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Electrified engine air intake system: modeling,optimization and controlMarinkov, S.

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Electrified engine air intake system:modeling, optimization and control

The research reported in this thesis is part of the research program of the Dutch

Institute of Systems and Control (DISC). The author has successfully completed the

educational program of the Graduate School DISC.

This research was financially supported by the EUREKA program – project

WETREN #5765.

Electrified engine air intake system: modeling, optimization and control – PhD thesis

by Sava Marinkov, Eindhoven University of Technology.

A catalogue record is available from the Eindhoven University of Technology Library.

ISBN: 978-90-386-3997-0

Typeset by the author using the pdf LATEX documentation system.

Cover design: Sava Marinkov.

Reproduction: CPI Koninklijke Wohrmann, Zutphen, The Netherlands.

Copyright © 2015 by Sava Marinkov. All rights reserved.

Electrified engine air intake system:modeling, optimization and control

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. F.P.T. Baaijens, voor een

commissie aangewezen door het College voor

Promoties, in het openbaar te verdedigen

op woensdag 27 januari 2016 om 16.00 uur

door

Sava Marinkov

geboren te Novi Sad, Servie

Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de

promotiecommissie is als volgt:

voorzitter: prof.dr. L.P.H. de Goey

promotor: prof.dr.ir. M. Steinbuch

copromotor: dr.ir. A.G. de Jager

leden: prof.dr. C. Onder (ETH Zurich, Switzerland)

prof.dr. J. Sjoberg (Chalmers University of Technology, Sweden)

prof.dr.ir. N. van de Wouw

adviseur: dr.ir. M. Boot

Het onderzoek dat in dit proefschrift wordt beschreven is uitgevoerd in overeenstemming

met de TU/e Gedragscode Wetenschapsbeoefening.

Contents

Societal summary ix

1 Introduction 1

1.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Electric supercharging . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Regenerative throttling . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Switched Reluctance Machines . . . . . . . . . . . . . . . . . . . . 7

1.2 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Contributions and outline . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Interconnections between topics of research . . . . . . . . . . . . . . . . . 14

1.5 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Convex modeling and sizing of electrically supercharged internal com-

bustion engine powertrains 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 The powertrain sizing problem . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Quasistatic vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Wheels & brakes . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.3 Gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.4 Mechanical power link . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.5 ICE & air-fuel control . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.6 Air system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.7 Alternator and MCU motor . . . . . . . . . . . . . . . . . . . . . 28

2.3.8 Electric power link . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.9 Electric buffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Optimization problem formulation . . . . . . . . . . . . . . . . . . . . . . 32

2.4.1 Convex optimization problem . . . . . . . . . . . . . . . . . . . . 33

v

vi Contents

2.4.2 Gear selection strategy . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Case study results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5.1 Optimal component sizes . . . . . . . . . . . . . . . . . . . . . . . 36

2.5.2 Optimal state and control trajectories . . . . . . . . . . . . . . . . 37

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Convex modeling and optimization of a vehicle powertrain equipped

with a generator-turbine throttle unit 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Quasistatic vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.2 Wheels & brakes . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.3 Gearbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.4 Mechanical power link . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.5 Air system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.6 ICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.7 Alternator and GTU generator . . . . . . . . . . . . . . . . . . . 49

3.2.8 Electric power link . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.9 Electric buffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Optimization problem formulation . . . . . . . . . . . . . . . . . . . . . . 53

3.3.1 Gear selection strategy . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.2 Convex optimization problem . . . . . . . . . . . . . . . . . . . . 55

3.4 Case study results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.1 Optimal engine operating points . . . . . . . . . . . . . . . . . . . 57

3.4.2 Optimal state and control trajectories . . . . . . . . . . . . . . . . 57

3.4.3 The effect of varying the ICE displacement volume on the GTU

performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Four-Quadrant speed control of 4/2 Switched Reluctance Machines 63

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 SRM modeling and operation . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.2 Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.3 Four-quadrant operation . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 SRM control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3.1 Speed and position estimation . . . . . . . . . . . . . . . . . . . . 78

4.3.2 Supervisor for startup and change of rotational direction . . . . . 78

Contents vii

4.3.3 Speed control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.4 Current reference parametrization . . . . . . . . . . . . . . . . . . 81

4.3.5 Current control and commutation . . . . . . . . . . . . . . . . . . 82

4.4 Hardware and software implementation . . . . . . . . . . . . . . . . . . . 82

4.4.1 Hardware configuration . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Model predictive voltage control of high-speed Switched Reluctance

Generators 89

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2 SRG modeling and operation . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.1 Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2.3 Analytic commutation angles . . . . . . . . . . . . . . . . . . . . 93

5.2.4 Optimized commutation angles . . . . . . . . . . . . . . . . . . . 95

5.3 SRG control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3.1 Model predictive voltage control . . . . . . . . . . . . . . . . . . . 98

5.3.2 Current control and commutation . . . . . . . . . . . . . . . . . . 99

5.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6 Speed control of high-speed Switched Reluctance Machines using only

the DC-link measurements 105

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 SRM modeling and operation . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2.1 Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2.2 Electromagnetic properties . . . . . . . . . . . . . . . . . . . . . . 108

6.2.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.3 SRM position-sensorless control design . . . . . . . . . . . . . . . . . . . 109

6.3.1 Open-loop phase flux-linkage and current estimation . . . . . . . 109

6.3.2 Closed-loop speed and position estimation . . . . . . . . . . . . . 111

6.3.3 Speed control, current control and commutation . . . . . . . . . . 113


6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7 Auto-calibration of a generator-turbine throttle unit 119

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

viii Contents

7.2 Extremum-Seeking Control with disturbance-based optimal input

parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.2.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.2.2 Input error estimation . . . . . . . . . . . . . . . . . . . . . . . . 124

7.2.3 Input parameter estimation . . . . . . . . . . . . . . . . . . . . . 124

7.3 Application of proposed ESC scheme to generator-turbine throttle unit

auto-calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.3.1 GTU modeling and optimal operation . . . . . . . . . . . . . . . . 126

7.3.2 GTU speed control and mechanical power estimation . . . . . . . 128

7.3.3 ESC implementation . . . . . . . . . . . . . . . . . . . . . . . . . 129


7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8 Conclusions and recommendations 135

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.2 Recommendations for future research . . . . . . . . . . . . . . . . . . . . 138

A MCU case study: parameter specification 141

B GTU case study: parameter specification 145

C SRM advance angle scheduling signal 149

D SRG average DC-link current 151

Summary 169

Acknowledgements 171

Curriculum Vitae 173

Societal summary

Electrified engine air intake system: modeling, optimization and control

A major part of the current world energy supply comes from burning of fossil fuels. This

releases pollutants which have a negative impact on the environment. Consequently, the

countries around the world have introduced a series of evermore stringent regulations

promoting efficient fossil fuel use. Such trend is especially evident in the ground

transportation sector which continues to rely on fossil fuel-driven internal combustion

engines (ICEs). As a result, over the past few decades, the ICE has seen steady

improvements – both in terms of emissions and fuel-economy. One of the key facilitators

of the undergoing ICE evolution is the advent of powerful microprocessors enabling

the engine electrification and their more versatile control. In this context, this thesis

investigates the emerging electrification of the engine air intake system, i.e., its capacity

to further improve the ICE fuel-economy.

The conducted research focuses on two air intake electrification technologies: electric

supercharging and regenerative throttling. The first enables temporary engine over-

powering and thus creates a possibility to replace a larger engine with a smaller, more

efficient one. The second enables fuel-efficient electricity production using the energy

extracted from the engine intake airflow. The analysis of both of these technologies

has resulted in a range of modeling, optimization and control problems that have been

formulated and solved. This includes optimization problems supporting the design of

ICE powertrains with an electrified engine air intake system, derivation of several novel

control strategies for switched reluctance machines (being the enabling technology for

the two considered applications) and a development of an auto-calibration scheme for

regenerative throttling devices. The presented results show that the electrification of

the engine air intake system can help reduce the vehicle fuel consumption.

ix

x

Chapter 1

Introduction

Abstract In this Chapter a general introduction to the research presented in this thesis is

provided, the research objectives are formulated and the corresponding thesis contributions

are outlined and discussed.

1.1 General introduction

The present world population of 7.3 billion people is projected to increase by almost

one billion within the next twelve years – reaching as many as 9.6 billion by 2050 [1].

Almost all of the additional 2.3 billion people will enlarge the population of developing

countries whereas, in contrast, the population of more developed regions will experience

a minimal change [1]. As a consequence, the anticipated population increase will have

profound effects on the future political stability, food security and energy consumption

of the world as a whole [2].

A major part of the current world energy supply comes from fossil fuels: oil, coal and

natural gas. As a result, these non-renewable resources are being rapidly depleted.

Due to their increased consumption, driven by economic and/or population growth in

the dominating energy markets, it is estimated that the only fossil fuel remaining after

2042 will be coal [3]. On the other hand, it is also widely accepted that the burning

of fossil fuels releases pollutants which contribute to the global warming and overall

have a negative impact on the environment [4]. In general, a long-term solution to the

ensuing environmental problems is seen in the advancement of technology while it is

recognized that the success of this solution will strongly depend on our ability to reduce

1

2 1 Introduction

the influence of materialistic values1 on the society [5], [6].

In a quest for technology-based solutions, an increasing environmental awareness

and dwindling fossil fuel supplies continue to push the policymakers to promote the

renewable and clean energy sources, such as solar and wind, and to penalize the fossil

fuel use. The pressure towards more-efficient and less-polluting utilization of fossil fuels

is hardly anywhere more evident than in the road transport sector [7]. Over the past two

decades, in the European Union (EU) alone, the transport sector has been exposed to a

series of evermore stringent regulations regarding vehicle exhaust gas emissions (EURO

I to EURO VI). As a result, between 1990 and 2013, the emissions of transport-based

nitrogen oxides NOx, i.e., gases responsible for the formation of smog and acidic rain,

were reduced by 56% [8].

To meet ever-tightening expectations on emissions and fuel-economy, the automotive

industry has responded with a multitude of technological advancements in vehicle

mechanics, materials and manufacturing, along with an electrification of virtually every

vehicle component [9]. The electrification is not only represented by the development

of hybrid and fully electric vehicles, which currently comprise only a small fraction of

new car sales (1.4% in the EU in 2013 [10]), but also by an introduction of a variety

of electronic sensors and actuators to conventional internal combustion engine (ICE)

powertrains. The key facilitator of the electronic revolution, which is shaping the

ICE future, has been the advent of powerful microprocessors and the accompanying,

dedicated ICE control and optimization algorithms. Hence, similar to computers and

smartphones, the operation of majority of modern-day cars became governed by millions

of lines of computer code [11].

However, there are numerous scientific challenges lying ahead, which prevent the auto-

motive industry from fully exploiting the electrification potential to reduce the negative

impact of the ICE road transport on the environment. Examples of such challenges

include energy-efficient sizing and control of newly-introduced vehicle components, in

the presence of their rising complexity, multi-domain nonlinear dynamics and tight

physical constraints. Furthermore, powertrain calibration is often viewed as one of the

most expensive, time-consuming tasks in the vehicle development. For this reason, a

derivation of an accurate and automated calibration procedure, applicable to a varying

operating condition setting – remains a major scientific endeavor.

These and other issues impeding the improvement of the ICE vehicle fuel economy are

addressed in this thesis, with a focus on two emerging vehicle technologies: electric

1Refers to excessive concerns regarding personal comfort, wealth and material possessions.

1.1 General introduction 3

supercharging and regenerative throttling. These technologies belong to the domain

of the ICE air intake system electrification and are investigated using mathematical

modeling, numerical optimization and control design tools. An introduction to these

topics is given in Sections 1.1.1 and 1.1.2, whereas Section 1.1.3 provides a case for the

Switched Reluctance Machine as a suitable electric machine candidate for the selected

two applications.

1.1.1 Electric supercharging

The ICE supercharging (boosting) refers to the practice of increasing the pressure or

density of air supplied to the engine to provide it with more oxygen. This enables the

engine to burn more fuel and do more work. By facilitating engine overpowering, the

supercharging allows one to replace a larger engine with a smaller one while maintaining

or even enhancing the resulting engine torque/power output. The fact that the reduction

in engine size often leads to lower carbon emissions and a better fuel economy [12] has

led the automotive industry to pursue the path of engine downsizing. The downsizing

trend is evident in the case of all major vehicle manufacturers, especially in the last few

years, see Fig. 1.1.

2002 2004 2006 2008 2010 2012

1400

1600

1800

2000

2200

2400

Time [year]

Enginedisplacement[cm

3]

All brandsFiatRenaultFordPeugeot

CitroenOpel

VWAudiMercedes-BenzBMW

Figure 1.1: Engine displacement by passenger car brand [10].

4 1 Introduction

The engine supercharging (boosting) is achieved by using specialized air compressors,

called superchargers. Most commonly, these devices are powered mechanically, e.g.,

via chain or a belt directly coupled to the engine crankshaft, or by means of a turbine

propelled by the engine exhaust-gas flow. Compared to their more expensive, belt-

driven counterparts, the turbochargers have a considerably higher boosting efficiency,

partly because they exploit the exhaust-gas energy that would otherwise be wasted.

However, due to their reliance on the exhaust-gas flow, the turbochargers are known

to suffer from the “turbo-lag” and “acceleration-surge” phenomena. Specifically, at

low engine speeds these devices are (to a greater or a lesser extent) characterized by a

boost delay caused by the lack of sufficient exhaust-gas flow which limits the turbine

acceleration [13]. On the other hand, at higher engine speeds, the turbine is affected by

a “surge” of acceleration as a higher boost causes an even larger exhaust-gas production

– spinning the turbine ever faster. From the perspective of the driver, these effects are

often perceived in terms of a degraded accelerator pedal response.

One way to efficiently and cost-effectively eliminate engine boosting problems is to

electrify the turbocharger, i.e., augment or replace its turbine with a high-speed electric

motor. Owing to the electric machine’s ability to produce torque almost instantly,

the turbocharger electrification promises to deliver a far more responsive boosting

performance [14]–[18].

A standalone electric supercharger is obtained by directly coupling the motor to a com-

pressor, whereas the turbine is entirely omitted, see Fig. 1.2. Recent investigations have

shown that by placing such a device up or downstream of a conventional turbocharger

even the competing 2-stage-turbo configuration can be outperformed, especially during

the transient, vehicle acceleration intervals [19].

(a) Turbine-driven (Garrett©) (b) Belt-driven (Lysholm©) (c) Motor-driven (Valeo©)

Figure 1.2: Supercharger types.

A greater proliferation of (purely) standalone electric superchargers has so far been

hindered by a partial inadequacy of current vehicle electric systems to meet their

transient energy requirements. This is expected to change in the near future as more-


efficient, 48V architectures replace conventional, 12V vehicle electrical systems and as

low-cost, high-power-density batteries and (ultra-) capacitors appear on the market.

The prospective advantages of standalone, electric supercharging over its alternatives

can be summarized as:

1. Simplified packaging and installation.

2. Reduced manufacturing costs.

3. Instantaneous throttle response.

4. Programmable and efficient boosting.

The first two are a consequence of the electric supercharger’s lack of the engine

exhaust connection which eliminates the need for materials capable of withstanding

high exhaust-gas pressures and temperatures. The second two reflect its potential to

quickly, precisely and when necessary, deliver the air-mass required for the optimal fuel

combustion [20].

Due to these beneficial properties, electric supercharging is seen as a key enabler of the

engine downsizing in the near future. The main difficulty facing such a prospect lies

in the optimal sizing and utilization of the components constituting the electrically

supercharged ICE powertrain. Unlike the turbocharger, the electric supercharger

introduces both a coupling and a trade-off between the sizes of the ICE and the vehicle

electric components, such as a battery and an alternator. Clearly, the resolution of

this trade-off depends on an envisioned vehicle daily usage, which therefore has to be

accounted for as well when attempting to improve the vehicle fuel economy, via the

described ICE downsizing mechanism. This topic is treated in more detail as a part of

the research presented in this thesis.

1.1.2 Regenerative throttling

A throttle is a valve used to regulate the amount of air entering the gasoline ICE in

response to the driver’s accelerator pedal input. Thus, by assuming that a relatively

constant air-fuel ratio is maintained, the throttle actuation (throttling) also indirectly

determines the amount of fuel burned in each engine cycle. When the throttle is fully

open, the air in the engine intake manifold is at approximately ambient atmospheric

pressure. Otherwise, when it is partially closed, the (intake manifold) air pressure drops

below the ambient value, i.e., a partial vacuum develops.

6 1 Introduction

Negative work done by the engine to inhale and exhale gases is known as a pumping

(throttling) loss, which constitutes a significant portion of the total energy losses of

throttled gasoline ICEs, see Fig. 1.3. This is because in such engines, due to the

throttling, the piston is required to counteract the developed pressure differential

between the intake manifold and the engine crankcase in order to draw the air into

the cylinder.

Power loop

Pumping loop

VTDC VBDC

pAM

pIM

Volume

Pressure

Figure 1.3: PV diagram for a throttled gasoline ICE.

To minimize the pumping loss, an increasing number of modern-day drive-by-wire2

gasoline ICEs are designed to operate with a wide open throttle, under various load

conditions. This is made possible by a combination of different engine technologies

such as gasoline direct injection [21], exhaust gas recirculation [22] and variable valve

actuation [23], [24]. However, as in most cases dramatic changes to the engine design

become necessary, this raises concerns regarding the durability and cost-effectiveness of

such solutions.

Regenerative throttling represents an alternative and yet simple way to deal with the

engine pumping loss. Instead of trying to eliminate it, this approach actually exploits

the loss, i.e., the inherent pressure differential associated with it, to do useful work.

Regenerative throttling can be achieved by replacing the throttle valve with a turbo-

expander (turbine) equipped with variable stator vanes [25]. The turbine is used to

extract the potential energy from the intake airflow and convert it into kinetic energy

for driving an electric generator. In this way, electricity needed for powering a growing

number of vehicle electric auxiliary loads can be produced [26]. As an additional benefit,

the cool air mass exiting the turbine can be utilized to assist the vehicle air conditioning

2Refers to the use of an electrical, instead of a traditional, purely mechanical linkage between the

accelerator pedal and the throttle.


system via a dedicated heat-exchanger, see Fig. 1.4.

Figure 1.4: Regenerative throttling: Waste-Energy Driven Air-Conditioning

System concept [25].

Apart from the added cooling capacity, the advantage of using a regenerative throttling

device instead of, or in addition to, the conventional (Lundell [27], [28]) car alternator

is a potentially more efficient vehicle electricity production. The higher operational

efficiency stems from the fact that, unlike the alternator, a regenerative throttling device

(i.e., a generator-turbine throttle unit) is not connected to the engine’s crankshaft. This

offers the opportunity to independently control its speed such that it maximizes the

generator power output.

However, the computation of the optimal turbine speed is not at all trivial – the optimal

value varies with the conditions present in the ICE air intake system and, in addition,

depends on a range of turbine parameters. This creates a necessity for a specialized

turbine calibration procedure, which adds to the cost and limits the functionality of the

resulting device. Motivated by this problem, a part of this thesis is dedicated to the

research and development of methods for relieving the engineering systems, e.g., the

generator-turbine throttle unit, from excessive calibration requirements. In addition,

this thesis also presents the techniques for investigating the effect of the (calibrated)

regenerative throttling device on the vehicle fuel economy.

1.1.3 Switched Reluctance Machines

One of the prerequisites for a wider adoption of the electric supercharging and regener-

ative throttling technologies is the availability of an electric machine endowed with the

8 1 Introduction

following characteristics:

1. Low rotor inertia.

2. Low material costs.

3. High efficiency and power density.

4. Robust operation in a wide speed range.

The first property is critical to ensure a superior throttle response of the electric

supercharger, i.e., an accurate maximum power point tracking capability of the turbine-

driven generator. The second and third relate to the cost-effectiveness of these devices

whereas the fourth reflects an ever-present need for the vehicle durability and safety.

These properties can be found in the rotary, radial-field Switched Reluctance Machines

(SRM) [29].

The SRM is the earliest brushless motor known. It was first used as a locomotive traction

drive by Davidson in the mid-19th century, which employed mechanical switching of

currents from one phase to another (commutation) [30]. However, this motor technology

was quickly abandoned in favor of DC and later AC machines, as it resulted in a

pronounced torque pulsation. Its “reinvention” began in 1969 when the term switched

reluctance was first coined [31]. The SRM revival is primarily attributed to the advent of

inexpensive, high-power, electronic switching devices and programmable logic required

for their actuation [32].

The SRM operation is based on two physical principles. The first is that a magnetic field

causes a magnetic flux to follow the path of least magnetic reluctance. The second is that

the concentration of flux in low-reluctance materials, such as iron, causes mechanical

forces that tend to align them with the applied magnetic field (i.e., to move them

towards regions of higher flux). In the SRM, both a rotor and a stator are characterized

by salient (protruding) poles, i.e., regions of low magnetic reluctance. The windings,

however, are present only at the stator and are connected to form several electromagnets

(phases), see Fig. 1.5. As an outcome of such a design and due to the principles described

above, whenever the phase is energized the rotor poles will be pulled into alignment with

a corresponding stator pole pair. Therefore, the desired rotor movement can be simply

obtained by switching the phase excitation in an appropriate (sequential) manner.

Evidently, the SRM does not require permanent magnets for its operation. This

contributes to its low manufacturing cost as it renders it Rare-Earth-free. The

Rare-Earths represent a series of chemical elements found in the Earth’s crust (e.g.,


1

1

2 2

statorpole

rotorpole

statorwindings

+

Figure 1.5: Cross section of a 2-phase 4/2 SRM.

dysprosium and neodymium), which are vital to many modern technologies such as

consumer electronics, clean energy, health care, national defense and many others [33],

[34]. Since mid-1990s China has emerged as a dominant Rare-Earth supplier and as such

uses its near-monopoly to restrict their availability and dictate the price, see Fig. 1.6.

In the long run, this considerably adds to the attractiveness of the SRM, w.r.t. its

alternatives, e.g., Permanent Magnet Synchronous Machine (PMSM) [35].

Due to its lack of magnets and windings, the SRM rotor is highly mechanically robust,

small in size and has a low moment of inertia. At low speeds (e.g., below 10000rpm) the

power density of the SRM is comparable to that of the induction motor and somewhat

lower than the PMSM, whereas at higher speeds it is equivalent or even larger [29].

These and other favorable characteristics make the SRM well-suited to the vehicle

electrification purposes, including the electric supercharging and regenerative throttling

applications. In an effort to help harvest the full potential of these two technologies a

part of this thesis is devoted to the research on high-speed (low-cost) SRMs.

The “standard” SRM control problems concern the minimization of acoustic noise,

torque ripple, energy losses and the elimination of position/speed sensors. Furthermore,

as a consequence of its simple mechanical design and the switched reluctance principle

of operation, the SRM is also characterized by an inherently nonlinear switched system

dynamics, which renders a treatment of the “standard” issues challenging. For this

reason, a particular emphasis in this thesis has been put on the dynamical modeling,

optimization and control of the SRMs.

10 1 Introduction

1950 1960 1970 1980 1990 20000

10

20

30

40

50

60

70

80

90

Time [year]

Rare

EarthOxideproduction[kt]

Rest of the worldUSAChina

2008 2009 2010 2011 2012 2013 20140

500

1000

1500

2000

2500

Time [year]

%ofJanu

ary

2008price

GoldDysprosium

Neodymium

Figure 1.6: Left: global production of Rare Earth Oxides [36]. Right: the price

of dysprosium and neodymium compared with gold (source Bloomberg). Dashed

vertical line indicates a publishing date of the article titled “China tightens grip

on output of rare earths”, in Financial Times [37].

1.2 Research objectives

The main goal of this thesis is to deepen the understanding of the electric supercharging,

regenerative throttling and Switched Reluctance Machine technologies from the perspec-

tive of optimization and control, while taking into consideration recent developments in

these fields. In particular, this goal has been translated into the following main research

objectives:

1. Theoretical investigation of methods for evaluating the potential of engine down-

sizing via electric supercharging to improve the vehicle fuel economy.

2. Theoretical investigation of methods for evaluating the potential of regenerative

throttling to improve the vehicle fuel economy.

3. Theoretical investigation of methods for the Switched Reluctance Machine speed

and voltage control, which can alleviate the difficulties posed by the SRM nonlinear

dynamics, tight physical constraints and position/speed sensing requirements.

1.3 Contributions and outline 11

4. Theoretical investigation of methods for relieving the engineering systems, such as

the generator-turbine throttle unit, from excessive calibration requirements.

In response to these objectives, a number of research contributions has been made.

They are listed in the following Section.

1.3 Contributions and outline

This thesis contains six research chapters, i.e., Chapters 2-7. The first objective is

addressed in Chapter 2, the second in Chapter 3, the third in Chapters 4 to 6 and

the fourth in Chapter 7. A short summary of the main contributions of each of these

chapters is given below.

In Chapter 2, a method for sizing of an electrically supercharged ICE powertrain is

presented. The main contributions of Chapter 2 are:

• Detailed modeling of the electrically supercharged ICE vehicle powertrain. Apart

from the ICE, the model also includes a standalone electric supercharger used to

help the engine during short-duration high-power demands (and thus allows it to

be downsized), as well as an electric energy buffer which provides the supercharger

with sufficient electric energy/power to operate.

• Development of a computational method for finding the optimal ICE and buffer

size, for the case of the described vehicle powertrain. The sizing of both

components is performed by minimizing the sum of the vehicle operational (fuel)

and component (engine and buffer) costs. In general, the resulting optimization

problem constitutes a non-convex, nonlinear and a mixed-integer dynamic pro-

gram, where the ICE and buffer are optimally sized only when the vehicle is

also optimally controlled (on a studied driving cycle). The problem is handled

by first decoupling the integer decisions, i.e., the gear selection strategy (decided

heuristically), and then by formulating the remaining problem as a convex second-

order cone program, which can be solved efficiently with the help of dedicated

numerical tools.

• A representative, simulation-based case study is performed providing a solution of

the problem defined above, where the optimal engine and the electric buffer sizes

are computed for a specific, electrically supercharged ICE vehicle. The results

show that the (standalone) electric supercharging can support considerable ICE

12 1 Introduction

downsizing – yielding up to 10% savings in fuel costs over a specific driving cycle,

w.r.t. a baseline, naturally-aspirated engine scenario.

In Chapter 3, a regenerative throttling potential to improve the gasoline ICE vehicle

fuel economy is investigated. The main contributions of Chapter 3 are:

• Detailed modeling of the vehicle powertrain equipped with a generator-turbine

throttle unit, which is used to complement the car alternator while powering the

vehicle electric auxiliaries.

• Development of a computational method for finding the optimal buffer size, for

the case of the described vehicle powertrain. In particular, the minimization of the

sum of the vehicle operational (fuel) and component (buffer) costs is considered.

The resulting optimization problem is casted into a semi-definite convex program

using a series of convex (model) relaxation steps, whereas a transmission gear, an

integer variable, is decided outside the convex optimization.

• A representative, simulation-based case study is performed that provides a solu-

tion of the problem defined above for several different driving cycles and engine

sizes. The presented results show that the use of the generator-turbine throttle

unit can reduce the total operational (fuel) and component (buffer) costs by

typically 2-4% or even more than 4% in selected cases, depending on factors such

as the engine size and the choice of a driving cycle.

In Chapter 4, a method for four-quadrant speed control of 4/2 Switched Reluctance

Machines is presented. The main contributions of Chapter 4 are:

• Development of a model-based, open-loop average torque control scheme in the

form of a nonlinear mapping between the 4/2 SRM operating point (defined by

the desired average torque, rotational speed and the applied DC-link voltage) and

a set of low-level current reference parameters.

• Development of a four-quadrant 4/2 SRM speed controller that can cope with the

identified, speed and voltage-dependent, average torque bounds.

• Development of a supervisory control algorithm for supporting the startup and

change of rotational direction of 4/2 SRMs.

In Chapter 5, a method for Model Predictive voltage Control (MPC) of high-speed

Switched Reluctance Generators (SRG) is developed. The main contributions of

Chapter 5 are:

1.3 Contributions and outline 13

• Development of a linear, explicit MPC law that enforces the desired SRG average

DC-link current and voltage bounds and enables tracking of the specified DC-link

voltage reference, in the presence of an unknown electrical load.

• Development of a model-based, parametric SRG commutation strategy based on

its measured electromagnetic characteristics. In this context, the commutation

rules are parametrized by the desired average DC-link current, rotational speed

and the DC-link voltage.

In Chapter 6, a method for speed control of high-speed Switched Reluctance Ma-

chines, using only the DC-link measurements, is presented. The main contribution of

Chapter 6 is:

• Development of a novel position-sensorless speed control strategy for high-speed

Switched Reluctance Machines. Its key component is an algorithm for rotor po-

sition and speed estimation using the DC-link voltage and current measurements

only. This algorithm eliminates a need for a number of hardware components

related to position, speed, phase current and phase voltage sensing. It thus allows

the SRM electric system’s costs to be lowered and its reliability increased.

In Chapter 7, a method for auto-calibration of the generator-turbine throttle unit is

described. The main contributions of Chapter 7 are:

• Development of a novel, parametric Extremum-Seeking Control (ESC) algorithm,

with a disturbance-based optimal input parametrization, which is suitable for

tracking an unknown, time-varying extremum. The proposed ESC scheme is

applicable to situations where the disturbances leading to changes in the optimal

input are known/measurable. Its main advantage over other ESC approaches is

that it identifies a mapping between the disturbances and optimal inputs. Once

found, the constructed mapping can be directly employed in real-time, without

the need for further extremum-seeking.

• Application of the developed ESC algorithm for purposes of the generator-turbine

throttle unit auto-calibration. In this context, the presented solution is used to

find an unknown relationship between the disturbances (turbine pressure ratio

and vanes position) and the optimal input (turbine reference speed), in an initial,

automated calibration step.

Finally, in Chapter 8, conclusions are drawn and recommendations for future research

are presented.

14 1 Introduction

1.4 Interconnections between topics of research

Chapters 2 and 3 employ a similar powertrain modeling and optimization methodology.

Namely, in these chapters, the common vehicle powertrain components, such as wheels,

brakes, gearbox and electric buffer, are modeled in the same way. Also, the formulation

of the underlying optimization problems and related cost functions is, in certain aspects,

shared as well (e.g., the electric buffer and fuel costs are considered in both cases).

However, as these chapters treat the two different air intake system electrification

technologies, the crucial differences between them are in the model and functionality

of the related air intake devices. This also results in distinct fuel-saving mechanisms

(engine downsizing via electric supercharging vs. regenerative throttling) and air intake

manifold conditions (above vs. below ambient pressure). Moreover, even though both

applications implicitly assume the use of high-speed electric machines, Chapter 2

explicitly refers to their motoring and Chapter 3 to their generating operation, i.e.,

discharging and charging of the electric buffer.

The 4/2 SRM four-quadrant controller, proposed in Chapter 4, can be used for both

electric supercharging and regenerative throttling purposes. This is because it allows

the 4/2 SRM to operate in either direction of rotation, both as a motor and as a

generator. Hence, the developed algorithm could be even applied to control the speed

of a hypothetical, hybrid device, which would combine the properties of the standalone

electric supercharger and generator-turbine throttle unit. The practicality of such a

device, however, remains to be researched. In contrast, the SRM controllers, discussed

in Chapters 5 and 6, are intended for unidirectional applications only. These and other

key characteristics of the presented SRM algorithms are summarized in Table 1.1.

