The Pennsylvania State University The Graduate School DIRECT FIELD-ORIENTED CONTROL OF AN INDUCTION MACHINE USING AN ADAPTIVE ROTOR RESISTANCE

Transcription

1 The Pennsylvania State Univesity The Gaduate School DIRECT FIELD-ORIENTED CONTROL OF AN INDUCTION MACHINE USING AN ADAPTIVE ROTOR RESISTANCE ESTIMATOR A Thesis in Electical Engineeing by David M. Reed c 009 David M. Reed Submitted in Patial Fulfillment of the Requiements fo the Degee of Maste of Science August 009

2 The thesis of David M. Reed was eviewed and appoved by the following: Heath F. Hofmann Associate Pofesso of Electical Engineeing Thesis Adviso Jeffey L. Schiano Associate Pofesso of Electical Engineeing W. Kenneth Jenkins Pofesso of Electical Engineeing Head of the Depatment of Electical Engineeing Signatues ae on file in the Gaduate School.

3 Abstact When field oiented-contol techniques ae applied to induction machines, a DC machine-like toque esponse may be achieved, making the induction machine a viable candidate fo applications peviously dominated by othe electic machines. Some of these applications, howeve, demand high pefomance ove a ange of opeating conditions. This becomes an issue as the field-oiented techniques needed to achieve the desied pefomance also have a well-known sensitivity to vaiations in machine paametes. While methods exist to account fo the vaiation in magnetic paametes due to satuation, compensation fo vaiations in oto esistance, which vaies with tempeatue and fequency, emains an open eseach topic. This thesis pesents a new adaptive oto esistance estimato, fo which stability and convegence ae igoously poven, along with oto flux linkage estimatos deived using the tapezoidal ule fo impoved accuacy and stability. These estimatos ae then incopoated into a diect field-oiented contolle and thooughly tested on expeimental hadwae. The esulting contolle achieves an acceptable level of pefomance ove a ange of opeating conditions. iii

4 Table of Contents List of Figues List of Tables List of Commonly Used Symbols Acknowledgments vi viii ix xi Chapte Field-Oiented Contol of Induction Machines. Basic Dive System and Backgound Field-Oiented Contol Basic Pinciples of Field-Oiented Contol Shotcomings of Field Oiented Contol Roto Resistance Compensation In Liteatue Thesis Contibutions and Oganization Chapte Induction Machine Model 5. Smooth Aigap Dynamic d q Model Obsevability Analysis State Space Model Obsevability Test Simulink R Model Chapte 3 Contolle Design 5 3. Diect Field-Oiented Contolle iv

5 3. Roto Flux Estimato Design Deivation using Tapezoidal Integation Voltage-Based Estimato Cuent-Based Estimato Dead-Time Effect and Compensation Adaptive Roto Resistance Estimato Deivation and Lyapunov Stability Poof Steady State Analysis Simulink Simulation Chapte 4 Implementation and Expeimental Veification Induction Machine Paamete Estimation Paamete Estimation Technique Expeimental Setup and Results Dead Time Measuement Contolle Expeimental Veification Implementation Expeimental Veification Chapte 5 Discussion and Futue Wok Contolle Pefomance and Limitations Dependence on Opeating Point Flux Estimato Pefomance Suggestions fo Futue Wok Bibliogaphy 76 v

6 List of Figues. Dive system achetectue Roto tempeatue incease with espect to time fo a.5 kw at ated opeating conditions [4] Pak tansfom vecto diagam Simulated toque esponses fo a N-m command with R mismatch using diect field-oiented contol with flux estimation fo toque egulation Smooth ai-gap -phase (d q) induction machine coss-section Induction machine magnetics model Simulink block diagam fo the cuent/flux elationships Simulink block diagam fo the electical dynamics Simulink block diagam fo the mechanical dynamics Tansfomation of electical vaiables into the oto flux linkage efeence fame Diect Field-Oiented Contolle Fequency esponse compaison of integato appoximations Pole locus compaison of Fowad Eule and Tapezoidal methods Invete IGBT totem pai Invete wavefoms with dead-time effect Illustation of the fist hamonic of the dead-time voltage Simulation esults Simulink block diagam of esistance estimato and efeence fame calculato Simulink block diagam of voltage-based oto flux linkage estimato Simulink block diagam of cuent-based oto flux linkage estimato Simulink block diagam of the diect field-oiented contolle Simulink block diagams of the Modified (left) and Invese (ight) Clak tansfoms vi

7 3.4 Simulink block diagams of the Pak (left) and Invese Pak (ight) tansfoms Simulink block diagam of a PI egulato Stato cuent locus as a function of slip fequency in the stato flux linkage efeence fame Expeimental setup Simulink block diagam fo paamete estimation expeiment Compaison of expeimental data and fitted stato cuent locus (left) with nomalized eo (ight) Dead time measuement expeimental setup Dead time measuement fo a 0 pecent duty cycle IGBT Bidge Configuation dspace DS04 block diagam [30] Simulink block diagam of contolle implementation fo dspace code geneation Simulink block diagam of duty cycle geneato Simulink block diagam of second ode integato appoximation fo the voltage-based estimato Simulink block diagam of coefficient calculation fo the second ode integato appoximation Simulink block diagam of cuent-based estimato Simulink block diagams of the cuent-based estimato coefficients a[k] (left) and b[k] (ight) Simulink block diagams of the cuent-based estimato coefficients c[k] (left) and d[k] (ight) Simulink block diagam of the cuent-based estimato coefficient e[k] Adaptive estimato expeimental tansient esponses Expeimental esponse to a toque step fom N-m to 3 N-m Compaison of toque steps fo mismatched and tuned oto esistances Steady-state oto esistance estimation at diffeent oto velocities as a function of slip fequency Open-loop chaacteization of ageement between the voltage and cuent-based oto flux linkage estimatos Induction machine model incopoating magnetic satuation effects Block diagam of the poposed method fo tuning the esistance estimato ON and OFF vii

8 List of Tables 3. Paametes used fo pole locus compaison Simulation paametes Manufactue machine atings Estimated test machine paametes viii

