# A new platooning model for connected and autonomous vehicles to improve
string stability

Shouwei Hui<EMAIL_ADDRESS>H.M. Zhang<EMAIL_ADDRESS>
###### Abstract

This paper introduces a novel idea of coordinated vehicle platooning such that
platoon followers inside the platoon communicates only to the platoon leader.
A novel dynamic model is proposed to take driving safety into account when
there is communication delay. Some general results of linear stability are
proved mathematically, and numerical simulations are conducted to show the
effect of model parameters for both ring road with an initial disturbance and
infinite road with a periodic disturbance. The simulation results are
consistent with theoretical analysis, and demonstrate that the proposed look-
to-the-leader platooning strategy is far superior than the follow-one-vehicle-
ahead or follow-two-vehicle-ahead conventional car-following (CF) strategies
in stabilizing traffic flow. This paper provides a new perspective for the
organization of platoons of autonomous vehicles.

###### keywords:

Platoon , Connected and Autonomous Vehicles , Car-following model , Linear
stability

††journal: Physica A

[1]organization=Department of Mathematics, University of California Davis,
city=Davis, state=CA, postcode=95616, country=United States

[2]organization=Department of Civil and Environmental Engineering, University
of California Davis, city=Davis, state=CA, postcode=95616, country=United
States

##  1 Introduction

In recent years, Connected and Autonomous Vehicles (CAV) have attracted
extensive attention around the world. With the fast development of CAV
technologies, and its supporting infrastructures, cooperative car-following
(CF), e.g. platooning, can be implemented in the near future to improve
traffic. Platooning is a coordinated driving method for a group of vehicles
and it can be analysed as a systematical longitudinal traffic control system
[1]. Such a system can be modelled by microscopic traffic models, and in
particular, ordinary differential equation (ODE) based car-following models.

Pipes [2] firstly introduced car-following models to describe a string of
cars’ behavior in 1953, and since then many car-following models with
adjustable parameters have been proposed. Gazis et al [3] proposed a nonlinear
follow-the-leader model and investigated its equilibrium states. Bando et al.
[4, 5] proposed an optimal velocity model (OVM) that uses an optimal velocity
function to replace velocity of front vehicle in Pipes’ model. Moreover, OVM
is able to capture traffic instabilities on a ring road without external
disturbance. Treiber et al. [6] proposed the intelligent driver model (IDM)
that considered the difference of the front vehicle’s velocity, and was shown
to have several advantages compared to Bando’s model. These models are
commonly used in traffic simulation and traffic control design, since they are
capable of describing typical traffic phenomena with relatively simple forms,
but these models only consider the interaction between a pair of vehicles in a

Various extensions have been developed based on the aforementioned models.
Lenz et al. [7] proposed a multi-following model based on OVM to follow
multiple vehicles ahead and Nakayama et al. [8] proposed a backward-looking
model based on OVM to capture the influence of the vehicle behind on the
subject vehicle. Jiang et al. [9] proposed full velocity difference model
(FVDM) that consider both positive and negative relative velocity to eliminate
unrealistic acceleration and deceleration. Yu et al. [10] extended FVDM by
considering the acceleration difference. Lazar et al. [11] gave a thorough
review of OVM based models from 1995 to 2016. Treiber et al. [12] extended IDM
to follow multiple vehicles ahead and considered effects of reaction time and
estimation error. Derbel et al. [13] proposed a modified version of IDM with
improvement of vehicles’ capability and safety. Zhang and Kim [14] proposed a
car-following model based on Pipes’ model that explained capacity drop and
traffic hysteresis.

Car-following models can be combined with various control designs to assess
the impact of CAVs in mixed traffic of human driven and autonomous vehicles.
Zhu and Zhang [15] modified OVM with smoothing factor to model autonomous
vehicles (AV) and analysed mixed traffic of AV and human driven vehicles
(HDV). Jia and Ngoduy [16] proposed a platoon-controlled car following model
based on realistic inter-vehicle communication and designed a consensus-based
control system for multiple platoons moving together. Zhang et al. [17]
extended Zhu’s model for platooned CAVs by combining multi-following with
weighted relative velocity differences and analysed the linear stability of
this model. Wang et al [18] proposed Leading Cruise Control (LCC), a mixed
traffic flow control strategy for CAVs to both adaptively follow the vehicle
in front and pace the following vehicles. Zhao et al. [19] proposed safety-
critical traffic control (STC), for which CAVs maintain safety relative to
both the preceding vehicle and following HDVs. For Wang and Zhao’s paper, HDVs
are model by OVM. Zhou et al. [20, 21] proposed a platoon-based intelligent
driver model (P-IDM) together with an autonomous platoon formation strategy
(APFS), to stabilize CAV platoons under periodic disturbance. Jin and Meng
[22] proposed a model that combined OVM with a time-delayed feedback
controller and stability can be increased by analyzing the transcendental
characteristic equation to eliminate unstable eigenvalues. Sun et al. [23]
analysed the relationship of string instability of IDM based controllers and
traffic oscillations and found promising CAV parameters to improve stability
by forming finite-sized platoons.

