MODELING AND CONTROL OF DUAL MECHANICAL PORT ELECTRIC MACHINE DISSERTATION

Transcription

1 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 By 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

2 Copyright by Haiwei Cai 2015

3 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

4 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. iii

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

6 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. v

7 Vita September 2006 July B.S. Electrical Engineering & Automation, South China University of Technology, Guangzhou, China September 2010 August Master s Student, Graduate Fellow, The Ohio State University, Columbus, Ohio May 2015 August 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 , Sept 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 , Sept 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 , Oct 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 Sept vi

8 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 , Sept 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): 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, 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 , Sept 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 , Sept Fields of Study Major Field: Electrical and Computer Engineering vii

9 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 Background of research Literature review...4 Chapter 2: Permanent Magnet Dual Mechanical Port Electric Machine Modeling of PMDMP machine viii

10 2.1.1 Introduction of PMDMP machine Three phase model of PMDMP machine Model in synchronous reference frame Design of PMDMP machine Simplified machine model PMDMP machine design flowchart Dimensions, parameters and specifications of prototype PMDMP machine Open circuit test result of prototype PMDMP machine Control of PMDMP machine Field oriented control Position sensorless control of outer rotor Operation modes of PMDMP machine Power flow analysis Multi-operational modes of the PMDMP machine Simulation of the multi-operation modes Chapter 3: Squirrel-Cage Dual Mechanical Port Electric Machine Modeling of SCDMP machine Three phase model of SCDMP machine Model in synchronous reference frame ix

11 3.1.3 Model in stationary reference frame Validation of the proposed model by finite element method Independent control of the two rotors of SCDMP machine Introduction of SCDMP machine control algorithm Current command calculation module Outer rotor flux observer Simulation of the proposed FOC algorithm Operational modes of the SCDMP machine Power flow analysis of the SCDMP machine Simplified driving cycle of hybrid vehicle Chapter 4: Conclusions and Future Work Conclusions Future work References Appendix A: Model Derivation of the PMDMP Machine Appendix B: Model Derivation of the SCDMP Machine Appendix C: Parameter Measurement of the SCDMP Machine x

12 List of Tables Table 2. 1 Dimensions of the prototype PMDMP machine Table 2. 2 Open circuit test result of the prototype PMDMP machine Table 2. 3 Flux linkage of the prototype PMDMP machine Table 2. 4 Power relationships of the constant power mode of the PMDMP machine Table 3. 1 Specifications of sample SCDMP machine Table 3. 2 Validation of the proposed steady state machine model by FEM Table 3. 3 Rated fluxes and inudctances of the sample SCDMP machine Table C. 1 Measurement of SCDMP machine parameters xi

13 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) Figure 2. 2 Components of PMDMP machine Figure 2. 3 Conceptual cross-section of double-layer PMDMP machine Figure 2. 4 Flux line distribution of the PMDMP machine with stator current alone Figure 2. 5 Flux line distribution of the PMDMP machine with inner rotor current alone Figure 2. 6 The d-axis equivalent circuit of the PMDMP machine Figure 2. 7 The q-axis equivalent circuit of the PMDMP machine Figure 2. 8 Simplified d-axis equivalent circuit of the PMDMP machine Figure 2. 9 Simplified q-axis equivalent circuit of the PMDMP machine Figure Design flowchart for the PMDMP machine xii

14 Figure Cross-section of the prototype PMDMP machine Figure Open circuit flux line distribution of the prototype machine Figure Open circuit flux density distribution of the prototype machine Figure Stator winding line to line back EMF and harmonic analysis Figure Inner rotor winding line to line back EMF and harmonic analysis Figure Control block diagram for the PMDMP machine Figure Matlab model of the PMDMP machine and controller Figure Actual speed (black) and reference speed (red) of the DMP machine Figure Torque production of the DMP machine Figure Phase current waveforms of the DMP machine Figure Test setup for sensorless control of PMDMP machine Figure Effect of high frequency boltage on salient pole PMSM Figure Procedures to extract rotor position from stator currents Figure Polarity identification based on magnetic saturation Figure Block diagram for outer rotor sensorless control at zero and low speeds Figure Simulation result of high frequency injection rotor position estimation Figure Experiment result of high frequency injection rotor position estimation Figure Saturation function for SMO Figure Block diagram for the rotor position SMO Figure Block diagram for outer rotor sliding mode sensorless control at medium and high speeds Figure Simulation result of sliding mode rotor position observer xiii

15 Figure Experimental result of sliding mode rotor position observer Figure Constant power mode of the HEV based on PMDMP machine Figure Power flow of the constant power mode of the PMDMP machine Figure Low power mode of the HEV based on PMDMP machine Figure High power mode of the HEV based on PMDMP machine Figure Pure electric vehicle mode of the HEV based on DMP machine Figure Highway (pure ICE) mode of the HEV based on PMDMP machine Figure Reference and actual rotor speeds in different operational modes Figure Torque productions of both rotors in different operational modes Figure Current waveforms of stator and inner rotor windings in different modes Figure 3. 1 Conceptual cross-section of the SCDMP machine Figure 3. 2 Reference frames for the SCDMP machine Figure 3. 3 Flux line distribution of the SCDMP machine with stator current alone Figure 3. 4 Flux line distribution of the SCDMP machine with inner rotor current alone Figure 3. 5 The d-axis equivalent circuit of the SCDMP machine Figure 3. 6 The q-axis equivalent circuit of the SCDMP machine Figure 3. 7 Steady state equivalent circuit of the SCDMP machine Figure 3. 8 SCDMP machine operated as squirrel-cage induction machine Figure 3. 9 SCDMP machine operated as doubly-fed induction machine Figure Cross-section of the sample SCDMP machine xiv

16 Figure Comparison between transient responses of proposed model (left) and FEM model (right) Figure Reference frames for the SCDMP machine Figure Current command calculation module of the SCDMP machine Figure Proposed outer rotor flux observer Figure Control block diagram of the SCDMP machine Figure Flux, speed and torque production of the SCDMP machine Figure Current of the SCDMP machine Figure Comparison between estimated and actual outer rotor flux angles Figure Simplified driving cycle of hybrid vehicle Figure C. 1 Steady state equivalent circuit of the SCDMP machine Figure C. 2 SCDMP machine operated as squirrel-cage induction machine Figure C. 3 SCDMP machine operated as doubly-fed induction machine Figure C. 4 Equivalent SCIM when stator windings open xv

