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POLITECHNIKA WARSZAWSKA

WARSAW UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering

ROZPRAWA DOKTORSKA Ph.D. Thesis

Marek Jasiński, M. Sc.

Direct Power and Torque Control of AC/DC/AC Converter-Fed Induction Motor Drives

WARSZAWA 2005

Warsaw University of Technology

Faculty of Electrical Engineering Institute of Control and Industrial Electronics

Ph.D. Thesis

Marek Jasiński, M. Sc.

Direct Power and Torque Control of AC/DC/AC Converter-Fed Induction

Motor Drives

Thesis supervisor Prof. Dr Sc. Marian P. Kaźmierkowski

Warsaw – Poland, 2005

Acknowledgements

The work presented in the thesis was carried out during author’s Ph.D. studies at

the Institute of Control and Industrial Electronic at the Warsaw University of

Technology, Faculty of Electrical Engineering and grant of the Ministry of Science

and Information Society Technologies. Some parts of the work were realized in

cooperation with foreign Universities and scientific organization:

University of Aalborg at Institute of Energy Technology, Denmark (Prof.

Frede Blaabjerg)

Nordic Network for Multi Disciplinary Optimised Electric Drives,

Denmark (Prof. Ewen Ritchie)

Politecnico di Bari (Prof. Marco Liserre)

First of all, I would like to express gratitude Prof. Marian P. Kaźmierkowski for the

continuous support and help. His precious advice and numerous discussions

enhanced my knowledge and scientific inspiration.

I am grateful to Prof. Andrzej Sikorski from the Białystok Technical University

and Prof. Roman Barlik from the Warsaw University of Technology for their interest

in this work and holding the post of referee.

Furthermore, I thank my colleagues from the Intelligent Control Group in Power

Electronics for their support and friendly atmosphere. Especially, to Dr. Mariusz

Malinowski, Dr Marcin Żelechowski, Dariusz Świerczyński M.Sc., and Patrycjusz

Antoniewicz M.Sc.

Finally, I am very grateful for my wife Agnieszka, daughter Maja, and son Mateusz

for their love, patience and faith. I would also like to thank to my whole family,

particularly my parents for their care over the years.

Contents

Page

1. Introduction............................................................................................................ 1 1.1. AC/DC/AC Converters ..................................................................................... 1

2. Voltage Source Converters – VSC...................................................................... 11 2.1. Introduction..................................................................................................... 11 2.2. Space Vector Based Description of VSC........................................................ 11 2.3. Operation of Voltage Source Converter – VSC.............................................. 12 2.4. Mathematical Model of the VSI - Fed Induction Motor (IM) ........................ 14

2.4.1. IM Mathematical Model in Rotating Coordinate System with Arbitrary Angular Speed.................................................................................................... 16 2.4.2. IM Model in Stationary αβ Coordinates ................................................ 16 2.4.3. IM Model in Synchronous Rotating dq Coordinates - RFOC ................ 18 2.4.4. IM Model in Synchronous Rotating xy Coordinates - SFOC................. 19

2.5. Operation of Voltage Source Rectifier – VSR................................................ 22 2.5.1. Operation Limits of the Voltage Source Rectifier – VSR ....................... 24 2.5.2. VSR Model in Three-Phase ABC Coordinates........................................ 27 2.5.3. VSR Model in Stationary αβ Coordinates............................................... 29 2.5.4. VSR Model in Synchronously Rotating xy Coordinates ......................... 30

2.6. Summary ......................................................................................................... 32 3. Vector Control Methods of AC/DC/AC Converter-Fed Induction Motor Drives – A Review .................................................................................................... 33

3.1. Introduction..................................................................................................... 33 3.2. Control Methods of VSI-Fed Induction Motor ............................................... 34

3.2.1. Field Oriented Control – FOC ................................................................. 34 3.2.2. Direct Torque Control – DTC.................................................................. 38 3.2.3. Direct Torque Control with Space Vector Modulation – DTC-SVM...... 43

3.3. Control Methods of VSR ................................................................................ 45 3.3.1. Virtual Flux Oriented Control – V-FOC.................................................. 45 3.3.1.1. Line Current Controllers ....................................................................... 47 3.3.2. VF based Direct Power Control – VF-DPC............................................. 52 3.3.3. Direct Power Control with Space Vector Modulator – DPC-SVM ......... 57 3.3.3.1. Line Power Controllers ......................................................................... 57 3.3.3.2. DC-link Voltage Controller .................................................................. 65

3.4. Conclusion ...................................................................................................... 68 4. Direct Power and Torque Control with Space Vector Modulation – DPTC-SVM........................................................................................................................... 69

4.1. Introduction..................................................................................................... 69 4.2. Model of the AC/DC/AC Converter-Fed Induction Motor Drive with Active power feedforward ................................................................................................. 69

4.2.1. Analysis of the Power Response Time Constant ..................................... 71

Contents

II

4.2.2. Energy of the DC-link Capacitor ............................................................. 71 4.2.2.1. Transfer Function of the AC/DC/AC Converter-Fed IM Drive with DC-link Voltage Feedback only – 0PF .................................................................... 74 4.2.2.2. Transfer Function of the AC/DC/AC Converter-Fed IM Drive with DC-link Voltage Feedback and Active Power Feedforward Calculated Based on Mechanical Speed, Commanded Torque, and Power Losses – ΩPF ................ 75 4.2.2.3. Transfer Function of the AC/DC/AC converter-Fed IM Drive with DC-link Voltage Feedback and Active Power Feedforward Calculated From Commanded Stator Voltage and Actual Stator Current - UIPF ......................... 75

4.3. Simulation Study............................................................................................. 76 4.3.1. Steady State Performances....................................................................... 76 4.3.2. AC/DC/AC Converter-Fed IM Drive Operated with Closed Torque Control Loop ...................................................................................................... 79 4.3.3. AC/DC/AC Converter-Fed IM Drive Operated with Closed Speed Control Loop ...................................................................................................... 83

4.4. Conclusion ...................................................................................................... 91 5. Passive Components Design – DC-link Capacitor ............................................ 93

5.1. Introduction..................................................................................................... 93 5.2. Selection of Filter Components....................................................................... 93

5.2.1. Nominal Voltage of the DC-link Capacitor ............................................. 93 5.2.2. Ripple Current Consideration .................................................................. 95 5.2.3. Ratings of the DC-link Capacitor............................................................. 97 5.2.3.1. Consideration of Operation with Reduced DC-link Capacitor ........... 102

5.3. Conclusion .................................................................................................... 104 6. Simulation an Experimental Results ................................................................ 105

6.1. Introduction................................................................................................... 105 6.2. Steady States Operation ................................................................................ 105 6.3. Active and Reactive Power Controllers ........................................................ 108 6.4. AC/DC/AC Converter-Fed IM Drive Operated with Closed Torque Control Loop ..................................................................................................................... 109 6.5. AC/DC/AC Converter-Fed IM Drive Operated with Closed Speed Control Loop ..................................................................................................................... 112 6.6. Conclusion .................................................................................................... 122

7. Summary and Conclusion ................................................................................. 123 References ............................................................................................................... 126 Symbols Employed................................................................................................. 136

Main Symbols ...................................................................................................... 136 Rectangular Coordinates System ......................................................................... 139 Indices .................................................................................................................. 140 Mathematical symbols ......................................................................................... 140 Abbreviations ....................................................................................................... 140

A. Appendices ......................................................................................................... 141 A.1. Space vector in coordinate systems.............................................................. 141

A.1.1. Fixed System of Coordinates - αβ ....................................................... 141 A.1.2. Rotating System of Coordinates............................................................ 141 A.1.3. Model of the Induction Motor in Natural ABC Coordinates ................ 143

A.2. Coordinate Transformation .......................................................................... 145 A.2.1. Three-Phase to Two-Phase Conversion (ABC/αβ ) ............................. 145 A.2.2. Two-Phase to Three-Phase Conversion (αβ /ABC)............................. 145

Contents

III

A.2.3. Rectangular to Rectangular Coordinate Conversion (αβ /xy) and (xy/αβ )............................................................................................................ 145

A.3. Apparent, Active, and Reactive Power ........................................................ 146 A.3.1. Complex Representation of the Power.................................................. 146

A.4. Simulation Model and Laboratory setup...................................................... 148 A.4.1. Saber Model .......................................................................................... 148 A.4.2. Matlab Simulink Power Toolbox Model............................................... 149 A.4.3. Laboratory setup.................................................................................... 150 A.4.4. List of Equipment.................................................................................. 154

Chapter 1

1. Introduction

1.1. AC/DC/AC Converters

AC/DC/AC converters are part of a group of AC/AC converters. Generally

AC/AC converters take power from one AC system and deliver it to another with

waveforms of different amplitude, frequency and phase. Those systems can be single

phase or three phase. The major application of voltage source AC/AC converters are

adjustable speed drives – ASD [15], [63], [65], [140].

