto ǫ = .
6; some noise has been added to the leader to initiate oscillations); middle row: strong acceleration noise Q = .
32 m / s at marginal stability ( ǫ =
0) and without action points ( ∆ a max = ∆ a max = . / s , Q =
0) atmarginal stability. The remaining IDM parameters of the followers are given in the main text.0
M. Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 a good agreement is reached for the first platoon vehicles as well while the IDM with acceleration noisesystematically overestimates the speed standard deviation of these vehicles.We conclude that the instability mechanism alone is not able to reproduce the observations, even in thepresence of a fluctuating leader. In contrast, both action points in the form of model (3) and white noise canreproduce the observations with best results obtained near marginal stability. Remarkably, action points andwhite noise are essentially interchangeable with action points giving marginally better results.The question arises if these results depend on the underlying model, namely the IDM, or if other, possiblysimpler, car-following models can be used as well. In order to test this proposition, we have performedsimulations with the stochastic Full Velocity Di ff erence Model (SFVDM) and also with the PCF model [19]of Laval et al which essentially implements the acceleration noise mechanism at marginal stability withoutinteraction points. Instead of the optimal velocity (OV) function of the original FVDM [22], we apply anOV function corresponding to a tridiagonal fundamental diagram which resembles the IDM and has thesame desired speed, desired time gap and minimum gap parameters. The resulting acceleration function ofthe FVDM reads f ( s , v , v l ) = β (cid:16) v opt ( s ) − v (cid:17) + λ ( v l − v ) , v opt ( s ) = max (cid:18) , min (cid:18) v , s − s T (cid:19)(cid:19) (16)with values of the steady-state parameters v =
30 m / s, T = . s = T = τ = T but also with the reaction delay time τ [8]; furthermore, the wave-speed parameter ofthis model can be identified by w = ( l veh + s ) / T where l veh is the vehicle length which, however, does notplay a role in the platoon simulations).The top left panel of Fig. 8 shows an instance run of the SFVDM simulations with the calibrated dy-namical parameters 1 /β =
10 s, λ = .
52 s − , and the white-noise intensity Q = .
25 m / s . The top rightpanel displaying the speed standard deviations along the platoon for 10 realisations can be compared withthat of the SIDM: Generally, we observe a good agreement. However, for most realisations, the simulatedspeed standard deviation of the first platoon vehicles is too high. Simulating the FVDM with action points(not shown) gives similar results as for the IDM.The middle row gives a typical run for the PCF model with the recommended relaxation parameter1 /β =
16 s, and the calibrated noise intensity Q = .
05 m / s (only for the followers) while the staticparameters v lead0 = . / s, v =
30 m / s, T = τ = . s = /β =
16 s and a scaled noise intensity ˜ σ = Q / ( β v ) = . (Section 4) and˜ σ = . (Section 5). With v =
30 m / s, this gives Q = .
68 m / s and Q = .
27 m / s , respectively,which are of the same order of magnitude. (Notice that, for a given value of the scaled noise intensity,the physical noise intensity Q depends strongly on the desired speed v while only a weak dependency isplausible for the experiments.) Both values of the noise intensity are significantly higher as that calibrated forthe stochastic IDM and FVDM. This is caused by the fact that the stochasticity of the PCF model is restrictedto the free regime while, in the interacting regime (the speed is restricted by the leader), the PCF modelreduces to Newell’s model, i.e., the drivers follow deterministically the leader’s trajectories with a constantspace and time shift and, consequently, have the same constant variance as that of the leader. This applieswhenever the realisation of the stochastic free displacement is greater than the deterministic displacementobtained from Newell’s model since the minimum of the displacements is taken in the model. The chanceof deterministic car-following increases with decreasing Q , increasing v , and increasing β . For su ffi cientlylow acceleration noise, this is always the case in our experiment and the PCF model reverts to deterministiccar-following according to Newell’s model. This switching between a deterministic and a stochastic modelseems also to be the reason why, although only implementing the acceleration noise mechanism, the PCFmodel fits the observations better than the SIDM and the SFVDM (and equally well as the IDM with actionpoints): For the first platoon vehicles, the leaders exhibit only low-amplitude oscillations increasing thechance that the deterministic part of the PCF model applies. This reduces the e ff ective acceleration noise ofthe first followers relative to that of the vehicles further behind thereby reducing the speed standard deviation . Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 S peed s t anda r d de v i a t i on [ m / s ] Vehicle index FVDMData 0 0.5 1 1.5 2 2.5 0 5 10 15 20 25 S peed s t anda r d de v i a t i on [ m / s ] Vehicle index PCFData 0 0.5 1 1.5 2 0 5 10 15 20 25 S peed s t anda r d de v i a t i on [ m / s ] Vehicle index PCFData 0 5 10 15 20 25 30 35 40 45 50 0 50 100 150 200 250 300 350 400 S peed [ k m / h ] Time [s] Veh 1Veh 5Veh 10Veh 15Veh 25 0 5 10 15 20 25 30 35 40 45 50 0 50 100 150 200 250 300 350 400 S peed [ k m / h ] Time [s] Veh 1Veh 5Veh 10Veh 15Veh 25 0 5 10 15 20 25 30 35 40 45 0 50 100 150 200 250 300 350 400 S peed [ k m / h ] Time [s] Veh 1Veh 5Veh 10Veh 15Veh 25 full noise (b)(d)(f)(a)(c)(e)