17 Nomenclature Subscripts s, or, ir l, m a, b, c d, q, 0 α, β, γ Stator, outer and inner rotor variables, respectively. Leakage and mutual inductance indicator, respectively. Phases a, b, c variables, respectively. Variables in d-axis, q-axis, 0-axis, respectively. Variables in α-axis, β-axis, γ-axis, respectively. dq0 Column vector in [d q 0] T format. For example: λ dq0s = [λ ds λ qs λ 0s ] T. abc Column vector in [a b c] T format. For example: λ abcs = [λ as λ bs λ cs ] T. Greek symbols v, λ, i, R Voltage, flux linkage, current and resistance, respectively. P, T, ω, J Power, torque, angular speed and inertia of rotor, respectively. λ PMor λ PMir θ PM θ d Stator flux linkage provided by outer rotor permanent magnets. Inner rotor flux linkage provided by outer rotor permanent magnets. Electrical angle between permanent magnet flux and stator phase a-axis. Electrical angle between d-axis and stator phase a-axis. xvi

18 θ io Electrical angle between outer rotor phase a-axis and inner rotor phase a- axis. θ ir θ or Electrical angle between inner rotor phase a-axis and stator phase a-axis. Electrical angle between outer rotor phase a-axis and stator phase a-axis. Abbreviations DFIM DMP EV EVT FEM FOC HEV HFI ICE ID/OD PM PMSM Doubly Fed Induction Machine Dual Mechanical Port Electric Vehicle Electric Variable Transmission Finite Element Method Field Oriented Control Hybrid Electric Vehicle High Frequency Injection Internal Combustion Engine Inner Diameter/ Outer Diameter Permanent Magnet Permanent Magnet Synchronous Machine PMDMP Permanent Magnet Dual Mechanical Port SCDMP SCIM SMO Squirrel Cage Dual Mechanical Port Squirrel Cage Induction Machine Sliding Mode Observer xvii

19 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 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

20 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 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. 2

21 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. 3

22 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 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 rings and brushes Outer Rotor Figure 1. 3 Structure of the DMP machine. 4

23 As shown in Figure 1. 4, the generator-motor-planetary gear set in the existing powersplit 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: Inner Rotor ICE Mechanical Port 2: Outer Rotor - Wheels Gasoline ICE Wheels Electrical port 1: Inner rotor windings & slip rings- Inverter 1 Inverter Electrical port 2: Stator windings Inverter 2 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

24 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 of PMs One Layers of PMs Two Layers of 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]. 6

25 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

26 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

27 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 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. 9

28 Chapter 2: Permanent Magnet Dual Mechanical Port Electric Machine 2.1 Modeling of PMDMP machine 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

29 11 Figure 2. 2 Components of PMDMP machine. 11

30 2.1.2 Three phase model of PMDMP machine The cross-section of the double layer PMDMP machine is shown in Figure 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. Magnets a s axis or es Poles /2 d axis ir air axis a ir N ir d PM bs S b ir c ir cs a axis ( axis) s Figure 2. 3 Conceptual cross-section of double-layer PMDMP machine. 12

31 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 selfinductance. 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. 13

32 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. L sir L abcsir = [ L ss L irs L irir ]6X6 (2. 1) [ λ abcs λ abcir ] = L abcsir [ i abcs i abcir ] + [ λ PMabcs λ PMabcir ] (2. 2) [ v abcs v abcir ] = [ R s 0 0 R ir ] [ i abcs i abcir ] + d dt [ λ abcs λ abcir ] (2. 3) 14

33 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. { λ ds = L ds i ds + L dsir i dir + λ PMor λ qs = L qs i qs +L qsir i qir { λ dir = L dir i dir + L dsir i ds + λ PMir λ qir = L qir i qir + L qsir i qs v ds = R s i ds + dλ ds { dt ω esλ qs v qs = R s i qs + dλ qs dt + ω esλ ds (2. 4) (2. 5) (2. 6) v dir = R ir i dir + dλ dir S { dt ir ω es λ qir v qir = R ir i qir + dλ qir + S dt ir ω es λ dir (2. 7) T eir = Poles [λ qiri dir λ dir i qir ] = Poles [λ PMiri qir + (L dir L qir )i qir i dir + L dsir i ds i qir L qsir i qs i dir ] (2. 8) T eor = Poles [λ PMiri qir + λ PMor i qs + (L dsir L qsir )(i qs i dir + i ds i qir ) (2. 9) +(L ds L qs )i qs i ds + (L dir L qir )i qir i dir ] 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

34 Note that S ir = 1 ω or ω ir is the slip percentage of the inner rotor. ω or and ω ir are mechanical angular speeds of the outer and the inner rotors, respectively. + v - ds R s i L L S es qs ds dsir Ldir Ldsir ir es qir i ds dir L dsir Figure 2. 6 The d-axis equivalent circuit of the PMDMP machine. R ir + v dir - + v qs - R s i qs es ds L qs L qsir L qsir i qir Figure 2. 7 The q-axis equivalent circuit of the PMDMP machine. L qir L qsir S ir es dir R ir + v qir - 16

35 2.2 Design of PMDMP machine 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 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. + v - R s ds i es qs L ds L - dir Sir es qir ds i dir Figure 2. 8 Simplified d-axis equivalent circuit of the PMDMP machine. R ir + v dir - 17

36 + v qs - R s i qs es ds L qs L qir S ir es dir i qir Rir + v qir - Figure 2. 9 Simplified q-axis equivalent circuit of the PMDMP machine. 18

37 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 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. 19

38 Figure Design flowchart for the PMDMP machine. 20

39 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 Table 2. 1 Dimensions of the prototype PMDMP machine. Poles 6 Stack length mm 70.0 Stator OD mm Slot area 75 mm 2 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 mm 2 Inner rotor ID mm 34.2 Outer PM width mm 2.9 3PMs/ pole Outer PM length mm PMs/ pole Inner PM width mm PM/pole Inner PM Radian rad deg/pole λ PMor Web λ PMir Web The cross-section of the prototype machine is shown in Figure The open circuit flux line and flux density distributions of the prototype machine is shown in Figure and Figure 2. 13, respectively. 21