9 List of Commonly Used Symbols λs = [λ sd, λ sq ] T λ = [λ d, λ q ] T i s = [i sd, i sq ] T i = [i d, i q ] T v s = [v sd, v sq ] T v = [v d, v q ] T R s R L s L M σ = L s L M τ e τ 3ph τ l P ω e ω ω e = P ω Stato Flux-Linkage Vecto. Roto Flux-Linkage Vecto. Stato Cuent Vecto. Roto Cuent Vecto. Stato Voltage Vecto. Roto Voltage Vecto. Stato Resistance. Roto Resistance. Stato Self-Inductance. Roto Self-Inductance. Mutual Inductance. Leakage Tem. Electomagnetic Toque. Thee-Phase Electomagnetic Toque. Load Toque. Numbe of Poles. Electical Fequency. Roto Angula Velocity. Electical Roto Angula Velocity. ix

10 ω se = ω e ω e H B [ ] 0 J = 0 [ ] 0 I = 0 [ ] = 0 0 Electical Slip Fequency. Moment of Inetia. Mechanical Damping. 90 Rotation Matix. The Identity Matix. The Zeo Matix. x

11 Acknowledgments I would like to thank D. Heath Hofmann fo his guidance thoughout this poject. I have leaned a geat deal fom him, and with him, in the past yea. I would also like to thank D. Jeff Schiano fo seving on my committee, and teaching me a geat deal about contol theoy and design. Additionally, I wish to thank D. Jack Mitchell, Pof. Mak Whaton, D. Jeff Maye, D. Javie Gomez-Caldeon, and all the teaches who have inspied me thoughout my education. I have been fotunate to have had so many excellent teaches. Finally, I would like to thank my family and fiends fo all of thei love and suppot. Because of you, I can safely say that I have come out of gad school with my sanity intact. xi

12 Chapte Field-Oiented Contol of Induction Machines Field-oiented contol (also efeed to as vecto contol) is often used in highpefomance dive applications. The technique, conceived in the ealy 970s, achieves DC machine-like pefomance fom induction machines; and while computationally intensive, the availability of cheap, poweful micopocessos has made field-oiented contol a viable choice fo moden high pefomance dive systems.. Basic Dive System and Backgound The tem electic dive geneally efes to the powe electonics, contolle and electical sensos equied to opeate an electic machine in specific applications whee contol ove toque, speed and opeating points is desied. While electic dives can be used with a vaiety of electic machines (AC and DC), the focus of this thesis will be the contol of induction machines using an electic dive. The basic moto dive system achitectue, shown in Figue., gives us contol ove the fequency of the AC voltage used to dive the moto.

13 (dspace) v a v b v c Figue.. Dive system achetectue. It should be noted that, while the focus of this thesis is the contol of 3- phase induction machines, the contol techniques used ae based on a two-phase equivalent model which will be discussed in the following chapte. The use of twophase models is common pactice as it leads to educed ode models which ae easie to wok with, and may be used with highe-than-thee-phase machines as well. Fo thee-phase machines, the electical vaiables (a,b,c) can be tansfomed into thei equivalent two-phase values (d,q,0) using the Clak tansfom (.). x d /3 /3 /3 x q = 0 3/3 3/3 /3 /3 /3 x 0 x a x b x c = T 3 x a x b x c (.) Likewise, the invese Clak tansfom is given by x a 0 x b = / 3/ / 3/ x c x d x q x o = T 3 x d x q x o = T 3 x d x q x o (.)

14 3 A simplification can be made by taking the zeo sequence component (x o ) to be equal to zeo. This common assumption, which will be used thoughout this thesis, leads to the following expession known as the Modified Clak tansfom (.3). x d = x q x a (.3) x b Electic dives can be used in a vaiety of applications anging fom motion contol (sevo dives) to vehicle populsion (taction dives). The inceasing affodability of the necessay electonics continues to make electic dives a cost-effective altenative fo applications once dominated by mechanical systems. Hydaulic lines and mechanical linkages/cables can be eplaced by electic wies and motos []. Fo example, many new cas now employ electic thottle and tansmission linkages. While thee ae pactical concens with such systems (loss of electic powe means loss of electical systems) they also have advantages ove thei mechanical altenatives. The dynamics of electical systems ae typically much faste than those of mechanical systems, and electical powe is often easie to tanspot than fuel o hydaulic fluid. Also, when designed popely, electical systems can pove to be moe eliable than mechanical systems. Induction machines in paticula ae enowned fo thei low maintenance and duability. Recently, the ising cost of enegy and envionmental concens has spaked inteest in hybid vehicles such as the Toyota Pius, and electic vehicles such as the Tesla Roadste. The use of electic dives fo vehicle populsion pesents an inteesting challenge, paticulaly in the case of an all-electic vehicle such as the Tesla Roadste. Such applications equie pefomance ove a vey wide speed ange and unde a vaiety

15 4 of opeating conditions (loads, tempeatues, etc.). Achieving an acceptable level of pefomance unde these cicumstances is no easy task. This is especially tue when using an induction machine fo populsion. Machine paametes can vay with electical fequency, flux levels and tempeatue. These vaiations tend to detune the electic dive s contol system and degade its pefomance. Electical fequency affects efficiency, esistance, and contolle pefomance. The losses due to hysteesis and eddy cuents incease with fequency, and depending on the chaacteistics of the conductos in a machine, the skin effect can significantly incease the effective oto and stato esistances. Contolle pefomance is affected by fequency as a esult of necessay appoximations made when implementing the contolle on the dspace micopocesso. One concen in paticula is the ability to accuately estimate flux linkages ove wide speed anges. Howeve, this issue has been addessed by Jansen, Thompson and Loenz in [7]. The magnetic paametes, while modeled as linea in ou wok, ae actually nonlinea in natue. This nonlineaity is due to the satuation of magnetic mateials in the machine unde high magnetic field stengths. The esult is that the inductances, which ae modeled as constant unde the assumption of lineaity, ae actually a nonlinea function of magnetic field stength. Thee ae, howeve, methods of accounting fo this nonlineaity. One appoach is to make the mutual inductance between the stato and oto, a nonlinea function of the ai-gap flux linkage magnitude []. Natually, tempeatue plays a majo ole in the vaiation of paametes in an induction machine. The most significantly affected paametes ae the stato and oto esistances. In paticula, vaiations in the oto esistance have a moe damatic effect on contolle pefomance. Fo this eason, many attempts at accounting fo this vaiation have been made [9]-[8]. This vaiation in oto