Various other theoretical and experimental approaches can be utilized to
assess the effectiveness of CAV and CAV platooning. From an theoretical
approach, Li [24, 25] proposed stochastic dynamic model for vehicle platoons
and investigated its statistical characteristics. Gong et al. [26] proposed a
multi-agent dynamic system to model platoons and the control scheme is
formulated as a constrained optimization problem. Zhou et al. [27] proposed a
series of distributed model predictive control strategies for coordinated car-
following of CAVs and gave strategies for guaranteed local, $L_{2}$, and
$L_{\infty}$ stabilities. Amoozadeh et al. [28] developed a platoon management
protocal for Connected and adaptive cruise control (CACC) vehicles based on
wireless communication through Vehicle Ad-hoc Network (VANET). Validity and
effectiveness are shown by simulations of different platoon settings. For
experimental methodologies, Stern et al [29] performed a ring road experiment
of over 20 vehicles and showed that with a controlled autonomous vehicle stop-
and-go waves can be dampened and total fuel consumption can be reduced. Lee et
al [30] implemented a large-scale field experiment with 100 CAVs as
MegaController on a freeway network to dampen stop-and-go waves, which is by
far the largest scale of field experiment of CAVs. Tsugawa et al. [31] tested
a platoon of $3$ trucks on a test track along an expressway and showed that
fuel consumption can be reduced by $14$ percent. Additionally, for a
systematic review of research method on platooning, Lesch et al. [32] did an
literature review of over $1600$ papers on state of the art in platooning
coordination research for both cars and trucks.

Overall, previous studies on CAV platoon control have shown good potentials of
CAVs in stabilizing traffic, but few are showing that platooned CAVs can be
always stable with a combination of rigorous mathematical proof and numerical
simulation. Moreover, only few studies prioritize following the leading
vehicle of a platoon system. The main contributions of this paper are: (1)
Presented a platoon control model with a combination of designed leading
vehicle and platoon following strategies; (2) Verified linear stability of the
proposed models theoretically and numerically.

The remaining part of this paper is organized as follows. In section section
2, the car-following models for human-driven vehicles and platoon-controlled
CAVs together with a transition phase model are proposed. In section 3, linear
stability criteria of the proposed models are presented and proved. In section
4, numerical simulations of ring road and infinite road scenarios are
presented and performance of different model parameters are evaluated. Lastly
in section 5, conclusion and future direction of research are given.

##  2 The Non-Platoon and Platoon CF Models

Without loss of generality we assume that a string of $N$ vehicles are moving
on a single lane road without any overtaking. We are also assuming the initial
position of the rear bumper of the first vehicle is at $0$m of the road and
the leading vehicle is the $N$-th vehicle. In particular, if the single-lane
road is a ring road, then the $(N+1)$-th vehicle is the same as the $1$st
vehicle. To show the effectiveness of platoon-controller for connected and
autonomous vehicles (CAV), a base model for human driven vehicles (HDV) is
proposed in Section section 2.1, and a model for platoon-controlled CAVs is
given in Section section 2.2, then a transition phase model is given in
Section section 2.3.

###  2.1 Optimal velocity model

To model HDVs, one common choice of car-following model is the optimal
velocity model (OVM) [4]. OVM has the form

$\ddot{x}_{i}(t)=a\left[V(x_{i}(t)-x_{i+1}(t))-\dot{x}_{i}(t)\right],$ (1)

where $x_{i}(t)$, $i=1,2,\dots N$ is the position of $i$-th vehicle at time
$t$ on the single-lane road without overtaking. $x_{i}(t)-x_{i+1}(t)\triangleq
h_{i}(t)$ is the headway between the $i$-th and $i+1$-th vehicle.
$\dot{x}_{i}(t)$, $\ddot{x}_{i}(t)$ are velocity and acceleration of $i$-th
vehicle at time $t$, $V(h)$ is the optimal velocity function of headway (head
to head distance) $h$, and $a$ is a sensitivity constant. An example of
optimal velocity function is given in [18], which is equivalent to the form

