# Exact solutions to traffic density estimation problems involving the Lighthill-Whitham-Richards traffic flow model using Mixed Integer Programming

**ABSTRACT** This article presents a new mixed integer programming formulation of the trafﬁc density estimation problem in highways modeled by the Lighthill Whitham Richards equation. We ﬁrst present an equivalent formulation of the problem using an Hamilton-Jacobi equation. Then, using a semi-analytic formula, we show that the model constraints resulting from the Hamilton-Jacobi equation result in linear constraints, albeit with unknown integers. We then pose the problem of estimating the density at the initial time given

incomplete and inaccurate trafﬁc data as a Mixed Integer Program. We then present a numerical implementation of the method using experimental ﬂow and probe data obtained during Mobile Century experiment.

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**ABSTRACT:**We propose a general traffic network flow model suitable for analysis as a dynamical system, and we qualitatively analyze equilibrium flows as well as convergence. Flows at a junction are determined by downstream supply of capacity as well as upstream demand of traffic wishing to flow through the junction. This approach is rooted in the celebrated Cell Transmission Model for freeway traffic flow. Unlike related results which rely on certain system cooperativity properties, our model generally does not possess these properties. We show that the lack of cooperativity is in fact a useful feature that allows traffic control methods, such as ramp metering, to be effective. Finally, we leverage the results of the paper to develop a linear program for optimal ramp metering.09/2014; - [Show abstract] [Hide abstract]

**ABSTRACT:**This article presents a new optimal control framework for transportation networks in which the state is modeled by a first order scalar conservation law. Using an equivalent formulation based on a Hamilton-Jacobi (H-J) equation and the commonly used triangular fundamental diagram, we pose the problem of controlling the state of the system on a network link, in a finite horizon, as a Linear Program (LP). We then show that this framework can be extended to an arbitrary transportation network, resulting in an LP or a Quadratic Program. Unlike many previously investigated transportation network control schemes, this method yields a globally optimal solution and is capable of handling shocks (i.e., discontinuities in the state of the system). As it leverages the intrinsic properties of the H-J equation used to model the state of the system, it does not require any approximation, unlike classical methods that are based on discretizations of the model. The computational efficiency of the method is illustrated on a transportation network.Control of Network Systems, IEEE Transactions on. 03/2014; 1(1):28-39. -
##### Conference Paper: Optimal traffic control in highway transportation networks using linear programming

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**ABSTRACT:**This article presents a framework for the optimal control of boundary flows on transportation networks. The state of the system is modeled by a first order scalar conservation law (Lighthill-Whitham-Richards PDE). Based on an equivalent formulation of the Hamilton-Jacobi PDE, the problem of controlling the state of the system on a network link in a finite horizon can be posed as a Linear Program. Assuming all intersections in the network are controllable, we show that the optimization approach can be extended to an arbitrary transportation network, preserving linear constraints. Unlike previously investigated transportation network control schemes, this framework leverages the intrinsic properties of the Halmilton-Jacobi equation, and does not require any discretization or boolean variables on the link. Hence this framework is very computational efficient and provides the globally optimal solution. The feasibility of this framework is illustrated by an on-ramp metering control example.2014 European Control Conference (ECC); 06/2014

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Exact solutions to traffic density estimation problems involving the

Lighthill-Whitham-Richards traffic flow model using Mixed Integer

Linear Programming

Edward S. Canepa, Christian G. Claudel, Member, IEEE

Abstract—This article presents a new mixed integer linear

programming formulation of the traffic density estimation prob-

lem in highways modeled by the Lighthill Whitham Richards

equation. We first present an equivalent formulation of the

problem using an Hamilton-Jacobi equation. Then, using a

semi-analytic formula, we show that the model constraints

resulting from the Hamilton-Jacobi equation result in linear

constraints, albeit with unknown integers. We then pose the

problem of estimating the density at the initial time given

incomplete and inaccurate traffic data as a Mixed Integer

Linear Program. We then present a numerical implementation

of the method using experimental flow and probe data obtained

during Mobile Century experiment.

I. INTRODUCTION

Control and estimation of the state in a distributed param-

eter system such as the transportation network requires the

fusion of traffic flow data with traffic flow models, which are

typically formulated as partial differential equations (PDEs).

They provide an efficient manner of modeling physical

phenomena in a mathematically compact way, in which the

distributed features of the systems are integrated. For the case

in which in the dynamics of the system is modeled by a PDE,

the derivation of the model constraints is a very complex

problem. This is the case for the Lighthill-Whitham-Richards

(LWR) partial differential equation [18], [22] commonly used

to model urban and highway traffic. The LWR equation is a

first order scalar hyperbolic conservation law that computes

the evolution of a density function, corresponding to the

density of vehicles on a highway section. Most estimation

techniques such as Extended Kalman Filtering [1] (EKF) or

Ensemble Kalman Filtering [25] or Particle Filtering (PF)

rely on approximations to determine the model constraints,

either through linearization or sampling. Their objective is

to estimate the current state of the system, as well as the

uncertainty around this state.