The most used voltage source AC/AC converters utilize a DC-link between the

two AC systems as presented in Fig.1. 1a,b, and provide direct power conversion as

in Fig.1. 1c.

~3 ~3 ~3

Fig.1. 1. Chosen AC/AC converters for adjustable speed drives – ASD; a) with diode rectifier, b) with voltage source rectifier – VSR, c) direct converter (matrix or cycloconverter) [67].

Where VSI – voltage source inverter, IM – induction motor, PWM – pulse width modulation [47]

In AC/DC/AC converter the input AC power is rectified into a DC waveform and

then is inverted into the output AC waveform. A capacitor (and/or inductor) in DC-

1. Introduction

2

link stores the instantaneous difference between the input and output powers. AC/DC

and DC/AC converters can be controlled independently.

The matrix converter (cycloconverter) avoids the intermediate DC-link by

converting the input AC waveforms directly into the desired output waveforms

(Fig.1. 1c) [21], [58].

Although a three-phase induction motor was introduced more than one hundred

years ago, the research and development – R&D in this area appears to be never-

stopping. Moreover, the new power semiconductor devices and power electronics

converters are developing in last twenty/thirty years even faster. The introduction of

IGBTs in the mid of 80s was an important milestone in the history of power

semiconductor devices. Similarly, digital signal processors – DSP developed in 90s

were a milestone in implementation and applications of advanced control strategies

for power converter drives [1], [15], [25], [27], [70], [98], [104], [131], [151]. As a

result ASD systems are widely used in applications such as pumps, fans, paper and

textile mills, elevators, electric vehicles and underground traction, home appliances,

wind generation systems, servo drives and robotics, computer peripherals, steel and

cement mills, ship propulsion, etc. [15]. Nowadays, most of ASD consist of

uncontrollable diode rectifier (Fig.1. 1a) or a line commutated phase controlled

thyristor bridge. Although both these converters offer a high reliability and simple

structure, they also have serious disadvantages. The DC-link voltage of the diode

rectifier is uncontrolled and pulsating; therefore bulky DC-link capacitor and usually

DC-choke are needed. Moreover, the power flow is unidirectional and the input

current (line current) is strongly distorted [36], [42], [43], [106], [135]. The last

drawback is very important because of standard regulation such as IEEE Std 519-

1992 in the USA and IEC 61000-3-2/IEC 61000-3-4 in UE. Even small power ASD

can cause a total harmonics distortion – THD problem for a supply line when a large

number of nonlinear loads are connected to one point of common coupling – PCC

[8], [54]. Tab. 1. 1 lists the harmonic current limits based on the size of a load with

respect to the size of line power supply. The ratio of LmSC I/I is the ratio of short-

circuit current SCI available at the PCC, to the maximum fundamental load current

LmI . It is recommended that the load current LmI , should be calculated as the average

current of the maximum demand over a year [54].

1. Introduction

3

Tab. 1. 1. Current Distortion Limits for General Distribution Systems (up to 69 kV) Where:

TDD – is the total demand distortion (root-sum-square – RSS) [54]

The recommended voltage distortion limits, usually expressed by THD index, is

shown in Tab. 1. 2. Where, THD – is total (root-sum-square – RSS) harmonic

voltage in percent of nominal fundamental frequency voltage. This term has come

into common usage to define either voltage or current distortion factor – DF (Eq.(1.

1)). The DF: is the ratio of the RSS of the harmonic content to the root-mean-square

– RMS value of the fundamental quantity, expressed as a percent of the fundamental

[54]:

( )

( )%THD

L

hL

100U

U2

1

50

2h

2∑== (1. 1)

Tab. 1. 2. Voltage distortion limits [54]

Some types of electronic receiver can be affected by transmission of AC supply

harmonics through the equipment power supply or by electromagnetic coupling of

harmonics into equipment components (electromagnetic interference – EMI

problem). Computers and associated equipment such as programmable controllers

frequently require AC sources that have no more distortion than a 5% THD, with the

largest single harmonic being no more than 3% of the fundamental. Higher levels of

harmonics result in erratic, sometimes subtle malfunctions of the equipment that can,

1. Introduction

4

in some cases, have serious consequences [108]. Also, instruments can be affected

similarly. Perhaps the most serious of these are malfunctions in medical instruments.

Consequently, many medical instruments are provided with special power electronics

devices (line-conditioners). Here is a wide application field, especially, for

AC/DC/AC converters, such as uninterruptible power supplies – UPS systems. Less

dramatic interference effects of harmonics can be observed in audio and video

devices [54].

Therefore, a lot of methods for elimination of harmonics distortion in the power

system are developed and implemented [100], [123]. Moreover, several blackouts in

recent years (USA and Canada (New York, Detroit, Toronto) in 08.2003, Russia

(Moscow) in 05.2005, USA (Los Angeles) 09.2005), and high prices of the oil shows

that the idea of “clean power” is more and more up-to-date.

Harmonics reduction methods can be divided into two main groups (Fig.1. 2):

a) passive filters and active filters – harmonics reduction of the already

installed nonlinear loads,

b) multi-pulse rectifiers and VSR (active rectifiers) – power-grid friendly

converters (with limited THD) [8], [73].

Fig.1. 2. Chosen harmonics reduction techniques; where CSR – is current source rectifier

Furthermore, the energy saving is important because, VSR assures regenerating

braking with energy saving capability [71] as well as after minor modification active

filtering function can be implemented [2], [3], [24], [154], [166].

Typical application of the VSR is like in Fig.1. 1b. Thanks to, systematical cost

reduction of the IGBTs and DSPs there have appeared on the market serially

produced VSR from few kVA up to MVA range. An individual VSR can provide the

1. Introduction

5

DC-link voltage to several VSI-fed IM (for cost reduction) [148]. Moreover, VSR

can compensate the nonlinear load’s current connected in parallel with VSR to PCC.

In the thesis author is focused on three-phase AC/DC/AC converter consisted of

two identical voltage source converters – VSC, insulated gate bipolar transistors –

IGBT bridges as in Fig.1. 1b. First of them (at the line side) works as an voltage

source rectifier – VSR feeding the DC-link circuit, whereas the second (at the motor

side) operates as an voltage source inverter – VSI feeding induction motor – IM.

Sometimes, VSR is called active rectifier, PWM rectifier or active-front-end [15].

Generally the high-performance frequency controlled AC/DC/AC converter fed

IM drive should offer following features and abilities:

On the VSI-fed IM side:

• four-quadrant operation,

• fast flux and torque response,

• maximum output torque available in wide range of speed operation,

• constant switching frequency,

• uni-polar voltage PWM, thus lower switching losses,

• low flux and torque ripple,

• wide range of speed control,

• robustness to parameter variations,

On the VSR-fed DC-link side:

• bi-directional power flow,

• nearly sinusoidal input current (low THD typically below 5%),

• controllable reactive power (up to unity power factor – UPF),

• controllable DC-link voltage (well stabilized at desired level),

• reduction of DC-link capacitor,

• insensitivity to line voltage variations [88], [122], [139],

• reduction of transformer and cable cost due to UPF.

These features depend mainly on the applied control strategy. The main goal of

the chosen control strategy is to provide optimal parameters of ASD concurrently,

with reduction of the cost and maximal simplification of the whole system.

Moreover, robustness of the control system is very important.

IM control methods can be divided into scalar and vector control. The general

classification of the variable frequency methods is presented in Fig.1. 3. Following

1. Introduction

6

the definition from [65] we may say that: “in scalar control, which is based on

relationships valid for steady state, only magnitude and frequency (angular speed) of

voltage, current and flux linkage space vectors are controlled. As result, the scalar

control does not act on space vector position during transients. Contrarily, in vector

control, which is based on relations valid for dynamic states, not only magnitude and

frequency (angular speed) but also instantaneous positions of voltage, current and

flux space vectors are controlled. Thus, the vector control acts on the positions of the

space vectors and provides their correct orientation both in steady state and during

transients”.

Fig.1. 3. Classification of IM control methods; where NFO is the natural field orientation [65]

Therefore, vector control is a general control concept that can be implemented in

many different ways. The most known method, called field oriented control – FOC

[13], [26], [100], [125] or vector control [15], has been proposed by Hasse (Indirect

FOC) and Blaschke (Direct FOC) [11] (see also [15], [63], [140]), and gives the

induction motor good performance [33]. In the FOC the IM equations are

transformed into rotor flux vector oriented coordinate system [15], [63], [62], [134],

[140]. In rotor flux vector oriented coordinates (assumed constant rotor flux

amplitude) there is a linear relationship between current vector components and

motor torque. Moreover, like in a DC motor, the flux reference amplitude is reduced

in the field-weakening range in order to limit the stator voltage typically at higher

1. Introduction

7

then nominal speed [15], [63], [121], [140]. IM equations represented in the flux

vector oriented coordinates have a good physical basis because they correspond to

the decoupled torque generation in separately excited DC motor. Nevertheless, from

the theoretical point of view, another types of mathematical transformations can be

chosen to achieve decoupling and linearization of IM equations. That methods are

known as modern nonlinear control [58]. Marino et al. and Krzeminski (see

Kazmierkowski [65]) have proposed a nonlinear transformation of the motor state

variables so that, in the new coordinates, the speed and rotor flux amplitude are

decoupled by feedback; the method is called feedback linearization control – FLC

[56]. Also, method based on the variation theory and energy shaping, called passivity

based control – PBC has been recently investigated [61].

