#  Comparison of gap-based and flow-based control strategies using a new
controlled stochastic cellular automaton model for traffic flow

Kayo Kinjo  Graduate School of Science and Engineering, Saitama University,
255 Shimo-Okubo, Sakura-ku, Saitama 338-8570, Japan<EMAIL_ADDRESS>Akiyasu Tomoeda  Faculty of Informatics, Kansai University, 2-1-1 Ryozenji-
cho, Takatsuki-shi, Osaka 569-1095, Japan

###### Abstract

Autonomous vehicles are widely considered an imperative element of future
transportation systems, and their adoption is expected to reduce traffic
congestion. What is an effective vehicle control to reduce it? This study
explores the impact of different vehicle control strategies on traffic flow
through simulations. To achieve this, we introduce a novel stochastic traffic
flow model called the controlled stochastic optimal velocity (CSOV) model,
which incorporates vehicle control effects. Two distinct control strategies
are applied within the CSOV model: gap-based control (GC) and flow-based
control (FC). The GC strategy regulates the velocity to equalize the gap
between the front and rear vehicles. Conversely, the FC strategy regulates the
velocity to maintain a consistent front and rear flow rate. The results show
that there were density regions where the GC strategy improved the flow rate.
However, only the GC strategy with the weak control resulted in a lower flow
rate compared to when there was no control. Conversely, the FC strategy
consistently improved the flow rate regardless of control strength, yielding
more robust results. Furthermore, when the two controls achieve similar flow
rates, the FC strategy provided a more robust velocity distribution for
density change than the GC strategy.

###### keywords:

Traffic flow, Traffic congestion , Stochastic cellular automaton , Vehicle
control, Autonomous vehicles, Fundamental diagram

###### MSC:

[2010] 00-01, 99-00

††journal: Journal of LaTeX Templates

##  1 Introduction

Modern life is heavily dependent on automobile-based transportation. This
automobile-centered transportation system is subject to traffic congestion,
which has various adverse effects on our living environment, such as reduced
transportation efficiency and an increased environmental burden due to exhaust
emissions. For example, [1] reported that in 2014, urban Americans drove 6.9
billion extra hours due to traffic congestion, consumed 3.1 billion gallons of
oil, and suffered economic losses of 160 billion U.S. dollars due to the
additional time and oil consumed. Hence, easing traffic congestion has become
an imperative mission for modern society.

The phenomenon of traffic flow, including traffic congestion, has attracted
considerable research interest not only in the fields of traffic and
transportation engineering but also in the natural sciences [2, 3, 4, 5, 6,
7]; This is because traffic flow is not only a subject of practical importance
for easing traffic congestion but also because it can be considered a typical
self-driven many-particle system that is far from equilibrium. Research on
traffic flow has a long history, dating back to the first half of the 20th
century. For example, Greenshield [8] conducted an experimental study on
traffic flow by measuring actual flows; he suggested a linear relation between
traffic density and velocity, which is still sometimes used for analytical
investigations. Starting with this pioneering study, research on traffic flow
has developed significantly around sophisticated mathematical models, such as
hydrodynamic models (e.g., see [9] for a brief overview on recent relevant
studies), car-following models (e.g., see [10] for a review on existing
studies on car-following models), and cellular automaton models (e.g., see
[11] for a review on CA models). In the last decade or so, as the performance
and convenience of measurement equipment have improved and accurate traffic
data have become more readily available, experimental studies have been
vigorously conducted using a circular road in homogeneous lane conditions on
flat ground [12, 13, 14, 15], an open road section in a suburban area [16, 17,
18], and the runway and taxiway of an old airport [19]. These significant
contributions to traffic flow research have deepened our understanding of the
dynamics of traffic flow.

One of the key contributions of these traffic flow studies was the elucidation
of the mechanism of phantom traffic jams, which occur spontaneously without
any apparent reasons, such as bottlenecks, merging, and lane changes. Phantom
traffic jams were clarified to be the collective effect of a many-particle
system that induces destabilization of the free flow state due to enhancement
of fluctuations; when the average vehicle density exceeds a certain critical
value, the transition to a jamming state occurs spontaneously [12, 13, 14].
One way to ease traffic congestion is to change the driving behavior of each
driver as the origin of phantom traffic jams lies within the collective motion
of vehicles. Driving strategies such as jam-absorption driving have also been
proposed and developed [20, 21, 22, 23].

Meanwhile, the traffic flow landscape has begun to change dramatically in
recent years with the advent of autonomous vehicles. Autonomous vehicles are
expected to lead the next paradigm shift in the field of transportation, and
the widespread use of autonomous vehicles is highly anticipated to alleviate
traffic congestion. However, the characteristics of mixed traffic between
human-driven vehicles (HVs) and autonomous vehicles (AVs) have to be
investigated and understood because realization of a fully-automated driving
society will take time and mixed traffic will be a normalcy in the near
future. Numerous studies have been conducted on mathematical models of
autonomous vehicles and mixed traffic flow with HVs [24, 25, 26, 27, 28]
(e.g., see [28] for a review of mathematical models for autonomous vehicles).

