Autonomous Driving in the Framework of Three-Phase Traffic Theory

Appendices

Appendix A. Kerner–Klenov Microscopic Stochastic Traffic Flow Model

In this Appendix, we make explanations of the Kerner–Klenov stochastic microscopic three-phase model for human driving vehicles (Kerner and Klenov 2002, 2003, 2009) and model parameters used for simulations of mixed traffic flow presented in the main text.

Update Rules of Vehicle Motion

In a discrete model version of the Kerner-Klenov stochastic microscopic three-phase model used in all simulations presented in the main text, rather than the continuum space coordinate (Kerner and Klenov 2002), a discretized space coordinate with a small enough value of the discretization space interval δx is used (Kerner and Klenov 2009). Consequently, the vehicle speed and acceleration (deceleration) discretization intervals are δv = δx/τ and δa = δv/τ, respectively, where τ is time step. Because in the discrete model version discrete (and dimensionless) values of space coordinate, speed, and acceleration are used, which are measured, respectively, in values δx, δv, and δa, and time is measured in values of τ, value τ in all formulas is assumed below to be the dimensionless value τ = 1. In the discrete model version used for all simulations, the discretization cell δx = 0.01 m is used.

A choice of δx = 0.01 m made in the model determines the accuracy of vehicle speed calculations in comparison with the initial continuum in space stochastic model of (Kerner and Klenov 2002). We have found that the discrete model exhibits similar characteristics of phase transitions and resulting congested patterns at highway bottlenecks as those in the continuum model at δx that satisfies the conditions

$$ \delta x/{\tau}^2\ll b,\, a,\, {a}^{\left(\mathrm{a}\right)},\, {a}^{\left(\mathrm{b}\right)},\, {a}^{(0)}, $$

(15)

where model parameters for driver deceleration and acceleration b, a, a^(a), a^(b), and a⁽⁰⁾ will be explained below.

Update rules of vehicle motion in the discrete model for identical drivers and identical vehicles moving in a road lane are as follows (Kerner and Klenov 2009):

$$ {v}_{n+1}=\max \left(0,\min \left({v}_{\mathrm{free}},{\tilde{v}}_{n+1}+{\xi}_n,{v}_n+ a\tau, {v}_{\mathrm{s},n}\right)\right), $$

(16)

$$ {x}_{n+1}={x}_n+{v}_{n+1}\tau, $$

(17)

where the index n corresponds to the discrete time t_n = τn, n = 0, 1, ..., v_n is the vehicle speed at time step n, a is the maximum acceleration, and ${\tilde{v}}_n$ is the vehicle speed without speed fluctuations ξ_n:

$$ {\tilde{v}}_{n+1}=\min \left({v}_{\mathrm{free}},{v}_{\mathrm{s},n},{v}_{\mathrm{c},n}\right), $$

(18)

$$ {v}_{\mathrm{c},n}=\left\{\begin{array}{ll}{v}_n+{\Delta}_n& \mathrm{at}{g}_n\le {G}_n\\ {}{v}_n+{a}_n\tau & \mathrm{at}{g}_n>{G}_n,\end{array}\right. $$

(19)

$$ {\Delta}_n=\max \left(-{b}_n\tau, \, \min \left({a}_n\tau, {v}_{\mathrm{\ell},n}-{v}_n\right)\right), $$

(20)

$$ {g}_n={x}_{\mathrm{\ell},n}-{x}_n-d. $$

(21)

The subscript ℓ marks variables related to the preceding vehicle, v_s,n is a safe speed at time step n, v_free is the free flow speed in free flow, ξ_n describes speed fluctuations, g_n is a space gap between two vehicles following each other, and G_n is the synchronization space gap; all vehicles have the same length d. The vehicle length d includes the mean space gap between vehicles that are in a standstill within a wide moving jam. Values a_n ≥ 0 and b_n ≥ 0 in (19) and (20) restrict changes in speed per time step when the vehicle accelerates or adjusts the speed to that of the preceding vehicle.

Synchronization Space Gap and Hypothetical Steady States of Synchronized Flow

Equations (19) and (20) describe the adaptation of the vehicle speed to the speed of the preceding vehicle, i.e., the speed adaptation effect in synchronized flow. This vehicle speed adaptation takes place within the synchronization gap G_n: At

$$ {g}_n\le {G}_n $$

(22)

the vehicle tends to adjust its speed to the speed of the preceding vehicle. This means that the vehicle decelerates if v_n > v_ℓ,n and accelerates if v_n < v_ℓ,n.

In (19), the synchronization gap G_n depends on the vehicle speed v_n and on the speed of the preceding vehicle v_ℓ,n:

$$ {G}_n=G\left({v}_n,{v}_{\mathrm{\ell},n}\right), $$

(23)

$$ G\left(u,w\right)=\max \left(0,\left\lfloor k\tau u+{a}^{-1}u\left(u-w\right)\right\rfloor \right), $$

(24)

where k > 1 is constant; ⌊z⌋ denotes the integer part of z.

