Virtual Rings on Highways: Traffic Control by Connected Automated Vehicles

Consider the scenario in Fig. 1, in which vehicles follow each other on a single lane of a straight road. We number the vehicles with indices increasing in the direction of motion. We denote the set of all vehicle indices by $\mathcal{N}$ that comprises of the indices ${\mathcal{N}=\{\mathcal{N}_{\rm HV},\mathcal{N}_{\rm CHV},\mathcal{N}_{\rm AV},\mathcal{N}_{\rm CAV}\}}$ representing four vehicle types defined previously. We distinguish a so-called ego vehicle of interest by index $0$. We denote the length of vehicle $n$ by $l_{n}$, the position of its rear bumper by $s_{n}$ and its speed by $v_{n}$, ${n\in\mathcal{N}}$.

We model the motion of vehicle $n$ by a delayed double integrator with saturation:

$\displaystyle\begin{split}\dot{s}_{n}(t)&=v_{n}(t)\,,\\ \dot{v}_{n}(t)&={\rm sat}(u_{n}(t-\tau_{n}))\,,\end{split}$

(1)

${\forall n\in\mathcal{N}}$, where $u_{n}$ is the desired acceleration of vehicle $n$, selected by the human driver for HVs and CHVs (${n\in\{\mathcal{N}_{\rm HV},\mathcal{N}_{\rm CHV}\}}$) and prescribed by the longitudinal controller of AVs and CAVs (${n\in\{\mathcal{N}_{\rm AV},\mathcal{N}_{\rm CAV}\}}$). We assume that each vehicle realizes the desired acceleration by the help of human action or low-level controllers, respectively, unless this acceleration is above the acceleration capability $a_{\rm max}$ or below the braking limit $-a_{\rm min}$ of the vehicle. This is captured by the saturation function:

$${\rm sat}(u)={\rm min}\{{\rm max}\{-a_{\rm min},u\},a_{\rm max}\}\,.$$

(2)

Furthermore, we incorporate the time delay $\tau_{n}$ into the model, that involves the response time of the vehicle, as well as the driver reaction time for HVs and CHVs, and feedback delays for AVs and CAVs. For simplicity, we assume identical delays for HVs and CHVs throughout this study with ${\tau_{n}=\tau}$, ${n\in\{\mathcal{N}_{\rm HV},\mathcal{N}_{\rm CHV}\}}$, and we distinguish the delay of the ego vehicle $0$ by the notation ${\sigma=\tau_{0}}$.

To capture human driver behavior, we use simple car-following models. Namely, our human driver model (HDM) assumes that vehicle $n$ responds to vehicle ${n+1}$ ahead considering the headway (range) ${h_{n}=s_{n+1}-s_{n}-l_{n}}$ and the speed difference (range rate) ${\dot{h}_{n}=v_{n+1}-v_{n}}$ between them:

$$u_{n}=f_{\rm H}(h_{n},\dot{h}_{n},v_{n})\,,$$

(3)

where the specific expression of $f_{\rm H}$ depends on the choice of the model. Examples for HDM include the optimal velocity model (OVM) Bando1998 and the intelligent driver model (IDM) Treiber2000 , which were shown to capture human driver behavior observed in experimental data Avedisov2018 ; Dollar2021 . For simplicity of exposition, throughout this paper we assume that human drivers are identical in their driving behaviors and $f_{\rm H}$ is the same for each ${n\in\{\mathcal{N}_{\rm HV},\mathcal{N}_{\rm CHV}\}}$.

Now we consider vehicle control for (connected) automated vehicles where AVs and CAVs influence the traffic behind them without explicitly responding to its state. Consider the scenario in Fig. 2(a)-(c). We assume that the ego vehicle (vehicle $0$) is automated (an AV in Fig. 2(a)-(b) and a CAV in Fig. 2(c)), and it is followed by $N$ human-driven vehicles (HVs) constituting the traffic influenced by vehicle control.

Our goal is to design a control input $u_{0}$ for the ego vehicle based on the traffic ahead and observe its effects on the behavior of the traffic behind. If traffic conditions are free-flowing and there are no vehicles ahead of the AV such as in Fig. 2(a), then the AV may use cruise control (CC) to track a reference speed $v_{\rm ref}(t)$ such that its speed $v_{0}(t)$ approaches $v_{\rm ref}(t)$:

$$u_{0}=f_{\text{CC}}(v_{0},\underbrace{v_{\rm ref}}_{\text{reference}})\,.$$

(6)

The specific form of $f_{\text{CC}}$ depends on the controller type. The reference speed $v_{\rm ref}(t)$ may be constant, or time-varying, or it may even depend on the position $s_{0}(t)$ — an example for the latter case is when vehicles optimize their set speed based on road elevation He2016 .

