Maggi_Seminario

Definition and analysis of model
predictive control schemes for
freeway traffic including capacity
drop phenomena
Università Degli Studi di Genova – Scuola Politecnica

Traffic congestion is a major issue in modern motorway
systems in and around metropolitan areas
The dramatic expansion of car-ownership has led to the daily
appearance of recurrent and nonrecurrent freeway congestions
Researchers are looking for methods and technologies that seek
to manage, operate, and maintain freeway facilities
The context

In the literature some
macroscopic traffic-flow
simulation models have
been criticized because they
do not include the capacity
drop phenomenon
In recent years, some
researchers focused their
studies on the mechanism
concerning the capacity
drop phenomenon and
unveiled its main features
The purpose of this thesis is related to integrate
capacity drop phenomena into some control
schemes and analyze the performances
The starting point

The background
The Cell Transmission Model
Model Predictive Control
schemes
Analyses and results
Conclusions
Computational analysis of FHOCPs
Simulation analysis of MPC schemes
Agenda
Simulation analysis of CTM models

The background
The Cell Transmission Model
Model Predictive Control
schemes
Analyses and results
Conclusions
Agenda
Computational analysis of FHOCPs
Simulation analysis of MPC schemes
Simulation analysis of CTM models

Overview of traffic controls
Ramp Metering Variable Speed Limits
Dynamic Routing
Vehicle-based traffic
control

What are bottlenecks and capacity drop phenomena?
By the literature, a bottleneck is a point in a stretch of freeway
system which has a reduction of capacity
An active freeway bottleneck is a point on the network upstream of which one
finds a queue and downstream of which one finds freely flowing traffic

Existing methodologies to estimate the drop
The traditional methodology The Phase Diagram

May geometry features influence this mechanism?
Merge bottleneck
Reduction in
travel lanes
Horizontal curve
• Vehicle density and its normalized version relate to capacity drop
• Capacity drop is recovered once densities near the bottleneck diminish
• Capacity drop is entirely avoided when densities remain sufficiently low
Capacity drops might be averted with traffic control schemes that regulate
density and prevent it from exceeding some specified threshold

Ramp metering and capacity drop: which are the benefits?
Postponing and sometimes eliminating bottleneck
activation
Accommodating higher flows during the pre-
queue transition period than without metering
Increasing queue discharge flow rates after
breakdown
73%
+2%
+3%

Cell Transmission Model (CTM) - Introduction
Macroscopic
Discrete
Given two cells, the flow across these
cells is determined by comparing:
• vehicles demand from the previous
cell
• vehicles supply at the next cell

CTM1 – Main variables
(1) Daganzo, C.F., The cell transmission model. Part I: A simple dynamic representation of highway
traﬃc, California Partners for Advanced Transit and Highways (PATH). UC Berkeley: California Partners
for Advanced Transit and Highways (PATH), 1993

CTM – The dynamic model
Density and queue length state
equations
Demand and supply functions
Mainstream flows and exiting flow
updating equations

CTM – Demand and Supply
Demand function Supply function

CTM – The dynamic model: the merge
The merge connection model

CTM – The closed-loop case
The optimal flow which has
been computed by a given
controller

CTM – How to include the capacity drop in CTMs?
CTM with a change in the
demand function2
CTM originated from the
discretization of the
Fundamental Diagram3
• Easy to apply
• The effect of the capacity
drop linearly increases with
the density
• The capacity drop mechanism
is incorporated in the demand
function
• More sophisticated
• A ‘’drop’’ is really taken into
account
• The capacity drop mechanism
is incorporated in the supply
function
(3) Srivastava, A., Geroliminis, N., Empirical
observations of capacity drop in freeway merges
with ramp control and integration in a ﬁrst order
model, Transportation Research C, 30, 2013,
161−177
(2) Roncoli, C., Papageorgiou, M., Papamichail, I.,
Optimal control for multi−lane motorways in
presence of vehicle automation and
communication system, 19th IFAC World
COngress at Cape Town, South Africa, 2014,
4178−4183

