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DATE DUE DATE DUE DATE DUE 5/08 K:/Proj/Acc&Pres/ClRC/DateDue.indd ANALYSIS AND CONTROL OF AN IPl\/ISL\"I FOR TRACTION APPLICATIONS By Carlos Eduardo N ino-Baron A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Electrical Engineering 2010 ABSTRACT ANALYSIS AND CONTROL OF AN IPMSM FOR TRACTION APPLICATIONS By Carlos Eduardo Nine-Baron This thesis presents methodologies to optimally design controllers of Interior Per- manent Magnet Synchronous Motors (IPMSM’S) for traction applications. These controllers include: optimal current reference calculation, Optimal torque-speed tra- jectory design and sensorless control. The design of the current controller reference requires an accurate machine model; the selected model was a cross saturated model. Cross-magnetization and saturation produce a reduction of the maximum power of the machine; their accurate calculation allows us to predict the generated torque and the stator voltage in the machine under field weakening. The current controller reference is based on maximum torque per ampere and field weakening; this current reference is designed such that the motor can operate with minimal losses over the whole speed and torque range while satisfying voltage and current limits. Optimal trajectory design is used to generate the speed and torque references for two subsystems in a series Hybrid Electric Vehicles (HEV): traction motors and engine-generator. 'The trajectory optimization of the traction motors is designed to minimize losses while achieving the speed requirements. The trajectory of the engine- generator subsystem is designed to produce a requested amount of energy in a given period of time while minimizing the energy losses. Sensorless control deals with the issues in practical impler‘nentation of the high- frequency injection methods for control of IPMSM machines without shaft position sensors. The controller is based on the high-frequency injection method, the injection is combined with an sliding mode observer eliminating the requirements of low-pass filters and improving the performance of the torque controller and hence improving the machine efficiency. Simulation and experimental results are presented to validate the proposed con- trollers. DEDICATION I dedicate this thesis to my wife Solimar Reyes Rodriguez, who offered me uncon- ditional love and support throughout the course of this thesis. To my parents Jose Armando Nino and Maria del Carmen Baron, my sister Monica Andrea Nine and niece Lina Maria Camargo, for their love and unconditional support all the way since the beginning of my studies. iv ACKNOWLEDGMENT Dr. Elias Strangas has been the ideal thesis adviser and great friend. His sage advice, insightful criticisms, and patient encouragement aided the writing of this thesis in innumerable ways. I wish to thank my committee members who were more than generous with their expertise and precious time: Dr Hassan K. Khalil , Dr Guoming (George) Zhu, Dr Fang Z. Peng and Ranjan Mukherjee. I would also like to thank all my friends in the electrical and machines laboratory Sajjad, Tariq, Sinisa, Wes, John, and many other for their help and time. TABLE OF CONTENTS List of Tables ................................. viii List of Figures ................................ x Introduction . . ......................... 1 1.1 Contributions ............................... 1 1.1.1 Cross Saturation Analysis .................... 2 1.1.2 Trajectory Optimization ..................... 2 1.1.3 Sensorless Control ........................ 3 1.2 Thesis Organization ............................ 3 Background ........................... 5 2.1 Permanent Magnet AC Machines Configuration ............ 5 2.1.1 Location of the Permanent Magnets ............... 6 2.1.2 Winding Distribution ....................... 7 2.2 2—Axis Model ............................... 8 2.3 Vector Control .............................. 10 2.3.1 Maximum Torque Per Ampere .................. 11 2.3.2 Direct Flux Weakening Control ................. 13 2.3.3 Machine Operation ........................ 14 Machine Modeling ........................ 15 3.1 Introduction ................................ 15 3.2 Flux Linkages Calculation and Machine Modeling ........... 17 3.2.1 Flux Linkages Calculation .................... 19 3.3 Cross Magnetization: Concept ...................... 20 3.4 Cross Magnetization: Machine Model and Calculation ........ 28 3.4.1 Machine Model .......................... 28 3.4.2 Model Calculation ........................ 29 3.4.3 Experimental Model Calculation ................ 35 Trajectory Optimization for the Operation of Traction Motors . 44 4.1 Problem Description ........................... 46 4.2 Maximum Torque Calculation ...................... 46 4.3 Optimal ’Itajectory Calculation for the Operation of Traction Motors 49 4.3.1 General Trajectory Optimization Algorithm .......... 50 4.3.2 Continuous Time Ttajectory Optimization of the Traction Mo- tor System Controller ...................... 53 4.3.3 Numerical Optimization ..................... 54 vi 4.4 Simulated and Experimental Results .................. 58 4.4.1 Simulation Results ........................ 58 4.4.2 Real Time Optimization ..................... 60 4.4.3 Experimental Setup ....................... 67 Trajectory Optimization for Engine-Generator Operation of a Series Hybrid Electric Vehicle .............. . . . . . . . 70 5.1 Introduction ................................ 71 5.2 System Description ............................ 72 5.3 Trajectory Optimization ......................... 73 5.3.1 'Iiajectory Optimization for the Operation of Engine—generator System ............................... 74 5.3.2 Numerical Optimization ..................... 77 5.3.3 Simulation Results ........................ 83 5.4 Approximation to Optimal Trajectory ................. 87 5.4.1 Mode 1: Low Energy Requirement With Enough Time . . . . 87 5.4.2 Mode 2: High Energy Requirement With Enough Time . . . . 89 5.4.3 Mode 3: Fast Energy Production ................ 90 5.5 Comparison ................................ 91 High Performance Low Speed Sensorless Control of Interior Perma- nent Magnet Synchronous Motor ................ 94 6.1 Introduction ................................ 95 6.2 High Frequency Injection Architectures ................. 97 6.2.1 Filtered Controller ........................ - 98 6.2.2 High Performance Controller ................... 102 6.3 Experimental Results ........................... 109 Conclusions and Anticipated Phture Work . . . . . . . . . . . 121 7.1 Cross Saturation Analysis ........................ 121 7.2 Trajectory Optimization for the Operation of Traction Motors . . . . 122 7.3 Tiajectory Optimization for the Operation of Engine-generator Sub- system ................................... 122 7.4 Sensorless Control ............................. 123 Experimental Setup . . ..... . . . . . . . . . . . . . . . . 125 Al Real Time Controller: Rt-Linux and FPGA .............. 127 A2 National Instrument Based Data Acquisition System ......... 128 A3 Sensors: Voltage, Current, Rotor Position and Torque ......... 129 A4 IPMSM .................................. 132 A5 Inverter .................................. 132 A6 Dynamometer ............................... 133 Bibliography . ..... . . . . . . . . . . . . . . . . . . . . 136 vii 3.1 4.1 4.2 4.3 4.4 4.5 4.6 4.7 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4 LIST OF TABLES Machine parameters ........................... Weight sets ho h1 h2 ........................... Constants for on—line optimization approximation for t f > 5 3 Error in the cost function for t f > 5 s ................. Time intervals for t f < 53 ........................ Constants for on-line optimization approximation for t f < 53 ..... Error in the cost function for t f < 5(3) ................. Maximum number of iterations ..................... Simulation parameters .......................... Parameters of the approximated optimal trajectory .......... Controllers comparison, Efficiency ................... Controller comparison, normalized final energy Wt f/ Wire f ...... Experiment setup parameters ...................... Steady state error in the rotor position estimation ........... Rotor position error for a current ramp ................ Steady state error in the rotor position ................. viii 58 64 65 65 66 83 89 93 111 113 6.5 Rotor position error for a current ramp ix LIST OF FIGURES 2.1 General configuration of a PMSM .................... 6 2.2 Permanent magnet location ....................... 7 2.3 Winding configuration of a PMSM ................... 7 2.4 Phasor diagram of a PSMS ........................ 8 2.5 Schematic of the PMSM-drive control strategy ............. 11 3.1 Machine’s stator configurations ..................... 21 3.2 Direct axis flux AAd(id, iq) ....................... 22 3.3 Quadrature axis flux AAqu, iq) ..................... 23 3.4 IPMSM with 4SPP and with sinusoidally distributed windings . . . . 23 3.5 Rotor flux linkages for a 4SPP machine He: 900 ........... 24 3.6 IPMSM with HSPP and with concentrated windings ......... 24 3.7 Rotor flux linkages for a HSPP machine 06: 900 and 06: 750 . . . . 25 3.8 Stator flux linkages for HSPP machine 63: 900 ............ 26 3.9 Space vector diagram of flux linkages .................. 27 3.10 Stator flux linkages as function of 68 .................. 27 3.11 Self inductances Ld and Lq ....................... 30 3.12 Mutual inductances qu and qu ................... 31 X 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 Self Inductance calculation Lq(z'd, iq) .................. Mutual Inductance calculation M dq(i d? iq) ............... Torque calculation including cross saturation .............. Torque components calculation ..................... Field weakening curve for the configurations analyzed ......... Schematic of the experimental setup .................. Calculated experimental Flux linkages using V* ............ Calculated experimental Flux linkages using Vmes ........... Three phase Inverter ........................... Switching pattern and actual voltage in the machine .......... Calculated Flux linkages using V* and dead time compensation Calculated experimental inductances using V* ............. Series HEV configuration ......................... Torque as function of 6 for rated and half of rated current ...... Torque as function of speed for maximum current ........... Torque as function of speed corrected by the iron losses ........ Optimal current commands ....................... Bilinear interpolation ........................... Optimal algorithm flowchart ....................... Cost vs. iteration number ........................ Simulated speed wm ........................... Simulated trajectories .......................... 33 34 35 36 38 38 39 4O 41 43 46 49 50 55 59 4.11 4.12 4.13 4.14 4.15 4.16 4.17 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 6.1 6.2 Simulated torque Te ........................ . . . . . 62 Simulated current is ........................... 62 Controller topology ............................ 63 Experiment setup ............................. 67 Comparison: experimental and simulated speed ............ 68 Comparison: experimental and simulated torque ............ 68 Comparison: experimental and simulated current ........... 69 Series HEV configuration ......................... 73 Engine efficiency map contours ..................... 74 Generator efficiency map contours ................... 74 Grid data example used to describe the bilinear interpolation . . . . 79 Minimization algorithm ......................... 82 Cost Function ............................... 83 Engine torque ............................... 84 Generator torque ............................. 84 Mechanical speed ............................. 85 Generator output energy ......................... 85 Total energy losses ............................ 86 Torque—speed points over combined efficiency map contours ...... 86 Block diagram of approximated optimal trajectory ........... 90 Filtered rotor position estimation block diagram ............ 98 Block diagram of rotor position estimation by synchronous frame of reference filters .............................. 101 xii 6.3 High frequency equivalent model of the motor and controller ..... 103 6.4 Block diagram of rotor position estimation by the use of a high current observer .................................. 109 6.5 Proposed rotor position estimation block diagram ........... 109 6.6 Experimental Setup ............................ 110 6.7 Rotor position estimation for controller with filters .......... 114 6.8 Speed estimation for controller with filters ............... 115 6.9 Rotor position estimation for the high performance controller ..... 116 6.10 Speed estimation for the high performance controller ......... 117 6.11 Machine current for the controller with filters ............. 118 6.12 Machine current for the high performance controller .......... 119 6.13 Dynamic performance comparison of the estimator .......... 120 A1 Motor and dynamorneter ......................... 125 A2 Controller and inverter .......................... 126 A3 Schematic of the experimental setup .................. 126 A4 RT-Linux ................................. 128 A5 FPGA and I/O .............................. 128 A6 NI USE-6009 ............................... 130 A7 NI USB-6259 ............................... 130 A8 Labview with the virtual instrument .................. 130 A9 Current and voltage sensors ....................... 131 A10 Torque sensor ............................... 132 A11 IPMSM .................................. 133 xiii A. 12 Inverter .................................. A.13 Dynamometer ............................... xiv 134 Chapter 1 Introduction Hybrid Electric Vehicles (HEV’S) combine a conventional propulsion system (that includes an internal combustion engine), electric drives, and an energy storage system. The combination of these devices improves the efficiency of the vehicle. The efficiency increase is due to a more efficient operation of the drivetrain, the recovery of the energy during the braking and the engine size reduction. The braking energy is captured by the traction motors, converted through inverters and stored in batteries. During the acceleration, some energy is provided by the batteries, which makes the power requirements of the engine to be less. The flow of energy back and forth produces losses in each element, therefore, an efficient design and control of each device and the integration of all of them improves the efficiency of the whole system. 1 . 1 Contributions This thesis present methodologies to design controllers used for Interior Permanent Magnet Synchronous Motors (IPMSM) for traction applications. These controllers are explained and their performance is evaluated and compared to other controllers. The contributions of this thesis to the proposed controllers design are summarized below. 1.1.1 Cross Saturation Analysis The cross saturation is a phenomenon in ac electric machines, where the saturation of the iron core causes direct and quadrature axis current components to affect per- pendicular fluxes to them. The contributions related to this issue are: 0 Develop and apply a methodology to calculate the self and the quasi-mutual inductances and their dependencies on the currents using flux estimation. These inductances are used to model the cross saturation effect. 0 Develop and validate a methodology to calculate the optimal current commands to control the IPMSM over the whole operation range with minimal losses by the inclusion of cross saturation effect. 1.1.2 Trajectory Optimization This thesis present two cases where trajectory optimization was used: Optimal speed control of a traction motor and optimal torque-speed control of engine-generator sub- system, both applied to a series hybrid electric vehicle. The trajectory optimization was used to transfer the system from one operating point to another with minimal energy losses. The main contributions in trajectory optimization are: 0 Apply the trajectory optimization to the speed control of an IPMSM by trans- ferring the system from one torque speed point to another with minimal energy losses. Develop an approximation to the optimal trajectory to use it in real time. 0 Apply the trajectory optimization, in the subsystem engine-generator, to deter- mine the optimal torque and speed commands in order to produce a requested energy in a fixed time, while minimizing the energy losses. Develop an approx- imation to the optimal trajectory to use it in real time. 1.1.3 Sensorless Control Rotor position sensing requires resolvers or encoders. An alternative is to operate the machine with a rotor position estimator. High frequency injection for sensorless control is used for low speed operation. The main contribution in sensorless control of IPMSM is: 0 Design a new controller based on the high-frequency injection method that elim- inates the need for low-pass filtering for sensorless operation while improving the motor performance. 1 .2 Thesis Organization The thesis presented here has the following content: Chapter 2 presents the basic configurations of the Permanent Magnet Synchronous Motor (PMSM): rotor struc- tures, magnets location and windings distribution. The concepts of two axis model and field oriented control are presented. The basic criteria to determine the currents commands, based on maximum torque per ampere and field weakening, is described. Chapter 3 describes the cross saturation and presents the machine model which includes this effect. This model is the most accurate and suitable for control. First the flux linkages calculation is shown in sections 3.1 and 3.2; the results of these sections justify the cross saturation analysis. Section 3.3 explains the cross saturation concept and its effect in the machine performance. Section 3.4 describes the methodology used to extract the machine parameters from the flux linkages calculation. The pro- posed methodology is validated with FEA simulations, and the model of a prototype machine is calculated. Chapter 4 presents the Optimization of the speed reference for the traction n10- tor controller in series HEV. Section 4.1 describes the Optimization problem. The trajectory calculation is done by a multistage Optimization. Section 4.2 presents the Optimization of the current commands to Operate the machine at a desired torque and speed point while minimizing its losses. Section 4.3 presents the numerical trajectory optimization to determine speed command. This trajectory defines the torque and speed points in time to transfer the system from one torque and speed point to another while minimizing energy losses. Simulation and experimental results are presented in section 4.4. Chapter 5 presents the Optimization of the torque-speed reference for the con- trollers Of the engine-generator subsystem. Section 5.2 presents the Optimization problem and the engine and generator efficiency maps. Section 5.3 describes the tra- jectory Optimization scheme. The minimization problem is described in continuous time and the problem is solved using numerical Optimization in discrete time. Sim- ulation results Of the proposed trajectory Optimization are presented. Section 5.4 presents an approximation to the Optimal trajectory. Section 5.5 shows the resulting efficiency and it is compared to non-optimized control techniques. Chapter 6 presents the design Of a high performance sensorless controller for IPMSM. Section 6.2 is divided in two subsections. Subsection 6.2.1 shows the classical approach, where low pass and high pass filters are used for current control and rotor position estimation respectively. Subsection 6.2.2 describes the proposed method- ology. Section 6.3 presents and discusses experimental results, where the filtered demodulation and the proposed controller are compared. Chapter 7 presents the concluding remarks and expected future work. The chapter is subdivided in sections describing the work done in the topics presented in chapters 3, 4, 5 and 6. Chapter 2 Background As high efficient electric drives are becoming more important, Permanent Magnet Synchronous Machines are rapidly replacing induction machines.PMSM’s offer high power density and lower losses, so they are more suitable in the applications requiring volume reduction e. g. automotive, aircraft, and portable generator industries. 