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VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ BRNO UNIVERSITY OF TECHNOLOGY FAKULTA ELEKTROTECHNIKY A KOMUNIKAČNÍCH TECHNOLOGIÍ ÚSTAV VÝKONOVÉ ELEKTROTECHNIKY A ELEKTRONIKY FACULTY OF ELECTRICAL ENGINEERING AND COMMUNICATION

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VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ BRNO UNIVERSITY OF TECHNOLOGY FAKULTA ELEKTROTECHNIKY A KOMUNIKAČNÍCH TECHNOLOGIÍ ÚSTAV VÝKONOVÉ ELEKTROTECHNIKY A ELEKTRONIKY FACULTY OF ELECTRICAL ENGINEERING AND COMMUNICATION DEPARTMENT OF POWER ELECTRICAL AND ELECTRONIC ENGINEERING Control of salient PM machine using d-q frame machine model and Matlab Simulink DIPLOMOVÁ PRÁCE MASTER S THESIS AUTOR PRÁCE AUTHOR Bc. Jiří Dušek BRNO 2 VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ BRNO UNIVERSITY OF TECHNOLOGY FAKULTA ELEKTROTECHNIKY A KOMUNIKAČNÍCH TECHNOLOGIÍ ÚSTAV VÝKONOVÉ ELEKTROTECHNIKY A ELEKTRONIKY FACULTY OF ELECTRICAL ENGINEERING AND COMMUNICATION DEPARTMENT OF RADIO ELECTRONICS Control of salient PM machine using d-q frame machine model and Matlab Simulink DIPLOMOVÁ PRÁCE MASTER S THESIS AUTOR PRÁCE AUTHOR Bc. Jiří Dušek VEDOUCÍ PRÁCE SUPERVISOR doc. Ing. Čestmír Ondrůšek, CSc. BRNO, 2 BRNO UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering and Communication Department of Power Electrical and Electronic Engineering Diploma thesis master's study field Power Electrical and Electronic Engineering Student: Bc. Jiří Dušek ID: 7836 Year of study: 2 Academic year: 29/ TITLE OF THESIS: Control of salient PM machine using d-q frame machine model and Matlab Simulink INSTRUCTION:. Provide the literature search. 2. Design accurate torque control for wide torque and speed range. 3. Field weekening optimization in torque and speed range. 4. Efficiency calculation. 5. Evaluate the results. REFERENCE: Assigment deadline:..29 Submission deadline: Head of thesis: Consultant: doc. Ing. Čestmír Ondrůšek, CSc. doc. Ing. Čestmír Ondrůšek, CSc. Subject Council chairman WARNING: The author of this diploma thesis claims that by creating this thesis he/she did not infringe the rights of third persons and the personal and/or property rights of third persons were not subjected to derogatory treatment. The author is fully aware of the legal consequences of an infringement of provisions as per Section and following of Act No 2/2 Coll. on copyright and rights related to copyright and on amendments to some other laws (the Copyright Act) in the wording of subsequent directives including the possible criminal consequences as resulting from provisions of Part 2, Chapter VI, Article 4 of Criminal Code 4/29 Coll. Abstrakt Tato práce se zabývá synchronním motorem s permanentními magnety na rotoru (PMSM), jeho modelováním a návrhu regulační struktury. V práci jsou uvedeny způsoby a výhody použití permanentních magnetů v elektrických motorech. Dále se práce zabývá transformací třífázové soustavy do dq. Pomocí Parkovy transformace jsou v práci odvozeny rovnice stroje v dq souřadnicovém systému a vytvořeny náhradní schémata stroje v dq osách. Rovnice i schémata zahrnují jak ztráty v mědi, tak ztráty v železe. Náhradní schémata jsou popsány elektrickými a mechanickými rovnicemi a následně překresleny do grafické podoby v programu Matlab Simulink. Vytvořeny jsou dva modely PMSM, jeden s uvažováním ztrát v železe a druhý bez těchto ztrát. Pro oba dva modely je zde popsán postup návrhu regulátorů proudu a otáček. Pro model, u kterého jsou uvažovány ztráty v železe je navíc použito více druhů řídicích strategií a tyto strategie jsou mezi sebou navzájem porovnány. Abstract This thesis deals with permanent magnet synchronous machine (PMSM) and its modeling. There are mentioned ways of the advantages and use of permanent magnets in electric machines and machines which uses permanent magnets. The transformation of three-phase system to the dq reference frame using the Park s transformation is described and used for description of