Fig. 8. Speed time series (left column) and speed standard deviations (right column) for the same initial configuration and the sameleader’s speed profile as in Fig. 7. Top row: stochastic Full Velocity Di ff erence Model (SFVDM) with a triangular fundamental diagramfor calibrated model parameters; middle row: Parsimonious Car-Following Model (PCF model) [19] with the same fundamentaldiagram as the SFVDM. Bottom row: PCF model with the same parameters but stochasticity also turned on in the interacting regime.The values of the model parameters are given in the main text.2 M. Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 Lo c a t i on [ m ] Time [s]
IDM with noiseExperiment
Fig. 9. Left: experimental trajectories of bicycles on a closed ring course of 140 m circumference; right: stochastic IDM simulationwith acceleration noise Q = . / s . The IDM parameters corresponding to a subcritical situation are given in the main text. of the first vehicles which is in line with the observation.However, it is obviously unrealistic to switch from stochastic to deterministic driving when enteringthe car-following regime. Therefore, a straightforward generalization of the PCF model consists in addingthe same acceleration noise to the car-following situation as well and also relax the rigid following rule ofNewell’s model by introducing the same relaxation parameter β as for the free part. It can be shown that theresulting “Full-noise PCF model” (FPCF model) is mathematically equivalent to (1) where f is given by thetime-delayed OVM with tridiagonal fundamental diagram, f ( s n ( t − T ) , v n ( t − T )) = β (cid:16) v opt ( s ( t − T )) − v ( t − T ) (cid:17) , (17)with v opt ( s ) given by FVDM. With the significantly reduced noise intensity Q = . / s , this modelreproduces the data similarly well as the stochastic IDM and FVDM models which is to be expected sinceit utilizes the noise mechanism while having no action points and being at marginal stability (in Newell’smodel, oscillations neither decay nor grow).We conclude that the IDM is not necessary for our general findings and can be replaced by other under-lying car-following models. In contrast, the mechanism matters. While the instability mechanism cannotreproduce the data even qualitatively, the action-point and noise mechanisms reproduce the data nearlyquantitatively with the action-point mechanism and the selective-noise mechanism of the PCF model givingmarginally better results than unconditional acceleration noise.
5. Bicycle Tra ffi c on a Ring The purpose of this section is twofold: Firstly, we demonstrate that, by a suitable change of the IDMparameters (particularly the vehicle length and the desired speed), the IDM can also be used to realisticallysimulate bicycle tra ffi c. Secondly, we show that, for low-speed tra ffi c, acceleration noise alone can lead tofully developed stop-and-go waves even in the subcritical regime.Figure 9 shows simulations (right) of bicycle experiments (left) on a ring road of 140 m circumfer-ence [27]. The bicycle length is assumed to be 1.67 m and the IDM parameters have been set to v = / s, T = . s = . a = . / s , and b = . / s . A comparison of the simulated trajectories (rightpanel) with Fig. 3(c) of Ref. [27] (left panel) reveals a nearly quantitative agreement, at least in the sta-tistical sense. The nearly triangular and symmetrical fundamental diagram (not shown) is consistent withthe observations as well. Remarkably, the simulation is well in the subcritical regime ( ǫ = − .
2) which ismainly caused by the low speeds. Nevertheless, the acceleration noise alone ( Q = . / s ) leads to fully . Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14 S pa c e gap [ m ] Time [s] Bicycle 2Bicycle 4Bicycle 6 S pa c e gap [ m ] Time [s] Bicycle 2Bicycle 4Bicycle 6 deterministic IDM
IDM with noise
Fig. 10. Top panel: gap time series of the simulation of Fig. 9; bottom panel: time series for the deterministic simulation with zeroacceleration noise, Q =
0. The remaining parameters are that of Figure 9. developed stop-and-go waves which can also be seen by the gap time series of selected bikers (Fig. 10, toppanel). To validate this, we also run simulations without noise and unchanged parameters, otherwise (bottompanel). We observe that the initial transients quickly dissipate which is consistent with string stability.
6. Conclusion
In this contribution, we have proposed a minimal general model for simultaneously considering threepossible mechanisms to tra ffi c flow oscillations: string instability, external white acceleration noise, andindi ff erence regions implemented by action points. Each of these mechanisms can be activated and con-trolled independently from the others by varying a single model parameter per mechanism, ǫ , Q , and ∆ a max ,respectively. The model is based on existing deterministic time-continuous car-following models with afundamental diagram. However, the action-point mechanism introduces an indi ff erence region and de facto converts this model into one consistent with the three-phase theory of Kerner [9].By analytical means and numerical simulations, we have shown that white acceleration noise as well asthe action points leads to highly spatiotemporally correlated fluctuations of speeds and gaps that “anticipate”the tra ffi c waves produced by linear instabilities even well in the linearly stable region ( ǫ < ffi c oscillations. Additionally, at least marginal stability ( ǫ =
0) or a mild form oflinear instability ( ǫ slightly positive) is necessary. In contrast, for low-speed tra ffi c such as bicycle tra ffi c,acceleration noise (or action points) alone can lead to fully developed and realistic tra ffi c waves. M. Treiber, A. Kesting / ISTTT Procedia 00 (2017) 1–14
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