The speed adaptation effect within the synchronization distance is related to the hypothesis of the three-phase theory: Hypothetical steady states of synchronized flow cover a 2D region in the flow–density (Fig. 18a). Boundaries F, L, and U of this 2D region shown in Fig. 18a are, respectively, associated with the free flow speed in free flow, a synchronization space gap G, and a safe space gap g_safe. A speed function of the safe space gap g_safe(v) is found from the equation

Fig. 18

Steady speed states for the Kerner–Klenov traffic flow model in the flow–density (a, b) and in the space gap–speed planes (c). In (a, b), L and U are, respectively, lower and upper boundaries of 2D regions of steady states of synchronized flow. In (b), J is the line J whose slope is equal to the characteristic mean velocity v_g of the downstream front of a wide moving jam; in the flow–density plane, the line J represents the propagation of the downstream front of the wide moving jam with time-independent velocity v_g. F, free flow; S, synchronized flow

$$ v={v}_{\mathrm{s}}\left({g}_{\mathrm{s}\mathrm{afe}},v\right). $$

(25)

Respectively, as for the continuum model (see Sec. 16.3 of the book (Kerner 2004)), for the discrete model hypothetical steady states of synchronized flow cover a 2D region in the flow–density plane (Fig. 18a, b). However, because the speed v and space gap g are integers in the discrete model, the steady states do not form a continuum in the flow–density plane as they do in the continuum model. The inequalities

$$ v\le {v}_{\mathrm{free}},\, \, g\le G\left(v,v\right),\, \, g\ge {g}_{\mathrm{safe}}(v) $$

(26)

define a 2D region in the space gap–speed plane (Fig. 18c) in which the hypothetical steady states exist for the discrete model, when all model fluctuations are neglected.

In (26), we have taken into account that in the hypothetical steady states of synchronized flow vehicle speeds and space gaps are assumed to be time-independent, and the speed of each of the vehicles is equal to the speed of the associated preceding vehicle: v = v_ℓ. However, due to model fluctuations, steady states of synchronized flow are destroyed, i.e., they do not exist in simulations; this explains the term “hypothetical” steady states of synchronized flow. Therefore, rather than steady states, some nonhomogeneous in space and time traffic states occur. In other words, steady states are related to a hypothetical model fluctuation-less limit of homogeneous in space and time vehicle motion that does not realized in real simulations. Driver time delays are described through model fluctuations. Therefore, any application of the Kerner–Klenov stochastic microscopic three-phase traffic flow model without model fluctuations has no sense. In other words, for the description of real spatiotemporal traffic flow phenomena, model speed fluctuations incorporated in this model are needed.

Model Speed Fluctuations

In the model, random vehicle deceleration and acceleration are applied depending on whether the vehicle decelerates or accelerates or else maintains its speed:

$$ {\xi}_n=\left\{\begin{array}{ll}{\xi}_{\mathrm{a}}& \mathrm{if}\, {S}_{n+1}=1\\ {}-{\xi}_{\mathrm{b}}& \mathrm{if}\, {S}_{n+1}=-1\\ {}{\xi}^{(0)}& \mathrm{if}\, {S}_{n+1}=0.\end{array}\right. $$

(27)

State of vehicle motion S_{n + 1} in (27) is determined by formula

$$ {S}_{n+1}=\left\{\begin{array}{ll}-1& \mathrm{if}\ {\tilde{v}}_{n+1}<{v}_n\\ {}1& \mathrm{if}\ {\tilde{v}}_{n+1}>{v}_n\\ {}0& \mathrm{if}\ {\tilde{v}}_{n+1}={v}_n.\end{array}\right. $$

(28)

In (27), ξ_b, ξ⁽⁰⁾, and ξ_a are random sources for deceleration and acceleration that are as follows:

$$ {\xi}_{\mathrm{b}}={a}^{\left(\mathrm{b}\right)}\tau \Theta \left({p}_{\mathrm{b}}-r\right), $$

(29)

$$ {\xi}^{(0)}={a}^{(0)}\tau \left\{\begin{array}{ll}-1& \mathrm{if}\ r<{p}^{(0)}\\ {}1& \mathrm{if}\ {p}^{(0)}\le r<2{p}^{(0)}\\ {}0& \mathrm{otherwise},\end{array}\right.\, \, \mathrm{and}\ {v}_n>0 $$

(30)

$$ {\xi}_{\mathrm{a}}={a}^{\left(\mathrm{a}\right)}\tau \Theta \left({p}_{\mathrm{a}}-r\right), $$

(31)

p_b is the probability of random vehicle deceleration, p_a is the probability of random vehicle acceleration, p⁽⁰⁾ and a⁽⁰⁾ ≤ a are constants, r = rand(0, 1), Θ(z) = 0 at z < 0 and Θ(z) = 1 at z ≥ 0, and a^(a) and a^(b) are model parameters (see Table 1), which in some applications can be chosen as speed functions a^(a) = a^(a)(v_n) and a^(b) = a^(b)(v_n).

Table 1 Model parameters of vehicle motion in road lane used in simulations of the main text

Stochastic Time Delays of Acceleration and Deceleration

To simulate time delays either in vehicle acceleration or in vehicle deceleration, a_n and b_n in (20) are taken as the following stochastic functions

$$ {a}_n=a\Theta \left({P}_0-{r}_1\right), $$

(32)

$$ {b}_n=a\Theta \left({P}_1-{r}_1\right), $$

(33)

$$ {P}_0=\left\{\begin{array}{ll}{p}_0& \mathrm{if}\ {S}_n\ne 1\\ {}1& \mathrm{if}\ {S}_n=1,\end{array}\right. $$

(34)

$$ {P}_1=\left\{\begin{array}{cc}{p}_1& \mathrm{if}\ {S}_n\ne -1\\ {}{p}_2& \mathrm{if}\ {S}_n=-1,\end{array}\right. $$

(35)

r₁ = rand(0, 1), p₁ is constant, and p₀ = p₀(v_n) and p₂ = p₂(v_n) are speed functions (see Table 1).

