Optimal Control for Platooning in Vehicular Networks

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OPTIMAL CONTROL FOR PLATOONING IN VEHICULAR

NETWORKS

A PREPRINT

Thiago S. Gomides

School of Computer Science

Carleton University

Ottawa, ON, Canada

thiagodasilvagomides@cmail.carleton.ca

Evangelos Kranakis

School of Computer Science

Carleton University

Ottawa, ON, Canada

kranakis@scs.carleton.ca

Ioannis Lambadaris

Department of Systems and Computer Engineering

Carleton University

Ottawa, ON, Canada

ioannis@sce.carleton.ca

Yannis Viniotis

Department of Electrical and Computer Engineering

North Carolina State University

Raleigh, NC, USA

candice@ncsu.edu

ABSTRACT

As the automotive industry is developing autonomous driving systems and vehicular networks,

attention to truck platooning has increased as a way to reduce costs (fuel consumption) and improve

efﬁciency in the highway. Recent research in this area has focused mainly on the aerodynamics,

network stability, and longitudinal control of platoons. However, the system aspects (e.g., platoon

coordination) are still not well explored. In this paper, we formulate a platooning coordination

problem and study whether trucks waiting at an initial location (station) should wait for a platoon

to arrive in order to leave. Arrivals of trucks at the station and platoons by the station are modelled

by independent Bernoulli distributions. Next we use the theory of Markov Decision Processes to

formulate the dispatching control problem and derive the optimal policy governing the dispatching of

trucks with platoons. We show that the policy that minimizes an average cost function at the station

is of threshold type. Numerical results for the average cost case are presented. They are consistent

with the optimal ones.

Keywords Truck platooning, Optimal control of queues.

1 Introduction

Truck platooning is the practice of virtually connecting two or more automated trucks forming convoys, where trucks

follow one another closely. This practice has recently gained attention as the automotive industry develops toward

autonomous driving systems and vehicular networks Adler et al. (2020). It holds great potential to make trafﬁc more

efﬁcient and clean. In particular, allowing close-distance driving mitigates the effects of aerodynamic drag, which in

turn leads to a substantial reduction in fuel consumption Zabat (1995). Platooning also optimizes highway use, reduces

travel times and enhances transportation safety.

The beneﬁts of platooning may vary due to the complex and dynamic behaviour of trucks and the resulting trafﬁc. For

instance, this potential depends on several aspects, such as the inter-vehicle gap in a platoon, the travel speed, the cost

of platooning formation, and others Zabat (1995). Therefore, it is crucial to study and understand platooning under

concrete mathematical models.

arXiv:2210.04297v3 [eess.SY] 12 Nov 2022

Optimal Control for Platooning in Vehicular Networks A PREPRINT

1.1 Related Work

Most of the research efforts so far have been concerned with studying the aerodynamic aspects of platooning Zabat

(1995); Vohra et al. (2018), the cooperative longitudinal control of trucks Tsugawa et al. (2016); Hu et al. (2020);

Milanes et al. (2014), and the stability of the platoons from a network perspective Wang et al. (2021); Petrillo et al.

(2021). However, the system aspects (e.g., platoon coordination) are still not well explored Adler et al. (2020), especially

concerning optimal control.

Previous works on platooning coordination considered the dispatching control of trucks waiting in a station/hub Adler

et al. (2020); Zhang et al. (2017); A. Johansson et al. (2020). In these works, trucks arrive at the station following a

random process (e.g., Poisson or Bernoulli), and the station decides whether they should wait to form platoons. If the

station holds them, it may build platoons with many trucks, which reduces fuel consumption. However, forcing many

trucks to wait at the station incurs high transportation delay cost. These works investigated the optimal dispatching

control at the station that minimizes an average cost function.

In Zhang et al. (2017), the authors studied optimal platoon coordination at a highway junction (hub). Their model

consisted of two trucks arriving at the hub with stochastic arrival times. If they arrive at the same time, they form a

platoon. One truck may have to wait for the other when their arrival times differ, which incurs a waiting cost. The

authors proved that it is optimal to build a platoon only when the arrival time of each truck differs by less than a

threshold.