In the mid 1980s, there was a trend toward the standardization of the control

systems on the basis of the FOC methodology. However, Depenbrock (see [18]),

Takahashi and Nogouchi [126] have presented a new strategy, which abandon an

idea of mathematical coordinate transformation and the analogy with DC motor

control. These authors proposed to replace the averaging based decoupling control

with the instantaneous bang-bang control, which very well corresponds to on-off

operation of the VSI semiconductor power devices [129]. These strategies are known

as direct torque control – DTC. Since 1985 the DTC has been continuously

developed and improved by many researchers [5], [6], [18], [34], [35], [74], [75],

[102], [127], [140], [149]. Among the main advantages of DTC scheme are: simple

structure, good dynamic behavior and is inherently a motion-sensorless control

method. However, it has a very important drawbacks i.e. variable switching

frequency [132], [133], high torque pulsation, unreliable start up, and low speed

operation performance [80], [144]. Therefore, to overcome these disadvantages a

space vector modulator – SVM was introduced to DTC structure [18], [38], [101]

giving DTC-SVM control scheme [124]. In this method disadvantages of the

classical DTC are eliminated [152].

However, it should be pointed that no commonly shared terminology exists

regarding DTC and DTC-SVM. From the formal considerations DTC-SVM can also

be called as stator field oriented control – SFOC, (Fig.1. 3). In the thesis DTC, and

DTC-SVM scheme will refer to control schemes operating with closed torque and

flux loops without current controllers [18].

1. Introduction

8

Control of the VSR can be considered as a dual problem with vector control of an

induction motor (see Fig.1. 4). The simples scalar control is based on current

regulation in three-phase system (AC waveforms) [15], [109].

Fig.1. 4. Classification of VSR control methods

Like for IM, vector control of VSR is a general control philosophy that can be

implemented in many different ways. The most popular method, known as voltage

oriented control – VOC [73], [83], [84], [85], [86], [87], [93], [97] gives high

dynamic and static performances via internal current control loops. In the VOC the

VSR equations are transformed in a line voltage vector oriented coordinate system.

In line voltage vector oriented coordinates there is a linear relationship between

current vector control components and power flow. To improve the robustness of

VOC scheme a virtual flux – VF concept was introduced by Duarte (see Malinowski

[93]). However, from the theoretical point of view, other types of mathematical

coordinate transformations can be defined to achieve decoupling and linearization of

the VSR equations. This has originated the methods known as nonlinear control.

Jung [56] and Lee [78] have been proposed a nonlinear transformation of VSR state

variables so that, in the new coordinates, the DC-link voltage and line current are

decoupled by feedback; this method is called also feedback linearization control –

FLC like for induction motor. Moreover, a passivity based control – PBC, as for IM,

was also investigated in respect to VSR [57], [65], [111].

In the mid of 1990s Manninen [95] and in the second part of 1990s, Nogouchi at

al. [105] have expanded the idea of DTC for VSR called direct power control – DPC

1. Introduction

9

[69]. From that time it has been continuously improved ([65], [93], [107]). However,

these control principles are very similar to DTC schemes for IM and have the same

drawbacks. Therefore, to overcome that disadvantages a space vector modulator –

SVM [50] was introduced to DPC structure [92] giving new DPC-SVM control

scheme [157]. Hence, presented DPC-SVM and DTC-SVM joins important

advantages of SVM (e.g. constant switching frequency, unipolar voltage pulses),

with advantages of DPC, and DTC (e.g. simple and robust structure, lack of internal

current control loops, good dynamics, etc.). However, when control structure of the

VSR operates independently from control of the IM, the DC-link voltage

stabilization is not sufficiently fast and, as a consequence a large DC-link capacitor is

required for instantaneous power balancing. Therefore, for speed up the DC-link

voltage dynamic an additional active power feedforward – PF loop from the VSI-fed

IM side to VSR-fed DC-link control is required. As result a direct power and torque

control with space vector modulation – DPTC-SVM scheme was obtained. This new

control scheme with PF loop allows to significant reduction of the DC-link capacitor

keeping fast instantaneous power balancing. In contrast to well discussed in literature

current control loops based AC/DC/AC converter control schemes (among other

VOC-FOC) [9], [10], [22], [28], [23], [32], [39], [40], [53], [66], [68], [78], [81],

[86], [89], [90], [120] the new DPTC-SVM scheme is not well known and published

literature is very limited. However, it seems to be very attractive for industrial

application. Therefore, this dissertation is devoted to analyze and study of DPTC-

SVM scheme with PF loop.

The author has formulated the following thesis: “The very convenient control

scheme for line power friendly adjustable speed AC/DC/AC converter-fed

drives from the point of view of industrial manufacturing is direct power and

torque control with space vector modulation DPTC-SVM scheme. Moreover, by

adding an active power feedforward – PF loop, the DC-link capacitor can be

considerably reduced”.

In order to proof the above thesis author had used an analytical and simulation

based approach, as well as experimental verification on the laboratory setup with 3

kW induction motor fed by 5kVA IGBT AC/DC/AC converter.

In the author’s opinion the following parts of the thesis are his original

contributions:

1. Introduction

10

• development (in Matlab Simulink as well as in a professional package Saber)

simulation algorithm for control and investigation of AC/DC/AC converters-

fed induction motor,

• elaboration and experimental verification of an active and reactive power

controllers design for DPC-SVM (see Chapter 3),

• elaboration and experimental verification of a novel active power

feedforward estimators (see Chapter 4),

• construction and experimental verification of the laboratory setup based on

mixed RISC/DSP (PowerPC 604/TMS320F240) controller, and 5kVA as

well as 7.5 kVA AC/DC/AC power converter-fed 3kW induction motor

drive.

The thesis consists of seven chapters. Chapter 1 is an introduction. Chapter 2 is

devoted to presentation of the voltage source converters – VSC. The mathematical

models of VSR and induction motor and operation description are also presented. In

Chapter 3 basic vector control methods of VSI-fed IM as well as VSR-fed DC-link

are reviewed. Moreover, analysis and synthesis of the controllers for VSR (for VOC

and DPC-SVM) is given. Chapter 4 presents the analysis and synthesis of the DPTC-

SVM control techniques. Also two active power feedforward concepts are described

and investigated. Chapter 5 discusses passive components design. Chapter 6 contains

experimental results and its study. Finally, Chapter 7 presents summary and

conclusion. The thesis is supplemented by Appendices consisted of space vector

principles (A.1), coordinate transformations (A.2), apparent, reactive and active

power definitions (A.3), simulation model, and laboratory setup description (A.4).

Chapter 2

2. Voltage Source Converters – VSC

2.1. Introduction

As can be seen from literature study the mathematical modeling of the controlled

object is very important for control structure [13], [19], [31], [46], [49], [48], [51],

[52], [55], [61], [77], [96], [110], [119], [141], [145], [146], [147], [150].

Therefore, in this chapter some principles of VSC operation will be discussed.

Also, mathematical models of voltage source rectifier - VSR as well as of voltage

source inverter - VSI leading to the whole AC/DC/AC converter model will be

presented.

2.2. Space Vector Based Description of VSC

Symmetric three-phase system is represented by phase quantities (natural

coordinate’s), such as voltages, currents and flux linkages. However, in such system

can be represented by one space vector of voltage, current, and flux linkage,

respectively [63], [99].

A

B

C

Re

Im

)t(kA

)t(kBa

)t(kC2a

k23

ka

2a

1

Fig. 2. 1. Construction of space vector according to definition Eq. (2. 1)


12

( ) ( ) ( )( )tktktk CBAdf

2

32 aak ++= (2.1)

Where:32 - normalization factor, 2 aa1 ,, - complex unity vectors, with phase shift,

( ) ( ) ( )tk,tk,tk CBA - denotes arbitrary phase quantities in a system of natural coordinates satisfying the condition:

( ) ( ) ( ) 0=++ tktktk CBA (2.2) In Fig. 2. 1 is shown a graphical representation of the space vector described by

Eq. (2.1).

An advantage of space vectors is possibility of their representation in various

systems of coordinates. Therefore, space vectors are very convenient mathematical

tool to describe three phase systems (see also Appendix A.1).

2.3. Operation of Voltage Source Converter – VSC

Let consider the VSC as in Fig. 2. 2. Main circuit of the bridge converter consists

of three legs with two insulated gate bipolar transistors – IGBT transistors with anti-

parallel diodes. Transistor is “on” when gate signal is “1” and “off” when gate signal

is “0”.

