an o ine dynamic programming technique for autonomous ... · proach to hybrid vehicle supervisory...

An Offline Dynamic Programming Technique for

Autonomous Vehicles with Hybrid Electric

Powertrain

AN OFFLINE DYNAMIC PROGRAMMING TECHNIQUE FOR

AUTONOMOUS VEHICLES WITH HYBRID ELECTRIC

POWERTRAIN

BY

BRYNN VADALA, B.Sc.

a thesis

submitted to the department of mechanical engineering

and the school of graduate studies

of mcmaster university

in partial fulfilment of the requirements

for the degree of

Master of Applied Science

c© Copyright by Brynn Vadala, April 2018

All Rights Reserved

Master of Applied Science (2018) McMaster University

(Mechanical Engineering) Hamilton, Ontario, Canada

TITLE: An Offline Dynamic Programming Technique for Au-

tonomous Vehicles with Hybrid Electric Powertrain

AUTHOR: Brynn Vadala

B.Sc., (Mathematics and Engineering)

McMaster University, Hamilton, Canada

SUPERVISOR: Dr. Ali Emadi

NUMBER OF PAGES: xvii, 170

ii

To my family

Abstract

There has been an increased necessity to search for alternative transportation meth-

ods, mainly driven by limited fuel availability and the negative impacts of climate

change and exhaust emissions. These factors have lead to increased regulations and a

societal shift towards a cleaner and more efficient transportation system. Automotive

and technology companies need to be looking for ways to reshape mobility, enhance

safety, increase accessibility, and eliminate the inefficiencies of the current transporta-

tion system in order to address such a shift. Hybrid vehicles are a popular solution

that address many of these goals. In order to fully realize the benefits of hybrid ve-

hicle technology, the power distribution decision needs to be optimized. In the past,

global optimization techniques have been dismissed because they require knowledge

of the journey of the vehicle in advance, and are generally computationally extensive.

Recent advancements in technologies, like sensors, cameras, lidar, GPS, Internet of

Things, and computing processors, have changed the basic assumptions that were

made during the vehicle design process. In particular, it is becoming increasingly

possible to know future driving conditions. In addition to this, autonomous vehicle

technology is addressing many safety and efficiency concerns.

iv

This thesis considers and integrates recent technologies when defining a new ap-

proach to hybrid vehicle supervisory controller design and optimization. The dy-

namic programming algorithm has been systematically applied to an autonomous

vehicle with a power-split hybrid electric powertrain. First, a more realistic driv-

ing cycle, the Journey Mapping cycle, is introduced to test the performance of the

proposed control strategy under more appropriate conditions. Techniques such as

vectorization and partitioning are applied to improve the computational efficiency of

the dynamic programming algorithm, as it is applied to the hybrid vehicle energy

management problem. The dynamic programming control algorithm is benchmarked

against rule-based algorithms to substantively measure its benefits. It is proven that

the DP solution improves vehicle performance by at least 9 to 17% when simulated

over standard drive cycles. In addition, the dynamic programming solution improves

vehicle performance by at least 32 to 39% when simulated over more realistic condi-

tions. The results emphasize the benefits of optimal hybrid supervisory control and

the need to design and test vehicles over realistic driving conditions. Finally, the dy-

namic programming solution is applied to the process of adaptive control calibration.

The particle swarm optimization algorithm is used to calibrate control variables to

match an existing controller’s operation to the dynamic programming solution.

v

Acknowledgements

This research was undertaken, in part, thanks to funding from the Canada Excellence

Research Chairs (CERC) Program.

Thank you to my supervisor, Dr. Ali Emadi, for granting me the opportunity to

pursue this research and for ensuring that my graduate studies were insightful and

enriching.

Thank you to all my peers and fellow researchers the researchers at the McMaster

Automotive Resource Center for sharing your professional expertise, guidance, and

support. In particular, Joel Roeleveld and Jeremy Lempert were instrumental in my

success throughout this process. Thank you to David Henry and Jordan Vadala for

reading my thesis and providing their feedback.

Finally, I would like to thank my family for all their love and support throughout

my studies. I owe my achievements to my parents and brother for their endless

encouragement and advice.

vi

Notation and Abbreviations

ABC Artificial Bee Colony

API Application Programming Interface

BEV Battery Electric Vehicle

CAGR Compound Annual Growth Rate

CO2 Carbon Dioxide

DARPA Defense Advanced Research Projects Agency

DOF Degree of Freedom

DP Dynamic Programming

DUC DARPA Urban Challenge

ECMS Equivalent Consumption Minimization Strategies

EM Electric Machine

EPA Environmental Protection Agency

EV Electric Vehicle

EVT Electric Variable Transmission

FTP Federal Test Procedure

vii

GA Genetic Algorithm

GHG Greenhouse Gas

GHO Global Health Organization

GM General Motors

GPS Global Positioning System

HEV Hybrid Electric Vehicle

HSD Hybrid Synergy Drive

HWFET Highway Fuel Economy Test

ICE Internal Combustion Engine

LIDAR Light Detection and Ranging

LUUDC Loughborough University Urban Drive Cycle

MG Motor Generator

MPG Miles per Gallon

MPGe Miles per Gallon Equivalent

NEC Net Energy Change

NEDC New European Driving Cycle

NHTSA National Highway Traffic Safety Administration

PGS Planetary Gear Set

PHEV Plug-in Hybrid Electric Vehicle

PMP Pontryagin’s Minimum Principle

viii

PMSM Permanent Magnet Synchronous Motor

PSO Particle Swarm Optimization

SDP Stochastic Dynamic Programming

SOC State of Charge

UDDS Urban Dynamometer Driving Schedule

ix

Contents

Abstract iv

Acknowledgements vi

Notation and Abbreviations vii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Fundamentals of Hybrid Electric Powertrains 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 HEV Powertrain Architectures . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Series Configuration . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Parallel Configuration . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Series-Parallel Configuration . . . . . . . . . . . . . . . . . . . 14

3 Autonomous Vehicles 16

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

x

3.2 History of Autonomous Vehicles . . . . . . . . . . . . . . . . . . . . . 20

3.3 Levels of Automation . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Level 1 - Function-Specific Automation . . . . . . . . . . . . . 22

3.3.2 Level 2 - Combined Function Automation . . . . . . . . . . . 23

3.3.3 Level 3 - Limited Self-Driving Automation . . . . . . . . . . . 23

3.3.4 Level 4 - Full Self-Driving Automation . . . . . . . . . . . . . 23

3.4 Autonomous Vehicle Control . . . . . . . . . . . . . . . . . . . . . . . 23

4 Journey Mapping 28

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Standard Drive Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Improving the Standard Drive Cycle . . . . . . . . . . . . . . . . . . 35

4.4 Proposed Improved Drive Cycle . . . . . . . . . . . . . . . . . . . . . 39

4.5 Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Vehicle Control and Energy Management 44

5.1 Energy Management Problem . . . . . . . . . . . . . . . . . . . . . . 44

5.2 Control System Formulation . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Optimal Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.4 HEV Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 54

6 Representative Vehicle Model 59

6.1 Vehicle Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.1.1 Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.1.2 Road Load Model . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.1.3 Final Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

xi

6.1.4 Planetary Gear Set . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1.5 Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.1.6 Electric Machines . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.1.7 Battery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2 Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2.1 Clutches and Operating Modes . . . . . . . . . . . . . . . . . 73

6.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3 Simulation Environment . . . . . . . . . . . . . . . . . . . . . . . . . 78

7 Dynamic Programming for Energy Management 79

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.3 Optimal Control problem . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.4 Theory of Dynamic Programming . . . . . . . . . . . . . . . . . . . . 84

7.5 Power Split Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.5.2 EV Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.5.3 EVT Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.5.4 EV to EVT Mode . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.5.5 EVT to EV Mode . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.5.6 Interpolation Method . . . . . . . . . . . . . . . . . . . . . . . 108

7.6 Optimal Vehicle Operation Points . . . . . . . . . . . . . . . . . . . . 109

8 Benchmarking 120

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

xii

8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.2.1 Rule-Based Control . . . . . . . . . . . . . . . . . . . . . . . . 125

8.2.2 Genetic Algorithm Rule-Based Control . . . . . . . . . . . . . 134

8.2.3 Algorithm Comparison . . . . . . . . . . . . . . . . . . . . . . 144

9 Conclusion and Future Work 146

9.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

9.2.1 Adaptive Control Calibration . . . . . . . . . . . . . . . . . . 148

9.2.2 Technical Challenges . . . . . . . . . . . . . . . . . . . . . . . 160

xiii

List of Figures

2.1 Fuel efficiency improvement based on degree of electrification. . . . . 10

2.2 Series hybrid vehicle configuration. . . . . . . . . . . . . . . . . . . . 12

2.3 Parallel hybrid vehicle configuration. . . . . . . . . . . . . . . . . . . 13

2.4 Planetary gear set configuration. . . . . . . . . . . . . . . . . . . . . 14

2.5 Block diagram of Toyota Hybrid Synergy Drive (HSD). . . . . . . . . 15

3.1 The hierarchy of decision making processes in an autonomous vehicle

control system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Pseudo-steady-state driving cycle - EMPA T115 cycle . . . . . . . . . 31

4.2 Standard driving cycle - NEDC cycle . . . . . . . . . . . . . . . . . . 31

4.3 Representative Real-word driving cycle - ARTEMIS Urban cycle . . . 32

4.4 North American Standard Cycle - FTP75 cycle . . . . . . . . . . . . 33

4.5 North American Standard Cycle - HWFET cycle . . . . . . . . . . . 33

4.6 North American Standard Cycle - US06 cycle . . . . . . . . . . . . . 34

4.7 North American Standard Cycle - SC03 cycle . . . . . . . . . . . . . 34

4.8 North American Standard Cycle - UDDS cycle at -6.7C (20F) . . . . 35

4.9 Velocity over time profile for the proposed journey mapping cycle. . 40

4.10 Vehicle grade over time for the proposed journey mapping cycle. . . 40

4.11 Velocity over time profile for the Google maps cycle. . . . . . . . . . 43

xiv

4.12 Vehicle grade over time for the Google maps cycle. . . . . . . . . . . 43

5.1 Generic structure of a hybrid vehicle supervisory controller. . . . . . 45

5.2 Power split configuration and power flow diagram. . . . . . . . . . . 47

6.1 Free body diagram of vehicle linear dynamics. . . . . . . . . . . . . . 60

6.2 Lever diagram of planetary gear set. . . . . . . . . . . . . . . . . . . 64

6.3 Engine fuel map from Autonomie. . . . . . . . . . . . . . . . . . . . 65

6.4 Motor A efficiency map from Autonomie. . . . . . . . . . . . . . . . 68

6.5 Motor B efficiency map from Autonomie. . . . . . . . . . . . . . . . 69

6.6 Equivalent circuit battery model. . . . . . . . . . . . . . . . . . . . . 71

6.7 All possible clutch locations for an input-split configuration. . . . . . 73

6.8 A possible input-split configuration that achieves all four modes. . . 75

6.9 Configuration of vehicle model used for simulation. . . . . . . . . . . 77

7.1 Dynamic programming. . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.2 An example of a transition matrix, J, at a time step k. . . . . . . . . 94

7.3 An example of the minimum cost to go matrix, Jopt, at a time step k. 95

7.4 An example of the control input matrix, Uopt, at a time step k. . . . 96

7.5 Block Diagram of DP system inputs and outputs. . . . . . . . . . . . 98

7.6 Nearest neighbour interpolation example. . . . . . . . . . . . . . . . 109

7.7 State of charge over time for the FTP75 city cycle. . . . . . . . . . . 111

7.8 Vehicle mode (or engine on/off) over time for the FTP75 city cycle. 111

7.9 Torque split between the engine, motor A, and motor B over time for

the FTP75 city cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.10 Angular speed of the engine, motor A, and motor B over time for the

FTP75 city cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

xv

7.11 Power of the engine, motor A, and motor B over time for the FTP75

city cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.12 State of charge over time for the highway cycle. . . . . . . . . . . . . 114

7.13 Mode over time for the highway cycle. . . . . . . . . . . . . . . . . . 114

7.14 Torque over time for the highway cycle. . . . . . . . . . . . . . . . . 115

7.15 Speed over time for the highway cycle. . . . . . . . . . . . . . . . . . 115

7.16 Power over time for the highway cycle. . . . . . . . . . . . . . . . . . 116

7.17 State of charge over time for the journey mapping cycle. . . . . . . . 117

7.18 Mode over time for the journey mapping cycle. . . . . . . . . . . . . 117

7.19 Torque over time for the journey mapping cycle. . . . . . . . . . . . 118

7.20 Speed over time for the journey mapping cycle. . . . . . . . . . . . . 118

7.21 Power over time for the journey mapping cycle. . . . . . . . . . . . . 119

8.1 State of charge over time for the city cycle. . . . . . . . . . . . . . . 126

8.2 Mode over time for the city cycle. . . . . . . . . . . . . . . . . . . . 126

8.3 Torque over time for the city cycle. . . . . . . . . . . . . . . . . . . . 127

8.4 Speed over time for the city cycle. . . . . . . . . . . . . . . . . . . . 127

8.5 Power over time for the city cycle. . . . . . . . . . . . . . . . . . . . 128








xvi




8.16 State of charge over time for the city cycle. . . . . . . . . . . . . . . 135

8.17 Mode over time for the city cycle. . . . . . . . . . . . . . . . . . . . 136

8.18 Torque over time for the city cycle. . . . . . . . . . . . . . . . . . . . 136

8.19 Speed over time for the city cycle. . . . . . . . . . . . . . . . . . . . 137

8.20 Power over time for the city cycle. . . . . . . . . . . . . . . . . . . . 137











9.1 Autonomie model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.2 Battery SOC over time for the highway cycle. . . . . . . . . . . . . . 158


9.4 Engine power over time for the highway cycle. . . . . . . . . . . . . 159

9.5 Motor A power over time for the highway cycle. . . . . . . . . . . . 159

9.6 Motor B power over time for the highway cycle. . . . . . . . . . . . . 160

xvii

Chapter 1

Introduction

1.1 Motivation

Concerns of environmental deterioration and the increase in emissions are important

problems in global development. The transportation sector is responsible for a sig-

nificant contribution to the global greenhouse gas emissions, and has been responsive

in terms of legislation and consumer demand. Safety is also a large concern when

evaluating the current transportation system. Millions of people every year lose their

lives or become physically impaired or disabled as a result of motor vehicle collisions.

It is estimated that this number will continue to increase under current conditions

[1]. As such, it is important to consider how to reduce the frequency and severity

of motor vehicle accidents. Moreover, this thesis explores how to move towards a

smarter, safer, and greener transportation system.

With increasing regulations on emissions, the electrification of vehicles is becom-

ing more attractive in the automotive industry. Hybrid electric vehicles (HEVs) have

1

M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering

received considerable attention in the past few years due to their numerous bene-

fits compared to conventional fossil fuel powered vehicles. Hybrid vehicles provide a

more sustainable, cleaner, and greener transportation alternative. Hybrid solutions

allow for low energy consumption, lower emissions, high fuel economy, and are fea-

sible for mass production. Electrified vehicle technology advancements are typically

focused on four major areas: power electronic drive systems, battery technology, elec-

tric control systems, and materials and body structures [2]. This thesis will focus on

the power-split HEV and how to integrate an optimal electric control system with

smarter, safer, and greener technologies.

Hybrid vehicles typically consist of an internal combustion engine (ICE), a bat-

tery, and electric machines. The addition of supplemental sources of energy allows

for the opportunity to optimize the use of the power sources while still delivering the

required power. The power distribution decision is determined by the hybrid super-

visory controller. Many control strategies have been explored in practice and in liter-

ature, including rule-based strategies and optimization based strategies. Rule-based

strategies are based on heuristics, and are generally easier to implement. However,

these strategies are not optimal. Model-based optimization methods with meaningful

objective functions are widely used to obtain an improved energy controller.

Literature often states that unless the future driving conditions can be predicted

during real-time operations, global optimization techniques cannot be implemented

directly [3, 4, 5]. But with the constant advancement of both vehicle and outside

technologies, it is becoming increasingly possible to predict future driving conditions.

2


The advancement of technologies such as sensors, cameras, lidar, GPS, and the Inter-

net of things are changing the transportation system. These technologies can allow

for real-time traffic information, such as traffic light information, road conditions,

vehicle speeds, and optimal route planning, to be known in advance. All of this infor-

mation can be combined to determine future driving conditions. With such significant

changes in the transportation system, it is important to revisit global optimization

techniques that would have been seen as impractical years ago. As such, this thesis

explores applying the Dynamic Programming (DP) global optimization technique to

the control of the hybrid electric powertrain.

Optimal control of the hybrid powertrain components addresses the green com-

ponent of creating a sustainable transportation system. Next, it is critical to explore

how to create a smarter and safer transportation system. Vehicle control techniques

are generally tested in a simulation environment on standard drive cycles. This test-

ing occurs during the design phase in order to predict the performance of the vehicle.

This means that the optimality of the hybrid supervisory controller strategy is gen-

erally validated over standard drive cycles. Accurately predicting the performance of

a vehicle in all environments will help to increase the safety of the vehicle. Standard

drive cycles are not representative of real driving conditions, as real world driving

tends to be faster, more aggressive, and much more unpredictable [6]. The standard

drive cycle omits many important real-world conditions, such as weather, traffic, and

terrain. Since vehicle performance predictions are only as accurate as their testbeds,

it is crucial to test and measure performance on testbeds that are representative of

real driving conditions. Thus, it is important to study the impact of designing vehicle

3


control systems over such unrealistic driving conditions.

When assessing the sustainability of the transportation system as a whole, it is

important to look beyond the vehicle efficiency. Some of the major concerns with the

current system include traffic congestion, accidents resulting from human error, and

roadway infrastructure. As mentioned, there are many new technologies that can aid

in the advancement of the transportation system. Autonomous vehicle technology

can address many of these concerns. Autonomous vehicle technology and popularity

is progressing rapidly. It is predicted that autonomous vehicles will reduce collisions,

energy consumption, and pollution considerably [2]. Moreover, this thesis considers

a dynamic programming technique for autonomous vehicles with a HEV powertrain

as a means of creating a smarter, safer, and greener transportation system.

1.2 Thesis Contributions

This thesis considers and integrates recent technologies in order to define a new ap-

proach to HEV supervisory controller design and optimization. The first layer of the

design process to consider is the testing method. In order to improve the accuracy of

vehicle performance predictions, the standard drive cycle is replaced. A more realis-

tic driving cycle, the Journey Mapping cycle, is defined to test vehicle performance

on. The journey mapping cycle is defined as a vehicles journey from an origin to a

destination that is influenced by terrain, vehicle aerodynamic conditions, and traffic.

Although this definition does not include all of the conditions that a vehicle is subject

to, it is still an improvement from the standard drive cycle and is simple enough to

test complex control strategies over.

4


Next, global optimization techniques are revisited with a new lense. These tech-

niques are becoming more realistic with the advancement of technologies that allow

for the prediction of future driving conditions. A systematic approach for applying

the dynamic programming algorithm to an autonomous vehicle with a power-split

HEV powertrain is presented. Vectorization and partitioning techniques are applied

when developing this strategy to improve the computational efficiency of the algo-

rithm. The problem space is reduced from typical applications of the DP algorithm

to HEVs. Apart from [7], most formulations have three control inputs: engine on/off,

engine torque, and motor torque. This formulation reduces this to two control inputs:

engine on/off and engine torque. The DP approach presented can be considered for

real-time application, or can be used in the design process for the benchmarking of

other control techniques. This thesis also shows how to apply the DP solution for

adaptive control calibration. Adaptive control calibration methods are not widely

considered.

