Direct Power and Torque Control of AC/DC/AC Converter

Transkrypt

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
z
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:
¾ University of Aalborg at Institute of Energy Technology, Denmark (Prof.
Frede Blaabjerg)
¾ Nordic Network for Multi Disciplinary Optimised Electric Drives,
Denmark (Prof. Ewen Ritchie)
¾ 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 – DPTCSVM........................................................................................................................... 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
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 DClink Voltage Feedback only – PF0 .................................................................... 74
4.2.2.2. Transfer Function of the AC/DC/AC Converter-Fed IM Drive with DClink Voltage Feedback and Active Power Feedforward Calculated Based on
Mechanical Speed, Commanded Torque, and Power Losses – PFΩ ................ 75
4.2.2.3. Transfer Function of the AC/DC/AC converter-Fed IM Drive with DClink Voltage Feedback and Active Power Feedforward Calculated From
Commanded Stator Voltage and Actual Stator Current - PFUI ......................... 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
II
Contents
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
III
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
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 neverstopping. 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 5191992 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 I SC / I Lm is the ratio of shortcircuit current I SC available at the PCC, to the maximum fundamental load current
I Lm . It is recommended that the load current I Lm , should be calculated as the average
current of the maximum demand over a year [54].
2
1. Introduction
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]:
50
THD =
∑U ( )
h =2
2
L h
U 2L (1)
100%
(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,
3
1. Introduction
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
4
1. Introduction
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
5
1. Introduction
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
6
1. Introduction
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].
7
1. Introduction
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
8
1. Introduction
[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 DPTCSVM 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:
9
1. Introduction
•
development (in Matlab Simulink as well as in a professional package Saber)
simulation algorithm for control and investigation of AC/DC/AC convertersfed 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 DPTCSVM 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).
10
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].
B
3
k
2
Im
a 2 kC ( t )
k
a
ak B ( t )
1
k A( t )
Re
A
a2
C
Fig. 2. 1. Construction of space vector according to definition Eq. (2. 1)
(
)
2
(2.1)
k A (t ) + ak B (t ) + a 2 kC (t )
3
2
Where: - normalization factor, 1, a , a 2 - complex unity vectors, with phase shift,
3
df
k=
k A (t ), k B (t ), kC (t ) - denotes arbitrary phase quantities in a system of natural
coordinates satisfying the condition:
k A (t ) + k B (t ) + kC (t ) = 0
(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 antiparallel 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
12
states construct zero vectors. The converter AC side voltage can be represented as a
complex space vector as follows:
π
j ( n −1)
2
3
U c( n ) = U dce
, n = 1,...,6,
3
(2.3)
U c( n ) = 0 ,
(2.4)
n = 0,7.
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
1
2
and
of the DC-link voltage
3
3
U dc . Zero vectors apply zero voltage to the converter AC side (all AC side phases
are connected to “+” or “-“ DC-link).
13
β
α
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,
14
•
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
Motoring
a)
Regenerating
b)
jLSσ I S
RSr I S
Us
E
Ψ
jLSσ I S
E
Ψ
Is
Us
Is
Fig. 2. 6. Pictorial phasor diagrams for VSI-fed IM drive
15
RSr I S
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:
U SK = RS I SK +
U rK = Rr I rK +
dΨ SK
+ jΩ K Ψ SK ,
dt
(2.5)
dΨ rK
+ j (Ω K − pbΩ m )Ψ rK ,
dt
(2.6)
Flux-currents equations:
Ψ SK = LS I SK + LM I rK ,
(2.7)
Ψ rK = Lr I rK + LM I SK ,
(2.8)
And motion equation:
dΩ m 1
=
dt
J
(
)
⎡ mS
⎤
*
⎢ pb 2 Im Ψ SK I SK − M L ⎥
⎣
⎦
(2.9)
2.4.2. IM Model in Stationary αβ Coordinates
Adopting a system of coordinates rotating with the angular speed Ω K = 0 , the set
of induction motor vector equations (2.5) and (2.6) may be rewritten as:
U S = RS I S +
dΨ S
,
dt
(2.10)
U r = Rr I r +
dΨ r
− jpb Ω m Ψ r ,
dt
(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:
σ =1−
L2M
L L − L2M
w
1
= S r
=
=
LS Lr
LS Lr
LS Lr gLS Lr
Where:
16
(2.12)
g=
1
1
=
w LS Lr − L2M
(2.13)
Equations (2.7) and (2.8) can be rewritten:
IS =
Lr
L
ΨS − M Ψr ,
w
w
(2.14)
Lr
L
Ψr − M ΨS ,
(2.15)
w
w
In the case of a squirrel cage motor U r = 0 . Therefore, the block diagram in
Ir =
stationary αβ coordinates can be constructed as in Fig. 2. 7. [16].
RS
U Sα
+
−
1 Ψ Sα
LR
+ −
s
g
I Sα
LM
−
U Sβ
+
−
1
s
Ψ Sβ
LM
−
LR
+
g
I Sβ
ms
pb
2
RS
STATOR
−
−
Me
Rr
1 Ψ rα
s
LS
+ −
g
ML
−
1
sJ
I rα
Ωm
LM
pb
LM
+
−
1
s
Ψ rβ
LS
−
+
g
I rβ
g=
Rr
1
1
=
w Ls Lr − L2M
ROTOR
Fig. 2. 7. Model of an induction motor - in stationary
17
αβ
coordinates
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 Ω K = ΩΨ r such that:
Ψ r =Ψ r =Ψ rd ,
(2.16)
it is convenient to analyze dynamic states of induction motors. In the case of a
squirrel-cage rotor motor:
U rdq = U rd = U rq = 0,
(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 = Rr I rdq +
(
)
dΨ rdq
+ j ΩΨ r − pb Ω m Ψ rdq ,
dt
(2.18)
Ψ Sdq = LS I Sdq + LM I rdq ,
(2.19)
Ψ rdq = Lr I rdq + LM I Sdq ,
(2.20)
dΩ m 1
=
dt
J
(
)
⎡ mS
⎤
*
⎢ pb 2 Im Ψ Sdq I Sdq − M L ⎥
⎣
⎦
(2.21)
From Eq. (2.18) and Eq. (2.19):
I rdq =
L
1
Ψ rdq − M I Sdq ,
Lr
Lr
(2.22)
and substituting Eq. (2.21) into Eq. (2.18) one obtains:
0=−
dΨ rdq
Rr LM
R
I Sdq + r Ψ rdq +
+ j ΩΨ r − pbΩ m Ψ rdq ,
Lr
Lr
dt
(
)
(2.23)
After decomposition into real d and imaginary q parts:
0=−
Rr LM
R
dΨ r
I Sd + r Ψ r +
,
Lr
Lr
dt
0=−
Rr LM
I Sq + ΩΨ r − pb Ω m Ψ r ,
Lr
(
(2.24)
)
(2.25)
Moreover, eliminating from (2.21) the stator flux Ψ*Sdq vector, a motion equation
can be rewritten in the form:
18
⎤
⎞
dΩ m 1 ⎡ mS ⎛ LM *
Ψ rdq I Sdq ⎟⎟ − M L ⎥ ,
= ⎢ pb
Im⎜⎜
dt
J⎣
2
⎝ Lr
⎠
⎦
(2.26)
where, the electromagnetic torque developed by the IM is expressed by:
M e = pb
m L
mS LM
Ψ r I Sq = pb S M Ψ r I S sin γ I s
2 Lr
2 Lr
(2.27)
Additionally, an angular slip frequency can be described as:
Ω r = ΩΨ − pb Ω m
(2.28)
r
Equations (2.23), (2.24), (2.25) and (2.26) form the block diagram of Fig. 2. 8.