As mentioned in Section 1.1.3, the “standard” SRM control problems concern the

minimization of acoustic noise, torque ripple, energy losses and the elimination of posi-

tion/speed sensors. The SRM energy loss problem is, at least indirectly, handled in both

Chapter 4 and 5, by means of an optimized phase current reference parametrization.

Clearly, position-sensorless control is treated in Chapter 6, whereas the reduction of

acoustic noise/torque ripple has not been explicitly addressed in this thesis.

The role of the auto-calibration algorithm, derived in Chapter 7, is to maximize the

energy recovery potential of a given regenerative throttling device. To fulfill this role,

an accurate tracking of a computed turbine speed reference is required. For this purpose,

the controllers derived in Chapter 4 and 6 can be employed, assuming the generator-

turbine throttle unit utilizes the (4/2) SRM technology.

1.5 List of publications 15

Table 1.1: Comparison of proposed SRM controllers

Chapter SRM controller Operation Speed range Sensors

Chapter 4 speed mot/genneg/pos, DC-link voltage,

low/high position, phase currents

Chapter 5 DC-link voltage genpos, DC-link voltage/current

high position, speed

Chapter 6 speed mot/genpos,

DC-link voltage/currenthigh

1.5 List of publications

Journal articles

• S. Marinkov, N. Murgovski and B. de Jager. “Convex modeling and sizing of

electrically supercharged internal combustion engine powertrain.” accepted for

publication, IEEE Transactions on Vehicular Technology. – Chapter 2

• S. Marinkov, N. Murgovski and B. de Jager. “Convex modeling and optimization

of a vehicle powertrain equipped with a generator-turbine throttle unit.” submit-

ted, under review. – Chapter 3

• S. Marinkov and B. de Jager. “Four-Quadrant speed control of 4/2 Switched

Reluctance Machines.” submitted, under review. – Chapter 4

Proceedings & Conference Contributions

• S. Marinkov, B. de Jager, and M. Steinbuch, “Model predictive control of a high

speed switched reluctance generator system,” in European Control Conference,

Zurich, Switzerland, 2013. – Chapter 5

• S. Marinkov, B. de Jager, and M. Steinbuch, “Extremum seeking control with

data-based disturbance feedforward,” in American Control Conference, Portland

OR, USA, 2014. – Chapter 7 (in part)

• S. Marinkov, B. de Jager, and M. Steinbuch, “Extremum seeking control with

adaptive disturbance feedforward,” in The 19th IFAC World Congress, Cape

Town, South Africa, 2014. – Chapter 7 (in part)

16 1 Introduction

• S. Marinkov and B. de Jager, “Control of a high-speed Switched Reluctance

Machine using only the DC-link measurements,” in IEEE International Conference

on Industrial Technology, Seville, Spain, 2015. – Chapter 6

• N. Murgovski, S. Marinkov, D. Hilgersom, B. de Jager, M. Steinbuch, and J.

Sjoberg, “Powertrain Sizing of Electrically Supercharged Internal Combustion

Engine Vehicles,” in The 4th IFAC Workshop on Engine and Powertrain Control,

Simulation and Modeling, Columbus OH, USA, 2015.

Supervised projects

• E. Hoedemaekers, “SRG vehicle charging system: design and implementation of

the test setup”, BSc thesis, Eindhoven University of Technology, February-May

2012.

• E. Stamatopoulos, “Observer-based control of a high-speed Switched Reluctance

Machine”, MSc thesis, Eindhoven University of Technology, January-August 2014.

• N. Strous, “Modeling and measurement of the acoustic noise produced by a

Switched Reluctance Machine”, BSc thesis, Eindhoven University of Technology,

September 2014-January 2015.

• D. Hilgersom, “Potential of an Add-On Electric Supercharger for Internal Combus-

tion Engines”, MSc thesis, Eindhoven University of Technology, February 2014-

August 2015.

• D. Shanbhag, “Extremum Seeking Control tuning of Switched Reluctance Ma-

chines”, research project, Eindhoven University of Technology and University of

Queensland, Australia, March-June 2015.

Chapter 2

Convex modeling and sizing of

electrically supercharged internal

combustion engine powertrains

Abstract This Chapter investigates a concept of an electrically supercharged internal

combustion engine powertrain. A supercharger consists of an electric motor and a compressor.

It draws its power from an electric energy buffer (e.g., a battery) and helps the engine during

short-duration high-power demands. Both the engine and the buffer are sized to reduce the

sum of the vehicle operational (fuel) and component (engine and buffer) costs. For this

purpose, a convex, driving cycle-based vehicle model is derived, enabling the formulation of

an underlying optimization problem as a second-order cone program. Such a program can be

efficiently solved using dedicated numerical tools (for a given gear selection strategy), which

provides not only the optimal engine/buffer sizes but also the optimal vehicle control and state

trajectories (e.g., compressor power and buffer energy). Finally, the results obtained from a

representative, numerical case study are discussed in detail.

2.1 Introduction

Recent years have shown high interest in the reduction of energy consumption and

pollutant emissions of ground transportation. With the goal of improving energy

efficiency and employing renewable energy sources, vehicle manufacturers are currently

introducing several types of electrified vehicles. Nevertheless, internal combustion

engines (ICE) are expected to remain the dominant force in the automotive market

for the next decade [38].

17

182 Convex modeling and sizing of electrically supercharged internal


To meet the ever-tightening expectations on the vehicle fuel economy, the automotive

industry has pursued the path of engine downsizing [12]. The engine downsizing has

been typically followed by the ICE overpowering, e.g., by means of torque boosting [39],

[40], to improve the vehicle drivability. In general, the application of the ICE downsizing

and overpowering results in lower carbon emissions and a better fuel economy, w.r.t.

the original, large engine situation – due to the reductions in the engine weight, friction

and pumping losses [19]. The ICE overpowering can be also achieved using intake air

boosting, with the help of a turbocharger (driven by hot exhaust gases) or a supercharger

(driven mechanically by a crankshaft via a chain or a belt). In both cases, a compressor

is utilized to increase (boost) the pressure/density of air supplied to the engine and thus

provide it with more oxygen. This allows more fuel to be injected and burned, thereby

rising the ICE maximum torque and power limits.

However, the turbocharged ICEs exhibit a relatively poor torque capability at low engine

speeds, which compromises the vehicle drivability and acceleration performance [41].

Namely, at low speed, the downsized ICEs suffer from insufficient exhaust gas-flow

to adequately propel the turbocharger from the moment the gas pedal is pressed,

resulting in a well-known turbo-lag [12]. The belt-driven supercharger, on the other

hand, does not experience this phenomenon but is less fuel economic, as it increases

the engine parasitic losses. One way to efficiently provide the required low-end torque

and at the same time eliminate the turbo-lag is to electrify the supercharger, i.e., to

replace its mechanical power source (prime mover) with an electric motor [16]–[18]. The

resulting device, an electric supercharger, i.e., a motor-compressor unit (MCU), follows

a popular automotive trend of vehicle electrification – which has already proven capable

of enhancing the efficiency and performance of numerous systems such as steering, water

pump and air conditioning [9].

Historically, a lack of compact, high-power/energy-density electric sources and of light-

weight, high-speed, high-power-density electric motors prohibited the proliferation of

the MCUs throughout the automotive sector. The widely used 12 V battery system is

at the limit of providing sufficient power for the electrical boost [40]. Besides, the high

power surges from the MCU may incur high battery losses.

Today the situation regarding electric storage elements is somewhat different as a

plethora of high-power batteries and high-energy capacitors has appeared on the market.

However, the choice of the electric buffer technology and optimal buffer size, in terms

of its power rating and energy density, is still an open question.

This Chapter presents a method for computing the buffer size that provides sufficient

electric power and energy to run the supercharger. The supercharger is used to help

2.2 The powertrain sizing problem 19

the engine during short-duration high-power demands, which could potentially allow it

to be downsized. Specifically, the sizing of both the ICE and the buffer is performed

by minimizing the sum of the vehicle operational (fuel) and component (engine and

buffer) costs. This optimization problem constitutes a dynamic program, where the

ICE and buffer are optimally sized only when the vehicle is also optimally controlled on

a studied driving cycle. In addition, this problem is also a non-convex, nonlinear and

a mixed-integer dynamic program, where both plant design and control parameters act

as optimization variables.

The plant design and control problem is typically handled by decoupling the plant and

the controller, and then optimizing them sequentially or iteratively [42]–[47]. However,

sequential and iterative strategies generally fail to achieve global optimality [48]. An

alternative is a nested optimization strategy, where an outer loop optimizes the system

objectives over a set of feasible plants, and an inner loop generates optimal controls

for plants chosen by the outer loop [45]. This approach delivers a globally optimal

solution, but may incur heavy computational burden (when, e.g., dynamic programming

is used to optimize the energy management [49]), or may require substantial modeling

approximations [50]–[52].

This Chapter addresses the plant design and control problem by first decoupling the

integer decisions, i.e., the gear selection strategy, and then by formulating the remaining

problem as a convex second-order cone program (SOCP) [53]. The integer signals are

decided outside of the convex program, by means of two simple heuristic strategies –

one designed to promote the ICE downsizing and one that aims to maximize the ICE

efficiency. Finally, a case study is provided where the optimal engine and the electric

buffer sizes are computed for a specific, MCU-equipped vehicle.

This Chapter is organized as follows. Section 2.2 provides background to the electrically

supercharged ICE configuration and states a verbal problem formulation. The math-

ematical modeling is provided in Section 2.3 and the convex optimization problem is

formulated in Section 2.4. Section 2.5 presents a use-case study. Conclusions are drawn

in Section 2.6.

2.2 The powertrain sizing problem

The block diagram of the electrically supercharged ICE is illustrated in Fig. 2.1. The

MCU, which is placed in the ICE air intake along with a bypass valve, enables more

power to be delivered by the ICE, e.g., while overtaking or when starting-off at the



BV TV

A

MC

IM EM

ICEMCU

B

AUe

G

AUm

W

BRE. link

electrical

mechanicalpneumatic

Figure 2.1: Illustration of the components considered in the proposed power-

train sizing problem – arrows indicate a power flow direction. The source of air

flow feeding an internal combustion engine (ICE) intake (IM) and exhaust (EM)

manifolds is determined using a bypass (BV) and a throttle valve (TV). The

ICE is equipped with a stand-alone motor-compressor unit (MCU) consisting of

a compressor (C) and an electric motor (M). The motor, as well as other electric

auxiliary loads (AUe), draws its power from an electric buffer (B). The buffer

is charged by a conventional car-alternator (A) that is mechanically coupled to

the ICE crankshaft along with mechanical auxiliary loads (AUm). A clutch and

a gearbox (G) connect the ICE with wheels (W) and brakes (BR).

traffic lights. The supercharging (SC) refers to a situation when the excess power is

needed, i.e., when the bypass valve is closed. In contrast, during naturally-aspirated

(NA) operation the bypass valve is open.

The bursts of mechanical MCU power have to be matched by the power ratings of

the electric buffer that feeds the MCU. However, deciding the optimal buffer energy

requirement is not trivial since it depends on the typical daily usage of the vehicle. A

common way of representing the vehicle daily usage is by recording the vehicle speed

and acceleration time profiles, and then by constructing a driving cycle that contains

both the vehicle speed and road topography as functions of time. An example of one

such cycle is the Class 3 World Harmonized Light Vehicle Test Procedure1 (WLTP3),

which is used here as a proof of concept for realization of the method being proposed.

The vehicle is required to exactly follow the speed demanded by the driving cycle,

thus ensuring that a possible downsizing of the powertrain does not compromise the

demanded performance. To have a fair comparison, the buffer is required to sustain

its initial charge at the end of the driving cycle, meaning that any energy used for

supercharging has to be put back in the buffer at some point, through the use of a

conventional car alternator driven by the ICE. This may require high utilization of the

electric buffer, making it beneficial to increase its size. However, a larger buffer increases

1http://www.dieselnet.com/standards/cycles, March 2015.

2.3 Quasistatic vehicle model 21

Table 2.1: Optimization problem for powertrain components sizing and energy

management.

Minimize:

Operational + component cost,

Subject to:

Driving cycle constraints,

Energy conversion and balance constraints,

Buffer dynamics,

Physical limits of components,

...

(For all time instances along the driving cycle).

the cost of the vehicle. Then, to keep the cost down, the possibility of downsizing

the ICE is also considered, such that the optimal trade-off is reached between the

components cost and the operational cost within the lifetime of the vehicle.

The resulting optimization problem is verbally stated in Table 2.1, whereas its mathe-

matical description is deferred to Section 2.4.

2.3 Quasistatic vehicle model

In the remainder, a power-based [54], quasistatic model of a 4-stroke ICE is provided.

The ICE is downsized by scaling its displacement volume while keeping its bore-stroke

and compression ratios constant. Specifically, the ICE displacement volume is defined

as VE = sEVE, where VE denotes the baseline volume and sE ∈ (0, 1] the engine scaling

coefficient. The ICE is equipped with the MCU, which provides the possibility to

enhance its torque capacity by means of supercharging. To match the MCU power

requirements the vehicle electric energy buffer is sized as well. For this purpose, the

buffer is considered to be built out of nB = sBnB cells connected in series, where sB > 0

represents the buffer scaling coefficient and nB the baseline cell count. Both sE and sB

are treated as real optimization variables.



PmW

Vehicle

sE sB

ωW

PmG2

ωG

PmG1

ICEωE

PmE

ωA PmA

sE

P cE

Gearbox

sE

P eA

systemAir

ωC PmC

P eC

sBP eB2

EeB

P eAU

derived

optimized

predetermined

sE

g

bufferElectric

Motor

P eB1

Ambient

pAM TAM

Brakes

ωW PmBR

Electricaux.

Mech.aux.

ωE PmAU

ωE

α

aV

vV

Wheels

cycleDrive Alter.

& p.e. & p.e.

ωW

Electriclink

Mech.link

Figure 2.2: Quasistatic model of an ICE equipped with a stand-alone MCU –

arrows indicate component inputs and outputs.

2.3.1 Vehicle

The vehicle is modeled as a system with a point mass

mV = mEsE + mBsB + mV, (2.1)

where mE denotes the baseline engine mass, mB the baseline buffer mass and mV the

baseline vehicle mass (excluding mE and mB). The mass mB is further defined as

mB = mcnB, with mc being the baseline buffer cell mass.

If the driving cycle provides the demanded vehicle speed vV ≥ 0, the acceleration aV

and the road slope α, and assuming that the driving occurs in still air, the vehicle wheel

speed ωW and the power at the wheels PmW can be computed as

ωW =vV

rW

, (2.2)

PmW = aVλVmVvV + ag sin(α)mVvV + agcr cos(α)mVvV (2.3)

+1

2ρAMcdAfv

3V.

The terms on the righthand side of (2.3) respectively represent the inertial driving,

driving on a slope, rolling resistance and aerodynamic drag power components. Here,

rW denotes the wheel radius, λV the equivalent vehicle mass ratio, ag the gravitational


acceleration, cr the rolling resistance coefficient, ρAM the ambient air density, cd the

drag coefficient and Af the vehicle’s frontal area. Due to (2.1) the power PmW is affine

in the optimization variables, i.e.,

PmW = γW0 + γW1sE + γW2sB, (2.4)

where

γW2 = (aVλV + ag sin(α) + agcr cos(α)) mBvV, (2.5)

γW1 = (aVλV + ag sin(α) + agcr cos(α)) mEvV,

γW0 =1

2ρAMcdAfv

3V + (aVλV + ag sin(α) + agcr cos(α)) mVvV.

2.3.2 Wheels & brakes

The required mechanical power at the wheel side of the gearbox is given by

PmG2 = Pm

W + PmBR, (2.6)

where PmBR ≥ 0 is an optimization variable representing the power of the brakes.

2.3.3 Gearbox

The speed and power at the engine side of the gearbox are given by the following

expressions

ωG = λG(g)ωW, (2.7)

PmG1 = γG0 + Pm

G2, (2.8)

where

γG0 = cG0 + cG1ωG + cG2ω2G, (2.9)

with λG(g) denoting the gear ratio corresponding to the gear g ∈ 1, . . . , 5 and γG0 ≥0 the gearbox drag losses. The gear is treated as an optimization variable decided

separately and prior to the rest, see Section 2.4.2.

2.3.4 Mechanical power link

To be able to distinguish between regular and idling engine operation, define

γML =

1, for ωG ≥ ωE,min,

0, otherwise.(2.10)



Then the engine and the alternator speed exiting the mechanical power link follow

ωE = ωGγML + ωE,min(1− γML), (2.11)

ωA = λAωE, (2.12)

while the corresponding engine mechanical power balance reads

PmE = Pm

G1γML + PmA + Pm

AU. (2.13)

Here, λA denotes the alternator speed ratio, ωE and ωE,min the ICE rotational and idle

speed, ωA the alternator speed, and PmE , P

mA , P

mAU ≥ 0 the mechanical power of the

ICE, alternator and the remaining mechanical auxiliary units (assumed constant). The

power PmA is one of the optimization variables.

2.3.5 ICE & air-fuel control

The ICE mechanical behavior can be described with the following relationship [55], [56]:

pE = ηEpφ − pEf − pEg, (2.14)

where ηE denotes the effective ICE efficiency, pE and pφ the engine brake and fuel mean

effective pressures, and pEf and pEg the brake mean effective pressure losses due to engine

friction and pumping work. Assuming that the ratio λec, the ignition/injection timing

angles, the burnt gas fraction and the bore-stroke ratio are kept constant, the effective

efficiency ηE can be treated as a function of the engine speed only [55], i.e.,

ηE = cη0 + cη1ωE + cη2ω2E, (2.15)

where cηi, i ∈ 0, 1, 2 denote constant parameters. The two pressures, pE and pφ,

further read

pE =4πτE

VEsE

(2.16)

pφ =4πP c

E

VEωEsE

, (2.17)

while the related loss components may be modeled2 [55] as

pEf = cf0 + cf2ω2E, (2.18)

pEg = cg0 + cg1τE

τE,max

, (2.19)

2The adopted friction loss model is pessimistic as lower friction losses can be expected in the case

of downsized engines.


with τE,max = τE,max(ωE, sE) denoting the maximum achievable torque τE of a downsized

ICE during its NA operation, cf0, cf2, cg0 and cg1 some constant parameters. The ICE

chemical (fuel) power P cE is given by

P cE = mφHl, (2.20)

with Hl being the fuel lower heating value and mφ the ICE fuel mass flow. Assuming

that the fuel controller maintains a constant, stoichiometric air-fuel ratio λαφ, the flow

mφ is related to the required engine air mass flow as

mφ =mα

λαφ. (2.21)

Since from the air system perspective the engine acts as a volumetric pump, the air

mass flow fulfills

mα = ηvolpAMsEVEωEλΠ

4πcRTIM

, (2.22)

where λΠ represents the ratio between the intake manifold and the ambient air pressures,

pIM and pAM, satisfying 0 ≤ λΠ ≤ λΠ,max > 1, with λΠ,max being its stoichiometric

combustion knock limit value. Furthermore, cR is the specific gas constant of air,

TIM the intake manifold air temperature and ηvol = ηvol(ωE, λΠ) the engine volumetric

efficiency. The efficiency ηvol is frequently modeled as a multilinear function [55] of the

pressure ratio λΠ and the speed ωE, i.e.,

ηvol = ηvol,ωηvol,Π, (2.23)

where

ηvol,ω = cvol0 + cvol1ωE + cvol2ω2E, (2.24)

ηvol,Π = 1 +1

λec

(1−

(pEM

pAM

)1/λκ

λ−1/λκΠ

),

with pEM being the (constant) exhaust manifold pressure, λec the engine compression

ratio, λκ the specific heat ratio of air and cvoli, i ∈ 0, 1, 2 some constant parameters.

Using (2.16) to (2.21) the expression (2.14) can be rewritten in terms of torque, as

follows

τE = ηEmαHl

λαφωE

− VEsE

4π

(pEf + cg0 + cg1

τE

τE,max

). (2.25)

Since during the ICE NA operation at a wide-open throttle it holds τE = τE,max(ωE, sE),

λΠ ≈ 1 and TIM ≈ TAM, where TAM denotes the ambient air temperature – by



substituting these values and the flow given by (2.22) in (2.25), the expression for

torque τE,max can be derived. This yields

τE,max =

(ηEηvol

pAMVEHl

4πcRTAMλαφ− VE(pEf + cg0 + cg1)

4π

)sE, (2.26)

= τE,maxsE. (2.27)

where τE,max = τE,max(ωE) denotes the maximum achievable torque of a baseline (non-

downsized) NA ICE and ηvol = ηvol(ωE, 1). By multiplying both sides of (2.25) with ωE

and using (2.27), one can further obtain the relationship between the ICE mechanical

PmE = τEωE and the fuel power P c

E, i.e.,

P cE =

4πτE,max + cg1VE

4πτE,maxηE

PmE +

VEωE(pEf + cg0)

4πηE

sE, (2.28)

P cE = γE1P

mE + γE,minsE, (2.29)

where γE1 > 1 and γE,min > 0. The power P cE is treated as an optimization variable.

2.3.6 Air system

During the ICE SC operation it holds τE > τE,max(ωE, sE), when pIM ≈ pC > pAM and

TIM ≈ TC > TAM, where pC, TC denote the compressor outlet pressure and temperature.

Thus one may approximate the compressor pressure ratio pC/pAM by the intake manifold

pressure ratio λΠ and express the temperature TC, as

TC = TAM +TAM

ηC

(λλκ−1λκ

Π − 1

), (2.30)

where ηC is the isentropic compressor efficiency (assumed constant). The equa-

tions (2.22) and (2.30) uniquely determine the values of TC and λΠ for each given mα

and ωE. Although the underlying relations are nonlinear in λΠ, in its narrow range of

interest, i.e., for λΠ ∈ [1, λΠ,max], they can be approximated by suitable affine functions.

To achieve this, the expressions (2.22) and (2.30) are first rewritten as

TC = λTTAM, (2.31)

mα = λmγmsE,

where λT = 1 + 1ηC

(λλκ−1λκ

Π − 1

), λm =

ηvol,ΠλΠ

λTand γm =

pAMVEωEηvol,ω

4πcRTAM. Then, the best

affine fits λT ≈ λT and λm ≈ λm (w.r.t. λΠ, in the least-squares sense) are constructed


1 1.2 1.4 1.60

0.5

1

1.5

λΠ [-]

λ[-]

λT

λT

λm

λm

λΠ,min,max

Figure 2.3: Affine approximation of compressor temperature TC and engine air

mass flow mα dependence on the intake manifold pressure ratio λΠ.

while enforcing λT (1) = λT (1) and λm(1) = λm(1), to preserve continuity. This yields

λT = cT0 + cT1λΠ, (2.32)

λm = cm0 + cm1λΠ, (2.33)

where cT i > 0 and cmi > 0, i ∈ 0, 1 are constant parameters, see Fig. 2.3.

From (2.31), (2.33), (2.21) and (2.20) the pressure ratio limit λΠ ≤ λΠ,max can be

expressed as the engine size-dependent limit on the fuel power P cE. Specifically, it

follows

P cE ≤

γmHl(cm0 + cm1λΠ,max)

λαφsE = γE,maxsE. (2.34)

Furthermore, the mechanical MCU power PmC ≤ cC,max, is given by

PmC = max (mαcp(TC − TAM), 0) , (2.35)

where cp denotes the specific heat of air at constant pressure and cC,max the maxi-

mum compressor mechanical power (assumed constant). Using (2.31), (2.32), (2.21)

and (2.20), the power PmC can be computed as

PmC = max

(cC1P

cE + γC2

P c2E

sE

, 0

), (2.36)

with coefficients

cC1 =λαφcpTAM(cT0cm1 − cT1cm0 − cm1)

Hlcm1

, (2.37)

γC2 =λ2αφcpTAMcT1

H2l cm1γm

.



Since γC2 > 0, ∀ωE, the expression γC2Pc2E + cC1P

cE is convex in P c

E. Moreover, as

sE > 0, a perspective function [53] corresponding to this expression is given by the

first argument of the maximum function in (2.36), which implies that this argument

is convex in both P cE and sE. Because the maximum of two convex functions is itself

convex, the same also holds for the compressor power PmC .

Furthermore, it is assumed that for the air mass flow/pressure ratio range of interest

the MCU can be always chosen such that the compressor surge/choke phenomena do

not occur. The surge condition can be expressed as [55]:

ωC ≤ ωC,surge ≈ csurge0 + csurge1mα, (2.38)

where ωC denotes the compressor speed, ωC,surge(mα) the compressor surge speed limit

(for a given air mass flow) and csurge0, csurge1 > 0 some constant fitting coefficients.

2.3.7 Alternator and MCU motor

The electric machine electric power can be modeled as a second-order polynomial of its

mechanical power, with a constant term representing a speed-dependent drag loss [56].

In this context, the alternator electric power can be formulated as

P eA = γA0 + cA1P

mA + cA2P

m2A , (2.39)

where

γA0 = cA00 + cA01ωA + cA02ω2A > 0, (2.40)

with cA0i, i ∈ 0, 1, 2, cA02 ≥ 0, −1 < cA1 < 0 and cA2 ≥ 0 being constant coefficients,

see Fig. 2.5. The signs of cA1 and cA2 ensure that the electric alternator power P eA is

non-positive for a non-negative mechanical alternator power PmA . This is in accordance

with a general convention (in the energy management of electrified vehicles) which

states that the electric power should be positive when the electric machine acts to

discharge the electric buffer (i.e., during motoring) and negative when it charges it (i.e.,

during generating). Furthermore, the coefficient cA1 is bounded to reflect the physical

limitations of the alternator. Namely, if one would let cA1 >= 0 this would imply that

the alternator can sometimes operate as a motor, which is not considered in this study.

If, however, cA1 <= −1, then it could happen that the alternator electrical power output

is absolutely larger than its mechanical power input, which is not physically possible.

The mechanical power PmA is constrained both above and below, i.e., 0 ≤ γA,min ≤

PmA ≤ cA,max, where

γA,min =−cA1 −

√c2

A1 − 4γA0cA2

2cA2

(2.41)


2 4 6

0.5

1

1.5

2

2.5

nE [krpm]

Pm C[kW]

BL & NADS & NADS & SCideal

2 4 6

50

60

70

80

90

100

nE [krpm]

τE[N

m]

2 4 60

50

100

nE [krpm]

Pc E[kW]

0 20 400

50

100

1500

4000

4000

6000

6000

P mE [kW]

Pc E[kW]

Figure 2.4: Top left: the constrained compressor power PmC needed to achieve

the maximum pressure ratio λΠ,max = 1.58. Top right: the maximum ICE

torque τE of the baseline NA, the downsized NA and the downsized SC ICE

(with sE = 0.8, λΠ,max = 1.58). Bottom left: the corresponding maximum ICE

chemical power P cE. Bottom right: the ICE chemical power P c

E vs. the ICE

mechanical power PmE . The speed nE represents the ICE speed ωE expressed in

(thousands of) rotations per minute.

and cA,max is a constant parameter.

Similarly to the alternator, the MCU electric power P eC can be modeled as

P eC = Pm

C0 + cM1PmC + cM2P

m2C , (2.42)

where the speed-dependent motor drag loss is given by

PmC0 = cM00 + cM01ωC + cM02ω

2C > 0, (2.43)

with cM0i, i ∈ 0, 1, 2, cM02 ≥ 0, cM1 > 1 and cM2 ≥ 0 being constant coefficients [56].

The motor drag loss PmC0 can be upper-bounded by setting ωC = ωC,surge, which

from (2.20), (2.21) and (2.38) yields

PmC0 = cM00 + cM01P

cE + cM02P

c2E , (2.44)



with the coefficients

cM00 = cM00 + cM01csurge0 + cM02c2surge0, (2.45)

cM01 =λαφHl

(cM01csurge1 + 2cM02csurge0csurge1) ,

cM02 =λ2αφ

H2l

cM02c2surge1.

Note that since cM02 ≥ 0 the drag loss PmC0 is convex w.r.t. the fuel power P c

E. In

the same way, the compressor electric power P eC is convex w.r.t. both its mechanical

power PmC and fuel power P c

E, and the alternator electric power P eA is convex w.r.t. its

mechanical power PmA .

0 0.5 1 1.5 2

−1.5

−1

−0.5

0

2250

2250

6000

6000

9000

9000

PmA [kW]

Pe A[kW]

P eA

P eA

ideal

Figure 2.5: Convex approximation of the alternator electric power P eA as a

function of its mechanical power PmA and speed ωA.

2.3.8 Electric power link

The electric buffer terminal power follows from the power balance at the electric power

link, given by

P eB2 = P e

A + P eC + P e

AU, (2.46)

where P eAU ≥ 0 is the power consumed by electric auxiliary devices (assumed constant).

2.3.9 Electric buffer

The electric buffer cells are considered to be either lithium-ion batteries, or supercapac-

itors. Each cell is modeled as a cell open circuit voltage uc with a constant resistance Rc


connected in series. The voltage uc is modeled as an affine function of state-of-charge

soc ∈ [0, 1], i.e.,

uc =Qc

cc1soc+ cc0, (2.47)

where Qc is the cell capacity, and cc1 and cc0 are the resulting fitting coefficients. Such

a model is suitable for lithium-ion battery technology, see Fig. 2.6, where low/high soc

operation is avoided due to battery longevity reasons [57].

As nB cells are connected in series, the buffer terminal voltage uB2 reads

uB2 = nBuc − nBRcic, (2.48)

where ic denotes the cell current, which determines the evolution of the cell state-of-

charge, i.e.,

dsoc

dt= − ic

Qc

. (2.49)

0 20 40 60 80 1002

2.5

3

3.5

soc [%]

uc[V

]

uc

ucsocmin,max

Figure 2.6: Battery cell open circuit voltage.

In the following, convex modeling steps of [58] were employed, where instead of using

uc and ic the electric buffer is modeled in terms of its energy EeB, internal power P e

B1,

dissipative power P eB0 and terminal power P e

B2. In this context P eB1 and Ee

B are treated

as the additional (although constrained) optimization variables. Specifically, the energy

readsdEe

B

dt= −P e

B1, (2.50)

meaning that P eB1 > 0 results in buffer discharge and vice versa. The internal and

dissipative buffer powers are defined as

P eB1 = nBucic, (2.51)

P eB0 = nBRci

2c .



This implies

P eB1 = P e

B2 + P eB0. (2.52)

Furthermore, from (2.49), (2.50) and (2.51), it follows

EeB = nBQc

soc∫

0

uc(s)ds =nBcc1

2(u2

c − c2c0). (2.53)

Thus the pack losses may be expressed as

P eB0 =

Rccc1Pe2B1

2EeB + nBcc1c2

c0

, (2.54)

which, as a quadratic-over-linear, with a strictly positive denominator, is a convex

function [53] of P eB1, Ee

B and nB (and therefore also sB). Note that when cc0 = 0 the

adopted model, (2.53) and (2.54), describes a capacitor with a capacitance cc1.