As the strategies of autonomous vehicles target the movements of individual
vehicles, they are more suitable for microscopic models, such as car-following
and cellular automaton models, where the focus is explicitly on each
individual. Examples of previous studies built on the car-following model are
as follows: Cui et al. [24] presented a theoretical analysis on the potential
of a single connected and autonomous vehicles (CAV) in a ring-road scenario;
Zhu et al. [25] proposed a new car-following model with adjustable sensitivity
and smooth factor to describe the autonomous vehicle’s movement to balance the
front and back headway and analyzed their impacts on traffic flow; Yao et al.
[26] studied the stability of mixed traffic flows comprising CAVs and HVs and
showed that under certain conditions, mixed traffic flows would not be
affected by the velocity for a certain penetration rate of the CAVs and a
desired time headway. Conversely, Ye and Yamamoto [27] proposed an extended CA
model that incorporates CAVs in the heterogeneous traffic flow and numerically
investigated the possible impact of CAVs on road capacity under different
penetration rates.

Although research on mathematical models for autonomous vehicles is
increasingly active, most models describe the control based on velocity or
distance; none have considered the flow rate or incorporated it explicitly
into the model. The essence of congestion mitigation lies in increasing the
number of vehicles passing through a measurement point per unit of time and
maximizing the flow rate. As this flow rate can be obtained by the product of
velocity and density, the strength of focusing on flow rate is to maximize
flow rate by balancing both velocity and distance between vehicles. For
instance, even if a series of vehicles keep uniform spacing, if they are
driven so slowly that the flow rate is lowered, we cannot say this is a
solution to traffic congestion.

The car-following model has an advantage in theoretical analyses such as
stability analysis, but as described in [11], the cellular automaton model
also has advantages, such as computational efficiency, simplicity of control
algorithms, and flexible scenario reproducibility. These advantages are
effective and important factors in large-scale simulations. Therefore, this
study proposes a new model based on the stochastic optimal velocity (SOV)
model [29] that encompasses the totally asymmetric simple exclusion process
(TASEP) [30] and the zero range process (ZRP) [31]. The model can obtain
analytically important exact solutions and reproduce metastability, which is
an essential feature of traffic flows. While maintaining the framework of the
stochastic CA model, we propose a new model that incorporates the effect of
control. Moreover, we clarify the characteristics of traffic flows with a
mixture of controlled vehicles.

The remainder of this paper is organized as follows. In Section II, we briefly
review the SOV model and then propose a new controlled SOV model (CSOV) model
with two control strategies, gap-based control (GC) and flow-based control
(FC). The numerical results for the CSOV model are shown in Section III, and
Section IV is devoted to the concluding discussions.

##  2 Controlled SOV (CSOV) model

We propose a novel model of controlled vehicles called the CSOV model to
explore the difference between vehicle control strategies and their impact on
mixed traffic through simulations. The model builds upon the SOV model [29],
which describes the movement of HVs as a hopping probability of moving to the
next cell. As most vehicles in use today have a vehicle control system as a
driving support rather than an autonomous driving system, this study is
modeled based on the concept of a HV with control strategies. Thus, the CSOV
model proposed in this study incorporates an additional term for vehicle
control in the SOV model.

###  2.1 HVs in the SOV model

Let us start by briefly introducing the SOV model. The SOV model is a
stochastic cellular automaton model; hence, both time and position are
discretized: the road is divided into $L$ identical cells such that each cell
can accommodate at most one vehicle at one time step, and the velocity of a
vehicle is represented as a hopping probability of moving to the next cell.
Figure 1 (a) shows the schematic illustration of the SOV model. When the $i$th
vehicle is located at position $x_{i}^{t}\in\\{1,2,3,\cdots,L\\}$ at time step
$t$, and its velocity is $v_{i}^{t}\in[0,1]$, the velocity of the $i$th
vehicle at the next time step $t+1$ is given by

$\displaystyle v_{i}^{t+1}$ $\displaystyle=(1-\alpha)v_{i}^{t}+\alpha V(\Delta
x^{t}_{i}).$ (1)

The above expression can be regarded as the weighted average between the
current velocity $v_{i}^{t}$ and an ideal velocity for a given front gap
$\Delta x^{t}_{i}:=x_{i+1}^{t}-x_{i}^{t}-1$, which is given by an optimal
velocity (OV) function, in a proportion of $(1-\alpha):\alpha$. Here, as in
the previous study [29], we employ the OV function $V(\Delta x)\in[0,1]$,
which is given by

$\displaystyle V(\Delta x)$ $\displaystyle:=\dfrac{\tanh(\Delta
x-c)+\tanh(c)}{1+\tanh(c)},$ (2)

where the constant $c$ was set to 1.0 throughout this paper. Figure 1(b) plots
the OV function of Equation (2).

Figure 1: (a) Schematic illustration of the SOV model and (b) OV function. As
the headway is discretized in the SOV model, the values of the OV function are
represented by the blue points.

The SOV model has notable features to describe the HVs. First, it can
reproduce the metastable state commonly found in practical traffic, such as
highway traffic. In the metastable state, a subtle perturbation of flow is
sufficient to trigger the transition into the congested state. Figures 2 (a)
and (b) show a fundamental diagram of the measured traffic flow on a highway
and that of the SOV model, respectively. Within the low-density region, the
flow increases as the density increases (Figure 2(a)). However, beyond a
certain density, called the critical density, a further increase in the
density leads to a discontinuous reduction of the stationary flow. It is also
reported that this characteristic is also observed in the SOV model (Figure
2(b)). Moreover, the SOV model includes two solvable stochastic processes: the
TASEP [30] and the ZRP [31]. When $\alpha=0$ in Equation (1), the SOV model
reduces to the TASEP, where the velocity $v_{i}$ remains constant as
$v_{i}=v_{i}^{0}$ for $i=1,\cdots,N$. Conversely, when $\alpha=1$, the
velocity depends only on the second term of Equation (1), which represents a
function in terms of the front gap. This configuration corresponds to the ZRP.