16 5 esistance is the pimay motivation fo the wok pesented in this thesis. The esistance change with tempeatue, in this case fo the oto, can be descibed by the following equation [4]: R = R,cold ( + α ϑ p ) (.4) whee α is the tempeatue coefficient (typically fo aluminum, α Al = 0.004) and C ϑ p = ϑ oto ϑ ambient is the oto tempeatue incease. In [4] the authos used a fist ode equivalent RC cicuit to model the oto tempeatue incease: C t d ϑ t dt = p loss ϑ t R t (.5) whee C t and R t ae the themal capacitance and esistance espectively, and the subscipt t efes to the themal model. Least-squaes was used to estimate the themal capacitance and esistance fom expeimental data fo a.5 kw induction machine unning at the manufactues ated conditions. The esulting themal time constant, T t = R t C t, was found to be 490 seconds. Thei plot, evaluating the accuacy of the themal model (.5) has been povided in Figue.. Inspection of these esults yields two impotant obsevations as it concens this thesis. Fist, using the data in Figue. and (.4), we find that the oto esistance would have inceased appoximately 5%. Secondly, inspection of Figue. and the themal time constant, T t = 490s, eveals that the oto esistance will change vey slowly with espect to time. This will allow us to make an impotant assumption when deiving ou adaptive oto esistance estimato in a late chapte.

17 DATA OF THE INVESTIGATED INDUCTION MACHINES 6 istance fo vaious ion loss components as a actual ion losses ae epesented by the best al cuve (p = 0:054 fo the investigated machine). he efeence tempeatue ise. e total losses (0) ae equied in ode paametes in (4). A two-dimensional s and enables the minimization eo of the estimated tempeatue dif- Fig. 7. Estimated oto tempeatue with espect to the paamete model Figue.. Roto (# ) and tempeatue with espect incease to the themal withmodel espect (# ) tofotime the investigated fo a.5 kw at ated opeating (3) conditions.5-kw induction [4]. machine.. Field-Oiented Contol tained measuing points in the time do- amete model fom the tempeatue estimated by the themal model is less than 3 C fo all thee motos. The units of the themal esisto and the themal capacito ae As mentioned (33) mixed ealie, p.u. the andinteest SI units. in Usually, field-oiented the physical contol unit ofstems the themal fom the ability to esisto is. In this case the moto losses ae deived as f oto tempeatue with espect obtaintoa the nealy p.u. instantaneous quantities. Theefoe, toquethe esponse unit of the (i.e. themal DC machine-like esisto this pefomance) d in Figs. 7 9 fo the investigated moal paametes and as in well induction as the machines. can be explained This pefomance with the help of is the achieved steady-state by exploiting tempeatue the natual de- pape is only. The physical meaning of the themal esisto of (33) coupling which esults when the chaacteistic equations fo an induction machine (34) (35) ae witten in the oto-flux efeence fame. hines ae summaized in Table II. The If the p.u. total losses ae, e.g., 5% of the efeence powe (5), the the tempeatue estimated by the pa- steady-state oto tempeatue is 5% of. The physical unit of a.. Basic Pinciples of Field-Oiented Contol se limited to: Penn State Univesity. Downloaded on June 4, 009 at 5:49 fom IEEE Xploe. Restictions apply. A fundamental featue of field-oiented contol is the tansfomation of electical vaiables into a otating synchonous efeence fame [Figue.3]. This tansfomation is often efeed to as the Pak tansfom (.6), named fo R.H. Pak who published the fist papes in 99, detailing the application of efeence fame theoy to the analysis of AC machines [3].

18 7 syn q syn q syn x d syn syn d Figue.3. Pak tansfom vecto diagam. x syn = xsyn d x syn q = cos (θ syn) sin (θ syn ) x d = e Jθsyn x (.6) sin (θ syn ) cos (θ syn ) Fo completeness, the invese Pak tansfom is given by x = x d = cos (θ syn) sin (θ syn ) sin (θ syn ) cos (θ syn ) x q xsyn d x syn q x q = e Jθsyn x syn (.7) It has been shown [] that the toque expessions fo vaious AC machines simplify when expessed in cetain synchonous efeence fames. Fo this eason Pak tansfoms ae used in field-oiented contol, despite the added complexity. Once tansfomed into a synchonous efeence fame, standad contol techniques can be applied. The output is then tansfomed back into a stationay efeence fame using the invese Pak tansfom (.7) []. A common analogy used to descibe the opeation of field-oiented AC dives,

19 8 is that of a sepaately excited DC moto dive []. The geneal toque expession fo a DC machine is given by τ e = P λ af (i f ) i a, (.8) whee P is the numbe of poles, λ af (i f ) is the amatue flux linkage expession which is a function of the field winding cuent, i f, in a sepaately excited DC machine, and i a is the amatue cuent. Inspection of (.8) eveals that i f can be used to egulate the amatue flux linkage, and i a can be used to egulate toque. This decoupling is inheent in DC machines because the fields poduced by i a and i f ae othogonal by design. Howeve, it will late be shown that this decoupling and a simila toque expession can be achieved fo induction machines by tansfoming the electical vaiables into the oto flux linkage efeence fame. In geneal, thee ae two types of field-oiented contolles fo induction machines which ae based on the oto flux linkage efeence fame, indiect (o feedfowad) and diect (o feedback). The latte method will seve as the basis fo the contolle poposed in this thesis... Shotcomings of Field Oiented Contol Despite the high pefomance which can be achieved using field-oiented contol, thee ae shotcomings as well. Thee ae two pimay methods of estimating flux linkages fo use in contol algoithms, which will be discussed in detail late. If a flux linkage estimato is used in the contol algoithm, pefomance at eithe low fequencies o high fequencies will suffe depending on the type of estimato used. These estimatos can also suffe fom paamete vaiations, and if speed-sensoless contol is being pefomed, the loss of obsevability at DC excitation [6] will likely