2.1 Permanent Magnet AC Machines Configura- tion The permanent magnet synchronous machine is basically an ordinary AC machine with windings in the stator slots so that the flux created by stator current is approx- imately sinusoidal, and magnets in the rotor that produce a constant flux. The work presented here concentrates on Permanent Magnet Synchronous Machines (PMSM’s) with a sinusoidal flux distribution. Rotating permanent magnet synchronous ma- chines used today corne in two different configurations: interior and exterior rotor as shown in Figure 2.1. (a) Exterior Rotor (b) Interior Rotor Figure 2.1: General configuration of a PMSM 2.1.1 Location Of the Permanent Magnets The rotor magnets are the most important elements in the machine; they affect the characteristics of the machine, and their location in the rotor would change maximum torque, maximum speed and power density. Based on the rotor magnets configuration, the PMSM can be: Surface PMSM, Inset PMSM and Interior PMSM as shown in Figure 2.2. In a surface magnet motor the magnets are usually magnetized radially. Due to the use of low permeability rare-earth magnets the synchronous inductances in the d-axis and q-axis may be considered to be equal which can be helpful while designing the motor. The construction of the motor is less expensive and simpler than the other configurations, because the magnets can be attached to the rotor surface. Inset Permanent Magnet (PM) rotor is desirable over surface PM, because it gets an additional component of the torque due to saliency. Inset has a construction more complex than surface PM. The embedded or interior magnet motor has permanent magnets embedded in the deep slots. In the case Of an embedded magnet motor, the stator synchronous inductance in the q-axis is Often greater than the synchronous inductance in the d-axis, this saliency produces an additional reluctance torque that makes it suitable for traction applications where high torque is required. 6 ewe ewe; O m (a) Surface PM (b) Inset PM (c) Interior PM Figure 2.2: Permanent magnet location 2.1.2 Winding Distribution The stator windings play an important role in the machine, and their distribution modifies the efficiency and size. Figure 2.3 shows two of the basic configurations: concentrated and distributed windings. Concentrated windings are usually designed for machines with smaller size, however nowadays they have been used for mid power configurations. Their advantage is basically the reduction of end windings, hence the copper losses and leakage flux is reduced. Distributed windings configuration has longer end turns that means more copper losses, however their distribution is used to have a more sinusoidal back-emf and hence this topology allow us to reduce the iron losses. (a) Concentrated (b) Distributed Figure 2.3: Winding configuration of a PMSM 2.2 2-Axis Model The PMSM model can be developed in the stator frame of reference or in the rotor frame or reference. The model in the rotor frame is preferred because it is simpler. It allows to decouple the flux into magnetization and torque components; similarly the current can be decomposed in the same fashion. These simplifications make this model suitable for vector control. The phasor diagram of a PMSM is shown in Figure 2.4. The rotor flux linkages are assumed constant and aligned with the direct axis, while the flux component in the quadrature axis is taken as zero. The core losses are neglected. The stator flux is also represented by two components: direct and quadrature fluxes. q axis Vs 1;, I. A r Vqs ¢ 6 g d axis <—+ S w—»——> r VJ. 9e Ids ’Id—PM Stator Frame Of Reference Figure 2.4: Phasor diagram Of a PSMS The stator voltage equations in the rotor frame of reference are: 8 aid ’Ud = Rsld-— Lag/\q + at (2.1) v—Ri+' A+£A—q (22) q_ 8"] (.06 d at ' where R3 is the stator resistance, v and z' are the stator voltage and current and the subscripts d and q refer to the direct and quadrature axes. /\ is the the stator flux linkages and we is the electrical frequency. The direct and quadrature axis fluxes in the rotor frame of reference are: Ad = “\d—PM + Laid (23) where Ad—. P .M is the stator flux linkages due to the PM. The inductances Ld and Lq model the stator flux change due to the stator currents. In general these inductances are not equal, because the reluctance of direct and quadrature paths are not the same. In the IPM of Figure 2.2c, the direct axis reluctance is bigger than quadrature axis reluctance, this produces that Ld < Lq. The electrical torque Te is given by: 3 P . . Te = a? (Adlq — Aqu) (2.5) where P is the number Of poles. By substituting equations 2.3 and 2.4 in 2.5 the torque is calculated as: 3P . . . Te = 53 (Ad—PMZQ + (Ld - Lq) 141(1) (2-6) The first component of the torque, Tm, is called “PM Torque”. This component is due to the interaction between the rotor flux and the quadrature axis current. The 9 second component, Tr, is produced due to the relative change of the reluctance of the machine with the rotor position called “Reluctance Torque”. 3 P . m = 53 (Ad—prim) (2.7) 3P . . Te 2 Tm + Tr (2.9) 2.3 Vector Control The PMSM vector control is based on the 2-axis model. The Operation of the machine is restricted to mechanical and electrical limits. From control systems point of view the electrical limits of the machine are the maximum current and the maximum voltage. The maximum current is determined by the capability to remove heat from the windings and depends on the cooling system; it is usually fixed after the machine design. The maximum voltage is restricted to the peak voltage available from the inverter connected to the machine. The basic Objectives in the motor controller design are to have high dynamic performance, low Operational energy losses and Operate it within voltage and current limits. The operation in the machine is divided basically in two regions: constant torque and constant power. These regions are separated by the base speed; for speeds below the base, the induced voltage is lower than the maximum available from the inverter. The only restriction there is the current. In this region the criteria commonly selected is to use the maximum torque available restricted by the maximum current. This is called “maximum torque per ampere”. When the machine speed is higher than the base speed, the induced voltage that 10 is the product Of the flux and the speed reaches its maximum. At this point it is necessary to control the flux in order to Operate the machine at the voltage limit. The stator current is controlled in a such way that one component id reduces the direct axis flux and hence the induced voltage. Under this condition the current limit should be also satisfied. The component iq is chosen in such way that the current limit is satisfied and the torque produced at this point is maximum but lower than the maximum torque per ampere for the same total current. This control technique is called “direct flux weakening control”. Figure 2.5 shows the block diagram used to control the PMSM. * a: a) z" 1---- _ _ J - _ -. m ; . 7 SI- x” T‘\ ——>"<;>I Speed Hi Constant '4 Pr in ?__. Current I. .L,,,/ I *—~: “ - Control i ’L rcu _ T . _-_ I >\ // ‘ m. . j . m‘ ,l ; Inverter ; l s\_//, "' W—FTK “- “\ \f I 1 _ ___ a I lbs - > “" I LOAD I S” (maX) __ ’l Flux ___ , . /,« V I Weakening e ‘\.\,.,--/ ‘ sn(max) H’ Control l l’ , I IT, . CS l 4. a) I »~— ——~ ~ \ :5 m f ‘ --- I 6 Encoder \7/ __ _ _ __ -_ Signal ‘1. V; *j 3 _fi .1, _ __j Conditioning M.._— Figure 2.5: Schematic Of the PMSM-drive control strategy 2.3.1 Maximum Torque Per Ampere The Maximum Torque Per Ampere (MTPA) control Operates the motor in the region where the induced voltage is lower than the maximum voltage. Then equation 2.6 is satisfied under the restriction of maximum current and to maximum torque pro- duction. Lets consider the case where the current is equal to the maximum current Imax, this current can be decomposed in terms of Id and z'q as: 11 2d = 1111axCOS(6) and Zq = IIII‘ch 8111(6) (2.10) were (5 is the angle Of the current space vector. By substituting equation 2.10 in 2.6 the torque is given by: 3 P . - 2 . - Te = 5:2— (Id—PMI max 8111(0) + (Ld — La) 1 max SIIIIQ‘II) (2'11) The relation Of torque per ampere is given by 7'6 , 3 P . . . .1— : 22 (Ad—P1111 5111(6) + 0.5 * (Ld — Lq) Irnax sm(26)) (2.12) The angle at where the maximum torque occurs is given by: —— (Ad_PMsin(6) + 0.5 * (Ld — Lq)1maxsin(26))) = 0 (2.13) solving for (5 2 2 2 —/\d_P]U+\//\d_PM + 81ma.X(Ld — Lq) 4(Ld — Lq) Imax 1 6 = cos— (2.14) The range of (5 should be between 900 and 1800 in order to have positive reluctance torque when Ld < Lq, which is the common case in IPMSM’S. For an specified value Of stator current 13, the angle is calculated from equation 2.14 and the current commands are calculated from equation 2.10. 12 2.3.2 Direct Flux Weakening Control The direct flux weakening control operates the motor in the region defined by the base speed and above. There, the machine has reached the maximum voltage and the current space vector angle 6 should be increased as the speed increases. The value of 6 is selected such that it maximizes the output torque and keeps the motor in or below the voltage limit. It is clear that in this region equation 2.14 is not longer satisfied. By combining equations 2.1 -2.2 and neglecting the voltage drop in the resistor the maximum voltage in steady state can be written as: Vrfiax Z V(12+ Vq2 —_— (weLqu)2 + (we (Ad—PA’I+LdId))2 in this region it is also required to produce the maximum torque available, then it is desirable to use the maximum current. The maximum voltage can be written as: 2 v3m=w§ [L3 (13m — 13) + (A d_ P M+Ld1d) J (2.16) in a compact form equation 2.14 is VmX = weflja + bId + A3_PM + c with 2 2 a = L — L “I q (2.17) b = QAd—PMLd C = ”\d—PIWQ + 1431121161.): For fixed input voltage, speed and stator current, the value Of Id is calculated 13 2.3.2 Direct Flux Weakening Control The direct flux weakening control operates the motor in the region defined by the base speed and above. There, the machine has reached the maximum voltage and the current space vector angle 6 should be increased as the speed increases. The value of 6 is selected such that it maximizes the output torque and keeps the motor in or below the voltage limit. It is clear that in this region equation 2.14 is not longer satisfied. By combining equations 2.1 -2.2 and neglecting the voltage drop in the resistor the maximum voltage in steady state can be written as: V3,,1X = V3 + V3 = (.....qu + (and)2 (2.15) = (weLqIQIZ + (“6 (’\ci—iDiv1+Lc11al))2 in this region it is also required to produce the maximum torque available, then it is desirable to use the maximum current. The maximum voltage can be written as: 2 vgaxwg [L3 (1,2max — 13) + (A d_ P M+Ld1d) ] (2.16) in a compact form equation 2.14 is anax = we «[30. + 01d + APP-PILL + C with. 2 2 a = L — L ‘1 q (2.17) b = 2/\d—PMLai C = Ad—PM2 + Lilrilax For fixed input voltage, speed and stator current, the value of Id is calculated 13 2.3.2 Direct Flux Weakening Control The direct flux weakening control operates the motor in the region defined by the base speed and above. There, the machine has reached the maximum voltage and the current space vector angle 6 should be increased as the speed increases. The value of 6 is selected such that it maximizes the output torque and keeps the motor in or below the voltage limit. It is clear that in this region equation 2.14 is not longer satisfied. By combining equations 2.1 -2.2 and neglecting the voltage drop in the resistor the maximum voltage in steady state can be written as: Vn21ax 1' de “I“ Vq2 = (wet)? + (wad)? (2.15) = (weLqIQ)2 + (“6 ("cl—10M+Ldlalll2 in this region it is also required to produce the maximum torque available, then it is desirable to use the maximum current. The maximum voltage can be written as: 2 V3m=w§ [L3 (1,2mix — 13) + (,\d_PM+LdId) ] (2.16) in a compact form equation 2.14 is Vmax = we\/13a + bId + 13PM + c with 2 2 a = L — L d ‘1 (2.17) b = ”cl—PMLd 2 2 2 c = )‘d—PII'I + qumax For fixed input voltage, speed and stator current, the value of Id is calculated 13 from equation 2.17, and [(1 is calculated by: Iq = \/(Ir2nax -— 13). 2.3.3 Machine Operation The torque controller Of the machine determines the current commands from the corn- manded torque, the machine speed and maximum voltage. The control methodology will be subdivided in two regions as stated before. For speed below than the corner, the relation between 15, 6 and re can be precal- culated an use it tO calculate the current commands. The stator current I 3 is varied in fixed steeps from 0 to Imam; equation 2.14 is used to determine the corresponding angle 6 for MTPA. For each set Of Is and 6, equation 2.6 is used to determine the torque produced by the machine. From this analysis the relation of 13, 6 and Te is determined, that is for a specific current value IS , there is a corresponding set of 6 and re. A numerical inversion Of the function re (I 3) is used to determine the current command for a requested torque command T6. The value of 6 is calculated by the relation established between it and Is. For the corner speed and beyond, the relation between I d and we is precalculated and used as a command independently Of the motor torque. For defined values of maximum voltage, maximum current and machine speed the direct axis current com- ponent can be calculated from 2.17. The speed is varied from corner to maximum in fixed steeps; and for every speed value a correspondent I d is calculated. I d is used as command, and the stator current component Iq is changed in real time to produce the required torque, that is a PI controller will adjust Iq to produce the required torque or speed. 14 Chapter 3 Machine Modeling 3. 1 Introduction Applications Of electric vehicles (EVs) require traction motors with high efficiency, high power density and extended field weakening capability. The Permanent Magnet Synchronous Motor (PMSM) has high efficiency and high power density. The Interior Permanent Magnet Synchronous Motor (IPMSM) has an extended field-weakening re- gion due the additional torque Obtained by the saliency; its accurate torque calculation is important to Operate the machine efficiently and within its designed Operational limits. The developed torque in a PMSM has three components: permanent magnet, re- luctance and cross-saturation. As seen earlier, the permanent magnet torque is due to the interaction between the flux produced by the rotor magnets and the stator current. The reluctance torque is produced by the interaction between the stator current and the reluctance change in the machine when the rotor moves. The cross saturation torque is the component that represents the change Of the total torque due to the coupling between the direct and quadrature axis currents with their perpendic- ular fluxes caused by saturation. This cross saturation component becomes relevant 15 in high power density machines, where the simplified model presented in section 2.2 produce: significant error in the torque calculation, controller detuning and hence efficiency reduction. The PMSM model presented here includes self inductances (L d and Lq) and quasi- mutual inductances (qu and qu). The self-inductance, Ld, relates the change of direct axis flux linkages due to the change in the direct axis current, and Lq represents the same relation in the quadrature axis. The cross magnetization is modeled by the inclusion Of two quasi-mutual inductances, qu and qu. qu models the change of quadrature axis flux due to change Of the direct axis current and M dq models the change of direct axis flux due to change in the quadrature axis current. The generated torque in IPMSM is calculated from the estimated flux in "the machine and directly from Finite Element Analysis (F EA). The torque calculated directly by FEA is not useful enough, because it does not allow us to determine the percentage of PM-torque, reluctance torque and cross magnetization torque, which are important to Optimize the machine design. The torque computation from flux estimation using the 2-axis model presented in [1] does not include the saturation effect. TO make an accurate estimation Of the torque, it is necessary to consider the saturation and the consequent reduction Of the flux in the machine [2, 3, 4, 5]. Additional considerations Of the cross-magnetization effect include the dependence of the inductances on the currents as Ld(z'd, z'q) and Lq(id,iq)[6, 7, 8]. In [9, 10, 11] the inclusion of mutual inductances in the machine model was presented; [9, 10] assume that the direct and quadrature flux can be modeled by superposition, where Ld and qu depend only on fd , and Lq and qu depend only on iq. There is a coupling between the direct and quadrature currents and their perpen— dicular fluxes due to cross magnetization and saturation. In [11] this effect is described and the methodology of a square wave voltage injection is applied to calculate self and mutual inductances, the inductances depend on the stator current components 16 as follows: Ld(id, iq), Lq(z'd, iq), Aqu(id, iq) and Arfqdfid, z'q). This methodology was validated by comparing simulated and experimentally calculated flux linkages, how- ever, the main disadvantage is that the test has to be made at standstill neglecting entirely the dependance of the flux on the rotor position. Rahman et a1. [12] pro- posed a methodology to calculate the flux linkages while the machine is operating at constant speed, however the results were not used for the inductances calculation. The methodologies presented above for the model calculation vary from a constant parameter to a cross saturated model, however the methodologies could not validated experimentally or the models are not calculated under the normal operation of the machine. This chapter describes the cross saturation effect and presents a methodology to calculate the self and mutual inductances using flux estimation. These inductances are used to calculate the voltage, torque and the field weakening profile Of the ma- chine. The proposed methodology was validated by FEA analisys and experiments. Four different machine configurations were simulated: Half Slot per Pole per Phase (HSPP), 1 Slot per Pole per Phase (SPP), 2 SPP and 4 SPP. The experimentally determined parameters of a prototype IPMSM with 2 SPP are presented. 3.2 Flux Linkages Calculation and Machine Mod- eling The two axis model in the rotor frame Of reference Of a PMSM described by the flux, current, voltage and torque equations is summarized by equations 3.1 to 3.7. Ad 2 ALI—Pill + Ldid (3.1) 17 , 80d 'Ud = R31d — (fie/\q + —()t_ , at, ’qu = R32q + weAd + W 3 P . Tm = g; (Id—13ml) 3P . . Tr = f2" ((1305 ‘ La) Idzq) Te=7m+7r the currents, and hence the inductances are not constant. stator flux linkages is described below. 