Simulations of Slow-To-Start Rule

In the model, simulations of the well-known effect of the driver time delay in acceleration at the downstream front of synchronized flow or a wide moving jam known as a slow-to- start rule (Barlović et al. 1998; Takayasu and Takayasu 1993) are made as a collective effect through the use of Eqs. (19) and (20) and a random value of vehicle acceleration (32). Eq. (32) with P₀ = p₀ < 1 is applied only if the vehicle did not accelerate at the former time step (S_n ≠ 1); in the latter case, a vehicle accelerates with some probability p₀ that depends on the speed v_n; otherwise P₀ = 1 (see formula (34)).

The mean time delay in vehicle acceleration is equal to

$$ {\tau}_{\mathrm{de}1}^{\left(\mathrm{acc}\right)}\left({v}_n\right)=\frac{\tau }{p_0\left({v}_n\right)}. $$

(36)

From formula (36), it follows that the mean time delay in vehicle acceleration from a standstill within a wide moving jam (i.e., when in formula (36) the speed v_n = 0) is equal to

$$ {\tau}_{\mathrm{de}1}^{\left(\mathrm{acc}\right)}(0)=\frac{\tau }{p_0(0)}. $$

(37)

The mean time delay in vehicle acceleration from a standstill within a wide moving jam determines the parameters of the line J in the flow–density plane (Fig. 18b).

Probability p₀(v_n) in (34) is chosen to be an increasing speed function (see Table 1). Because the speed within synchronized flow is larger than zero, the mean time delay in vehicle acceleration at the downstream front of synchronized flow that we denote by

$$ {\tau}_{\mathrm{de}1,\mathrm{syn}}^{\left(\mathrm{acc}\right)}={\tau}_{\mathrm{de}1}^{\left(\mathrm{acc}\right)}\left({v}_n\right),{v}_n>0 $$

(38)

is shorter than the mean time delay in vehicle acceleration at the downstream front of the wide moving jam ${\tau}_{\mathrm{de}1}^{\left(\mathrm{acc}\right)}(0)$:

$$ {\tau}_{\mathrm{de}1,\mathrm{syn}}^{\left(\mathrm{acc}\right)}<{\tau}_{\mathrm{de}1}^{\left(\mathrm{acc}\right)}(0). $$

(39)

Safe Speed

In the model, the safe speed v_s,n in (16) is chosen in the form

$$ {v}_{\mathrm{s},n}=\min \left({v}_n^{\left(\mathrm{safe}\right)},{g}_n/\tau +{v}_{\mathrm{\ell}}^{\left(\mathrm{a}\right)}\right), $$

(40)

${v}_{\mathrm{\ell}}^{\left(\mathrm{a}\right)}$ is an “anticipation” speed of the preceding vehicle that will be considered below, and the function

$$ {v}_n^{\left(\mathrm{safe}\right)}=\left\lfloor {v}^{\left(\mathrm{safe}\right)}\left({g}_n,{v}_{\mathrm{\ell},n}\right)\right\rfloor $$

(41)

in (40) is related to the safe speed v^(safe)(g_n, v_ℓ,n) in the model by Krauß et al. (1997), which is a solution of the Gipps’s Eq. [32]

$$ {v}^{\left(\mathrm{safe}\right)}\tau +{X}_{\mathrm{d}}\left({v}^{\left(\mathrm{safe}\right)}\right)={g}_n+{X}_{\mathrm{d}}\left({v}_{\mathrm{\ell},n}\right), $$

(42)

where X_d(u) is the braking distance that should be passed by the vehicle moving first with the speed u before the vehicle can come to a stop.

The condition (42) enables us to find the safe speed v^(safe) as a function of the space gap g_n and speed v_ℓ,n provided X_d(u) is a known function. In the case when the vehicle brakes with a constant deceleration b, the change in the vehicle speed for each time step is −bτ except the last time step before the vehicle comes to a stop. At the last time step, the vehicle decreases its speed at the value bτβ, where β is a fractional part of u/bτ. According to formula (17) for the displacement of the vehicle for one time step, the braking distance X_d(u) is (Krauß et al. 1997)

$$ {X}_{\mathrm{d}}(u)=\tau \left(u- b\tau +u-2 b\tau +\dots +\beta b\tau \right). $$

(43)

From (43), it follows (Krauß et al. 1997)

$$ {X}_{\mathrm{d}}(u)=b{\tau}^2\left(\alpha \beta +\frac{\alpha \left(\alpha -1\right)}{2}\right), $$

(44)

α = ⌊u/bτ⌋ is an integer part of u/bτ.