Figure 2: Fundamental diagrams of (a) measurement data and (b) SOV model for a
system length $L$ of 200, $\alpha=0.01$, under the periodic boundary
condition.

Furthermore, we confirmed that the SOV model exhibits car-following behavior
that is consistent with empirical experiments [16]. As a vehicle in a platoon
is positioned further behind the leading vehicle, there is a concave increase
in the standard deviation of the time series of its velocities, revealed for
the first time by the empirical experiments in [16]. This characteristic is
not reproduced by traditional deterministic car-following models, such as the
optimal velocity model. The details of the simulation are as follows: We
started by placing a platoon of 25 vehicles on a 1000-cell circuit. While the
leading vehicle accelerated and its hopping probability reached 1.0, we
evaluated the standard deviation of the time series of velocity for each
vehicle. To ensure accuracy, we conducted this simulation three times and used
the average of the three standard deviations for the plot shown in Figure 3.
In Figure 3, the black points represent the means of the three standard
deviations for each vehicle, and the green filled band corresponds to the
error of the evaluated standard deviations.

Figure 3: Standard deviation of the time series of the hopping probability of
each vehicle for the SOV model. The black points are the simulation results,
the green filled band corresponds to the error, and the blue line is the
approximate curve.

###  2.2 Controlled vehicles in the CSOV model

In this paper, CVs aim for the ideal velocity at each time step through
controls; this ideal velocity is referred to as the target velocity. The CV’s
velocity is calculated as a weighted average between the velocity of the HV as
given in Equation (1) and the target velocity specified by a control strategy.
Here, we propose two vehicle-control strategies for the CSOV model: the gap-
based control (GC) strategy and the flow-based control (FC) strategy.
Hereinafter, we refer to vehicles with the GC and the FC as GCVs and FCVs,
respectively.

To ensure that the velocity $v_{i}^{t+1}$ remains within the range $[0,1]$ of
stochastic models, we introduce a weight parameter $\beta$ that regulates the
proportion of the two factors: the velocity given in Equation (1) and the
target velocity. Thus, we define the velocity of the next time step, given as
a weighted average in the following form

$\displaystyle v_{i}^{t+1}$
$\displaystyle=(1-\beta)\Bigl{(}(1-\alpha)v_{i}^{t}+\alpha V(\Delta
x_{i}^{t})\Bigr{)}+\beta C(\cdot),$ (3)

where $C(\cdot)$ represents a function that determines the target velocity by
each of the controls described in the following Subsections 2.3 and 2.4. In
the case when $\beta=0$, which corresponds to the scenario where no control is
applied, Equation (3) is reduced to the SOV model, as represented by Equation
(1). Hereinafter, we denote the argument of the control function $C(\cdot)$ as
$\Delta\tilde{x}_{i}^{t+1}$, which is the gap for the controls, meaning the
target gap at the next time step.

###  2.3 Gap-based control (GC)

The GC strategy is a control strategy for regulating the motion of a GCV by
smoothing the gap between the front and rear of the GCV to reduce the
unevenness of the distances between the vehicles, which is based on the idea
of [25]. The target velocity given by the control function $C(\cdot)$ is
defined as the value of the optimal velocity function evaluated as the mean of
the front and rear gaps:

$\displaystyle C_{G}(\Delta\tilde{x}_{i}^{t+1})$
$\displaystyle=V(\Delta\tilde{x}_{i}^{t+1})=V\left(\frac{\Delta
x_{i}^{t}+\Delta x_{i-1}^{t}}{2}\right),$ (4)

where $\Delta x_{i}^{t}$ and $\Delta x_{i-1}^{t}$ denote the front and rear
gaps for the $i$th vehicle at time step $t$, respectively, and
$\Delta\tilde{x}_{i}^{t+1}$ is the mean of these two gaps, which can be a
rational number. The variable of the OV function can be accepted to have
rational numbers as its arguments. Note that, to distinguish the control
strategies clearly, the control function $C(\cdot)$ for the GC strategy is
denoted as $C_{G}(\cdot)$.

###  2.4 Flow-based control (FC)

We define the local flow first to introduce the FC strategy. The local flow of
the $i$th vehicle $Q_{i}^{t}$ at time step $t$ is defined as the product of
the local density $\rho_{i}^{t}$ and its velocity $v_{i}^{t}$ in terms of the
gap $\Delta x_{i}^{t}$:

$\displaystyle Q_{i}^{t}=\rho_{i}^{t}v_{i}^{t}=\frac{1}{\Delta
x_{i}^{t}+1}v_{i}^{t}.$ (5)

Note that the density used in the calculation of the local flow is the
reciprocal of the front gap. To determine the velocity at the next time step,
we assume that an FCV will adjust its velocity so that its local flow at the
next time step will be equal to the average flow of the vehicles in front
$Q_{i}^{t}$ and behind it $Q_{i-1}^{t}$:

$\displaystyle Q_{i}^{t+1}=\frac{Q_{i}^{t}+Q_{i-1}^{t}}{2}.$ (6)