Unlike standard estimation methods, the framework pre-

sented in this article does not require any approximation of

the model. It can be used whenever one wants to determine

the ranges of initial densities (or any convex function of the

boundary data) compatible with the traffic model as well as

measurement data. It can also compute an estimate of the

state of the system minimizing a given convex functional.

E. Canepa is a Ph.D. Student, Deparment of Electrical Engineering,

King Abdullah University of Science and Technology, Saudi Arabia (email:

edward.canepa@kaust.edu.sa). Corresponding Author

C. Claudel is an Assistant Professor, Deparment of Electrical Engineering,

King Abdullah University of Science and Technology, Saudi Arabia (email:

christian.claudel@kaust.edu.sa).

This exact estimation technique is based on the Moskowitz

function [20], [21], also known as the cumulative number of

vehicles function (CVN), which is used here as an interme-

diate computational abstraction. The CVN function can be

understood as the integral form of the density function, and

solves an Hamilton-Jacobi (HJ) PDE, whereas the density

function itself solves the LWR PDE. The main advantage

of using the HJ PDE is that its solutions can be expressed

semi analytically [8], which enables us to derive the model

constraints explicitly. In addition, the CVN function is a

natural framework for incorporating Lagrangian (probe) data

into the traffic flow model [7], [6], [10].

We introduced in [9] a Linear Programming framework

for solving data assimilation and reconciliation problems on

highway sections, which we now extend to the problem

of initial density estimation. This extension requires the

derivation of model compatibility constraints arising from

the initial conditions themselves, which was not considered

earlier. In this article, we show that these compatibility

conditions do not break the convexity of the problem, and

that the problem of estimating a linear function of the initial

density given inaccurate boundary flows still results in a LP.

We also present an extension of this approach to problems

involving Lagrangian data, which result in a Mixed Integer

Linear Program (MILP) in general. The resulting estimates

are guaranteed to be exact, unlike estimates obtained using

standard nonlinear optimization techniques. We illustrate

the effectiveness of the proposed initial density estimation

algorithm on a dataset [26] obtained during the Mobile

Century experiment [17].

The rest of this article is organized as follows. Section II

defines the solution to the LWR PDE and its equivalent

formulation as a HJ PDE. In section III, we recall the

analytical expressions of the solutions to HJ PDEs for the

triangular flux functions investigated in this article, and

show that the LWR PDE constraints correspond to convex

constraints in the unknown initial and boundary condition

parameters. The problem of estimating unknown initial and

boundary conditions from measurement data under LWR

model constraints is posed as a Linear Program (LP) in

section IV, and is shown to become a Mixed Integer Linear

Program (MILP) when mobile data is considered. We then

present a numerical implementation of the density estimation

problem in section V, using experimental data from the

Mobile Century dataset, freely available from [26].

Page 2

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II. BACKGROUND

A. The Lighthill-Whitham-Richards traffic flow model

In the remainder of the article, we will assume that the

spatial domain which represents the highway section is [ξ,χ],

where ξ and χ respectively represent the upstream and

downstream boundaries of the domain. Traffic flow on this

section can be described by the density function, traditionally

denoted ρ(·,·). The density function represents an aggregated

number of vehicles per space unit, and can be modeled by

the Lighthill-Whitham-Richards (LWR) PDE:

∂ρ(t,x)

∂t

+∂ψ(ρ(t,x))

∂x

= 0

(1)

The function ψ(·) is called flux function. It depends on

several empirical parameters, such as the number of lanes,

the drivers habits and the capabilities of their vehicles.

Several models have been proposed for ψ, in particular

the triangular model defined below, and widely used in the

literature [12], [13], [14].

?

In the remainder of this article, we assume that the flux

function is given by (2). While other concave flux functions

could be used and would also yield convex constraints,

the instantiation of model constraints as linear inequalities

requires piecewise linear flux functions, such as (2).

ψ(ρ) =

vρ

w(ρ − km)

if ρ ≤ kc

otherwise

(2)

B. Hamilton-Jacobi equation

Equivalently, the state of traffic can be described by

a scalar function M(·,·) of both time and space, known

as Moskowitz function [20], [21]. The Moskowitz function

is a macroscopic description of traffic flow, which appears

naturally in the context of traffic. It can be thought of as

follows: let consecutive integer labels be assigned to vehicles

entering the highway at location x = ξ. The Moskowitz

function M(·,·) satisfies ?M(t,x)? = n where n is the label

of the vehicle located in x at time t [13], [14], and is assumed

to be continuous.

The density function ρ(·,·) is related [21] to the spatial

derivative of the Moskowitz function M(·,·) as follows:

ρ(t,x) = −∂M(t,x)

If the density function is to be modeled by the LWR PDE,

the Moskowitz function satisfies an Hamilton-Jacobi (HJ)

PDE obtained [2], [7] by integration of the LWR PDE:

?