The safe speed v^(safe) as a solution of Eq. (42) at the distance X_d(u) given by (44) has been found by Krauß et al. (1997)

$$ {v}^{\left(\mathrm{safe}\right)}\left({g}_n,{v}_{\mathrm{\ell},n}\right)= b\tau \left({\alpha}_{\mathrm{safe}}+{\beta}_{\mathrm{safe}}\right), $$

(45)

where

$$ {\alpha}_{\mathrm{safe}}=\left\lfloor \sqrt{2\frac{X_{\mathrm{d}}\left({v}_{\mathrm{\ell},n}\right)+{g}_n}{b{\tau}^2}+\frac{1}{4}}-\frac{1}{2}\right\rfloor, $$

(46)

$$ {\beta}_{\mathrm{safe}}=\frac{X_{\mathrm{d}}\left({v}_{\mathrm{\ell},n}\right)+{g}_n}{\left({\alpha}_{\mathrm{safe}}+1\right)b{\tau}^2}-\frac{\alpha_{\mathrm{safe}}}{2}. $$

(47)

The safe speed in the model by Krauß et al. (1997) provides collisionless motion of vehicles if the time gap g_n/v_n between two vehicles is greater than or equal to the time step τ, i.e., if g_n ≥ v_nτ (Krauß 1998). In the model, it is assumed that in some cases, mainly due to lane changing or merging of vehicles onto the main road within the merging region of bottlenecks, the space gap g_n can become less than v_nτ. In these critical situations, the collision-less motion of vehicles in the model is a result of the second term in (40) in which some prediction $\left({v}_{\mathrm{\ell}}^{\left(\mathrm{a}\right)}\right)$ of the speed of the preceding vehicle at the next time step is used. The related “anticipation” speed ${v}_{\mathrm{\ell}}^{\left(\mathrm{a}\right)}$ at the next time step is given by formula

$$ {v}_{\mathrm{\ell}}^{\left(\mathrm{a}\right)}=\max \left(0,\min \left({v}_{\mathrm{\ell},n}^{\left(\mathrm{safe}\right)},{v}_{\mathrm{\ell},n},{g}_{\mathrm{\ell},n}/\tau \right)- a\tau \right), $$

(48)

where ${v}_{\mathrm{\ell},n}^{\left(\mathrm{safe}\right)}$ is the safe speed (41) and (45, 46, 47) for the preceding vehicle and g_ℓ,n is the space gap in front of the preceding vehicle. Simulations have shown that formulas (40), (41), and (45), (46), (47), (48) lead to collision-less vehicle motion over a wide range of parameters of the merging region of on-ramp bottlenecks (Appendix D).

In hypothetical steady states of traffic flow (Fig. 18a), the safe space gap g_safe is determined from condition v = v_s. In accordance with Eqs. (40, 41, and 42), at a given v in steady traffic states v = v_ℓ for the safe speed v_s, we get

$$ {v}_{\mathrm{s}}={g}_{\mathrm{s}\mathrm{afe}}/\tau, $$

(49)

and, therefore,

$$ {g}_{\mathrm{safe}}= v\tau . $$

(50)

Thus, for hypothetical steady states of traffic flow τ_safe = τ = 1 s.

In (Kerner 2017a) it has been shown that the Kerner–Klenov model (16)–(21), (23), (24), (27)–(35), (40), (41), and (45)–(48) is a Markov chain: At time step n + 1, values of model variables v_{n + 1}, x_{n + 1}, and S_{n + 1} are calculated based only on their values v_n, x_n, and S_n at step n.

It should be noted that after the Kerner–Klenov car-following model with indifference zones in car-following based on the three-phase theory (dashed region in Fig. 3) has been introduced (Kerner and Klenov 2002) (as mentioned, in this review entry a discrete version of this model of Ref. (Kerner and Klenov 2009) has been used), a number of different traffic flow models incorporating some of the hypotheses of the three-phase theory have been developed, and many results with the use of these models have been found (e.g., (Borsche et al. 2012; Davis 2004a, b, 2006a, b, 2007, 2008, 2014; Gao et al. 2007, 2009; Hausken and Rehborn 2015; He et al. 2010; Hoogendoorn et al. 2008; Jiang and Wu 2004, 2005, 2007a; Jiang et al. 2007, 2014, 2015, 2017; Jiang and Wu 2007b; Jin et al. 2010, 2015; Jin and Wang 2011; Kerner 2015b; Kerner and Klenov 2003, 2006, 2018; Kerner et al. 2002, 2006a, b, 2007, 2011, 2013a, b, 2014; Klenov 2010; Gasnikov et al. 2013; Kokubo et al. 2011; Knorr and Schreckenberg 2013; Lee et al. 2004; Lee and Kim 2011; Li et al. 2007; Neto et al. 2011; Qian et al. 2017; Pottmeier et al. 2007; Rehborn and Klenov 2009; Rehborn et al. 2011a, b, 2017; Rehborn and Koller 2014; Siebel and Mauser 2006; Tian et al. 2009, 2012, 2016a, b, c; Wang et al. 2007; Wu et al. 2008; Xiang et al. 2013; Yang et al. 2013, 2018; Zhang et al. 2012)). A review of some of these models that are related to the class of cellular automation models that can be found in this Encyclopedia has been recently done by Tian et al. (2018).