Finally, the benefits of applying a global optimization technique are realized

through benchmarking exercises. The DP algorithm is measured against a rule-based

technique and an intelligent rule-based technique. The importance of testing over

a realistic drive cycle is also emphasized through the benchmarking activity. There

are significant deviations in performance between the DP solution and non-optimal

methods over the journey mapping cycle. Overall, this thesis emphasizes the need

to take a more holistic view of the transportation system when designing and testing

vehicles.

5


1.3 Thesis Outline

This thesis is divided into nine different chapters. The first chapter provides an

introduction to the problem and highlights the novel contributions of the research.

Chapter 2 explores the fundamental concepts of hybrid electric vehicle technology.

A brief history of the electrification of vehicles is provided and the classification of

powertrain configurations is discussed. Chapter 3 introduces autonomous vehicles,

and explores their benefits and current limitations. The progression of autonomous

vehicle technologies and the varying levels of autonomy is explored as well. Finally,

autonomous vehicle control is considered as it relates the problem at hand. Chapter 4

discusses the limitations of standard drive cycles and proposes the Journey Mapping

cycle. Chapter 5 reviews the energy management problem by defining the vehicle

control system mathematically, defining the notion of optimal behaviour, and consid-

ering multiple control strategies. Chapter 6 defines the representative vehicle model

considered, and its components. Chapter 7 introduces the theory of the dynamic pro-

gramming algorithm and systematically applies it to the autonomous HEV. Chapter

8 introduces two rule-based algorithms for benchmarking purposes and substantively

proves the optimality of the proposed DP solution. Finally, Chapter 9 presents the

conclusions to be drawn from this work and suggests future work. In particular, the

application of the DP algorithm to adaptive control calibration is considered.

6

Chapter 2

Fundamentals of Hybrid Electric

Powertrains

2.1 Introduction

The automotive industry has had a significant impact on the development of modern

society by satisfying the need for mobility. Enhanced mobility is integral to both eco-

nomic and global development. Conventional transportation technology is powered

by internal combustion engines (ICEs) and requires fossil fuels as the energy source.

This dependency on fossil fuels is a major threat to our societies and to our quality

of life. The burning of fossil fuels emits gases, such as carbon dioxide, that are a ma-

jor contributor to Green House Gas (GHG) emissions. There is scientific consensus

that the rising GHG levels are contributing to global warming [8]. Since fuel burnt

for transportation makes up approximately one third of global GHG emissions, the

current transportation system is not environmentally sustainable.

7


With growing concerns of environmental deterioration leading to increased reg-

ulations on emissions, there is pressure on automakers to produce alternatives to

conventional, fossil fuel powered vehicles. This has lead to increased interest in vehi-

cle electrification, or the addition of electric capabilities to vehicles.

The concept of the electric vehicle is not new, as the first successful electric car

in the U.S. debuted around 1890 [9]. Electric vehicles were the top selling vehicle

in the year 1900, representing 28% percent of the market [10]. The market shifted

away from electric vehicles shortly thereafter, as the ICE offered increased driving

range and performance capabilities. The high availability and low costs of fuel also

catalyzed the rise of the petrol-powered car and halted electric vehicle development

and production by 1935 [10].

Towards the end of the 20th century, increasing oil prices and pollution lead to re-

newed interest in the electrification of vehicles. Legislation in governments around the

world was introduced encouraging electric vehicles as a means of reducing greenhouse

gas emissions. In addition, programs with the aim of EV research and development

were launched globally. In 1996, General Motors began production of the EV1 elec-

tric car. In 1997, Toyota released the Prius in Japan. The Prius became the first

mass-produced hybrid car. This momentum slowed through the early 2000s until the

Battery Electric Vehicle (BEV) Nissan Leaf was launched in 2010. At this time, the

public and private sectors recommit to vehicle electrification.

In 2012, the plug-in hybrid electric vehicle (PHEV) Chevrolet Volt was launched

8


and outsold half of the car models on the U.S. market. The issues that are of con-

cern to potential electric vehicle owners are range, performance, and cost. The Tesla

Model S alleviated some of these concerns, thus generating major consumer interest.

The Tesla proved that electric cars can have the range and performance capabilities

that consumers desire, initiating the mainstream popularity of electric vehicles. EVs

are expected to gain more than 35% market share by 2035 [10]. The Wall Street Jour-

nal now reports that the mainstream popularity of electric cars will reduce gasoline

demand by 5% to 20% over the next two decades [10].

To create a more sustainable transportation system, higher efficiency vehicles with

significantly lower fuel consumption are required. The use of electrical energy to power

propulsion and non-propulsion loads in vehicles can provide these higher efficiencies

[11]. Modern electric vehicles range in levels of electrification and include battery elec-

tric vehicles (BEVs), hybrid electric vehicles (HEVs), plug-in hybrid electric vehicles

(PHEVs), etc.. The level of hybridization in vehicles also comes in various degrees.

There are three different degrees of hybridization: full, assist and mild hybrid electric

vehicles. A full hybrid is capable of running completely on the engine, on the battery,

or a combination of the two. An assist hybrid uses the engine for the base load, and

only utilizes the battery for engine start and torque boost during acceleration. A

mild hybrid is most similar to a conventional vehicle. Mild hybrids are equipped with

an oversized starting motor which allows the engine to be turned off when coasting,

breaking or stopped, and to restart quickly [12].

In general, electrified vehicles provide a more sustainable, cleaner and greener

9


transportation alternative. Such solutions allow for lower energy consumption, lower

emissions, and improved fuel economy. The relative fuel-efficiency improvement for

different electrification levels can be seen in Figure 2.1.3

-10

%

Deg

ree

of

Elec

trif

icat

ion

[%

]

100 kW

30-80 kW

20-50 kW

12-20 kW

8-15 kW

3-10 kW

Fuel Efficiency Improvement [%]

2-5

%

3-7 kW

8-1

5%

12

-2

0%

20

-5

0%

30

-8

0%

10

0%

BEV

PHEV

Full Hybrid

HV Mild HybridLV Mild

HybridMicro Hyrbid

Start/stop

100%

Figure 2.1: Fuel efficiency improvement based on degree of electrification.

HEVs have received considerable attention over the past few years due to their

numerous benefits over traditional ICE-based vehicles and their extended range ca-

pabilities over fully electric vehicles. Hybrid vehicles typically consist of an internal

combustion engine, a battery, and an electric machine. Hybrid powertrain topologies

are classified as series, parallel, and series-parallel configurations. The selection of

powertrain topology is application dependent and considers factors such as vehicle

10


size and weight, driving cycles, and performance requirements. All topologies are

characterized by several energy sources and the effective management of the energy

flow between these sources dictates the fuel efficiency of the vehicle.

2.2 HEV Powertrain Architectures

HEV powertrain configuration classification is based on how the engine and electric

motors are connected. The following subsections describe the three classes of config-

urations, namely, series, parallel, and series-parallel. For reference, the generator is

commonly referred to as Motor A, MG1, and EM1 in literature and throughout this

thesis. Similarly, the second motor is commonly referred to as Motor B, MG2, and

EM2.

2.2.1 Series Configuration

The design of the series powertrain configuration was inspired by the electric vehicle

[13]. The objective was to overcome the disadvantages of the EV and extend the

drive range by adding an engine/generator system to charge the batteries. In a series

configuration, an electric motor is used to supply the tractive energy for propulsion.

The ICE powers an electric generator, which either charges the batteries or powers

the tractive motor. Series hybrid power flow follows a single path, fuel to electric

then electric to mechanical power [14]. A series powertrain configuration can be seen

in Figure 2.2.

Under light load conditions, the engine/generator is used to charge the battery.

11


Engine

Inverter Battery

Motor B

Power Flow

Motor A

Figure 2.2: Series hybrid vehicle configuration.

Under large load conditions, the engine/generator helps the battery power the tractive

motor. In times of deceleration, some of the braking energy can be recovered through

the process of regenerative braking. The electric motor acts as a generator and is

used to charge the batteries.

2.2.2 Parallel Configuration

In a parallel configuration, both the ICE and EM are mechanically connected to

the wheel drive. Thus a parallel hybrid can use the engine and electric motor si-

multaneously to supply the tractive force necessary to drive the vehicle. The ICE

12


and EM are connected to the wheels through defined differential gear ratios. Any

combination of torque split between the two components can be used to provide the

requested driver torque. A parallel powertrain configuration can be seen in Figure 2.3.

Engine

Inverter Battery

Motor

Power Flow

Transmission

Figure 2.3: Parallel hybrid vehicle configuration.

Parallel hybrids generally operate on the principle that the engine provides the

base load and the traction motor provides the addition load requirement [13]. How-

ever, this is dependent on the level of hybridization.

13


2.2.3 Series-Parallel Configuration

A series-parallel, or power-split, configuration uses a combination of series and parallel

power flow. A planetary gear set is used to split the engine power between the

generator to produce electricity and the mechanical gear system to drive the wheels.

The engine, generator, and motor speeds are decoupled, allowing for a variable output

torque and speed. A planetary gear set consists of 3 gear types: sun, planet, and ring.

The sun gear is circled by three planet gears on a carrier. These are then enclosed by

the ring gear. The planetary gear set assembly can be seen in Figure 2.4.

S

C

R

Sun Gear

Ring Gear

Planetary Gears

Planetary Carrier

Figure 2.4: Planetary gear set configuration.

Here, the ICE is connected to the planet carrier, the generator motor is directly

coupled to the sun gear, and the tractive motor is coupled with the ring gear. The

combination of torque provided by the ring gear and the tractive motor powers the

vehicle wheels. During low speeds, the tractive motor supplements the power split

14


torque. During deceleration, the tractive motor acts as a generator to power the

battery. There are a variety of vehicle configurations based on the planetary gear set.

The most common is the Toyota Hybrid Synergy Drive (HSD), found in the Toyota

Prius. The block diagram of the Toyota HSD can be seen in Figure 2.5.

S

C

R

Output Shaft

Engine

Motor A

Motor B

Gear Ratio

Figure 2.5: Block diagram of Toyota Hybrid Synergy Drive (HSD).

15

Chapter 3

Autonomous Vehicles

3.1 Introduction

When assessing how to develop a more sustainable transportation system, it is im-

portant to look beyond the efficiency of the vehicle itself. There are multiple outside

factors that influence the overall efficiency of the transportation system. Some of

these factors include traffic congestion, human error, roadway infrastructure, and ve-

hicle storage. The current infrastructure is neither efficient, nor sustainable.

Dated technologies in the transportation sector have many negative impacts. Traf-

fic congestion is a global problem that leads to wasted time and fuel. In 2010 it was

estimated that 4.8 billion hours of individuals time and 1.9 billion gallons of fuel were

wasted as a result of traffic congestion [1]. It is also a widespread belief among traffic

safety professionals that increased congestion leads to more accidents [15].

The impact of the current transportation system on safety is also notable. Human

16


judgement and reaction time are unreliable, making vehicles dangerous to operate.

Many additional human failures including distraction, alcohol impairment, drug im-

pairment, and fatigue are common causes of motor accidents. The Global Health

Organization (GHO) estimates that there are 1.2 million road deaths a year [1]. Ad-

ditional research in the United States found that 93% of car accidents are primarily

the result of human error [16].

In addition, the current infrastructure does not utilize individual vehicles in an

efficient manner. It is estimated that the average vehicle is in operation for approx-

imately 4% of its lifetime [1]. When the vehicle is not in use, it may be using space

inefficiently by sitting in a garage or driveway, or it may be parked in an expensive lot.

Overall, this is not an effective system in terms of time, space, and economics. Thus,

there is motivation to sustain the positive benefits and mitigate the negative impacts

of mobility. Autonomous vehicle development looks at addressing these issues.

Autonomous, or self-driving, vehicle technology is progressing rapidly. The global

autonomous vehicle market is expected to grow at a compound annual growth rate

(CAGR) of 39.6% between 2017 and 2027 [17]. This growth is, in part, due to the

environmental and safety benefits of increased automation. The introduction of more

autonomous vehicles into the transportation system is expected to increase the effi-

ciency of the system through several mechanisms. Autonomous vehicles are expected

to reduce traffic congestion significantly, if not completely. First, an autonomous

vehicle can sense the acceleration and deceleration of a vehicle in front of it, and

respond with smoother and more efficient speed adjustments. This would decrease

17


emissions at the level of the vehicle, and reduce traffic propagating events. Increased

automation can also reduce the gap necessary between cars, allowing vehicles to utilize

space more efficiently on the roadway. This would be especially beneficial at traffic

lights, as more vehicles could utilize a green signal. The combination of autonomous

vehicles and traffic monitoring systems will help with efficient route planning. As

autonomous vehicles become more prevalent, other parts of the transportation sys-

tem can be advanced to further decrease congestion. For example, signal control and

autonomous intersection management could be enabled by autonomous vehicles. In

theory, autonomous vehicles can also operate at higher speeds in a safer manner. All

of these factors working in conjunction will significantly reduce traffic congestion, and

therefore reduce emissions. However, these benefits will not be fully realized if only

a small number of vehicles on the road are autonomous.

Autonomous vehicles also address many of the safety concerns with driving, remov-

ing the possibility of human error. The onboard computers calculate exact distances,

speeds, and accelerations required in each situation and react faster than humans

can. This will eliminate accidents due to fatigue, distractions, and impairment. Au-

tomated vehicles can also be programmed to follow speed limits and obey traffic laws,

which would reduce the amount of accidents due to speeding and aggressive driving.

However, there are still some safety issues as it is difficult to design a system that

can operate safely in every condition. Human and foreign object recognition tech-

nology needs to be advanced to improve detection in complex environments. People

can be multiple shapes and sizes and can be performing many different actions such

as walking, sitting, or biking. It is thus difficult to always perform accurate sensor

18


recognition of humans. Finally, there are many significant social applications of au-

tonomous vehicles. For example, in times of disaster, unmanned vehicles can be used

in dangerous situations to transport supplies.

Although there are many benefits of autonomous vehicles, there are also multiple

concerns, both technological and social, that need to be addressed. A key concern is

the current lack of legislation applicable to the technology. New legislation needs to

be put in place that is applicable to the technology. For example, laws would need to

address whether or not a driver must be present in a fully autonomous vehicle. From

a technological standpoint, the environment that autonomous vehicles must navigate

is extremely complex and variable. In an ideal world, roads would have embedded

sensors working with the technology. However, it is impractical and likely impossi-

ble to upgrade the current infrastructure on such a mass scale, as many social and

economic constraints exist. In addition, the overall cost of the technology may not

be affordable both on a personal and global level. One of the most difficult problems

to face will be human sentiment. Public opinion is a major constraint to progress.

This is because so much infrastructure revolves around the automobile, such as gas

stations, car washes, car dealerships, drive-throughs, parking garages, car insurance,

car loans, etc. Changes to the current transportation infrastructure will affect all

such stakeholders, and are destined to have some pushback.

There are many economic, political, and technological factors influencing the rate

of growth of the autonomous vehicle market, in both a positive and negative manner.

In spite of this, many advancements have been made in vehicle automation in recent

19


years, and continue to be made.

3.2 History of Autonomous Vehicles

Fully autonomous driving has been a major goal for automotive manufacturers and

the emerging technology sector in recent years. To date, autonomous transmissions,

automatic cruise control, automatic parking, and lane shifting assist are widely com-

mercially available in vehicles.

The first example of automation dates back to the 1500s when Leonardo da Vinci

invented the self-propelled cart. Much later, in 1933, the first autopilot system was

designed for long-range aircrafts [18]. Next, the first cruise control system was in-

vented in 1945 with the use of a mechanical throttle to smooth the ride. During

the space race in the 1960s, an autonomous cart was developed at Stanford with

the intention of operating on the moon. The cart was outfitted with cameras and

was programmed to detect and follow a white line on the ground. In 1977, the first

autonomous passenger vehicle was developed in Japan. Two cameras enabled this

vehicle to recognize street markings while traveling approximately 20 miles per hour

[18]. Autonomous robot and vehicle development continued to progress, with the

addition of cameras and microprocessing modules for detection in the 1980s to the

emergence of drones in the 1990s.

A pivotal point in the development of autonomous vehicles was the DARPA chal-

lenges from 2004 to 2013. The United States Department of Defense’s research wing

20


(DARPA) held a series of challenges with the intention of pushing autonomous tech-

nologies forward [18]. In the DARPA Urban Challenge (DUC) in 2007, the course

was an urban environment with paved roads, intersections, rotaries, winding roads,

highways, and traffic [19]. The traffic was simulated by 70 vehicles, some robotic and

some human operated. The vehicles were required to follow traffic laws, such as obey-

ing the speed limit and yielding to the correct cars at intersections. This competition

resulted in the development and integration of many new technologies in planning,

control, and sensing [19].

There are generally two approaches to autonomous vehicle development today.

The first approach, which most car manufacturers take [20], is an iterative approach

that adds autonomous capabilities to the current vehicle. This is done by adding

sensors and cameras that enable additional control to existing vehicles in order to

slowly transition vehicles to become more autonomous. Many large automotive com-

panies including Mercedes, GM, Ford, Nissan, and BMW have announced that they

are working towards selling driverless cars [20]. Alternatively, technology companies

tend to take a software based approach to autonomous technology and design the

vehicles from scratch. An example of this is the Google Car that was developed in

2010 [20]. Another significant event in autonomous vehicle development happened

in 2015 when Tesla introduced its semi-autonomous autopilot feature with a single

software update to the Model S [18]. This feature is capable of lane control with

autonomous steering, braking, and speed limit adjustment. The autopilot system

allows the vehicle to perform almost unassisted while highway driving.

21


In 2015, both Google and Uber announced their autonomous car technology pro-

grams in the same week. Both companies have had many significant developments

and impediments since then. Uber’s initial fleet of cars consisted of 20 Ford Fusions

that were equipped with cameras, lasers, a GPS, radar, and lidar. These cars al-

ways had drivers present for safety reasons. Google provided the world’s first fully

driverless ride on public roads in 2015. This vehicle did not have pedals or a steering

wheel. Each company continued to release more autonomous vehicles, and offered

ride sharing services with these vehicles. However, both companies have been under

public scrutiny as they have both experienced collisions. Many of these collisions

were determined to be the fault of human error.

3.3 Levels of Automation

The National Highway Traffic Safety Administration (NHTSA) has classified the level

of automation of a vehicle [21]. This classification is described in the following sec-

tions.

3.3.1 Level 1 - Function-Specific Automation

This is the lowest level of automation. This level includes vehicles that feature au-

tomation of specific control functions, such a cruise control, lane guidance, and parallel

parking. In this case, drivers are fully responsible for the control of the vehicle and

must be completely engaged. Hands on the wheel and feet on the pedal are required

at all times.

22


3.3.2 Level 2 - Combined Function Automation

Combined function automation encompasses vehicles that feature automation of mul-

tiple and integrated control functions. This may include adaptive cruise control with

lane centering. In this case, the driver must monitor the roadway and be available

for control at all times. However, there are circumstances where the driver may have

their hands off the wheel and feet off the petal simultaneously.

3.3.3 Level 3 - Limited Self-Driving Automation

Vehicles of this level have the ability to function without driver monitoring. However,

a driver is required to be present and control can be transitioned to the driver for all

safety-critical functions under certain conditions. This type of vehicle will monitor

the changes in such conditions and notify the driver.

3.3.4 Level 4 - Full Self-Driving Automation

These vehicles can perform all driving functions and monitor roadway conditions for

an entire trip. There is no driver required, and thus can operate with or without

human occupants.

3.4 Autonomous Vehicle Control

Autonomous vehicle control systems differ in complexity and function depending on

the level of automation present. There is extensive research into autonomous vehicle

control with many proposed systems and techniques [22, 23, 24]. To limit the scope

for purposes of this thesis, a high level overview of the control decisions that the

23


system must make is presented with a focus on Level 3 and 4 vehicle automation.