18 (3.2) (3.3) (3.4) (3.6) (3.7) The model described above assumed constant inductances, that is not entirely true in high power density machines, where the saturation in the iron core produces that. the flux in the machine changes nonlinearly with the stator currents. The machine is usually operated under saturation; therefore the flux is not linearly dependent on The calculation Of the stator flux linkages demonstrate the dependence of direct and quadrature axis fluxes on the stator currents; torque, voltage and inductances calculation are based on the calculated flux. The general procedure to calculate the 3.2.1 Flux Linkages Calculation The test proposed to calculate the flux linkages and the analysis are described in this section. This methodology is similar to [12], however some modifications have been made in order to Obtain the machine voltage from experiments. The experimental calculation of the flux linkages requires the measurement or estimation Of the induced voltage. One approach requires for example filter and then measure the voltage, then extract the first harmonic and correct it for the shift and amplitude reduction due to the filter [12]. The methodology proposed here uses the commanded voltage, a bandstop filter is used to remove the Pulse With Modulation (PWM) content of the power signal. The test used tO calculate the flux linkages is presented below: 1. Connect the machine to a dynamometer operated in speed mode, that is the dynamometer will keep the speed value constant independently of the load. 2. Command zero stator current and measure / estimate the induced voltage in the stator. 3. Integrate the voltages with respect to time, and calculate the stator fluxes in the stator frame Of reference. With the transformation matrix Of equation 3.8 convert the stator fluxes from the stator to the rotor frame of reference. The direct and quadrature axis fluxes linkages calculated are ”\d— PII/I and Aq__ PZI/I' 003(06) cos(6’e + 27r/3) cos(96 — 27r/3) Me) = swag) sin(6le + 27r/3) 51an — 27r/3) (3.8) 0.5 0.5 0.5 4. Command a rotating, fixed—magnitude current space vector aligned with the quadrature axis. Measure the induced voltage and subtract the drop Of voltage in the resistance. Follow the procedure described above to calculate the total 19 flux in the direct and quadrature axis “\dfldt Iq) and Aq(1d,lq). 5. Repeat step 4, for magnitudes of the current space vector from 0.05 to 1 per unit (pu) and values Of 6 from 900 to 1800 electrical degrees. This means a variation of the current space vector from the quadrature axis to the negative direct axis. 6. The flux linkages components due to stator current are given by: AAdUdi Iq) = Muir Iq) — Ail—PM (39) Four machines with the same rotor and with different stator configurations were evaluated: HSPP, 18PP, 2SPP and 4SPP, Figure 3.1. Figures 3.2-3.3 show the cal- culated AAdUd, iq) and AAd(id, z'q). The calculated flux linkages components, direct and quadrature, depend on the direct and quadrature axis currents. Although, it is clear that the direct axis flux depends more on id and quadrature axis flux depends more on iq, there exists a dependence of the fluxes on their perpendicular currents. This effect is called “cross saturation”. 3.3 Cross Magnetization: Concept Lets consider the interior permanent magnet synchronous motor with 4 SPP shown in Figure 3.4. This machine has 4 poles and sinusoidally distributed windings. The direct and quadrature axis Of the machine are presented in Figure 3.4-a. The windings of phase “a” are distributed as shown in Figure 3.4-b. The equivalent flux linkages due to current in this phase are aligned in the direction described by the arrow. The same convention is used for phases “b” and “c” with a shift Of 1200 and 2400 electrical 20 \ / ~.\‘ / \\ ,/ \ I \~ / \ / , \ \ / , ~ - / K T I ‘\ \ / l \x /<( l .“ \ ./\ l ( , , 1 I. I ' / ‘ f . ." /_1 t i' ,' \ / \ \ \ l , z' / .' \_ . \ ,' x ,5' . / .\ \V,//’,’— _\ I‘V" (- /\ \‘ j ‘4 - _:\ . ,’ / \ 5 5*" '7 ’ “s. ,. A. \ r ‘ \z _ . /. V/rry...‘ ' ‘ .\\\\/, 1 \\ 7 I 7 [ \‘I h ‘1 .,'/ ‘J - \ 'T\:\\ \I I I I I / "I 1 / , y c?\ {3‘ \ L \\\\\«'./ \T 1.7“ \ v / / 2 / . a. ,3 z _ 1 ' ~. \\ , i \ . \ ‘ , \i/ / \ \_ .‘r s, ». I / . ~. / i N~ —‘ / \ .I/ \/ . \ I L A, / \ I , ’ ’ \ x . , —*"\ K ‘1 ”/4, v // \ \,x \ 1‘ ; ‘ \_\ /’_ /\/ / \ \ \\ 7"- ‘ ( ‘ ,/’ / I ‘ a _ /" \ r“ L J ,1 - lk ' #—~ . i l m _. /"’ f “’\ — d- ” T -‘ ‘5. // \\\\ /,,// \ \ / (= fl {a .\ \ / ,.- ,- R x 3 \ ,. . A . l 3'». \ ("x \ .\ 1‘ . x» / ’\ / ‘ . l ‘ .‘z \ , \ .. . / c 4 \ 1“\:\ \I‘I‘II‘ I“. s ‘1 .\ v ' I 3‘ I/ L , \ , , . . .’ ‘...- 7‘,‘ . h N _ K‘x \.I‘/ '5 f \’i'.' U ) / . r '5'“ [I r» 1 .r‘ " \ '. \ ' A . i , . ix? ; 1 ’ . ‘ . . ‘ ‘A\ ‘ / l ) I, K _1 -’ ~ ' .' ,7 ‘ ‘- . ‘ a ‘v . 4 _ .. I ‘\ \\ .\ 7. 1. I. .' 1 «\r‘ c. ./ 43 ‘ , k‘ a : ‘. ..I" , _ .‘ ‘ , i. , . a" , < \. q \, ’ . . f , t 1 "fl \‘ 5?’ . “.». I, . .h I r 1 v4. .- . . .1 1 .; \~. \, x , ..- -,. ~( ~. / ,9! j) , . A.» ~ , - , V ,' r .4“ . x ' ’ ~ , ,3 \ \‘ \‘\‘ g . ,\ /\ \ , . / 1‘ i I ~,:\ ‘ . , 3 \ . '. ‘3 . . 7.7,. t \ “-\‘~ K . I \ .\ Lr I. _.//« ., .‘ , p . 7._.‘_-, / ,r/_. \\ ’ _, z]. _ 5 . . . \ _ .. ,,- . , 1. r ‘ 1‘ . .~-.' ‘ (I ,, .5 ., V‘- . . _..-‘ . I .t2;:~-—.:\‘ ] z” //- -‘ I“ II ,-»::‘/A‘ I ‘I '5" .' ,. ..1 \\ I " . . ' TL I] ( __ x :11; . ’,~_\. I‘d, ~35 51:19:“ ‘ ~ ['3/ ,, N ""'V.:7:;7~'~:' ‘ \_‘u._i" r-—’_'I"_.~' /, \\ \.\T;“—I=; , ..-.”r,» 1 f 7 ‘. / \\ [:>\ < —- 3 -— fl 1 ’ .“ l . f 7 _ . - - " 7 -4 |_ r- I ’ I i . 1 r \ H‘ #1 _ _ ___L‘,11’_.._‘ L. EJT—ah.—__.__L. ___. i A;_L;T—-:A _. a Figure 3.1: Machine’s stator configurations degrees ( 600 and —600 mechanical degrees). The flux linkages components in the stator are given by equation 3.11. In this equation, the stator flux linkages depends on the rotor position He and on the pro jec- tion of the flux linkages Of each phase over the direct and quadrature axis. A = Ad + qu = 8198(1, + AbejQW/3 + Acej47I/3) (3.11) Figure 3.5 shows the machine at rotor position Of 900. With no stator current, the rotor flux only links the phases “b” and “c”. The direct axis flux linkages consist of the sum of the projections of the flux linkages of phases “b” and “c”. The quadrature axis flux linkages is zero because the projection Of the flux linkages of phases “b” and “c” cancel each other. When the rotor position changes, the direct and quadrature flux linkages components Of each phase change, however, the stator flux linkages have only a direct axis component and their magnitude remains almost constant. Now let us consider a machine with stator saliency like the machine with HSPP whose stator and rotor are shown in Figure 3.6. 21 ’ ‘0'1 y ~o-05/ -— "025 -10 7V ' , _1of’T'0-25 / —\_—— —0_55/ a0.45 A _\ ___... A ’—“ 3:. -2o » ‘0-7 g —20» . _ \ - W 50.65 " —3o - “0'95 ‘ -30 KN . ..0-8 ...— I‘ ‘0.95 _40 \ . _40 . 1 1O 20 30 40 10 20 30 40 lq (A) la (A) (a) HSPP (b) ISPP 4105/ / 4105’- / / -10 a, —o.25 - -10 r’ ‘0-25 43.45 . e f/ a E ‘20 ‘ 2 '20 ’ -065/ f/ — .65/ —30 H 0 ‘ -3o - _0_8/ m -O.8 —-—— 10 20 30 40 1o 20 30 4o lq (A) la (A) (c) 28PP (d) 4SPP Figure 3.2: Direct axis flux AAdUd, iq) Figure 3.7 a—b shows the machine and the equivalent flux diagram at a rotor position of 900, with no stator currents. The stator flux linkages have only direct axis components. When the rotor moves, Figure 3.7 c-d, the stator slots modify the shape the rotor flux linkages, creating a quadrature flux linkages component oscillatory with the rotor position, however, its average over one revolution is zero. In a similar way, when the rotor moves, the direct axis flux linkages change, but its average over one revolution is similar to the value at 66 = 00. A similar analysis can be done when a stator current is applied. Let us consider the case when the current space vector is aligned with the quadrature axis, at rotor 22 ld (A) Id (A) l I N —L e e ..I/___ ld(A) | l N _l O O M A -30 o _o ) .2 , 1 -30 90;: N O ) .P 'A NU'Im -40) .\\-\\ _40lll 1 10 20 30 4O 10 20 3O 40 Iq(A) Iq(A) l l N _s O O 30 fit/m 14 7 . . :’\ ld(A) I l N _s O O 75 17 -30 o p o _o 01 on N co 10 20 30 4O 10 20 30 40 la (A) la (A) (c) 2SPP (d) 4SPP Figure 3.3: Quadrature axis flux AAq(id, iq) (a) rotor (b) stator Figure 3.4: IPMSM with 4SPP and with sinusoidally distributed windings 23 (a) Flux distribution (b) Equivalent phasor dia- gram Figure 3.5: Rotor flux linkages for a 4SPP machine 03: 900 [e d- O) “x 419% (9/ 1M 3% a (a) rotor (b) stator Figure 3.6: IPMSM with HSPP and with concentrated windings position Of 900, as shown in Figure 3.8-a. Under no saturation, the stator flux linkages are equal to the sum of the flux linkages due to the rotor magnets and the quadrature axis current. However, when the saturation is taken in account this statement is no longer true. TO explain the saturation effect, let us calculate the flux density in two pointsm “ m” and “n ” symmetrically located around the quadrature axis. The rotor flux linkages define the Operation point of the machine, then, with no stator currents, the points In and “n ” have the same flux density 3. 8-b. When the quadrature axis stator current is applied, the flux linkages due to the current add to the rotor flux linkages 24 R \ \ \ \ \ \ \ \ \ I / A"9/\’~b @ © (Ad (b) Equivalent phasor dia— gram 682 900 “d (c) Flux distribution (95: 750 ((1) Equivalent phasor dia- gram 082 750 Figure 3.7: Rotor flux linkages for a HSPP machine 96: 900 and Be: 750 in the point “It” and subtract in the point “m” as shown in Figure 3.8-c. Under saturation, the flux density in “n” is lower than the linear sum of the rotor flux linkages and the stator flux linkages due to the stator currents; the flux density in m is more than the difference of these fluxes because the flux moves away from saturation. The previous analysis can be used to determine the total flux linkages at the same rotor position. The rotor flux linkages and the flux linkages due the stator currents are added in the tooth of phase “b” (like in the point n), this means that under 25 - - Rotor flux linkage; — Stator flux linkages @ ( due to stator current (a) Flux distribution 1.5 1 1-51 —‘ —+ E \4 8 En _: 8 En T " " f: Bm : t: Bm“ m r m a : Bin” lofinIHIIOSI 16/HIH\05 I] III ' an I] m HI,1 ' H(kA/m) H(kA/m) (b) Rotor flux linkages in m and n (c) Stator flux linkages in m and n Figure 3.8: Stator flux linkages for HSPP machine 68: 900 saturation the total flux linkages are lower than the linear sum. In the tooth of phase “c” (like in the point m), the total flux linkages is the difference between the rotor flux linkages and the flux linkages due the stator currents. At this rotor position the direct axis component of the flux is lower than the rotor flux linkages, and the quadrature axis flux linkages is lower that the flux due to the stator currents as shown in Figure 3.9-c. When the rotor moves, the saturation and the saliency produce pulsation in the stator flux linkages as shown in Figure 3.10. The saturation produces a decrease in the stator flux linkages as shown in Figures 3.10 a-b. This analysis shows how the direct 26 (a) Stator flux linkages due to PM (b) Stator flux linkages due to stator current f. \ \ ----> Linear case @‘. —’ Including . \ saturatlon (c) Total flux linkages in the stator Figure 3.9: Space vector diagram of flux linkages axis current produces a decrement in the rotor linkages flux. Permanent axis flux and quadrature axis current component are orthogonal one to each other, then exist cross-magnetization. In a general case the saturation nonlinearly with the stator currents. Then the cross level Of the machine depends magnetization effect depends nonlinearly on the direct and quadrature axis components of the stator currents. l 80 90 219 "* '* \11 50 120/..-» ‘"\.§0 // ...,.v""90...~ . \\\ //_./ .~."'....'099..'.‘ \..\ 240:" ’90‘ /’ \\\ ' / PIX T... h 1 20 1 50 A", .9. -> I (1:0 I q=O v . t .3 I .3 ' v. 7. " . 3‘ , v , 9 ' V o, = = 3 / ' - zt '. i ”I. ' A” q I d O Iq 20 . ‘1 . . v' 270. 5,1" 0.5 -90 180; 2 1 .. ----------- s. I. 0 . ...; \..\\ :3 I[ ’2 E . 9 . .... I A"(:1 . to: , 1 z, '9'. \\ 6.. ,\ // . \AK ‘ /: ..I' ‘ V. .0 f, 300 ”v. ° . ' ' 60 210 ‘x ‘~ ~ ~" 330 \\ T'O.N!,.,cw'v /‘ \\ ”fovy. " ,// \ \x / \\ ' ”Vtvt‘. ,,/ \ / \ \ // 330 \1 _O / 30 240 1; 1 300 (a) Direct axis flux linkages (b) Quadrattu‘e axis flux linkages Figure 3.10: Stator flux linkages as function of (98 27 3.4 Cross Magnetization: Machine Model and Cal- culation The 2-axis model presented in section 3.2 does not include the cross-magnetization effect. Figure 3.2 shows that AAd changes with iq and AN, with id, Figure 3.3. The model that includes the dependence of the flux linkages on Id and iq is presented below, and the methodology to calculate the parameters is explained and applied to the machines presented in section 3.3 . 3.4. 1 Machine Model A more detailed model that includes cross-magnetization was presented in [9, 10]. This model includes mutual inductances in the flux, voltage and torque equation as follows: Ad = .\(I_PM + Ldid + 111,),qu (3.12) /\q = Aq_PM + Lqrq + 11le (3.13) ”d = Rsid — weAq + 33%! (3.14) ‘vq = Rsiq + wekd + (3% (3.15) v? = 213 + 113 (3.16) Tm = 3; (Ad—PAM - Aq—Pririai) (53-17) T. = g; ((Ld _ Lq) idiq) (3.18) rc = 3; (—qur3 + A» (1,113) (3.19) T=Tm+Tr+Tc (3'20) 28 This model includes the dependence of ’\d on iq by including the quasi-mutual inductance A/qu; in the same manner, it includes the dependence Of /\q on fd by the mutual inductance qu . An additional torque component Tc is produced due to cross saturation, which is dependant on the currents and the mutual inductances. 3.4.2 Model Calculation The model parameters are stator resistance, rotor flux linkages, self inductances and mutual inductances. The resistance can be measured and its value will be considered constant for the whole Operation Of the machine. The rotor flux linkages will be assumed constant with a value equal to the flux linkages when the stator currents are zero; any change in the stator flux linkages will be taken in account by the inductances. Theses inductances are calculated as the ratios of the flux linkages AA due to stator currents and the stator currents components that produces the flux. Let us consider two particular cases: 1. The stator current only has direct axis component. The inductances Ld, qu can be calculated as follows: MW» 0) — Act—PM Id (3.21) LdUd, 0) = Aq(1d,0)— )‘q—PM M Id qdudi 0) : (3.22) 2. The stator current only has quadrature axis component. The inductances Lq, qu can be calculated as follows: Aq(0,1q) - Aq—PM [q (3.23) quoilq) = 29 AmJ)—A_, 1., (3.24) The calculated self and mutual inductances are presented in Figures 3.11 and 3.12. 0.12 \ - . - 0.12 or \\ . dt‘\\ A \ A 1—2 \ 2;. \ \ a: 0.08’ \ a: 0.08» \ 8 \ \ 8 \ 3 \ g ‘ \ 3 0.06: \ \ 3 0.06- \ 'o \ 'o \ \ E ....... , AAAAA _ \ g \ 0.04~ 0.04— 0.02 . . . 0.02 . . ...... . ........ 0 10 20 30 40 0 10 20 30 40 Current (A) Current (A) (a) HSPP (b) ISPP 0.1 . . . 0.14 f ~ . \ \ \ 012 .. ..... Ld ("Id) \ - ’ \ \ — — —- L I ‘ AQ%_ \ , A \ QM) 5 \ £(M- \ s ‘ \ s ‘ . g 0.06 \ \ g 0.08: \ \ g \ \ g \ \ '2 \ \ g 0.06: \ \ - — 0.04 ' I — \ ~ 0.04- . 0.02 ....... . '''''''' . IIIIIIII . _______ 0.02 ....................... - ........ 0 10 20 30 40 0 10 20 30 40 Current (A) Current (A) (0)2SPP (d)4SPP Figure 3.11: Self inductances Ld and Lq These figures show that the inductances Ld and IIqu are almost independent of I d, and Lq, qu change considerably with Iq. The basic approach to calculate the machine parameters with cross-magnetization is to include the dependence Of self and mutual inductances on both Id and Iq . However, to calculate the inductances we have two equations (312,313) and four unknowns (Ld, Lq, 1).]qu and Alqd), hence 30 A O: A 0: :1: I .5, é a) -2. a) -2. 8 , ’ \ E - (U ’ \ cu ‘ ~ *5 -4 \ 1 ‘5 ‘4' \ 3 \ j \ E \ \ E ‘ . - -6* \ c — '6’ T ‘ s I _8 . . . _8 . . . 0 10 20 3O 40 0 10 20 30 40 Current (A) Current (A) (a) HSPP (b) ISPP 2 - 2 1 e 0’ I I 3? 0. g, g. ...mgdr'dw a) -2 . a) q ( Cl) 0 \ 0 _4 \ g \ ‘- E \ \ ”5 '4' \ 73 \ D \ 3 ‘6 ’ \ 'o \ U E _6 . \ E. \ \ \ \ \ _8 \ \ \ — — _— ._8 1 . . _10 . . x 0 10 20 3O 40 0 10 20 30 40 Current (A) Current (A) (c) 2SPP (d) 4SPP Figure 3.12: Mutual inductances .4qu and Ilqu it is necessary to make some simplifications. The previous results show that Ld remains almost independent of currents (it changes about 5% for 18PP, 2SPP and 4SPP and 21% for HSPP) and the inductance qu is the smallest compared with other inductances and it is almost independent Of I d' Then, the dependence of these parameters on Iq should be small. The assumptions proposed here are to consider that Ld and 1)qu are independent of Iq, and the flux components due to these inductances are equal tO the product Of them with I (1' Any additional change in the stator flux linkages will be included by considering that Lq and Aqu are functions Of both I d 31 and Iq. The procedure used to calculate the inductances is described below: 1. Calculate the flux linkages following the procedure described in section 3.2. 2. Calculate Ld(Id, 0) and 111(1(1(Id,0) by using equations (3.21) and (3.22) . 3. Calculate Lq(Id, Iq) and rlquUd, Iq) from equations (3.25) and(3.26). /\q(1d» 1(1) T A( —P.’lf _ Afrrlfldtofld qulqalq) = 1 lg 1 (3-25) ’\d(1dvI€1)‘ Ari—PM - LdUd» OIId 1 q quud, Iq) = (3.26) The procedure described above was used to calculate the inductances Of the ma- chines analyzed. TLd and [Ilqd are the same as shown in Figures 3.11-3.12 and Lq and qu with cross magnetization are shown in Figures 3.13-3.14. To determine the accuracy Of the proposed methodology and the improvements in the flux and torque calculation over the classical 2-axis model, the torque as function Of the angle of the current space vector is calculated. The cases analyzed are: 1. Classical 2-axis model of equations (3.1-3.7), and include the saturation effect: Ld(Id,0) , Lq(0, 1(1) , 6’qu = 0 and Alqd = 0. 2. Cross-magnetization model of equations (3.12-3.20), include the saturation ef- fect Of each inductance but not the dependence Of them on both currents I d and [qt LdIId’O) , Lq(0,1q) , A’qu(0,1q) and AlquId’O)‘ 3. Use the cross-magnetization model of equations (3.12-3.20), and include the sat- uration effect Of each inductance and the dependence of them on both currents c Id and [qt LdIId’O) , Lq(Id,Iq), Aqu(ll,1q) and IWQCIIICI’O)‘ 4. The FEA model; the torque is calculated directly from F EA. 32 . I. L -10» 8 -1o- 6: S. 2‘ _ Q5 2‘ _. 9‘3 E ’20 L— 699 Z: ’20’ 3° —o.09 -3o ”0.08 -30» g g 0' o1 __, 07 _40 0‘ . . _40 . I 1 I 10 20 30 4o 10 20 30 40 la (A) In (A) (a) HSPP (b) 1SPP I. . -10. a, -10» g 3 -20 7‘99“?) 3 -20. P- 2 -30- .o s .0 4 —30» .o g o 2 2. o' 8 3% o' 8 8 10 20 30 4o 10 20 30 4o '0 (A) '0 (A) (c) 2SPP (d) 4SPP Figure 3.13: Self Inductance calculation Lq(id, iq) The cases described above were used to calculate the total torque Figure 3.15. This figure shows the calculated torque as a function of the current space vector angle. The most accurate torque calculation is the one produced by F EA analysis, and the torque error is calculated with respect to the value Of F EA. The model presented in the numeral 1 has an error of about 15% for a current space vector angle Of 900. The model described in the numeral 2 has an error of 15% at maximum torque, while the proposed model in item 4 has a maximum error less than 2%. The torque calculation shows the accuracy of the methodology to determine the model parameters. After validating the results it is possible to accurately decompose 33 Id (A) do :1: O 0 900° 00 § /‘?/ Id (A) do :1: 9 O \000 I g o O a / / - 1o 20 3o 40 1o 20 30 40 la (A) la (A) (a) HSPP (b) lSPP K.0006 \ -10 —10» -0.006-« _\ ‘0.004.n A A —o.o ——« s -20 . 4 5 -2o 04 . E 0 -o 002 E ~o —3o ; 0 v . -30 r 0 . \ /—\o \ o\ -40 ‘ ‘ - -4o . - - 1o 20 30 40 1o 20 30 4o Iq (A) la (A) (c) ZSPP (d) 4SPP Figure 3.14: Mutual Inductance calculation qulid’ z'q) this torque in: PM torque, reluctance and cross magnetization. This decomposition gives the tool to redesign and optimize the machine according to the requirements. Figure 3.16 shows that different stators can achieve different torque components. For example with 4SPP it is possible to achieve more PM torque and reluctance torque, however due to the cross magnetization the total torque is reduced. For the configuration of HSPP, the reluctance torque is the smallest, however the total torque is not negligible compared with the other stator configurations. Finally, after selecting the maximum voltage of the power supply the field weaken- ing profile is calculated. Figure 3.17 presents the field weakening curves for maximum 34 200» 200» 1 1 150;. /’ 7: .