Ψ rd
RR
LSR
R
I Sd
Rr LM
Lr
−
+
1
s
Ψ rd
ms LM
pb
2 Lr
Me
+
I Sq
Rr LM
Lr
−
÷
(Ω
Ψr
− pb Ω m
)
ML
1
sJ
Ωm
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 Ω K = ΩΨ S such that:
Ψ S = Ψ S = Ψ Sx ,
(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:
19
U rxy = U rx = U ry = 0,
(2.30)
The set of equations (2.5)-(2.9) can be written as follows:
U Sxy = RS I Sxy +
0 = Rr I rxy +
dΨ Sxy
dΨ rxy
dt
+ jΩΨ S Ψ Sxy ,
dt
(
(2.31)
)
+ j ΩΨS − pb Ω m Ψ rxy ,
(2.32)
Ψ Sxy = LS I Sxy + LM I rxy ,
(2.33)
Ψ rxy = Lr I rxy + LM I Sxy ,
(2.34)
dΩ m 1 ⎡ mS
⎤
= ⎢ pb
Im Ψ*Sxy I Sxy − M L ⎥
2
dt
J⎣
⎦
(
)
(2.35)
Rearranging Eq. (2.33) and (2.34) the equation for currents can be obtained as:
I Sxy =
Lr
L
Ψ Sxy − M Ψ rxy ,
w
w
(2.36)
I rxy =
LS
L
Ψ rxy − M Ψ Sxy ,
w
w
(2.37)
and based on Eq. (2.29) the equation (2.35) can be rewritten as:
dΩ m 1 ⎡ mS
⎤
Ψ S I Sy − M L ⎥
= ⎢ pb
2
dt
J⎣
⎦
(2.38)
Where the torque is described by:
M e = pb
mS
Ψ Sx I Sy
2
(2.39)
Model in synchronous coordinates ( xy ) rotating concurrently with stator flux
linkage vector can be constructed as shown in Fig. 2. 9.
20
RS
U Sx −
+
1 Ψ Sx
LR
+ −
s
g
I Sx
LM
I Sy
U Sy
+
Ω SΨ
÷
−
S
ms
pb
2
RS
Me
STATOR
+
−
+
Rr
1 Ψ rx
s
LS +
−
g
ML
−
1
sJ
I rx
Ωm
LM
+
−
pb
− LM g
−
−
1
s
Ψ ry
LS g
Rr
I ry
g=
1
1
=
w Ls Lr − L2M
ROTOR
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:
M e = pb
⎞
mS ⎛ LM 1 *
Im⎜⎜
Ψ rxy Ψ Sxy ⎟⎟
2
⎠
⎝ Lr σLs
(2.40)
And it gives the relation:
M e = pb
m L 1
mS LM 1
Ψ ryΨ S = pb S M
Ψ r Ψ S sin γΨ
2 Lr σLS
2 Lr σLS
(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:
U Sx = RS I Sx +
dΨ S
,
dt
(2.42)
U Sy = RS I Sy + ΩΨ S Ψ S ,
(2.43)
Rearranging Eq. (2.39) into form:
21
I Sy = M e
2
pb mSΨ S
(2.44)
Substituting I Sy to equation (2.43) electromagnetic torque can be calculated:
(
M e = U Sy − ΩΨ SΨ Sx
) p 2mRΨ
b
S
Sx
,
(2.45)
S
Therefore, the simplified model presented in (Fig. 2. 10) can be constructed.
I Sx
RS
U Sx
−
+
Ω SΨ
U Sy
Ψ Sx
1
s
S
−
+
ms
pb
2 RS
Me
+
ML
−
1
sJ
Ωm
Fig. 2. 10. Simplified model of VSI-fed IM - in SFOC synchronous xy coordinates for DTCSVM 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, U L is a line voltage space
vector, I L is a line current space vector, U p is the VSR input voltage space vector,
and U i is a space vector of voltage drop on the input (AC line side) choke L and it
resistance R .
The U p voltage is controllable and depends on switching signals pattern and DClink voltage level. Thanks to control magnitude and phase of the U p voltage, the line
current can be controlled by changing the voltage drop on the input choke - U i .
Therefore, inductances between line and AC side of the VSR are indispensable. They
22
create a current source and provide boost feature of the VSR. Through controlling
the converter AC side voltage in its phase and amplitude U p , the phase and
amplitude of the line current vector I L is controlled indirectly.
a)
AC- side
ULA
VSR
R
L
DC- side
Iload
Idc
Ic
UpA
ULB
UpB
ULC
UpC
C
Rload
b)
UL
IL
L
R
Up
Ui
Fig. 2. 11. Voltage source rectifier topology: a) three phase system; b) single phase equivalent
circuit
Rectifying
a)
Inverting
b)
RI L
UL
IL
j ωLI L
Up
j ωLI L
Up
RI L
UL
IL
c)
d)
UL
RI L
IL
Up
j ωL I L
j ωLI L
Up
RI L
IL
UL
Fig. 2. 12. Pictorial phasor diagrams for VSR: a,b) non unity power factor; c,d) unity power
factor operation
23
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 U p is higher during
regeneration than in rectifying mode. With assumption of a stiff line power (i.e., U L
is a pure voltage source with zero internal impedance) terminal voltage of VSR U p
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 DClink 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:
U dc > 2 3U LRMS
(2.46)
(For example: for U LRMS =230V; U dc > 2.45 ⋅ 230 = 564V ).
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 ∆I Lxy :
∆I Lxy =I Lxyc −I Lxy
(2.47)
The direction and velocity of the line current vector changes are described by
derivative of that current L
dI Lxy
dt
. It can be represented by equations in synchronous
rotating xy coordinates (Section 2.5.4, Eq.(2.74)):
L
dI Lxy
dt
+ R(I Lxyc − ∆I Lxy ) = U Lxy − U dcS1xy + jωL L(I Lxyc − ∆I Lxy )
(2.48)
With assumptions that resistance of the input chokes R ≅ 0 and actual current is
close to commanded value ( ∆I Lxy ≅ 0 ), Eq. (2.48) can be simplified to:
L
dI Lxy
dt
= U Lxy − U dcS1xy + jωL LI Lxyc
24
(2.49)
Based this equation the direction and velocity of the line current vector changes
depends on:
•
values of input chokes L ,
•
line voltage vector U Lxy ,
•
line current vector I Lxy ,
•
value of the DC-link voltage U dc ,
•
switching states of the VSR S1xy .
Let us consider that six active vectors ( U pc( 1− 6 ) ) of the VSR rotate clockwise in
synchronous xy coordinates. For each voltage vector ( U pc( 0− 7 ) ) the current
derivatives multiplied by L are denoted as ( PU( 0− 7 ) ) [117], [118]. Graphical
illustration of the Eq. (2.49) is shown in Fig. 2. 13.
y
Upc6
U pc1
PU 1
PU 6
ε
U Lxy
− jω L L I Lxyc
PU 5
PU 0,7
PU 2
U pc2
U pc5
U pc0,7
PU 3
x
PU 4
U pc3
Upc4
Fig. 2. 13. Graphical illustration of the Eq. (2.49) – instantaneous position of vectors [117]
Command current I Lxyc is in phase with line voltage vector U Lxy and it lies on the
axe x . The difference between actual current I Lxy and commanded I Lxyc is defined
by Eq. (2.47) and is illustrated in Fig. 2. 14.
25
y
I Lxyc
PU 1
PU6
UL
∆I Lxy ε
x
I Lxy
PU 0,7
PU 2
PU 3
PU 5
PU4
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 ε = π . Fig. 2. 13
shows that for such case ε created by PU1 , PU 2 , U pc1 , and U pc 2 vectors, are the
arms of the equilateral triangle. Therefore, based on the equation for its altitude the
boundary condition can be defined as:
U Lxy + jωL LI Lxyc =
3
U pxy
2
(2.50)
Assuming that: U Lxy = U Lm , I Lxyc = I Lmc and U pxy = 23 U dc the following expression
can be derived:
2
U Lm
+ ( ωL LI Lmc )2 =
32
U dc
2 3
(2.51)
After rearranging one obtains dependence for minimum DC-link voltage:
(
2
U dc min = 3 U Lm
+ ( ωL LI Lmc )2
)
(2.52)
(For example with parameters as: U Lm = 230 2 V, ω L = 2π 50 , L = 0.01 H ,
I Lmc = 10 A , then U dc min ≥ 566 V ).
Based on this relation the maximum value of the input inductance can be
calculated as:
26
Lm =
1 2
2
U dc − (U Lm )
3
ωL I Lmc
(2.53)
(For example with parameters as: U Lm = 230 2 V, ωL = 2π 50 , I Lmc = 10 A , and
U dc min = 566 V then maximum input line inductance is: Lm = 0.01 H ).
