Modeling and Control of Dual Mechanical Port Electric Machine

MODELING AND CONTROL OF DUAL MECHANICAL PORT ELECTRIC MACHINE

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Haiwei Cai

Graduate Program in Electrical and Computer Science

The Ohio State University

2015

Dissertation Committee:

Dr. Longya Xu, Advisor

Dr. Jin Wang,

Dr. Mahesh Illindala

Haiwei Cai

2015

Abstract

The Dual Mechanical Port (DMP) electric machine has two rotors that can be controlled to rotate in different speeds and directions. Compared with conventional electric machines with only one rotor, the DMP machine provides higher torque density and much better control flexibility. However, the DMP machine also has relatively complex structure, which bring challenges to the modeling and control of the machine. The existing model and control algorithms for single rotor electric machines cannot be applied to the DMP machine directly.

In this work, the application of DMP machine on hybrid electric vehicle will be used as an example to explain the electromagnetic characteristics and functionality of the DMP machine. The model and the control algorithms for two different DMP machines are investigated. The first DMP machine uses two layers of Permanent Magnets in the outer rotor; it is called the PMDMP machine. The second DMP machine uses single layer of

Squirrel Cage in the outer rotor; it is called the SCDMP machine. The study on modeling and control for the SCDMP machine is the major contribution of this work.

The PMDMP machine stands out for its high torque density and high efficiency when compared with other DMP machines. Detail model derivation for the PMDMP machine is presented. The independent control of its two rotors is investigated and verified by simulations and experiments. To overcome the problems brought by the position sensors,

ii position sensorless control algorithms for the PMDMP are also investigated. High frequency injection and sliding mode sensorless control algorithms are applied to the

PMDMP machine at low speed and high speed, respectively. The performance of the sensorless control algorithms in experiments matches well with the simulation results. To verify the functionality of the DMP machine in power split hybrid application, the power flow pattern in various operational modes are discussed and simulated.

In order to avoid using the high cost rare earth permanent magnets, the SCDMP machine is proposed. This DMP machine replaces the permanent magnets in the outer rotor with a squirrel cage. Since this DMP machine has a squirrel-cage outer rotor, it is named as SCDMP machine. First, the electromagnetic characteristic of the SCDMP machine is analyzed. Then, the transient model and steady-state model of the SCDMP machine are derived. The proposed machine models are verified by finite element method and simulation. The results show that the proposed models accurately represent the unique electromagnetic characteristics of the SCDMP machine.

Due to its unique electromagnetic characteristics, control algorithms for conventional machines cannot be applied to the SCDMP machine. The methods to calculate the correct current commands and to estimate the outer rotor flux position are proposed. Based on these two methods, a control algorithm for the SCDMP machine is proposed and estimated by simulation. The results show that the proposed control algorithm is able to independently control the torque productions and the flux levels of the two rotors of the

SCDMP machine.

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Dedication

This document is dedicated to my family and all the people that I love.

Acknowledgments

I would like to express my deepest gratitude to my adviser, Dr. Longya Xu, for his insightful academic guidance and consistent funding support during my graduate study. Dr.

Xu set a great example of an excellent researcher for me and helped me develop critical and independent thinking skills, which will be beneficial throughout my life. I also want to thank Dr. Xu for providing me the chance to freely explore different research topics.

Without his encouragement and patience, this dissertation would not have been possible.

I would like to thank Dr. Jin Wang and Dr. Mahesh Illindala for being committee members of my graduate study. They provided me many insightful comments and constructive suggestions in the review of my research proposal and dissertation. My special thanks go to Dr. Vadim Utkin as well for his invaluable advice in my Candidacy Exam.

My thanks are extended to my fellow colleagues Dr. Dakai Hu, Dr. Yazan Alsmadi,

Dr. Yu Liu, Dr. Kaichien Tsai, Dr. Zhendong Zhang, Dr. Bo Guan, Mr. Ying Xiao, Mr.

Feng Qi, Mr. Miao Wang, Mr. Jianyu Pan, Mr. Alejandro Pina Ortega, Mr. Qi Chen and

Mr. Han Yang for their generous help and friendship during my study at the Ohio State

University.

Vita

September 2006 – July 2010 ...... B.S. Electrical Engineering & Automation, South China University of Technology, Guangzhou, China

September 2010 – August 2012 ...... Master’s Student, Graduate Fellow, The Ohio State University, Columbus, Ohio

May 2015 – August 2015 ...... Systems Engineer, Summer Intern, Nexteer Automotive, Saginaw, Michigan

September 2012 – present ...... PhD Student, Graduate Research Associate, The Ohio State University, Columbus, Ohio

Publications

H. Cai and L. Xu, "Modeling and Control for Cage Rotor Dual Mechanical Port Electric Machine–Part I: Model Development," in Energy Conversion, IEEE Transactions on , vol.30, no.3, pp.957-965, Sept. 2015.

H. Cai and L. Xu, "Modeling and Control for Cage Rotor Dual Mechanical Port Electric Machine—Part II: Independent Control of Two Rotors," in Energy Conversion, IEEE Transactions on , vol.30, no.3, pp.966-973, Sept. 2015.

H. Cai, B. Guan and L. Xu, "Low-Cost Ferrite PM-Assisted Synchronous Reluctance Machine for Electric Vehicles," in Industrial Electronics, IEEE Transactions on , vol.61, no.10, pp.5741-5748, Oct. 2014.

H. Cai and L. Xu, "Control principle of Dual Mechanical Port electric machine with Squirrel-Cage outer rotor," in Transportation Electrification Asia-Pacific (ITEC Asia- Pacific), 2014 IEEE Conference and Expo , vol., no., pp.1-6, Aug. 31 2014-Sept. 3 2014.

H. Cai and L. Xu, "Modeling of dual mechanical port machine with squirrel-cage outer rotor for hybrid electric vehicles," in Energy Conversion Congress and Exposition (ECCE), 2014 IEEE , vol., no., pp.4888-4895, 14-18 Sept. 2014

H. Cai, B. Guan, L. Xu and W. Choi, "Optimal design of synchronous reluctance machine: A feasible solution to eliminating rare earth permanent magnets for vehicle traction applications." COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 33.5 (2014): 1569-1586.

Q. Ahmed, H. Cai, G. Rizzoni and L. Xu, "Modeling and Control of a Novel Power Split Hybrid Electric Vehicle." ASME 2014 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2014.

H. Cai, L. Xu, "Maximum Torque Control of Induction Machine in Deep Flux Weakening Region," in Energy Conversion Congress and Exposition (ECCE), 2015 IEEE , vol., no., pp.3928-3933, 20-24 Sept. 2015

A. Pina, H. Cai, Y. Alsmadi, L. Xu, “Analytical Model for the Minimization of Torque Ripple in Permanent Magnets Assisted Synchronous Reluctance Motors Through Asymmetric Rotor Poles”, in Energy Conversion Congress and Exposition (ECCE), 2015 IEEE , vol., no., pp.5609-5615, 20-24 Sept. 2015

Fields of Study

Major Field: Electrical and Computer Engineering

vii

Table of Contents

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... vi

Publications ...... vi

Fields of Study ...... vii

Table of Contents ...... viii

List of Tables...... xi

List of Figures ...... xii

Nomenclature ...... xvi

Chapter 1: Introduction ...... 1

1.1 Background of research ...... 1

1.2 Literature review ...... 4

Chapter 2: Permanent Magnet Dual Mechanical Port Electric Machine ...... 10

2.1 Modeling of PMDMP machine ...... 10

viii

2.1.1 Introduction of PMDMP machine ...... 10

2.1.2 Three phase model of PMDMP machine ...... 12

2.1.3 Model in synchronous reference frame...... 15

2.2 Design of PMDMP machine ...... 17

2.2.1 Simplified machine model...... 17

2.2.2 PMDMP machine design flowchart...... 19

2.2.3 Dimensions, parameters and specifications of prototype PMDMP machine. ... 21

2.2.4 Open circuit test result of prototype PMDMP machine...... 24

2.3 Control of PMDMP machine ...... 28

2.3.1 Field oriented control...... 29

2.3.2 Position sensorless control of outer rotor ...... 33

2.4 Operation modes of PMDMP machine ...... 49

2.4.1 Power flow analysis...... 49

2.4.2 Multi-operational modes of the PMDMP machine...... 53

2.4.3 Simulation of the multi-operation modes...... 57

Chapter 3: Squirrel-Cage Dual Mechanical Port Electric Machine ...... 60

3.1 Modeling of SCDMP machine...... 60

3.1.1 Three phase model of SCDMP machine ...... 60

3.1.2 Model in synchronous reference frame ...... 62

3.1.3 Model in stationary reference frame ...... 69

3.1.4 Validation of the proposed model by finite element method...... 74

3.2 Independent control of the two rotors of SCDMP machine ...... 82

3.2.1 Introduction of SCDMP machine control algorithm...... 82

3.2.2 Current command calculation module...... 82

3.2.3 Outer rotor flux observer...... 88

3.2.4 Simulation of the proposed FOC algorithm...... 90

3.3 Operational modes of the SCDMP machine ...... 98

3.3.1 Power flow analysis of the SCDMP machine...... 98

3.3.2 Simplified driving cycle of hybrid vehicle...... 102

Chapter 4: Conclusions and Future Work ...... 105

4.1 Conclusions ...... 105

4.2 Future work ...... 107

References ...... 109

Appendix A: Model Derivation of the PMDMP Machine ...... 115

Appendix B: Model Derivation of the SCDMP Machine...... 120

Appendix C: Parameter Measurement of the SCDMP Machine ...... 124

List of Tables

Table 2. 1 Dimensions of the prototype PMDMP machine...... 21

Table 2. 2 Open circuit test result of the prototype PMDMP machine...... 24

Table 2. 3 Flux linkage of the prototype PMDMP machine...... 27

Table 2. 4 Power relationships of the constant power mode of the PMDMP machine. .... 51

Table 3. 1 Specifications of sample SCDMP machine...... 75

Table 3. 2 Validation of the proposed steady state machine model by FEM...... 80

Table 3. 3 Rated fluxes and inudctances of the sample SCDMP machine...... 90

Table C. 1 Measurement of SCDMP machine parameters...... 126

List of Figures

Figure 1. 1 Conventional Internal Combustion Engine (ICE) based vehicle...... 1

Figure 1. 2 Existing power-split HEV system topology...... 2

Figure 1. 3 Structure of the DMP machine...... 4

Figure 1. 4 Power-split HEV based on the DMP machine...... 5

Figure 1. 5 Different outer rotor designs for the DMP machine...... 6

Figure 1. 6 One layer of squirrel-cage windings (the SCDMP machine)...... 7

Figure 2. 1 Sideview of PMDMP machine (two layers of PMs)...... 10

Figure 2. 2 Components of PMDMP machine...... 11

Figure 2. 3 Conceptual cross-section of double-layer PMDMP machine...... 12

Figure 2. 4 Flux line distribution of the PMDMP machine with stator current alone...... 13

Figure 2. 5 Flux line distribution of the PMDMP machine with inner rotor current alone.

...... 14

Figure 2. 6 The d-axis equivalent circuit of the PMDMP machine...... 16

Figure 2. 7 The q-axis equivalent circuit of the PMDMP machine...... 16

Figure 2. 8 Simplified d-axis equivalent circuit of the PMDMP machine...... 17

Figure 2. 9 Simplified q-axis equivalent circuit of the PMDMP machine...... 18

Figure 2. 10 Design flowchart for the PMDMP machine...... 20 xii

Figure 2. 11 Cross-section of the prototype PMDMP machine...... 22

Figure 2. 12 Open circuit flux line distribution of the prototype machine...... 22

Figure 2. 13 Open circuit flux density distribution of the prototype machine...... 23

Figure 2. 14 Stator winding line to line back EMF and harmonic analysis...... 25

Figure 2. 15 Inner rotor winding line to line back EMF and harmonic analysis...... 26

Figure 2. 16 Control block diagram for the PMDMP machine...... 30

Figure 2. 17 Matlab model of the PMDMP machine and controller...... 31

Figure 2. 18 Actual speed (black) and reference speed (red) of the DMP machine ...... 31

Figure 2. 19 Torque production of the DMP machine...... 32

Figure 2. 20 Phase current waveforms of the DMP machine...... 32

Figure 2. 21 Test setup for sensorless control of PMDMP machine...... 33

Figure 2. 22 Effect of high frequency boltage on salient pole PMSM...... 35

Figure 2. 23 Procedures to extract rotor position from stator currents...... 36

Figure 2. 24 Polarity identification based on magnetic saturation...... 37

Figure 2. 25 Block diagram for outer rotor sensorless control at zero and low speeds. .... 38

Figure 2. 26 Simulation result of high frequency injection rotor position estimation...... 39

Figure 2. 27 Experiment result of high frequency injection rotor position estimation. .... 40

Figure 2. 28 Saturation function for SMO...... 44

Figure 2. 29 Block diagram for the rotor position SMO...... 45

Figure 2. 30 Block diagram for outer rotor sliding mode sensorless control at medium and high speeds...... 45

Figure 2. 31 Simulation result of sliding mode rotor position observer...... 47

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Figure 2. 32 Experimental result of sliding mode rotor position observer...... 47

Figure 2. 33 Constant power mode of the HEV based on PMDMP machine...... 51

Figure 2. 34 Power flow of the constant power mode of the PMDMP machine...... 52

Figure 2. 35 Low power mode of the HEV based on PMDMP machine...... 53

Figure 2. 36 High power mode of the HEV based on PMDMP machine...... 54

Figure 2. 37 Pure electric vehicle mode of the HEV based on DMP machine...... 55

Figure 2. 38 Highway (pure ICE) mode of the HEV based on PMDMP machine...... 56

Figure 2. 39 Reference and actual rotor speeds in different operational modes...... 58

Figure 2. 40 Torque productions of both rotors in different operational modes...... 58

Figure 2. 41 Current waveforms of stator and inner rotor windings in different modes. .. 59

Figure 3. 1 Conceptual cross-section of the SCDMP machine...... 61

Figure 3. 2 Reference frames for the SCDMP machine...... 62

Figure 3. 3 Flux line distribution of the SCDMP machine with stator current alone...... 65

Figure 3. 4 Flux line distribution of the SCDMP machine with inner rotor current alone.

...... 66

Figure 3. 5 The d-axis equivalent circuit of the SCDMP machine...... 67

Figure 3. 6 The q-axis equivalent circuit of the SCDMP machine...... 68

Figure 3. 7 Steady state equivalent circuit of the SCDMP machine ...... 71

Figure 3. 8 SCDMP machine operated as squirrel-cage induction machine...... 72

Figure 3. 9 SCDMP machine operated as doubly-fed induction machine...... 73

Figure 3. 10 Cross-section of the sample SCDMP machine...... 74

xiv

Figure 3. 11 Comparison between transient responses of proposed model (left) and FEM model (right)...... 77

Figure 3. 12 Reference frames for the SCDMP machine...... 84

Figure 3. 13 Current command calculation module of the SCDMP machine...... 87

Figure 3. 14 Proposed outer rotor flux observer...... 89

Figure 3. 15 Control block diagram of the SCDMP machine...... 91

Figure 3. 16 Flux, speed and torque production of the SCDMP machine...... 94

Figure 3. 17 Current of the SCDMP machine...... 96

Figure 3. 18 Comparison between estimated and actual outer rotor flux angles...... 97

Figure 3. 19 Simplified driving cycle of hybrid vehicle...... 104

Figure C. 1 Steady state equivalent circuit of the SCDMP machine ...... 124

Figure C. 2 SCDMP machine operated as squirrel-cage induction machine...... 125

Figure C. 3 SCDMP machine operated as doubly-fed induction machine...... 125

Figure C. 4 Equivalent SCIM when stator windings open...... 126

Nomenclature

Subscripts

푠, 표푟, 푖푟 Stator, outer and inner rotor variables, respectively.

푙, 푚 Leakage and mutual inductance indicator, respectively.

푎, 푏, 푐 Phases a, b, c variables, respectively.

푑, 푞, 0 Variables in 푑-axis, 푞-axis, 0-axis, respectively.

훼, 훽, 훾 Variables in 훼-axis, 훽-axis, 훾-axis, respectively.

푇 푇 푑푞0 Column vector in [푑 푞 0] format. For example: 휆푑푞0푠 = [휆푑푠 휆푞푠 휆0푠] .

푇 푇 푎푏푐 Column vector in [푎 푏 푐] format. For example: 휆푎푏푐푠 = [휆푎푠 휆푏푠 휆푐푠] .

Greek symbols

푣, 휆, 푖, 푅 Voltage, flux linkage, current and resistance, respectively.

푃, 푇, 휔, 퐽 Power, torque, angular speed and inertia of rotor, respectively.

휆푃푀표푟 Stator flux linkage provided by outer rotor permanent magnets.

휆푃푀𝑖푟 Inner rotor flux linkage provided by outer rotor permanent magnets.

휃푃푀 Electrical angle between permanent magnet flux and stator phase a-axis.

휃푑 Electrical angle between d-axis and stator phase a-axis.

xvi

휃𝑖표 Electrical angle between outer rotor phase a-axis and inner rotor phase a- axis.

휃𝑖푟 Electrical angle between inner rotor phase a-axis and stator phase a-axis.

휃표푟 Electrical angle between outer rotor phase a-axis and stator phase a-axis.

Abbreviations

DFIM Doubly Fed Induction Machine

DMP Dual Mechanical Port

EV Electric Vehicle

EVT Electric Variable Transmission

FEM Finite Element Method

FOC Field Oriented Control

HEV Hybrid Electric Vehicle

HFI High Frequency Injection

ICE Internal Combustion Engine

ID/OD Inner Diameter/ Outer Diameter

PM Permanent Magnet

PMSM Permanent Magnet Synchronous Machine

PMDMP Permanent Magnet Dual Mechanical Port

SCDMP Squirrel Cage Dual Mechanical Port

SCIM Squirrel Cage Induction Machine

SMO Sliding Mode Observer

xvii

Chapter 1: Introduction

1.1 Background of research

Most of the vehicles nowadays depend solely on the mechanical power from the

Internal Combustion Engine (ICE) to provide traction. The system topology can be simplified to the one shown in Figure 1. 1. The ICE is capable of converting the chemical energy from the gasoline to the mechanical energy at the wheels. This kind of system is relatively simple and the technologies it requires are mature enough to allow low manufacturing cost.

Gasoline ICE Wheels

Figure 1. 1 Conventional Internal Combustion Engine (ICE) based vehicle.

However, the system shown in Figure 1. 1 has its own drawbacks even though it has been very popular over the past 100 years. The first problem is that the ICE cannot always be operated at the highest fuel efficiency point due to the varying driving needs at the wheels. The second problem is that the kinetic energy of the vehicle simply becomes heat 1 during the braking of the vehicle, which is a waste of energy. The third problem is that the

ICE efficiency is low during the starting of the vehicle because the ICE provides very low torque when its speed is low. In order to overcome these problems, the Hybrid Electric

Vehicle (HEV) was proposed.

As indicated by its name, the HEV combines the two energy sources, the electrical energy from the battery and the mechanical energy from the ICE, to meet the varying driving needs. There are several types of HEV structures, such as the series structure, parallel structure and the series-parallel structure. With the help of the HEV systems, fuel economic of vehicles has been greatly improved. In this work, the serial-parallel hybrid vehicle is selected as the research topic [1]-[3]. The series-parallel HEV structure is also called the power-split HEV structure. A typical power-split HEV system topology is shown in Figure 1. 2.

Planetary Gasoline ICE Motor Wheels Gear

Mechanical port 1 Mechanical Port 2 Generator

Inverter

Electrical port 1: Battery Electrical port 2

Figure 1. 2 Existing power-split HEV system topology.

As shown in Figure 1. 2, the exiting power-split HEV system requires a motor, a generator, a mechanical power split device (usually a planetary gear set) and inverters.

With these extra components inserted between the ICE and the wheels, the difference between the driving needs at the wheel and the highest fuel efficiency point of the ICE can be compensated by power provided by the battery. Thus, the ICE can always be operated at the highest fuel efficiency point. The power-split HEV system has solved the problems of the conventional vehicle, but it introduces new problems. First, the power-split HEV system requires extra components, which complicate the system structure and increase manufacturing cost. Second, the overall system size could be larger due to these extra components.

If the motor-generator-planetary gear set is represented by a block as indicated by the dash line in Figure 1. 2, it is easy to find out that this block has two mechanical ports and two electrical ports. The two mechanical ports are connected to the shaft of the ICE and the driving shaft of the wheels, respectively. The two electrical ports are connected two independent sets of inverters, respectively. If this block can be replaced by a single unit, which should also have two mechanical ports and two electrical ports, the overall system structure will be simplified and the system size will be reduced.

1.2 Literature review

In order to solve the problems brought by the exiting HEV system shown in Figure 1.

2 without sacrificing the benefits, the Dual-Mechanical-Port (DMP) electric machine was proposed [4]. The structure of the DMP machine is shown in Figure 1. 3. The DMP machine has three parts: the stator, the outer rotor and the inner rotor. The outer rotor and the inner rotor are mechanically coupled with the driving shaft of the vehicle and the shaft of the ICE, respectively. Hence, the machine can be named as Dual Mechanical Port (DMP) machine in order to distinguish it from conventional single-rotor machines. The stator and the inner rotor windings are connected to two sets of inverters sharing the same dc bus.

