# MPC-Based Emergency Vehicle-Centered Multi-Intersection Traffic Control

Mehdi Hosseinzadeh, , Bruno Sinopoli, , Ilya Kolmanovsky, ,
and Sanjoy Baruah This research has been supported by National Science
Foundation under award numbers ECCS-1931738, ECCS-1932530, and ECCS-2020289.M.
Hosseinzadeh and B. Sinopoli are with the Department of Electrical and Systems
Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA
(email<EMAIL_ADDRESS>bsinopoli@wustl.edu).I. Kolmanovsky is
with the Department of Aerospace Engineering, University of Michigan, Ann
Arbor, MI 48109, USA (email: ilya@umich.edu).S. Baruah is with the Department
of Computer Science and Engineering, Washington University in St. Louis, St.
Louis, MO 63130, USA (email:baruah@wustl.edu).

###### Abstract

This paper proposes a traffic control scheme to alleviate traffic congestion
in a network of interconnected signaled lanes/roads. The proposed scheme is
emergency vehicle-centered, meaning that it provides an efficient and timely
routing for emergency vehicles. In the proposed scheme, model predictive
control is utilized to control inlet traffic flows by means of network gates,
as well as configuration of traffic lights across the network. Two schemes are
considered in this paper: i) centralized; and ii) decentralized. In the
centralized scheme, a central unit controls the entire network. This scheme
provides the optimal solution, even though it might not fulfil real-time
computation requirements for large networks. In the decentralized scheme, each
intersection has its own control unit, which sends local information to an
aggregator. The main responsibility of this aggregator is to receive local
information from all control units across the network as well as the emergency
vehicle, to augment the received information, and to share it with the control
units. Since the decision-making in decentralized scheme is local and the
aggregator should fulfil the above-mentioned tasks during a traffic cycle
which takes a long period of time, the decentralized scheme is suitable for
large networks, even though it may provide a sub-optimal solution. Extensive
simulation studies are carried out to validate the proposed schemes, and
assess their performance. Notably, the obtained results reveal that traveling
times of emergency vehicles can be reduced up to $\sim$50% by using the
centralized scheme and up to $\sim$30% by using the decentralized scheme,
without causing congestion in other lanes.

###### Index Terms:

Traffic control, multi-intersection control, emergency vehicle, model
predictive control, centralized control, decentralized control.

##  I Introduction

Traffic congestion is one of the most critical issues in urbanization. In
particular, many cities around the world have experienced 46–70% increase in
traffic congestion [1]. Congested roads not only lead to increased commute
times, but also hinder timely deployment of emergency vehicles [2]. Hence,
emergency vehicles often fail to meet their target response time [3].
According to $\sim$240 million emergency calls every year in the U.S. [4],
such hindering greatly affects hospitalization and mortality rates [5].

The common practice by regular vehicles (i.e., non-emergency vehicles) in the
presence of an emergency vehicle is to pull over to the right (in two-way
roads) or to the nearest shoulder (in one-way roads) [6], and let the
emergency vehicle traverse efficiently and timely. This is not always
possible, as in dense areas the edges of the roads are usually occupied by
parked/moving vehicles.

The chance of an emergency vehicle getting stuck is even higher when it has to
traverse intersections with cross-traffic [7]. Note that the majority of
incidents involving emergency vehicles happen within intersections [8]. One
possible way to cope with this problem is to use traffic lights at
intersections to detect emergency vehicles and facilitate their fast and
efficient travel. For this purpose, traffic lights in most parts of the U.S.
are equipped with proper detectors (e.g., 3M Opticom™ [9]), and emergency
vehicles are equipped with emitters which broadcast an infrared signal. When
the receiver on a traffic light detects a recognized signal, the traffic light
changes to allow priority access to the emergency vehicle. In this context,
the “green wave” method has been proposed to reduce emergency vehicles’
traveling time [10]. In the “green wave” method, a series of traffic lights
are successively set to ‘green’ to allow timely passage of emergency vehicles
through several intersections [11]. The main issue with the “green wave”
method is that it leads to prolonged red lights for other lanes [12], meaning
that it may cause congestion in other lanes.

A different method for controlling the traffic in the presence of an emergency
vehicle is to convert the traffic control problem to a real-time scheduling
problem [13, 3]. The core idea of this method is to model the vehicles and
traffic lights as aperiodic tasks and sporadic servers, respectively, and then
to utilize available task scheduling schemes to solve the resulting problem.
Other existing traffic control methods either do not consider emergency
vehicles [14, 15, 16, 17, 18, 19] or require vehicle to vehicle connectivity
[20, 21, 22, 23, 24, 25]. Note that the presence of 100% of connected vehicles
is not expected until 2050 [26], making these methods inapplicable to the
current traffic systems.

The aim of this paper is to propose control algorithms to manipulate traffic
density in a network of interconnected signaled lanes. The core idea is to
integrate the Cell Transmission Model (CTM) [27, 28] with Model Predictive
Control (MPC) [29]. Our motivation to use MPC is that it solves an optimal
control problem over a receding time window, which provides the capability of
predicting future events and taking actions accordingly. Note that even though
this approach is only sub-optimal, in general [30], it works very well in many
applications; our numerical experiments suggest that MPC yields very good
performance in traffic control applications. Two schemes are developed in this
paper: i) centralized; and ii) decentralized. In the centralized scheme,
assuming that the control inputs are inlet traffic flows and configuration of
the traffic lights across the network, a two-step control scheme is proposed.
In a normal traffic mode, the proposed centralized scheme alleviates traffic
density in all lanes, ensuring that traffic density in the entire network is
less than a certain value. When an emergency vehicle approaches the
network—this condition is referred as an emergency traffic mode—the control
objective is to clear the path for the emergency vehicle, without causing
congestion in other lanes. It is shown that our proposed centralized scheme
provides the optimal solution, even though its computation time may be large
for large networks. In the decentralized scheme, inlet traffic flows and
configuration of the traffic lights at each intersection are controlled by a
local control unit, while the control units share data with each other through
an aggregator. In the decentralized scheme, the aggregator should receive and
send the data during every traffic light state (i.e., ‘red’ or ‘green’). Since
the traffic cycle ranges from one minute to three minutes in real-world
traffic systems [31], the smallest duration of traffic light states is 30
seconds; thus, the maximum allowable communication delay is around 30 seconds,
which is achievable even with cheap communication technologies. Thus, the
decentralized scheme is more suitable for large networks, even though it
yields a sub-optimal solution. Note that the robustness and tolerance of the
decentralized scheme to uncertainty in communication delay and communication
failures are out of the scope of this paper, and will be considered as future
work.

The key contributions of this paper are: i) we develop a traffic control
framework which provides an efficient and timely emergency vehicle passage
through multiple intersections, without causing congestion in other lanes; ii)
we propose a centralized scheme for small networks and a decentralized scheme
for large networks that addresses scalability issues in integrating CTM and
MPC; and iii) we validate our schemes via extensive simulation studies, and
assess their performance in different scenarios. The main features of the
proposed framework are: i) it is general and can be applied to any network of
interconnected signaled lanes; and ii) it does not require vehicle to
everything (V2X) connectivity, and hence it can be utilized in the currently
existing traffic systems; the only communication requirement is between the
emergency vehicle and the central control unit in the centralized scheme, and
with the aggregator in the decentralized scheme. Note that this paper
considers only macroscopic characteristics of traffic flow; it is evident that
the existence of V2X connectivity can not only be exploited to further improve
efficiency at the macro-level, but it can also be leveraged to ensure safety
and avoid collisions.

The key innovations of this paper with respect to prior work are: i)
formulating the traffic density control problem in both normal and emergency
modes as MPC problems; ii) developing a two-step optimization procedure
implementable in the current traffic systems; and iii) deriving centralized
and decentralized schemes for traffic networks with different size and
communication capacity.

The rest of the paper is organized as follows. Section II describes
macroscopic discrete-time model of the traffic flow in the network. Section
III discusses the design procedure of the centralized traffic control scheme.
The decentralized scheme is discussed in Section IV. Section V reports
simulations results and compares the centralized and decentralized schemes.
Finally, Section VI concludes the paper.

##### Notation

$\mathbb{R}$ denotes the set of real numbers, $\mathbb{R}_{\geq 0}$ denotes
the set of non-negative real numbers, $\mathbb{Z}$ denotes the set of integer
numbers, and $\mathbb{Z}_{\geq 0}$ denotes the set of non-negative integer
numbers. For the matrix $X$, $X^{\top}$ denotes its transpose, $\rho(X)$
denotes its spectral radius, and $\left\|X\right\|_{1}=\sup_{y\neq
0}\frac{\left\|Xy\right\|_{1}}{\left\|y\right\|_{1}}$ with
$\left\|\cdot\right\|_{1}$ as the $\ell_{1}$-norm. For the vector $y$,
$[y]_{+}$ is the element-wise rounding to the closest non-negative integer
function. For given sets $X,Y$, $X\oplus Y:\\{x+y:x\in X,y\in Y\\}$ is the
Minkowski set sum. TABLE I lists the essential notation of this paper.

TABLE I: List of calligraphic, Greek, Latin, and subscript and superscript symbols. Type | Symbol | Description
---|---|---
Calligraphic | $\mathcal{N}$ | Set of lanes
| $\mathcal{M}$ | Set of intersections
| $\mathcal{G}$ | Traffic network
| $\mathcal{E}$ | Edge of traffic graph
| $\mathcal{X}$ | Constraint set
| $\mathcal{D}$ | Disturbance set
Greek | $\Lambda$ | Set of possible commands by traffic lights
| $\lambda$ | Configuration of traffic lights
| $\gamma$ | Prioritizing parameter
Latin | $x$ | Traffic density
| $y$ | Traffic inflow
| $z$ | Traffic outflow
| $u$ | Inlet flow
| $d$ | Disturbance input
| $t$ | Discrete time instant
| $k$ | Prediction time instant
Subscript and | $in$ | inlet
Superscript | $df$ | Disturbance-free
| $nom$ | Nominal
| $n$ | normal condition
| $e$ | Emergency condition
| $t$ | Computed at time $t$

##  II System Modelling

In this section, we formulate the traffic control problem for a general
traffic network.