Appendix B. Discrete Version of Classical ACC Model

In simulations of the classical ACC model (4), as in the model of human driving vehicles (Appendix A), we use the discrete time t = nτ, where n = 0, 1, 2, ..; τ = 1 s is time step. Therefore, the space gap to the preceding vehicle is equal to g_n = x_ℓ,n − x_n − d, and the relative speed is given by Δv_n = v_ℓ,n − v_n, where x_n and v_n are coordinate and speed of the ACC-vehicle, x_ℓ,n and v_ℓ,n are coordinate and speed of the preceding vehicle, and d is the vehicle length that is assumed the same one for autonomous driving and human driving vehicles. Correspondingly, the classical model of the dynamics of ACC-vehicle (4) can be rewritten as follows (Kerner 2016, 2017a):

$$ {a}_n^{\left(\mathrm{ACC}\right)}={K}_1\left({g}_n-{v}_n{\tau}_{\mathrm{d}}^{\left(\mathrm{ACC}\right)}\right)+{K}_2\left({v}_{\mathrm{\ell},n}-{v}_n\right). $$

(51)

The ACC-vehicles move in accordance with Eq. (51) where, in addition, the following formulas are used:

$$ {v}_{\mathrm{c},n}^{(ACC)}={v}_n+\tau \max \left(-{b}_{\mathrm{max}},\min \left(\left\lfloor {a}_n^{\left(\mathrm{ACC}\right)}\right\rfloor, {a}_{\mathrm{max}}\right)\right), $$

(52)

$$ {v}_{n+1}=\max \left(0,\min \left({v}_{\mathrm{free}},{v}_{\mathrm{c},n}^{\left(\mathrm{ACC}\right)},{v}_{\mathrm{s},n}\right)\right), $$

(53)

⌊z⌋ denotes the integer part of z. Through the use of formula (52), acceleration and deceleration of the ACC-vehicles are limited by some maximum acceleration a_max and maximum deceleration b_max, respectively. Owing to the formula (53), the speed of the ACC-vehicle v_{n + 1} at time step n + 1 is limited by the maximum speed in free flow v_free and by the safe speed v_s,n to avoid collisions between vehicles. The maximum speed in free flow v_free and the safe speed v_s,n are chosen, respectively, the same as those in the microscopic model of human driving vehicles (Appendix A). It should be noted that the model of ACC-vehicle merging from the on-ramp onto the main road is similar to that for human driving vehicles (see Appendix D).

Simulations show that the use of the safe speed in formula (53) does not influence on the dynamics of the ACC-vehicles (4) in free flow outside the bottleneck. However, due to vehicle merging from the on-ramp onto the main road, time headway of the vehicle to the preceding vehicle can be considerably smaller than ${\tau}_{\mathrm{d}}^{\left(\mathrm{ACC}\right)}$. Therefore, formula (53) allows us to avoid collisions of the ACC-vehicle with the preceding vehicle in such dangerous situations. Moreover, very small values of time headway can occur in congested traffic; formula (53) prevents vehicle collisions in these cases also.

Appendix C. Discrete Version of TPACC Model

The model for human driving vehicles (Kerner and Klenov 2002, 2003, 2009) used in all simulations is discrete in time (Appendix A). Therefore, we simulate TPACC model (9) with discrete time t_n = τn, n = 0, 1, .... Respectively, TPACC model (9) should be rewritten as follows:

$$ {a}_n^{\left(\mathrm{TPACC}\right)}=\left\{\begin{array}{ll}{K}_{\Delta \mathrm{v}}\left({v}_{\mathrm{\ell},n}-{v}_n\right)& \mathrm{at}{g}_n\le {G}_n\\ {}{K}_1\left({g}_n-{v}_n{\tau}_{\mathrm{p}}\right)+{K}_2\left({v}_{\mathrm{\ell},n}-{v}_n\right)& \mathrm{at}{g}_n>{G}_n,\end{array}\right. $$

(54)

where G_n = v_nτ_G and τ_p < τ_G.

Safety Conditions

When g_n < g_safe,n, the TPACC-vehicle should move in accordance with some safety conditions to avoid collisions between vehicles (Fig. 3). A collision-free TPACC-vehicle motion is described as made in (Kerner 2016) for the classical model of ACC:

$$ {v}_{\mathrm{c},n}^{\left(\mathrm{TPACC}\right)}={v}_n+\tau \max \left(-{b}_{\mathrm{max}},\min \left(\left\lfloor {a}_n^{\left(\mathrm{TPACC}\right)}\right\rfloor, {a}_{\mathrm{max}}\right)\right), $$

(55)

$$ {v}_{n+1}=\max \left(0,\min \left({v}_{\mathrm{free}},{v}_{\mathrm{c},n}^{\left(\mathrm{TPACC}\right)},{v}_{\mathrm{s},n}\right)\right), $$

(56)

where the TPACC acceleration and deceleration are limited by a_max and b_max, respectively; the speed v_{n + 1} (56) at time step n + 1 is limited by the maximum speed v_free and by the safe speed v_s,n that have been chosen, respectively, the same as those in the model of human driving vehicles; ⌊z⌋ denotes the integer part of z.