20 9 need to be addessed [8]. Fo simplicity in ou wok, we allowed ouselves the use of a position encode, fom which oto speed was deived. While diect field-oiented contolles can be implemented by diectly measuing the magnitude and position of the aig-gap flux in the machine using eithe pickup coil o hall-effect sensos, this method is not favoed as it equies modifications to the machine to place the sensos in the ai-gap, and will still be pone to eo. Theefoe, diect field-oiented contolles ae often implemented using estimatos to detemine the oto flux linkage. Both the indiect and diect methods tend to lack obustness, having simila sensitivities to vaiations in machine paametes [5]. In fact, when a flux estimato is used with the diect method, the paamete vaiation of concen fo both methods is that of the oto esistance, which is a function of tempeatue and fequency. This is because a change in the oto esistance, which may exceed 50% [], causes a change in what is efeed to as the oto time constant, T = L /R. The accuacy of this paamete is citical fo the slip fequency computation pefomed when using the indiect method. Likewise, it is of impotance in the flux estimatos commonly used to implement the diect method. A mismatch between the actual oto esistance and the value used in a fieldoiented contolle can lead to significant steady-state eos in toque egulation if the mismatch is sevee enough. To demonstate the potential seveity of such a mismatch, Simulink simulations wee pefomed fo a N-m toque command using a diect field-oiented contolle with ±50% vaiations in the value of R [Figue.4]. Inspection of the toque esponses in Figue.4 eveals that while the steady-state eo due to oto esistance mismatches can be athe lage, thee ae also oscillations in the esponse that indicate that the stability of the system could be degaded as well. Fo these easons, a geat deal of eseach has been

21 0.6.4 R = + 50% τ 3ph [N-m] R = 0% R = - 50% Time [Seconds] Figue.4. Simulated toque esponses fo a N-m command with R mismatch using diect field-oiented contol with flux estimation fo toque egulation. done to find a pactical method fo updating the value of the oto esistance in field-oiented contol algoithms...3 Roto Resistance Compensation In Liteatue Addessing oto esistance vaiation in induction machines has been a popula eseach topic fo the past two decades. Despite the vaiety of appoaches [9]- [8], no one method emeged as a widely accepted technique. The Model Refeence Adaptive System (MRAS) schemes poposed in [, ] ae based on the indiect field-oiented contol technique. Howeve, the scheme poposed in [] does not include a stability poof, and the contolle in [] has yet to be tested on an expeimental setup. Anothe MRAS scheme poposed in [3] is based on the diect field-oiented contol technique. Howeve, thei poposed oto time constant

22 update (.9) will esult in a singulaity as the slip fequency goes to zeo, ω sl 0. T = ˆω ω ω sl ˆT (.9) Othe adaptive techniques have been poposed as well [4]-[8]. Howeve, the contolle poposed in [4] uses adaptation laws deived fom a dynamic model of the induction machine which is based on the stato cuent efeence fame. Since the electical vaiables used in the adaptive estimato ae constant in such a efeence fame, the system lacks pesistency of excitation [0] unde steadystate opeating conditions, and so it must ely on tansients to tune the esistance estimation. A non-linea adaptive sliding mode contolle was deived in [5]; howeve, it has not been tested on hadwae. The othe adaptive estimation papes [6]-[8] pesent the evolution of a speed-sensoless diect field-oiented contolle with adaptive oto time constant estimation. Thei fist attempt [6] estimated the oto esistance, ˆR, as being popotional to thei stato esistance adaptation: d dt ˆR d = R sn dt ˆR s = λ (e ids î ds + e iqs î qs ) (.0) whee e ids = i ds î ds, e iqs = i qs î qs, λ > 0 is a contol gain, and R sn is the atio of the nominal values of the stato and oto esistance. This method is flawed howeve, as the stato and oto esistances will vay quite diffeently fo a numbe of easons. Fist, the stato windings consist of coppe wie while the oto cage consists of aluminum bas. Secondly, the oto and stato ae sepaated by an ai-gap which makes it unlikely that the stato and oto tempeatues will be the same. Theefoe, the elation between the stato esistance and oto esistance cannot be descibed by a simple constant. The next evolution of the contolle [7] poposes a new oto time constant

23 estimato given by: d { dt (/ˆτ ) = λ /L e ids ( ˆφ d Mî ds ) + e iqs ( ˆφ } q Mî qs ) (.) whee e ids = i ds î ds, e iqs = i qs î qs, and λ is an abitay positive gain. The authos diect eades to a pio confeence pape [9] in which the estimato was poposed and stability veified using the Lyapunov stability citeia. Thei poof howeve, only yields a negative semi-definite esult, as stated by the authos. This esult only poves that the estimato is stable in the sense of Lyapunov, it does not guaantee that the oto time constant estimation will convege to the tue value (i.e. e τ 0 as t ). Late evolutions of this estimato [8] ae all deived fom the oiginal adaptation law (.) by ewiting it in the electical efeence fame and in tems of oto esistance. Theefoe, the convegence of these late incanations is also unpoven. Othe techniques fo updating the oto time constant have been poposed as well [9, 0]. The method poposed in [9], uses a modified switching scheme to extend the zeo cossing of a paticula phase cuent. This foces the cuent in one of the phases to be zeo, simplifying the dynamic equations fo the induction machine and allowing a diect calculation of the oto time constant using known and measuable paametes. Howeve, the accuate convegence of this estimation technique is unpoven and would likely be difficult to pove. Additionally, the modified switching technique used to implement the estimato is fo Cuent Regulated Pulse-Width-Modulated (CRPWM) convetes and it is unclea how it could affect the pefomance of the field-oiented contolle. Lastly, the authos only povide expeimental esults fo the implementation of the switching scheme and not the oto time constant estimation.