In Adler et al. (2020), the authors studied platooning coordination of multiple trucks at a station, where the arrivals

are Poisson distributed. The station decides whether to hold trucks in order to build platoons. The authors compared

different truck dispatching policies governing the station under energy-delay tradeoff considerations. They proved the

optimality of threshold policies to control station dispatching, where the station dispatches all trucks whenever the

number of waiting trucks in the station grows above the threshold. In A. Johansson et al. (2020), the authors proposed a

similar model to the one in Adler et al. (2020) under the assumption that arrivals of trucks are i.i.d and their distribution

is known by the station. They proved that the optimal policy is of the type "one time-step look-ahead".

1.2 Model Novelty and Main Contributions

In this paper, we study the dispatching and formation of platoons from a novel perspective. In particular, like Adler

et al. (2020), A. Johansson et al. (2020) (and unlike Zhang et al. (2017)), in our model we consider a waiting station

that can hold multiple trucks. However, unlike Adler et al. (2020) and A. Johansson et al. (2020), we assume that trucks

arrive at the station, while platoons arrive alongside the station. Therefore, we dispatch trucks with arriving platoons, as

opposed to forming platoons among waiting trucks. Our cost function is also different.

We formalize the dispatching actions at the station as an optimal control problem. We then use dynamic programming

to analyze it. The main contributions of this work are as follows:

•

We derive the optimal dispatching policy governing the system under ﬁnite and inﬁnite horizon discounted

cost criteria. We show that the policies are of threshold type with a ﬁnite threshold.

• We use Lippman’s Lippman (1973) results to derive the optimal policy for the average cost criterion.

• We present numerical results for the average cost case.

The paper is organized as follows. In Section 2, we formulate the dispatching trucks to arriving platoons problem. We

also present the Markov Decision Problem. We characterize the optimal control policy for the discounted cost (with

ﬁnite and inﬁnite horizons) in Section 3. In Section 4, we derive the optimal policy for the average cost problem. We

present some numerical results for the average cost problem in Section 5. Our conclusion and suggestions for future

work are presented in Section 6.

2 Model and Control Problem Formulation

We consider the platooning system shown in Figure 1.

Deﬁne the station as a location where trucks arrive and can wait to join platoons. Platoons arrive alongside the station.

The station decides how trucks are dispatched from the station. We assume that only one truck at a time can be

dispatched.

The system operates in discrete time. Truck arrivals are Bernoulli with parameter p. Trucks join the same queue upon

arrival. Platoon arrivals are Bernoulli with parameter

. In order to avoid trivialities, we assume that

0<p<1

and

0< q < 1.

Optimal Control for Platooning in Vehicular Networks A PREPRINT

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Station

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…

Figure 1: System model.

We assume truck and platoon arrivals are a sequence of ordered events in one time slot. A truck arrival (if any) always

takes place earlier than a platoon arrival (if any). This assumption guarantees that if the station is empty and a truck

arrives, it can join a platoon if any arrives.

2.1 The Markovian Decision problem

Let

denote the number of trucks waiting at the station at slot

n∈ {1,2, . . . }

is the state of the system and

X={0,1,2, . . . }is the state space of the system.

Deﬁne the events as the combinations of arrivals (of a truck or a platoon) that may occur during a time slot. Each event

has a given probability and a state operator (mapping

into

). Transitions among the states are described in Table 1.

Events State Operators Probabilities

No arrivals. Z(x) = x. (1 −p)(1 −q)

A platoon arrives and no truck arrives. P(x) = x. (1 −p)q

A truck arrives and no platoon arrives. V(x) = x+ 1. p(1 −q)

A platoon and a truck arrive. B(x) = x+ 1. pq

Table 1: Events, State Operators, and Probabilities.

Deﬁne the action space

A={H, D}

, where the action operator

represents the action of holding a truck at the station,

and Ddenotes the dispatching action. We have:

H(x) = x, dom H=X.

D(x) = x−1,dom D={x∈X:x≥1}.

A(xn)represents the available actions for a given state xnat time slot n.

Dispatching trucks with platoons reduces energy (fuel) consumption. Holding trucks at the station (when waiting for

platoons) incurs transportation delay costs. Trucks dispatched without platoons pay the entire transportation cost.

We assume all arriving platoons provide the same energy reduction. It simpliﬁes the model as the cost of dispatching

a truck by platooning (or not) becomes deterministic. We introduce a real-valued constant

representing the cost of

dispatching a truck without a platoon.

