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THESIS FOR THE DEGREE OF LICENCIATE OF ENGINEERING

Vector Control of a Double-Sided PWM

Converter and Induction Machine Drive

ROLF OTTERSTEN

Department of Electric Power EngineeringCHALMERS UNIVERSITY OF TECHNOLOGY

Goteborg, Sweden 2000


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Vector Control of a Double-Sided PWM Converter

and Induction Machine Drive

ROLF OTTERSTEN

c ROLF OTTERSTEN, 2000.

Technical Report no 368LDepartment of Electric Power EngineeringChalmers University of TechnologySE-412 96 GoteborgSwedenTelephone + 46 (0)31-772 1000

Chalmers Bibliotek, ReproserviceGoteborg, Sweden 2000


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Abstract

This thesis describes high-performance control methods for a double-sidedPWM converter and induction machine drive. A control scheme is developedthat uses stator flux orientated vector control for the induction machine andgrid flux orientated vector control for the grid side PWM converter. Thederivation of PI-controllers for synchronous current control, direct voltagecontrol and stator flux control is described. The thesis also describes someimportant properties of stator flux orientation and deals with decoupling thestator flux from the torque during stator flux orientated vector control. Sev-eral stator flux observers of various complexity are described and a Luenbergerobserver with nearly constant poles is developed. Furthermore, a feed-forwardterm that improves the direct voltage control during large torque transientsis developed and eight variants of pulse width modulation are studied.

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Acknowledgements

This thesis is funded by the European Commission, in the framework of theNon Nuclear Energy Programme Joule III, and the Swedish National En-ergy Administration. The Joule project is entitled Low Cost Variable SpeedSystem for Stall Wind Turbines, Contract JOR3-CT98-0311, and is a co-operative venture between Ecotecnia S.C.C.L. (Spain), Quest AB (Goteborg,Sweden), and the Department of Electric Power Engineering at ChalmersUniversity of Technology.

I would like to thank my main supervisor Dr. Ola Carlsson for helping outwith EC bureaucracy.

I am very thankful Dr. Torbjorn Thiringer and Robert Karlsson for buildingfirst-class PWM converters. I would especially like to thank Dr. Thiringerfor also providing me with measurement systems, technical discussions andlaboratory help.

I would like to acknowledge Ph.D. candidate Andreas Petersson, who con-tributed with the theoretical work of what eventually became Paper I in thisthesis.

Thanks to Dr. Jan Svensson for discussions and proof reading.

Finally, I would also like to thank all my colleagues at the department whohave assisted me during the work on this thesis.

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Table of Contents

Abstract iii

Acknowledgements v

Table of Contents vii

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objective of the Thesis . . . . . . . . . . . . . . . . . . . . . . 21.3 Related Research . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Report . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.2 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.3 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Double-Sided PWM Converter 72.1 Name Convention . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Characteristics of Double-Sided PWM Converter . . . . . . . . 7

2.2.1 Regenerative Braking . . . . . . . . . . . . . . . . . . . 72.2.2 Fast Speed Dynamics . . . . . . . . . . . . . . . . . . . 92.2.3 Boosted and Controlled DC-Voltage . . . . . . . . . . . 92.2.4 Arbitrary Power Factor on the Grid Side . . . . . . . . 92.2.5 Grid Current Harmonics . . . . . . . . . . . . . . . . . 102.2.6 Radio Frequency Interference . . . . . . . . . . . . . . 10

2.2.7 Disadvantages . . . . . . . . . . . . . . . . . . . . . . . 112.3 Semiconductor Switches . . . . . . . . . . . . . . . . . . . . . 112.4 Sensors and Control Components . . . . . . . . . . . . . . . . 122.5 Control of the Double-Sided PWM Converter . . . . . . . . . 14

2.5.1 Two Induction Machine Control Principles . . . . . . . 152.5.2 Two Control Strategies for a Grid Connected PWM

Converter . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Stator Flux Orientated Vector Control of an Induction Ma-chine 21

3.1 Transient Model of an Induction Machine . . . . . . . . . . . . 213.1.1 Transient T-Model . . . . . . . . . . . . . . . . . . . . 223.1.2 Transient Inverse -Model . . . . . . . . . . . . . . . . 23

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3.2 Internal Model Control . . . . . . . . . . . . . . . . . . . . . . 243.3 Some Characteristics of Stator Flux Orientation . . . . . . . . 26

3.3.1 Cross Coupling between Active Current and Stator Flux 263.3.2 Torque Pull-Out . . . . . . . . . . . . . . . . . . . . . 27

3.3.3 Operation in Field Weakening Region . . . . . . . . . . 303.3.4 Low Speed Operation . . . . . . . . . . . . . . . . . . . 313.4 Stator Flux Orientated Stator Flux Controller . . . . . . . . . 313.5 Generation of Active Current Reference . . . . . . . . . . . . . 333.6 Stator Flux Control Without Reactive Current Control . . . . 343.7 Stator Flux Orientated Current Controller . . . . . . . . . . . 34

4 Stator Flux Observers 374.1 Open-Loop Stator Flux Observers . . . . . . . . . . . . . . . . 37

4.1.1 Voltage Model . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.2 Improved Integrator Algorithms for the Voltage Model 394.1.3 Current Model . . . . . . . . . . . . . . . . . . . . . . 414.2 Closed-Loop Stator Flux Observers . . . . . . . . . . . . . . . 42

4.2.1 Current and Voltage Model . . . . . . . . . . . . . . . 424.2.2 Luenberger Observer with Nearly Constant Poles . . . 44

4.3 Euler Forward Discretization . . . . . . . . . . . . . . . . . . . 50

5 Current Control and Control of the DC-Voltage for a GridConnected PWM Converter 515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Synchronization to the Grid . . . . . . . . . . . . . . . . . . . 51

5.3 DC-Voltage Controller . . . . . . . . . . . . . . . . . . . . . . 525.3.1 Direct Voltage Filtering . . . . . . . . . . . . . . . . . 55

5.4 Open-loop Reactive Power Control . . . . . . . . . . . . . . . 555.5 Design of a Current Controller for a Grid Connected PWM

Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Carrier Based Modulation 576.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2 Regular Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 616.3 Modulation Index . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.4 Triangular PWM with Zero Sequence Injection . . . . . . . . . 626.5 Some Zero Sequence Signals . . . . . . . . . . . . . . . . . . . 636.6 Avoiding Short Time Duration Switching States . . . . . . . . 726.7 Summary of Triangular PWM with Zero Sequence Injection . 72

7 Simulations 757.1 Software Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.1.1 Changes in the Control System . . . . . . . . . . . . . 757.1.2 Simulation Environment . . . . . . . . . . . . . . . . . 77

7.2 Safety Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.2.1 Speed Limit . . . . . . . . . . . . . . . . . . . . . . . . 777.2.2 Stator Current Limits . . . . . . . . . . . . . . . . . . 787.2.3 Voltage Limit . . . . . . . . . . . . . . . . . . . . . . . 80

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7.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 81

8 Experiments 938.1 Hardware Description . . . . . . . . . . . . . . . . . . . . . . . 93

8.1.1 Double-Sided PWM Converter . . . . . . . . . . . . . . 938.1.2 Induction Machine . . . . . . . . . . . . . . . . . . . . 968.1.3 Control Computer . . . . . . . . . . . . . . . . . . . . 968.1.4 Pulse Width Modulation . . . . . . . . . . . . . . . . . 978.1.5 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.1.6 Measurement System . . . . . . . . . . . . . . . . . . . 98

8.2 Software Description . . . . . . . . . . . . . . . . . . . . . . . 988.3 Laboratory Experiences with Stator Flux Observers . . . . . . 998.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 100

9 Conclusions 109

10 Future Work 111

References 113

A Glossary of Symbols, Subscripts, Superscripts and Abbrevi-ations 119

B Space Vectors 123B.1 Space Vector Notation . . . . . . . . . . . . . . . . . . . . . . 123

B.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . 124B.3 Torque in Terms of Space Vectors . . . . . . . . . . . . . . . . 125

C Base Values 127

D Parameter Sensitivity of Stator Flux Observers 129D.1 Open-Loop Observers . . . . . . . . . . . . . . . . . . . . . . . 129

D.1.1 Voltage Model . . . . . . . . . . . . . . . . . . . . . . . 129D.2 Closed-Loop Observers . . . . . . . . . . . . . . . . . . . . . . 130

D.2.1 Current and Voltage Model . . . . . . . . . . . . . . . 130

D.2.2 Luenberger Observer . . . . . . . . . . . . . . . . . . . 130E Induction Machine Data 133

F Publications 135

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Chapter 1

Introduction

1.1 Background

Electrical machine systems consume approximately 50 % of the producedelectrical energy in the western world [1] and are used in all kinds of kindsindustrial processes, transportation systems and household appliances. Themarket value of electric machine systems was $70.5B in 1997 [2] and thevalue was growing at a rate of 7 % per year [2]. About 75-80 % of installedelectrical machine systems operate at constant speed while 20-25 % operate atadjustable speed [1]. The use of adjustable speed machine systems is predictedto increase further due to increased industrial needs for better torque, speed

and position control [2]. Furthermore, adjustable speed machine systems arealso motivated because they can reduce energy consumption for quadratictorque loads, eliminate inrush currents and reduce mechanical wear.

