Distributed Control of Dangerous Goods Flows
Claudio Roncoli, Chiara Bersani and Roberto Sacile
DELAB – Laboratory forLogistics and Safety, University of Genova, via Opera Pia 13, Genova, Italy
Keywords: Dangerous Good Transport, Optimisation Problem, Decentralised Control, Linear Quadratic Regulator.
Abstract: A risk-based approach to managing dangerous goods (DG) transport flows by road is proposed, solving a
real-time flow assignment problem. The model assumes the planned scheduling of the fleets and the
medium planned speed for vehicles known a priori. The objective is to plan in the vehicle tour schedules in
base on DG and general vehicle flows data on the infrastructures acquired in real time. The model
minimises both the total risk on the road network and the gap between the real delivery times with respect to
the planned ones. The first objective is a social intent of a National Authority and the second one represents
the main important cost minimisation for DG carriers. The proposed model is formulated according to an
original distributed control approach, based on the decomposition of the original centralised linear quadratic
problem.
1 INTRODUCTION
Currently, the main important Dangerous Goods
(DG) transportation companies use Intelligent
Transport Systems (ITS) to implement DG
information systems in order to monitor and manage
their fleet during the tours (Benza at al., 2010)
From a legislative viewpoint, recently, the
European Commission emanated directives to
impose to the DG transportation companies the
adoption of new ITS aiming to improve safety and
security on road infrastructure. The Directive
2010/40/EU of the European Parliament, on the
framework for the deployment of Intelligent
Transport Systems in the field of road transport and
for interfaces with other modes of transport, has
entered into force on August 28, 2010. The EU
Commission has recognised that ITS would
significantly help traffic management and enable
various users to be better informed and make safer,
more coordinated and "smarter" use of transport
networks. Besides, it asserts also that ITS should
integrate telecommunications, electronics, and
information technologies with transport engineering
in order to plan, design, operate, maintain and
manage transport systems.
In this paper a risk-based approach to managing
DG transport flows by road is proposed, solving a
real-time flow assignment problem. The model
assumes that the planned scheduling of deliveries
and the average planned speed for vehicles are
known a priori. The objective is to plan the vehicle
tour schedules depending on DG and general vehicle
flows data on the infrastructures, acquired in real
time. The innovative aspect of the proposed
approach is to balance two different objectives
which usually are referred to different subjects
involved in DG transportation: the model minimises
both the total risk on the road network and the gap
between the real delivery times with respect to the
planned ones. The first objective is a social intent of
a National Authority and the second one represents
the most important cost for DG carriers.
A similar approach has already been presented
by Roncoli et al. (2012) assuming that a central DM
takes his decisions minimising both the risk due to
eventual accidents, and the cost due to delay in
deliveries. In this paper, a model with similar targets
is presented, however assuming a set of
decentralised DMs, allowing a significant reduction
of the information exchanged in the network, which
is limited to neighbouring nodes only.
This paper introduces a model, formulated
following a game theory framework, presenting the
mathematical formulation, and a functional approach
to solve it. Moreover, a case study is presented,
illustrating the feasibility and the effectiveness of the
solution.
325
Roncoli C., Bersani C. and Sacile R..
Distributed Control of Dangerous Goods Flows.
DOI: 10.5220/0004042803250328
In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 325-328
ISBN: 978-989-8565-21-1
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
2 MODEL FORMULATION
2.1 Mathematical Description
The model is formulated considering a logistic
network represented by a graph made up of nodes
(set ) and directed links (set ). Each node, that
represents a region involved in DG shipment, is
considered as an autonomous DM, and it may be an
origin node, a destination node, or simply a
transition node. Links does not have a physical
meaning, but simply represent the movement of
product from a node to another. The topology of the
graph is described by two subsets, defined for each
node:
including the links entering the
node;
including the links leaving the node.
The model is defined in a discrete time domain,
considering a time window .
The following variables associated to each node
are defined:
dimension of the node;
state variables related to the DG mass
quantity present at the node at instant
directed to destination ;
input variable related to a time-
dependent value of risk;
input variable related to the planned
quantity present at the node directed to
destination at time .
The variables
and
are assumed to be
positive only at the origin and at the destination
nodes, being zero at transition nodes.
Links are characterised by:
input variable related to the forecasted
reference (maximum) speed (expressed as
the space covered in a unitary time
interval) for the DG transported along link
directed to destination during time
interval
. These values may
represent the speed limit over the link, as
well as the speed due to potential traffic
congestion during a specific time interval.
The control variables are
and
,
where:
control variables related to the DG flow
leaving the node through link
directed to destination during the
unitary time interval
;
control variables related to the DG flow
entering in the node through link
directed to destination during the
unitary time interval
.
Multipliers
are also introduced in order to
guarantee a global convergence of the problem.
Variables introduced above permit to specify the
linear dynamic of the system as:
(1)
Where values
are initialised as:
(2)
The cost function to be minimised at each node
is represented by a weighted sum of quadratic
functions:
(3)
The equation (3) could be divided in three main
parts: the inventory-risk part, the flow-inventory
part, and the flow-multipliers part. The first one,
weighted by coefficient when and when
, has a double purpose, depending on the
typology of node. In fact, in an intermediate node,
according to the definition of
, it raises that
, thus the risk is weighted
only by the density of DG product at the node.