\ 150;_.~=’3,/" A\ ’2‘ / , .. ’ x E c x Z _ __ , , \ g // ‘ o 100. . a, 100/ P P 50» . . 50» C . \_ \ \. . o A A A A o A A A A 100 120 140 160 180 100 120 140 160 180 6(Deg) 8(Deg) (a) HSPP (b) ISPP 200 200» 5 ‘ / ff. ’ . /" /" // / / ‘ ‘- . 15o» / /”‘ 150~ // \, A - / / . A / . E o ’ \ E ,2 / ...... I Z // , Z ./ sat1 31007 3100» ___, P P sat2 50* 50» — cross TFEA 1 . r i 0 . . i A 100 120 140 160 180 100 120 140 160 180 8(Deg) 5(099) (c) ZSPP (d) 4SPP Figure 3.15: Torque calculation including cross saturation torque per ampere; it shows that although the configuration of 4 SPP has the highest maximum torque it has lower field weakening range and lower maximum speed com- pared with ISPP and 2SPP. Although the configuration of HSPP has the maximum inductance Ld and hence more capability to weaken the rotor flux linkages, the field weakening range is limited because it has the lower maximum torque. 3.4.3 Experimental Model Calculation The cross saturated model parameters were calculated for an IPMSM with the pro- cedure described in the previous sections. The experimental setup is shown and 35 ‘t e (Nm) 1: rel (Nm) 200 150’ 100 50’ 120 100’ 80’ 60’ 40‘ 20 100 120 (a) Total Torque 140 5 (Deg) 160 180 100 120 140 8 (Deg) 160 (c) Reluctance Torque 200 0 0 180 1 m (Nm) I cross (Nm) 150 . 100 50’ 100 120 140 8 (Deg) (b) PM torque 160 180 HSPP —-— 1SPP . — —-—2$PP 4SPP 100 120 140 6 (Deg) 160 180 (d) Cross Magnetizing torque Figure 3.16: Torque components calculation # - ~A-HSPP —A—A1SPP — —-—2SPP 4SPP ~_ “I‘ 4 2000 4000 a) m (r/min) 6000 8000 10000 12000 14000 F igurc 3.17: Field weakening curve for the configurations analyzed 36 described in Apendix A, for reference, the schematic of the experimental setup is shown in Figure 3.18. Experimental calculation of the flux linkages requires the mea- surement or estimation of the induced voltage. Power RT linux controller + FPGA _ supply ' (Controller) 7 + V T Labview V “ Control ‘ l 66 Controller i VDC Signals . . 6 Encoder C I l l lbs ZCS e a) 1525 25 ’" 25 2525 [I] . J7 /’J/ Inverter ”l7 Torque-meter J7 PMAC motor Dynamometer Tachometer PMDC motor Figure 3.18: Schematic of the experimental setup The initial approach was to consider that the commanded voltage is equal to the machine’s terminal voltages. This commanded voltages were used calculate the flux linkages as shown in Figure 3.19. Figure 3.19-a shows a significant dependence of ’\d on iq. In Figure 3.19-b it is shown that Aq changes with id when iq = 0A, which can not be explained by the cross saturation model. The second approach uses the procedure described in [12], where a low pass filter and a data acquisition system sampling with a sampling frequency of 10 times the switching frequency where used to record the voltage signals. The calculated flux linkages using this method are shown in Figure 3.20. The results of Figure 3.19 and Figure 3.20 have significant difference caused by the dead time, that is, the commanded voltage is different to the filtered-measured voltage. Let us consider the three phase inverter presented in Figure . To determine 37 _ . , _ , l / / / 5 5 L 5; ° 3- 3- " -10 . ‘0-05 < -10 { 3 -15 /\ ‘0 q 3’ ‘15 E '75 E \—< -20 ,_/\ ‘ . -20 025— CI) -25 *_—/\ -25 N ~ -0.35— l -30 ‘ ‘ ‘ -30 -20 0 20 -20 0 20 '0 (A) 'q (A) (a) AAd (b) mg I . . 3 ('3‘ , \ .1 J. n 0 0‘1 970-*—‘* 920-—* 0 025“ 045W NOSSE M 0?5\/ —99°o- ,ofb .— 5 _20 ___/\A/ 09°?) , -20 . '3“ 0 -2o —10 0 10 2‘0 —2‘ —1‘0 0 1o 20 In (A) In (A) (a) AAd (b) mg Figure 3.20: Calculated experimental Flux linkages using Vmes the dead time effect in the machine voltage, the voltage in one phase leg is analyzed, Figure -b. The ideal switching commands are shown in Figure -a. The ideal pulses need to be modified to provide dead time to avoid a short circuit in the phase leg, Figure -b. During the dead time, the phase voltage is not zero, because due to the current, the diodes conduct during this time connecting the phase terminal either to Vdc/ 2 or to —Vdc/ 2. By neglecting the on time and off time of the switches and diodes it is possible to determine the increment or decrement in the phase voltage, Figures -c and -d. 38 $2112} S1 Vdc/2__ ‘{ Vdc/2 :1:— T 1 S4 1311:31— (a) Three phase inverter configura- tion (b) Configuration of one phase leg Figure 3.21: Three phase Inverter The increment the three phases voltages due to the three phase currets is calcu— lated as follows: T 'Ava=— at [23ign(z “E Avb= :[TfiZl/Ifc 23ign(z’b Avc— — is%Q[ [25ign(ic a) — sign(ib) — 2sign(ic)] — signed) — 28ign(ic)] (3.27) — sign(ia) — 28ign(ib)] where Tsw is the period associated to the switching frequency and Td is the dead time and Vdc is the DC-link voltage. U; , v; and '05 are the commanded voltages. These differences produces that the commanded voltage differ from the machine terminal voltages. Dead time compensation were used to reduce the error difference between both commanded and measured voltages. Several approaches have been proposed to compensate for dead time [13, 14, 15], where dead time, on—time off-time of the power 39 S1 S4 p——-—---— ——-——-—— Td/2 _. .— _. ._Td/2 S1 S4 —; s—Td/2 Td/2—S .— ______ Td/2 Td/2 (d) Van for id < 0 Figure 3.22: Switching pattern and actual voltage in the machine devices and their associated drop-off voltage where included to corrected to for this effect. A simple approach is used here to correct for dead time. The drop of voltage in the power electronics devices and the associated on and off times were neglected. The commanded voltages were corrected as follows: T V . . . . . . va_ [NV 2 v; + 7L—%Q[ 2szgn(za ' — szgn(zb) — 2s'z.gn('z.c)] vb— INV_ _ 11* + -Td——1g—C[2sign(ib — sign(ia,) — 2sign(ic)] (328) vc—INV — v25 + £3‘Klggl [2szgn(ic — sign(ia) — 2sign(ib)] 40 where va_ I NV, vb_ I NV and vb_ I NV are the voltage used to calculate the switching times in the inverters. v; , v; and U: are the output voltages of the PI current controllers that are used to calculate the flux linkages. The calculated flux linkages with dead time compensation are presented in Figure 3.23. ‘0.07 W “DA \ _.——4 o. ..- 0y 97'0- ——1 SZ'O- —— 0.25 —~—‘ 0.45 —1 0.62 *— _5 -5 g p 1 W’W 8 d s, -o.2 S, 2 ’15/\/:o \ 32 _15. N -25\ \ cow—~03; —20» m N~os4— '3 —25 A A A A 4 —25 A- ~ -20 -10 o 10 20 -2o -10 0 1o 20 la (A) In (A) (a) mid (b) AAq Figure 3.23: Calculated Flux linkages using V* and dead time compensation A better agreement was obtained between the calculated flux linkages of Figure 3.20 and Figure 3.23. Phase voltages where compared and the maximum error is less than 5% and the maximum phase shift was about 3 electrical degrees. The previous analysis verified the accuracy of the proposed approach for dead time compensation. The flux linkages shows the effect of cross saturation, where, the direct axis flux linkages changes with the quadrature axis current. Taking into account the cross sat- uration effect and dead time compensation, the parameters calculated for an IPMSM prototype are shown in Figures 3.24. In addition to the cross saturated model complexity, the iron losses were included in the torque equation by subtracting a torque component due to the iron loss from the electrical torque. This component is proportional to the speed with a slope of k as described By Senjyu et a1. [16], however, the dependence of the iron i-loss losses on the stator current was neglected here. The iron loss coefficient. of the motor 41 is 15,4035 2 0.0013 N ms. The remaining parameters of the motor drive are presented in Table 3.1. Table 3.1: l\~lachine parameters Parameter Value Units Number of pole pairs 2 Stator resistance 0.4 Q Rotor flux linkages .4652 llll’b Rated peak voltage per phase 120 V Rated peak current 20 - A Rotor inertia Jm 0.1938 kgm2 Viscous friction coefficient Bm 0.0043 N ms Iron loss coefficient k-i— loss 0.0013 Nms 42 F’_—.—F'TTW_I 0.02 . . 0.04 0.03 ' i: 0.018 g 8 8 0.02 ’ g 0.016’ g g g 0.01 * '2 2 0.014 0 0.012 ‘ -0.01 ‘ ‘ -30 -20 -10 0 -30 -20 -10 0 Current (A) Current (A) (a) La (1)) qu O -5~ o b D _5. Q 8 l t m D )6 061’ 8. -1o- . —1o~ ,0. <3 3 -15~ o 3 —15» E o E C” I -20 i v- -20 ' 0 <1" 0 N O —25» c5 —25 :3, 8 l I ‘I’ -30 ‘ A -30 m . 1 . -20 0 20 -20 0 20 la (A) Iq (A) (c) Lq (d) qu Figure 3.24: Calculated experimental inductances using V* 43 Chapter 4 Trajectory Optimization for the Operation of Traction Motors Motor efficiency and hence the motor controller performance depend on the model ac- curacy used for the controller design. In high power and high power density machines saturation and cross saturation significantly affects the flux linkages and torque; in the same manner current commands calculated from the cross saturated model differs significantly from those calculated using the classical two-axis model with constant inductances. Here the cross—saturated model [9, 17, 11] of the IPMSM was used for the algorithm design. Several approaches have been developed to design the speed, torque and current controllers for PMSM. Jeong et al. [18], Shinn—Ming et al. [19] and Cheol et al. [20] calculated the optimized currents commands for torque control, however, cross saturation was not considered and the trajectory calculation for the speed control was not analyzed. In [21] the online calculated current commands depend onthe predefined motor losses; this control scheme allows the operation of the motor with fixed losses for a wide range of speed, but cross saturation was neglected. Molavi et al. [22], Mademlis et al. [23] and Cavallaro et al. [24] designed optimal speed 44 controls without including the saturation effect in the inductances. In [22] the current commands were calculated using a suboptimal controller by considering the motor as a linear system with constant parameters. In [23, 24] a current controller with minimal losses was presented but the speed control was not analyzed or optimized. Panahi et al. [25], Schlemmer et a1. [26] and Choi et al. [27] presented optimal time trajectory references calculation, where motor losses were not optimized. In [25] a simpler algorithm is presented in order to apply it in a low cost controller. In [26, 27] the limitations of the actuator were included to get better accuracy. Nifio et al. [28] presented an optimal speed controller of an IPMSM, however the iron losses were neglected and the details for an on-line implementation were not presented. The contributions presented in this chapter are to develop an algorithm to calcu- late the speed command for the speed controller and an approximation to the optimal controller command to be used in real time. The speed command is calculated by a trajectory optimization. This trajectory has to achieve a predefined speed in a fixed time, minimizes energy losses and keeps good dynamic performance of the drive. The cross saturation effect was included to improve the solution accuracy, where the al- gorithm and modeling complexity are justified for high power density machines like the motor prototype presented in this thesis. The optimization presented in this chapter, to determine a trajectory for the speed controller of the traction motors in a series HEV, was developed in two stages. In the first stage the optimal currents I s and I d are determined to operate the machine at a specified torque and speed point with minimum losses. In the second stage the optimal trajectory, between two torque and speed points, is determined. 45 4. 1 Problem Description The optimization problem presented in this chapter results in the calculation of the speed reference for the controller of the traction motors in a series HEV as shown in Figure 4.1. In Series HEV the speed command for the traction motor is given by the main controller, and the transition from one operation point to another is decided by the motor controller. The optimization goal is to produce a transition between two operation points efficiently, that is minimizing the motor losses. Supervisory Control DC converter Traction and energy motor+ -— storage I; inverter Generator En in g e + Inverter Figure 4.1: Series HEV configuration The subsystem of traction motor and the wheels are simulated by the motor de- scribed in section 3.4.3 and a dynamometer was used to simulate the load (equivalent inertia of the vehicle and the additional torque due to the grade). The experimental setup is the same as shown Figure 3.18 and its details are described in Apendix A. 4.2 Maximum Torque Calculation The calculation of the optimal currents commands was developed by the numerical inversion of the model shown in equations (3.12-3.20). In order to invert the model, 46 a control strategy should be defined; the control strategy was based on the maximum efficiency operation of the motor. The algorithm used to calculate the current corn- mands I d and Iq that operates the PMSIVI at a specific torque and speed point with minimum losses is presented below. 1. The system of equations (3.12) to (3.20) is solved to determine the relationship between torque and speed with stator current components. The fist part re- quires to calculate the torque from known values of stator current components I d and Iq. It is unfeasible to analyze all operation points of the motor, a repre- sentative combinations of I d and Iq where analyzed. For simplicity the torque is evaluated in the points located in a mesh of Is and (5 (that is equivalent to evaluate combinations of I d and Iq). The points in the mesh are equally spaced in 13 in a range from Is = 0 to Is = I may: and equally spaced in 6 for a range from 6 = 900 to 6 = 1800. 2. The torque is evaluated for a selected value of I 3, the angle of the Current Space Vector (CSV) ‘6’ is varied from 900 to 1800 and the machine output torque is calculated. As an example, two typical curves of the torque as a function of 6 are shown in Figure 4.2 for stator current values of Is = Imam and Is = Imam / 2. For the selected stator current 13, the angle 00(13) for maximum torque was calculated, in this case 00(Imaw) = 1280 and a0(1ma$ / 2) = 1160. Below the corner speed, the machine was operated at maximum torque per ampere, hence from zero to the corner speed the current CSV angle is equal to (5 = 00. 3. The corner speed was calculated by solving equations (3.12-3.16) in ‘steady state’. The unknown values of Vd, Vq and we are calculated from the motor model and with the known values of v3 = Vs—max (that is the maximum available voltage produced by the inverter) and the stator current components I d and Iq (calculated from the selected current Is and its angle for maximum 47 torque per ampere 6 = (10). For the example presented above and for is = Imam, the calculated corner speed is 630 ’r/min. . Above the corner speed the machine operates at the maximum voltage of the inverter (not at maximum torque per ampere), and the angle of the current space vector was increased. The CSV angle is increased in fixed steps, from 6 = 010 to (5 = 1800. For the sets of I dr Iq (generated by changing 6 and for the selected 13) and V5, equations (3.14), (3.15) and (3.16) were solved for Vd, Vq and cum. . The previous steps allow us to determine the relation between the angle of the current space vector 6 and the speed, hence the relation between torque and speed was established. Figure 4.2 was modified and the horizontal axis was changed to speed, as shown in Figure 4.3. . The torque calculation described above did not include the iron losses. To account for them, the torque at every speed was decreased by the torque loss due to the iron losses. Figure 4.4 shows the corrected curve. For the machine presented in section 3.4.3 the iron losses were not significant within the speed range, this is why Figure 4.3 is similar Fig. 4.4. The procedure described in the previous items is used to determine the relation between: Te and cum with Id and Iq. . The previous process was repeated for Is from Imam to zero. With the cal- culated values it was possible to create maps of 2's and id as shown in Figure 4.5. 48 ls=lmax E 30 ' ----- Is=lmax/2 33 2o~ , —— ---------- ~ - 0 1 r r r 1 r 1 r 90 100 110 120 130 140 150 160 170 180 8 (Deg) Figure 4.2: Torque as function of 6 for rated and half of rated current 50 r . r e (Nm) on O N O 0 500 1000 1500 o) m (r/min) Figure 4.3: Torque as function of speed for maximum current 4.3 Optimal Trajectory Calculation for the Oper- ation of Traction Motors The second stage of the work address the speed reference calculation by a numerical optimization. The trajectory calculation problem presented in section 4.1 uses the currents commands calculated in section 4.2 to determine the speed reference. The 49 50 r . 1? e (Nm) w o N O 0 l r 0 500 1000 1500 (n m (r/min) Figure 4.4: Torque as function of Speed corrected by the iron losses trajectory was calculated by minimizing a cost function, this minimization was donee by a numerical constrained optimization, based on the steepest descent method [29]. Trade of between losses minimization and high dynamic performance is the criteria used to determine the trajectory. 4.3.1 General Trajectory Optimization Algorithm In this subsection, a general minimization problem is presented and the necessary conditions are shown. The performance cost J is a function of the input u(t), the state variables :r(t), the initial time ti) and final time t f: t f J = f (mm) + / g<.u.t>dt, (4.1) tr where f is a function that depends of the state variables at the final time and g is a function that depends on the inputs, outputs, and time. The state equation (4.2) describes the dynamics of the system. 50 50 Figure 4.5: Optimal current commands é o - « Q) 65 P _. 5 / l/ 10 15 _50 llllll 1‘Ill lllll 1 lllllllll 1 lllllllll 1lllJ20 1 1 0 200 400 600 800 1000 1200 1400 (o m (r/min) (a) 2's 50 T l I r r r """"""""""""""" \\\ .9 r— _ ' o 10 \75 M -5 —-X§ E ~5 E 0 F 2 F (D P _— —5 -5—/ L -10 -10 _50 ....... 1 ''''''' 1 IIIIIIII 1 """""" 1 """""" 1 ' ' ' ' I | ‘1' 2 0 200 400 600 800 1000 1200 1400 a) m (r/min) 0?) id with the initial condition To minimize the cost function, the control functions 11(t) have to be detern‘rined. The minimization method used is the minimal principle as described by Athans and Falb [30]. The calculation of the control function with the minimal principle method requires the calculation of the Hamiltonian shown in equation (4.4). From the Hamil— tonian, the differential equations used calculate costate function P(t) and the bound- ary conditions are determined. Finally, by solving the state and the costate equations, the control function is calculated. The Hamiltonian, ”H, is defined as: 71 (110.1(0). 13(0) = g (1130). 110)) + P100 (“0.140) (4.4) The costate function has to satisfy: Pft) = _37‘1(;:t()t,)u(t)) (4'5) with the boundary condition: 0f (:L(tf)) ___—811%) — P (tf) = 0 (4.6) Besides satisfying the previous equations, the system has to satisfy the following necessary condition: aa(:r(t>.u(t>.P> 0 = 0u(t) (4.7) 52 4.3.2 Continuous Time Trajectory Optimization of the Trac- tion Motor System Controller The general algorithm for trajectory optimization in subsection 4.3.1 was applied to the calculation of torque and speed references for the traction motor. Initial and reference speed, load torque and final time are assumed to be known before the optimization starts. The performance measure function ‘J’ was designed to give a tradeoff between dynamic performance of the drive and the motor losses. The performance measure function, also called cost function, is shown below: tf J = f(w7n(tf), if) ‘1' g (Wyn, is, t)dt ti = h.0(wrn(tf) — wref)2 (4.8) if 2 3 _. 2 2 + hl ((4)77) — Lure-f) + ’12 51213 + Ali—loss”!!! (It t); where ho is the weight used to lead the algorithm to reach the final speed, hl is the weight for the time required to reach the reference speed, and fig is the weight of the motor losses. The selection of these gains leads to a tradeoff between dynamic speed response of the controller and energy losses. The mechanical equation that describes the dynamics of the motor and the load is shown below: 8w . Jm 0:71 = Te (lsawm) — Tl " Bmwm (4-9) where Jm is the inertia of the system, T) is the load torque and Bm, is the viscous friction coefficient. The mechanical system can be rearranged as: 53 aw"). 