If the traffic is more dense, the AV may need to respond to the vehicle ahead; see Fig. 2(b). This can be achieved by sensing the position and speed of the vehicle ahead via on-board sensors (radar, lidar, camera or ultrasonics) and using adaptive cruise control (ACC) that takes into account the positions and speeds of the AV and the vehicle ahead:

$$u_{0}=f_{\text{ACC}}(s_{0},v_{0},\underbrace{s_{1},v_{1}}_{\text{vehicle ahead}})\,.$$

(7)

Finally, if the ego vehicle is equipped with a communication device, i.e., it is a CAV, then it may respond to connected human-driven vehicles (CHVs) farther ahead, as shown by Fig. 2(c). This leads to connected cruise control (CCC) that may potentially take into account the positions and speeds of $M$ vehicles ahead:

$$u_{0}=f_{\text{CCC}}(s_{0},v_{0},\underbrace{s_{1},v_{1},\ldots,s_{M},v_{M}}_{\text{multiple vehicles ahead}})\,.$$

(8)

If a vehicle ahead is not connected to or sensed by the CAV, its state can be omitted from $f_{\text{CCC}}$. An aggregated response to multiple vehicles ahead facilitates smooth driving by making the CAV more clairvoyant about upcoming changes in traffic conditions Ge2018 .

Now consider the scenario illustrated in Fig. 3(a)-(c), where the ego vehicle is a CAV and at least one vehicle behind it is connected, e.g., a CHV. Once the CAV syncs with the CHV, it can actively regulate the traffic behind, since it uses traffic state feedback when responding to the CHV. When there are no vehicles ahead of the CAV to respond to, such as in Fig. 3(a), the cruise control (6) can be extended to traffic control (TC) in response to the positions and speeds of the connected vehicles behind the ego CAV:

$$u_{0}=f_{\text{TC}}(\underbrace{s_{-N},v_{-N},\ldots,s_{-1},v_{-1}}_{\text{traffic behind}},s_{0},v_{0},\underbrace{v_{\rm ref}}_{\text{reference}})\,.$$

(13)

The quantities ${s_{-N},v_{-N},\ldots,s_{-1},v_{-1}}$ represent the state of the traffic behind the CAV. We assume that not all vehicles behind the ego CAV are connected, thus, only some of these states (corresponding to connected vehicles) may be available to the CAV and affect $u_{0}$. Accordingly, $-N$ denotes the index of the farthest connected vehicle behind communicating with the ego CAV. This scenario is ideal for traffic control in the sense that the CAV does not have to respond to the traffic ahead in order to minimize collision risks. Such large control freedom enables the CAV to stabilize long vehicle chains behind it.

However, when the CAV is traveling in denser traffic environments like the one in Fig. 3(b), it has to respond to the vehicle ahead and has less freedom to regulate the traffic behind it. In such a setup, the adaptive cruise control (7) can be extended with feedback from the traffic behind to adaptive traffic control (ATC):

$$u_{0}=f_{\text{ATC}}(\underbrace{s_{-N},v_{-N},\ldots,s_{-1},v_{-1}}_{\text{traffic behind}},s_{0},v_{0},\underbrace{s_{1},v_{1}}_{\text{vehicle ahead}})\,.$$

(14)

Finally, if vehicles ahead of the CAV are also connected, as illustrated by Fig. 3(c), we may establish connected traffic control (CTC) that responds to multiple vehicles ahead while also taking into account its effect on the traffic behind:

$$u_{0}=f_{\text{CTC}}(\underbrace{s_{-N},v_{-N},\ldots,s_{-1},v_{-1}}_{\text{traffic behind}},s_{0},v_{0},\underbrace{s_{1},v_{1},\ldots,s_{M},v_{M}}_{\text{multiple vehicles ahead}})\,.$$

(15)

First, we study a baseline scenario without automation. We simulate (1, 3) for human driver model (4) in Example 1 with typical human driver parameters selected from Avedisov2018 . The simulation results are shown in Fig. 1(a). In the simulation, a lead vehicle (black) sequentially decelerates, accelerates and cruises at constant speed, followed by 11 HVs (gray). The following HVs tend to reduce their speed more the farther they are from the lead vehicle as speed perturbations amplify along the vehicle chain. Consequently, the tail vehicle brakes noticeably more than the lead (purple highlight). This is called string unstable behavior and often leads to the formation of stop-and-go traffic jams for large enough number of string unstable drivers.

