Exact solutions to traffic density estimation problems involving the LighthillWhithamRichards 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 HamiltonJacobi equation. Then, using a semianalytic formula, we show that the model constraints resulting from the HamiltonJacobi 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.

Conference Paper: A wireless computational platform for distributed computing based traffic monitoring involving mixed EulerianLagrangian sensing
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ABSTRACT: This paper presents a new wireless platform designed for an integrated traffic monitoring system based on combined Lagrangian (mobile) and Eulerian (fixed) sensing. The sensor platform is built around a 32bit ARM Cortex M4 microcontroller and a 2.4GHz 802.15.4 ISM compliant radio module, and can be interfaced with fixed traffic sensors, or receive data from vehicle transponders. The platform is specially designed and optimized to be integrated in a solarpowered wireless sensor network in which traffic flow maps are computed by the nodes directly using distributed computing. A MPPT circuitry is proposed to increase the power output of the attached solar panel. A selfrecovering unit is designed to increase reliability and allow periodic hard resets, an essential requirement for sensor networks. A radio monitoring circuitry is proposed to monitor incoming and outgoing transmissions, simplifying software debug. An ongoing implementation is briefly discussed, and compared with existing platforms used in wireless sensor networks.Industrial Embedded Systems (SIES), 2013 8th IEEE International Symposium on; 01/2013 
Conference Paper: Spoofing Cyber Attack Detection in Probebased Traffic Monitoring Systems using Mixed Integer Linear Programming
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ABSTRACT: Traffic sensing systems rely more and more on user generated (insecure) data, which can pose a security risk whenever the data is used for traffic flow control. In this article, we propose a new formulation for detecting malicious data injection in traffic flow monitoring systems by using the underlying traffic flow model. The state of traffic is modeled by the LighthillWhithamRichards traffic flow model, which is a first order scalar conservation law with concave flux function. Given a set of traffic flow data, we show that the constraints resulting from this partial differential equation are mixed integer linear inequalities for some decision variable. We use this fact to pose the problem of detecting spoofing cyberattacks in probebased traffic flow information systems as mixed integer linear feasibility problem. The resulting framework can be used to detect spoofing attacks in real time, or to evaluate the worstcase effects of an attack offline. A numerical implementation is performed on a cyberattack scenario involving experimental data from the Mobile Century experiment and the Mobile Millennium system currently operational in Northern California.IEEE International Conference on Computing, Networking and Communications; 01/2013  SourceAvailable from: Edward Canepa
Conference Paper: A Framework for Privacy and Security Analysis of Probebased Traffic Information Systems
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ABSTRACT: Most large scale traffic information systems rely on fixed sensors (e.g. loop detectors, cameras) and user generated data, this latter in the form of GPS traces sent by smartphones or GPS devices onboard vehicles. While this type of data is relatively inexpensive to gather, it can pose multiple security and privacy risks, even if the location tracks are anonymous. In particular, creating bogus location tracks and sending them to the system is relatively easy. This bogus data could perturb traffic flow estimates, and disrupt the transportation system whenever these estimates are used for actuation. In this article, we propose a new framework for solving a variety of privacy and cybersecurity problems arising in transportation systems. The state of traffic is modeled by the LighthillWhithamRichards traffic flow model, which is a first order scalar conservation law with concave flux function. Given a set of traffic flow data, we show that the constraints resulting from this partial differential equation are mixed integer linear inequalities for some decision variable. The resulting framework is very flexible, and can in particular be used to detect spoofing attacks in real time, or carry out attacks on location tracks. Numerical implementations are performed on experimental data from the Mobile Century experiment to validate this framework.2nd ACM International Conference on High Confidence Networked Systems, Philadelphia, PA; 04/2013
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1
Exact solutions to traffic density estimation problems involving the
LighthillWhithamRichards 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 HamiltonJacobi equation. Then, using a
semianalytic formula, we show that the model constraints
resulting from the HamiltonJacobi 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 LighthillWhithamRichards
(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 HamiltonJacobi (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
2
II. BACKGROUND
A. The LighthillWhithamRichards 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 LighthillWhithamRichards (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. HamiltonJacobi 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 HamiltonJacobi (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 (BJ/F) solutions [4], [15]. For the prob
lem investigated in this article, these solutions are equivalent,
and can be computed implicitly using a LaxHopf formula.
∂x
(3)
∂M(t,x)
∂t
− ψ
−∂M(t,x)
∂x
?
= 0
(4)
C. BarronJensen/Frankowska solutions to Hamilton Jacobi
equations
In order to characterize the BJ/F solutions, we first need
to define the LegendreFenchel transform of the Hamiltonian
ψ(·) as follows.
Definition 2.1: [LegendreFenchel 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 LaxHopf formula [2],
[7].
[p · u + ψ(p)]
(5)
Proposition 2.3: [LaxHopf formula] Let ψ(·) be a con
cave Hamiltonian, and let ϕ∗(·) be its LegendreFenchel
transform (5). Let c(·,·) be a lower semicontinuous value
condition, as in Definition 2.2. The BJ/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 BJ/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 HamiltonJacobi
equations
The structure of the LaxHopf formula (6), implies the fol
lowing important property, known as infmorphism property.
The infmorphism property can be formally derived through
capture basins, such as in [2].
Proposition 2.4: [Infmorphism 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
3
(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 infmorphism 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 semiexplicit. 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 LaxHopf 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 LaxHopf 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 N95
cellphones as mobile traffic sensors in 2008, and was a joint
UCBerkeley/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
LighthillWhithamRichards 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 N95 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
LaxHopf 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 smallscale 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 worstcase
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