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:
U L = Ui + U p
(2.54)
Where, voltage drop vector U i on the line choke is defined as:
Ui = L
dI L
+ RI L
dt
(2.55)
Moreover, AC side voltage of the VSR can be described by Eq. (2.3) and Eq. (2.4)
or by:
1 C
⎛
⎞
U p = U dc ⎜ Sk − ∑ S k ⎟
3 k=A ⎠
⎝
(2.56)
Where, S k = “0”, or “1”, are switching states of the VSR ( k = A, B ,C ) for
appropriate line phase.
From the other hand the DC-link capacitor current equation can be expressed as:
C
dU dc
= Ic ,
dt
(2.57)
The capacitor current can be calculated as the difference between DC current I dc
of the VSR and input DC current I load of the VSI:
I c = I dc − I load ,
(2.58)
Moreover, I dc current can be calculated as a sum of a product of the phase current
and appropriate switching states:
27
I dc = I LA S A + I LB S B + I LC SC ,
(2.59)
Hence, AC side of the VSR voltage equations in the three phase system can be
expressed as:
L
dI LA
1 C
1
⎛
⎞
⎛
⎞
+ RI LA = U LA − U dc ⎜ S A − ∑ Sk ⎟ = U LA − U dc ⎜ S A − (S A + S B + SC )⎟
dt
3 k=A ⎠
3
⎝
⎠
⎝
(2.60)
L
1 C
1
dI LB
⎛
⎞
⎛
⎞
+ RI LB = U LB − U dc ⎜ S B − ∑ S k ⎟ = U LB − U dc ⎜ S B − (S A + S B + SC )⎟
3 k=A ⎠
3
dt
⎝
⎠
⎝
(2.61)
L
dI LC
1 C
1
⎛
⎞
⎛
⎞
+ RI LC = U LC − U dc ⎜ SC − ∑ S k ⎟ = U LC − U dc ⎜ SC − (S A + S B + SC )⎟
3 k=A ⎠
3
dt
⎝
⎠
⎝
(2.62)
Where the line voltages are expressed by:
U LA = U Lm sin(ω Lt ), U LB = U Lm sin(ωLt − 2π / 3), U LC = U Lm sin(ωLt + 2π / 3)
(2.63)
Also DC-link side equation of the VSR can be modeled by:
C
C
dU dc
= ∑ I Lk S k − I load = I LA S A + I LB S B + I LC SC − I load
dt
k=A
(2.64)
The above expressions can be written as:
(L
C
1 C
d
⎛
⎞
+ R )I Lk = U Lk − U dc ⎜ S k − ∑ S k ⎟
3 k=A ⎠
dt
⎝
C
dU dc
= ∑ I Lk S k − I load
dt
k=A
(2.65)
(2.66)
The equations (2.65) and (2.66) can be represented as a block diagram in Fig. 2. 15
[13], [16].
28
U LA
+
−
U pA
SA
U LB
+
−
U pB
SB
1
sL + R
I LA
fA
+
1
sL + R
I LB
I load
−
+
1
sC
U dc
−
+
+
+
+
+
fB
+
+
1
3
−
U LC
+
−
SC
1
sL + R
U pC
I LC
+
fC
−
Fig. 2. 15. Block diagram of the VSR in three-phase ABC coordinates
Where:
1
1
1
⎛
⎞
⎛
⎞
⎛
⎞
f A = ⎜ S A − (S A + S B + SC )⎟ , f B = ⎜ S B − (S A + S B + SC )⎟ , fC = ⎜ SC − (S A + S B + SC )⎟
3
3
3
⎝
⎠
⎝
⎠
⎝
⎠
(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:
L
dI L
+ RI L = U L − U dcS1
dt
C
dU dc 3
*
= Re I LS1 − I load
dt
2
[
(2. 68)
]
(2. 69)
Further, those equations can be decomposed in α and β components:
29
L
L
C
dI Lα
+ RI Lα = U Lα − U dc Sα ,
dt
dI Lβ
dt
(2.70)
+ RI Lβ = U Lβ − U dc S β ,
(2.71)
β
3
dU dc
= ∑ I Lk S k − I load = (I Lα Sα + I Lβ S β ) − I load .
2
dt
k =α
(2.72)
Where, appropriate switching states are expressed as:
1
1
⎛
⎞
(S B − SC )
Sα = f A = ⎜ S A − (S A + S B + SC )⎟ , S β =
3
3
⎝
⎠
(2.73)
Equations (2.70) - (2.72) can be represented as a block diagram in stationary αβ
coordinates as in Fig. 2. 16.
U pα
U Lα + −
U Lβ +
−
U pβ
Sα
I load
1
sL + R
I Lα
1
sL + R
I Lβ
+
3
+ 2
+
−
1
sC
U dc
Sβ
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:
L
dI Lxy
dt
+ RI Lxy = U Lxy − U dcS1xy + jωL LI Lxy
30
(2.74)
C
[
]
dU dc 3
= Re I LxyS*1xy − I load
dt
2
(2.75)
And after decomposition into x and y components yields:
L
L
C
dI Lx
+ RI Lx = U Lx − U dc S x − ωL LI Ly
dt
dI Ly
dt
(2.76)
+ RI Ly = U Ly − U dc S y + ωL LI Lx
(2.77)
dU dc 3 y
= ∑ I Lk S k − I load = I Lx S x + I Ly S y − I load
2 k=x
dt
(2.78)
Further, based on equations (2.76) - (2.78) a block diagram can be constructed as
in Fig. 2. 17.
U px
U Lx + −
U Ly +
I Lx
1
sL + R
−
ωL
Sx
I load
+
+
3 +
2
−
1
sC
U dc
L
+
−
U py
1
sL + R
I Ly
Sy
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 U L or as in Section 3.3 with line virtual flux vector – Ψ L . Therefore,
for VF oriented xy coordinates can be written: U px = 0 , U py = U Lm .
Note that U pxy = U dcS1xy .
31
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 U dc 0 ) is equal to commanded value of this variable
e.g U dc 0 = U dcc .
32
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].
UL
VM
VSR
Control of VSR
V-FOC
DPC
DPC-SVM
U dc I dc
I
C I c load
FOC
DTC
DTC-SVM
VSI
Control of VSI
IM 3 ~ U S
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
•
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 Ω K = ΩΨ r angular speed are selected. In this case the coordinate system is
oriented along d rotor flux linkage component (Fig. 3. 2) such as:
Ψ r = Ψ r = Ψ rd ,
(3. 1)
34
β
q
IS
I Sq
B
Ψ rq = 0
ΩΨ
d
γI
S
I Sd
γΨ r
r
Ψ r = Ψ r = Ψ rd
α
A
C
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 LM
⎜⎜
⎝ Lr
⎞
dΨ rd Rr
⎟⎟ I Sd =
+ Ψr
dt
Lr
⎠
(3. 2)
⎛ Rr LM
⎜⎜
⎝ Lr
⎞
⎟⎟ I Sq = (ΩΨr − pb Ω m )Ψ r
⎠
(3. 3)
dΩ m 1 ⎡⎛ ms LM
(Ψ r I Sq )⎞⎟⎟ − M L ⎤⎥
= ⎢⎜⎜ pb
dt
J ⎣⎝
2 Lr
⎠
⎦
(3. 4)
The block diagram of the IFOC is presented in Fig. 3. 3. The commanded
electromagnetic torque M ec , is delivered from outer PI speed controller, based on
mechanical speed error eΩ m . From Eq. (2.27) the q component of the command
stator current can be calculated as:
I Sqc =
2 Lr
M ec
pb mS LM Ψ rdc
(3. 5)
Assuming steady state conditions, from Eq. (3. 2) the d component of the
command stator current can be computed as:
I Sdc =
1
Ψ rdc
LM
(3. 6)
Then, command values I Sdc and I Sqc are compared with actual values of current
component I Sd and I Sq respectively. It should be stressed that (for steady state) I Sd
35
is equal to the magnetizing current, while the torque both dynamic and steady states
is proportional to I Sq .The current errors eI Sd and eI Sq are fed to two PI controllers,
which generate commanded stator voltage components U Sqc , and U Sdc , respectively.
Further, commanded voltages are converted from rotating dq coordinates into
stationary αβ coordinates using rotor flux vector position angle γΨ r . So obtained
voltage vector U Sc is delivered to space vector modulator – SVM which generates
appropriate switching states vector S 2 ( S2 A , S 2 B , S2C ) for control power transistors of
the VSI.