The decision making architecture of an autonomous vehicle has many components

with specific responsibilities. Generally, an autonomous vehicle observes environmen-

tal information through the use of a perception system. The perception system is typ-

ically a combination of cameras, sensors, GPS units, LIDARs, and other instruments

used to measure the vehicles surrounding environment. The observed information of

the vehicle surroundings must be combined with prior knowledge of the road network,

traffic laws, vehicle dynamics, and sensor models to make an appropriate decision of

vehicle motion [22].

A hierarchical control structure is commonly used in the design of control systems

for autonomous vehicles [25]. The decision making can be divided into four major

components, as shown in Figure 3.1. At the highest level, a route is planned based

on a user specified destination and available road network data. Next, a behaviour

layer exists where local driving tasks are determined based on the environment and

rules of the road. Motion planning then determines a continuous path for the vehicle

to follow based on vehicle position and orientation, as well as collision free space. Fi-

nally, a controller is used to execute the planned motion and determines the necessary

steering, throttle, and brake commands.

The predictive path control structure discussed above does not consider compo-

nent specific decisions. Powertrain operation is rarely discussed in literature in the

24


design of autonomous vehicle control systems. Depending on the powertrain archi-

tecture, the control system may need to make an additional level of decisions. The

powertrain operation decision layer must determine how the vehicle components will

satisfy the demanded acceleration or deceleration. For a fully electric vehicle, the

acceleration and deceleration values can be directly linked to component operations

through the use of actuators. The power demand can be directly related to the

torque demand since there is only one power source. On the other hand, hybrid ve-

hicle powertrain architectures have more components and thus more complex power

flow diagrams. In HEV powertrain architectures, the power demand is satisfied by

a combination of the battery and engine. Thus, an additional layer to the control

architecture is necessary to determine the power split between the engine and the

battery. Moreover, the predictive path controller acts as a supervisory controller over

a core powertrain controller.

To effectively increase the efficiency of the transportation system, it is important

to consider all aspects of vehicle efficiency. It is predicted that hybrid technology

will overlap with autonomous vehicles and thus this is an important area of research.

Simply implementing autonomous vehicle technology and electrifying components of

a powertrain does not guarantee optimal efficiency without ensuring that the con-

trol architecture governing the component operation is optimal or near-optimal. An

ineffective control system could result in high component power losses, undesirable

drivability characteristics, and minimal fuel efficiency improvements.

Thus, optimal operation of the powertrain components of an autonomous vehicle

25


is still important. Metrics to define optimal operation should focus on increasing fuel

economy, decreasing component power loss, and maintaining acceptable drivability

characteristics. The optimization of an autonomous vehicle with a hybrid electric

powertrain is similar to that of a conventional HEV. However, under such a condi-

tion, some of the constraints change. Focusing on Level 3 and Level 4 autonomous

vehicles, the driver demand and behaviour is excluded from the optimization prob-

lem. Speed and traffic estimations and data become increasingly important in the

controller design. This means that the controller must be designed on a testbed that

is representative of real driving conditions.

26


Route Planning

Behavioural Layer

Motion Planning

Control

Sequence of waypoints through road network

Motion specification

Reference path or trajectory

User specified destination

Steering, throttle, and brake commands

Road network data

Perceived agents, obstacles, signage, and road rules

Estimated position and orientation and collision free space

Estimated vehicle state

· Lane following· Intersections· Parking · Lane changes· Unstructured

environments

Figure 3.1: The hierarchy of decision making processes in an autonomous vehiclecontrol system.

27

Chapter 4

Journey Mapping

4.1 Introduction

Hybrid vehicles are sensitive to the conditions that they are operated under. That

is, the performance of an HEV is largely dependent on the environment that it is in.

It is important to accurately predict the performance and behaviour of an HEV in

all environments. This becomes increasingly important as the automation of vehicles

progresses. Vehicle performance is currently simulated and tested on standardized

drive cycles, as instructed by the government. It is impossible to test all vehicles on

the road in the conditions that they will be driven. This is why the standardized

drive cycle was defined; to simulate general driving conditions. If all vehicles are

tested under identical conditions, then a consistent measure of performance can be

produced and measured against.

In the United States and Canada, the governments use a 5-cycle testing system

for vehicle certification and performance rating. This 5-cycle testing system uses

28


standardized drive cycles that are generalized velocity-time profiles. Thus, the per-

formance predictions of vehicles in North America are only as good as the standard

drive cycles they are tested on. Real-world driving typically does not reflect these

cycles, as it is much more unpredictable [6]. Real-world driving tends to be more ag-

gressive; speeds are generally greater and faster and more frequent changes in speed

occur. In addition, a velocity-time profile is simply not a full enough picture of the

actual conditions that affect driving. The velocity profile is be subject to varying

conditions over time, such as weather, traffic, terrain, road, driver behaviour and so

on.

The omission of these conditions is clearly demonstrated by the significant devia-

tions between the EPA labels for fuel economy and energy consumption and the true

values measured [26]. Greater fuel consumption than quoted by the manufacturer

means higher CO2 emissions than expected. As a result, the consumer and manu-

facturer have a skewed perspective of the performance and environmental benefits of

vehicles. It is important to note that this problem is not unique to hybrid vehicles,

and is prevalent to conventional vehicles as well.

The consequence of using such standards to test and certify vehicles is that vehicle

design is largely based on these standards. This means that vehicle operation is being

optimized over unrealistic conditions. Inaccurate vehicle performance prediction may

also be a contributing factor in accidents that occur due to unknown driving con-

ditions. Moreover, there exists a large need for re-defining testing standards in our

transportation system. It may be impossible to simulate all driving conditions, but it

29


is important to continually take steps to improve the current system. In this Chapter,

an approach to improve the definition of the standard drive cycle is proposed.

4.2 Standard Drive Cycles

Vehicle manufacturers test their own vehicles using standardized testing and ana-

lytical procedures defined by the pertinent regulator. There are over 200 standard-

ized test cycles that are used in legislation for emissions and performance regulation

[27]. These cycles can be grouped into three major categories: U.S., European, and

Japanese. Standard test cycles can be further classified by applicable vehicle type.

In particular, these standards are designed specifically for cars, vans, trucks, buses,

and motorcycles.

Cycles can also be broadly divided into steady-state or transient cycles [27]. This

definition is based on the character of the speed and engine load changes. A steady-

state cycle is a sequence of constant speed and engine load modes. These cycles do

not represent achievable driving conditions. The reality of maintaining a constant

speed is illustrated by Figure 4.2, which shows a pseudo-steady-state driving cycle

[27].

30


Figure 4.1: Pseudo-steady-state driving cycle - EMPA T115 cycle

On the other hand, transient cycles represent real driving pattern, as vehicle speed

and engine load are changing continuously. Some of these cycles are more represen-

tative of real-life driving than others. For example, Figures 4.2 and 4.3 show an

unrealistic transient drive cycle and a more realistic drive cycle, respectively.

0 200 400 600 800 1000 1200

Time [s]

0

20

40

60

80

100

120

NE

DC

Cyc

le V

eloc

ity [k

m/h

]

Figure 4.2: Standard driving cycle - NEDC cycle

31


0 100 200 300 400 500 600 700 800 900 1000

Time [s]

0

10

20

30

40

50

60A

RT

EM

IS U

rban

Cyc

le [k

m/h

]

Figure 4.3: Representative Real-word driving cycle - ARTEMIS Urban cycle

The five cycles used in North America to test and certify hybrid vehicles for fuel

economy are shown in Figures 4.4 through 4.8. The first cycle shown is the FTP-75

cycle; it is a representative city driving cycle. The second cycle shown, the HWFET-

75 cycle, is a representative highway driving cycle. Next, the aggressive driving cycle,

US06, is shown. The last two cycles account for extreme temperatures. The SC03-

95 cycle shown in Figure 4.7 is intended to account for air conditioner use in high

temperatures. Lastly, the UDDS-20 cycle is used to represent cold temperature opera-

tion. This cycle is just the classic UDDS cycle at a lower temperature of -6.7 C (20 F).

32


0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time [s]

0

20

40

60

80

100F

TP

75 C

ycle

[km

/h]

Figure 4.4: North American Standard Cycle - FTP75 cycle

0 100 200 300 400 500 600 700 800

Time [s]

0

20

40

60

80

100

HW

FE

T C

ycke

[km

/h]

Figure 4.5: North American Standard Cycle - HWFET cycle

33


0 100 200 300 400 500 600

Time [s]

0

20

40

60

80

100

120

140U

S06

Cyc

le [k

m/h

]

Figure 4.6: North American Standard Cycle - US06 cycle

0 100 200 300 400 500 600

Time [s]

0

10

20

30

40

50

60

70

80

90

SC

03 C

ycle

[km

/h]

Figure 4.7: North American Standard Cycle - SC03 cycle

34


0 200 400 600 800 1000 1200 1400

Time [s]

0

20

40

60

80

100U

DD

S C

ycle

[km

/h]

Figure 4.8: North American Standard Cycle - UDDS cycle at -6.7C (20F)

4.3 Improving the Standard Drive Cycle

Multiple studies have been performed that demonstrate the need to improve these

standard drive cycles. For example, in [6] a study is done to compare real world driv-

ing to the ECE-15 and FTP-75 (or UDDS) drive cycles. A Toyota Prius is equipped

with a data logger and driven in an urban environment over a 9 month period. The

data collected is used to develop a new, more realistic drive cycle that was named

the Loughborough University Urban Drive Cycle (LUUDC). This vehicle was then

tested on a dynamometer on the LUUDC, ECE-15, FTP-75, and other drive cycle

cycles. It was determined that LUUDC predicted the miles per gallon 16.7% better

than the ECE-15 cycle and 31.4% better than the FTP-75 cycle. The LUUDC is a

better measure of urban driving, but there is still a significant amount of error from

the actual mpg. This was determined to be mainly due to the absence of road grade

in the definition of the drive cycle [6].

35


In another study [26], a 2012 Ford Focus Electric and a Toyota Prius 2006 are

driven over a 10 month span. The resulting real world driving conditions are com-

pared with several standard drive cycles, including the UDDS, NEDC, JC08, FTP75,

and US06 cycles. In addition, the fuel economy results are compared with EPA rated

value. The two vehicles are equipped with a CAN data logger and driven in an urban

environment. Autonomie is used to predict the MPGe and MPG of the standard

drive cycles listed above for each vehicle. Results for the Ford Focus Electric show a

percent error ranging from 27.85% to 90.72% between MPG predicted by the drive

cycles to the average actual measured MPG. The percent error between EPA rated

MPG and the average actual measured MPG is 28.23%. Similarly, results for the

Toyota Prius show a percent error ranging from 27.34% to 138.06% between MPG

predicted by the drive cycles to the average actual measured MPG. The percent error

between EPA rated MPG and the average actual measured MPG is 39.00%.

The study in [26] proposes a new concept called Journey Mapping that aims to

re-define drive cycles. Journey Mapping defines a vehicles drive cycle as the journey

of that particular vehicle from its origin to destination on a particular road which is

affected by various conditions; some of which are terrain, weather, road conditions,

traffic, driver behavior, vehicle condition, etc. [26]. The study parameterizes these

conditions in order to more accurately measure a vehicle’s environment. For example,

terrain is represented by road grade and weather conditions are represented by wind

speed, air density, ambient temperature, and so on. A more complete list of vehicle

condition parameterizations is shown in Table 4.1. It is found that the Journey Map-

ping model predicts energy consumption accurately within about 5% error on average

36


when compared to the true consumption [26].

The above examples highlight the weaknesses of the current standard drive cycles

in predicting energy consumption. It is important to asses which factors contribute

to these discrepancies. In [26], it is determined that road grade, auxiliary power,

and traffic conditions have the largest impact on energy consumption. In [28], it is

seen that driver characteristics (aggressive driving, driving at excessive speeds) and

route selection (road type, grade, and congestion) have the biggest impact on energy

consumption.

These findings help to prioritize factors to include in defining a drive cycle, as it

is difficult to parameterize and measure all driving conditions.

37


Table 4.1: Parameterizing Journey Conditions

Condition Parameterizations

Weather Wind SpeedAir DensityAmbient TemperatureAltitude of ObservationAlbedoCloud CoverLocation SettingAir Penetration

Traffic CongestionSignalsSpeed Limits

Terrain Grade Profile

Aerodynamic Longitudinal SlipVehicle MassMass DistributionWheel InertiaTire WidthTire HeightWheel Rim DiameterAir Penetration CoefficientVehicle Active Area for Aerodynamic Drag

Road Longitudinal SlipCoulomb FrictionViscous FrictionStictionTire slip

Driver Behaviour AgeExperienceMoodReflexesAggression

38


4.4 Proposed Improved Drive Cycle

When using drive cycles as test beds for vehicle performance, it is not feasible to

include all of the driving conditions. For example, including road grade, road curva-

ture, congestion, ambient temperature, wind speed, air density, altitude, etc. when

optimizing the supervisory controller of a vehicle in simulation would increase the

complexity of the problem significantly. This would impact the computation time

when evaluating a control algorithm for the powertrain. For this reason, the journey

mapping definition has been simplified from the definition proposed in [26].

It is generally seen in literature that one of the main parameters that has a large

impact on fuel consumption is road grade. As such, the journey mapping definition

has been simplified to the journey of a vehicle from an origin to a destination that is

influenced by terrain, vehicle aerodynamic conditions, and traffic. This eliminates the

complexity of variable weather conditions. In addition, factors that are dependent on

driver behaviour are excluded as the focus is on autonomous vehicles.

Thus, we define the journey mapping cycle as a velocity-time and grade profile.

Note that the proposed journey mapping cycle is taken from real-world driving con-

ditions found in [26]. The velocity-time profile for the new journey mapping cycle

can be seen in Figure 4.9 and the grade profile can be seen in 4.10.

39


0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time [s]

0

5

10

15

20

25

Jour

ney

Map

ping

Cyc

le [m

/s]

Figure 4.9: Velocity over time profile for the proposed journey mapping cycle.

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time [s]

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Gra

de [r

ad]

Figure 4.10: Vehicle grade over time for the proposed journey mapping cycle.

Journey mapping has been introduced to test the vehicle in simulation because

it is important to have an accurate journey prediction for autonomous vehicles. In

40


addition, hybrid vehicle performance is sensitive to the environment that it is in.

Moreover, designing and testing the control strategy over realistic conditions, like

the journey mapping solution proposed, will allow for the performance of the control

strategy to be more accurately depicted.

Journey mapping is also used to eliminate the complexities of simulating the entire

autonomous vehicle control system. It more accurately simulates the outputs of the

autonomous vehicle control system that is governing the hybrid supervisory controller

than standard cycles.

4.5 Next Steps

The proposed cycle is limited in the sense that it is not a complete picture of driving

conditions. This is because it is difficult to collect all of the data necessary to define

a journey map as it is defined in [26]. In addition, it is an example of a journey on

a specific road from point A to point B and is again not representative of all road

conditions. To overcome these limitations, many improvements could be made in the

future to include additional driving conditions.

When it comes to autonomous driving, the inclusion of such conditions becomes

increasingly important. The vehicle controller is responsible for safely navigating the

vehicle while accounting for the vehicle’s environment. A vehicle supervisory con-

troller for an autonomous vehicle could take historical and current traffic data, speed

limits, and road inclination into account. This data can be collected in many ways,

41


including the use of global positioning system (GPS) data. Vehicle navigation ap-

plications, such as Google Maps and Waze, use a combination of historical data and

current vehicle conditions to predict the velocity profile of a vehicle from point A

to point B. Google Maps Application Programming Interfaces (APIs) also allow a

user to collect the latitude and longitude at each point along a road, and therefore

the grade profile of a vehicle’s route can be determined. This information can be

combined with current weather data such as wind speed and temperature to create a

fairly accurate representation of the conditions that a vehicle will experience.

For example, the Google Maps APIs were used to determine the vehicle path

from Hamilton, Ontario to Toronto, Ontario. The Google Maps APIs returned the

latitude, longitude, and elevation of the points along the path. The elevation allowed

for the gradient of the path to be determined. In addition, a duration for each section

of the trip is given. This allows for an estimated speed to be determined for each leg

of the trip. The velocity and gradient profiles developed based off of this information

are shown in Figures 4.11 and 4.12, respectively. It is clear that the velocity profile

is not realistic, but it gives a general idea of the speeds that are to be expected over

the journey.

42


0 500 1000 1500 2000 2500 3000

Time [s]

0

5

10

15

20

25

30

Spe

ed [m

/s]

Figure 4.11: Velocity over time profile for the Google maps cycle.

0 500 1000 1500 2000 2500 3000

Time [s]

-15

-10

-5

0

5

10

15

20

Gra

dien

t

Figure 4.12: Vehicle grade over time for the Google maps cycle.

43

Chapter 5

Vehicle Control and Energy

Management

Energy management control strategies are crucial in the design of an efficient hybrid

vehicle. The goal of the vehicle’s supervisory controller is to minimize fuel consump-

tion and emissions while maintaining vehicle performance and safety. To achieve

overall optimality, it is important to optimize the vehicle architecture, components,

and control strategy. Thus, a considerable amount of research has been done on

energy management control strategies.

5.1 Energy Management Problem

Regardless of the powertrain configuration, the challenge of energy management in

a hybrid vehicle is to assure optimal use and regeneration of the total energy in the

vehicle. The control strategy must determine the power distribution between the

primary energy converter and the renewable electrical storage system. In topologies

44


with multiple components, additional power distribution between components must

be determined. The distribution is generally constrained by two factors. First, the

power requested by the vehicle to output the necessary vehicle speed must be satisfied

up to a known limit. Second, the state of charge of the battery must be maintained

within particular limits. The basic structure of a hybrid vehicle control system can

be seen in Figure 5.1 [29].

Supervisory Controller

Torque Demand

Vehicle Speed

Vehicle Acceleration

Battery SOCComponent Constraints

ICEPower Split

DeviceMotor Generator

Battery Control Unit

Figure 5.1: Generic structure of a hybrid vehicle supervisory controller.

The supervisory controller has the following inputs: torque demand, current ve-

hicle speed, vehicle acceleration, battery state of charge, and component constraints.

With these inputs, an efficient strategy must be used to control the ICE, motor, gener-

ator, transmission, and battery control unit. The primary decision of the supervisory

controller is the power split between the ICE and the battery. The ideal controller

will do this in such a way that the overall system losses are minimized and the most

fuel efficient operation is achieved. Moreover, the goal of the controller is to satisfy

the power demand and battery limits while:

45


1. Maximizing fuel economy

2. Minimizing emissions

3. Minimizing system losses

4. Meeting performance and drivability criteria

The configuration of the powertrain dictates the power flow and thus influences

the control strategy. The number of degrees of freedom (DOF) in the power flow

diagram of the vehicle defines the dimension of the control vector u(t) in the energy

management controller. In addition to this, the size and specifications of the vehicle

components influences the control strategy. For example, adding a larger battery

would increase the electric range of the vehicle. However, the engine and regenerative

braking may not be able to recharge the battery enough to meet the charge sustaining

constraints. In this case, the battery would need to be plugged in to an electrical

outlet to recharge. An appropriate control strategy for a vehicle with a battery that

can be plugged in to recharge would not be suited to a configuration where the bat-

tery can not be plugged in. Thus it is important to design the hardware and control

strategy together.

The energy management controller must have a control vector with dimension

equal to the number of DOFs of the power flow diagram. The selection of an ap-

propriate control vector is dependent on the configuration and design constraints.