Fig. 2. 2. Equivalent scheme of a voltage source converter – VSC. Rectifier operation (as VSR) –

energy flows from AC to DC side. Inverter operation (as VSI) energy flows from DC to AC side

The converter AC side voltage is constructed by eight possible switching states as

shown in Fig. 2. 3. Six switching states construct active vectors and two switching


13

states construct zero vectors. The converter AC side voltage can be represented as a

complex space vector as follows:

( ),,...,n,eU

nj

dc)n(c 61 32 31 ==

−π

U (2.3)

.,n,)n(c 70 0 ==U (2.4)

Graphical representation of eight converter switching states is shown in Fig. 2. 4.

For the sake of the converter structure, each VSC leg can be represented by an ideal

switch.

Fig. 2. 3. Switching states for voltage source converter – VSC

Active states correspond to phase voltage equal 31 and

32 of the DC-link voltage

dcU . Zero vectors apply zero voltage to the converter AC side (all AC side phases

are connected to “+” or “-“ DC-link).


14

α

β

Fig. 2. 4. VSC AC side voltage represented as a space vector

2.4. Mathematical Model of the VSI - Fed Induction Motor (IM)

To present basic control methods of VSI-fed IM (see Fig. 2. 5 and Fig. 2. 6), the

space vector based IM mathematical model will be presented and discussed in this

Subsection. The fundamental-wave IM model is developed under following idealized

assumptions [63], [134], [140]:

• the object is a symmetrical, three-phase motor,

• only the basic harmonics are considered while the higher harmonics of the

spatial field distribution and magnetomotive force – MMF in the air gap are

disregarded,

• the spatially distributed stator and rotor windings are represented by a virtual

so-called concentrated coil,

• the effects of anisotropy, magnetic saturation, iron losses and eddy currents

are neglected,

• the coil resistances and reactances are assumed to be constant,


15

• the current and voltage are taken to be sinusoidal ( in many cases, especially

when considering steady states),

Fig. 2. 5. VSI with IM equivalent circuit: a) three phase system; b) single phase equivalent circuit

ΨE

sI

SSjL Iσ

SSrR IsU

ΨE

sI

SSjL Iσ SSrR I

sU

a) Motoring Regenerating

b)

Fig. 2. 6. Pictorial phasor diagrams for VSI-fed IM drive


16

2.4.1. IM Mathematical Model in Rotating Coordinate System with Arbitrary Angular Speed

The model of the IM in natural ABC coordinates is very complicated. Therefore,

in order to reduce the set of equations (Appendix A.1.3) from 12 to 4, the complex

space vectors are used. Moreover, based on transformation into a common rotating

coordinate system with arbitrary angular speed ΩK and referring rotor quantities to

the stator circuit, a following set of equations can be written [63]:

Voltage equations:

,SKKSKSKSSK jdtdR ΨΨIU Ω++= (2.5)

( ) ,rKmbKrKrKrrK pjdtdR ΨΨIU ΩΩ −++= (2.6)

Flux-currents equations:

,rKMSKSSK LL IIΨ += (2.7)

,SKMrKrrK LL IIΨ += (2.8)

And motion equation:

( ) ⎥⎦⎤

⎢⎣⎡ −= LSKSK

Sb

m MmpJdt

d IΨ*Im2

1Ω (2.9)

2.4.2. IM Model in Stationary αβ Coordinates

Adopting a system of coordinates rotating with the angular speed 0=KΩ , the set

of induction motor vector equations (2.5) and (2.6) may be rewritten as:

,dt

dR SSSSΨIU += (2.10)

,rmbrrrr jpdtdR ΨΨIU Ω−+= (2.11)

While, flux-current equations (2.7) and (2.8), and motion equation (2.9) remain

unchanged.

Knowing that, total leakage factor is expressed as:

rSrSrS

MrS

rS

M

LgLLLw

LLLLL

LLL 11

22

==−

=−=σ (2.12)

Where:


17

2

11

MrS LLLwg

−== (2.13)

Equations (2.7) and (2.8) can be rewritten:

,rMSrS wL

wL ΨΨI −= (2.14)

,SMrrr wL

wL ΨΨI −= (2.15)

In the case of a squirrel cage motor 0=rU . Therefore, the block diagram in

stationary αβ coordinates can be constructed as in Fig. 2. 7. [16].

SR

s1

RL

ML

g

211

MLLLw

grs −

==

SR

RL

rR

SL

rR

SL

ML

βΨS

+

pb

bs pm

2

sJ1

LM

mΩ

eM

ML

g

ML

ROTOR

STATOR

g

g

αSU

βSU

αΨS αSI

βSI

−

−

−

−

−

−

−

−

−

−βΨr

αΨr−

αrI

βrI

+

+

+

+

+

+

s1

s1

s1

Fig. 2. 7. Model of an induction motor - in stationary αβ coordinates


18

2.4.3. IM Model in Synchronous Rotating dq Coordinates - RFOC

In a system of coordinates rotating concurrently with the rotor flux linkage

angular speed rK Ψ

ΩΩ = such that:

,rdrr ΨΨ ==Ψ (2.16)

it is convenient to analyze dynamic states of induction motors. In the case of a

squirrel-cage rotor motor:

,0=== rqrdrdq UUU (2.17)

When the stator current control loop is used (e.g. indirect field oriented control –

IFOC) the block diagram of the motor can be simplified by omitting the stator circuit

voltage equation (2.5). Therefore, the set of equations for current controlled VSI-fed

IM can be written:

( ) ,0 rdqmbrdqrdqr pjdtd

Rr

ΨΨ

I ΩΩΨ −++= (2.18)

,rdqMSdqSSdq LL IIΨ += (2.19)

,LL SdqMrdqrrdq IIΨ += (2.20)

( ) ⎥⎦⎤

⎢⎣⎡ −= LSdqSdq

Sb

m MmpJdt

d IΨ*Im2

1Ω (2.21)

From Eq. (2.18) and Eq. (2.19):

,1 Sdqr

Mrdq

rrdq L

LL

IΨI −= (2.22)

and substituting Eq. (2.21) into Eq. (2.18) one obtains:

( ) ,0 rdqmbrdqrdqr

rSdq

r

Mr pjdt

dLR

LLR

rΨ

ΨΨI ΩΩΨ −+++−= (2.23)

After decomposition into real d and imaginary q parts:

,dt

dΨΨLRI

LLR r

rr

rSd

r

Mr ++−=0 (2.24)

( ) ,ΨpILLR

rmbSqr

Mrr

ΩΩΨ −+−=0 (2.25)

Moreover, eliminating from (2.21) the stator flux *SdqΨ vector, a motion equation

can be rewritten in the form:


19

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= LSdqrdq

r

MSb

m MLLmp

Jdtd IΨ*Im

21Ω , (2.26)

where, the electromagnetic torque developed by the IM is expressed by:

sISrr

MSbSqr

r

MSbe sinIΨL

LmpIΨLLmpM γ

22== (2.27)

Additionally, an angular slip frequency can be described as:

mbr pr ΩΩΩ Ψ −= (2.28)

Equations (2.23), (2.24), (2.25) and (2.26) form the block diagram of Fig. 2. 8.

SR

s1

br

Ms pLLm

2

sJ1

LM

mΩ

eM

SdI rdΨ−

−

r

Mr

LLR

+

R

R

LR

SqI

r

Mr

LLR

( )mbpr ΩΩΨ −

+

÷

rdΨ

Fig. 2. 8. Model of current controlled VSI-fed IM in rotor flux oriented synchronous dq coordinates

2.4.4. IM Model in Synchronous Rotating xy Coordinates - SFOC

Adopting a system of coordinates rotating concurrently with the stator flux

linkage angular speed SK Ψ

ΩΩ = such that:

,SxSS ΨΨ ==Ψ (2.29)

it is convenient to analyze a dynamic states of induction motors when direct

torque control with space vector modulation – DTC-SVM is applied. In the case of a

squirrel-cage rotor motor:


20

,0=== ryrxrxy UUU (2.30)

The set of equations (2.5)-(2.9) can be written as follows:

,SxyΨSxy

SxySSxy Sj

dtd

R ΨΨ

IU Ω++= (2.31)

( ) ,0 rxymbrxyrxyr pjdtd

RS

ΨΨ

I ΩΩΨ −++= (2.32)

,rxyMSxySSxy LL IIΨ += (2.33)

,SxyMrxyrrxy LL IIΨ += (2.34)

( ) ⎥⎦⎤

⎢⎣⎡ −= LSxySxy

Sb

m MmpJdt

d IΨ*Im2

1Ω (2.35)

Rearranging Eq. (2.33) and (2.34) the equation for currents can be obtained as:

,rxyMSxyrSxy wL

wL ΨΨI −= (2.36)

,SxyMrxySrxy wL

wL ΨΨI −= (2.37)

and based on Eq. (2.29) the equation (2.35) can be rewritten as:

⎥⎦⎤

⎢⎣⎡ −= LSyS

Sb

m MIΨmpJdt

d2

1Ω (2.38)

Where the torque is described by:

SySxS

be IΨmpM2

= (2.39)

Model in synchronous coordinates ( xy ) rotating concurrently with stator flux

linkage vector can be constructed as shown in Fig. 2. 9.