The stator voltage equations in dq coordinates is expressed as:
U Sdq = Lσ
dI Sdq
dt
+ Rim I Sdq + jΩΨ r Lσ I Sdq −
LM
Lr
⎞
⎛ Rr
⎜⎜ − jpb Ω m ⎟⎟Ψ rdq , (3. 7)
⎠
⎝ Lr
and can be decomposed into the d and q components:
U Sd = Lσ
U Sq = Lσ
dISd
L R
+ Rim I Sd − ΩΨ r Lσ I Sq − M 2 r Ψ rd ,
dt
Lr
dI Sq
dt
+ Rim I Sq + ΩΨ r Lσ I Sd +
LM pb Ω m
Ψ rd ,
Lr
(3. 8a)
(3. 8b)
where:
L2r RS + L2M Rr
Lσ = σLS , and Rim =
.
L2r
(3.9)
It can be seen that U Sdc and U Sqc are coupled each other.
The part of Eq. (3.7) with rotor flux Ψ rdq 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 , L r ),
•
the d and q voltage components are coupled each other, therefore
decoupling in control circuit is required,
•
coordinate transformations are needed,
36
•
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).
V-FOC
I Ly
I Lx
K Pc =
γΨ
2
3ωLΨ Lx
I Lxc −e
U pxc
I Lx
+
PI
2
3ω LΨ Lx
U dcc eU dc
Pc
PI
+
U dc −
Ω mc
eΩm
+
Ωm −
K Pc
Ψ rc
PI
1
LM
M ec
K Isqc
I Lyc
+
−
eI Ly
eI Sd
I Sdc +
−
eI Sq
I Sqc +
U pc
PI
DN
PI
U Sqc
VSR
S1
SVM
αβ
U pyc
U Sdc
UL
IL
xy
DN
PI
VM
U dc
D1
Qc = 0
IFOC
L
Current
Transformation
& Virtual Flux
Estimation
C1
C2
dq
U Sc
SVM
VSI
S2
αβ
−
γΨ
r
I Sd
I Sq
2 Lr
K Isqc =
pb ms LMΨ rdc
M ec
Current
Transformation
& Rotor Flux
Angle
Estimation
Ωm
IS
IM
U dc
Fig. 3. 3. Virtual flux oriented control – V-FOC and indirect field oriented control – IFOC; where
DN is decoupling network
37
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 ( Ω K = 0 ):
Ψ S = Ψ S = Ψ Sx ,
y
(3. 10)
β
ΩΨ
ΨS
B
γΨ
A
α
Ψr
C
S
x
Sector 1
ΩΨ
r
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 M ec is
delivered from outer PI speed controller. Then, M ec and commanded stator flux
Ψ Sc amplitudes are compared with estimated values of M e and Ψ S respectively. The
torque eM and flux eψ errors are fed to two hysteresis comparators.
From predefined switching table, based on digitized error signals S M and SΨ ,
and the stator flux position γΨ S the appropriate voltage vector is selected. The
outputs from the predefined switching table are switching states S 2 for the VSI.
Please consider that for electromagnetic torque the hysteresis is defined as:
SM = 1
for eM > H M
(3. 11)
SM = 0
for eM = 0
(3. 12)
S M = −1
for eM < − H M
(3. 13)
And for stator flux the hysteresis is described as:
38
SΨ = 1
for eΨ > HΨ
(3. 14)
SΨ = 0
for eΨ < − HΨ
(3. 15)
Where H M and HΨ 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].
a)
b)
SΨ
SM
eM
eψ
HΨ
HM
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:
π
j ( n −1)
2
3
U Sc( n ) = U dc e
, n = 1,...,6,
3
U Sc( n ) = 0,
(3. 16)
n = 0,7.
(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:
Sector 2:
Sector 3:
−
π
6
π
2
π
6
Sector 4:
5π
5π
< γΨ S < −
,
6
6
,
Sector 5:
−
5π
,
6
Sector 6:
−
< γΨ S <
< γΨ S <
< γΨ S <
π
2
π
6
,
39
π
5π
< γΨ S < − ,
6
2
π
2
< γΨ S < −
π
6
,
β
α
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:
M e = pb
mS LM 1
Ψ r Ψ S sin γΨ ,
2 Lr σLS
(3. 18)
where, the angle between stator and rotor flux vectors is defined as below:
γΨ = γΨ − γΨ
S
(3. 19)
r
From Eq. (3.18) it can be seen that the electromagnetic torque depends on
amplitudes of stator and rotor fluxes and angle between them γΨ . Thanks to long
rotor time constant the angle γ Ψ can be controlled by fast change of stator flux
vector position (Fig. 3. 4). Under assumption that the stator resistance Rs is zero, the
stator flux can be easy expressed as a function of a stator voltage:
dΨ S
= US .
dt
(3. 20)
Or in the form:
Ψ S = ∫ U S dt ,
(3. 21)
Under appropriate active voltage vector stator flux vector position change (the
angle γ Ψ ) in forward direction causing increase of the torque. During zero voltage
40
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.
y
β
USc0 USc7
B
USc3
USc2 x
USc4 Ψ S
USc1
USc5
A
Ψr
USc6
α
Sector 1
C
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 γ Ψ can be
increased by selecting vectors U Sc 2 , and U Sc 3 or decreased by vectors U Sc 5 , and U Sc 6 .
When any zero-voltage U Sc 0 or U Sc 7 is applied the stator flux is not changed. When
the angle is increasing then the torque M e is increasing. When the angle is
decreasing then the toque M e 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 ( Ω m > 0.2Ω N ) [34] the increasing of the
angle γ Ψ is slower then the decreasing.
41
Tab. 3. 1. Optimal switching table.
SΨ
SM
1
1
0
-1
1
0
0
-1
Sector 1
Sector 2
Sector 3
Sector 4
Sector 5
Sector 6
USc2
(1,1,0)
USc7
(1,1,1)
USc6
(1,0,1)
USc3
(0,1,0)
USc0
(0,0,0)
USc5
(0,0,1)
USc3
(0,1,0)
USc0
(0,0,0)
USc1
(1,0,0)
USc4
(0,1,1)
USc7
(1,1,1)
USc6
(1,0,1)
USc4
(0,1,1)
USc7
(1,1,1)
USc2
(1,1,0)
USc5
(0,0,1)
USc0
(0,0,0)
USc1
(1,0,0)
USc5
(0,0,1)
USc0
(0,0,0)
USc3
(0,1 0)
USc6
(1,0,1)
USc7
(1,1,1)
USc2
(1,1,0)
USc6
(1,0,1)
USc7
(1,1,1)
USc4
(0,1,1)
USc1
(1,0,0)
USc0
(0,0,0)
USc3
(0,1,0))
USc1
(1,0,0)
USc0
(0,0,0)
USc5
(0,0,1)
USc2
(1,1,0)
USc7
(1,1,1)
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.
42
VF-DPC
P
Power
& Virtual Flux
Estimation
Q
γΨ
+
U dcc eU dc
+
U dc −
PI
Qc = 0
Pc
+
+
Ψ Sc
eΩm
+
Ωm −
eQ
ep
Sector
Selection
U dc
Switching
Table
S1
IL
SQ
SP
UL
VSR
C1
DTC
Ω mc
−
L
VM
PI
M ec
eψ
−
eM
C2
SΨ
SM
−
Sector
Selection
γΨ
ΨS
Me
VSI
S2
Switching
Table
IS
S
Torque
& Stator Flux
Estimation
Ωm
IM
U dc
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 ( Ω K = ΩΨS ). In this case the coordinate system is
oriented with x stator flux linkage component (Fig. 3. 9) such as:
43
Ψ S = Ψ S = Ψ Sx ,
(3. 22)
β
y
ΩΨ
S
x
B
Ψ Sy = 0
γΨ S
Ψ S = Ψ S = Ψ Sx
α
A
C
Fig. 3. 9. Stator flux oriented xy coordinates
The commanded electromagnetic torque M ec is delivered from outer PI speed
controller (Fig. 3. 10). Then, M ec and commanded stator flux Ψ Sc amplitudes are
compared with estimated actual values of M e and Ψ S . The torque eM and flux eψ
errors are fed to two PI controllers. The output signals are the command stator
voltage components U Syc , and U Sxc respectively.