“Indifference Zone” in Car-Following

In accordance with Eq. (56), condition ${v}_{\mathrm{c},n}^{\left(\mathrm{TPACC}\right)}\le {v}_{\mathrm{s},n}$ is equivalent to condition g_n ≥ g_safe,n. Under this condition, from the TPACC model (54)–(56), it follows that when time headway ${\tau}_n^{\left(\mathrm{net}\right)}={g}_n/{v}_n$ of the TPACC-vehicle to the preceding vehicle is within the range

$$ {\tau}_{\mathrm{safe},n}\le {\tau}_n^{\left(\mathrm{net}\right)}\le {\tau}_{\mathrm{G}}, $$

(57)

the acceleration (deceleration) of the TPACC-vehicle does not depend on time headway. In (57), τ_safe,n = g_safe,n/v_n is a safe time headway, and it is assumed that v_n > 0.

In accordance with (57), for the TPACC model (54)–(56), there is no fixed desired time headway to the preceding vehicle (Fig. 3). This means that in the TPACC model (54)–(56), there is “indifference zone” in the choice of time headway in car-following. This is in contrast with the classical ACC model (4) for which there is a fixed desired time headway in car-following. It should be noted that formula (57) for the indifference zone in time headway of TPACC is a discrete version of formula (13) for the indifference zone in time headway of TPACC discussed in the main text.

Operating Points

From formula for the safe speed v_s,n in (56) that is given in Appendix A, we find that the safe time headway τ_safe,n in (13) for the operating points of TPACC model (54), (55), and (56) is a constant value that is equal to τ_safe = τ = 1 s. In operating points of TPACC model (54), (55), and (56), a^(TPACC) = 0; respectively, v = v_ℓ, g_safe(v) ≤ g ≤ G(v), and v = v_free at g > G(v), where g_safe(v) = vτ and G(v) = vτ_G. The operating points of the TPACC model (54), (55), and (56) cover a 2D region in the space gap–speed plane (dashed 2D region in Fig. 19a). The inequalities v ≤ v_free, g ≤ G(v), and g ≥ g_safe(v) define a 2D region in the space gap–speed plane (Fig. 19a) in which the operating points exist for the discrete version of TPACC model (54), (55), and (56).

Fig. 19

Operating points of TPACC model (54)–(56) presented in the space gap–speed (a), flow–density (b), speed–flow (c), and speed–density (c) planes. Model parameters τ_G = 1.4 s, τ_p = 1.3 s, τ_safe = 1 s, v_free = 30 m/s (108 km/h), vehicle length (including the mean space gap between vehicles stopped within a wide moving jam) d = 7.5 m

It should be noted that the speed v and space gap g are integer in the discrete version of TPACC model (54), (55), and (56). Therefore, the operating points do not form a continuum in the space gap–speed plane as they do in the continuum version of TPACC model (9).

From Fig. 19, we can see that under conditions g_safe(v) ≤ g ≤ G(v) for each given speed v > 0 of TPACC, there is no fixed time headway to the preceding vehicle in operating points of the TPACC model (dashed 2D regions in Fig. 19), as explained in section “ACC in Framework of Three-Phase Theory (TPACC)” (Fig. 3).

The discretization interval of TPACC acceleration (deceleration) made in TPACC model (54), (55), and (56) is chosen to be an extremely small value that is equal to δa = 0.01 m/s² (see Appendix A). Therefore, the maximum value of a small round down of ${a}_n^{\left(\mathrm{TPACC}\right)}$ in Eqs. (55) and (56) through the application of the floor operator $\left\lfloor {a}_n^{\left(\mathrm{TPACC}\right)}\right\rfloor$ is less than 0.01 m/s², and it is, therefore, negligible. We have tested that no conclusions about physical features of TPACC dynamic behavior have been changed, when the continuum in space model of human driving vehicles of Ref. (Kerner and Klenov 2003) and, respectively, the continuum in space TPACC model version (without the floor operator) are used (the reason for the use of the discrete in space model for human driving vehicles of Ref. (Kerner and Klenov 2009) that leads to Eqs. (55) and (56) has been explained in (Kerner and Klenov 2009) as well as in Appendix A of the book (Kerner 2017a)).

Simulations show that the use of the safe speed in formula (56) does not influence on the dynamics of the TPACC-vehicles (54) in free flow outside the bottleneck. However, formula (56) allows us to avoid collisions of the TPACC-vehicle with the preceding vehicle in dangerous situations that can occur at the bottleneck as well as in congested traffic.

Appendix D. Model of On-Ramp Bottleneck

An on-ramp bottleneck consists of two parts (Fig. 20):

• (i) The merging region of length L_m where vehicle can merge onto the main road from the on-ramp lane.

• (ii) A part of the on-ramp lane of length L_r upstream of the merging region where vehicles move in accordance with the model of Appendix A. The maximal speed of vehicles is v_free = v_{free on}.

  Fig. 20

  

  Model of on-ramp bottleneck on single-lane road

At the beginning of the on-ramp lane ($x={x}_{\mathrm{on}}^{\left(\mathrm{b}\right)}$), the flow rate to the on-ramp q_on is given through boundary conditions that are the same as those that determine the flow rate q_in at the beginning of the main road (see Appendix E below).