###  II-A Traffic Network

Consider a traffic network with $N$ lanes and $M$ intersections. There are
$N_{in}<N$ inlets through which vehicles enter the network. We denote the set
of lanes by $\mathcal{N}=\\{1,\cdots,N\\}$, the set of intersections by
$\mathcal{M}=\\{1,\cdots,M\\}$, and the set of inlets by
$\mathcal{N}_{in}\subset\mathcal{N}$.

The considered traffic network can be represented by a graph
$\mathcal{G}(\mathcal{N},\mathcal{E})$, where
$\mathcal{E}\subset\mathcal{N}\times\mathcal{N}$ defines the edge of graph.
The edge $(i,j)\in\mathcal{E}$ represents a directed connection from lane $i$
to lane $j$. Since all lanes are assumed to be unidirectional (note that two-
way roads are modeled as two opposite-directional lanes), if
$(i,j)\in\mathcal{E}$, we have $(j,i)\not\in\mathcal{E}$. Also, we assume that
U-turns are not allowed, i.e., $(i,j),(j,i)\not\in\mathcal{E}$, if lanes $i$
and $j$ are opposite-directional lanes on a single road.

Note that we assume that the traffic graph
$\mathcal{G}(\mathcal{N},\mathcal{E})$ remains unchanged; that is we do not
consider graph changes due to unexpected events (e.g., changes in the edge
$\mathcal{E}$ as a result of lane blockages due to accidents). We leave the
developments of strategies for rerouting in the case of a change in the
traffic graph to future work.

###  II-B Action Space By Traffic Lights

Suppose that all lanes, except outlets, are controlled by traffic lights which
have three states: ‘red’, ‘yellow’, and ‘green’. The vehicles are allowed to
move when the light is ‘yellow’ or ‘green’, while they have to stop when the
light is ‘red’. This means that there are practically two states for each
traffic light.

Let $\lambda_{j}(t)$ be the configuration of traffic lights at intersection
$j\in\mathcal{M}$ at time $t$. We denote the set of all possible
configurations at intersection $j$ by
$\Lambda_{j}=\\{\lambda_{j,1},\cdots,\lambda_{j,\mu_{j}}\\}$, where
$\mu_{j}\in\mathbb{Z}_{\geq 0}$. Indeed, the set $\Lambda_{j}$ represents the
set of all possible actions that can be commanded by the traffic lights at
intersection $j$. Therefore, the set of all possible actions by traffic lights
across the network is $\Lambda=\Lambda_{1}\times\cdots\times\Lambda_{M}$, and
the $M$-tuple
$\lambda(t)=\big{(}\lambda_{1}(t),\cdots,\lambda_{M}(t)\big{)}\in\Lambda$
indicates the action across the network at time $t$.

###  II-C Macroscopic Traffic Flow Model

The traffic density in each lane is a macroscopic characteristic of traffic
flow [32, 33], which can be described by the CTM that transforms the partial
differential equations of the macroscopic Lighthill-Whitham-Richards (LWR)
model [34] into simpler difference equations at the cell level. The CTM
formulates the relationship between the key traffic flow parameters, and can
be cast in a discrete-time state-space form.

Let traffic density be defined as the total number of vehicles in a lane at
any time instant, then the traffic inflow is defined as the total number of
vehicles entering a lane during a given time period, and traffic outflow is
defined as the total number of vehicles leaving a lane during a given time
period. We use $x_{i}(t)\in\mathbb{Z}_{\geq 0}$, $y_{i}(t)\in\mathbb{R}_{\geq
0}$, and $z_{i}(t)\in\mathbb{R}_{\geq 0}$ to denote the traffic density,
traffic inflow, and traffic outflow in lane $i$ at time $t$, respectively. The
traffic dynamics [35, 36] in lane $i$ can be expressed as

$\displaystyle x_{i}(t+1)=\left[x_{i}(t)+y_{i}(t)-z_{i}(t)\right]_{+},$ (1)

where the time interval $[t,t+1)$ is equivalent to $\Delta T$ seconds. Since
$x_{i}(t)$ is defined as the number of existing vehicles in each lane, we use
the rounding function in (1) to ensure that $x_{i}(t)$ remains a non-negative
integer at all times. Given $\Delta T$, $y_{i}(t)$ and $z_{i}(t)$ are equal to
the number of vehicles entering and leaving the lane $i$ in $\Delta T$
seconds, respectively.

The traffic outflow $z_{i}(t)$ can be computed as [19]

$\displaystyle z_{i}(t)=p_{i}\big{(}\lambda(t)\big{)}x_{i}(t),$ (2)

where $p_{i}\big{(}\lambda(t)\big{)}$ is the fraction of outflow vehicles in
lane $i$ during the time interval $[t,t+1)$, satisfying

$\displaystyle
p_{i}\big{(}\lambda(t)\big{)}\left\\{\begin{array}[]{ll}=0,&\text{if traffic
light of lane }i\text{ is `red'}\\\ \in[0,1],&\text{if traffic light of lane
}i\text{ is `green'}\end{array}\right.;$ (5)

in other words, $p_{i}\big{(}\lambda(t)\big{)}$ is the ratio of vehicles
leaving lane $i$ during the time interval $[t,t+1)$ to the total number of
vehicles in lane $i$ at time instant $t$. It is noteworthy that even though
the impact of lane blockage or an accident in lane $i$ can be modeled by
adjusting $p_{i}\big{(}\lambda(t)\big{)}$, this paper does not aim to deal
with such unexpected events.

###### Remark II.1

We assume that outlet traffic flows are uncontrolled, i.e., there is no
traffic light or gate at the end of outlets. This assumption is plausible, as
any road connecting the considered traffic network to the rest of the grid can
be divided at a macro-level into an uncontrollable outlet inside the
considered network and a lane outside the considered network (possibly
controlled with a traffic light or a network gate). The extension of the
proposed methods to deal with controlled outlet flows is straightforward by
modifying (2) and all presented optimization problems to account for outlet
flow (similar to what we do for inlet flow $u_{i}(t)$); thus, to simplify the
exposition and subsequent developments, we will not discuss controlled
outlets.

The traffic inflow $y_{i}(t)$ can be computed as

$\displaystyle y_{i}(t)=\left\\{\begin{array}[]{ll}u_{i}(t),&\text{if
$i\in\mathcal{N}_{in}$}\\\
\sum\limits_{j=1}^{N}q_{j,i}\big{(}\lambda(t)\big{)}z_{j}(t),&\text{otherwise}\end{array}\right.,$
(8)

where $u_{i}(t)\in\mathbb{Z}_{\geq 0}$ is the inlet flow which is defined as
the number of vehicles entering the traffic network through inlet $i$ during
the time interval $[t,t+1)$. The computed optimal inflows can be implemented
by means of network gates, i.e., ramp meters [37, 38] for highways and
metering gates [39] for urban streets). In (8),
$q_{j,i}\big{(}\lambda(t)\big{)}$ is the fraction of outflow of lane $j$
directed toward lane $i$ during the time interval $[t,t+1)$, which is

$\displaystyle
q_{j,i}\big{(}\lambda(t)\big{)}\left\\{\begin{array}[]{ll}=0,&\text{(if
traffic light of lane $i$ is `red')}\\\ &\text{OR (if
}(j,i)\not\in\mathcal{E}\text{)}\\\ \in[0,1],&\text{(if traffic light of lane
$i$ is `green')}\\\ &\text{AND (if
}(j,i)\in\mathcal{E}\text{)}\end{array}\right.,$ (13)

and satisfies $\sum_{i=1}^{N}q_{j,i}\big{(}\lambda(t)\big{)}=1$ for all
$j\in\mathcal{N}$. More precisely, $q_{j,i}\big{(}\lambda(t)\big{)}$ is the
ratio of vehicles leaving lane $j$ and entering lane $i$ during the time
interval $[t,t+1)$ to the total number of vehicles leaving lane $j$ during the
time interval $[t,t+1)$.

From (1)-(13), traffic dynamics of the entire network can be expressed as

$\displaystyle
x(t+1)=\left[\bar{A}\big{(}\lambda(t)\big{)}x(t)+B\bar{U}(t)\right]_{+},$ (14)

where $x(t)=[x_{1}(t)~{}\cdots~{}x_{N}(t)]^{\top}\in\mathbb{Z}_{\geq 0}^{N}$,
$\bar{A}:\Lambda\rightarrow\mathbb{R}^{N\times N}$ is the so-called traffic
tendency matrix [40], $B\in\mathbb{R}^{N\times N_{in}}$, and
$\bar{U}(t)\in\mathbb{Z}_{\geq 0}^{N_{in}}$ is the boundary inflow vector. It
should be noted that the $(i,j)$ element of $B$ is 1 if lane $i$ is the $j$-th
inlet, and 0 otherwise.