Further, voltage components in rotating xy system of coordinates are transformed
into αβ stationary coordinates using γΨS flux position angle. Obtained voltage
vector U Sc is delivered to space vector modulator – SVM which generates
appropriate switching states vector S 2 ( S 2 A , S 2 B , S 2C ) for the VSI.
An exhausting description of the DTC-SVM can be found in [152].
44
DPC-SVM
P
Power
& Virtual Flux
Estimation
Q
γΨ
U dcc eU dc
PI
+
−
U dc
Pc
+
U dc
+
Ωm −
m
PI
ep
PI
PI
U pqc
pq
U pc
U ppc
SVM
VSR
S1
αβ
C1
DTC-SVM
Ω mc eΩ
−
eQ
UL
IL
L
D1
+
Qc = 0
−
VM
Ψ Sc
M ec
+
+
eψ
−
eM
PI
PI
U Sxc
C2
xy
U Sc
U Syc
SVM
VSI
S2
αβ
−
γΨ
ΨS
Me
D2
IS
S
Torque
& Stator Flux
Estimation
Ωm
IM
U dc
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
45
UPF condition is fulfill when the line current vector, I L = I Lx + jI Ly , is aligned with
the phase voltage vector, U L = U Lx + jU Ly , 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 Ψ L is used (Fig. 3. 11). The
vector of VF lags the voltage vector by 90o . For the UPF condition, the command
value of the direct component current vector I Lxc , is set to zero. Command value of
the I Lyc 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 γΨL . Switching signals vector S1 , for the VSR is generated
by a space vector modulator.
β
y
UL
IL
I Ly
B
Ψ Ly = 0
ΩΨ
x
ϕ
I Lx
γΨ L
L
Ψ L =Ψ L =Ψ Lx
α
A
C
Fig. 3. 11. Synchronous rotating reference frame xy with line virtual flux angular
frequency ΩΨ L
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].
46
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.
I Lxc
I Lx
−
eI Lx
U Lx = 0
U Lx = 0
PI
U px − +
U ixc − + U pxc
−
+
1
sL + R
I Lx
1
sL + R
I Ly
DN
ωL
I Lyc
eI Ly
PI
ωL
L
U iyc − − U pyc
U py −
+
+
U Ly = U Lm
+
U Ly = U Lm
−
I Ly
L
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 U pxyc = U dc S xy would be generated as follows:
U pxc = U Lx − L
U pyc = U Ly − L
dI Lx
− RI Lx + ωL LI Ly
dt
dI Ly
dt
− RI Ly − ω L LI Lx
(3. 23)
(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.
47
U px −
U Lx = 0
+
U py −
+
1
sL + R
I Lx
1
sL + R
I Ly
U Ly = U Lm
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 TS . It could be done by sample & hold – S&H block. Moreover, the
statistical delay of the PWM generation TPWM = 0.5TS should be taken into account
(block VSC). In the literature [13], [83] the delay of the PWM is approximated from
zero to two sampling periods TS . Further, K C = 1 is the VSC gain, τ 0 is a dead time
of the VSC ( τ 0 = 0 for ideal converter).
U Ldist
I Lcc
1
sTpf + 1
I Lc +
1
−
sTS + 1
K Pi1 (sTIi1 + 1)
sTIi1
K c e − sτ 0
+
sTPWM + 1
−
1
sL + R
IL
Fig. 3. 14. Block diagram for a decoupled current control loop in the synchronous rotating
reference frame
48
The block S&H and VSC could be joined in one block S&H+VSC. Where, sum of
their time constants is expressed by:
τ Σi = TS + TPWM .
(3. 25)
So, the model of Fig. 3. 14 can be simplified as shown in Fig. 3. 15.
I Lcc
I Lc +
1
sTpf + 1
−
K Pi1 (sTIi1 + 1)
sTIi1
U Ldist
− sτ 0
Kce
sτ Σ + 1
+
−
K RL
sTRL + 1
IL
Fig. 3. 15. Modified block diagram of Fig. 3. 14
Please note that, τ Σi is a sum of small time constants, TRL = L / R is a large time
constant and K RL = 1 / R is a gain of input choke. Hence, TRL gives a dominant pole.
Between several methods of analysis, there are two simple way for the controller
parameters design: modulus optimum - MO and symmetry optimum – SO [134]. With
assumption that disturbance U L = const . , open loop transfer function can be derived
as:
GOi (s ) =
K Pi1 (1 + sTIi1 )K RL
sTIi1 (sτ Σi + 1)(sTRL + 1)
(3. 26)
With simplification (sTRL +1) ≈ sTRL [63] gives following closed-loop transfer
function:
GZi (s ) =
K Pi1 K RL (1 + sTIi1 )
K Pi1 K RL (1 + sTIi1 ) + s 2TIi1TRL + s 3TIi1τ ΣiTRL
(3. 27)
For the Eq. (3.27) the proportional gain and integral time constant of the PI
current controller can be calculated:
K Pi1 =
TRL
2τ Σi K RL
(3. 28)
TIi1 = 4τ Σi
(3. 29)
which, substituted in Eq. (3.26), yields open loop transfer function of the form:
49
(1 + s 4τ Σi )K RL
TRL
2τ Σi K RL s 4τ Σi (sτ Σi + 1)(sTRL + 1)
GOi (s ) =
(1 + s 4τ Σi ) = (1 + s 4τ Σi ) .
T
≈ RL
2τ Σi s 4τ Σi (sτ Σi + 1)sTRL s 2 8τ Σ2i + s 3 8τ Σ3i
(3. 30)
For the closed loop transfer function:
GCi (s ) =
1 + s 4τ Σi
.
1 + s 4τ Σi + s 2 8τ Σ2i + s 3 8τ Σ3i
(3. 31)
To compensate for the forcing element in the numerator, the first order prefilter
can be used:
G pfi (s ) =
1
1 + sT pf
(3. 32)
Where, Tpf = 4τ Σi .
So that Eq. (3.31) takes a form:
GCif (s ) = GCi (s )G pfi (s ) =
1
.
1 + s 4τ Σi + s 8τ Σ2i + s 3 8τ Σ3i
2
(3. 33)
This relation can be approximated by first order transfer function as:
GCif (s ) ≅
1
.
1 + s 4τ Σi
(3. 34)
For the comparison parameters of the PI current controller calculated based on
MO are defined as:
K Pi1 =
TRL
2τ Σi K RL
(3. 35)
TIi1 = TRL ,
(3. 36)
GOi (s ) =
=
1
2τ Σi
(1 + sTRL )K RL
TRL
2τ Σi K RL sTRL (sτ Σi + 1)(sTRL + 1)
1
1
=
s (sτ Σi + 1) s 2τ Σi + s 2 2τ Σ2i
(3. 37)
This relation can be approximated by first order transfer function as:
GCif (s ) ≅
1
.
1 + s 2τ Σi
(3. 38)
Tuning of the regulators based on Eqs. (3.37) and (3.38) (MO) gives good
response to a step change of reference (4% overshoot [65]). But the answer for the
step of a disturbance signal is not satisfactory (about 3 TRL [64]) whereas, the SO has
50
much better capabilities in transient for disturbance signal step. Therefore, it is better
to choice the SO [64] for further consideration.
In Fig. 3. 16 is shown reference step at t = 0.1 [s] and disturbance step at
t = 0.11 [s] .
1.5
1.5
1
1
0.5
0.5
0
0.098
0.1
0.102
0.104
0.106
0.108
0.11
0.112
0.114
0.116
0
0.096
0.098
0.1
0.102
0.104
0.106
0.108
0.11
0.112
0.114
0.116
Fig. 3. 16. Simulated (Matlab Simulink) active current response for controller parameters
designed according to: 1) symmetry optimum – SO, and 2) modulus optimum – MO;
a) without prefilter, b) with prefilter
Simulations in Saber show that the answer is little bit different. Fig. 3. 17 presents
response to a step change of reference ( t = 0.4 [s] ) in complete Saber model without
decoupling in control system. In time t = 0.41 [s] the answer to the disturbance step
change is shown. The difference is caused because of presence of the coupling
between x and y axes it real VSR and in Saber model.
Fig. 3. 17. Simulated (Saber) active power response without decoupling;
a) without prefilter, b) with prefilter (used in practice)
From the top: command and estimated active power, command and estimated reactive power
51
Therefore, decoupling in current control feedback should be introduced as
mentioned before. The answer of the system with implemented decoupling in control
system is presented in Fig. 3. 18.
