Physics of automated driving in framework of three-phase traffic theory

We have revealed physical features of automated driving in the framework of the three-phase traffic theory for which there is $\mathit{no}$ fixed time headway to the preceding vehicle. A comparison with the classical model approach to automated driving for which an automated driving vehicle tries to reach a fixed (desired or “optimal”) time headway to the preceding vehicle has been made. It turns out that automated driving in the framework of the three-phase traffic theory can exhibit the following advantages in comparison with the classical model of automated driving: (i) The absence of string instability. (ii) Considerably smaller speed disturbances at road bottlenecks. (iii) Automated driving vehicles based on the three-phase theory can decrease the probability of traffic breakdown at the bottleneck in mixed traffic flow consisting of human driving and automated driving vehicles; on the contrary, even a single automated driving vehicle based on the classical approach can provoke traffic breakdown at the bottleneck in mixed traffic flow.

Figure 1

Qualitative explanation of operating points of TPACC. Qualitative presentation of a part of 2D region for operating points of TPACC in the space gap–speed plane: At a given speed of TPACC vehicle there are the infinity of operating points of TPACC [69]. $G$ is a synchronization space gap, $g_{\mathrm{safe}}$ is a safe space gap.

Figure 2

Operating points of TPACC model (5, 6, 7) presented in the space gap–speed (a), flow-density (b), speed-flow (c), and speed-density (d) planes. Model parameters ${\tau }_{\mathrm{G}}=1.4$ s, ${\tau }_{\mathrm{p}}=1.3$ s, ${\tau }_{\mathrm{safe}}=1$ s, $v_{\mathrm{free}}=30\mathrm{m}$/s (108 km/h), vehicle length (including the mean space gap between vehicles stopped within a wide moving jam) $d=7.5$ m.

Figure 3

Simulations of string instability of the classical ACC model (1) (see Appendix pp2) (a) and string stability of TPACC model (5, 6, 7) (b) in traffic flow consisting of $100\%$ of automated driving vehicles on a single-lane road with on-ramp bottleneck located at $x_{\mathrm{on}}=10$ km (see Appendix pp3); speed in space and time. Simulation parameters of ACC (a) and TPACC (b) are identical ones: the on-ramp inflow rate $q_{\mathrm{on}}=320$ vehicles/h and the flow rate upstream of the bottleneck $q_{\mathrm{in}}=2002.6$ vehicles/h, ${\tau }_{\mathrm{d}}^{\left(\mathrm{ACC}\right)}={\tau }_{\mathrm{p}}=1.3$ s, ${\tau }_{\mathrm{G}}=1.4$ s, $K_1=0.3{\text{s}}^{-2}$, and $K_2=K_{\mathrm{\Delta }\mathrm{v}}=0.3{\text{s}}^{-1}$; $a_{\max }=b_{\max }=3\mathrm{m}/{\mathrm{s}}^2, v_{\mathrm{free}}=30\mathrm{m}$/s (108 km/h), vehicle length $d=7.5$ m. In accordance with the desired time headway ${\tau }_{\mathrm{d}}^{\left(\mathrm{ACC}\right)}=1.3$ s of the ACC vehicles, the sum flow rate $q_{\mathrm{sum}}=q_{\mathrm{in}}+q_{\mathrm{on}}=2322.6$ vehicles/h is related to time headway 1.3 s between vehicles in free flow.

Figure 4

Speed disturbances at on-ramp bottleneck in stable free flow with $100\%$ automated driving vehicles for classical ACC model (1) (see Appendix pp2) [(a)–(c)] and for TPACC model (5, 6, 7) [(d)–(g)]: (a), (d) Speed in space and time. (b), (e) Fragments of vehicle trajectories; (b) is related to (a) and (e) is related to (d). (c), (f), (g) Microscopic speeds along vehicle trajectories shown by, respectively, the same numbers as in (b), (e); (c) is related to (b); (f), (g) are related to (e); (f) and (g) show the same speed in different speed scales. Simulation parameters of ACC [(a)–(c)] and TPACC [(d)–(g)] are identical ones. $K_2=K_{\mathrm{\Delta }\mathrm{v}}=0.6{\text{s}}^{-1}$. In (b, e), the on-ramp merging region that is within road locations $x_{\mathrm{on}}\le x\le x_{\mathrm{on}}^{\left(\mathrm{e}\right)}$ (see Appendix pp3) is labeled by “on-ramp”. Other model parameters are the same as those in Fig. 3.