Constraints on the state soc and the current ic can be translated into constraints on

the energy EeB and the power P e

B1, i.e.,

EeB ∈

nBcc12

([u2c(socmin), u2

c(socmax)]− c2

c0

)(2.55)

= [EeB,min, E

eB,max]sB,

P eB1 ∈ [ic,min, ic,max]

√nB

(2Ee

B

cc1+ nBc2

c0

), (2.56)

where socmin and socmax represent the minimum and the maximum state-of-charge,

ic,min < 0 and ic,max > 0 the minimum and the maximum cell current and EeB,min

and EeB,max the baseline minimum and the maximum buffer energy. Notice that the

geometric mean in (2.56) is a concave function [53] of EeB and nB.

2.4 Optimization problem formulation

The optimization problem formulated in Table 2.1 is revisited here by providing

mathematical meaning to constraints and the objective function. In this context, the

optimization goal is defined as finding the minimum of a weighted sum of operational and

component costs. The former is simply represented by the cost of consumed petroleum

while the latter consists of a sum of the ICE and the electric buffer cost. Each specific

cost component is weighted by its respective weighting coefficient, wφ, wE or wB, so

2.4 Optimization problem formulation 33

that its contribution is expressed in currency per distance. Using the approach outlined

in [59] these coefficients can be computed as

wφ = µφ1

Hlρφd, (2.57)

wE = µE

PmE,max

dyear

tyear

tlife

(1 + εE,year

tlife + 1

2

),

wB = µBEe

B,max

dyear

tyear

tlife

(1 + εB,year

tlife + 1

2

).

where µφ, µE and µB are fuel, engine and buffer price expressed respectively in currency

per volume, power and energy. Furthermore, d =∫ tend

0vVdt is the total drive cycle

length, dyear the average distance a vehicle travels during one year tyear, ρφ the fuel

density, PmE,max = maxωE

τE,maxωE the maximum baseline NA ICE mechanical power,

tlife the duration of the expected vehicle life-cycle, and εE,year and εB,year the yearly engine

and buffer interest rates.

2.4.1 Convex optimization problem

Based on the equations derived in the previous section the related optimization problem

is summarized as follows

min J = wφ

tend∫

0

P cEdt+ wEsE + wBsB, (2.58)

s.t. (2.55), (2.56),

0 < sB, 0 < sE ≤ 1, 0 ≤ PmBR, Pm

C ≤ cC,max,

γA,min ≤ PmA ≤ cA,max, γE,minsE ≤ P c

E ≤ γE,maxsE,

EeB = −P e

B1,

EeB(0) = Ee

B(tend),

P cE = γE,minsE + (Pm

A + PmAU

+ (γG0 + PmBR + γW0 + γW1sE + γW2sB) γML) γE1,

PmC = max

(γC2

P c2E

sE

+ cC1PcE, 0

),

P eB1 =

Rccc1Pe2B1

2EeB + sBnBcc1c2

c0

+ γA0 + cA1PmA + cA2P

m2A

+ cM00 + cM01PcE + cM02P

c2E + cM1P

mC + cM2P

m2C + P e

AU,

with the constraints imposed ∀t ∈ [0, tend], where tend is the time when the trip ends.

Note that the values of all the varying coefficients γ can be pre-computed for the entire



range of the driving cycle. This is because the variations originate from the changes in

vehicle speed vV, acceleration aV or slope α that are given, or from one of the rotational

speeds ωG, ωE or ωA, which can be computed using the knowledge of vV and of the gear

ratio trajectory λG(t). The λG(t) trajectory is thus assumed fixed prior to solving the

convex optimization problem.

In (2.58) the last two equality constraints can be relaxed with inequalities by replacing

“=” with “≥” sign. The relaxation changes the original formulation by creating a

convex superset of the non-convex set. However, it can be logically reasoned that

the resulting two constraints hold with equality at the optimum – otherwise energy

would be unnecessarily wasted. Hence, the solutions of the relaxed and the non-relaxed

problem are the same. For a detailed proof see [60]. This implies that the compressor

power PmC may be treated as an additional optimization variable, yielding in total six

time dependent, P cE, P

mBR, P

mA , P

mC , P

eB1, E

eB, and two scalar ones, sE and sB. Also strict

inequalities describing lower bounds of sizing parameters can be relaxed by means of a

small constant ε > 0. In other words, (2.58) may be rewritten as

min J = wφ

tend∫

0

P cEdt+ wEsE + wBsB, (2.59)

s.t. (2.55), (2.56),

ε ≤ sB, ε ≤ sE ≤ 1, 0 ≤ PmBR,

γA,min ≤ PmA ≤ cA,max, γE,minsE ≤ P c

E ≤ γE,maxsE,

EeB = −P e

B1,

EeB(0) = Ee

B(tend),

P cE = γE,minsE + (Pm

A + PmAU

+ (γG0 + PmBR + γW0 + γW1sE + γW2sB) γML) γE1,

max

(γC2

P c2E

sE

+ cC1PcE, 0

)≤ Pm

C ≤ cC,max

P eB1 ≥

Rccc1Pe2B1

2EeB + sBnBcc1c2

c0

+ γA0 + cA1PmA + cA2P

m2A

+ cM00 + cM01PcE + cM02P

c2E + cM1P

mC + cM2P

m2C + P e

AU.

All the optimization variables in (2.59) are further scaled with their expected maximal

values, the resulting problem is discretized using a zero-order hold with a sample time

δt and then casted into a standard convex second-order cone program (SOCP) form.


The SOCP is given by

min J = fTx (2.60)

s.t. ||Aix+ ei||2 ≤ cTi x+ di, i = 1, .., .m,

Fx = g

where x ∈ Rn are optimization variables, Ai ∈ Rni×n, F ∈ Rp×n, and || · ||2 is Euclidean

norm. Constraints of the type z ≥ x2/y are written as∣∣∣∣∣

∣∣∣∣∣

(2x

y − z

)∣∣∣∣∣

∣∣∣∣∣2

≤ y + z. (2.61)

The SOCP (2.60) is specified and solved using the CVX optimization modeling lan-

guage [61], [62], in combination with the SDPT3 solver [63].

2.4.2 Gear selection strategy

The discrete-time gear trajectory g(tk), with tk = kδt and k ≥ 0, is decided prior to and

outside of the convex optimization. In particular, two different gear selection strategies

have been implemented. The first searches for a gear that results in the largest difference

between the approximate and the maximum operating torque of a baseline NA ICE,

at every time instant. This is expected to promote the ICE downsizing. The second

however, at every time instant finds a gear that maximizes the estimated ICE efficiency,

so that a lower vehicle fuel consumption can be achieved.

These two gear selection strategies are implemented using the functions

g(1)(tk) = arg maxg∈1,...5

(τE,max(tk)− ˆτE(tk)), (2.62)

g(2)(tk) = arg maxg∈1,...5

ηE(tk),

where ˆτE and ηE denote the baseline NA ICE operating torque and efficiency estimates,

computed as

ˆτE =Pm

E

ωE

, (2.63)

ηE =Pm

E

γE1PmE + γE,min

.

with PmE being the estimated baseline NA ICE mechanical power, given by

PmE = max

(Pm

A + PmAU + γG0 + γW0 + γW1 + γW2, 0

)(2.64)



and PmA the estimated alternator mechanical power, i.e.,

PmA = Re

(−cA1 −

√c2

A1 − 4 (γA0 + P eAU) cA2

2cA2

). (2.65)

The use of torque ˆτE, efficiency ηE, power PmE and Pm

A estimates, instead of their optimal

values, is required since the computation of gear selection strategy precedes the convex

optimization – which renders the optimal values unavailable. However, an alternative,

more sophisticated gear selection strategies, employing iterative solutions of a convex

problem (2.59), could be applied here as well. For details, see [47], [64], [65].

2.5 Case study results

This section provides the case study results related to the sizing of electrically su-

percharged ICE powertrain. The purpose of the study is mainly to demonstrate the

proposed modeling and optimization methodology. For this reason, the standard

WLTP3 driving cycle has been used. Note that cycles, such as the WLTP3, are

more often employed for evaluating the vehicle fuel consumption than for component

sizing. Thus, for purposes a specific, real-world powertrain sizing application, one should

consider replacing the WLTP3 with a different (more demanding) cycle, which would

better suit the intended vehicle driving scenario. The values of the relevant model

parameters, used in this particular study, are listed in Appendix A.

2.5.1 Optimal component sizes

Table 2.2 summarizes the results of 6 different optimization runs. They correspond to

3 distinct engine scenarios: baseline NA ICE, downsized NA ICE and downsized SC

ICE, each for 2 defined gear selection strategies: g(1) and g(2). From the presented data

it is apparent that the electric supercharging has led to a substantial decrease in both

the engine size (1 − sE) and the dominant fuel cost (Jφ). As expected, the engine size

reduction is more prominent in the case of g(1) strategy, where gears are chosen such

that they maximize the distance between the engine operating torque and its maximum

torque line. In this case, the use of the MCU has resulted in 41.57% of engine volume

decrease, when the downsized SC ICE is compared to the baseline NA ICE. On the other

hand, more fuel is saved with g(2) strategy, which is reflected in the downsized SC ICE

fuel cost of 6.53 ¢/km vs. the baseline NA ICE fuel cost of 7.22 ¢/km. This is expected

since g(2) strategy maximizes the ICE efficiency, instead of the torque difference, thereby

also promoting a lower ICE fuel consumption.

2.6 Conclusions 37

Table 2.2: Optimization results

Case Parameter g(1) g(2)

Baseline &

naturally-

aspirated

J 8.98 ¢/km 7.25 ¢/km

Jφ 8.95 ¢/km 7.22 ¢/km

PmE,max 45.44 kW 45.44 kW

EeB,max 0.03 kWh 0.03 kWh

1− sE 0 % 0 %

Downsized &

naturally-

aspirated

J 8.29 ¢/km 7.19 ¢/km

Jφ 8.25 ¢/km 7.14 ¢/km

PmE,maxsE 38.54 kW 44.89 kW

EeB,maxsB 0.07 kWh 0.06 kWh

1− sE 15.18 % 1.21 %

Downsized &

supercharged

J 7.36 ¢/km 6.63 ¢/km

Jφ 7.26 ¢/km 6.53 ¢/km

PmE,maxsE 26.55 kW 31.87 kW

EeB,maxsB 0.23 kWh 0.21 kWh

1− sE 41.57 % 29.85 %

2.5.2 Optimal state and control trajectories

Apart from the optimal component sizes, the solution of the problem (2.59) provides

also the optimal control and state trajectories for the studied driving cycle. These are

shown in Fig. 2.7 for the case of the downsized SC ICE employing the first gear selection

strategy. Indeed, it can be observed that the MCU is mostly activated during short-

duration, high-power demands present near the end of the cycle. As a consequence, this

reduces the load on the ICE and allows it to be downsized. Furthermore, it can be seen

that the energy stored in the buffer at the end of the cycle is the same as the one stored

in the beginning. This implies that all the energy consumed by the MCU is ultimately

compensated by the work of the alternator.

2.6 Conclusions

This Chapter presented convex modeling steps for the problem of optimal ICE and

electric energy storage buffer sizing, for the case of the electrically supercharged ICE

powertrain concept. In this context, the electric supercharging was used to help

the engine during short-duration, high-power demands. The underlying optimization



0 5 10 15 20 25 300

20

40

60

80

100

120

vV[km/h]

0 5 10 15 20 25 300

20

40

60

80

100

P[kW]

P cE

PmBR

0 5 10 15 20 25 30−1

0

1

2

3

P[kW]

P eB

PmA

PmC

0 5 10 15 20 25 30

0.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

t [min]

Ee B[kWh]

Figure 2.7: Optimal control and state trajectories for a vehicle with the

downsized SC ICE employing the gear selection strategy that maximizes the

difference between the approximate and the maximum operating torque of a

baseline NA ICE.

problem was formulated as the minimization of the vehicle operational (fuel) and

component (the ICE and electric buffer) costs on a given driving cycle. The optimization

problem was solved for the case of the WLTP3 cycle, which delivered not only the

optimal component sizes but also the optimal control (e.g., engine fuel power) and state

(e.g., buffer energy) trajectories. In the analyzed scenario, the fuel cost savings of up

to 10% were obtained, showing that the engine downsizing via electric supercharging

could constitute a promising fuel-saving mechanism.

Chapter 3

Convex modeling and optimization of

a vehicle powertrain equipped with a

generator-turbine throttle unit

Abstract This Chapter investigates an internal combustion (gasoline) engine throttled by a

generator-turbine unit. Apart from throttling, the purpose of this device is to complement

the operation of a conventional car alternator by introducing an additional source of energy

for the electric auxiliaries. Its energy recovery potential is examined by employing a novel,

convex approach to modeling and optimization of the resulting vehicle powertrain. For a

given gear-shifting strategy, the proposed method allows the computation of optimal control

trajectories, e.g., the optimal engine fuel, alternator and turbine power, as well as of optimal

design parameters, e.g., the optimal battery size. The conducted numerical case study shows

that the use of a generator-turbine throttle unit has a potential to reduce the total operational

(fuel) and component (battery) costs by typically 2-4% or even more than 4% in selected cases

– depending on factors such as the engine size and the choice of a driving cycle.

3.1 Introduction

Although recent years have brought significant advances in electric vehicle technology,

the internal combustion engine (ICE) is still seen as a key enabler of ground trans-

portation for the foreseeable future. This can be primarily attributed to the superior

energy density and storage properties of carbon-based fuels, compared to the alternative

energy carriers (e.g., electric batteries). The ICE, however, suffers from high energy

losses, whereas the loss reduction remains a considerable challenge for the automotive

39

403 Convex modeling and optimization of a vehicle powertrain equipped with a


industry.

In [66]–[69] the authors have attempted to address the ICE energy loss problem by

introducing an exhaust heat recovery device as a substitute for a poorly-performing,

conventional car alternator. The device consists of a high-speed radial turbo-expander

which captures the energy from the exhaust air and uses it to (directly) drive a

permanent magnet or a switched reluctance electric generator. Compared to the

alternator such a device offers substantially more efficient vehicle electricity production

and a greater power density – at the cost of potentially increased exhaust back-pressure

and the need for water-cooling.

The energy recovery devices of this type can be also applied to the ICE air intake

system with the added benefit of easier installation due to a lack of external cooling

requirements. For instance, in the case of a spark-ignited (SI) gasoline engine a similar

generator-turbine unit (GTU) has been used to recover otherwise wasted intake throt-

tling losses [25], [26]. Here, the authors have proposed to replace a throttle valve with

the electric generator directly coupled to a variable geometry turbine (with adjustable

stator vanes), so that the intake airflow can be controlled while simultaneously producing

electricity. In [25], it has been also noted that in such a configuration the cool air after

the turbine can have both a negative (poor evaporation) and a positive (reduced risk of

knock) effect on the ensuing combustion phase. For this reason, the authors have argued

that the resulting lower intake air temperatures can be in addition used to assist the

air conditioning system, by cooling the working fluid via a dedicated heat exchanger.

In this Chapter, a numerical study is performed to evaluate a fuel-saving potential of

the GTU-based gasoline engine throttle. The main contributions of this research are:

1. Detailed modeling of the GTU assisted powertrain.

2. Convex relaxation steps that allow the combined optimal control and buffer sizing

problem to be formulated as a semi-definite convex program, where transmission

gear is considered as an input signal decided outside the convex optimization.

Moreover, a case study is performed providing a solution of the problem defined above

for several different driving cycles and engine sizes. The presented results show that the

use of the GTU-based throttle can potentially reduce the total operational (fuel) and

component (battery) costs by typically 2-4% or even more than 4% in selected cases,

depending on factors such as the engine size and the choice of a driving cycle.

This Chapter is organized as follows. The mathematical modeling is presented in

Section 3.2 and the optimization problem is formulated in Section 3.3. Section 3.4


BV

A

GE

TB

IM EM

ICE

GTUB

AUe

G

AUm

W

BRE. link

electrical

mechanicalpneumatic

VG

Figure 3.1: Illustration of the components considered in the proposed electric

buffer sizing problem – arrows indicate a power flow direction. The air flow

feeding an internal combustion engine (ICE) intake (IM) and exhaust (EM)

manifolds is determined by either the bypass valve (BV) or a variable turbine

geometry (VG) being a part of a stand-alone generator-turbine unit (GTU).

The GTU also consists of a turbine (TB) and an electric generator (GE), used

alongside a conventional car-alternator (A), to charge an electric buffer (B)

and power electric auxiliary loads (AUe). The ICE crankshaft is connected to

mechanical auxiliary loads (AUm) and a clutch and a gearbox (G) which further

link it to wheels (W) and brakes (BR).

discusses the results of the conducted numerical case study while the conclusions are

drawn in Section 3.5.

3.2 Quasistatic vehicle model

In the remainder, a model of a vehicle powertrain is developed which includes a 4-stroke

gasoline ICE equipped with the GTU at the intake, see Fig. 3.1. The GTU is used to

throttle the engine while extracting energy from the intake airflow to produce additional

power for electric auxiliary loads. Its power and energy requirements are matched by

sizing the vehicle electric energy buffer (e.g., a battery or a supercapacitor).

The vehicle powertrain model is illustrated in Fig. 3.2. It is a power-based [54],

quasistatic [57], backward simulation model [70]. This modeling approach implies

a reversed direction of computation, i.e., backward from the physical outputs (e.g.,

vehicle speed vV) to the control inputs (e.g., alternator mechanical power PmA ). The

main benefit of the backward simulation is that it does not require feedback control

implementation [70]. This allows us to start from the knowledge of the driving cycle

without the need for a separate driver model or inclusion of states for engine/vehicle

speed. In fact, the resulting model only has one state, used to capture the battery

energy (i.e., state-of-charge) dynamics. The backward simulation approach is widely



PmW

Vehicle

sB

ωW

PmG2

ωG

PmG1

ICEωE

PmE

ωA PmA

P cE

P eA

systemAir

ωTB PmTB

P eTB

sBP eB2

EeB

P eAU

derived

optimized

predetermined

g

bufferElectric

Gener.

P eB1

Ambient

pAM TAM

Brakes

ωW PmBR

Electricaux.

Mech.aux.

ωE PmAU

ωE

α

aV

vV

Wheels

cycleDrive Alter.

& p.e. & p.e.

ωW

Electriclink

λmMech.link

Gearbox

Figure 3.2: Power-based model of an ICE equipped with a stand-alone GTU –

arrows indicate component inputs and outputs.

applied for energy management of hybrid electric vehicles [57].

Note that certain model variables (colored blue in Fig. 3.2) are neither predetermined

nor can they be directly derived from the rest. Thus, there is some freedom left to choose

them, e.g., by minimizing the total fuel cost on a given driving cycle. Among others, the

resulting optimal problem formulation needs to decide upon an integer gear trajectory

g. Since obtaining the optimal g leads to a non-convex program, the approach taken in

this Chapter is to decouple the optimization problem into two subproblems. The first

subproblem optimizes gear for a baseline powertrain (fixed buffer size), while the second

subproblem decides on the buffer size and the remaining control signals. To solve the

second subproblem, the vehicle powertrain model has been constructed according to

the rules and conventions of the so-called disciplined convex programming [71]. The

adopted approach effectively reveals the convex structure of the second subproblem and

allows the exploitation of its attractive theoretical properties (e.g., if a local optimum

exists, it is also a global optimum). Finally, it also enables the use of efficient, reliable

numerical algorithms for finding its solution, such as, e.g., [63].


3.2.1 Vehicle

The vehicle is modeled as a system with a point mass

mV = mBsB + mV, (3.1)

where mB denotes the baseline buffer mass, mV the baseline vehicle mass (excluding

the mass of the electric buffer) and sB ∈ R>0 the buffer scaling coefficient (nominally

equal to 1). The mass mB is further defined as mB = mcnB, with mc being the baseline

buffer cell mass and nB the baseline cell count. Here, it is considered that the buffer is

built out of nB = sBnB cells connected in series.

Provided that the driving that occurs in still air is described by the driving cycle with

the demanded vehicle speed vV ≥ 0, acceleration aV and road slope α, the vehicle wheel

speed ωW and its power at the wheels PmW can be computed as

ωW =vV

rW

, (3.2)

PmW = aVλVmVvV + ag sin(α)mVvV+ (3.3)

agcr cos(α)mVvV +1

2ρAMcdAfv

3V.

The power components on the righthand side of (3.3) represent the inertial driving,

driving on a slope, rolling resistance and aerodynamic drag, respectively. Here, rW

denotes the wheel radius, λV the equivalent vehicle mass ratio, ag the gravitational

acceleration, cr the rolling resistance coefficient, ρAM the ambient air density, cd the

drag coefficient and Af the vehicle’s frontal area. Due to (3.1) the power PmW is affine

in the scalar optimization variable sB, i.e.,

PmW = γW0 + γW1sB, (3.4)

where

γW0 = (aVλV + ag sin(α) + agcr cos(α)) mVvV (3.5)

+1

2ρAMcdAfv

3V,

γW1 = (aVλV + ag sin(α) + agcr cos(α)) mBvV (3.6)

are vectors/constants that are fully determined by the driving cycle.

3.2.2 Wheels & brakes

The required mechanical power at the wheel side of the gearbox is given by

PmG2 = Pm

W + PmBR, (3.7)



where PmBR ≥ 0 is an optimization variable representing the power applied by the brakes.

3.2.3 Gearbox

The speed and power at the engine side of the gearbox are given by the following

expressions

ωG = λG(g)ωW, (3.8)

PmG1 = γG0 + Pm

G2, (3.9)

where λG(g) denotes the gear ratio corresponding to the gear g ∈ 1, . . . , 5 and γG0

the gearbox drag losses, modeled as

γG0 = cG0 + cG1ωG + cG2ω2G. (3.10)

The gear g is treated as an optimization variable found separately and prior to the rest.

The description of the related optimization subproblem is differed to Section 3.3.1.

3.2.4 Mechanical power link

The distinction between a regular and an idling engine operation is made using the

idling variable

γidle =

1, for ωG < ωE,min,

0, otherwise,(3.11)

which is fully determined at each time instance for a given gear trajectory g. Thus, the

engine and the alternator speed read

ωE = ωG(1− γidle) + ωE,minγidle, (3.12)

ωA = λAωE, (3.13)

while the corresponding engine mechanical power balance is given by

PmE = Pm

G1(1− γidle) + PmA + Pm

AU. (3.14)

Here, λA denotes the alternator speed ratio, ωE and ωE,min the ICE rotational and idle

speed, ωA the alternator speed, and PmE , P

mA , P

mAU ≥ 0 the mechanical power of the ICE,

alternator and the remaining mechanical auxiliary units. The power PmAU is modeled as

a constant, whereas PmA is treated as an optimization variable.


3.2.5 Air system

In the remainder, two different engine throttling scenarios are analyzed and distin-

guished using a binary variable q ∈ 0, 1. They arise by fully opening (q = 0) or

closing (q = 1) the bypass valve (BV), i.e., disabling or enabling the GTU. The intake

manifold is assumed to have a large surface-to-volume ratio, so that in a quasi-steady

state the temperatures before and after it can be considered to be the same. Also the

bypass valve is considered to behave as an isothermal orifice and that the pressure drop

over the air filter can be neglected. Then the first situation, when the BV is open, is in

the quasi-steady state described by the intake manifold temperature TIM equal to the

ambient air temperature TAM. The second situation however, when the BV is closed, is

in the quasi-steady state characterized by TIM that is equal to the temperature at the

turbine outlet, i.e.,

TTB = TAM − TAMηTB

(1− λ

λκ−1λκ

Π

), (3.15)

with ηTB being the turbine thermodynamic efficiency, λκ the specific heat ratio of air

and λΠ the ratio between the intake manifold pressure pIM and the ambient air pressure

pAM. As a result, the intake manifold-to-ambient temperature ratio is defined as

λT =TIM

TAM

= 1− qηTB

(1− λ

λκ−1λκ

Π

). (3.16)

The efficiency ηTB can be formulated [55] as

ηTB = max

(ηTB,max

[2λbsrλ∗bsr

−(λbsrλ∗bsr

)2], 0

), (3.17)

with

λbsr =rTBωTB√

2cpTAM

(1− λ

λκ−1λκ

Π

) (3.18)

representing the ratio between the turbine blade tip speed and the speed of isentropically

expanded air [72], λ∗bsr the optimal λbsr value, ηTB,max the maximal turbine efficiency,

rTB the turbine blade radius, cp the specific heat of air at constant pressure and ωTB

the turbine speed. In general, both λ∗bsr and ηTB,max depend on the turbine variable

nozzle area [73], [74] which itself is a function of the required engine air mass flow.

However, for simplicity here it is assumed that the resulting variation is small enough

to be neglected, thus yielding constant λ∗bsr and ηTB,max.



The mechanical turbine power is given by

PmTB = qmαcpTAMηTB

(1− λ

λκ−1λκ

Π

), (3.19)

where mα represents the engine air mass flow. If the engine fuel controller maintains a

constant air-fuel ratio λαφ, it holds

mα = mφλαφ, (3.20)

where mφ denotes the ICE fuel mass flow, i.e.,

mφ =P c

E

Hl

, (3.21)

with Hl being the lower heating value of the fuel and P cE the chemical fuel power used

by the engine. As the engine acts as a volumetric pump the air mass flow also satisfies

mα =ηvolpAMVEωEλΠ

4πcRTAMλT, (3.22)

with VE being the ICE displacement volume, cR the specific gas constant of air and

ηvol = ηvol(ωE, λΠ) the engine volumetric efficiency, defined for λΠ ∈ [λΠ,min, 1], where

λΠ,min is the pressure ratio satisfying ηvol(ωE, λΠ,min) = 0. The efficiency ηvol can be

modeled as a multilinear function [55] of the pressure ratio λΠ and the engine speed

ωE, i.e.,

ηvol = ηvol,ωηvol,Π, (3.23)

where

ηvol,ω = cvol0 + cvol1ωE + cvol2ω2E, (3.24)

and assuming an isentropic process with an ideal gas

ηvol,Π = 1 +1

λec

(1−

(pEM

pAM

)1/λκ

λ−1/λκΠ

), (3.25)

with pEM being the constant exhaust manifold pressure, λec the constant engine

compression ratio and cvoli, i ∈ 0, 1, 2 the constant fitting coefficients.

If a turbine speed controller uses the freedom to choose the GTU speed as

ωTB =λ∗bsrrTB

√2cpTAM

(1− λ

λκ−1λκ

Π

), (3.26)


then from (3.17) and (3.18) it follows that the turbine will be operated on the optimal

efficiency line, i.e., ηTB = ηTB,max while the turbine power may be written as

PmTB = qηTB,maxcpTAMγmγTB, (3.27)

where

γTB = λm

(1− λ

λκ−1λκ

Π

), (3.28)

λm =ηvol,ΠλΠ

1− qηTB,max

(1− λ

λκ−1λκ

Π

) , (3.29)

γm =pAMVEωEηvol,ω

4πcRTAM

. (3.30)

Note that the air mass flow coefficient γm and the ratio λm satisfy

mα = γmλm. (3.31)

Now, λm is treated as an optimization variable while, e.g., the ratio λΠ is derived from

it. For this purpose, let the function λΠ = λΠ(λm, q) denote the inverse of (3.29) for

λm ∈ [0, ηvol,Π] and q ∈ 0, 1, where ηvol,Π = ηvol,Π(λΠ = 1). Then, the function γTB =

γTB(λΠ(λm, 1)) can be accurately approximated by a concave 4th-order polynomial with

constant coefficients cTBi, i ∈ 1, . . . , 4, i.e.,

γTB = cTB1λm + cTB2λ2m + cTB3λ

3m + cTB4λ

4m, (3.32)

such that it passes through the end points (λm, γTB) ∈ (0, 0), (ηvol,Π, 0). The function

γTB and the resulting fit γTB are shown in Fig. 3.3.

Finally, to simplify the derivations that follow consider a polynomial approximation of

λΠ = λΠ(λm, q), given by

λΠ = cΠ0 + cΠ1λm + cΠ2λ2m, (3.33)

where cΠi = cΠi(q), i ∈ 0, 1, 2 are the resulting fitting coefficients. The functions λΠ

and λΠ are illustrated in Fig. 3.4.

3.2.6 ICE

The ICE mechanical behaviour can be described with the following relationship [55],

[56]:

pE = ηE1pφ − ηE2p2φ − pEf − pEg, (3.34)



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.02

0.04

0.06

0.08

λm [-]

γTB[-]

γTB

γTB

Figure 3.3: The 4th-order polynomial approximation of the function γTB(λm)

for q = 1.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.2

0.4

0.6

0.8

1

λm [-]

λΠ[-]

λΠ, q = 0

λΠ, q = 0λΠ, q = 1

λΠ, q = 1

Figure 3.4: The 2nd-order polynomial approximation of the pressure ratio

λΠ(λm, q).

where 0 ≤ ηE1 ≤ 1 and 0 ≤ ηE2 denote the effective ICE efficiency coefficients, pE and

pφ the engine brake and fuel mean effective pressures, and pEf and pEg the brake mean

effective pressure losses due to engine friction and pumping work. If the compression

ratio λec, the ignition/injection timing angles, the burnt gas fraction and the bore-stroke

ratio are kept constant, the efficiency ηE1 can be treated as a function of the engine speed

only [55], i.e.,

ηE1 = cη0 + cη1ωE + cη2ω2E, (3.35)

where cηi ≥ 0, i ∈ 0, 1, 2 denote some constant parameters. The efficiency ηE2 is

modeled as a constant.


The two pressures, pE and pφ, read

pE =4πτE

VE

(3.36)

pφ =4πP c

E

VEωE

, (3.37)

with τE being the ICE torque. The related loss components are modeled [55], [75] as

pEf = cf0 + cf1ωE + cf2ω2E, (3.38)

pEg = pEM − pAMλΠ,

where cf0i ≥ 0, i ∈ 0, 1, 2 are some constant parameters. Using (3.33) the ICE

pumping loss becomes

pEg = pEM − pAMcΠ0 − pAMcΠ1λm − pAMcΠ2λ2m. (3.39)

By inserting the expressions for pressures in (3.34) and by multiplying both sides withVEωE

4π, it follows

PmE = γE0 + γE1λm + γE2λ

2m, (3.40)

with PmE = τEωE the mechanical ICE power and

γE0 =VEωE(−cf0 − cf1ωE − cf2ω

2E − pEM + pAMcΠ0)

4π, (3.41)

γE1 =VEωEpAMcΠ1

4π+ηE1Hlγmλαφ

,

γE2 =VEωEpAMcΠ2

4π− ηE2H

2l γ

2m

λ2αφ

.

Note that only if γE2 ≤ 0, the expression (3.40) has physical sense, as it results in a

convex relationship between the mechanical PmE and the chemical power P c

E, as shown

in Figure 3.5. Thus, under this condition (3.40) is a concave function of the λm ratio.

3.2.7 Alternator and GTU generator

The electric machine electric power can be modeled as a second-order polynomial of its

mechanical power, with a constant term representing a speed-dependent drag loss [56].

In this context, the alternator electric power can be formulated as

P eA = γA0 + cA1P

mA + cA2P

m2A , (3.42)



1000

2000

2000

3000

3000

3000

4000

4000

4000

4000

5000

5000

5000

5000

PmE [kW]

Pc E[kW]

0 10 20 30 40 50 600

50

100

150

200

Figure 3.5: The ICE chemical power P cE vs. the ICE mechanical power Pm

E , for

different ICE speeds ωE expressed in (thousands of) rotations per minute.

where

γA0 = cA00 + cA01ωA + cA02ω2A > 0, (3.43)

with cA00, cA01, cA02 ≥ 0, −1 < cA1 < 0 and cA2 > 0 being constant coefficients. The

mechanical alternator power PmA is constrained both above and below, i.e., 0 ≤ Pm

A ≤γA,max, where γA,max represents a combined maximum torque and power bound of the

alternator, given by

γA,max = cAmax0

(1− e−ωA/cAmax1

), (3.44)

with cAmax0, cAmax1 ≥ 0 being some constant coefficients. Such a bound is not considered

in the case of the GTU. Namely, it is assumed that the GTU device is chosen such that

it its power/torque limits are broad enough to allow the available mechanical turbine

energy to be fully exploited.