CTM – The CTM model with a change in the demand
In case of congestion, the
demand flow linearly
decreases according to a
fixed slope

FD
CTM – The CTM originated from a discretization of the FD (I)
What is the Fundamental Diagram?
Density
Flow

CTM – The CTM originated from a discretization of the FD (II)

CTM – The CTM originated from a discretization of the FD (III)
Demand function Supply function
CongestedUncongested

CTM – The CTM originated from a discretization of the FD (IV)
The demand function
The supply function
The updating state parameter function

Model Predictive Control (MPC) scheme: Introduction
Prediction
The future response of the controlled
plant is predicted using a dynamic
model over the prediction horizon
Optimization of the FHOCP
The predictive control feedback law is
computed by minimizing a predicted
performance cost which lets finding the
optimal input control sequence
Receding horizon
implementation
The first element of the optimal
predicted control sequence is used as
input of the system
The system is controlled by using Ramp Metering and the
optimal input control sequence consists of the predicted
optimal entering flows at the on-ramps

MPC scheme: The FHOCP
Which models have been used in the formulation of the FHOCP?
The standard CTM
The CTM with a
change in the
demand
Reformulation and simplification
Mixed Logical Dynamical form
FHOCPs to be optimized in the reciding horizon
implementation
Some auxiliary binary
variables have been
included to make the
problem be linear

MPC scheme: The performance costs
Cost function ConstraintsVariables
A FHOCP is characterized by
CF1 CF2

Simulation analysis of CTMs – Introduction
What is the focus?
• The focus was understanding how the different CTMs
work
• In order to define how they differ, some datasets have
been tested
Which models are
compared against?
• The standard CTM
• The CTM with a change in the demand function
• The CTM originated from a discretization of the FD (it is
used as reference model)
How do the datasets
vary?
• Mainstream demand before the first cell
• Suppy after the last cell
• On-ramp demands
How have the models
been implemented?
C# programming language was used and Microsoft Visual
Studio 2010 was adopted as the development environment

Simulation analysis of CTMs – The methodology
The comparing
methodology
• Visual comparison evaluating the density profiles
• Analytical comparison by defining some indices:
1. |σm| is the average absolute value gap between
two density profiles without including zero values
2. σm is the average value gap between two density
profiles
3. M is the maximum positive gap between two
density profiles
4. m is the maximum negative gap between two
density profiles
5. ϒ is an adherence factor

Simulation analysis of CTMs – The selected datasets

Simulation analysis of CTMs – An example
The standard CTM
The CTM originated from the FD discretization
The CTM with a change in the demand

Simulation analysis of CTMs – The results (I)
The obtained indices from the comparison between
• the standard CTM
• the CTM originated from the discretization of the FD

Simulation analysis of CTMs – The results (II)
The obtained indices from the comparison between
• the CTM with a change in the demand function
• the CTM originated from the discretization of the FD

What is the focus?
The focus of this analysis was the solution of a single FHOCP
to appreciate the computational time needed by the solver
to find the optimal solution. From the perspective of an
online application of the MPC scheme, short solution time
are required.
What is compared
against?
• The standard CTM and the CTM with a change in the
demand function
• Two different cost functions: CF1 and CF2
How do the datasets
vary?
• Dimension of the problem: the number of sections (N)
and the prediction horizon (Kp)
• Level of the congestion:
• initial densities
• demand from the cell before the first one
• supply after the last cell
• on ramp demands
Computational analysis of FHOCPs - Introduction

Computational analysis of FHOCPs – The methodology (I)
The methodology
• 4 different types of FHOCP have been analyzed:
1. Standard CTM and CF1
2. Standard CTM and CF2
3. CTM with a change in demand and CF1
4. CTM with a change in demand and CF2
• 9 groups of instances which differ in the dimension have
been selected
• 3 different datasets have been chosen
• 5 random instances have been tested in each case
Some computational
aspects
• IBM ILOG CPLEX Optimization Studio 12.3 has been
utilized as optimization software
• Personal computer: Intel(R) Core(TM) i5 CPU M460 @
2.53 GHz and installed memory RAM 4.00 GB

Computational analysis of FHOCPs – The methodology (II)
The 9 selected groups are characterized by an
increasing number of cells and finite-horizon
The 5 instances for each dataset are randomly
selected in the intervals showed in the table
A time limit equal to 60 seconds has been set
The intervals are characterized by an uniform
distribution.