A modern adjustable speed ac machine system is equipped with an adjustablefrequency drive (AFD). The AFD is a power electronic device for torque/speedcontrol of an electric machine. The AFD controls the torque/speed of theelectric machine by converting the fixed voltage and frequency of the grid toadjustable values on the machine side. An AFD converter for an electric ma-

chine can be designed with various power electronic principles. The six pulsediode rectifier/pulse width modulated (PWM) inverter with an intermediatevoltage stiff direct voltage link is perhaps the most widespread AFD technol-ogy in industrial applications. The six pulse diode rectifier/PWM inverteris a cost-effective and high-efficiency AFD that is hard to beat. However,alternative rectifier/inverter topologies can be attractive in certain ap-plications since the six pulse diode rectifier/PWM inverter only permits onepower flow direction and injects low order current harmonics to the grid. Thedouble-sided PWM converter (Fig. 1.1) that uses PWM converters on boththe grid and machine side is one such alternative converter topology. Thedouble-sided PWM converter offers a bi-directional power flow, full control ofthe direct voltage and the grid current, and reduced grid current distortion.

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MGrid

" % ' ( ')

' " % ' ( '

DC-link

AC machine

Figure 1.1: Double-sided PWM converter and ac machine.

1.2 Objective of the Thesis

The objective of this thesis is to study fast and accurate torque control of adouble-sided PWM converter and induction machine drive in the medium/high

speed range. Stator flux orientated vector control of the induction machineand grid flux orientated vector control of the grid side PWM converter havebeen chosen for this purpose.

1.3 Related Research

Numerous researchers have carried out extensive work on the double-sidedPWM converter. The following presents a selection of achievements thathave been of great inspiration for the author.

Eguchi and Imura were among the first who proposed vector current controlof the grid connected voltage source converter in 1986 [3]. It was shown thatvector current control is superior to amplitude-phase control. An experimen-tal 24-pulse forced commutated bridge, consisting of four forced commutatedthyristor inverters, of 20 kW was used and the lowest order voltage harmonicwas the 11th. The output voltages of the four 6-pulse forced commutatedthyristor inverters were phase controlled in order to obtain the desired volt-age vector. It was shown that the inverters could operate at lagging, leading,unity power factors and good controllability of the active and the reactivepower were also shown.

Kohlmeier et al. studied high performance GTO-based double-sided PWMconverters in 1987 [4],[5]. The reference currents were generated in rotor fluxorientated and grid flux orientated reference frames for the machine side andgrid side PWM converters, respectively. Predictive current control, i.e. theswitching instants were determined by a circular boundary around the currentreference vector [6], was used in order to obtain a switching frequency below1.5 kHz. Several digital signal processors were used to control the converter.The converter had a high power factor on the grid side, low grid currentdistortion and bi-directional power flow. Furthermore, it was found that afeed-forward term that is proportional to the induction machine loading could

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improve the direct voltage control.

Svensson studied a transistor-based grid connected PWM converter and pulsewidth modulation in his Ph.D. thesis [7] from 1988, along with several other

topics. Very accurate dead-beat response and low current distortion wasobtained with a switching frequency of 1 kHz.

Blaabjerg et al. [8] presented a IGBT-based double-sided PWM converterwith 4.8 kHz switching frequency in 1993. The entire control system, mon-itoring and communication were implemented in one microcontroller. Spacevector modulation was used for both PWM converters. The induction ma-chine was Volts-per-Hertz controlled while the grid connected PWM converterused a vector current controller. An outer direct voltage control loop gener-ated the active current reference for the grid connected PWM converter while

the reactive current reference was set to zero. The power factor droppedslightly at low loads, due to non-linearities in voltage transducers and in thegrid side PWM converter, but was otherwise near unity. Similar to Kohlmeieret al., a feed-forward term was used to improve the direct voltage control.

Several research papers on the double-sided PWM converter during the 1990swere focused on minimizing the direct voltage link electrolyte capacitor bank[9],[10]. Carlsson found in 1998 that the electrolytic capacitor bank could bereplaced by a small plastic capacitor bank and a braking chopper [11].

Hansen et. al [12] recently showed that it is possible to eliminate the grid

voltage of a double-sided PWM converter if the grid is balanced. However,the grid current became distorted for an unbalanced and distorted grid volt-age. Grid voltage sensors together with a distortion compensating algorithmwere therefore recommended for a standard industrial grid connected PWMconverter. Furthermore, [12] also recommended discontinuous pulse widthmodulation for the grid connected PWM converter in order to reduce switch-ing losses or, alternatively, reduce the size of the grid filter.

The historical development of vector control for induction machines is tooextensive to be fully covered here. Only the inventors of vector control alongwith research that has made a significant impact on this thesis are mentionedin the following.

Vector control is a popular high performance control principle for inductionmachine regulation that was developed by Blaschke [13] and Hasse [14] inthe late 1960s. Vector control has since then been constantly developedand there are several variants of vector control today. This thesis focuses onvector control that uses stator flux orientation and synchronous PI currentcontrol. Xu et. al. have written several research papers, spanning from1988 to 1992, on stator flux orientated vector control [15],[16],[17],[18]. Thesepapers describe the characteristics of stator flux orientation, operation in thefield weakening region, propose a current decoupling term and treat sensorless

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operation. The papers are highly recommended reading for studies on statorflux orientated vector control.

The excellent Ph.D. thesis [19] by Lennart Harnefors, presented in 1997, has

also been a huge source of inspiration for this thesis. The thesis representsstate-of-the-art in induction machine control and is highly recommended.This report in this thesis has especially adopted internal model control forthe design of PI-controllers from [19].

1.4 Outline of the Thesis

This thesis consists of three parts. These three parts are a technical reportand two papers, Paper I and Paper II. Thus, the following make up the present

thesis

Report: Rolf Ottersten, Vector Control of a Double-Sided PWM Converterand Induction Machine Drive, Department of Electric Power Engineering,Chalmers University of Technology, Goteborg, Sweden, May, 2000.

Paper I: Jan Svensson, Rolf Ottersten, Shunt Active Filtering of VectorCurrent-Controlled Voltage Source Converter at a Moderate Switching Fre-quency, IEEE Transactions on Industry Applications, vol. 35, no. 5, pp.1083-1090, Sep./Oct. 1999.

Paper II: Rolf Ottersten, Jan Svensson, Vector Current Controlled VoltageSource Converter - Deadbeat Control and Saturation Strategies, in Proc.IEEE Nordic Workshop on Power and Industrial Electronics, Espoo, Finland,pp. 65-70, August 26-27, 1998.

The main part of the thesis consists of the technical report. Paper I andPaper II are included in Appendix F of the report. The following subsectionsprovide a content description of the technical report and the two papers.

1.4.1 Report

The report describes vector control of the double-sided PWM converter andinduction machine drive and also investigates some different variants of pulsewidth modulation. The report is divided into ten chapters.

Chapter 2 presents some important characteristics of the double-sided PWMconverter. The characteristics are in some cases compared with the char-acteristics of the commonly used six pulse diode rectifier/PWM inverter.Some common control schemes for induction machine control and thecontrol of a grid connected PWM converter are also described.

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Chapter 3 derives controllers for stator flux regulation and for current con-trol during stator flux orientation. Transient models of the inductionmachine and some characteristics of stator flux orientation are also de-scribed.

Chapter 4 describes some open-loop and closed loop stator flux observers.One problem with a direct open-loop stator flux observer is its sensitivityto measurement bias. Recently presented methods for handling measure-ment bias are described. Two closed-loop observers are described. Thefirst closed-loop observer is the well known closed-loop observer [20] thatcombines the best properties of the current and voltage models. A secondclosed-loop observer, a Luenberger observer with nearly constant poles,is developed in this thesis.

Chapter 5 develops a vector control scheme for the current control and thedirect voltage control of a grid connected PWM converter. A feed-forward term is used to improve direct voltage regulation in order toimprove direct voltage control during heavy torque transients.

Chapter 6 deals with some features of carrier based modulation and espe-cially triangular PWM with zero-sequence injection. Both conventionaland discontinuous zero-sequence signals are described. The differencesbetween the zero-sequence signals concerning the resulting current dis-

tortion and switching losses are theoretically analyzed.Chapter 7 describes software implementation and presents simulation re-

sults of a double-sided PWM converter. In addition, some software safetylimits are implemented and described.