21

SR

s1

RL

ML

g

211

MLLLw

grs −

==

SR

rR

SL

rR

pb

bs pm

2

sJ1

LM

mΩML

ROTOR

STATOR

g

SxU

SyU

SxΨ SxI

SyI

−

−

−

−

−

+

−ryΨ

rxΨ−

rxI

ryI

SSΨΩ

÷

+

+

gLS

gLM−

−

+

−

eM

+

+

+

s1

s1

Fig. 2. 9. Model of VSI-fed IM - in stator flux oriented synchronous xy coordinates

Based on Eq. (2.9) the relation between stator and rotor fluxes can be derived as:

⎟⎟⎠

⎞⎜⎜⎝

⎛= Sxy

*rxy

sr

MSbe LL

LImmpM ΨΨσ

12

(2.40)

And it gives the relation:

ΨγΨΨσΨΨ

σsin

LLLmp

LLLmpM Sr

Sr

MSbSry

Sr

MSbe

12

12

== (2.41)

When constructing a block diagram of the IM a simplification can be made by

omitting the rotor circuit voltage equation (2.32). After decomposition the Eq. (2.31)

into real x and imaginary y part:

,dt

dΨIRU SSxSSx += (2.42)

,ΨΩIRU SΨSySSy S+= (2.43)

Rearranging Eq. (2.39) into form:


22

SSbeSy Ψmp

MI 2= (2.44)

Substituting SyI to equation (2.43) electromagnetic torque can be calculated:

( ) ,RΨmpΨΩUMS

SxSbSxΨSye S 2

−= (2.45)

Therefore, the simplified model presented in (Fig. 2. 10) can be constructed.

SR

s1

sJ1

LM

mΩ

SxU

SyU

SxΨ

SxI

−

−

SSΨΩ

+

eM

+

−+ b

S

s pR

m2

Fig. 2. 10. Simplified model of VSI-fed IM - in SFOC synchronous xy coordinates for DTC-

SVM control needs

2.5. Operation of Voltage Source Rectifier – VSR

VSR can be described in different coordinate system. Basic scheme of the VSR

with AC input choke and output DC side capacitor is shown in Fig. 2. 11a, while Fig.

2. 11b shows it’s single-phase representation. Where, LU is a line voltage space

vector, LI is a line current space vector, pU is the VSR input voltage space vector,

and iU is a space vector of voltage drop on the input (AC line side) choke L and it

resistance R .

The pU voltage is controllable and depends on switching signals pattern and DC-

link voltage level. Thanks to control magnitude and phase of the pU voltage, the line

current can be controlled by changing the voltage drop on the input choke - iU .

Therefore, inductances between line and AC side of the VSR are indispensable. They


23

create a current source and provide boost feature of the VSR. Through controlling

the converter AC side voltage in its phase and amplitude pU , the phase and

amplitude of the line current vector LI is controlled indirectly.

C

ULA

ULB

ULC

Rload

RL

UL

Ui

UpRL

a)

b)

IL

Idc Iload

Ic

AC- side VSR DC- side

UpA

UpB

UpC

Fig. 2. 11. Voltage source rectifier topology: a) three phase system; b) single phase equivalent

circuit

Up

UL

LLj Iω

LI

LRI

c)

UpUL

LLj Iω

LI

LRI

d)

Up

UL

LLj Iω

LI

LRI

Up

UL

LLj IωLI

LRI

Rectifying Invertinga) b)

Fig. 2. 12. Pictorial phasor diagrams for VSR: a,b) non unity power factor; c,d) unity power factor operation


24

Further, in Fig. 2. 12 are shown both motoring and regenerating phasor diagrams

of VSR. From this figure can be seen that the magnitude of pU is higher during

regeneration than in rectifying mode. With assumption of a stiff line power (i.e., LU

is a pure voltage source with zero internal impedance) terminal voltage of VSR pU

can differ up to about 3% between motoring and regenerating modes [117].

2.5.1. Operation Limits of the Voltage Source Rectifier – VSR

In Fig. 2. 12 is indicated that for VSR is a load current limit for fixed line and DC-

link voltage, as well as input choke. Beyond that limit, the VSR is not able to operate

and maintain a unity power factor requirement. Lower line inductance and higher

voltage reserve (between the line voltage and the DC side voltage) can increase that

limits. However, there is a limitation for the minimum DC-link voltage defined as:

LRMSdc UU 32> (2.46)

(For example: for LRMSU =230V; V.Udc 564230452 =⋅> ).

This limitation is introduced by freewheeling diodes in VSR which operate as a

diode rectifier. However, in the literature exists other limitation [93], [117], [118]

which takes into account the input power (value of the current) of the VSR.

Let consider that commanded value of the line current differ from actual current

by LxyI∆ :

LxyLxycLxy III −=∆ (2.47)

The direction and velocity of the line current vector changes are described by

derivative of that current dt

dL Lxy

I. It can be represented by equations in synchronous

rotating xy coordinates (Section 2.5.4, Eq.(2.74)):

( ) ( )LxyLxycLxydcLxyLxyLxycLxy LjURdtd

L IISUIII

∆ω∆ −+−=−+ 1 (2.48)

With assumptions that resistance of the input chokes 0≅R and actual current is

close to commanded value ( 0≅LxyI∆ ), Eq. (2.48) can be simplified to:

LxycLxydcLxyLxy LjU

dtd

L ISUI

ω+−= 1 (2.49)


25

Based this equation the direction and velocity of the line current vector changes

depends on:

• values of input chokes L ,

• line voltage vector LxyU ,

• line current vector LxyI ,

• value of the DC-link voltage dcU ,

• switching states of the VSR xy1S .

Let us consider that six active vectors ( )(pc 61−U ) of the VSR rotate clockwise in

synchronous xy coordinates. For each voltage vector ( )(pc 70−U ) the current

derivatives multiplied by L are denoted as ( )(P 70−U ) [117], [118]. Graphical

illustration of the Eq. (2.49) is shown in Fig. 2. 13.

y

x

Upc0,7

ULxy

Upc1

Upc2

Upc3 Upc4

Upc5

Upc6

LxycL Lj Iω−

PU1

PU2 PU0,7

PU6

PU3 PU4

PU5

ε

Fig. 2. 13. Graphical illustration of the Eq. (2.49) – instantaneous position of vectors [117]

Command current LxycI is in phase with line voltage vector LxyU and it lies on the

axe x . The difference between actual current LxyI and commanded LxycI is defined

by Eq. (2.47) and is illustrated in Fig. 2. 14.


26

ULLxycI

y

xLxyI

PU1

PU2 PU0,7

PU6

PU3 PU4

PU5

εLxyI∆

Fig. 2. 14. Error area of the line current vector [117]

Full current control is possible when the current is kept in desired error area (Fig.

2. 14.). Critical operation of the VSR is when the angle achieves πε = . Fig. 2. 13

shows that for such case ε created by 21 UU P,P , 21 and pcpc , UU vectors, are the

arms of the equilateral triangle. Therefore, based on the equation for its altitude the

boundary condition can be defined as:

pxyLxycLLxy Lj UIU 23

=+ ω (2.50)

Assuming that: LmcLxycLmLxy I,U == IU and dcpxy U32=U the following expression

can be derived:

dcLmcLLm U)LI(U 32

2322 =+ ω (2.51)

After rearranging one obtains dependence for minimum DC-link voltage:

( )223 )LI(UU LmcLLmmindc ω+= (2.52) (For example with parameters as: H 010 502 V, 2230 ,.L,U LLm === πω

A 10=LmcI , then V 566≥mindcU ).

Based on this relation the maximum value of the input inductance can be

calculated as:


27

( )

LmcL

Lmdc

m I

UUL

ω

22

31

−= (2.53)

(For example with parameters as: A 10 502 V, 2230 === LmcLLm I,U πω , and

V 566=mindcU then maximum input line inductance is: H 010.Lm = ).

2.5.2. VSR Model in Three-Phase ABC Coordinates

Assuming that the system of Fig. 2. 11a is balanced three phase system without

neutral connection, and the power switches are ideal, following equations for VSR’s

input circuit can be derived:

piL UUU += (2.54)

Where, voltage drop vector iU on the line choke is defined as:

LL

i RdtdL IIU += (2.55)

Moreover, AC side voltage of the VSR can be described by Eq. (2.3) and Eq. (2.4)

or by:

⎟⎠

⎞⎜⎝

⎛−= ∑

=

C

Akkkdcp SSU 3

1U (2.56)

Where, kS = “0”, or “1”, are switching states of the VSR ( C,B,Ak = ) for

appropriate line phase.