###### Remark II.2

At any $t$, the $(i,i)$ element of the traffic tendency matrix
$\bar{A}\big{(}\lambda(t)\big{)}$ is $1-p_{i}\big{(}\lambda(t)\big{)}$. Also,
its $(i,j)$ element ($i\neq j$) is
$q_{j,i}\big{(}\lambda(t)\big{)}p_{j}\big{(}\lambda(t)\big{)}$. As a result,
since $\sum_{i=1}^{N}q_{j,i}\big{(}\lambda(t)\big{)}=1,~{}\forall
j\in\mathcal{N}$, the maximum absolute column sum of the traffic tendency
matrix is less than or equal to 1. This means that at any $t$, we have
$\left\|\bar{A}\big{(}\lambda(t)\big{)}\right\|_{1}\leq 1$, which implies that
$\rho\Big{(}\bar{A}\big{(}\lambda(t)\big{)}\Big{)}\leq 1$. Therefore,
$\rho\Big{(}\bar{A}\big{(}\lambda(t)\big{)}\bar{A}\big{(}\lambda(t+1)\big{)}\bar{A}\big{(}\lambda(t+2)\big{)}\cdots\Big{)}\leq
1$, which means that the unforced system (i.e., when $\bar{U}(t)=0$) is
stable, although trajectories may not asymptotically converge to the origin.
This conclusion is consistent with the observation that in the absence of new
vehicles entering to lane $i$, the traffic density in lane $i$ remains
unchanged if the corresponding traffic light remains ‘red’.

###### Remark II.3

In general, system (14) is not bounded-input-bounded-output stable. For
instance, the traffic density in lane $i$ constantly increases if $y_{i}(t)>0$
at all times and the corresponding traffic light remains ‘red’.

Given the action $\lambda(t)$, the traffic dynamics given in (14) depend on
the parameters $p_{i}\big{(}\lambda(t)\big{)}$ and
$q_{j,i}\big{(}\lambda(t)\big{)},~{}\forall i,j$, as well as the boundary
inflow vector $\bar{U}(t)$. These parameters are, in general, a priori
unknown. We assume that these parameters belong to some bounded intervals, and
we can estimate these intervals from prior traffic data. Thus, traffic
dynamics given in (14) can be rewritten as

$\displaystyle x(t+1)=\Big{[}\Big{(}A\big{(}\lambda(t)\big{)}$
$\displaystyle+\Delta A(t)\Big{)}x(t)$ $\displaystyle+B\big{(}U(t)+\Delta
U(t)\big{)}\Big{]}_{+},$ (15)

where $A\big{(}\lambda(t)\big{)}\in\mathbb{R}^{N\times N}$ is the traffic
tendency matrix computed by nominal values of $p_{i}$ and $q_{j,i},~{}\forall
i,j$ associated with the action $\lambda(t)$, $\Delta
A(t)\in\mathbb{R}^{N\times N}$ covers possible uncertainties,
$U(t)\in\mathbb{Z}_{\geq 0}^{N_{in}}$ is the boundary inflow vector at time
$t$, and $\Delta U(t)\in\mathbb{Z}_{\geq 0}^{N_{in}}$ models possible inflow
uncertainties.

###### Remark II.4

The boundary inflow $U(t)$ is either uncontrolled or controlled. In the case
of uncontrolled inlets, $U(t)$ represents the nominal inflow learnt from prior
data, which, in general, is time-dependent, as it can be learnt for different
time intervals in a day (e.g., in the morning, in the evening, etc). In this
case, $\Delta U(t)$ models possible imperfections. In the case of a controlled
inlet traffic flows, $U(t)$ is the control input at time $t$. Note that $U(t)$
determines the available throughput in inlets, i.e., an upper-bound on
vehicles entering the network through each inlet. However, traffic demand
might be less than the computed upper-bounds, meaning that utilized throughput
is less than the available throughput. In this case, $\Delta U(t)$ models
differences between the available and utilized throughput.

Finally, due to the rounding function in (II-C), the impact of the uncertainty
terms $\Delta A(t)$ and $\Delta U(t)$ can be expressed as an additive integer.
More precisely, traffic dynamics given in (II-C) can be rewritten as

$\displaystyle x(t+1)=\max\Big{\\{}$
$\displaystyle\Big{[}A\big{(}\lambda(t)\big{)}x(t)+BU(t)\Big{]}_{+}+d(t),0\Big{\\}},$
(16)

where $d(t)=[d_{1}(t)~{}\cdots~{}d_{N}(t)]^{\top}\in\mathcal{D},~{}\forall t$
is the disturbance that is unknown but bounded, with
$\mathcal{D}\subset\mathbb{Z}^{N}$ as a polyhedron containing the origin. Note
that $d_{i}(t)$ also models vehicles parking/unparking in lane $i$.

##  III Emergency Vehicle-Centered Traffic Control—Centralized Scheme

In this section, we will propose a centralized scheme whose algorithmic
flowchart given in Fig. 1. As seen in this figure, a central control unit
determines the optimal inlet flows and configuration of all traffic lights.
This implies that the data from all over the network should be available to
the central unit at any $t$.

In this section, we will use the following notations. Given the prediction
horizon $[t,t+T_{f}]$ for some $T_{f}\in\mathbb{Z}_{\geq 0}$, we define
$U_{t:t+T_{f}-1}^{t}=[U^{t}(t)^{\top}~{}\cdots~{}U^{t}(t+T_{f}-1)^{\top}]^{\top}\in\mathbb{Z}_{\geq
0}^{T_{f}N_{in}}$, where $U^{t}(t+k)\in\mathbb{Z}_{\geq 0}^{N_{in}}$ is the
boundary inflow vector for time $t+k$ (with $k\leq T_{f}-1$) computed at time
$t$. Also,
$\lambda_{t:t+T_{f}-1}^{t}=\\{\lambda^{t}(t),\cdots,\lambda^{t}(t+T_{f}-1)\\}\in\Lambda^{T_{f}}$,
where $\lambda^{t}(t+k)$ is the configuration of all traffic lights for time
$t+k$ (with $k\leq T_{f}-1$) computed at time $t$. Note that $\ast$ is added
to the above-mentioned notations to indicate optimal decisions.

Figure 1: Algorithmic flowchart of the proposed centralized traffic control
scheme. This algorithm should be run at every $t$ in the central control unit.

###  III-A Normal Traffic Mode

The normal traffic mode corresponds to traffic scenarios in which there is no
emergency vehicle. Given the prediction horizon $[t,t+T_{f}]$, the control
objective in a normal traffic mode is to determine boundary inflows and
configurations of traffic lights over the prediction horizon such that traffic
congestion is alleviated in all lanes. This objective can be achieved through
the following two-step receding horizon control; that is, the central unit
computes the optimal boundary inflows and configuration of traffic lights over
the prediction horizon by solving the associated optimization problems at
every time instant $t$, but only implements the next boundary inflows and
configuration of traffic lights, and then solves the associated optimization
problems again at the next time instant, repeatedly.

####  III-A1 Step 1

Consider $\\{\lambda_{t:t+T_{f}-2}^{t-1,\ast},\lambda(t+T_{f}-1)\\}$, where
$\lambda_{t:t+T_{f}-2}^{t-1,\ast}$ is the optimal
solution111$\lambda_{0:T_{f}-2}^{-1,\ast}$ should be selected randomly from
the action space $\Lambda^{T_{f}-1}$. of (19) obtained at time $t-1$ and
$\lambda(t+T_{f}-1)$ is selected randomly from the action space $\Lambda$.
Then, the optimal boundary inflows over the prediction horizon $[t,t+T_{f}]$
(i.e., $U_{t:t+T_{f}-1}^{t,\ast}$) can be obtained by solving the following
optimization problem:

$\displaystyle\min\limits_{U}\;\sum\limits_{k=0}^{T_{f}-1}\Big{(}\left\|\hat{x}_{df}(k|t)\right\|_{\Gamma_{n}}^{2}+\left\|U(t+k)-U_{nom}(t+k)\right\|_{\Theta}^{2}\Big{)},$
(17a) subject to $\displaystyle\hat{x}(k|t)\subseteq\hat{\mathcal{X}},$
$\displaystyle~{}k=1,\cdots,T_{f},$ (17b) $\displaystyle
U(t+k)\in\mathbb{Z}_{\geq 0}^{N_{in}},$ $\displaystyle~{}k=0,\cdots,T_{f}-1,$
(17c)

where $\Theta=\Theta^{\top}\geq 0$ ($\in\mathbb{R}^{N_{in}\times N_{in}}$) is
a weighting matrix, $\hat{\mathcal{X}}\subset\mathbb{Z}_{\geq 0}^{N}$ is a
polyhedron containing the origin222The upper-bound on the traffic density of
each lane can be specified according to the capacity of the lane. See [41] for
a comprehensive survey., and

$\displaystyle\hat{x}(k+1|t)\in\Big{(}A\big{(}\lambda^{t-1,\ast}(t+k)\big{)}\hat{x}(k|t)+BU(t+k)\Big{)}\oplus\mathcal{D},$
(18)

with initial condition $\hat{x}(0|t)=x(t)$, and
$\lambda^{t-1,\ast}(t+T_{f}-1)=\lambda(t+T_{f}-1)$ which is selected randomly
from the action space $\Lambda$. Note that to account for the disturbance
$d(t)$, (18) uses the Minkowski set-sum of nominal predictions plus the set of
all possible effects of the disturbance $d(t)$ on the traffic density. The
subscript “df” in (17a) stands for disturbance-free, and $\hat{x}_{df}(k|t)$
can be computed via (18) by setting $\mathcal{D}=\\{\textbf{0}\\}$. The
$U_{nom}(t)$ is the nominal boundary inflow at time $t$, which can be
estimated based on prior traffic data. In (17),
$\Gamma_{n}=\text{diag}\\{\gamma_{1}^{n},\cdots,\gamma_{N}^{n}\\}$, where
$\gamma_{i}^{n}\geq 0,~{}\forall i\in\\{1,\cdots,N\\}$ is a design parameter
that can be used to prioritize lanes. As suggested by the U.S. Department of
Transportation [42], the prioritizing parameters can be determined according
to total crashes and congestion over a specified period of time (e.g., over a
5-year period); the higher the prioritizing parameter is, the higher priority
is given to the density alleviation.

In summary, Step 1 computes the optimal boundary inflows by solving the
optimization problem (17), which has $T_{f}\times N_{in}$ integer decision
variables constrained to be non-negative, and has $T_{f}\times N$ inequality
constraints on traffic density.