Because the inner rotor windings are connected to the inverter via slip rings and brushes, it is important to improve the system reliability by adopting high quality slip rings and brushes, and the development of the uncluttered variable transmission machine also provides a good solution to the issue [5] - [7].

Stator with three phase windings

Inner rotor with three phase windings, slip Outer Rotor rings and brushes

Figure 1. 3 Structure of the DMP machine.

As shown in Figure 1. 4, the generator-motor-planetary gear set in the existing power- split HEV system is replaced by a single electric machine. The DMP machine also has two electrical ports and two mechanical ports. Hence, the DMP machine turns out to be a very promising substitute for the existing power-split HEV structure. It’s gaining more and more attention for its compact size and flexibility in control.

To achieve the best fuel economic performance, if the battery pack has sufficient power to drive the DMP machine, the ICE should be turned off; otherwise, the ICE should be controlled to operate at the highest fuel efficiency point of the ICE whenever possible.

The highest fuel efficiency point is a fixed torque-speed point of the ICE, which is also called the “Sweet Spot”. With the help of the DMP machine, the fuel consumption is significantly reduced [8].

Mechanical port 1: Mechanical Port 2: Inner Rotor – ICE Outer Rotor - Wheels

Gasoline ICE Wheels

Electrical port 1: Electrical port 2: Inner rotor windings Inverter Stator windings – Inverter 2 & slip rings- Inverter 1

Battery

Figure 1. 4 Power-split HEV based on the DMP machine.

Different outer rotor designs for the DMP machine have been proposed [4], [9] - [12].

As shown in Figure 1. 5 (a), the DMP machine proposed in [9] uses two layers of permanent 5 magnet (PM) in the outer rotor (PMDMP machine). The single-layer PM outer rotor shown in Figure 1. 5 (b) is discussed in [4]. In [10], the Electric Variable Transmission (the EVT) is proposed. As shown in Figure 1. 5 (c), the EVT is built up from two concentric induction machines with a combined relatively thin yoke. The outer rotor in the EVT has two layers of squirrel-cage windings.

Two Layers One Layers Two Layers of of PMs of PMs Squirrel Cages

(a) (b) (c)

Figure 1. 5 Different outer rotor designs for the DMP machine.

The purpose of Chapter 2 is to conduct a systematic study on modeling, control and operational-modes of the PMDMP machine (Figure 1. 5 (a)). The modeling and the field oriented control algorithms of the PMDMP machine have been discussed in [9], [15] - [16] .

The operational-modes of the DMP machine is discussed in [18]. These research topics related to the PMDMP machine are presented as necessary background information in this work. On the other hand, the sensorless control algorithms for the PMDMP machine has not been fully investigated, except a sensorless control algorithm for the inner rotor is reported in [19].

Different position-sensorless control algorithms have been proposed for conventional single rotor Permanent Magnet Synchronous Machines (PMSMs) [19] - [24]. Based on the electromagnetic characteristics of the PMDMP machine, it will be shown in this work that these sensorless control methods for PMSMs can also be applied to the PMDMP machine.

In Chapter 3, the DMP machine with a single-layer Squirrel-Cage outer rotor

(SCDMP) is proposed. As shown in Figure 1. 6, in terms of its mechanical structure, the

SCDMP machine can be regarded as squirrel-cage induction machine with a wound rotor sitting inside the squirrel-cage rotor. Compared with the EVT [10], the size of the SCDMP machine is inherently smaller because only one set of squirrel-cage winding is needed in the outer rotor. Compared with DMP machines using expensive permanent magnets [4] [9], the cost for the machine can be significantly reduced. Also, the SCDMP machine has much better thermal robustness because the squirrel-cage rotor can tolerate higher temperature than permanent magnet rotors.

Figure 1. 6 One layer of squirrel-cage windings (the SCDMP machine).

However, the single-layer squirrel-cage rotor also brings challenge to the modeling of the machine. First, since the outer rotor of the proposed machine does not have a yoke, a

7 significant amount of flux will have to travel through the two layers of airgap and link all the windings of the machine. In other words, the SCDMP machine should be modeled as one single machine, rather than two magnetically irrelevant conventional machines (for example, designs shown in Figure 1. 5 (a) and Figure 1. 5 (c)). Second, compared with the constant flux provided by PM outer rotors, the flux generated by the outer rotor of the

SCDMP machine varies as the operational condition changes, which greatly increases the complexity of the machine model.

A validated electromagnetic machine model is a prerequisite for advanced machine control algorithms. To the best of the author’s knowledge, model of the SCDMP machine has not been presented by other researchers. In this work, the unique electromagnetic characteristics of the SCDMP machine is studied and the models for the machine are proposed.

The control techniques for conventional induction machines have been widely studied

[25] - [31]. Field Oriented Control (FOC) has been one of the most popular control methods for decades [32] - [35]. It is well-known that the accurate estimation of rotor flux position is the key to achieve a high performance FOC. The FOC can be divided in to two types, namely, Direct FOC (DFOC) and Indirect FOC (IFOC). DFOC calculates rotor flux position from airgap flux [36]. IFOC calculates rotor flux position by integrating the sum of rotor mechanical speed and slip frequency [37] [38]. In IFOC, the rotor mechanical speed can be obtained by either position sensor or sensorless control [39] [40], and the expression of slip frequency is derived from the mathematical model of the machine. The current model flux observer is usually adopted by IFOC. Though it is sensitive to rotor

8 time constant, the current model flux observer has shown better performance at low speed when compared with the voltage model flux observer [41]-[43].

The success of FOC in induction machine application relies on the fact that the torque and the flux of the rotor can be fully decoupled. To be more specific, the rotor torque is controlled by the q-axis component of stator current and the rotor flux is controlled by the d-axis stator current. However, this decoupling method is not valid for the SCDMP machine, which will be explained in Section 3.2.2.

Though the SCDMP machine has higher torque density when compared with the EVT and better thermal performance when compared with the PMDMP machine, the unique electromagnetic characteristics of the squirrel-cage outer rotor introduces challenges to the control. First, the magnetic coupling between its stator and inner rotor windings is significant. The SCDMP machine cannot be modeled as two magnetically independent machines. So the control methods for the PMDMP machines are not suitable for the

SCDMP machine. Second, the flux of the squirrel-cage rotor is controlled by both the stator and the inner rotor currents. As a result, the control methods for PMDMP machines, where the outer rotor flux is determined by the PMs, also fail to match the electromagnetic characteristics of the SCDMP machine.

Thus far, the control algorithm that is able to deal with the strong magnetic coupling and the variability of the outer rotor flux has not been found in any literature published by other researchers. This work is the first to discuss the control algorithm for the SCDMP machine.

Chapter 2: Permanent Magnet Dual Mechanical Port Electric Machine

2.1 Modeling of PMDMP machine

2.1.1 Introduction of PMDMP machine

The DMP machine use two layers of Permanent Magnets (PM) in the outer rotor

(PMDMP) is investigated in this chapter [4] [9]. As shown in Figure 2. 1 and Figure 2. 2, the PMDMP machine consists of a stator, PM outer rotor and wound inner rotor with brushes and slip rings. With the help of the high performance rare earth PMs, the PMDMP machine is able to achieve high torque density and high efficiency.

Figure 2. 1 Sideview of PMDMP machine (two layers of PMs). 10

Figure 2. 2 Components of PMDMP machine.

2.1.2 Three phase model of PMDMP machine

The cross-section of the double layer PMDMP machine is shown in Figure 2. 3.

Because the magnetic permeability of permanent magnets are close to that of the air and the outer rotor yoke provides a return path for the flux, the flux generated by the stator currents hardly links the inner rotor windings, and vice versa. It means the mutual inductance between the stator and the inner rotor windings is very small. The flux line distribution of the PMDMP machine is analyzed as follows.

   es or Poles /2  axis d axis Magnets  a ir s air  axis N

air d PM ir

cir bir as  axis ()  axis S cs bs

Figure 2. 3 Conceptual cross-section of double-layer PMDMP machine.

1) Stator current excitation alone: As shown in Figure 2. 4, the magnets are removed from the outer rotor to show the flux generated by the stator current alone. The flux line distribution shows that most of the flux lines link only the stator windings and only a very small part of the flux lines link the inner rotor windings. Hence, the mutual inductance between the stator and inner rotor windings is very small when compared with the self- inductance.

Figure 2. 4 Flux line distribution of the PMDMP machine with stator current alone.

2) Inner rotor current excitation alone: As shown in Figure 2. 5, stator currents are zero and only inner rotor current is provided. The flux line distribution shows that most of the flux lines link only the inner rotor windings and only a very small part of the flux lines link the stator windings.

Figure 2. 5 Flux line distribution of the PMDMP machine with inner rotor current alone.

Based on the above analysis, the three phase PMDMP machine model can be expressed by (2. 1)-(2. 3). Details of the model can be found in Appendix A. Note that (2.

1) is a 6 by 6 matrix. All values are referred to the stator side.

퐿푠푠 퐿푠𝑖푟 퐿푎푏푐푠𝑖푟 = [ ] (2. 1) 퐿𝑖푟푠 퐿𝑖푟𝑖푟 6푋6

휆푎푏푐푠 푖푎푏푐푠 휆푃푀푎푏푐푠 [ ] = 퐿푎푏푐푠𝑖푟 [ ] + [ ] (2. 2) 휆푎푏푐𝑖푟 푖푎푏푐𝑖푟 휆푃푀푎푏푐𝑖푟 푣 푅 0 푖 푑 휆 [ 푎푏푐푠 ] = [ 푠 ] [ 푎푏푐푠 ] + [ 푎푏푐푠 ] (2. 3) 푣푎푏푐𝑖푟 0 푅𝑖푟 푖푎푏푐𝑖푟 푑푡 휆푎푏푐𝑖푟

2.1.3 Model in synchronous reference frame.

Based on the three phase machine model for the PMDMP machine, the machine model in d-q reference frame (synchronous reference frame) can be derived. As shown in

Figure 2. 3, the d-axis is aligned with the north pole of the magnets. To make the machine model more general, the two equivalent PM rotors are assumed to be salient rotors, i.e., interior PM rotors. The model for the PMDMP machine in d-q reference frame is expressed by (2. 4)-(2. 9). The detail derivation is shown in Appendix A.

휆 = 퐿 푖 + 퐿 푖 + 휆 { 푑푠 푑푠 푑푠 푑푠𝑖푟 푑𝑖푟 푃푀표푟 (2. 4) 휆푞푠 = 퐿푞푠푖푞푠+퐿푞푠𝑖푟푖푞𝑖푟 휆 = 퐿 푖 + 퐿 푖 + 휆 { 푑𝑖푟 푑𝑖푟 푑𝑖푟 푑푠𝑖푟 푑푠 푃푀𝑖푟 (2. 5) 휆푞𝑖푟 = 퐿푞𝑖푟푖푞𝑖푟 + 퐿푞푠𝑖푟푖푞푠

푑휆푑푠 푣푑푠 = 푅푠푖푑푠 + − 휔푒푠휆푞푠 { 푑푡 (2. 6) 푑휆 푣 = 푅 푖 + 푞푠 + 휔 휆 푞푠 푠 푞푠 푑푡 푒푠 푑푠

푑휆푑𝑖푟 푣푑𝑖푟 = 푅𝑖푟푖푑𝑖푟 + − 푆𝑖푟휔푒푠휆푞𝑖푟 { 푑푡 (2. 7) 푑휆 푣 = 푅 푖 + 푞𝑖푟 + 푆 휔 휆 푞𝑖푟 𝑖푟 푞𝑖푟 푑푡 𝑖푟 푒푠 푑𝑖푟 푃표푙푒푠 3 푇푒𝑖푟 = [휆푞𝑖푟푖푑𝑖푟 − 휆푑𝑖푟푖푞𝑖푟] 2 2 (2. 8) 푃표푙푒푠 3 = − [휆 푖 + (퐿 − 퐿 )푖 푖 + 퐿 푖 푖 − 퐿 푖 푖 ] 2 2 푃푀𝑖푟 푞𝑖푟 푑𝑖푟 푞𝑖푟 푞𝑖푟 푑𝑖푟 푑푠𝑖푟 푑푠 푞𝑖푟 푞푠𝑖푟 푞푠 푑𝑖푟 푃표푙푒푠 3 푇 = [휆 푖 + 휆 푖 + (퐿 − 퐿 )(푖 푖 + 푖 푖 ) 푒표푟 2 2 푃푀𝑖푟 푞𝑖푟 푃푀표푟 푞푠 푑푠𝑖푟 푞푠𝑖푟 푞푠 푑𝑖푟 푑푠 푞𝑖푟 (2. 9) +(퐿푑푠 − 퐿푞푠)푖푞푠푖푑푠 + (퐿푑𝑖푟 − 퐿푞𝑖푟)푖푞𝑖푟푖푑𝑖푟] Based on the d-q reference frame machine model, the equivalent circuit in d-q reference frame can be obtained as shown in Figure 2. 6 and Figure 2. 7.

휔표푟 Note that 푆𝑖푟 = 1 − is the slip percentage of the inner rotor. 휔표푟 and 휔𝑖푟 are 휔푖푟 mechanical angular speeds of the outer and the inner rotors, respectively.

S  Rs es qs LLds dsir LLdir dsir ir es qir R - + + - ir + + ids idir vds Ldsir vdir - -

Figure 2. 6 The d-axis equivalent circuit of the PMDMP machine.

S  Rs es ds LLqsqsir LLqir qsir ir es dir R + - - + ir + i + iqs qir L vqir vqs qsir - -

Figure 2. 7 The q-axis equivalent circuit of the PMDMP machine.

2.2 Design of PMDMP machine

2.2.1 Simplified machine model.

It has be pointed out in Subsection 2.1 that the mutual inductance between the stator and inner rotor windings is very small. For simplicity of machine design, the mutual inductance is neglected in the following discussion. However, it is important to make sure the outer rotor yoke is not saturated. Otherwise, the mutual inductance will increase significantly.

The simplified machine model is shown in Figure 2. 8 and Figure 2. 9. The result shows that the PMDMP machine can be simplified as two magnetically irrelevant PM machines – the first one is the outer PM machine, which consists of the stator, the outer layer of PMs and the outer rotor yoke; the second one is the inner PM machine, which consists of the inner rotor, the inner layer of PMs and the outer rotor yoke.

S  Rs es qs Lds Ldir ir es qir Rir - + + - + + ids idir vds vdir - -

Figure 2. 8 Simplified d-axis equivalent circuit of the PMDMP machine.

L S  Rs es ds Lqs qir ir es dir Rir + - - + + i i + qs qir v vqs qir - -

Figure 2. 9 Simplified q-axis equivalent circuit of the PMDMP machine.

2.2.2 PMDMP machine design flowchart.

Based on the simplified machine model, the PMDMP machine can be designed as two separate machines. If the vehicle output requirement and the ICE power are known, the design procedure can be summarized as shown in Figure 2. 10. Since the two equivalent

PM machines share the same outer rotor yoke, the design of PMDMP machine needs to consider the flux density of the outer rotor yoke. If the outer rotor yoke is saturated, the assumption that the magnetic coupling between the stator and the inner rotor winding is negligible will no longer be true.

Figure 2. 10 Design flowchart for the PMDMP machine.

2.2.3 Dimensions, parameters and specifications of prototype PMDMP machine.

A prototype PMDMP machine is built to verify the machine model and the control algorithms, the dimensions of the machine is shown in Table 2. 1.

Table 2. 1 Dimensions of the prototype PMDMP machine.

Poles 6 Stack length mm 70.0 Stator OD mm 148.0 Slot area 75 푚푚2 Stator ID mm 93.0 Outer airgap length mm 0.6 Outer rotor OD mm 91.8 Outer rotor ID mm 70.0 Inner airgap length mm 0.4 Inner rotor OD mm 69.2 Slot area 43 푚푚2 Inner rotor ID mm 34.2 Outer PM width mm 2.9 3PMs/ pole Outer PM length mm 11.3 3PMs/ pole Inner PM width mm 2.95 1 PM/pole Inner PM Radian rad 1.028 58.9deg/pole 휆푃푀표푟 Web 0.0727 휆푃푀𝑖푟 Web 0.0889

The cross-section of the prototype machine is shown in Figure 2. 11. The open circuit flux line and flux density distributions of the prototype machine is shown in Figure 2. 12 and Figure 2. 13, respectively.

Figure 2. 11 Cross-section of the prototype PMDMP machine.

Figure 2. 12 Open circuit flux line distribution of the prototype machine.

Figure 2. 13 Open circuit flux density distribution of the prototype machine.

2.2.4 Open circuit test result of prototype PMDMP machine.

Test result of the resistance and back EMF are summarized in Table 2. 2. Note that the inner rotor phase resistance is 0.29 Ω if measured at the slip rings; it will be 0.40Ω if it is measured at the inverter terminals. The resistance increases due to long connection wire between the inverter and brushes.

Table 2. 2 Open circuit test result of the prototype PMDMP machine.

Stator phase resistance 푅푎푠 0.08 Ω Inner rotor phase resistance 푅푎𝑖푟 0.29/0.40 Ω Stator d-axis inductance 퐿푑푠 1.15 mH Stator q-axis inductance 퐿푞푠 1.85 mH Inner rotor d-axis inductance 퐿푑𝑖푟 1.5 mH Inner rotor q-axis inductance 퐿푞𝑖푟 1.5 mH

The open circuit back EMF of the prototype machine is also measured. The waveforms and harmonic analysis of the stator winding and the inner rotor back EMFs are shown in Figure 2. 14 and Figure 2. 15, respectively.

Figure 2. 14 Stator winding line to line back EMF and harmonic analysis.

Figure 2. 15 Inner rotor winding line to line back EMF and harmonic analysis.

From the back EMF analysis results, Table 2. 3 regarding the line to neutral flux linkage of the prototype machine can be obtained. It should be pointed out that the line to neutral back EMF of the stator and the inner rotor windings actually contain third order harmonic. However, since the third order harmonic in the back EMF will not result in any torque or current, it’s not listed in Table 2. 3. 26

Table 2. 3 Flux linkage of the prototype PMDMP machine.

Harmonic order Stator PM flux linkage [Wb] Inner rotor PM flux linkage [Wb] 1 0.0837 0.0901 5 0.0023 0.0139 11 0.0069 0

2.3 Control of PMDMP machine

The finite element analysis result shows that the mutual coupling between the stator and the inner rotor windings can be neglected as long as the outer rotor yoke is not highly saturated. Hence, the PMDMP machine can be controlled as two independent PM machine as shown in Figure 2. 8 and Figure 2. 9. The mathematical model of the machine can still be expressed by (2. 4) - (2. 9), except 퐿푑푠𝑖푟 = 0 and 퐿푞푠𝑖푟 = 0. Besides, Table 2. 2 shows the inner PM machine is a non-salient machine because 퐿푑𝑖푟 = 퐿푞𝑖푟. Hence, the machine model can be simplified as (2. 10) - (2. 17).

휆 = 퐿 푖 + 휆 { 푑푠 푑푠 푑푠 푃푀표푟 (2. 10) 휆푞푠 = 퐿푞푠푖푞푠 휆 = 퐿 푖 + 휆 { 푑𝑖푟 푑𝑖푟 푑𝑖푟 푃푀𝑖푟 (2. 11) 휆푞𝑖푟 = 퐿푑𝑖푟푖푞𝑖푟

푑휆푑푠 푣푑푠 = 푅푠푖푑푠 + − 휔푒푠휆푞푠 { 푑푡 (2. 12) 푑휆 푣 = 푅 푖 + 푞푠 + 휔 휆 푞푠 푠 푞푠 푑푡 푒푠 푑푠

푑휆푑𝑖푟 푣푑𝑖푟 = 푅𝑖푟푖푑𝑖푟 + − 푆𝑖푟휔푒푠휆푞𝑖푟 { 푑푡 (2. 13) 푑휆 푣 = 푅 푖 + 푞𝑖푟 + 푆 휔 휆 푞𝑖푟 𝑖푟 푞𝑖푟 푑푡 𝑖푟 푒푠 푑𝑖푟 푃표푙푒푠 3 푇 = − 휆 푖 (2. 14) 푒𝑖푟 2 2 푃푀𝑖푟 푞𝑖푟 푃표푙푒푠 3 푇 = [휆 푖 + 휆 푖 + (퐿 − 퐿 )푖 푖 ] (2. 15) 푒표푟 2 2 푃푀𝑖푟 푞𝑖푟 푃푀표푟 푞푠 푑푠 푞푠 푞푠 푑푠 푑휔 퐽 표푟 = 푇 − 푇 − 푇 (2. 16) 표푟 푑푡 푒표푟 푙표푎푑−표푟 푓푟𝑖푐푡𝑖표푛−표푟 푑휔 퐽 𝑖푟 = 푇 − 푇 − 푇 (2. 17) 𝑖푟 푑푡 푒𝑖푟 푙표푎푑−𝑖푟 푓푟𝑖푐푡𝑖표푛−𝑖푟

2.3.1 Field oriented control.

The Field Oriented Control (FOC) has been a popular control technique for electric machines for many years. One of the key factor to guarantee the performance of FOC is to find the right orientation angle. In PMSMs, the position of the permanent magnets (usually north pole) is often selected as the orientation angle for the FOC. The control block diagram for the PMDMP machine is shown in Figure 2. 16.