40 Figure Cross-section of the prototype PMDMP machine. Figure Open circuit flux line distribution of the prototype machine. 22

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

42 2.2.4 Open circuit test result of prototype PMDMP machine. Test result of the resistance and back EMF are summarized in Table 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 R as 0.08 Ω Inner rotor phase resistance R air 0.29/0.40 Ω Stator d-axis inductance L ds 1.15 mh Stator q-axis inductance L qs 1.85 mh Inner rotor d-axis inductance L dir 1.5 mh Inner rotor q-axis inductance L qir 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 and Figure 2. 15, respectively. 24

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

44 Figure 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

45 Table 2. 3 Flux linkage of the prototype PMDMP machine. Harmonic order Stator PM flux linkage [Wb] Inner rotor PM flux linkage [Wb]

46 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 The mathematical model of the machine can still be expressed by (2. 4) - (2. 9), except L dsir = 0 and L qsir = 0. Besides, Table 2. 2 shows the inner PM machine is a non-salient machine because L dir = L qir. Hence, the machine model can be simplified as (2. 10) - (2. 17). { λ ds = L ds i ds + λ PMor λ qs = L qs i qs { λ dir = L dir i dir + λ PMir λ qir = L dir i qir v ds = R s i ds + dλ ds { dt ω esλ qs v qs = R s i qs + dλ qs dt + ω esλ ds (2. 10) (2. 11) (2. 12) v dir = R ir i dir + dλ dir S { dt ir ω es λ qir v qir = R ir i qir + dλ qir + S dt ir ω es λ dir (2. 13) T eir = Poles λ PMiri qir (2. 14) T eor = Poles [λ PMiri qir + λ PMor i qs + (L ds L qs )i qs i ds ] (2. 15) J or dω or dt J ir dω ir dt = T eor T load or T friction or (2. 16) = T eir T load ir T friction ir (2. 17) 28

47 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 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 The performance of the proposed controller is simulated. The simulation results are shown in Figure Figure 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. 29

48 30 * ir * or * i dir PI * i ds PI ir or * i qir * i qs Speed Cal Speed PI PI PI PI i dir i qir i ds i qs ir abc to dq dq to abc dq to abc abc to dq Cal ( ) d PM d Position Sensor Current Sensor ir PWM PWM Current Sensor Position Sensor i air, i bir 3_Phase Inverter 3_Phase Inverter i as, i bs engine DMP Machine wheel Figure Control block diagram for the PMDMP machine. 30

49 Figure Matlab model of the PMDMP machine and controller. Speed [RPM] Speed [RPM] Outer Rotor Reference and Actual Speed Inner Rotor Reference and Actual Speed Time [s] Figure Actual speed (black) and reference speed (red) of the DMP machine 31

50 Torque [Nm] Torque [Nm] Outer Rotor Electromagnetic Torque Inner Rotor Electromagnetic Torque Time [s] 4 5 Figure Torque production of the DMP machine. Current [A] Current [A] Stator Phase Current Inner Rotor Phase Current Time [s] Figure Phase current waveforms of the DMP machine. 32

51 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 Brushes & Slip Rings Position Sensor Terminals Prototype PMDMP Machine Stator Winding Terminals Inner Rotor Winding Terminals Stator Inverter Rotor Inverter S BUS DC bus R Power System Variac Rectifier Figure 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

52 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 Besides the d-q reference frame, the stationary reference frame, i.e., α β reference frame, is also commonly used in machine modeling. As shown in Figure 2. 3, the α-axis is aligned with the a s -axis. The voltage and flux equations in α β reference frame are very similar to that in d q reference frame. To be more specific, if the subscripts d and q are replaced with α and β, and let ω es = 0, (2. 4) - (2. 9) will then be valid equations in α β reference frame. Hence, the machine equations in α β reference frame are not derived in this work. It should be pointed out that the variables in d q reference frame are constant values (dc) in steady state, but they are sinusoidal values in α β reference frame. 34

53 q axis axis es N d axis q axis axis es N d axis d d S H axis V H S H I H axis Figure Effect of high frequency boltage on salient pole PMSM. As shown in Figure 2. 22, when a high frequency ( ω H ) voltage (V H ) is injected to the machine windings, a high frequency current (I H ) will be generated. Because of the saliency of the rotor, the trajectory of I H on the stationary reference frame (α β) will be an ellipse. The relationship between V H and I H can be expressed by (2. 18). Note that the synchronous frequency components are not shown in (2. 18). L s (2θ d ) is expressed by (2. 19). = [ V H = v αβs = R s i αβs + dl s(2θ d )i αβs dt L s (2θ d ) L ds + L qs + L ds L qs cos(2θ d ) 2 L ds L qs 2 2 sin(2θ d ) L ds + L qs 2 = R s I H + dl s (2θ d )I H dt L ds L qs sin(2θ 2 d ) ] cos(2θ d ) L ds L qs 2 (2. 18) (2. 19) Since R s ω H L s, the first term (R s i αβs ) in (2. 18) can be neglected. Considering V H = V HMag [ sinω Ht ], (2. 20) can be derived. cosω H t 35

54 I H = [L s (2θ d )] 1 V H dt where L ds + L qs = V HMag [ 2 L ds L qs ω H L ds + L qs 2 cosω H t L ds L qs 2 sinω H t L ds L qs 2 cos (2θ d ω H t) ] sin (2θ d ω H t) (2. 20) [L s (2θ d )] 1 L ds + L qs = 1 [ 2 L ds L qs L ds L qs 2 L ds L qs 2 sin(2θ d ) cos(2θ d ) L ds L qs sin(2θ 2 d ) ] cos(2θ d ) L ds + L qs 2 + L ds L qs 2 It is obvious in (2. 20) that the d-axis angle (2θ d ) is carried by the high frequency current. The stator current has three components with different frequencies synchronous frequency ( ω es ), positive sequence high frequency (ω H ) and negative sequence high frequency ( ω H ). Figure shows the mathematical procedures to extract the d-axis angle (2θ d ) from the ω H current component. i abcs dq i s H t Filter H Component dq High Pass Filter 2 H t Filter es Component Low Pass Filter tan 1 2 2k d 0.5 d k Figure Procedures to extract rotor position from stator currents. Note that k in Figure 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