We formalize these assumptions mathematically with

c(xn, an)

, the instantaneous cost as a function of the system state

xnwhen taking the action anat time slot n. More speciﬁcally, satisﬁes:

c(xn, an) = 





xnif an=H.

xn−1if a platoon arrives and an=D.

xn−1 + κif no platoon arrives and an=D.

(1)

Optimal Control for Platooning in Vehicular Networks A PREPRINT

We complete the speciﬁcation of our Markov Decision Problem (MDP) by deﬁning the transition probability function

as follows:

P(xn+1|xn=x, an) =











(1 −p)(1 −q)if xn+1 =Z(x), an=H.

(1 −p)(1 −q)if xn+1 =x−1, an=D.

(1 −p)qif xn+1 =P(x), an=H.

(1 −p)qif xn+1 =x−1, an=D.

p(1 −q)if xn+1 =V(x), an=H.

p(1 −q)if xn+1 =x, an=D.

pq if xn+1 =B(x), an=H.

pq if xn+1 =x, an=D.

(2)

Our goal is to choose the control actions to minimize the expected ﬁnite horizon discounted cost, as

n=1

βnc(xn, an),(3)

where βis a discount factor 0< β < 1, and Nis the time horizon.

Since

c(xn, an)

grows linearly

∀x∈X

and

is ﬁnite, it is well known that there exists an optimal stationary

(time-independent) policy that minimizes the cost (3) and is the unique solution for the MDP.

Deﬁne

Jβ

N(x)

as the minimum expected discounted cost for

(3)

with

steps to go and initial state

x0=x

Jβ

N(x)

satisﬁes:

Jβ

N(x) = min

a∈A(x){c(x, a) + β[(1 −p)(1 −q)Jβ

N−1(Z(x)) + (1 −p)qJβ

N−1(P(x)) +

p(1 −q)Jβ

N−1(V(x)) + pqJβ

N−1(B(x))]},(4)

where the initial condition is Jβ

1(x) = min{c(x, a)|a∈ A(x)}.

From the Dynamic Programming (DP) equation

(4)

, we immediately see that with

n+ 1

steps to go and initial state

x0=x, the optimal action ais given by the difference Jβ

n+1(H(x)) −Jβ

n+1(D(x)), as:











H, if (1 −p)(1 −q)[Jβ

n(x)−Jβ

n(x−1)] + (1 −p)q[Jβ

n(x)−Jβ

n(x−1)]

+p(1 −q)[Jβ

n(x+ 1) −Jβ

n(x)] + pq[Jβ

n(x+ 1) −Jβ

n(x)] ≤−1 + κ

β.

D, if (1 −p)(1 −q)[Jβ

n(x)−Jβ

n(x−1)] + (1 −p)q[Jβ

n(x)−Jβ

n(x−1)]

+p(1 −q)[Jβ

n(x+ 1) −Jβ

n(x)] + pq[Jβ

n(x+ 1) −Jβ

n(x)] >−1 + κ

β.











(5)

We will use (5) to characterize the optimal policy for (4) in the next section.

3 Characterization of the Optimal Policy

In this section, we ﬁrst prove some properties of the optimal cost function; we then use them to characterize the optimal

policy as a threshold policy.

The following Lemma (which we can show using Equations

(1)

and

(4)

) simpliﬁes the search for an optimal policy,

since we can disregard the holding actions with probabilities pq and (1 −p)qin Equation (2).

Lemma 3.1. Dispatching a truck with an arriving platoon is always optimal.

Proof. We use induction on nto prove that when a platoon arrives

Jβ

n(D(x)) ≤Jβ

n(H(x)),∀x∈dom D. (6)

Base case (n= 1).

This proof is immediate since

Jβ

1(x)

is the instantaneous cost

(1)

and when the system state is

xn≥1and a platoon arrives, c(xn, D)< c(xn, H)for all x∈dom D.

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OPTIMALCONTROLFORPLATOONINGINVEHICULARNETWORKSAPREPRINTThiagoS.GomidesSchoolofComputerScienceCarletonUniversityOttawa,ON,Canadathiagodasilvagomides@cmail.carleton.caEvangelosKranakisSchoolofComputerScienceCarletonUniversityOttawa,ON,Canadakranakis@scs.carleton.caIoannisLambadarisDepartmentofSystemsa...

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