Based on the simplified machine model and the parameters of the prototype, the model for the PMDMP machine is built in Matlab/Simulink. The machine model and the controller for the PMDMP machine are shown in Figure 2. 17. The performance of the proposed controller is simulated. The simulation results are shown in Figure 2. 18 - Figure

2. 20.

As shown in Figure 2. 18, the outer rotor and the inner rotor are controlled to rotate in different directions. The actual speeds are tracking the references as expected. Hence, the performance of the proposed FOC for the DMP machine is satisfying.

Speed Position Cal Sensor ir i ir dir Current abc to dq Sensor i iiair, bir qir  *  d ir DMP idir PI 3_Phase Machine * dq to abc PWM  Inverter wheel ir PI   i* PI qir engine  * ids PI 3_Phase * dq to abc PWM Inverter

30  or PI * PI i qs ii, ids Current as bs abc to dq Sensor Speed iqs Position Cal Sensor  d() PM or

Figure 2. 16 Control block diagram for the PMDMP machine.

Figure 2. 17 Matlab model of the PMDMP machine and controller.

Outer Rotor Reference and Actual Speed 500 0 -500

Speed [RPM] Speed -1000 Inner Rotor Reference and Actual Speed 1000 500 0 -500

Speed [RPM] Speed -1000 0 1 2 3 4 5 Time [s]

Figure 2. 18 Actual speed (black) and reference speed (red) of the DMP machine

Outer Rotor Electromagnetic Torque 10 0 -10

Torque [Nm] Torque -20 Inner Rotor Electromagnetic Torque 20 10 0 -10

Torque [Nm] Torque -20 0 1 2 3 4 5 Time [s]

Figure 2. 19 Torque production of the DMP machine.

Stator Phase Current 40 20 0 -20

Current [A] Current -40 Inner Rotor Phase Current 40 20 0 -20

Current [A] Current -40 0 1 2 3 4 5 Time [s]

Figure 2. 20 Phase current waveforms of the DMP machine.

2.3.2 Position sensorless control of outer rotor

The position sensorless control of the outer rotor (salient pole rotor) includes two parts.

The first part is the zero speed starting and low speed position sensorless control. The second part is the medium and high speed position sensorless control.

The test setup to verify the effectiveness of the field oriented control and position sensorless control algorithm is shown in Figure 2. 21.

Brushes & Slip Rings Prototype PMDMP Machine Position Sensor Terminals

Stator Winding Terminals Stator Rotor Inverter Inverter Inner Rotor Winding Terminals

S R BUS DC bus

Power System Variac Rectifier

Figure 2. 21 Test setup for sensorless control of PMDMP machine.

A. Zero speed starting and low speed sensorless control.

The zero speed starting has always been the difficult part of the sensorless control.

The High Frequency Voltage Injection (HFI) method has been widely used to estimate rotor positions. Based on the superposition principle of electric circuit, the effects of the

33 injected high frequency voltage and the synchronous frequency voltage can be considered separately if the high frequency component does not change the saturation level of the machine core.

If the machine reluctance is position dependent (salient pole machine or saturation variation), the variation of rotor reluctance can be estimated by the high frequency current resulting from the injected high frequency voltage. Based on the variation of current magnitude, the rotor position can be estimated. The effect of the high frequency voltage is illustrated by Figure 2. 22.

Besides the d-q reference frame, the stationary reference frame, i.e., 훼 − 훽 reference frame, is also commonly used in machine modeling. As shown in Figure 2. 3, the 훼-axis is aligned with the 푎푠-axis. The voltage and flux equations in 훼 − 훽 reference frame are very similar to that in 푑 − 푞 reference frame. To be more specific, if the subscripts “푑” and “푞” are replaced with“훼” and “훽,” and let 휔푒푠 = 0, (2. 4) - (2. 9) will then be valid equations in 훼 − 훽 reference frame. Hence, the machine equations in 훼 − 훽 reference frame are not derived in this work. It should be pointed out that the variables in 푑 − 푞 reference frame are constant values (dc) in steady state, but they are sinusoidal values in 훼 − 훽 reference frame.

 axis  axis qaxis qaxis es es daxis daxis N N

d d   axis   axis V IH S H S  H H

Figure 2. 22 Effect of high frequency boltage on salient pole PMSM.

As shown in Figure 2. 22, when a high frequency (−휔퐻) voltage (푉퐻) is injected to the machine windings, a high frequency current (퐼퐻) will be generated. Because of the saliency of the rotor, the trajectory of 퐼퐻 on the stationary reference frame (훼 − 훽) will be an ellipse. The relationship between 푉퐻 and 퐼퐻 can be expressed by (2. 18). Note that the synchronous frequency components are not shown in (2. 18). 퐿푠(2휃푑) is expressed by (2.

19).

푑퐿푠(2휃푑)푖훼훽푠 푑퐿푠(2휃푑)퐼퐻 푉 = 푣 = 푅 푖 + = 푅 퐼 + (2. 18) 퐻 훼훽푠 푠 훼훽푠 푑푡 푠 퐻 푑푡

퐿푠(2휃푑) 퐿푑푠 + 퐿푞푠 퐿푑푠 − 퐿푞푠 퐿푑푠 − 퐿푞푠 + 푐표푠(2휃푑) 푠푖푛(2휃푑) (2. 19) = [ 2 2 2 ] 퐿 − 퐿 퐿 + 퐿 퐿 − 퐿 푑푠 푞푠 푠푖푛(2휃 ) 푑푠 푞푠 − 푑푠 푞푠 푐표푠(2휃 ) 2 푑 2 2 푑

Since 푅푠 ≪ 휔퐻퐿푠, the first term (푅푠푖훼훽푠) in (2. 18) can be neglected. Considering

−푠푖푛휔퐻푡 푉퐻 = 푉퐻푀푎푔 [ ], (2. 20) can be derived. 푐표푠휔퐻푡

−1 퐼퐻 = [퐿푠(2휃푑)] ∫ 푉퐻 푑푡

퐿푑푠 + 퐿푞푠 퐿푑푠 − 퐿푞푠 (2. 20) 푉 푐표푠휔퐻푡 − cos (2휃푑 − 휔퐻푡) = 퐻푀푎푔 [ 2 2 ] 퐿 퐿 휔 퐿 + 퐿 퐿 − 퐿 푑푠 푞푠 퐻 푑푠 푞푠 푠푖푛휔 푡 − 푑푠 푞푠 sin (2휃 − 휔 푡) 2 퐻 2 푑 퐻 where

−1 [퐿푠(2휃푑)] 퐿푑푠 + 퐿푞푠 퐿푑푠 − 퐿푞푠 퐿푑푠 − 퐿푞푠 1 − 푐표푠(2휃푑) − 푠푖푛(2휃푑) = [ 2 2 2 ] 퐿 퐿 퐿 − 퐿 퐿 + 퐿 퐿 − 퐿 푑푠 푞푠 − 푑푠 푞푠 푠푖푛(2휃 ) 푑푠 푞푠 + 푑푠 푞푠 푐표푠(2휃 ) 2 푑 2 2 푑

It is obvious in (2. 20) that the d-axis angle (2휃푑) is carried by the high frequency current. The stator current has three components with different frequencies – synchronous frequency (휔푒푠 ), positive sequence high frequency (휔퐻 ) and negative sequence high frequency (−휔퐻). Figure 2. 23 shows the mathematical procedures to extract the d-axis angle (2휃푑) from the −휔퐻 current component.

 t Filter 2 t Filter H H H  es Component Component 22d  k High Pass Low Pass 1 i i   dq  Filter dq   Filter  tan 0.5d  k abcs  s

Figure 2. 23 Procedures to extract rotor position from stator currents.

Note that “k” in Figure 2. 23 can be any integer. As a result, the HFI method cannot identify the polarity of the rotor. Some polarity identification methods based on the magnetic saturation have been proposed. By imposing the same voltage pulse on both directions (0 degree, k=0 and 180 degree, k =1) of the d-axis, the stator current will increase

36 in different magnitudes. Note that the voltage pulse should last long enough to cause the saturation of the core. As shown in Figure 2. 24, it is obvious that the direction with lower current magnitude is the d-axis position needed for Field Oriented Control (FOC) of the rotor.

d axis d axis  v dt  v t  L  i d pulse pulse d d N N d 1  S S d1 o dd12   d 2 2 iidd12   id1 (1) (2) id 2 id  Stator current flux Magnet flux

Figure 2. 24 Polarity identification based on magnetic saturation.

Figure 2. 25 shows the block diagram for the HFI sensorless control with the polarity identification algorithm.

DMP Machine Feedback Inner Rotor PWM 3_Phase Signals Control Inverter wheel vpulse ICE VH * i  3_Phase ds PI to abc * dq to PWM Inverter or PI * PI iqs 38 ii, i as bs qs Current dq to abc to  Sensor  ids Polarity or Identification NS/? d() PM HFI Position Estimation

Figure 2. 25 Block diagram for outer rotor sensorless control at zero and low speeds.

To start the rotor from zero speed, the sensorless control algorithm needs to go through three steps. 1) The HFI algorithm is used to estimate the d-axis of the outer rotor at zero speed; 2) Based on the d-axis angle position, the control will be switched to the polarity identification algorithm to find out the north-pole and the south-pole of the d-axis; 3) The control will be switched back to the HFI algorithm to apply FOC to the outer rotor.

The simulation result is shown in Figure 2. 26. From 0 to 0.5 s, the initial rotor position is estimated; after that, the rotor speed is increased from 0 to 40 RPM. It is obvious that the estimation error can be controlled within 3 electrical degrees. Figure 2. 27 shows that experiment result of the HFI method, the angle error can be controlled to be smaller than

10 electrical degrees. Hence, the performance of the HFI algorithm is satisfying.

] ActuActualal R Rotoroto Positionr Po [deg]sition g 400 e 400 d [ 200 e 200 l g

n 0 0 A

] HFI EHFIst Estimatedimate Rotord R Positionotor [deg]Position g 400

e 400 d [ 202000 e l g

n 0 0 A EstEstimationimati oErrorn [deg]Error

] 4040 g

e 2020 d [ 0 0 e l

g -20-20 n 00 11 22 33 44 55 A Time [s]

Figure 2. 26 Simulation result of high frequency injection rotor position estimation. 39

] HHighFI EFrequencystima tInjectioned Ro Estimatedtor Pos Angleition g 400

e 400 d [

e 202000 l g

Angle [Deg] Angle n

A 0 0 0 1 2 3 4 5 HIFEst iEstimationmaTimetio [s]n E rErrorror ] 2020 g e

d 0

[ 0

e l

Angle [deg] Angle

g -2-200 n 00 11 22 33 44 5 A Time [s] Tim e [s]

Figure 2. 27 Experiment result of high frequency injection rotor position estimation.

B. Medium and high speed sensorless control.

When the rotor speed increases, the back EMF resulting from the rotor magnets becomes large enough to be used to estimate the position of the rotor. The machine model

(outer PM machine only) in stationary reference frame can be expressed by (2. 21).

푑휆훼 퐿푑푠 + 퐿푞푠 푑푖훼푠 푣훼푠 = 푖훼푠푅푠 + = 푖훼푠푅푠 + + 푒훼 + ℎ훼 { 푑푡 2 푑푡 (2. 21) 푑휆훽 퐿 + 퐿 푑푖훽푠 푣 = 푖 푅 + = 푖 푅 + 푑푠 푞푠 + 푒 + ℎ 훽푠 훽푠 푠 푑푡 훽푠 푠 2 푑푡 훽 훽 where

푒 = −휔 휆 푠푖푛휃 { 훼 푒푠 푃푀표푢푡 푃푀 (2. 22) 푒훽 = 휔푒푠휆푃푀표푢푡푐표푠휃푃푀

퐿 − 퐿 푑푖 cos (2휃 ) 푑푖훽sin (2휃푃푀) ℎ = 푑푠 푞푠 [ 훼 푃푀 + ] 훼 2 푑푡 푑푡 (2. 23) 퐿푑푠 − 퐿푞푠 푑푖훽cos (2휃푃푀) 푑푖훼sin (2휃푃푀) ℎ훽 = [− + ] { 2 푑푡 푑푡

Applying extended back EMF (푒훼푒푥푡 and 푒훽푒푥푡) theory to the machine model, (2. 21) can be derived as (2. 24).

푑푖훼푠 퐿푞푠 = 푣훼푠 − 푖훼푠푅푠 − 푒훼푒푥푡 { 푑푡 (2. 24) 푑푖훽푠 퐿 = 푣 − 푖 푅 − 푒 푞푠 푑푡 훽푠 훽푠 푠 훽푒푥푡 where

퐿푑푠 − 퐿푞푠 푑푖훼푠 푒훼푒푥푡 = 푒훼 + ℎ훼 − { 2 푑푡 (2. 25) 퐿 − 퐿 푑푖훽푠 푒 = 푒 + ℎ − 푑푠 푞푠 훽푒푥푡 훽 훽 2 푑푡 Hence, considering (2. 22)， (2. 23)，(2. 24) and 푖훼 = 퐼푠푐표푠휃푒, 푖훽 = 퐼푠푠푖푛휃푒, (2. 26) and (2. 27) can be derived.

푒 −휔 휆 + (퐿 − 퐿 )퐼 cos(휃 − 휃 ) 푠푖푛휃 푠푖푛휃 훼푒푥푡 = 푒푠 푃푀표푢푡 푑푠 푞푠 푠 푃푀 푒 푃푀 = − 푃푀 (2. 26) 푒훽푒푥푡 휔푒푠휆푃푀표푢푡 − (퐿푑푠 − 퐿푞푠)퐼푠 cos(휃푃푀 − 휃푒) 푐표푠휃푃푀 푐표푠휃푃푀 푒 푠푖푛휃 푒 ⇒ 훼 = − 푃푀 = 훼푒푥푡 (2. 27) 푒훽 푐표푠휃푃푀 푒훽푒푥푡 It is obvious that if the machine parameters, stator currents and voltages are known,

푒훼푒푥푡 and 푒훽푒푥푡 can be directly calculated by (2. 24). Then the rotor position can be obtained by solving (2. 27).

However, this method requires accurate parameter estimation. Besides, the differential terms can be easily affected by the noises of the system. In order to overcome these problems, the sensorless control algorithm based on sliding mode theory has been widely adopted.

In this dissertation, the sliding mode method is used to observe the stator current in stationary reference frame. A sliding mode observer model with control 푈훼/푈훽 is used to estimate the system states as shown in (2. 28). The estimated values are indicated by “ ̂ ”.

푑푖̂훼푠 퐿푞푠 = 푣훼푠 − 푖̂훼푠푅푠 + 푈훼 { 푑푡 (2. 28) 푑푖̂훽푠 퐿 = 푣 − 푖̂푅 + 푈 푞푠 푑푡 훽푠 훽푠 푠 훽 If the difference between the estimated and actual values of the stator currents are chosen to be the sliding surfaces as expressed by (2. 29), the Sliding Mode Observer (SMO) for stator currents can be derived by subtracting (2. 24) from (2. 28). The result is (2. 30).

푆 = 푖̂ − 푖 { 훼 훼푠 훼푠 (2. 29) 푆훽 = 푖̂훽푠 − 푖훽푠

푑푆훼 푅푠 1 = − 푆훼 + (푒훼푒푥푡 + 푈훼) 푑푡 퐿푞푠 퐿푞푠 (2. 30) 푑푆훽 푅푠 1 = − 푆훽 + (푒훽푒푥푡 + 푈훽) { 푑푡 퐿푞푠 퐿푞푠

As expressed by (2. 31), the control input 푈훼 and 푈훽 are simple functions decided by the signs of 푆훼 and 푆훽, respectively. 푘푠푙푑 is called the sliding mode gain. If 푆훼/푆훽 doesn’t equal to zero, 푆훼/푆훽 will be forced to be zero by the control input 푈훼/푈훽.

푈 = −푘 푠푖푔푛푆 { 훼 푠푙푑 훼 (2. 31) 푈훽 = −푘푠푙푑푠푖푔푛푆훽

Even though 푈훼 and 푈훽 are high frequency sign functions, they can be decomposed into two parts as express in (2. 32).

푈훼 = 푈훼푙표푤 + 푈훼ℎ𝑖푔ℎ = −푒̂훼푒푥푡 + 푈훼ℎ𝑖푔ℎ { (2. 32) 푈훽 = 푈훽푙표푤 + 푈훽ℎ𝑖푔ℎ = −푒̂훽푒푥푡 + 푈훽ℎ𝑖푔ℎ

The low frequency components 푒̂훼푒푥푡 and 푒̂훽푒푥푡 are the estimated back EMFs. During steady state, these two estimated values will be forced to track the actual values. If the sign functions are applied directly to a system with high frequency filters (inductors in a machine), the high frequency components will be filtered out by the system itself. However, since the estimated back EMF will be used to further calculate the rotor position, these high

42 frequency components will bring significant noise to the result. Hence, a low pass filter as expressed by (2. 33) is needed to filter out the high frequency components.

푘 푙푝푓 푒̂훼푒푥푡 = − 푈훼 푘푙푝푓 + 휔푐푢푡표푓푓 (2. 33) 푘푙푝푓 푒̂훽푒푥푡 = − 푈훽 { 푘푙푝푓 + 휔푐푢푡표푓푓 To verify the existence of the sliding mode, the Lyapunov function (2. 34) is considered.

1 푇 푌훼 = 푆훼 푆훼 { 2 (2. 34) 1 푌 = 푆푇푆 훽 2 훽 훽 푑푌 To guarantee the existence of the sliding mode, 푑푌훼 and 훽 should be negative values. 푑푡 푑푡

푑푌훼 푇 푑푆훼 푇 푅푠 푇 1 = 푆훼 = −푆훼 푆훼 + 푆훼 (푒훼푒푥푡 + 푈훼) < 0 푑푡 푑푡 퐿푞푠 퐿푞푠 ⇒ (2. 35) 푑푌훽 푇 푑푆훽 푇 푅푠 푇 1 = 푆훽 = −푆훽 푆훽 + 푆훽 (푒훽푒푥푡 + 푈훽) < 0 { 푑푡 푑푡 퐿푞푠 퐿푞푠 푇 푅푠 푇 푅푠 Obviously, −푆훼 푆훼 ≤ 0 and −푆훽 푆훽 ≤ 0. Hence, if the condition in (2. 36) can 퐿푞푠 퐿푞푠 be satisfied, the sliding mode exists.

푆푇(푒 + 푈 ) < 0 훼 훼푒푥푡 훼 (2. 36) { 푇 푆훽 (푒훽푒푥푡 + 푈훽) < 0

If 푆훼 > 0, 푈훼 = −푘푠푙푑 ⇒ 푒훼푒푥푡 + 푈훼 = 푒훼푒푥푡 − 푘푠푙푑 < 0 ⇒ 푒훼푒푥푡 < 푘푠푙푑;

If 푆훼 < 0, 푈훼 = 푘푠푙푑 ⇒ 푒훼푒푥푡 + 푈훼 = 푒훼푒푥푡 + 푘푠푙푑 > 0 ⇒ −푒훼푒푥푡 < 푘푠푙푑.

⇒ 푘푠푙푑 > |푒훼푒푥푡|. Similarly, 푘푠푙푑 > |푒훽푒푥푡|.

The result indicates that the SMO requires high gain 푘푠푙푑 to guarantee the tracking of the actual currents. As the rotor speed increases, the back EMF magnitude increases. Hence, high 푘푠푙푑 is required at high speed. 43

When the system states (currents and back EMFs) are forced to oscillate around the sliding mode surfaces, the chattering issue occurs. To mitigate the chattering behavior, the sign functions of 푈훼 and 푈훽 are replaced by saturation functions as shown in Figure 2. 28.

UU/ ksld

E E chat chat SS/

k sld

Figure 2. 28 Saturation function for SMO.

The block diagram for the SMO to estimate rotor position is summarized in Figure 2.

29. The SMO Current Estimation block is built based on (2. 28). The Low Pass Filter block is built based on (2. 33). The sensorless control algorithm based on SMO is shown in

Figure 2. 30.

i  s Compensation

UU/ ˆ  k v SMO Current i s  sld U Low Pass eˆ s e   s  tan1 (s ) Echat Echat SS/  Estimation  Filter PM k e  sld  s  U

Figure 2. 29 Block diagram for the rotor position SMO.

DMP Machine Feedback Inner Rotor PWM 3_Phase

Signals Control Inverter wheel

ICE * ids PI 3_Phase * dq to to abc PWM Inverter or PI * PI iqs v s iias, bs iqs  i dq to Current or ds abc to  Sensor i s PM Speed Cal Rotor Position SMO

Figure 2. 30 Block diagram for outer rotor sliding mode sensorless control at medium and high speeds. 45

The simulation result for the sliding mode sensorless control algorithm is shown in

Figure 2. 31. When the rotor speed is low, the back EMF is too low to be accurately estimated. So the sliding mode method discussed in the dissertation cannot be used to estimate the rotor position. An open loop control is applied to the machine to ramp up the rotor speed to 100RPM in 0.5 second. After that, the control is switched to the SMO method and the motor speed is increased from 200 RPM to 1000 RPM.

The simulation result shows that the rotor position estimation error can be kept within

3 electrical degrees. The experimental result for the sliding mode sensorless control algorithm is shown in Figure 2. 32. When the rotor speed is controlled to be 500 RPM, the experimental result shows that the estimated error is less than 5 electrical degrees. Hence, the performance of the sliding mode sensorless control for the PMDMP machine is satisfying.