Figure 5

Effect of small share of automated driving vehicles in mixed traffic flow on the flow-rate function of the probability $P^{\left(\mathrm{B}\right)}\left(q_{\mathrm{sum}}\right)$ of traffic breakdown at on-ramp bottleneck on single-lane road, where $q_{\mathrm{sum}}=q_{\mathrm{in}}+q_{\mathrm{on}}$. $P^{\left(\mathrm{B}\right)}\left(q_{\mathrm{sum}}\right)$ is calculated through the change in the on-ramp inflow rate $q_{\mathrm{on}}$ at a given flow rate $q_{\mathrm{in}}=$ 2000 vehicles/h. Curve 1 is related to traffic flow without automated driving vehicles as well as to mixed traffic flow with $2\%$ of TPACC vehicles. Curves 2 and 3 are related to mixed traffic flow with $2\%$ of ACC vehicles, respectively, with ${\tau }_{\mathrm{d}}^{\left(\mathrm{ACC}\right)}=1.3$ s (curve 2) and 1.6 s (curve 3). Other model parameters for ACC vehicles and TPACC vehicles are, respectively, the same as those in Fig. 4. For calculation of $P^{\left(\mathrm{B}\right)}\left(q_{\mathrm{sum}}\right)$, at each given value $q_{\mathrm{sum}}$ different simulation realizations (runs) $N_{\mathrm{r}}=$ 40 during the same time interval for the observing of traffic flow $T_{\mathrm{ob}}=$ 30 min have been made. The different realizations have been performed at the same set of model parameters, however, at different values of the initial values of random function rand( ) in the traffic flow model (see Appendices pp1-s4 and pp1-s5). Then, $P^{\left(\mathrm{B}\right)}\left(q_{\mathrm{sum}}\right)=n_{\mathrm{r}}/N_{\mathrm{r}}$, where $n_{\mathrm{r}}$ is the number of realizations in which traffic breakdown has occurred during the time interval $T_{\mathrm{ob}}$ (a more detailed explanation of calculation of the function $P^{\left(\mathrm{B}\right)}\left(q_{\mathrm{sum}}\right)$ has been given in Ref. [35]).

Figure 6

Explanation of the effect of a single automated driving vehicle on the probability of traffic breakdown at on-ramp bottleneck in mixed traffic flow associated with Fig. 5; speed disturbances occurring at on-ramp bottleneck through a single TPACC vehicle [(a)–(c)] and a single ACC vehicle [(d)–(f)]: (a), (d) Speed in space and time. (b), (e) Fragments of vehicle trajectories. (c), (f) Microscopic speeds along vehicle trajectories shown by the same numbers in (b), (e), respectively. In (b), (c), (e), (f), vehicles 1 and 2 are human driving vehicles whereas vehicle 3 is TPACC vehicle in (b), (c) and ACC vehicle in (e), (f). Mixed traffic flow with $2\%$ of automated driving vehicles; $q_{\mathrm{in}}=2000$ vehicles/h, $q_{\mathrm{on}}=280$ vehicles/h; other model parameters for ACC vehicles and TPACC vehicles are, respectively, the same as those in Fig. 4. In (a), (d): F, free flow; S, synchronized flow.

Figure 7

Probability of traffic breakdown at on-ramp bottleneck as a function of the flow rate $q_{\mathrm{sum}}=q_{\mathrm{in}}+q_{\mathrm{on}}$ at a given on-ramp inflow rate $q_{\mathrm{on}}=$ 320 vehicles/h in mixed traffic flow with $20\%$ automated driving vehicles: Curve 1 is related to traffic flow without automated driving vehicles. Curves 2 and 3 are related to mixed traffic flow with TPACC vehicles (curve 2) and ACC vehicles (curve 3). Simulation parameters of ACC and TPACC are, respectively, the same as those in Fig. 4.