Model of Vehicle Merging at Bottleneck

Vehicle Speed Adaptation Within Merging Region of Bottleneck

For the on-ramp bottleneck, when a vehicle is within the merging region of the bottleneck, the vehicle takes into account the space gaps to the preceding vehicles and their speeds both in the current and target lanes. Respectively, instead of formula (19), in (18) for the speed v_c,n, the following formula is used:

$$ {v}_{\mathrm{c},n}=\left\{\begin{array}{ll}{v}_n+{\Delta}_n^{+}& \mathrm{at}{g}_n^{+}\le G\left({v}_n,{\widehat{v}}_n^{+}\right)\\ {}{v}_n+{a}_n\tau & \mathrm{at}{g}_n^{+}>G\left({v}_n,{\widehat{v}}_n^{+}\right),\end{array}\right. $$

(58)

$$ {\Delta}_n^{+}=\, \max \left(-{b}_n\tau, \min \left({a}_n\tau, {\widehat{v}}_n^{+}-{v}_n\right)\right), $$

(59)

$$ {\widehat{v}}_n^{+}=\max \left(0,\min \left({v}_{\mathrm{free}},{v}_n^{+}+\Delta {v}_r^{(2)}\right)\right), $$

(60)

$\Delta {v}_r^{(2)}$ is constant (see Table 2).

Table 2 Parameters of model of on-ramp bottleneck used in simulations of the main text

Superscripts + and − in variables, parameters, and functions denote the preceding vehicle and the trailing vehicle in the “target” (neighboring) lane, respectively. The target lane is the lane into which the vehicle wants to change.

The safe speed v_s,n in (16) and (18) for the vehicle that is the closest one to the end of the merging region is chosen in the form

$$ {v}_{\mathrm{s},n}=\left\lfloor {v}^{\left(\mathrm{safe}\right)}\left({x}_{\mathrm{on}}^{\left(\mathrm{e}\right)}-{x}_n,0\right)\right\rfloor $$

(61)

(see Table 2).

Safety Conditions for Vehicle Merging

Vehicle merging at the bottleneck occurs, when safety conditions (∗) or safety conditions (∗∗) are satisfied.

Safety conditions (∗) are as follows:

$$ {\displaystyle \begin{array}{l}{g}_n^{+}>\min \left({\widehat{v}}_n\tau, G\left({\widehat{v}}_n,{v}_n^{+}\right)\right),\\ {}{g}_n^{-}>\min \left({v}_n^{-}\tau, G\left({v}_n^{-},{\widehat{v}}_n\right)\right),\end{array}} $$

(62)

$$ {\widehat{v}}_n=\min \left({v}_n^{+},{v}_n+\Delta {v}_r^{(1)}\right), $$

(63)

${\Delta v}_r^{(1)}>0$ is constant (see Fig. 20 and Table 2).

Safety conditions (∗∗) are as follows:

$$ {x}_n^{+}-{x}_n^{-}-d>{g}_{\mathrm{target}}^{\left(\min \right)}, $$

(64)

where

$$ {g}_{\mathrm{target}}^{\left(\min \right)}=\left\lfloor {\lambda}_{\mathrm{b}}{v}_n^{+}+d\right\rfloor, $$

(65)

λ_b is constant. In addition to conditions (64), the safety condition (∗∗) includes the condition that the vehicle should pass the midpoint

$$ {x}_n^{\left(\mathrm{m}\right)}=\left\lfloor \left({x}_n^{+}+{x}_n^{-}\right)/2\right\rfloor $$

(66)

between two neighboring vehicles in the target lane, i.e., conditions

$$ {\displaystyle \begin{array}{l}{x}_{n-1}<{x}_{n-1}^{\left(\mathrm{m}\right)}\ \mathrm{and}\ {x}_n\ge {x}_n^{\left(\mathrm{m}\right)}\\ {}\mathrm{or}\\ {}{x}_{n-1}\ge {x}_{n-1}^{\left(\mathrm{m}\right)}\ \mathrm{and}\ {x}_n<{x}_n^{\left(\mathrm{m}\right)}.\end{array}} $$

(67)

should also be satisfied.

Speed and Coordinate of Vehicle after Vehicle Merging

The vehicle speed after vehicle merging is equal to

$$ {v}_n={\widehat{v}}_n. $$

(68)

Under conditions (∗), the vehicle coordinates x_n remains the same. Under conditions (∗∗), the vehicle coordinates x_n is equal to

$$ {x}_n={x}_n^{\left(\mathrm{m}\right)}. $$

(69)

Merging of ACC-Vehicle or TPACC-Vehicle at On-Ramp Bottleneck

Here we consider rules of the merging of an ACC-vehicle at the on-ramp bottleneck presented in (Kerner 2017a) and used in simulations. The same rules have also been used in simulations of the merging of a TPACC-vehicle from the on-ramp lane onto the main road at the bottleneck.

In the on-ramp lane, an ACC-vehicle or a TPACC-vehicle moves in accordance with the ACC model (51), (52), and (53) or in accordance with the TPACC model (54), (55), and (56), respectively. The maximal speed of the ACC-vehicle or the TPACC-vehicle in the on-ramp lane is v_free = v_{free on}. The safe speed v_s,n in (53) for the ACC-vehicle and in (56) for the TPACC-vehicle that is the closest one to the end of the merging region is the same as that for human driving vehicles that is given by formula (61).