$V(h)=\begin{cases}v_{\max},&\text{if }h\geq h_{\max}\\\
\frac{v_{\max}}{2}\left(1-\cos\left(\frac{h-h_{\min}}{h_{\max}-h_{\min}}\right)\right),&\text{if
}h_{\min}\leq h\leq h_{\max}\\\ 0,&\text{if }h\leq h_{\min}\end{cases}$ (2)

where $h_{\min}$ is the minimum headway, $h_{\max}$ is the maximum headway,
$v_{\max}$ is the maximum headway and $l$ is the length of each vehicle.
Figure fig. 1 is an example plot of (eq. 2) and the corresponding fundamental
diagram (density-flow diagram).

(a) Optimal velocity function

(b) Fundamental diagram

Figure 1: Plot of an optimal velocity function with $l=5$m and
$v_{\max}=20$m/s and the corresponding fundamental diagram.

###  2.2 Platoon controller: A coordinated car following strategy

Now we consider a system of CAVs such that every following vehicle is
connected to the leading autonomous vehicle. In other words, the following
vehicles know the position and speed of the leading vehicle. In the proposed
platoon model, we only use the position information of the leading vehicle and
propose that the target (optimal) velocity of a following vehicle in the
platoon is based on its distance from the leading vehicle:

$\ddot{x}_{i}(t)=a\left[V(\frac{x_{N}(t)-x_{i}(t)}{N-i})-\dot{x}_{i}(t)\right]$
(3)

where $x_{N}$ is the controlled leading vehicle. We refer (eq. 3) as the
platoon controlled OVM (P-OVM) model. In this model, scaling is applied to the
spacing between the leader of the platoon ($N$-th vehicle) and the $i$-th
following vehicle used in the optimal velocity function, which is equivalent
to consider the average spacing of vehicles between the leader of the platoon
and the $i$-th following vehicle. Figure fig. 2 is an illustration of this
platoon-controlled car-following configuration. Under this configuration, the
influence of the immediate front vehicle is not taken into account, which
could lead to collisions if the errors in position measurement and delays in
communicaitons between the platoon leader and the following vehicles inside
the platoon reach a certain level.

Figure 2: Illustration of the platoon-controlled car-following system. Leading
CAV is connected to all followers of the platoon and followers are directly
following the leading CAV

###  2.3 A transition phase model

We can address the previously mentioned safety issue by adding a local, within
platoon vehicle-to-vehicle control to the P-OVM model. This local control can
be any feedback control that eliminates vehicle collisions, and for the
convenience of presentation and analysis, we adopt the OVM model as the local
control model and called the combined model as the transition phase optimal
velocity model (T-OVM). The model is of the form

$\ddot{x}_{i}=a(V(x_{i+1}-x_{i})-\dot{x}_{i})+b\left(V\left(\frac{x_{N}-x_{i}}{N-i}\right)-\dot{x}_{i}\right)$
(4)

where $a$ is the sensitivity to vehicle in front and $b$ is the sensitivity to
the leading vehicle. The total sensitivity is defined as sum of the
sensitivities $(a+b)$. Figure fig. 3 is an illustration of the car following
pattern of T-OVM. With two parts of the model each vehicle inside the platoon
will balance between the optimal velocity to the vehicle in front and the
optimal velocity to the leading vehicle, to avoid collision and increase
stability simultaneously.

Figure 3: Illustration of the transition phase car-following system. Leading
CAV is connected to all followers inside the platoon and followers are
following the leading CAV and the front CAV simultaneously

##  3 Stability analysis of the platoon models

If the string of $N$ vehicles are on a ring road with length $L$ and the
leading vehicle is following the last vehicle of the string by the original
OVM (eq. 1), the position of $i-th$ vehicle is the same for both OVM (eq. 1)
and P-OVM (eq. 5) under steady state:

$e_{i}(t)=hi+V(h)t$ (5)

where $h=L/N$ is the equilibrium headway. For the classic optimal velocity
model, we have the following linear stability criterion:

######  Theorem 3.1.

The optimal velocity model (eq. 1) is linearly stable if

$a>2V^{\prime}(h).$ (6)

A proof slightly different from [4] is given in Appendix appendix A. Then with
the platoon controller applied, it turns out that no stability criteria is
required:

######  Theorem 3.2.

In the $N$-vehicle platoon where the leader vehilce follows the last vehicle
of the platoon on a ring road according to OVM (eq. 1), the platoon-controlled
system P-OVM (eq. 3) is always linearly stable.