String instability can be mitigated by vehicle control using AVs or CAVs. To demonstrate how vehicle control influences traffic, we simulate (1, 3, 7) for a heterogeneous chain of HVs and AVs using ACC controller (10) in Example 2. The simulation results shown by Fig. 1(b) indicate that the AVs can successfully mitigate the onset of a traffic congestion: the tail vehicle brakes as much as the lead vehicle and not more (purple highlight). We remark that congestions are often caused by the reaction time of human drivers: a delayed reaction forces drivers to overreact and brake or accelerate more. As such, automated vehicles must be designed carefully to reduce their response delays, and their controllers must be tuned by considering the presence of delays. There exist strategies for automated vehicles that specifically aim to eliminate the effect of delays BekiarisLiberis2018 ; Molnar2018its ; Qi2021 ; Wang2018a ; Zhang2020 . Still, even with well-designed automated vehicles it may require a relatively large penetration of AVs in the traffic flow to mitigate congestions if human drivers are string unstable (the penetration is 25% in our example, i.e., every fourth vehicle is automated). Connectivity and a CCC strategy could lead to further benefits: some level of connectivity may reduce the required penetration of automation to stabilize traffic Avedisov2021 . Yet, with vehicle control it is often hard to achieve stability for CAV penetrations around or below 10%.

The benefits of connectivity are further exploited by traffic control strategies. Simulation of (1, 3, 14) with ATC controller (18) in Example 3 is shown in Fig. 1(c). Here a single CAV responds to the vehicle ahead and to a single CHV ${N=10}$ vehicles behind. With ATC strategy, the ego CAV is able to stabilize the traffic flow, and each vehicle including the CHV brakes less than the lead vehicle. This is achieved with about 8% penetration of automation and 17% penetration of connectivity (one automated and two connected vehicles out of 12 vehicles). The required penetration of automation is reduced (compared to when traffic is influenced by vehicle control) thanks to the extra connectivity — which is a significant benefit since automation implies more cost and requirements than communication. Moreover, the CHV has incentive to be connected to the CAV as it can slow down less.

The performance with and without connectivity is further explored in Fig. 5, where the influence of a single ACC vehicle on traffic is compared to traffic control by ATC. These scenarios are illustrated in Fig. 2(b) and Fig. 3(b); their only difference is that the vehicle $-N$ is connected in the ATC setup and the ego CAV responds to it. It can be seen that ATC results in less speed reduction and smoother driving for the whole vehicle chain. The price is that the headway of the CAV executing ATC fluctuates more than that for ACC. However, it does not compromise safety, the vehicle is still far from collision. In the meantime, smooth driving leads to less energy consumption, which is discussed next.

We evaluate the energy consumption for each vehicle by the following measure He2020 :

$$w_{n}(t)=\int_{t_{0}}^{t}v_{n}(\theta)g\big{(}\dot{v}_{n}(\theta)+p(v_{n}(\theta))\big{)}{\rm d}\theta\,,$$

(21)

that is, the energy over unit mass consumed during the time interval $[t_{0},t]$. It is an integral of the power over unit mass obtained from the product of speed and commanded acceleration. The latter involves the vehicle acceleration and the resistance terms represented by $p$ that vehicles need to overcome. These can be modeled for example by

$$p(v)=a_{\rm r}+c_{\rm r}v^{2}\,,$$

(22)

where $a_{\rm r}$ accounts for rolling resistance, while $c_{\rm r}v^{2}$ represents air drag. Furthermore, function $g$ in (21) describes how the commanded acceleration is related to energy consumption. For example, if we take

$$g(x)=\max\{0,x\}\,,$$

(23)

then we assume energy consumption during acceleration only and no energy consumption or regeneration during braking.

The energy measure (21, 22, 23) was evaluated for the simulations in Fig. 5; see the bottom of the plot. The energy consumption stays constant during braking, increases linearly during constant speed cruising, and increases at a higher rate during acceleration. Therefore, it is critical for the vehicles to drive smoothly and avoid decelerations after which they need to accelerate to recover the speed.