]

g ActuaActuall R Rotoroto Positionr Po [deg]sition e 404000 d [

e 200 l 200 g n

A 0 0

] SMO ESMOsti Estimatedmate Rotord R Positionoto r[deg] Position g 400

e 400 d [ 200 e 200 l g

n 0

A 0 EstiEstimationmatio Errorn [deg]Error ] 2020 g e

d 0

[ 0

e l

g -20-20 n 0 0.5 11 1.5 22 2.5 33 3.5 44 4.5 55 A Time [s]

Figure 2. 31 Simulation result of sliding mode rotor position observer.

SMO Estimated Angle 400 200

Angle [deg] Angle 0 0 0.1 0.2 0.3 0.4 SMO EstimationTime [s] Error 10 0 -10 Angle [deg] Angle 0 0.1 0.2 0.3 0.4 Time [s]

Figure 2. 32 Experimental result of sliding mode rotor position observer. 47

The sensorless control of inner rotor is similar to that of the outer rotor. The difference is the inner layer permanent magnets of the outer rotor is surface mounted, thus the control of the inner rotor is the same as that of a Surface mounted Permanent Magnet (SPM) machine. Since the inner rotor is mechanically coupled to the ICE, it is acceptable to start the inner rotor with open loop control. When the inner rotor speed is high enough, the sliding mode method discussed in this chapter can be applied directly to control the inner rotor.

2.4 Operation modes of PMDMP machine

2.4.1 Power flow analysis.

The PMDMP machine has very high control flexibility because its dual-mechanical- port and dual-electrical-port structure provides many operational possibilities. If the power losses are neglected, the relationship between the mechanical and electrical power can be represented by (2. 37). The subscripts “e” and “m” indicates electrical and mechanical powers, respectively.

푃푒푠 + 푃푒𝑖푟 + 푃푚퐼퐶퐸 − 푃푚푊ℎ푒푒푙 = 0 (2. 37)

푃푒푠 is the electrical power provided by the stator windings; 푃푒𝑖푟 is the electrical power provided by the inner rotor windings; 푃푚퐼퐶퐸 is the mechanical power provided by the ICE, which is equal to the mechanical power provided by the inner rotor; 푃푚푊ℎ푒푒푙 is the mechanical output power of the outer rotor. Since the outer rotor is mechanically coupled to the driving shaft of the wheels, the subscription “Wheel” is adopted.

Theoretical speaking, all the powers in (2. 37) can be bidirectional. However, some of the scenarios are not likely to occur during steady state operation. For example, the mechanical power from the ICE (푃푚퐼퐶퐸) is not likely to be negative, because the ICE cannot consume mechanical power.

Considering 푃푒푠 = (푇푒표푟 + 푇푒𝑖푟)휔표푟, 푃푒𝑖푟 = −푇푒𝑖푟(휔표푟 − 휔𝑖푟 ), 푃푚퐼퐶퐸 = 푇퐼퐶퐸휔𝑖푟 =

−푇푒𝑖푟휔𝑖푟 and 푃푚푊ℎ푒푒푙 = 푇푒표푟휔표푟, (2. 38) can be easily derived from (2. 37).

(푇푒표푟 + 푇푒𝑖푟)휔표푟 − 푇푒𝑖푟(휔표푟 − 휔𝑖푟) − 푇푒𝑖푟휔𝑖푟 − 푇푒표푟휔𝑖푟 = 0 (2. 38)

The total electrical input power from both the stator and the inner rotor windings equals to the power provided by the battery, thus the battery output power can be expressed by (2. 39).

푃 = 푃 + 푃 = (푇 + 푇 )휔 − 푇 (휔 − 휔 ) 푏푎푡푡푒푟푦 푒푠 푒𝑖푟 푒표푟 푒𝑖푟 표푟 푒𝑖푟 표푟 𝑖푟 (2. 39) = 푇푒표푟휔표푟 + 푇푒𝑖푟휔𝑖푟 = 푃푚푊ℎ푒푒푙 − 푃푚퐼퐶퐸 Three important observations can be obtained from (2. 39). First, the two energy sources of the system, i.e., the battery and the ICE, work together to drive the vehicle

( 푃푏푎푡푡푒푟푦 + 푃푚퐼퐶퐸 = 푃푚푊ℎ푒푒푙 ). Second, the inner rotor windings provides positive electrical power to the system when the outer rotor speed is higher than the inner rotor speed, and this part of power is called the slip power (if 휔표푟 > 휔𝑖푟 , then 푃푒𝑖푟 =

−푇푒𝑖푟(휔표푟 − 휔𝑖푟) = 푇퐼퐶퐸(휔표푟 − 휔𝑖푟) > 0). Third, the mechanical power from the ICE is transferred to the wheels directly without flowing into the battery.

To better explain the power flow of the PMDMP machine, the constant power mode is selected as an example. The constant power mode means the mechanical input power from the ICE equals to the mechanical output power of the vehicle (푃푚퐼퐶퐸 = 푃푚푊ℎ푒푒푙 ).

Note that power losses are neglected for simplicity of discussion. Thus, (2. 40) is satisfied.

푃푏푎푡푡푒푟푦 = 푃푒푠 + 푃푒𝑖푟 = 푃푚퐼퐶퐸 − 푃푚푊ℎ푒푒푙 = 0 (2. 40)

The ICE sweet spot (푃푚퐼퐶퐸) and the expected output from the wheel (푃푚푊ℎ푒푒푙 ) are indicated in Figure 2. 33 by a dot and a star, respectively.

Torque Constant Power Curve

PmICE ICE Sweet Spot TTICE eir 1 3 PmWheel T eor 2 1 2 4 Speed ir() ICE or

Figure 2. 33 Constant power mode of the HEV based on PMDMP machine.

As shown in Figure 2. 33, even though 푃푚퐼퐶퐸 = 푃푚푊ℎ푒푒푙 , they are actually two different operational points – the expected output has higher speed but lower torque when compared with the sweet spot of the ICE (휔표푟 > 휔𝑖푟 and 푇푒표푟 < 푇퐼퐶퐸 = −푇푒𝑖푟). As a result,

푃푒𝑖푟 = 푇퐼퐶퐸(휔표푟 − 휔𝑖푟) > 0, 푃푒푠 = (푇푒표푟 + 푇푒𝑖푟)휔표푟 < 0. Hence, it is clear that the inner rotor winding is providing electrical power to drive the outer rotor and the stator winding is recovering the same amount of electrical power. The relationship between different powers are summarized in Table 2. 4.

Table 2. 4 Power relationships of the constant power mode of the PMDMP machine.

Power Input Output Mechanical 푃푚퐼퐶퐸 = 푇퐼퐶퐸휔𝑖푟 푃푚푊ℎ푒푒푙 = 푇푒표푟휔표푟 = 푃1 + 푃2 = 푃2 + 푃4 Electrical 푃푒푠 = (푇푒표푟 − 푇퐼퐶퐸)휔표푟 푃푒𝑖푟 = 푇퐼퐶퐸(휔표푟 − 휔𝑖푟) = 푃1 + 푃3 = 푃3 + 푃4

푃1, 푃2, 푃3 and 푃4 in Table 2. 4 represents the amount of power indicated by “1”, “2”,

“3” and “4” in Figure 2. 33, respectively. Note that 푃1 + 푃2 = 푃2 + 푃4 ⇒ 푃1 = 푃4. Define

푃𝑖푟 = 푃푚퐼퐶퐸 + 푃푒𝑖푟, then 푃𝑖푟 = 푃1 + 푃2 + 푃3 + 푃4. The power flow of the constant power mode can be illustrated by Figure 2. 34.

ir or ICE P Wheels PPPmICE 12 ir PPPmWheel 24

Inverter Slip power PPP eir 34Battery PPPes 13

Figure 2. 34 Power flow of the constant power mode of the PMDMP machine.

2.4.2 Multi-operational modes of the PMDMP machine.

Besides the constant power mode, the operation of the PMDMP machine can be characterized by different modes based on the State Of Charge (SOC) of the battery pack and the relationship between the operational points of 푃푚퐼퐶퐸 and 푃푚푊ℎ푒푒푙 . These operational modes include low power mode, high power mode, pure Electric Vehicle (EV) mode and highway (pure ICE) mode.

As shown in Figure 2. 35, if the outer rotor requires lower power when compared with the sweet spot of the ICE, the PMDMP machine is in low power mode. Depending on the relationship between the ICE speed (휔𝑖푟 ) and the outer rotor speed (휔표푟), three different operational points of the outer rotor are indicated by A, B and C in Figure 2. 35.

Torque Low Power Mode

A PmICE ICE Sweet Spot B C PmWheel Speed

Figure 2. 35 Low power mode of the HEV based on PMDMP machine.

If the outer rotor is operating at point A, the stator winding will have to provide extra torque to compensate the different between the ICE torque and the outer rotor torque.

Hence, the stator winding is discharging the battery. Because the ICE speed is higher than the outer rotor speed, the inner rotor winding is recovering the slip power and charging the battery. Because 푃푚퐼퐶퐸 > 푃푚푊ℎ푒푒푙−퐴, the net power goes into the battery will be positive; the battery is being charged in this mode. Similar analysis can be done to Point B and Point

Figure 2. 36 shows the high power mode of the PMDMP machine. The high power mode can be analyzed in a similar way as that was done on the lower power mode. The most important difference is the battery is being discharged in the high power mode.

High Power Mode Torque

PmWheel

PmICE

Speed

Figure 2. 36 High power mode of the HEV based on PMDMP machine.

Even though the constant power mode, low power mode and high power mode are different operational modes, they share something in common – the two energy sources

(the ICE and the battery pack) are working at the same time. They all require the battery to stay in a healthy conditions – the battery state of charge is within a reasonable range.

However, when the battery is not in healthy condition, the HEV based on the DMP machine

54 will have to keep running with only one energy source. Thus, the pure Electric Vehicle

(EV) mode and the highway mode are introduced.

As shown in Figure 2. 37, the ICE is shut down and the battery provides all the requested power. This happens when the battery is already fully charged and the ICE is still providing more power than needed. Another scenario to use the pure EV mode is the starting of the vehicle.

Torque Pure Electric Vehicle Mode

PmWheel Speed

Figure 2. 37 Pure electric vehicle mode of the HEV based on DMP machine.

As shown in Figure 2. 38, the ICE becomes the only energy source of the vehicle during the highway mode. When requested output power is higher than the power at the sweet spot and the battery state of charge is at a low level, the operational point of the ICE will have to deviate from the sweet spot. This operational mode requires a mechanical component (for example, a clutch) to directly connect the inner rotor and the outer rotor.

In this mode, the HEV actually degrades to a conventional gasoline-only vehicle.

Highway (Pure ICE) Mode Torque

PmICE

ICE Sweet Spot

Speed

Figure 2. 38 Highway (pure ICE) mode of the HEV based on PMDMP machine.

2.4.3 Simulation of the multi-operation modes.

Based on the power flow and operational mode analysis, a multi-operational mode simulation is conducted to verify the functionality of the PMDMP machine in HEV application. As shown in Figure 2. 39 - Figure 2. 41, the simulation is divided into the following stages.

(1) 0-2 s, starting of vehicle (pure EV mode). The outer rotor speed is increased from

0 to 750 RPM. The inner rotor (ICE) does not provide any mechanical power.

(2) 2-3 s, engine starting. The inner rotor (ICE) speed is increased from 0 to 1000

RPM, but the ICE does not provide any torque to the system. The PMDMP machine is an engine starter and the DMP is still in pure EVE mode.

(3) 3-8 s, hybrid operation. The engine is working together with the battery to provide power to the system. Both low power and high power modes occur.

(4) 8-9 s, engine shuts down. The engine is shut down and the system runs on pure

EV mode.

(5) 9-10 s, braking. The vehicle speed decreases to zero. The kinetic energy of the vehicle is returned to the battery.

Outer Rotor Reference and Actual Speed [RPM] 2000

1000

-1000 Inner Rotor Reference and Actual Speed [RPM] 1500

1000

500

-500 0 1 2 3 4 5 6 7 8 9 10 Time [S]

Reference speed Actual speed

Figure 2. 39 Reference and actual rotor speeds in different operational modes.

Outer Rotor Electromagnetic Torque [Nm] 100

-50 Inner Rotor Electromagnetic Torque [Nm] 5

-5

-10

-15 0 1 2 3 4 5 6 7 8 9 10 Time [S]

Figure 2. 40 Torque productions of both rotors in different operational modes.

Stator Phse Current [A] 200

100

-100

-200 Inner Rotor Phase Current [A] 40

-20

-40 0 1 2 3 4 5 6 7 8 9 10 Time [S]

Figure 2. 41 Current waveforms of stator and inner rotor windings in different modes.

Chapter 3: Squirrel-Cage Dual Mechanical Port Electric Machine

3.1 Modeling of SCDMP machine

3.1.1 Three phase model of SCDMP machine

The conceptual cross section of the proposed SCDMP machine is shown in Figure 3.

1. As shown, the SCDMP machine has 3 sets of three-phase windings (the squirrel-cage rotor windings are treated as equivalent three-phase windings), so it has nine different phase windings. In general, each phase winding has its own self-inductance and mutual- inductance with any other phase windings. The inductance matrix of the SCDMP machine is presented in (3. 1). Note that (3. 2) is a 9 by 9 matrix.

Figure 3. 1 Conceptual cross-section of the SCDMP machine.

퐿푠푠 퐿푠표푟 퐿푠𝑖푟 퐿푠표푟𝑖푟 = [퐿표푟푠 퐿표푟표푟 퐿표푟𝑖푟] (3. 1) 퐿𝑖푟푠 퐿𝑖푟표푟 퐿𝑖푟𝑖푟 The flux and voltage equations for the SCDMP machine are summarized in (3. 2) and

(3. 3). Note that 푣푎푏푐표푟 = 0.

휆푎푏푐푠 푖푎푏푐푠 [휆푎푏푐표푟] = 퐿푠표푟𝑖푟 [푖푎푏푐표푟] (3. 2) 휆푎푏푐𝑖푟 푖푎푏푐𝑖푟 푣 푎푏푐푠 푅푠푖푎푏푐푠 푑 휆푎푏푐푠 [푣푎푏푐표푟] = [푅 푖 ] + [휆 ] (3. 3) 표푟 푎푏푐표푟 푑푡 푎푏푐표푟 푣푎푏푐𝑖푟 푅𝑖푟푖푎푏푐𝑖푟 휆푎푏푐𝑖푟 All the values are referred to the stator side. Details of the three-phase model and meanings of the variables are explained in Appendix B and Nomenclature.

3.1.2 Model in synchronous reference frame

A. Three-phase to d-q reference frame.

Based on the three-phase flux and voltage equations presented in Section 3.1.1, the machine model in synchronous reference frame, i.e., d-q reference frame, can be derived and the details are shown in Appendix B. As shown in Figure 3. 2, an arbitrary d-axis rotating at synchronous speed (휔푒푠) is selected.

Figure 3. 2 Reference frames for the SCDMP machine.

(3. 4) - (3. 9) are the voltage equations for the machine.

푑휆 푣 = 푅 푖 + 푑푠 − 휔 휆 (3. 4) 푑푠 푠 푑푠 푑푡 푒푠 푞푠 푑휆 푣 = 푅 푖 + 푞푠 + 휔 휆 (3. 5) 푞푠 푠 푞푠 푑푡 푒푠 푑푠 푑휆 0 = 푅 푖 + 푑표푟 − 푆 휔 휆 (3. 6) 표푟 푑표푟 푑푡 표푟 푒푠 푞표푟 푑휆 0 = 푅 푖 + 푞표푟 + 푆 휔 휆 (3. 7) 표푟 푞표푟 푑푡 표푟 푒푠 푑표푟

푑휆 푣 = 푅 푖 + 푑𝑖푟 − 푆 휔 휆 (3. 8) 푑𝑖푟 𝑖푟 푑𝑖푟 푑푡 𝑖푟 푒푠 푞𝑖푟 푑휆 푣 = 푅 푖 + 푞𝑖푟 + 푆 휔 휆 (3. 9) 푞𝑖푟 𝑖푟 푞𝑖푟 푑푡 𝑖푟 푒푠 푑𝑖푟

푆표푟 and 푆𝑖푟 are defined by (3. 10) and (3. 11), respectively.

휔푒푠 − 휔표푟푃표푙푒푠/2 푆표푟 = (3. 10) 휔푒푠 휔 − 휔 푃표푙푒푠/2 푒푠 𝑖푟 (3. 11) 푆𝑖푟 = 휔푒푠 where 휔표푟 and 휔𝑖푟 (in rad/s) are the mechanical rotating speeds of the outer and inner rotors, respectively. Poles is the number of poles of the machine.

Note that mechanical rotating speeds of the outer and inner rotors are 푛 = 60휔표푟 and 표푟 2휋

푛 = 60휔푖푟 (in Round Per Minute: RPM), respectively. 𝑖푟 2휋

The flux linkages of the SCDMP machine are expressed by (3. 12) - (3. 17).

휆푑푠 = 퐿푠푖푑푠 + 푀푠표푟푖푑표푟 + 푀푠𝑖푟푖푑𝑖푟 (3. 12)

휆푞푠 = 퐿푠푖푞푠 + 푀푠표푟푖푞표푟 + 푀푠𝑖푟푖푞𝑖푟 (3. 13)

휆푑표푟 = 푀푠표푟푖푑푠 + 퐿표푟푖푑표푟 + 푀𝑖푟표푟푖푑𝑖푟 (3. 14)

휆푞표푟 = 푀푠표푟푖푞푠 + 퐿표푟푖푞표푟 + 푀𝑖푟표푟푖푞𝑖푟 (3. 15)

휆푑𝑖푟 = 푀푠𝑖푟푖푑푠 + 푀𝑖푟표푟푖푑표푟 + 퐿𝑖푟푖푑𝑖푟 (3. 16)

휆푞𝑖푟 = 푀푠𝑖푟푖푞푠 + 푀𝑖푟표푟푖푞표푟 + 퐿𝑖푟푖푞𝑖푟 (3. 17) Torque productions of the outer rotor and the inner rotor are represented by (3. 18) and (3. 19), respectively.

푃표푙푒푠 3 푇 = (휆 푖 − 휆 푖 ) (3. 18) 푒표푟 2 2 푞표푟 푑표푟 푑표푟 푞표푟 푃표푙푒푠 3 푇 = (휆 푖 − 휆 푖 ) (3. 19) 푒𝑖푟 2 2 푞𝑖푟 푑𝑖푟 푑𝑖푟 푞𝑖푟

B. Physical meanings of inductances.

The physical meanings of 퐿푠, 퐿표푟, 퐿푟푟, 푀푠표푟 , 푀푠𝑖푟 and 푀𝑖푟표푟 are explained in this section. Note that the inductances discussed here are different from the phase variables (see

Appendix B).

1) Stator excitation alone: Figure 3. 3 shows the flux line distribution when current is supplied to the stator windings while inner and outer rotor windings are open.

It is observed that the flux lines can be divided into three parts.

The first part links only the stator windings. The inductance accounting for this part of flux is the stator self-leakage inductance (퐿푠).

The second part links both the stator and the outer rotor windings, but it does not link the inner rotor windings; the corresponding inductance is called “stator-outer rotor mutual- leakage inductance” (푀푙푠).

The third part links all the windings; the corresponding inductance is actually the stator and inner rotor mutual inductance (푀푠𝑖푟).

From the above analysis, it is easy to understand that the mutual inductance between the stator and outer rotor and the self-inductance of the stator are 푀푠표푟 = 푀푙푠 + 푀푠𝑖푟 and

퐿푠 = 퐿푙푠 + 푀푙푠 + 푀푠𝑖푟 = 퐿푙푠 + 푀푠표푟, respectively.

Figure 3. 3 Flux line distribution of the SCDMP machine with stator current alone.

2) Inner rotor excitation alone: Figure 3. 4 shows the flux line distribution when current is supplied to the inner rotor windings alone.

Again the flux lines can be divided into three parts.

The first part links only the inner rotor windings. The inductance accounting for this part of flux is the inner rotor self-leakage inductance (퐿푙𝑖푟).

The second part links both the inner rotor and the outer rotor windings, but it does not link the stator windings; the corresponding inductance is called “inner rotor-outer rotor mutual-leakage inductance” (푀푙𝑖푟).

The third part links all the windings; the corresponding inductance is 푀푠𝑖푟

Thus, the mutual inductance between the inner rotor and the outer rotor is 푀𝑖푟표푟 =

푀푙𝑖푟 + 푀푠𝑖푟 and the self-inductance of the inner rotor is 퐿𝑖푟 = 퐿푙𝑖푟 + 푀푙𝑖푟 + 푀푠𝑖푟 = 퐿푙𝑖푟 +

푀𝑖푟표푟, respectively.

Figure 3. 4 Flux line distribution of the SCDMP machine with inner rotor current alone.

3) Outer rotor excitation alone: Similar to the analysis in the former two cases, the outer rotor self-leakage inductance (퐿푙표푟) accounts for the part of flux linking only the outer rotor windings. The self-inductance of the outer rotor is 퐿표푟 = 퐿푙표푟 + 푀푙푠 + 푀푙𝑖푟 + 푀푠𝑖푟 =

퐿푙표푟 + 푀푠표푟 + 푀𝑖푟표푟 − 푀푠𝑖푟.