55 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. S N d axis (1) (2) Stator current flux Magnet flux S N d axis v dt v t L i d1 d 2 d pulse pulse d d d 1 i d 2 2 o i d1 i d1 d2 id i d1 d2 Figure Polarity identification based on magnetic saturation. Figure shows the block diagram for the HFI sensorless control with the polarity identification algorithm. 37

56 v pulse Feedback Signals Inner Rotor Control PWM 3_Phase Inverter DMP Machine wheel 38 * or or * i ds PI * i qs V H PI PI i qs i ds ( ) d dq to dq to PM to abc Polarity Identification N / S? HFI Position Estimation PWM abc to 3_Phase Inverter i as, i bs Current Sensor ICE Figure Block diagram for outer rotor sensorless control at zero and low speeds. 38

57 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 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 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. Angle [deg] Angle [deg] Angle [deg] Actual Rotor Position Position [deg] HFI Estimated HFI Rotor Rotor Position [deg] Position Estimation Error [deg] Error Time [s] Figure Simulation result of high frequency injection rotor position estimation. 39

58 Angle [deg] Angle [deg] Angle [deg] Angle [Deg] HFI High Estimated Frequency Injection Rotor Estimated Position Angle Estimation Error HIF Estimation Time [s] Error Time [s] Time [s] Figure 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). where v αs = i αs R s + dλ α { dt = i αsr s + L ds + L qs di αs 2 v βs = i βs R s + dλ β dt = i βsr s + L ds + L qs 2 dt + e α + h α di βs dt + e β + h β (2. 21) h α = L ds L qs 2 h β = L ds L qs { 2 { e α = ω es λ PMout sinθ PM e β = ω es λ PMout cosθ PM [ di αcos (2θ PM ) dt [ di βcos (2θ PM ) dt + di βsin (2θ PM ) ] dt + di αsin (2θ PM ) ] dt (2. 22) (2. 23) 40

59 Applying extended back EMF (e αext and e βext ) theory to the machine model, (2. 21) can be derived as (2. 24). where di αs L qs { dt = v αs i αs R s e αext di βs L qs dt = v βs i βs R s e βext (2. 24) e αext = e α + h α L ds L qs { 2 dt e βext = e β + h β L (2. 25) ds L qs di βs 2 dt Hence, considering (2. 22), (2. 23),(2. 24) and i α = I s cosθ e, i β = I s sinθ e, (2. 26) and (2. 27) can be derived. e αext = ω esλ PMout + (L ds L qs )I s cos(θ PM θ e ) sinθ PM = sinθ PM (2. 26) e βext ω es λ PMout (L ds L qs )I s cos(θ PM θ e ) cosθ PM cosθ PM 41 di αs e α = sinθ PM = e αext (2. 27) e β cosθ PM e βext It is obvious that if the machine parameters, stator currents and voltages are known, e αext and e βext 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 U α /U β is used to estimate the system states as shown in (2. 28). The estimated values are indicated by.

60 L di αs qs { dt = v αs i αs R s + U α (2. 28) di βs L qs dt = v βs i R βs s + U β 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). { S α = i αs i αs (2. 29) S β = i βs i βs ds α dt = R s S L α + 1 (e qs L αext + U α ) qs ds β { dt = R s S L β + 1 (e qs L βext + U β ) qs (2. 30) As expressed by (2. 31), the control input U α and U β are simple functions decided by the signs of S α and S β, respectively. k sld is called the sliding mode gain. If S α /S β doesn t equal to zero, S α /S β will be forced to be zero by the control input U α /U β. { U α = k sld signs α U β = k sld signs β (2. 31) Even though U α and U β are high frequency sign functions, they can be decomposed into two parts as express in (2. 32). { U α = U αlow + U αhigh = e αext + U αhigh (2. 32) U β = U βlow + U βhigh = e βext + U βhigh The low frequency components e αext and e βext 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

61 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. k lpf e αext = U k lpf + ω α cutoff k lpf e βext = U { k lpf + ω β cutoff (2. 33) To verify the existence of the sliding mode, the Lyapunov function (2. 34) is considered. To guarantee the existence of the sliding mode, dy α dt Y α = 1 { 2 S α T S α Y β = 1 (2. 34) 2 S β T S β and dy β dt dy α dt = S ds α T α dt = S R α T s S L α + S 1 T α (e qs L αext + U α ) < 0 qs dy β { dt = S ds β T β dt = S β T R s T S L β + S 1 β (e qs L βext + U β ) < 0 qs should be negative values. (2. 35) Obviously, S α T R s L qs S α 0 and S β T R s L qs S β 0. Hence, if the condition in (2. 36) can be satisfied, the sliding mode exists. { S α T (e αext + U α ) < 0 S β T (e βext + U β ) < 0 (2. 36) If S α > 0, U α = k sld e αext + U α = e αext k sld < 0 e αext < k sld ; If S α < 0, U α = k sld e αext + U α = e αext + k sld > 0 e αext < k sld. k sld > e αext. Similarly, k sld > e βext. The result indicates that the SMO requires high gain k sld to guarantee the tracking of the actual currents. As the rotor speed increases, the back EMF magnitude increases. Hence, high k sld is required at high speed. 43

62 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 U α and U β are replaced by saturation functions as shown in Figure k sld U / U E chat Echat S / S k sld Figure Saturation function for SMO. The block diagram for the SMO to estimate rotor position is summarized in Figure 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

63 v s U ˆ s SMO Current i Estimation i s k sld U / U E chat Echat S / S k sld U Low Pass Filter eˆ s e e 1 s tan ( ) Compensation s PM Figure Block diagram for the rotor position SMO. Feedback Signals Inner Rotor Control PWM 3_Phase Inverter DMP Machine wheel 45 * or or * i ds PI * i qs i ds Speed Cal i qs PI PI PM dq to dq to to abc v s Rotor Position SMO i s PWM abc to 3_Phase Inverter i ICE, i as bs Current Sensor Figure Block diagram for outer rotor sliding mode sensorless control at medium and high speeds. 45

64 The simulation result for the sliding mode sensorless control algorithm is shown in Figure 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 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. 46

65 Angle [deg] Angle [deg] Angle [deg] Actual Actual Rotor Position Position [deg] SMO Estimated SMO Rotor Rotor Position [deg] Position Estimation Error [deg] Error Time [s] Figure Simulation result of sliding mode rotor position observer. Angle [deg] Angle [deg] SMO Estimated Angle SMO Estimation Time [s] Error Time [s] Figure Experimental result of sliding mode rotor position observer. 47