Computational analysis of FHOCPs – The results (I)
The standard CTMCF1 CF2

Computational analysis of FHOCPs – The results (II)
The CTM with a change in the demand functionCF1 CF2

Computational analysis of FHOCPs – The results (III)

Computational analysis of FHOCPs – The results (IV)

Computational analysis of FHOCPs – Final considerations
Dimension
Model
Cost function
The length of the prediction horizon and the state of congestion
widely influence the solving time while, according to the
obtained data, the number of cells does not
It may be concluded that the attempt to include capacity drop
in the FHOCP reasonably leads to longer solving times
Despite a larger number of variables and constraints, the first
cost function lets the solver get the optimal solution in a shorter
time than the second one

Simulation analysis of the MPC schemes - Introduction
What is the focus?
• The focus is evaluating the effectiveness of the MPC
schemes resulting from embedding the different FHOCP
previously analyzed
• The MPC schemes have been compared trough
simulation with the open-loop case, i.e. when no control
is applied
What is compared
against?
The open-loop case against the closed-loop cases
Open-loop
VS
Closed-loop
• Both of them are characterized by the CTM originated
from the discretization of the FD as simulation model
• In the open-loop case, the system runs without any
control, i.e. the MPC scheme is not applied
• In the closed-loop case, the system is controlled by ramp
metering using a MPC scheme

Simulation analysis of the MPC schemes – The methodology
How do datasets
vary?
Two different datasets have been selected in this part of the
analysis:
• dataset 1, which causes light traffic condition
• dataset 2, which causes heavy traffic condition
Datasets 1 and 2 show different on-ramp rates
Which indices are
taken into account?
The following indices have been utilized in order to evaluate
the performances:
1. CF1 or CF2 calculated over the whole simulation period
2. Total Time Spent (TTS)
How are the
performances
compared?
• The performances are calculated by comparing the
indices calculated in the open-loop and closed-loop
respectively
• To be specific, the obtained percentage reduction by
using the MPC scheme has been evaluated

Simulation analysis of the MPC schemes – The analysis
An example - dataset 1 The resulted profile density in the open-loop case (I)

An example - dataset 1 The resulted profile density in the close-loop case (II)
Standard CTM
CF1

An example - dataset 1 The resulted profile density in the close-loop case (III)
Standard CTM
CF2

An example - dataset 1 The resulted profile density in the close-loop case (IV)
CTM with a
change in the
demand
CF1

An example - dataset 1 The resulted profile density in the close-loop case (V)
CTM with a
change in the
demand
CF2

An example - dataset 1 The resulted queue length at the on-ramps
On-ramp 3 On-ramp 6

Simulation analysis of the MPC schemes – Results
Index values in the
open-loop case
Index values in the
close-loop case and
performance improvements

Simulation analysis of the MPC schemes – Considerations
General
considerations
• The obtained results are satisfying
• The presence of the control improves the performances
Cost functions
comparison
• When CF1 is adopted, the percentage cost reduction is higher
• By comparing the TTS improvement , the two cost functions are
comparable
Models
comparison
Adopting a MPC scheme in which the capacity drop is included does
not seem to provide performance improvements

Conclusions
The researches on bottlenecks and capacity drop phenomena
have been conducted in recent years and some further aspects
may be analyzed in the next years
MPC schemes which are capable to capture these phenomena
are necessary to ensure a better development of this branch of
research
The presence of available data from a real freeway system may
refine the analysis and offer concrete possibilities to real application
on the field

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