It should be pointed out that the main flux of the SCDMP machine does not encounter high reluctance in the outer rotor core, though it does have to go through two layers of airgaps. The inductances of the SCDMP machine are actually close to that in a Doubly-

Fed Induction Machine (DFIM), rather than that in a PM machine. When the inner rotor windings are removed, the SCDMP machine is equivalent to a conventional Squirrel-Cage

Induction Machine (SCIM), only the airgap is divided into two parts.

C. Equivalent circuits in d-q reference frame.

Replacing the inductances in (3. 12) - (3. 17) with the inductances presented in Section

3.1.2.B, the flux equations for the SCDMP machine can also be rewritten as (3. 20) - (3.

25).

휆푑푠 = 퐿푙푠푖푑푠 + 푀푙푠(푖푑푠 + 푖푑표푟) + 푀푠𝑖푟(푖푑푠 + 푖푑표푟 + 푖푑𝑖푟) (3. 20)

휆푞푠 = 퐿푙푠푖푞푠 + 푀푙푠(푖푞푠 + 푖푞표푟) + 푀푠𝑖푟(푖푞푠 + 푖푞표푟 + 푖푞𝑖푟) (3. 21)

휆푑표푟 = 퐿푙표푟푖푑표푟 + 푀푠𝑖푟(푖푑푠 + 푖푑표푟 + 푖푑𝑖푟) + 푀푙푠(푖푑푠 + 푖푑표푟 ) + 푀푙𝑖푟(푖푑표푟 + 푖푑𝑖푟)

(3. 22)

휆푞표푟 = 퐿푙표푟푖푞표푟 + 푀푠𝑖푟(푖푞푠 + 푖푞표푟 + 푖푞𝑖푟) + 푀푙푠(푖푞푠 + 푖푞표푟) + 푀푙𝑖푟(푖푞표푟 + 푖푞𝑖푟)

(3. 23)

휆푑𝑖푟 = 퐿푙𝑖푟푖푑𝑖푟 + 푀푠𝑖푟(푖푑푠 + 푖푑표푟 + 푖푑𝑖푟) + 푀푙𝑖푟(푖푑표푟 + 푖푑𝑖푟) (3. 24)

휆푞𝑖푟 = 퐿푙𝑖푟푖푞𝑖푟 + 푀푠𝑖푟(푖푞푠 + 푖푞표푟 + 푖푞𝑖푟) + 푀푙𝑖푟(푖푞표푟 + 푖푞𝑖푟) (3. 25) Based on (3. 4) - (3. 9) and (3. 20) - (3. 25), the equivalent circuits of the SCDMP machine in d-q reference frame are derived and shown in Figure 3. 5 and Figure 3. 6.

Figure 3. 5 The d-axis equivalent circuit of the SCDMP machine.

Figure 3. 6 The q-axis equivalent circuit of the SCDMP machine.

3.1.3 Model in stationary reference frame

A. Three-phase to 훼 − 훽 reference frame.

As shown in Figure 3. 2, the 훼 - axis is aligned with the 푎푠 - axis. To obtain the 훼 − 훽 reference frame machine model from the 푑 − 푞 reference frame model, the easiest way is to replace the subscripts “푑” and “푞” with“훼” and “훽,” and let 휔푒푠 = 0, then all the equations in the 푑 − 푞 reference frame model will be valid in the 훼 − 훽 reference frame.

Voltage equations for the SCDMP machine in the stationary (훼 − 훽) reference frame are expressed by (3. 26) - (3. 31).

푑휆 푣 = 푅 푖 + 훼푠 (3. 26) 훼푠 푠 훼푠 푑푡

푑휆훽푠 푣 = 푅 푖 + (3. 27) 훽푠 푠 훽푠 푑푡 푑휆 푃표푙푒푠 0 = 푅 푖 + 훼표푟 + 휔 휆 (3. 28) 표푟 훼표푟 푑푡 2 표푟 훽표푟

푑휆훽표푟 푃표푙푒푠 0 = 푅 푖 + − 휔 휆 (3. 29) 표푟 훽표푟 푑푡 2 표푟 훼표푟 푑휆 푃표푙푒푠 푣 = 푅 푖 + 훼𝑖푟 + 휔 휆 (3. 30) 훼𝑖푟 𝑖푟 훼𝑖푟 푑푡 2 𝑖푟 훽𝑖푟

푑휆훽𝑖푟 푃표푙푒푠 푣 = 푅 푖 + − 휔 휆 (3. 31) 훽𝑖푟 𝑖푟 훽𝑖푟 푑푡 2 𝑖푟 훼𝑖푟 The flux linkages of the SCDMP machine in stationary reference frame are expressed by (3. 32) - (3. 37).

휆훼푠 = 퐿푠푖훼푠 + 푀푠표푟푖훼표푟 + 푀푠𝑖푟푖훼𝑖푟 (3. 32)

휆훽푠 = 퐿푠푖훽푠 + 푀푠표푟푖훽표푟 + 푀푠𝑖푟푖훽𝑖푟 (3. 33)

휆훼표푟 = 푀푠표푟푖훼푠 + 퐿표푟푖훼표푟 + 푀𝑖푟표푟푖훼𝑖푟 (3. 34)

휆훽표푟 = 푀푠표푟푖훽푠 + 퐿표푟푖훽표푟 + 푀𝑖푟표푟푖훽𝑖푟 (3. 35)

휆훼𝑖푟 = 푀푠𝑖푟푖훼푠 + 푀𝑖푟표푟푖훼표푟 + 퐿𝑖푟푖훼𝑖푟 (3. 36)

휆훽𝑖푟 = 푀푠𝑖푟푖훽푠 + 푀𝑖푟표푟푖훽표푟 + 퐿𝑖푟푖훽𝑖푟 (3. 37) Torque productions of the outer rotor and the inner rotor are represented by (3. 52) and (3. 53), respectively.

푃표푙푒푠 3 푇 = (휆 푖 − 휆 푖 ) (3. 38) 푒표푟 2 2 훽표푟 훼표푟 훼표푟 훽표푟 푃표푙푒푠 3 푇 = (휆 푖 − 휆 푖 ) (3. 39) 푒𝑖푟 2 2 훽𝑖푟 훼𝑖푟 훼𝑖푟 훽𝑖푟

B. Steady state model in complex vector form.

If the complex vector is defined as 푓 = 푓훼 + 푗푓훽 , where “j” is a 90 degree phase operator, the machine equations in steady state can be expressed by (3. 40) - (3. 47).

휆푠 = 퐿푠푖푠 + 푀푠표푟푖표푟 + 푀푠𝑖푟푖𝑖푟 (3. 40)

휆표푟 = 푀푠표푟푖푠 + 퐿표푟푖표푟 + 푀𝑖푟표푟푖𝑖푟 (3. 41)

휆𝑖푟 = 푀푠𝑖푟푖푠 + 푀𝑖푟표푟푖표푟 + 퐿𝑖푟푖𝑖푟 (3. 42)

푣푠 = 푅푠푖푠 + 푗휔푒푠휆푠 (3. 43)

푅표푟 0 = 푖표푟 + 푗휔푒푠휆표푟 (3. 44) 푆표푟

푣𝑖푟 푅𝑖푟 = 푖𝑖푟 + 푗휔푒푠휆𝑖푟 (3. 45) 푆𝑖푟 푆𝑖푟 푃표푙푒푠 3 푇 = 퐼푚푎푔[휆 푐표푛푗(푖 )] (3. 46) 푒표푟 2 2 표푟 표푟 푃표푙푒푠 3 푇 = 퐼푚푎푔[휆 푐표푛푗(푖 )] (3. 47) 푒𝑖푟 2 2 𝑖푟 𝑖푟

C. Steady state equivalent circuit in 훼 − 훽 reference frame.

Based on (3. 40) - (3. 45), the equivalent circuit of the SCDMP machine can be derived and is shown in Figure 3. 7. The inductances used in (3. 20) - (3. 25) are applied again here.

Though the SCDMP machine is different from conventional induction machines (squirrel- cage and doubly-fed), it can be operated as a conventional induction machine in special conditions.

Figure 3. 7 Steady state equivalent circuit of the SCDMP machine

1) Squirrel-cage induction machine operation: When the inner rotor windings are open, the steady state equivalent circuit becomes the one shown in Figure 3. 8. The SCDMP machine actually degrades to a SCIM. The only difference is the main flux will have to travel through two layers of airgaps. The HEV with the SCDMP machine will become a pure electric vehicle mode in this kind of operation.

Figure 3. 8 SCDMP machine operated as squirrel-cage induction machine.

2) Doubly-fed induction machine operation: When the squirrel-cage rotor rotates at synchronous speed, 푆표푟 = 0. The outer rotor current becomes zero. As shown in Figure 3.

9, the equivalent circuit of the SCDMP machine is identical to that of a DFIM in such case.

Hence, the SCDMP machine can also be operated as a DFIM.

Figure 3. 9 SCDMP machine operated as doubly-fed induction machine.

3.1.4 Validation of the proposed model by finite element method.

In order to prove the effectiveness of the proposed machine models, a sample SCDMP machine is designed and evaluated by the FEM software (Ansys/Maxwell). The cross- section of the sample machine is shown in Figure 3. 10 and Table 3. 1 lists its main dimensions, performance and parameters.

Figure 3. 10 Cross-section of the sample SCDMP machine.

Table 3. 1 Specifications of sample SCDMP machine.

Unit Value Poles 4 Stator OD/ID mm 270/180 Outer rotor OD/ID mm 178.5/156.5 Inner rotor OD/ID mm 155/80 Stack length mm 84 Rated speed of outer/inner rotor RPM 2000/2000 Rated 휆표푟/ 휆𝑖푟 Wb 0.40/0.42 Rated 푇푒표푟 / 푇푒𝑖푟 Nm 100/-100 Rated 푃표푟푚푒푐ℎ/ 푃𝑖푟푚푒푐ℎ kW 21/-21 Rated efficiency % 85 퐿푠/퐿표푟/퐿𝑖푟 mH 2.15/2.30/2.32 푀푠표푟/푀𝑖푟표푟/푀푠𝑖푟 mH 1.97/2.03/1.88

It is shown in (3. 20) - (3. 25) that the fluxes are decided by the currents. By trial and error, proper currents are selected and supplied to the FEM model of the sample machine to attain the rated rotor flux levels. In this way, the six unknown inductances can be obtained by solving (3. 20) - (3. 25). It should be pointed out that the combination of currents is not unique. Hence, in order to calculate the inductances under rated fluxes, as least two sets of current combinations are required. The results in Table 3. 1 show that

퐿푠 > 푀푠표푟 > 푀푠𝑖푟, 퐿𝑖푟 > 푀𝑖푟푠표푟 > 푀푠𝑖푟 and 퐿표푟 > 푀푠표푟(푀𝑖푟표푟). These relationships are consistent with the analysis in Section 3.1.4.

Because the SCDMP machine cannot be modeled as two separate electric machines, the definition of efficiency for conventional machines is not valid for the SCDMP machine.

Hence, a new efficiency (휂) definition is introduced.

If 푃푏푎푡푡 > 0, 휂 = 푃표푟푚푒푐ℎ/ (−푃𝑖푟푚푒푐ℎ + 푃푏푎푡푡);

if 푃푏푎푡푡 < 0, 휂 = (푃표푟푚푒푐ℎ − 푃푏푎푡푡)/ (−푃𝑖푟푚푒푐ℎ);

푃표푟푚푒푐ℎ: Outer rotor mechanical output power.

푃𝑖푟푚푒푐ℎ: Inner rotor mechanical output power.

푃푏푎푡푡: Power provided by battery/DC bus.

A. Transient performance validation

Based on the parameters shown in Table 3. 1, the proposed transient model of the

SCDMP machine is discretized and built in Matlab/Simulink. In order to compared the simulation results with the FEM calculation results, the d,q-axis variables are transformed back to three-phase variables. Friction and windage are neglected. The simulation is divided into the following four intervals and results are shown in Figure 3. 11.

1) No-load starting of outer rotor: A balanced three-phase voltage ( 푣푎푠 =

100 cos(120휋푡) 푉) is applied to the stator windings at time zero and the inner rotor windings are open. As shown in Figure 3. 11 (left), a large inrush current occurs at the stator windings, which generates a starting torque to accelerate the outer rotor. The inrush current decreases to no-load magnetizing current once the outer rotor reaches synchronous speed (1800 RPM) at steady state. The outer rotor current also decays to zero at the same time. This process is the same as the starting of a squirrel-cage induction machine.

Figure 3. 11 Comparison between transient responses of proposed model (left) and FEM model (right). 77

Traces in Figure 3. 11 from top to bottom:

푣푎푠 - stator phase A to neutral voltage;

푣푎𝑖푟 - inner rotor phase A to neutral voltage;

푖푎푠 - stator phase A current;

푖푎표푟 - outer rotor phase A current;

푖푎𝑖푟 - inner rotor phase A current;

푇퐿표푟 - outer rotor load torque;

푇푒표푟 - outer rotor electromagnetic torque;

푇퐿𝑖푟 - inner rotor load torque;

푇푒𝑖푟 - inner rotor electromagnetic torque;

푛표푟 - outer rotor mechanical speed;

푛𝑖푟 - inner rotor mechanical speed.

2) No-load starting of inner rotor: A balanced three-phase voltage ( 푣푎𝑖푟 =

40 cos(40휋푡) 푉) is applied to the inner rotor windings at t = 0.5 s. A large inrush current occurs at the inner rotor windings, which leads to inrush current at the stator windings and current oscillation at the outer rotor windings. The speed of outer rotor drops slightly and then goes back to synchronous speed after the inner rotor reaches its steady state speed

(1200 RPM).

3) Sudden load change of outer rotor: A 50 Newton meter (Nm) load is suddenly applied to the outer rotor at t = 1.5 s and lasts one second. As a result, the outer rotor speed slows down slightly and the outer rotor electromagnetic torque increases from 0 to 50 Nm to counter balance the load torque. It is shown in Figure 3. 11 (left) that the stator current increases and thus electrical power can be transformed to mechanical power. It can be observed that the variation of inner rotor current and speed is much smaller than that of the outer rotor.

4) Sudden load change of inner rotor: A -50 Nm load is suddenly applied to the inner rotor at t = 3.0 s and lasts one second. The inner rotor electromagnetic torque responses to the load change quickly. The electromagnetic torque of inner rotor bounces around -50 Nm.

Figure 3. 11 (right) shows the FEM results of the same operation process. It can be observed that the FEM results match well with simulation results. Since the nonlinearity of FEM model, such as slot-tooth effect, winding method, material saturation, are not considered in the proposed linear model, their differences in results are understandable.

B. Steady state performance validation

Based on the machine model proposed in Section 3.1.4 and the parameters listed in

Table 3. 1, the required stator and inner rotor currents in different operational conditions are calculated and supplied to the finite-element model of the machine. Then, the steady state torque productions of the machine are computed by the FEM. The results are summarized in Table 3. 2.

Table 3. 2 Validation of the proposed steady state machine model by FEM..

Proposed Model Calculation Results FEM Results ∗ ∗ ∗ ∗ Case 휆표푟 휆𝑖푟 푇푒표푟 푇푒𝑖푟 휆표푟 휆𝑖푟 푇푒표푟 푇푒𝑖푟 [Wb] [Wb] [Nm] [Nm] [Wb] [Wb] [Nm] [Nm] 1 0.40 0.42 100 -100 0.40 0.42 100 -97 2 0.40 0.42 -100 -100 0.39 0.42 -96 -95 3 0.40 0.42 100 0 0.40 0.42 100 0 4 0.40 ir-open 100 0 0.39 ir-open 95 0 5 0.40 0.37 100 -100 0.36 0.30 83 -91 6 0.35 0.42 100 -100 0.38 0.45 116 -117 6* 0.35 0.42 100 -100 0.35 0.42 102 -99 6** 0.35 0.42 50 -100 0.35 0.41 49 -98

In case 1, 2 and 3, the flux levels of the two rotors are maintained at rated values. The rotor flux levels calculated by the FEM are very close to the predicted ones. The differences between the FEM calculated torque productions and the predicted ones are within the range of ±5% .

In case 4, the SCDMP machine is reduced to a squirrel-cage induction machine because the inner rotor windings are open. The results show that the calculated values still closely follow the predicted values.

In order to extend the speed range of the rotors, it is necessary to weaken the rotor fluxes at high speed. The goal of case 5/case 6 are to weaken the outer/inner rotor flux while maintaining the inner/outer rotor flux. The FEM results show that the rotor fluxes are weakened, but the output torque productions present a highest discrepancy of 17 % when compared with the predicted values. These deviations can be explained as follows.

Since the proposed machine model uses constant inductances, the computed results may not be accurate if the parameters are not accurately estimated. Due to the non-linearity of the core material, if the flux levels change, the magnetic field distribution and hence the inductances of the machine will also vary. Because the inductances listed in Table 3. 1 are calculated under rated flux levels, it is reasonable that these inductances should be modified if the flux levels are not rated. Therefore, the performance of the proposed machine model can be further improved if the non-linearity is taken into consideration.

In case 6* and case 6**, the inductances in the flux weakening condition are calculated by the FEM and adopted in the model. As expected, the results show that the discrepancies in cases 6* and 6** (±2%) are greatly decreased when compared with that in case 6 (±17%)

3.2 Independent control of the two rotors of SCDMP machine

3.2.1 Introduction of SCDMP machine control algorithm.

This subchapter proposes an independent control algorithm for the two rotors of the

SCDMP machine. The four important variables of the machine, which are the outer rotor flux 휆표푟, the inner rotor flux 휆𝑖푟, the outer rotor torque production 푇푒표푟 and the inner rotor torque production 푇푒𝑖푟, can be independently controlled with the proposed algorithm. Here

“independent” means the control over any one of these four variables will not have effect on the other three variables. A current model flux observer is used to estimate the slip frequency of the outer rotor.

3.2.2 Current command calculation module.

The goal of rotor flux FOC for a squirrel-cage induction machine is to satisfy the output torque request while maintaining a constant rotor flux. In a typical FOC control, if the d-axis of an induction machine is aligned with the rotor flux, it can be derived from the machine model that the torque production of the machine is in direct proportion to the q- axis current of the stator in steady state, while the rotor flux is indirect proportion to the d- axis stator current. Hence, the calculation of stator current reference is relatively straightforward. The rotor speed can then be controlled by using a speed regulator (for example, a PI controller) that generates a q-axis stator current reference from the difference between reference speed and actual speed.

Following the same logic, if an equivalent stator current can be found, the outer rotor of the SCDMP machine can also be controlled in the same way. It is evident from the transient model ((3. 4) - (3. 19)) that the flux, the current and thus the torque production of the outer rotor can be controlled by the stator and the inner rotor currents. The equivalent stator current represents the total effect of stator and inner rotor currents on the outer rotor and can be easily derived from (3. 14) and (3. 15).

푀𝑖푟표푟 푖푑푠−푒푞 = 푖푑푠 + 푖푑𝑖푟 (3. 48) 푀푠표푟

푀𝑖푟표푟 푖푞푠−푒푞 = 푖푞푠 + 푖푞𝑖푟 (3. 49) 푀푠표푟 Compared with the squirrel-cage induction machine, the SCDMP machine has one more controllable current source - the inner rotor current. (3. 48) and (3. 49) shows that the inner rotor current of the SCDMP machine provides one extra degree of freedom in the control of the outer rotor. As long as the equivalent stator currents remain the same, different combinations of stator and inner rotor currents will result in the same outer rotor torque and flux.

However, it should be kept in mind that the inner rotor, which is mechanically coupled to the shaft of the engine, also requires high performance control. The control freedom brought by the inner rotor current makes it possible to independently control the flux and torque production of the inner rotor while controlling those of the outer rotor.

The method for the determination of the proper current references, i.e., current commands, is discussed here. Since the current commands are targeting in achieving certain steady state performances, the differential terms in (3. 4) - (3. 9) accounting for the transient

83 response are neglected in the following discussion. The effect of this simplification on the control performance will be discussed in Section 3.2.4.

In order to simplify the control algorithm, the d-axis of the machine should be aligned with the rotor flux; and the two rotors of the SCDMP machine offer two options. 1) If the d-axis is aligned with the outer rotor flux, then 휆푑표푟 = 휆표푟 and 휆푞표푟 = 0. 2) If the d-axis is aligned with the inner rotor flux, then 휆푑𝑖푟 = 휆𝑖푟 and 휆푞𝑖푟 = 0. It should be pointed out that there is usually a non-zero angle between the fluxes of the two rotors due to the existence of leakage inductances. In other words, the d-axes in these two cases are usually not aligned with each other. The first option is selected for the discussion in this paper.

As shown in Figure 3. 12, the d-axis of the SCDMP machine is aligned with the outer rotor flux (휃푑 = 휃휆표푟); the d2-axis is aligned with the inner rotor flux. 휃𝑖푟−표푟 is the angle of the d2-axis with respect to the d-axis. 휃휆표푟 is the angle of the outer rotor flux with respect to the axis of phase a of stator winding (푎푠-axis).

Figure 3. 12 Reference frames for the SCDMP machine.

The following equations can be easily derived. Note that 휆푑2𝑖푟 = 휆𝑖푟 and 휆푞2𝑖푟 = 0.