PMSM. The equations in dq reference frame and equivalent circuit of PMSM in dq system are established. These equations takes in consideration copper and iron losses.. Due to derived equations and circuit, PMSM is described and model in Matlab Simulink is realized. For both of models of machines are designed currents and revolutions controllers. For model of PMSM where losses are taken in consideration is more kinds of control strategies discussed and used. These strategies are compared between each other in this thesis. Also results of simulations of different states are given. Klíčová slova Synchronní motor s permanentními magnety; matematický model; dq souřadný systém; Matlab-Simulink; Parkova transformace, návrh regulátoru, řídicí strategie. Keywords Permanent magnet synchronous machine, mathematical model, dq reference frame, Matlab-Simulink, Park s transformation, controller design, control strategies Bibliografical citation Dušek, J. Control of salient PM machine using d-q frame machine model and Matlab Simulink, Brno: FEEC BUT in Brno, 2. 6 p. Prohlášení Prohlašuji, že svou diplomovou práci na téma Control of salient PM machine using d-q frame machine model and Matlab Simulink jsem vypracoval samostatně pod vedením vedoucího diplomové práce a s použitím odborné literatury a dalších informačních zdrojů, které jsou všechny citovány v práci a uvedeny v seznamu literatury na konci práce. Jako autor uvedené diplomové práce dále prohlašuji, že v souvislosti s vytvořením této diplomové práci jsem neporušil autorská práva třetích osob, zejména jsem nezasáhl nedovoleným způsobem do cizích autorských práv osobnostních a jsem si plně vědom následků porušení ustanovení a následujících autorského zákona č. 2/2 Sb., včetně možných trestněprávních důsledků vyplývajících z ustanovení 52 trestního zákona č. 4/96 Sb. V Brně dne Podpis autora.. Acknowledgments I would really like to say thanks to my family for support that they have given me during my life and my studies. Also I want to say thanks to my girlfriend, who has stayed with me across all difficulties that appears during writing this thesis. Last but not least I want to say huge thanks to my supervisor doc. Ing. Čestmír Ondrůšek, CSc. for his great advices and for all experiences which he gave me. Thank you all very, very much. V Brně dne Podpis autora.. 7 CONTENTS INTRODUCTION PERMANENT MAGNET SYNCHRONOUS MACHINE ADVANTAGES OF PMSM CONSTRUCTION TRANSFORMATION OF THREE PHASES SYSTEM ADVANTAGES OF PARK S TRANSFORMATION PARK S TRANSFORMATION DESCRIPTION OF PMSM TRANSFORMATION STATOR VOLTAGES TO DQ ROTOR REFERENCE FRAME LOSSES AT PMSM COPPER LOSSES EDDY CURRENTS LOSSES HYSTERESIS LOSSES CORE LOSSES STRAY LOSSES EQUIVALENT CIRCUITS OF PMSM WITH LOSSES CONTROL STRATEGIES ZERO D AXIS CURRENT CONTROL STRATEGY MAXIMUM TORQUE PER AMPERE CONTROL STRATEGY (MTPA) MAXIMUM EFFICIENCY CONTROL STRATEGY (ME) CONTROLLER DESIGN FOR NO LOSSES PMSM CURRENT CONTROLLER DESIGN REVOLUTIONS CONTROLLER DESIGN CONTROLLER DESIGN FOR PMSM WITH LOSSES CONTROL MODE SWITCHING BLOCK DESIGN MAIN PURPOSE OF SWITCHING BLOCK DESIGN SWITCHING BLOCK EFFICIENCY CALCULATION SIMULATIONS START UP OF PMSM START UP OF PMSM AND ELEVATION OF BRAKING TORQUE SWITCH FROM MTPA TO FW EFFICIENCY COMPARING CONCLUSION... 36 8 REFERENCES LIST OF APPENDIXES APPENDIX A: IDEAL MODEL OF PMSM APPENDIX B: MODEL OF PMSM WITH LOSSES... 4 APPENDIX C: SWITCHING BLOCK APPENDIX D: ELECTRIC DRIVE WITH PMSM APPENDIX E: TRANSFER FUNCTIONS AND CONSTANTS FREQUENCY INVERTER REVOLUTIONS SENSOR APPENDIX F: CONSTANT LOAD RESULTS APPENDIX G: VARIABLE LOAD RESULTS... 5 APPENDIX H: SWITCH FROM MTPA TO FW CTRL. STGY APPENDIX I: CONTROL STRATEGIES EFFICIENCIES APPENDIX J: CONFIGURATION M-FILE... 59 9 LIST OF FIGURES Fig. 2.: Cross section of used PMSM... 5 Fig. 3.: Simplified cross section of synchronous machine with wound rotor [9]... 7 Fig. 4.: Equivalent circuit of PMSM at d-axis... 9 Fig. 4.2: Equivalent circuit of PMSM at q-axis... 2 Fig. 4.3: Equivavlent d axis curcuit of PMSM with losses... 