By the definition of the local flow in Equation (5), the FCV can reach the
ideal flow with the gap $\Delta\tilde{x}_{i}^{t+1}$:

$\displaystyle
Q_{i}^{t+1}=\frac{1}{\Delta\tilde{x}_{i}^{t+1}+1}V(\Delta\tilde{x}_{i}^{t+1}),$
(7)

where $\Delta\tilde{x}_{i}^{t+1}$ can be obtained by solving Equation (7)
numerically, since the left-hand side of Equation (7) can be calculated by the
flow rate at time step $t$ from Equation (6). Provided that the gap
$\Delta\tilde{x}_{i}^{t+1}$ is given, the target velocity for the FCV is
determined by evaluating the optimal velocity function, that is,

$\displaystyle\begin{split}C_{F}\left(\Delta\tilde{x}_{i}^{t+1}\right)=V\left(\Delta\tilde{x}_{i}^{t+1}\right).\end{split}$
(8)

Similarly, the control function $C(\cdot)$ for the FC strategy is denoted as
$C_{F}(\cdot)$.

We will remark on the choice of the gap
$\Delta\tilde{x}_{i}^{t+1}=-1+1/\tilde{\rho}$ determined by solving Equation
(7) of the density $\tilde{\rho}$. The equation leads to two possible cases,
depending on the flow $Q$ on the left-hand side of the equation. The first
case is when there are any solutions to the equation, and the second case is
when there is no solution. Figure 4 (a) and (b) schematically depict the
density to be chosen with the green points for the above two cases. In the
former case, a line $Q=Q_{a}$ on the left-hand side of Equation (7) meets the
curve in terms of $\rho$ on the right-hand side of Equation (7), as shown in
Figure 4 (a). Though we have two points of intersection, we choose the left-
side intersection $\rho=\tilde{\rho}$ to calculate the gap. This is because it
is more reasonable to aim for a low density and free flow condition than a
high density and jamming flow condition to achieve the same flow rate. In the
latter case, we choose the density $\tilde{\rho}_{M}$ that maximizes the
right-hand side to calculate the gap.

Figure 4: Two cases of Equation (7). (a) When there are two points of
intersection, the left-side intersection is adopted for determining
$\Delta\tilde{x}_{i}^{t+1}=-1+1/\tilde{\rho}$. (b) When there is no point of
intersection, we adopt $\Delta\tilde{x}_{i}^{t+1}$ as $-1+1/\tilde{\rho}_{M}$.

##  3 Simulations

We conducted computer simulations of complete CV and mixed traffic flow. The
initial positions of the vehicles were randomly assigned on a one-lane and
200-cell circuit with the periodic boundary condition; once the number of
vehicles on the circuit was determined, no vehicles were allowed to enter or
exit the circuit, that is, the global density of the system is fixed. The
simulations were terminated after 50,000 time steps. As depicted in Figure 1,
the rear gap for the $i$th vehicle is $\Delta x_{i-1}^{t}$. As we are using
periodic boundary conditions, the front gap of the first vehicle and the rear
gap of the last vehicle are the same. Denoting the number of vehicles by $N$,
this gap is given by $x_{1}^{t}-x_{N}^{t}+L-1$.

To investigate the impact of vehicle placement on the performance of the
control strategies, we considered two different patterns of CV placement for
the mixed traffic flow: uniform placement, in which the CVs were evenly
distributed among vehicles on the circuit, and block placement, in which the
CVs were grouped together as a block.

###  3.1 Complete CV traffic

Figure 5: Fundamental diagrams for the complete CV traffic. (a) Fundamental
diagram when the CVs are implemented on the GC strategy. (b) Fundamental
diagram when the CVs are implemented on the FC strategy. The black points
represent the fundamental diagram for the SOV model for comparison. Figure 6:
Time–space diagrams for (a) SOV model and (b) CSOV model with the GC strategy
when the density $\rho=0.2$. The vehicles are placed as a platoon.

Figures 5 (a) and (b) show the fundamental diagrams of complete CV traffic
cases where the GC and FC strategies are applied, respectively. In each panel,
two scenarios when the control parameter $\beta=0.1$ (weak control effect) and
0.9 (strong control effect) are plotted with empty squares, along with the
fundamental diagram of the SOV model with black points for comparison. In the
free flow state when the density $\rho\in[0,0.1]$, the flow of both control
strategies coincides with that of the SOV model regardless of the strength of
the control parameter $\beta$ because each term in Equation (3) approaches 1
in the free flow state. In the other densities, the complete CV traffic with
both control strategies results in higher flow than the SOV model, except in
the case of the GC strategy with a weak control parameter $\beta=0.1$, where
the density $\rho\in[0.17,0.25]$.

In that density region $\rho\in[0.17,0.25]$, the SOV model surprisingly shows
higher flow than the CSOV model with the GC strategy for $\beta=0.1$, plotted
in the green squares in Figure 5 (a). This is likely due to the GCVs slowing
down when the GCVs leave the region of traffic congestion. We conducted an
additional simulation to observe such behavior of the GCVs. Figures 6 (a) and
(b) show the time–space diagrams for the SOV and CSOV models with the GC
strategy, respectively, when the density $\rho=0.2$. The time–space diagrams
depict the trajectory of each vehicle, with the color indicating the hopping
probability. In this additional simulation, the vehicles are placed in a
platoon with velocity $v=0.5$ at the initial time. First, the velocities of
the GCVs in the platoon decrease to around 0, indicated in red in Figure 6 (b)
because the GCVs take account of both front and rear gaps. After the GCVs
leave the traffic congestion, they drive faster, maintaining sufficient gaps
compared to the HVs. However, as shown in Figure 6 (b), the traffic congestion
remains even at the time when the congestion is eased in the SOV model.