휆푑𝑖푟 = 휆푑2𝑖푟 푐표푠휃𝑖푟−표푟 = 휆𝑖푟 cos 휃𝑖푟−표푟 (3. 50)

휆푞𝑖푟 = 휆푑2𝑖푟 푠푖푛휃𝑖푟−표푟 = 휆𝑖푟 sin 휃𝑖푟−표푟 (3. 51)

푖푑𝑖푟 = 푖푑2𝑖푟 푐표푠휃𝑖푟−표푟 − 푖푞2𝑖푟 sin 휃𝑖푟−표푟 (3. 52)

푖푞𝑖푟 = 푖푞2𝑖푟 푐표푠휃𝑖푟−표푟 + 푖푑2𝑖푟 sin 휃𝑖푟−표푟 (3. 53)

Neglecting the differential term in (3. 6) and considering 휆푞표푟 = 0 yields (3. 54).

푖푑표푟 = 0 (3. 54)

Substituting 휆푑표푟 with 휆표푟 and 휆푞표푟 with 0 in (3. 18), (3. 55) is derived. Substituting

(3. 50) - (3. 53) into (3. 19), (3. 56) is obtained.

2 2 푇푒표푟 푖푞표푟 = − (3. 55) 3 푃표푙푒푠 휆표푟

2 2 푇푒𝑖푟 푖푞2𝑖푟 = − (3. 56) 3 푃표푙푒푠 휆𝑖푟

Equations (3. 57) and (3. 58) are obtained by solving (3. 14) - (3. 17) and (3. 50) - (3.

53). Note that 휆푑표푟 = 휆표푟 and 휆푞표푟 = 0.

1 푖 = (푖 − √푖2 + 푎2푖2 − 푏2푖2 ) (3. 57) 푑2𝑖푟 푏 휆𝑖푟 휆표푟 푞표푟 푞2𝑖푟

휃𝑖푟−표푟 = 휃 < 푖휆표푟, 푎푖푞표푟 > −휃 < 푖휆𝑖푟 − 푏푖푑2𝑖푟 , −푏푖푞2𝑖푟 > (3. 58) where

휆표푟 푖휆표푟 = (3. 59) 푀푠표푟

휆𝑖푟 푖휆𝑖푟 = (3. 60) 푀푠𝑖푟

푀 퐿 푎 = 𝑖푟표푟 − 표푟 (3. 61) 푀푠𝑖푟 푀푠표푟

퐿 푀 푏 = 𝑖푟 − 𝑖푟표푟 (3. 62) 푀푠𝑖푟 푀푠표푟

The angle calculation operator “휃 < 푚, 푛 >” is defined as follows.

푛 arcsin , 푖푓 푚 ≥ 0 √푚2 + 푛2 휃 < 푚, 푛 >= { 푛 (3. 63) 휋 − arcsin , 푖푓 푚 < 0 √푚2 + 푛2

Based on the computed 푖푞2𝑖푟 , 푖푑2𝑖푟 and 휃𝑖푟−표푟, the inner rotor current commands can be easily calculated by (3. 52) and (3. 53). The stator current commands expressed by (3.

64) and (3. 65) are derived from (3. 14) and (3. 15).

푀𝑖푟표푟 푖푑푠 = 푖휆표푟 − 푖푑𝑖푟 (3. 64) 푀푠표푟

퐿표푟 푀𝑖푟표푟 푖푞푠 = − 푖푞표푟 − 푖푞𝑖푟 (3. 65) 푀푠표푟 푀푠표푟

The calculation of stator and inner rotor current commands is summarized in Figure

3. 13. The results show that there is no one-to-one correspondence between the four input currents (푖푑푠, 푖푞푠, 푖푑𝑖푟 and 푖푞𝑖푟) and four output variables (휆표푟, 휆𝑖푟, 푇푒표푟 and 푇푒𝑖푟). Actually, all the four input currents have to be adjusted when the command for any of the four output variables changes. Though the FOC for induction machine cannot be directly applied to the SCDMP machine for its dual-rotor structure, this section shows that the general idea of simplifying the control algorithm by decoupling the flux and the torque is still effective.

* * iqor * (3.64) or (3.55) * ids * iq2 ir * (3.56) (3.65) ir * iqs * i * T (3.57) d2 ir (3.52) eor idir * ir_ or * T (3.58) (3.53) eir iqir

Figure 3. 13 Current command calculation module of the SCDMP machine.

3.2.3 Outer rotor flux observer.

The position of the outer rotor flux (휃휆표푟) shown in Figure 3. 12 is treated as a known value in the previous discussion, so the three-phase stator and inner rotor currents can be transformed to the correct synchronous reference frame. However, when flux sensors are not installed in the machine, the direct access to the rotor fluxes is not available. The observer for the outer rotor flux is proposed to estimate its position.

The steady state slip frequency of the outer rotor can be derived from (3. 7).

푖푞표푟푅표푟 푖푞표푟푅표푟 푆표푟휔푒푠 = − = − (3. 66) 휆푑표푟 휆표푟 where 푖푞표푟 and 휆푑표푟 can be derived from (3. 48) - (3. 49) and (3. 64) - (3. 65),

1 푖푞표푟 = − 푀푠표푟푖푞푠−푒푞 (3. 67) 퐿표푟

휆표푟 = 푀푠표푟푖푑푠−푒푞 (3. 68)

Thus, the slip frequency can be expressed by (3. 69).

푅표푟 푖푞푠−푒푞 푆표푟휔푒푠 = (3. 69) 퐿표푟 푖푑푠−푒푞

It is shown in (3. 69) that the slip frequency of the outer rotor is in direct proportion to the outer rotor time constant (퐿표푟 ), hence the accuracy of rotor parameters will have great 푅표푟 impact on the performance of the proposed flux observer.

In order to overcome the parameter sensitivity issue in the flux observer for convention induction machines, different approaches have been proposed [44] - [46]. Due to the similarity between the outer rotor of the SCDMP machine and the rotor of induction

88 machine, it is reasonable to believe that these approaches can also be applied to the SCDMP machine if they are modified properly. Further improvement on the proposed outer rotor flux observer can be referred to these approaches, but this paper will still use the simplest observer as proposed and it is shown in Figure 3. 14. Note that the outer rotor mechanical speed (휔표푟) is required.

i ids ds eq (3.48) (3.68) or idir

iqs eq Sor es  iqs (3.49) (3.69) Integrator i or qir   Speed or Pole sensor pairs

Figure 3. 14 Proposed outer rotor flux observer.

3.2.4 Simulation of the proposed FOC algorithm.

The proposed control block diagram for the SCDMP machine is shown in Figure 3.

15. The simulation model of the SCDMP machine is built according to the transient model presented in Section 3.1. The torque production references are calculated by speed regulators. Besides the two speed regulators, four current regulators are required. In the simulation, all the six regulators are Proportional - Integral (PI) controllers. The way to tune the PI coefficients can be referred to [47] and [48].

Note that the outer rotor is connected to the driving shaft of the vehicle and the inner rotor is connected to the ICE. This means the equivalent inertia of outer rotor is large, while the inertia of inner rotor is relatively small.

The rated flux level and inductances of the SCDMP machine model are listed in Table

3. 3.

Table 3. 3 Rated fluxes and inudctances of the sample SCDMP machine.

휆표푟 [Wb] 0.40 휆𝑖푟 [Wb] 0.42 퐿푠 [mH] 2.15 푀푠표푟 [mH] 1.97 퐿표푟 [mH] 2.30 푀𝑖푟표푟 [mH] 2.03 퐿𝑖푟 [mH] 2.32 푀푠𝑖푟 [mH] 1.88

ii, i ds qs abc to dq abcs ** * ii, or ds qs vvds, qs PI dq to SVPWM Inverter1 * Current ir Command DC ICE * Calculation ** bus PI vv,  * Module iidir, qir dir qir or PI SVPWM Inverter2 Teor dq to * PI *

 ir T n eir n r o r o o i o i s i t s abcir t n ii, i n i e

dir qir s 91 e abc to dq s S o S o P

Outer Rotor or    Flux Observer P ir ir or Speed Calculation  Speed Calculation or

Figure 3. 15 Control block diagram of the SCDMP machine.

The simulation is divided into five intervals, and each interval lasts one second. The operational speeds and fluxes of the two rotors will be established in the first interval. Then in each of the following four intervals, one of the four variables (휆표푟, 휆𝑖푟, 푇푒표푟 and 푇푒𝑖푟) will be changed while the other three variables are kept constant.

1) Interval I - Initialization of Operation: The first interval starts from time zero and lasts one second. The frictions are neglected.

Figure 3. 16 shows that the flux levels of the two rotors are pushed up to rated values

(휆표푟 = 0.4푊푏, 휆𝑖푟 = 0.42푊푏) in less than 0.2 s. In order to demonstrate the zero speed staring capability of the outer rotor (vehicle starting), a step change of outer rotor speed command is given at Time = 0.2 s. The torque reference for outer rotor is limited under

350 Nm. This value (350 Nm) is arbitrarily selected in the simulation. In practical application, the maximum torque is limited by the machine design. The speed of the outer rotor is increased to 2000 RPM in 0.5 s.

At Time = 0.8 s, the inner rotor speed command is issued a slope change. The inner rotor speed is thus increased to 1500 RPM. The fast speed response is due to the low inertia value of the inner rotor. The starting of the inner rotor is actually the starting of the vehicle

ICE, so the SCDMP can also operates as an engine starter.

2) Interval II - Outer Rotor Torque Production Control: In the second interval (1.0 s

- 2.0 s), a 150 Nm step change to the load torque is given to the outer rotor at 1.0 s and the outer rotor load torque is stepped down to 100 Nm at 1.5 s. Figure 3. 16 shows that the outer rotor torque production is able to follow the step change of load torque with the help of the speed regulator. It can be noticed that the speeds and flux levels of the two rotors

92 are controlled at reference values despite the slight transient responses. However, the flux angle difference is no longer zero in this interval.

This interval shows that the proposed controller is able to satisfy the outer rotor torque production request while maintaining the rotor flux levels and the torque production of the inner rotor. The vehicle with the SCDMP machine in this interval is similar to a pure electric vehicle because the ICE does not provide any power to drive the outer rotor.

3) Interval III - Inter Rotor Torque Production Control: The purpose of this interval

(2.0 s-3.0 s) is to show the independent control capability of the controller over the inner rotor torque production. The inner rotor load torque is stepped down to -100Nm at 2.0 s and increased to -50Nm at 2.5 s.

As shown in Figure 3. 16, the inner rotor torque production quickly responses to the step changes of inner rotor load torque without affecting the outer rotor torque production and the flux levels of both rotors.

Outer Rotor Flux 0.5 ]

b 0.4 W [

0.3 x u

l 0.2 F 0.1 0 Inner Rotor Flux 0.5

] 0.4 b W

[ 0.3

u 0.2 l F 0.1 0 Angle between Inner Rotor Flux and Outer Rotor Flux 10 ] g e

d 5 [

e l

g 0 n A -5 Outer Rotor Speed 2500 ] M

P 2000 R

[ 1500

e 1000 e p

S 500 0 Inner Rotor Speed 2500 ]

M 2000 P R

[ 1500

e 1000 e p

S 500 0 Outer Rotor Torque Production 400 ]

m 300 N [

200 e u

q 100 r o

T 0 -100 Inner Rotor Torque Production 50 ] m

N 0 [

e -50 u q r

o -100 T -150 0 1 2 3 4 5 Time [s]

Figure 3. 16 Flux, speed and torque production of the SCDMP machine. 94

Since the inner rotor shares the same shaft with the ICE of the vehicle, the load of the inner rotor is actually the ICE. When the inner rotor has negative torque production and positive speed, the ICE is actually acting as the prime mover and provides energy to the

SCDMP machine. Because the ICE and the battery are working together to drive the vehicle, the vehicle is operating at hybrid mode in this interval.

4) Interval IV - Outer Rotor Flux Level Control: The flux level of the outer rotor is weakened in this interval (3.0 s -4.0 s). A step change of the inner rotor flux reference is issued at 3.0 s to decrease the flux level from 0.40 Wb to 0.30 Wb.

Instead of following the step change of its reference, the results in Figure 3. 16 show that the outer rotor flux gradually decreases from 0.40 Wb to 0.3 Wb. This can be explained as follows.

Since the d-axis is aligned with the outer rotor flux, the outer rotor flux has only d- axis component. From (3. 6), (3. 70) can be easily derived.

푑휆 푑휆 푑표푟 = 표푟 = −푖 푅 (3. 70) 푑푡 푑푡 푑표푟 표푟

As mentioned in Section 3.2.2, the differential term in (3. 70) is neglected for simplicity in the previous discussion. Hence, 푖푑표푟 is assumed to be zero in the control. This simplification will not have any effect on the control performance in steady states.

However, when the outer rotor flux level varies, it will introduce errors. Figure 3. 16 shows that these errors affect not only the outer rotor flux, but also the inner rotor flux. Figure 3.

17 shows that after the outer rotor reaches steady state (0.30Wb) at Time = 3.2 s, 푖푑표푟 drops back to near zero. Theoretically speaking, the errors can be eliminated if the estimated outer rotor flux is fed back to a flux regulator. 95

Figure 3. 17 Current of the SCDMP machine.

5) Interval V - Inter Rotor Flux Level Control: The independent control of the inner rotor flux is presented in this interval (4.0 s-5.0 s). The inner rotor flux reference is decreased from 0.42Wb to 0.32Wb at 4.0 s. The result in Figure 3. 16 shows that the inner rotor flux can closely follow the step change of its reference.

The actual outer rotor flux angle and its estimated value are compared in Figure 3. 18.

The result shows that the difference is smaller than one degree at rated flux operation.

During the transient of flux weakening, the angle error increases beyond three degrees, but it decreases to less than two degrees quickly. So the overall performance of the proposed flux observer is satisfactory.

Estimated Outer Rotor Flux Angle 400 ] g

e 300 d [

e 200 l

g zoom in

n 100 A

0 0 0.1 0.2 0.3 0.4 0.5 Actual Outer Rotor Flux Angle 400 ] g

e 300 d [

200 e l g

n 100 A 0 0 0.1 0.2 0.3 0.4 0.5 Error of Estimated Outer Rotor Flux Angle 4

] 3 g e

d 2 [

1 e l

g 0 n

A -1 -2 0 1 2 3 4 5 Time [s]

Figure 3. 18 Comparison between estimated and actual outer rotor flux angles.

3.3 Operational modes of the SCDMP machine

3.3.1 Power flow analysis of the SCDMP machine.

Similar to the analysis on the power flow of the PMDMP machine, the power flow of the SCDMP machine is analyzed. It should be pointed out that the PMDMP machine does not have much power loss on the PM outer rotor because its outer rotor rotates at synchronous speed in steady state. Hence, it is reasonable to neglect all the power losses of the PMDMP machine in power flow analysis. On the contrary, the power loss of the squirrel-cage outer rotor of the SCDMP machine is not always negligible, which is explained in the following paragraph.

It is well-known that the total power transferred across the airgap from the stator in a conventional SCIM becomes two different parts – the first part becomes the mechanical power output of the squirrel-cage rotor; the second part simply becomes conduction power loss of the squirrel-cage windings. The ratio of the power loss over the total power is the slip percentage. As discussed in Section 3.2, the total effect of the stator and the inner rotor on the outer rotor can be represented by the equivalent stator currents as expressed by (3.

48) and (3. 49). Hence, the squirrel-cage rotor of the SCDMP machine shares the same electromagnetic characteristics with the rotor of a conventional SCIM. In other words, the conduction power loss of the outer rotor of the SCDMP is in proportion to the total power transferred across both airgaps from the stator and the inner rotor.

If all the other power losses are neglected except the conduction loss of the squirrel- cage rotor, the relationship between the mechanical and electrical power can be represented 98 by (3. 71). The subscripts “e” and “m” indicates electrical and mechanical powers, respectively.

푃푒푠 + 푃푒𝑖푟 + 푃푚퐼퐶퐸 − 푃푚푊ℎ푒푒푙 − 푃표푟푙표푠푠 = 0 (3. 71)

푃푒푠 is the electrical power provided by the stator windings; 푃푒𝑖푟 is the electrical power provided by the inner rotor windings; 푃푚퐼퐶퐸 is the mechanical power provided by the ICE, which is equal to the mechanical power provided by the inner rotor; 푃표푟푙표푠푠 is the conduction power loss of the outer rotor; 푃푚푊ℎ푒푒푙 is the mechanical output power of the outer rotor.

The torque production equations for the outer and the inner rotors can be further derived based on (3. 14)-(3. 17), (3. 48) and (3. 49). The result is shown in (3. 72) and (3.

73).

푃표푙푒푠 3 푇 = (휆 푖 − 휆 푖 ) 푒표푟 2 2 푞표푟 푑표푟 푑표푟 푞표푟 푃표푙푒푠 3 = [(퐿 푖 + 푀 푖 + 푀 푖 )푖 − (퐿 푖 + 푀 푖 + 푀 푖 )푖 ] 2 2 표푟 푞표푟 푠표푟 푞푠 𝑖푟표푟 푞𝑖푟 푑표푟 표푟 푑표푟 푠표푟 푑푠 𝑖푟표푟 푑𝑖푟 푞표푟 푃표푙푒푠 3 = [푀 (푖 푖 − 푖 푖 ) + 푀 (푖 푖 − 푖 푖 )] 2 2 푠표푟 푞푠 푑표푟 푑푠 푞표푟 𝑖푟표푟 푞𝑖푟 푑표푟 푑𝑖푟 푞표푟

= 푇푒−푠표푟 + 푇푒−𝑖푟표푟

(3. 72) 푃표푙푒푠 3 푇 = (휆 푖 − 휆 푖 ) 푒𝑖푟 2 2 푞𝑖푟 푑𝑖푟 푑𝑖푟 푞𝑖푟 푃표푙푒푠 3 = [(퐿 푖 + 푀 푖 + 푀 푖 )푖 − (퐿 푖 + 푀 푖 + 푀 푖 )푖 ] 2 2 𝑖푟 푞𝑖푟 푠𝑖푟 푞푠 𝑖푟표푟 푞표푟 푑𝑖푟 𝑖푟 푑𝑖푟 푠𝑖푟 푑푠 𝑖푟표푟 푑표푟 푞𝑖푟 푃표푙푒푠 3 = [푀 (푖 푖 − 푖 푖 ) + 푀 (푖 푖 − 푖 푖 )] 2 2 푠𝑖푟 푞푠 푑𝑖푟 푑푠 푞𝑖푟 𝑖푟표푟 푞표푟 푑𝑖푟 푑표푟 푞𝑖푟

= 푇푒−푠𝑖푟 + 푇푒−표푟𝑖푟

(3. 73) 99

푇푒−푠표푟, 푇푒−𝑖푟표푟, 푇푒−푠𝑖푟 and 푇푒−표푟𝑖푟 are defined as follows.

푃표푙푒푠 3 푇 = 푀 (푖 푖 − 푖 푖 ) (3. 74) 푒−푠표푟 2 2 푠표푟 푞푠 푑표푟 푑푠 푞표푟 푃표푙푒푠 3 푇 = 푀 (푖 푖 − 푖 푖 ) (3. 75) 푒−𝑖푟표푟 2 2 𝑖푟표푟 푞𝑖푟 푑표푟 푑𝑖푟 푞표푟 푃표푙푒푠 3 푇 = 푀 (푖 푖 − 푖 푖 ) (3. 76) 푒−푠𝑖푟 2 2 푠𝑖푟 푞푠 푑𝑖푟 푑푠 푞𝑖푟 푃표푙푒푠 3 푇 = 푀 (푖 푖 − 푖 푖 ) (3. 77) 푒−표푟𝑖푟 2 2 𝑖푟표푟 푞표푟 푑𝑖푟 푑표푟 푞𝑖푟

As shown in (3. 74) and (3. 75), 푇푒−푠표푟 involves only stator and outer rotor currents; and 푇푒−𝑖푟표푟 involves only inner rotor and outer rotor currents. It can be concluded that

푇푒−푠표푟 and 푇푒−𝑖푟표푟 represent the contributions of the stator current and the inner rotor current on the torque production of the outer rotor, respectively. Similarly, 푇푒−푠𝑖푟 and

푇푒−표푟𝑖푟 represent the contributions of the stator current and the outer rotor current on the torque production of the inner rotor, respectively. It is easy to see that 푇푒−𝑖푟표푟 = −푇푒−표푟𝑖푟.

푃 and 푃 can be derived from the above equations. Define 휔 = 휔푒푠 . 푒푠 푒𝑖푟 푒푠푚 푃표푙푒푠/2

푃푒푠 = (푇푒표푟 + 푇푒𝑖푟) 휔푒푠푚 = (푇푒−푠표푟 + 푇푒−푠𝑖푟) 휔푒푠푚 (3. 78)

푃푒𝑖푟 = −푇푒𝑖푟( 휔푒푠푚 − 휔𝑖푟) = −(푇푒−푠𝑖푟 + 푇푒−표푟𝑖푟)( 휔푒푠푚 − 휔𝑖푟 ) (3. 79)

The mechanical power 푃푚퐼퐶퐸 and 푃푚푊ℎ푒푒푙 are

푃푚퐼퐶퐸 = 푇퐼퐶퐸휔𝑖푟 = −푇푒𝑖푟휔𝑖푟 = −(푇푒−푠𝑖푟 + 푇푒−표푟𝑖푟)휔𝑖푟 (3. 80)

푃푚푊ℎ푒푒푙 = 푇푒표푟휔표푟 = (푇푒−푠표푟 + 푇푒−𝑖푟표푟)휔표푟 (3. 81) Then (3. 82) can be easily derived from (3. 71).