Furthermore, the GTU electric power P eTB is approximated with an affine relationship

P eTB = qP e

TB0 + cGE1PmTB, (3.45)

where −1 < cGE1 < 0 and the GTU drag loss reads

P eTB0 = cGE00 + cGE01ωTB + cGE02ω

2TB ≥ 0, (3.46)

with cGE00 ≥ 0, cGE01, cGE02 ≥ 0 being some constant coefficients.

By expressing the turbine speed as a function of the ratio λm, i.e., ωTB =

ωTB(λΠ(λm, 1)), the loss P eTB0 can be approximated with a convex 4th-order polynomial

P eTB0 = cTB01λm + cTB02λ

2m + cTB03λ

3m + cTB04λ

4m, (3.47)


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

20

40

60

80

100

ωTB[krpm]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

40

60

80

100

λm [-]

Pe TB0[W

]

P eTB0

P eTB0

Figure 3.6: Top: turbine speed ωTB in (thousands of) rpm. Bottom: Convex

polynomial approximation of the GTU electric power drag losses P eTB0.

where cTB0,i, i ∈ 1, . . . , 4, are some constant coefficients. The fitting result is shown

in Fig. 3.6.

Note that since P eTB0 is a convex and Pm

TB a concave function of λm, see Fig. 3.3, and

since cGE1 < 0, it can be deduced that the GTU electric power P eTB is a convex function

of the λm ratio.

3.2.8 Electric power link

The electric buffer terminal power follows from the power balance at the electric power

link, given by

P eB2 = P e

A + P eTB + P e

AU, (3.48)

where P eAU ≥ 0 is the power consumed by electric auxiliary devices (modeled as a

constant) and P eB2 the electric buffer terminal power, defined in the following section.



3.2.9 Electric buffer

The electric buffer cells are considered to be either lithium-ion batteries, or supercapac-

itors. Each cell is modeled as a cell open circuit voltage uc with a constant resistance

Rc connected in series. The battery voltage uc is modeled as an affine function of

state-of-charge soc, i.e.,

uc =Qc

cc1soc+ cc0, (3.49)

where Qc is the cell capacity, and cc1 and cc0 are the resulting fitting coefficients. Such

a model is suitable for lithium-ion battery technology where low/high soc operation is

avoided due to battery longevity reasons [57], as well as for supercapacitors.

As nB cells are connected in series, the buffer terminal voltage uB2 reads

uB2 = nBuc − nBRcic, (3.50)

where ic denotes the cell current, which determines the evolution of the cell state-of-

charge, i.e.,

dsoc

dt= − ic

Qc

. (3.51)

In the following, convex modeling steps of [58] are employed, where instead of using uc

and ic the electric buffer is modeled in terms of its energy EeB, internal power P e

B1 and

dissipative power P eB0. Specifically, the following holds

dEeB

dt= −P e

B1, (3.52)

implying that P eB1 > 0 results in the buffer discharge and vice versa. The internal and

dissipative buffer powers are defined as

P eB1 = nBucic, (3.53)

P eB0 = nBRci

2c .

This implies

P eB1 = P e

B2 + P eB0. (3.54)

Furthermore, from (3.51), (3.52) and (3.53), the following derives

EeB = nBQc

soc∫

0

uc(s)ds =nBcc1

2(u2

c − c2c0). (3.55)


Thus the pack losses may be expressed as

P eB0 =

Rccc1 (P eB1)2

2EeB + nBcc1c2

c0

, (3.56)

which as a quadratic-over-linear with a strictly positive denominator is a convex

function [53] of P eB1, Ee

B and nB (and therefore convex in sB as well). Note that when

cc0 = 0 the adopted model, (3.55) and (3.56), describes a capacitor with a capacitance

cc1.

Constraints on the state-of-charge soc ∈ [socmin, socmax] and the current ic ∈[ic,min, ic,max] can be translated into constraints on the energy Ee

B and the power P eB1,

i.e.,

EeB ∈ sB

( nBcc12

([u2c(socmin), u2

c(socmax)]− c2

c0

))(3.57)

= sB[EeB,min, E

eB,max],

P eB1 ∈[ic,min, ic,max]

√sBnB

(2Ee

B

cc1+ sBnBc2

c0

), (3.58)

where 0 ≤ socmin < socmax ≤ 1 represent the minimum and the maximum state-of-

charge, ic,min < 0 and ic,max > 0 the minimum and the maximum cell current and

EeB,min and Ee

B,max the baseline minimum and maximum buffer energy. Notice that the

geometric mean in (3.58) is a concave function [53] of EeB and nB. In this context Ee

B is

treated as an optimization variable, although it can also be treated as a state variable

that depends on the control variable P eB1.

3.3 Optimization problem formulation

The optimization goal is to find a minimum of a weighted sum of operational and

component costs. The operational costs are simply represented by the cost of consumed

gasoline while the component costs are given by the cost of the electric buffer. Each

cost component is weighted by its respective weighting coefficient, wφ or wB, so that its

contribution is expressed in currency per distance. Using the approach outlined in [59]

these coefficients are computed as

wφ = µφ1

Hlρφd, (3.59)

wB = µBEe

B,max

dyear

tyear

tlife

(1 + εB,year

tlife + 1

2

),



where µφ and µB are fuel and buffer price expressed respectively in currency per volume

and usable energy content (i.e., at a specified maximum state-of-charge). Furthermore,

d =∫ tend

0vVdt is the total drive cycle length, dyear the average distance a vehicle travels

during one year tyear, ρφ the fuel density, tlife the duration of the expected vehicle life-

cycle and εB,year the yearly buffer interest rate.

3.3.1 Gear selection strategy

For a driving cycle specified by the vehicle speed vV, the acceleration aV and the slope α,

a feasible gear trajectory g is found by minimizing the fuel power of a baseline powertrain

(fixed buffer size), i.e., when the ICE is throttled using a bypass valve (q = 0).

First, the expressions (3.2), (3.5), (3.6), (3.8) and (3.10)-(3.13) are evaluated at each

time instant tk = kδt, k ≥ 0 for all gears g(tk) ∈ 1, . . . , 5, where δt denotes the driving

cycle sample time. Second, the obtained values are used to estimate the powers PmA ,

PmE and P c

E, and the ratio λm, according to

PmA =

−cA1 −√c2

A1 − 4 (γA0 + P eAU) cA2

2cA2

, (3.60)

PmE = max

(Pm

A + PmAU + (γG0 + γW0 + γW1) (1− γidle), 0

),

P cE = λm

Hlγmλαφ

,

λm =

−γE1 +

√γ2

E1 − 4(γE0 − Pm

E

)γE2

2γE2

.

Third, at each time instant tk, the gear g(tk) ∈ F(tk) yielding the lowest fuel power P cE

is selected. Here F(tk) ⊂ 1, . . . , 5 denotes a feasible set of gears at the moment tk, i.e.,

the gears resulting in a feasible ratio λm ∈ [0, ηvol,Π] ⊂ R and power PmA ⊂ R. Finally,

to limit the maximum number of consecutive gear up-shifts gup,max, the computed gear

trajectory g is processed as follows

g(tk) =

g(tk), if F(tk) = ∅,ming∈F(tk)

g, otherwise,(3.61)

where F(tk) = g ∈ F(tk)|g(tk−1) < g ≤ g(tk−1)+gup,max denotes a set of feasible gears

at the moment tk satisfying the imposed gear shifting criterion. The gear trajectory

obtained using the described procedure is shown in Fig. 3.7, for the case of the US

Environmental Protection Agency Federal Test Procedure driving cycle (FTP75).


5 10 15 20 25 300

20

40

60

80

vV[km/h]

5 10 15 20 25 301

2

3

4

5

g[-]

t [min]

Figure 3.7: Top: the speed profile vV corresponding to the FTP75 driving

cycle. Bottom: the computed gear trajectory g.

3.3.2 Convex optimization problem

Based on the equations derived in the previous sections the convex optimization problem

is formulated as follows

minsB,P

mBR,P

mA ,EeB,λm

J = wφ

tend∫

0

P cEdt+ wBsB, subject to (3.62)

ε ≤ sB, 0 ≤ PmBR, 0 ≤ Pm

A ≤ γA,max, P eA ≤ 0, P e

TB ≤ 0,

0 ≤ λm ≤ ηvol,Π,

EeB ∈ [Ee

B,min, EeB,max]sB,

P eB1 ∈ [ic,min, ic,max]

√sBnB

(2Ee

B

cc1+ sBnBc2

c0

),

0 ≤ PmE ≤ γE0 + γE1λm + γE2λ

2m,

P eB0 + γA0 + cA1P

mA + cA2P

m2A + qP e

TB0 + cGE1PmTB + P e

AU ≤ P eB1,

dEeB

dt= −P e

B1,

EeB(0) = Ee

B(tend),



where

P eB0 =

Rccc1 (P eB1)2

2EeB + sBnBcc1c2

c0

, (3.63)

P eTB0 = cTB01λm + cTB02λ

2m + cTB03λ

3m + cTB04λ

4m,

PmTB = qηTB,maxcpTAMγm

(cTB1λm + cTB2λ

2m + cTB3λ

3m + cTB4λ

4m

),

PmE = Pm

A + PmAU + (γG0 + Pm

BR + γW0 + γW1sB) (1− γidle),

P cE =

Hlγmλαφ

λm,

with the constraints imposed ∀t ∈ [0, tend], where tend is the time when the trip ends

and ε > 0 is some small constant. Note that the values of all the varying coefficients

γ can be pre-computed for the entire range of the driving cycle. This is because they

only depend on the vehicle speed vV, acceleration aV and/or slope α or the rotational

speeds ωG, ωE and ωA, i.e., on the gear trajectory g.

In (3.62) the electrical and mechanical power balance equalities are relaxed with an

inequality. The relaxation changes the original model formulation by creating a convex

superset of a non-convex set. However, it can be logically reasoned that the resulting two

constraints hold with equality at the optimum – otherwise energy would be unnecessarily

wasted. Hence, the solutions of the relaxed and of the non-relaxed problem are the same.

For a more detailed discussion, see [60].

The problem (3.62) has in total four time dependent, λm, PmBR, P

mA , E

eB, and one scalar

optimization variable, sB. Before the problem is solved all the optimization variables are

scaled with their expected maximal values. Then (3.62) is discretized using a zero-order

hold (with a sample time δt) and casted into a standard convex semi-definite program

(SDP) form [76]. The SDP is implemented and solved using the CVX optimization

modeling language [61], [62], in combination with the SDPT3 solver [63].

3.4 Case study results

In this Section the solution of (3.62) is discussed for the case of 4 different driving cycles,

i.e.,

• the US Environmental Protection Agency Federal Test Procedure driving cycle

(FTP75),

• the Worldwide harmonized Light vehicles Test Procedure Class 3 driving cycle

(WLTP3),

3.4 Case study results 57

• the New European Driving Cycle (NEDC) and

• the National Renewable Energy Laboratory driving cycle (VAIL2NREL).

The used vehicle parameters are listed in Appendix B.

The FTP75 can be best described as an urban driving cycle while the WLTP3 mimics

the “real world” driving, thereby covering a wide range of vehicle speeds. The NEDC

is a highly stylised mostly urban cycle with only a quarter of its duration reserved for

highway. Finally, the VAIL2NREL represents a high-demanding mountain driving cycle

and is the only one providing also the slope information.

3.4.1 Optimal engine operating points

Figure 3.8 shows the location and the relative density of the optimal ICE torque-speed

operating points for different driving cycles. Clearly, the largest differences exist between

the VAIL2NREL and the rest. Specifically, it can be seen that during this cycle the

engine mostly operates at around 3000rpm while producing low-to-medium torque. In

contrast, idling dominates the WLTP3, FTP75 and NEDC engine operation as indicated

by the dark red areas at the minimal engine speed. The “real world” inspiration behind

the WLTP3 can be inferred from a large spread of its resulting operating points, e.g.,

when compared to the urban FTP75 which is narrowly placed in the low-to-medium

speed region. The segmentation of the NEDC operating area is a direct consequence of

its simple and a highly stylised form.

3.4.2 Optimal state and control trajectories

Figures 3.9a and 3.9b depict the optimal control (P cE, Pm

BR and PmA ) and state (Ee

B)

trajectories for the VAIL2NREL and the FTP75 driving cycle. It can be seen that in

both cases the energy stored in the buffer at the end of the trip equals the one at the

beginning. This implies that all the energy consumed by the electric auxiliaries has been

compensated by the combined work of the alternator and the GTU. However, it can

also be observed that the GTU generates substantially more electric power P eTB during

the VAIL2NREL cycle. In fact, this is also true when comparing the VAIL2NREL with

other cycles under study. These results can be attributed to the specific distribution of

the engine operating points associated with the VAIL2NREL cycle, as seen in Fig. 3.8.

This leads to a conclusion that a low-to-medium load, medium-to-high speed type of

driving is especially suited for the GTU application. Note that in general such situations



are also characterized by increased engine throttling (pumping) losses as they necessarily

imply partially closed engine air-path.

Table 3.1 summarizes the optimization results while Fig. 3.10 provides an illustration.

Here, the total vehicle operational and component costs J are presented for all driving

cycles, both when the GTU is disabled (q = 0) and when it is enabled (q = 1).

Specifically for a baseline, 1.6 liter engine, the GTU use delivered cost reductions ranging

from 0.17 ¢/km (FTP75) to 0.28 ¢/km (VAIL2NREL), i.e., 2.24% to 3.86% respectively.

1000 1500 2000 2500 3000 3500 4000 4500 5000

20

40

60

80

100

120

140

FTP75

ωE [rpm]

τE[N

m]

1000 1500 2000 2500 3000 3500 4000 4500 5000

20

40

60

80

100

120

140

NEDC

ωE [rpm]

τE[N

m]

1000 1500 2000 2500 3000 3500 4000 4500 5000

20

40

60

80

100

120

140

VAIL2NREL

ωE [rpm]

τE[N

m]

1000 1500 2000 2500 3000 3500 4000 4500 5000

20

40

60

80

100

120

140

WLTP3

ωE [rpm]

τE[N

m]

Figure 3.8: The ICE torque-speed diagram showing the relative density of the

optimal engine operating points for different driving cycles. To emphasize the

active driving w.r.t. engine idling operation, a logarithmic color scale is used

with the high density areas colored in red.

The data also shows that in all cases the addition of the GTU led to an increase in

the battery size, e.g., from 6.57 Wh to 15.82 Wh for the case of the FTP75 driving

cycle. The location of its battery operating points is presented in Fig. 3.11. Notice that

some of them are located on the buffer power P eB1 and energy Ee

B bounds. The activity

of one or more of these constraints is expected as this is what ultimately determines

the required buffer size. Nevertheless, it can be seen that the entire battery energy

content was not actually used. Namely, it appears that a battery with the same power

capability but half of the energy content could have been employed just as well. As

3.4 Case study results 59

0 20 40 60 800

50

100

150

vV[km/h]

−5

0

5

α[deg]

0 20 40 60 800

50

100

150

P[kW]

P cE

PmBR

0 20 40 60 80

−1

−0.5

0

0.5

P[kW]

P eB

P eA

P eTB

0 20 40 60 800

0.005

0.01

0.015

0.02

0.025

0.03

t [min]

Ee B[kWh]

(a) VAIL2NREL

0 5 10 15 20 25 300

50

100

150

vV[km/h]

−5

0

5

α[deg]

0 5 10 15 20 25 300

50

100

150

P[kW]

P cE

PmBR

0 5 10 15 20 25 30

−1

−0.5

0

0.5

P[kW]

P eB

P eA

P eTB

0 5 10 15 20 25 300

0.005

0.01

0.015

0.02

0.025

0.03

t [min]

Ee B[kWh]

(b) FTP75

Figure 3.9: Optimal control and state trajectories for a vehicle equipped with

the GTU.

the required battery is small (tens of Wh) compared to conventional car batteries, it

may be argued that the introduction of the GTU would have a rather small impact

on the existing vehicle energy storage. Also, the resulting small buffer sizes can be

attributed to the fact that the power of the electric auxiliaries P eAU was modeled as

a constant, i.e., without including short-duration, high-power draws typical for engine

starting. In future studies however, this and similar phenomena can be analyzed using

the same methodology, e.g., by adopting a more realistic, driving cycle-based model of

the required P eAU power.



VAIL2NREL FTP75 WLTP3 NEDC0

2

4

6

8

10

12

J[cent/km]

3.86%2.24% 2.96% 2.47%

GTU enabled

reduction

Figure 3.10: The effect of using the GTU at the place of a throttle valve in

terms of the total operational and component costs J for different driving cycles.

Table 3.1: Optimization results

Driving cycle Parameter disabled GTU enabled GTU

VAIL2NRELJ 7.19 ¢/km 6.91 ¢/km

sBEeB,max 5.83 Wh 20.22 Wh

FTP75J 7.77 ¢/km 7.59 ¢/km


WLTP3J 8.10 ¢/km 7.86 ¢/km


NEDCJ 8.26 ¢/km 8.05 ¢/km


3.4.3 The effect of varying the ICE displacement volume on the

GTU performance

The effect of varying the ICE displacement volume on the total cost J associated with

the VAIL2NREL driving cycle, is shown in Fig. 3.12. The cost reduction caused by the

GTU use ranges from 3.6% for a 1.5 liter to 4.5% for a 2 liter engine. Thus, although

larger engines induce greater driving costs they also yield greater benefits from the

GTU-based throttle technology, w.r.t. their smaller counterparts. This relates to the

fact that engine throttling losses become more prominent when the engine is oversized.

3.5 Conclusions 61

−0.3 −0.2 −0.1 0 0.1 0.2 0.30

0.005

0.01

0.015

8888

9090

90 9090

909292

92 9292

929494

94 9494

949696

96 9696

969898

9898

9898

P eB [kW]

Ee B[kWh]

Figure 3.11: The electric buffer power P eB vs. energy Ee

B, for the case

of the FTP75 driving cycle. The contour lines represent the buffer charg-

ing/discharging efficiency.

1.5 1.6 1.7 1.8 1.9 26.8

7

7.2

7.4

7.6

7.8

Cost

[cent/km]

VE [liter]

disabled GTUenabled GTU

Figure 3.12: The effect of varying the ICE displacement volume on the total

operational and component costs for the case of the VAIL2NREL driving cycle.

3.5 Conclusions

This Chapter provided a method for numerical evaluation of the fuel-saving potential of

the generator-turbine throttle unit. For this purpose, the disciplined convex modeling

and optimization methodology was employed to derive a convex model of the underlying

vehicle powertrain. The derived model was used to formulate a convex optimization

problem considering the minimization of the total vehicle operational (fuel) and com-

ponent (electric buffer) costs. The analysis of the optimal problem solution has shown

that the GTU-based throttle can yield major fuel benefits during the high-demand

highway driving and for vehicles equipped with oversized ICEs. In such circumstances,

the GTU use was capable of reducing the total costs by more then 4%.



The optimal problem solution also provided the optimal control and state trajectories

which minimize the vehicle operational costs, on a given driving cycle. Thus, due to its

convex properties, the proposed optimization algorithm (excluding the electric buffer

sizing part) could also be applied for online vehicle state/control prediction purposes.

However, as this inherently implies the use of driving cycle estimates, the fuel-economy

improvements might be lower.

Chapter 4

Four-Quadrant speed control of 4/2

Switched Reluctance Machines

Abstract The main challenge in designing a 4Q speed control algorithm for a 4/2 SRM lies

in the existence of “dead” torque zones where the machine’s capability to produce torque is

considerably reduced. This makes its startup challenging and has so far hindered its four-

quadrant control. This Chapter provides a solution for the 4/2 SRM startup problem and

uses it to enable its four-quadrant operation. The proposed control algorithm respects the

varying average torque bounds and explicitly enforces the maximum phase current limit.

A parametric, open-loop, average torque control scheme is derived and employed to ensure

efficient production of torque required to track the desired speed reference. The effectiveness

of the proposed solution is verified experimentally on a high-speed hardware setup.

4.1 Introduction

The Switched Reluctance Machine (SRM) is an electric machine often attributed with

high reliability, fault tolerance and power density, as well as the ability to operate in

harsh environments. It has salient (protruding) poles both on a rotor and a stator while

its windings consist exclusively of coils wound around the stator poles [77]. Due to its

lack of brushes and permanent magnets the SRM is a machine of a simple design and low

manufacturing costs, capable to operate in a wide speed range. Since most of its losses

arise on the stator it is also easy to cool and maintain. This places it in a favorable

position for many industrial applications, especially within the automotive [17], [67],

[78] and aerospace industries [79], [80].

63

64 4 Four-Quadrant speed control of 4/2 Switched Reluctance Machines

Some of the commonly reported disadvantages of the SRM include a larger acoustic

noise [81]–[83] and a larger torque ripple [84], when compared to other electric machines.

Many different control techniques have been proposed to address these issues, most

commonly represented by the instantaneous torque control (ITC) methods. The ITC

is mainly focused on regular SRMs [85], where multiple phases can simultaneously

produce the torque of the same sign. Thus, in such machines, the production of

the desired instantaneous torque can be shared between the phases. This property

has been exploited in [86], which describes a method to reduce the torque ripple

using offline-constructed phase torque sharing functions. Instead, in [87], the torque

ripple is minimized by directly optimizing the phase current profiles. In either case,

the employed ITC scheme requires phase current shaping, w.r.t. rotor position, which

becomes increasingly difficult with increasing speed [88]. For this reason, such methods

are best applicable for low-speed, high-precision torque control applications [88].

In contrast to the ITC, the SRMs can be also controlled by means of the average torque

control (ATC). Typically, in this case, the reference currents are kept constant during

the phase excitation period, which yields a wider speed range capability. One such

algorithm is reported in [89]. It enables closed-loop adjustment of the low-level control

parameters (e.g., reference phase current) using an online estimate of the average torque.

In this way the performance of the underlying feedforward ATC can be improved at the

cost of the additional phase voltage measurement needed for torque estimation.

The ATC is also applicable to irregular SRMs, which lack the torque sharing capability,

e.g., described by a 4-stator/2-rotor pole, 2-phase configuration shown in Fig. 4.1. The

4/2 SRMs have an advantage of a reduced core loss and a simpler power converter

topology compared to other SRMs (with more than two phases) [90], [91]. They

have been applied in unidirectional, high-speed applications, typically where a machine

efficiency/cost is often more of a concern than the emitted noise and the torque

ripple [92]–[94]. This is also the case in this work, which presents a novel, 4/2 SRM,

four-quadrant (4Q) speed control algorithm based on the open-loop ATC.

The 4Q operation refers to the electric machine’s ability to work in both directions of

rotation, either as a motor or as a generator. So far, such operation has been rarely

investigated in the SRM context [95] and especially not in the case of the irregular

machine configurations. In fact, to the best of the author’s knowledge this Chapter

describes the first 4Q speed control algorithm dedicated to the conventional 4/2 SRMs.

The main contributions of this research are:

1. Development of a model-based, open-loop ATC scheme in the form of a nonlinear

mapping between the SRM operating point (defined by the desired average torque,

4.1 Introduction 65

1

1

2 2

statorpole

rotorpole

statorwindings

+

Figure 4.1: Cross section of a 2-phase 4/2 SRM.

rotational speed and the applied DC-link voltage) and a set of low-level current

reference parameters.

2. Development of a four-quadrant speed controller that can cope with the identified

speed and voltage-dependent average torque bounds.

3. Development of a supervisory control algorithm for supporting a startup and a

change of the SRM direction of rotation.

The key advantage of the proposed ATC method is that it is parametric and thus simple

to implement. Furthermore, the derived current reference parametrization is applicable

to all four quadrants of machine operation and is characterized by a smooth transition

between the ”constant torque“ and the “constant power” regions of the SRM operation.

Being open-loop, however, implies that its effectiveness strongly depends on the quality

of the utilized SRM model. Still, a suitable model can be easily derived from the

measured static electromagnetic characteristics, as demonstrated in this work.

The main challenge in designing the 4Q speed control algorithm for the 4/2 SRM lies in

the existence of “dead” torque zones where the machine’s capability to produce torque

is considerably reduced. Thus at these locations rotor may get “stuck” when starting

from a standstill, e.g., at startup or during a change of rotational direction. To ensure

that the rotor can always leave such “stuck” positions so that a normal 4Q operation

can be resumed – a supervising state-machine has been designed. The effectiveness of

the proposed control solution is verified on a high-speed experimental setup.

This Chapter is organized as follows. Section 4.2 presents the control-oriented model

of the 4/2 SRM as used in this study, based on its measured, static electromagnetic

characteristics. It further discusses the SRM actuation and provides the details behind


the proposed 4Q commutation and current control strategy. Section 4.3 describes the

derived 4Q speed controller, the speed/position estimation and the proposed 4/2 SRM

startup procedure. Section 4.4 discusses some representative experimental results while

conclusions are drawn in Section 4.5.

4.2 SRM modeling and operation

4.2.1 Dynamics

The control-relevant dynamics of a 4/2 SRM can be described as1

dipdt

=1

Lp(ip, θ)(vp −Rpip −Mp(ip, θ)ω) , (4.1)

dθ

dt= ω, (4.2)

dω

dt=

1

J

(2∑

p=1

τe,p(ip, θ)− τl −Bω), (4.3)

where ip, vp and Rp denote the current, voltage and resistance of the phase winding

p ∈ 1, 2, B and J the friction and inertia coefficients, ω and θ the rotor speed and

absolute position, and τl the load torque. It is considered that the rotor poles are

fully aligned with the phase 2 stator pole pair whenever θ = kπ, k ∈ Z. The phase p

incremental inductance, back e.m.f. and electromagnetic torque, Lp(ip, θ), Mp(ip, θ)ω

and τe,p(ip, θ), are related to the flux linked in its winding ψp(ip, θ), as follows

Lp(ip, θ) =∂ψp(ip, θ)

∂ip, (4.4)

Mp(ip, θ) =∂ψp(ip, θ)

∂θ, (4.5)

τe,p(ip, θ) =∂Wc,p(ip, θ)

∂θ, (4.6)

where Wc,p(ip, θ) represents the phase magnetic co-energy, given by

Wc,p(ip, θ) =

ip∫

0

ψp(i, θ)di. (4.7)

The phase p flux-linkage function ψp(ip, θ) is periodic w.r.t. the position θ with a period

equal to the rotor pole pitch angle θrpp = 2πNr

= π, where Nr = 2 is the rotor pole

1Assuming negligible mutual couplings between phases.

4.2 SRM modeling and operation 67

number, i.e.,

ψp(ip, θ) = ψp (ip, θ + kπ) , k ∈ Z. (4.8)

Furthermore, the two phase flux-linkage functions are shifted versions of each other2 by

a stroke angle θsk = 2πmNr

= π2, where m = 2 is the phase count, i.e.,

ψ1(i, θ) = ψ2

(i, θ +

π

2

), ∀i, θ. (4.9)

The periodicity and shifting properties (4.8) and (4.9) imply that both ψ1(i, θ) and

ψ2(i, θ) can be fully described using the knowledge of only one of them on a restricted

position interval, e.g., θ ∈ [0, π], ∀i. For this purpose, define the flux-linkage function

ψ(i, θ) = ψ1(i, θ), for θ ∈ [0, π], (4.10)

and the rotor position w.r.t. phase winding p ∈ 1, 2, as

θp = mod(θ + (p− 1)

π

2, π), (4.11)

where

mod(x, y) =

x− y · floor(x/y), for y 6= 0,

x, otherwise,(4.12)

denotes the modulus function and floor a function that rounds a real number to the

nearest integer towards minus infinity. Then it follows

ψp(ip, θ) = ψ(ip, θp), p ∈ 1, 2. (4.13)

In the same way, it holds

Lp(ip, θ) = L(ip, θp), (4.14)

Mp(ip, θ) = M(ip, θp), (4.15)

τe,p(ip, θ) = τe(ip, θp), (4.16)

where p ∈ 1, 2 and L(i, θ) = L1(i, θ), M(i, θ) = M1(i, θ) and τe(i, θ) = τe,1(i, θ), for

θ ∈ [0, π].

The flux-linkage function ψ(i, θ) can be obtained, e.g., by employing finite-element

or analytical calculations. However in this work, ψ(i, θ) of a particular 4/2 SRM

is determined using the measurement-based DC-excitation approach outlined in [96].

2Assuming same electromagnetic properties for both phases.


Specifically, the SRM phase 1 was energized with a DC voltage while the rotor was

mechanically locked at a desired position θ ∈ [0, π]. The instantaneous phase voltage

v1 and current i1 were measured and the difference v1 − R1i1 was integrated (and

filtered) to obtain ψ1(i1, θ) = ψ(i1, θ). Subsequently, the functions L(i, θ), M(i, θ)

and τe(i, θ), θ ∈ [0, π], were obtained by numerically evaluating the expressions (4.4)

to (4.7). Figure 4.2 shows the resulting flux-linkage ψ(i, θ) and torque τe(i, θ) functions.

0.010.01

0.01

0.02

0.020.02

0.02

0.03

0.03

0.03

0.04

0.04

Flux-linkage ψ(i, θ) [Vs]

Currenti[A

]

0 45 90 135 1800

5

10

15

20

−0.3−0.2

−0.2

−0.1

−0.1

00

0.1

0.1

0.2

0.2 0.3

0.4Electromagnetic torque τe(i, θ) [Nm]

Position θ [deg]

Currenti[A

]

0 45 90 135 1800

5

10

15

20

Figure 4.2: 4/2 SRM flux-linkage ψ(i, θ) and electromagnetic torque τe(i, θ)

functions obtained using the DC-excitation method [96].

4.2.2 Actuation

The 4/2 SRM is actuated using the Asymmetric Half Bridge Converter (AHBC)

equipped with two switches, Sup and Slp, per phase p ∈ 1, 2, see Fig. 4.3. The AHBC

converter allows application of different voltages to each phase winding. Its functionality

is described in Table 4.1, where 0 is used to denote an open and 1 a closed switch, and vdc

the applied DC-link voltage. The switch states are manipulated in accordance with their

PWM duty cycles dup ∈ [0, 1] for Sup and dlp ∈ [0, 1] for Slp, to allow the exploitation of a

full DC link voltage range [−vdc, vdc]. Here, the duty cycle value d implies that during

each PWM period 1/fpwm a switch is closed d · 100% of the time, where fpwm = 60kHz


v1 v2

Su1 Su

2

Sl1 Sl

2

i1 i2

L1 L2+vdc

idc

Figure 4.3: 4/2 SRM AHBC circuit.

Table 4.1: Phase voltage vs. the AHBC control input

Sup Slp vp

0 0 −vdc1 0 0

0 1 0

1 1 vdc

represents the adopted PWM switching frequency. The duty cycles are specified using

the phase control input up ∈ [−1, 1], as follows

dup =

1, for up ≥ 0,

0, otherwise,(4.17)

dlp =

up, for up ≥ 0,

1 + up, otherwise,

which in average (over the PWM period) yields the phase voltage

vp = upvdc, p ∈ 1, 2. (4.18)

Note that for a more balanced AHBC phase switch utilization the roles of the upper

and the lower switch may be swapped, e.g., once per rotation.