Making the assumption that the product is to be at
origin and destination nodes, it is clear that, in these
circumstances, only the difference between the real
product and the planned one is to be considered. The
second part, weighted by coefficient , aims to
minimise the gap between the computed speed and
ICINCO 2012 - 9th International Conference on Informatics in Control, Automation and Robotics
326
the planned one, according to:
(4)
Equation (4) is valid also for
, considering
. This part of the cost function have also the
purpose to avoid possible “jumps” of nodes during
the dispatching of DG products. The last part,
weighted by coefficient , represents the core of the
decentralised model: the multipliers
are
adjusted between neighbouring nodes in order to
achieve a global convergence.
The solutions of this minimisation problem,
performed at each node, when put together may not
constitute a feasible schedule since coupling
constraints have been relaxed by the multipliers. In
fact, the problem that arises is a non-cooperative
game among several players, and a further
optimisation part has to be added. Introducing the
notation
and
for variables
and
computed at node , the problem that
must be performed for each link shared by
neighbouring nodes is:
(5)
The multipliers are thus iteratively adjusted
based on the degree of constraint violations: hence,
the links represent the “market makers”, who adjust
values
taking advantage on the gap between
and
. Under these assumption, the Nash
equilibrium of this game represents therefore the
solution of the complete problem (Rantzer, 2009).
2.2 Solving Approaches
The minimisation problem at nodes could be
rewritten in matrix form, considering
as the
vector of state variables and
as the vector of
control variables, generating a quadratic cost
function:
(6)
Subject to a linear state equation:
(7)
This problem is assumed to be a Linear
Quadratic Regulator (LQR), and it could be easily
solved finding the closed loop optimal control, given
by following equations, as described by Shaiju and
Petersen (2008):
(8)
Values for matrices
in (8) are calculated via
the Riccati recursion (DARE):
(9)
The maximisation problem for links could be
handled applying a gradient ascent method: defining
an appropriate step , the values of
are
updated at each step according to:
(10)
3 CASE STUDY
In order to show the effectiveness of the proposed
methodology, a case study has been realised
considering a network made up of 8 nodes and 10
links, as shown in Figure 1. The time window
considered is , simulating an entire day
of planning, and thus assuming a time interval of
one hour. Planned speeds are set considering that
some links could be affected by variation of traffic
during the day causing a slowdown, whereas risk
values are set assuming that some areas could be
more crowded in some time intervals. Moreover, a
set of 5 deliveries of 20 units each is planned.
Figure 1: The network considered for the case study.
A calibration process was necessary to determine
proper values for coefficients , , , and in order
to respect problem purposes and to guarantee a
convergence of the algorithm. An important aspect
of the model is the computation of the speed over
Distributed Control of Dangerous Goods Flows
327
links, which guarantees the physical feasibility of
the solution. In fact, if the speed computed by the
model is higher than the planned one, the solution
may not be applicable to a real network. Figure 2
shows that the speed computed by this model is
always lower than the planned one, and moreover
their behaviour is very similar.
Figure 2: A comparison between the forecasted speed and
the real one over a link.
The success in the dispatch is another important
aspect to be pointed out. Results show a complete
delivery of product at the destinations at least at the
last time instant, as illustrated in Figure 3.
Figure 3: The trend of the planned inventory and the
computed one at a destination node.
Figure 4: The density computed for an intermediate node
according to a time-varying value of risk.
Assuming that the previous illustrated targets have
been achieved, the main part of this methodology is
thus the minimisation of risk. As a matter of facts,
this model aims to avoid, or at least decrease, the
quantity of products in a specific area (node) during
time intervals characterised by an high risk value.
The Figure 4 highlights that the value of density at a
node decreases when risk value at the same node
raises and, on the contrary, computed density raises
when risk is lower.
4 CONCLUSIONS
This paper presents an innovative methodology for a
time-dependent dispatching of hazardous materials.
This method considers decentralised Decision
Makers, allowing a moderate exchange of
information in the whole network. This assumption
make this methodology attractive for very large
scale networks. Besides, the problem to be
performed by each DM is formulated as a LQR,
making it computationally efficient. A case study
applied on a medium size network provided results
showing a proper behaviour with respect to initial
goals. A possible critical aspect of this methodology
is the elevated number of iterations that could be
necessary to achieve a full convergence of the
model. Further investigations could be the increase
of the information exchanged between neighbouring
nodes, for example transferring also the planned
inventory values.
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Benza M., Bersani C., D’Incà M., Roncoli C., Sacile R.
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Rantzer, A., 2009. Dynamic dual decomposition for
distributed control. In American Control Conference.
Roncoli, C., Bersani, C., Sacile, R., 2012. A system of
systems control model for the risk-based management
of dangerous goods transport flows by road. Submitted
to IEEE Systems Journal.
Shaiju, A. J., Petersen I. R., 2008. Formulas for Discrete
Time LQR, LQG, LEQG and Minimax LQG Optimal
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