Similarly, there are many state variables in a hybrid powertrain. The dimension and

selection of the state vector x(t) is dependent on the required accuracy [8].

46


Power split configurations have more energy flow paths than a series or parallel

configuration, and thus there is more complexity in the controller. The power flow

diagram of a power split powertrain configuration can be seen in Figure 5.2.

Engine

Motor A Inverter Battery

PGS Motor B

Power Flow

Figure 5.2: Power split configuration and power flow diagram.

The power flow has several different operating modes that can be realized by the

overall control strategy. The basic operating modes are summarized as follows:

1. Electric-only Mode: At low speeds, the vehicle is typically powered by the

electric motor(s) only.

2. Cruising Mode: In cruising mode, the engine power is distributed between the

wheels and the generator. The generator runs backwards to provide electricity

to the motor, and the motor provides additional torque at the drive shaft.

47


3. Motor-Assist Mode: If the vehicle requires more power, then both the battery

and ICE provide power. This is also referred to as passing mode or accelerating

mode.

4. Charging Mode: In this mode, the engine power is again distributed between

the wheels and the generator. If the battery SOC is low, then the generator

provides electrical energy to charge the battery.

5. Braking Mode: Regenerative braking is used to convert kinetic energy from

the wheels into electrical energy. The motor acts as a generator to charge the

battery.

The operating modes are ultimately determined by the control system. The fol-

lowing section will set up the control problem at hand. In order to design an effective

energy management controller, the following three key steps must be taken: (1) De-

fine the control system; (2) Define the notion of optimal behaviour; (3) Select an

appropriate optimization algorithm or control strategy.

5.2 Control System Formulation

A power-split HEV is an example of a hybrid dynamical system [30]. A hybrid dy-

namical system, H, is a system where continuous and discrete dynamics interact.

Definition (Hybrid Dynamical System): A hybrid dynamical system or hy-

brid automata, H, is a collection:

H = (X ,Q,U ,Y , f, Init, Inv, E ,G,R)

48


where

X − is the continuous state space, such that X ∈ Rn

Q− is a finite set of discrete states, such that Q = {q1, q2...qk}

U − is a finite collection of input variables

Y − is a finite collection of output variables

f − is a set of vector fields describing the continuous dynamics for all q ∈ Q

Init− is the set of initial states, such that Init ⊆ Q×X

Inv− is the invariants of each discrete state q ∈ Q

E − is a collection of discrete transitions, such that E ⊂ Q×Q

φ− is a set of guards prescribing when a discrete state transition occurs

R− is the reset map

The following will describe how the control system of a power-split HEV can be

described as a hybrid dynamical system. This is done similar to [30]. The power-split

HEV configuration considered can operate in 2 modes: engine on or engine off. These

operating modes represent a set of discrete variables q = {q1, q2} ∈ Q. Here, let q1

denote power-split or engine on mode and q2 denote two motor EV or engine off mode.

The system must transition from engine on mode to engine off mode, and vice-

versa. These transitions make up the collection of discrete transitions E . Let us define

our discrete transitions as follows: e1 : q1 → q1 represents the transition from engine

off to engine off, e2 : q1 → q2 represents the transition from engine off to engine on,

49


e3 : q2 → q1 represents the transition from engine on to engine off, and e4 : q2 → q2

represents the transition from engine on to engine on.

The state of the HEV system is represented by the engine speed, ring gear speed,

and generator speed. Thus, the set of continuous state variables X are represented

by x = {ωeng, ωr, ωgen}.

The finite collection of input variables consists of both continuous and discrete

inputs, where U = Uc × Ud. The continuous inputs to the system are the engine

torque, the motor torque, and the generator torque. Thus, the continuous inputs are

uc = {Teng, Tmot, Tgen}. The discrete input to the system is the engine on/off decision

u1, such that ud = {u1}.

The finite collection of output variables Y consists of the vehicle speed, battery

SOC, engine speed, and generator speed. As such, y = {v, SOC, ωeng, ωgen}.

The dynamics of the system are different depending on the discrete state. This

is because the mechanics of the HEV system are different when the engine is on and

when the engine is off. The set of vector fields f are input dependent as a result. The

invariant set for each q ∈ Q is described as follows:

50


Inv(q1) =

ωeng = 0

|ωring| ≤ ωring,max

Teng = 0

u1 = Engine off

and

Inv(q2) =

ωeng,min ≤ ωeng ≤ ωeng,max

|ωring| ≤ ωring,max

|ωgeb| ≤ ωgen,max

The components of the HEV system have operating limits in each operating mode.

These operating constraints apply to the engine speed and torque, motor speed and

torque, generator speed and torque, battery SOC, and battery power. The set of

guards φ assigns the set of admissible inputs for each state.

In terms of the system model described, the goal of the energy management con-

troller is to find the optimal input control sequence U and transition rule E such that

the optimal design objective is achieved [30]. To do this, it is important to define an

appropriate optimal control law to achieve the design objective.

51


5.3 Optimal Control Law

In order to formulate an optimal control law it is necessary to define the notion of

optimal behaviour. This is illustrated through the use of a performance index or

objective function. The controller must provide a control vector, u(t), that is the

dimension of the number of degrees of freedom in the energy flow diagram. The

degrees of freedom of the system is dependent on the topology of the powertrain in

question. Here, we will begin to define an objective function, J , to optimize the control

law over. The primary goal of the energy-management controller is to minimize the

total fuel consumption of the vehicle over a journey from origin to destination. Thus,

the objective function should minimize the overall fuel mass consumed, mF , over the

trip time, T .

J =

∫ T

0

mF (t, u(t))dt (5.1)

Other performance criteria is typically included in the objective function. These

are mainly factors that account for pollutant emissions and drivability concerns. For

example, emission rates of regulated pollutants, anti-jerk factors, smoothness, and

mode-switching factors are all parameterized in the objective functions found in lit-

erature [8, 31, 32]. For this reason, the objective function will be defined in more

general terms, as follows.

J =

∫ T

0

L(t, u(t))dt (5.2)

where L() is the cost function.

52


It is clear that the vehicle operation that minimizes Equation 5.1 in pure electric

mode. This becomes an issue if the energy recovered from regenerative braking is not

enough to maintain the battery charge within the appropriate limits. This will leave

the battery depleted at the end of a journey, which is not ideal for hybrids that are not

PHEVs. The vehicle certification process requires charge sustaining operation. This

behaviour constraint can be taken into account in two different ways. First, a penalty

factor on the final state can be added to the objective function. This will penalize

the performance index if the final state of charge deviates from the initial state of

charge. This penalty function can be added to the objective function as follows.

J = γ(SOC(T )) +

∫ T

0

L(t, u(t))dt (5.3)

where γ is the penalty function and SOC(T ) is the state of charge of the battery at

the final time, T . In this case, γ(SOC(T )) is often applied as a hard constraint on

the control problem that requires the initial state of charge to be equal to the final

state of charge of the battery.

Alternatively, the charge sustaining penalty can be included in the performance

criteria L, as shown in Equation 5.4.

J =

∫ T

0

{L(t, u(t)) + α

[SOC(t)− SOC(0)

]}dt (5.4)

where α is a weighting factor. The value of α is a positive constant that is typically

determined through experimentation.

53


The optimal control law can be extended further with the addition of various soft

and hard constraints. For example, operational limits of the powertrain components

must be included to ensure that the solution to the problem is feasible. Moreover,

the optimal control law selected should reflect the overall goals and constraints of the

problem at hand.

Minimizing fuel consumption and satisfying the charge sustaining criterion are the

key objectives of this work. As such, the charge sustaining criteria is incorporated

as both a hard final state constraint and in the performance criteria. The charge

sustaining performance index is incorporated to discourage rapid charging and dis-

charging of the battery. The optimal control law used throughout experimentation

will be as follows:

J = γ(SOC(T )) +

∫ T

0

{mF (t, u(t)) + α ˙SOC(t) + γ(t)

}dt (5.5)

Now that the notion of optimal behaviour has been defined, an appropriate ap-

proach to evaluating the control law must be determined.

5.4 HEV Control Strategies

Many energy management strategies have been proposed in literature. This section

discusses the various approaches that can be taken to evaluate the optimal control

law. The approaches are divided into two main categories: rule-based strategies and

optimization based strategies.

54


Early energy management strategies take heuristic approaches. The most common

strategy used is a rule-based approach, which is based on the expected powertrain

behaviour. For example, it is common to have an HEV operate in pure electric mode

until a particular speed threshold. At this speed, the electric motor reaches its torque

limit and the engine is turned on. This is done because ICEs are not typically ef-

ficient at low speeds; they have low torque thresholds at low speeds. On the other

hand, electric motors have a high maximum torque limit at low speeds. This intuition

allows the components to operate in their efficient ranges. Similar heuristics can be

applied to generate an overall rule-based control strategy. Rule-based strategies are

often based on the concept of load leveling [5]. This is where the electric machine is

used to force the ICE to operate at its peak efficiency at all times during the driving

cycle. Rule-based techniques can be further categorized into deterministic or fuzzy

rule-based methods.

Deterministic rule-based approaches are based off the analysis of power flow and

heuristics. Efficiency maps and lookup tables are used to force components to operate

in their efficient ranges. Engineering intuition and experience is often used to meet

drivability constraints. The issue with rule-based approaches is that the parameters,

such as the speed threshold, are highly dependent on the vehicle architecture, com-

ponents selected, and driving conditions. In order to acquire accurate parameters,

extensive experimental calibration activities would need to be performed.

Efforts to improve rule-based approaches have been made with the use of Fuzzy

55


Logic [33, 34, 35] and Neural Networks [29, 36]. Fuzzy logic controllers are an ex-

tension of the rule-based approach that removes the need for binary outcome type

rules. Fuzzy logic is typically seen as advantageous when dealing with non-linear

systems as it is robust. Several fuzzy logic strategies have been discussed in litera-

ture [37, 38]. Rule-based methods have also been combined with Neural Networks in

literature [29, 39]. Neural Networks use the concepts of machine learning to improve

the already existing rule-based controller. The main advantage of such rule-based

approaches is the effectiveness in real time. However, these methods still require a

considerable amount of calibration and prior knowledge to design. In addition to

this, rule-based methods do not guarantee overall system efficiency. If a component

is operating in its most efficient range, this does not mean that the entire system is

operating optimally.

To resolve such problems, a more rigorous mathematical approach can be taken.

Model-based optimization methods with meaningful objective functions are widely

used to obtain an improved energy controller. Various static, numerical, analytical,

and equivalent-consumption minimization optimization strategies have been explored.

Optimization based methods can be further categorized into global optimization tech-

niques and real time optimization techniques. Global optimization techniques typi-

cally need to know the vehicle velocity profile a priori and are generally computation-

ally extensive. Whereas, real-time methods optimize based on instantaneous driving

conditions.

56


Equivalent Consumption Minimization Strategies (ECMS) are based on the prin-

ciple of defining engine consumption and electric machine consumption on the same

scale [29]. When solving a power management problem with the overall goal of mini-

mizing fuel, it is necessary to assign a cost to the electrical power so that the optimal

behaviour is not to deplete the battery. Thus, ECMS techniques assign an equivalent

fuel consumption factor for the electric machine power, thereby creating a single cost

function to apply conventional optimization techniques to. This allows for a near-

optimal solution to be found. Other real-time optimization based techniques include

model predictive control [40], robust control [41], and decoupling control approaches

[42].

Global optimization techniques aim at minimizing the performance index (typi-

cally based on energy losses, fuel consumption, and emissions) throughout an entire

cycle. Dynamic Programming is seen as the most suitable solution for this type of

optimal control problem, as it guarantees the globally optimal solution. The DP tech-

nique is discrete and requires gridding of the state and time variables. As a result,

there is a trade-off between accuracy and computation time, as a smaller grid means

longer computation time. Many adaptations of the Dynamic Programming have been

made to improve computation time. For example, [43] uses the Stochastic Dynamic

Programming (SDP) technique, where the problem is posed as an infinite horizon

stochastic optimization problem. Here, the power demand is treated as a Markov

process, which means that the next step solely depends on the current vehicle state

and not previous ones. The control law is computed offline and it is implemented on-

line as a state feedback controller. Other simplifications to DP have also been made,

57


including breaking up the cycle into segments and solving each segment as its own

optimization problem [44]. All of these simplifications result in suboptimal solutions.

Approximations of the original optimization problem are often also made. For

example, in [45, 46, 47] the cost function is linearized and Linear Programming is

used. Similarly, the control problem can be simplified to a quadratic cost function

with linear constraints. This is seen in [48, 49], where Quadratic Programming is used.

There are also many alternate techniques that have been explored in literature.

Genetic Algorithm (GA) solutions and adaptations are often proposed. However, it is

generally seen that GA techniques are not well suited to the HEV energy management

problem [29]. Optimization approaches based on Pontryagin’s minimum principle

(PMP) are also used. According to PMP, minimizing the cost function is equivalent to

minimizing the Hamiltonian. This is a generalization of the Euler-Lagrange equations.

PMP is an instantaneous optimization approach, but again can result in a suboptimal

global solution.

58

Chapter 6

Representative Vehicle Model

A quality dynamic vehicle model is essential for the development of an effective con-

trol strategy. This section outlines the vehicle dynamics of the primary components of

the selected hybrid powertrain. Key system components discussed include the vehicle

dynamics, planetary gear set, ICE, electric motor/generator, battery, and final drive.

The main model is derived from the HEV Power Split Midsize Gasoline model from

the Autonomie rev15sp1 software package. This model has a power-split architecture

with a single planetary gear set. There are many possible configurations that utilize

a single planetary gear set. This configuration resembles the Toyota Hybrid Synergy

Drive (HSD) with the addition of a torque coupling on the electric motor.

Although the components of the representative vehicle model are based on the

HEV Power Split Midsize Gasoline model from Autonomie, a simplified vehicle model

has been built in MATLAB. This was done to allow for more flexibility in the design

of the hybrid supervisory controller. A transient powertrain model is assumed. Com-

ponent data (e.g. efficiency maps, fuel maps, etc.) is collected from the Autonomie

59


rev15sp1 software package, wherever possible.

6.1 Vehicle Components

6.1.1 Vehicle

Let us describe the vehicle by its longitudinal dynamics. That is, for modeling pur-

poses, the description of the roadway is simplified to a straight, flat plane with variable

slope. A free body diagram of the linear dynamics of the vehicle with velocity, v, is

shown in Figure 6.1.

Fa

Ft

Fr

Fg

v, a

mg θ

Figure 6.1: Free body diagram of vehicle linear dynamics.

Newton’s second law is applied to yield the basic vehicle dynamics [13]. Forward

driving is assumed. The relationship between vehicle acceleration, a, and the forces

acting on the vehicle body are as follows:

60


ma = Ft − Fa − Fg − Fr (6.1)

where m is the vehicle mass, Ft is the total tractive force generated by the powertrain,

Fa is the aerodynamic drag force, Fg is the grading resistance force, and Fr is the

rolling resistance force.

The aerodynamic drag force is described by Equation 6.2.

Fa =1

2ρAfCdv

2 (6.2)

where ρ is the density of air, Af is the effective frontal area of the vehicle, and Cd is

the coefficient of drag. The aerodynamic drag coefficient is a constant value that is

dependent on the design of the vehicle body.

Grading resistance is the force of gravity acting downward on the vehicle. It

opposes forward motion on an incline and aids forward motion on a decline. For a

vehicle on a grade with an angle, θ, the grading force is described with Equation 6.3.

Fg = mg sin(θ) (6.3)

where g is the acceleration due to gravity (9.81m/s2).

Rolling resistance occurs at the contact point of the tire and roadway. It is the

normal component of the weight multiplied by the rolling resistance coefficient fr as

seen in Equation 6.4.

61


Fr = mgfr cos(θ) (6.4)

For a known drive cycle, the total tractive force required can be determined as

follows with Equation 6.5.

Ft =1

2ρAfCdv

2 +mg sin(θ) +mgfr cos(θ) +ma (6.5)

6.1.2 Road Load Model

For a given drive cycle with velocity v, acceleration a, and grade θ, the wheel speed

ωwheel and wheel torque Twheel can be determined at each time instant k ∈ {1, 2, ..., N}

as follows:

ωwheel(k) =v(k)

rwheel(6.6)

Twheel(k) = Ft(k) · rwheel (6.7)

where rwheel is the wheel radius.

62


6.1.3 Final Drive

The final drive model is a reduction gear with a speed and torque dependent power

loss. The final drive speed ωfd and final drive torque Tfd are determined as follows:

ωfd = Kωwheel (6.8)

Tfd =TwheelKηfd

(6.9)

where K is the final drive ratio, and ηfd is the associated power loss. The power loss

at the reduction gear is determined from a map within Autonomie that is indexed by

the angular speed and torque at the wheel. This is done to account for the increased

friction and thus higher gearbox losses at higher speeds.

6.1.4 Planetary Gear Set

A single planetary gear set acts as the power split device in this configuration. Other

power split hybrid vehicle configurations may have have multiple planetary gear sets.

A planetary gear set and its associated lever diagram can be seen in Figure 6.2.

As illustrated in the lever diagram, the speeds of the ring gear, sun gear, and

carrier must satisfy the following constraint.

ωsS + ωrR = ωc(S +R) (6.10)

where S and R are the number of teeth on the sun gear and ring gear, respectively.

The torque relation of the planetary gear set is as follows.

63


S

R

Tr, ωr

Tc, ωc

Ts, ωs

Ring Gear

Sun Gear

Carrier

ωs

ωc

ωr

Figure 6.2: Lever diagram of planetary gear set.

Tc = −S +R

RTr = −S +R

STs (6.11)

Each node shown in the lever diagram is connected to one or more powertrain

components. An input-split configuration is selected where the second electric ma-

chine is connected to the output shaft. The configuration chosen has the ring gear

connected to the motoring EM, the sun gear connected to the generator EM, and the

carrier connected to the engine.

6.1.5 Engine

A generic internal combustion engine with spark-ignition is modeled. The engine is

directly connected to the carrier gear. The engine model focuses on the characteristics

64


of fuel consumption and torque output. The fuel rate model used is based on the brake

specific fuel consumption (BSFC) map shown in Figure 6.3. Thus, for a given drive

cycle, the fuel consumption mfuel is determined at each instant k as follows:

mfuel(k) =MAP(ωeng(k), Teng(k)) (6.12)

where ωeng and Teng denotes the engine speed and torque, respectively.

150 200 250 300 350 400 450

Engine Speed [rad/s]

0

50

100

150

En

gin

e T

orq

ue

[Nm

]

Engine Fuel Map [kg/s]

0.35 0.35

0.350.35

0.36 0.36

0.360.36 0.370.37

0.37 0.370.370.38

0.38

0.33 0.33

0.330.33

0.34 0.34

0.340.34

0.35 0.35

0.350.35

0.36 0.36

0.360.36 0.370.37

0.37 0.370.370.38

0.38

Fuel Map ContourMax Torque Curve

Figure 6.3: Engine fuel map from Autonomie.

The engine is only allowed to operate within defined limits at all times. The stall

speed of the engine, ωeng,min, is the minimum speed at which the engine can generate

65


torque. The maximum speed at which the engine can generate torque is denoted by

ωeng,max. The engine’s torque production capability is limited by its inherent speed

limits. That is, the maximum engine torque Teng,max is speed dependent.