Several classes of weak solutions to equation (4) ex-

ist, such as viscosity solutions [11], [3] or Barron-

Jensen/Frankowska (B-J/F) solutions [4], [15]. For the prob-

lem investigated in this article, these solutions are equivalent,

and can be computed implicitly using a Lax-Hopf formula.

∂x

(3)

∂M(t,x)

∂t

− ψ

−∂M(t,x)

∂x

?

= 0

(4)

C. Barron-Jensen/Frankowska solutions to Hamilton Jacobi

equations

In order to characterize the B-J/F solutions, we first need

to define the Legendre-Fenchel transform of the Hamiltonian

ψ(·) as follows.

Definition 2.1: [Legendre-Fenchel transform] For an

upper semicontinuous Hamiltonian ψ(·), the Legendre-

Fenchel transform ϕ∗(·) is given by:

ϕ∗(u) :=

sup

p∈Dom(ψ)

Solving the HJ PDE (4) requires the definition of value

conditions, which encode the traditional concepts of initial,

boundary and internal conditions.

Definition 2.2: [Value condition] A value condition

c(·,·) is a lower semicontinuous function defined on a subset

of [0,tmax] × [ξ,χ].

In the remainder of the article, a value condition can

encode an initial condition, an upstream boundary condition

or a downstream boundary condition. Each of these functions

is defined on a subset of R+× [ξ,χ].

For each value condition c(·,·), we define the partial solu-

tion [19] to the HJ PDE (4) using the Lax-Hopf formula [2],

[7].

[p · u + ψ(p)]

(5)

Proposition 2.3: [Lax-Hopf formula] Let ψ(·) be a con-

cave Hamiltonian, and let ϕ∗(·) be its Legendre-Fenchel

transform (5). Let c(·,·) be a lower semicontinuous value

condition, as in Definition 2.2. The B-J/F solution Mc(·,·)

to (4) associated with c(·,·) can be algebraically repre-

sented [2], [7] by:

Mc(t,x) = inf

(u,T)∈Dom(ϕ∗)×R+

(c(t − T,x + Tu) + Tϕ∗(u))

(6)

Equation (6) implies the existence of a B-J/F solution

Mc(·,·) for any value condition function c(·,·). However,

the solution itself may be incompatible with the value

condition that we imposed on it, i.e. we do not necessarily

have ∀(t,x) ∈ Dom(c),Mc(t,x) = c(t,x).

D. Properties of the solutions to scalar Hamilton-Jacobi

equations

The structure of the Lax-Hopf formula (6), implies the fol-

lowing important property, known as inf-morphism property.

The inf-morphism property can be formally derived through

capture basins, such as in [2].

Proposition 2.4: [Inf-morphism property] Let the value

condition c(·,·) be minimum of a finite number of lower

semicontinuous functions:

∀(t,x) ∈ [0,tmax] × [ξ,χ], c(t,x) := min

j∈Jcj(t,x)

(7)

The solution Mc(·,·) associated with the above value

condition can be decomposed [2], [7], [8] as:

∀(t,x) ∈ [0,tmax]×[ξ,χ], Mc(t,x) = min

In the present work, the value conditions cj(·,·) are not

known exactly, either because of measurement uncertainty

j∈JMcj(t,x) (8)

Page 3

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(case of the upstream and downstream boundary condition)

or because of the lack of measurements (case of the initial

condition). However, even if the real values of cj(·,·) are not

known exactly, they cannot be arbitrary as they have to apply

in the strong sense (see [24] for a mathematical definition)

to be compatible with the LWR model. In the remainder of

this article, we define the model constraints as the set of

constraints that applies on the value conditions cj(·,·) to

ensure that all value conditions apply in the strong sense.

The inf-morphism property is critical for the derivation of

the LWR PDE model constraints, allowing us to instantiate

these constraints as inequalities.

Proposition 2.5: [Model compatibility constraints for

block value conditions] Let c(·,·) = min

and let Mc(·,·) be defined as in (6). The value condition

c(·,·) satisfies ∀(t,x) ∈ Dom(c),Mc(t,x) = c(t,x) if and

only if the following inequality constraints are satisfied:

j∈Jcj(·,·) be given,

Mcj(t,x) ≥ ci(t,x) ∀(t,x) ∈ Dom(ci), ∀(i,j) ∈ J2

(9)

The proof of this proposition is available in [9]. Note that

equation (9) represents an important improvement, as the

model constraints are now semi-explicit. In order to solve the

problem completely, we still need to evaluate the functions

Mcj(·,·) explicitly. These explicit solutions were derived

in [8] for affine initial and boundary conditions blocks, and

are presented in section III.

III. EXPLICIT SOLUTIONS TO PIECEWISE AFFINE INITIAL

AND BOUNDARY CONDITIONS

Multiple types of value conditions can be incorporated into

the estimation problem. In the present article, we restrict

ourselves to initial conditions, upstream and downstream

boundary conditions. These value conditions are typically

measured (with some error) using fixed sensors, such as

inductive loop detectors, magnetometers or traffic cameras.