To highlight the difference between the total energy consumption of ACC and ATC, in Fig. 6 we plot the final value ${w_{0}(t_{\rm f})}$ of the energy consumed by the CAV over the simulation interval $[t_{0},t_{\rm f}]$. Specifically, the simulations were repeated and the energy consumption was calculated for a grid of $\beta$ and $\beta_{\rm B}$ control gains representing various control designs. The corresponding energy consumption levels are shown by the colored contours. It can be seen that increasing $\beta_{\rm B}$ leads to a lower energy level. One should, however, always keep safety in mind and choose $\beta_{\rm B}$ small enough to maintain a safe headway similarly to Fig. 5.

The gains corresponding to Fig. 5 are indicated by point A for ACC (${\beta_{\rm B}=0}$) and point B for ATC (${\beta_{\rm B}\neq 0}$) in Fig. 6. These two cases are compared on the right of Fig. 6 for various values of $N$. It can be seen that ATC consistently saves around 2-3% energy for the ego CAV compared to ACC if ${N\geq 5}$. The energy consumption $w_{-N}(t_{\rm f})$ of the CHV is also indicated. By deciding to stay connected, the CHV saves significant energy, around 6-8% if ${N\geq 14}$. The trend is that the farther the CHV is from the CAV (i.e., the larger $N$ is), the more energy it saves — this is a consequence of a longer string unstable chain of HVs being more sensitive to the CAV’s motion.

We remark that the energy contours in Fig. 6 are specific to the velocity profile of the lead vehicle and to the choice (21) of the energy measure. Besides, extensive numerical simulations were needed to obtain these contours and to identify which controller parameters are energy efficient. Finding energy-optimal parameters through theoretical analysis is a challenging task He2020 . On the other hand, energy consumption is associated with the smoothness of driving, thus it is related to the stability of the traffic flow. String stable chains of vehicles attenuate speed fluctuations, drive smoother, and hence tend to consume less energy. Stability analysis is more straightforward than finding energy optima. Hence, below we analyze the dynamics and stability of traffic to drive the controller parameter selection.

To analyze stability, first we consider traffic flows in equilibrium (often referred to as uniform flow) where all vehicles drive at the same constant speed. The equilibrium speed is dictated by how fast the lead vehicle travels — in practice it can be considered as the average speed of the lead vehicle — and the following vehicles adjust their speed to the lead. Afterwards, we consider deviations from this equilibrium to quantify how much speed perturbations amplify. This will also be used to carry out stability analysis in the next section. We conduct our analysis in the frequency domain by linearization and by constructing transfer functions that allow us to characterize the response of individual vehicles and the overall traffic behavior.

We start with formalizing the equilibrium solution as

$$v_{n}(t)\equiv v^{*}\,,\quad s_{n}(t)=s_{n}^{*}(t)=v^{*}t+s_{n}(0)\,,$$

(24)

${n\in\mathcal{N}}$, which is associated with constant speed $v^{*}$ that is identical for each vehicle, and constant headway ${h_{n}^{*}=s_{n+1}^{*}-s_{n}^{*}-l_{n}}$ that may be different for each vehicle. This equilibrium can be found by substituting (24) into (1, 3) and (7) for ACC or (14) for ATC, and solving for $v^{*}$ and $h_{n}^{*}$.

We consider perturbations around the equilibrium in the form

$$v_{n}(t)=v^{*}+\tilde{v}_{n}(t)\,,\quad s_{n}(t)=s_{n}^{*}(t)+\tilde{s}_{n}(t)\,,$$

(25)

and collect position and speed perturbations into the state vector $\mathbf{x}$ given by

$$\mathbf{x}_{n}(t)=\begin{bmatrix}\tilde{s}_{n}(t)\\ \tilde{v}_{n}(t)\end{bmatrix}=\begin{bmatrix}\tilde{v}_{n}(0)+\int_{0}^{t}\tilde{v}_{n}(\theta){\rm d}\theta\\ \tilde{v}_{n}(t)\end{bmatrix},$$

(26)

from which the speed fluctuations can be recovered by

$$\tilde{v}_{n}(t)=\mathbf{c}\mathbf{x}_{n}(t),\quad\mathbf{c}=\begin{bmatrix}0&1\end{bmatrix}.$$

(27)