Fig. 3. 18. Simulated (Saber) active current response with decoupling.
a) without prefilter, b) with prefilter(used in practice)
The response is closer to continuous response in Fig. 3. 16 obtained in Matlab.
However, there is still a difference. It is caused by not fully decoupled signals and
sampling time of discrete control system.
It could be concluded that the prefilter for SO is needed because significantly
reduction of the overshoot in transient to reference step but the answer for an
disturbance step is unchanged. (More about prefilter please see Subsection 3.3.3.1).
3.3.2. VF based Direct Power Control – VF-DPC
Direct power control – DPC proposed in [105] assures the instantaneous active
and reactive power control technique directly by selecting the optimum switching
state of the VSR. Moreover, line voltage sensors are eliminated. In a case of large
power, when rectifiers are installed in the plant, it is desirable to omit adjustment of
the voltage sensors and feedback signal lines. In the DPC there are no current
controllers and line voltage sensors. The control is performed by selecting an
optimum switching state of the VSR, so that the active and reactive powers errors are
restricted into appropriate hysteresis bands. It is possible by using switching table
and hysteresis comparators. Moreover, this method deals with instantaneous active
and reactive powers in obtaining the voltages, it is possible to estimate not only a
fundamental component, but also harmonic components of the voltage waveforms
[105].
52
Further, for DPC needs it is assumed that R is negligible small and the switching
devices in the VSR are ideal (i.e. they do not need dead-time and there is no forward
voltage drops).
From Fig. 3. 8 it can be seen that there are two power loops: for active P , and
reactive Q ones. Command active power Pc is controlled by DC-link voltage loop,
while the command reactive power Qc is given from the outside of the control
scheme. Usually reactive power is set to be zero, to obtain a unity power factor
operation. The DC-link voltage is maintaining to be constant by appropriate active
power adjustment. Estimated values of the active power P and reactive power Q are
compared with commanded values. The power errors eP and eQ are input signal to
hysteresis comparators. At the output of the comparators are digitized signals S P and
SQ .
In classical (voltage based) DPC from predefined switching table, based on
signals S P , and SQ , and position of the line voltage γ U L , the appropriate voltage
vector is selected. In virtual flux based VF-DPC instead of γ U L the position of the
virtual flux γΨ L are utilized in control algorithm. The outputs from the predefined
switching table (Tab. 3. 2) are the switching states S1 for the VSR. The powers
hysteresis is defined in the same way for active and reactive power, therefore only
for active power is given:
SP = 1
for eP > H P
(3. 39)
SP = 0
for eP < − H P
(3. 40)
Where, H P is a hysteresis band of the active power (Fig. 3. 19).
SP
HP
eP
Fig. 3. 19. Power hysteresis controller - two level
53
For DPC (similarly like DTC) the VSR input voltage is constructed base on
appropriate space vector (detailed described in Section 2.3). The voltage source
rectifier AC side voltage can be represented as a complex space vector as follows:
π
j ( n −1)
2
3
U pc( n ) = U dc e
, n = 1,...,6,
3
(3. 41)
U pc( n ) = 0,
(3. 42)
n = 0,7.
For this purpose, the stationary αβ coordinates are divided into six sectors, as
shown in Fig. 3. 20.
β
β
α
α
Fig. 3. 20. Voltage space vector plane divided into 6 sectors for VSR; a) DPC, b) VF-DPC
The sectors could be defined in different manner [93], [105] here is divided
similarly as in Subsection 3.2.2
The instantaneous apparent power S can be expressed by different manners (see
[65], [123] and Appendix A.3).
Tab. 3. 2. Switchnig table for DPC
SP
SQ
1
1
0
1
0
0
Sector 1
UPc7
(1,1,1)
UPc6
(1,0,1)
UPc1
(1,0,0)
UPc6
(1,0,1)
Sector 2
UPc0
(0,0,0)
UPc1
(1,0,0)
UPc2
(1,1,0)
UPc1
(1,0,0)
Sector 3
UPc7
(1,1,1)
UPc2
(1,1,0)
UPc3
(0,1,0)
UPc2
(1,1,0)
54
Sector 4
UPc0
(0,0,0)
UPc3
(0,1,0)
UPc4
(0,1,1)
UPc3
(0,1,0)
Sector 5
UPc7
(1,1,1)
UPc4
(0,1,1)
UPc5
(0,0,1)
UPc4
(0,1,1)
Sector 6
UPc0
(0,0,0)
UP5
(0,0,1)
UPc6
(1,0,1)
UPc1
(0,0,1)
The instantaneous active power P is a scalar product between the line voltages
and currents instantaneous space vectors, whereas the instantaneous reactive power
Q is a vector product between them, and they can be expressed in complex form as:
{
}
(3. 43)
{
}
(3. 44)
P=
3
3
3
*
Re U L I L = (U Lα I Lα + U Lβ I Lβ ) = U L ⋅ I L
2
2
2
Q=
3
3
3
*
Im U L I L = (U Lβ I Lα − U Lα I Lβ ) = U L × I L
2
2
2
There is possibility to estimate the line voltages by adding the input voltage
U p = U dcS1 of the VSR to the voltages drops on the input choke U I . Therefore,
active and reactive power of the line can be calculated in line voltage sensorless
manner as follows:
dI ⎞
dI ⎞
⎛
⎛
P = ⎜U dc S A + L LA ⎟ I LA + ⎜U dc S B + L LB ⎟ I LB +
dt ⎠
dt ⎠
⎝
⎝
dI ⎞
⎛
+ ⎜U dc SC + L LC ⎟ I LC
dt ⎠
⎝
(3. 45)
⎧ ⎛ dI LA
⎫
dI
⎞
I LC − LC I LA ⎟ +
1 ⎪3L⎜
⎪
Q=
dt
⎠
⎨ ⎝ dt
⎬
3⎪
⎪
⎩− U dc [S A (I LB − I LC ) + S B (I LC − I LA ) + SC (I LA − I LB )]⎭
(3. 46)
Such calculated power can be used as a feedback signals for DPC scheme. Please
consider that, power losses on the resistance of the input choke R are neglected
because they have low value in comparison to total active power.
Unfortunately, such calculation causes some problems in DSP implementation.
The differential operations of the currents are performed on the basis of finite
differences and gives very noisy signals. So, to suppress the current ripples a
relatively large inductance is needed. Moreover, calculation finite differences of the
currents should be as accurate as possible (about ten times per a switching period)
and should be avoided at the moment of the switching [105].
To avoid this problem a virtual flux – VF of the line has been introduced in [93],
[95]. The voltage in the line can be expressed with the formula:
dΨ L
= UL
dt
(3. 47)
After integration the VF can be expressed as:
55
Ψ L = ∫ U L dt + Ψ L 0
(3. 48)
Further, when the frequency of the rotating VF is constant, also the length of VF
and voltage are proportional to each other. Moreover, a phase position between VF
and voltage is 90o (lagging).
From analogy with IM the instantaneous active power can be expressed as:
P = MωL
(3. 49)
Where, M is an instantaneous virtual torque – VT and can be expressed as [63]:
M=
{
3
Im Ψ*L I L
2
}
(3. 50)
Then instantaneous active power is described by:
P=
{
}
3
Im Ψ*L I L ωL
2
(3. 51)
Moreover, instantaneous reactive power can be derived from following equation:
Q=
{
}
3
Re Ψ*L I L ωL
2
(3. 52)
After calculation in stationary αβ coordinates instantaneous active and reactive
power can be calculated as:
(
)
(3. 53)
(
)
(3. 54)
3
P = ωL Ψ Lα I Lβ −Ψ Lβ I Lα
2
3
Q = ωL Ψ Lα I Lα +Ψ Lβ I Lβ
2
Note that this estimation method is based on instantaneous variables. So, it gives
possibility to estimate harmonic components and as well as a fundamental
components. That implies a total power factor improvement.
The DPC is interesting for researcher thanks to its advantages as follows:
•
excellent dynamic performance of power control,
•
no current control loops,
•
no coordinates transformations,
•
pulse width modulator is not needed,
However, main known drawbacks of the DPC are:
•
high and variable switching frequency, which produce high VSR power
losses,
•
high input choke inductance required,
•
difficulties of EMI (LC input) filter design,
56
•
differentiation of the currents (in classical solution [105]),
•
violence of polarity consistency rules (+1 switching over DC link voltage).