푃표푟푙표푠푠 = (푇푒−푠표푟 + 푇푒−푠𝑖푟) 휔푒푠푚 − (푇푒−푠𝑖푟 + 푇푒−표푟𝑖푟)( 휔푒푠푚 − 휔𝑖푟) − (푇푒−푠𝑖푟 + 푇푒−표푟𝑖푟)휔𝑖푟 − (푇푒−푠표푟 + 푇푒−𝑖푟표푟)휔표푟

= (푇푒−푠표푟 + 푇푒−𝑖푟표푟)(휔푒푠푚 − 휔표푟) (3. 82) 휔 = 푇 ( 푒푠 − 휔 ) 푒표푟 푃표푙푒푠 표푟 2 100

휔푒푠 푃 푃 − 휔표푟 표푟푙표푠푠 표푟푙표푠푠 푃표푙푒푠/2 (3. 83) ⇒ = 휔 = 휔 = 푆표푟 푃표푟−푔푎푝 푇 푒푠 푒푠 푒표푟 푃표푙푒푠/2 푃표푙푒푠/2 The result shown in (3. 83) matches the expectation that the ratio of the power loss over the total airgap power from the stator and the inner rotor equals to the slip percentage.

The total electrical input power from both the stator and the inner rotor windings equals to the power provided by the battery, thus the battery output power can be expressed by (3. 84).

푃푏푎푡푡푒푟푦 = 푃푒푠 + 푃푒𝑖푟

= (푇푒−푠표푟 + 푇푒−푠𝑖푟) 휔푒푠푚 − (푇푒−푠𝑖푟 + 푇푒−표푟𝑖푟)( 휔푒푠푚 − 휔𝑖푟 )

= 푇푒−푠표푟휔푒푠푚 + 푇푒−푠𝑖푟휔𝑖푟 − 푇푒−표푟𝑖푟( 휔푒푠푚 − 휔𝑖푟) (3. 84) = (푇푒−푠표푟 − 푇푒−표푟𝑖푟)휔푒푠푚 + (푇푒−푠𝑖푟 + 푇푒−표푟𝑖푟)휔𝑖푟

= 푇푒표푟휔푒푠푚 + 푇푒𝑖푟휔𝑖푟

= 푇푒표푟휔푒푠푚푆표푟 + 푇푒표푟휔표푟 + 푇푒𝑖푟휔𝑖푟

= 푃표푟푙표푠푠 + 푃푚푊ℎ푒푒푙 − 푃푚퐼퐶퐸 It can be observed from (3. 84) that the inner rotor windings provides positive electrical power to the system when the synchronous speed is higher than the inner rotor

휔표푟 speed (if 휔푒푠푚 > 휔𝑖푟 , then 푃푒𝑖푟 = −푇푒𝑖푟(휔푒푠푚 − 휔𝑖푟 ) = 푇퐼퐶퐸 ( − 휔𝑖푟 ) > 0). This 1−푆표푟 indicates that even if the mechanical speeds of the two rotors are the same, the inner rotor windings can still provide electric power to the system.

101

3.3.2 Simplified driving cycle of hybrid vehicle.

The following operations are simulated. The results are shown in Figure 3. 19.

(1) 0-5 s, starting of vehicle (outer rotor). The outer rotor speed is increased from 0 to

4000 RPM (rated speed). Note that the acceleration and deceleration of the outer rotor are directly decided by the outer rotor torque production. The outer rotor flux level is controlled to be rated value (0.4 Wb).

(2) 5-10 s, vehicle runs at rated speed. The outer rotor speed and flux level are maintained at rated values.

(3) 6-7.8 s, starting of inner rotor (ICE). The inner rotor speed is increased from 0 to

3000 RPM without firing on the ICE.

(4) 9-37 s, ICE provides mechanical power to the vehicle. The ICE is fired on and acts as the prime mover of the inner rotor. Then, the inner rotor generates negative torque to counter balance the torque from the ICE and transforms the mechanical energy from the

ICE to electrical form.

(5) 10-30 s, vehicle high speed operation. The outer rotor speed is increased from

4000 RPM to 6000 RPM Figure 3. 19 shows that the stator phase voltage increases correspondingly when the outer rotor speed increases.

(6) 15-25 s, outer rotor flux weakening. The outer rotor flux level is decreased to 0.3

Wb. As a result, the stator phase to neutral voltage decreases from 500 V to 255 V.

(7) 30-40 s, vehicle runs at rated speed. The outer rotor speed and flux levels are back to rated values.

(8) 40-45 s, braking. The outer rotor speed is decreased to 0 RPM. 102

It can be observed that, with the proposed independent control algorithm, the SCDMP machine is able to satisfy various driving needs. However, it should be pointed out that the outer rotor flux weakening does not guarantee the decline of stator phase voltage.

103

Outer Rotor Speed 8000

] (5) m 6000 p r (2) (8) [ 4000 d

e 2000 e (7) p 0 S (1) -2000 Outer Rotor Torque Production 200 ] m

N 100 [

e 0 u q r

o -100 T -200 Outer Rotor Flux 0.5

] (6) b 0.4 W [

0.3 x u 0.2 l F 0.1 0 Stator Phase A to Neutral Voltage 800

] 600 (6) V

[ 400

e 200 g a 0 t l -200 o

V -400 -600 Inner Rotor Speed 4000 ] m

p 3000 r [

2000 d e (3) e

p 1000 S 0 Inner Rotor Torque Production 50 ] m

N 0 (4) [

e -50 u q r

o -100 T -150 0 10 20 30 40 50 Time [s]

Figure 3. 19 Simplified driving cycle of hybrid vehicle. 104

Chapter 4: Conclusions and Future Work

4.1 Conclusions

This dissertation investigated the modeling and control algorithms for the Permanent

Magnet Dual Mechanical Port (PMDMP) machine and the Squirrel Cage Dual Mechanical

Port (SCDMP) machine. The following conclusions can be drawn from the dissertation.

 A complete PMDMP machine model analysis is presented. Based on the flux line

distribution of the PMDMP machine, the analytical machine model and the equivalent

circuit are presented. A simplified machine model is proposed for control purpose.

 The field oriented control (FOC) algorithms for both rotors of the PMDMP machine

are studied, both with and without position sensors. Independent control of both rotors

are realized by the FOC algorithms. Position sensorless control for the outer rotor is

achieved by high frequency injection and sliding mode in low speed and high speed,

respectively. The effectiveness of the algorithms are verified by simulation and

experiments. Based on the FOC algorithm, power flow and operational modes of the

PMDMP machine are analyzed.

 A new type of electric machine, the SCDMP machine, is proposed. The SCDMP

machine stands out for its low cost, thermal robustness and compact size when

compared with other DMP machine types.

 The following models for the SCDMP machine have been derived: 1) the three-phase

machine model; 2) the transient machine model in 푑 − 푞 reference frame and

105

corresponding equivalent circuits; 3) the transient machine model in 훼 − 훽 reference

frame; 4) steady state machine model in complex vector form and corresponding

equivalent circuits. It has been validated by the FEM that both the transient model and

steady state model accurately represent the unique electromagnetic characteristics of

the SCDMP machine.

 Based on transient model of the SCDMP machine, the current command calculation

module and outer rotor flux observer for the SCDMP machine are derived. The

independent control algorithm for the two rotors is proposed and simulated. The results

show that the outer rotor flux, the inner rotor flux, the outer rotor torque production

and the inner rotor torque production are independently controlled without affecting

each other. A simplified driving cycle is simulated to verify the functionality of the

SCDMP machine and to further confirm the effectiveness of the proposed control

algorithm.

106

4.2 Future work

Future research may focus on the following issues.

 Research related to the brushes and the slip rings. One of the drawbacks of the DMP

machine is the need for brushes and slip rings. Brushes and slip rings not only increase

the size of the machine, they also introduce manufacturing and maintenance

difficulties. Possible solution are proposing brushless DMP design and installing high

quality brushes and slip rings.

 Research related to the mechanical strength of the outer rotor and the heat dissipation

of the inner rotor. Because the outer rotor is sandwiched by two airgaps and it rotates

at high speed, the mechanical strength of the outer rotor should be designed to meet

the deformation requirement. Because the inner rotor sits inside the outer rotor, the

heat generated by the inner rotor current has to go through two layers of airgaps to

reach the stator core. The thermal resistance of this path is very high. Thermal analysis

of the DMP machine should be studied to provide an optimal solution.

 The proposed control algorithm for the SCDMP machine has some deficiencies. One

of them is the error of the flux observer. Even though the estimated flux angle is close

to actual value, the error increases as transients occur. Also, the proposed control

algorithm does not provide enough information to predict the torque/speed envelopes

of the machine. Flux weaning control algorithm is needed to fully utilize the potential

of the SCDMP machine. Research on these topics are also recommended.

107

 Install the DMP machine in a prototype vehicle to test the actual performance of the

DMP machine in a vehicle system.

 The design of the SCDMP machine should be an important topic of future research. A

prototype SCDMP machine should be designed and built to verify the proposed model

and control algorithms by experiment.

108

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Appendix A: Model Derivation of the PMDMP Machine

The three-phase abc to dq0 transformation matrix is defined as A. 1.

2휋 2휋 푐표푠휃 cos (휃 − ) cos (휃 + ) 3 3 2 2휋 2휋 푇(휃) = −푠푖푛휃 − sin (휃 − ) − sin (휃 + ) A. 1 3 3 3 1 1 1 [ 2 2 2 ]

The inverse of 푇(휃) is dq0 to abc transformation matrix.

푐표푠휃 −푠푖푛휃 1 2휋 2휋 −1 cos (휃 − ) −sin (휃 − ) 1 푇 (휃) = 3 3 A. 2 2휋 2휋 cos (휃 + ) − sin (휃 + ) 1 [ 3 3 ]

Assume the self- and mutual- inductances of the stator windings and the inner rotor windings are outer rotor position dependent, the inductances of the PMDMP machine in

퐿푠푠 퐿푠𝑖푟 (2. 1) 퐿푎푏푐푠𝑖푟 = [ ] can be expressed as follows. 퐿𝑖푟푠 퐿𝑖푟𝑖푟 6푋6

퐿푎푠푎푠 푀푎푠푏푠 푀푎푠푐푠 퐿푠푠 = [푀푏푠푎푠 퐿푏푠푏푠 푀푏푠푐푠] A. 3 푀푐푠푎푠 푀푐푠푏푠 퐿푐푠푐푠 where stator phase self-inductances and mutual inductances are

퐿푎푠푎푠 = 퐿0푠 + 퐿푙푠 + 퐿푔푠cos (2휃푃푀),

퐿 = 퐿 + 퐿 + 퐿 cos (2휃 + 2휋), 푏푠푏푠 0푠 푙푠 푔푠 푃푀 3

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퐿 = 퐿 + 퐿 + 퐿 cos (2휃 − 2휋) 푐푠푐푠 0푠 푙푠 푔푠 푃푀 3

푀 = 푀 = −0.5퐿 + 퐿 cos (2휃 − 2휋), 푎푠푏푠 푏푠푎푠 0푠 푔푠 푃푀 3

푀 = 푀 = −0.5퐿 + 퐿 cos (2휃 + 2휋), 푎푠푐푠 푐푠푎푠 0푠 푔푠 푃푀 3

푀 = 푀 = −0.5퐿 + 퐿 cos (2휃 − 2휋). 푏푠푐푠 푐푠푏푠 0푠 푔푠 푃푀 3

퐿푎𝑖푟푎𝑖푟 푀푎𝑖푟푏𝑖푟 푀푎𝑖푟푐𝑖푟 퐿𝑖푟𝑖푟 = [푀푏𝑖푟푎𝑖푟 퐿푏𝑖푟푏𝑖푟 푀푏𝑖푟푐𝑖푟 ] A. 4 푀푐𝑖푟푎𝑖푟 푀푐𝑖푟푏𝑖푟 퐿푐𝑖푟푐𝑖푟 where inner rotor phase self-inductances and mutual inductances are

퐿푎𝑖푟푎𝑖푟 = 퐿0𝑖푟 + 퐿푙𝑖푟 + 퐿푔𝑖푟cos (2휃푃푀 − 2휃𝑖푟 ),

퐿 = 퐿 + 퐿 + 퐿 cos (2휃 − 2휃 + 2휋), 푏𝑖푟푏𝑖푟 0𝑖푟 푙𝑖푟 푔𝑖푟 푃푀 𝑖푟 3

퐿 = 퐿 + 퐿 + 퐿 cos (2휃 − 2휃 − 2휋), 푐𝑖푟푐𝑖푟 0𝑖푟 푙𝑖푟 푔𝑖푟 푃푀 𝑖푟 3

푀 = 푀 = −0.5퐿 + 퐿 cos (2휃 − 2휃 − 2휋), 푎𝑖푟푏𝑖푟 푏𝑖푟푎𝑖푟 0𝑖푟 푔𝑖푟 푃푀 𝑖푟 3

푀 = 푀 = −0.5퐿 + 퐿 cos (2휃 − 2휃 + 2휋), 푎𝑖푟푐𝑖푟 푐𝑖푟푎𝑖푟 0𝑖푟 푔𝑖푟 푃푀 𝑖푟 3

푀 = 푀 = −0.5퐿 + 퐿 cos (2휃 − 2휃 − 2휋). 푏𝑖푟푐𝑖푟 푐𝑖푟푏𝑖푟 0𝑖푟 푔𝑖푟 푃푀 𝑖푟 3

푀푎푠푎𝑖푟 푀푎푠푏𝑖푟 푀푎푠푐𝑖푟 푇 퐿푠𝑖푟 = 퐿𝑖푟푠 = [푀푏푠푎𝑖푟 푀푏푠푏𝑖푟 푀푏푠푐𝑖푟] A. 5 푀푐푠푎𝑖푟 푀푐푠푏𝑖푟 푀푐푠푐𝑖푟 where stator-inner rotor phase mutual-inductances

푀푎푠푎𝑖푟 = 퐿0푠𝑖푟 cos(휃𝑖푟 ) + 퐿푔푠𝑖푟 cos(2휃푃푀 − 휃𝑖푟 ),

푀 = 퐿 cos (휃 + 2휋) + 퐿 cos (2휃 − 휃 − 2휋), 푎푠푏𝑖푟 0푠𝑖푟 𝑖푟 3 푔푠𝑖푟 푃푀 𝑖푟 3

푀 = 퐿 cos (휃 _ 2휋) + 퐿 cos (2휃 − 휃 + 2휋), 푎푠푐𝑖푟 0푠𝑖푟 𝑖푟 3 푔푠𝑖푟 푃푀 𝑖푟 3

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푀 = 퐿 cos (휃 − 2휋) + 퐿 cos (2휃 − 휃 − 2휋), 푏푠푎𝑖푟 0푠𝑖푟 𝑖푟 3 푔푠𝑖푟 푃푀 𝑖푟 3

푀 = 퐿 cos(휃 ) + 퐿 cos (2휃 − 휃 + 2휋), 푏푠푏𝑖푟 0푠𝑖푟 𝑖푟 푔푠𝑖푟 푃푀 𝑖푟 3

푀 = 퐿 cos (휃 + 2휋) + 퐿 cos(2휃 − 휃 ), 푏푠푐𝑖푟 0푠𝑖푟 𝑖푟 3 푔푠𝑖푟 푃푀 𝑖푟

푀 = 퐿 cos (휃 + 2휋) + 퐿 cos (2휃 − 휃 + 2휋), 푐푠푎𝑖푟 0푠𝑖푟 𝑖푟 3 푔푠𝑖푟 푃푀 𝑖푟 3

푀 = 퐿 cos (휃 − 2휋) + 퐿 cos(2휃 − 휃 ), 푐푠푏𝑖푟 0푠𝑖푟 𝑖푟 3 푔푠𝑖푟 푃푀 𝑖푟

푀 = 퐿 cos(휃 ) + 퐿 cos (2휃 − 휃 − 2휋). 푐푠푐𝑖푟 0푠𝑖푟 𝑖푟 푔푠𝑖푟 푃푀 𝑖푟 3

In (2. 2), 휆푃푀푎푏푐푠 and 휆푃푀푎푏푐𝑖푟 are expressed as follows.

2휋 2휋 푇 휆 = 휆 [cos (휃 ) cos (휃 − ) cos (휃 + )] A. 6 푃푀푎푏푐푠 푃푀표푟 푃푀 푃푀 3 푃푀 3

2휋 2휋 푇 휆 = 휆 [cos (휃 ) cos (휃 − ) cos (휃 + )] A. 7 푃푀푎푏푐𝑖푟 푃푀𝑖푟 푃푀 푃푀 3 푃푀 3

휆 퐿 퐿 푖 휆 Apply A. 1 to (2. 2) [ 푎푏푐푠 ] = [ 푠푠 푠𝑖푟 ] [ 푎푏푐푠 ] + [ 푃푀푎푏푐푠 ] and (2. 3) 휆푎푏푐𝑖푟 퐿𝑖푟푠 퐿𝑖푟𝑖푟 푖푎푏푐𝑖푟 휆푃푀푎푏푐𝑖푟

푣푎푏푐푠 푅푠 0 푖푎푏푐푠 푑 휆푎푏푐푠 [ ] = [ ] [ ] + [ ]; and choose 휃 = 휃푑 = 휃푃푀 and 휃 = 휃푑 − 휃𝑖푟 푣푎푏푐𝑖푟 0 푅𝑖푟 푖푎푏푐𝑖푟 푑푡 휆푎푏푐𝑖푟 for stator and inner rotor, respectively. The following equations can be obtained.

푇 푖푑푞0푠 = 푇(휃푑)푖푎푏푐푠 = 푇(휃푑)[푖푎푠 푖푏푠 푖푐푠] A. 8

푇 푖푑푞0𝑖푟 = 푇(휃푑 − 휃𝑖푟)푖푎푏푐𝑖푟 = 푇(휃푑 − 휃𝑖푟 )[푖푎𝑖푟 푖푏𝑖푟 푖푐𝑖푟] A. 9

푇 휆푃푀푑푞0푠 = 푇(휃푑)휆푃푀푎푏푐푠 = [휆푃푀표푟 0 0] A. 10

푇 휆푃푀푑푞0𝑖푟 = 푇(휃푑 − 휃𝑖푟 )휆푃푀푎푏푐𝑖푟 = [휆푃푀𝑖푟 0 0] A. 11

푇 푣푑푞0푠 = 푇(휃푑)푣푎푏푐푠 = 푇(휃푑)[푣푎푠 푣푏푠 푣푐푠] A. 12

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푇 푣푑푞0𝑖푟 = 푇(휃푑 − 휃𝑖푟 )푣푎푏푐𝑖푟 = 푇(휃푑 − 휃𝑖푟)[푣푎𝑖푟 푣푏𝑖푟 푣푐𝑖푟] A. 13

푇 휆푑푞0푠 = 푇(휃푑)휆푎푏푐푠 = [휆푑푠 휆푞푠 휆0푠]

= 푇(휃푑)(퐿푠푠푖푎푏푐푠 + 퐿푠𝑖푟푖푎푏푐𝑖푟 + 휆푃푀푎푏푐푠) A. 14

= 퐿푑푞0푠푠푖푑푞0푠 + 퐿푑푞0푠𝑖푟푖푑푞0𝑖푟 + 휆푃푀푑푞0푠

푇 휆푑푞0𝑖푟 = 푇(휃푑 − 휃𝑖푟 )휆푎푏푐𝑖푟 = [휆푑𝑖푟 휆푞𝑖푟 휆0𝑖푟 ]

= 푇(휃푑 − 휃𝑖푟 )(퐿𝑖푟푠푖푎푏푐푠 + 퐿𝑖푟𝑖푟푖푎푏푐𝑖푟 + 휆푃푀푎푏푐𝑖푟 ) A. 15

= 퐿푑푞0푠𝑖푟 푖푑푞0푠 + 퐿푑푞0𝑖푟𝑖푟푖푑푞0𝑖푟 + 휆푃푀푑푞0𝑖푟

Hence, a set of new inductance can be obtained. Note that 휃푑 = 휃푃푀.

퐿 퐿 퐿 = [ 푑푞0푠푠 푑푞0푠𝑖푟 ] 퐷푄0 퐿 퐿 A. 16 푑푞0𝑖푟푠 푑푞0𝑖푟𝑖푟 6푋6 where

퐿푑푠 0 0 −1 퐿푑푞0푠푠 = 푇(휃푑)퐿푠푠푇 (휃푑) = [ 0 퐿푞푠 0 ] A. 17 0 0 퐿0푠

⇒ 퐿 = 3 퐿 + 퐿 + 3 퐿 , 퐿 = 3 퐿 + 퐿 − 3 퐿 , 퐿 = 퐿 . 푑푠 2 0푠 푙푠 2 푔푠 푞푠 2 0푠 푙푠 2 푔푠 0푠 푙푠

퐿푑𝑖푟 0 0 −1 퐿푑푞0𝑖푟𝑖푟 = 푇(휃푑 − 휃𝑖푟 )퐿𝑖푟𝑖푟푇 (휃푑 − 휃𝑖푟 ) = [ 0 퐿푞𝑖푟 0 ] A. 18 0 0 퐿0𝑖푟

⇒ 퐿 = 3 퐿 + 퐿 + 3 퐿 , 퐿 = 3 퐿 + 퐿 − 3 퐿 , 퐿 = 퐿 . 푑𝑖푟 2 0𝑖푟 푙𝑖푟 2 푔𝑖푟 푞𝑖푟 2 0𝑖푟 푙𝑖푟 2 푔𝑖푟 0𝑖푟 푙𝑖푟

퐿푑푠𝑖푟 0 0 푇 −1 퐿푑푞0푠𝑖푟 = 퐿푑푞0𝑖푟푠 = 푇(휃푑)퐿푠𝑖푟푇 (휃푑 − 휃𝑖푟) = [ 0 퐿푞푠𝑖푟 0] A. 19 0 0 0

⇒ 퐿 = 3 퐿 + 3 퐿 , 퐿 = 3 퐿 − 3 퐿 . 푑푠𝑖푟 2 0푠𝑖푟 2 푔푠𝑖푟 푞𝑖푟 2 0푠𝑖푟 2 푔푠𝑖푟

Differential terms in voltage equations can be transformed.