Note that an increasing proportion of traffic data is generated

by mobile sensors onboard vehicles, which generate internal

conditions [7]. We incorporate such internal conditions to

solve an estimation problem in section V.

A. Definition of affine initial, upstream and downstream

boundary conditions

The formal definition of initial, upstream and downstream

boundary conditions associated with the HJ PDE (4) is the

subject of the following definition.

Definition 3.1: [Affine initial, upstream and down-

streamboundaryconditions] Let us define K

{0,...,kmax} and N = {0,...,nmax}. For all n ∈ N and

m ∈ M, we define the following functions, respectively

called initial, upstream and downstream boundary conditions:

=

Mk(t,x)=

−?k−1

+∞

i=0ρ(i)X

−ρ(k)(x − kX)if t = 0

and x ∈ [kX,(k + 1)X]

otherwise

(10)

γn(t,x)=

?n−1

+∞

i=0qin(i)T

+qin(n)(t − nT)if x = ξ

and t ∈ [nT,(n + 1)T]

otherwise

(11)

βn(t,x)=

?n−1

−?kmax

+∞

i=0qout(i)T

+qout(n)(t − nT)

k=0

ρ(k)X if x = χ

and t ∈ [nT,(n + 1)T]

otherwise

(12)

Note that the affine initial, upstream and downstream

boundary conditions defined above for the HJ PDE (4)

are equivalent to constant initial, upstream and downstream

boundary conditions for the LWR PDE (1). The domains of

definitions of these functions are illustrated in Figure 1.

Fig. 1.

conditions. The block upstream and downstream boundary conditions

respectively denoted by γn(·,·) and βn(·,·) are defined on line segments

corresponding to the upstream and downstream boundaries of the physical

domain. In contrast, the block initial conditions Mk(·,·) are defined on line

segments corresponding to the initial time. Note that the actual problem

involves block initial conditions covering the entire physical domain [ξ,χ],

and block boundary conditions covering the temporal domain [0,tmax], but

that these functions are unknown (or only partially known).

Domains of the initial, upstream and downstream boundary

B. Analytical solutions to affine initial, upstream and down-

stream boundary conditions

Given the affine initial, upstream and downstream bound-

ary conditions defined above, the corresponding solutions

MMk(·,·), Mγn(·,·) and Mβn(·,·) are given [9], [19] by

the following formulas:

Page 4

4

MMk(t,x) =

+∞

−?k−1

if x ≤ kX + wt

or x ≥ (k + 1)X + vt

i=0ρ(i)X

+ρ(k)(tv + kX − x)if kX + tv ≤ x

(k + 1)X + tv ≥ x

ρ(k) ≤ kc

and

and

−?k−1

i=0ρ(i)X

+kc(tv + kX − x)if kX + tv ≥ x

kX + tw ≤ x

ρ(k) ≤ kc

and

and

−?k−1

−kmtw

i=0ρ(i)X

+ρ(k)(tw + kX − x)

if kX + tw ≤ x

(k + 1)X + tw ≥ x

ρ(k) ≥ kc

and

and

−?k

−kmtw

i=0ρ(i)X

kc(tw + (k + 1)X − x)

if (k + 1)X + tv ≥ x

(k + 1)X + tw ≤ x

ρ(k) ≥ kc

and

and

(13)

Mγn(t,x) =

+∞

?n−1

if t ≤ nT +x−ξ

v

i=0qin(i)T

+qin(n)(t −x−ξ

v

− nT)if nT +x−ξ

and t ≤ (n + 1)T

+x−ξ

v

v

≤ t

?n

i=0qin(i)T

+kcv(t − (n + 1)T −x−ξ

v) otherwise

(14)

Mβn(t,x) =

+∞

−?kmax

−km(x − χ)

if t ≤ nT

+x−χ

w

k=0

ρ(k)X +?n−1

w

i=0qout(i)T

− nT)

+qout(n)(t −x−χ

if nT

+x−χ

t ≤ (n + 1)T

+x−χ

w

w

≤ t

and

−?kmax

k=0

ρ(k)X +?n

i=0qout(i)T

)

+kcv(t − (n + 1)T −x−χ

v

otherwise

(15)

C. Properties of the affine initial, upstream and downstream

boundary conditions

Inthissection, we

modelconstraints(9)are

?

...,qout(nmax)

.

Proposition 3.2: [Linearity property of the initial,

upstream and downstream boundary conditions] Let

us fix (t,x)

∈

and downstream boundary condition functions Mk(t,x),

γn(t,x) and βn(t,x) are linear functions of the co-

efficients (ρ(1),...,ρ(kmax),qin(1),...,qin(nmax),qout(1),

...,qout(nmax)).

The proof of this proposition is straightforward and fol-

lows directly from equations (10), (11) and (12).

Proposition 3.3: [Concavity property of the solution

associated with the initial condition] Let us fix (t,x) ∈

show

convex

that

in

theLWR

variablethe

ρ(1),ρ(2),...,ρ(kmax),qin(1),...,qin(nmax),qout(1),

?