From the other hand the DC-link capacitor current equation can be expressed as:

,Idt

dUC cdc = (2.57)

The capacitor current can be calculated as the difference between DC current dcI

of the VSR and input DC current loadI of the VSI:

,III loaddcc −= (2.58)

Moreover, dcI current can be calculated as a sum of a product of the phase current

and appropriate switching states:


28

,SISISII CLCBLBALAdc ++= (2.59)

Hence, AC side of the VSR voltage equations in the three phase system can be

expressed as:

( )⎟⎠⎞

⎜⎝⎛ ++−−=⎟

⎠

⎞⎜⎝

⎛−−=+ ∑

=CBAAdcLA

C

AkkAdcLALA

LA SSSSUUSSUURIdt

dIL31

31

(2.60)

( )⎟⎠⎞

⎜⎝⎛ ++−−=⎟

⎠

⎞⎜⎝

⎛−−=+ ∑

=CBABdcLB

C

AkkBdcLBLB

LB SSSSUUSSUURIdt

dIL31

31

(2.61)

( )⎟⎠⎞

⎜⎝⎛ ++−−=⎟

⎠

⎞⎜⎝

⎛−−=+ ∑

=CBACdcLC

C

AkkCdcLCLC

LC SSSSUUSSUURIdt

dIL31

31

(2.62)

Where the line voltages are expressed by:

( ) ( ) ( )32 32 /tsinUU,/tsinUU,tsinUU LLmLCLLmLBLLmLA πωπωω +=−== (2.63)

Also DC-link side equation of the VSR can be modeled by:

loadCLCBLBALA

C

AkloadkLk

dc ISISISIISIdt

dUC −++=−= ∑=

(2.64)

The above expressions can be written as:

⎟⎠

⎞⎜⎝

⎛−−=+ ∑

=

C

AkkkdcLkLk SSUUI)Rdt

dL(31 (2.65)

∑=

−=C

AkloadkLk

dc ISIdt

dUC (2.66)

The equations (2.65) and (2.66) can be represented as a block diagram in Fig. 2. 15

[13], [16].


29

RsL +1

RsL +1

RsL +1

LAU

AS

LBU

BS

LCU

CS

LAI

LBI

LCI

pAU

Af

Bf

Cf

31

sC1

loadIdcU

pBU

pCU

−

+

−

+

+

+

+

+

+

+

−

+

−

+

−

+

−

+

−

+

Fig. 2. 15. Block diagram of the VSR in three-phase ABC coordinates

Where:

( ) ( ) ( )⎟⎠⎞

⎜⎝⎛ ++−=⎟

⎠⎞

⎜⎝⎛ ++−=⎟

⎠⎞

⎜⎝⎛ ++−= CBACCCBABBCBAAA SSSSf,SSSSf,SSSSf 3

131

31

(2.67)

2.5.3. VSR Model in Stationary αβ Coordinates

In some studies is useful to present the VSR model in two axis coordinates

system. Equations (2.60) - (2.62) and Eq. (2.64) after transformation into stationary

αβ coordinates (Appendix A.2) can be described using the complex space vector

notation as:

1SUII

dcLLL UR

dtdL −=+ (2. 68)

[ ] load*Ldc IRedtdUC −= 12

3 SI (2. 69)

Further, those equations can be decomposed in α and β components:


30

,SUURIdt

dIL dcLLL αααα −=+ (2.70)

,SUURIdt

dIL dcLL

Lβββ

β −=+ (2.71)

( ) .ISISIISIdt

dUC loadLLk

loadkLkdc −+=−= ∑

=ββαα

β

α 23 (2.72)

Where, appropriate switching states are expressed as:

( ) ( )CBCBAAA SSS,SSSSfS −=⎟⎠⎞

⎜⎝⎛ ++−==

31

31

βα (2.73)

Equations (2.70) - (2.72) can be represented as a block diagram in stationary αβ

coordinates as in Fig. 2. 16.

RsL +1

RsL +1

αLU

αS

βLU

βS

αLI

βLI

loadI

−+

−

+

23

−+

sC1 dcU+

+

αpU

βpU

Fig. 2. 16. Model of a three-phase VSR in stationary αβ coordinates

2.5.4. VSR Model in Synchronously Rotating xy Coordinates

The two-phase model in stationary αβ coordinates (Eqs. (2.70) - (2.72)), can be

transformed into a two-phase model in synchronously rotating xy coordinates using

the appropriate transformation (see Subsection A.2.3.). Therefore, xy model using

the complex space vector notation can be expressed as:

LxyLxydcLxyLxyLxy LjUR

dtd

L ISUII

ω+−=+ 1 (2.74)


31

[ ] load*xyLxydc IRedtdUC −= 12

3 SI (2.75)

And after decomposition into x and y components yields:

LyLxdcLxLxLx LISUURI

dtdIL ω−−=+ (2.76)

LxLydcLyLyLy LISUURI

dtdI

L ω+−=+ (2.77)

loadyLyxLx

y

xkloadkLk

dc ISISIISIdt

dUC −+=−= ∑=2

3 (2.78)

Further, based on equations (2.76) - (2.78) a block diagram can be constructed as

in Fig. 2. 17.

23LxU

yS

−+

+RsL +1

xS loadI

sC1−++LxI dcU

Lω L

−

++

−

LyURsL +

1

LyI

pxU

pyU

Fig. 2. 17. Model of a three-phase VSR - in synchronously rotating xy coordinates

Remark:

The model of VSR in xy coordinates can be oriented (synchronized) with line

voltage vector LU or as in Section 3.3 with line virtual flux vector – LΨ . Therefore,

for VF oriented xy coordinates can be written: 0=pxU , Lmpy UU = .

Note that xydcpxy U 1SU = .


32

2.6. Summary

In this Chapter space vector based mathematical model of voltage source

converter – VSC operated as voltage source rectifier – VSR-fed DC-link and voltage

source inverter – VSI-fed induction motor – IM has been presented and discussed.

These models create basis for further chapters of the thesis, especially Chapter 3

and 4 where specific control methods for AC/DC/AC converter-fed IM drive will be

discussed and studied. However, presented models are nonlinear. Therefore, it is not

straightforward to analyze such systems. One way to avoid the nonlinearities is to

linearize around the operating point. Hence, for further considerations will be

assumed that model is linearized aroud the setpoint i.e. the steady state of the

controlled variable (initial value 0dcU ) is equal to commanded value of this variable

e.g dccdc UU =0 .

Chapter 3

3. Vector Control Methods of AC/DC/AC Converter-Fed Induction Motor Drives – A Review

3.1. Introduction

In this chapter a main high performance control methods for VSR and VSI will be

presented and briefly described. A couple of them will be chosen for control of an

AC/DC/AC converter-fed IM drive for further investigation needs.

Control of the VSR can be considered as a dual problem with vector control of an

induction motor (Fig. 3. 1) [65], [146], [154].

V-FOC

FOC DTC

DPC DPC-SVM

DTC-SVM

Control of VSR

Control of VSI

VSR

C

IM SU

VM

~3

LU

dcI

loadIcI

dcU

VSI

Fig. 3. 1. Relationship between control methods of VSR and VSI – fed IM

Besides of classification as in Chapter 1 control techniques for VSR can be

classified in respect to voltage and virtual flux – VF bases. Overall, four types of

these techniques can be distinguished:

3. Vector Control Methods of AC/DC/AC Converter-Fed IM Drives – A Review

34

• voltage oriented control – VOC ,

• voltage based direct power control – DPC ,

• virtual flux oriented control – V-FOC,

• virtual flux based direct power control – VF-DPC.

All this methods are very well described in the literature [65], [93], [95], [94],

where superiority of VF based methods is clearly shown. Therefore, in this chapter

only virtual flux based method will be described.

This chapter has been performed with two main goals: presenting theoretical

background of each control technique and comparative analysis, and choosing most

interesting control methods (in author opinion) for VSR, as well as for VSI, for

further investigation.

3.2. Control Methods of VSI-Fed Induction Motor

3.2.1. Field Oriented Control – FOC

First publications about inverter vector control (field oriented control – FOC) was

published 30 years ago [11], and from that time it has been widely used in industry

[25]. As have been mentioned in Chapter 1 the FOC can be divided into direct field

oriented control – DFOC and indirect field oriented control – IFOC. The second one

seems to be more attractive because of lack of the flux estimator. Thanks to this

ability it is easier in implementation. Therefore, for further consideration IFOC is

chosen.