0t = Mix/"rm is: t) . 4. 10 Te (is, Wm) Tl B mwm ( ) Jm Jm J’NL with the state :c(t) =wm and the control input u(t)= 15(t). The nonlinear relation between T1, cum and is was established in 4.3, where the information presented in Figure 4.5 was stored in lookup tables. The value of is (wm. T6) is calculated by using a bilinear interpolation described in the next subsection. Equations (4.8 to 4.10) were used to create the Hamiltonian: 4.11 + P(t (43 — fl- — Bmwm) ( ) Jm Jm Jm The costate function is rewritten as: P — —2h1(edm '— wref) — 2h2fti_losswrn — 'Tn‘ (525,-; — Bm) (4.12) Ptf = 2h0(wm(tf) -— w'f‘ef) (4.13) with the boundary condition: 0 2) R' + P(t)( 1 078 (411) = r ,2) —— . r 2 5 Jm 0233 4.3.3 Numerical Optimization The continuous time optimization problem of subsection 4.3.2 was solved numerically. State and costate equations have to be solved in discrete time and the calculation of 54 the control input have to be done iteratively. First we rewrite the state equation in discrete time as: 21 ,Az' 1'. 2' - r 1.. w. , 2. e ,5» m, T B. . 7+1 < ) 1 "twin 2. _ w") :At —— _ — _ —_ ‘1' Wyn (fl-15) Jm Jm. Jm where At is the sampling time, and 2' is the sample number. The control input is is related with re in a nonlinear form. To determine re for a given value of is and am a bilinear interpolation can be used. Figure 4.6 shows a representative grid of the data stored as a lookup table, equation (4.16) is used to determine T8 (is, com) ls] Te (ls 9 mm) 1' // s 32 Figure 4.6: Bilinear interpolation . ~ Te (isI’wml) . Te (Z‘Siw'fll) ~ AisAwm (232 - 25) (”mg — Wm.) M082, W1) —i 1) (w 2 _ w") s m ' l fi’SA:m( (4.16) i31,w 2)% . AzsAwm 2'5) (”m - ”('11) Mm(i32’ 2().S . AzsAwm —281) (ram _ (“U-ml) with Awm = 01th — “ml and A13 = 2'32 — 2'31. The costate and its final condition 55 equation in discrete time are given by: i—l 2' 2' Pi 376 2 P = — —2h,1(w.,,, — emf) — 211.2k,_ 10881.)", — T"; at} — Bm At + P m (4.17) Nt P.Ntf = 2h0<¢drn f — wref) (4.18) where the partial derivative can be calculated as follows: Ti ,1 wi ~ T€(282’wm1)_78(281’wm1) .2' _ .2; + i -z' 2' 8 3 m1 ~ .r)‘ -2' 26' 2.91 Te zsl’wml 232 T 2.31 1,2 2' ~ Te (ZSQ’wTrLQ) —T€ (zsl’wm2) .z' .7' z" .z' i 76 (Ls’wm2) N -z' _ .i (7’5 _ 2'51) + T6 (zsl’wm2) 232 231 an; (gain) .. 7.3 ((64.2) _ 7:; (2:13am) awfn Awm (4.19) Equation (4.14)becornes: - - 1 at 0 = 2112121; + Pz —— T? (4.20) Jm 8273 56 where the partial derivative can be calculated as follows: 1' 1 i _ i 1 ,,1 2' z' 1 ~76 (isl’wm2) (lsl’wml> i 1' 3: I.) 1 Te islawm ~ wi —wf “Tn-”ml +T€ l‘sl’wml 112 ml '1'. -1 1 '27 -i 1 T 1 — T 1 e ( s2’w1‘1'12) ‘3 ( s2’w' 111) 1 1 1 .1 1 7, i (w.,,L—w,".,1 +76 ‘82’wml w —-w, m2 ml '1' xi ,1 1' 1 ,1 1' '1'. ,2' 8T8 ($310411) Te (lsgewm) " Te (151700111) awfn A23 1' -i i N Te (332159111) ~ (4.21) The steepest descent method was chosen to solve the optimization problem. The control function is calculated iteratively, which leads to the iterative minimization of the performance measure function. The steepest descent method [29] is used to calculate the control function iteratively. The control input is updated every iteration, equation (4.14) is modified to create an update for 11(1) as follows: 3,1 21.14.1 _ 1,1 0710; ,P. am) ‘8 — 1,1 018 ‘8 + Tadap 1.1 .271 1 1 3T6 : 23 + Tadap (2h2Rls + P (jg-07,1 2's where l is the iteration number and “r, (4.22) 11111;) is the adaptation step. The value of 1:351 can be greater than the maximum allowed, a basic saturation is used to limit is. The general algorithm flowchart do determine the speed reference is presented in Figure 4.7. 57 4.4 Simulated and Experimental Results 4.4.1 Simulation Results The preposed algorithm was tested for three sets of weights ho, 11.1, and 112 to demon- strate the performance of the algorithm. The selection criteria for the weights was a tradeoff between the speed response of the controller and the energy losses. The sets of weights are shown in table 4.1. The final time selected was t f = 63, the initial speed was Lum(tz-) = 500 r/min, the reference speed was wm(tf) = 1200 1: / min, and a load torque was Tl(t) = 17Nm. Table 4.1: Weight sets ho 111 112 Set Number hO hl 112 Set 1 20000 50 2000 Set 2 20000 100 1333 Set 3 20000 400 333 First the convergence of the algorithm was investigated. Figure 4.8 shows how the cost changes with the number of the iterations. The rate of convergence of the speed depends on the adaptation step Ta and the initial guess of the trajectory. (lap) This initial trajectory was a straight line between the initial and final points, however a more refined approach can be used to reduce the number of iterations required to reach the minimum, as will be presented in the next section. Figure 4.9 presents the trajectories resulting from the optimization algorithm for the three sets of weights. For illustration the trajectories were plotted over the contour curves of equal current 13. Figures 4.10, 4.11 and 4.12 show the change of the speed wm, torque T8 and current is as functions of time. The energy losses are 524.] , 698J and 924.] for sets 1, 2 and 3 respectively. These results show that although all trajectories reach the final speed in the specified final time, different trajectories have different energy losses, e. g. trajectory 3 has 60% more 58 C START D 1 Define initial conditions arma. ). the load torque Z' i) and the reference speed are) [=1 . Create a initial approximation for input: 2' 1a i 1g, for i =1 ...Ntf. ’1 i=1. Initialize mm ’1 Calculate mm," (415,416). Initialize P " from (4.18). 3 i=i-1. Calculate P i" from (417,419). no . 1:1 yes i=i+1. Calculate z' “”3 from (4.21.422). no yes 1=1+1. yes C END D Figure 4.7: Optimal algorithm flowchart 59 14. a e 12: 12* 10* 2° 2° ‘0' o 3. o ‘>'< 3? 8’ 175 6’ '55 8 8 6’ 4 4» 2.] 2+; 0 100 200 300 400 500 0 100 200 300 400 500 It. number It. number (a) Set 1 (b) Set 2 20’ g 15- 3'2 8 10» o 5'! 0 100 200 300 400 500 It. number (c) Set 3 Figure 4.8: Cost vs. iteration number energy losses than trajectory 1. Figure 4.13 shows the proposed control scheme, where the trajectory resulting from the optimization process is used as a reference for the feedback controller. 4.4.2 Real Time Optimization The optimization discussed so far cannot be easily implemented in real time, due to the long computational time required and the difficulty to the backward calculation of the costate function. A modification is discussed here to calculate the trajectory. After the off-line minimization, an approximation to the trajectory was made by cre- 60 50 r r r r r r r $90 :"'l"'Set1 40- E30“ 5 a) P 20. 10 \ 0 1 1 1 1 1 400 500 600 700 800 900 1000 1 100 1200 1300 a) m (r/min) Figure 4.9: Simulated speed wm 1200. , __________________ 1 / ’2 / ._'_._. .—."/.. A - I I" _._ —.—.—._I—'— .1 .E 1000 /." s .1 _ ----------------------------------------- 800- I." . E [f ------- Set1 /.-‘ -—---Set2 6001 ---Set3 1 400 ' . . 1 1 0 1 2 3 4 5 6 Time (s) Figure 4.10: Simulated trajectories ating a continuous function consisting of three sections. The functions were calculated by developing the optimization process for different sets of t f’ wm (til) wm (ff) and Tl and by making a curve fitting over the resulting data. The three sets of gains pre— sented in the previous sections were further analyzed in order to verify the accuracy 61 r e (Nm) Is (A) Figure 4.11 shows the torque trajectory as function of the time; this trajectory 50 45 ' -1 f1 40 [1‘ - 3511,1‘ ------- Set 1 . f....\\ "-'-Set2 301. ".'\ \ - - -Set3 I 25 [ '\.\ .- f '-.\\ , / 20 - I:§M*m._.._mn_—n.—-.—.—.—m_—uH:_:v—f 15 ‘ ‘ 1 . 1 0 1 2 3 4 5 Time (s) Figure 4.11: Simulated torque Te 20 ~ - - ll: \\ '''''' Set 1 _- 181"- ( -—---Set2 1 ll \ ---Set3 :1 16 t; \. \ 5,. - f \ \ 4 141 ', '\ \ ~ —————————————— 4’ 4 i -. x, ..I 12 i \ l \ ~ _________________ a I n 10 [ ................................ 4 8 1 1 1 1 1 0 1 2 3 4 5 Time (s) Figure 4.12: Simulated current is of the approximated analytic function that models the optimal trajectory. can be approximated by a sequence of three functions: The first is an exponentially decaying function, the second is a constant, and the third one is a exponentially increasing function. The coefficients depend on the initial and final speed, the final time and the final torque. The optimal trajectory calculation presented in the previous 62 Optimization algorithm l a) m i Speed reference Figure 4.13: Controller topology ,*_.____ - + PI ls Current // -\\ '< >" 7 Lookup 7 Control . 4.x” controller , l m table 7 + —: Motor ' ,1. inverter . ‘A..\_ ,./ "T T 3‘ ' Id ‘ Tlesl \ f w J [LOADV w -_,_. i m ] Signal [A] r CS /.\1 [ LandItlonlrlgE [ En coder\ -2 section for t f > 5 3 shows that the sum of the time of the two exponential functions is almost constant, which make the analysis for t f > 53 simpler than for t f < 53. As shown in Figure 4.11 for t f > 53, the duration of the second segment is equal to the difference between final time and 53. The general form of the functions are shown in equations (4.23),(4.24),(4.25) and (4.26). Te__1(t) fO’I‘ 0 S t < t1 Te(t) Te_2(t) for 11 g t < 12 Te_3(t) f0?” t2 S t < if 76—1“) = 0311+ k12 (wm (1f) — cum (1,)» e_k13t + 6‘14 *T1 Te_2(t) = k21=k Tl 63 (42a (4.24) (4.25) T6—3(t) = 031 + 4532 (Wm (if) — wm (t,))) e+k33t (4.26) + (1634 * Tl + 6835 (wm (tf) — cam, (ti))) where t1 = 3.55, and t f — t2 = 1.53. This means that the approximation is valid for t f > 5.9. The constants of each function were calculated for a fixed set of weights of the cost function, i.e. for each set of weights of the cost function (110,111,112) there is a different set of gains k1, j" Table 4.2 shows the calculated parameters for the three sets of weights analyzed in the previous sections. Table 4.2: Constants for on-line optimization approximation for t f > 5 s Functions Set parameters number k," j Set 1 Set 2 Set 3 kll 3.39 2.825 -2.61 1612 .02158 .028125 .088 A713 1.3 1.4 3 1614 .982 1 1.012 1:21 1.012 1.012 1.03 1931 -0.11667 -0.11667 -0.11667 1632 1.166763 1.16678—3 1.16678-3 1633 1.79 1 .9 A334 1.025 1.025 1.025 1635 -1.1667e—3 -1.1667e—3 -1.1667e-3 The approximate trajectories were evaluated for representative points with differ- ent initial and final torque-speed points and different final times. These representative trajectories were computed for each set of weights (h.0,h1,h.2) described in the pre- vious sections. The error in the cost function was calculated by using equation (4.27) J (wm(tf),wm,is) — J (w.,,),(tf),wfn,1§) J (wrnftflatfiwmtis) error = :1: 100 (4.27) 64 where ‘J ’ is defined in equation (4.8) and wfn is calculated by using the approximated torque of equation (4.23) and solving the differential equation (4.10); ‘1;’ is calculated by using the relation described in subsection 4.3 between ‘13:? and a; ’. Table 4.3 summarizes the results where optimal and subOptimal trajectory are compared in terms of the cost function error. Table 4.3: Error in the cost function for t. f > 5 s Gain Set number Cost function error % Set 1 Set 2 Set 3 Minimum 0.94 0.47 3.33 Maximum 17.17 12.25 30.88 Mean 6.55 5.21 10.97 For t f < 58 the approximation is more complex and the time intervals t1, and t f — t2 are not constant, moreover the function parameters are different than for t f > 53. The torque curve is composed by three functions as described in equation (4.23) and the time intervals are given in table 4.4. For t f different than the values presented in table 4.4 a linear interpolation was used to determine the time intervals. The segments are again two exponentials and one straight line as shown in equations (4.24), (4.25), and (4.26), however the coefficients are slightly different, and are summarized in table 4.5. Table 4.4: Time intervals for t f < 53 Final time Time intervals 1f 11 12 1(8) 1 0 2(3) 1.4 0 3(3) 1.4 .8 4(3) 1.7 1.5 The performance of the approximation was evaluated. A summary of the analysis in presented in table 4.6. 65 Table 4.5: Constants for on-line optimization approximation for t f < 56' Functions Set parameters number 16M Set 1 Set 2 Set 3 kll -1.25 -1.75 3.8 1:12 .0375 .0525 .0756 (($13 1.5 2.3 3 .1114 1.02 1.05 .97 1:21 1.01 1.02 1.03 [:31 -0.035 -0.11667 -0.11667 k32 1.05e-3 1.1667e—3 1.1667e—3 11:33 2.8 1.6 .9 1:34 1.06 1.048 1.0186 .1635 -1.75e—3 -1.1667e-3 -1.1667e-3 Table 4.6: Error in the cost function for t f < 5(3) Gain Set number Cost function error % Set 1 Set 2 Set 3 Minimum .1428 .9254 0.01 Maximum 27.38 17.41 14.18 Mean 10.87 7.35 4.83 The approximate function can also be used as an initial guess for the optimization. The cases presented in the previous section, were further analyzed and the online approximation was used as an initial guess for the trajectory. The minimum value of the cost function is assumed to be the value after 1000 iterations. The effect of the initial guess on the number of iterations was also considered. Two cases were evaluated: a. straight line that connects initial and final torque—speed points, and the proposed curve approximation. The criterion to end the trajectory optimization was that the the cost function is within an error of 2% of the minimum value. Table 4.7 summarizes the results. From the previous results it is clear that the use of the approximation curve as initial guess for the iterative optimization reduces the number of iterations required to achieve the optimal trajectory to about half. 66 Table 4.7: Maximum number of iterations Cases analyzed Set number Initial guess Set 1 Set 2 Set 3 Linear trajectory 440 576 421 Approximated curve 132 179 78 4.4.3 Experimental Setup The proposed algorithm was tested experimentally. The experimental setup is shown and described in Apendix A, for reference, the schematic of the experimental setup is shown in Figure 4.14. The speed from the optimization algorithm were used as reference for the controller, as shown in Figure 4.13. Any trajectory can be selected to validate the simulated results; the selected trajectory is the one corresponding to the weight set 3. The dynamometer was used as a load with constant torque. The experimental results are presented in Figures 4.15, 4.16 and 4.17 and the experimental motor energy losses are 875.5]. Experimental results show a good agreement with the simulated values. The error in the energy losses was about 5.4%. Power RT linux controller + FPGA supply (Controller) 7 + T Labview 1 Control A A i ’ ec Controller - V Encoder 1 DC Signals . , 0 ()———— l l 1 lbs 165 e (0 25 25 z: m 25 z: 25 T A7 [ //7 Inverter J] Torque-meter #7 PMAC motor Dynamometer Tachometer PMDC motor Figure 4.14: Experiment setup 67 1200 - _l O O 0 co m (r/min) 600 400 0 800 - Experimental Simulated l 3 4 5 6 Time (s) P b- Figure 4.15: Comparison: experimental and simulated speed I I I Experimental ''''' Simulated l 3 4 5 6 Time (s) b- Figure 4.16: Comparison: experimental and simulated torque 68 Is (A) I 25 T . Experimental . - ' - - Simulated [- 10 1 1 1 0 2 3 4 5 6 Time (s) Figure 4.17: Comparison: experimental and simulated current 69 Chapter 5 Trajectory Optimization for Engine-Generator Operation of a Series Hybrid Electric Vehicle This chapter presents a methodology of calculating the optimal torque and speed commands for the engine—generator system of a series Hybrid Electric Vehicle (HEV). In series HEVs the engine-generator subsystem provides electrical energy to the DC link. This chapter proposes an optimal control strategy of the engine-generator sub- system to generate a desired amount of energy within a given period of time. The optimization algorithm based on trajectory optimization determines the torque and speed reference signals for the engine-generator subsystem that achieve maximum efficiency with the desired energy. A simplified version of the controller is also pre— sented for online implementation. The proposed control strategy is compared with non-optimized control techniques and simulation results show the improvements in energy efficiency. 70 5. 1 Introduction Series configuration is often used for high power hybrid electric vehicles [31], where the internal combustion engine converts the fuel energy into mechanical, and the gerr- erator converts the mechanical energy into electricity. The electrical energy is either stored in the batteries or consumed by the traction motors [32]. Several control strate- gies have been proposed to optimize the fuel consumption in electric vehicles [33, 34]. These include global optimization (such as linear programming, control theory, dy- namic programming and stochastic digital programming), and online optimization (equivalent fuel consumption, robust control, fuzzy logic and optimal predictive con- trol). The hybrid vehicle supervisory control system optimizes the engine-generator op- eration as a single DC source [35, 36, 37] at steady state, while the transient operation is usually neglected. The torque and speed references for the engine-generator result from the optimization of the overall system; the supervisory controller takes into ac- count the efficiency of the devices that comprise the traction system and determines the power distribution of each subsystem [38]. Similar considerations with respect to the generator-engine subsystem were made for the power distribution in series Hybrid Electrical Vehicle (HEV) [39]; the opti- mization algorithm used there was dynamic programming. The objective function was designed to minimize the energy losses over a driving cycle, while achieving the wheel speed requirements. The power distribution algorithm did not consider the subsystem limitations such as torque and speed, and the mechanical system dynam- ics were neglected. Simulation results showed the power distribution and the fuel consumption. Ceraolo et a1. [40] proposed a methodology that optimized the operation of the engine-generator subsystem. The subsystem torque and speed were determined by 71 the requested power for a certain period, and the optimization scheme found the optimal operational speed and load conditions that provide the desired power with maximum efficiency. The resulting speed and torque were used as references for the engine and generator controllers. This optimization process does not consider the operational efficiency during the transition between two operational conditions, which is not necessary optimal. This chapter presents an optimal control strategy of the engine-generator subsys- tem. Starting from a requested electrical energy to be delivered from the generator in a given time interval, the trajectory optimization scheme determines the optimal torque and speed references required to satisfy this request with minimal energy loss. The advantage of this optimization is that it is based on the fact that the power provided by the engine-generator subsystem is not constant. The chapter is organized as follows: section 5.2 presents the optimization problem and the engine-generator efficiency maps. Section 5.3 describes the trajectory opti- mization scheme. The minimization problem is described in continuous time domain and the problem is solved using numerical optimization in discrete time. Simulation results of the proposed trajectory optimization are presented. Section 5.4 presents an approximation to the Optimal trajectory. Section 5.5 shows the resulting efficiency and compared it to that of non-optimized control techniques. 5.2 System Description The optimization problem presented in this chapter results in the calculation of the torque and speed references for the engine-generator subsystem. In series HEVs the system of engine and generator is shown in Figure 5.1. The goal of the optimization is to produce the requested electrical energy at the generator terminals in a given period of time, while minimizing the energy losses. 