Chapter 8 presents experimental results of the developed control scheme forstator flux orientated vector control of induction machines. The directvoltage link is charged by a dc-machine during the experiments.

Chapter 9 contains the conclusions of this thesis.Chapter 10 provides recommendations for future work on the double-sided

PWM converter.

The report also contains five appendices:

Appendix A lists the symbols, subscripts, superscripts and abbreviationsthat are used in this thesis.

Appendix B describes some fundamentals of space vector theory.Appendix C presents the per-unit system that is used in this thesis.

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Appendix D provides the frequency response functions of the stator fluxobservers that are studied in this thesis.

Appendix E lists the parameters of the induction machine that is used in

the theoretical and simulation parts of this thesis.Appendix F contains Paper I and Paper II, which are described in more

detail below.

1.4.2 Paper I

The grid current of double-sided PWM converter is fully controlled. Thispermits to use the grid connected PWM converter of the double-sided PWMconverter as a parallel active power filter. A parallel active power filter iscapable of cancelling low order current harmonics from non-linear loads. Theparallel active power must be able measure the load current and to detectthe current harmonics in real time/nearly real time. Paper I compares fourtime delayed methods to detect current harmonics. Stationary conditions areassumed so the detection methods are sufficient. A vector current controllerwith deadbeat gain is used to track current harmonics.

1.4.3 Paper II

A vector current controller with deadbeat gain is suitable to use for an par-allel active power filter. Deadbeat gain permits a high bandwidth and anaccurate cancellation of current harmonics. One drawback of using deadbeatgain is that the current controller easily saturates due to limited dc-voltage.Paper II presents control methods for reducing the negative influence fromvoltage saturation. Furthermore, a one sample calculation delay may appearin the current control due to the implementaion of the control computer. Thepaper proposes the use of a Smith predictor in order to compensate for the

calculation delay.

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Chapter 2

Double-Sided PWM Converter

2.1 Name Convention

The double-sided PWM converter and its two sub PWM converters are knownby various names in technical literature. The author prefers the names gridside or grid connected PWM converter and machine side PWM converter forthe two sub PWM converters. These names clarify the locations of the twosub PWM converters and indicate that the power flow can be reversed atany time instant. Furthermore, two three-phase PWM converters with anintermediate stiff direct voltage and with only two possible phase potentiallevels are referred to as a double-sided PWM converter.

2.2 Characteristics of Double-Sided PWM Converter

Fig. 2.1 shows three important characteristics of the double-sided PWM con-verter. These three characteristics are operation within all four quadrants ofthe torque-speed domain, nearly sinusoidal and fully controlled currents onboth the machine side and the grid side, and a unity power factor is possible onthe grid side. The four quadrant operation permits bi-directional power flowand hence four quadrant operation. The fully controlled grid current gives

full control of the direct voltage and full control of the grid power factor,preferably set to unity. Finally, the fully controlled grid current also enablesthe use of a the grid connected PWM converter as a parallel active powerfilter. The following sections discuss the characteristics of the double-sidedPWM converter in more detail. Some differences/similarities between thedouble-sided PWM converter and the six pulse diode rectifier/PWM inverterare also discussed.

2.2.1 Regenerative Braking

An electric machine can normally operate both as a motor and as a generator.When an adjustable speed machine operates as a generator, for instance dur-

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Forwardmotor

operation1

43

2

Generator7 8 @ 8 B C E

operation

Reversemotor

operation

Generator7 8 @ 8 B C E

operation

eT

F B G H I P E E R

Max.transient

torque

Max.cont.

torque

Max.transientpower

b

b

Max.cont.power

(a)

0.02 0.025 0.03 0.035 0.04 0.045 0.051.5

1

0.5

0

0.5

1

1.5

Time (s)

Current,vol

tage(pu)

is1

ig1

eg1

(b)

Figure 2.1: (a) Four-quadrant torque-speed diagram. (b) Simulated grid voltage eg1, grid

current ig1 and induction machine current is1 with 4.8 kHz switching frequency and 10 %grid filter impedance.

ing braking, the regenerated power must flow from the machine to the directvoltage link which raises the voltage on the direct voltage link. In some cases,the regenerated energy is small and can be absorbed by the direct voltagelink without problems. During braking though, overvoltage may occur on thedirect voltage link which causes the drive to trip and the capacitor bank isthen exposed to additional stress. Applications that require frequent accel-

eration and deacceleration may therefore require regenerative braking. Thebraking power can be transferred back to the grid (regenerative braking) orit can be dissipated by a braking chopper on the direct voltage link (dynamicbraking). The braking power can be calculated from the load cycle profile andthe load torque characteristic [21]. Regenerative or dynamic braking shouldbe considered if Pbr/Pnom > 0.1, where Pbr is the regenerated power at topspeed and Pnom is the nominal power of the drive [21].

The double-sided PWM converter, with its grid connected PWM converter, is

a four quadrant converter and have regenerative braking built in. The powerflow between the grid and direct voltage link can therefore be reversed at anytime instant. However, a standard six pulse diode rectifier/PWM inverteris non-generative converter and cannot reverse the power flow. The directvoltage link is therefore normally equipped with a braking chopper if dynamicbraking is required. The braking chopper transfers the regenerated power intoheat. Additional cooling may be required due to the dissipated heat, which inthat case introduces additional cost. As an alternative to the braking chopper,dc-braking [1] can also be used to obtain some, low-performance, electricbraking. To sum up, the double-sided PWM converter can be very cost-competitive for regenerative applications. The double-sided PWM converteralso places less stress on the dc-capacitor for such applications.

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The braking capability of an electric machine can also be obtained with amechanical brake. However, a mechanical brake leads to wear on mechanicalparts, possibly less smooth braking and a waste of energy. Regenerativebraking is therefore often a better choice. Note, though, that a mechanical

brake is sometimes necessary as an emergency backup to the electrical brake.Some AFD applications which are likely to require regenerative braking arelifts, ramped stopping of flywheels or large mechanical arms, and certainmaterial handling and packaging lines [21].

2.2.2 Fast Speed Dynamics

The four-quadrant operation permits fast speed dynamics. The fast speeddynamics are closely related to the regenerative braking capability, which

permits faster acceleration and deacceleration compared to a six pulse dioderectifier/PWM inverter without braking chopper.

2.2.3 Boosted and Controlled DC-Voltage

The controlled direct voltage of a double-sided PWM converter permits a lowdependency on machine loading and grid voltage sags [23]. Furthermore, theoperation of the grid side PWM converter is similar to a dc/dc boost converterand ac inductors are therefore required. The direct voltage must be at least

equal to the peak of the grid phase-to-phase voltage for successful operationand in practice it must be higher. The boosted direct voltage permits fasttorque response. Alternatively, the boosted direct voltage also permits theuse of a machine with higher nominal voltage. Unfortunately, the boosteddirect voltage may also increase dv/dt problems [24].

The direct voltage is charged with short-duration current pulses during normaloperation. The short-duration pulses permit the use of a smaller capacitorbank compared to a six pulse diode rectifier/PWM inverter, which requires arather bulky dc-capacitor in order obtain a smooth direct voltage.

2.2.4 Arbitrary Power Factor on the Grid Side

The grid connected PWM converter permits arbitrary power factor on thegrid side. This permits the use of the grid connected PWM converter as areactive power compensator. Often, though, unity power factor is desired.Unity power factor minimizes the grid current RMS value. A unity powerfactor can also be obtained with a six pulse diode rectifier/PWM inverter.However, the low order current harmonics of the diode rectifier increase theRMS value of the current compared to when using a grid connected PWMconverter. Both the six pulse diode rectifier and the grid connected PWMconverter reduce the current RMS value compared to a directly connected

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induction machine, without phase compensation though, since the grid nolonger has to supply reactive power.

2.2.5 Grid Current HarmonicsNo stand-alone grid connected converter/rectifier emits a purely sinusoidalgrid current. The grid current is always more or less distorted. The gridcurrent of a grid connected PWM converter is nearly sinusoidal but containshigh order current harmonics at essentially the multiples of the switching fre-quency. The six pulse diode rectifier on the other hand emits low order cur-rent harmonics such as 5th, 7th, 11th... The low order harmonics may causetransformer overheating, motor failure, fuse blowing, capacitor failures [25].Relevant standards for grid harmonics are [26]:

IEC 1000-3-2: Sets harmonic current limits for smaller equipment, typicalhousehold appliances or similar below 16 A/phase.

IEC 1000-3-4: Sets harmonic current limits for larger customers, i.e. indus-try, for loads between 16-75 A. Deals with both individual equipmentand the whole system installation and gives a consideration of the gridshort circuit ratio.

IEEE 519: Limits current and voltage harmonics at the point of commonconnection. Applicable mainly to larger consumers and for continuousoperation and in USA.