###### Proof.

From previous introduction, P-OVM has the same equilibrium solution of the
optimal velocity model as $x_{n}^{0}(t)=e_{n}(t)$ for any car $n$ inside the
platoon. Now suppose that all the vehicles are deviated from the initial
position with a small disturbance $y_{n}(t)$, that is,

$x_{n}(t)=e_{n}(t)+y_{n}(t),\;|y_{n}|\ll 1$ (7)

To linearize the original system, we can do a Taylor expansion of the optimal
velocity function term $V(\Delta x_{n})$ and neglect the higher order terms to
get

$\ddot{y}_{n}(t)=\begin{cases}a\left[(V^{\prime}(h)\frac{y_{N}(t)-y_{n}(t)}{N-n}-\dot{y}_{n}(t)\right],\;\text{if
}n\neq N\\\
a\left[(V^{\prime}(h)(y_{1}(t)-y_{N}(t))-\dot{y}_{N}(t)\right],\;\text{if
}n=N\end{cases}$ (8)

Then by the formation of the controller, $y_{1}$ and $y_{N}$ forms a system of
2 linear ODEs, and the solution in vector form is:

$\mathbf{Y}=\sum_{n=1,\;N}C_{n}e^{\lambda_{n}t}\mathbf{V}_{n}$ (9)

Where $Y=[y_{1},y_{N}]^{T}$ is the vector form, $\lambda_{n}$ and $V_{n}$ are
the eigenvalues and eigenvectors of the system, and $C_{n}$ are constants
determined by the initial condition. Now suppose that $\lambda$ is an
eigenvalue of the system, and $\xi_{n}$ is the coefficient of $y_{n}$ with the
term $e^{\lambda t}$, then simplified from (eq. 8), $\lambda$ satisfies

$\lambda^{2}+a\lambda-aV^{\prime}(h)(\frac{\xi_{1}}{\xi_{N}}-1)=0,$ (10)

and

$\lambda^{2}+a\lambda-
aV^{\prime}(h)(\frac{\xi_{N}}{(N-1)\xi_{1}}-\frac{1}{N-1})=0$ (11)

Then by comparing these equations we can calculate that $\xi_{1}=\xi_{N}$ or
$\xi_{1}=-\xi_{N}/(N-1)$. And the first case is only true if we have
$y_{1}=y_{N}$ and the two cars just stay at the same distance from the
original equilibrium. For the other case we have

$\lambda^{2}+a\lambda-aV^{\prime}(h)(-\frac{1}{N-1}-1)=0$ (12)

Then, by solving the quadratic equation, we have

$\lambda=\pm\frac{-a+\sqrt{a^{2}+4aV^{\prime}(h)(-\frac{1}{N-1}-1)})}{2}$ (13)

For the system to be stable, we need to have the real parts of both $\lambda$
to be negative. This is true if $V^{\prime}(h)>0$ which is always true for any
well-defined $V(h)$. Now we just need to check if the same property holds for
the remaining vehicles. By (eq. 8) we can have a initial guess that the
nonlinear part is $y_{j}=\zeta y_{N}$ for $j\neq 1$, then plug this into (eq.
8) and compare the coefficients we have

$\zeta=N-j+\frac{j-1}{N-1}$ (14)

Thus the other cars also have the same eigenvalue for the nonlinear part and
the system will always be linearly stable if the problem is well defined. ∎

As for the transition phase model, stability clearly depends on both $a$, $b$
and $N$. For $N$ sufficiently large we have the following stability criterion:

######  Theorem 3.3.

Assuming that the leading vehicle follows the last vehicle by OVM (eq. 1) with
sensitivity constant $a+b$. Then the transition phase model T-OVM (eq. 4) is
linearly stable if

$\frac{(a+b)^{2}}{a}>2V^{\prime}(h)$ (15)

The proof is given in Appendix appendix B. Figure fig. 4 is a plot of neutral
stability lines for different sensibilities and the area above the lines is
the stable and below is unstable. And from this figure it is clear that with
larger values of $b$ the unstable area is smaller and eventually vanished if
$a\to 0$.

Figure 4: neutral stability lines of the transition phase model for selected
percentages of leading vehicle sensitivity $b/(a+b)=0\%,20\%,40\%,60\%,80\%$.
Above each line the region is stable and below is unstable.