ωeng,min ≤ ωeng(k) ≤ ωeng,max (6.13)

0 ≤ Teng(k) ≤ Teng,max(ωeng(k)) (6.14)

The effective power of the ICE is determined using the following relation:

Peng = Tengωeng (6.15)

6.1.6 Electric Machines

The system uses two permanent magnet synchronous motors (PMSM). The two mo-

tors have several names in literature. While both electric machines (EMs) can function

as a motor and/or generator, one is commonly known as the generating EM and the

other as the motoring EM. The generator is commonly referred to as Motor A, MG1,

and EM1. Similarly, the second motor is commonly referred to as Motor B, MG2,

and EM2. From now on the generator will be referred to as Motor A and the motor

will be referred to as Motor B. The efficiency map for the motor and inverter as well

as the maximum torque curves were taken from the Autonomie power-split hybrid

model/

Motor A is directly connected to the sun gear of the planetary gear set (PGS).

66


Motor A is the smaller, generating motor and can generate up to 30 [kW] of mechan-

ical power. The motor power Pa can be determined from the effective motor torque

and motor speed as shown in Equation 6.16.

Pa = Taωaη−sgn(Ta)a (6.16)

where ηa is the efficiency of motor A. The effective motor torque is determined by the

efficiency of the EM and the sign of the torque. The motor A efficiency map, which

is indexed by torque and speed, is shown in Figure 6.4. In other words, the efficiency

is determined as follows:

ηa(k) =MAP(ωa(k), Ta(k)) (6.17)

The motor power is limited by its upper and lower torque limits Ta,min and Ta,max,

respectively, as well as its maximum speed ωa,max.

−ωa,max ≤ ωa(k) ≤ ωa,max (6.18)

Ta,min(ωa(k)) ≤ Ta(k) ≤ Ta,max(ωa(k)) (6.19)

Motor B is coupled to the ring gear of the planetary gear set with a torque coupling

ratio rtc of 2.5. Motor B can produce up to 105 [kW] of power. The electric motor

speed ωB can be determined as shown in Equation 6.20.

ωB = rtcωr (6.20)

67


-1000 -800 -600 -400 -200 0 200 400 600 800 1000

Motor A Speed [rad/s]

-150

-100

-50

0

50

100

150

Mo

tor

A T

orq

ue

[Nm

]Motor A Efficiency Map

0.60.6 0.6

0.60.6

0.60.6

0.60.65

0.65 0.650.65

0.650.65

0.650.65

0.70.7 0.7

0.70.7

0.7

0.70.7

0.75

0.75 0.75

0.750.75

0.75

0.75

0.75

0.80.8 0.8

0.80.8

0.8

0.80.8

0.85

0.85 0.85

0.850.85

0.85

0.85

0.85

0.9

0.9

0.90.9

0.9

0.9

0.9

0.9

0.9 0.9

0.90.9

Efficiency ContourMax Torque CurveMin Torque Curve

Figure 6.4: Motor A efficiency map from Autonomie.

Motor B power is determined by its effective torque and speed, as shown in Equa-

tion 6.21.

Pb = Tbωbη−sgn(Tb)b (6.21)

where ηb is the efficiency of motor B. The efficiency map for motor B is indexed by

torque and speed and can be seen in Figure 6.5. Thus, the efficiency of motor B is

determined at each time step as follows.

ηb(k) =MAP(ωb(k), Tb(k)) (6.22)

68


-1000 -500 0 500 1000

Motor B Speed [rad/s]

-250

-200

-150

-100

-50

0

50

100

150

200

250

Mo

tor

B T

orq

ue

[Nm

]Motor B Efficiency Map

0.6

0.6

0.6

0.6

0.7

0.7

0.7

0.7

0.8

0.8

0.8

0.8

0.85

0.85

0.85

0.85

0.85

0.850.

85

0.85

0.850.850.

90.9

0.9

0.9

0.9

0.9

0.9

0.90.

9

0.9

0.9

0.9 Efficiency Contour

Max Torque CurveMin Torque Curve

Figure 6.5: Motor B efficiency map from Autonomie.

Again, the motor power is limited by its inherent upper and lower torque limits

Tb,min and Tb,max, and its maximum speed ωb,max.

0 ≤ ωb(k) ≤ ωb,max (6.23)

Tb,min(ωb(k)) ≤ Tb(k) ≤ Tb,max(ωb(k)) (6.24)

Alternatively, the motor efficiencies ηmot can be determined as a ratio of the motor

power losses Pmot loss and the total motor power Pmot, if the necessary information is

69


available.

ηmot =Pmot − Pmot loss

Pmot(6.25)

6.1.7 Battery

The battery is more difficult to model as there are many factors that impact the

performance of the battery. For example, temperature, age, and battery state of

charge (SOC) all have nonlinear effects on battery voltage [13]. In practice, the

battery performance is dependent on the battery management system, which includes

a state of charge estimator. The battery used is assumed to be a Lithium-ion battery

with known data. For simplicity, the battery is modeled as an equivalent circuit with

an open circuit voltage VOC in series with an internal resistance Rint [13]. This open

circuit model is shown in Figure 6.6. The open circuit voltage and internal resistance

are determined by maps that are indexed by SOC.

Voc =MAP(SOC) (6.26)

Rint =MAP(SOC) (6.27)

The battery current can be determined as follows:

Ib(k) = ebattVOC −

√V 2OC − 4RintPelec(k)

2Ri

(6.28)

where ebatt is the coulombic efficiency and Pelec is the electrical power consumed by the

battery. The coulombic efficiency is assumed to be a constant of 0.9 when charging

and 1.0 when discharging. The electrical power is a combination of the motor powers

70


Rint

Voc V

+

-

I

Figure 6.6: Equivalent circuit battery model.

and the auxilary power Paux. The auxilary power is the constant power consumed by

the electric auxiliary units.

Pelec = Pa + Pb + Paux (6.29)

This equivalent circuit model allows the following conclusion to be made for the

battery state of charge:

SOC(k + 1) = SOC(k)− Ib(k)

3600Qbatt

·∆t (6.30)

71


where SOC(k) is the state of charge of the battery at time k, Qbatt is the maximum

capacity of the battery, Ib is the battery current, and ∆t is a time sampling unit

selected for simulation.

The battery is constrained by its minimum and maximum currents, Ib,min and

Ib,max, as well as its maximum charging and discharging power limits, Pbatt ch and

Pbatt disch. Note that positive power represents discharging. The battery SOC is

also constrained by its user defined SOC limits, SOCmin and SOCmax. The battery

constraints are summarized as follows:

Pbatt disch ≤ Pbatt ≤ Pbatt ch (6.31)

Ib,min ≤ Ib(k) ≤ Ib,max (6.32)

SOCmin ≤ SOC(k) ≤ SOCmax (6.33)

If the appropriate information is available then the battery efficiency ηbatt can be

determined from the measured battery losses Pbatt loss and the total battery power

Pbatt, as displayed in Equation 6.34.

ηbatt =Pbatt − Pbatt loss

Pbatt(6.34)

6.2 Vehicle Dynamics

The governing dynamics of all of the powertrain components are dependent on the

clutches and the resulting operating modes. The following will briefly outline all

possible clutch configurations and the useful modes.

72


6.2.1 Clutches and Operating Modes

The addition of clutches in various locations impacts the functionality of the trans-

mission. Engaging or disengaging a clutch allows the transmission to switch modes.

As mentioned, power-split hybrids can operate in several different modes depending

on the configuration. The following Figure 6.7 illustrates all the possible locations

for the clutches in an input-split configuration.

Motor A

Engine

Motor B Wheels

Clutch 3' Clutch 3

Clutch 2

Clutch 1

Clutch 2'

Clutch 1'

K

Sun Gear

Ring Gear

Carrier

Final Drive

Figure 6.7: All possible clutch locations for an input-split configuration.

There are eight possible clutch states and operating modes of an input-split con-

figuration. Some of these clutch states are either infeasible or equivalent, reducing the

practical number of modes to four [50]. A possible clutch configuration that achieves

all four operating modes can be seen in Figure 6.8. The resulting operating modes

are summarized below in reference to the configuration shown in Figure 6.8 [50].

73


1. Mode 1 (EV1): This is pure electric mode. In this mode, Clutch 1′ and Clutch 2′

are closed and Clutch 1 is open. The vehicle is driven by the generator motor

(Motor A) only.

2. Mode 2 (EV2): This is also pure electric mode. In this mode, Clutch 1 and

Clutch 2′ are closed and Clutch 1′ is open. So the engine is disconnected and

the carrier gear is grounded. The vehicle is driven by both motors (Motor A

and Motor B).

3. Mode 3 (Series): This mode is equivalent to series operation. In this mode,

Clutch 1′ is closed and Clutch 1 and Clutch 2′ are open. The generator motor

(Motor A) and engine are connected to the PG to charge the battery. The

vehicle is only driven by Motor B mechanically.

4. Mode 4 (Power Split): This is power split mode. In this mode Clutch 1 is closed

and Clutch 1′ and Clutch 2′ are open. So the engine, Motor A, and Motor B

are all connected to the PG. The vehicle is driven by all three components.

74


Motor A

Engine

Motor B WheelsClutch 1

Clutch 2'

Clutch 1'

K

Sun Gear

Ring Gear

Carrier

Final Drive

Figure 6.8: A possible input-split configuration that achieves all four modes.

The dynamic equations for the four output modes can be derived for the config-

uration shown in Figure 6.8. The governing Equations are summarized in Equations

6.35 to 6.38. Equation 6.35 describes the dynamics of the Mode 1 (EV1).

(Mr2

wheel

K2+ Ib)ωb = Tb −

1

KTwheel (6.35)

Equation 6.36 describes the dynamics of Mode 2 (EV2).

(Mr2wheel

K2 + Ib) 0 −R

0 Ia −S

−R −S 0

ωb

ωa

F

=

Tb − 1

KTwheel

Ta

0

(6.36)

The dynamic equations for Mode 3 are the same as the equations for a a series

hybrid. This is illustrated in Equation 6.37 below.

75


(Mr2wheel

K2 + Ib)ωb = Tb − 1KTwheel

Ieng 0 R + S

0 Ia −S

R + S −S 0

ωeng

ωa

F

=

Teng

Ta

0

(6.37)

The dynamic equations for Mode 4 are equivalent to those of power-split operation.

This can be seen in Equation 6.38.

Ieng 0 0 R + S

0 (Mr2wheel

K2 + Ib) 0 −R

0 0 Ia −S

R + S −R −S 0

ωeng

ωb

ωa

F

=

Teng

Tb − 1KTwheel

Ta

0

(6.38)

where M is the vehicle mass, rwheel is the wheel radius, and K is the final drive

ratio. Ieng, Ia, Ib are the inertias for the engine, motor A, and motor B, respectively.

Similarly, Teng, Ta, and Tb are the respective torques for the engine, motor A, and

motor B. Twheel is the resistive torque due to rolling resistance and aerodynamic drag

during driving. This is also known as the road load torque and is defined at the

output shaft. ωeng, ωa, and ωb are speeds of the engine, motor 1, and the output

shaft, respectively. Motor B is directly connected to the output shaft, and thus the

motor B speed is determined from the output shaft speed ωout. The force F is the

internal force acting in the planetary gear set.

76


6.2.2 Summary

The vehicle model in question is an input-split powertrain with a single clutch. The

configuration can be seen in Figure 6.9. This vehicle configuration operates in modes

2 and 4. This means that Equations 6.36 and 6.38 apply.

Motor A

Engine

Motor B Wheels

Clutch

K

Sun Gear

Ring Gear

Carrier

Final Drive

Figure 6.9: Configuration of vehicle model used for simulation.

The demanded power is satisfied by the engine, motor, and generator as described

in Equation 6.39.

Pdem = Tengωeng + Taωa + Tbωb (6.39)

Finally, the power-split decision is simulated with a dynamic programming vehicle

controller, which is proposed in Chapter 7. Two rule-based controllers are used for

benchmarking purposes and are described in Chapter 8.

77


6.3 Simulation Environment

A simplified model is used where the component dynamics are modeled using MAT-

LAB R2016a. Component efficiency maps and operating ranges are taken from Au-

tonomie for the 2004 Toyota Prius. The vehicle controllers take drive cycle inputs

and use this information along with vehicle information to determine the powertrain

control. The powertrain plant model takes in the energy input and powertrain control

and outputs the component power losses. The vehicle chassis model determines the

vehicle loads.

.

78

Chapter 7

Dynamic Programming for Energy

Management

7.1 Introduction

Dynamic programming is a global optimization algorithm that is commonly consid-

ered in the design and optimization of hybrid electric powertrains. The main goal of

this section is to present a detailed procedure on how to apply the theory of DP to

an autonomous vehicle model with a power split powertrain.

The energy management problem is commonly cast as an optimal control problem,

as discussed in Section 5.4. Many strategies have been explored and implemented to

solve the control problem, such as rule-based approaches, equivalent consumption

minimization strategies, and global optimization techniques. It is agreed that DP

provides the best solution as it outputs the global optimum. However, DP is not

79


commonly put into practice. DP is often dismissed as it can be extremely compu-

tationally extensive. In addition to the computational constraints, there are other

limitations to the DP algorithm that lead to its dismissal. The specifics of these

limitations will be discussed throughout the problem formulation. One of the main

limitations is that the algorithm requires the vehicle journey to be known in advance.

Important insight can be gained through knowing the globally optimal solution

and thus it is worth studying the DP solution. Since the algorithm is commonly

dismissed, there is little detail in literature on the actual implementation in its appli-

cation to HEV models. This chapter considers the limitations of the algorithm and

develops a solution that looks to minimize the impact of these limitations.

Despite the limitations discussed, the DP algorithm is well suited for this problem

for many reasons. Mainly, autonomous vehicles know their journey in advance and it

is beneficial to use the journey map to inform the controller. Dynamic programming

will use this information to output the optimal operation of the powertrain compo-

nents.

The formulation of the DP algorithm presented in this chapter could also be ap-

plied to non-autonomous vehicles. However, this would require an accurate prediction

of the cycle. Many drive cycle prediction methods are proposed in literature. These

prediction methods account for characteristics, such as time of day, day of week, cur-

rent location of the vehicle, and historical driver behaviour. These characteristics are

used to estimate the future driving cycle of the vehicle. This estimate could then

80


be used to inform a dynamic programming control algorithm. Drive cycle prediction

methods often accumulate prediction error, and thus are not always accurate. As

such, the dynamic programming algorithm is more suited to an autonomous vehicle

application.

This chapter is organized as follows. First, Section 7.2 presents a brief literature

review of the work that has been done in dynamic programming. In Section 7.3

the principles of optimal control law explained earlier are applied to a simplified

vehicle model to define the optimal control problem at hand. Section 7.4 discusses

the mathematical theory behind the DP algorithm. Section 7.5 develops a procedure

that aims to structure the solution method in a way that reduces the computational

complexity and increases the accuracy. Finally, Section 7.6 presents the results of the

DP solution.

7.2 Literature Review

Dynamic programming has been explored and applied to the energy management

problem in literature. It is often used to explore the limitation of the performance

of the powertrain. The DP solution is also used as a baseline to evaluate the per-

formance of other control strategies against. Moreover, the DP algorithm provides

useful information for the design of HEV powertrains and control strategies.

The dynamic programming algorithm has been applied to many HEV vehicle

architectures. In [51], the procedures for implementing DP to a series, parallel, and

power-split powertrain are explained. [51] implements DP in a linear manner, which is

81


highlighted in the flow charts of speed/torque control based on DP procedures shown.

[5] and [52] both apply dynamic programming to a parallel HEV. Both papers use the

dynamic programming strategy to aid in developing a new control strategy. Similarly,

[3] presents a formalization of the energy management problem in HEVs and com-

pares PMP and ECMS control strategies to the DP strategy. All control strategies

in [3] are implemented on an series HEV powertrain.

In [53] the issues related to the implementation of dynamic programming for op-

timal control are presented. This paper applies the DP to the energy management

problem for a parallel HEV. The numerical issues that arise during implementation

discussed in [53] are considered throughout the implementation presented in Section

7.5.

There are fewer cases of the application of dynamic programming to a power-split

HEV architecture, as the power-split decision is more complicated than with a series

or parallel powertrain configuration. In [54] the dynamic programming algorithm

is applied to a series-parallel HEV powertrain with an electric variable transmission

(EVT). The EVT does not use a mechanical planetary gear to perform the power split

function. EVTs are used as an energy converter in the HEV powertrain to decouple

the engine from the wheel speed to allow the engine to operate its optimal efficiency

points [54].

A few cases in literature outline the implementation of dynamic programming

to the power-split vehicle model. In [50], the DP algorithm is used to compare the

82


performance of the Toyota Prius and Chevrolet Volt, which are both power-split

HEVs with a single planetary gear set. The state variables used in DP for the Toyota

Prius, which we are considering in this thesis, are the engine speed and battery state

of charge. The control variables used in the DP for the Toyota Prius are the engine

torque, MG1 torque, and the mode decision. In [55], the same state variables and

control inputs as in [50]. However, [55] increases the computational efficiency of

the DP algorithm by vectorizing the states and inputs. The most comprehensive

application of DP to the power-split model is found in [7]. There were no cases found

in literature of dynamic programming as it is applied to an autonomous vehicles for

hybrid supervisory control. As such, this thesis will present a systematic approach

for applying the DP algorithm to an autonomous vehicle with a power-split HEV

powertrain.

7.3 Optimal Control problem

The energy management problem for a HEV can be formulated to a problem in the

class of optimal control problems with a fixed final time and partially constrained final

states. This problem has both state and input constraints, and the state disturbances

83


are assumed to be known. The optimal control problem is summarized as follows.

minimizeu(t)

J(u(t))

subject to x = F (x(t), u(t), t)

x(0) = x0

x(N) ∈ [xN,min, xN,max]

x(t) ∈ X (t)

u(t) ∈ U(t)

where

J(u(t)) = φ(x(N)) +

∫ N

0

L(x(t), u(t), t)dt (7.1)

where J is the cost function, x is the state, and u is the control input.

Considering the vehicle model discussed in Chapter 6, the control problem can be

reformulated with more detail. First, let us look at the theory behind DP to ensure

that the problem is reformulated efficiently.

7.4 Theory of Dynamic Programming

This section briefly explains the mathematical principle behind dynamic program-

ming. Interested readers can refer to literature for more detail on the theory of DP.

DP is a numerical algorithm that solves optimal control problems where decisions

are made at each stage. Since the problem at hand involves a continuous time state

84


vehicle model, the model must be discretized to apply the DP algorithm. The system

can be expressed discretely as follows.

xk+1 = f(xk, uk) (7.2)

where xk ∈ Xk and uk ∈ Uk. Let us assume that the dynamic optimization problem is

over the control sequence π = [u0u1...uN−1] that minimizes the cost function J . The

discretized cost function is of the following form.

Jπ = gN(xN) + φN(xN) +N−1∑k=0

Lk(xk, uk) + φk(xk) (7.3)

where Jπ is the aggregated cost and gN(xN) +φN(xN) is the terminal cost, Lk(xk, uk)

is the instantaneous transition cost at step k, and φk(xk) is the penalty function that

enforces the state constraints.

The optimal control policy π∗ is the policy that minimizes J .

J∗ = minπ∈Π

Jπ (7.4)

where Π is the set of all admissible control sequences.

Bellman’s principle of optimality [56] defines an optimal policy as follows:

Definition (Optimal Policy): An optimal policy has the property that what-

ever the initial state and initial decisions are, the remaining decisions must constitute

85


an optimal policy with regard to the state resulting from the first decision.

The DP algorithm uses this principle by partitioning the problem into a set of

smaller sub-problems and solving recursively. In other words, the sub-problem in-

volving the last stage is solved first, then the sub-problem involving the last two

stages is solved, then the last three stages,..., etc. Recursively evaluating the optimal

cost-to-go function Jk(xi) at every node in the discretized space until the entire prob-

lem is solved will yield the optimal solution. This ensures that for a particular initial

state decision, the outcome will be known and optimal. For example, the algorithm

begins with the final cost calculation step.