It is often claimed that the use of double-sided PWM converters are motivateddue to better power quality than the six pulse diode rectifier/PWM inverter.However, a dilemma for the double-sided PWM converter is that the dioderectifier/PWM inverter offers cost-effective solutions to reduce the grid cur-rent harmonics [27],[28],[29]. One interesting solution is to mix groups of sixpulse diode rectifier/PWM inverters with one double-sided PWM converterthat has active power filtering capability [29]. It must also be mentioned that

the grid current of the grid connected PWM converter is not purely sinusoidalbut contains high order current harmonics. The high order current harmonicscan introduce voltage harmonics that disturb senstive electronics [30]. Filter-ing of the high order current harmonics is somehow simpler than low orderharmonics, though. The high order current harmonics can be filtered witha rather small notch- or LCL-filter likely combined with common mode andradio frequency interference filters.

2.2.6 Radio Frequency Interference

The switchings of a PWM converter in conjunction with various stray capaci-tances and conductor inductances cause charging/discharging currents during

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each switching transient. The charging/discharging currents can be dividedinto common mode and differential mode components and represent dampedoscillations with very high frequencies. The damped oscillations create a wide-band radio frequency interference (RFI) that may interfere with the control

equipment and other electric equipment. The European standard EEF 82/499sets tolerable levels for RFI emission over an AC grid. The common modecurrent creates an RFI that seems to be especially troublesome since the com-mon mode current may take various long paths back to the PWM converter.Special RFI filters [31] possibly on both the grid side and machine side con-verter are probably the only successful solution for reducing the RFI and tocomply with standards. Shielded and properly grounded power cables andcontrol cables are also useful for reducing RFI problems.

2.2.7 Disadvantages

Two main disadvantages of the double-sided PWM converter are a higherprice and higher losses as compared to a six pulse diode rectifier/PWM in-verter. The grid connected PWM converter is rather rare on the market todayand the converter require more semiconductor components, more sensors anda more complicated control system. An increased cost of about 40-50 % com-pared to the six pulse diode rectifier/PWM inverter is possible [29]. Thehigher losses of a PWM converter are because both switching and conduc-tion losses are present and the conduction losses of an IGBT are higher thanthat of a diode. A reasonable assumption is also that a double-sided PWMconverter produces more RFI than a six pulse diode rectifier/PWM invertersince a double-sided PWM converter has twice as many RFI sources.

2.3 Semiconductor Switches

PWM converters require semiconductor switches that are capable of switch-ing on/off at arbitrary time instants. Typical switching frequencies are 2-

8 kHz. Fig. 2.2 shows ratings of single semiconductor switches as they werein 1997 [6]. Note that this figure is likely to be outdated since new and im-proved devices are entering the market continuously. The MOSFET is thepreferred semiconductor valve for low power/low voltage applications. TheMOSFET has low conduction losses and low switching losses. The MOS-FET also has a parasitic anti-parallel diode that may be used as a built-infree-wheeling diode if its recovery time is acceptable. However, the on-stateresistance of a normal MOSFET has a strong dependency on the blockingvoltage. This makes the IGBT the preferred choice for applications in themid/high voltage range. The IGBT was introduced in the 1980s and has,today, nearly replaced the bipolar junction transistor (BJT) as the preferredsemiconductor switch in adjustable frequency drives. Both the MOSFET and

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0.2

1

3

10

200

12

34

56

0

2

4

6

Umax

(kV)

fsw

(kHz)

Imax

(kA

)

Mosfet

BJT

IGBT

GTO

Figure 2.2: Ratings of semiconductor switches as they were in 1997.

the IGBT are voltage controlled which allows simple driver circuits. TheGTO has high switching losses and requires a complicated driver since it iscurrent controlled. The GTO is therefore mainly used for high power appli-cations beyond 2-3 MW [6]. The material in this thesis is not focused on anyspecific semiconductor switch although IGBT:s are assumed and have beenused during experiments.

2.4 Sensors and Control Components

A double-sided PWM converter consists of two PWM converters with anintermediate voltage stiff direct voltage link. The grid side PWM convertermaintains constant direct voltage and handles the reactive power exchangewith the grid while the machine side PWM converter controls the torque andflux of the induction machine.

A traditional implementation of a double-sided PWM converter requires alarge amount of sensors. All electrical sensors should normally have galvanicinsulation between the power electronics and the low voltage electronics. Itmay not be required to anti-alias filter the sampled variables but at leastlow-pass filters are useful for suppressing switching noise. Fig. 2.3 shows apossible sensor implementation which consists of

Current sensors on the grid side, required for current control. Current sensors on the machine side, required for current control. Voltage sensors on the grid side. These sensors are required for synchro-

nization to the grid.

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ge

Sample&hold

S T U V W X Y ` W a Y b c S U e X f Y g h Y gi

g W a ` W a Y b c S U e X f Y g h Y g

DC-link

Induction machine

dcu

~

gi

Driver

Control

suModulator

gu3

3

36 2

Grid

si r

q e s t u Y v ` W a Y a b c S U e X f Y g h Y g

Driver Sample&hold

Modulator

3

Figure 2.3: Double-sided PWM converter with sensors and control components.

Voltage sensor on the direct voltage link, required for control of the directvoltage.

Shaft sensor for rotor position/speed, required for position/speed control.It is desirable to reduce the amount of sensors since sensors represent an in-creased risk of system failure and an increased cost. Speed sensorless systemsare common today for less critical applications. Furthermore, the current sen-sors on the ac side can be moved to the direct voltage link [32] since two of thephase currents appear twice on the direct voltage link during each switchingperiod. There are also recent research efforts to eliminate the three voltagesensors on the grid side [12].

The control algorithms of the back-to-back system are often implementedin a digital signal processor (DSP). The DSP operates with a sampling fre-quency of some kHz and the double-sided PWM converter is therefore a crit-ical real-time application. The reference voltages of the controller are sentto modulators which determine the switching instants of the semiconductorswitches. The realization of the switching instants are handled by separatedrivers. Blanking time must be added to the driver signals in order to avoidshort-circuits in the phase-legs. The drivers should be galvanically separatedfrom the control system. Optical fibers are often used for this purpose.

It is sound to let a diagnostic program monitor the operation of the double-sided PWM converter. The diagnostics program can for instance detect sensorfailure, grid disturbances, overspeed, overcurrents and overvoltage on the di-rect voltage link. The monitor program can take suitable actions in case ofoverloading or faults or it can trip the converter. The diagnostic program canalso check the condition of the sensors before starting the system. Smallermeasurement offsets can be stored and compensated by the control program.In addition to the diagnostic program, fast backup hardware for overvoltage

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and overcurrent protection is also useful. The monitor program may also havea user-friendly self-commissioning feature. With the self-commissioning fea-ture, the control system automatically identifies the induction machine andgrid filter parameters before actual operation starts.

In addition to sensors and control components, several passive componentsstand for a significant part and cost of the double-sided PWM converter. Thepassive components are, for instance filters, fuses and overvoltage protection.The passive components are treated very briefly, or not at all, in this thesis.Instead, this thesis focuses on the control of the double-sided PWM converter.

2.5 Control of the Double-Sided PWM Converter

Fig. 2.4 shows the block diagram of the vector control system for a double-sided PWM converter driving an induction machine, which is the main themeof this thesis. This figure is greatly simplified and does not display all detailsof the control system. Both the grid connected PWM converter and theinduction machine are vector controlled and have multi loop cascaded controlsystems of a very similar structure. Table 2.1 summarizes the similaritiesof the two control systems. The current control loop is the inner controlloop for both systems. The main difference is how the active and reactivecurrent references are generated. The induction machine control may have an

additional outer loop for position control while, of course, the speed controlloop is not used during torque control.

A cascaded control system, such as vector control, is a form of state feedback.One important advantage of state feedback is that the inner control loop canbe made very fast. For vector control, current control is the inner controlloop. The fast inner current control nearly eliminates the influence fromparameter variations, cross coupling, disturbances and minor non-linearitiesin the control process [33]. A cascaded control system such as vector controlis therefore in many cases superior to simpler control principles without aninner current control loop. Two such control methods are Volts-per-Hertzcontrol for the induction machine and amplitude-phase control for the gridconnected PWM converter. These two control methods do not have an innerloop for current control.

Table 2.1: Similarities of vector control of an induction machine and vector control of a gridconnected PWM converter.

Control loop Induction machine Grid connected PWM converterInner loop Current control

Current control

Active loop Speed control Direct voltage controlReactive loop Flux control Reactive power control

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x y y

x y

Mechanical

subsystem

Fluxobserver

si

si

Currentcontrol

su

j k m o

k m o

k m o

Open-loopcontrol

n

n

eT

PI

Fluxcontrol

x y y

x y

DC-link

subsystem

gi

gi

gu

Currentfeedback

dcu

dcu

DC-voltagecontrol

Open-loop

control

y

k m o

si

Q

Induction

machine

Currentcontrol

k

software

y y y k y

y y y k

y

x y

Possibleco-ordination:

y

y y

modulation

Coordinatetransform.

y

y

Grid

gfGridvoltage

Coordinatetransform.