2 Fig. 4.4: Equivavlent q axis curcuit of PMSM with losses... 2 Fig. 6.: PMSM model without losses Fig. 6.2: Transfer function of full system with current controller Fig. 6.3: Revolutions controller loop Fig. 6.4: PMSM with revolutions and current controllers (i d = control strategy) Fig. 8.: Ideal ratio of i d and i q for MTPA Fig. 8.2: Flow chart of switching block... 33 LIST OF TABLES Table 2.: Parameters of equivalent circuit of PMSM... 5 LIST OF SHORTCUTS AND SYMBOLS Shortcut of symbol Definition AC Alternating current [-] CM Control method [-] DC Direct current [-] dq Direct-quadrature-zero [-] EMF Electro-motive force [V] F Ci Transfer function of curent controller [-] F Cω Transfer function of revolutions controller [-] Fig. Figure [-] F S Transfer function of [-] F SL Transfer function of system with losses [-] F wi Transfer function of [-] HEV Hybrid electric vehicle [-] I a Armature current [A] I am Maximal possible I a [A] i d, i q d-, q-axis current respectively [A] i d, i q Current in shunt branch to resistance represents core loss in d-, q-axis [A] i dc, i qc Current through resistance Rc in d-, q-axis [A] i df * Wanted i d due to FW control strategy [A] i dt * Wanted i d due to MTPA control strategy [A] I nom Rated rms line phase current [A] Unit J Moment of inertia of the entire rotor assembly [kgm 2 ] k d, k q, k Transformation constants for direct-, quadrature-axis and zero sequence [-] K FI Frequency inverter constant [-] L d, L q d- and q-axis inductances respectively [mh] ME Maximum efficiency [-] mmf Magneto-motive force [A] MTPA Maximal torque per ampere [-] p Laplace operator [-] P Output power [W] P Cu Copper losses [W] P E Electrical losses [W] P Fe Iron losses [W] P L Total losses [W] PM Permanent magnet [-] P M Mechanical losses [W] PMSM Permanent magnet synchronous machine [-] 2 Shortcut of symbol Definition p p Number of pole pairs [-] R a Phase resistance [Ω] R c Resistance represents core loss [Ω] R eddy Resistance represents eddy current loss [Ω] R h Resistance represents hysteresis loss [Ω] R hb Hysteresis resistance at base speed [Ω] Unit T D Dynamic torque [Nm] T e Electromagnetic torque [Nm] T Z Load torque [Nm] U Constraint voltage [V] U m Maximal constraint voltage [V] U a Terminal voltage [V] u a, u b, u c Stator voltage at phase a, b, c respectively [V] U am Maximal terminal voltage [V] u d, u q Votalge at d- and q-axis respectively [V] u d, u q Voltage at shunt branch to resistance represents core loss in d-, q-axis [V] u dc, u qc Voltage at resistance Rc in d-, q-axis [V] U nom Rated rms line phase voltage [V] η Efficiency [-] θ Initial angle between rotor direct axis and stator phase-a [rad] θ e Angle between rotor direct axis and stator phase-a [rad] ν nom Nominal winding temperature [ C] ρ Saliency coefficient [-] τ FI Time constant of frequency inverter [s] τ q, τ d Time constant of PMSM in d or q axis respectively [s] τ SAMP Time constant of frequency inverter sampler [s] τ SAMP2 Time constant of revolutions sampler [s] τ σ, τ Σ Sumary time constans [s] Ψ a Flux linkage due stator phase-a [Wb] ψ a Maximum flux linkage due to permanent magnet per phase [Wb] ψ d, ψ q, ψ Flux linkage due d-,q-axis and zero sequence respectively [Wb] ψ PM Flux linkage due permanent magnets [Wb] ω b Rotor electrical angular base speed [rad.s - ] ω e Rotor electrical angular speed [rad.s - ] ω m Rotor angular speed [rad.s - ] 3 INTRODUCTION At last few years the Permanent Magnets Synchronous Machines (PMSM) became more attractive. There are many reasons for this fact. One of the biggest reasons for that is that price of rare-earth magnets is dropping. Also PMSMs efficiency is very high in compare with electric rotary machines without PM. [] Due to advantages, that PMSM has, it fit well for hybrid electric vehicles (HEV). [2] With massive developing in this area, requirements for drive control grow up. There are many ways to control drives with PMSM. Due this, there are some control strategies for different operation