Assuming the accelerations of vehicles do not saturate, the linearized dynamics of HVs and CHVs can be derived from (1, 3) in the form

$$\dot{\mathbf{x}}_{n}(t)=\mathbf{a}\mathbf{x}_{n}(t)+\mathbf{a}_{\rm H}\mathbf{x}_{n}(t-\tau)+\mathbf{b}_{\rm H}\mathbf{x}_{n+1}(t-\tau)\,,$$

(28)

${n\in\{\mathcal{N}_{\rm HV},\mathcal{N}_{\rm CHV}\}}$. That is, it contains the delay-free state of vehicle $n$ and the delayed states of vehicles $n$ and $n+1$ with coefficient matrices

$$\mathbf{a}=\begin{bmatrix}0&1\\ 0&0\end{bmatrix},\quad\mathbf{a}_{\rm H}=\begin{bmatrix}0&0\\ -\left.\dfrac{\partial f_{\rm H}}{\partial h_{n}}\right|_{*}&\left.\dfrac{\partial f_{\rm H}}{\partial v_{n}}\right|_{*}-\left.\dfrac{\partial f_{\rm H}}{\partial\dot{h}_{n}}\right|_{*}\end{bmatrix},\quad\mathbf{b}_{\rm H}=\begin{bmatrix}0&0\\ \left.\dfrac{\partial f_{\rm H}}{\partial h_{n}}\right|_{*}&\left.\dfrac{\partial f_{\rm H}}{\partial\dot{h}_{n}}\right|_{*}\end{bmatrix},$$

(29)

where star indicates that partial derivatives are evaluated at the equilibrium. For AVs performing ACC the linearized dynamics become

$$\dot{\mathbf{x}}_{0}(t)=\mathbf{a}\mathbf{x}_{0}(t)+\mathbf{a}_{\rm F}\mathbf{x}_{0}(t-\sigma)+\mathbf{b}_{\rm F}\mathbf{x}_{1}(t-\sigma)\,,$$

(30)

with

$$\mathbf{a}=\begin{bmatrix}0&1\\ 0&0\end{bmatrix},\quad\mathbf{a}_{\rm F}=\begin{bmatrix}0&0\\ \left.\dfrac{\partial f_{\text{ACC}}}{\partial s_{0}}\right|_{*}&\left.\dfrac{\partial f_{\text{ACC}}}{\partial v_{0}}\right|_{*}\end{bmatrix},\quad\mathbf{b}_{\rm F}=\begin{bmatrix}0&0\\ \left.\dfrac{\partial f_{\text{ACC}}}{\partial s_{1}}\right|_{*}&\left.\dfrac{\partial f_{\text{ACC}}}{\partial v_{1}}\right|_{*}\end{bmatrix}.$$

(31)

whereas CAVs executing ATC are described by

$$\dot{\mathbf{x}}_{0}(t)=\mathbf{a}\mathbf{x}_{0}(t)+\mathbf{a}_{\rm FB}\mathbf{x}_{0}(t-\sigma)+\mathbf{b}_{\rm F}\mathbf{x}_{1}(t-\sigma)+\mathbf{b}_{\rm B}\mathbf{x}_{-N}(t-\sigma)\,,$$

(32)

with

$\displaystyle\begin{split}&\mathbf{a}=\begin{bmatrix}0&1\\ 0&0\end{bmatrix},\quad\mathbf{a}_{\rm FB}=\begin{bmatrix}0&0\\ \left.\dfrac{\partial f_{\text{ATC}}}{\partial s_{0}}\right|_{*}&\left.\dfrac{\partial f_{\text{ATC}}}{\partial v_{0}}\right|_{*}\end{bmatrix},\\ &\mathbf{b}_{\rm F}=\begin{bmatrix}0&0\\ \left.\dfrac{\partial f_{\text{ATC}}}{\partial s_{1}}\right|_{*}&\left.\dfrac{\partial f_{\text{ATC}}}{\partial v_{1}}\right|_{*}\end{bmatrix},\quad\mathbf{b}_{\rm B}=\begin{bmatrix}0&0\\ \left.\dfrac{\partial f_{\text{ATC}}}{\partial s_{-N}}\right|_{*}&\left.\dfrac{\partial f_{\text{ATC}}}{\partial v_{-N}}\right|_{*}\end{bmatrix}.\end{split}$

(33)

In summary, vehicle control influencing traffic is described by (28, 30) at the linear level, while traffic control is characterized by (28, 32). For the ring configuration, one also has periodicity given by (20) that yields

$$\mathbf{x}_{-N}(t)=\mathbf{x}_{1}(t)\,.$$

(34)

We analyze the linearized dynamics in Laplace domain (assuming zero initial perturbations). Our ultimate goal is to analyze the stability of the system, in particular, whether speed fluctuations amplify or decay as they propagate along the chain of vehicles. This amplification property can be well represented by means of transfer functions describing the response of vehicles to the neighboring traffic.