Therefore, in the literature there is a lot of work which have a goal to improve the
features of the DPC, for instance another definition of switching table. As mentioned
earlier using a virtual flux – VF concept the drawback of current differentiation can
be eliminated. Moreover, reduction of the sampling frequency [93], and line
inductances less then 5% [95] in per unit values are obtained. All mentioned
difficulties can be successfully overcome when, instead of the switching table, a
PWM modulator is applied.
3.3.3. Direct Power Control with Space Vector Modulator – DPC-SVM
Direct power control with space vector modulation – DPC-SVM [157] guarantees
high dynamics and static performance via an internal power control loops. It is not
well known in the literature. This method joins the concept of DPC and V-FOC. The
active and reactive power is used as control variables instead of the line currents.
The DPC-SVM with constant switching frequency uses closed active and reactive
power control loops (Fig. 3. 10). The command active power Pc are generated by
outer DC-link voltage controller, whereas command reactive power Qc is set to zero
for unity power factor operation. These values are compared with the estimated P
and Q values respectively. Calculated errors e p and eQ are delivered to PI power
controllers. Voltages generated by power controllers are DC quantities, what
eliminates steady state error (PI controllers features), as well as in V-FOC. Then after
transformation to stationary αβ coordinates (Appendix A.2), the voltages are used
for switching signals generation by SVM block. The proper design of the power
controller parameters is very important. Therefore, analysis and synthesis will be
described in the followed Subsection.
3.3.3.1. Line Power Controllers
The assumptions are similar to that like in current controllers design in Subsection
3.3.1.1. In Fig. 3. 21 a block diagram for a simplified power control loop in the
synchronous xy rotating coordinates (Fig. 2. 17) is presented. Since, the same block
diagram applies to both P and Q power controllers, description only for P active
power control loop will be presented.
57
U Ldist
Pc +
K PP (sTIP + 1)
sTIP
1
−
sTS + 1
K c e − sτ 0
sτ Σp + 1
+
−
K RL
sTRL + 1
3
U L = const.
2
P
Fig. 3. 21. Block diagram for a simplified active power control loop in the synchronous rotating
reference frame
The model of Fig. 3. 21 can be modified as shown in Fig. 3. 22, where sum of the
small time constants is defined by:
τ Σp = TS + TPWM
Pc +
−
P
(3. 55)
K PP (sTIP + 1)
sTIP
K c e − sτ 0
sτ Σ p + 1
U Ldist
−
+
K RL
sTRL + 1
3
UL
2
Please note that, τ Σp is a sum of small time constants, TRL is a large time constant
of the input choke. The similar simplifications as in Subsection 3.3.1.1 are taken into
account. From several methods of design, symmetry optimum - SO is chosen because
its good response to a disturbance U Ldist step. For U L = const . the following open
loop transfer function can be derived:
GOP (s ) =
K RL K PP (1 + sTIP ) 3
UL
sTIP (sτ Σp + 1)(sTRL + 1) 2
With simplification
(sTRL +1) ≈ sTRL
(3. 56)
[63] gives following closed loop transfer
function for power control loop:
58
GZP (s ) =
3
K RL K PP (1 + sTIP )
UL
2
3
K RL K PP (1 + sTIP ) + s TIPTRL + s TIPτ ΣpTRL 2
(3. 57)
For this relation the proportional gain and integral time constant of the PI current
controller can be calculated as:
K PP =
TRL
2
2τ Σp K RL 3 U L
(3. 58)
TIP = 4τ Σp
(3. 59)
GOP (s ) =
K RL K PP (1 + sTIP ) 3
UL
sTIP (sτ Σp + 1)(sTRL + 1) 2
GOP (s ) =
K RL (1 + s 4τ p )
2TRL
3
UL
2τ Σp K RL 3U L s 4τ Σp (sτ Σp + 1)(sTRL + 1) 2
(1 + s 4τ Σp ) = (1 + s4τ Σp ) .
T
≈ RL
2τ Σi s 4τ Σp (sτ Σp + 1)sTRL s 2 8τ Σ2p + s 3 8τ Σ3p
(3. 60)
(3. 61)
For the closed loop transfer function:
GCP (s ) =
1 + s 4τ Σp
1 + s 4τ Σp + s 2 8τ Σ2p + s 3 8τ Σ3p
.
(3. 62)
Tuning of the regulators based on Eqs. (3.58) and (3.59) gives power tracking
performance with more then 40% overshoot as shown in Fig. 3. 24 caused by the
forcing element in the numerator (Eq. (3.62)). Therefore, for decreasing the overshot
(compensate for the forcing element in the numerator) a first order prefilter on the
reference signal can be used:
G pfp (s ) =
1
1 + sT pfp
(3. 63)
59
Where, Tpfp usually equals to a few τ Σp [64]. In further investigation a time delay of
the prefilter is set to 4τ Σp . So that Eq. (3.62) takes a form:
GCPf (s ) = GCP (s )G pfp (s ) =
1 + s 4τ Σp
1
.
+ s 8τ Σ2p + s 3 8τ Σ3p
(3. 64)
2
Hence, the block diagram of the control loop takes a form as in Fig. 3. 23. Relation
(3.64) can be approximated by first order transfer function as:
GCpf (s ) ≅
Pcc
1
sTpfp + 1
1
.
1 + s 4τ Σp
Pc +
−
`
(3. 65)
K PP (sTIP + 1)
sTIP
− sτ 0
Kce
sτ Σp + 1
U Ldist
−
+
K RL
sTRL + 1
3
UL
2
P
Fig. 3. 23. Power control loop with prefilter
Comparison of step answer in control loop without (a) and with (b) prefilter is shown
in Fig. 3. 24, and Fig. 3. 25. First one shows response (in Matlab Simulink) to a step
change of active power reference step at time t = 0.1 [s] whereas in t = 0.11 [s] the
disturbance step is applied.
60
Fig. 3. 24. Active power tracking performance (simulated in Matlab Simulink) controller
parameters designed according to SO; a) without prefilter, b) with prefilter
Discrete simulations (in Saber) show that the answer is little bit different.
Fig. 3. 25 presents response to a step change of reference ( t = 0.4 [s] ) in complete
Saber model. At time t = 0.41 [s] disturbance step is applied. The difference is
caused because of presence of the nonlinear coupling.
500.0
a)
500.0
b)
450.0
450.0
400.0
400.0
350.0
350.0
300.0
300.0
250.0
250.0
200.0
200.0
150.0
150.0
100.0
100.0
50.0
50.0
0.0
0.0
-50.0
-50.0
-100.0
-100.0
-150.0
-150.0
0.396 0.398 0.4
0.402 0.404 0.406 0.408 0.41 0.412
0.396
0.398
0.4
0.402
0.404
0.406
0.408
0.41
0.412
Fig. 3. 25. Active power tracking performance (simulated in Saber) without decoupling;
a) without prefilter, b) with prefilter
Therefore, decoupling in power control feedbacks should be introduced. Hence,
based on Eqs. (2.76) and (2.77) it could be clearly seen that command line voltages
should be generated as follows:
L
L
dI Lx
+ RI Lx + ω L LI Ly + U dc S x = U Lxc
dt
dI Ly
dt
+ RI Ly − ωL LI Lx + U dc S y = U Lyc
61
(3. 66)
(3. 67)
The step answer of the system with implemented decoupling in power control loop is
presented in Fig. 3. 26 (reference step at t = 0.4 [s] and disturbance step at
t = 0.41 [s] ).
500.0
a)
500.0
b)
450.0
450.0
400.0
400.0
350.0
350.0
300.0
300.0
250.0
250.0
200.0
200.0
150.0
150.0
100.0
100.0
50.0
50.0
0.0
0.0
-50.0
-50.0
-100.0
-100.0
-150.0
-150.0
0.396
0.398
0.4
0.402
0.404
0.406
0.408
0.41
0.412
0.396
0.398
0.4
0.402
0.404
0.406
0.408
0.41
0.412
Fig. 3. 26. Active power tracking performance (simulated in Saber) with decoupling;
without prefilter, b) with prefilter
The response is closer to ideal one obtained in Matlab. However, there is still
difference. It is caused by not fully decoupled signals and effect of sampling time TS
of discrete control system.