R+ × [ξ,χ]. The initial, upstream

R+× [ξ,χ]. The solution MMk(t,x) associated with the

initial condition (10) is a concave function of the coefficients

ρ(·).

Proof —The Lax-Hopf formula (6) associated with the

solution MMk(·,·) can be written [7], [8] as:

MMk(t,x) =

inf

u∈Dom(ϕ∗) s. t. (x+tu)∈[kX,(k+1)X]

−ρ(k)(x − kX) + tϕ∗(u)

?

−

?

k−1

?

i=0

ρ(i)X

(16)

Let us fix u × Dom(ϕ∗). The function f defined as

f(ρ(1),...,ρ(kmax)) = −?k−1

MMk(t,x) is a concave function of (ρ(1),...,ρ(kmax)),

since it is the infimum of concave functions [5], [23].

Proposition 3.4: [Concavity property of the solutions

associated with upstream and downstream boundary

conditions] Let us fix (t,x) ∈ R+× [ξ,χ]. The solutions

Mγn(t,x) and Mβn(t,x) respectively associated with the

upstream and downstream boundary conditions (11) and (12)

are concave functions of the coefficients ρ(·), qin(·) and

qout(·).

Proof —The Lax-Hopf formula (6) associated with the

solution Mγn(·,·) can be written [7], [8] as:

i=0ρ(i)X − ρ(k)(x − kX) +

tϕ∗(u) is concave (indeed, affine). Hence, the function

?

Mγn(t,x) =inf

s∈R+∩[t−(n+1)T,t−nT]

n−1

?

i=0

qin(i)T

+qin(n)(t − s) + sϕ∗(ξ−x

Let us fix s

The function d defined as d(qin(1),...,qin(nmax))

?n−1

function of (qin(1),...,qin(nmax)), since it is the infi-

mum of concave functions [5], [23]. The same property

applies for Mβn(t,x), which is a concave function of

(ρ(1),...,ρ(kmax),qout(1),...,qout(nmax))

Propositions 3.2, 3.3 and 3.4 thus imply the following

convexity property:

Proposition 3.5: [Convexity property of model con-

straints] The model constraints (9) are convex functions

of

...,qout(nmax)?.

written as:

s)

(17)

∈

R+ ∩ [t − (n + 1)T,t − nT].

=

i=0qin(i)T + qin(n)(t − s) + sϕ∗(ξ−x

deed, affine). Hence, the solution Mγn(t,x) is a concave

s) is concave (in-

?

?ρ(1),ρ(2),...,ρ(kmax),qin(1),...,qin(nmax),qout(1),

Proof —The set of inequality constraints (9) can be

Mci(t,x) ≥ cj(t,x), ∀(t,x) ∈ Dom(cj)

∀j ∈ I such that (t,x) ∈ Dom(cj), ∀i ∈ I

Notethat Proposition

cj(t,x) in (18) is a linear function (labeled lj,t,x(·))

of

...,qout(nmax)?. In addition, by Propositions 3.3 and 3.4,

of

...,qout(nmax)?. Hence, the equality (18) can be written

(18)

3.2impliesthattheterm

?ρ(1),ρ(2),...,ρ(kmax),qin(1),...,qin(nmax),qout(1),

the term Mci(t,x) is a concave function (labeled ci,t,x(·))

?ρ(1),ρ(2),...,ρ(kmax),qin(1),...,qin(nmax),qout(1),

as:

Page 5

5

−ci,t,x

qout(1),...,qout(nmax)

?

ρ(1),ρ(2),...,ρ(kmax),qin(1),...,qin(nmax),

?

qin(1),...,qin(nmax),qout(1), ...,qout(nmax)

∀j ∈ I,∀(t,x) ∈ Dom(cj), ∀i ∈ I

+ lj,t,x

?

ρ(1),ρ(2),...,ρ(kmax),

?

≤ 0,

(19)

This last inequality is a convex inequality [5] in

?ρ(1),ρ(2),...,ρ(kmax),qin(1),...,qin(nmax),qout(1),

where f(·) is a convex function.

The above property is very important, and can be thought

of as follows. Consider the vector space V of all parameters

of the initial, upstream and downstream boundary conditions.

Each point of this vector space corresponds to a known value

condition (encompassing initial, upstream and downstream

boundary conditions). However, the solution to the LWR

PDE (1) associated with this arbitrary value condition will

satisfy the value condition itself on its boundaries if and

only if the model constraints (9) are satisfied. Proposition 3.5

essentially states that the set of value conditions compatible

with the LWR PDE model is convex.

...,qout(nmax)?, that is, an inequality of the form f(·) ≤ 0

?

IV. FORMULATION OF THE ESTIMATION PROBLEM AS A

MIXED INTEGER LINEAR PROGRAM

1) Decision

lined

?ρ(1),ρ(2),...,ρ(kmax),qin(1),...,qin(nmax),qout(1),

problem, and will be defined as the decision variable of our

optimization framework.