####  III-A2 Step 2

Given $U_{t:t+T_{f}-1}^{t,\ast}$ as the optimal solution of (17) obtained at
time $t$, the optimal configuration of all traffic lights over the prediction
horizon $[t,t+T_{f}]$ (i.e., $\lambda_{t:t+T_{f}-1}^{t,\ast}$) can be
determined by solving the following optimization problem:

$\displaystyle\min\limits_{\lambda}\;\sum\limits_{k=0}^{T_{f}-1}\left\|\tilde{x}_{df}(k|t)\right\|_{\Gamma_{n}}^{2},$
(19a) subject to $\displaystyle\tilde{x}(k|t)\subseteq\tilde{\mathcal{X}},$
$\displaystyle~{}k=1,\cdots,T_{f},$ (19b)
$\displaystyle\lambda(t+k)\in\Lambda,$ $\displaystyle~{}k=0,\cdots,T_{f}-1,$
(19c)

where $\tilde{\mathcal{X}}\subset\mathbb{Z}_{\geq 0}^{N}$ is a polyhedron
containing the origin, and

$\displaystyle\tilde{x}(k+1|t)\in\max\Big{\\{}\Big{[}A\big{(}\lambda$
$\displaystyle(t+k)\big{)}\tilde{x}(k|t)$
$\displaystyle+BU^{t,\ast}(t+k)\Big{]}_{+}\oplus\mathcal{D},0\Big{\\}},$ (20)

with the initial condition, $\tilde{x}(0|t)=x(t)$. Note that
$\tilde{x}_{df}(k|t)$ can be computed via (III-A2) by setting
$\mathcal{D}=\\{\textbf{0}\\}$. Note that similar to (18), a set-valued
prediction of traffic density by taking into account all possible realizations
of the disturbance $d(t)$ is considered in (III-A2) to account for the
disturbance $d(t)$.

In summary, Step 2 determines the optimal configuration of traffic lights
across the network by solving the optimization problem (19) which has $T_{f}$
decision variables (each one is an $M$-tuple representing the configuration of
traffic lights) constrained to belong to the set $\Lambda$ (see Subsection
II-B), and has $T_{f}\times N$ inequality constraints on traffic density.

###### Remark III.1

The cost function in (17) has two terms. The first term penalizes traffic
density in all lanes of the network, and the second term penalizes the
difference between the inlet traffic flows and their nominal values. It should
be noted that a sufficiently large matrix $\Theta$ guarantees that vehicles
will never be blocked behind the network gates. A different method [19] to
ensure that vehicles will not be blocked is to constrain the total boundary
inflow to be equal to a certain amount, i.e.,
$\sum_{i\in\mathcal{N}_{in}}u_{i}(t)=\bar{u},~{}\forall t$, where $\bar{u}$
can be determined based upon prior traffic data. It is noteworthy that the
computed optimal inflows can be implemented by means of network gates, i.e.,
ramp meters [37, 38] for highways and metering gates [39] for urban streets).

###### Remark III.2

The prediction given in (18) provides an approximation to system (16), and the
traffic density may take non-integer and/or negative values. However, as will
be shown later, this approximation is efficient in ensuring optimality. The
main advantage of using such an approximation is that the integer programming
as in (17) can be easily solved by available tools.

###### Remark III.3

The optimization problem (19) can be solved by using the brute-force search
[43] (a.k.a. exhaustive search or generate&test) algorithm. Note that the size
of the problem (19) is limited, since $\Lambda^{T_{f}}$ and $\mathcal{D}$ are
finite. However, there are some techniques to reduce the search space, and
consequently speed up the algorithm. For instance, if the configuration
$\lambda_{t:t+T_{f}-1}^{t}$ is infeasible and causes congestion at time $t+k$
(with $0\leq k\leq T_{f}-1$), all configurations with the same first $k-1$
actions will be excluded from the search space. Our simulation studies show
that this simple step can largely reduce the computation time of the
optimization problem (19) (in our case, from 10 seconds to 8 milliseconds).

###### Remark III.4

In the case of uncontrolled boundary inflow, the proposed scheme for normal
traffic mode reduces to solving only the optimization problem (19) based upon
learnt nominal boundary inflows.

###### Remark III.5

We assume that constraints on the traffic density are defined such that the
resulting optimization problems are feasible. However, in the case of
infeasibility, we can use standard methods (e.g., introducing slack variables)
to relax constraints.

###  III-B Emergency Traffic Mode

Suppose that:

  * •

At time $t=t_{e}$, a notification is received by the central control unit
indicating that an emergency vehicle will enter the network in $T_{a}^{t}$
time steps. Note that for $t<t_{e}$ the condition of the network was normal.

  * •

Given the entering and leaving lanes, let $\mathcal{P}$ represents the set of
all possible paths for the emergency vehicle. Once the notification is
received, i.e., at time $t=t_{e}$, based on the current and predicted traffic
conditions, the optimal emergency path $I_{e}^{\ast}$ should be selected by
the central control unit (see Remark III.10) and be given to the emergency
vehicle. We assume that the emergency vehicle will follow the provided path.

  * •

The emergency vehicle should leave the network in maximum $T_{s}^{t}$ time
steps.

  * •

Once the emergency vehicle leaves the network, the traffic density in all
lanes should be recovered to the normal traffic mode in $T_{r}^{t}$ time
steps. This phase will be referred as the recovery phase in the rest of the
paper.

###### Remark III.6

$T_{a}^{t}$, $T_{s}^{t}$, and $T_{r}^{t}$ are specified at time $t$. These
values can be computed by leveraging connectivity between the emergency
vehicle and the roadside infrastructure. Note that these variables are time-
variant, as they should be recomputed based on the traffic condition and
position of the emergency vehicle at any $t$. For instance, once the emergency
vehicle enters the network, $T_{a}^{t}$ should be set to zero, and once the
emergency vehicle leaves the network $T_{s}^{t}$ should be set to zero. Also,
when the recovery phase ends, $T_{r}^{t}$ will be zero.

The control objective in an emergency traffic mode is to shorten the traveling
time of the emergency vehicle, i.e., to help the emergency vehicle traverse
the network as quickly and efficiently as possible. Given the emergency path
with length $L$, the traveling time of the emergency vehicle can be estimated
[44, 45] as

$\displaystyle\text{Traveling Time}=\frac{L}{V_{d}}+\beta\times\text{Traffic
Density on the Path},$ (21)

for some constant $\beta>0$, where $V_{d}$ is the desired traverse velocity.
This relationship indicates that for fixed $L$ and $V_{d}$, to shorten the
traveling time of the emergency vehicle one would need to reduce the traffic
density on the emergency path.

Therefore, in an emergency traffic mode, given the prediction horizon
$[t,t+T_{f}]$ with $T_{f}\geq T_{a}^{t}+T_{s}^{t}+T_{r}^{t}$, the control
objective can be achieved by determining boundary inflows and configuration of
all traffic lights such that: i) during the time interval
$[t,t+T_{a}^{t}+T_{s}^{t}]$ traffic density in emergency path should be
reduced as much as possible, while traffic density in other lanes is less than
a certain amount; ii) during the time interval
$[t+T_{a}^{t}+T_{s}^{t},t+T_{a}^{t}+T_{s}^{t}+T_{r}^{t}]$ the traffic density
in all lanes should be recovered to the normal traffic mode; and iii) during
the time interval $[t+T_{a}^{t}+T_{s}^{t}+T_{r}^{t},T_{f}]$ the traffic
density in all lanes should satisfy constraints of normal mode.

We propose the following two-step receding horizon control approach to satisfy
the above-mentioned objectives. In this approach, the central unit computes
the optimal boundary inflows and configuration of traffic lights over the
prediction horizon by solving the associated optimization problems at every
time instant $t$, but only implements the next boundary inflows and
configuration of traffic lights, and then solves the associated optimization
problems again at the next time instant, repeatedly.

####  III-B1 Step 1

Consider $\\{\lambda_{t:t+T_{f}-2}^{t-1,\ast},\lambda(t+T_{f}-1)\\}$, where
$\lambda_{t:t+T_{f}-2}^{t-1,\ast}$ is the optimal solution333Since the traffic
condition was normal for $t<t_{e}$, $\lambda_{0:T_{f}-2}^{t_{e}-1,\ast}$ is
the optimal solution of (19) at time $t_{e}-1$. of (26) obtained at time $t-1$
and $\lambda(t+T_{f}-1)$ is selected randomly from the action space $\Lambda$.
Then, the optimal boundary inflows over the prediction horizon $[t,t+T_{f}]$
(i.e., $U_{t:t+T_{f}-1}^{t,\ast}$) can be computed by solving the following
optimization problem:

$\displaystyle\min\limits_{U}\;\sum\limits_{k=0}^{T_{f}-1}\Big{(}\left\|\hat{x}_{df}(k|t)\right\|_{\Gamma_{e}}^{2}+\left\|U(t+k)-U_{nom}(t+k)\right\|_{\Theta}^{2}\Big{)},$
(22a) subject to $\displaystyle\hat{x}(k|t)\subseteq\hat{\mathcal{X}}^{+},$
$\displaystyle~{}k=1,\cdots,T_{a}^{t}+T_{s}^{t}+T_{r}^{t}$ (22b)
$\displaystyle\hat{x}(k|t)\subseteq\hat{\mathcal{X}},$
$\displaystyle~{}k=T_{a}^{t}+T_{s}^{t}+T_{r}^{t}+1,\cdots,T_{f}$ (22c)
$\displaystyle U(t+k)\in\mathbb{Z}_{\geq 0}^{N_{in}},$
$\displaystyle~{}k=0,\cdots,T_{f}-1,$ (22d)

where $\hat{x}(k|t)$ is as in (18),
$\hat{\mathcal{X}}^{+}\supset\hat{\mathcal{X}}$ is the extended constraint set
(see Remark III.8), and
$\Gamma_{e}=\text{diag}\\{\gamma_{1}^{e}(k),\cdots,\gamma_{N}^{e}(k)\\}$ (see
Remark III.9) with

$\displaystyle\gamma_{i}^{e}(k)=\left\\{\begin{array}[]{ll}\bar{\gamma}_{e},&\text{if
$i\in I_{e}^{\ast}$ and $k\leq T_{a}^{t}+T_{s}^{t}$}\\\
\gamma_{i}^{n},&\text{otherwise}\end{array}\right.,$ (25)

with $\bar{\gamma}_{e}\gg\max_{i}\\{\gamma_{i}^{n}\\}$, and $I_{e}^{\ast}$ is
the selected emergency path (see Remark III.10). The prioritizing parameters
as in (25) ensure that the traffic density in the lanes included in the
emergency path will be alleviated with a higher priority in the emergency
traffic mode.