Figure 8

Traffic stream flow characteristics for free flow on single-lane road with on-ramp bottleneck in traffic without automated driving vehicles and in mixed traffic with $2\%$ of TPACC vehicles: (a) Flow-density relationship (fundamental diagram). (b) A part of speed-flow relationship for larger flow rates. (c), (d) Parts of speed-flow (c) and flow-density (d) relationships for larger flow rates versus the breakdown probability $P^{\left(\mathrm{B}\right)}\left(q_{\mathrm{sum}}\right)$; function $P^{\left(\mathrm{B}\right)}\left(q_{\mathrm{sum}}\right)$ is curve 1 from Fig. 5. To compare stream flow characteristics with simulations shown in Figs. 5 and 6, at $q_{\mathrm{sum}}\le 2000$ vehicles/h there is no on-ramp inflow ($q_{\mathrm{on}}=0$); at $q_{\mathrm{sum}}=q_{\mathrm{in}}+q_{\mathrm{on}}>2000$ vehicles/h, the increase in $q_{\mathrm{sum}}$ has been achieved through increase in $q_{\mathrm{on}}$ at constant $q_{\mathrm{in}}=2000$ vehicles/h. Traffic stream flow characteristics have been calculated as follows. At each given flow rate $q_{\mathrm{sum}}$ (black points on the characteristics), 5-min averaged data for the speed, density, and flow rate have been measured with the use of a virtual road detector installed at the end of the on-ramp merging region $x=10.3$ km. The data have been measured only during time interval within which free flow has been observed in a simulation realization. Then, as by the calculation of $P^{\left(\mathrm{B}\right)}\left(q_{\mathrm{sum}}\right)$ in Fig. 5, $N_{\mathrm{r}}=40$ different realizations have been simulated for each of the chosen flow rates $q_{\mathrm{sum}}$ (see caption to Fig. 5). This allows us to make a statistical analysis of the average speed and density in the traffic stream. Black points on the speed-flow and flow-density relationships are related to the average values of the speed and density derived from this statistical analysis. Other model parameters are the same as those in Fig. 4. Calculated values: $q_{\mathrm{th},\mathrm{TPACC}}^{\left(\mathrm{B}\right)}=q_{\mathrm{th}}^{\left(\mathrm{B}\right)}=2290$ and $C_{\max ,\mathrm{TPACC}}=C_{\max }=2360$ vehicles/h.

Figure 9

Comparison of traffic stream flow characteristics for free flow on single-lane road with on-ramp bottleneck in mixed traffic with $2\%$ of automated driving vehicles. Parts of flow-density (a) and speed-flow (b) relationships for larger flow rates. Solid curves “TPACC” are related to TPACC vehicles. Dashed curves “ACC” are related to classical ACC vehicles. Stream flow characteristics have been calculated as explained in caption to Fig. 8. Other model parameters are the same as those in Fig. 4. Calculated values: $q_{\mathrm{th},\mathrm{TPACC}}^{\left(\mathrm{B}\right)}=2290$ and $C_{\max ,\mathrm{TPACC}}=2360$ vehicles/h; $q_{\mathrm{th},\mathrm{ACC}}^{\left(\mathrm{B}\right)}=2265$ and $C_{\max ,\mathrm{ACC}}=2330$ vehicles/h.

Figure 10

Comparison of traffic stream flow characteristics for free flow on single-lane road with on-ramp bottleneck in mixed traffic with $20\%$ of automated driving vehicles. Parts of flow-density (a) and speed-flow (b) relationships for larger flow rates. Solid curve “TPACC” is related to TPACC vehicles. Dashed curve “ACC” is related to classical ACC vehicles. Stream flow characteristics have been calculated as explained in caption to Fig. 8. Other model parameters are the same as those in Fig. 4. Calculated values: $q_{\mathrm{th},\mathrm{TPACC}}^{\left(\mathrm{B}\right)}=2308$ and $C_{\max ,\mathrm{TPACC}}=2371$ vehicles/h; $q_{\mathrm{th},\mathrm{ACC}}^{\left(\mathrm{B}\right)}=2050$ and $C_{\max ,\mathrm{ACC}}=2147$ vehicles/h.

Figure 11

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 $v_{\mathrm{g}}$ 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 $v_{\mathrm{g}}$. F, free flow; S, synchronized flow.

Figure 12

Model of on-ramp bottleneck on single-lane road.