24 3.3 Thesis Contibutions and Oganization Sensitivity to paamete vaiations is the leading dawback to the use of fieldoiented contol techniques with induction machines. Despite the lage eseach effot [9]-[8], the poblem of compensating fo oto esistance vaiations emains an open topic. While pevious attempts have had some success, vey few sufficiently exploed the pactical limitations of thei estimatos. Slip fequency, integato appoximations and unmodeled effects such as dead-time and satuation can significantly affect the pefomance of oto esistance estimatos. This thesis addesses the oto esistance vaiation using a new adaptive oto esistance estimato fo which convegence is igoously poven using Lyapunov s stability citeion, Babalat s lemma, and two-time-scale theoy []. Roto flux linkage estimatos ae deived using tapezoidal integation to povide accuacy and stability ove a sufficiently wide speed ange. The esulting oto flux and esistance estimatos ae incopoated into a diect field-oiented contolle and tested ove a ange of oto speed and slip fequency. To summaize, the contibutions of this thesis ae the following: Roto Resistance Estimato: A new oto esistance estimato is pesented along with a thoough analysis of its stability and convegence. Roto Flux Linkage Estimatos: Deivations of the standad voltagebased and cuent-based oto flux linkage estimatos using tapezoidal integation ae pesented. Thoough Evaluation of Estimato Limitations: The poposed oto esistance estimato and flux linkage estimatos ae thooughly tested on expeimental hadwae to eveal limitations.

25 4 The oganization of this thesis is as follows. The standad smooth aigap twophase induction machine model is pesented in Chapte in ode to familiaize the eade with the model and assumptions which will seve as the basis of ou contol design. The following sections of Chapte pesent an obsevability analysis of the electical vaiables necessay to ou contol design, followed by the implementation of the induction machine model in Simulink. The theoetical details of ou contol design ae pesented in Chapte 3. We begin by intoducing the basic diect field-oiented contol theoy followed by the deivation of ou oto flux linkage estimatos using tapezoidal integation. A subsection discussing the dead-time effect and a method fo its compensation is included as well. The next section details the deivation and stability poof of ou oto esistance estimato, and includes a steady-state analysis as well. The chapte concludes with a Simulink implementation and simulation esults fo an idealized implementation of the esulting contolle. Chapte 4 begins with a desciption of ou induction machine paamete estimation technique along with the expeimental setup and esulting machine paametes. We then discuss ou expeimental measuement of the dead time, and conclude with the expeimental veification of ou diect field-oiented contolle and adaptive oto esistance estimato. Finally, Chapte 5 discusses the pefomance of ou contolle ove a ange of oto speed and slip fequencies. The limitations which wee pedicted and discoveed ae summaized as well. The chapte concludes with ecommendations fo futue wok and suggestions fo a pactical implementation of the esulting contolle.

26 Chapte Induction Machine Model Thee ae a vaiety of ways to model the opeation of an induction machine. Natually, the design of high pefomance dive systems equies the use of a dynamical model fo the induction machine. The model which will be used fo the emainde of this document is efeed to as the Smooth Aigap Dynamic d q Model. This model will be pesented along with an obsevability analysis and Simulink implementation of the model.. Smooth Aigap Dynamic d q Model To simplify analysis, we have modeled the induction machine as having a smooth aigap, and so slot hamonics can be neglected. Additionally, the following assumptions have been made: Linea magnetics model (Neglect satuation and hysteesis). Lossless coe (Neglect hysteesis and eddy cuent losses). Balanced constuction with sinusoidally distibuted magneto motive foce (mmf).

27 6 : effective tuns atio. These assumptions ae typical and appopiate unde nomal opeating conditions. A coss-sectional depiction of the smooth ai-gap d q model is povided in Figue.. Figue.. Smooth ai-gap -phase (d q) induction machine coss-section. As mentioned ealie, the dynamic d q (two-phase) model of the induction machine seves as the basis fo the field-oiented contol techniques which will be used in the poposed contolle. This model descibes the dynamical behavio of a 3-phase induction machine as an equivalent mmf -phase machine. The esulting -phase (d q) paametes (voltage, cuent, etc.) can then be tansfomed into equivalent 3-phase values using the invese Clak tansfom (.). Note that unde balanced opeation the zeo sequence paamete x o is equal to zeo, and so can

28 7 be neglected. The ode eduction achieved by using the d q model can now be appiciated as the chaateistic equations fo a two-phase equivalent induction machine model [Figue.] ae pesented. s s ls l s s e Figue.. Induction machine magnetics model. The stato and oto flux linkages (designated by λ) in an abitay efeence fame (designated by the supescipt x) ae given by the following elationships: λ x s = L s i x s + M i x, (.) λ x = M i x s + L i x, (.) whee L s = L ls + M and L = L l + M, ae the stato and oto self-inductances, i x s and i x ae the stato and oto cuents, M is the mutual inductance, L ls and L l ae leakage inductances. The stato voltage in the stationay efeence fame can be found using Kichoff s Voltage Law (KVL) and Faaday s law: v s = R s i s + d λ s dt (.3)

29 8 Similaly, the oto voltage expessed in the oto electical efeence fame (denoted by the supescipt e), is given by the same elationship as above: v e = R i e + d λ e dt (.4) This elationship can then be expessed in the stationay efeence fame by ewiting the pevious equation as follows: v = e Jθe v e = e Jθe [ R i e ] + d λ e dt = R e Jθe e Jθe i + e d Jθe dt e Jθe λ { } = R i + e Jθe Jω e e Jθe λ + e d λ Jθe dt (.5) Finally, we aive at ou desied expession fo the oto electical dynamics: v = R i ω e J λ + d λ dt = 0 (.6) It should be noted that in ou analysis, we have modeled the induction machine as having a squiel-cage oto winding and so the oto voltage is equal to zeo (the teminals ae shoted togethe). Additionally, the machine used in ou expeimental testing was in fact a squiel-cage type induction machine. To deive the toque expession fo an induction machine, we will use the concept of co-enegy [, 3, 4]. It can be shown that the expession fo co-enegy is given by W c = L s i s + L i e + i T s M (θ e ) i e (.7)