Similar to (17), the optimization problem (22) has $T_{f}\times N_{in}$
integer decision variables constrained to be non-negative, and has
$T_{f}\times N$ inequality constraints on traffic density.

####  III-B2 Step 2

Given $U_{t:t+T_{f}-1}^{t,\ast}$ as the optimal solution of (22) obtained at
time $t$, the optimal configurations of the traffic lights over the prediction
horizon $[t,t+T_{f}]$ (i.e., $\lambda_{t:t+T_{f}-1}^{t,\ast}$) can be
determined by solving the following optimization problem:

$\displaystyle\min\limits_{\lambda}\;\sum\limits_{k=0}^{T_{f}-1}\left\|\tilde{x}_{df}(k|t)\right\|_{\Gamma_{e}}^{2},$
(26a) subject to
$\displaystyle\tilde{x}(k|t)\subseteq\tilde{\mathcal{X}}^{+},$
$\displaystyle~{}k=1,\cdots,T_{a}^{t}+T_{s}^{t}+T_{r}^{t},$ (26b)
$\displaystyle\tilde{x}(k|t)\subseteq\tilde{\mathcal{X}},$
$\displaystyle~{}k=T_{a}^{t}+T_{s}^{t}+T_{r}^{t}+1,\cdots,T_{f}$ (26c)
$\displaystyle\lambda(t+k)\in\Lambda,$ $\displaystyle~{}k=0,\cdots,T_{f}-1$
(26d)

where $\tilde{x}(k|t)$ is as in (III-A2), and
$\tilde{\mathcal{X}}^{+}\supset\tilde{\mathcal{X}}$ is the extended set (see
Remark III.8). Similar to (19), the optimization problem (26) has $T_{f}$
decision variables (each one is an $M$-tuple representing the configuration of
traffic lights) constrained to belong to the set $\Lambda$ (see Subsection
II-B), and has $T_{f}\times N$ inequality constraints on traffic density.

###### Remark III.7

The optimization problem (22) can be solved by mixed-integer tools, and the
optimization problem (26) can be solved by using the brute-force search
algorithms.

###### Remark III.8

We assume that constraints on the traffic density can be temporarily relaxed.
This assumption is reasonable [46, 47], as in practice, constraints are often
imposed conservatively to avoid congestion. In mathematical terms, by
relaxation we mean that traffic density should belong to extended sets
$\hat{\mathcal{X}}^{+}\supset\hat{\mathcal{X}}$ and
$\tilde{\mathcal{X}}^{+}\supset\tilde{\mathcal{X}}$. This relaxation enables
the control scheme to put more efforts on alleviation of traffic density in
emergency path. This relaxation can last up to maximum
$T_{a}^{t_{e}}+T_{s}^{t_{e}}+T_{r}^{t_{e}}$ time steps.

###### Remark III.9

$\Gamma_{e}$ as in (25) prioritizes alleviating traffic density in lanes
included in the emergency path $I_{e}^{\ast}$ during the time interval in
which the emergency vehicle is traversing the network, i.e., the time interval
$[t,t+T_{a}^{t}+T_{s}^{t}]$.

###### Remark III.10

Once the emergency notification is received by the central control unit (i.e.,
at time $t=t_{e}$), the optimization problems (22) and (26) should be solved
for all possible paths, i.e., for each element of $\mathcal{P}$. Then: i)
according to (21), the optimal emergency path $I_{e}^{\ast}$ should be
selected as

$\displaystyle
I_{e}^{\ast}=\arg\;\min\limits_{I_{e}\in\mathcal{P}}\;\sum\limits_{k=1}^{T_{a}^{t_{e}}+T_{s}^{t_{e}}}\sum\limits_{i\in
I_{e}}x_{i,df}(k|t_{e});$ (27)

and ii) the boundary inflow and configuration of traffic lights at time
$t=t_{e}$ will be the ones associated with the optimal emergency path
$I_{e}^{\ast}$.

###### Remark III.11

Once the recovery phase ends, the traffic condition will be normal, and the
boundary inflow vector and configuration of traffic lights should be
determined through the two-step control scheme presented in Subsection III-A

###### Remark III.12

In the case of uncontrolled boundary inflow, the proposed scheme for emergency
traffic mode reduces to solving only the optimization problem (26) based upon
learnt nominal boundary inflows.

##  IV Emergency Vehicle-Centered Traffic Control—Decentralized Scheme

In this section, we will develop a decentralized traffic control scheme whose
algorithmic flowchart is depicted in Fig. 2. In decentralized scheme, there is
a control unit at each intersection, which controls configuration of the
traffic lights at that intersection, as well as the traffic flow in the
corresponding inlets. During each sampling period, an aggregator receives data
from all control units, augments data, and shares across the network. This is
reasonable for real-time applications even with cheap and relatively high-
latency communication technologies, as the duration of the traffic light
states is large (e.g., 30 seconds). In Section V, we will characterize the
optimality of the developed decentralized scheme in our numerical experiments
in different traffic modes in comparison with the centralized scheme.

The main advantage of the decentralized scheme is that the size of resulting
optimization problem is very small compared to that of centralized scheme, as
it only needs to determine the configuration of traffic lights and inlet
traffic flows at one intersection. This greatly reduces the computation time
for large networks, even though it may slightly degrade performance. This will
be discussed in Section V.

In this section, we use ${}^{j}x(t)\in\mathbb{R}^{N_{j}},~{}j\in\mathcal{M}$
(with $N_{j}\leq N$) to denote traffic density in lanes controlled by Control
Unit#$j$. Also,
${}^{j}U_{t:t+T_{f}-1}^{t}=[^{j}U^{t}(t)^{\top}~{}\cdots~{}^{j}U^{t}(t+T_{f}-1)^{\top}]^{\top}\in\mathbb{Z}_{\geq
0}^{T_{f}N^{j}_{in}},~{}j\in\mathcal{M}$, where
${}^{j}U^{t}(t+k)\in\mathbb{Z}_{\geq 0}^{N^{j}_{in}}$ is the traffic flows in
inlets associated with intersection $I_{j}$ for time $t+k$ (with $k\leq
T_{f}-1$) computed at time $t$, and
${}^{j}\lambda_{t:t+T_{f}-1}^{t}=\\{\lambda_{j}^{t}(t),\cdots,\lambda_{j}^{t}(t+T_{f}-1)\\}\in\Lambda_{j}^{T_{f}},~{}j\in\mathcal{M}$,
where $\lambda_{j}^{t}(t+k)$ is the configuration of traffic lights at
intersection $I_{j}$ for time $t+k$ (with $k\leq T_{f}-1$) computed at time
$t$. Note that $\sum_{j}N_{in}^{j}=N_{in}$, and $\ast$ in the superscript of
the above-mentioned notations indicates optimal decisions.

Figure 2: Algorithmic flowchart of the proposed decentralized traffic control
scheme. This algorithm should be run at every $t$ in the Control Unit#$j$.

###  IV-A Normal Traffic Mode

As discussed in Subsection III-A, the control objective in a normal traffic
mode is to alleviate traffic density across the network. During the time
interval $[t-1,t)$, all control units receive
${}^{i}\lambda_{t-1:t+T_{f}-2}^{t-1,\ast}$ and
${}^{i}U_{t-1:t+T_{f}-2}^{t-1,\ast}$ for all $i\in\mathcal{M}$, $x(t-1)$,
$\\{U_{nom}(t),\cdots,U_{nom}(t+T_{f}-1)\\}$, and $p_{i}$ and
$q_{g,i},~{}i,g\in\mathcal{N}$ from the aggregator. At any $t$, the Control
Unit#$j,~{}j\in\mathcal{M}$ follows the following steps to determine the inlet
traffic flows and the configuration of the traffic lights at intersection
$I_{j}$ in a normal traffic mode:

  1. 1.

Compute $x(t|t-1)$ based on the shared information by the aggregator, and
according to (16) with $d(t-1)=0$.

  2. 2.

Update traffic density at local lanes (i.e., ${}^{j}x(t)$), and replace
corresponding elements in $x(t|t-1)$ with updated values.

  3. 3.

Compute $\\{^{i}\lambda_{t:t+T_{f}-2}^{t-1,\ast},\lambda_{i}(t+T_{f}-1)\\}$
for all $i\in\mathcal{M}$, where ${}^{i}\lambda_{t:t+T_{f}-2}^{t-1,\ast}$ is
the optimal solution444${}^{i}\lambda_{0:T_{f}-2}^{-1,\ast}$ should be
selected randomly from the action space $\Lambda_{i}^{T_{f}-1}$. of Control
Unit#$i$ obtained at time $t-1$ and $\lambda_{i}(t+T_{f}-1)$ is selected
randomly from the action space $\Lambda_{i}$.

  4. 4.