JN(xi) = gN(xi) + φN(xi) (7.5)

Next, these methods can be propagated back in time for k = N − 1 to 0.

Jk(xi) = minuk∈Uk

{lk(xi, uk) + φk(xi) + Jk+1(Fk(x

i, uk))} (7.6)

The optimal control is the one that minimizes the preceding expression. The result

is a map with a set of paths that are known to be optimal, or an optimal control

signal map. This map can then be used in a forwards simulation of the model. If an

initial state of the system, x0, is chosen such that it exists in the state space, then

the optimal path is known. The control signal map is limited to the points on the

state-space grid. Thus, an interpolation method is necessary if the actual state does

not exist on the state-space grid.

86


Jk+1(xi,k+1) = min [lk+1(xi,uk,j+1,i) + Jk(xj+1,k), lk+1(xi,uk,j,i) + Jk(xj,k), lk+1(xi,uk,j-1,i) + Jk(xj-1,k), lk+1(xi,uk,j-3,i) + Jk(xj-3,k)]

Jk+1(xi-3,k+1) = min [lk+1(xi-3,uk,j-3,i-3) + Jk(xj-3,k), lk+1(xi-3,uk,j-4,i-3) + Jk(xj-4,k), lk+1(xi-3,uk,j-5,i-3) + Jk(xj-5,k)]

Jk(xj+2)

Jk(xj-2)

Jk(xj-1)

Jk(xj-3)

Jk(xj-4)

Jk(xj-5)

Jk(xj)

Jk(xj+1)uk,j+1,i+2

uk,j+1,i+3

uk,j+1,i+1

uk,j+1,i

uk,j,i

uk,j-1,i

uk,j-3,i

uk,j-3,i-3

uk,j-5,i-3

uk,j-4,i-3

k k+1. . . . . . Step

State

Figure 7.1: Dynamic programming.

7.5 Power Split Model

This section outlines how to apply the theory of dynamic programming to a power

split vehicle model. The process of applying the theory to a model is rarely discussed

in literature beyond the formulation an appropriate control problem. The most de-

tailed process is a flow chart shown in [51], which shows a linear approach to a DP

procedure.

87


First, a hybrid vehicle can be described by a set of piece-wise difference equations.

x(k + 1) = f(x(k), u(k)) (7.7)

Here, x(k) is the state vector and represents the vehicle speed, engine speed, motor

speeds, and battery SOC. The control vector is represented by u(k) and includes

engine torque, motor torque, generator torque, and the mode decision. Here, we

are considering a model that is capable of operating in modes 2 and 4, as discussed

in Section 6.2.1. Thus the mode decision is equivalent to that of an engine on/off

decision. The mode decision is parameterized such that mode ∈ {0, 1}, where mode =

0 denotes that the engine is off (EV2 mode) and mode = 1 denotes that the engine

is on (power-split mode). The complete parameterization of the state and control

vectors is illustrated in Equations 7.8 and 7.9, respectively.

x(k) =

[v(k) ωeng(k) ωa(k) ωb(k) SOC(k)

](7.8)

u(k) =

[Teng(k) Ta(k) Tb(k) mode(k)

](7.9)

As discussed, the DP algorithm requires that the vehicle model is discretized. The

state and control vectors consist of multiple dependent variables and thus can be re-

duced. It is important to select the independent variables to be discretized carefully.

The DP algorithm is limited by the discretization of the independent variables in

several ways.

First, the cost-to-go function is evaluated and stored at discrete state points. This

means that the values that each independent state can take on must be meshed into

88


a grid. The density of the mesh determines the accuracy of the algorithm. If the

vehicle dynamics output a state that is not on the state mesh then the algorithm

interpolates to the closest state on the mesh. This interpolation can result in inaccu-

rate dynamics and thus skewed results. For example, let the SOC be discretized and

coarsely meshed such that SOC ∈ [0 : 0.1 : 1]. Suppose that the SOC at time k is

0.50 and that the battery dynamics dictate that the next SOC at time k+1 is 0.49.

The SOC at k+1 would be interpolated to 0.5 on the mesh space. This inaccurately

displays the battery dynamics and the algorithm would think that charge has been

sustained, effectively resulting in free energy. Ultimately, the finer the mesh density

the more accurate the results. However, meshing states too finely will limit the DP

algorithm computationally. Selecting the mesh density is trade-off between accuracy

and computation time.

Increasing the dimension of the discretized space increases the memory and com-

putation time required exponentially. Minimizing the amount of discretization will

allow for a more accurate representation of the problem and can reduce the problem

size. Thus the goal is to select the independent states and control inputs such that

discretization is minimized and to find an appropriate mesh density. The following

aims to prove that the most computationally efficient control variables are selected.

First, dynamic programming requires the disturbances to be known, a priori. In

other words, the power-split decision is optimized over a predetermined drive cycle

with known velocity v and acceleration a. Since the vehicle velocity and acceleration is

known, the speed of the output shaft ωout and subsequently the speed and acceleration

89


of the the ring gear can be determined. Assuming that there is no tire slip at the

road, then the following is true:

ωring =K

Rtire

v (7.10)

ωring =K

Rtire

a (7.11)

As seen in Equation 6.20, the speed of the electric motor ωb can be directly

calculated from the ring gear speed. The power demand Pdem of the vehicle can also

be determined from the driving force Ft and velocity profile v, where the driving force

is determined as shown in Equation 6.5.

Pdem = Ftv (7.12)

The demanded power must be satisfied by a combination of the electric motor, gen-

erator, and engine. Moreover, applying the conservation of power to the vehicle

components yields the following relation:


This means that there are now only 3 unknown state variables (ωeng, ωa, and

SOC) and 4 unknown input variables (Teng, Ta, Tb, and mode).

First, the SOC must be selected as a state variable. This is because the battery’s

dynamics are separate from the mechanical function and therefore cannot be related

to the mechanical states. Next, the engine speed is selected as a state variable. This

90


was done as the engine speed also functions to store the mode decision, eliminating

the need for an engine on/off state variable. If the engine speed is zero, then the

vehicle is in EV mode and the engine is off. If the engine speed is non-zero, then the

vehicle is in EVT mode and the engine is on. Alternatively, the selection of motor

A speed or motor B speed as a state would result in the need for an engine on/off

state. This would effectively double the size of the state mesh. This is proven later

in Table 7.2. In addition, the selection of SOC and ωeng in combination is logical as

it ties the state to both the battery and engine dynamics. This directly relates to the

overarching power-split decision between the battery and engine.

The control vector is reduced to the mode (engine on/off) and the engine torque.

The engine torque was selected as it is directly related to the power-split decision

between the engine and the battery. Selecting the engine speed and torque in com-

bination allows the engine power to be determined. This means that the power split

between the battery and engine is known. Typically, either the torque of motor

A or motor B is selected as a control input in combination with the engine torque

[50]. However, the control policy decision of the power split between the battery

and engine is already known. This eliminates the need to include the motor and/or

generator torques as a control input. These torques can be directly calculated with

the information available, which will be demonstrated in Sections 7.5.2 through 7.5.5.

Eliminating the need for an additional control input significantly decreases the size

of the mesh.

The effect of the choice of state and input variables on problem size can be realized

91


by looking at the resultant size of mesh space. To illustrate this, let us assume that

the DP parameters are discretized as shown in Table 7.1. The mesh density N of

each variable is arbitrarily selected for the purposes of this study.

Table 7.1: DP Parameters

Variable Type Min Max N

mode q 0 1 2SOC x 0.4 0.6 201ωeng x 0, 104.72 rad/s 471.23 rad/s 50ωa x -1047.2 rad/s 1047.2 rad/s 50Teng u 0 Nm 155.91 Nm 50Ta u -153.4 Nm 153.4 Nm 50

The possible state and input combinations are shown in Tables 7.2 and 7.3. The

size of the resultant mesh space Smesh space is determined using Equation 7.14. Thus,

it is clear that state and control input combination selected generates the smallest

mesh space.

Smesh space = nx1 × nx2 × nx3 × nu1 × nu2 × nu3 (7.14)

Table 7.2: State Space Mesh Sizes

Variables Calculation Size

SOCωeng

201*50 10,050

SOCωamode

201*50*2 20,100

92


Table 7.3: Control Input Mesh Space

Variables Calculation Size

Tengmode

50*2 100

TengTamode

50*50*2 5000

In order to further increase computational efficiency, the mesh space can be vec-

torized. This allows for multiple sets of state and input variables to be simulated in

parallel. Instead of performing the model calculations for every possible combination

of states and control inputs iteratively at each time step, a single matrix calculation

can be performed. This reduces the number of operations performed in the model by

a factor of Smesh space.

The output of the vectorized calculations for every possible state and control is

the transition cost matrix J that stores the cost-to-go. A transition matrix (shown

in Figure 7.2) is determined at each time step.

93


Size U

Size X

Mode = 0, Teng = 0

Mode = 1, Teng = 0

Mode = 1, Teng = [Teng,min, Teng,max]

weng = 0, SOC = [SOCmin, SOCmax]

weng = [weng,min, weng,max], SOC = [SOCmin, SOCmax]

EV Mode EVT to EV Mode

EVT Mode

EVT ModeJ =

Figure 7.2: An example of a transition matrix, J, at a time step k.

It is clear that the storage such a transition cost matrix would require considerable

memory. For this reason, the problem has been strategically partitioned to reduce

memory usage without increasing computation. The governing dynamics change de-

pending on the mode of the vehicle. Thus the problem is organized so that the

necessary equations and constraints are only evaluated when applicable.

The model has been partitioned into the 4 possible discrete transition cases: EV

mode (e1), EV to EVT (e2), EVT to EV (e3), and EVT mode (e4). These transition

cases are defined in Table 7.4. Each transition model consists of the equations and

constraints pertaining to the specific transition case. This allows for 4 smaller tran-

sition cost matrices to be determined and cleared to reduce the required memory. It

is possible to further partition the problem if necessary, however it will increase the

number of necessary operations.

94


Table 7.4: Mesh Space Partition

Transition Description Mesh Space Matrix

e1 : q1 → q1 EV to EV [ ωeng = 0 SOC = meshed mode = 0 Teng = 0 ]

e2 : q1 → q2 EV to EVT [ ωeng = 0 SOC = meshed mode = 1 Teng = 0 ]

e3 : q2 → q1 EVT to EV [ ωeng = meshed SOC = meshed mode = 0 Teng = 0 ]

e4 : q2 → q2 EVT to EVT [ ωeng = meshed SOC = meshed mode = 1 Teng = meshed ]

The result of partitioning the problem in this way are the 4 transition cost matri-

ces outlined in blue in Figure 7.2 above.

The minimum cost-to-go and its corresponding control input for each state must

be stored at every time step. This is done by finding the minimum of the transition

cost matrix J . The final result is an optimal transition matrix and an optimal control

policy matrix. The dimensions of these are as seen in Figures 7.3 and 7.4. Note that

the size of X, nX , is equal to the mesh density of the SOC points (nSOC) times the

mesh density of engine speed points (nweng). The control input uopt,1 corresponds to

the minimum cost Jmin(x1), and so on.

Size X (nX)

Jopt = Jmin(x1) Jmin(xnX)… … …

Figure 7.3: An example of the minimum cost to go matrix, Jopt, at a time step k.

95


Size X (nX)

Uopt = Uopt,1 Uopt,nX… … …

Figure 7.4: An example of the control input matrix, Uopt, at a time step k.

The minimum cost-to-go at each state is determined follows:

Jk(xi) = minuk∈Uk

{mF (xi, uk)+α|SOCi

k+1−SOCik|+φk(x

i)+Jk+1(Fk(xi, uk))

}(7.15)

In each transition case model, the states and inputs are subject to a set of con-

straints that represent the limitations of the components. The infeasible points are

accounted for in the cost function in the φ term in Equation 7.15. Infeasible points

should have infinite cost, however there are numerical issues that can arise with this

[53]. Again, this limitation is a product of the discretization of the state space. Inter-

polating between an infinite cost-to-go and a finite cost-to-go will result in an infinite

cost-to-go. If this is propagated backwards, then the infeasible range will grow. For

this reason, infeasible states are given an arbitrarily high cost of 107. These physical

constraints that define the infeasible space are summarized below.

96


SOCmin ≤ SOC ≤ SOCmax

ωeng min ≤ ωeng ≤ ωeng max

ωa min ≤ ωa ≤ ωa max

ωb min ≤ ωb ≤ ωb max

Teng min ≤ Teng ≤ Teng max

Ta min ≤ Ta ≤ Ta max

Tb min ≤ Tb ≤ Tb max

Peng min ≤ Peng ≤ Peng max

Pa min ≤ Pa ≤ Pa max

Pb min ≤ Pb ≤ Pb max

Pbatt min ≤ Pbatt ≤ Pbatt max

(7.16)

7.5.1 Summary

A block diagram summarizing the DP model is shown in Figure 7.5. The procedure

to implement DP for the power split HEV model is outlined by the pseudo code

shown in Algorithm 1. The procedure begins by defining the system inputs. At each

time step, the forces and demanded power are calculated, and the PGS dynamics

are then applied. First, the cost-to-go matrix for transition case EV to EV, JEV , is

determined. Next, the cost-to-go matrix for transition case EV to EVT, JEV 2EV T , is

evaluated. After this, the cost-to-go matrix for transition case EVT to EV, JEV T2EV ,

is determined. Finally, the cost-to-go matrix for transition case EVT to EVT, JEV T ,

is evaluated. The complete cost-to-go matrix, J , at that time step is then created

97


and minimized to output the minimum cost-to-go matrix, Jopt, and its corresponding

control input matrix, Uopt. This process is repeated until the end of the cycle. In the

end, the optimal control input at each step can be combined to create the optimal

control signal map. This optimal control signal map shows the optimal path forward

for every starting condition, x0. The transition models are described in more detail

in the sections below.

Dynamic Model

mode Teng

ωeng(k)

SOC(k)

v(k)

ωeng(k+1)

SOC(k+1)

θ(k)

Transition Cost

Figure 7.5: Block Diagram of DP system inputs and outputs.

98


Algorithm 1 DP for Power Split HEV Model

procedure DP ProcedureDefine X

Define U

Define v, a

for k=1:T doFind FtFind PdemFind ωring, ωringFind ωb, ωbEV model . Output is JEVstore JEVEV to EVT model . Output is JEV 2EV T

store JEV 2EV T

EVT to EV model . Output is JEV T2EV

store JEV T2EV

EVT model . Output is JEV Tminimize JEV Tstore JEV T min

clear JEV TDetermine Jopt(k) and Uopt(k)Store Jopt(k) and Uopt(k)clear JEV , JEV 2EV T, JEV T2EV , JEV T

end forOutput Jopt and Uopt

end procedure

Optimal Control Problem Reformulation

Considering the theory of DP and the application to the power split HEV model

discussed above, the optimal control problem stated in Section 7.3 can be reformulated

with more detail. The reformulated problem is posed as follows:

99


minimizeu(t)

J(u(t))

subject to x = F (x(t), u(t), t)

x(0) = x0

x(N) ∈ [xN,min, xN,max]

x(t) ∈ X (t)

u(t) ∈ U(t)

where

100


x(t) ,

SOC(t)

ωeng(t)

u(t) ,

Teng(t)mode(t)

X (t) ,

SOCmin ≤ SOC ≤ SOCmax

ωeng min ≤ ωeng ≤ ωeng max

ωa min ≤ ωa ≤ ωa max

x : ωb min ≤ ωb ≤ ωb max

Peng min ≤ Peng ≤ Peng max




U(t) ,

Teng min ≤ Teng ≤ Teng max

Ta min ≤ Ta ≤ Ta max

Tb min ≤ Tb ≤ Tb max

u : Peng min ≤ Peng ≤ Peng max




J(u(t)) = φ(SOC(N)) +

∫ T

0

{mF (u(t)) + α ˙SOC(u(t)) + φ(x(t))

}dt

101


7.5.2 EV Mode

When the vehicle is in EV mode Equation 6.36 applies. This means that the engine

is off and the carrier in the PGS is locked. A summary of the equations that apply is

shown below in 7.17 through 7.20.

Pdem = Taωa + Tbωb (7.17)

0 = ωaS + ωbR (7.18)[Mr2

K2+ Ib

]ωb = Tb −

1

KTroad +RF (7.19)

Iaωa = Ta − SF (7.20)

Since ωout and ωout and subsequently ωb and ωb are known, there are 4 equations

and 4 unknowns and the entire system of equations can be solved.

First, ωa and ωa can be determined through the PGS relation in Equation 7.18.

Next, Equations 7.19 and 7.20 are rearranged to solve for Ta and Tb, respectively.

Tb =[Mr2

K2+ Ib

]ωb −RF +

1

KTroad (7.21)

Ta = Iaωa + SF (7.22)

Now, Equations 7.21 and 7.22 can be substituted into Equation 7.17 to solve for

F . This is only applicable when v > 0 since the reaction force is zero when the vehicle

is not moving. Thus F = 0 when v = 0 and F can be determined by Equation 7.23

102


when v > 0.

F =Iaωaωa +

[Mr2

K2 + Ib

]ωbωb + 1

KTroadωb − Pdem

−Rωb + Sωa(7.23)

Ta and Tb are now determined by substituting F back into Equations 7.21 and

7.22. Now that all of the unknowns have been solved for the transition cost to stay

in EV mode can be evaluated.

The maximum and minimum torques and powers for motor A and B must be

determined from their respective component maps, as discussed in Section 6.1.6. The

present state component constraints are evaluated based on these values to determine

φk(xi) at each point. The engine speed and SOC at the next time step must be

determined to complete the cost-to-go evaluation and to ensure that the next state is

feasible. The engine speed at the next time step is approximated using forward Euler

integration:

ωeng(k + 1) = ωeng(k) + ωeng(k) · dt (7.24)

where dt is the time step. The next SOC is determined by applying the battery

dynamics model discussed in Section 6.1.7. If the pair (SOC(k+ 1), ωeng(k+ 1)) does

not exist in the state space mesh, then the nearest neighbour interpolation method

described in Section 7.5.6 is used. The next SOC is also used to evaluate the charge

sustaining performance index α|SOC(k+ 1)− SOC(k)|. The fuel use in EV mode is

zero and thus mF = 0 at each state. Finally, all the information needed to determine

the cost-to-go at each state is known. The output is the transition cost matrix JEV 2EV .

103


Since there is only one control input that yields the EV mode to EV mode transition,

JEV 2EV is the minimum cost to remain in EV mode and must be stored for now.

7.5.3 EVT Mode

When the engine is on, the carrier ring in the PGS is now able to rotate. The dynamics

are now governed by Equation 6.38. A summary of the equations that apply is shown

below in Equations 7.25 through 7.29.


ωeng(R + S) = ωaS + ωbR (7.26)

Iengωeng = Teng − (R + S)F (7.27)[Mr2

K2+ Ib

]ωb = Tb −

1

KTroad +RF (7.28)

Iaωa = Ta + SF (7.29)

Again, ωb and ωb are known. The engine speed is known as it is the state space mesh

vector x1, where ωeng ∈ x1 , [ωeng min : sx1 : ωeng max]. Here, sx1 is defined so that

the length of control input is as defined length(x1) = nx1. Teng is also known as it is

the input control mesh vector. where Teng ∈ u1 , [Teng min : su1 : Teng max]. Similarly,

su1 is defined so that the length of control input is as defined length(u1) = nu1. Thus,

the system has 5 equations and 5 unknowns and can be solved analytically.

First, ωa and ωa can be determined by Equation 7.26. Equations 7.27 and 7.29

must be rearranged to solve for ωeng, ωa, respectively.