30 9 whee M (θ e ) = Me Jθe. This expession may look familia as it is stikingly simila to the equation fo enegy stoed in a tansfome. The electomagnetic toque of a thee-phase machine is then given by τ 3ph = 3 W c = 3 W c θ e (.8) θ θ e θ whee θ e θ = P. Substituting (.7) into (.8) yields the following toque expession (.9). τ 3ph = 3P 4 is T MJe Jθe i e (.9) While the toque expession above (.9) is valid, it is not paticulaly useful due to its dependence on the oto electical angula displacement (θ e ). This dependence can be eliminated howeve, by ecognizing that i e we obtain ou desied esult: = e Jθe i, and so τ 3ph = 3P 4 M i T s J i (.0) Lastly, the mechanical dynamics ae given by dω dt = H [(τ 3ph τ l ) Bω ] (.) whee H is the combined moment of inetia of the oto and load, B is the mechanical damping on the oto, and τ l is the load toque applied to the oto.

31 0. Obsevability Analysis As mentioned ealie, diect field-oiented contolles ae typicially implemented using a oto flux linkage estimato. The estimato is needed to tansfom the electical vaiables into and out of the oto flux linkage efeence fame. Additionally, an estimation of the oto flux linkage magnitude is computed as well, which is then used in the contol algoithm. Given the impotance of the oto flux linkage estimato, an obsevability analysis is well justified... State Space Model The obsevability analysis pesented in this thesis will be limited to the electical dynamics of the induction machine. While the oto velocity, whose dynamics ae given by (.), will be pesent in the state equations, we will ague that the time scale associated with them is consideably slowe than that of the electical dynamics. Theefoe, the oto velocity will be teated as a constant in ou analysis, since in eality it is changing much moe slowly than the electical vaiables. Additionally, we will be measuing the oto velocity with an optical encode in ou expeiments, and so an instantaneous measuement will always be available. The same agument will be used in the following chapte to deive flux estimatos using tapezoidal integation. By making this assumption we not only educe the ode of the state space equations, but also obtain a linea model. d dt x e = A x e + B v s e i e s = C x (.) The state space equations in the electical efeence fame (denoted by the supescipt e) fo the smooth aigap d q model of an induction machine [6] can

32 be deived using the equations pesented in the pevious section. The esulting vectos and matices ae given as follows: (R sl + R M ) I ω A = σ e J L x e i s e = (.3) λ e ( ) M R I ω σ e J L R M I R I (ω e ω e ) J L L L (.4) B = σ I (.5) 0 [ ] C = I 0 (.6) whee σ = L s L M. The electical efeence fame is used so that the electical fequency (ω e ) appeas in the equations. This way, the dependency of the system s obsevability on electical fequency, if any, can be analyzed... Obsevability Test To veify the obsevability of the electical vaiables ( i e s and λ e ), we will use the standad test fom linea systems theoy [7, 8]. The obsevability matix can then be fomed as follows: C CA Q = CA CA 3 (.7)

33 Even with the eduction in model ode, the esult is a athe lage and complex 8- by-4 matix. Howeve, fom linea algeba we know that the ank of a matix ρ (Q), which is the maximum numbe of linealy independent columns, must be less-thano-equal-to the minimum dimension. Since the matix Q has a minimum dimension of 4, the ank of the matix must be less-than-o-equal-to 4. Additionally, the ank of a matix is also equivalent to the numbe of linealy independent ows []. In ou analysis, we seek to pove that the electical vaiables ae completely obsevable by showing that the obsevability matix has full ank (i.e. ρ (Q) = 4) fo all values of ω e and ω e. Fotunately, this can be poven by showing that the fist fou ows of the obsevability matix ae linealy independent. These ows ae given by Q 4 = C = CA I 0 (R sl + R M ) σ L I ω e J M σ ( R L I ω e J ) (.8) Inspection of (.8) eveals that the only way the bottom two ows could be a linea combination of the top two ows is when w e and w e both equal zeo, along with eithe M o R. While the fist condition (w e = w e = 0) is not only possible, but a vey likely opeating condition, the second condition (M o R = 0) is physically impossible. Theefoe, the matix Q has full ank and so the electical vaiables i e s and λ e ae completely obsevable fo all non-zeo values of w e and w e..3 Simulink R Model Simulations wee pefomed in ode to test ou contolle s stability and pefomance befoe implementation with hadwae. The softwae package Simulink was

34 3 chosen fo its familiaity, ease of use and its simple inteface with dspace. Howeve, befoe a contolle could be simulated, we fist needed to implement ou induction machine model in Simulink. Ou simulink model of an induction machine consists of thee layes. The bottom laye [Figue.3] implements the cuent/flux linkage elationships given by i x s i x = L σ λ x s M σ λ x (.9) = L s σ λ x M σ λ x s (.0) lambda _sd L/sig^ i_sd lambda _sq L/sig^ M/sig^ Add i_sq M/sig^ M/sig^ 3 lambda _d M/sig^ 3 i_d Ls/sig^ 4 lambda _q Ls/sig^ Add 4 i_q Figue.3. Simulink block diagam fo the cuent/flux elationships. These elationships can be deived fom the flux linkage expessions (..) pesented ealie. The electical dynamics, which ae given by equation (.3) and (.6), ae implemented in the next laye [Figue.4]. The final laye [Figue.5] implements the mechanical dynamics (..0) and pefoms the necessay Clak tansfoms.