4.2.3 Four-quadrant operation

Average torque control

Figure 4.4 shows the phase torques τe,1 and τe,2, w.r.t. the rotor position θ when i1 =

i2 = imax = 20A, where imax denotes the maximum allowed phase current. Clearly, the

torque varies significantly with position. Also there exists almost no overlap between

the torque regions of the same sign that belong to different phases. Thus sharing the

torque production between them (a method often used in the case of SRMs with m > 2)

is not possible. This implies that this SRM cannot produce constant torque, i.e., the

torque ripple will always characterize its output.

0 45 90 135 180−0.4

−0.2

0

0.2

0.4

Position θ [deg]

Torq

ueτe,p[N

m]

90 92 94−0.05

−0.025

0

0.025

0.05

Position θ [deg]

p = 1

p = 2

Figure 4.4: Phase torques τe,1 and τe,2 as a function of rotor position θ for

i1, i2 = imax = 20A. Enlarged region around the phase 1 aligned position is

presented on the right-hand side.

For this reason, the approach taken in this work is to treat the SRM as a device that

can produce a desired constant average torque τ ∗e , instead of the instantaneous one.

Furthermore, since a certain torque τ ∗e can be achieved using a variety of different

current profiles, the reference phase currents i∗p(θp), p ∈ 1, 2 have been constrained

to a constant level i∗(τ ∗e ) ∈ [0, imax] during the active phase interval (up > −1).

The duration and timing of phase activation is determined by the adopted phase

commutation strategy.


Fixed commutation strategy

Each phase activation cycle begins by setting up = 1 to magnetize the windings and

ends with up = −1 to demagnetize them. The phase positions where the magnetization

and demagnetization start are denoted with θon(Q) and θoff(Q), where Q ∈ 1, . . . , 4represents the machine operating quadrant. The quadrants are defined in Table 4.2,

along with the nominal phase activation/deactivation position values, θon(Q) and

θoff(Q), which are listed in the last column. The θon(Q) and θoff(Q) positions directly

follow from the 4/2 SRM torque characteristics shown in a lower part of Fig. 4.2,

and together describe a nominal, fixed four-quadrant phase commutation strategy.

Figure 4.5 illustrates the phase torque τe,p, in the case when ip > 0 within the θp

interval delimited by θon(Q) and θoff(Q).

Table 4.2: Quadrant specification

Q Operation Direction Condition (θon, θoff)

1 motoring positive τ ∗e ≥ 0 ∧ ω ≥ 0 (0, π/2)

2 generating negative τ ∗e ≥ 0 ∧ ω < 0 (π/2, 0)

3 motoring negative τ ∗e < 0 ∧ ω < 0 (π, π/2)

4 generating positive τ ∗e < 0 ∧ ω ≥ 0 (π/2, π)

θp

τe,p

0

θp

τe,p

0

θp

τe,p

0 ππ2

θp

τe,p

0

τ∗e

ω0

Q = 1Q = 2

Q = 3 Q = 4

ππ2

ππ2

ππ2

Figure 4.5: Phase torque τe,p as a function of the phase position θp for the case

of a nominal phase commutation strategy.

Based on this commutation scheme the nominal (constant) phase current level i∗(τ ∗e )


can be computed by solving

mini∗

i∗, subject to (4.19)

2

π

b∫

a

τe,1(i∗, θ)dθ = τ ∗e ,

i∗ ∈ [0, imax],

where (a, b) = (0, π/2) when τ ∗e ≥ 0 and (a, b) = (π/2, π) when τ ∗e < 0. In other words,

the aim of (4.19) is to find the lowest (constant) phase current, not greater than imax,

such that it yields the desired average electromagnetic torque. Notice that the average

torques outside of the range [τe,min0, τe,max0] = [−0.284, 0.287]Nm are infeasible due to

the imposed current constraint, see Fig. 4.6.

−0.2 −0.1 0 0.1 0.2 0.30

5

10

15

20

τe,max0 →← τe,min0

Currenti∗

[A]

Torque τ∗

e [Nm]

Figure 4.6: Nominal reference current level i∗ vs. the desired average torque τ ∗e .

Varying commutation strategy

The fixed commutation strategy is effective only when the (abs.) speed is low. Namely,

with higher |ω| values commutation needs to be adapted to compensate for the effects

of current dynamics (4.1). In particular, as the speed increases so does the relative

influence of the back e.m.f. Mpω on the phase current rise/fall, w.r.t. the phase voltage

vp. From (4.5)-(4.7) it can be deduced that sign(Mpω) = sign(τe,pω), implying that a

larger |Mpω| will yield a slower phase current rise during motoring (since Mpω > 0)

and a slower phase current fall during generating (since Mpω < 0). In both cases the

ideal phase current profile becomes distorted which leads to the loss of average torque,


either because the desired current level is reached too late or because the current does

not extinguish on time.

To address these issues, the phase activation/deactivation positions, θon(Q) and θoff(Q),

are advanced3 w.r.t. their nominal values θon(Q) and θoff(Q). The effect of the

commutation angle advance on the phase current/torque is shown in Fig. 4.7. Clearly,

the current/torque profiles become less distorted and the average torque is higher in the

case of commutation advancement.

0 90 180 2700

10

Current[A

]

i1 nom.

i∗1 nom.

i1 adv.

i∗1 adv.

0 90 180 270−0.1

0

0.1

0.2

Torque[N

m]

Position θ [deg]

τe,1 nom.

τe nom.

τe,1 adv.

τe adv.

Figure 4.7: The effect of commutation angle advance on the phase 1 cur-

rent/torque production at vdc = 50V and ω = 15000rpm.

Since the resulting θon(Q) and θoff(Q) can lie outside of the admissible θp interval [0, π],

their ”wrapped“ versions are defined as

[θ′, θ′′]T

=

mod([θon, θoff ]T , π), for ω ≥ 0,

mod([θoff , θon]T , π), otherwise.(4.20)

3Shifted in the direction opposite to that of rotation.


They are used to compute the phase commutation (activation) signal

σp =

1, for (θ′ < θ′′ AND θp ∈ [θ′, θ′′]) OR

(θ′′ < θ′ AND θp /∈ [θ′′, θ′]),

0, otherwise,

(4.21)

which further defines the phase reference current

i∗p = σpi∗. (4.22)

Optimal commutation angles

The optimal θon and θoff commutation angles and the optimal current level i∗ are found

for a set of SRM operating points given by the triplets (τ ∗e , ω, vdc), τ∗e ∈ [τe,min0, τe,max0],

ω ∈ [−ωmax, ωmax], vdc ∈ [vdc,min, vdc,max], by solving

minθon,θoff ,i∗

i∗, subject to (4.23)

τe(θon, θoff , i∗, ω, vdc) = τ ∗e ,

θon, θoff ∈ [−π, 3π],

|θoff − θon| ∈ [π/2, π],

i∗ ∈ [0, imax],

where ωmax = 40000rpm is the maximal considered SRM speed, vdc,min = 50V and

vdc,max = 100V the minimal and the maximal DC-link voltage and τe the steady-

state average of the total electromagnetic torque2∑p=1

τe,p(ip, θ), produced by the SRM

during a time period T = π/|ω|. The total torque is computed by simulating the phase

current (4.1) and position (4.2) dynamics at a specified speed ω and voltage vdc, while

the phase currents ip are forced to track the reference (4.22) defined by θon, θoff and

i∗. The currents are controlled using a proportional controller with output saturation,

given by

up = sat(κcc[i∗p − ip

],−1, 1

), (4.24)

where κcc = 5 represents a proportional tuning gain and

sat(x, xmin, xmax) =

xmin, for x ≤ xmin,

xmax, for x ≥ xmax,

x, otherwise,

(4.25)


the classical linear saturation function. The closed-loop simulation was conducted

with the help of Simulinkr while Matlabr (global) optimization toolbox was used to

implement and solve the constrained nonlinear programming problem (4.23). Notice

that by minimizing current i∗ in (4.23), for each specific speed ω, voltage vdc and torque

τ ∗e , the proposed approach puts an emphasis on high-efficiency operation of the SRM.

To support easier real-time implementation the optimal θon, θoff and i∗ have been

parameterized by the variables defining the SRM operating point. In particular, the

optimal current level i∗ is represented as

i∗(Q, τ ∗e , ω, vdc) = i∗(τ ∗e e

(ω/ωb(Q,vdc))2), (4.26)

where ωb(Q, vdc) denotes the quadrant-dependent base speed which separates the

constant torque (|ω| ≤ ωb) and constant power (|ω| > ωb) regions of the SRM operation.

The base speed is related to the DC-link voltage, i.e.,

ωb = κb(Q)vdc, (4.27)

where κb(Q) is a constant coefficient. Furthermore, the commutation angles are

described as

θon(Q, q) = θon(Q)− αon(Q, q), (4.28)

θoff(Q, q) = θoff(Q)− αoff(Q, q),

where αon(Q, q) and αoff(Q, q) denote the optimal advance angles parameterized using

the scheduling signal4

q =i∗

imax

|ω|ωmax

vdc,minvdc

, (4.29)

where ωmax = 40000rpm is the maximum considered SRM speed and vdc,min = 50V

the minimal expected DC-link voltage. Moreover, because of similarity between the

positive and negative phase torque characteristics (in the absolute sense, see the lower

plot in Fig. 4.2), similar optimization results were obtained in both directions of rotation.

In other words, the following identities hold

i∗(3, τ ∗e , ω, vdc) ≈ i∗(1, τ ∗e , ω, vdc), (4.30)

i∗(2, τ ∗e , ω, vdc) ≈ i∗(4, τ ∗e , ω, vdc),

αon(3, q) ≈ −αon(1, q),

αon(2, q) ≈ −αon(4, q),

ωb(3, vdc) ≈ −ωb(1, vdc),ωb(2, vdc) ≈ −ωb(4, vdc).

4Motivation behind this choice is explained in the Appendix C.


The final optimal commutation angles are shown in Fig. 4.8.

0 0.5 1

−45

0

45

90Q = 1

0 0.5 10

45

90

135

Q = 2Comm.angles[deg]

0 0.5 190

135

180

225Q = 3

Scheduling signal q [-]

Comm.angles[deg]

0 0.5 1

45

90

135

180Q = 4

Scheduling signal q [-]

θon

θoff

Figure 4.8: Optimal commutation angles θon and θoff as a function of the

scheduling signal q in different operating quadrants Q.

Speed-dependent torque limits

For certain SRM operating points (τ ∗e , ω, vdc) the problem (4.23) becomes infeasible.

In fact, at every given speed ω and the DC-link voltage vdc there exist limit torques,

τe,min(ω, vdc) and τe,max(ω, vdc), such that τe ∈ [τe,min, τe,max] for all feasible commutation

angles and current levels. These limits can be specified in parametric form by

solving (4.26) for τ ∗e when i∗ = imax, which yields

τe,min(ω, vdc) = τe,min0e−(ω/ωb(4,vdc))

2

, (4.31)

τe,max(ω, vdc) = τe,max0e−(ω/ωb(1,vdc))

2

,

when ω ≥ 0,

τe,min(ω, vdc) = τe,min0e−(ω/ωb(1,vdc))

2

, (4.32)

τe,max(ω, vdc) = τe,max0e−(ω/ωb(4,vdc))

2

.

4.3 SRM control design 77

i.e., when ω < 0. The comparison between the torque bounds obtained using the

nominal commutation strategy and the varying commutation strategies with the optimal

and with the parameterized reference current parameters (θon, θoff , i∗), is shown

in Fig. 4.9.

−40 −30 −20 −10 0 10 20 30 40

−0.2

−0.1

0

0.1

0.2

Speed ω [krpm]

Torqueτe,min,m

ax[N

m]

nom.opt.par.

Figure 4.9: Average torque bounds in different operating quadrants Q for

vdc = 50V and imax = 20A, for the case of the nominal, varying-optimal and

varying-parameterized commutation strategy.

4.3 SRM control design

The proposed 4/2 SRM four-quadrant control system is illustrated in Fig. 4.10. The

speed controller calculates the electromagnetic torque needed for tracking of the rotor

speed reference ω∗. Its output τ ∗e is translated into a set of low-level control parameters,

i.e., the reference current level i∗ and the (wrapped) commutation angles, θ′ and θ′′.

They are passed to a low-level current control and commutation component which

implements the 4Q commutation logic and computes the appropriate phase control

inputs up, i.e., the duty cycles dup and dlp. Furthermore, a supervising state-machine is

introduced to enable the machine startup and the change of its direction of rotation,

under all circumstances. The rotor speed ω and phase position θp estimates, p ∈ 1, 2,are derived from the measured rotary encoder position θenc. The implementation details

are presented in the remainder of this Section.


ip

dup , dlp

mode

vdc

θp

controlSpeed

ω Speed &position

ω∗ SRMθ′, θ′′

i∗

Param.

Supervisor

τ∗e

tsc tcc

θenc

Currentcontrol &

commutation

Figure 4.10: Structure of the proposed four-quadrant SRM control system.

4.3.1 Speed and position estimation

The position θ is measured using an incremental rotary encoder with a resolution θres =0.25π180

rad, which yields the encoder position θenc. The encoder is mounted on the SRM

shaft such that its index (absolute reference) position coincides with the phase 2 aligned

position, i.e., θ2 = π2

and θ1 = 0. Absolute position measurement is achieved by applying

a hardware-reset of θenc to 0 whenever the index pulse is detected. This yields θenc in

the range [−2π, 2π]. The phase position θp estimates are found as

θp = mod(θenc + (p− 1)

π

2, π), p = 1, 2. (4.33)

Furthermore, at the time instant k ∈ Z≥0 with the sampling time tcc = 5 · 10−5s, the

rotor speed estimate ω(k) is computed from the difference δ(k) = θenc(k)− θenc(k − 1).

First, δ(k) is processed to remove its erroneous values (spikes of ±2π, see Fig. 4.11)

occurring when θenc transits from −2π or 2π to 0 due to index reset. This yields

δ(k) =

δ(k)− 2πsign

(δ(k)

), for |δ(k)| ≥ δmax,

δ(k), otherwise,(4.34)

where δmax = ωmaxtcc denotes the maximum position difference. Then, the result δ(k)

is filtered using a discretized version of a low-pass filter Glp(s) = 1tcc

1κlps+1

with the filter

time constant κlp = 1/10π. This provides the speed estimate

ω(z) = Glp(z)δ(z). (4.35)

4.3.2 Supervisor for startup and change of rotational direction

In a 4/2 SRM, the ability of each phase to produce torque is considerably reduced when

the rotor is (nearly) aligned to a pair of stator poles, see the right-hand side of Fig. 4.4.


0 0.05 0.1 0.15 0.20

90

180

270

360

Position[deg]

θenc

θ1

0 0.05 0.1 0.15 0.2−360

−270

−180

−90

0

Difference[deg]

T ime t [s]

δ

δ

Figure 4.11: The measured encoder output (in degrees) at ω = 600rpm.

As a consequence, the rotor may get “stuck” in place when starting from a standstill

position. This can occur either at startup or when the speed ω is required to change its

sign. In such cases, the torque produced by the active phase is insufficient to move the

rotor past the nearest alignment point. The rotor is instead further aligned with the

phase, i.e., pulled towards a location with an even lower torque. This eventually causes

it to stop moving.

The remedy that can be applied in such situations is to deactivate the phase winding

closest to the rotor and activate the opposite one5. In this manner, the rotor can be

pulled away from a “stuck” position since the torque in this direction increases with

each small displacement. Once the rotor has moved far enough (or after a certain time

interval/speed level) the original phase activation strategy can be restored, thereby

yielding the rotation in the desired direction. In other words, the rotor can be let loose

by swinging it backwards.

The “stuck” and “loose” situations are detected by a supervising state-machine shown

in Fig. 4.12. The Supervisor monitors the estimated speed ω and compares it to the

speed thresholds ωstop and ωrun, where 0 ≤ ωstop < ωrun. It outputs a mode ∈ −1, 1signal which is used to switch between the “normal” and the “swing” operation of a

speed controller.

5Or, if the phase mutual couplings are not negligible, one could activate both phases simultaneously

to provide the rotor with an initial offset w.r.t. the nearest aligned position, see [97].


|ω| > ωrun|ω| < ωstop

mode = 1

Run

Wait

after(twait)

after(tswing)

Swing

mode = −1

Figure 4.12: Supervising state-machine.

The Supervisor is initialized in the Run state where “normal” controller operation is

enforced by setting mode = 1. However, as soon as the (abs.) speed becomes lower than

the threshold ωstop a switch to the Wait state is made. The entry to this state implies

that the rotor is potentially unable to move. Thus, if this state persists for at least twait

seconds, i.e., if during this time interval the (abs.) speed does not become larger than

ωrun, a switch to the Swing state occurs. In this state mode = −1, which causes the

speed controller to enter “swing” operation by inverting the sign of its output τ ∗e for

the duration of tswing seconds. As explained, this pulls the rotor towards the opposite

phase, thereby letting it loose. Finally, a transition back to the Run state is made, so

that the “normal” speed controller operation can continue.

4.3.3 Speed control

The speed controller is designed to compute the desired average electromagnetic torque

τ ∗e so that the speed error ω∗ − ω is driven to zero. In particular, a standard back-

calculation-based, anti-windup PID control law is applied [98]. The PID controller is

parameterized by the proportional κP , integral κI , derivative κD, derivative filter κN

and anti-windup κA tuning gains, see Fig. 4.13. Their values have been tuned manually

to ensure stability of the speed dynamics (4.3).

The application of anti-windup helps to improve the controller responsiveness in the

presence of the input constraint τ ∗e ∈ [τe,min, τe,max]. It ensures that the PID’s internal

integrator discharges once the controller output τ ∗e reaches the specified torque limits.


At the given speed and the DC-link voltage, the torque limits τe,min and τe,max are

computed using (4.31) and (4.32).

The output of the saturation block shown in Fig. 4.13 is multiplied with the mode signal

produced by the Supervisor. This changes its sign during the “swing” operation so that

the rotor can be temporarily pulled in the direction opposite to the one of the reference

speed ω∗. In this way, the rotor can get loose from a “stuck” position as explained in

the previous section.

The speed controller (as well as the Supervisor and the Parametrization block) is

implemented at the discrete sampling time tsc = 5 · 10−4s.

1s sat(·)+

κP

κI

κN

κA

1s

−+

+

+

−

−+

ω∗

ωτe,min(·)

τe,min, τe,max

vdc

×

mode

κD

τ∗e

τe,max(·)

Figure 4.13: PID speed controller with a back-calculation integrator anti-

windup.

4.3.4 Current reference parametrization

Based on the desired speed controller torque τ ∗e , speed estimate ω and the DC-link

voltage vdc, the desired SRM operating quadrant Q is determined as in Table 4.2, the

phase current reference level i∗ as in (4.26), the scheduling signal q as in (4.29), the

commutation angles, θon and θoff , as in (4.28) and the wrapped commutation angles, θ′

and θ′′ as in (4.20). The current reference parameters i∗, θ′ and θ′′ are then passed to

the low-level current controller and commutation component.


4.3.5 Current control and commutation

The control parameters θ′, θ′′ are used to compute the binary signal σp, as in (4.21).

The signal σp defines the active phase region, w.r.t. the estimated phase position θp, and

is multiplied with i∗ to produce the phase current reference i∗p, as in (4.22). The phase

current is controlled using a simple proportional control law (4.24), which produces the

phase input up. Finally, up is translated into a set of AHBC switch duty cycles dup and

dlp according to (4.17).

The current control and commutation (as well as the speed and position estimation) is

implemented at the discrete sampling time tcc = 5 · 10−5s. Thus, due to the difference

between the speed and the current controller sampling rates, the signals ω, θ′, θ′′ and

i∗ are passed through rate transition blocks when transiting between the two domains.

4.4 Hardware and software implementation

4.4.1 Hardware configuration

Figures 4.14 and 4.15 show the experimental hardware arrangement and the SRM test-

bed used in this project. The test-bed consists of two identical 4/2 SRMs (manufactured

by Dyson Ltd.), coupled via a Magtrol TMHS-304 torque sensor. Speed ω is controlled

using SRM1 while the load torque τl is controlled using SRM2. For this purpose,

the proposed 4Q controller is used to control the SRM1, whereas its simplified version,

without the Speed Control and the Supervisor components, is used to control the SRM2.

A custom-made measurement board enabled the measurement of the phase and the

DC-link currents and voltages. The currents ip, p ∈ 1, 2 were measured using

galvanically isolated LEM LA 55-P current transducers with a bandwidth of approx.

200kHz while the voltage vdc was measured using a galvanically isolated LEM LV 25-P

voltage transducer with a bandwidth of approx. 200kHz. The voltage measurement

is subsequently low-passed using an analog low-pass filter with a bandwidth of 5Hz.

Furthermore, a custom-made power board implemented the AHBC shown in Fig. 4.3.

The phase switches were realized using insulated-gate bipolar transistors (IGBTs).

The data acquisition and real-time control implementation was performed using a

dSPACE DS1103 PPC controller board with MATLABr/Simulinkr installed on a

standard personal computer (PC) running a 64-bit Windows 7 operating system. The

real-time hardware input/output (IO) access was obtained through the use of Simulinkr

CoderTM blocks.

4.4 Hardware and software implementation 83

The SRM’s windings were powered by the Delta Elektronika SM 300-20 programmable

power supply capable of delivering DC voltage of up to 300V and current of up to 20A.

Due to torque sensor limitations the speed was limited to 40000rpm.

1 5

4

32

Figure 4.14: Experimental setup – hardware arrangement: 1) DS1103 PPC 2)

power and measurement board 3) SRM test-bed, 4) Windows 7 PC 5) power

supply.

2

1 3

Figure 4.15: Experimental setup – SRM test-bed: 1) SRM1 2) torque sensor

3) SRM2.

4.4.2 Experimental results

Table 4.3 lists the system and the controller design parameters and their values. The

experimental results demonstrating the effectiveness of the proposed 4Q SRM control

algorithm are shown in Figures 4.16a to 4.17.


Table 4.3: System and controller design parameter values

Type Parameter Value

SRM

Nr 2

m 2

Pdc,rated 500W

vdc 50V

La 6.3 · 10−3H

Lu 0.3 · 10−3H

R1 = R2 0.45Ω

Speed estimation κlp1

10πs

Supervisor

ωstopπ3rad/s

ωrun2π3

rad/s

twait 0.1s

tswing 0.02s

Speed control

κP 1.2 · 10−3

κI 3.1 · 10−3

κD 0.2 · 10−3

κN 1.5

κA 2.5

tsc 5 · 10−4

Parametrization

[vdc,min, vdc,max] [50, 100]V

ωmax4000π

3rad/s

imax 20A

[κb(1), κb(4)] [62.7, 96.5]rad/Vs

[τe,min0, τe,max0] [−0.284, 0.287]Nm

Current control

fpwm 6 · 104Hz

κcc 5

tcc 5 · 10−5s

Sinusoidal speed reference tracking

In the first plot from the top of Fig. 4.16a it can be seen that the desired average

torque τ ∗e is at all times within the imposed time-varying bounds specified by τe,min

and τe,max. Also it is visible that its shape is nearly sinusoidal, i.e., it resembles that

of the tracked speed reference ω∗. The similarity between the two implies that the

nonlinearities inherent to the SRM dynamics have been for the most part compensated

by the proposed current reference parametrization method. As a result, good speed

4.4 Hardware and software implementation 85

tracking quality is achieved in both directions as it can be seen from the second plot

from the top of Fig. 4.16a. The third one from the top, presents the SRM operating

quadrant Q. As expected, when ω > 0, the reference is tracked by alternating between

Q = 1 (mot+) and Q = 4 (gen+), whereas when ω < 0 the quadrant switches between

Q = 3 (mot-) and Q = 2 (gen-). Finally, a bottom plot of Fig. 4.16a shows the

scheduling of the commutation angles according to the operating quadrant, i.e., the

SRM operating point.

0 5 10 15 20

−0.2

−0.1

0

0.1

0.2

0.3

Torque[N

m]

τ∗

e

τe,min,max

0 5 10 15 20−15

−10

−5

0

5

10

15

Speed[krpm]

ω∗

ω

0 5 10 15 20

−1

0

1

2

3

4

Quadrant/mode[-]

Qmode

0 5 10 15 20

0

45

90

135

180

Comm.angles[deg]

Time t [s]

θonθoff

(a) Sinusoidal

0 5 10 15 20

−0.2

−0.1

−0.05

0

0.06

0.1

Torque[N

m]

τ∗

eτl

0 5 10 15 20

8

11.5

15

Speed[krpm]

ω∗

ω

0 5 10 15 20

1

2

3

4

Quadrant/mode[-]

Q

mode

0 5 10 15 20

0

45

90

135

180

Comm.angles[deg]

Time t [s]

θon

θoff

(b) Block

Figure 4.16: Tracking of a speed reference.


Block speed reference tracking under load

Figure 4.16b shows the tracking of a block (step) speed reference in the presence of load

torque τl 6= 0. In the top plot it can be seen that the adopted algorithm is successful in

rejecting the load torque disturbance irrespective of its sign. In particular, when the load

changes from 0.06Nm to −0.05Nm the controller quickly compensates for the change by

switching the machine operating quadrant Q from 1 (mot+) to 4 (gen+). At t ≈ 12s

the speed reference is abruptly reduced from 15krpm to 8krpm while τl = 0.06Nm.

However, it can be seen that accurate tracking is still maintained, even under load.

Supervised startup

Figure 4.17 zooms on the beginning of the first (sinusoidal reference tracking) exper-

iment where the rotor was initially placed at a position θ1 ≈ 90deg. As initially the

reference speed was positive and the speed was zero, the controller started operation

in the quadrant Q = 1 (mot+). Since θ1 < 90deg the adopted commutation strategy

resulted in phase 1 activation. However, due to the inability of the phase 1 to produce

sufficient torque and thus move the rotor past its alignment point – the rotor got “stuck”.

This caused it to vibrate in place, with its speed alternating between positive and

negative (which eventually caused a switch to Q = 2, gen-). However, after twait = 0.1s

had elapsed with ω < ωstop condition true, the “stuck” situation was detected by the

Supervisor and the Swing state was activated. This can be seen from the change in the

mode signal from 1 to −1, i.e., the resulting change in the sign of the desired torque

τ ∗e . The negative torque request was fulfilled by switching the quadrant to Q = 3

(mot-), i.e., by engaging the phase 2. As a consequence, during twait = 0.02s the rotor

was pulled away from the “stuck” position after which the regular operation had been

resumed.

4.5 Conclusions

In this Chapter, a novel four-quadrant control scheme for the 4/2 SRM was proposed.

The control algorithm was designed to explicitly handle the physical torque and current

constraints and to allow the SRM to start/resume operation even when the rotor gets

“stuck” near or at one of the aligned positions. A dedicated commutation strategy

was also derived, which supports the compensation of the inherent torque loss in a

wide range of the SRM operating conditions. Experimental results were provided to

demonstrate the feasibility of the presented solution.

4.5 Conclusions 87

0.65 0.7 0.75 0.8 0.85

−0.2

−0.1

0

0.1

0.2

0.3Torq

ue[N

m]

τ∗

eτe,min,max

0.65 0.7 0.75 0.8 0.8575

90

105

120

Position[deg]

θ1

0.65 0.7 0.75 0.8 0.85

−50

0

50

Speed[rpm]

ω

0.65 0.7 0.75 0.8 0.85

−1

0

1

2

3

Quadra

nt/

mode[-]

T ime t [s]

Qmode

Figure 4.17: 4/2 SRM startup from the “stuck” position.

88

Chapter 5

Model predictive voltage control of

high-speed Switched Reluctance

Generators 1

Abstract This Chapter presents a predictive voltage control strategy for high-speed Switched

Reluctance Generators, derived using an average model of the SRG electrical system. The

controller computes an average DC-link current needed to track a desired DC-link voltage

reference, in the presence of an unknown electrical load. The production of the required DC-

link current is ensured by means of single-pulse commutation and control. For this purpose,

the turn-on and turn-off commutation angles are chosen such that the resulting peak phase

flux-linkage and, indirectly, the machine iron loss, are minimized. The proposed control scheme

explicitly enforces the average DC-link current and voltage bounds. This is also verified in

closed-loop simulations.

5.1 Introduction

The Switched Reluctance Generator (SRG) is an electric machine characterized by high

reliability, fault tolerance and power density [99]–[101]. It has salient poles both on a

rotor and a stator whereas it lacks brushes, rotor windings and permanent magnets.

Due to its simple and rugged structure the SRG is easy to cool and maintain, and can

be operated in a wide range of speeds.

Specifically, at speeds higher than the base speed the SRG is often controlled in a Single-

1This chapter is based on [94].

89


Generators

Pulse (SP) mode where the turn-on and turn-off commutation angles are the only control

parameters used to adjust the amount of produced DC-link current. This approach is

mainly motivated by a large phase back-emf which in such cases prevents an effective use

of phase current regulation and/or by the need to eliminate the hardware costs related

to current sensing. When compared to the actual rotor position, the commutation

angles provide information when the phase current pulses should occur. The generating

operation is achieved when these pulses are synchronized with a decreasing phase

inductance [102].

So far, numerous approaches for computation of optimal commutation angles have been

reported. In general, one can distinguish between online and offline methods, depending

on whether the angles are adapted during the SRG operation or determined and stored

prior to it. In either case, various optimality criteria have been employed. For instance,

in [78], [102] the optimal turn-on and turn-off commutation angles were found through

a series of steady-state experiments by minimizing the RMS phase current. A similar

approach was used in [99] where instead experiments were conducted to maximize the

ratio of the average output and input power. The extensive experimentation required

by these methods is however often impractical, especially when one considers variable

speed/voltage applications. Thus an alternative, offline solution, based on the SRG

Finite Element Model (FEM), was proposed in [103]. Here, the optimal commutation

angles were derived directly from FEM data by maximizing the SRG electro-mechanical

efficiency, i.e., its average produced torque. In contrast, an analytic approach was

pursued in [104]. Based on simplified SRG electromagnetic characteristics, this work

establishes a relationship between the commutation angles, peak phase flux-linkage,

applied DC-link voltage and rotational speed for the SRG with a regular machine

geometry [85]. In contrast, an online, search-based algorithm was proposed in [101]

that focuses on minimization of the DC-link current ripple. The reported algorithm

however does not take into account the changes in the SRM operating point (DC-link

voltage, speed), which limits its application.

Irrespective of the adopted commutation angle selection strategy, the SRM output

voltage is most commonly regulated using a PI control law [78], [102]–[104]. Thus

typically, the bounds on the DC-link voltage/current are not directly enforced. This

is in spite of the fact that the SRG loses its self-excitation capacity as soon as the

DC-link voltage drops to zero [105]–[108]. Recently several papers have described the

use of Model Predictive Control (MPC) for purposes of Switched Reluctance Motor

(SRM) torque control [109]–[111]. One of the main advantages of using the MPC over

conventional motor control techniques, such as the PID, is the possibility to enforce

current and voltage limits [112].