Figure 2.4: Block diagram of a possible vector control system for a double-sided PWMconverter.

Vector control often uses PI-controllers with various feed-forward terms inorder to improve dynamic response and to reduce cross coupling betweenflux/torque and active power/reactive power. These feed-forward terms areoften in the current control and flux control and are referred to as voltageand current decoupling, respectively. Feed-forward terms may also be usedto obtain a better co-ordinated control of the double-sided PWM converter.Especially the direct voltage control of the grid connected PWM converter canbe greatly improved during large torque transients by using a feed-forwardterm that is proportional to the induction machine loading [4],[8].

2.5.1 Two Induction Machine Control Principles

Volts-per-Hertz Control

Volts-per-Hertz control, or V/f-control, is a widely used induction machinecontrol principle. The control performance of Volts-per-Hertz control is sat-isfactory for many applications. The underlying principle of Volts-per-Hertzcontrol is that the stator frequency is varied close to the motor speed. Thestator flux can be maintained at its rated value by varying the stator voltageamplitude linearly with the stator frequency since the stator flux is propor-tional to the stator voltage divided with the stator frequency in steady state.A speed sensor is seldom required for applications with moderate performancedemands. Slip frequency compensation is often used to obtain the desired

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(fixed)

d

q

k

k

sd

sdi

sqi

sisu

Figure 2.5: Rotating two-axis flux orientated reference frame.

speed regardless of the loading torque.

The torque is proportional to the slip if the flux is maintained constant.However, the slip frequency must not exceed the pull-out slip frequency inorder to maintain stable operation. The induction machine equations were

linearized in [34] and it was shown that there are regions in the magnetizingcurrent-stator frequency plane and in the stator voltage-stator frequency planewhere Volts-per-Hertz control becomes unstable if [34] m/r < 0.5, where mis the mechanical time constant and r is the open circuit rotor time constant.Due to the instability problems, Volts-per-Hertz control is mainly suitable forapplications with slow dynamic loads, although a higher value of the leakageinductance L tends to reduce the instability problem [34]. Furthermore,Volts-per-Hertz control is likely to provide a reduced peak torque for speedsless than 5 Hz [1]. This is due to the difficulty in maintaining constant stator

flux at low speeds due to stator resistance voltage drop. A voltage boost isnormally added in order to attempt to compensate for the stator resistancevoltage drop.

Vector Control

The principle of vector control was presented by Blasche [13] and Hasse [14]in the late 1960s. Vector control is a relatively complex control methodand requires more computational power and more and better sensors than

open-loop Volts-per-Hertz control. Vector control exists in many differentvariants. Generally though, vector control implies that the torque and theflux of the induction machine are controlled in a manner that is similar to adc-machine with a separate field winding. Vector control is implemented ina rotating two-axis reference frame, see Fig. 2.5, that is synchronized withone of the induction machine flux linkages. The rotating two-axis referenceframe is often referred to as a flux orientated reference frame. Vector controlregulates the length of the flux vector and the length and position of the statorcurrent vector in the flux orientated reference frame. The stator current canbe divided into two orthogonal components which correspond to currentsthat produce flux and torque. The two current components are similar tothe field current and the armature current of a separately magnetized dc-

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machine. The orthogonal current components allow independent, or nearlyindependent, control of torque and flux.

The current controller in a vector control scheme is often implemented in the

flux orientated reference frame as well. This is because the fundamental ACvariables in the stationary reference frame become DC variables in the fluxorientated reference frame in steady state. The DC variables allow accuratetracking of the current reference signals and the performance of the currentcontroller becomes independent of frequency. Fig. 2.6 shows the typical blockdiagram of a PI current controller implemented in a flux orientated referenceframe, i.e. a synchronous PI-controller. The current control is the fast innerloop in a cascaded vector control system. Flux control and speed control areslower outer loops that generate the reactive and active current references,respectively.

Flux orientation can be accomplished by using either direct or indirect fluxorientation. Direct flux orientation methods use an observer to estimate theflux and directly track the flux position. Indirect rotor flux orientation re-quires, at least theoretically, no flux observer and tracks the rotor flux positionindirectly from the rotor position and the slip angle.

Rotor flux orientation and stator flux orientation are two common flux ori-entation methods. Only rotor flux orientation has the advantage of havingfully decoupled control of the torque and rotor flux during perfect rotor flux

orientation, i.e., a torque step does not affect the rotor flux and the rotor fluxcan be controlled via an open loop. For stator flux orientation, there existsa cross coupling between the torque and the stator flux even during perfectflux orientation. The cross coupling in stator flux orientation can be handledwith a decoupler. Stator flux orientation has a slightly higher complexitythan rotor flux orientation due to the cross coupling and decoupled controlof torque and flux cannot be guaranteed for stator flux orientation. Further-more, stator flux orientation with simple stator flux feedback often has poorlow speed properties due to high sensitivity to the stator resistance estimate

3i2i1i

123

dqi

'qu

'du

Voltage

decoupling

qu

du

dq

123

iqi

+

+

u

u

1u

2u

3u

di

qi

di

)cos(

)sin(

PI

PI

Three-phase

system

(fixed)

Two-phase

system

(fixed)

z { } ~ } } { ~ }

{ } ~ }

~

Figure 2.6: PI current controller implemented in a flux orientated reference frame.

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and sensitivity to measurement offsets. However, stator flux orientation canstill be desirable due to low parameter sensitivity at medium to high speedsand good torque capability in the field weakening region [18].

The success of vector control depends on the accuracy of the feedback flux sig-nal. The feedback flux signal consists of the flux position and the feedback canalso contain the flux amplitude. Errors in the estimated flux amplitude de-grade the accuracy of torque control and affect the losses of the machine. Thelosses are affected since an incorrect flux amplitude estimate either increasesthe core losses or the resistive losses in the machine due to higher slip [ 35].An incorrect flux position feedback decreases the dynamic performance of thetorque control and flux and torque are then no longer decoupled.

Very early vector control schemes [13] obtained the flux feedback by measur-ing the flux linkage with Hall sensors in the airgap or by using separate fluxcoils in the induction machine. However, these two methods resulted in amore complicated machine design and reduced reliability. The flux feedback,today, is therefore estimated from machine equations instead. The accuracyof the estimation depends on the accuracy of the induction machine param-eters. The parameters are likely to vary due to saturation and temperaturevariations and some kind of on-line parameter correction may be required,at least for accurate torque control. Indirect rotor flux orientation is, above

all, dependent on the open circuit rotor time constant. On the other hand,the traditional direct stator flux orientation method has a dependency onthe stator resistance estimate. The dependency on the stator resistance ishigh at low frequencies but low at high frequencies. Therefore, direct statorflux orientation usually does not operate well at low frequencies. To overcomethese problems, indirect and direct methods can be combined by using specialalgorithms to obtain better accuracy [20], [19] in a wide operating region. Ifthe flux feedback signal is sufficiently accurate, though, vector control offersfaster and more accurate torque control, higher peak torque at low speeds,wider speed range and greater stability than Volts-per-Hertz control.

2.5.2 Two Control Strategies for a Grid Connected PWM Con-verter

Amplitude-Phase Control

Amplitude-phase control is based on the stationary circuit in Fig. 2.7. Con-trolling the converter voltage amplitude Ug and phase provides a methodfor controlling the active and reactive power since

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30/145

a faster and less oscillative dynamic response than power angle control sinceactive power and reactive power are controlled independently [3].

(fixed)

d

q

gf

gf""

gd

gdigqi

gigqegu

Figure 2.8: Rotating two-axis grid flux orientated reference frame.

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Chapter 3

Stator Flux Orientated Vector Control

of an Induction Machine

The objective of this chapter is to derive controllers for stator flux regulationand for current control during stator flux orientation. The stator flux con-troller and the current controller become PI-controllers that are implementedin the stator flux orientated reference frame. Current and voltage decouplingterms for the controllers are also described. Transient models of the inductionmachine, internal model control [19] and characteristics of stator flux orienta-tion are described in order to prepare for the design of the controllers. In thefollowing, all variables refer to the stator side of the induction machine andmotor references are used. The parameters and base values of the induction

machine in Appendix E are used in all graphs. Appendix C describes the perunit system that is used in this thesis.

3.1 Transient Model of an Induction Machine

The induction machine is the most common of electric machines, mainly ow-ing to the fact that machines with cage rotor windings are simple and rugged,and therefore require less maintenance. The induction machine has also ahigh torque per inertia ratio and permits fast torque control. Induction ma-chines used in demanding adjustable speed applications can be expected tobe started, stopped, braked and possibly reversed frequently. A transientmodel of the induction machine is therefore required in order to obtain goodmodeling accuracy. In the following, two equivalent transient models will bedescribed. The transient performance is described by using the space vectortheory [36]. It is assumed that the following simplifications are valid

1. Space harmonics of the flux linkage distribution can be neglected.

2. Linear magnetization characteristic or constant saturation.3. No iron losses in the machine.

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4. No temperature nor frequency dependency of the resistances and induc-tances.