66 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. 48

67 2.4 Operation modes of PMDMP machine Power flow analysis. The PMDMP machine has very high control flexibility because its dual-mechanicalport 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. P es + P eir + P mice P mwheel = 0 (2. 37) P es is the electrical power provided by the stator windings; P eir is the electrical power provided by the inner rotor windings; P mice is the mechanical power provided by the ICE, which is equal to the mechanical power provided by the inner rotor; P mwheel 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 (P mice ) is not likely to be negative, because the ICE cannot consume mechanical power. Considering P es = (T eor + T eir )ω or, P eir = T eir (ω or ω ir ), P mice = T ICE ω ir = T eir ω ir and P mwheel = T eor ω or, (2. 38) can be easily derived from (2. 37). (T eor + T eir )ω or T eir (ω or ω ir ) T eir ω ir T eor ω ir = 0 (2. 38) 49

68 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). P battery = P es + P eir = (T eor + T eir )ω or T eir (ω or ω ir ) (2. 39) = T eor ω or + T eir ω ir = P mwheel P mice 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 ( P battery + P mice = P mwheel ). 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 ω or > ω ir, then P eir = T eir (ω or ω ir ) = T ICE (ω or ω ir ) > 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 (P mice = P mwheel ). Note that power losses are neglected for simplicity of discussion. Thus, (2. 40) is satisfied. P battery = P es + P eir = P mice P mwheel = 0 (2. 40) The ICE sweet spot (P mice ) and the expected output from the wheel (P mwheel ) are indicated in Figure by a dot and a star, respectively. 50

69 Torque Constant Power Curve T ICE T eir T eor ICE Sweet Spot 3 4 ( ) ir P mice ICE or P mwheel Speed Figure Constant power mode of the HEV based on PMDMP machine. As shown in Figure 2. 33, even though P mice = P mwheel, 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 (ω or > ω ir and T eor < T ICE = T eir ). As a result, P eir = T ICE (ω or ω ir ) > 0, P es = (T eor + T eir )ω or < 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 Table 2. 4 Power relationships of the constant power mode of the PMDMP machine. Power Input Output Mechanical P mice = T ICE ω ir = P 1 + P 2 P mwheel = T eor ω or = P 2 + P 4 Electrical P es = (T eor T ICE )ω or = P 1 + P 3 P eir = T ICE (ω or ω ir ) = P 3 + P 4 51

70 P 1, P 2, P 3 and P 4 in Table 2. 4 represents the amount of power indicated by 1, 2, 3 and 4 in Figure 2. 33, respectively. Note that P 1 + P 2 = P 2 + P 4 P 1 = P 4. Define P ir = P mice + P eir, then P ir = P 1 + P 2 + P 3 + P 4. The power flow of the constant power mode can be illustrated by Figure mice ir or ICE P ir Wheels PmWheel P2 P4 P P P 1 2 Slip power P P P Inverter Battery eir 3 4 Pes P1 P 3 Figure Power flow of the constant power mode of the PMDMP machine. 52

71 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 P mice and P mwheel. 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 (ω ir ) and the outer rotor speed (ω or ), three different operational points of the outer rotor are indicated by A, B and C in Figure Torque Low Power Mode A PmICE ICE Sweet Spot B C P mwheel Speed Figure 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. 53

72 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 P mice > P mwheel A, 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 C. Figure 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. Torque High Power Mode P mwheel P mice Speed Figure 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

73 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 P mwheel Speed Figure 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. 55

74 Torque Highway (Pure ICE) Mode P mice ICE Sweet Spot Speed Figure Highway (pure ICE) mode of the HEV based on PMDMP machine. 56

75 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 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. 57

76 2000 Outer Rotor Reference and Actual Speed [RPM] Inner Rotor Reference and Actual Speed [RPM] Time [S] Reference speed Actual speed Figure Reference and actual rotor speeds in different operational modes. 100 Outer Rotor Electromagnetic Torque [Nm] Inner Rotor Electromagnetic Torque [Nm] Time [S] Figure Torque productions of both rotors in different operational modes. 58

77 200 Stator Phse Current [A] Inner Rotor Phase Current [A] Time [S] Figure Current waveforms of stator and inner rotor windings in different modes. 59

78 Chapter 3: Squirrel-Cage Dual Mechanical Port Electric Machine 3.1 Modeling of SCDMP machine Three phase model of SCDMP machine The conceptual cross section of the proposed SCDMP machine is shown in Figure 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 mutualinductance 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. 60

79 Figure 3. 1 Conceptual cross-section of the SCDMP machine. L ss L sor L sir L sorir = [ L ors L oror L orir ] (3. 1) L irs L iror L irir The flux and voltage equations for the SCDMP machine are summarized in (3. 2) and (3. 3). Note that v abcor = 0. v abcs λ abcs i abcs [ λ abcor ] = L sorir [ i abcor ] (3. 2) λ abcir i abcir R s i abcs [ v abcor ] = [ R or i abcor ] + d λ abcs v abcir R ir i dt [ λ abcor ] (3. 3) abcir λ abcir 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. 61

80 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 (ω es ) is selected. Figure 3. 2 Reference frames for the SCDMP machine. (3. 4) - (3. 9) are the voltage equations for the machine. v ds = R s i ds + dλ ds dt ω esλ qs (3. 4) v qs = R s i qs + dλ qs dt + ω esλ ds (3. 5) 0 = R or i dor + dλ dor dt 0 = R or i qor + dλ qor dt S or ω es λ qor (3. 6) + S or ω es λ dor (3. 7) 62