First, we formulate the so-called link transfer functions Zhang2016 associated with vehicle pairs. For human drivers, the link transfer function $T_{\rm H}(s)$ relates the speed fluctuation of an HV to that of the vehicle ahead as

$$V_{n}(s)=T_{\rm H}(s)V_{n+1}(s)\,,$$

(38)

where $T_{\rm H}(s)$ can be obtained from (27, 28) in the form

$$T_{\rm H}(s)=\mathbf{c}\Big{(}s\mathbf{I}-\mathbf{a}-\mathbf{a}_{\rm H}{\rm e}^{-s\tau}\Big{)}^{-1}\mathbf{b}_{\rm H}{\rm e}^{-s\tau}\begin{bmatrix}\frac{1}{s}\\ 1\end{bmatrix}.$$

(39)

Similarly, in ACC where the automated vehicle responds to the vehicle immediately ahead only, the response can be characterized by a single link transfer function $T(s)$:

$$V_{0}(s)=T(s)V_{1}(s)\,,$$

(40)

whose expression can be derived from (30) as

$$T(s)=\mathbf{c}\Big{(}s\mathbf{I}-\mathbf{a}-\mathbf{a}_{\rm F}{\rm e}^{-s\sigma}\Big{)}^{-1}\mathbf{b}_{\rm F}{\rm e}^{-s\sigma}\begin{bmatrix}\frac{1}{s}\\ 1\end{bmatrix}.$$

(41)

On the other hand, in ATC the ego CAV responds to multiple vehicles, associated with multiple link transfer functions. For example, when responding to a single vehicle ahead and a single vehicle behind, we obtain

$$V_{0}(s)=T_{\rm F}(s)V_{1}(s)+T_{\rm B}(s)V_{-N}(s)\,,$$

(42)

where $T_{\rm F}(s)$ and $T_{\rm B}(s)$ are the link transfer functions characterizing the response forward and backward. These are obtained from (32):

$\displaystyle\begin{split}T_{\rm F}(s)&=\mathbf{c}\Big{(}s\mathbf{I}-\mathbf{a}-\mathbf{a}_{\rm FB}{\rm e}^{-s\sigma}\Big{)}^{-1}\mathbf{b}_{\rm F}{\rm e}^{-s\sigma}\begin{bmatrix}\frac{1}{s}\\ 1\end{bmatrix},\\ T_{\rm B}(s)&=\mathbf{c}\Big{(}s\mathbf{I}-\mathbf{a}-\mathbf{a}_{\rm FB}{\rm e}^{-s\sigma}\Big{)}^{-1}\mathbf{b}_{\rm B}{\rm e}^{-s\sigma}\begin{bmatrix}\frac{1}{s}\\ 1\end{bmatrix}.\end{split}$

(43)

Notice that the denominators of $T_{\rm F}(s)$ and $T_{\rm B}(s)$ are the same, ${\rm D}(T_{\rm F}(s))={\rm D}(T_{\rm B}(s))$, since the same matrix is inverted.

Using the link transfer functions, we can analyze the overall response of traffic. For a chain of $N$ human drivers, the overall response

$$V_{-N}(s)=\Gamma(s)V_{0}(s)\,.$$

(44)

is characterized by $\Gamma(s)$ given by

$$\Gamma(s)=T_{\rm H}(s)^{N}\,.$$

(45)

Note that this corresponds to identical human drivers; for the case of a heterogeneous chain of HVs it should be replaced with ${\Gamma(s)=\prod_{n=-N}^{-1}T_{{\rm H},n}(s)}$.