5.2 SRG modeling and operation 91

This Chapter presents a novel SRG voltage controller in the form of a computationally

cheap explicit model predictive control (MPC) law [113]. The proposed solution is

constructed on the basis of an average DC-link capacitor dynamics. The SRG is

controlled in the SP mode with the commutation angles found offline by means of open-

loop simulations. Specifically, they have been computed by minimizing the simulated

peak phase flux-linkage and thus, indirectly, the SRG iron loss. The derived control

algorithm is primarily intended for high-speed and/or low-cost SRG applications.

The main contributions of this research are:

1. Development of a linear MPC law that explicitly enforces the SRG average DC-link

current and voltage bounds and enables tracking of the DC-link voltage reference,

in the presence of an unknown electrical load. In this way, the SRG self-excitation

capability can be preserved2.

2. Development of a model-based commutation angle selection strategy based on

measured SRG electromagnetic characteristics. The resulting SP mode commu-

tation angles are parametrized by the desired average DC-link current, rotational

speed and the DC-link voltage.

This paper is organized as follows. Section 5.2 provides the nonlinear continuous time

equations describing the physical behavior of the SRG electrical generation system. It

also introduces a simplified, high-speed model of the system used in the design of the

proposed control law. In Section 5.3 the MPC voltage controller and the procedure for

finding the optimized excitation angles are given. Section 5.4 contains a representative

numerical example, and conclusions are presented in Section 5.5.

5.2 SRG modeling and operation

5.2.1 Actuation

The SRG phase voltage vp, p ∈ 1, . . . ,m, where m denotes a phase count, is often

regulated using an Asymmetric Half Bridge Converter (AHBC), see Fig. 5.1. The AHBC

provides two controllable switches per phase, Sup and Slp, which are manipulated in

unison during the SRG single-pulse (SP) mode of operation. In particular, when both

switches are closed vp = vdc, when they are both open vp = −vdc, where vdc denotes the

DC-link capacitor voltage. The opening/closing of Sup and Slp is performed by setting

2Assuming no load faults.


Generators

C

idcil

+v1 v2

Su1 Su

2

Sl1 Sl

2

i1 i2

L1 L2

vdc

Figure 5.1: Two-phase 4/2 SRG AHBC circuit.

their corresponding inputs dup and dlp to zero/one. This occurs at precise rotor locations

specified by the turn-on (θon) and turn-off (θoff) commutation angles. To capture the

described functionality it is useful to define the phase input

up =

1 for θon ≤ θp < θoff,

−1 otherwise,(5.1)

where

θp = mod (θ + (p− 1)θspp, θrpp) , (5.2)

is the rotor position w.r.t. phase p, with θ being the absolute rotor position (θ = 0

corresponds to the unaligned position of phase 1), θspp = 2πNs

the stator pole pitch angle,

θrpp = 2πNr

the rotor pole pitch angle and Ns and Nr the stator and the rotor pole counts.

In this context, the phase voltage reads

vp = vdcup (5.3)

and the produced DC-link current is given by

idc =m∑

p=1

ipup, (5.4)

where ip represents the current in the phase winding p. Furthermore, the switch inputs

can be formulated as

dup = dlp =up + 1

2. (5.5)


5.2.2 Dynamics

The voltage equation for the SRG phase p reads

dψpdt

= upvdc −Rpip (5.6)

and the DC-link capacitor dynamics can be formulated as

dvdcdt

= − 1

C(idc + il), (5.7)

with ψp denoting the phase flux-linkage, Rp the phase winding resistance, il the unknown

electrical load current and C the DC-link capacitor capacitance. Note that due to a

presence of diodes in the AHBC circuit it holds ψp ≥ 0 and ip = ip(ψp, θ) ≥ 0. The

expressions (5.6) and (5.7) capture the SRG electrical system dynamics. Its mechanical

part, is described by

dθ

dt= ω, (5.8)

where ω is the rotor speed.

Equation 5.7 implies that an instantaneous DC-link voltage vdc can be controlled using

an instantaneous DC-link current idc as a control input. However, this would require

precise idc control which is impossible in the single-pulse mode, where only the beginning

and the end of phase excitation can be specified (and not the shape of the current in

between). Instead, an average DC-link current idc can be used to design a controller

for an average DC-link voltage vdc. For this purpose, an average model of capacitor

dynamics is introduced by replacing idc with idc, vdc with vdc and il with an average

load current il in (5.7), i.e.,

dvdcdt

= − 1

C(idc + il). (5.9)

5.2.3 Analytic commutation angles

To gain insight into the relationship between the DC-link current, voltage, speed and

the commutation angles, an analytic expression for idc has been derived by averaging idc

during one electrical period of length θrpp, see Appendix D. For this purpose, a number

of simplifying assumptions has been made, i.e.,

• All SRG phases have the same electromagnetic properties (such as inductance,

resistance, etc.).


Generators

• During one electrical period the voltage vdc and speed ω vary slowly enough that

they can be considered constant.

• The phase ohmic voltage drop Rpip in (5.6) can be neglected since |Rpip| vdc.

• The SRG operates under non-saturating conditions yielding Lp(ip, θ) = Lp(θ), i.e.,

ip = ψpLp(θ)

, where Lp > 0 represents the phase inductance.

• In its decreasing region, given by θ ∈[θrpp

2, θrpp

], the phase inductance L1(θ) can

be approximated by an affine function, i.e., L1(θ) ≈ kLθ + nL.

• The commutation angles satisfy θon ≥ θrpp2

and θe = 2θoff − θon ≤ θrpp.

These simplifications give rise to Lp, ψp, up and idc profiles (w.r.t. position θ) illustrated

in Fig. 5.2, i.e., to the average DC-link current

idc ≈mvdcθrppωk2

L

[2∆kL − Lon ln

(Loff

Lon

)+ Le ln

(Loff

Le

)], (5.10)

where kL < 0 is the slope of Lp when it is decreasing, ∆ = θoff− θon the dwell angle and

Lon, Loff and Le the inductance Lp values at the angles θon, θoff and θe, respectively.

LonLoffLe

vdcω

∆

1

−1

Lp

ψp

up

idc

θon

θoffθe θ

0

0

θ

θ

θ

Figure 5.2: Simplified phase inductance Lp, flux-linkage ψp, input up and the

DC-link current idc profiles, w.r.t. rotor position θ.

Equation (5.10) can be rewritten as

q ≈ mvdc,minθrppidc,minωmaxk2

L

(5.11)

·[2∆kL − Lon(θoff ,∆) ln

(Loff(θoff)

Lon(θoff ,∆)

)+ Le(θoff ,∆) ln

(Loff(θoff)

Le(θoff ,∆)

)],

≈ f(∆, θoff),


where

q =idcω

vdc

vdc,minidc,minωmax

, (5.12)

will be used as the commutation angle scheduling signal, vdc,min the minimal allowed

average DC-link voltage, ωmax the maximal expected speed and idc,min < 0 the minimal

allowed average DC-link current. Thus, e.g., by fixing the turn-off angle to θoff = 3θrpp4

,

as this value yields the largest feasible dwell angle range ∆ ∈ [0, θrpp4

], an inverse of (5.11)

can be found numerically. However, this only holds for a limited range of q ∈ [0, qub].

The angles corresponding to q outside of this range can be obtained by extrapolating

the resulting relationship ∆ = f−1(q, θoff = 3θrpp

4

)using the function ∆ = a1q

b1 , a1 > 0,

b1 ∈ (0, 1). The described procedure has been applied to a case of a two-phase 4/2 SRG

which resulted in commutation angles shown in Fig. 5.3.

0 0.2 0.4 0.6 0.8 10

45

90

135

180

q [-]

Comm.angles[deg]

θoff

∆

∆

Figure 5.3: 4/2 SRG commutation angles obtained using the analytic approach.

5.2.4 Optimized commutation angles

Equation (5.11) implies that multiple combinations of commutation angles can yield the

same value of the scheduling signal q. This creates space for angle optimization, e.g., by

minimizing root-mean-square DC-link current or peak phase flux-linkage. In this work,

the minimization of the peak phase flux-linkage has been pursued. However, due to its

limited validity, the established analytic relationship between q and the commutation

angles, given by (5.11), was not used for this purpose. Instead, the commutation angles

were found by means of numerical optimization.


Generators

Specifically, the optimal ∆∗ and θ∗off were computed for a range of desired average DC-

link current values i∗dc ∈ [idc,min, 0], at the DC-link voltage vdc = vdc,min and the SRG

rotor speed ω = ωmax, by solving

min∆∗,θ∗off

ψ1,max, subject to (5.13)

idc(∆∗, θ∗off , ω, vdc) = i∗dc,

∆∗ ∈ [0, θ∗rpp],

θ∗off ∈ [θrpp/2, θrpp],

where idc denotes the steady-state average of the DC-link current idc produced by

the SRG and ψ1,max the steady-state maximum of the phase 1 flux-linkage ψ1, within

one electrical period of length θrpp. The current idc was computed by simulating the

phase flux-linkage (5.6) and position (5.8) dynamics, i.e., by evaluating (5.4), for each

particular ∆∗ and θ∗off. For this purpose, the phase current functions ip = ip(ψp, θ) have

been numerically derived from measured phase flux-linkage characteristics.

Subsequently, by inserting i∗dc, vdc = idc,min and ω = ωmax in (5.12) the desired scheduling

signal q∗ was calculated for each optimized operating point. The result was used to

parametrize ∆∗ and θ∗off, yielding

∆ = a2qb2 , (5.14)

θoff = c0 + c1q + c2q2,

with a2 > 0, b2 ∈ (0, 1), c0 > 0, c1 ∈ R and c2 ≥ 0 being some fitting coefficients,

and ∆ the nominal (optimized) dwell angle value. The outcome of the optimization is

illustrated in Fig. 5.4.

Remark: The main SRG losses are the copper and the iron (core) losses, PCu and

PFe, with the iron loss PFe being the dominant loss component at high speeds [114].

The iron loss consists of hysteresis Phy and eddy current losses Ped, which can be

approximated [115] by

Phy = k1fψk2 , (5.15)

Ped = k3f2ψ,

where f is the flux-linkage excitation frequency and k1, k2, k3 > 0 are some unknown

empirical constants. Equation (5.15) implies that the iron loss can be reduced by

minimizing the (phase) flux-linkage ψ. This motivates the choice of objective function

in (5.13).

5.3 SRG control design 97

0 0.2 0.4 0.6 0.8 10

45

90

135

180

q [-]

Comm.angles[deg]

θ∗

off

θoff

∆∗

∆

Figure 5.4: 4/2 SRG commutation angles obtained by minimizing the peak

phase flux-linkage.

5.3 SRG control design

The proposed SRG control scheme is shown in Fig. 5.5. The model predictive voltage

dup , dlp

vdc

ω

v∗dc

SRG∆, θoff

Param.i∗dc

idcCurrent

control &commutation

filterKalman

ˆvdcîl

θ

MPCVoltage

Figure 5.5: Proposed SRG control system.

controller (MPC) is designed based on the average capacitor dynamics (5.9). The MPC

calculates the average DC-link current i∗dc required to enable tracking of the average

voltage reference v∗dc while compensating for the estimated average load current îl.

For this purpose, it receives both the average DC-link voltage ˆvdc and the current îl

from the employed Kalman filter. Apart from voltage tracking, the MPC ensures that

both i∗dc and ˆvdc reside within the pre-determined bounds – assuming a regular SRG

operation given by il ∈ [0, |idc,min|]. The computed MPC output i∗dc is translated into

a commutation angle pair (∆, θoff) using (5.12) and (5.14). The nominal dwell angle ∆

is used within the current control and commutation component to compute the actual


Generators

dwell angle ∆ such that the difference between the desired i∗dc and the estimated average

DC-link current îdc is reduced. Finally, the resulting commutation angles are compared

with the rotor position θ to obtain the AHBC phase switch inputs dup and dlp.

5.3.1 Model predictive voltage control

Due to a time-discrete character of the MPC the dynamics (5.9) is first discretized

(using exact discretization). If x = [vdc, il]T denotes the state, u = i∗dc the input and

y = vdc the output of such a system, its time-discrete state-space representation then

reads

x(k + 1) =

[1 −T

C

0 1

]

︸︷︷︸Ac

x(k) +

[−TC

0

]

︸︷︷︸Bc

u(k) + ρ(k), (5.16)

y(k) =[1 0

]

︸︷︷︸Cc

x(k) + η(k),

where k ∈ N0 is used to represent the discrete time instant tk = kT , with T being

the adopted MPC sampling time. The signals ρ and η denote white zero-mean state

and measurement noise described by their covariance matrices W1 and W2, respectively.

Since (5.16) is observable a Kalman filter was employed for estimation of the state x

from the measured output y and the input u. It was designed using MATLABr kalman

routine, where the matrices W1 and W2 were treated as tuning parameters. For more

information regarding Kalman filter design, see, e.g., [116].

Remark: Equation (5.16) includes a load current disturbance il as a part of the state.

The augmentation of the plant with a disturbance model is a standard practice in the

MPC, inspired by the Internal Model Principle (IMC) [117]. This particular choice

of the disturbance model is commonly used to remove a steady-state offset due to a

persistent disturbance [118].

By representing the predicted state and the output at time instant k + j, j ≥ 1, given

the state and the output at time k, as xk+j|k and yk+j|k, the MPC optimization problem

5.3 SRG control design 99

at a time k is defined as:

minU,u(k),...,u(k+N−1)

J =N∑

j=1

[Ru2

k+j−1 +Qy(yk+j|k − rk+j|k)2]

(5.17)

subject to:

xk|k = x(k),

xk+j|k = Acxk+j−1|k +Bcuk+j−1, yk+j|k = Ccxk+j|k, rk+j|k = r(k), j = 1 . . . N,

uk ∈ [umin, 0], yk ∈ [ymin, ymax], xk ∈ [xmin, xmax], ∀k,

where N denotes the prediction horizon, r = v∗dc the voltage reference, R and Qy the

state, input and output weighting matrices, umin = idc,min the minimal allowed (desired)

average DC-link current, ymin = vdc,min and ymax = vdc,max the minimal and maximal

allowed average DC-link voltage and xmin = [vdc,min, 0] and xmax = [vdc,max,−idc,min]

the corresponding state bounds.

A standard, online MPC requires that the problem (5.17) is solved at each time instant

k. Then, only the first entry of the resulting optimal input vector U is applied to the

system. This is known as the receding horizon principle. However in this work, an

alternative, explicit MPC [119] approach is pursued. In particular, the problem (5.17)

is rewritten in a form of a multi-parametric Quadratic Program (mp-QP) and solved

offline, which yields an explicit MPC. This controller is represented by a piecewise-affine

continuous-state feedback defined on a finite number of state-space contiguous regions,

given by

uj(k) = Fjx(k) +Gj, (5.18)

where j represents an active region index and Fj and Gj matrices resulting from the

mp-QP solution. In this way, an online computation of the control input u(k) is reduced

to an active region search and a simple linear function evaluation. The explicit MPC

law was computed using the multiparametric toolbox (MPT) [113] for MATLABr that

also provided the routines for state-space region partitioning and online active region

search [112]. The resulting regions are shown in Fig. 5.6.

5.3.2 Current control and commutation

The proposed current control and commutation scheme is shown in Fig. 5.7. The MAF

component represents a variable-period Moving Average Filter used for estimation of


Generators

|

Figure 5.6: The explicit MPC state-space region partitioning.

the average DC-link current îdc. It is defined as

îdc =ω

θrpp

t∫

t− θrppω

idc(τ)dτ. (5.19)

Any difference between its output îdc and the desired average DC-link current is

corrected using a PI controller GPI(s) = κP s+κIs

, where κP and κI denote the related PI

tuning gains. This correction is achieved by adding a small increment δ to the supplied

nominal dwell angle ∆ before it is used in commutation. The turn-on angle thus reads

θon = θoff − ∆, where ∆ = ∆ + δ. Finally, the switch inputs dup and dlp are found

by comparing the phase rotor position θp with the resulting commutation angles, i.e.,

from (5.1) and (5.5) it follows

dup = dlp =

1 for θon ≤ θp < θoff,

0 otherwise.(5.20)

5.4 Simulation results

The system and design parameter values used in simulations are listed in Table 5.1.

Figure 5.8 shows the tracking of the average voltage reference v∗dc in the presence of

varying average load current il and rotor speed ω. It can be seen that both the estimated

5.4 Simulation results 101

dup , dlpω

∆

i∗dc

idc

θ

PI+

−MAF îdc

θoff

+∆ −+

θon

Interval test

mod(θ + (p − 1)θspp, θrpp)θp

0

1

θon θoff

θp

dup , dlp

δ

Figure 5.7: Average DC-link current control and commutation.

Table 5.1: System and design parameter values


SRG

Nr 2

Ns 4

m 2

R1 = R2 0.45Ω

kL −3.8 · 10−3Hs/rad

nL 1.3 · 10−2H

C 6 · 10−4F

[vdc,min, vdc,max] [150, 360]V

idc,min −3 A

ωmax6000π

3rad/s

Parametrization

[a1, b1] [1.25, 0.32]

[a2, b2] [1.08, 0.41]

[c0, c1, c2] [2.68,−0.55, 0.18]

Speed control

T 10−3s

N 8

Qy 1

R 10

Kalman filterW1 [1, 0; 0, 1]

W2 105

Current controlκP 10−2

κI 20

voltage ˆvdc and computed i∗dc reside within their prescribed bounds, at all times. The

positive and negative step changes of the average load current il occur at t = 0.2s and

t = 0.8s. It is visible however that in both cases they are quickly compensated by a


Generators

suitable change in the average DC-link current. Nevertheless, when the speed abruptly

rises/falls, at t = 1s and t = 1.2s, the desired average DC-link current i∗dc remains

the same. This is because the changes in the speed and the average DC-link voltage

are directly compensated by the adopted commutation angle scheduling strategy, i.e.,

by modifying the scheduling signal q. This can be observed from the left plot from

the bottom. Furthermore, the resulting turn-on and turn-off commutation angles are

shown in Fig. 5.9, along with the output δ of the average DC-link current controller.

A comparably small size of δ (w.r.t. θon and θoff), demonstrates the effectiveness of

the proposed commutation angle selection method to yield the desired average DC-link

current i∗dc.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6150

200

250

300

350

Voltage[V

]

vdcv∗dcˆvdc

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−4

−3

−2

−1

0

1

2

3

4

5

Current[A

]

îdc

il

i∗dc

îl

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

t [s]

Schedulingsignal[-]

q

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

40

45

50

55

60

65

t [s]

Speed[krpm]

ω

Figure 5.8: Tracking of the average voltage reference v∗dc in the presence of

varying average load current il and rotor speed ω.

5.5 Conclusions 103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.690

135

180

t [s]

Comm.angles[deg]

θonθoff

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−1

−0.5

0

0.5

1

1.5

2

2.5

t [s]Dwellincrem

ent[deg]

δ

Figure 5.9: The turn-on and turn-off commutation angles (left) and the current

controller output (right).

5.5 Conclusions

In this Chapter a novel SRG voltage control scheme was proposed based on the explicit

Model-Predictive Control methodology. The SRG was considered to operate in the

Single-Pulse mode, with the turn-on and turn-off commutation angles being the only

current control parameters. For this purpose, two different commutation angle selection

strategies were introduced: one based on analytic calculations and one derived from

open-loop simulations of the relevant SRG dynamics. In the simulation-based case, the

peak phase flux-linkage was minimized and thus, indirectly, also the SRG iron loss.

The proposed control solution is implementable at high speeds (e.g., 50-100 krpm) and

capable of keeping the average DC-link current and voltage within their prescribed

bounds. Simulations, conducted for the case of two-phase 4/2 SRG, have shown that

the resulting control system was capable of smooth average voltage tracking in the

presence of both the varying average load current and rotor speed.

104

Chapter 6

Speed control of high-speed Switched

Reluctance Machines using only the

DC-link measurements 1

Abstract This Chapter presents a novel position-sensorless speed control strategy for high-

speed Switched Reluctance Machines. The main contribution of this work is an algorithm

for rotor position and speed estimation using the DC-link voltage and current measurements

only. This eliminates a number of hardware components related to position, speed, phase

current and phase voltage sensing. In this way, the electrical system’s costs can be further

lowered and its reliability increased. The effectivness of the derived solution is demonstrated

in closed-loop simulations.

6.1 Introduction

The Switched Reluctance electrical Machine (SRM) is characterized by the absence of

permanent magnets, rotor windings and brushes. It is robust and cheap to produce,

achieves high efficiency and is capable of operating in a wide speed range. This makes

it suitable for applications in automotive [121], aircraft [80], wind power [122] and home

appliance industry [123].

Due to its lack of brushes the SRM is electronically commutated which requires

accurate knowledge of either phase inductance or rotor position. The inductance-based

1This chapter is based on [120].

105

1066 Speed control of high-speed Switched Reluctance Machines using only the

DC-link measurements

commutation [122], [124] is achieved by comparing the instantaneous phase inductance

(measured or estimated) with its computed turn-on and turn-off values. This determines

the moments when each phase should be magnetized/demagnetized. However, as

inductance sensing is often difficult to perform the SRMs are more commonly position-

commutated. This is achieved by energizing each SRM phase in a rotor position interval

specified by turn-on and turn-off angles. Typically, an absolute rotary encoder is used

to measure the rotor position. This not only adds to the cost and complexity of the

system but also reduces its reliability. As a consequence, a lot of interest has emerged

for position-sensorless commutation, both in industry [123] and academia [125]. The

related control methods can be divided into hardware and software-based.

A hardware-based method is proposed in [126]. Here, an analog circuit is designed for

detecting a change in a phase current gradient that occurs when rotor and stator poles

begin or cease to overlap. The resulting gradient detection signal is used to obtain high

frequency position pulses by feeding it to a Phase-Locked Loop (PLL). However, apart

from a need for the additional hardware, this method is not suitable for applications

involving large load torque transients.

Software-based methods typically require knowledge of the relationship between phase

flux-linkage, current and position. These methods do not require any extra hardware

such as analog differentiators, inductance sensors, PLLs, etc. They can be further

split into open- and closed-loop methods. In an open-loop case [127]–[129], the

measured phase current and the computed flux-linkage are directly mapped to the

rotor position. Popular mapping choices constitute either look-up tables or certain

neural network/fuzzy logic-based approximators. As the map is constructed (trained)

off-line, these methods bear low real-time computational requirements at the cost of

increased sensitivity to modeling and measurement errors. The closed-loop, observer-

based methods [130]–[133], utilize a dynamic model of the SRM to estimate its position

and speed. This practically implies the use of (sliding-mode) observers based on either

phase flux-linkage or current error. When compared to others, observer-based methods

provide superior estimation accuracy and as a result are becoming a preferred choice in

many sensorless applications [134]. Their main drawback is an increased computational

burden and a reliance on both phase voltage and current measurements.

The contribution of this Chapter is a derivation of a position-sensorless SRM control

algorithm which does not require phase voltage/current sensing. Specifically, a novel

SRM speed control strategy is presented using the measurements of DC-link voltage and

current only. In this context, rotor position and speed are estimated using a nonlinear

observer. The proposed method is intended for high-speed SRM applications.


This Chapter is organized as follows. In Section 6.2 a physical model of the SRM

electrical system is provided. Section 6.3 presents the structure of the observer and

the closed-loop controller, along with the analysis of the observer stability. Section 6.4

discusses results of some representative closed-loop simulations whereas conclusions are

drawn in Section 6.5.

6.2 SRM modeling and operation

6.2.1 Actuation

The SRM phase voltage vp, p ∈ 1, . . . ,m, with m being the total phase count, is

typically regulated using an Asymmetric Half Bridge Converter (AHBC), see Fig. 6.1.

v1 v2

Su1 Su

2

Sl1 Sl

2

i1 i2

L1 L2+vdc

idc

Figure 6.1: Two-phase 4/2 SRM AHBC circuit.

The switch states are manipulated in accordance with their PWM duty cycles dup ∈ [0, 1]

for Sup and dlp ∈ [0, 1] for Slp, to allow the exploitation of a full DC-link voltage vdc range

[−vdc, vdc]. Here, the duty cycle value d implies that during each PWM period 1/fpwm

a switch is closed d · 100% of the time, where fpwm represents the PWM switching

frequency. The duty cycles are specified using the phase control input up ∈ [−1, 1], as

follows

dup =

1, for up ≥ 0,

0, otherwise,(6.1)

dlp =

up, for up ≥ 0,

1 + up, otherwise,



which in average (over the PWM period) yields the phase voltage

vp = upvdc, p ∈ 1, 2. (6.2)

and a DC-link current

idc =m∑

p=1

upip, (6.3)

where ip denotes a phase current in a phase winding p.

6.2.2 Electromagnetic properties

The SRM is typically described by its phase flux-linkage characteristics ψp = ψp(ip, θ),

where θ denotes an absolute rotor position defined w.r.t. a fully unaligned position of

phase 1. A time-derivative of ψp reads

dψpdt

(ip, θ) = Lp(ip, θ)dipdt

+Mp(ip, θ)dθ

dt, (6.4)

where Lp(ip, θ) = ∂ψp∂ip

(ip, θ) and Mp(ip, θ) = ∂ψp∂θ

(ip, θ). In general, both Lp and Mp

are bounded. Since an incremental phase inductance Lp is strictly positive, ψp(ip) is

strictly monotonic and there exists an inverse mapping ip = ip(ψp, θ). Then, from (6.4),

it follows

dipdt

=1

Lp

dψpdt− Mp

Lp

dθ

dt, (6.5)

i.e.,

dipdψp

(ψp, θ) =1

Lp(ip(ψp, θ), θ)= Np(ψp, θ), (6.6)

dipdθ

(ψp, θ) = −Mp(ip(ψp, θ), θ)

Lp(ip(ψp, θ), θ)= Kp(ψp, θ). (6.7)

Both ip and ψp are periodic in θ with a period equal to the rotor pole pitch angle

θrpp = 2πNr

, with Nr denoting the rotor pole count, i.e.,

ip(ψp, θ) = ip(ψp, θk), (6.8)

where

θk = θ + kθrpp, k ∈ Z. (6.9)

Thus, from the perspective of phase commutation and/or speed control the positions

θk, k ∈ Z, are mutually equivalent.

6.3 SRM position-sensorless control design 109

6.2.3 Dynamics

The following set of equations can be used to describe the SRM electrical system

dynamics:

dψpdt

= zpvp, (6.10)

vp = upvdc −Rpip,

where Rp represents the phase winding resistance. The logic function zp = zp(ψp, vp)

models the fact that both ip and ψp must be nonnegative (≥ 0) due to the presence of

diodes in the AHBC circuit. It is defined as

zp = (ψp > 0) ∨ (vp > 0). (6.11)

Furthermore, the SRM mechanical system dynamics may be formulated as

dθ

dt= ω, (6.12)

dω

dt=

1

J

(m∑

p=1

τe,p − τl −Bω), (6.13)

where

τe,p(ip, θ) =∂

∂θ

ip∫

0

ψp(i, θ)di, (6.14)

is the phase torque, τl the load torque, B the viscous friction coefficient and ω the

rotor speed. Note that in non-saturating conditions, i.e., when Lp(ip, θ) = Lp(θ) and

ψp(ip, θ) = Lp(θ)ip, one can simplify (6.14) to obtain

τe,p(ip, θ) =1

2

dLp(θ)

dθi2p. (6.15)

6.3 SRM position-sensorless control design

6.3.1 Open-loop phase flux-linkage and current estimation

As a consequence of phase commutation and the fact that ψp ≥ 0, the SRM phase

flux-linkage is characterized by alternating intervals of positive and zero values. The



interval when ψp > 0 is referred to as a phase p cycle, see Fig. 6.2. The cyclic reset of

ψp motivates its estimation using an open-loop estimator, given by

dψpdt

= zp ˆvp, (6.16)

ˆvp = upvdc −Rpip,

zp = zp(ψp, ˆvp)

where ψp denotes the phase flux-linkage estimate, ip = ip(ψp, θ) the phase current

estimate and θ the rotor position estimate. It can be assumed that the adopted phase

flux-linkage estimation law results in eψp = 0 at the beginning of every cycle. This is

because both (6.10) and (6.16) are dominated by a common upvdc voltage term (w.r.t.

to the Ohmic voltage drop Rpip), which ensures that both ψp and ψp are reset almost

simultaneously.

2 4 6 8 10

0

0.01

0.02

0.03

0.04

0.05

Fluxlinka

ge[W

b]

Position θ [rad]

ψp

ψp

eψp

cycle

Figure 6.2: Typical ψp, ψp and eψp waveforms as a function of θ.

Define phase current, position and speed estimation errors, respectively as

eψp = ψp − ψp, (6.17)

eip = ip − ip,eθj = θj − θ,eω = ω − ω,

with j ∈ Z such that

|θj − θ| ≤ |θk − θ|, ∀k ∈ Z, (6.18)

where ω is the rotor speed estimate. It can be reasoned that during each phase cycle

the largest |eψp | is reached when zp = zp = 1. This is because if (zp, zp) = (0, 1) then


ψp = 0 and dψpdt

< 0, which implies that |eψp | is decreasing. The same holds for the

opposite case. If however both are zero, then also ψp = ψp = 0. The dynamics of eψpfor zp = zp = 1 can be derived from (6.10) and (6.16), i.e.,

eψp = −Rpeip . (6.19)

Consider the linearization of eip at (ψp, θ), given by

eip ≈ Npeψp + Kpeθj , (6.20)

where Np = Np(ψp, θ) and Kp = Kp(ψp, θ). Then, it follows

eψp ≈ −RpNpeψp −RpKpeθj . (6.21)

Since Np is always positive (as Lp > 0), the largest |eψp | is obtained if the first term on

the right-hand side of (6.21) is approx. zero, i.e., if the second term has the same sign

throughout the cycle. The second term can be bounded as

|RpKpeθj | < RpKmaxθrpp2, (6.22)

where Kmax = maxψp,θ|Kp(ψp, θ)| and θrpp/2 = max eθj . It thus follows

∣∣∣∣deψpdθ

∣∣∣∣ < RpKmaxθrpp2ω

, (6.23)

i.e.,

|eψp | <θc,max∫

0

RpKmaxθrpp2ω

dθ ≈ RpKmaxθrppθc,max2ω

, (6.24)

where θc,max ≥ 0 denotes a maximal angular span of the cycle, during which it can be

assumed that ω ≈ const. The last inequality implies that the maximum phase flux-

linkage error can be lowered by either shortening the duration of phase excitation or by

operating the SRM at high speed.

6.3.2 Closed-loop speed and position estimation

Consider the DC-link current error eidc given by

eidc = idc −m∑

p=1

upip =m∑

p=1

upeip , (6.25)

≈m∑

p=1

upNpeψp +m∑

p=1

upKpeθj .



The closed-loop observer for the rotor position/speed estimation is derived under the

assumption that the SRM speed is sufficiently high such that the second term in the

last equation dominates the first one (i.e., |eψp | is sufficiently small). Initially, the SRM

can be brought to such a high-speed state, e.g., by means of open-loop control or by

applying an external torque to its shaft. Under this assumption (6.25) simplifies to

eidc ≈m∑

p=1

upKpeθj = Ωeθj , (6.26)

where Ω =m∑p=1

upKp. The observer is then given by

˙θ = ω + κθsign (eidcΩ) , (6.27)

˙ω = κωsign (eidcΩ) , (6.28)

where κω > 0 and κθ > 0 denote some tuning parameters, see Fig. 6.3.

vdc1s

up

θ

ωψp

Rp

×0

ip

Kp

×

Σ×

idcΣ

+

−+

−

idc

×eidc

κωs

1s

++

κθΩ

ip

LUT sign(·)

Figure 6.3: The SRM speed and position observer based on the measurements

of vdc and idc only.