81 v dir = R ir i dir + dλ dir dt S ir ω es λ qir (3. 8) v qir = R ir i qir + dλ qir + S dt ir ω es λ (3. 9) dir S or and S ir are defined by (3. 10) and (3. 11), respectively. S or = ω es ω or Poles/2 ω es (3. 10) S ir = ω es ω ir Poles/2 ω es (3. 11) where ω or and ω ir (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 n or = 60ω or 2π n ir = 60ω ir 2π (in Round Per Minute: RPM), respectively. The flux linkages of the SCDMP machine are expressed by (3. 12) - (3. 17). λ ds = L s i ds + M sor i dor + M sir i dir (3. 12) λ qs = L s i qs + M sor i qor + M sir i qir (3. 13) λ dor = M sor i ds + L or i dor + M iror i dir (3. 14) λ qor = M sor i qs + L or i qor + M iror i qir (3. 15) λ dir = M sir i ds + M iror i dor + L ir i dir (3. 16) λ qir = M sir i qs + M iror i qor + L ir i qir (3. 17) Torque productions of the outer rotor and the inner rotor are represented by (3. 18) and (3. 19), respectively. T eor = Poles (λ qori dor λ dor i qor ) (3. 18) T eir = Poles (λ qiri dir λ dir i qir ) (3. 19) and 63

82 B. Physical meanings of inductances. The physical meanings of L s, L or, L rr, M sor, M sir and M iror 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 (L s ). 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 mutualleakage inductance (M ls ). The third part links all the windings; the corresponding inductance is actually the stator and inner rotor mutual inductance (M sir ). 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 M sor = M ls + M sir and L s = L ls + M ls + M sir = L ls + M sor, respectively. 64

83 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 (L lir ). 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 (M lir ). The third part links all the windings; the corresponding inductance is M sir Thus, the mutual inductance between the inner rotor and the outer rotor is M iror = M lir + M sir and the self-inductance of the inner rotor is L ir = L lir + M lir + M sir = L lir + M iror, respectively. 65

84 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 (L lor ) accounts for the part of flux linking only the outer rotor windings. The self-inductance of the outer rotor is L or = L lor + M ls + M lir + M sir = L lor + M sor + M iror M sir. 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. 66

85 C. Equivalent circuits in d-q reference frame. Replacing the inductances in (3. 12) - (3. 17) with the inductances presented in Section B, the flux equations for the SCDMP machine can also be rewritten as (3. 20) - (3. 25). λ ds = L ls i ds + M ls (i ds + i dor ) + M sir (i ds + i dor + i dir ) (3. 20) λ qs = L ls i qs + M ls (i qs + i qor ) + M sir (i qs + i qor + i qir ) (3. 21) λ dor = L lor i dor + M sir (i ds + i dor + i dir ) + M ls (i ds + i dor ) + M lir (i dor + i dir ) (3. 22) λ qor = L lor i qor + M sir (i qs + i qor + i qir ) + M ls (i qs + i qor ) + M lir (i qor + i qir ) (3. 23) λ dir = L lir i dir + M sir (i ds + i dor + i dir ) + M lir (i dor + i dir ) (3. 24) λ qir = L lir i qir + M sir (i qs + i qor + i qir ) + M lir (i qor + i qir ) (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 Figure 3. 5 The d-axis equivalent circuit of the SCDMP machine. 67

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

87 3.1.3 Model in stationary reference frame A. Three-phase to α β reference frame. As shown in Figure 3. 2, the α - axis is aligned with the a s - axis. To obtain the α β reference frame machine model from the d q reference frame model, the easiest way is to replace the subscripts d and q with α and β, and let ω es = 0, then all the equations in the d q reference frame model will be valid in the α β reference frame. Voltage equations for the SCDMP machine in the stationary (α β) reference frame are expressed by (3. 26) - (3. 31). v αs = R s i αs + dλ αs dt (3. 26) v βs = R s i βs + dλ βs dt 0 = R or i αor + dλ αor dt 0 = R or i βor + dλ βor dt v αir = R ir i αir + dλ αir dt v βir = R ir i βir + dλ βir dt 69 (3. 27) + Poles 2 ω orλ βor (3. 28) Poles ω 2 or λ αor (3. 29) + Poles ω 2 ir λ βir (3. 30) Poles ω 2 ir λ αir (3. 31) The flux linkages of the SCDMP machine in stationary reference frame are expressed by (3. 32) - (3. 37). λ αs = L s i αs + M sor i αor + M sir i αir (3. 32) λ βs = L s i βs + M sor i βor + M sir i βir (3. 33) λ αor = M sor i αs + L or i αor + M iror i αir (3. 34) λ βor = M sor i βs + L or i βor + M iror i βir (3. 35)

88 λ αir = M sir i αs + M iror i αor + L ir i αir (3. 36) λ βir = M sir i βs + M iror i βor + L ir i βir (3. 37) Torque productions of the outer rotor and the inner rotor are represented by (3. 52) and (3. 53), respectively. T eor = Poles (λ βori αor λ αor i βor ) (3. 38) T eir = Poles (λ βiri αir λ αir i βir ) (3. 39) B. Steady state model in complex vector form. If the complex vector is defined as f = f α + jf β, where j is a 90 degree phase operator, the machine equations in steady state can be expressed by (3. 40) - (3. 47). λ s = L s i s + M sor i or + M sir i ir (3. 40) λ or = M sor i s + L or i or + M iror i ir (3. 41) λ ir = M sir i s + M iror i or + L ir i ir (3. 42) v s = R s i s + jω es λ s (3. 43) 0 = R or S or i or + jω es λ or (3. 44) v ir S ir = R ir S ir i ir + jω es λ ir (3. 45) T eor = Poles Imag[λ orconj(i or )] (3. 46) T eir = Poles Imag[λ irconj(i ir )] (3. 47) 70

89 C. Steady state equivalent circuit in α β reference frame. Based on (3. 40) - (3. 45), the equivalent circuit of the SCDMP machine can be derived and is shown in Figure The inductances used in (3. 20) - (3. 25) are applied again here. Though the SCDMP machine is different from conventional induction machines (squirrelcage 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 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. 71

90 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, S or = 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. 72

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

92 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 crosssection of the sample machine is shown in Figure and Table 3. 1 lists its main dimensions, performance and parameters. Figure Cross-section of the sample SCDMP machine. 74

93 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 λ or / λ ir Wb 0.40/0.42 Rated T eor / T eir Nm 100/-100 Rated P ormech / P irmech kw 21/-21 Rated efficiency % 85 L s /L or /L ir mh 2.15/2.30/2.32 M sor /M iror /M sir 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 L s > M sor > M sir, L ir > M irsor > M sir and L or > M sor (M iror ). These relationships are consistent with the analysis in Section 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 (η) definition is introduced. If P batt > 0, η = P ormech / ( P irmech + P batt ); if P batt < 0, η = (P ormech P batt )/ ( P irmech ); 75