35 4 Rs Rs Rs Rs Vsd Vsq s Integato 3 s Integato lambda_sd lambda_sq i_sd i_sq i_sd i_sq 3 w_e Poduct - J s Integato lambda_d lambda_q i_d i_q 3 i_d 4 i_q Cuent / Flux Relations Poduct J s Integato R R R R Figue.4. Simulink block diagam fo the electical dynamics. v_a v_b a b d q Vsd Vsq i_sd i_sq a d b q c -3 Phase Convesion i_a i_b 3 i_c 3 v_c c 0 Teminato abc -dq 0 Convesion w_e i_d i_q Poduct Poduct Add 3*P*M/4 3*P*M/4 tau_3ph Add /H /H s Integato w_ 4 Roto _Speed IM _dq _model 4 Load _Toque w_e P/ B B P/ Figue.5. Simulink block diagam fo the mechanical dynamics.

36 Chapte 3 Contolle Design Until now, we have kept the discussion of ou contol algoithm vey geneal, gadually building upon the basic concepts. In this chapte howeve, we will begin pesenting the detailed deivation and design of ou contolle. The diect fieldoiented contol technique will be fist be discussed along with the deivation of ou oto flux linkage estimatos. Aftewads, the deivation and stability poof of ou adaptive oto esistance estimato will be pesented as well as simulations veifying the stability and pefomance of the esulting contolle. 3. Diect Field-Oiented Contolle As mentioned ealie in chapte one, the diect field-oiented contol technique which we ae using is based on the tansfomation of electical vaiables into and out of the oto flux linkage efeence fame. The diect axis of this efeence fame is aligned with the oto flux vecto, as shown in Figue 3.. The toque expession in the oto flux linkage efeence fame can be deived using the equations fom chapte two as follows:

37 6 syn syn q λ d λ λ syn Figue 3.. Tansfomation of electical vaiables into the oto flux linkage efeence fame. τ 3ph = 3P 4 M i T s = 3P 4 M i s T J = 3P 4 = 3P 4 = 3P 4 J i ( λ L M L i s ) M ( is T J L ) λ M i s T J i s M ( is T J L ) λ M (λ d i sq λ q i sd ) (3.) L Taking the electical vaiables to be in the oto flux linkage efeence fame, the expession above (3.) can be ewitten as follows: τ 3ph = 3P 4 M ( L λ cos (θ λ θ syn ) i λ sq ) λ sin (θ λ θ syn ) i λ sd = 3P M 4 L λ i λ sq (3.)

38 7 Inspection of (3.) eveals pomising similaities to the toque expession fo a sepaately excited DC machine (.8) given in chapte one. By tansfoming electical vaiables into the oto flux linkage efeence fame, we can egulate the toque of an induction machine by egulating the oto flux linkage magnitude, λ, and quadatue stato cuent, i λ sq. Conceptually, inspection of Figue 3. eveals that, since the diect stato cuent vecto will be aligned with the oto flux linkage vecto in the oto flux linkage efeence fame, it may be used to egulate the oto flux linkage magnitude. This can be shown analytically as follows: d dt λ = R i + ω e J λ ( ) λ = R M i s + ω e J L L λ [ = ω e J R ] I L Noting that in the oto flux linkage efeence fame λ = the following expession λ + R M L i s (3.3) [ λ 0] T we aive at d dt λ = R L λ + R M L i λ sd (3.4) Theefoe, i λ sd can be used as a contol input to egulate and shape the dynamics of the oto flux linkage magnitude, while i λ sq can be used to egulate toque fo a give oto flux linkage magnitude [Figue 3.]. One challenge which emains to be addessed is the estimation of the oto flux linkage. As it has been shown, the oto flux linkage is needed to pefom the necessay efeence fame tansfoms. Since it is impossible to diectly measue

39 8 3P M 4 L a b c λ λ Figue 3.. Diect Field-Oiented Contolle. the oto flux linkage, estimatos must be used to compute it based on electical paametes which ae known o can be measued. 3. Roto Flux Estimato Design Thee ae two pimay methods of estimating the oto flux linkage in an induction machine. The fist method, sometimes efeed to as the stato voltage-based estimato, diectly solves fo the stato flux linkage by integating the stato voltage minus the esistive dop (3.5). The stato flux linkage can then be used along with the stato cuent to calculate the oto flux linkage (3.6). ˆ λ s = ( v s R s i s ) dt (3.5) ˆ λ = L ˆ λ s σ M M i s (3.6) In pactical implementations, this pue integation is toublesome as a DC offset eo can build up ove time, leading to instability. Theefoe, a decay constant K is often added to the stato flux linkage estimato (3.7), implementing what is

40 9 sometimes efeed to as a leaky integato. The value of this decay constant can be adjusted to eliminate the DC dift poblems, unfotunately it will also make the estimato uneliable at low fequencies []. Theefoe, the value should be chosen such that it is lage enough to eliminate DC dift, but small enough to allow eliable opeation at as low a fequency as possible. ˆ λ s = ( v s R s i s Kˆ λs ) dt (3.7) The second method of calculating the oto flux linkage, often efeed to as the cuent-based estimato, can be implemented by integating the equation fo the oto electical dynamics given by (3.3). The esulting expession has been povided below. ˆ λ = ([ ω e J R ] I ˆ λ + R M L L i s ) dt (3.8) It can be shown that the poles of this estimato ae located at p, = R L ± jω e (3.9) by viewing the estimato as a state-space equation whee the system matix is given by A = [ ω e J R ] I L (3.0) Theefoe, the cuent-based estimato is natually stable and so it tends to yield highe accuacy estimations compaed to the stato voltage-based estimato at low fequencies. Howeve, it will be shown in the following section that discetetime implementations of this estimato may suffe fom accuacy and ultimately stability issues at high speeds depending on the integation method used.