104


ωeng =1

Ieng[Teng − (R + S)F ] (7.30)

ωa =1

Ia[Ta + SF ] (7.31)

These can now be substituted into Equation 7.26. The result can be rearranged

in terms of motor A torque.

Ta =

[IaIeng

(R + S)

S

]Teng −

[IaR

S

]ωb −

[IaIeng

(R + S)2

S+ S

]F (7.32)

Equation 7.28 is then rearranged to solve for motor B torque.

Tb =[Mr2

K2+ Ib

]ωb +

1

KTroad −RF (7.33)

Now Equations 7.32 and 7.33 are substituted into the power demand equation to

solve for the reaction force F .

F =

[IaIeng

(R+S)S

]Tengωa −

[Ia

RS

]ωbωa +

[Mr2

K2 + Ib

]ωbωb + 1

KTroadωb + Tengωeng

Rωb +[IaIeng

(R+S)2

S+ S

]ωa

(7.34)

Ta and Tb are now determined by substituting F back into Equations 7.32 and

7.33, respectively. Finally, ωeng and ωa are evaluated by substituting F back into

Equations 7.30 and 7.31.

The cost to stay in engine on mode can now be determined in a similar way to

105


EV mode. The maximum and minimum torque and power constraint values must

be determined for the engine and two motors. This is done using the maximum and

minimum torque and power curves for each component. Each curve is indexed by

the respective component speed. The present state constraints are now evaluated to

determine φk(xi) at each point. The next state is then determined using a forward

Euler approximation for engine speed and the battery model for SOC. Interpolation

is used to ensure that the state lies on the state space mesh. The charge sustaining

performance index is then evaluated. The fuel consumption is determined using the

engine fuel map discussed in Section 6.1.5. The cost-to-go at each state can now be

evaluated. The output is the transition cost matrix JEV T . Since JEV T has dimension

nu1 by nx1 ∗ nx2, it can be minimized to increase available memory. It is necessary

to store the control input associated with the minimum cost-to-go JEV T min at each

point in the state. Let the minimum control input for EVT mode be denoted by

UEV T min. Thus, this model outputs two vectors JEV T min and UEV T min.

7.5.4 EV to EVT Mode

The carrier is free to rotate during engine start up. As a result, the same dynamics

apply as in the EVT model. During start up, the engine speed is zero and it does

not produce any torque. It is assumed that the engine can reach its idle speed

ωeng idle during the time step dt selected for simulation. This means that the engine

acceleration can be determined as follows:

ωeng =ωeng idledt

(7.35)

ωa and ωa can now be determined from the PGS dynamics. Next, the reaction

106


force can be determined by substituting Teng and ωeng into Equation 7.27.

F = −Iengωeng(R + S)

(7.36)

Ta and Tb are now determined by substituting F back into Equations 7.29 and 7.28,

respectively.

The maximum and minimum torques and powers of the two motors are determined

to evaluate the component constraints. The battery model is then used to determine

the next SOC. The next state is interpolated if it does not exist on the mesh space.

To mimic realistic conditions a fuel penalty must be incorporated. For simplicity, the

maximum fuel rate in the engine’s fuel table is taken. Finally, the cost-to-go at each

state is calculated and the model output is the transition cost matrix JEV 2EV T . Since

there is only one control input that yields the EV to EVT mode transition JEV 2EV T

is the minimum cost to turn the engine on and must be stored for now.

7.5.5 EVT to EV Mode

The carrier is free to rotate during engine shutdown. Thus, the same dynamics

apply as in the EVT and EV to EVT models. Since the engine is originally on the

engine speed is known and is the state space mesh vector x1. However, during engine

shutdown the engine is not producing any torque. It is assumed that the engine can

reduce its speed to zero during the time step dt selected for simulation. As a result,

the engine deceleration can be determined as follows:

ωeng = −ωengdt

(7.37)

107


ωa and ωa can now be determined using the PGS relation. The reaction force is

evaluating as shown in Equation 7.36. Ta and Tb are now determined by substituting

F back into Equations 7.29 and 7.28, respectively.

Again, the maximum and minimum torques and powers for the two motors are

determined using their respective maps. The component constraints, next state, and

charge sustaining performance index are determined in the same way as the other

models. A fuel penalty must also be applied to the engine shutdown model to reflect

realistic conditions. This fuel penalty should be less than the engine start up penalty.

For simplicity, the fuel rate at the engine’s idle speed is taken. The output of this

model is the transition cost matrix JEV T2EV . There is only one control input that

results in engine shutdown. This means that JEV T2EV is the minimum cost to turn

off the engine and must be stored for now.

7.5.6 Interpolation Method

Nearest neighbour interpolation is used to ensure that the state values remain on the

grid. In this interpolation method, each point is set to that of its closest neighbour.

For example, say we wish to find the nearest neighbour of point P which is at location

(u,v). Suppose that point P has four neighbouring points A, B, C, and D at locations

(i,j), (i,j+1), (i+1, j), and (i+1, j+1), respectively. The distance between point P

and each neighbour would be determined, and point P would take the value of the

point with the shortest distance to it. In the case of Figure 7.6, point P would take

the value of point A.

108


P(u,v)

A(i,j) B(i, j+1)

C(i+1,j) D(i+1,j+1)

Figure 7.6: Nearest neighbour interpolation example.

This interpolation method is suited to the DP problem as the state is limited to

the values on the meshed state matrix, X.

7.6 Optimal Vehicle Operation Points

The backwards process defined above outputs the optimal path at every possible state.

The parameters in Table 7.5 were used in all the backwards DP simulations. The

process must then be run forwards with an initial state to output the optimal vehicle

operation points (torque split, SOC profile, mode switching, etc.). The parameters in

Table 7.6 were used in all the forwards DP simulations. The DP algorithm was run

on three cycles: the FTP75 city cycle, the highway cycle, and the proposed journey

mapping cycle. The DP solution for the three cycles is shown below.

109


Table 7.5: DP Backwards Simulation Parameters

Variable Min Max N

mode 0 1 2

SOC 0.4 0.6 1811

ωeng 0, 104.72 rad/s 471.23 rad/s 57

Teng 0 Nm 155.91 Nm 51

Table 7.6: DP Forwards Simulation Parameters

Parameter Value

SOC(0) 0.50

ωeng(0) 0

α 0

dt 0.5

Mode Penalty 1

City Cycle Solution

First, the output of FTP75 standard cycle is shown. Figures 7.7 to 7.11 show the SOC,

mode, torque split, angular speeds, and power split, respectively, over time. The DP

algorithm output an mpg rating 60.5 for this cycle. Note that this is for the city cy-

cle where SOC0 = SOCN = 0.50 on a simplified model with a 2004 Prius engine map.

110


0 500 1000 1500 2000 2500 3000

Time [s]

0.48

0.5

0.52

0.54

0.56

0.58

0.6

SO

C [%

]

Figure 7.7: State of charge over time for the FTP75 city cycle.

0 500 1000 1500 2000 2500 3000

Time [s]

0

0.2

0.4

0.6

0.8

1

Mod

e

Figure 7.8: Vehicle mode (or engine on/off) over time for the FTP75 city cycle.

111


0 500 1000 1500 2000 2500 3000

Time [s]

-40

-20

0

20

40

60

80

100

120

140

Tor

que

[Nm

]

engine

motor A

motor B

Figure 7.9: Torque split between the engine, motor A, and motor B over time for the

FTP75 city cycle.

0 500 1000 1500 2000 2500 3000

Time [s]

-1000

-500

0

500

1000

Spe

ed [r

ad/s

]

engine

motor A

motor B

Figure 7.10: Angular speed of the engine, motor A, and motor B over time for the

FTP75 city cycle.

112


0 500 1000 1500 2000 2500 3000

Time [s]

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Pow

er [W

]#104

engine

motor A

motor B

Figure 7.11: Power of the engine, motor A, and motor B over time for the FTP75

city cycle.

Highway Cycle Solution

Next, the output of the dynamic programming algorithm over the standard highway

cycle is determined. Figures 7.12 to 7.16 show the SOC, mode, torque split, angular

speeds, and power split, respectively, over time. The DP algorithm output an mpg

rating 61.8 for this cycle. Note that this is for the highway cycle where SOC0 =

SOCN = 0.50 on a simplified model with a 2004 Prius engine map.

113


0 200 400 600 800 1000 1200 1400 1600

Time [s]

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

SO

C [%

]

Figure 7.12: State of charge over time for the highway cycle.

0 200 400 600 800 1000 1200 1400 1600

Time [s]

0

0.2

0.4

0.6

0.8

1

Mod

e

Figure 7.13: Mode over time for the highway cycle.

114


0 200 400 600 800 1000 1200 1400 1600

Time [s]

-40

-20

0

20

40

60

80

100

120

Tor

que

[Nm

]

engine

motor A

motor B

Figure 7.14: Torque over time for the highway cycle.

0 200 400 600 800 1000 1200 1400 1600

Time [s]

-1000

-500

0

500

1000

Spe

ed [r

ad/s

]

engine

motor A

motor B

Figure 7.15: Speed over time for the highway cycle.

115


0 200 400 600 800 1000 1200 1400 1600

Time [s]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Pow

er [W

]#104

engine

motor A

motor B

Figure 7.16: Power over time for the highway cycle.

Journey Mapping Cycle Solution

Finally, the output of the dynamic programming algorithm over the defined journey

mapping cycle is determined. Figures 7.17 to 7.21 show the SOC, mode, torque split,

angular speeds, and power split, respectively, over time. The DP algorithm output

an mpg rating 54.2 for this cycle. Note that this is for the journey mapping cycle

where SOC0 = SOCN = 0.50 on a simplified model with a 2004 Prius engine map.

116


0 500 1000 1500 2000 2500 3000 3500 4000

Time [s]

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

SO

C [%

]

Figure 7.17: State of charge over time for the journey mapping cycle.

0 500 1000 1500 2000 2500 3000 3500 4000

Time [s]

0

0.2

0.4

0.6

0.8

1

Mod

e

Figure 7.18: Mode over time for the journey mapping cycle.

117


0 500 1000 1500 2000 2500 3000 3500 4000

Time [s]

-150

-100

-50

0

50

100

150

200

Tor

que

[Nm

]

engine

motor A

motor B

Figure 7.19: Torque over time for the journey mapping cycle.

0 500 1000 1500 2000 2500 3000 3500 4000

Time [s]

-800

-600

-400

-200

0

200

400

600

800

Spe

ed [r

ad/s

]

engine

motor A

motor B

Figure 7.20: Speed over time for the journey mapping cycle.

118


0 500 1000 1500 2000 2500 3000 3500 4000

Time [s]

-4

-3

-2

-1

0

1

2

3

4P

ower

[W]

#104

engine

motor A

motor B

Figure 7.21: Power over time for the journey mapping cycle.

119

Chapter 8

Benchmarking

It is important to benchmark the dynamic programming results obtained, as a sub-

stantive measure of the accuracy of the results. Performance metrics have been de-

fined in Section 5.3. In this chapter, the performance of the dynamic programming

controller is compared to a rule based controller and Genetic Algorithm enhanced

rule-based hybrid controller. First, Section 8.1 summarizes the two algorithms that

are compared against and the expected outcome based on literature. Next, Section

8.2 summarizes the results obtained through simulation.

8.1 Introduction

As discussed in Section 5.4, rule-based algorithms are most commonly used in prac-

tice. This is because they are simple to implement in real-time. For the purposes of

benchmarking, the DP solution is compared against two rule-based techniques. Both

techniques follow the same process, but the second has been enhanced by the Genetic

Algorithm.

120


The general process that these controllers follow can be seen in Algorithm 2. If the

engine is off and the demanded power exceeds the engine on power threshold Peng on

or if the battery SOC is lower than the minimum allowable value SOCmin then the

engine will be turned on. On the other hand, if the engine is on and the demanded

power is below the engine off power threshold Peng off and the battery SOC is greater

than the minimum allowable value SOCmin then the engine will turn off. Similarly,

if the engine is on and the battery SOC is greater than the maximum allowable value

SOCmax then the engine will shut off. Finally, if none of the conditions mentioned

are true then the engine will remain in the same mode. The engine speed and torque

is determined by its efficiency map. This means that the engine is operating at its

most efficient points for each demanded power.

It is difficult to attain charge balanced operation with such rule-based approaches.

As such, an SOC controller has been incorporated into the model. This simple PI

controller has three parameters that require tuning: ess soc target, ess soc ki, and

ess soc offset. In addition to these, suitable parameters must be chosen for the

engine on/off thresholds. For the first rule-based approach, these parameters were

chosen using engineering intuition along with trial and error. For the second rule-

based approach, which we will call the Genetic Algorithm Rule-Based Approach, the

SOC controller parameters and the power thresholds are determined using the Ge-

netic Algorithm.

The Genetic Algorithm is based on the idea of natural selection or survival of the

121


fittest. Natural selection is a process by which the individuals in a population with

superior traits or genes reproduce to create the next generation of individuals. Here,

a population represents a possible set of solutions for a given optimization problem.

Each individual in this population has a fitness value, which is determined by an

objective function. This objective function dictates which traits are determined to

be superior and is determined by the user based on the desired result. The objective

function in this case aims to minimize the fuel, as well as the change in SOC. Parents

are selected based on each individual’s fitness value with the intention of combining

their genes to produce offspring with an improved fitness value.

The Genetic Algorithm creates offspring for the next generation using three pro-

cesses: elite, crossover, and mutation. Elite children are individuals in the current

generation with the best fitness values. Crossover children are created by randomly

combining the genes of a pair of parents. Mutation children are created by introduc-

ing random changes to the genes of an individual parent. This mutation could be used

to inhibit premature convergence. When the new generation of offspring vary little

from those in previous generations, the algorithm has converged to a set of solutions.

122


Algorithm 2 Rule-Based Algorithm

procedure RB ProcedureDefine [SOCmin, SOCmax]Define Peng onDefine Peng offDefine mode(0) = 0Define SOC(0)

for k=1:T doFind Pdem(k)if mode(k − 1) = 0 and (Pdem(k) > Peng on or SOC(k) < SOCmin) then

mode(k) = 1else if mode(k − 1) = 1 and Pdem(k) < Peng off and SOC(k) > SOCmin

thenmode(k) = 0

else if mode(k − 1) = 1 and SOC(k) > SOCmax thenmode(k) = 0

elsemode(k) = mode(k − 1)

end ifweng(k) = w∗eng(Pdem) ·mode(k)if weng(k) = 0 then

Teng(k) = 0else

Teng(k) = min(Pdem(k)weng(k)

, Teng max)

end ifRun vehicle model and output SOC(k + 1)

end forend procedure

123


8.2 Results

Both algorithms were run over the city, highway, and journey mapping cycles. The

resulting fuel consumption for each cycle is summarized and compared to the GA

results in Section 8.2.3. Since the two rule-based algorithms do not allow for hard

constraints on the final SOC, charge balancing behaviour must be regulated by the

objective function as well as some key control variables. As a result, a charge sus-

taining term was added to the objective function. As seen in Table 8.1, the charge

sustaining term coefficient α that was set to zero in the DP solution is now set to

0.01. The control variables that impact the battery behaviour are ess soc target,

ess soc ki, and ess soc offset. The values of these variables for each simulation are

highlighted later. In addition to this, the engine on/off thresholds impact the charge

sustaining characteristics of the battery. These thresholds are also indicated above

each simulation.

Table 8.1: Rule-Based Simulation Parameters

Parameter Value

SOCmin 0.40

SOCmax 0.60

α 0.01

dt 0.5

Mode Penalty 1

SOC(0) 0.50

ωeng(0) 0

124


8.2.1 Rule-Based Control

The results from the rule-based control algorithm for all three cycles are shown be-

low. The simulation parameters were determined using trial and error with the in-

tention of reaching charge sustaining operation. With multiple attempts at reaching

charge-sustaining operation, it was not always achieved. This is because it requires

a significant amount of time and investment to calibrate a control system to achieve

this operation. It is possible that there may not be enough control over the com-

plex system with only five variables to calibrate. This is important to consider when

comparing the results.

City Cycle Solution

The control parameters used in the city cycle simulation and their respective values

can be seen in Table 8.2. Figures 8.1 to 8.5 show the vehicles operating characteristics

for the city cycle.

Table 8.2: Rule-Based City Cycle Simulation Parameters

Parameter Value

ess soc target 0.50

ess soc ki 3.5673

ess soc offset 0

Peng on 10,000 [W]

Peng off 4,000 [W]

125


0 500 1000 1500 2000 2500 3000Time [s]

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

SO

C [%

]

Figure 8.1: State of charge over time for the city cycle.

0 500 1000 1500 2000 2500 3000Time [s]

0

0.2

0.4

0.6

0.8

1

Mod

e

Figure 8.2: Mode over time for the city cycle.

126


0 500 1000 1500 2000 2500 3000Time [s]

-100

-50

0

50

100

150

Tor

que

[Nm

]

enginemotor Amotor B

Figure 8.3: Torque over time for the city cycle.

0 500 1000 1500 2000 2500 3000Time [s]

-1000

-500

0

500

1000

Spe

ed [r

ad/s

]


Figure 8.4: Speed over time for the city cycle.

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0 500 1000 1500 2000 2500 3000

Time [s]

-4

-3

-2

-1

0

1

2

3

Pow

er [W

]

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Figure 8.5: Power over time for the city cycle.


The Rule-Based highway cycle results are shown below. The control parameters used

in the highway cycle simulation and their respective values can be seen in Table 8.3.

Figures 8.6 to 8.10 show the SOC, mode, torque split, angular speeds, and power

split, respectively, over time.

Table 8.3: Rule-Based Highway Cycle Simulation Parameters

Parameter Value

ess soc target 0.50

ess soc ki 3.5673

ess soc offset 0

Peng on 10,000 [W]

Peng off 6,500 [W]

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0 200 400 600 800 1000 1200 1400 1600

Time [s]

0.46

0.47

0.48

0.49

0.5

0.51

0.52

SO

C [%

]


0 200 400 600 800 1000 1200 1400 1600

Time [s]

0

0.2

0.4

0.6

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e


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0 200 400 600 800 1000 1200 1400 1600Time [s]

-150

-100

-50

0

50

100

150

Tor

que

[Nm

]



0 200 400 600 800 1000 1200 1400 1600Time [s]

-1000

-500

0

500

1000

Spe

ed [r

ad/s

]



130


0 200 400 600 800 1000 1200 1400 1600Time [s]

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er [W

]

#104




The Rule-Based Journey Mapping cycle results are shown below. The control param-

eters used in the journey mapping cycle simulation and their respective values can

be seen in Table 8.4. Figures 8.11 to 8.15 show the SOC, mode, torque split, angular

speeds, and power split, respectively, over time.

Table 8.4: Rule-Based Journey Mapping Cycle Simulation Parameters

Parameter Value

ess soc target 0.90

ess soc ki 7.5673

ess soc offset 5

Peng on 8,000 [W]

Peng off 900 [W]

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0.44

0.45

0.46

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C [%

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0 500 1000 1500 2000 2500 3000 3500 4000

Time [s]

-150

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-50

0

50

100

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200

250

300

Tor

que

[Nm

]

engine

motor A

motor B


0 500 1000 1500 2000 2500 3000 3500 4000

Time [s]

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-600

-400

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Spe

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engine

motor A

motor B


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]

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engine

motor A

motor B


8.2.2 Genetic Algorithm Rule-Based Control

Next, the Genetic Algorithm Rule-Based results for the city, highway, and journey

mapping cycles are shown below. Instead of trial and error, the GA Rule-Based

control strategy determines the parameters with an evolutionary algorithm. The

objective function aims to minimize fuel consumption. A charge balancing term is

also included to promote charge sustaining operation. This operation was not always

achieved due to the limitations of such a simple controller.