For the IFOC presentation the coordinate system rotating concurrently with the

rotor flux rK Ψ

ΩΩ = angular speed are selected. In this case the coordinate system is

oriented along d rotor flux linkage component (Fig. 3. 2) such as:

,rdrr ΨΨ ==Ψ (3. 1)


35

A

B

C

α

β

rΨΩ

q

d

rΨγrdrr ΨΨ ==Ψ

0=rqΨ

SIγ

SI

SdI

SqI

Fig. 3. 2. Space vector representation in rotor flux oriented coordinates dq

More detailed description of the mathematical model in dq coordinates is given

in Section 2.4.3. From Eqs. (2.18-2.21), the set of equations for induction motor can

be written as:

rr

rrdSd

r

Mr

LR

dtdI

LLR ΨΨ +=⎟⎟

⎠

⎞⎜⎜⎝

⎛ (3. 2)

( ) rmbrSqr

Mr pILLR ΨΩΩΨ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛ (3. 3)

( ) ⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= LSqr

r

Msb

m MILLmp

Jdtd ΨΩ

21 (3. 4)

The block diagram of the IFOC is presented in Fig. 3. 3. The commanded

electromagnetic torque ecM , is delivered from outer PI speed controller, based on

mechanical speed error m

eΩ . From Eq. (2.27) the q component of the command

stator current can be calculated as:

ecrdcMSb

rSqc MΨLmp

LI 2= (3. 5)

Assuming steady state conditions, from Eq. (3. 2) the d component of the

command stator current can be computed as:

rdcM

Sdc ΨLI 1= (3. 6)

Then, command values SdcI and SqcI are compared with actual values of current

component SdI and SqI respectively. It should be stressed that (for steady state) SdI


36

is equal to the magnetizing current, while the torque both dynamic and steady states

is proportional to SqI .The current errors SdIe and SqIe are fed to two PI controllers,

which generate commanded stator voltage components SqcU , and SdcU , respectively.

Further, commanded voltages are converted from rotating dq coordinates into

stationary αβ coordinates using rotor flux vector position angle rΨ

γ . So obtained

voltage vector ScU is delivered to space vector modulator – SVM which generates

appropriate switching states vector )S,S,S( CBA 2222S for control power transistors of

the VSI.

The stator voltage equations in dq coordinates is expressed as:

,jpLR

LLLjR

dtd

L rdqmbr

r

r

MSdqSdqim

SdqSdq r

ΨIII

U ⎟⎟⎠

⎞⎜⎜⎝

⎛−−++= ΩΩ σΨσ (3. 7)

and can be decomposed into the d and q components:

,ΨL

RLILIRdt

dILU rdr

rMSqSdim

SdSd r 2

−−+= σΨσ Ω (3. 8a)

,ΨLpLILIR

dtdI

LU rdr

mbMSdSqim

SqSq r

ΩΩ σΨσ +++= (3. 8b)

where:

.L

RLRLR,LLr

rMSrimS 2

22

and +== σσ (3.9)

It can be seen that SdcU and SqcU are coupled each other.

The part of Eq. (3.7) with rotor flux rdqΨ may be treated as a distortion and can

be omitted [121]. Any change of d stator voltage component has an influence not

only for d but also for q current component (the same apply to q component). So,

separate control of the current components is unrealizable. Therefore, decoupling

network in the control circuit is necessary (see Subsection 3.3.1.1).

Among main known drawbacks of the IFOC are:

• dependency on rotor parameter ( rr ,R L ),

• the d and q voltage components are coupled each other, therefore

decoupling in control circuit is required,

• coordinate transformations are needed,


37

• separate modulator block is needed.

However, the IFOC is used very widely thanks to the following advantages:

• no problems with start,

• flux estimator is not needed, (only position of the flux is calculated in

feedforward manner),

• no steady states operation error,

• operation at fixed switching frequency (defined by SVM block).

SVM

PI

Current Transformation & Virtual Flux

EstimationLΨγ

Current Transformation & Rotor Flux

Angle Estimation

rΨγ

pxcU

−

−PI

pycUdccU

dcU

−

−+

++

+

−

−

mcΩ

mΩ

rcΨ

ecM

mΩ dcU

dcU

LUVM

LI

1S

IM

2S

SI

VSR

VSI

1C

2C+

PI

PI

LyI

LxI

pcU

LycI

LxcI1D

xy

αβ

SdcU

SqcU

PI

PI

SVM

dq

αβ

ScU

SdI

SqI

SdcI

ML1

LxIe

LyIe

SdIe

SqIe

SqcI +

ecMrdcMsbr

Isqc LmpLK

Ψ2

=

IsqcKm

eΩ

dcUe

PcKcP

LxLΨω320=cQ

LxLPcK Ψω3

2=

V-FOC

IFOC

DN

DN

Fig. 3. 3. Virtual flux oriented control – V-FOC and indirect field oriented control – IFOC; where

DN is decoupling network


38

3.2.2. Direct Torque Control – DTC

For needs of the direct torque control – DTC a mathematical model of IM

represented in a stationary system of coordinates αβ has been chosen. Stationary

coordinates ( 0=KΩ ):

,SxSS ΨΨ ==Ψ (3. 10)

A

B

C

α

SΨ

β

SΨΩ

y

x

Ψγ

rΨrΨ

Ω

Sector 1

Fig. 3. 4. Stator and rotor flux vectors and angle between them in direct torque control - DTC

Direct Torque Control was proposed by Takahashi [126]. The block diagram of

the method is presented in Fig. 3. 8. The commanded electromagnetic torque ecM is

delivered from outer PI speed controller. Then, ecM and commanded stator flux

ScΨ amplitudes are compared with estimated values of eM and SΨ respectively. The

torque Me and flux ψe errors are fed to two hysteresis comparators.

From predefined switching table, based on digitized error signals MS and ΨS ,

and the stator flux position SΨ

γ the appropriate voltage vector is selected. The

outputs from the predefined switching table are switching states 2S for the VSI.

Please consider that for electromagnetic torque the hysteresis is defined as:

1=MS for MM He > (3. 11)

0=MS for 0=Me (3. 12)

1−=MS for MM He −< (3. 13)

And for stator flux the hysteresis is described as:


39

1=ΨS for ΨΨ He > (3. 14)

0=ΨS for ΨΨ He −< (3. 15)

Where MH and ΨH are a hysteresis band of the torque and the flux respectively

(Fig. 3. 5). The hysteresis bands are chosen by consideration of the switching loses in

the VSI and low harmonic copper losses in the motor [34].

MHΨHψe

MS

Me

ΨS)a )b

Fig. 3. 5. Hysteresis controllers a) two level; b) three level

For DTC the VSI output voltage is constructed base on appropriate space vector

(detailed described in Section 2.3). The voltage source inverter AC side voltage can

be represented as a complex space vector as follows:

( ),,...,n,eU

nj

dc)n(Sc 61 32 31 ==

−π

U (3. 16)

.,n,)n(Sc 70 0 ==U (3. 17)

Then voltage space vector plane for the DTC needs is divided into six sectors as

in Fig. 3. 6. The sectors could be defined in different manner [63]. For control

method proposed by Takahashi [126] the sectors are defined as follows:

Sector 1: 66πγπ Ψ


40

α

β

Fig. 3. 6. Voltage space vector plane divided into six sectors

DTC is based on controlling the stator flux vector position in respect to rotor flux

vector position based on expression:

,sinLL

LmpM SrSr

MSbe ΨγΨΨσ

12

= (3. 18)

where, the angle between stator and rotor flux vectors is defined as below:

rS ΨΨΨγγγ −= (3. 19)

From Eq. (3.18) it can be seen that the electromagnetic torque depends on

amplitudes of stator and rotor fluxes and angle between them Ψγ . Thanks to long

rotor time constant the angle Ψγ can be controlled by fast change of stator flux

vector position (Fig. 3. 4). Under assumption that the stator resistance sR is zero, the

stator flux can be easy expressed as a function of a stator voltage:

.dt

dS

S UΨ = (3. 20)

Or in the form:

,dtSS ∫= UΨ (3. 21) Under appropriate active voltage vector stator flux vector position change (the

angle Ψγ ) in forward direction causing increase of the torque. During zero voltage


41

vector the flux is kept constant but the torque is reduced. It can be clearly seen that in

DTC exists natural decoupling between the stator flux and the torque control.

A

B

C

α

SΨ

βy

x

rΨSector 1

USc2USc3

USc4

USc5 USc6

USc0 USc7

USc1

Fig. 3. 7. Voltage vectors applied to control of stator flux vector in direct torque control - DTC

The switching time of the zero vectors are specified by permitted torque

pulsations. While the switching time of the active vectors are depended on values of

the torque and the stator flux.

Let consider that the stator flux position is as in Fig. 3. 7. The angle Ψγ can be

increased by selecting vectors 32 and , ScSc UU or decreased by vectors 65 and , ScSc UU .

When any zero-voltage 70 or ScSc UU is applied the stator flux is not changed. When

the angle is increasing then the torque eM is increasing. When the angle is

decreasing then the toque eM is decreasing. Hence, to chose the appropriate voltage

vector the optimal switching table Tab. 3. 1 is defined as in [126].

It should be noted that the angle change depends on the rotor speed. That means

for the middle and high-speed operation ( Nm ΩΩ 2.0> ) [34] the increasing of the

angle Ψγ is slower then the decreasing.