Compute $\\{^{i}U_{t:t+T_{f}-2}^{t-1,\ast},^{i}U_{nom}(t+T_{f}-1)\\}$ for all
$i\in\mathcal{M}$ and $i\neq j$, where ${}^{i}U_{t:t+T_{f}-2}^{t-1,\ast}$ is
the optimal solution555${}^{i}U_{0:T_{f}-2}^{-1,\ast}$ is
$\\{^{i}U_{nom}(0),\cdots,^{i}U_{nom}(T_{f}-2)\\}$. of Control Unit#$i$
obtained at time $t-1$.

  5. 5.

Solve the following optimization problem to determine the inlet traffic flows
at intersection $I_{j}$ over the prediction horizon $[t,t+T_{f}]$ (i.e.,
${}^{j}U_{t:t+T_{f}-1}^{t,\ast}$):

$\displaystyle\min\limits_{{}^{j}U}\;\sum\limits_{k=0}^{T_{f}-1}\Big{(}$
$\displaystyle\left\|{}^{j}\hat{x}_{df}(k|t)\right\|_{{}^{j}\Gamma_{n}}^{2}$
$\displaystyle+\left\|{}^{j}U(t+k)-~{}^{j}U_{nom}(t+k)\right\|_{{}^{j}\Theta}^{2}\Big{)},$
(28a) subject to ${}^{j}\hat{x}(k|t)\subseteq~{}^{j}\hat{\mathcal{X}},$
$\displaystyle~{}k=1,\cdots,T_{f},$ (28b) ${}^{j}U(t+k)\in\mathbb{Z}_{\geq
0}^{N_{in}^{j}},$ $\displaystyle~{}k=0,\cdots,T_{f}-1,$ (28c)

where ${}^{j}\Gamma_{n}=~{}^{j}\Gamma_{n}^{\top}\geq 0$
($\in\mathbb{R}^{N_{j}}$) and ${}^{j}\Theta=~{}^{j}\Theta^{\top}\geq 0$
($\in\mathbb{R}^{N_{in}^{j}\times N_{in}^{j}}$) are weighting matrices,
${}^{j}\hat{x}(k|t)$ can be computed via (18) with initial condition
$x(t|t-1)$, and ${}^{j}\hat{\mathcal{X}}\subset\mathbb{R}_{\geq 0}^{N_{j}}$ is
a polyhedron containing the origin. The optimization problem (28) has
$T_{f}\times N_{in}^{j}$ integer decision variables constrained to be non-
negative, and has $T_{f}\times N_{j}$ inequality constraints on traffic
density.

  6. 6.

Given ${}^{j}U_{t:t+T_{f}-1}^{t,\ast}$ as the optimal solution of (28)
obtained at time $t$, solve the following optimization problem to determine
the configuration of traffic lights at intersection $I_{j}$ over the
prediction horizon $[t,t+T_{f}]$ (i.e.,
${}^{j}\lambda_{t:t+T_{f}-1}^{t,\ast}$):

$\displaystyle\min\limits_{\lambda_{j}}\;\sum\limits_{k=0}^{T_{f}-1}\left\|{}^{j}\tilde{x}_{df}(k|t)\right\|_{\Gamma_{n}^{j}}^{2},$
(29a) subject to ${}^{j}\tilde{x}(k|t)\subseteq~{}^{j}\tilde{\mathcal{X}},$
$\displaystyle~{}k=1,\cdots,T_{f},$ (29b)
$\displaystyle\lambda_{j}(t+k)\in\Lambda_{j},$
$\displaystyle~{}k=0,\cdots,T_{f}-1,$ (29c)

where ${}^{j}\tilde{x}(k|t)$ can be computed via (III-A2) with initial
condition $x(t|t-1)$, and ${}^{j}\tilde{\mathcal{X}}\subset\mathbb{R}_{\geq
0}^{N_{j}}$ is a polyhedron containing the origin. The optimization problem
(29) has $T_{f}$ decision variables constrained to belong to the set
$\Lambda_{j}$ (see Subsection II-B), and has $T_{f}\times N_{j}$ inequality
constraints on traffic density.

Note that the above-mentioned scheme is receding horizon control-based; that
is the Control Unit#$j,~{}j\in\mathcal{M}$ computes the optimal inlet traffic
flows and configuration of the traffic lights at intersection $I_{j}$ over the
prediction horizon by solving the associated optimization problems at every
time instant $t$, but only implements the next inlet traffic flows and
configuration of traffic lights, and then solves the associated optimization
problems again at the next time instant, repeatedly.

###### Remark IV.1

The optimization problem (28) can be solved by mixed-integer tools, and the
optimization problem (29) can be solved by using the brute-force search
algorithms.

###### Remark IV.2

In decentralized scheme, Control Unit#$j,~{}j\in\mathcal{M}$ estimates the
traffic density at time $t$ across the network by assuming $d(t-1)=0$. Thus,
in general, $x(t|t-1)\neq x(t)$. Also, Control Unit#$j$ determines the optimal
decisions over the prediction horizon based upon the optimal decisions of
other control units at time $t-1$. As a result, the decentralized scheme is
expected to provide a sub-optimal solution. This will be shown in Section V.

###  IV-B Emergency Traffic Mode

Consider the assumptions mentioned in Subsection III-B regarding the arriving,
leaving, and recovery times. The control objective in an emergency traffic
mode is to shorten the traveling time of the emergency vehicle, without
causing congestion in other lanes. Given $T_{a}^{t}$, $T_{s}^{t}$, and
$T_{r}^{t}$ by the aggregator, the Control Unit#$j,~{}j\in\mathcal{M}$
executes the following steps to determine the inlet traffic flows and
configuration of the traffic lights at intersection $I_{j}$ in an emergency
traffic mode. Note that the following scheme is receding horizon control-
based; that is the Control Unit#$j,~{}j\in\mathcal{M}$ computes the optimal
inlet traffic flows and configuration of the traffic lights at intersection
$I_{j}$ over the prediction horizon by solving the associated optimization
problems at every time instant $t$, but only implements the next inlet traffic
flows and configuration of traffic lights, and then solves the associated
optimization problems again at the next time instant, repeatedly.

  1. 1.

Compute $x(t|t-1)$ based on the shared information by the aggregator, and
according to (16) with $d(t-1)=0$.

  2. 2.

Update traffic density at local lanes (i.e., ${}^{j}x(t)$), and replace
corresponding elements in $x(t|t-1)$ with updated values.

  3. 3.

Compute $\\{^{i}\lambda_{t:t+T_{f}-2}^{t-1,\ast},\lambda_{i}(t+T_{f}-1)\\}$
for all $i\in\mathcal{M}$, where ${}^{i}\lambda_{t:t+T_{f}-2}^{t-1,\ast}$ is
the optimal solution of Control Unit#$i$ obtained at time $t-1$ and
$\lambda_{i}(t+T_{f}-1)$ is selected randomly from the action space
$\Lambda_{i}$.

  4. 4.

Compute $\\{^{i}U_{t:t+T_{f}-2}^{t-1,\ast},^{i}U_{nom}(t+T_{f}-1)\\}$ for all
$i\in\mathcal{M}$ and $i\neq j$, where ${}^{i}U_{t:t+T_{f}-2}^{t-1,\ast}$ is
the optimal solution of Control Unit#$i$ obtained at time $t-1$.

  5. 5.

Solve the following optimization problem to determine the inlet traffic flows
at intersection $I_{j}$ over the prediction horizon $[t,t+T_{f}]$ (i.e.,
${}^{j}U_{t:t+T_{f}-1}^{t,\ast}$):

$\displaystyle\min\limits_{{}^{j}U}\;\sum\limits_{k=0}^{T_{f}-1}\Big{(}$
$\displaystyle\left\|{}^{j}\hat{x}_{df}(k|t)\right\|_{{}^{j}\Gamma_{e}}^{2}$
$\displaystyle+\left\|{}^{j}U(t+k)-~{}^{j}U_{nom}(t+k)\right\|_{{}^{j}\Theta}^{2}\Big{)},$
(30a) subject to ${}^{j}\hat{x}(k|t)\subseteq~{}^{j}\hat{\mathcal{X}}^{+},$
$\displaystyle~{}k=1,\cdots,T_{a}^{t}+T_{s}^{t}+T_{r}^{t},$ (30b)
${}^{j}\hat{x}(k|t)\subseteq~{}^{j}\hat{\mathcal{X}},$
$\displaystyle~{}k=T_{a}^{t}+T_{s}^{t}+T_{r}^{t}+1,\cdots,T_{f},$ (30c)
${}^{j}U(t+k)\in\mathbb{Z}_{\geq 0}^{N_{in}^{j}},$
$\displaystyle~{}k=0,\cdots,T_{f}-1,$ (30d)

where ${}^{j}\hat{\mathcal{X}}^{+}\supset~{}^{j}\hat{\mathcal{X}}$ is the
extended set (see Remark III.8), and
${}^{j}\Gamma_{e}=~{}^{j}\Gamma_{e}^{\top}\geq 0$ ($\in\mathbb{R}^{N_{j}}$) is
the weighting matrix (see Remark III.9). Similar to (28), the optimization
problem (30) has $T_{f}\times N_{in}^{j}$ integer decision variables
constrained to be non-negative, and has $T_{f}\times N_{j}$ inequality
constraints on traffic density.

  6. 6.