Definition 4.1: [Decision variable] Let us consider a

finite set of, initial, upstream and downstream boundary

conditions be defined as in (10) (11) and (12). The decision

variable v associated with this finite set of value conditions

is defined by:

variable: As

the

out-

in Proposition3.5, variable

...,qout(nmax)?

plays an important role in our estimation

v :=

?

ρ(1),ρ(2),...,ρ(kmax),qin(1),...,qin(nmax),

?

We denote by V the vector space of the decision variables

v defined by equation (20).

2) Model and data constraints: Let v denote the value

of the decision variable associated with the true state of the

system (which is not known in practice, and can only be

estimated). Because of model and data constraints, v must

satisfy the set of constraints outlined in Propositions 3.5

and 4.3 below.

Proposition 4.2: [Model constraints] The model con-

straints (9) can be expressed as the following finite set of

convex inequality constraints:

qout(1),...,qout(nmax)

(20)

MMk(0,xp) ≥ Mp(0,xp)

MMk(pT,χ) ≥ βp(pT,xp)

MMk(χ−xk+1

v

βp(χ−xk+1

v

,χ)

∀(k,p) ∈ K2

∀(k,p) ∈ K2l

,χ) ≥

∀(k,p) ∈ K2s. t.

χ−xk+1

v

∈ [pT,

(p + 1)T]

∀(k,p) ∈ K2

∀(k,p) ∈ K2s. t.

ξ−xk

w

∈ [pT,(p + 1)T]

MMk(pT,ξ) ≥ γp(pT,ξ)

MMk(ξ−xk

w

,ξ) ≥ γp(ξ−xk

w

,ξ)

(21)

Mγn(pT,ξ) ≥ γp(pT,ξ)

Mγn(pT,χ) ≥ βp(pT,χ)

Mγn(nT +χ−ξ

∀(n,p) ∈ N2

∀(n,p) ∈ N2

∀(n,p) ∈ N2s. t. nT+

χ−ξ

v

∈ [pT,(p + 1)T]

v,χ) ≥ βp(nT +χ−ξ

v,χ)

(22)

[0,nmax],Dom(Mk) ∩ Dom(Mγn) = ∅ and that ∀(k,n) ∈

[0,kmax]×[0,nmax],Dom(Mk)∩Dom(Mβn) = ∅. Thus, the

set of inequality constraints (9) can be written in the case of

initial, upstream and downstream boundary conditions as:

Mβn(pT,ξ) ≥ γp(pT,ξ)

Mβn(nT +ξ−χ

∀(n,p) ∈ N2

∀(n,p) ∈ N2s. t. nT+

ξ−χ

w

∈ [pT,(p + 1)T]

w,ξ) ≥ γp(nT +ξ−χ

w,ξ)

Mβn(pT,χ) ≥ βp(pT,χ)

Proof

—

∀(n,p) ∈ N2

(23)

Notethat

∀(k,n)

∈

[0,kmax] ×

MMk(0,x) ≥ Mp(0,x)

MMk(t,χ) ≥ βp(t,xp)

MMk(t,ξ) ≥ γp(t,ξ)

Mγn(t,ξ) ≥ γp(t,ξ)

Mγn(t,ξ) ≥ βp(t,ξ)

Mβn(t,ξ) ≥ γp(t,ξ)

Mβn(t,ξ) ≥ βp(t,ξ)

The conditions (24) all involve checking that a function

of (t,x) is greater than another function of (t,x) on a

line segment of R+× [ξ,χ]. Yet, because of the affine

structure of the initial and boundary conditions (10), (11)

and (12) as well as the piecewise affine structure of their

solutions (13), (14) and (15), the inequalities of the form

∀(t,x) ∈ Dom(ci),

lent to a finite number of inequalities of the form ∀p ∈

{0,...,pmax}, Mcj(tp,xp) ≥ ci(tp,xp). This arises from

the fact that a piecewise affine function is positive on all

points of a segment if and only if it is positive on each

extremity of the segment, and on the finite number of points

of the segment on which the function is not differentiable. In

the present case, this property implies the equivalence of (24)

and of (21), (22) and (23).

Hypothesis 4.3: [Data constraints] In the remainder of

our article, we assume that the data constraints are convex

in the decision variable v.

Different choices of error models yield convex data con-

straints, such as the two examples outlined below.

Example of convex data constraints (1) —

sensor measuring the boundary flows (qin(0),...qin(nmax))

∀x ∈ [pX,(p + 1)X],∀(k,p) ∈ K2

∀t ∈ [pT,(p + 1)T],∀(k,p) ∈ K2

∀t ∈ [pT,(p + 1)T],∀(k,p) ∈ K2

∀t ∈ [pT,(p + 1)T],∀(n,p) ∈ N2

∀t ∈ [pT,(p + 1)T],∀(n,p) ∈ N2

∀t ∈ [pT,(p + 1)T],∀(n,p) ∈ N2

∀t ∈ [pT,(p + 1)T],∀(n,p) ∈ N2

(24)

Mcj(t,x) ≥ ci(t,x) are equiva-

?