Appendix B describes the construction of a space vector as well as coordinate

transformations between stationary and rotating reference frames.

3.1.1 Transient T-Model

Fig. 3.1 shows a transient equivalent circuit of an induction machine. Thismodel is commonly referred to as the transient T-model since the three induc-tances form a T. The electrical equations of the T-model in space vectorform and in an arbitrary reference frame k are given by

u

k

s = Rsi

k

s +

dks

dt + jkk

s (3.1)

0 = Rrikr +

dkrdt

+ j(k )kr (3.2)

ks = Lsi

ks + Lmi

kr (3.3)

kr = Lmi

ks + Lri

kr (3.4)

In addition, the mechanical subsystem can be written as

Te =3

2

pIm

{

sis}

=

3

2

pIm

{

rir}

(3.5)

J

p

d

dt= Te Tl (3.6)

where the latter equation assumes a stiff rotor shaft. The electro-mechanicaltorque can be written in several other ways. Only the two expressions thatcan be derived directly from (3.1) and (3.2) are included here. Appendix Bderives (3.5) analytically.

The T-model is more complex than necessary for modelling purposes. Since

the induction machine can be represented by two complex state variables,

sR

ksu

ksi

rR kri

kmi

dt

d kr

dt

dLm

krk )(j

dt

d ks

kskj dt

dLs

dt

dLr

+ +

Figure 3.1: T-form transient equivalent circuit in an arbitrary reference frame.

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three inductances (derivatives) is one too many. It is therefore suitable totransform the T-model into equivalent but less complex models that onlyuse two inductances. In the opinion of the author, the equivalent modelswith only two inductances are easier to work with analytically and provide a

simpler representation of the induction machine. Two commonly used modelsare the -model and the inverse -model [37]. Only the inverse -model willbe described in the following since this model seems to be especially suitableto work with.

3.1.2 Transient Inverse -Model

The underlying idea of the inverse -model, shown in Fig. 3.2, is that actualrotor variables are not required. The non-actual rotor variables can therefore

be defined by choosing an arbitrarily stator-to-rotor turns ratio. When usingthe inverse -model, a stator-to-rotor turns ratio is chosen so that the rotorand stator leakage inductances are combined into one equivalent stator leakageinductance. The model is named for the fact that the two inductances forman inverse . The following turns ratio is introduced in order to derive theinverse -model

=LmLr

(3.7)

This choice of turns ratio gives the following rotor flux linkage for the inverse

-model kR = kr (3.8)

Although actual rotor variables are not required, the torque of the T- andthe inverse -models must be the same. This implies that the rotor currentof the inverse -model is

ikR =1

ikr (3.9)

The mutual inductance of the inverse -model is defined by

k

R = (Lmik

s + Lrik

r) = Lm(ik

s + ik

R) = LM(ik

s + ik

R) (3.10)

sR

ksu

ksi

kMi

RR kRi

dt

dL

dt

dLM

dt

d ks

kskj

kRk )(j

dt

dkR

+ +

Figure 3.2: Inverse transient equivalent circuit in an arbitrary reference frame.

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The stator inductance should not change between the two models, hence, theleakage inductance becomes

L = Ls

LM = Ls = Ls + Lr

Ls + Lr (3.11)

The rotor resistance RR of the inverse -model remains to be determined.Starting with the rotor voltage equation of the T-form circuit, the rotor re-sistance is defined by

0 = Rrikr +

dkrdt

+ j(k )kr = 2RrikR +dkR

dt+ j(k )kR

= RRikR +

dkRdt

+ j(k )kR (3.12)

In summary, the transformation from the T-model to the inverse -model is

=LmLr

(3.13)

kR =

kr (3.14)

ikR =ikr

(3.15)

L = Ls = Ls + Lr (3.16)

LM = Lm (3.17)

RR = 2Rr (3.18)

The stator voltage, stator current, stator resistance and stator inductanceremain the same for both models. This gives the inverse -model in the spacevector form and in an arbitrary reference frame as

uks = Rsiks +

dksdt

+ jkks (3.19)

0 = RR

ikR

+dkR

dt+ j(

k )k

R(3.20)

ks = Lsi

ks + LMi

kR (3.21)

kR = LM(i

ks + i

kR) =

ks Liks (3.22)

Te =3

2pIm{sis} =

3

2pIm{RiR} (3.23)

3.2 Internal Model Control

Harnefors [19, pages 4752] has proposed using internal model control forcurrent regulation of induction machines. The following text provides a veryshort description of internal model control. In the following sections, internal

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model control will be used for designing current and stator flux controllers.The structure of internal model control is shown in Fig. 3.3. In this figure, Fis a classical controller, C is the so called internal model controller while Gand G are the plant and plant models, respectively. It is assumed that G isa 22 matrix with no zeros in the right half plane. The closed-loop transferfunction for a perfect plant model, G = G, becomes

Gc|G=G = GC (3.24)An idea that is not far-fetched would be to choose the optimal internal modelcontroller C = G1, resulting in a closed-loop transfer function of Gc =I. However, this choice has little practical use since it introduces a highsensitivity to plant model errors, and the control signals u will be very large.In addition, implementing the optimal controller C = G1 requires that theorder of the G1 denominator is higher than the order of the G1 nominator,i.e., G1 must be proper. Luckily, a proper controller can be obtained bydetuning the optimal controller with a low-pass filter Glp as

C = G1Glp = G1

1p + 1

0

02

p + 2

(3.25)The following controller is obtained if 1 and 2 are chosen equal

F = [I CG]1C = 1 p +

1 p +

G1 = pG1 (3.26)

The closed-loop transfer function for a perfect plant model, G = G, butdetuned optimal controller now becomes

Gc = G

G1Glp|G=G =

p +

1 00 1

(3.27)

Eqs. (3.26) and (3.27) are highly interesting since they imply that it is possibleto design a current controller that is directly given by machine parameters

+

G

y yue

+

+

F

C G

Figure 3.3: Structure of internal model control.

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and the desired closed-loop system bandwidth . It is therefore believed thatone of the main advantages of internal model control is the ease with which acontroller can be designed. However, manual tuning or control modificationsmay be required since internal model control is dependent on plant model

accuracy.

For a system with no complex pole-pairs, a diagonal controller is obtainedwhen implementing internal model control. For a system with complex pole-pairs, such as an induction machine modeled in a flux orientated referenceframe, a non-diagonal controller is obtained. This means that the controlleris attempting to cancel the complex pole pair. It was shown in [19] thatexact cancellation is not possible and the controller can introduce currentoscillations. Decoupled internal model control was proposed as a remedyfor this. Decoupled internal model control implies in essence that the crosscoupling between axes is decoupled by a separate feedback loop. The crosscoupling can therefore be neglected when applying internal model control anda diagonal controller is obtained.

3.3 Some Characteristics of Stator Flux Orientation

This section describes four important characteristics of stator flux orienta-tion. These four characteristics are cross coupling between active current andstator flux, torque pull-out, field weakening operation and low speed opera-tion. The characteristics of stator flux orientation are also compared with thecharacteristics of rotor flux orientation.

3.3.1 Cross Coupling between Active Current and Stator Flux

When performing rotor flux orientated vector control, the torque and rotorflux are ideally truly decoupled. For stator flux orientated vector control,

though, a cross coupling between the active stator current and stator flux ispresent. A decoupler can be used to minimize the cross coupling [16]. Even-tually, this section derives a stator flux PI-controller and a current decouplerof slightly simpler design than in [16].

The rotor flux and the rotor current are expressed in terms of stator currentand stator flux in order to investigate the cross coupling between torque andstator flux

R = s

Lis (3.28)

iR =R

LM is = s

LM1 + L

LM

is =

s

LM Ls

LMis (3.29)

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Inserting these two equations into the rotor voltage equation (3.20) gives

dsfsdt

+ jslsfs +

1

r

sfs = L

disfsdt

+ jslisfs +

Lsr

isfs (3.30)

The slip speed sl and the open circuit rotor time constant r are defined by

sl = sf (3.31)r =

LrRr

=LMRR

(3.32)

Solving (3.30) in steady state gives the following expressions for the reactiveand active stator currents

isfsd =1 + (slr)

2

1 + (slr)2

sfsd

Ls(3.33)

isfsq =(1 )slr1 + (slr)2

sfsdLs

(3.34)

where the total leakage factor is equal to

= 1 L2m

LsLr= 1 LM

Ls(3.35)

It can be shown that when combining (3.33) and (3.34), and observing that

s/sp = rsl, the following stator current will be obtained in steady state [38]

isfs = isfsd + ji

sfsq =

sfsdLs

1 +

2 1 +

2ej2atan

ssp

(3.36)

This is the equation for the circle diagram of an induction machine. A con-siderable amount of information can be extracted from the circle [ 39]. Thediagram should be used with some care though since saturation is not consid-ered. The reactive current, required to maintain a constant stator flux linkage,is plotted against the active current for different slip speeds in Fig. 3.4. The

parameters in Appendix E are used and it is assumed that sf

sd=1 p.u. Thenominal slip speed of this machine is sl,n = 0.012 p.u. It can be observedthat the cross coupling is rather strong and a stator flux controller is recom-mended to maintain constant stator flux. Open-loop regulation of the flux,which is possible for rotor orientation [40], is probably not suitable for sta-tor flux orientation. Stator flux orientation therefore has a slightly highercomplexity than rotor flux orientation.