##  4 Numerical simulation

To have a better visualization of how stability is improved by applying
platoon control, several numerical experiments are conducted for different
road settings and model parameters. All the numerical simulations are
performed on MATLAB 2023b with simulation time step $\Delta t=0.1$s. To update
solutions of each time step, forward Euler is applied for velocity variable
$\dot{x}_{i}$ and a modified Euler scheme (Heun’s method) is applied for
position variable $x_{i}$:

$\dot{x}_{i,j+1}=\dot{x}_{i,j}+\ddot{x}_{i,j}\Delta
t;\;x_{i,j+1}=x_{i,j}+\frac{\dot{x}_{i,j}+\dot{x}_{i,j+1}}{2}\Delta t$ (16)

where $x_{i,j}$ is referring to the position of $i$-th car at $j$-th time step
of simulation. This is equivalent to the discretizaiton method in [15].

###  4.1 Ring road with initial disturbance

In this subsection simulations are performed on a ring road. The total length
of the ring road is $L=264$m with a total of 12 vehicles. As for model
parameters we choose the optimal velocity function given by (eq. 2) and set
$h_{\min}=7$m, $h_{\max}=37$m, $v_{\max}=20$m/s and $l=5$m. Then the
equilibrium headway is $h=L/N=22$m and the equilibrium velocity is $V(h)=10$
m/s. The initial position and velocity of the $i$-th vehicle are deviated from
the equilibrium states $(e_{i},V(h))$ with random perturbation uniformly
distributed on the interval $[0,5]$. The initial condition of the model can be
written as

$\begin{cases}x_{i}(0)=&e_{i}(0)+r_{i},\\\
\dot{x}_{i}(0)=&V(h)+\bar{r}_{i}\end{cases}$ (17)

where $r_{i},\bar{r}_{i}$ are random numbers generated from uniform
distribution on $[0,5]$, and $e_{i}(0)=hi$ can be calculated from (eq. 5).

Simulation 1.1: Comparison of OVM and P-OVM

To show the stability improvement of P-OVM, for sensitivity constant $a$, we
tested for $a=0.4,\;0.8\;,1.6,\;2.4$ for both OVM and P-OVM with the same
initial condition. Simulation results are shown for the original OVM and
platoon-controlled OVM in Figures[fig. 5-fig. 8]. Figures [fig. 5, fig. 6] are
the 3-D plots of headways of all vehicles under OVM and P-OVM with different
values of $a$ respectively . Figures [fig. 7, fig. 8] are the velocity
profiles of the $6$th vehicle under OVM and P-OVM respectively with different
values of $a$.

(a) $a=0.4$

(b) $a=0.8$

(c) $a=1.6$

(d) $a=2.4$

Figure 5: Headway profile of OVM with sensitivity constant
$a=0.4,\;0.8\;,1.6,\;2.4$

(a) $a=0.4$

(b) $a=0.8$

(c) $a=1.6$

(d) $a=2.4$

Figure 6: Headway profile of P-OVM with sensitivity constant
$a=0.4,\;0.8\;,1.6,\;2.4$

(a) $a=0.4$

(b) $a=0.8$

(c) $a=1.6$

(d) $a=2.4$

Figure 7: Velocity profile of $6$th vehicle of OVM with sensitivity constant
$a=0.4,\;0.8\;,1.6,\;2.4$

(a) $a=0.4$

(b) $a=0.8$

(c) $a=1.6$

(d) $a=2.4$

Figure 8: Velocity profile of $6$th vehicle of P-OVM with sensitivity constant
$a=0.4,\;0.8\;,1.6,\;2.4$

From the simulation results, for $a=0.4,\;0.8\;,1.6$ OVM is unstable. For
$a=0.4$ negative headway can be generated which means collisions can happen
with low sensitivity, and for $a=2.4$ OVM is stable but the initial
acceleration is unrealistically large for the selected vehicle. As for P-OVM,
for all selected values of $a$ the model is stable and $a$ is only affecting
the convergence speed: for greater values of $a$ the flow converge to
equilibrium state faster and for $a$ sufficiently large the improvement in
convergence is minor since the original OVM is already stable.

Simulation 1.2: Test parameters of T-OVM

To test the performance of the transition phase model, four cases of
sensitivity pair: $(a,b)=(0.5,0.1),$ $(0.1,0.5),$ $(1,0.2),$ $(0.6,0.6)$ are
tested with same initial condition. Extreme cases of $a=0$ or $b=0$ are not
considered since they are already presented in the previous section.
Simulation results are in Figures [fig. 9-fig. 10]. Figure fig. 9 is the
headway profile of all 12 vehicles in 3-D plot with different pairs of
sensitivity constants and Figure fig. 10 is the velocity profile of the $6$th
vehicle.