72 Supervisory Control Wheel ——- DC converter —- Traction and energy motor+ ~m storage —- inverter Generator En in g e + Inverter Wheel Figure 5.1: Series HEV configuration The method was applied to a specific engine-generator system. The efficiency maps for the engine and the generator of this system are shown in Figure 5.2 and Figure 5.3 respectively. Figure 5.2 shows the efficiency map of a 6 cylinder Cummings engine rated at 194 kW. Figure 5.3 shows the efficiency map of an IPMSM rated 130kW connected to an inverter. Engine and generator efficiency maps presented are for the useful speed range of 1100 — 2200 r/mz'n; the torque range as a function of speed is presented by solid bold lines. Subsystem loses are calculated for each operational condition based upon these two efficiency maps. Inertia and viscous friction coefficient of the system are also used for the loss calculation. 5.3 Trajectory Optimization This section describes the trajectory optimization. The trajectory, consisting of the points in time of torque and speed, was calculated by minimizing a cost function. The numerical constrained optimization used was based on the steepest descent method [29]. The general minimization problem and its necessary conditions were presented in subsection 4.3. 1. 73 800 ~ ' ' a (v a 6 PA 600 ~ 0 - / ‘ D: 400 ~\/\/—/ 0 . q \ 0.25-f“ ,//:..._... 1200 1400 1600 1800 2000 2200 a) m (r/min) 1 e (Nm) Figure 5.2: Engine efficiency map contours 700 " l 600 ' 500 ' 300 200 - o 4 0.92 fl \ 09.x i” 1200 1400 1600 1800 2000 2200 a) m (r/min) r e (Nm) # O O I Figure 5.3: Generator efficiency map contours 5.3.1 Trajectory Optimization for the Operation of Engine- generator System The general algorithm for trajectory optimization in subsection 4.3.1 was applied to the calculation of torque and speed references for the engine-generator subsystem. Ini- tial speed and final command time are assumed to be known before the optimization 74 starts. The final energy can be obtained by modifying the supervisory control system to provide energy instead of torque and speed. The performance measure function, J, was designed so that its minimization allows to reach the desired final energy and minimize the energy losses. The performance measure function here becomes: 2 . 2 J : h’l (VVGEN '— Wref) + h2(wloss) (5'1) where hl is the weight used to lead the algorithm to reach the desired final energy, and h2 is the weight used to minimize the energy losses in the system. Wre f and PVC E N are reference and the output energy of the generator and W108 s is the energy loss of the overall system. The equations that describe the dynamics of the system are: 81.0 Jm—atT-r-l = TENG —TGEN —Bmwm (5.2) aWGEN a, = TGENwmnGEN (5-3) aW709); —‘ " = T Nwm (1 - "7 ) + TENG wm (l/T’ENG '- 1) + Brno)?” with the initial conditions: Wm (ti) = wm—tOv LVGEN (t1) = 0’ ”/1035 (t?) = 0 (5'5) where Jm is the inertia of the system, cam is the mechanical speed, TENG is the engine torque at the shaft, 70 E N is the counter torque of the generator and Bm is 75 the viscous friction coefficient. 77 E NG and 776' EN are the efficiencies of engine and and generator respectively. The state equations (5.2) to (5.5) are rearranged as: [ a1 817(t) _ at _ “2 _ a3 1 TE- N G 7C E N B 111 Wm 11 - —— — _— _ _— Jm Jm Jm (5'6) 62 = TGENw’m-"GEN a3 = 3112an+ TGENw-m (1 — ’IGEN) + TENGw‘m (1/"7ENG - 1) with the state .r(t) =[w-m, WC E N) W103 3]T and the control input 11(1): [TG E N, TE NGlT- The Hamiltonian is: 7" = h1(VVGEN _ ”(ref)2 (11 . 2 “12011039 +P(1)T a2 103 The costate function is calculated by solving equation(5.8) with the final condition (5.9): F 011] (9119 (9113 - Wm awm Wm 0 0 0 0 0 O 76 001 _ Byn awm Jm 30.2 aWm 8(13 (9mm + TENG ( 1 _1_ 877GEN = TGEN'IGEN + TGENWm awm 877 r : TGEN (1 _ 77G'EN “ Lam. 0(5):: ) + QBmwm "ENG 0 Equation (2.7) becomes: Ba, 1 BTENG 8a3 Bal BTGEN 80,2 3TGEN 8a 3 aTGEN 3am ] 753 N G (Ll 8a2 wm 377mm) 2h1 (VI/GEN — l’VT‘ef) (5'9) 2h2l/V loss 8a 5’ TENG 3a3 P(t) aTGEN ETGEN aTGEN 1 Jm 1 = Wm , — ( ”ENG _ 1 J m. = Wm ("GEN + TGEN = wm (1- ”GEN 5.3.2 Numerical Optimization 1 _ TENG aTIENG) 77125 NC UTENG (5.10) anGEN ) aTGEN a77GEN ) _ TGEN BTGEN Subsection 5.3.1 presented the continuous time optimization. This section presents the solution of the problem in discrete time. Since the information of the engine 77 losses and generator losses is usually given in lookup tables, this section presents a procedure to implement the numerical differentiation required for some calculations. First we rewrite equations (5.6) and (5.8) in discrete time as: [ w flil i all “in 2+1 = 2' 2' WGEN ”2 N + WGEN (5'11) 2+1 2' z' I/Vloss J _ a3 _ _ 1Vloss P 8a 8 8 - —1—At+ 1 film Jim - aWm a’m 8mm - p’l‘l = P2 O 1 0 0 0 1 8&1 : _Br__r_L 8w77én Jm 8u§ 8nGEN a: ,2: — TCEN’lGEN + TGENwm—a 2: Wm Wm (5 12) 2 . ‘9“3 __zT 1_ _, zLGEN +23 2' i — GEN ”GEN “NI—_— mwm 8mm 8wm ' 1 wig (W 2 m EN G 1. 8wm 7] 1,2,2 ENG ’/ E NG where At is the sampling time, and 2' is the sample number. When the losses are stored as lookup tables, the values of 71%, NC (win, TENG) and 17%. E N (win, TE; E N) are calculated using a bilinear interpolation summarized in equation(5.13): 77 (o, 7) ~ 77:23) (W2 - w) (1’2 -— T) + WZWBA?) (w W1) (72 —‘ T) (5 13) + "MT” (wg —w> + ——————”‘“’2"’2) (w-w1)(T-T1) AwAT 78 T1 n(r¢v) ...—I’d...) T 72 at a) a5 Figure 5.4: Grid data example used to describe the bilinear interpolation with Au) 2 w? — W1 and AT 2 72 — T1. The values of 877ENG (0);”, TEA/G) /8w.}i’n and 811814: N (Win: T8 E N) /8w$n are calculated numerically in (5.14): )~ 77('w'1,72)- 77(u11271) ( n(w1,T ~ 72-71 T-T1)+77(W1v71) w ,T — w 17- 71(w2,7)z M 2 2) M 2 1)(T—T1)+77(w2i71) (5'14) 72—71 80 (cam) z 77(w2J) — n(w1,T) 8w Aw The control function is calculated iteratively, which leads to the iterative min- imization of the performance measure function. The steepest descent method [29] determines the updated value of the control function iteratively by modifying condi- tion of equation (2.15): _ T 8mm), u(t)l, P(t)) adap 8n (t ) l mnH1=mnl (am) where l is the iteration number and Ta is the adaptation constant that defines the dap convergence rate of the minimization algorithm. Equation (5.10) can be substituted in (5.15) and discretized as (5.16). 79 - 8ai 8ai 2',z+1 ——l— 0 —=—3— TENG _ BTENG BTENG pi 2' ,z+1 _ — “(lap 62 621' 0 (EN 1 2 “3 aTGEN aTGEN aTGEN - 2'1 +TE1VG 2,1 TGEN 8&1 _ 1 aTENG Jm 5.03 : win ( . 1 _1_ ENG anENG Z Z aTENG 77ENG 77E’2NG BTENG Z 3a Z __1. 2.22) 878 E N Jm 8ai - 871i 2 ENG 827? — Wm ("ENG + TGEN 022 > GEN GEN 8a?“ - - . 8772 3 _ z z GEN W- ”m (1 “ "GEN — TGEN 672 ) GEN GEN The values 0f ‘9"7ENG (win) TENG) (BTENG and 8’7EEN (“Ev TEEN) MTEEN are calculated numerically by equation (5.17): 'I](Ld2,T1)-—- T](Ld1,7'1)( 7)(w,71)z w2_ “’1 w— w1)+n(w1,71) , ... 71012272) “77(011272) , 77 (22,72) ~ 212 __ “’1 (w w1l+ 77001272) (9726027) ‘Ufwe‘rQl ‘Mwafil 87' A7" (5 7) The calculation of the updated control function equation (5.16) may produce 80 values of Tall-3+]; or 72711:;(1; outside the intervals 0 < Tz‘ljité' < Tgfié~max (cal) and O < Tag—1%,. < ngllf—max (LUZ) with z' = 1"‘Ntf' In order to to keep them within these intervals, saturation blocks were used, as shown in equations (5.18) and (5.19). In addition, if (5.11) produces a value of 223;,“ above the upper limit, 72’}ij is decreased by KG E N and TglE N is increased by KC E N' A similar procedure is used if (.03;th 1 has a value below the lower limit, but in this case KG EN is replaced by —K G E N' Figure 5.5 summarizes the optimization algorithm. r . . If TEENS > TENG—max (WEI) TENG = ENG—max (win) TEZvNG = i If TszG < TENG-min (WEN) (5-18) TENG = TENG-min (win) othemuise 7'ch = TéENG If TEEN > TGEN—max (win) _ TEEN = GEN—max (win) TEEN = If TEEN < TGEN—min (win) (5'19) TEEN = TGEN—min (win) otherwise TgEN = TgEN If WEI, > “max TEEN Z TEEN‘LKGEN' 72:2le Z TE_AiG"KEN G (520) i. If Wm < wmin '—1 “—1 '—1 :-1 TEEN = TEEN‘KGEM TENG = Ti§3NG+KENG 81 l START ,‘ EH~_ . ,_E ‘__:_ L‘V:_t : ___:4 Define initial conditions (0mm- ) WGENfli) and WIossfli ) and required energy WGE~(tf). : Em , , ,2 L _, _ L ":7 _,_____, l=1.Createanmltlalapproxnnatlon for mputs: 2' ENG T GEN. forzzl...Ntf. T i ___?“ :L ______ _ l l i=1. Initalize a) 2m W IGEN and W 1/038 l l ’“ .--:umj; i" _ ___. i Saturate T lENG and TIGEN by (5.18) and (5.19) j ;; V a Calculate a) ”1m W’HGEN and WNW from (5.11). i ‘" Ee—CEICHEE: “E‘— a’/'B/O// i +1 < \M\\2\ .i inputs by (5.20) ‘ “\afgfli <0 m (0,29,2/ l \ yes \\ *1, ,, i _ t A yes InitializeP’. from (5.9). 2'=2'-1. Calculate P” from (5.12). ', , , i./+1 2',1+1 _,, _' gz=z+1. Calculate 2' ENG and ‘Z' GEN from (5.16). l i fi fi :\.\ ‘\ (ZNtf ”/I‘ML) \\v/ yes 7 7 E no ,/-/--'“""T‘”“/!\\-~-‘\ ‘ I :1 +1 - l‘ ""'T~4—41-§:-\\_> I =Iteralionmax //:_:::.—’ _ i A l x\\\\-v/’T” _ _ I _ yes END Figure 5.5: Minimization algorithm 82 5.3.3 Simulation Results The prOposed algorithm was applied for the Operational conditions and parameters specified in Table 5.1. The optimal trajectory was calculated iteratively; the cost function was plotted against the iteration number for wm(ti) = 1800 r/min, Figure 5.6. The calculated cost function of Figure 5.6 shows the convergence of the algorithm. Table 5.1: Simulation parameters Parameter Value Units Required energy Wre f 4.5e5 Joules Maximum speed Wmagj 2200 r/min Minimum speed “min 1100 r/min Inertia Jm 3.1 kng Viscous coefficient Bm 0.0043 Nms hl 1000 h2 30 Sample time At 0.1 Final time t f 6 iterationmag; 2000 Tadap 85e-012 17 N 2— 16.5 ‘ 9. x 16 ~ a _, ,5 15.5 - - 8 S 15 ~ - LL g 14.5 ~ 0 14 ~ * o 100 200 300 _ 400 500 Iteration number Figure 5.6: Cost Function The final calculated functions of control inputs TENG and TGEN, and the outputs 83 cum, WC E N and VII are presented in Figure 5.7, Figure 5.8, Figure 5.9, Figure , loss 5.10 and Figure 5.11. The simulations show the results for two initial conditions (Urn/(ti) = 1100 r/mz‘n and wm(ti) = 1800 r/mz’n. The speed was kept within the predefined limits and the generator energy requirement was satisfied. 800 t t ,. ..... ‘ _ _ _ .\ ----- wm(to)=1100 ' (0 m(to)=1800 600 - _ E E (9 400 r a z PLIJ 200 - 0 1 1 1 fi- 0 1 2 3 4 5 6 Time (s) Figure 5.7: Engine torque 800 T l l r 1 T "_ 600 ‘. l E i Z, . UZJ 400 - ' | ..0 -- . - - (om(to)=1100 ‘_ 0 1 a m 1 1 0 1 2 3 4 5 6 Time (s) Figure 5.8: Generator torque The torque and speed points of engine 0 and generator 0 were plotted over 84 2000 f I I r r -— - - - u) m(to)=1100 (1) m(to)=1800 1800 E E 1600 E 3 1400 1200 - V 1000 L ‘ ‘ ' ‘ 0 1 2 3 4 5 6 Time (s) Figure 5.9: Mechanical speed 5 r I r I T I , -4 _ I. E? /'/. o . r; 3 ~ I.’ 3 ./' @2' _x" ----- mm(to)=1100 1 a , . ’ (o m(to)=1800 1 - _/‘ - . / .I 0 ‘ ' 1 1 1 1 0 1 2 3 4 5 6 Time (s) Figure 5.10: Generator output energy the combined efficiency contours of engine-generator subsystem in Figure 5.12 for wm(ti) = 1100. Engine and generator torque were kept within their limits, were the combined maximum torque is presented by a solid bold line. The maximum combined efficiency is n = 38.8% which corresponds to T = 690 N m and w = 1800 T/min. This means when the subsystem operates at its optimal condition, the maximum possible efficiency is 38.8%. As shown in Figures 5.7 to 5.12 the subsystem moves to the point 85 8 I I I I l :5“ 6" . a ./ 3? .’ 3 4~ - 33 ,f‘ 8 ,‘ ----- (I) m(to)=1100 3 2- /'/ (om(to)=1800‘ ./‘ / 0 1 1 1 1 1 o 1 2 3 4 5 6 Time(s) Figure 5.11: Total energy losses of highest efficiency, and stays there until it satisfies the energy requirement. Then it moves to a point with low torque and speed to minimize the losses in the remaining time. 800- \ \ \/——0.4’/ ‘ 700 V r 600 A 5’ g 500 o‘ a, 400 0.37 ‘ P V," 300 036 a 200 0.31% -—_—_____________——-J 100_ , ,o o //o.24 _ P( ‘ l 1 (MT—___“ 1200 1400 1600 1800 2000 2200 m m (r/min) Figure 5.12: Torque-speed points over combined efficiency map contours: engine points 0 and generator points 0 86 5.4 Approximation to Optimal Trajectory The previous section described the algorithm to determine the optimal trajectory of the engine-generator subsystem. In this section an approximation to the opti- mal trajectory is presented that can be used in real time. Three possible operation modes were identified and a controller command for each one is proposed; these three operation modes are defined by the requested energy and time. 0 Mode 1: Low energy requirement with enough time to produce it optimally. This operation will be the most common operation. The subsystem has plenty of time to reach the most efficient point 13 = “Pub, produce the requested energy, and then go to a point with minimum speed and torque and hence low losses. 0 Mode 2: High energy requirement with enough time to produce it. The system has enough time to produce the requested energy, but not enough to produce it by operating at the most efficient point. A point corresponding to higher power with lower efficiency than the optimal is determined. The system will operate at this point and keep there to produce the requested energy. 0 Mode 3: Fast energy production. This operation will be the least common operation, the system is required to produce energy in a short period of time. Under this condition the energy optimization is not the most important criterion to operate the subsystem. The system is free to speed up by imposing maximum torque in engine and generator (when maximum engine torque is higher than the generator torque). 5.4.1 Mode 1: Low Energy Requirement With Enough Time This common operation mode is based entirely in the trajectory optimization results presented in the previous section. Let us summarize the behavior of the optimization 87 as follows: The trajectory starts from initial speed, the subsystem speeds up until it reaches the point of maximum efficiency. The subsystem stays at this most efficient point to supply almost all the required energy. The subsystem slows down to produce minimal losses. Torque of the generator follows the speed profile to produce the required energy. Torque of the engine is varied to allow the subsystem to speed up, keep the system at the optimal point, and then slow down. Acceleration and deceleration times are kept constant, moreover torque and speed trajectory during the transients are the same for different energy require- ments and final time values. These conditions were used to develop an approximation to the optimal trajec- tory. For the system presented in the previous section, the final time and energy requirement should satisfy the following conditions: 1. t f 2 2 s : final time should be greater than the time required to accelerate from any initial speed to the most efficient point plus the time required to decelerate from the most efficient point to a point with minimum speed and torque. . VVre f 2 185 J: final energy should be greater than the energy produced during the acceleration and deceleration described in the previous item. ref — 265)/P +1 .3. This condition ensures that the system can provide the energy while operating at the most efficient power point “I3 = fab”, where + = 690 Nm, a; = 1800 r/min and 13 = 130 kW. 88 When the previous conditions are satisfied, engine-generator speed and generator torque can be approximated by trapezoidal functions as shown in Figures 5.7 to 5.9. The trapezoidal function and its time dependence is shown in equation (5.21). A low pass filter is used to smooth the trapezoidal function as shown in equation (5.22). t1 < t < t2 :L‘O t xtTa-p (t) — 0 t—IO p (5.21) 152 S t < t3 2 3 (t t2) + $opt t3 < t < tf xmm H( — 1 (5 22) _ kfifs +1 . The time intervals and the gains for the trapezoidal functions and filters are sum- marized in table 5.2. The block diagram of the approximated optimal trajectory is shown in Figure 5.13. Table 5.2: Parameters of the approximated optimal trajectory Parameter wrefm 77.6fm t1 0.3 s 0.7 s W W t2 — t1 3.3 + 1.7 (jg-EL — 2.25) 3.1+ 1'7(7%(5£ — 2.25) t3 — t2 0.9 s 0.7 s mopt 1800 r/mz’n 690 N m 55min 1100 r/min 0 Nm kfz'l 0.0429 0.0538 5.4.2 Mode 2: High Energy Requirement With Enough Time This mode occurs when the first two conditions of Mode 1 of subsection 5.4.1 are satisfied, but the last one is not, i.e. tf Z 2 s, Wref Z 105 J and tf S (Vi/ref —- 2- 105)/P* + 1 s. We assume that the requested energy can be produced by operating 89 Xtrap(t) H(S) t —> D— ——> D_ —> ‘CENG*(t) Xtrap(t) I"(3) t —> D —> I:— —> COGEN*(t) Figure 5.13: Block diagram of approximated optimal trajectory the subsystem at one point (P 79 P) for the specified time (otherwise the energy command is wrong and it should be adjusted as Wm f = Pmaa: :1: tf). The power point P of operation can be determined by a local optimization as described in [40]. The procedure to determine P and the torque and speed commands is summarized below. 0 The energy command and final time allowed to determine the required power as P = Ill/Tef/tf' o The values of f and J) (P = “f * Q) are determined by searching among the torque-speed points that satisfy the power requirement P and selecting the combination that produces the lowest losses. 0 f and J) are used as a references for torque and speed controller in the engine— generator subsystem. 5.4.3 Mode 3: Fast Energy Production This mode of operation is the least common; it occurs when the first two conditions Mode 1 are not satisfied, i.e. the energy needs to be produced in a short period of time, and hence the efficiency is not the main criterion to determine the controller 90 references. This operation model is difficult to determine because the energy vs time relationship is more complex than in modes 1 or 2. The transients in engine and generator should be synchronized to properly follow a specified profile. A simple controller is proposed: 0 When the maximum torque that the engine can provide is higher than the maximum counter torque of the generator, torques are commanded to their maximum values, allowing the production of power from the beginning. This allows the system to speed up. 0 When the maximum engine torque is lower than the maximum torque of the generator, the engine torque is commanded to its maximum value and the gen- erator torque is commanded to a value 10% less than the one from the engine. This strategy allows production of power from the beginning, and allows the system to speed up. 0 When the maximum speed is achieved, engine and generator torque commands are set to be equal. Both torque commands are set to the minimum of the maximum engine torque and the maximum generator torque. 5.5 Comparison This section presents a comparison between two classical control strategies [40, 41] and the optimal trajectory and the approximated optimal trajectory proposed here. The main purpose of the optimization algorithm is to provide the requested energy while maximizing efficiency. Generally, initial and final speeds are not the same, therefore the kinetic energy stored in the engine-generator rotational inertia changes, which affects the efficiency calculation. The efficiency formula used in this paper 91 compensates for this change in speed as follows: 7 2 ,2 IlGEN — 0’5Jm (writ—t0 _ (”m—tf) n = f 2 2 7 ll GEN — 051m (“m—t0 “ wm—tf) + ”1088 The two classical control strategies are summarized below: 0 The first control strategy of “constant power” proposed by Ceraolo et a1. [40] determines the torque and speed references by the procedure summarized in “Mode 2” but it is used for all power values (higher and lower than the power at the most efficient point). 