42

Tab. 3. 1. Optimal switching table.

ΨS MS Sector 1 Sector 2 Sector 3 Sector 4 Sector 5 Sector 6

1 USc2 (1,1,0) USc3

(0,1,0) USc4

(0,1,1) USc5

(0,0,1) USc6

(1,0,1) USc1

(1,0,0)

0 USc7 (1,1,1) USc0

(0,0,0) USc7

(1,1,1) USc0

(0,0,0) USc7

(1,1,1) USc0

(0,0,0) 1

-1 USc6 (1,0,1) USc1

(1,0,0) USc2

(1,1,0) USc3

(0,1 0) USc4

(0,1,1) USc5

(0,0,1)

1 USc3 (0,1,0) USc4

(0,1,1) USc5

(0,0,1) USc6

(1,0,1) USc1

(1,0,0) USc2

(1,1,0)

0 USc0 (0,0,0) USc7

(1,1,1) USc0

(0,0,0) USc7

(1,1,1) USc0

(0,0,0) USc7

(1,1,1) 0

-1 USc5 (0,0,1) USc6

(1,0,1) USc1

(1,0,0) USc2

(1,1,0) USc3

(0,1,0)) USc4

(0,1,1)

The main known drawbacks of the DTC are:

• high and variable switching frequency, which produces high VSI power

losses,

• violence of polarity consistency rules,

• start and low speed operation problems,

• steady states operation error,

• torque pulsation,

• flux and torque estimation problems.

Therefore, in the literature there is a lot of work which have a goal to improve the

features of the DTC, for instance, modified switching table. Other method produces

additional active vector as a sum of the nearest one. Finally, another kind of three

levels of hysteresis controller is proposed as in [18], [34], [60].

In spite of these disadvantages the DTC is very interesting for researcher thanks to

its advantages as follows:

• simple control structure,

• independent of rotor parameter,

• inherently motion-sensorless,

• excellent dynamic performance of torque control loop,

• no current control loops,

• no coordinates transformations,

• separate modulator is not needed.


43

Switching Table

PI

Power & Virtual Flux

Estimation

Switching Table

LΨγ

Torque & Stator Flux

Estimation

SΨγ

QS

ψe

MS−

−

MePI

Sector Selection

ΨS

PS0=cQ

cPdccU

dcU

−+

++

+

−

−

mcΩ

mΩ

eM

SΨ

ScΨ

ecM

mΩ dcU

dcUSector Selection

LUVM

LI

1S

IM

2S

SI

VSR

VSI

1C

2C

P

Q

+

meΩ

dcUe

Qe

pe

VF-DPC

DTC

Fig. 3. 8. Conventional switching table based direct power control – DPC and direct torque

control – DTC

3.2.3. Direct Torque Control with Space Vector Modulation – DTC-SVM

To avoid the drawbacks of switching table based DTC (described in Section 3.2.2)

instead of hysteresis controllers and switching table the PI controllers with the SVM

block were introduced like in IFOC (described in Section 3.2.1). Therefore, DTC

with SVM (DTC-SVM) joins DTC and IFOC features in one control structure as in

Fig. 3. 10.

For needs of the DTC-SVM method a mathematical model of IM in a xy rotating

system of coordinates are chosen (SK Ψ

ΩΩ = ). In this case the coordinate system is

oriented with x stator flux linkage component (Fig. 3. 9) such as:


44

,SxSS ΨΨ ==Ψ (3. 22)

A

B

C

α

β

SΨΩ

y

x

SΨγSxSS ΨΨ ==Ψ

0=SyΨ

Fig. 3. 9. Stator flux oriented xy coordinates

The commanded electromagnetic torque ecM is delivered from outer PI speed

controller (Fig. 3. 10). Then, ecM and commanded stator flux ScΨ amplitudes are

compared with estimated actual values of eM and SΨ . The torque Me and flux ψe

errors are fed to two PI controllers. The output signals are the command stator

voltage components SycU , and SxcU respectively.

Further, voltage components in rotating xy system of coordinates are transformed

into αβ stationary coordinates using SΨ

γ flux position angle. Obtained voltage

vector ScU is delivered to space vector modulator – SVM which generates

appropriate switching states vector )S,S,S( CBA 2222S for the VSI.

An exhausting description of the DTC-SVM can be found in [152].


45

SVM

PI

Power & Virtual Flux

EstimationLΨ

γ

Torque & Stator Flux

Estimation

pqcU

−

−PI

ppcUdccU

dcU

−

−+

++

+

−

−

mcΩ

mΩ

ScΨ

ecM

mΩ dcU

dcU

LUVM

LI

1S

IM

2S

SI

VSR

VSI

1C

2C

+

PI

PI

pcU

1D

pq

αβ

SxcU

SycU

PI

PI

SVM

xy

αβ

ScU

Qe

pe

+m

eΩ

dcUe

P

Q

0=cQ

cP

SΨγ

eM

SΨ

2D

ψe

Me

DPC-SVM

DTC-SVM

Fig. 3. 10. Direct power control with space vector modulation – DPC-SVM and direct torque

control with space vector modulation – DTC-SVM

3.3. Control Methods of VSR

3.3.1. Virtual Flux Oriented Control – V-FOC

Voltage oriented control – VOC guarantees high dynamics and static performance

via an internal current control loops. It has become very popular and has

consequently been developed and improved [93]. Therefore, VOC is a basis for

virtual flux oriented control – V-FOC which is shown in Fig. 3. 3.

The goal of the control system is to maintain the DC-link voltage Udc, at the

required level, while currents drawn from the power system should be sinusoidal like

and in phase with line voltage to satisfy the unity power factor – UPF condition. The


46

UPF condition is fulfill when the line current vector, LyLxL jII +=I , is aligned with

the phase voltage vector, LyLxL jUU +=U , of the line.

The idea of VF has been proposed to improve the VSR control under distorted

and/or unbalanced line voltage conditions, taking the advantage of the integrator’s

low-pass filter behavior [93], [95].

Therefore, a rotating reference frame aligned with LΨ is used (Fig. 3. 11). The

vector of VF lags the voltage vector by o90 . For the UPF condition, the command

value of the direct component current vector LxcI , is set to zero. Command value of

the LycI is an active component of the line current vector. After comparison

commanded currents with actual values, the errors are delivered to PI current

controllers. Voltages generated by the controllers are transformed to αβ coordinates

using VF position angle LΨγ . Switching signals vector 1S , for the VSR is generated

by a space vector modulator.

A

B

C

α

β

LΨΩ

y

x

LΨγLxLL ΨΨ ==Ψ

0=LyΨ

ϕ

LI

LxI

LyI

LU

Fig. 3. 11. Synchronous rotating reference frame xy with line virtual flux angular frequency

LΨΩ

V-FOC guarantees high dynamics and static performance via an internal current

control loops. However, the final performance of the V-FOC largely depends on the

quality of the applied current control strategy [65]. Therefore, analysis and synthesis

of the current controllers will be shortly describes [16], [31], [65].


47

3.3.1.1. Line Current Controllers

The model presented in Section 2.5.4 is very convenient to use in synthesis and

analysis of the current regulators for VSR. However, presence of coupling requires

an application of decoupling network – DN as in Fig. 3. 12.

−+

RsL +1 LxI

Lω L

−

+

+

−RsL +

1 LyI

pxU

pyU

−+

LxcI

Lω L

+

−

+

−

0=LxU

pxcU

pycU

LxI

LxeIPI ixc

U

LycI

LyI

LyeIPI iyc

U

−

−

DN

LmLy UU =

0=LxU

LmLy UU =

Fig. 3. 12. Current control with decoupling network – DN of VSR controlled by V-FOC

Hence, based on Eqs. (2.76) and (2.77) it could be clearly seen that decoupled

command rectifier voltage xydcpxyc U SU = would be generated as follows:

LyLLxLx

Lxpxc LIRIdtdILUU ω+−−= (3. 23)

LxLLyLy

Lypyc LIRIdtdI

LUU ω−−−= (3. 24)

Decoupling for the x and y axes reduces the synchronous rotating current

control plant to a first-order delay as in Fig. 3. 13.


48

−+

LxI

+

− LyI

pxU

pyU

LmLy UU =

0=LxU

RsL +1

RsL +1

Fig. 3. 13. Decoupled current loops of VSR in xy coordinates

It simplifies the analysis and enables the derivation of analytical expressions for

the parameters of current regulators. In Fig. 3. 14 a block diagram for a simplified

current control loop in the synchronous rotating coordinates are presented. Because

the same diagram applies to both the x and y axis regulators, the subscripts x and

y are omitted.

Control structure will operates in discontinuous environment (complete model in

Saber, and implementation in DSP) therefore, is necessary to take into account the

sampling period ST . It could be done by sample & hold – S&H block. Moreover, the

statistical delay of the PWM generation SPWM T.T 50= should be taken into account

(block VSC). In the

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