Introducing the ego vehicle (indexed $0$) and the vehicle ahead of it (lead vehicle $1$) into the human-driven traffic, like in Figs. 2(b) and 3(b), we can describe the overall response of the traffic flow from the head vehicle to the tail vehicle via the head-to-tail transfer function Zhang2016 given by

$$V_{-N}(s)=G(s)V_{1}(s)\,.$$

(46)

$G(s)$ depends on the human response $\Gamma(s)$ and the control strategy of the ego vehicle. For vehicle control with ACC, we write (40, 44) and obtain

$$G(s)=T(s)\Gamma(s)\,.$$

(47)

For the traffic control setup with ATC, we have (42, 44) that lead to

$$G(s)=\frac{T_{\rm F}(s)\Gamma(s)}{1-T_{\rm B}(s)\Gamma(s)}\,.$$

(48)

If traffic is considered on a ring road, we have the periodic boundary condition

$$V_{-N}(s)=V_{1}(s)\,,$$

(49)

that leads to

$$G(s)=1\,,$$

(50)

as the characteristic equation of the ring configuration.

For a chain of vehicles traveling on a straight road, the system is non-autonomous, forced by the speed perturbations of the lead vehicle. Thus, we consider the notions of plant stability and string stability Feng2019 ; Ploeg2014a ; Zhang2016 to analyze the behavior without speed perturbations and the response to the perturbations, respectively. For the ring configuration, the system evolves autonomously, and we apply a single notion of stability; for more details on stability definitions see Giammarino2019 .

String stability is particularly important in trarffic control. It has various mathematical definitions; see a comprehensive summary in Feng2019 . Now we consider linear input-to-output string stability with respect to $\mathcal{L}_{2}$ norm. This string stability notion is directly related to the magnitude ($\mathcal{H}_{\infty}$ norm) of the transfer function describing system, which makes it is easy to analyze and useful for control design. Other notions are also widely-used, such as $\mathcal{L}_{p}$ string stability for any value of $p$ Ploeg2014a . $\mathcal{L}_{\infty}$ string stability is particularly relevant for traffic safety, since it characterizes the overshoot of the response to perturbations. As this notion is related to the impulse response, it is more challenging to analyze it explicitly, and we rather consider $\mathcal{L}_{2}$ string stability.

Now let us take a deeper look into the relationship between the various vehicle configurations and their stability conditions. These relationships are summarized at the bottom of Fig. 4 for the ${s={\rm i}\omega}$ stability boundaries.

For vehicle control with (47), we have ${{\rm D}(G(s))={\rm D}(T(s)){\rm D}(\Gamma(s))}$, hence the plant stability condition becomes

$\displaystyle\begin{split}{\rm D}(T(s_{\ell}))&=0\quad\Rightarrow\quad\Re(s_{\ell})<0\,,\quad\forall\ell\in\mathbb{N}\,,\quad\text{and}\\ {\rm D}(\Gamma(s_{\ell}))&=0\quad\Rightarrow\quad\Re(s_{\ell})<0\,,\quad\forall\ell\in\mathbb{N}\,,\end{split}$

(65)

that is, both the AV and HVs must be plant stable individually. For string stability, we require

$$\left|T({\rm i}\omega)\right|\left|\Gamma({\rm i}\omega)\right|<1\,,\quad\forall\omega>0\,,$$

(66)

and at the stability boundary we have

$$T({\rm i}\omega)\Gamma({\rm i}\omega)={\rm e}^{-{\rm i}K}\,,$$

(67)

which depend both on the response of the AV and the chain of HVs. If the human-driven traffic is not considered, i.e., ${N=0}$, ${\Gamma({\rm i}\omega)=1}$, then we recover the string stability limit ${T({\rm i}\omega)={\rm e}^{-{\rm i}K}}$ for a single ACC vehicle.

Now consider a nonzero (and potentially large) number $N$ of HVs. This leads to one of the most interesting research questions, that several recent works have studied Cui2017 ; Qin2020 ; Stern2018 : how many AVs are needed to mitigate traffic congestions? If a human-driven vehicle chain behind an AV is long ($N$ is large), it results in a significant (exponential) increase of speed perturbations. Thus, the AV has to make considerable effort to dampen the perturbation in order to prevent or mitigate a congestion. Mathematically, the magnitude ${\left|T_{\rm H}({\rm i}\omega)\right|}$ of the individual human responses affects the overall response ${\left|\Gamma({\rm i}\omega)\right|=\left|T_{\rm H}({\rm i}\omega)\right|^{N}}$ more significantly for large $N$. Therefore, it may be more challenging to stabilize a long chain of HVs with a single AV.