City Cycle Solution

The values of the control parameters that the Genetic Algorithm determined can be

seen in Table 8.5. Figures 8.16 to 8.20 show the vehicles operating characteristics for

the city cycle.

134


Table 8.5: Genetic Algorithm Rule-Based City Cycle Simulation Parameters

Parameter Value

ess soc target 0.30

ess soc ki 2.97

ess soc offset 0.59

Peng on 10,345 [W]

Peng off 1,656 [W]

0 500 1000 1500 2000 2500 3000

Time [s]

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55

0.56

SO

C [%

]

Figure 8.16: State of charge over time for the city cycle.

135


0 500 1000 1500 2000 2500 3000

Time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mod

e

Figure 8.17: Mode over time for the city cycle.

0 500 1000 1500 2000 2500 3000

Time [s]

-100

-50

0

50

100

150

Tor

que

[Nm

]

engine

motor A

motor B

Figure 8.18: Torque over time for the city cycle.

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0 500 1000 1500 2000 2500 3000

Time [s]

-800

-600

-400

-200

0

200

400

600

800

1000

Spe

ed [r

ad/s

]

engine

motor A

motor B

Figure 8.19: Speed over time for the city cycle.

0 500 1000 1500 2000 2500 3000

Time [s]

-4

-3

-2

-1

0

1

2

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Pow

er [W

]

#104

engine

motor A

motor B

Figure 8.20: Power over time for the city cycle.

137



The Genetic Algorithm Rule-Based highway cycle results are shown below. The

values of the control parameters that the Genetic Algorithm determined can be seen

in Table 8.6. Figures 8.21 to 8.25 show the SOC, mode, torque split, angular speeds,

and power split, respectively, over time.

Table 8.6: Genetic Algorithm Rule-Based Highway Cycle Simulation Parameters

Parameter Value

ess soc target 0.55

ess soc ki 1.1077

ess soc offset 0.69

Peng on 13,053 [W]

Peng off 5,780 [W]

0 200 400 600 800 1000 1200 1400 1600

Time [s]

0.46

0.465

0.47

0.475

0.48

0.485

0.49

0.495

0.5

0.505

SO

C [%

]


138


0 200 400 600 800 1000 1200 1400 1600

Time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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e


0 200 400 600 800 1000 1200 1400 1600

Time [s]

-100

-50

0

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100

150

Tor

que

[Nm

]

engine

motor A

motor B


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Time [s]

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600

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1000

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ed [r

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motor A

motor B


0 200 400 600 800 1000 1200 1400 1600

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The Genetic Algorithm Rule-Based city cycle results are shown below. The values of

the control parameters that the Genetic Algorithm determined can be seen in Table

8.7. Figures 8.26 to 8.30 show the SOC, mode, torque split, angular speeds, and

power split, respectively, over time.

Table 8.7: Genetic Algorithm Rule-Based Journey Mapping Cycle Simulation Param-eters

Parameter Value

ess soc target 0.94

ess soc ki 8.7458

ess soc offset 3.83

Peng on 8,589 [W]

Peng off 531 [W]

0 500 1000 1500 2000 2500 3000 3500 4000

Time [s]

0.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.51

SO

C [%

]


141


0 500 1000 1500 2000 2500 3000 3500 4000Time [s]

0

0.2

0.4

0.6

0.8

1

Mod

e


0 500 1000 1500 2000 2500 3000 3500 4000

Time [s]

-150

-100

-50

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100

150

200

Tor

que

[Nm

]

engine

motor A

motor B


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ed [r

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motor B


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-1

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motor A

motor B


143


8.2.3 Algorithm Comparison

The fuel consumption for each algorithm and cycle case is summarized below. Note

that these values were normalized, as the simulations were performed on a simplified

model with charge sustaining criteria.

Table 8.8: Control Algorithm Performance Comparison

City Cycle Highway Cycle Journey Mapping Cycle

DP 1 1 1

GA RB 1.15 1.10 1.47

RB 1.20 1.13 1.63

It is clear that the DP solution outperforms the rule-based methods. The DP

algorithm improved vehicle performance in the city cycle by 13 − 17%, the highway

cycle by 9− 12%, and the journey mapping cycle by 32− 39% . It is also important

to highlight the difference in the final SOC between algorithms. The two rule-based

algorithms result in a final SOC that is not equal to the initial SOC. Whereas, the DP

solution allows for hard constraints on the battery SOC (i.e. the final SOC is equal

to the initial SOC). This means that the DP solution outperforms the rule-based

algorithm on a larger scale than what is shown in Table 8.8.

Another key finding is the scale by which the DP outperforms the rule-based al-

gorithms for the journey mapping cycle. This emphasizes the need to design and

optimize vehicle operation over more realistic drive cycles. There is significant room

for improvement of traditional rule-based control algorithms during real world driving.

144


An important issue in control design that is not well covered in literature is driv-

ability. Drivability is difficult to consider in the development of a control strategy

since it depends on the subjective judgments of the driver or passenger. Powertrain

control strategy has the most direct relation to drivability. One of the key metrics

used to quantify drivability in HEVs is mode switching [57]. During mode switching,

differences in the ICE and electric motor torque performances occur. This can cause

a large torque surge and impacts the performance of the HEV [57]. The vibrations

associated with mode switching degrade ride comfort, thus decreasing the drivability

of the HEV. The DP solution results in less mode switching than the rule-based al-

gorithms. In particular, the Journey Mapping cycle, which is a drive cycle derived

from real driving and includes both highway and city driving, has larger and more

frequent changes in speed. The Rule-Based algorithms result in significantly more

mode switches than the DP solution. This suggests that the DP solution would also

increase the drivability of the vehicle.

Finally, the rule-based results suggest that charge balancing operation is more

difficult to accomplish over more aggressive cycles. The simplified control system

considered did not have enough functionality to find calibration variables that resulted

in charge balanced results for all cycles. It requires an extensive amount of engineering

intuition and time to output a controller that achieves charge sustaining results over

all cycles. The DP solution eliminates this issue by allowing for hard constraints on

the final SOC of the battery.

145

Chapter 9

Conclusion and Future Work

9.1 Conclusion

The objective of this thesis is twofold:

1. To apply the dynamic programming technique to an autonomous vehicle with

a HEV powertrain

2. To highlight the importance of designing and optimizing vehicles over realistic

drive cycles

This thesis explores hybrid supervisory control in the context of autonomous vehi-

cles, and emphasizes the importance of considering this level of control. The dynamic

programming technique has been applied to an autonomous vehicle with a power-

split HEV powertrain. Many steps have been taken to reduce the complexity and

thereby improve the efficiency of the DP algorithm as it is applied to the autonomous

power-split HEV model. Vectorization and partitioning techniques were applied to

the problem in order to further reduce the complexity of the algorithm. This allowed

146


for more dense state and input meshes, and thus more accurate results.

The DP solution can also be used as a benchmark for other control strategies, and

can used as a tool in the design process. The design parameters of other strategies

can be tuned with the aim of reaching the optimal control policy. The DP solution is

suitable to multi-objective optimization problems. As autonomous vehicle technology

progresses, the additive cost function will allow for other constraints to be considered.

The DP solution is benchmarked against two rule-based algorithms to substan-

tively measure the accuracy of the results. A Journey Mapping cycle is introduced to

test the DP solution under more realistic driving conditions. This Journey Mapping

cycle is derived from real world driving measurements, and includes road grade in the

definition. This is because road grade is considered to have a significant impact on

the fuel consumption of a vehicle.

The results of the study show that the DP solution improves vehicle performance

by at least 9 − 17% when compared to commonly used rule-based techniques over

standard drive cycles. The DP technique is also applied over the proposed journey

mapping cycle, which represents more realistic driving conditions. It is seen that the

DP technique improves vehicle performance by at least 32 − 39% when compared

against rule-based techniques over the journey mapping cycle. This suggests that

it would be extremely beneficial to design and optimize the vehicle control strategy

over more realistic drive cycles. The DP solution also eliminates the complexity of

achieving charge sustaining operation, which is required by the EPA. In addition, the

147


results suggest that the DP control strategy improves the drivability of the HEV.

9.2 Future Work

There are many possible extensions of this work. This section will focus on using the

DP solution for adaptive control calibration.

9.2.1 Adaptive Control Calibration

One possible extension of this work is the process of adaptive control calibration. In

industry, it does not make sense to design a new controller for every vehicle config-

uration and variation as it is time consuming and expensive. As a result, extensive

time is put into calibrating existing controllers to maximize fuel economy and meet

performance constraints. One of the key performance constraints for conventional

and plug-in HEVs is that they are required to operate in a charge sustaining mode.

This means that the battery SOC must be maintained within certain limits. In par-

ticular, EPA standards require ±1% Net Energy Change (NEC) of fuel energy over

a cycle [58]. The NEC is defined as net battery energy delta, which is expressed as

percentage of the fuel energy consumed on a cycle. Thus, the development of a charge

sustaining control strategy is essential to the feasibility of a hybrid vehicle.

A common challenge in meeting SOC requirements is that the optimal solution for

one drive cycle may not be optimal or even charge sustaining for other drive cycles.

For conventional hybrid vehicles and PHEV charge sustaining operation, the EPA

148


tests the vehicle using either the derived 5-cycle (city/highway) method or the ve-

hicle specific 5-cycle (city/highway/US06/SC03/Cold temperature test) method [D.

Good]. In order to have a vehicle comply with these tests, a control strategy must

be implemented that maintains ±1% Net Energy Change (NEC) of fuel energy over

all test cycles. In practice, this is done through a series of offline controller cali-

brations where multiple control variables of a generic EMS controller are varied to

obtain charge balanced results. Because of the non-linearity and complexity of such

a simulation environment, this manual process is very time consuming and requires

extensive background knowledge and proper engineering intuition to produce results.

Moreover, this method is not conducive to obtaining the optimal solution.

The calibration process generally begins in a simulation environment, where a

subset of the calibration variables are used. This simulation environment allows for

a relatively low cost way to identify how a vehicle might perform. Once reason-

able values are found in simulation, the calibration process generally continues with

static calibrations in production. The calibration variables are modified, the vehicle

is driven to test the operation, and then the controller is re-calibrated from the test

findings, and so on. This process is repeated until acceptable values are found. This

method is subjective, requires extensive experience and prior knowledge, and does

not guarantee that the optimal operation is achieved.

In order to overcome these obstacles, the DP solution could be used as a target

state. In other words, the calibration process can be posed as an optimization problem

where the calibration variables are the control variables and the objective function is

149


the difference between the vehicle’s operation and the DP solution. Assuming that

the controller already exists, the proposed process attempts to match the DP solution

through calibration.

Literature often discusses active control, but does not explore active calibration.

This section proposes a method to actively calibrate a vehicle controller offline to

match the optimal operation. The idea is that this process can be applied to any

existing controller. In this case, a controller from Autonomie is used. This controller

uses a rule-based method to determine the power split. To begin the adaptive control

calibration process, the controller’s calibration variables must first be identified with

appropriate ranges. A learning algorithm is then selected to find the calibration

variables that output the closest operation to the global optimum. This means that

the cost function will be the difference in operation between the GA result and the

Autonomie controller result. This process has been applied over the highway cycle.

Autonomie Model

A power-split vehicle model from Autonomie has been selected for adaptive control

calibration. The simulation environment can be seen in Figure 9.1. The component

maps in the Autonomie model shown are the same as the maps used in the DP

model. The supervisory controller is similar to the rule-based algorithm presented

in Section 8.1, with some added complexity. For example, the engine on/off decision

also accounts for the time that the engine has been on or off. The engine will only

turn on if the demanded power is above a set threshold and the engine has been in

the off state for a minimum amount of time. An interested reader can look at the

150


”vpc prop split best eng” supervisory control model in Autonomie to gain a better

understanding of the controller operation.

Figure 9.1: Autonomie model.

After analyzing the input variables within the controller, 15 variables were chosen

for the adaptive control calibration process. The control input variables that have

been selected are as outlined in Table 9.1. The battery SOC controller calibration

variables are ess soc target, ess soc offset, and ess soc ki. There are six calibra-

tion variables that help to determine the engine on/off decision. These variables are

eng time min pwr dmd above thresh, eng time min pwr dmd below thresh,

eng time min stay off , eng time min stay on, mot2 kp engine on mode4, and

mot2 ki engine on mode. Finally, the last six control variables replace the engines

optimal operation map, which determines the engine speed from the requested power.

151


The new power to speed map is constructed from the calibration variables pwr2spd 1,

pwr2spd 2, pwr2spd 3, pwr2spd 4, pwr2spd 5, and pwr2spd 6.

Table 9.1: Autonomie Model Control Variables

Calibration Variable Minimum Maximum

vpc.prop.init.ess soc target 0.56 0.60

vpc.prop.init.ess soc offset 0.10 0.90

vpc.prop.init.ess soc ki 1 20

vpc.prop.init.eng time min pwr dmd above thresh 0.70 1.30

vpc.prop.init.eng time min pwr dmd below thresh 0.10 0.50

vpc.prop.init.eng time min stay off 7 150

vpc.prop.init.eng time min stay on 2 100

vpc.prop.init.mot2 kp engine on mode 0.05 3

vpc.prop.init.mot2 ki engine on mode 0.05 3

eng.plant.init.pwr2spd 1 104.72 144.70






Since the hyrbid vehicle model is a non-linear system, it is impossible to know the

direct impact each variable will have on the system as a whole. However, a sensitivity

analysis can be performed on the model to gain insight into how changing each control

or calibration variable will impact some key system outputs. This can be done by

holding all control variables constant except for the one in question and analyzing

the outputs. A sensitivity analysis was performed and helped to define the maximum

152


and minimum values for each control variable input.

Process

It is important to select an appropriate cost function to ensure that the learning al-

gorithm is minimizing the difference in operation between the DP solution and the

controller in question.

The cost function selected in this case is as follows:

J =

∫ N

0

(Peng dp − Peng)2 + (Pa dp − Pa)2 + (Pb dp − Pb))2dt (9.1)

where Peng dp, Pa dp, and Pb dp are the powers output by the dynamic program-

ming algorithm for the engine, motor A, and motor B, respectively. This objective

function determines the square difference of the component powers for the optimal

(DP) solution and the calculated solution.

Because of the complexity and non-linearity of the problem, the proposed system

can be viewed as a black box. That is, we know the inputs and outputs of the system,

but we do not know the internal workings. It is impossible to know how varying each

calibration variable will impact the outputs of the system. Thus, this problem can

be defined as a multi-variable black box optimization problem.

Traditional optimization techniques, such as dynamic programming and linear

programming, generally fail at solving large non-linear problems. As such, population-

based algorithms are typically used to solve black box optimization problems. These

153


algorithms are based on concepts from population biology, genetics, and evolution.

Some common population-based algorithms include genetic algorithms (GAs), parti-

cle swarm optimization (PSO), and artificial bee colony (ABC) [59]. Population-based

algorithms have a set of possible solutions with fitness values. The algorithms work

to move each individuals fitness value towards the individual with the best fitness.

PSO

Particle swarm optimization was used to find the calibration variables. The pseudo

code for the PSO algorithm can be seen in Algorithm 3. The concept of PSO orig-

inated from the behaviour exhibited by a swarm of birds [59]. Particle swarm op-

timization begins with a random initial population. Each potential solution in the

population is called a particle. The particles are then ”flown” through the problem

space and ”follow” the particle with the current best solution. Each particles position

is monitored and its best position, p best, is kept track of. In addition to this, the

best particle with the best overall fitness value is monitored. This is called the global

best, g best. At each step, the PSO moves each particle towards its best position and

the global best position by changing the velocity of the particle. The velocity, v, for

a particle i is updated as shown in Equation 9.2.

vi(k + 1) = wv(k) + c1r1(pi − xi) + c2r2(pg − xi) (9.2)

where w, c1, and c2 are constants that weight the current velocity, particle best, and

global best, respectively. The terms r1 and r2 are random numbers from [0,1] and

update at each iteration. Finally, xi is the particle’s current position. The particles

position is updated by it’s velocity as seen in Equation 9.3

154


xi(k + 1) = xi(k) + vi(k + 1) (9.3)

The algorithm continues until a stopping criteria is reached. This stopping criteria

can be either a defined fitness value or a maximum number of iterations. In the case

of the adaptive control calibration problem, each particle consists of the 15 calibration

variables selected. The fitness value is determined by Equation 9.1 and the algorithm

continues until the defined maximum number of iterations.

155


Algorithm 3 Particle Swarm Optimization Algorithm

procedure PSO AlgorithmDefine Xmin, Xmax

Define Vmin, Vmax

Initialize Population

for each particle i dofor each dimension d do

Initialize position xid ∈ [xmin, xmax]Initialize velocity vid

end forend for

Iteration k = 1while k < Maximum Iterations do

for each particle i doCalculate the fitness value

if The current fitness value is better than p bestid thenp bestid = current fitness value

end ifend forChoose particle with the best fitness value as g bestdfor each particle i do

for each dimension d doCalculate velocity

vid(k + 1) = wvid(k) + c1rand1(pid − xid) + c2rand2(pgd − xid)Update particle position

xid(k + 1) = xid(k) + vid(k + 1)end for

end fork = k + 1

end whileend procedure

156


Results

The results from the PSO algorithm are discussed below and compared to the orig-

inal DP solution. Note that the battery parameters used for these simulations are

different than those used in previous simulations and can be seen in Table 9.2.

Table 9.2: PSO Comparison DP Simulation Parameters

Parameter Value

SOCmin 0.40

SOCmax 0.70

α 0.001

dt 0.5

Mode Penalty 1

Figures 9.2 to 9.6 show the results of the PSO algorithm. The SOC, mode, engine

power, motor A power, and motor B power are shown for Autonomie and the DP

solution over time.

157


0 200 400 600 800 1000 1200 1400 1600

Time [s]

0.42

0.44

0.46

0.48

0.5

0.52

0.54

SO

C [%

]

Autonomie

DP

Figure 9.2: Battery SOC over time for the highway cycle.

0 200 400 600 800 1000 1200 1400 1600

Time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mod

e

Autonomie

DP


158


0 200 400 600 800 1000 1200 1400 1600

Time [s]

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Pow

er [W

]

#104

Autonomie

DP

Figure 9.4: Engine power over time for the highway cycle.

0 200 400 600 800 1000 1200 1400 1600

Time [s]

-4

-3

-2

-1

0

1

2

3

4

Pow

er [W

]

#104

Autonomie

DP

Figure 9.5: Motor A power over time for the highway cycle.

159


0 200 400 600 800 1000 1200 1400 1600

Time [s]

-4

-3

-2

-1

0

1

2

Pow

er [W

]

#104

Autonomie

DP

Figure 9.6: Motor B power over time for the highway cycle.

9.2.2 Technical Challenges

Overall, the PSO algorithm leads the system towards the optimal operation. How-

ever, there are some key technical challenges that prevent the operation from reaching

the DP solution. First, because of the nature of the Autonomie controller that we are

trying to calibrate, it is impossible to match the operation exactly. This is because

the engine will always turn on and off at specific thresholds or after a particular time

is reached. In the DP algorithm, the engine on/off state is not reliant on any specific

thresholds and turns on and off at many different requested power values and after

various time intervals. This difference in fundamental operational principles makes

it unlikely that the mode switches will match exactly between the Autonomie model

and the DP solution. This is made clear when analyzing the results.

160


It is also clear from the results shown that the calibration variables selected do

not give enough control over the electric machine power split. In the case shown, the

DP solution generally uses motor A less and consequently operates motor B at higher

power levels. The addition of control variables that influence the power split between

the two electric machines could help to overcome this problem. Future work would

be to identify calibration variables that impact the motor operation.

161

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