Consider a

Page 6

6

with 5% relative uncertainty, a loop detector measuring the

initial density ρ(3) with 10% absolute uncertainty, and no

downstream sensor. In this situation, the constraints are con-

vex inequalities (indeed, linear inequalities) in the decision

variable:

?

0.95qmeasured

in

ρ(3)measured− 0.1km≤ ρ(3) ≤ ρ(3)measured+ 0.1km

(n) ≤ qin(n) ≤ 1.05qmeasured

in

(n) ∀n ∈ [0,nmax]

(25)

?

Example of convex data constraints (2) —

sider two identical sensors measuring the boundary flows

(qin(0),...qin(nmax)), and (qout(1),...qout(nmax)) which are

characterized by a RMS relative error of 3%. In this situation,

the constraints are convex inequalities (quadratic convex

inequalities) in the decision variable:

Con-

In this situation the estimation problem becomes a

Quadratic Program.

nmax

?

nmax

?

n=0

?

?

qin(n) − qmeasured

in

?2

?2

≤ 0.03

nmax

?

nmax

?

n=0

?

?

qmeasured

in

?2

?2

n=0

qout(n) − qmeasured

out

≤ 0.03

n=0

qmeasured

out

(26)

?

V. IMPLEMENTATION

We now present an implementation of the estimation

framework presented earlier on an experimental dataset. The

dataset includes fixed sensor data (obtained from inductive

loop detectors in the present case) and mobile sensor data.

A. Experimental setup

In the following sections, the effectiveness of the

method is illustrated on different traffic flow estimation

problems which are formulated as LPs and MILPs, using

the Mobile Century [26], [25] dataset. The Mobile Century

field experiment demonstrated the use of Nokia N-95

cellphones as mobile traffic sensors in 2008, and was a joint

UC-Berkeley/Nokia project.

For the numerical applications, a spatial domain of

3.858 km is considered, located between the PeMS [27]

VDS stations 400536 and 400284 on the Highway I -

880 N around Hayward, California. The data used in this

implementation was generated on February 8th, 2008,

between times 18 : 30 and 18 : 55. In our scenario, we

only consider inflow and outflow data qmeasured

qmeasured

out

(·) generated by the above PeMS stations, i.e. we

do not assume to know any density data. Of course the

framework presented in this article allows incorporation of

density data, see for instance Example 1 in the previous

section. The layout of the spatial domain is illustrated in

Figure 2.

in

(·) and

For all the subsequent applications, we divided the spatial

domain into six segments of equal distance. We also set T=30

s as the aggregation time for the flow data (T is determined

by the granularity of PeMS data). All optimization programs

Fig. 2.

The upstream and downstream PeMS stations are delimiting a 3.858 km

spatial domain, outlined by a solid line. The direction of traffic flow is

represented by an arrow.

Spatial domain considered for the numerical implementation.

(LPs and MILPs) have been implemented using the CVX1

package [16] running in Matlab. The problems described

in this article are tractable: they typically involve hundreds

of variables and thousands of constraints, and are solved in

a few seconds (LPs) to a few minutes (MILPs) on a typical

desktop computer.

B. Initial density estimation on systems modeled by the

Lighthill-Whitham-Richards PDE

In this first scenario, our objective is to find the minimal

and maximal values of a function of the decision variables,

assuming that boundary flow data is available from the PeMS

sensors. For this specific application the objective function

is chosen as the total number of vehicles at initial time,

defined by?kmax

constraints are linear inequalities in

both model (21), (22) and (23) as well as data constraints.

For this specific application, the data constraints are (1 −

e)qmeasured

in/out

(n) ≤ qin/out(n) ≤ (1 + e)qmeasured

[0,nmax], where e = 0.05 = 5% is chosen the worst-

case relative error of the sensors. The maximal densities are

solutions to the following optimization program:

i=0ρ(i), though any convex piecewise affine

function of the decision variable would be acceptable. The

(20), and comprise

in/out

(n) ∀n ∈

Minimize

(respectively Maximize)?kmax

qout(n) ≤ (1 + e)qmeasured

For this implementation, we choose 50 pieces of upstream

and downstream flow data corresponding to 25 minutes of

data. The parameters of the triangular flux function are

set with standard values: v = 65mph, w = −10mph,

km = 120veh/(lane.mile). The optimal solutions to (27)

are illustrated in Figure 3.

i=0

ρ(i)

such that

(21)

(22)

(23)

(1 − e)qmeasured

qin(n) ≤ (1 + e)qmeasured

(1 − e)qmeasured

in

(n) ≤ qin(n) ∀n ∈ [0,nmax]

in

(n) ∀n ∈ [0,nmax]

(n) ≤ qout(n) ∀n ∈ [0,nmax]

out

(n) ∀n ∈ [0,nmax]

out

(27)

1The MILPs considered in this article are not convex. However, they can

be separated in a finite set of LPs, which can in turn be solved using CVX

Page 7

7

Fig. 3.

maximal/minimal number of initial vehicles, given boundary flow data.