0 The second control strategy ,“ON-OF F” proposed by Barsali et a1. [41], pro- poses to turn on and off the engine-generator subsystem according to the energy and / or power requirement. This control method requires to speed up and down the subsystem in zero time, which is not possible. This concept is adapted here to analyze the losses during the transient. To achieve the “ON-OFF” opera- tion, the system will be kept at constant speed with a value equal to w = adopt. The generator will operate at Thz- = Topt for time t hi to produce the requested energy, and for the remaining time, tlo’ it operates at Tlo = 40N m to produce low losses. Four control methods were compared. Tables 5.3 and 5.4 summarize the simula- tion results of the controllers with the same initial speeds wm—tO = 1100 r/mz'n and wm—tO = 1800 r/min and final fixed time t f = 6 s. The engine was operated in speed mode and the generator in torque mode. The PI speed controller of the engine had Kp = 45 and K,- = 100. The approximate optimal trajectory operates in mode 1 for the first 6 cases and in mode 2 for the last case. Neither of the controllers were tested in mode 3 due to the fact that there are not an adequate criteria to make a fair comparison between them. 92 The proposed optimal trajectory shows better efficiency than the other methods in almost all conditions, Table 5.3. For wm—tO = 1800 r/min and Wm f = 7.485 J the proposed method had a lower efficiency, but this was due to the fact that the non-optimized methods produced less energy. The approximate optimal trajectory has an efficiency comparable to the efficiency of the optimal trajectory and better that the two classical control strategies. The two proposed controllers satisfy the energy requirements as shown in Table 5.4, which is the main objective of the optimization. Table 5.3: Controllers comparison, Efficiency Control Strategy (Um—to erf ON Const. Proposed Proposed Approx. OFF Power Method Method 2.50E+05 0.342 0.364 0.379 0.377 1100 4.50E+05 0.350 0.374 0.382 0.382 6.50E+05 0.350 0.364 0.384 0.384 2.50E+05 0.347 0.360 0.381 0.377 1800 4.50E+05 0.375 0.376 0.384 0.380 6.50E+05 0.384 0.381 0.385 0.383 7.40E+05 0.385 0.388 0.378 0.388 Table 5.4: Controller comparison, normalized final energy It f / W16 f Control Strategy wm_t0 Wref ON Const. Proposed Proposed Approx. OFF Power Method Method 2.50E+05 1.730 0.998 0.998 0.988 1100 4.50E+05 1.257 1.001 1.003 1.002 6.50E+05 0.880 0.944 0.991 1.000 2.50E+05 1.078 1.005 1.004 0.996 1800 4.50E+05 0.986 1.009 1.005 1.004 6.50E+05 0.950 0.996 1.002 1.003 7.40E+05 0.908 0.962 0.980 0.962 93 Chapter 6 High Performance Low Speed Sensorless Control of Interior Permanent Magnet Synchronous Motor The work presented in this chapter deals with the issues in practical implementa- tion of high-frequency injection methods to control IPMSM’S without shaft position sensors. Implementing a high-frequency injection technique for rotor position estima- tion requires intense signal processing of the measured currents, as well as filtering of the signals to extract the rotor position information. These tasks complicate the control algorithm, but more importantly degrade the overall system performance by introducing delays, increasing response times and decreasing the bandwidth of the controller. Moreover, some specific filtering techniques limit the operation of the ma— chine to a very narrow range. These issues and their specific effects are discussed in this chapter, and mitigation procedures are proposed, implemented and verified through the experimental setups. Experimental results of a new high-performance 94 controller without rotor position sensors are presented as well. The major contribu- tion of this chapter is a new controller based on the high-frequency injection method that eliminates the need for low-pass filtering and improves the performance of the machine. 6. 1 Introduction Rotor position information is necessary for Field Oriented Control (FOC) of Per- manent Magnet Synchronous Machines. Rotor position sensing requires resolvers or encoders, an alternative is to operate the machine without shaft sensors. Back EMF methods are preferred at higher speed operation [42, 43, 44], while Pulse Width Mod- ulation(PWM) analysis [45, 46] and high frequency injection methods [47, 48, 49] have been used for low speed operation. Juliet and Holtz [45] investigated measuring the di/dt in the phase currents result- ing from the PWM modulation so no additional signal injection is required, however, as they noted, measuring the di/dt is very difficult and almost unfeasible for the implementation beyond laboratory setup. High frequency injection of current or voltage has inherent problems related to decoupling the injected signals and the power component signals, one used for the rotor position estimation and the other to control the machine torque respectively. The selection of filters and their parameter change significantly the motor performance [50]. The most common method of post-processing the signal obtained through the high-frequency injection is by using Synchronous Reference Harrie Filters (SRFF’s) [50]. They can remove a specific frequency, such as the injected frequency. The spectrum location of the resultant high frequency current (or voltage) depends on the frequency of the applied high frequency voltage (or current) and the electrical speed, which is unknown. Then, in the synchronous frame of reference the spectrum of the 95 resultant current (or voltage) is located at low frequency, requiring a low pass filter to remove it. A low-pass filter can be also used to isolate a given signal, but signals beyond the filter bandwidth will only be attenuated [51]. In addition, both methods will lower the overall system bandwidth. Vadstrup [52] used two cascaded Band Stop Filters (BSF’S); one to remove the positive and the other to remove the negative sequence currents, which would allow the fundamental component to pass with no phase-lag if the filters have the same bandwidth. Some attenuation however is introduced, de- pending on the required bandwidth of the filters and the spectral separation of the carrier signal components [53]. Moreover, it is not always possible to tune the two cascaded BSF’s to the same bandwidth, resulting in both phase error and attenu- ation at the fundamental frequency. Young-doe [54] used a square wave injection voltage to remove the Low Pass Filters (LPF’S) from the rotor position estimator path, improving the observer performance, however, this methodology still requires a low pass filter in the current controller, which decreases the bandwidth of the current controller. Shinnaka [55] proposed a new injection method that includes injection in direct and quadrature axis, where the quadrature axis voltage magnitude is pro- portional to the speed. This modification allows the controller to increase the speed range of the controller, but BSF’s and BPF’S are used in the observer and controller to separate power and high frequency signals. The issues of cross-magnetic saturation in high frequency injection methods have been presented by Zhu et al. [56]. They proposed mitigating the rotor position error due to saturation by calculating the value of the error angle and its dependence on the current. It is well documented that the error in the estimate of rotor position increases as a function of the load current [47]. Guglielrni et al. showed in [57] that an error exists in rotor position estimation due to cross-magnetic saturation between d— and q-axes. Stumberger et al. in [58] evaluated the effect of saturation and cross- 96 magnetization for IPMSM’S, but offered no method for compensating for the resulting error. Boldea [59] proposed the concept of active flux, which changes the controller frame of reference and no high frequency injection is required for sensorless control at low speed. The active flux is convenient for control purposes because the machines can be controlled as if they have no saliency. The response at low speed is validated by simulations. This chapter presents a new methodology to design a rotor position observer using high frequency injection with no filters in the current feedback loop. This simplification allows better performance of the current controller. The rotor position estimation is improved by removing the power frequency components of the signals due to the power-current control, from the high frequency signals due to injection. Experimental results are used to validate the proposed rotor position estimator and the cross saturation effect in the rotor position estimation is presented. This chapter is organized as follows: section 6.2 is divided in two subsections. Subsection 6.2.1 shows the classical approach where low pass and high pass filters are used for current control and rotor position estimation respectively. Subsection 6.2.2 describes the proposed methodology. Section 6.3 presents and discusses experimental results, where the filtered demodulation and the proposed controller are compared. 6.2 High Frequency Injection Architectures In this section two methodologies for rotor position estimation based on signal injec- tion along with the required demodulation are discussed. The first methodology uses Low Pass Filters (LPF’s) and High Pass Filters (HPF’S) to separate power compo- nent and high frequency injected signals [60, 61]. The second methodology, proposed here, uses a high gain current observer to indirectly extract the signal components due to high frequency injection. A transfer function to analyze the effect of the high 97 frequency injection over the power component is presented. This transfer function and a high gain observer are used instead of adding filters in the current control loop, this modification leads to a higher performance of the current controller. 6.2.1 Filtered Controller Several methods have been proposed for high frequency injection [62, 63]. For voltage injection, superposition is used to analyze the effect of the high frequency in the machine. High frequency and power frequency are separated with Low Pass Filters and High Pass Filters as shown in Figure 6.1. .=« vdii l + + Vd .* LCD—J PI 16+ —> Z _ 11 0Q] S .* n —>MTPA l V ABC wlnverter \[ IPMSM 6] + ‘1 Pl 7 —> _ 1 f 5 id ABC/ <4 q d LPF ‘ . DO 14 HPF : f lqh é Rotor position 3 observer Figure 6.1: Filtered rotor position estimation block diagram The block diagram in Figure 6.1 is composed of two basic subsystems: The power loop for the machine control, and the rotor position estimation loop. The power loop consists of two PI controllers in cascade with two low pass filters and the machine. In the power loop the low pass filter removes the high frequency content of the measured current, so that the inputs and outputs of the PI controllers are free of high frequency 98 content. The high frequency loop used to estimate the rotor position is described below: Consider the simplified machine model in the rotor frame of reference as shown in equation (6.1). 098 _ R5 + LC”) —weLq 2:18 USS WeLd R3 + 141]) 168 (6.1) 0 + “’eAd—PM where 1298 and 1153 are the direct and quadrature axis voltages, 2'98 and 1'68 are the currents, Ld and Lq are the inductances, R3 is the stator resistance, “\‘d— P M is the flux linkages due to the PM rotor flux and we is the electrical speed. Based on superposition, the machine model can be simplified to analyze the effect of the injected high frequency signals as shown in equation (6.2). 7' ' , -r L vqsh 0 R3 + Lqp zqsh For wh’ >> we the contribution of Ad— PM to the voltages and currents at the injected high frequency is zero. Equation (6.2) becomes simpler for an injected signal of constant magnitude and frequency. For slower changes in the motor speed with respect to the current changes, the machine model can be approximated by a linear system with the following state space representation: 99 ”fish R8 + J'de’h 0 fish T . .T‘ vqsh _ 0 R5 + quwh zqsh (6.3) Zdh. 0 Zita/t '7' In operation without rotor position sensor the mechanical rotor position is not known, instead the controller uses the estimated rotor position 96. To analyze this operation, the motor model is rewritten in the estimated rotor frame of reference 7‘. Equation (6.3) is rewritten in this estimated rotor position frame of reference, and the transformation matrix of equation (6.4) as function of de = 68 — (ale is used to transfer the motor model from 06 frame of reference to ée frame of reference as shown in equation(6.5). R(9~e)= cos(6l(j) sirl(de) (6.4) —sin(08) cos(68) of .. Z 0 ~ 23": €13" = R_1(9e) 0‘" Rte) 43” or 0 Z h ir h qsh (I qs j A (6.5) = ] Z + ZA COS(29€) ZA sin(268) ifish ] ZA sin(2ée) Z — ZA cos(29~e) A ifish where Z = R3 + jLwh and — Zdh+Zh Zdh—Zh Z=————2——q—,ZA=———2—q (5.5) Equation 6.6 shows that by eliminating the cross coupling terms, the error between 100 the estimated and the mechanical rotor position becomes zero. Voltage can be injected in the direct or in the quadrature flux axis. Direct axis voltage injection is preferred, because it produces less torque pulsations than quadrature axis voltage injection. For direct axis voltage injection the high frequency component of the voltage is: 1272, V cos w t fish = h. ( h. ) (6.7) vqsh 0 where Vh is the amplitude of the injected voltage and wh is the frequency of the injected signal. By substituting equation (6.7) in equation (6.6), and by neglecting the resistive component of the impedances, the quadrature axis current is written as in equation (6.8). ‘ VhLA sin (Witt) -T' =— ' 267 6.8 zqsh thqu sm( 6) ( ) The high frequency quadrature axis current is demodulated to obtain the rotor position information as shown in Figure 6.2. A PI regulator and an integrator are used to minimize the error function f (88) of equation (6.9), where, for smaller values of fig, sin(2fie) can be approximated as We The minimization of f (1935) leads to the minimization of the error in the rotor position. ~ V L - - fwe — —h—A—siu(2ae) a: Irma, (6.9) — 2thqu — sin(coht) flée) 0* ‘ A l LPF Pl 1/8 -> 6 qsh e L_. a; 8 Figure 6.2: Block diagram of rotor position estimation by synchronous frame of ref- erence filters 101 6.2.2 High Performance Controller In Figure 6.1, the power component of the measured current signal is obtained by re- moving the high frequency by LPF’s which have some inherent disadvantages: LPF’s isolate a given signal, but signals outside the filter bandwidth are attenuated, i.e. the LPF’s do not eliminate completely the high frequency signal. In the same manner, if the given signal has a high frequency component, these components will be attenu- ated and their phase will be shifted, which is undesirable. Current commands usually are composed by ramps or steps, which have high frequency content; in order to track these current commands properly, the LPF’S should have a bandwidth high enough so that the signals are not attenuated, but low enough to reject the injected high fre- quency. To properly filter the signals, the injected frequency is selected high enough, so that the stop band of the filter does not interfere with the power component. The required voltage to produce a desired high frequency current is proportional to the injected frequency, that means that higher the frequency for injection the higher the magnitude of the injected voltage. High magnitude of voltage produces higher losses and less available voltage to manage the transients of the current controller. In this subsection a new structure for rotor position estimation is presented. Power components and injected components are separated without the use of a LPF’s and HPF’s. In the current loop the LPF’S are removed; that is done by designing the de- modulation scheme based on the commanded high frequency voltage, which is known, rather than on the high frequency voltages and currents components at the machine terminals. This modification requires the calculation of the transfer function between commanded and applied high frequency components of the voltage. This transfer function is used to calculate the input voltage and to operate the motor within its voltage limits. For the extraction of the injected frequency components of the signals, the machine model is used to remove the power frequency components by the use of 102 a high gain current observer. In the first part of the analysis the relation between the applied and the com- manded high frequency voltage components is determined. Let us consider the high frequency model of the motor in steady state as shown in Figure 6.3. For direct axis voltage injection, the transfer function between the commanded high frequency voltage and the applied high frequency voltage is calculated in equation (6.10). K, + f Vd Q ' Kp + - Machine model Vd = RsId +jthdId K + K" vi ° _ p jot), Vq = RSIq +thLqu lq 1d Figure 6.3: High frequency equivalent model of the motor and controller 2 . vd(wh) : —thd + JUJhRs (6 10) vdh(‘*’h) (K,- — ngd) + j (ths + thp) In the second part of the rotor position estimation the high frequency component. of the signals due to the injection is extracted. The rotor position estimation is based on the commanded high frequency voltage vdh' A high gain observer is used indirectly to determine the effect of vdh on the machine signals. The high gain current observer is described below. For a low and almost constant speed operation, the state space machine model of 103 equation (6.1) can be rewritten as: ,T _ 1%, weLg ,,. 7“(13 g d 2(11.3 '.'r we _ R .7‘ 2<18 “If1 ‘17; Zqs 1 I; 0 0 0 ,,. (6.11) abl" .Q For rotor position detection, the high frequency commanded voltage is superim- posed on the main power voltage component t give 11 = ”up + ’Uh, where v is the input voltage to the machine, up is the voltage due to the power frequency and ”h is the high frequency commanded voltage. For an injected frequency relatively close to the fundamental, the inductances Ld and Lq do not change in this operation range, and can be assumed constant. By substituting v = up + ’Uh, the model of equation (6.11) can be expanded as equation (6.12). -. R weLq . zfis : — I; d 2its $53 - wel q! I _ %; {63 1 iv}; - 0 0 s-p IE 0 1 —)‘d—P]W ”qs—p ( ) I? T L “’6 1 0 ,r + . I; 7nits—h 7' 1 0 Z—q— UQS—h The system can be considered as a linear system with a disturbance, of the form: :13: A$+Bu+dh (6.13) 104 where :L' = 12:, u = [”25” 1158,1115] and (5), = U}, / L. The disturbance will be indirectly estimated by a high gain current observer. This disturbance, 6),, contains the infor- mation of the injected signal modulated by the rotor position. This term can be used to estimate the rotor position by the procedure described below. The high gain current observer is of the form: I k 3 hi; L(i‘) = 1 = ksgn(i‘) = 1 g ( 1) 12 k239n(5132) (6.14) 5’31 2 (id — 2.d) where 532 = (iq " iq) where id and 1' d are the measured values of the current and id and iq are the estimated values of the current. The estimated state becomes: a = A: + Bu. + L(*) (6.15) with a = .45: + (6(513) — L(5:)) (6.16) The stability of the observer is analyzed as follows. For a positive definite function: V = $511115: (6.17) with _ - 1 0 I? M = q 1 0 7 - La _ the derivative becomes 105 V zirTllIa“: = :ETMAi: + eTM (6(1) — L(:ir)) ,. - 0 O emf—l 1 0 (5 —l _2 2 2 - Ld 1 [...