A similar interpretation can be applied to the fundamental diagram shown in
Figure 5 (a) based on the additional simulation. When a GCV is located in
front of a traffic congestion, since the vehicle behind, which is stuck in the
traffic congestion, drives close to the GCV, the rear gap of the GCV is
considerably smaller than its front gap between the GCV and the vehicle in the
front, which is not stuck in the traffic congestion. Notably, a GCV determines
its velocity considering both front and rear gaps, whereas an HV determines
its velocity only by its front gap. Thus, the GCV is affected by its rear gap
and decelerates.

In contrast, the CSOV model with the FC strategy improved the flow in the
metastable and congestion state, as shown in Figure 5 (b), even when the
effect of the control is weak such as $\beta=0.1$. Moreover, the FC strategy
achieved a higher flow than the GC strategy, as shown by the two panels in
Figure 5. The shape of the fundamental diagram for the complete CV traffic,
shown in Figure 5 (b), is different from that of the measured fundamental
diagram in Figure 2 (b) because the FCVs are not affected by the velocities of
their rear vehicles and thus, exhibit almost constant velocities.

###  3.2 Mixed traffic

To investigate the impact of CV on mixed traffic flows, we conducted the
simulations for the CV penetration rates from 10 to 90% by 10%. The results
indicate that even with a low penetration rate of 30% and weak control
($\beta=0.1$), the traffic flow is improved compared to that of the SOV model.
Moreover, the characteristics of each control strategy are evident in the
fundamental diagrams and remain consistent up to 80% penetration rate. Beyond
80% penetration rate, the mixed traffic exhibits similar behavior to that of
complete CV traffic. Figure 7 (a), (b), and (c) display the fundamental
diagrams for CV penetration rates of 30%, 50%, and 70%, respectively. The
shape of the data points indicates the placement of the CVs: squares represent
block placement, and circles represent uniform placement.

Figure 7: Fundamental diagram of the mixed traffic flow with CVs and HVs at
different penetration rates of CVs. For both controls, the control parameter
$\beta$ is set as 0.1. The black points represent the SOV model case for
comparison.

Our simulations revealed that the placement of GCVs affects flow. For all CV
penetration rates in Figure 7, the block placement of the GC strategy extends
the density range of the metastable state to higher densities by comparing the
uniform placement of the GC strategy as observed in Figure 7 (a), (b) and (c).
While the uniform placement of the GC strategy does not extend the range of
the metastable state, it results in higher flow in the metastable state
compared to the SOV model and the block placement of the GC strategy.

Conversely, the flow of the mixed FCV traffic remains independent of their
placement, yielding almost identical flow rates. This is illustrated by the
red squares and yellow circles in Figure 7 (a), (b), and (c). The presence of
FCVs in the mixed traffic slightly extends the range of densities for the free
flow state more than that of GCVs. Particularly, the end density of the free
flow state is approximately $\rho\sim 0.115$ and 0.125 for penetration rates
of 30% and 50%, respectively, compared to $\rho=0.1$ in the SOV model. At a
70% penetration rate, discerning the boundary of the free flow state is
challenging. The flow, in this case, increases linearly within the free flow
state density range and smoothly connects to the flow of the metastable state.
The FC strategy expands the range of densities for the metastable state.
Furthermore, it provides a higher flow rate compared to the SOV model at 30%
and 50% penetration rates.

To further analyze these results, we plotted time–space diagrams to visualize
the flow obtained from our simulations. Figure 8 and 9 present the time–space
diagrams for a scenario in which 50 vehicles are on the circuit, and 30% of
them are controlled, which corresponds to the density of $\rho=0.25$ in Figure
7 (a). At this density, the flow of the block GCV placement is in the
metastable state, whereas the flow of the uniform GCV placement is in the
jamming state. The time–space diagrams depict the trajectory of each vehicle,
with the color indicating the hopping probability. While a single large
congestion is present in the uniform placement (see Figure 8 (a)), the
scenario of block placement with GC in Figure 8 (b) results in higher flow
compared to the uniform placement. This is due to the formation of smaller
congestions at various locations, which prevent the formation of a single
large congestion. These small congestions are caused solely by GCVs.
Conversely, in the case of the FC strategy, FCVs do not participate in the
congestion, resulting in the absence of congestion, as seen in Figure 9.

Figure 8: Time–space diagrams for mixed traffic flow with GCVs and HVs when
GCVs are put (a) uniformly and (b) in a block. The number of vehicles is 50 on
the circuit, and the penetration rate of the GCVs is 30 %.  Figure 9:
Time–space diagrams for mixed traffic flow with FCVs and HVs when FCVs are put
(a) uniformly and (b) in a block. The number of vehicles is 50 on the circuit,
and the penetration rate of the FCVs is 30 %.

####  3.2.1 Comparison of the two controls under similar performance

Figure 10: Fundamental diagram for the HVs traffic (black points) and the
mixed traffic with CVs and HVs (squares). The red empty squares represent the
case when the CVs are GCVs, and the blue ones represent the case when the CVs
are FCVs.