An ACC-vehicle or a TPACC-vehicle merges from the on-ramp lane onto the main road, when some safety conditions (∗) or safety conditions (∗∗) are satisfied for the ACC- vehicle or the TPACC-vehicle. Safety conditions (∗) for ACC-vehicles and TPACC-vehicles are as follows:

$$ {g}_n^{+}>{\widehat{v}}_n\tau, {g}_n^{-}>{v}_n^{-}\tau, $$

(70)

where ${\widehat{v}}_n$ is given by formula (63). Safety conditions (∗∗) are given by formulas (64), (65), (66), and (67), i.e., they are the same as those for human driving vehicles. Respectively, as for human driving vehicles, the ACC-vehicle speed and its coordinate or the TPACC-vehicle speed and its coordinate after the ACC-vehicle or the TPACC-vehicle has merged from the on-ramp onto the main road are determined by formulas (68) and (69).

A question can arise from the choice of the model time step τ = 1 s in Eqs. (54), (55), and (56) that have been used for numerical simulations of TPACC model (9): In TPACC model (54), (55), and (56), the time step τ = 1 s determines the safe space gap g_safe = vτ under hypothetical steady-state conditions in which all vehicles move at time-independent speed v. Contrarily to the TPACC model (54), (55), and (56), typical ACC controllers in vehicles that on the market have update time intervals τ of 100 ms or less. Indeed, there may be some very dangerous traffic situations in real traffic in which the safe time headway for an ACC-vehicle is quickly reached, and, therefore, the ACC-vehicle must decelerate strongly already after a time interval that is a much shorter than 1 s to avoid the collision with the preceding vehicle. Therefore, to avoid collisions, real ACC controllers must have update time intervals τ of 100 ms or less. However, at model time step τ = 1 s through the choice in the mathematical formulation of the safe speed in TPACC model (54), (55), and (56) and in the model of human driving vehicles, collision-less traffic flow is guaranteed in any dangerous traffic situation that can occur in simulations of traffic flow.

In other words, to disclose the physics of TPACC it is sufficient the choice of the update time τ = 1 s in simulations of TPACC behavior. To explain this, we should note that Eqs. (55) and (56) effect on TPACC dynamics only under condition g_n < g_safe,n, i.e., when the space gap becomes smaller than the safe one. This is because the physics of TPACC disclosed in this entry is solely determined by Eq. (54): Under condition g_n ≥ g_safe,n, Eqs. (55) and (56) do not change TPACC acceleration (deceleration) calculated through Eq. (54).

Appendix E. Boundary Conditions for Mixed Traffic Flow

Open boundary conditions are applied. At the beginning of the road, new vehicles are generated one after another in each of the lanes of the road at time instants

$$ {t}^{(m)}=\tau \left\lceil m{\tau}_{\mathrm{in}}/\tau \right\rceil, m=1,2,.\dots $$

(71)

In (71), τ_in = 1/q_in, q_in is the flow rate in the incoming boundary flow per lane, and ⌈z⌉ denotes the nearest integer greater than or equal to z. Human driving and autonomous driving vehicles are randomly generated at the beginning of the road with the rates related to chosen values of the flow rate q_in and the percentage γ of the autonomous driving vehicles in mixed traffic flow: (i) At time instant t^(m) (71), a new vehicle is human driving vehicle, when condition

$$ {r}_2\ge \gamma /100 $$

(72)

is satisfied where r₂ = rand(0, 1). (ii) At time instant t^(m) (71), a new vehicle is autonomous driving vehicle, when the opposite condition

$$ {r}_2<\gamma /100 $$

(73)

is satisfied. The same procedure of random generation of human driving and autonomous driving vehicles is applied at the beginning of the on-ramp lane. In this case, however, in (71) the flow rate q_in should be replaced by the on-ramp inflow rate q_on.

A new vehicle appears on the road only if the distance from the beginning of the road (x = x_b) to the position x = x_ℓ,n of the farthest upstream vehicle on the road is not smaller than the safe distance v_ℓ,nτ + d:

$$ {x}_{\mathrm{\ell},n}-{x}_{\mathrm{b}}\ge {v}_{\mathrm{\ell},n}\tau +d, $$

(74)

where n = t^(m)/τ. Otherwise, condition (74) is checked at time (n + 1)τ that is the next one to time t^(m) (71) and so on, until the condition (74) is satisfied. Then the next vehicle appears on the road. After this occurs, the number m in (71) is increased by 1.

The speed v_n and coordinate x_n of the new vehicle are

$$ {\displaystyle \begin{array}{l}{v}_n={v}_{\mathrm{\ell},n},\\ {}{x}_n=\max \left({x}_{\mathrm{b}},{x}_{\mathrm{\ell},n}-\left\lfloor {v}_n{\tau}_{\mathrm{in}}\right\rfloor \right).\end{array}} $$

(75)

The flow rate q_in is chosen to have the value v_freeτ_in integer. In the initial state (n = 0), all vehicles have the free flow speed v_n = v_free, and they are positioned at space intervals x_ℓ,n − x_n = v_freeτ_in.

After a vehicle has reached the end of the road, it is removed. Before this occurs, the farthest downstream vehicle maintains its speed and lane. For the vehicle following the farthest downstream one, the “anticipation” speed ${v}_{\mathrm{\ell}}^{\left(\mathrm{a}\right)}$ in (40) is equal to the speed of the farthest downstream vehicle.