From (6.12), (6.13), (6.27) and (6.28) the observer error dynamics becomes

eθj = eω − κθsign (eidcΩ) , (6.29)

eω = −κωsign (eidcΩ) ,

where it is assumed that κω is chosen large enough so that the right hand side of (6.13)

can be neglected, i.e., κω ω. Furthermore, introduce the Lyapunov function

V =1

2e2θj . (6.30)

Differentiation of V along the error system trajectory, yields

V = eθj (eω − κθsign (eidcΩ)) . (6.31)


Assuming that we choose κθ such that

|eω| < κθ, (6.32)

then it holds

sign(V ) = −sign(eθj)sign (eidcΩ) . (6.33)

Since (6.26) implies that sign(eidcΩ) ≈ sign(eθj), it follows

sign(V ) ≈ −1 (6.34)

i.e., eθj → 0. Once eθj = 0, the error dynamics becomes

eθj = 0, (6.35)

eω = −κωκθeω,

i.e., ω → ω at the rate given by the ratio of κω and κθ.

Remark: Typically, the applied SRM speed control law yields up > 0 in the phase

motoring region (Mp > 0) and up < 0 in the phase generating region (Mp < 0).

This implies that sign(upMp) ≈ 1, i.e., sign(upKp) ≈ −1, and as a result sign(Ω) ≈−1. In such control cases, the resulting property may be used to further simplify the

observer (6.27), at the cost of slightly degraded speed/position estimation performance.

6.3.3 Speed control, current control and commutation

Figure 6.4 illustrates the structure of the proposed SRM control system. The speed is

controlled using a PI controller C(s) = κP1 +κI1s

with a speed tracking error ω∗−ω as its

input and a desired average electromagnetic torque τ ∗e as its output. Here ω∗ represents

the speed reference and κP1 and κI1 the PI controller gains. In a subsequent current

reference parametrization step the torque τ ∗e is converted into a set of phase current

references i∗p, p ∈ 1, . . . ,m. Specifically, for each phase p a rectangular reference

i∗p ∈ 0, i∗ is used, with i∗p = σpi∗, where σp ∈ 0, 1 represents the phase commutation

signal. The level i∗ is computed as

i∗ = imax

√|τ ∗e |τe,max

, (6.36)

with imax denoting the maximum expected phase current and τe,max the maximum

average phase electromagnetic torque. The introduced torque-to-current mapping is



dup , dlp

vdc

θp

controlSpeed

ωObserver

ω∗

SRMθ′, θ′′

i∗

Param.τ∗e

idc

Currentcontrol &

commutation

ipup

Figure 6.4: Proposed SRM control system.

motivated by an inverse of the torque/current relationship in non-saturating condi-

tions (6.15).

The commutation signal σp enables synchronization of phase actuation with the rotor

position. It is defined using the turn-on and turn-off commutation angles, θon and θoff, as

σp =

1, for (θ′ < θ′′ AND θp ∈ [θ′, θ′′]) OR

(θ′′ < θ′ AND θp /∈ [θ′′, θ′]),

0, otherwise,

(6.37)

θp = mod(θ + (p− 1)θspp, θrpp),

θ′ = mod(θon, θrpp),

θ′′ = mod(θoff , θrpp).

where θspp = 2πNs

represents the stator pole pitch angle and Ns the stator pole count.

The commutation angles are scheduled in accordance with the SRM operating point

specified by a triplet (τ ∗e , vdc, ω). This is achieved by formulating an advance angle as

α = καi∗ω

vdc, (6.38)

with κα > 0 being some tuning gain, i.e., the commutation angle selection rules2 as

listed in Table 6.1. For details regarding rule-based commutation the reader is referred

to [95].

Lastly, the phase currents are controlled based on their estimated values ip(ψp, θ) using

a simple P controller with a saturated output, given by

up = sat(κP2(i∗p − ip),−1, 1

). (6.39)

where κP2 > 0 denotes the P controller gain.

2It is assumed that the SRM is rotating in a positive direction only.


Table 6.1: Commutation angle specification

Operation Condition θon θoff

motoring τ ∗e ≥ 0 −α θrpp/2− αgenerating τ ∗e < 0 θrpp/2− α θrpp − α


Closed-loop simulations have been conducted using a model of a commercial two-phase

4/2 SRM. Table 6.2 lists the related system and design parameter values. The functions

ip(ψp, θ) and τe,p(ip, θ) have been numerically derived from the measured ψp(ip, θ)

characteristics and stored in look-up tables. They are illustrated in Fig. 6.5. Note

that the same function ip(ψp, θ) has also been used to compute ip(ψp, θ) within the

speed/position observer.

2

22

2

4

4

4

46

6

6

6

8

88

1010

12121414 1616 1818

Current ip(ψp, θ) [A]

Flux-linkageψp[V

s]

0 45 90 135 1800

0.005

0.01

0.015

0.02

−0.3−0.2

−0.2

−0.1

−0.1

00

0.1

0.1

0.2

0.2 0.30.4

Electromagnetic torque τe,p (ip, θ) [Nm]

Position θ [deg]

Currenti p

[A]

0 45 90 135 1800

5

10

15

20

Figure 6.5: 4/2 SRM phase current ip(ψp, θ) and torque τe,p(ip, θ).

Figures 6.6 and 6.7 present the closed-loop simulation results demonstrating the

effectiveness of the proposed position-sensorless speed control algorithm. Figure 6.6

shows the data related to the speed/position observer. From the top and the middle



Table 6.2: System and design parameter values


SRM

Nr 2

Ns 4

m 2

R1 = R2 0.45Ω

B 10−6Nms/rad

J 1.5 · 10−5kgm2

vdc 300V

ω(0) 3000rad/s

θ(0) 890π180

rad

ψ1(0) = ψ2(0) 0Wb

Controller

κP1 3 · 10−3

κI1 3 · 10−2

κP2 5 · 10−1

κα 5 · 10−4

imax 20A

τe,max 0.29Nm

Observer

κθ 400

κω 1.6 · 105

ω(0) 3500rad/s

θ(0) 460π180

deg

ψ1(0) = ψ2(0) 0Wb

plot, on the left-hand side, it can be seen that in time both the rotor position and

DC-link current estimation error converge to zero. Their initial transient trajectories

(up to t = 0.02s) are shown on the right-hand side. Clearly, the position error

ejθ(0) = 70deg, corresponding to an initial difference between θ(0) = 890deg ≡ 530deg

and θ(0) = 460deg, is quickly removed. From the plots at the bottom one can also

observe good quality of speed estimation and tracking. The deviation of the true speed

(as well as the estimated one) from the reference, occurring at t = 0.6s, is attributed

to the change in load torque τl, from 0.05Nm to 0.15Nm. It can also be seen that

the controller has successfully managed to compensate for the load torque disturbance

effects, by restoring the desired reference speed level.

Figure 6.7 shows the control signal trajectories, where τe is used to denote the total

instantaneous SRM electromagnetic torque, i.e., τe =m∑p=1

τe,p. In the beginning, it

6.5 Conclusions 117

0 0.2 0.4 0.6 0.8 1

−20

0

20

40

60

80

Pos

ition

err

or [d

eg]

ej

θ

0 0.005 0.01 0.015 0.02

−20

0

20

40

60

80

ej

θ

0 0.2 0.4 0.6 0.8 1−10

0

10

20

30

40

Cur

rent

err

or [A

]

eidc

0 0.005 0.01 0.015 0.02−10

0

10

20

30

40

eidc

0 0.2 0.4 0.6 0.8 1

2.5

3

3.5

Time [s]

Spe

ed [k

rad/

s]

ω∗

ωω

0 0.005 0.01 0.015 0.02

2.5

3

3.5

Time [s]

ω∗

ωω

Figure 6.6: Speed estimation and tracking – observer signals. Left: complete

trajectories, right: their initial transient period.

can be seen that the SRM brakes, i.e., operates as a generator. This is because the

initial reference speed is lower than the estimated/true speed. Thus the controller first

outputs a negative torque τ ∗e , which results in the selection of commutation angles for

the generating operation. Once the speed set-point is reached, a switch to motoring

angles is performed to compensate for the positive load torque τl, see the upper plot

of Fig. 6.7. The tracking of the resulting phase current reference i∗ is shown in the

bottom plot.

6.5 Conclusions

In this Chapter a novel, position-sensorless speed control scheme was proposed, for

the high-speed SRMs. The main component of the presented design was the nonlinear



0 0.2 0.4 0.6 0.8 1−0.5

0

0.5T

orqu

e [N

m]

τe

τ∗

e

τl

0 0.2 0.4 0.6 0.8 1

0

90

180

Com

m. a

ngle

s [d

eg]

θonθoff

0 0.2 0.4 0.6 0.8 10

10

20

30

Time [s]

Cur

rent

[A]

i1i∗

Figure 6.7: Speed estimation and tracking – controller signals.

speed/position observer that relies solely on the DC-link voltage and current measure-

ments. The developed algorithm has shown good speed tracking performance, even in

the presence of the load torque disturbances. This was demonstrated in closed-loop

simulations.

Chapter 7

Auto-calibration of a generator-turbine

throttle unit1

Abstract This Chapter presents a method for auto-calibration of a generator-turbine throttle

unit (GTU). The GTU is a device consisting of a variable geometry turbine with adjustable

stator vanes and a high-speed electric generator. It is used to throttle the engine while

simultaneously producing electricity. To maximize its energy-recovery the turbine speed needs

to be suitably matched to the conditions present in the engine air-intake system, at all times.

However, often the exact dependency of the optimal turbine speed on its pressure ratio and

vanes position is either unknown or difficult to derive. This research proposes a non-model-

based solution to this problem using Extremum-Seeking Control with a disturbance-based

input parametrization. The presented algorithm enables the identification of the unknown

relationship between the disturbances (pressure ratio, vanes position) and the optimal input

(turbine reference speed) in an initial, automated calibration step. Once identified, the

resulting mapping can be employed for optimal, real-time reference speed generation. The

effectiveness of the proposed auto-calibration approach is demonstrated in simulations.

7.1 Introduction

Extremum-Seeking Control

In a wide variety of control applications the aim is to operate a physical system

or a process in the vicinity of an extremum of some performance function. The

1This chapter is based on [135], [136].

119

120 7 Auto-calibration of a generator-turbine throttle unit

performance function is often measurable but unknown to the designer, in terms of its

exact analytic dependency on the optimizing inputs. In such cases Extremum-Seeking

Control (ESC) techniques [137]–[140] can be used to achieve and maintain operation

of a dynamical system under optimal conditions. Numerous reports of successful ESC

implementation can be found in literature, e.g., for improving continuously variable

transmission efficiency [141] or Maximum Power Point Tracking (MPPT) in photovoltaic

(PV) [142], fuel cell [143] and wind energy systems [144].

The Extremum-Seeking Control was first investigated in the 1950s and 1960s as a

control framework for finding a minimum/maximum of a static map [140]. However, a

rigorous stability proof for the classical derivative-based ESC with a general nonlinear

dynamical plant arrived only at the beginning of the past decade [145]. Since then

there has been a revival of interest and a steady development in the field. Figure 7.1

shows the classification of various ESC schemes described in literature so far. On a

top level, a distinction between non-parametric (“black box”) and parametric (“gray

box”) ESC is made, based on whether the optimized system is assumed entirely or only

partially unknown. The non-parametric ESC can be further divided into derivative-

based, e.g., classical [145]–[147] and numerical [148], [149], and derivative-free, e.g.,

sliding-mode [150], [151] and direct-search [152], [153] methods, depending on the need

for performance function derivatives during optimization. In the parametric ESC the

knowledge of the structure of the performance function is typically assumed [138], [154]–

[156]. Specifically, the performance function is often parametrized by the optimizing

inputs and a set of unknown parameters that are estimated online, using the available

measurement data. The benefit of the parametric approach is that the optimal input

values can be computed analytically once the related parameters are identified.

ESC

Non-parametric

Parametric

Derivative-based

Derivative-free

Performance output(input)

Optimal input(disturbance)

- Classical- Numerical- etc.

- Sliding-mode- Direct-search- etc.

- Gradient- Least-Squares- etc.

- Gradient- Least-Squares- etc.

Figure 7.1: Classification of different ESC schemes. The path corresponding

to the proposed ESC algorithm is italicized.

In general, most of the reported ESC algorithms consider that the optimal input value

is constant. The tracking of an unknown, time-varying extremum is rarely investigated.

7.1 Introduction 121

However, in [157] this issue has been addressed for the case when the optimal input

variations can be represented as an output of a known, linear time-invariant system.

An alternative approach was presented in [158]. Instead of explicitly modeling the

optimal input time-dependency, in [158], the authors have treated the optimal input

variations as an uncertainty. As a result, they have shown that using robust tracking

techniques [159] the extremum of a static plant can be successfully followed.

This Chapter introduces a novel, parametric ESC algorithm, with a disturbance-

based optimal input parametrization, for tracking of the time-varying extremum. The

proposed ESC scheme is applicable to situations where the disturbances leading to

changes in the optimal input are known/measurable. Its main advantage over the

previously mentioned ESC approaches is that it allows one to identify the mapping

between the disturbances and the optimal inputs. Once found, the constructed mapping

can be directly employed in real-time, without the need for further extremum-seeking.

Generator-turbine throttle unit

The proposed ESC method is developed for purposes of auto-calibration of a generator-

turbine throttle unit (GTU). The GTU is a device consisting of a variable geometry

turbine with adjustable stator vanes and a high-speed electrical generator. It replaces a

throttle valve of a gasoline internal combustion engine (ICE) so that the intake airflow

can be controlled while simultaneously producing electricity [25], [26].

To maximize the GTU energy-recovery the turbine speed needs to be suitably matched

with the conditions present in the engine air-intake system, at all times. In this work,

the derived ESC scheme is employed to adaptively reconstruct the unknown relationship

between the optimal turbine speed, pressure ratio and vanes position. This results in a

disturbance-based optimal input mapping that can be used for real-time reference speed

generation, leading to an improved energy recovery. The effectiveness of the proposed

approach is demonstrated in simulations.

This Chapter is organized as follows. Section 7.2 introduces the proposed ESC scheme

whereas Section 7.3 describes its application for the GTU auto-calibration. Section 7.4

provides simulation results related to the proposed auto-calibration approach. The

conclusions are drawn in Section 7.5.


7.2 Extremum-Seeking Control with disturbance-based

optimal input parametrization

The proposed ESC scheme for stable closed-loop plants is shown in Fig. 7.2. The details

behind each component are provided in the remainder of this section.

y

û

q

+

a sin(ε1t)

u

u

ESC scheme

û = βε1aπ

∫ tt− 2π

ε1

y(τ) sin(ε1[τ − φ])dτ

Input error estimator

u = HT (q)θ

Input parameter estimator˙θ = −ε1ε2ΓH(q)û

fast

medium

slow

Stable closed-loop plant

x = fx(x, u, u∗, q)

y = hy(x, u, u∗, q)

u∗ = HT (q)θ

Figure 7.2: Proposed ESC scheme for stable closed-loop plants.

7.2.1 Problem description

Consider the following stable closed-loop plant:

x = fx(x, u, u∗, q) (7.1)

y = hy(x, u, u∗, q) (7.2)

u∗ = HT (q)θ, (7.3)

where x ∈ Bx ⊂ Rn is the plant state, u ∈ Bu ⊂ R the input, u∗ ∈ Bu ⊂ R the unknown

optimal input, q ∈ Bq ⊂ Rl the known disturbance, y ∈ R the known output and θ ∈Bθ ⊂ Rp the fixed unknown parameter vector. Furthermore, fx : Bx×Bu×Bu×Bq → Bxrepresents the unknown state function, hy : Bx×Bu×Bu×Bq → R the unknown output

function and H(q) = [h1(q), · · · , hp(q)]T the vector of known multivariate regressor

functions hi(q), i ∈ 1, . . . , p, such that HT (q)θ : Bq × Bθ → Bu.

7.2 Extremum-Seeking Control with disturbance-based optimal inputparametrization 123

Assuming the equation

0 = fx(x, u, u∗, q) (7.4)

has a unique solution x = l(u, u∗, q), define the equilibrium input-output map

Q(u, u∗, q) := hy(l(u, u∗, q), u∗, q). (7.5)

Furthermore, assume that the map Q(u, ·, ·) is smoothly differentiable sufficiently many

times and that for any q there exists an extremum2 at u = u∗. Moreover, assume that

the disturbance derivative exists and that it holds

|q| < ε1ε2, (7.6)

with 0 < ε1, 0 < ε2 ≤ 1 being some user-assignable (Lipschitz) constants. In the follow-

ing, the i-th derivative of Q(·, ·, ·), w.r.t. u, is denoted as DiQ(u, u∗, q) := ∂iQ∂ui

(u, u∗, q).

The aim of the proposed ESC scheme is to identify the unknown input parameters θ

which define the optimal input u∗, given by (7.3). The benefit of this ESC approach

is that, once identified, the parameters θ can be used to directly compute the input

u which maximizes the input-output map Q(u, u∗, q), for any value of the measured,

time-varying disturbance q and without the need for further extremum-seeking.

The proposed ESC algorithm contains two estimators: one for the unknown input error

u = u − u∗ and one for the unknown input parameters θ. As the input parameter

estimator relies on the estimated input error, it is designed to run on the slowest

timescale, whereas the input error estimator runs on the medium and the plant on

the fastest timescale. Note that for this purpose the disturbance is required to operate

(vary) at the slowest timescale, to allow the input error estimator to perceive the closed-

loop plant dynamics in terms of its equilibrium map Q(u, u∗, q).

Note that apart from the adopted optimal input structure (7.3), other model structures

may be also used. The advantage of the chosen one is that a plethora of suitable param-

eter estimators exists [161]. In the case of an alternative (nonlinear) parametrization the

input parameter estimation rule should be changed accordingly. The implementations

details related to the ESC scheme shown in Fig. 7.2 are in the following.

2Without loss of generality only maxima are considered; minima can easily treated in the same

manner by defining Q = −Q and then applying the theory to Q.


7.2.2 Input error estimation

Consider the following plant input:

u = u+ a sin(ε1t), (7.7)

and a moving-average filter:

D1Q =ε1aπ

t∫

t− 2πε1

y(τ) sin(ε1[τ − φ])dτ, (7.8)

where a > 0 denotes the tunable dither signal amplitude, u ∈ Bu the nominal plant

input and φ ≥ 0 an additional tuning parameter.

The expression (7.8) implies that by decreasing ε1 the dynamics of the moving-average

filter can be made arbitrarily slower than that of the plant. Thus from a perspective

of the moving-average filter, for a sufficiently small ε1, the plant (7.4) input-output

relationship will appear as the static mapping Q(u, u∗, q). As a consequence, the filter

output D1Q will become approximately equal to the gradient D1Q(u, u∗, q) (for details,

see [160]), i.e.,

D1Q ≈ D1Q(u, u∗, q). (7.9)

Consider a Taylor expansion of D1Q(u, u∗, q), w.r.t. u, in the vicinity of u∗:

D1Q(u, u∗, q) ≈ D1Q(u∗, u∗, q) +D2Q(u∗, u∗, q)u. (7.10)

As the first term on the right-hand side of (7.10) is zero at the extremum, it follows

u ≈ û = βD1Q(u, u∗, q), (7.11)

with û being the input error estimate and β < 0 the tuning parameter representing the

inverse of the Hessian3 of Q(u, ·, ·), at u = u∗.

7.2.3 Input parameter estimation

Based on (7.3), consider the following nominal input parametrization

u = HT (q)θ. (7.12)

3The Hessian and other higher-order derivatives of Q(u, ·, ·) can be also estimated automatically,

e.g., see [137], [154], [155].

7.3 Application of proposed ESC scheme to generator-turbine throttle unitauto-calibration 125

The estimate θ of the unknown parameter vector θ can be obtained, e.g., via the gradient

algorithm [161], given by

˙θ = −ε1ε2ΓH(q)u, (7.13)

where Γ = ΓT > 0 is a diagonal matrix of gains. Here it is assumed that H(q) is

persistently exciting [161], i.e., for any t ≥ 0 there exist µ, T such that

t+T∫

t

H(q(τ))HT (q(τ))dτ ≥ µI. (7.14)

From (7.13) it follows that by decreasing ε2 the parameter estimator dynamics can be

made arbitrarily slower compared to that of the moving-average filter. The resulting

timescale separation allows one to use the input error estimate û, given by (7.11), at

the place of the true input error u, in (7.13). Note that a smaller ε2 also implies the

need for a slower disturbance q, due to the assumption on its derivative.

7.3 Application of proposed ESC scheme to generator-

turbine throttle unit auto-calibration

The generator-turbine throttle unit (GTU) is a device consisting of a variable geometry

turbine with adjustable stator vanes and a high-speed electrical generator. It replaces a

throttle valve of a gasoline internal combustion engine (ICE), so that the intake airflow

can be controlled while simultaneously producing electricity [25], [26]. To maximize

the GTU energy-recovery the turbine speed needs to be suitably matched with the

conditions present in the engine air-intake system, at all times.

In the context of the GTU auto-calibration, the stable closed-loop plant refers to the

speed-controlled GTU with the input u being the turbine reference rotational speed ωr,

the output y the mechanical turbine power estimate Pt and the disturbance q the vector

[ut, λΠ]T , where ut ∈ [0, 1] is the turbine vanes position control signal and λΠ ∈ (0, 1)

the ratio of the pressure after and before the turbine. The goal of calibration is to

find the relationship between the speed ωr, which maximizes the power Pt, and the

disturbance pair [ut, λΠ]T . For this purpose, it is assumed that both λΠ and ut are

known (measurable).


ESC GTU

×

λΠ, ut

Pt

ωr ω

τt

Speedcontrol

Kalmanfilter

τe

Stable closed-loop plantProposed ESC

Figure 7.3: Proposed GTU auto-calibration control scheme.

7.3.1 GTU modeling and optimal operation

This section describes a model of the GTU used to evaluate the designed auto-calibration

procedure. Note that an analytic derivation and a subsequent parameter estimation

of variable-geometry turbine models is in general challenging [75], which justifies the

presented non-model based approach to its calibration.

The dynamics of the turbine speed ω reads

ω =1

J(τt(ω, ut, λΠ) + τe) , (7.15)

where J is the rotor inertia, τe the electric generator torque and τt(ω, ut, λΠ) ≥ 0 the

unknown turbine torque. For simplicity, the electric torque is treated as a control input

whereas the turbine torque is given by

τt =Pt(ω, ut, λΠ)

ω, (7.16)

where Pt denotes the turbine mechanical power after the friction loss, modeled as [55]:

Pt(ω, ut, λΠ) = mα(ut, λΠ)cpTAMηt(ω, λΠ)

(1− λ

λκ−1λκ

Π

)−Bω2, (7.17)

with mα being the turbine air mass flow, cp the specific heat of air at constant pressure,

TAM the ambient air temperature, ηt the turbine efficiency, λκ the specific heat ratio of

air and B the viscous friction coefficient. Furthermore, the turbine mass-flow can be

formulated as [75]:

mα(ut, λΠ) = At(ut)pAM√cRTAM

√1− λKtΠ (7.18)

where cR is the specific gas constant of air, pAM the ambient air pressure, Kt > 0 an

empirical fitting coefficient and

At(ut) = −c2u2t + c1ut + c0 (7.19)


the turbine effective flow area with the fitting coefficients c0, c1, c2 > 0. The efficiency

ηt reads [55]:

ηt(ω, λΠ) = ηt,max

[2λbsr(ω, λΠ)

λ∗bsr−(λbsr(ω, λΠ)

λ∗bsr

)2], (7.20)

with

λbsr(ω, λΠ) =rtω√

2cpTAM

(1− λ

λκ−1λκ

Π

) = kbsr(λΠ)ω (7.21)

representing the ratio between the turbine blade tip speed and the speed of isentropically

expanded air [72], λ∗bsr the optimal λbsr value, ηt,max the maximal turbine efficiency, rt

the turbine blade radius, and ωt the turbine speed. In general, both λ∗bsr and ηt,max

depend on the At area [73], [74] and thus on the vanes control signal ut. However, here

it is assumed that this dependency is negligible to simplify the exposition.

Define

a1 = mα(ut, λΠ)cpTAMηt,max2kbsr(λΠ)

λ∗bsr

(1− λ

λκ−1λκ

Π

), (7.22)

a2 = mα(ut, λΠ)cpTAMηt,max

(kbsr(λΠ)

λ∗bsr

)2(1− λ

λκ−1λκ

Π

)+B. (7.23)

Then

Pt(ω, ut, λΠ) = −a2(ut, λΠ)ω2 + a1(ut, λΠ)ω, (7.24)

i.e.,

τt(ω, ut, λΠ) = −a2(ut, λΠ)ω + a1(ut, λΠ). (7.25)

From (7.24) it follows that the optimal speed ω∗r , which maximizes the power Pt, is

given by

ω∗r(ut, λΠ) =a1(ut, λΠ)

2a2(ut, λΠ), (7.26)

whereas the corresponding power maximum reads

P ∗t (ut, λΠ) =a2

1(ut, λΠ)

4a2(ut, λΠ), (7.27)

The optimal speed and power are illustrated in Fig. 7.4 for parameter values listed

in Table 7.1. Specifically, in this study, the values of typically uncertain parameters B,

Kt, λ∗bsr, rTB, ηTB,max, c0, c1 and c2 have been obtained by reformulating the turbine

model, identified in [75], such that it admits the structure given by (7.17)-(7.21).


2 2 2

4

4 4 4

6

66 6

88 8

101010

Vanes position control signal ut [-]

Pressure

ratioλΠ[-]

Optimal turbine speed ω∗

r [krad/s]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.1

0.1

0.1

0.1 0.1

0.25

0.25

0.25

0.25

0.25

0.50.5

0.5

0.5

11

1

1

1.5

1.5

1.5

2

2

2

2.5

2.53

3

Vanes position control signal ut [-]Pressure

ratioλΠ[-]

Optimal turbine power P∗

t [kW]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 7.4: Optimal turbine speed ω∗r(ut, λΠ) and power P ∗t (ω∗r , ut, λΠ) (after

friction loss) w.r.t. the turbine vanes position control signal ut and the pressure

ratio over the turbine λΠ.

7.3.2 GTU speed control and mechanical power estimation

Speed control

The turbine speed ω is controlled using a simple proportional controller GC = κP , with a

proportional gain κP , based on a speed tracking error eω = ωr−ω. The controller output

is the electric torque τe. The gain κP has been tuned by considering a linearized version

of the turbine speed dynamics (7.15), given by the transfer function GT (s) = 1Js+a2

,

where a2 represents an estimate of the coefficient a2, introduced in (7.23). This resulted

in a closed-loop system GP (s) = κPJs+a2+κP

= ω(s)ωr(s)

, with ≈ 90deg phase margin, ≈ 10−3s

delay margin and infinite gain margin, i.e., with a bandwidth wbw,P ≈ 1500rad/s.

In this study, the coefficient a2 is computed as an average of a2, defined in (7.23), for

ut, λΠ ∈ [−1, 1]. However, in the case when certain turbine parameters are unknown an

alternative way of deriving a2 can be used. For instance, based on (7.23), a reasonable

a2 estimate can be given byˆmαcpTAM

10·ω2 , where ˆmα and ω represent the expected average

turbine mass flow and speed. In either case, the estimate of the Hessian inverse reads

β = − 12a2

, which is implied by the second derivative of (7.24), w.r.t. speed ω.


Power estimation

The turbine mechanical power Pt is not measured directly but instead estimated as a

product between the speed ω and the estimated turbine torque τt, which yields

Pt = τtω. (7.28)

The torque τt is obtained by filtering the controller output τe and the speed ω with a

Kalman filter. For this purpose, the following state-space representation of the turbine

speed dynamics (7.15) is used

d

dt

[ω

τt

]=

[0 1

J

0 0

]

︸︷︷︸Ac

[ω

τt

]+

[1J

0

]

︸︷︷︸Bc

τe + ρ1, (7.29)

ωm =[1 0

]

︸︷︷︸Cc

[ω

τt

]+ ρ2,

where the signals ρ1 and ρ2 denote white zero-mean state and measurement noise. The

properties of the noise signals, ρ1 and ρ2, are described by their respective covariance

matrices, W1 and W2, which are in this context treated as tuning parameters. The

Kalman filter is implemented using MATLABr kalman routine. This resulted in a

transfer function GK(s) = τt(s)ωr(s)

, from the reference speed ωr to the estimated turbine

torque τt, with a bandwidth of wbw,K ≈ 1530rad/s.

7.3.3 ESC implementation

Optimal input model

The proposed ESC scheme requires that the adopted disturbance-based optimal input

parametrization admits a linear regressor form (7.3). This requirement is fulfilled by

assuming that the optimal turbine speed ω∗r can be sufficiently well approximated by

a multivariate polynomial in q = 2q − 1, of order c and with unknown coefficients θ.

Here q denotes the “normalized” disturbance vector obtained by linearly mapping the

entries of q = [ut, λΠ]T to the interval [−1, 1]. Note that the disturbance normalization

is not strictly necessary. However, it has been observed that such practice improves

the quality of parameter estimation when the regressor functions hi(q), i ∈ 1, . . . , p,p = (l+c)!

l!c!, are chosen as monomial terms of the corresponding multivariate polynomial.

Specifically, for c = 4 and l = 2, they are given by

hi(q) = qD1,i

1 qD2,i

2 , i ∈ 1, . . . , 15, (7.30)


where the “degree” matrix D reads

D =

[0 0 0 0 0 1 1 1 1 2 2 2 3 3 4

0 1 2 3 4 0 1 2 3 0 1 2 0 1 0

]T. (7.31)

Accordingly, the nominal input u is computed as

u = HT (q)θ. (7.32)

Disturbances

To satisfy both the disturbance derivative (7.6) and the persistence of excitation

condition (7.14) w.r.t. the normalized disturbance q, the signals ut and λΠ are each

specified by filtering a uniform random noise defined on the interval [0, 1], using a low-

pass filter

Gq(s) =ω2q

s2 +√

2ωqs+ ω2q

, (7.33)

with ωq = 12ε1ε2.

To facilitate the necessary timescale separation between the input error estimator and

closed-loop system dynamics, ε1 has been set an order of magnitude smaller than the

lowest bandwidth of GP (s) and GK(s), i.e., ε1 = 115

min(wbw,P , wbw,K) = 100rad/s.

Accordingly, the ε2 was set to 1/15, enforcing further timescale separation between the

disturbances (as well as the parameter estimator) and the error estimator.


Based on the model (7.15)-(7.21) closed-loop simulations have been conducted to verify

the feasibility of the proposed GTU auto-calibration scheme. Table 7.1 lists the related

system and design parameter values.

Figure 7.5 shows the tracking of the optimal turbine power P ∗t and reference speed ω∗r .

