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
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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
8.6 State of charge over time for the highway cycle. . . . . . . . . . . . . 129
8.7 Mode over time for the highway cycle. . . . . . . . . . . . . . . . . . 129
8.8 Torque over time for the highway cycle. . . . . . . . . . . . . . . . . 130
8.9 Speed over time for the highway cycle. . . . . . . . . . . . . . . . . . 130
8.10 Power over time for the highway cycle. . . . . . . . . . . . . . . . . . 131
8.11 State of charge over time for the journey mapping cycle. . . . . . . . 132
8.12 Mode over time for the journey mapping cycle. . . . . . . . . . . . . 132
xvi
8.13 Torque over time for the journey mapping cycle. . . . . . . . . . . . 133
8.14 Speed over time for the journey mapping cycle. . . . . . . . . . . . . 133
8.15 Power over time for the journey mapping cycle. . . . . . . . . . . . . 134
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
8.21 State of charge over time for the highway cycle. . . . . . . . . . . . . 138
8.22 Mode over time for the highway cycle. . . . . . . . . . . . . . . . . . 139
8.23 Torque over time for the highway cycle. . . . . . . . . . . . . . . . . 139
8.24 Speed over time for the highway cycle. . . . . . . . . . . . . . . . . . 140
8.25 Power over time for the highway cycle. . . . . . . . . . . . . . . . . . 140
8.26 State of charge over time for the journey mapping cycle. . . . . . . . 141
8.27 Mode over time for the journey mapping cycle. . . . . . . . . . . . . 142
8.28 Torque over time for the journey mapping cycle. . . . . . . . . . . . 142
8.29 Speed over time for the journey mapping cycle. . . . . . . . . . . . . 143
8.30 Power over time for the journey mapping cycle. . . . . . . . . . . . . 143
9.1 Autonomie model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.2 Battery SOC over time for the highway cycle. . . . . . . . . . . . . . 158
9.3 Mode 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
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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
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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
M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering
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
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M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering
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.
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M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering
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.
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M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering
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.
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M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering
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
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M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering
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
M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering
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
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M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering
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
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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
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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.
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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
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M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering
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
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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
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M.A.Sc. Thesis - Brynn Vadala McMaster - Mechanical Engineering
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
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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
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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
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(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.
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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.
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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
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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
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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
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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.
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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
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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
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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].
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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
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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).
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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
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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
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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].
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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
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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.
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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
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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.
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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
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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,
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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.
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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
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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:
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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].
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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.
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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)
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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,
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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:
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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.
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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.
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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.
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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.
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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
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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.
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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,
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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
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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:
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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.
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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.
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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.
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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
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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
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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).
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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)
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-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)
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-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
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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
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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)
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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.
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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].
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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.
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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.
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(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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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
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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
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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:
Pdem = Tengωeng + Taωa + Tbωb (7.13)
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
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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
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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
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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.
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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.
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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.
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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.
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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
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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.
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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:
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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
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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
Pa min ≤ Pa ≤ Pa max
Pb min ≤ Pb ≤ Pb max
Pbatt min ≤ Pbatt ≤ Pbatt max
U(t) ,
Teng min ≤ Teng ≤ Teng max
Ta min ≤ Ta ≤ Ta max
Tb min ≤ Tb ≤ Tb max
u : Peng min ≤ Peng ≤ Peng max
Pa min ≤ Pa ≤ Pa max
Pb min ≤ Pb ≤ Pb max
Pbatt min ≤ Pbatt ≤ Pbatt max
J(u(t)) = φ(SOC(N)) +
∫ T
0
{mF (u(t)) + α ˙SOC(u(t)) + φ(x(t))
}dt
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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
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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 .
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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.
Pdem = Tengωeng + Taωa + Tbωb (7.25)
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.
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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
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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
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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)
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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
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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
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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]
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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.
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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
]
enginemotor Amotor B
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
]
#104
enginemotor Amotor B
Figure 8.5: Power over time for the city cycle.
Highway Cycle Solution
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 [%
]
Figure 8.6: 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 8.7: Mode over time for the highway cycle.
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0 200 400 600 800 1000 1200 1400 1600Time [s]
-150
-100
-50
0
50
100
150
Tor
que
[Nm
]
enginemotor Amotor B
Figure 8.8: Torque over time for the highway cycle.
0 200 400 600 800 1000 1200 1400 1600Time [s]
-1000
-500
0
500
1000
Spe
ed [r
ad/s
]
enginemotor Amotor B
Figure 8.9: Speed over time for the highway cycle.
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0 200 400 600 800 1000 1200 1400 1600Time [s]
-4
-3
-2
-1
0
1
2
3
Pow
er [W
]
#104
enginemotor Amotor B
Figure 8.10: Power over time for the highway cycle.
Journey Mapping Cycle Solution
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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0 500 1000 1500 2000 2500 3000 3500 4000
Time [s]
0.44
0.45
0.46
0.47
0.48
0.49
0.5
0.51
SO
C [%
]
Figure 8.11: 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 8.12: Mode over time for the journey mapping cycle.
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0 500 1000 1500 2000 2500 3000 3500 4000
Time [s]
-150
-100
-50
0
50
100
150
200
250
300
Tor
que
[Nm
]
engine
motor A
motor B
Figure 8.13: 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 8.14: Speed over time for the journey mapping cycle.
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0 500 1000 1500 2000 2500 3000 3500 4000
Time [s]
-4
-3
-2
-1
0
1
2
3
4
5
Pow
er [W
]
#104
engine
motor A
motor B
Figure 8.15: Power over time for the journey mapping cycle.
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.
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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.
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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
3
Pow
er [W
]
#104
engine
motor A
motor B
Figure 8.20: Power over time for the city cycle.
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Highway Cycle Solution
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 [%
]
Figure 8.21: State of charge over time for the highway cycle.
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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
Figure 8.22: Mode over time for the highway cycle.
0 200 400 600 800 1000 1200 1400 1600
Time [s]
-100
-50
0
50
100
150
Tor
que
[Nm
]
engine
motor A
motor B
Figure 8.23: Torque over time for the highway cycle.
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0 200 400 600 800 1000 1200 1400 1600
Time [s]
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
Spe
ed [r
ad/s
]
engine
motor A
motor B
Figure 8.24: Speed 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
Pow
er [W
]
#104
engine
motor A
motor B
Figure 8.25: Power over time for the highway cycle.
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Journey Mapping Cycle Solution
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 [%
]
Figure 8.26: State of charge over time for the journey mapping cycle.
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0 500 1000 1500 2000 2500 3000 3500 4000Time [s]
0
0.2
0.4
0.6
0.8
1
Mod
e
Figure 8.27: Mode over time for the journey mapping cycle.
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 8.28: Torque over time for the journey mapping cycle.
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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 8.29: Speed over time for the journey mapping cycle.
0 500 1000 1500 2000 2500 3000 3500 4000
Time [s]
-4
-3
-2
-1
0
1
2
3
4
5
Pow
er [W
]
#104
engine
motor A
motor B
Figure 8.30: Power over time for the journey mapping cycle.
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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.
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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.
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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
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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
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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
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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
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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
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”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.
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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
eng.plant.init.pwr2spd 2 144.71 204.70
eng.plant.init.pwr2spd 3 204.71 310.70
eng.plant.init.pwr2spd 4 310.71 390.63
eng.plant.init.pwr2spd 5 390.64 451.15
eng.plant.init.pwr2spd 6 451.16 471.24
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
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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
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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
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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.
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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
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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.
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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
Figure 9.3: Mode over time for the highway cycle.
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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.
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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.
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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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