In the previous subsection, we observed that the flow of traffic with FCVs was
higher than that with GCVs for the same control parameter $\beta$. However, it
remains unclear whether there are any differences between the two control
strategies when they perform similarly; that is, the fundamental diagrams for
the FC and the GC strategies show almost the same peak flow at the same
density. The FC for $\beta=0.01$ plotted in blue squares and the GC for
$\beta=0.9$ plotted in red squares have similar shapes of fundamental diagrams
for a mixed traffic system comprising 50% HVs and 50% CVs placed in a block
formation (Figure 10). Both control strategies produced higher flow rates than
the SOV model for $\rho>0.1$. When comparing the FC and the GC cases, we
identified three distinct density regions: (i) a region where the GC performed
slightly better than the FC, (ii) a region where both controls performed
similarly, and (iii) a region where the FC was in a metastable state but the
GC was in a jamming state.

We conducted a detailed analysis to investigate the traffic further by
plotting time–space diagrams and velocity distributions for each region. When
plotting the velocity distributions, we specifically considered velocities at
a fixed position during the time interval $t\in[5000,6500]$, representing a
period when the traffic had reached a steady state.

In region (i), as indicated by Figure 10, the GC leads to a higher flow
compared to the FC. The corresponding time–space diagram in Figure 11 (a)
reveals that the GCVs drive with their velocity $v\sim 1$, indicated by purple
lines with small gaps in the time–space diagram. The driving strategy of the
GCVs gives the HVs more space than that of the FCVs, as shown in Figure 11
(b), resulting in a higher total flow on the circuit. The velocity
distribution in Figure 11 (c) reflects this result; for the GCV mixed traffic,
60% of the vehicles on the circuit passing through a cell are in the velocity
range $v\in[0.95,1.0]$. Conversely, for the mixed FCV traffic case, although
the mode of the velocity distribution is $v\in[0.95,1.0]$, the velocity is
uniformly distributed in the range 0.7 to 0.95. This wider velocity
distribution for the mixed FCV traffic results in a lower flow rate than for
the mixed GCV traffic.

Figure 11: Comparison of the mixed GCV traffic and mixed FCV traffic. The CVs
with both control strategies are placed in a block, and the total number of
the vehicles is 30, which is in the region (i) of Figure 10. Time–space
diagrams when the CVs use (a) GC strategy and (b) FC strategy. Velocity
distribution of the vehicles in the circuit for (c) GCV mixed traffic and (d)
FCV mixed traffic. Figure 12: Comparison of the mixed GCV traffic and mixed
FCV traffic. The CVs with both control strategies are placed in a block, and
the total number of the vehicles is 50, which is in the region (ii) of Figure
10. Time–space diagrams when the CVs use (a) GC strategy and (b) FC strategy.
Velocity distribution of the vehicles in the circuit for (c) GCV mixed traffic
and (d) FCV mixed traffic.

However, notably, for the mixed GCV traffic, we observed stop-and-go waves
that do not affect all vehicles in the circuit. The traffic congestion is
indicated in red narrow bands in Figure 11 (a) and Figure 12 (a). We replotted
those time–space diagrams in the upper panels of Figure 13, which identify the
GCVs (red lines) and the HVs (black lines). The lower panels are corresponding
time–space diagrams indicating velocity in color. Note that the time–space
diagrams of each upper and lower panel are identically the same. As can be
seen in the upper panels in Figure 13 (a) and (b), the characteristic
congestions occur in the block of the GCVs.

These congestions are particularly prominent when the congestion formed by the
HVs, indicated in blue or green, passes through the platoon of the GCVs. This
observation suggests that the GCVs with a large control parameter, such as
$\beta=0.95$, tend to maintain high speeds with minimal front gaps.
Consequently, the congestion propagates through stop-and-go waves within the
platoon of the GCVs.

Figure 13: Time–space diagrams for (a) 30 vehicles and (b) 50 vehicles in the
circuit. The upper panels identify the HVs (black lines) and the GCVs (red
lines). The lower panels of (a) and (b) are identical to Figure 11 (a) and 12
(a), respectively. The time–space diagrams of each upper and lower panel are
identically the same.

Interestingly, even though the flow in the fundamental diagram is almost the
same, the distribution of the velocity is apparently different between the two
control strategies, as shown in Figure 12 (c). A wider velocity distribution
is associated with less throughput and more fuel consumption in [32]. However,
in the velocity distribution for the mixed GCV traffic, the velocities of the
vehicles are distributed more widely than in the mixed FCV traffic. For
example, the velocity distribution of Figure 12 (c) shows 10% of vehicles in
the circuit have velocity $v\in[0.0,0.05]$ as the GCVs trapped in the
characteristic traffic congestion halt.

In region (iii), the GCVs cannot drive as fast as in regions (i) and (ii); the
mixed GCV traffic is in the jamming state as shown in Figure 14 (a).
Conversely, for the FC case, the median velocity, which is the middle value in
the velocity distributions, were recorded as 0.88, 0.74, and 0.56 for regions
(i), (ii), and (iii), respectively. These values provide insight into the
overall traffic behavior and reveal decreasing median velocities from regions
(i) to (iii), suggesting a higher congestion level in region (iii).

Figure 14: Comparison of the mixed GCV traffic and mixed FCV traffic. The CVs
with both control strategies are placed in a block, and the total number of
the vehicles is 80, which is in region (iii) of Figure 10. Time–space diagrams
when the CVs use (a) GC strategy and (b) FC strategy. Velocity distribution of
the vehicles in the circuit for (c) GCV mixed traffic and (d) FCV mixed
traffic.