94 P ormech : Outer rotor mechanical output power. P irmech : Inner rotor mechanical output power. P batt : 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 ) No-load starting of outer rotor: A balanced three-phase voltage ( v as = 100 cos(120πt) V) is applied to the stator windings at time zero and the inner rotor windings are open. As shown in Figure (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. 76

95 77 Figure Comparison between transient responses of proposed model (left) and FEM model (right). 77

96 Traces in Figure from top to bottom: v as - stator phase A to neutral voltage; v air - inner rotor phase A to neutral voltage; i as - stator phase A current; i aor - outer rotor phase A current; i air - inner rotor phase A current; T Lor - outer rotor load torque; T eor - outer rotor electromagnetic torque; T Lir - inner rotor load torque; T eir - inner rotor electromagnetic torque; n or - outer rotor mechanical speed; n ir - inner rotor mechanical speed. 78

97 2) No-load starting of inner rotor: A balanced three-phase voltage ( v air = 40 cos(40πt) V) 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 (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 (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. 79

98 B. Steady state performance validation Based on the machine model proposed in Section 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 Case Table 3. 2 Validation of the proposed steady state machine model by FEM.. Proposed Model Calculation Results λ or λ ir T eor T eir λ or FEM Results λ ir T eor T eir [Nm] [Wb] [Wb] [Nm] [Nm] [Wb] [Wb] [Nm] ir-open ir-open * ** 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. 80

99 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%) 81

100 3.2 Independent control of the two rotors of SCDMP machine 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 λ or, the inner rotor flux λ ir, the outer rotor torque production T eor and the inner rotor torque production T eir, 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 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. 82

101 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). i ds eq = i ds + M iror M sor i dir (3. 48) i qs eq = i qs + M iror M sor i qir (3. 49) 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

102 response are neglected in the following discussion. The effect of this simplification on the control performance will be discussed in Section 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 λ dor = λ or and λ qor = 0. 2) If the d-axis is aligned with the inner rotor flux, then λ dir = λ ir and λ qir = 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 (θ d = θ λor ); the d2-axis is aligned with the inner rotor flux. θ ir or is the angle of the d2-axis with respect to the d-axis. θ λor is the angle of the outer rotor flux with respect to the axis of phase a of stator winding (a s -axis). Figure Reference frames for the SCDMP machine. 84

103 The following equations can be easily derived. Note that λ d2ir = λ ir and λ q2ir = 0. λ dir = λ d2ir cosθ ir or = λ ir cos θ ir or (3. 50) λ qir = λ d2ir sinθ ir or = λ ir sin θ ir or (3. 51) i dir = i d2ir cosθ ir or i q2ir sin θ ir or (3. 52) i qir = i q2ir cosθ ir or + i d2ir sin θ ir or (3. 53) Neglecting the differential term in (3. 6) and considering λ qor = 0 yields (3. 54). i dor = 0 (3. 54) Substituting λ dor with λ or and λ qor with 0 in (3. 18), (3. 55) is derived. Substituting (3. 50) - (3. 53) into (3. 19), (3. 56) is obtained. i qor = Poles i q2ir = Poles T eor λ or (3. 55) T eir λ ir (3. 56) Equations (3. 57) and (3. 58) are obtained by solving (3. 14) - (3. 17) and (3. 50) - (3. 53). Note that λ dor = λ or and λ qor = 0. i d2ir = 1 b (i 2 λir i λor + a 2 i2 qor b 2 2 i q2ir ) (3. 57) where θ ir or = θ < i λor, ai qor > θ < i λir bi d2ir, bi q2ir > (3. 58) i λor = λ or M sor (3. 59) i λir = λ ir M sir (3. 60) a = M iror M sir L or M sor (3. 61) 85

104 b = L ir M sir M iror M sor (3. 62) The angle calculation operator θ < m, n > is defined as follows. n arcsin, if m 0 θ < m, n >= { m 2 + n2 n (3. 63) π arcsin m 2 + n, if m < 0 2 Based on the computed i q2ir, i d2ir and θ ir or, 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). i ds = i λor M iror M sor i dir (3. 64) i qs = L or M sor i qor M iror M sor i qir (3. 65) The calculation of stator and inner rotor current commands is summarized in Figure The results show that there is no one-to-one correspondence between the four input currents (i ds, i qs, i dir and i qir ) and four output variables (λ or, λ ir, T eor and T eir ). 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. 86

105 * or * ir * T eor * T eir (3.55) (3.56) (3.57) (3.58) * i qor * i q 2 ir * i d 2 ir ir _ or (3.52) (3.53) (3.64) (3.65) * i ds * i qs * i dir * i qir Figure Current command calculation module of the SCDMP machine. 87

106 3.2.3 Outer rotor flux observer. The position of the outer rotor flux (θ λor ) shown in Figure 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). S or ω es = i qorr or λ dor = i qorr or λ or (3. 66) where i qor and λ dor can be derived from (3. 48) - (3. 49) and (3. 64) - (3. 65), i qor = 1 L or M sor i qs eq (3. 67) λ or = M sor i ds eq (3. 68) Thus, the slip frequency can be expressed by (3. 69). S or ω es = R or L or i qs eq i ds eq (3. 69) It is shown in (3. 69) that the slip frequency of the outer rotor is in direct proportion to the outer rotor time constant ( L or R or ), hence the accuracy of rotor parameters will have great 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

107 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 Note that the outer rotor mechanical speed (ω or ) is required. i ds i dir (3.48) ids eq (3.68) or i qs i qir (3.49) or iqs eq (3.69) S or es Integrator or Speed sensor Pole pairs Figure Proposed outer rotor flux observer. 89

108 3.2.4 Simulation of the proposed FOC algorithm. The proposed control block diagram for the SCDMP machine is shown in Figure 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 Table 3. 3 Rated fluxes and inudctances of the sample SCDMP machine. λ or [Wb] 0.40 λ ir [Wb] 0.42 L s [mh] 2.15 M sor [mh] 1.97 L or [mh] 2.30 M iror [mh] 2.03 L ir [mh] 2.32 M sir [mh]