3.3.2 Torque Pull-Out

It was pointed out in [16] that stator flux orientation has two torque break-down points. The following derives these two breakdown points.

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In steady state, or for a case in which the rotor flux is constant during rotorflux orientation, the rotor voltage equation (3.20) simplifies to

0 = RRirfR + j(s

)rfR (3.37)

Identifying the imaginary part of (3.37) yields the following relation betweenrotor flux and active rotor current.

irfRq = ssRR

rfRd (3.38)

where s is the slip, defined by

s =s

s(3.39)

Eqs. (3.23) and (3.38) imply that the steady state electro-mechanical torquein a rotor flux orientated reference frame can be written as

Te = 32prfRdi

rfRq = [steady state] =

3

2p

ssRR

(rfRd)2 =

3

2p

ssL2M

RR(irfsd)

2 (3.40)

The electro-mechanical torque for rotor flux orientation at a given statorfrequency has hence, ideally, a linear dependency on the slip. For stator fluxorientation, though, it is required to eliminate the rotor flux from ( 3.40) in

order to derive a similar expression to (3.40). Equation (3.37) can then berewritten as

0 =RRLsLLM

rfR

RRL

rfs + jss

rfR

= LMLs

rfs +

1 + jssr

rfR (3.41)

Eq. (3.41) gives the following relation between rotor flux and stator flux insteady state

(rfRd)2 = |rfR |2 =

L2ML2s|1 + jssr|2

|rfs |2 =L2M

L2s

1 + (ssr)2 |rfs |2 (3.42)

The steady state torque for stator flux orientation becomes

Te =3

2p

ssL2M

RRL2s

1 + (ssr)2 |s|2 (3.43)

Note that (3.42) has been derived based on the assumption of constant rotorflux, which is not necessarily true for stator flux orientation. However, (3.43)implies that stator flux orientation has two extreme values (pull-out torques)

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0 1 2 3 4 52.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

Reactive current (p.u.)

Activecurrent(p.u.)

0.040.03

0.02

0.01

0

0.01

0.02

0.03

sl=0.04 p.u.

slp

slp

Figure 3.4: The circle diagram of the induction machine given in Appendix E, plotted forstator flux orientation in steady state.

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 14

3

2

1

0

1

2

3

4

Rotor speed (p.u.)

Torque(p.u.)

Stator flux reg.Rotor flux reg.

Figure 3.5: Torque as a function of rotor speed curves for stator flux and rotor flux orientation,plotted in steady state and below base speed.

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that are not present in rotor flux orientation. The pull-out slip speeds aregiven by

sps = 1r

(3.44)

This implies that the pull-out torques for stator flux orientation are

Tep = 32p

LM2L2s

|s|2 (3.45)

The pull-out torque has no dependency on the frequency, thus, peak torqueis provided from zero speed up to base speed. Using numerical values for themachine given in Appendix E and |s|=1 p.u., pull-out torques of1.86 p.u.are obtained. For this stator flux, the peak torque must be set below the

torque pull-out in order to avoid torque breakdown for stator flux orientation.Fig. 3.5 shows the electro-mechanical torque versus rotor speed curves forrotor flux and stator flux orientation. The curves are plotted below basespeed. Parameters of the machine given in Appendix E are used and it isassumed that |s|=1 p.u. and |R|=LM/Ls p.u.Note that (3.43)-(3.45) are the same equations that can be derived for aninduction machine directly connected to the grid if the stator resistance canbe neglected. The difference is that the stator flux can be selected arbitrarilyin an adjustable frequency drive.

3.3.3 Operation in Field Weakening Region

The induction machine is both current limited and voltage limited when oper-ating in the field weakening region. Maximum torque capability requires thatthe current limit and voltage limit are properly balanced. It was shown in [18]that stator flux orientation has good torque capability, i.e., peak torque, whenoperating in the field weakening region. Varying the stator flux reference pro-portional to 1/ gives nearly optimal torque capability [18]. Furthermore,stator flux orientation is parameter insensitive in the field weakening regionsince the back-emf dominates and the stator resistance voltage drop can nearlybe neglected. For rotor flux orientation, the torque capability is reduced whenselecting the rotor flux reference proportional to 1/ in the field weakening re-gion [18]. This is because field weakening operation is governed by the statorflux limit

||maxs |u|maxsbase

(3.46)

The rotor flux becomes too low when selecting the rotor flux reference propor-tional to 1/ which gives reduced torque capability at low speeds in the fieldweakening region. However, the optimal rotor flux can be calculated from theoptimal stator flux but this calculation can be complicated [18].

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3.3.4 Low Speed Operation

Stator flux orientation with the conventional open-loop voltage model for sta-tor flux feedback has poor low speed properties. This is because the stator

resistance voltage drop dominates over the back-emf at low speeds and thesensitivity to an incorrect stator resistance estimate becomes high. Further-more, the conventional open-loop voltage model is sensitive to measurementoffset and is often replaced with a delay element. The stator flux feedbackbecomes highly erroneous when operating at frequencies that are below thecut-off frequency of the filter. It seems that the minimum frequency of theconventional voltage model, and variants thereof, is about 5 Hz [40].

3.4 Stator Flux Orientated Stator Flux Controller

This section derives a stator flux PI-controller with a decoupling term. Eq. (3.30)is suitable to work with when designing the stator flux controller. Rearrang-ing (3.30) yields

sfs =

(p + jsl)r + 1

Ls

r(p + jsl) + 1isfs (3.47)

For perfect stator flux orientation, the stator flux linkage is aligned with thed-axis and sfsq = 0. Taking the real part of (3.47) gives

sfsd =Ls

rp + 1

(rp + 1)i

sfsd slrisfsq

(3.48)

The influence of the derivative operator in the numerator of (3.48) is probablyminor since the total leakage factor typically varies between 0.05 and up to0.2. Therefore, the following approximation to (3.48) should be reasonable

sfsd Ls

rp + 1isfsd slrisfsq

(3.49)

Fig. 3.6 shows the step response of the exact and approximate transfer func-tions, with parameters from Appendix E and = 0.073. The approximativetransfer function is quite accurate, at least for a step response.

Eq. (3.49) shows that the stator flux response time to a step in reactive currentisd is r, and for no-load conditions the steady state value of the stator flux isLsisd. Unfortunately, there is a cross coupling between the active current isqand the stator flux sd under loaded conditions, which was also shown inSection 3.3.1. It is, therefore, not trivial to obtain a decoupled control ofthe torque and stator flux during stator flux orientation since the stator fluxresponds to steps in active and reactive currents with the same time constant.A current decoupler is normally used to decouple the active current from the

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0 1 2 3 4 50

0.005

0.01

0.015

0.02

0.025

Time (s)

Amplitude

a)

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1x 10

4

Time (s)

Amplitude

b)

Figure 3.6: Comparison of exact, (rs + 1)/(rs + 1) (solid), and approximative, 1/(rs + 1)(dashed), transfer functions. (a) Step response. (b) Impulse response.

stator flux [16]. The slip speed is rewritten from the imaginary part (3.47) inorder to derive a suitable decoupler expression as

sl =(rp + 1)Lsi

sfsq

r(sfsd Lisfsd)

(3.50)

It is assumed that the derivative operator of the numerator can be neglected,i.e., the assumption that was applied to (3.48). Combining (3.49) and (3.50)gives

sfsd Ls

rp + 1

isfsd

L(isfsq )

2

sfsd Lisfsd

= G

isfsd

L(isfsq )

2

sfsd Lisfsd

(3.51)

The second term of (3.51) represents the cross coupling between active currentand stator flux. The cross coupling can be compensated with a separate loop,i.e., a so called current decoupler. The stator flux controller is now derivedby placing the closed-loop pole in the desired bandwidth , hence

F G

1 + F G=

p + =

/p

1 + /p(3.52)

F =

pG1 =

Lsp (rp + 1) (3.53)The controller becomes a conventional PI-controller directly parameterized inmachine parameters. The proportional gain and the integration time of thecontroller are

kp =rLs Ti = r (3.54)

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Combining the PI-controller with the current decoupler gives the followingreactive current reference

i

sf

sd = kp1 + 1Tip (sfsd sfsd ) + L(isfsq )

2

sfsd Lisfsd (3.55)The current decoupler is similar to the one in [16] but the current decouplerthat is developed here is of a slightly simpler design. It should be observedthat the decoupler is sensitive to the leakage inductance L. This is a draw-back since the magnetic saturation may vary during steps in active current,i.e., when the decoupler is needed most. The decoupler may also introduceadditional noise from the current sensors.