##  4 Conclusion

We proposed the CSOV model, which incorporates the effect of vehicle control
into the well-known SOV model in the form of a stochastic cellular automaton.
By exploiting a notable characteristic of the SOV model, which reproduces the
metastable state, we conducted a comprehensive investigation into the effects
of vehicle control on both complete CV and mixed traffic scenarios,
particularly focusing on the region near the metastable state where the
density ranges from $\rho=0$ to $\rho=0.3$. This model enables us to explore
more realistic aspects of traffic flow with CVs, including changes in the
metastable state and flow characteristics in the presence of CVs.

In this study, we implemented two control strategies within the CSOV model:
the GC and FC strategies. The GC strategy regulates a CV’s velocity to
optimize the smoothing of its front and rear gaps, whereas the FC strategy
adjusts a CV’s velocity to ensure the smoothness of both the front and rear
flow rates. When constructing the FC strategy, our focus was on maximizing the
flow rate to ease traffic congestion by increasing the number of vehicles
passing through per unit of time.

For complete CV traffic, we observed that some GCVs tend to decelerate when
leaving a traffic congestion, particularly for weak strength parameter
$\beta$. This behavior can be attributed to the influence of the velocities of
rear vehicles in the congestion. However, the FC strategy, which considers not
only gaps but also velocities, prevents such deceleration when leaving
congestion. This difference is evident in the fundamental diagrams when the
two control strategies were applied to complete CV traffic, highlighting
variations in the density that achieves the maximum flow between the two
strategies.

In the case of mixed traffic involving CVs and HVs, we found that regardless
of the placement of CVs, the FC strategy expands the region of the metastable
state and achieves higher flow in this state compared to the GC strategy. When
GCVs are placed in a block, the formation of localized but prominent
congestions solely caused by GCVs prevents the emergence of a single large
congestion in the system. Although the two proposed control strategies achieve
similar flow, their configurations differ in terms of time–space diagrams and
velocity distributions. The GC strategy effectively maintains traffic flow
while exhibiting a wide velocity distribution among vehicles. Conversely, the
FC strategy maintains the width of the velocity distribution until reaching
the metastable state; this implies that the FC strategy may lead to lower fuel
consumption compared with the GC strategy. The characteristics of each control
strategy are evident in the fundamental diagrams and remain consistent up to
80% penetration rate.

In conclusion, the CSOV model proposed in this study shows promising potential
and warrants further exploration with additional control strategies. An
important question for future research is to determine the driving behavior
parameters and compare them with empirical experiments. For instance, studies
such as [33] utilize measured vehicle trajectory data from highways to assign
values to a mathematical model. This approach could be applied to the CSOV
model to anticipate the future traffic flow landscape in a scenario where
autonomous vehicles become more prevalent. We can enhance the model’s accuracy
and applicability by incorporating real-world data; this would provide
valuable insights for transportation planning and management.

## Acknowledgements

This research was partially supported by JSPS Grant-in-Aid for Scientific
Research (C) JP17K05147, JP23K11139, and the graduate program Science and
Technology for Global Leaders at Ochanomizu University. We would like to thank
Yuri Nakayama and Reika Matsunaga for useful discussions.