range. For example Zero D Axis Current controls method (CM), Maximum Torque Per Ampere CM, Maximum Efficiency CM, etc. These and other control methods will be discussed on in this thesis. [3] If some system has to be controlled, there must be description of this system. At this case, is necessary to describe PMSM mathematically. For this description is suitable to transform system from three phases abc reference frame to dq rotor reference frame. Reasons and advantages can be found at this thesis. After transformation to dq system, equivalent circuit for each axis must be established. From these circuits can be mathematical equations derived and due this equations can be PMSM described. Most of equations are electrical, but there must be also used mechanical equations [4]- [7]. Due this, there are enough equations to solve the equation system. These are basics of model creation in Matlab-Simulink of PMSM. Usually equivalent circuit in dq reference frame contains only resistance of stator winding, inductance in d and q axis respectively, cross couplings and back EMF from permanent magnets. For more accurately outcome, losses are in this thesis taken in consideration, due this are equivalent circuits little bit different and complicated against usual equivalent circuits of PMSMs [7]. In this thesis core losses are taken in consideration and are described. There is a lot of ways, how to design controller for PMSM depending on area of operation. For every purpose, that PMSM will be used for is better to use different control strategy. In this thesis are used Optimal Module (OM) and Symmetric Optimum (SO) methods for design of angular speed and current controller. After design of controllers, there is designed system extension which takes care about choosing most suitable control method for actual operation state. 4 2 PERMANENT MAGNET SYNCHRONOUS MACHINE 2. Advantages of PMSM There are a lot of advantages of use permanent magnets instead of electric excitation system: No electrical energy is absorbed by the field excitation systems and thus there are no excitation losses, what means substantial increase in efficiency. Higher torque and output power per volume then when using electromagnetic excitation. Better dynamic performance than motor with electromagnetic excitation (higher magnetic flux density in the air gap). Simplification of construction and maintence. Reduction of prices. Permanent magnets are improving the motor s steady-state performance and power density, dynamic performance and quality. 2.2 Construction In dependence on how the rotor magnets are mounted onto (or inside) the rotor: Surface-mounted magnets Inset-mounted magnets Interior-mounted magnets Rotor with buried magnets symmetrically distributed Rotor with buried magnets asymmetrically distributed Bread loaf magnets Decentered magnets Interior double-layer magnets In dependence on which magnets are used: Alnico magnets (Al, Ni, Co, Fe) Ferrite magnets Rare-earth materials (SmCo, NdFeB) Etc [] [6] In this thesis PMSM with embedded magnets placed in V position as is shown on Fig. 2. is used. At this rotor form, two permanent magnets magnetize the same pole. The pole shoes are shaped to produce sine-wave air-gap flux density. Advantages of rotor with magnets placed into V position are that machine produces more torque per rotor volume due to high air-gap flux density, compared to the rotor with surface mounted magnets. And there are other advantages like there isn t danger of magnets coming off. There is no problem with fixing the magnets to the DEPARTMENTT OF POWER ELECTRICAL AND ELECTRONIC ENGENEERING 5 rotor as is at surface rotor machine. The biggest advantage of this rotor is that air-gap flux density can be easily made sinusoidal which makes it possibilities to achieve a very low cogging torque. [8] Fig. 2.: Cross section of used PMSM In table 2. are parameters of equivalent circuit of used PMSM: U nom [V] 2 I nom [A] 23,7 p p [-] 4 R a [Ω],28 Ψ PM [Wb],883 L d [mh],3286 L q [mh],689 R hb [Ω] 95,73 R eddy [Ω] 82,2 J[kgm 2 ],47 ν nom [ C] 6 Table 2.: Parameters of equivalent circuit of PMSMM 6 3 TRANSFORMATION OF THREE PHASES SYSTEM 3. Advantages of Park s transformation Main reason, why to transform three phases a,b,c system to dq is to simplify description of system. Instead of equation for each phase, there will be only, two dimensional system with direct- and quadrature-axis. Main advantage of using dq system is easy analysis of the interaction of rotor and stator flux and mmf waves, independent of whether or not saliency effects are present. By transforming a,b,c quantities into equivalent quantities which rotate in synchronism with the rotor steady state conditions, these interaction become those of constant mmf- and flux-waves separated by a constant spatial angle. This indeed is a point of view, which correspond to that of an observer in the rotor reference frame. Park transformation is used to transform three phase time variable quantities (in this case for example u a, u b, u c )to direct-quadrature-axis and zero sequence component (in this case u d, u q and u ). The zero sequence component is important if three phase system is not balanced. If it is, the zero sequence components are neutral. In this thesis is only balanced three phase system take in consideration, so sequence components are not discussed in any detail. [9] 3.2 Park s transformation For transformation of any stator quantity (voltage, current, flux linkage, etc.) the transformation matrix is established. ( θ ) ( θ ) ( θ + ) ( θ ) ( θ ) ( θ ) X d kd cos e kd cos e 2 kd cos e 2 X a X q kq sin e kq sin e 2 kq sin e 2 X = + b X k k k X c And the inverse transformation as (3.) 2 2 cos( θe ) sin ( θe ) 3 kd 3 kq 3k X a 2 2 X d X b cos ( θe 2 ) sin ( θe 2 ) X = q 3 kd 3 kq 3k X c X 2 2 cos( θe + 2 ) sin ( θe + 2 ) 3 kd 3 kq 3k (3.2) At equation (3.2) X represent any stator quantity to be transformed. Constants k d, k q and k are transformation constants and they can be chosen absolutely different from each other, but there is no reason to choose them different. There are advantages to choose k d =k q =2/3 and k =/3 which will be show in next section. These values was used by Park because of simplification of equations (3.2). Fig. 3. shows simplified sketch of synchronous machine with wound rotor with one pole pair for simpler understand Park s transformation. 7 Fig. 3.: Simplified cross section of synchronous machine with wound rotor [9] 8 4 DESCRIPTION OF PMSM 4. Transformation stator voltages to dq rotor reference frame Equations for stator voltages of PMSM are: where u k dψ dt k = Rkik + (4.) k = a, b, c (4.2) The transformation of stator voltages to d-q voltages is proceeding from equation for flux linkage due to phase A 2 2 ψ a = ψ d cos( θe ) ψ q sin ( θe ) + ψ (4.3) 3 k 3 k 3 k d Now let s do the differentiation of equation (4.3) with respect to time when: Equation (4.3) will go to: q θ = ω + θ, θ = (4.4) e dψ a 2 ψ 2 2 ψ d q 2 = cos( θ ) ψ ω sin ( θ ) sin ( θ ) ψ ω cos( θ ) + dt 3 k dt 3 k 3 k 3 k 3k d e t e d e e e q e e (4.5) d q dt q dt From equation (4.) is evidently that: dψ ψ a = u a Rai (4.6) a dt So when substitute u a and i a from definition of Park s transformation into (4.6) it will be: dψ a = ( θ ) u ( θ ) + u R i ( θ ) i ( θ ) + dt 3 k 3 k 3k (4.7) 3 k 3 k 3 u d cos e q sin a d cos q sin i d q d q k Equation (4.5) is equal to (4.7) if coefficients with the same trigonometric function are the same. So if these coefficients are compared under condition that k d =k q =2/3 and k =/3 it s possible to find equations for voltages at d and q axis. dψ d ψ qωe = ud Raid dt dψ u = R i + ω ψ dt d d a d e q (4.8) Where DEPARTMENT OF POWER ELECTRICAL AND ELECTRONIC ENGENEERING ψ = L i + ψ d d d PM ψ = L i q q q 9 (4.9) Institution (4.9) into (4.8) leads to did ud = Raid + Ld ωelqiq (4.) dt Institution (4.9) into (4.) leads to dψ ω ψ = R i u dt dψ q uq = Raiq + + ωeψ d dt q e d a q q (4.) diq uq =

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