To highlight this, consider the limit ${N\to\infty}$. If the human drivers are string stable, i.e., ${\left|T_{\rm H}({\rm i}\omega)\right|<1}$, then ${\left|\Gamma({\rm i}\omega)\right|=0}$ for all ${\omega>0}$. Unless ${\left|T({\rm i}\omega)\right|\to\infty}$, which can only happen at the plant stability boundary ${{\rm D}(T({\rm i}\omega))=0}$ of the AV, (66) is automatically satisfied. Thus string stability is achieved for a long string stable chain of HVs and a plant stable AV. However, when the human drivers are string unstable — which is more likely the case in real traffic Molnar2021trc — we have ${\left|T_{\rm H}({\rm i}\omega)\right|>1}$ and ${\left|\Gamma({\rm i}\omega)\right|\to\infty}$ for some ${\omega>0}$ and ${N\to\infty}$. Thus, string stability in (66) cannot be achieved by any AV design (as long as ${T({\rm i}\omega)\neq 0}$, i.e., the AV responds to what is ahead). This shows the fundamental limitation of influencing traffic by vehicle control: string stability cannot be achieved with arbitrarily small penetration of AVs (i.e., in the limit ${N\to\infty}$) if the human drivers are string unstable.

If vehicle control is executed on a ring, the stability limit (64) becomes

$$T({\rm i}\omega)\Gamma({\rm i}\omega)=1\,.$$

(68)

Therefore, the stability boundary of the ring can be obtained as the ${K=0}$ special case of the string stability boundary (67) of the open chain. Physically, it means that perturbations propagate through the ring and arrive back to the same vehicle in the same phase due to the periodic boundary condition. Another special case to mention is when all vehicles on the ring are identical, achieved by the substitution ${T(s)=T_{\rm H}(s)}$. Then the characteristic equation (68) gives

$$T_{\rm H}({\rm i}\omega)^{N+1}=1\quad\iff\quad T_{\rm H}({\rm i}\omega)={\rm e}^{-{\rm i}\frac{2k\pi}{N+1}}\,,\quad k\in\{0,1,\ldots,N\}\,.$$

(69)

When ${N\to\infty}$, the exponent ${K=2k\pi/(N+1)}$ may take any value on $[0,2\pi)$ and we recover the string stability condition ${T_{\rm H}({\rm i}\omega)={\rm e}^{-{\rm i}K}}$ for individual human drivers. Thus, the stability of an infinitely long homogeneous ring is equivalent to the string stability of a homogeneous chain of vehicles. Conversely, the substitution of ${K=2k\pi/(N+1)}$ into the string stability limit ${T_{\rm H}({\rm i}\omega)={\rm e}^{-{\rm i}K}}$ of an individual human driver gives the stability limit (69) of the homogeneous ring.

Additionally, and most importantly, the traffic control setup as a virtual ring is related to both the open chain and ring configurations in terms of stability. Consider formula (48) of $G(s)$. Since ${\rm D}(T_{\rm F}(s))={\rm D}(T_{\rm B}(s))$, we have

$${\rm D}(G(s))={\rm D}(T_{\rm B}(s)\Gamma(s))\big{(}1-T_{\rm B}(s)\Gamma(s)\big{)}\,.$$

(70)

Hence, and the plant stability condition (55) leads to

$\displaystyle\begin{split}{\rm D}(T_{\rm B}(s_{\ell})\Gamma(s_{\ell}))&=0\quad\Rightarrow\quad\Re(s_{\ell})<0\,,\quad\forall\ell\in\mathbb{N}\,,\quad\text{and}\\ T_{\rm B}(s)\Gamma(s)&=1\quad\Rightarrow\quad\Re(s_{\ell})<0\,,\quad\forall\ell\in\mathbb{N}\,.\end{split}$

(71)

That is, it requires both the plant stability of the associated vehicle control setup and the stability of the associated virtual ring configuration.

The string stability boundary of ATC can be written in the form

$$\frac{T_{\rm F}({\rm i}\omega)\Gamma({\rm i}\omega)}{1-T_{\rm B}({\rm i}\omega)\Gamma({\rm i}\omega)}={\rm e}^{-{\rm i}K}\,.$$

(72)

This reduces to the string stability boundary of traffic flows influenced by vehicle control when there is no response to the traffic behind, i.e., ${T_{\rm B}({\rm i}\omega)=0}$, ${T_{\rm F}({\rm i}\omega)=T({\rm i}\omega)}$, which is achieved by ${\beta_{\rm B}=0}$ in our example. In this sense, the string stability of ATC is related to that of ACC while the plant stability of ATC is related to the stability of its virtual ring.