## References

  * [1] D. L. Schrank, B. Eisele, T. J. Lomax, J. Bak, 2015 urban mobility scorecard, 2015\.
  * [2] D. Chowdhury, L. Santen, A. Schadschneider, Statistical physics of vehicular traffic and some related systems, Physics Reports 329 (4) (2000) 199–329.
  * [3] D. Helbing, Traffic and related self-driven many-particle systems, Reviews of modern physics 73 (4) (2001) 1067.
  * [4] T. Nagatani, The physics of traffic jams, Reports on progress in physics 65 (9) (2002) 1331.
  * [5] K. Nagel, P. Wagner, R. Woesler, Still flowing: Approaches to traffic flow and traffic jam modeling, Operations research 51 (5) (2003) 681–710.
  * [6] A. Schadschneider, D. Chowdhury, K. Nishinari, Stochastic transport in complex systems: from molecules to vehicles, Elsevier, 2010.
  * [7] M. Treiber, A. Kesting, Traffic Flow Dynamics: Data, Models and Simulation,, Springer, 2013.
  * [8] B. D. Greenshields, J. Bibbins, W. Channing, H. Miller, A study of traffic capacity, in: Highway research board proceedings, Vol. 14, Washington, DC, 1935, pp. 448–477.
  * [9] A. Jafaripournimchahi, Y. Cai, H. Wang, L. Sun, B. Yang, Stability analysis of delayed-feedback control effect in the continuum traffic flow of autonomous vehicles without v2i communication, Physica A: Statistical Mechanics and its Applications 605 (2022) 127975.
  * [10] Z. Wang, Y. Shi, W. Tong, Z. Gu, Q. Cheng, Car-following models for human-driven vehicles and autonomous vehicles: A systematic review, Journal of Transportation Engineering, Part A: Systems 149 (8) (2023) 04023075.
  * [11] J. Tian, C. Zhu, R. Jiang, M. Treiber, Review of the cellular automata models for reproducing synchronized traffic flow, Transportmetrica A: Transport Science 17 (4) (2021) 766–800.
  * [12] Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. ichi Tadaki, S. Yukawa, Traffic jams without bottlenecks—experimental evidence for the physical mechanism of the formation of a jam, New Journal of Physics (2008).  doi:10.1088/1367-2630/10/3/033001.
  * [13] A. Nakayama, M. Fukui, M. Kikuchi, K. Hasebe, K. Nishinari, Y. Sugiyama, S.-i. Tadaki, S. Yukawa, Metastability in the formation of an experimental traffic jam, New Journal of Physics 11 (8) (2009) 083025.
  * [14] S.-i. Tadaki, M. Kikuchi, M. Fukui, A. Nakayama, K. Nishinari, A. Shibata, Y. Sugiyama, T. Yosida, S. Yukawa, Phase transition in traffic jam experiment on a circuit, New Journal of Physics 15 (10) (2013) 103034.
  * [15] R. E. Stern, S. Cui, M. L. D. Monache, R. Bhadani, M. Bunting, M. Churchill, N. Hamilton, R. Haulcy, H. Pohlmann, F. Wu, B. Piccoli, B. Seibold, J. Sprinkle, D. B. Work, Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transportation Research Part C-emerging Technologies (2018).  doi:10.1016/j.trc.2018.02.005.
  * [16] R. Jiang, M.-B. Hu, H. Zhang, Z.-Y. Gao, B. Jia, Q.-S. Wu, B. Wang, M. Yang, Traffic experiment reveals the nature of car-following, PloS one 9 (4) (2014) e94351.
  * [17] R. Jiang, M.-B. Hu, H. Zhang, Z.-Y. Gao, B. Jia, Q.-S. Wu, On some experimental features of car-following behavior and how to model them, Transportation Research Part B: Methodological 80 (2015) 338–354.
  * [18] C.-J. Jin, W. Wang, R. Jiang, H. Zhang, H. Wang, M.-B. Hu, Understanding the structure of hyper-congested traffic from empirical and experimental evidences, Transportation Research Part C: Emerging Technologies 60 (2015) 324–338.
  * [19] R. Jiang, C.-J. Jin, H. Zhang, Y.-X. Huang, J.-F. Tian, W. Wang, M.-B. Hu, H. Wang, B. Jia, Experimental and empirical investigations of traffic flow instability, Transportation research part C: emerging technologies 94 (2018) 83–98.
  * [20] R. Nishi, A. Tomoeda, K. Shimura, K. Nishinari, Theory of jam-absorption driving, Transportation Research Part B: Methodological 50 (2013) 116–129.
  * [21] Y. Taniguchi, R. Nishi, T. Ezaki, K. Nishinari, Jam-absorption driving with a car-following model, Physica A: Statistical Mechanics and its applications 433 (2015) 304–315.
  * [22] R. Nishi, Theoretical conditions for restricting secondary jams in jam-absorption driving scenarios, Physica A: Statistical Mechanics and its Applications 542 (2020) 123393.
  * [23] R. Nishi, T. Watanabe, System-size dependence of a jam-absorption driving strategy to remove traffic jam caused by a sag under the presence of traffic instability, Physica A: Statistical Mechanics and its Applications 600 (2022) 127512\.
  * [24] S. Cui, B. Seibold, R. Stern, D. B. Work, Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations, in: 2017 IEEE Intelligent Vehicles Symposium (IV), IEEE, 2017, pp. 1336–1341.
  * [25] W.-X. Zhu, H. Zhang, Analysis of mixed traffic flow with human-driving and autonomous cars based on car-following model, Physica A: Statistical Mechanics and its Applications 496 (2018) 274–285.
  * [26] Z. Yao, R. Hu, Y. Wang, Y. Jiang, B. Ran, Y. Chen, Stability analysis and the fundamental diagram for mixed connected automated and human-driven vehicles, Physica A: Statistical Mechanics and its Applications 533 (2019) 121931.
  * [27] L. Ye, T. Yamamoto, Modeling connected and autonomous vehicles in heterogeneous traffic flow, Physica A Statistical Mechanics and its Applications 490 (2018) 269–277.
  * [28] H. Yu, R. Jiang, Z. He, Z. Zheng, L. Li, R. Liu, X. Chen, Automated vehicle-involved traffic flow studies: A survey of assumptions, models, speculations, and perspectives, Transportation research part C: emerging technologies 127 (2021) 103101.
  * [29] M. Kanai, K. Nishinari, T. Tokihiro, Stochastic optimal velocity model and its long-lived metastability, Physical Review E 72 (3) (2005) 035102.
  * [30] B. Derrida, M. R. Evans, V. Hakim, V. Pasquier, Exact solution of a 1d asymmetric exclusion model using a matrix formulation, Journal of Physics A: Mathematical and General 26 (7) (1993) 1493.
  * [31] O. J. O’Loan, M. R. Evans, M. E. Cates, Jamming transition in a homogeneous one-dimensional system: The bus route model, Phys. Rev. E 58 (1998) 1404–1418.
  * [32] R. E. Stern, S. Cui, M. L. Delle Monache, R. Bhadani, M. Bunting, M. Churchill, N. Hamilton, H. Pohlmann, F. Wu, B. Piccoli, et al., Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transportation Research Part C: Emerging Technologies 89 (2018) 205–221.
  * [33] A. Malibari, A. Higatani, W. Saleh, Assessing the impacts of autonomous vehicles on road congestion using microsimulation, Sensors 22 (12) (2022).