109 91 * or * ir PI or * or * ir PI * T eor * T eir ir Current Command Calculation Module i i, i * * ds qs, i * * dir qir i dir, i Speed Calculation qir PI PI v v i ds ds dir Outer Rotor Flux Observer, i, v, v qs qs qir or abc to dq dq to dq to abc to dq SVPWM SVPWM i abcs Inverter1 i abcir P Speed Calculation DC bus Inverter2 ICE Position Sensor ir or Position Sensor Figure Control block diagram of the SCDMP machine. 91

110 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 (λ or, λ ir, T eor and T eir ) 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 shows that the flux levels of the two rotors are pushed up to rated values (λ or = 0.4Wb, λ ir = 0.42Wb) 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 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 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

111 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. 93

112 Flux [Wb] Flux [Wb] Outer Rotor Flux Inner Rotor Flux Angle between Inner Rotor Flux and Outer Rotor Flux Angle [deg] 5 0 Torque [Nm] Torque [Nm] Speed [RPM] Speed [RPM] Outer Rotor Speed Inner Rotor Speed Outer Rotor Torque Production Inner Rotor Torque Production Time [s] Figure Flux, speed and torque production of the SCDMP machine. 94

113 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 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. dλ dor dt = dλ or dt = i dor R or (3. 70) As mentioned in Section 3.2.2, the differential term in (3. 70) is neglected for simplicity in the previous discussion. Hence, i dor 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 shows that these errors affect not only the outer rotor flux, but also the inner rotor flux. Figure shows that after the outer rotor reaches steady state (0.30Wb) at Time = 3.2 s, i dor 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

114 Figure 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 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 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. 96

115 Angle [deg] Angle [deg] Estimated Outer Rotor Flux Angle zoom in Actual Outer Rotor Flux Angle Angle [deg] Error of Estimated Outer Rotor Flux Angle Time [s] Figure Comparison between estimated and actual outer rotor flux angles. 97

116 3.3 Operational modes of the SCDMP machine 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 squirrelcage rotor, the relationship between the mechanical and electrical power can be represented 98

117 by (3. 71). The subscripts e and m indicates electrical and mechanical powers, respectively. P es + P eir + P mice P mwheel P orloss = 0 (3. 71) P es is the electrical power provided by the stator windings; P eir is the electrical power provided by the inner rotor windings; P mice is the mechanical power provided by the ICE, which is equal to the mechanical power provided by the inner rotor; P orloss is the conduction power loss of the outer rotor; P mwheel 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). T eor = Poles (λ qori dor λ dor i qor ) = Poles [(L ori qor + M sor i qs + M iror i qir )i dor (L or i dor + M sor i ds + M iror i dir )i qor ] = Poles [M sor(i qs i dor i ds i qor ) + M iror (i qir i dor i dir i qor )] = T e sor + T e iror 99 (3. 72) T eir = Poles (λ qiri dir λ dir i qir ) = Poles [(L iri qir + M sir i qs + M iror i qor )i dir (L ir i dir + M sir i ds + M iror i dor )i qir ] = Poles [M sir(i qs i dir i ds i qir ) + M iror (i qor i dir i dor i qir )] = T e sir + T e orir (3. 73)

118 T e sor, T e iror, T e sir and T e orir are defined as follows. T e sor = Poles M sor(i qs i dor i ds i qor ) (3. 74) T e iror = Poles M iror(i qir i dor i dir i qor ) (3. 75) T e sir = Poles M sir(i qs i dir i ds i qir ) (3. 76) T e orir = Poles M iror(i qor i dir i dor i qir ) (3. 77) As shown in (3. 74) and (3. 75), T e sor involves only stator and outer rotor currents; and T e iror involves only inner rotor and outer rotor currents. It can be concluded that T e sor and T e iror represent the contributions of the stator current and the inner rotor current on the torque production of the outer rotor, respectively. Similarly, T e sir and T e orir 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 T e iror = T e orir. P es and P eir can be derived from the above equations. Define ω esm = ω es Poles/2. P es = (T eor + T eir ) ω esm = (T e sor + T e sir ) ω esm (3. 78) P eir = T eir ( ω esm ω ir ) = (T e sir + T e orir )( ω esm ω ir ) (3. 79) The mechanical power P mice and P mwheel are P mice = T ICE ω ir = T eir ω ir = (T e sir + T e orir )ω ir (3. 80) P mwheel = T eor ω or = (T e sor + T e iror )ω or (3. 81) Then (3. 82) can be easily derived from (3. 71). P orloss = (T e sor + T e sir ) ω esm (T e sir + T e orir )( ω esm ω ir ) (T e sir + T e orir )ω ir (T e sor + T e iror )ω or = (T e sor + T e iror )(ω esm ω or ) = T eor ( ω es ω Poles or ) (3. 82)

119 P orloss P or gap = T eor P orloss ω es Poles/2 = ω es Poles/2 ω or ω es Poles/2 = S or (3. 83) 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). P battery = P es + P eir = (T e sor + T e sir ) ω esm (T e sir + T e orir )( ω esm ω ir ) = T e sor ω esm + T e sir ω ir T e orir ( ω esm ω ir ) = (T e sor T e orir )ω esm + (T e sir + T e orir )ω ir (3. 84) = T eor ω esm + T eir ω ir = T eor ω esm S or + T eor ω or + T eir ω ir = P orloss + P mwheel P mice 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 speed (if ω esm > ω ir, then P eir = T eir (ω esm ω ir ) = T ICE ( ω or 1 S or ω ir ) > 0). This 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

120 3.3.2 Simplified driving cycle of hybrid vehicle. The following operations are simulated. The results are shown in Figure (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) 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) s, vehicle high speed operation. The outer rotor speed is increased from 4000 RPM to 6000 RPM Figure shows that the stator phase voltage increases correspondingly when the outer rotor speed increases. (6) 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) s, vehicle runs at rated speed. The outer rotor speed and flux levels are back to rated values. (8) s, braking. The outer rotor speed is decreased to 0 RPM. 102

121 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

122 Torque [Nm] Flux [Wb] Voltage [V] Torque [Nm] Speed [rpm] Speed [rpm] (1) (2) Outer Rotor Speed Outer Rotor Torque Production Outer Rotor Flux Stator Phase A to Neutral Voltage (3) (5) (6) (6) Inner Rotor Speed Inner Rotor Torque Production (4) Time [s] (7) (8) Figure Simplified driving cycle of hybrid vehicle. 104