118

휔푒푠휆푞푠 푑휆푎푏푐푠 푑휆푑푞0푠 푇(휃푑) = − [−휔 휆 ] = 푞1 A. 20 푑푡 푑푡 푒푠 푑푠 0

푆𝑖푟 휔푒푠휆푞𝑖푟 푑휆푎푏푐𝑖푟 푑휆푑푞0𝑖푟 푇(휃푑 − 휃𝑖푟 ) = − [−푆 휔 휆 ] = 푞2 A. 21 푑푡 푑푡 𝑖푟 푒푠 푑𝑖푟 0

Hence, the d-q machine model can be expressed as follows:

휆푑푞0푠 푖푑푞0푠 휆푃푀푑푞0푠 [ ] = 퐿퐷푄0 [ ] + [ ] A. 22 휆푑푞0𝑖푟 푖푑푞0𝑖푟 휆푃푀푑푞0𝑖푟

푣푑푞0푠 푅푠푖푑푞0푠 푞1 [ ] = [ ] + [ ] A. 23 푣푑푞0𝑖푟 푅𝑖푟푖푑푞0𝑖푟 푞2

Then (2. 4) - (2. 7) can be obtained by expanding A. 22 and A. 23.

Electromagnetic torque of outer rotor and inner rotor

푇 푃표푙푒푠 푑푊표푟 푃표푙푒푠 푖푎푏푐푠 휕퐿푠𝑖푟 푖푎푏푐푠 푇푒표푟 = = [ ] [ ] 2 푑휃표푟 4 푖푎푏푐𝑖푟 휕휃표푟 푖푎푏푐𝑖푟

푃표푙푒푠 3 A. 24 = [휆 푖 + 휆 푖 + (퐿 − 퐿 )(푖 푖 + 푖 푖 ) 2 2 푃푀𝑖푟 푞𝑖푟 푃푀표푟 푞푠 푑푠𝑖푟 푞푠𝑖푟 푞푠 푑𝑖푟 푑푠 푞𝑖푟

+(퐿푑푠 − 퐿푞푠)푖푞푠푖푑푠 + (퐿푑𝑖푟 − 퐿푞𝑖푟)푖푞𝑖푟푖푑𝑖푟]

푇 푃표푙푒푠 푑푊𝑖푟 푃표푙푒푠 푖푎푏푐푠 휕퐿푠𝑖푟 푖푎푏푐푠 푇푒𝑖푟 = = [ ] [ ] 2 푑휃𝑖푟 4 푖푎푏푐𝑖푟 휕휃𝑖푟 푖푎푏푐𝑖푟

푃표푙푒푠 3 A. 25 = − [휆 푖 + (퐿 − 퐿 )푖 푖 + 퐿 푖 푖 2 2 푃푀𝑖푟 푞𝑖푟 푑𝑖푟 푞𝑖푟 푞𝑖푟 푑𝑖푟 푑푠𝑖푟 푑푠 푞𝑖푟

− 퐿푞푠𝑖푟 푖푞푠푖푑𝑖푟]

119

Appendix B: Model Derivation of the SCDMP Machine

Inductances of the three-phase machine model of the SCDMP machine are presented.

The inductances matrix in (3. 1) is specified as follows.

퐿푎푠푎푠 푀푎푠푏푠 푀푎푠푐푠 퐿푠푠 = [푀푏푠푎푠 퐿푏푠푏푠 푀푏푠푐푠] B. 1 푀푐푠푎푠 푀푐푠푏푠 퐿푐푠푐푠 where stator phase self-inductances 퐿푎푠푎푠 = 퐿푏푠푏푠 = 퐿푐푠푐푠 = 퐿0푠 + 퐿푙푠 and mutual inductances 푀푎푠푏푠 = 푀푎푠푐푠 = 푀푏푠푎푠 = 푀푏푠푐푠 = 푀푐푠푎푠 = 푀푐푠푏푠 = −0.5퐿0푠.

퐿푎표푟푎표푟 푀푎표푟푏표푟 푀푎표푟푐표푟 퐿표푟표푟 = [푀푏표푟푎표푟 퐿푏표푟푏표푟 푀푏표푟푐표푟 ] B. 2 푀푐표푟푎표푟 푀푐표푟푏표푟 퐿푐표푟푐표푟 where outer rotor phase self-inductances 퐿푎표푟푎표푟 = 퐿푏표푟푏표푟 = 퐿푐표푟푐표푟 = 퐿0표푟 + 퐿푙표푟 and mutual-inductances 푀푎표푟푏표푟 = 푀푎표푟푐표푟 = 푀푏표푟푎표푟 = 푀푏표푟푐표푟 = 푀푐표푟푎표푟 = 푀푐표푟푏표푟 =

−0.5퐿0표푟.

퐿푎𝑖푟푎𝑖푟 푀푎𝑖푟푏𝑖푟 푀푎𝑖푟푐𝑖푟 퐿𝑖푟𝑖푟 = [푀푏𝑖푟푎𝑖푟 퐿푏𝑖푟푏𝑖푟 푀푏𝑖푟푐𝑖푟 ] B. 3 푀푐𝑖푟푎𝑖푟 푀푐𝑖푟푏𝑖푟 퐿푐𝑖푟푐𝑖푟 where inner rotor phase self-inductances 퐿푎𝑖푟푎𝑖푟 = 퐿푏𝑖푟푏𝑖푟 = 퐿푐𝑖푟푐𝑖푟 = 퐿0𝑖푟 + 퐿푙𝑖푟 and mutual inductances 푀푎𝑖푟푏𝑖푟 = 푀푎𝑖푟푐𝑖푟 = 푀푏𝑖푟푎𝑖푟 = 푀푏𝑖푟푐𝑖푟 = 푀푐𝑖푟푎𝑖푟 = 푀푐𝑖푟푏𝑖푟 =

−0.5퐿0𝑖푟.

120

푀푎푠푎표푟 푀푎푠푏표푟 푀푎푠푐표푟 푇 퐿푠표푟 = 퐿표푟푠 = [푀푏푠푎표푟 푀푏푠푏표푟 푀푏푠푐표푟] B. 4 푀푐푠푎표푟 푀푐푠푏표푟 푀푐푠푐표푟 where stator-outer rotor phase mutual-inductances 푀푎푠푎표푟 = 푀푏푠푏표푟 = 푀푐푠푐표푟 =

퐿 cos(휃 ), 푀 = 푀 = 푀 = 퐿 cos (휃 + 2휋), 푀 = 푀 = 0푠표푟 표푟 푎푠푏표푟 푏푠푐표푟 푐푠푎표푟 0푠표푟 표푟 3 푎푠푐표푟 푏푠푎표푟

푀 = 퐿 cos (휃 − 2휋). 푐푠푏표푟 0푠표푟 표푟 3

푀푎푠푎𝑖푟 푀푎푠푏𝑖푟 푀푎푠푐𝑖푟 푇 퐿푠𝑖푟 = 퐿𝑖푟푠 = [푀푏푠푎𝑖푟 푀푏푠푏𝑖푟 푀푏푠푐𝑖푟] B. 5 푀푐푠푎𝑖푟 푀푐푠푏𝑖푟 푀푐푠푐𝑖푟 where stator-inner rotor phase mutual-inductances 푀푎푠푎𝑖푟 = 푀푏푠푏𝑖푟 = 푀푐푠푐𝑖푟 =

퐿 cos(휃 ) , 푀 = 푀 = 푀 = 퐿 cos (휃 + 2휋) , 푀 = 푀 = 0푠𝑖푟 𝑖푟 푎푠푏𝑖푟 푏푠푐𝑖푟 푐푠푎𝑖푟 0푠𝑖푟 𝑖푟 3 푎푠푐𝑖푟 푏푠푎𝑖푟

푀 = 퐿 cos (휃 − 2휋). 푐푠푏𝑖푟 0푠𝑖푟 𝑖푟 3

푀푎표푟푎𝑖푟 푀푎표푟푏𝑖푟 푀푎표푟푐𝑖푟 푇 퐿표푟𝑖푟 = 퐿𝑖푟표푟 = [푀푏표푟푎𝑖푟 푀푏표푟푏𝑖푟 푀푏표푟푐𝑖푟 ] B. 6 푀푐표푟푎𝑖푟 푀푐표푟푏𝑖푟 푀푐표푟푐𝑖푟 where outer rotor-inner rotor phase mutual-inductances 푀푎표푟푎𝑖푟 = 푀푏표푟푏𝑖푟 = 푀푐표푟푐𝑖푟 =

퐿 cos(휃 ) , 푀 = 푀 = 푀 = 퐿 cos (휃 + 2휋) , 푀 = 0𝑖푟표푟 𝑖표 푎표푟푏𝑖푟 푏표푟푐𝑖푟 푐표푟푎𝑖푟 0𝑖푟표푟 𝑖표 3 푎표푟푐𝑖푟

푀 = 푀 = 퐿 cos (휃 − 2휋). (휃 = 휃 − 휃 ). 푏표푟푎𝑖푟 푐표푟푏𝑖푟 0𝑖푟표푟 𝑖표 3 𝑖표 𝑖푟 표푟

Apply A. 1 to (3. 2) and (3. 3), the following equations can be obtained.

푇 휆푑푞0푠 = 푇(휃푑)[휆푎푠 휆푏푠 휆푐푠] B. 7

푇 휆푑푞0표푟 = 푇(휃푑 − 휃표푟)[휆푎표푟 휆푏표푟 휆푐표푟] B. 8

푇 휆푑푞0𝑖푟 = 푇(휃푑 − 휃𝑖푟 )[휆푎𝑖푟 휆푏𝑖푟 휆푐𝑖푟] B. 9

푇 푖푑푞0푠 = 푇(휃푑)[휆푎푠 휆푏푠 휆푐푠] B. 10

121

푇 푖푑푞0표푟 = 푇(휃푑 − 휃표푟)[푖푎표푟 푖푏표푟 푖푐표푟] B. 11

푇 푖푑푞0𝑖푟 = 푇(휃푑 − 휃𝑖푟 )[푖푎𝑖푟 푖푏𝑖푟 푖푐𝑖푟] B. 12

푇 푣푑푞0푠 = 푇(휃푑)[푣푎푠 푣푏푠 푣푐푠] B. 13

푇 푣푑푞0표푟 = 푇(휃푑 − 휃표푟)[푣푎표푟 푣푏표푟 푣푐표푟] B. 14

푇 푣푑푞0𝑖푟 = 푇(휃푑 − 휃𝑖푟 )[푣푎𝑖푟 푣푏𝑖푟 푣푐𝑖푟] B. 15

Hence, a set of new inductance can be obtained.

퐿푑푞0푠푠 퐿푑푞0푠표푟 퐿푑푞0푠𝑖푟 퐿퐷푄0 = [퐿푑푞0표푟푠 퐿푑푞0표푟표푟 퐿푑푞0표푟𝑖푟] B. 16 퐿 퐿 퐿 푑푞0𝑖푟푠 푑푞0𝑖푟표푟 푑푞0𝑖푟𝑖푟 9푋9 where

퐿푑푠푠 0 0 −1 퐿푑푞0푠푠 = 푇(휃푑)퐿푠푠푇 (휃푑) = [ 0 퐿푞푠푠 0 ] B. 17 0 0 퐿0푠푠

⇒ 퐿 = 퐿 = 퐿 = 3 퐿 + 퐿 , 퐿 = 퐿 . 푑푠푠 푞푠푠 푠 2 0푠 푙푠 0푠푠 푙푠

퐿 = 푇(휃 − 휃 )퐿 푇−1(휃 − 휃 ) ⇒ 퐿 = 퐿 = 퐿 = 3 퐿 + 푑푞0표푟표푟 푑 표푟 표푟표푟 푑 표푟 푑표푟표푟 푞표푟표푟 표푟 2 0표푟

퐿푙표푟, 퐿0표푟 = 퐿푙표푟.

퐿 = 푇(휃 − 휃 )퐿 푇−1(휃 − 휃 ) ⇒ 퐿 = 퐿 = 퐿 = 3 퐿 + 푑푞0𝑖푟𝑖푟 푑 𝑖푟 𝑖푟𝑖푟 푑 𝑖푟 푑𝑖푟𝑖푟 푞𝑖푟𝑖푟 𝑖푟 2 0𝑖푟

퐿푙𝑖푟, 퐿0𝑖푟 = 퐿푙𝑖푟.

퐿 = 퐿푇 = 푇(휃 )퐿 푇−1(휃 − 휃 ) ⇒ 퐿 = 퐿 = 푀 = 3 퐿 . 푑푞0푠표푟 푑푞0푠표푟 푑 푠표푟 푑 표푟 푑푠표푟 푞푠표푟 푠표푟 2 0푠표푟

퐿 = 퐿푇 = 푇(휃 )퐿 푇−1(휃 − 휃 ) ⇒ 퐿 = 퐿 = 푀 = 3 퐿 . 푑푞0푠𝑖푟 푑푞0푠𝑖푟 푑 푠𝑖푟 푑 𝑖푟 푑푠𝑖푟 푞푠𝑖푟 푠𝑖푟 2 0푠𝑖푟

푇 −1 퐿푑푞0𝑖푟표푟 = 퐿푑푞0표푟𝑖푟 = 푇(휃푑 − 휃𝑖푟 )퐿푠𝑖푟푇 (휃푑 − 휃표푟) ⇒ 퐿푑𝑖푟표푟 = 퐿푞표푟𝑖푟 = 푀𝑖푟표푟 =

3 퐿 . 2 0𝑖푟표푟 122

Differential terms in voltage equations can be transformed.

휔푒푠휆푞푠 푑휆푎푏푐푠 푑휆푑푞0푠 푇(휃푑) = − [−휔 휆 ] = 푚1 B. 18 푑푡 푑푡 푒푠 푑푠 0

푆표푟휔푒푠휆푞표푟 푑휆푎푏푐표푟 푑휆푑푞0표푟 푇(휃푑 − 휃표푟) = − [−푆 휔 휆 ] = 푚2 B. 19 푑푡 푑푡 표푟 푒푠 푑표푟 0

푆𝑖푟휔푒푠휆푞𝑖푟 푑휆푎푏푐𝑖푟 푑휆푑푞0𝑖푟 푇(휃푑 − 휃𝑖푟) = − [−푆 휔 휆 ] = 푚3 푑푡 푑푡 𝑖푟 푒푠 푑𝑖푟 B. 20 0

Hence, the d-q machine model can be expressed as follows:

휆푑푞0푠 푖푑푞0푠

[휆푑푞0표푟] = 퐿퐷푄0 [푖푑푞0표푟] B. 21 휆푑푞0𝑖푟 푖푑푞0𝑖푟

푣푑푞0푠 푅푠푖푑푞0푠 푚1 [푣푑푞0표푟] = [푅표푟푖푑푞0표푟] + [푚2] B. 22 푣 푑푞0𝑖푟 푅𝑖푟 푖푑푞0𝑖푟 푚3

Electromagnetic torque of outer rotor and inner rotor

푇 푖푎푏푐푠 푖푎푏푐푠 푃표푙푒푠 푑푊표푟 푃표푙푒푠 휕퐿푠표푟𝑖푟 푇푒표푟 = = [푖푎푏푐표푟] [푖푎푏푐표푟] 2 푑휃표푟 4 푖 휕휃표푟 푖 푎푏푐𝑖푟 푎푏푐𝑖푟 B. 23 3 푃표푙푒푠 = (휆 푖 − 휆 푖 ) 2 2 푞표푟 푑표푟 푑표푟 푞표푟

푇 푖푎푏푐푠 푖푎푏푐푠 푃표푙푒푠 푑푊𝑖푟 푃표푙푒푠 휕퐿푠표푟𝑖푟 푇푒𝑖푟 = = [푖푎푏푐표푟] [푖푎푏푐표푟] 2 푑휃𝑖푟 4 푖 휕휃𝑖푟 푖 푎푏푐𝑖푟 푎푏푐𝑖푟 B. 24 3 푃표푙푒푠 = (휆 푖 − 휆 푖 ) 2 2 푞𝑖푟 푑𝑖푟 푑𝑖푟 푞𝑖푟

123

Appendix C: Parameter Measurement of the SCDMP Machine

Compared with a SCIM or a DFIM, the SCDMP machine has more parameters. The experimental method to obtain the resistances ( 푅푠, 푅표푟, 푅𝑖푟 ) and the inductances

(퐿푙푠, 퐿푙표푟, 퐿푙𝑖푟, 푀푙푠, 푀푙𝑖푟 , 푀푠𝑖푟) of the SCDMP machine will be explained in this appendix.

It should be pointed out the proposed procedure is based on the parameter measurement methods for a SCIM and a DFIM. In other words, it is assumed that the parameters of a conventional SCIM and a conventional DFIM can be readily obtained following existing procedures. For convenience of discussion, the steady state equivalent circuit of the SCDMP machine, Figure C. 1, is shown below again.

Figure C. 1 Steady state equivalent circuit of the SCDMP machine

As discussed in 3.1.3.C, the SCDMP machine can be treated as a SCIM when the inner rotor windings are open. The corresponding circuit, Figure C. 2, is also shown below.

124

It is clear that 푅푠, 푅표푟, 퐿푙푠, (퐿푙표푟 + 푀푙𝑖푟), and (푀푠𝑖푟 + 푀푙푠), can be obtained since the

SCDMP machine has degraded to a conventional SCIM when the inner rotor windings are open.

Figure C. 2 SCDMP machine operated as squirrel-cage induction machine.

When the outer rotor is controlled to rotate at synchronous speed by an external mover, the SCDMP machine degrades to a DFIM (Figure C. 3). Hence, 푅푠 , 푅𝑖푟 , (퐿푙푠 + 푀푙푠 ),

(퐿푙𝑖푟 + 푀푙𝑖푟), and 푀푠𝑖푟, can be obtained.

Figure C. 3 SCDMP machine operated as doubly-fed induction machine.

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As shown in Figure C. 4, when the stator windings are open and the inner rotor is locked (standstill), the SCDMP machine degrades to a SCIM again. Hence, 푅표푟 , 푅𝑖푟 ,

(퐿푙표푟 + 푀푙푠), 퐿푙𝑖푟, and (푀푠𝑖푟 + 푀푙𝑖푟 ) can be obtained.

Llor i Ror or Rir Sor M ls Llir Sir v + M iir ir sir M S lir ir-

Ror Rir

Sor Llor M ls Llir Sir

v + ior iir ir

MMsir lir Sir -

Figure C. 4 Equivalent SCIM when stator windings open.

Table C. 1 summarizes the proposed method to obtain the resistances and inductances of the SCDMP machine.

Table C. 1 Measurement of SCDMP machine parameters.

Machine Status Method Measured Parameters Inner rotor windings open. Equivalent SCIM 푅푠, 푅표푟, 퐿푙푠, (퐿푙표푟 + 푀푙𝑖푟), (푀푠𝑖푟 + 푀푙푠) Outer rotor rotates at Equivalent DFIM 푅푠, 푅𝑖푟, (퐿푙푠 + 푀푙푠), (퐿푙𝑖푟 + 푀푙𝑖푟), synchronous speed 푀푠𝑖푟 Stator windings open, Equivalent SCIM 푅표푟, 푅𝑖푟, (퐿푙표푟 + 푀푙푠), 퐿푙𝑖푟, inner rotor is locked. (푀푠𝑖푟 + 푀푙𝑖푟 )

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It is shown in Table C. 1 that the resistances 푅푠, 푅표푟, 푅𝑖푟 and the inductances

퐿푙푠, 퐿푙𝑖푟, 푀푠𝑖푟 can be measured by either equivalent SCIM or equivalent DFIM method.

Then 푀푙푠 and 푀푙𝑖푟 can be calculated because (푀푠𝑖푟 + 푀푙푠 ), (푀푠𝑖푟 + 푀푙𝑖푟 ) and 푀푠𝑖푟 are known. Last, 퐿푙표푟 can be calculated since 푀푙푠 and (퐿푙표푟 + 푀푙푠) are known.

Thus, all resistances (푅푠, 푅표푟, 푅𝑖푟) and inductances (퐿푙푠, 퐿푙표푟, 퐿푙𝑖푟, 푀푙푠, 푀푙𝑖푟, 푀푠𝑖푟) of the SCDMP machine can be obtained by existing parameter measurement methods for a

SCIM and a DFIM.

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