(a) $(a,b)=(0.5,0.1)$

(b) $(a,b)=(0.1,0.5)$

(c) $(a,b)=(1,0.2)$

(d) $(a,b)=(0.6,0.6)$

Figure 9: Headway profile of T-OVM with sensitivity constant
$(a,b)=(0.5,0.1),(0.1,0.5),(1,0.2),(0.6,0.6)$

(a) $(a,b)=(0.5,0.1)$

(b) $(a,b)=(0.1,0.5)$

(c) $(a,b)=(1,0.2)$

(d) $(a,b)=(0.6,0.6)$

Figure 10: Velocity profile of $6$th vehicle of T-OVM with sensitivity
constant $(a,b)=(0.5,0.1),(0.1,0.5),(1,0.2),(0.6,0.6)$

From the simulation results, we can see that with the increase of percentage
of leading vehicle sensitivity, the traffic stream is more stable. And for
higher total sensitivity $(a+b)$ less percentage of leading vehicle
sensitivity is required, which is consistent with theoretical results (see
figure fig. 4). Moreover, OVM and P-OVM can be considered as extreme cases of
T-OVM.

Simulation 1.3: Compare T-OVM and with two cars ahead following

To show the effectiveness of following the leader of platoon, we compare T-OVM
(eq. 4) with the front multi-following OVM (F-OVM) of two cars in [7]:

$\ddot{x}_{i}=a(V(x_{i+1}-x_{i})-\dot{x}_{i})+b\left(V\left(\frac{x_{i+2}-x_{i}}{2}\right)-\dot{x}_{i}\right)$
(18)

where the sensitivity $b$ is for the second vehicle in front. Two cases of
sensitivity pair: $(a,b)=(0.8,0.4),(0.2,0.4)$ are tested. Simulation results
are in Figures fig. 11, fig. 12. Figure fig. 11 is the headway profile of all
12 vehicles in 3-D plot with T-OVM and F-OVM and Figure fig. 12 is the
velocity profile of the $6$th vehicle.

(a) T-OVM with $(a,b)=(0.8,0.4)$

(b) F-OVM with $(a,b)=(0.8,0.4)$

(c) T-OVM with $(a,b)=(0.2,0.4)$

(d) F-OVM with $(a,b)=(0.2,0.4)$

Figure 11: Headway profile of T-OVM and F-OVM with sensitivity constant
$(a,b)=(0.8,0.4),(0.2,0.4)$

(a) T-OVM with $(a,b)=(0.8,0.4)$

(b) F-OVM with $(a,b)=(0.8,0.4)$

(c) T-OVM with $(a,b)=(0.2,0.4)$

(d) F-OVM with $(a,b)=(0.2,0.4)$

Figure 12: Velocity profile of $6$th vehicle of T-OVM and F-OVM with
sensitivity constant $(a,b)=(0.8,0.4),(0.2,0.4)$

From the simulation results we can see that for same set of sensitivity
parameters, T-OVM can suppress the disturbance and the platoon remians stable
while F-OVM cannot. Moreover, this stabilizing effect of T-OVM becomes more
prominent when the following vehicles repond more strongly to the leader in
T-OVM, while the destabilizing effect in the F-OVM becomes stronger when the
following vehicle respond more strongly to the two-vehicle ahead vehicle,
demonstrating the superiority of T-OVM in stabilizing vehicle platoons.

###  4.2 Infinite road with periodic disturbance

Other than initial disturbance, it is interesting to know if platoon control
can also increase stability under periodic disturbance. Suppose that a total
of $N=10$ vehicles are on a infinite road with free flow speed $v_{max}=30$m/s
and uniform length $l=5$m. The optimal velocity function $V_{I}(h)$ for the
infinite road simulation is equivalent to triangular fundamental diagram:

$V_{I}(h)=\begin{cases}v_{\max},&\text{if }\rho(h)\leq\rho_{c}\\\
\frac{v_{\max}*\rho_{c}(\rho(h)-\rho_{\max})}{\rho(h)(\rho_{c}-\rho_{\max})},&\text{if
}\rho_{c}\leq\rho(h)\leq\rho_{\max}\\\ 0,&\text{if
}\rho\geq\rho_{\max}\end{cases}$ (19)

where $\rho(h)=l/h$ is the occupancy of vehicles on the road: $\rho=0$ means
no vehicles and $\rho=1$ means full of vehicles. $\rho_{c}=5/37$ is the
critical occupancy such that flow is maximized and $\rho_{\max}=5/7$ is the
jam occupancy with $0$ speed. Figure fig. 13 is a plot of the optimal velocity
function and corresponding triangular fundamental diagram.