Given ${}^{j}U_{t:t+T_{f}-1}^{t,\ast}$ as the optimal solution of (30)
obtained at time $t$, solve the following optimization problem to determine
the configuration of traffic lights at intersection $I_{j}$ over the
prediction horizon $[t,t+T_{f}]$ (i.e.,
${}^{j}\lambda_{t:t+T_{f}-1}^{t,\ast}$):

$\displaystyle\min\limits_{\lambda_{j}}\;\sum\limits_{k=0}^{T_{f}-1}\left\|{}^{j}\tilde{x}_{df}(k|t)\right\|_{\Gamma_{e}^{j}}^{2},$
(31a) subject to
${}^{j}\tilde{x}(k|t)\subseteq~{}^{j}\tilde{\mathcal{X}}^{+},$
$\displaystyle~{}k=1,\cdots,T_{a}^{t}+T_{s}^{t}+T_{r}^{t},$ (31b)
${}^{j}\tilde{x}(k|t)\subseteq~{}^{j}\tilde{\mathcal{X}},$
$\displaystyle~{}k=T_{a}^{t}+T_{s}^{t}+T_{r}^{t}+1,\cdots,T_{f},$ (31c)
$\displaystyle\lambda_{j}(t+k)\in\Lambda_{j},$
$\displaystyle~{}k=0,\cdots,T_{f}-1,$ (31d)

where ${}^{j}\tilde{\mathcal{X}}^{+}\supset~{}^{j}\tilde{\mathcal{X}}$ is the
extended set (see Remark III.8). Similar to (29), the optimization problem
(31) has $T_{f}$ decision variables constrained to belong to the set
$\Lambda_{j}$ (see Subsection II-B), and has $T_{f}\times N_{j}$ inequality
constraints on traffic density.

###### Remark IV.3

The optimization problem (30) can be solved by mixed-integer tools, and the
optimization problem (31) can be solved by using the brute-force search
algorithms.

###### Remark IV.4

In decentralized scheme the emergency path $I_{e}^{\ast}$ is determined by the
emergency vehicle, and is shared with control units through the aggregator.

###### Remark IV.5

In this paper we assume that each control unit in the decentralized scheme
controls the inlet traffic flows and configuration of traffic lights at one
intersection. However, the decentralized scheme is applicable to the case
where a network is divided into some sub-networks, and there exist a control
unit in each sub-network controlling the entire sub-network.

##  V Simulation Results

Consider the traffic network shown in Fig. 3. This network contains 14
unidirectional lanes identified by the set $\mathcal{N}=\\{1,\cdots,14\\}$,
and 4 intersections identified by the set $\mathcal{M}=\\{1,\cdots,4\\}$.
Also, $\mathcal{N}_{in}=\\{2,7,8\\}$. The edge set is
$\mathcal{E}=\\{(2,3),(2,11),(7,12),(7,14),(7,6),(8,1),(8,10),(8,13),\allowbreak(10,3),(10,11),(11,4),(11,5),(12,1),(12,9),(12,10),\allowbreak(13,6),(13,14),(14,4),(14,5)\\}$.

Figure 3: Considered traffic network with 14 lanes and 4 intersections. An
emergency vehicle enters through lane 8 and leaves through lane 5.

Fig. 4 shows possible configurations of traffic lights at each intersection of
the traffic network shown in Fig. 3. As seen in this figure,
$\mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}=2$, and the possible configurations at each
intersection are: i) Intersection $I_{1}$: $\lambda_{1,1}$ corresponds to a
‘green’ light at the end of lane 8, and a ‘red’ light at the end of lane 12;
$\lambda_{1,2}$ corresponds to a ‘red’ light at the end of lane 8, and a
‘green’ light at the end of lane 12; ii) Intersection $I_{2}$: $\lambda_{2,1}$
corresponds to a ‘green’ light at the end of lane 10, and a ‘red’ light at the
end of lane 2; $\lambda_{2,2}$ corresponds to a ‘red’ light at the end of lane
10, and a ‘green’ light at the end of lane 2; iii) Intersection $I_{3}$:
$\lambda_{3,1}$ corresponds to a ‘green’ light at the end of lane 7, and a
‘red’ light at the end of lane 13; $\lambda_{3,2}$ corresponds to a ‘red’
light at the end of lane 7, and a ‘green’ light at the end of lane 13; and iv)
Intersection $I_{4}$: $\lambda_{4,1}$ corresponds to a ‘green’ light at the
end of lane 14, and a ‘red’ light at the end of lane 11; $\lambda_{4,2}$
corresponds to a ‘red’ light at the end of lane 14, and a ‘green’ light at the
end of lane 11.

Figure 4: Possible configurations of traffic lights at each intersection of
the considered traffic network. The action space is
$\Lambda=\Lambda_{1}\times\Lambda_{2}\times\Lambda_{3}\times\Lambda_{4}$,
where $\Lambda_{1}=\\{\lambda_{1,1},\lambda_{1,2}\\}$,
$\Lambda_{2}=\\{\lambda_{2,1},\lambda_{2,2}\\}$,
$\Lambda_{3}=\\{\lambda_{3,1},\lambda_{3,2}\\}$, and
$\Lambda_{4}=\\{\lambda_{4,1},\lambda_{4,2}\\}$.

The boundary inflow vector of the traffic network shown in Fig. 3 is
$U(t)=[u_{2}(t)~{}u_{7}(t)~{}u_{8}(t)]^{\top}\in\mathbb{Z}_{\geq 0}^{3}$. We
assume that $\Delta T=30$ seconds; this sampling period is appropriate to
address macroscopic characteristics of traffic flow [19, 48, 49], as the
traffic cycle ranges from one minute to three minutes in real-world systems
[31]. For intersection $I_{1}$ and for the action
$\lambda=(\lambda_{1,1},\lambda_{2},\lambda_{3},\lambda_{4})$, we have
$p_{8}(\lambda)\in[0,1]$, $p_{12}(\lambda)=0$,
$q_{8,1}(\lambda),q_{8,10}(\lambda),q_{8,13}(\lambda)\in[0,1]$, and
$q_{12,1}(\lambda)=q_{12,9}(\lambda)=q_{12,10}(\lambda)=0$. For intersection
$I_{1}$ and for the action
$\lambda=(\lambda_{1,2},\lambda_{2},\lambda_{3},\lambda_{4})$, we have
$p_{8}(\lambda)=0$, $p_{12}(\lambda)\in[0,1]$,
$q_{8,1}(\lambda),q_{8,10}(\lambda),q_{8,13}(\lambda)=0$, and
$q_{12,1}(\lambda)=q_{12,9}(\lambda)=q_{12,10}(\lambda)\in[0,1]$. For
intersection $I_{2}$ and for the action
$\lambda=(\lambda_{1},\lambda_{2,1},\lambda_{3},\lambda_{4})$, we have
$p_{10}(\lambda)\in[0,1]$, $p_{2}(\lambda)=0$,
$q_{10,3}(\lambda),q_{10,11}(\lambda)\in[0,1]$, and
$q_{2,3}(\lambda)=q_{2,11}(\lambda)=0$. For intersection $I_{2}$ and for the
action $\lambda=(\lambda_{1},\lambda_{2,2},\lambda_{3},\lambda_{4})$, we have
$p_{10}(\lambda)=0$, $p_{2}(\lambda)\in[0,1]$,
$q_{10,3}(\lambda),q_{10,11}(\lambda)=0$, and
$q_{2,3}(\lambda)=q_{2,11}(\lambda)\in[0,1]$. For intersection $I_{3}$ and for
the action $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3,1},\lambda_{4})$, we
have $p_{13}(\lambda)=0$, $p_{7}(\lambda)\in[0,1]$,
$q_{13,6}(\lambda),q_{13,14}(\lambda)=0$, and
$q_{7,12}(\lambda)=q_{7,14}(\lambda)=q_{7,6}(\lambda)\in[0,1]$. For
intersection $I_{3}$ and for the action
$\lambda=(\lambda_{1},\lambda_{2},\lambda_{3,2},\lambda_{4})$, we have
$p_{13}(\lambda)\in[0,1]$, $p_{7}(\lambda)=0$,
$q_{13,6}(\lambda),q_{13,14}(\lambda)\in[0,1]$, and
$q_{7,12}(\lambda)=q_{7,14}(\lambda)=q_{7,6}(\lambda)=0$. For intersection
$I_{4}$ and for the action
$\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4,1})$, we have
$p_{14}(\lambda)\in[0,1]$, $p_{11}(\lambda)=0$,
$q_{14,4}(\lambda),q_{14,5}(\lambda)\in[0,1]$, and
$q_{11,4}(\lambda)=q_{11,5}(\lambda)=0$. For intersection $I_{4}$ and for the
action $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4,2})$, we have
$p_{14}(\lambda)=0$, $p_{11}(\lambda)\in[0,1]$,
$q_{14,4}(\lambda),q_{14,5}(\lambda)=0$, and
$q_{11,4}(\lambda)=q_{11,5}(\lambda)\in[0,1]$.

For implementing the decentralized scheme, we assume
${}^{1}x(t)=[x_{1}(t)~{}x_{8}(t)~{}x_{9}(t)~{}x_{12}(t)]^{\top}\in\mathbb{Z}_{\geq
0}^{4}$,
${}^{2}x(t)=[x_{2}(t)~{}x_{3}(t)~{}x_{10}(t)]^{\top}\in\mathbb{Z}_{\geq
0}^{3}$,
${}^{3}x(t)=[x_{6}(t)~{}x_{7}(t)~{}x_{13}(t)]^{\top}\in\mathbb{Z}_{\geq
0}^{3}$, and
${}^{4}x(t)=[x_{4}(t)~{}x_{5}(t)~{}x_{11}(t)~{}x_{14}(t)]^{\top}\in\mathbb{Z}_{\geq
0}^{4}$. That is Control Unit#1 controls lanes 1, 8, 9, and 12; Control Unit#2
controls lanes 2, 3, and 10; Control Unit#3 controls lanes 6, 7, and 13; and
Control Unit#4 controls lanes 4, 5, 11, and 14. Also, ${}^{1}U(t)=u_{8}(t)$,
${}^{2}U(t)=u_{2}(t)$, and ${}^{3}U(t)=u_{7}(t)$. Thus, $N_{in}^{1}=1$,
$N_{in}^{2}=1$, $N_{in}^{3}=1$, and $N_{in}^{4}=0$.