The selection of the stator flux reference depends on the application andthere are several possibilities for selecting the reference [41],[42], althougha common strategy is to select the stator flux reference equal to its ratedvalue for speeds lower than base speed and proportional to 1/ in the fieldweakening region [18], where is the rotor speed.

3.5 Generation of Active Current Reference

The active current reference is generated from an outer control loop, whichoften is a speed control loop. The design of a speed controller will not becovered here, since it would depend on the load characteristics. See [43, pages8990], [19, pages 7678] for details concerning speed controllers. Instead, thissection discusses how to generate correct torque during varying stator flux.

If a current decoupler is not used, or if a considerably detuned current decou-pler is used, one can expect undesirable stator flux variations. Furthermore,it is desirable to vary the flux in the field weaking region and also when max-imizing the efficiency [35, pages 388-396]. The following is often applied in

order to account for stator flux variations and to obtain correct torque

isfsq =2

3p

Tesfsd (3.56)Thus, an outer control loop gives a torque reference and the active currentreference is given by dividing the torque reference by the estimated statorflux. This method ideally ensures that the requested torque is obtained, butof course only when the stator flux feedback is correct. The importance of(3.56) should not be exaggerated, however. There is no need to introduce theadditional, possibly a bit computational expensive, division in (3.56) if thestator flux can be considered to be fairly constant.

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3.6 Stator Flux Control Without Reactive Current Con-

trol

A stator flux controller without reactive current control was presented in [44].The controller is mentioned here since its design differs significantly from thestator flux controller in the previous section. In the paper, the stator fluxcontroller was derived directly from the stator voltage equation as

usfsd = kp

1 +

1

Tip

(sfsd sfsd ) (3.57)

This controller does not suffer if the magnetizing curve of the machine isstrongly non-linear, which may be troublesome for the stator flux controller

described above. In the paper [44], dead-beat gains were used and the ex-perimental results were excellent. However, over-current protection is morecomplicated to implement since the controller has no reactive current control.

3.7 Stator Flux Orientated Current Controller

The objective of this section is to derive a synchronous PI-controller for sta-tor flux orientation. A suitable model of the induction machine is requiredwhen designing a current controller. Combining (3.19) and (3.20) yields the

following equation

usfs + Rsisfs + jsfsfs + Ldisdt

RRisfR jslsfR = 0 (3.58)

The rotor flux and rotor current in (3.58) should be eliminated when designinga stator flux orientated current controller for an induction machine with cagerotor windings. These variables can be substituted by using (3.28) and (3.29).Equation (3.58) can now be written, with some minor modifications, in thefollowing way that is useful for the design of a stator flux orientated current

controller

Ldisfsdt

+ jslLisfs +

Lsr

isfs = usfs jsfs +

1

r

sfs (3.59)

It is convenient to exclude some terms from (3.59) before continuing the designof the current controller. It is possible to exclude three terms from (3.59). Thefirst term that can be neglected is sfs /r. This term is ideally constant duringstator flux orientation so it can be compensated by intregator action. Thesecond term that can be neglected is jslLi

sf

s

. This term has been observedto be weak because the slip speed is normally low. However, some schemesuse a separate decoupler for this term, which may be relevant for applications

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where the slip speed is high and changes frequently. The third term that canbe neglected is the induced emf jsfs . This term can be compensated bythe controller integrator based on an assumption of slowly varying stator fluxand slowly varying rotor speed, seen from the electric time perspective. On

the other hand, it is easy to design a voltage decoupler for the induced emfif the assumption of slowly varying rotor speed does not apply. Nevertheless,the following is possible to write when excluding these three terms

isfs 1

pL +Lsr

usfs = Gusfs (3.60)

Internal model control can be applied for the design of the current controller.Placing the closed-loop pole in the desired bandwidth yields

F G

1 + F G=

p + =

/p

1 + /p(3.61)

F =

pG1 = L + Lsrp (3.62)

The controller F is an ordinary PI-controller. The reference voltage vectorbecomes

usfs = kp1 + 1Tip (isfs isfs ) (3.63)kp = L Ti = r (3.64)

The term jsfs can be added to the reference voltage if one wishes to use avoltage decoupler for the induced emf.

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Chapter 4

Stator Flux Observers

This chapter describes some different stator flux observers. A stator flux ob-server is required for vector control in order to derive stator flux feedbackwithout measuring the flux directly. The stringent meaning of the word ob-server is an estimator that uses both inputs to integration models and feedbackcontrol for error correction. However, the word observer is also frequently usedfor an observer that is based on integration models only, although a morecorrect term for such an observer would be a real-time simulation. Forconvenience, observers with feedback are in this chapter referred to as closed-loop observers while observers that only employ process models are referredto as open-loop observers. Only directly parametrized closed-loop observersare described in this thesis. A directly parametrized stator flux observer has

either a constant gain matrix or else the gain matrix is given by a simplefunction of rotor speed.

4.1 Open-Loop Stator Flux Observers

4.1.1 Voltage Model

The voltage model is a common open-loop stator flux observer. The modelis based on direct integration of the stator equation in the stator orientated

reference frame, given byss = t0

(uss Rsiss)dt + ss(0) (4.1)|ss| = (ss)2 + (ss)2 (4.2)

cos(

sf) =

ss|

s|

, sin(

sf) =

ss|

s|

(4.3)

The voltage model is a direct flux observer since the stator flux position isderived directly from the stator flux estimate. The commonly referred draw-back of the voltage model is its high sensitivity to an error in the estimated

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stator resistance at low speeds, which is illustrated in Fig. 4.1. Expressions ofthe relative magnitude error and the phase shift are given in Appendix D. Atlow frequencies, the resistive voltage drop dominates so that even the slightesterror in stator resistance estimation may be disastrous. Otherwise the voltage

model is not dependent on any other parameter and at high speeds the pa-rameter sensitivity is nearly non-existent. However, this observer can seldombe directly implemented even for a correct estimation of the stator resistance.The voltage model has namely two additional disadvantages due to direct in-tegration. These two disadvantages are, first, that the stator flux estimationwill drift for biased input signals and, second, the model will not converge tocorrect stator flux if the initial value ss(0) is incorrect. To overcome thesetwo problems, the voltage model is often implemented as a delay element,introducing other problems such as magnitude scaling and phase shift. A

fourth drawback of this model is the difficulty to measure the stator voltageuss. This drawback is due to the fact that the voltage changes very rapidlywhen using pulse width modulation, and aliasing is likely to occur. The alias-ing problem may be especially problematic during low-speed operation sincethe voltage pulses become very narrow. An entire pulse may be missed orextended with the sampling period time when sampling the narrow pulses.According to [44], the aliasing problem is normally solved by using a highorder analog low-pass filter or by using the stator voltage reference insteadof the measured stator voltage. Both of these two methods have drawbacks.

The bandwidth of the observer is limited when using a low-pass filter, andwhen using the reference voltage, an accurate compensation of the blankingtime and the voltage drop of the semiconductor valves is required. As analternative to these two methods, a partly implemented analogue integrator

0.001 0.01 0.1 1 2 30.6

0.7

0.8

0.9

1

1.1

Rotor speed (p.u.)

Relativemagnitudeerror

a)

0.001 0.01 0.1 1 2 35

0

5

10

15

20

Rotor speed (p.u.)

Phaseshift(de

g.)

b)

Figure 4.1: Relative magnitude error and phase shift of the voltage model at rated load andRs = 1.5Rs.

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was presented in [44].

4.1.2 Improved Integrator Algorithms for the Voltage Model

Hu et al. [45] have presented three improved integrator algorithms for thevoltage model. The algorithms are equivalent to direct integration but nearlyeliminate the problems with bias and incorrect initial value. The underlyingidea of the algorithms are shown in Fig. 4.2. This block circuit, referred toas algorithm 1, is equivalent to an integrator since the transfer function from(uss Rsiss) to ss becomes (neglecting the saturation block)

ss =uss

Rsiss

p + c

1

1c

p + c

=uss

Rsiss

p

(4.4)

The benefit of this algorithm is that the low-frequency gain is decreased andthe feedback path of ss, allows the use of a saturation block. The saturationblock is especially useful when the voltage model is combined with a stator fluxcontroller. The stator flux controller maintains constant stator flux and theamplitude of the stator flux estimate should not deviate significantly from thereference. A second integrator algorithm, algorithm 2, is s

otterstenrolflic

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