Top: scenario satisfying the model and data constraints for which the initial

number of vehicles is the largest. Bottom: scenario satisfying the model

and data constraints for which the initial number of vehicles is the lowest.

Computation of the traffic scenarios corresponding to the

As one can see from Figure 3, the problem of estimating

the density given boundary flow data is severely under-

determined in general. There are considerable differences

between both subfigures, but these two figures nonetheless

both satisfy the LWR model constraints as well as the

measurement data constraints. Thus, an observer having the

knowledge of the LWR traffic flow model and knowing

only boundary flow data cannot decide which of these two

situations (or any intermediate situation) corresponds to the

current traffic flow situation. This illustrates the power of the

framework presented in this article: it allows one to quickly

and exactly determine what the worst case and best case

scenarios corresponding to a given dataset are, which is an

important asset for the practitioner.

C. Incorporation of probe data into the estimation process

The framework presented in this article can be extended to

take into account mobile data. In this article we only consider

one piece of mobile data for simplicity, though any number

of mobile data points can be added into the estimation

problem (at the expense of larger computational cost). Let

us define the mobile data points as (t1,x1) and (t2,x2),

where x1corresponds to the position of the probe vehicle at

time t1, while x2 corresponds to the position of the probe

vehicle at time t2. In this situation, we want the solution to

satisfy M(t1,x1) = M(t2,x2), since we want (t1,x1) and

(t2,x2) to be on an isoline [21] of M(·,·). Setting (k1,k2)

as two integer parameters representing the value condition

blocks that minimize M(t1,x1) and M(t2,x2), the above

inequality constraint can be written as:

Mck1(t1,x1) ≤ Mck(t1,x1)∀k ∈ [0,kmax+ 2nmax]

Mck2(t2,x2) ≤ Mck(t2,x2)∀k ∈ [0,kmax+ 2nmax]

Mck1(t1,x1) = Mck2(t2,x2)

(28)

Adding the constraints (28) into the estimation problem

yields a Mixed Integer Linear Program, since the integer

parameters (k1,k2) are unknown. In contrast to the previous

case, the feasible set will not be convex, but will consist in

an union of convex sets.

We illustrate in Figure 4 the addition of one block of mo-

bile data generated by a Nokia N-95 cellphone participating

to the Mobile Century experiment. Similarly to the previous

section, our objective is to find the scenarios corresponding

to the maximal total number of vehicles at initial time.

As can be seen from Figure 4, mobile data modifies

considerably the possible values of the initial densities, and

the extent to which this information alters the previous initial

condition scenarios can be determined exactly by solving the

above MILP.

Because of the integer nature of (k1,k2), a large number of

situations may arise in practice. The choice of the parameters

(k1,k2) has a great impact on the computed traffic scenario.

In terms of optimal value, the lowest subfigure corresponds

to a global maximum of the decision variable, while the

upper center and lower center subfigures correspond to local

maxima for specific values of (k1,k2).

VI. CONCLUSION

This article illustrates some applications of a new Mixed

Integer Linear Programming based estimation framework for

estimating traffic flow conditions on transportation systems

modeled by the Lighthill Whitham Richards PDE. Using a

Lax-Hopf formula, we show that the constraints arising from

the model, as well as the the measurement data result in

linear inequality constraints for a specific decision variable,

and that the problem of estimating a linear function of this

decision variable is a Linear Program. We also show that

the method can be extended to integrate mobile traffic data,

at the expense of additional integer variables, resulting in

a Mixed Integer Linear Program. An implementation of the

method on an experimental dataset containing fixed sensor

data as well as probe data is performed, and illustrates the

ability of the method to quickly and efficiently compute all

traffic scenarios compatible both with both the LWR model

and a given traffic dataset.

In contrast to other estimation methods (EnKF, particle

filtering, EKF), this method has the advantage of being exact

and efficient for small-scale problems. However, integrating

very large amounts of mobile data into the estimates is not

currently feasible since the complexity of the problem is

exponential in the number of mobile data points (each addi-

tional piece of data requires two additional integer variables).

Future work will be dedicated to improving this worst-case

performance by determining in advance the cases in which

the optimization problem in guaranteed to be infeasible,

using the properties of the solution functions.

Page 8

8

Fig. 4.

maximal number of initial vehicles, using probe data. In all subfigures,

we compute the scenario for which the initial number of vehicles is the

largest, given the boundary data as well as a block of probe data (black

segment) Top: Scenario without probe data. Top center: Scenario with

probe data, assuming k1= 2 and k2= 0. Bottom center: Scenario with

probe data, assuming k1 = 1 and k2 = 5. Bottom: Scenario with probe

data, assuming k1= 5 and k2= 6.

Computation of the traffic scenarios corresponding to the

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Optimal Control and

Networks and heterogeneous

CVX:Matlab

Available

software

from

Mobile Computing, IEEE

Weak formulation of boundary

International Journal of