The simulations are run on an Intel(R) Core(TM) i7-7500U CPU 2.70 GHz with
16.00 GB of RAM. In order to have a visual demonstration of the considered
traffic network, a simulator is generated (see Fig. 5). A video of operation
of the simulator is available at the URL: https://youtu.be/FmEYCxmD-Oc. For
comparison purposes, we also simulate the centralized scheme presented in [19]
and a typical/existing/usual/baseline traffic system (i.e., the system with
periodic schedule for traffic lights). TABLE II compares the mean Computation
Time (CT) of the proposed schemes per time step with that of the scheme
presented in [19], where the value for the scheme of [19] is used as the basis
for normalization. As can be seen from this table, the computation time of the
proposed centralized scheme is $\sim 1.5$ times less than that of the scheme
of [19]. The computation time of the proposed decentralized scheme is $\sim
1000$ times less than that of the scheme of [19], and is $\sim 800$ times less
than that of the proposed centralized scheme.

TABLE II: Comparing the mean computation time of the proposed schemes with that of scheme of [40]. | Centralized | Decentralized | Scheme of [19]
---|---|---|---
Mean CT (Norm.) | $0.734$ | $1.03\times 10^{-3}$ | $1$
Figure 5: A screenshot of the generated simulator shown in the accompanied
video (https://youtu.be/FmEYCxmD-Oc). The black circle at the top shows the
traffic mode in the network, which is either normal or emergency. The color of
each lane indicates the traffic density, which can be interpreted according to
the bar at the left. Yellow arrows show traffic direction at each lane, and
pink arrows show selected emergency path.

###  V-A Normal Traffic Mode

Let $\hat{\mathcal{X}}=\tilde{\mathcal{X}}=\\{x|x_{i}\leq
20,~{}i\in\\{1,\cdots,14\\}\\}$, and $d_{i}(t),~{}\forall i$ be selected
uniformly from $\\{-2,-1,0,1,2\\}$. The initial condition is
$x(0)=[15,16,15,12,12,17,18,10,10,14,12,10,16,10]^{\top}$, and the nominal
boundary inflow is $U_{nom}(t)=[6,6,8]^{\top}$. Also, $\Theta=50I_{N_{in}}$
and $\gamma_{i}^{n}=1,~{}\forall i$.

Simulation results are shown in Fig. 6. TABLE III compares the achieved
Steady-State Density (SSD) with the considered schemes, where the value for
the typical/existing/usual/baseline traffic system is used as the basis for
normalization. Note that the reports are based on results of 1000 runs.
According to TABLE III, all methods perform better than the
typical/existing/usual/baseline traffic system. The proposed centralized
scheme provides the best response. The proposed decentralized scheme
outperforms the scheme of [19], while as expected, it yields a larger SSD
compared to the proposed centralized scheme. More precisely, degradation in
the mean SSD by the decentralized scheme in comparison with the centralized
scheme in a normal traffic mode is 11.42% which is small and acceptable in
real-life traffic scenarios. Thus, the cost of using the decentralized scheme
instead of the centralized scheme in a normal traffic mode is very small.

Figure 6: Simulation results for the centralized and decentralized traffic
control schemes in a normal traffic mode. Figure (a): Traffic density in all
lanes. Figure (b): Inlet traffic flows.

###  V-B Emergency Traffic Mode

Suppose that at time $t=10$, the aggregator receives a notification that an
emergency vehicle will enter the network through lane 8 in two time steps, and
should leave the network in two time steps through lane 5. Also, suppose that
we have one time step to recover the traffic condition. We have
$\mathcal{P}=\\{I_{e}^{1},I_{e}^{2}\\}$, where $I_{e}^{1}=\\{8,13,14,5\\}$ and
$I_{e}^{2}=\\{8,10,11,5\\}$. Let
$\hat{\mathcal{X}}^{+}=\tilde{\mathcal{X}}^{+}=\\{x|x_{i}\leq 25\\}$.

Simulation results are shown in Fig. 7, and the results from comparison
analysis are reported in TABLE IV, that are computed based on results of 1000
runs. Note that the values for the typical/existing/usual/baseline traffic
system are used as nominal values for normalization. As seen in TABLE IV, both
schemes proposed in this paper perform better than the
typical/existing/usual/baseline traffic system in an emergency traffic mode.
In particular, the centralized and decentralized schemes reduce the mean SSD
by 23.73% and 14.58%, respectively. As expected, the decentralized scheme
yields a larger SSD compared to the centralized scheme. More precisely,
degradation in mean SSD by the decentralized scheme in comparison with the
centralized scheme is 11.98%. TABLE IV also reports that the centralized and
decentralized schemes reduce the mean Density in Emergency Path (DEP) by
47.97% and 30.42%, respectively. It is noteworthy that the degradation in the
mean DEP by the decentralized scheme in comparison with the centralized scheme
is 33.73%.

TABLE III: Comparing the mean steady-state density with the considered schemes in a normal traffic mode. The value for the typical/existing/usual/baseline traffic system is used as the basis for normalization. | Centralized | Decentralized | Scheme of [19]
---|---|---|---
Mean SSD (Norm.) | $0.7251$ | $0.8079$ | $0.9035$
TABLE IV: Comparing the mean steady-state density and density in emergency path with the proposed centralized and decentralized schemes in an emergency traffic mode. The values for the typical/existing/usual/baseline traffic system are used as bases for normalization. | Centralized | Decentralized
---|---|---
Mean SSD (Norm.) | 0.7627 | 0.8542
Mean DEP (Norm.) | 0.5203 | 0.6958
Figure 7: Simulation results for the centralized and decentralized traffic
control schemes in an emergency traffic mode. Figure (a): Traffic density in
all lanes. Figure (b): Inlet traffic flows.

###  V-C Sensitivity Analysis—Impact of Look-Ahead Horizon $T_{f}$

Fig. 8 shows how the prediction window size impacts the performance and
computation time of the developed centralized scheme, where the values for
$T_{f}=1$ are used as nominal values for normalization. From Fig. 8-left, we
see that as the look-ahead horizon increases, the performance of the
decentralized scheme improves as it takes into account more information of
future conditions. However, as we look further into the future, the
performance is degraded since prediction accuracy reduces. Fig. 8-right shows
that as the look-ahead horizon increases, the computation time of the proposed
decentralized scheme increases concomitantly with the size and complexity of
the associated optimization problems. In the simulation studies, we selected
$T_{f}=4$, as it yields the best performance with an affordable computing
time. Note that a similar behavior is observed for the centralized scheme;
that is $T_{f}=4$ provides the best performance for the centralized scheme.

Figure 8: Impact of the look-ahead horizon $T_{f}$ on the performance and
computation time of the proposed decentralized scheme, where the values for
$T_{f}=1$ are used as bases for normalization. As $T_{f}$ increases the
performance improves at the cost of increased computation time. For large
$T_{f}$ the performance degrades due to poor prediction accuracy.

##  VI Conclusion

This paper proposed an emergency vehicle-centered traffic control framework to
alleviate traffic congestion in a network of interconnected signaled lanes.
The aim of this paper is to integrate CTM with MPC, to ensure that emergency
vehicles traverse multiple intersections efficiently and timely. Two schemes
were developed in this paper: i) centralized; and ii) decentralized. It was
shown that the centralized scheme provides the optimal solution, even though
its computation time may be large for large networks. To cope with this
problem, a decentralized scheme was developed, where an aggregator acts as the
hub of the network. It was shown that the computation time of the
decentralized scheme is very small, which makes it a good candidate for large
networks, even though it provides a sub-optimal solution. Extensive simulation
studies were carried out to validate and evaluate the performance of the
proposed schemes.

Future work will aim at extending the developed schemes to deal with cases
where two (or more) emergency vehicles traverse a network. This extension is
not trivial, and requires addressing many technical and methodological
challenges. Also, future work should investigate robustness and tolerance of
the decentralized scheme to uncertainty in communication delay and
communication failures.

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|  Mehdi Hosseinzadeh received his Ph.D. degree in Electrical Engineering-
Control from the University of Tehran, Iran, in 2016. From 2017 to 2019, he
was a postdoctoral researcher at Université Libre de Bruxelles, Brussels,
Belgium. In 2018, he was a visiting researcher at University of British
Columbia, Canada. He is currently a postdoctoral research associate at
Washington University in St. Louis, MO, USA. His research interests include
nonlinear and adaptive control, constrained control, and safe and robust
control of autonomous systems.
---|---
|  Bruno Sinopoli received his Ph.D. in Electrical Engineering from the
University of California at Berkeley, in 2005. After a postdoctoral position
at Stanford University, he was the faculty at Carnegie Mellon University from
2007 to 2019, where he was full professor in the Department of Electrical and
Computer Engineering with courtesy appointments in Mechanical Engineering and
in the Robotics Institute and co-director of the Smart Infrastructure
Institute. In 2019 he joined Washington University in Saint Louis, where he is
the chair of the Electrical and Systems Engineering department. He was awarded
the 2006 Eli Jury Award for outstanding research achievement in the areas of
systems, communications, control and signal processing at U.C. Berkeley, the
2010 George Tallman Ladd Research Award from Carnegie Mellon University and
the NSF Career award in 2010. His research interests include the
modeling,analysis and design of Secure by Design Cyber-Physical Systems with
applications to Energy Systems, Interdependent Infrastructures and Internet of
Things.
---|---
|  Ilya Kolmanovsky is a professor in the department of aerospace engineering
at the University of Michigan,with research interests in control theory for
systems with state and control constraints, and in control applications to
aerospace and automotive systems. He received his Ph.D. degree in aerospace
engineering from the University of Michigan in 1995. Prof. Kolmanovsky is a
Fellow of IEEE and is named as an inventor on 104 United States patents.
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|  Sanjoy Baruah joined Washington University in St. Louis in September 2017.
He was previously at the University of North Carolina at Chapel Hill
(1999–2017) and the University of Vermont (1993–1999). His research interests
and activities are in real-time and safety-critical system design, scheduling
theory, resource allocation and sharing in distributed computing environments,
and algorithm design and analysis.
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