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




Conclusions

Agenda




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%

The background




Conclusions



Agenda


Cell Transmission Model (CTM) - Introduction

Macroscopic

DiscreteGiven 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 - Parameters

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

CongestedU

ncongested

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

The demand function

The supply function

The updating state parameter function

The background




Conclusions



Agenda


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 CTMThe 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 MLD form

MPC scheme: The performance costs

Cost function ConstraintsVariables

A FHOCP is characterized by

CF1 CF2

The background




Conclusions



Agenda


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

The background




Conclusions



Agenda


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 CF12. Standard CTM and CF23. CTM with a change in demand and CF14. 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

The background




Conclusions



Agenda


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 period2. 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

The background




Conclusions



Agenda


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

Thank you for your attention!

Maggi_Seminario

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