(a) Optimal velocity function

(b) Triangular fundamental diagram

Figure 13: Plot of an optimal velocity function with $l=5$m and
$v_{\max}=30$m/s and the corresponding triangular fundamental diagram.

During the simulation time, the leading vehicle was trying to maintain at a
given equilibrium speed but encountered sinusoidal disturbance, similar to
[20]. The equilibrium speed is $v_{0}=15$m/s and the equilibrium headway is
$h=29.5$m. And if we denote $p$ as duration of each period of the sinusoidal
disturbance and $A$ as the amplitude of the disturbance, then the velocity of
the leading vehicle can be written as:

$v_{N}=v_{0}+A\sin\left(\frac{2\pi}{p}t\right)$ (20)

where period duration $p=5,\;10,\;20,\;30$s and amplitude $A=5$m/s. OVM and
P-OVM with sensitivity constant $a=1.2,\;2.4$ are tested. The simulation
duration is $60$ seconds for all simulations. Simulation results are in Figure
[fig. 14-fig. 17]. Figure fig. 14, fig. 15 are the headway profiles of OVM and
P-OVM with sensitivity constant $a=1$ under different perturbation frequency.
Figure fig. 16, fig. 17 are the headway profiles of OVM and P-OVM with
sensitivity constant $a=2$ under different perturbation frequency.

(a) $p=20$

(b) $p=15$

(c) $p=10$

(d) $p=5$

Figure 14: Headway profile of OVM with sensitivity constant $a=1.2$ and period
parameter $p=5,\;10,\;20,\;30$

(a) $p=20$

(b) $p=15$

(c) $p=10$

(d) $p=5$

Figure 15: Headway profile of P-OVM with sensitivity constant $a=1.2$ and
period parameter $p=5,\;10,\;20,\;30$

(a) $p=20$

(b) $p=15$

(c) $p=10$

(d) $p=5$

Figure 16: Headway profile of OVM with sensitivity constant $a=2.4$ and period
parameter $p=5,\;10,\;20,\;30$

(a) $p=20$

(b) $p=15$

(c) $p=10$

(d) $p=5$

Figure 17: Headway profile of P-OVM with sensitivity constant $a=2.4$ and
period parameter $p=5,\;10,\;20,\;30$

From the simulation results, we can see that if the frequency of the
sinusoidal disturbance is too high ($p=5$), the amplitude of disturbance can
get smaller without applying platoon-control even when OVM is not stable
($a=1.2$), due to the frequent change of optimal velocity and the delayed
reaction property of OVM. With platoon-controller applied, headway is
stabilized around equilibrium state for all model parameters and have better
stability when OVM is stable ($a=2.4$). In one sentence, P-OVM can increase
string stability under periodic disturbance.

##  5 Conclusions

In this paper, a platoon model with coordinated following strategy (P-OVM) and
a transition phase model with adjustable parameters (T-OVM) are proposed.
P-OVM model is proved always linearly stable but the model has a limited
practical use due to safety considerations when both the control and state
measurements are subject to errors. Stability of the base model and transition
phase model are also analysed and stability conditions for both models are
provided. To verify the theoretical results, numerical simulations are
performed on a ring road with an initial disturbance and an infinite road with
a periodic disturbance. Multiple model parameters are tested and analysed.
Results of the numerical simulations showed that with the proposed platoon
control applied, linear stability is guaranteed for all sensitivity constants.
The constants themselves just determine how quickly the disturbance is
attenuated through the platoon. With the transition phase model for fixed
total sensitivity, stability increases with sensitivity to the leading
vehicle, which is consistent with theoretical analysis.

The proposed P-OVM and T-OVM models produce considerably better results in
suppressing disturbances and maintaining string stability compared to non-
controlled model OVM. These models are simple forms of platooning that multi-
following models tend to focus on. This is a significant improvement compared
to previous studies on multi-following models. With the fast development of
CAV related technologies, the proposed platoon control offers a promising way
to mitigate oscillations in traffic flow.

This paper can be extended in various directions for future research,
including: (1) Add communication delay to the proposed models and analyse
their impact on stability; (2) Consider more general road conditions, e.g.
multi-lane ring road with mixed CAVs and HDVs. For such scenarios the control
design of leading vehicle will be adjusted accordingly [18]; (3) Conduct field
experiments with CAVs to test the control design under various traffic and
road conditions.

## Declaration of competing interest

The authors declare that there are no source of financial interests or
personal relationships that could have any impacts of the work in this paper.