41 Deivation using Tapezoidal Integation Befoe pesenting the deivation of the estimatos using tapezoidal integation, we will fist pesent the Fowad Eule implementation of the cuent-based estimato along with a stability analysis to motivate the use of tapezoidal integation. As we did in the obsevability analysis, we will again make the assumption that the mechanical dynamics associated with the oto ae changing much moe slowly than the electical dynamics, and theefoe ω e will be teated as a constant in ou analysis. Additionally, the oto velocity will be measued and so an instantaneous value is available. The fowad Eule implementation of the cuent-based estimato can be deived as follows: ˆ λ [k + ] ˆ ( λ [k] = ω e J R ) I ˆ λ [k] + R M T s L L i s [k] ˆ λ [k + ] = ˆ λ [k] + T s (ω e J R ) R M I ˆ λ [k] + T s L L i s [k] [( ) ] R R M ˆ λ [k + ] = T s I + T s ω e J ˆ λ [k] + T s L L i s [k] (3.) whee k is the discete-time index and T s is the sampling peiod. To analyze the stability of this estimato we note that the equation is in a state-space fom whee the system matix is given by A = [( ) ] R T s I + T s ω e J L (3.) inspection of which eveals that the eigenvalues (o poles) of the estimato ae

42 3 located at p, = ( ) R T s ± jt s ω e (3.3) L Theefoe, the estimato will only be stable fo small values of ω e, eventually becoming unstable at high speeds when the poles move outside the unit cicle, which is the stability bounday fo discete-time systems. It can be shown that this will occu when ω e = ( T s R L ) T s (3.4) Fo this eason, and accuacy concens, the decision was made to use oto flux linkage estimatos deived using tapezoidal integation Voltage-Based Estimato Since the voltage-based estimato (3.5) is only maginally stable due to the pue integation theoetically equied to solve fo the stato flux linkage, stable appoximations to ideal integatos (3.7) ae typically used in pactice. While the use of a fist ode appoximation o leaky integato is typically sufficient, ou adaptive oto esistance estimation algoithm necessitates the use of a highe ode appoximation. This is due in pat to sensitivity of ou adaptive algoithm to phase eos, as well as the need to completely null DC offsets. In the s-domain, an ideal integato is given by, the fist ode appoximation s of which is given by. Ou poposed appoximation is given by s + K G int (s) = s s + ζω n + ω n (3.5) whee ζ is the damping coefficient and ω n is the natual fequency which gives the cone fequency of the filte, much like K in the fist ode appoximation. A

43 3 Bode plot has been povided [Figue 3.3] which compaes the fequency esponse of an ideal integato to the fist and second ode appoximations with cone fequencies of ad/sec. Inspection of Figue 3.3 eveals that the phase esponse Magnitude [db] Ideal Fist Ode (K = ) Second Ode (ω n =, ζ = 0.) Phase [deg] Fequency [ad/sec] Figue 3.3. Fequency esponse compaison of integato appoximations. of the second ode appoximation is capable of eaching the ideal shift of 90 degees befoe the fist ode appoximation. To obtain a tapezoidal fom of (3.5) fo implementation, we fist need to deive state space equations which descibe the tansfe function. This can be done by splitting the tansfe function as follows: [ ] [ ] [ ] Y (s) P (s) G int (s) = = [s] P (s) U(s) s + ζω n + ωn (3.6)

44 33 The following state equations can then be deived: ẋ = x (3.7) ẋ = u ω nx ζω n x (3.8) y = x (3.9) Next, the tapezoidal ule is applied: x [k + ] = x [k] + T s {x [k + ] + x [k]} (3.0) x [k + ] = x [k] + T s + T s { u[k + ] ω n x [k + ] ζω n x [k + ] } { u[k] ω n x [k] ζω n x [k] } (3.) y[k] = x [k] (3.) By assuming u[k + ] = u[k] the following equations can be found afte some substitutions and eaanging: x [k + ] = x [k + ] = { ( ) } T s + ζd + d u[k] + + ζd + d ωn Ts x [k] + T s x [k], (3.3) + ζd + d { Ts u[k] ω nt s x [k] + ( ζd d ) x [k] }, (3.4) y[k] = x [k], (3.5) T s whee d = ω n. Povided the tapezoidal ule was applied coectly, the stability of the filte is assued so long as ω n and ζ ae appopiately chosen. This is because tapezoidal integation, which is the basis of the bilinea tansfom, peseves the

45 34 stability of continuous-time systems by mapping left-half-plane poles to locations inside the unit cicle [9]. The continuous-time pole locations ae given by p, = ζω n ± jω n ζ (3.6) Theefoe, ou second ode appoximation will be stable so long as ω n and ζ have positive eal values. To apply this integato appoximation to ou voltage-based flux estimato we simply let y = λ sx and u = v sx R s i sx Cuent-Based Estimato The cuent-based oto flux linkage estimato can be deived using tapezoidal integation as follows: ˆ λ [k + ] = ˆ λ [k] + T s [( ω e J ( I) ˆ λ [k + ] + ω e J ) I ˆ λ [k] + M ] i s [k] T T T [( + T ) ] [( s I T s ω e J ˆ λ [k+] = ˆ λ [k]+t s ω e J ) I ˆ λ [k] + M i s [k] T T T [( ] ˆ λ [k + ] = ) + Ts T I + T s ω e J ( ) + Ts T + (Ts ω e ) [ [( T s T ) I + T s ω e J ] ˆ λ [k] + MT s i s [k] ˆ λ [k + ] = Aˆ λ [k] + B i s [k], (3.7) T ] ] whee A = [( 4 (T s ω e ) B = [ MTs ( ) ) ] T s T I + 4T s ω e J ( + Ts T ) + (Ts ω e ) (3.8) T ( T ) ] I + MT s ω e T J + Ts ( ) (3.9) + Ts T + (Ts ω e )

46 35 Note that the oto time constant, T = L /R, was used to slightly simplify the equations. Again, the eigenvalues of the system matix A ae given by p, = ( ) 4 (T s ω e ) T s T 4T s ω e ( ) ± j ( ) (3.30) + Ts T + (Ts ω e ) + Ts T + (Ts ω e ) Fo compaison, the eigenvalues/poles of the system matix wee plotted as a fuction of ω e, in Matlab, fo both the Fowad Eule and Tapezoidal implementations using typical paametes [Table 3.] Tapezoidal Fowad Eule Unit-Cicle 0.04 Imaginay Real Figue 3.4. Pole locus compaison of Fowad Eule and Tapezoidal methods. R [ohms] L [mh] T s [msec] ω e min [ad/sec] ω e max [ad/sec] Table 3.. Paametes used fo pole locus compaison.