—[£512. weL - [ ~ ~ ] Lqu Lqu 551 = CCl 12 weL R ~ Lng Lng x2 1 1 +—i (6 —l)+—:E((5 —l) 2 1 1 1 2 2 2 2 Lq Ld "—31: #2— - - Lqu d ‘1 ‘21 _ qud Lng _ 2 1 1 1 + 2$1(51-11)+—L—2‘I2(52—12) q d (6.18) Equation 6.14 defines 11 = klsgn (i1) and 12 = kgsgn (1172). By the selection of the gains k1 = 111 + [£12 and k2 = k21 + 1522 with kn > 51, 1:21 > 52, and k1, k2, 1311,1912, 11:21, 1:22 > 0 the derivative of the Lyapunov function becomes: 106 ~2 ~2 . .1:le $2R3 1 1 .. ~ 1 ' :— " V = ——L—L§ — T‘s—L? — p (111 [931] - 51131) — p (‘2 [I2] — 52352) < 558125 523123 k12 [531] k22 [5E2] — ‘ 2 " ’—§ “ 2 ‘ 2 Lqu LqLd Lq Ld This derivative function becomes negative definite proving the stability of the observer. This condition ensures that the observer will converge to the sliding surface in finite time. The low frequency component associated to the high gain observer compensates for the disturbance. In order to obtain the information of the disturbance it is necessary to filter the signal from the high gain observer. Hem (6.16) it is clear that when :ET = 0 then 6(a) = LPF(L(§:)). That is: 7' v LPF (klsgn(;ic1)) = ifijfl .Ur LPF (kzsgn(;i:2)) = —%—Q where a LPF’S is used to remove the undesired components of L(:i:) produced by the (6.20) sgn function, these frequency components are located at frequencies higher than the switching frequency. These low pass filters have a bandwith significantly wider than the SRFF’S, because their purpose is to remove the frequency components located at least one order of magnitude further than the injected high frequency signals. Using this observer, the power frequency can be removed entirely from the high frequency signal, moreover no assumptions are needed with respect to the stator resistance. The rotor position estimation can be done by a similar approach as described in the subsection 6.2.1. The disturbance can be written as: 107 r 67‘ 60,13) __ dSh 6T qsh : (6.21 ) 1 ,'r L; tdsh 0 1 t 0 [5 quh For injected voltage in the estimated rotor frame of reference, it is possible to get an equivalent expression for the rotor position error. ”fish. =R_1(ée) th 0 12(53) :(Esh 0 L h f ‘13 qt ‘M (6.22) = L + LA cos (296) LA sin (206) éfilsh LA sin (2198) L —— LA cos (206) afish where _ L. + L L — L L = $1117.92, LA = _flifl (6.23) For direct axis voltage injection of equation(6.7) and by substituting equation (6.21) in equation (6.22), the quadrature axis disturbance can be written as: , V L cos (w 1‘) ~ _4 ~ __ r __ _ h A h ' c LPF (L2 (22)) —— (quh —— Lqu sm(266) (6.21) The high frequency quadrature axis disturbance is demodulated to get the rotor position information as shown in Figure 6.4. A PI regulator and an integrator are used to minimize the error function f (53) as shown in equation (6.25). The minimization of f (53) leads to the minimization of the error in the rotor position. 108 1(a) — The overall system is shown in Figure 6.5. L2 (522) —cos(a),,t) _> . LPF 6*, (I LPF Pl V L . ~ ~ __ Ef—LfL—q Slll(29e) % Kerrge (6.25) Me) 1/8 A ”0 e A ‘60 V 6 Figure 6.4: Block diagram of rotor position estimation by the use of a high current observer —> MTPA ’3 IPMSM _. £3!) +lnverter _>. . + Current< ld A53] Obs. : A l I I j a), q 68 Rotor position observer Figure 6.5: Proposed rotor position estimation block diagram 6.3 Experimental Results The proposed rotor position estimator was validated experimentally and compared with the filtered controller. 109 The experimental setup is shown and described in Apendix A, for reference, the schematic of the experimental setup is shown in Figure 6.6. The dynamometer is set to operate in constant speed wm = 100 r/min. and the PMAC is Operated in current/ torque control. The motor and the injection pa- rameters are listed on Table 6.1. To operate the machine under the same conditions, the input high frequency voltage component should be the same for both topologies. As listed above for the proposed controller, the commanded high frequency voltage differs from the input high frequency voltage component. For a fixed input high fre- quency voltage component ”d = 20 V, equation (6.10) is used to calculate vdh’ the required commanded voltage vdh is equal to 27 V. Power RT linux controller + FPGA _ supply (Controller) + T Labview ‘ Control 1 l l 66 Controller i VDC Signals . . 6 Encoder q l 1 1 lbs 165 e 60 252525 ’" 2:23;: x7]> , /77 Inverter #7 Torque-meter /7)7 PMAC motor Dynamo meter Tachometer PMDC motor Figure 6.6: Experimental Setup The proposed methodology subsection 6.2.2 was compared with the one of sub— section 6.2.1 for two operating conditions: steady state and for a current ramp with rise time of trise = 0.1 .9. Filter parameters were kept constant. Tables 6.2 and 6.3 summarize the comparison of the two methodologies, for steady state error and transient respectively. The current commands are applied either in the quadrature axis, 13 = Iq, or in the current angle for maximum torque per ampere, IS = 10, and 110 Table 6.1: Experiment setup parameters Parameter Value Units Number of pole pairs 2 Stator resistance 0.4 Q Rotor flux linkages 0.4652 I'Vb Direct axis inductance Ld 0.01462 H Quadrature axis inductance Lq 0.04810 H Rated peak voltage per phase 120 V Rated peak current 20 A Injected frequency w), 250 Hz Mechanical speed com 100 r/rm’xn Injected Voltage Vh 20 V Proportional gain lap 20 Integral gain k, 80 with current increments every 25%. Table 6.2: Steady state error in the rotor position estimation Error in the rotor position estimation in electrical degrees Controller with filters High performance controller (A) Max Min P-P Mean Max Min P-P Mean Iq = 0 2.2 -3.1 5.3 0.0 2.5 -3.6 6.0 0.0 Iq = 5 1.0 -3.9 4.9 —1.6 0.6 -3.1 3.7 -1.4 Iq = 10 ~36 -8.9 5.3 -6.5 -3.2 -7.9 4.7 -5.8 [q = 15 -12.0 -17.7 5.7 -15.0 -11.2 -16.2 5.0 -13.9 10 = 0 2.2 -3.1 5.3 0.0 2.5 —3.6 6.0 0.0 10 = 5 1.4 -3.1 4.5 -1.0 2.4 -2.5 4.9 0.1 10 = 10 1.5 4.6 6.0 —1.9 1.4 -3.3 4.7 -0.9 [0 = 15 -2.1 -8.2 6.1 —4.5 -O.5 ~52 4.7 -3.2 Current Practical implementation of the proposed observer requires that the voltage in the machine terminals and the commanded voltage to be equals. The observer proposed here uses the commanded voltage to extract injected signal from measured currents making important to match commanded and applied voltage. Dead time compen- sation were used to reduce the error difference between both voltages [64]. Several approaches have been proposed to compensate for dead time [13, 14, 15], where dead time, on—time off-time of the power devices and their associated drop-off voltage where 111 Table 6.3: Rotor position error for a current ramp Current Error in the rotor position estimation in electrical degrees Controller with filters High performance controller (A) Min Max P-P Min Max P-P Iq = 0 —3.1 2.9 6.0 -3.6 2.5 6.0 Iq = 5 -6.2 6.7 12.9 -6.0 8.6 14.6 [CI = 10 -11.0 10.9 21.9 -12.9 8.7 21.6 Iq — 15 -25.8 13.6 39. 4 —24.7 7.5 32.2 10 = 0 -3.1 2.9 6. 0 -3.6 2.5 6.0 Io = 5 -4.2 6.6 10. 8 -5. 6 6.2 11.8 10 = 10 —11.8 13.6 25.4 -10. 6 13.8 24.4 IO = 15 -18.1 17.1 35.2 -15.2 17.5 32.7 included to corrected to for this effect. A simple approach is used to correct for dead time, where the drop of voltage in the power electronics devices where neglected. The commanded voltages were corrected as follows: ’Ua—INV = a2; vb—INV - v}; * %- [NV — vc ing times in the inverters, Tsw is the period associated to the switching frequency and Td is the dead time. The methodology of subsection 6.2.1 was compared with the preposed estimator with and without dead time compensation for the following operating conditions: steady state and for a current ramp with rise time of trise = 0.1 3. Tables 6.4 and 6.5 summarize the comparison, for steady state and transient error respectively. Figures 6.7, 6.8, 6.9 and 6.10 show the transient response of the rotor position estimation and speed estimation and Figures 6.11 and 6.12 show the reference and measured currents in the motor for a ramp in the current reference of 75%. 8 T +7131” + 7;?“ + 3 U) <1<1< 2112 —1?§Q[ [282'g'n(2b —%C[ [2sz'gn(2 where va_ I NV, vb_ I NV and vb_ I NV are the voltage used to calculate the switch- [sigr2(2 112 _ Sign/(rib) — 2sign(2'c)] — sign(2'a) — 2sign(2'c)] — sign(2'a) — 2sign(ib)] Table 6.4: Steady state error in the rotor position Error in the rotor position estimation in electrical degrees Current Controller Proposed Proposed observer with filters observer with T d compensation (A) P-P Mean P-P Mean P-P Mean 10 = 0 5.30 0.00 6.00 0.00 4.81 -0.02 Io = 5 4.50 —1.00 4.90 0.10 2.50 -1.03 [0 = 10 6.00 -1.90 4.70 -0.90 3.98 -1.74 10 = 15 6.10 -4.50 4.70 -3.20 6.10 -3.16 Table 6.5: Rotor position error for a current ramp Error in the rotor position estimation in electrical degrees Current Controller Proposed Proposed observer with filters observer with Td compensation (A) Min Max P-P Min Max P-P Min Max P-P 10 = 0 —3.10 2.90 6.00 -3.60 2.50 6.00 -1.85 2.24 4.09 10 = 5 -4.20 6.60 10.80 -5.60 6.20 11.80 -5.40 5.35 10.75 10 = 10 -11.80 13.60 25.40 -10.60 13.80 24.40 -9.27 8.07 17.34 Io = 15 -18.10 17.10 35.20 -15.20 17.50 32.70 -16.44 11.20 27.64 The performance of the two estimators were evaluated, by comparing the esti- The experimental results show uniform improvement in the steady state error cross saturation has significant effect in the rotor position estimation [65, 57], which values of 6. The cross saturation effect in the main torque has been described in [28]. 113 steady state faster than the one that used SRF F Figure 6.13-b. mated torque during the transient specified above, as shown in Figure 6.13. The proposed rotor position estimator has a faster response Figure 6.13—a and comes to and in the variation of the rotor position error. The preposed methodology works significantly better under high load conditions as shown in Tables 6.2 to 6.5. The is demonstrated by the experimental results, where the error in the rotor position is different for the same magnitude of the Current Space Vector (CSV) but for different Further analysis requires the inclusion of the cross saturation effect in the current observer or add techniques to correct the effect of cross saturation in the rotor position 150 - - 100 r . a ‘I '36 50 ' Q) E. 0 :3: -5o - -100 r -150 - O 0.5 1 1.5 2 2.5 Time (s) (a) Estimated rotor position 20 I T I r r 10 _ - F63 Q) a. 0 ~ on 1Q '10 r 1 _20 1 m 1 1 1 0 0.5 1 1.5 2 2.5 3 Time (s) (b) Error in the rotor position estimation Figure 6.7: Rotor position estimation for controller with filters estimation [66]. Similarly the speed dependance of the controller-observer needs to be further explored. The main focus of the the work presented in this chapter was to present an alternative to the BSFs and the BPFs, and improving the dynamic performance of the controller, [66, 55] can be used to correct the rotor position error due cross-saturation and speed change. 114 ..a. O) O ‘ 30 t V 60 - O 0.5 1 1.5 2 2.5 Time (s) (a) Estimated speed 40 30 - "E" 20 - E 10 \ 5.4 C.__l O 138 -10 -20 - -30 ‘ ‘ ‘ * J 0.5 1 1.5 2 2.5 Time (s) (b) Error in the speed estimation Figure 6.8: Speed estimation for controller with filters 115 I r [I 150- _,_ 98 . 100 —ée - ”EB 50 . G) .9. 0- l Q31 -5o~ - —100- ~ -15o- fl 0 0.5 1 1.5 2 2.5 3 Time(s) (a) Estimated rotor position 10f ‘ 06 [Deg] i l -10 I -15 0 0.5 1 1.5 2 2.5 3 Time (s) (b) Error in the rotor position estimation Figure 6.9: Rotor position estimation for the high performance controller 116 1 l I 0 0.5 1 1.5 2 2.5 3 Time (s) (a) Estimated speed l 1.5 2 2.5 3 Time (s) (b) Error in the speed estimation Figure 6.10: Speed estimation for the high performance controller 117 z'q [Al ore-b.0500 — ‘ _ 7tq—mes I — 1 l-l-IZ. p- P 0 0.5 1 1.5 2 2.5 Time (s) (b) 2'q Figure 6.11: Machine current for the controller with filters 118 F— 8 - i 2‘ . . —mes s. 6 t q ,3 l .- . - . 2'; 4 " I 1 .” 2 - I. ~ 0 “land—Ii . 1 1 O 0.5 1 1.5 2 2.5 3 Time (s) (b) 2'q Figure 6.12: Machine current for the high performance controller 119 30 r . 25 4 Proposed Estimator Pg 20 b ''''' SRFF Z 15 . 10 1 - E 5 ' 4 o -.. ”'4' - 1 1.02 1.04 1.06 1.08 1.1 Time (s) (a) Te 40 . . . . )l . .~ '3‘.- .0 -1.“3t' .wmtzs'swtsewwmaw.tt'mttt'th‘tawa... I . 30 - (l. E. in a . 1‘ 10 Proposed Estimator _ ----- SRFF (+10 Nm) 0 . 1 1.1 1.2 1.3 1.4 1.5 Time (s) (b) Te Figure 6.13: Dynamic performance comparison of the estimator 120 Chapter 7 Conclusions and Anticipated Future Work This chapter presents the concluding remarks and future work as follows: 7 .1 Cross Saturation Analysis The cross magnetization was described, characterized and its effect in the torque was calculated. The cross magnetization was included in the machine model by two quasi- mutual inductances. Torque and voltage equations including mutual inductances were used to determine torque and speed profile. FEA simulations in four machines with different stator configuration and experimental results in a prototype motor were used to validate the proposed analysis. 121 7 .2 Trajectory Optimization for the Operation of Traction Motors A methodology for the optimal control of an interior permanent magnet motor for traction applications was presented. The methodology to calculate the optimal cur- rent command for a given torque and speed point considering cross saturation was described, and the optimal trajectory calculation was presented. The use of cross saturation and optimal trajectory calculation in the control algorithm leads to the operation of the motor with minimal energy losses. This work included the modifica- tion of the algorithm to calculate the optimal trajectory online. This approximation significatively reduces the computational time, however the solution was not optimal. The use of the approximated curves as an initial guest was also explored. It allows the reduction of the iteration number and achieves an optimal trajectory control. Simulated and experimental results validated the proposed methodology. 7 .3 Trajectory Optimization for the Operation of Engine-generator Subsystem A torque and speed optimization scheme of the engine-generator subsystem of a series hybrid electric vehicle was presented. The optimal methodology allows the calculation of the torque and speed references of the engine-generator subsystem that minimize the energy loss for a given target energy. An approximate optimal trajectory was proposed to implement this controller in real time. The developed algorithms were evaluated under different energy requirements and initial engine speeds through sim- ulations. The simulation results demonstrated that the proposed optimal trajectory and its approximation improved the subsystem efficiency over conventional control techniques. Future work will consider the integration of the proposed optimization in 122 the overall optimization of the entire powertrain system. 7 .4 Sensorless Control Two methodologies for rotor position estimation based on voltage injection were pre- sented. A high performance controller was described and a methodology for rotor position estimation was explained. A high gain current observer was described and its stability was proved. This observer was used to extract the high frequency com- ponents of the signals required for the rotor position estimation. The rotor position estimator with high gain current observer approach showed improvements over the classical estimator, where transient and steady state error in the rotor position esti— mation where analyzed. Further development would include the analysis of the cross saturation effect on the rotor position estimation. 123 APPENDICES 124 APPENDIX A Experimental Setup The experimental setup used to validate the proposed control techniques is described in this Appendix. Figures A.1 and A2 shows the experimental setup and Figure A.3 shows the schematic. Tor que- Dynamometer .... 1 _,meter ,. . . IPMSM o....; . Figure A.1: Motor and dynamometer The setup developed for this thesis is composed of the following devices: 0 Real Time Controller: Rt-Linux and Field-Programmable Gate Array (F PGA). 0 National Instrument based Data Acquisition System. 0 Sensors: voltage, current, rotor position and torque. 125 RT-Iinux system FPGA Inverter Figure A2: Controller and inverter Power RT linux controller + F PGA supply (Controller) + z- Labview ‘ Control ‘ ec Controller - V Encoder 1 DC (>— 25 2: 251 60’" 2t 52:] /'/.’ /‘/ Inverter .77 Torque-meter J7 PMAC motor Dyn amo meter Tachometer PMDC motor Figure A.3: Schematic of the experimental setup 0 Interior Permanent Magnet Synchronous Motor. 0 Inverter. o Dynamometer. 126 A.1 Real Time Controller: Rt-Linux and FPGA A PC running RT—Linux was used as the controller for this project, Figure A.4. The PC is a good choice compared with a Digital Signal Processor (DSP) in terms of cost, CPU power, and memory capacity. A fundamental limitation of the PC, however, is the limited I / O capability. A custom Xilinx FPGA based 1/ O board was developed. Communication between the I / O board and the PC is via the EPP parallel port. The I/O—FPGA board has one encoder input, four D/ A outputs, twelve PWM outputs and twelve A/D input channels. The encoder inputs accept dual quadra- ture channels and an index pulse in either differential or single-ended configurations. The board processes the encoder counts and stores absolute position in a twelve-bit counter. There are twelve digital outputs which can be configured for PWM (2500 counts at 20kHz), four D/A outputs (12-bit, +/-5V range) and twelve analog inputs (12-bit, +/-10V range). The list of the measured and output signals are presented below: 0 Measure: DC-link voltage. Measure: two phase currents. Measure: one phase voltage. Measure: rotor position. Output: six PWM outputs. 127 Figure A.5: FPGA and I/O A.2 National Instrument Based Data Acquisition System Two data acquisition boards were used with Labview. The first one (NI USB-6009) was used to develop a supervisory control for the dynamometer and the second one (NI USE-6259) was used to acquire and store voltage, current, torque and position. In the supervisory control Labview were used to give the setpoints to de dy- namometer an measure its signals. The analog to digital converters and the digital to analog converters have a resolution of 14-Bit, with a maximum sampling rate of 48 KiloSamples per second (kS/s). The measured and commanded signals are listed below: 128 0 Measure: speed. Command: speed. Measure: armature current. Command: current. Command: torque or speed operation. Command: Activate Field weakening. The second data acquisition card was used to collect data for analysis (not required for the online controller). Due to the sampling rate limitations two signals and the encoder were collected at the same time. The minimum sampling rate obtained was lips with a resolution of 16 bits. The measured signals are presented below: 0 DC—link Voltage. 0 DC—link Current. Phase Voltages. 0 Torque. Rotor position. Speed. A.3 Sensors: Voltage, Current, Rotor Position and Torque A set of sensors were used to acquire signals for control and analysis. The following sensor were used: 129 . 4- 1:1“43 Figure A.8: Labview with the virtual instrument 130 Two phase currents were measured using current transducers LEM (la-100p) with rated accuracy of 0.45% and bandwidth of 0-200kHz. Two phase voltages were measured using voltage transducers LEM (iv—25p) with rated accuracy of 0.45% and bandwidth of 0—200kHz used to calibrate the rotor position sensor. A single current sensor LEM (la-100p) to measure the DC link current in the inverter. A single voltage sensor LEM (Iv-100p) to measure the DC link voltage in the inverter. A quadrature encoder BEI (H25) with 1024 counts per revolution (4096 for quadrature) and an index pulse, this encoder was used to measure the rotor position. Torque sensor PCB (STS 5100) with a peak torque of 110 N m with an amplifier PCB (SERIES 8159). Figure A.9: Current and voltage sensors 131 Figure A.10: Torque sensor A.4 IPMSM An IPMSM was designed and assembled to validate the proposed control. The ma- chine has the following characteristics: 0 Four poles. 0 Two slots per pole per phase. 0 Three phase. 0 Winding pitch of 1200. 0 Double layer winding with 30 turns per coil and 60 turns per slot of copper conductor AWG 14. Neodymium iron boron magnets. Figure A.11 shows the rotor, stator and windings configuration. A.5 Inverter An inverter was designed and assembled to operate the IPMSM. The Integrated Power Module (IPM) selected was a powerex (pm75cla120) with a rated voltage of 1200 V 132 (c) Windings (d) Assembly Figure A.11: IPMSM DC and a rated current of 75 A. The gate drive used was a powerex BP7B. A bank of capacitor of 240;: f was used for the DC—link. Figure A.12 shows the setup of the inverter used. A.6 Dynamometer A EMERSON dynamometer with a PMDC motor of 20H P was used in this work to control the speed (or torque) of the test machine. It is controlled by an analog com- mand sent from the Labview. The analog signal can be configured to be proportional to either torque or speed. The motor has a maximum current of 60A DC, maximum 133 Figure A. 12: Inverter voltage of 220V DC and maximum torque of 80N m. The motor operates at a rated speed with full field of 1750r/min, and optional field weakening command allows the motor to operate at a maximum speed od 22001~ /min. Figure A. 13: Dynamometer 134 BIBLIOGRAPHY 135 [ll [3] [4] [7i [8] BIBLIOGRAPHY J. 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