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##  Appendix A Proof of Theorem eq. 6

In this section we give the proof of Theorem eq. 6:

###### Proof.

Similar to proof of Theorem theorem 3.2, we consider small deviation $y_{i}$
from the equilibrium solution and neglect higher order. Then we have

$\ddot{y}_{i}(t)=a\left[(V^{\prime}(h)(y_{i+1}(t)-y_{i}(t))-\dot{y}_{i}(t)\right],\;y_{N+1}=y_{1}$
(21)

which is a linear ODE system with $N$ equations. Suppose that $\lambda$ is an
eigenvalue of the system, and $\xi_{i}$ is the coefficient of $y_{i}$ with the
term $e^{\lambda t}$, then simplified from (eq. 21), $\lambda$ satisfies

$\lambda^{2}+a\lambda-aV^{\prime}(h)(\frac{\xi_{i+1}}{\xi_{i}}-1)=0$ (22)

This means $\xi_{i+1}/\xi_{i}=r$ is fixed for given $\lambda$, and note that
$\prod_{i=1}^{N}(\xi_{i+1}/\xi_{i})=1$, we have

$r=e^{\frac{2\pi k}{N}i},\;k=1,2,\dots,N$ (23)

Therefore $\lambda$ satisfies

$\lambda^{2}+a\lambda-aV^{\prime}(h)(e^{\frac{2\pi k}{N}i}-1)=0$ (24)

for some $k=1,2,\dots,N$. Then, to have a stable system of $y_{n}$, the real
parts of any $\lambda$ should be negative. By considering the solution of
$\lambda$ with smaller real parts we have

$\frac{a-\sqrt{d+\sqrt{d^{2}+e^{2}}}}{2}<0,$ (25)

where $d=a^{2}+4aV^{\prime}(h)\cos\frac{2\pi k}{N}-4aV^{\prime}(h)$ and
$e=-4aV^{\prime}(h)\sin\frac{2\pi k}{N}$. For the system to be stable, we need
this condition to be satisfied for every $k$. Now let
$\theta\triangleq\frac{2\pi k}{N}$ then (eq. 25) can be simplified to

$-V^{\prime}(h)\cos^{2}\theta+a\cos\theta+V^{\prime}(h)-a<0$ (26)

(eq. 26) holds if (eq. 6) holds, otherwise for N sufficiently large we can let
$\cos\theta=a/2V^{\prime}(h)$ and (eq. 26) will fail. ∎

##  Appendix B Proof of Theorem eq. 15

In this section we give the proof of Theorem eq. 15:

###### Proof.

Again we consider small deviation $y_{i}$ from the equilibrium solution and
neglect higher order. Then we have

$\ddot{y}_{i}(t)=\begin{cases}a\left[V^{\prime}(h)(y_{i+1}(t)-y_{i}(t))-\dot{y}_{i}(t)\right]+b\left[V^{\prime}(h)\frac{y_{N}(t)-y_{i}(t)}{N-i}-\dot{y}_{i}(t)\right],\;\text{if
}i\neq N\\\
(a+b)\left[V^{\prime}(h)(y_{i+1}(t)-y_{i}(t))-\dot{y}_{i}(t)\right],\;\text{if
}i=N\end{cases}$ (27)

which is a linear ODE system with $N$ equations. Suppose that $\lambda$ is an
eigenvalue of the system, and $\xi_{i}$ is the coefficient of $y_{i}$ with the
term $e^{\lambda t}$, then simplified from (eq. 27), $\lambda$ satisfies

$\lambda^{2}+(a+b)\lambda-
aV^{\prime}(h)(\frac{\xi_{i+1}}{\xi_{i}}-1)-\frac{bV^{\prime}(h)}{N-i}(\frac{\xi_{N}}{\xi_{i}}-1)=0,\;\text{if
}i\neq N$ (28)

and

$\lambda^{2}+(a+b)\lambda-(a+b)V^{\prime}(h)(\frac{\xi_{1}}{\xi_{N}}-1)=0$
(29)

When N is very large ($N\to\infty$), $bV^{\prime}(h)/(N-i)$ is very small for
most i, and the last term of (eq. 28) has minimal effect. This means
$\xi_{i+1}/\xi_{i}$ will converge to eigenvalues given in (eq. 23). Then
$\lambda$ will converge to values given by

$\lambda^{2}+(a+b)\lambda-aV^{\prime}(h)(e^{\frac{2\pi k}{N}i}-1)=0$ (30)

Follow the same steps of previous proof we can find the stability criterion
(eq. 15). A more rigorous mathematical proof will be part of future work. ∎