The corresponding turbine vanes position control signal ut and the pressure ratio λΠ are

shown in the plot at the bottom. It can be seen that after an initial transient period,

of approximately 5s, both the power Pt and the reference speed ωr have converged to

their optimal values. This is even more apparent in Fig. 7.6 which depicts the power

and speed errors, P ∗t −Pt and ω∗r − ωr, as both approach zero quickly upon the start of

auto-calibration. By looking at the estimated input parameters θ, shown at the bottom



Ambient

pAM 103 kPa

TAM 298 K

λκ 1.4

cp 1005 J/kg/K

cR 287 J/kg/K

GTU

λ∗bsr 0.5

rTB 0.03 m

ηTB,max 60 %

J 1.5 · 10−5kgm2

B 10−6Nms/rad

Kt 2.89

[c0, c1, c2] [5 · 10−6, 1.9 · 10−4, 9.5 · 10−5]

a2 2.5 · 10−5

Speed control κP 2.3 · 10−2

Kalman filterW1 105 · I2×2

W2 1

ESC: error estimator

a 100rad/s

φ 0rad

β −2 · 104

ε1 100rad/s

ESC: parameter estimator

c 4

Γ Ip×p

η0 0p×1

ε2 1/15

Table 7.1: Parameter values

plot of Fig. 7.6, it can be argued that this is mainly caused by the convergence of θ1 to

the neighborhood of its steady-state value. As θ1 is associated with a constant regressor

function h1(q) = 1, this implies that large energy gains can already be made by simply

fixing the reference speed to, e.g., ωr = 5000rad/s. However, for an improved energy

recovery it is beneficial to wait until all the parameters have converged. This occurs

approximately after 125s. At this point, the sinusoidal speed perturbation as well as

the input parameter estimation were stopped. Nevertheless, it can be seen that neither

the optimal power nor the speed tracking were affected by this action.

Figure 7.7 shows the identified relationship between the turbine reference speed ωr, vanes


0 125 2500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Power

[kW]

P ∗

t

Pt

0 125 250

1

2

3

4

5

6

7

8

9

Speed[krad/s]

ω∗

rωr

0 125 2500

0.2

0.4

0.6

0.8

1

Time [s]

Disturbance

[-]

ut

λΠ

Figure 7.5: Tracking of the optimal turbine power P ∗t and reference speed ω∗r .

position control signal ut and pressure ratio λΠ. By comparing Fig. 7.4 and Fig. 7.7 one

can observe that the contour lines corresponding to speeds ωr(ut, λΠ) and ω∗r(ut, λΠ)

follow the same trend but do not match exactly. The discrepancy between the two

can be explained by the following: the adopted optimal speed model (7.32) is different

than the true one (7.26); only the estimate of the input error (7.11) is known; the

performance function Hessian D2Q is approximated by a tunable constant β, whereas

in fact it varies with the disturbance q; the HessianD2Q is relatively small and as a result

the performance function is rather flat near the extremum. This has a negative effect

on the speed and accuracy of the parameter convergence as it renders the performance


0 125 250

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Pow

ererror[kW]

P ∗

t − Pt

0 125 250−2

0

2

4

6

8

10

12

Speederror[krad/s]

ω∗

r − ωr

0 125 250

−3000

−2000

−1000

0

1000

2000

3000

4000

5000

6000

Time [s]

Parametersθ[-]

Figure 7.6: Optimal turbine power and speed errors and the speed parameter

estimates θ. In the bottom plot, the colors are linearly distributed between the

entries of θ such that the darkest color corresponds to θ1 and the lightest to θ15.

function output (power Pt) rather insensitive to the changes in its input (reference speed

ωr). However clearly, even under such conditions the resulting power map Pt(ωr, ut, λΠ)

matches the optimal one P ∗t (ω∗r , ut, λΠ) almost perfectly, compare the right-hand sides

of Fig. 7.4 and Fig. 7.7.


22 2

44 4

66 6

88

8

10

Vanes position control signal ut [-]

Pressure

ratioλΠ[-]

Reference turbine speed ωr [krad/s]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.1

0.1

0.1

0.1 0.1

0.25

0.25

0.25

0.25

0.25

0.50.5

0.5

0.5

11

1

1

1.5

1.5

1.5

22

2

2.5

2.53

3

Vanes position control signal ut [-]Pressure

ratioλΠ[-]

Turbine power Pt(ωr, ut, λΠ) [kW]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 7.7: Turbine reference speed ωr(ut, λΠ) and power Pt(ωr, ut, λΠ) (after

friction loss) w.r.t. the turbine vanes position control signal ut and the pressure

ratio over the turbine λΠ.

7.5 Conclusions

In this Chapter a novel Extremum Seeking Control scheme, with disturbance-based

optimal input parametrization, was developed for purposes of the auto-calibration of

the generator-turbine throttle unit. The proposed ESC scheme allowed the identification

of the mapping between the turbine vanes position control signal, pressure ratio and

reference speed that yields the maximum GTU energy recovery. This has been verified


The usefulness of the presented auto-calibration method extends beyond this particular

application. This ESC solution can be applied to any similar situation where the

disturbances leading to the changes in the optimal inputs can be measured or estimated.

Furthermore, the derived ESC algorithm can be easily extended with the derivative

and/or parameter estimation schemes different than the ones described in this work, as

well as to plants involving more than one optimizing input.

Chapter 8

Conclusions and recommendations

Abstract This Chapter dicusses the main conclusions of this thesis and presents certain

recommendations for future directions of research.

8.1 Conclusions

This thesis was set out to obtain deeper understanding of the electric supercharg-

ing, regenerative throttling and Switched Reluctance Machine technologies, from the

perspective of optimization and control. In this context, the following main research

objectives were formulated:

1. Theoretical investigation of methods for evaluating the potential of engine down-

sizing via electric supercharging to improve the vehicle fuel economy.

2. Theoretical investigation of methods for evaluating the potential of regenerative

throttling to improve the vehicle fuel economy.

3. Theoretical investigation of methods for the Switched Reluctance Machine speed

and voltage control, which can alleviate the difficulties posed by the SRM nonlinear

dynamics, tight physical constraints and position/speed sensing requirements.

4. Theoretical investigation of methods for relieving the engineering systems, such as

the generator-turbine throttle unit, from excessive calibration requirements.

The first objective was addressed in Chapter 2, the second in Chapter 3, the third in

Chapters 4 to 6 and the fourth in Chapter 7. The conclusions of each of these chapters

are summarized in the following.

135

136 8 Conclusions and recommendations

The capacity of the electric supercharging to improve the vehicle fuel economy was

investigated in Chapter 2. For this purpose, a detailed model of the electrically

supercharged ICE was derived. The model was used to define a convex optimization

problem considering the minimization of the vehicle operational (fuel) and component

(ICE and electric energy storage buffer) costs. The optimal solution of this problem was

found using state-of-the-art numerical tools, for the case of the WLTP3 driving cycle.

The provided case study has shown that during short-duration high-power demands the

(standalone) electric supercharger can effectively increase the pressure of air supplied to

the engine and thereby considerably rise its maximum torque bound – especially at low

engine speeds. This helps facilitate engine downsizing and reduce the fuel consumption.

Specifically, in the analyzed case, fuel cost savings of up to 10% were obtained.

Chapter 3 addressed the regenerative throttling technology. To this end, a mathematical

model of the gasoline engine powertrain equipped with the generator-turbine throttle

unit was developed. The model was constructed in accordance with the disciplined

convex modeling methodology, for purposes of the convex optimization. In particular,

the considered optimization problem was formulated as the minimization of the total

vehicle operational (fuel) and component (electric buffer) costs. The analysis of the

results, obtained for several different driving cycles (FTP75, WLTP3, NEDC and

VAIL2NREL), has shown that regenerative throttling was the most beneficial for the

case of the VAIL2NREL cycle, which corresponds to high-demand highway driving,

as well as for vehicles equipped with oversized ICEs. In such circumstances, the

regenerative throttling technology offered the possibility to reduce the total costs by

more then 4%.

In Chapter 4, a novel Four-Quadrant (4Q) speed control scheme was developed for

the 4/2 SRM. The proposed algorithm adopted a cascade structure: the outer closed-

loop speed controller computed the desired torque reference and passed it to the inner

open-loop average torque controller (ATC), which then translated the torque command

into the suitable phase current reference pair – for the innermost closed-loop current

controller. For this purpose, a parametric, model-based, open-loop ATC was derived on

the basis of the measured, static electromagnetic SRM characteristics. The 4Q controller

was designed to explicitly handle the identified, speed and voltage-dependent, average

torque bounds, as well as to impose the desired phase current limit. Since the 4/2 SRM

is prone to getting “stuck” near or at one of the phase aligned positions – a supervising

state-machine was also constructed to enable the machine to start/resume operation

in “stuck” situations. The effectiveness of the proposed approach was demonstrated

experimentally, in terms of sinusoidal and block speed reference tracking (under load).

Due to the employed parametrization with the machine operating point, the proposed

8.1 Conclusions 137

4/2 SRM 4Q control scheme can be applied in a wide range of applications, including

electric supercharging and regenerative throttling.

Chapter 5 presented the Switched Reluctance Generator (SRG) voltage control scheme

utilizing the explicit Model-Predictive Control (MPC) methodology. The proposed

voltage controller was intended for high-speed (e.g., 50-100 krpm), low-cost applications,

such as regenerative throttling. Due to these requirements, the SRG was considered to

operate in the Single-Pulse mode, where the turn-on and turn-off commutation angles

were the only current control parameters. Two different commutation angle selection

strategies were derived: the first based on analytic calculations and the second on open-

loop simulations of the relevant SRG dynamics. In the case of the second, the angles were

found by minimizing the peak phase flux-linkage and thus also, indirectly, the SRG iron

loss. Furthermore, the voltage MPC was developed such to keep the average DC-link

current and voltage within their prescribed bounds. This was demonstrated in closed-

loop simulations, which are conducted for the case of the 4/2 SRG. The simulations

have also shown that the designed controller can ensure smooth voltage tracking, in the

presence of both varying load current and rotor speed.

In Chapter 6, a novel position-sensorless speed control scheme was proposed for the

high-speed SRMs. The main component of the presented design was the nonlinear

speed/position observer which relied solely on the DC-link voltage and current measure-

ments, whereas the measurements of phase voltage, current, rotor position and speed

were not used. Due to fewer sensors required, the derived speed control algorithm

can help reduce the associated hardware costs and increase the reliability of the SRM

electrical system. The developed control algorithm has shown good speed tracking

performance, even in the presence of load torque disturbances. This was demonstrated


Finally, Chapter 7 treated the development of a real-time optimization and control

method for the auto-calibration of the turbine-driven generator, used for regenerative

throttling purposes. The proposed method utilized a novel Extremum Seeking Control

(ESC) scheme with a disturbance-based optimal input parametrization. The ESC was

designed to enable tracking of a varying, unknown extremum of a dynamical plant,

under the assumption that the disturbances leading to changes in the optimal input are

known/measurable. In the considered application, the derived ESC scheme allowed

the identification of the mapping between the disturbances, i.e., the turbine vanes

position signal and pressure ratio, and the optimal input, i.e., the turbine reference

speed which maximizes the turbine-driven generator power output. Simulation results

were presented to verify the effectiveness of the derived auto-calibration scheme.

138 8 Conclusions and recommendations

8.2 Recommendations for future research

In control engineering, especially when dealing with real physical systems, it is fairly

common to adopt a series of modeling approximations, as well as simplifying assump-

tions, to arrive with a practically useful control solution. Furthermore, while performing

state-of-the-art research on a certain topic, some questions might remain partly or

even completely unanswered. However, this often represents a good starting point for

further scientific investigations. In this context, the remainder of this section provides

an overview of the recommended directions for future research, based on the insights

obtained during the work conducted on this thesis.

In Chapter 2 it was shown how to scale the electric energy storage buffer as well as the

engine displacement volume, in the case of the electrically supercharged ICE vehicle

powertrain. This research could be extended in at least two different directions. The

first would be to consider also the sizing of the vehicle alternator and gearbox, so

that they match the downsized engine and electric supercharger more precisely. The

second would be to evaluate the investigated powertrain concept on a more-demanding

driving cycle, or a few of them, consisting of prolonged high-load intervals. To this end,

it could be beneficial to also adapt the developed convex modeling and optimization

method towards an electrified turbocharger powertrain topology.

The regenerative throttling study, described in Chapter 3, could also be extended

to include the alternator sizing. Potentially, this could bring further fuel economy

benefits, as a consequence of the reduced alternator weight/cost. Moreover, due to its

convex properties, the proposed optimization algorithm (excluding the electric buffer

sizing part) could also be applied for online vehicle state/control prediction purposes.

However, as this inherently implies the use of driving cycle estimates, the fuel-economy

improvements might be lower. The analysis of the cooling potential of the regenerative

throttling, within the same, convex framework, constitutes yet another topic for future

research.

In Chapter 4, the cascaded 4/2 SRM 4Q speed control scheme was proposed, which

employed a novel open-loop average torque controller (ATC). The ATC was derived

using the measured, static SRM electromagnetic characteristics. To compensate for the

inevitable modeling errors, the first direction of research could be to extend the proposed

ATC towards parametric, closed-loop ATC. Second, the performance of the innermost

current control loop could be improved by means of a model-based feedforward control.

The feedforward controller could be derived based on the knowledge of the SRM phase

current dynamics and the employed commutation strategy. The third direction of

8.2 Recommendations for future research 139

research could be to replace the expensive, high-resolution encoder, with a cheap, low-

resolution one (1-2 pulses per rotation), e.g., to enable event-triggered 4/2 SRM 4Q

speed control.

Chapter 5 concerned the development of the SRG explicit voltage MPC. The controller

choice was motivated by its straightforward ability to enforce the constant, average DC-

link voltage and current bounds. However, to increase the control system performance

and extend the machine operating range, one could consider identifying and imposing

varying DC-link current constraints instead. This could be achieved, e.g., via an online

Nonlinear Model Predictive Control. Furthermore, the PWM current control could also

be employed to allow the SRG to operate at low rotor speeds. Naturally, experimental

validation of the proposed solution is of interest as well.

In Chapter 6, the position-sensorless SRM speed controller was described. The

developed closed-loop controller required that the SRM is brought to a sufficiently high

speed before being engaged. In the absence of an external rotor torque, this implies

the need for a dedicated SRM startup procedure. Its development, as well as the

experimental verification of the complete solution, could be a topic for future research.

Chapter 7 considered the auto-calibration of the generator-turbine throttle unit. For this

purpose, the optimal turbine reference speed was modeled as a multivariate polynomial

in the turbine vanes position signal and pressure ratio. Although simple to implement,

this model/basis choice is almost certainly not optimal. Therefore, the first extension

of this work could be to improve the parametrization of the turbine optimal reference

speed (e.g., using the obtained analytical insights), yielding a more accurate/faster

parameter convergence. The second extension could treat the design of the auto-

calibration experiment, i.e., of the mechanism for generation of suitable disturbance

inputs (such as the turbine pressure ratio). The third extension could focus on the

analysis and generalization of the underlying ESC scheme, e.g., by considering other

parameter/derivative estimators, neural networks, etc.

Finally, as a step towards further integration of the presented research topics, the

developed ESC algorithm could also be utilized for the SRM real-time tuning purposes,

e.g., to minimize the acoustic noise, torque ripple and/or energy losses. In such cases, the

ESC disturbances could be interpreted as the signals defining the SRM operating point,

whereas the commutation angles, or any other suitable low-level controller parameters,

could be selected as the ESC optimizing inputs.

140

Appendix A

MCU case study: parameter

specification


Drive cycle

δt 1 s

tend 30 min

tyear 1 year

tlife 15 year

dyear 12 000 km

Ambient

ρAM 1.184 kg/m3

pAM 98 kPa

TAM 298 K

ag 9.81 m/s2

λκ 1.4

cp 1005 J/kg/K

cR 287 J/kg/K

Vehicle

mV 970 kg

λV 1.3

rW 29.57 cm

cd 0.32

cr 0.01

Af 2.07 m2

Table A.1: MCU case study: model and design parameter values

141

142 A MCU case study: parameter specification


Gearbox[cG0, cG1, cG2] [700, 3, 0]

λG [15.2, 8.2, 5.3, 4.0, 3.3]

Mec. power linkλA 1.5

PmAU 500 W

ICE

mE 62.67 kg

VE 1000 cm3

[cvol0, cvol1, cvol2] [3.6 · 10−1, 3.0 · 10−3,−3.3 · 10−6]

[cη0, cη1, cη2] [4.0 · 10−1, 1.1 · 10−4,−1.7 · 10−7]

[cf0, cf2] [3.1 · 104, 6.0 · 10−1]

[cg0, cg1] [9.0 · 104,−7.6 · 104]

λec 10

pEM 108 kPa

ωE,min 60 rad/s

ωE,max 680 rad/s

εE,year 5 %

µE 0.67 e/kW

Air-fuel control

Hl 42.7 MJ/kg

ρφ 758.8 kg/m3

λαφ 14.7

µφ 1.216 e/l

Air system

ηC 60 %

λΠ,max 1.58

[cT0, cT1] [0.59, 0.41]

[cm0, cm1] [0.43, 0.56]

[csurge0, csurge1] [1.55 · 105, 3.72 · 103]

MCU

[cM00, cM01, cM02] [33.2, 1.4 · 10−3, 4.8 · 10−7]

[cM1, cM2] [1.09, 5.01 · 10−6]

cC,max 2.6 kW

Alternator

[cA00, cA01, cA02] [70.8,−2.0 · 10−1, 5.5 · 10−4]

[cA1, cA2] [−8.0 · 10−1, 6.8 · 10−5]

cA,max 2.4 kW


143


El. power link P eAU 300 W

Battery

nB 4

mc 80 g

Qc 2.3 Ah

cc0 3.27 V

cc1 58 kF

Rc 11.5 mΩ

ic,min/max ∓35 A

socmin/max 20, 80%

εB,year 5 %

µB 500 e/kWh


144

Appendix B

GTU case study: parameter

specification


Drive cycle

δt 2 s

tyear 1 year

tlife 8 year

dyear 12 000 km

Ambient

ρAM 1.184 kg/m3

pAM 103 kPa

TAM 298 K

ag 9.81 m/s2

λκ 1.4

cp 1005 J/kg/K

cR 287 J/kg/K

Vehicle

mV 970 kg

λV 1.3

rW 29.57 cm

cd 0.32

cr 0.01

Af 2.07 m2

Table B.1: GTU case study: model and design parameter values

145

146 B GTU case study: parameter specification


Gearbox

[cG0, cG1, cG2] [700, 3, 0]

λG [15.2, 8.2, 5.3, 4.0, 3.3]

gup,max 1

Mec. power linkλA 1.5

PmAU 1100 W

ICE

VE 1600 cm3

[cvol0, cvol1, cvol2] [3.6 · 10−1, 2.9 · 10−3,−2.3 · 10−6]

[cη0, cη1, cη2] [3.9 · 10−1, 1.16 · 10−4,−1.7 · 10−7]

[cf0, cf1, cf2] [2.51 · 104, 0.00, 0.45]

[cΠ0, cΠ1, cΠ2] for q = 0 [0.044, 1.000,−0.045]

[cΠ0, cΠ1, cΠ2] for q = 1 [0.032, 0.770, 0.200]

ηE2 7 · 10−7

λec 10

pEM 108 kPa

ωE,min/max [70, 680]rad/s

εE,year 5 %

µE 0.67 e/kW

Air-fuel control

Hl 42.7 MJ/kg

ρφ 758.8 kg/m3

λαφ 14.7

µφ 1.216 e/l

Air system

λΠ,min 0.0384

λ∗bsr 0.6

rTB 0.03 m

ηTB,max 60 %

[cTB1, . . . , cTB4] [0.54,−0.95, 0.62,−0.21]

GTU

[cGE00, cGE01, cGE02] [24.0, 1.0 · 10−3, 7.3 · 10−7]

[cTB0, . . . , cTB4] [117,−246, 388,−395, 160]

cGE1 −0.92

Alternator

[cA00, cA01, cA02] [102.5,−0.2, 5.5 · 10−4]

[cA1, cA2] [−0.7, 6.8 · 10−5]

[cA,max0, cA,max1] [1800, 60]


147


El. power link P eAU 800 W

Battery

nB 4

mc 80 g

Qc 2.3 Ah

cc0 3.27 V

cc1 58 kF

Rc 11.5 mΩ

ic,min/max ∓35 A

socmin/max 20, 80%

εB,year 5 %

µB 500 e/kWh


148

Appendix C

SRM advance angle scheduling signal

The finite difference approximation of (4.1) and (4.2) yields

∆ip ≈∆t

Lp(ip, θ)(vp −Rpip −Mp(ip, θ)ω), (C.1)

∆θ ≈ ω∆t.

By neglecting the Ohmic losses and the back e.m.f. term, it follows

∆ip ≈vp∆θ

ωLp(ip, θ). (C.2)

Then by setting |vp| = vdc and |∆ip| = i∗,

∆θ ∼ i∗ω

vdc∼ i∗

imax

ω

ωmax

vdc,minvdc

=: q. (C.3)

In the context of commutation, ∆θ can be interpreted as the required change in

duration/start of the active phase interval (by means of commutation angle advance)

needed to compensate for the change in the variable q. This motivates the use of q as

the advance angle scheduling signal.

149

150

Appendix D

SRG average DC-link current

The average DC-link current is defined as

idc =1

θrpp

θon+θrpp∫

θon

idcdθ =1

θrpp

m∑

p=1

θon+θrpp∫

θon

ipupdθ (D.1)

If all phases share the same electromagnetic properties, it follows

idc =m

θrpp

θon+θrpp∫

θon

i1u1dθ. (D.2)

Furthermore, assume that the phase ohmic voltage drop Rpip can be neglected, and

both ω and vdc are constant during one electrical period of length θrpp. Then, in the

SP mode, with θon and θoff commutation angles, the phase flux linkage extinguishes at

the angle θe = θoff + ∆, where ∆ = θoff − θon is the dwell angle. Moreover, if the SRG

operates in non-saturating conditions, it holds Lp(ip, θ) = Lp(θ), i.e., ip = ψp/Lp. This

yields

idc ≈m

θrpp

θoff∫

θon

ψ1

L1

dθ +

θe∫

θoff

−ψ1

L1

dθ

. (D.3)

151

152 D SRG average DC-link current

If the inductance L1 in its decreasing region can be approximated by an affine function

L1(θ) ≈ kLθ + nL, then it follows

idc ≈mvdcθrppω

θoff∫

θon

θ − θon

kLθ + nLdθ +

θe∫

θoff

θ − θe

kLθ + nLdθ

, (D.4)

≈ mvdcθrppωkL

[2∆−

(θon +

nLkL

)ln

(θoff + nL

kL

θon + nLkL

)−(θe +

nLkL

)ln

(θe + nL

kL

θoff + nLkL

)],

≈ mvdcθrppωk2

L

[2∆kL − Lon ln

(Loff

Lon

)+ Le ln

(Loff

Le

)],

with Lon = L1(θon), Loff = L1(θoff) and Le = L1(θe). Note that the derived expression

for idc is defined for θon ≥ θrpp2

and θe = 2θoff − θon ≤ θrpp.

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Summary

Electrified engine air intake system: modeling, optimization and control

Despite recent advances in electric and fuel cell vehicle technology, the internal com-

bustion engine is still seen as a key facilitator of ground transportation for the next

decade. This is mainly due to superior energy density and storage properties of carbon-

based liquid fuels compared to, e.g., electric batteries. However, increasing societal

concerns for natural resource depletion and environmental pollution put ever-tightening

constraints on the vehicle fuel economy. This thesis addresses the improvement of the

vehicle fuel economy using two novel engine technologies: electric supercharging and

regenerative throttling. The research conducted within this domain resulted in the

following main developments.

The first new development is a theoretical investigation of a downsized engine concept,

where short-duration, high-power demands are delivered by means of a standalone

electric supercharger. In this context, the supercharger consists of a compressor and a

high-speed electric motor, which is powered from a car battery. This research presents

a novel, convex method for the battery and engine sizing such that they match the

supercharger energy requirements and power-enhancing capabilities. A simulation-

based case study is also provided, showing that, over a specific driving cycle, the

investigated powertrain configuration can yield up to 10% savings in fuel costs, w.r.t. a

naturally-aspirated engine powertrain scenario.

The second new development concerns a theoretical investigation of regenerative throt-

tling for gasoline engines. Namely, by replacing a throttle valve with a high-speed

generator-turbine throttle unit (GTU), the engine intake airflow can be controlled while

simultaneously producing electricity. In this development a new computational method

is proposed for studying the effect of such a device on the vehicle fuel consumption.

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170 Summary

The analysis has shown that regenerative throttling has a potential to deliver 2-4% fuel

cost savings compared to a conventional throttle valve situation.

The third new development relates to Switched Reluctance Machine (SRM) control.

The SRM is an electric machine free of brushes, rotor windings and permanent magnets.

Its simple, low-cost design and high-speed capability make it suitable for both electric

supercharging and regenerative throttling applications. However, the same design also

imposes a considerable challenge for control, as it results in inherently nonlinear switched

system dynamics. This issue has been addressed by means of three novel control

strategies. The first introduces a four-quadrant speed tracking controller for 4-stator/2-

rotor pole SRMs. The second provides an Explicit Model Predictive Controller for

the SRM output voltage tracking. The third, however, explores the possibility of the

SRM speed control using only the DC-link voltage and current measurements, i.e.,

without speed, position, and phase voltage and current sensors. The effectiveness of all

presented control algorithms has been verified in simulations, whereas the first has also

been validated experimentally.

Finally, the fourth new development is a method for the GTU auto-calibration. To

maximize the GTU fuel-saving potential the turbine speed needs to be suitably matched

to the conditions present in the engine air-intake system, at all times. However, often the

exact optimal speed value, which yields maximal energy recovery, is unknown or difficult

to derive. This research proposes a non-model-based solution to the problem of finding

the optimal turbine rotational speed. The algorithm is based on a novel Extremum

Seeking Control (ESC) law, with a disturbance-based optimal input parametrization.

The proposed method allows adaptive reconstruction of the unknown relationship

between the measured disturbance signals, i.e., the turbine pressure ratio and its vanes

position, and the optimal turbine speed – in an initial, automated calibration step.

The usefulness of the presented auto-calibration scheme, however, extends beyond this

particular application. It can be applied to any similar ESC situation, where the

disturbances leading to the changes in the optimal inputs can be measured or estimated.

Acknowledgements

Over the past four years I have faced many challenges while working on this PhD thesis.

Overcoming them often required reaching beyond my original abilities and venturing

into the unknown. This lengthy endeavour would’ve been surely far more difficult for

me if it wasn’t for the help and support of many people, some of which I wish to thank

in particular.

First of all, I would like to state my sincere appreciation to my promotor Maarten

Steinbuch. Dear Maarten, thank you for giving me the opportunity to do this PhD

research, for supporting my attendance at numerous conferences and for many efficient

and stimulating conversations we had. Talking with you helped me keep the stress low

and focus sharp, for which I thank you the most.

I would also like to thank my co-promotor Bram de Jager. Dear Bram, I am grateful

for having the opportunity to work with you. Thank you for your thoughtful guidance

and support, for always being open for discussion, for your critical reading of all my

manuscripts and for providing me with an abundance of valuable suggestions on how

to improve my work. You have taught me the art of conducting a consistent and a

productive research process while keeping an eye on the detail. For this and for many

other things, I am immensely thankful.

I also want to express my gratitude to my other committee members: Christopher

Onder, Jonas Sjoberg, Nathan van de Wouw and Michael Boot, for taking the time

to read my thesis and for their constructive comments. I owe my special thanks to

Michael, for our regular discussions which helped me engage with new ideas and see the

big picture more clearly.

I would further like to thank Nikolce Murgovski for his help on convex modeling and

optimization, and for all the technical and non-technical discussions we had. Nikolce,

you have considerably contributed to my PhD thesis, both in a direct and an indirect

way. Thank you for being not only a fellow researcher but a friend as well.

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172 Acknowledgements

I also wish to thank my BSc and MSc students: Erik Hoedemaekers, Noud Strous,

Evangelos Stamatopoulos, Devavrat Shanbhag and Daniel Hilgersom, for their efforts

and contributions. My special thanks goes to Daniel for helping me resolve many

practical issues I’ve encountered while performing the lab experiments, for our casual

discussions and for embarking on the whole startup adventure with me. Moreover, I

would like to thank Wietse Loor for helping me with the specification and the assembly

of the experimental setup, and for explaining me all the details behind the accompanying

electronics.

Thanks to my office mates Tom Gommans, Niek Borgers and Benjamin Biemond for all

the fun talks we had, in and out of our office, and especially for tolerating my annoying

lack of coffee/tea drinking habits. In addition, I want to thank all my colleagues from the

-1 floor: Nick Bauer, Menno Lauret, Behnam Assadi, Masoud Dorosti, Xi Luo, Victor

Dolk, Elise Moers, Dennis Heck, Thijs Vromen, Bert Maljaars, Robbert van Herpen,

Eelco van Horssen, Nikolaos Kontaras, Alejandro Morales Medina, Isaac Castanedo

Guerra and many others, for making my stay at the TU/e very enjoyable. Special

thanks to Emilia Silvas, Cesar Lopez Martinez and Emanuel Feru for all the laughs we

had and for all those lengthy conversations about life and the meaning of everything.

Furthermore, I wish to thank Frank Boeren, Michiel Beijen, Bas van Loon and Niek

Borgers for an amazing US road trip that I will never forget. I would also like to thank

Petra Aspers, Geertje Janssen-Dols and all the women from the ME HR Services for

solving my administrative problems in a smooth and friendly manner.

Furthermore, I want to thank my closest friends: Nenad, Boris, Ivan, Brano, Nikola,

Mita, Branko, Saja, Zarko, Tijana, Durda, Maarten, Guido, Daniel and others not

explicitly mentioned, for supporting me and for being there for me.

Finally, I would like to express my deepest gratitude to my parents Dimitrije and Ruzica,

sister Milana, girlfriend Nevena and my extended family: cousins Milica, Ljubica,

Verica, Dragana, Boba, Branislav, aunt Ruzica, uncle Zika and others. Their limitless

love, caring advices and constant encouragement helped me endure through the toughest

of times.

Mama, tata, Milana i Nevena, bezgranicno vam hvala na vasoj ljubavi, podrsci i

strpljenju. Moj uspeh je i vas uspeh, jer vi ste moje najvece blago..

Sava Marinkov,

Eindhoven, November 2015.

Curriculum Vitae

Sava Marinkov was born in Novi Sad, Serbia in 1986. He received

his B.Sc. degree (best in class) from the department of Electrical

and Computer Engineering at the University of Novi Sad, Serbia,

in 2009. He continued his education at the Eindhoven University of

Technology, The Netherlands, where he obtained his M.Sc. degree

(cum laude) in Systems and Control from the department of Me-

chanical Engineering, in 2011. His master thesis concerned motion

control of an anthropomorphic robotic arm. During his Master studies he also joined

ASML, The Netherlands, for a three-month internship on wafer stage motion control.

In January 2012, Sava started his PhD research within the Control Systems Technology

group of the department of Mechanical Engineering at the Eindhoven University of

Technology, under the guidance of prof.dr.ir. M. Steinbuch and dr.ir. A.G. de Jager.

His research was financially supported by the EUREKA program, project WETREN,

focusing on a reduction of vehicle fuel consumption using high-speed electric turbine

and compressor devices. The main results of his research are presented in this thesis.

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Electrified engine air intake system: modeling ... · Electrified engine air intake system: modeling, optimization and ... of high-speed Switched Reluctance Machines ... system: modeling,