For better comparison with experimental results, the test under distorted line
voltage was performed. Command power has been changed from 1kW to 2.5 kW as
on the laboratory setup (see Chapter 6). The simulation result for this case is shown
in Fig. 3. 27 - Fig. 3. 28.
3500
3000
2500
2000
1500
1000
500
0
-500
0.094
0.096
0.098
0.1
0.102
0.104
0.106
0.108
0.11
0.112
0.114
Fig. 3. 27. Active power tracking performance (simulated) without prefilter; 1) command active
power, 2) estimated active power, 3) command reactive power, 4) estimated reactive power;
a) simulation in Matlab Simulink b) simulation in Saber
62
3500
3000
2500
2000
1500
1000
500
0
-500
0.094
0.096
0.098
0.1
0.102
0.104
0.106
0.108
0.11
0.112
0.114
Fig. 3. 28. Active power tracking performance (simulated) with prefilter; 1) command active
power, 2) estimated active power, 3) command reactive power, 4) estimated reactive power;
a) simulation in Matlab Simulink b) simulation in Saber
Please take into account the oscillations in Fig. 3. 27 - Fig. 3. 28. There are
generated by modeled line voltage distortion ( THDU L = 4 % of 5-th harmonics). This
harmonics after coordinate transformation to rotating coordinates gives AC
components with frequency six times higher then line voltage frequency (300 Hz)
and with amplitude U m 6 = 6.9 V. Hence, the question is appearing: how the sampling
frequency takes impact on the control parameters, and on the design of the power
controllers? Therefore, please take into consideration the following simulated results
presented in Fig. 3. 29 - Fig. 3. 30. The Fig. 3. 29 shows active and reactive power
tracking performance at different sampling frequency: (a,b) f s = 2.5kHz , and (c,d)
f s = 5kHz . The Fig. 3. 30 presents the same results at different sampling frequency:
(a,b) f s = 10kHz , and (c,d) f s = 20kHz respectively. The obtained results correspond
very well with experimental investigation (see Fig. 6. 5 and Fig. 6. 6).
63
a)
b)
3000.0
3000.0
2000.0
2000.0
1000.0
1000.0
0.0
0.0
0.392
0.4
0.408
0.416
0.424
0.392
3500.0
c)
3500.0
d)
3000.0
3000.0
2500.0
2500.0
2000.0
2000.0
1500.0
1500.0
1000.0
1000.0
500.0
500.0
0.0
0.0
-500.0
0.4
0.408
0.416
0.424
-500.0
0.388 0.392 0.396 0.4
0.404 0.408 0.412 0.416 0.42 0.424
0.388 0.392 0.396 0.4
0.404 0.408 0.412 0.416 0.42 0.424
Fig. 3. 29. Active and reactive power tracking performance (simulated in Saber) at different
sampling frequency: f s = 2.5kHz a) active power step, and b) reactive power
step; f s = 5kHz c) active power step and d) reactive power step
From the top: (a,c) command and estimated active power, command and estimated reactive power, (b,d)
command and estimated reactive power, command and estimated active power
Parameters of the power controllers derived according to SO for different values
of sampling frequency are shown in Tab. 3. 3.
Tab. 3. 3. Parameters of active and reactive power controllers
ULRMS= 141V;
K PP
f s [kHz]
2.5
0.0279
5
0.0557
10
0.11
20
0.22
50
0.44
TIP [s]
0.0024
0.0012
0.0006
0.0003
0.00015
Based on this comparison it can be concluded that for higher sampling frequency
distortion with fifth harmonics decay. Therefore, even for distorted line voltage the
line current will be very close to sinusoidal.
64
3500.0
a)
3500.0
b)
3000.0
3000.0
2500.0
2500.0
2000.0
2000.0
(W)
(W)
1500.0
1000.0
1000.0
500.0
500.0
0.0
0.0
-500.0
-500.0
0.388 0.392 0.396 0.4
0.404 0.408 0.412 0.416 0.42 0.424
0.388 0.392 0.396 0.4
3500.0
c)
3500.0
d)
3000.0
3000.0
2500.0
2500.0
2000.0
2000.0
(W)
(W)
1500.0
1500.0
1500.0
1000.0
1000.0
500.0
500.0
0.0
0.0
-500.0
0.404 0.408 0.412 0.416 0.42 0.424
-500.0
0.388 0.392 0.396 0.4
0.404 0.408 0.412 0.416 0.42 0.424
0.388 0.392 0.396 0.4
0.404 0.408 0.412 0.416 0.42 0.424
Fig. 3. 30. Active and reactive power tracking performance (simulated in Saber) at different
sampling frequency: f s = 10kHz a) active power step, and b) reactive power step;
f s = 20kHz c) active power step and d) reactive power step
From the top: (a,c) command and estimated active power, command and estimated reactive power, (b,d)
command and estimated reactive power, command and estimated active power
3.3.3.2. DC-link Voltage Controller
For DC-link voltage controller design, the inner current or power control loop can
be modeled with the first order transfer function (see Subsections 3.3.1.1 and
3.3.3.1).
The power control loop of voltage source rectifier – VSR can be approximated in
further consideration by first order block with equivalent time constant TIT .
G pz (s ) =
1
1 + sTIT
(3. 68)
Where, TIT = 2τ Σp for power controllers designed by MO criterion or TIT = 4τ Σp for
power controllers designed by SO criterion. Therefore, the DC-link voltage control
loop can be modeled as in Fig. 3. 31.
65
U dcc = const.
U dccc
K PU (sTIU + 1)
sTIU
U dcc +
1
sTpfu + 1
−
U dcf
1
1 + sTIT
Pload
PVSR −
+
Pcap
1
sCU dcc
U dc
1
sTU + 1
Fig. 3. 31. Block diagram for a simplified DC-link voltage control loop
The block diagram of Fig. 3. 31 can be modified as shown in Fig. 3. 32.
U dccc
U dcc +
1
sTpfu + 1
−
U dcf
K PU (sTIU + 1)
U dcc
sTIU
Pload
PVSR −
1
+
sTUT + 1
Pcap
1
sCU dcc
U dc
For simplicity it can be assumed that:
TUT = TU + TIT
(3. 69)
Where, TU is DC-link voltage filter time constant, TUT is a sum of small time
constants and CU dcc is an equivalent of integration time constant. So, the open loop
transfer function can be derived:
GUo (s ) =
K PU (sTIU + 1)
sTIU (sTUT + 1)sCU dcc
(3. 70)
This gives following closed loop transfer function:
GUz (s ) =
K PU + sTIU K PU
K PU (sTIU + 1)
+ s 2TIU CU dcc + s 3TIU CTUTU dcc
(3. 71)
The method of symmetrical optimum [134] is used to synthesize the DC-link voltage
controller. Therefore, square of the module of Eq. (3.71) takes a form:
66
(
)
K PU ω 2TIU2 + 1
GUz (ω ) =
M z (ω )
2
(3. 72)
Where:
2
M z (ω ) = K PU
+ ω 2TIU K PU (TIU K PU − 2CU dcc ) + ω 4TIU2 CU dcc (CU dcc − 2 K PU TUT ) +
+ ω 6 (TIU CU dccTUT )
2
ω –
frequency domain.
Hence, proportional gain K PU and integral time constant TIU of the DC-link voltage
controller can be calculated as follows:
K PU =
C
U dcc
2TUT
(3. 73)
TIU = 4TUT
(3. 74)
Please take into consideration that value of the DC-link voltage filter time constant
TU has to be determined. Theoretically it could be equal to one sampling period TS .
However, in practice the low pass filter in DC-link voltage loop is needed. Therefore,
for further consideration TU = 0.003 [s] .
660
640
620
600
(V)
580
560
540
520
500
480
460
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
Fig. 3. 33. Voltage disturbance compensation performance (simulated) for controller parameters
calculated according SO. Transient in load from zero to nominal (at t = 1.0 s ), and from
[]
[]
nominal to zero (at t = 1.1 s ); a) simulation in Matlab Simulink b) simulation in Saber
67
3.4. Conclusion
Based on discussion presented in this Chapter a brief comparison of control
techniques for AC/DC/AC converter-fed IM dives is given in Tab. 3. 4.
Among discussed control methods DTC-SVM and DPC-SVM seems to be most
attractive for further consideration. Because these methods connect well known
advantages of FOC and V-FOC with attractiveness of novel strategies such as
hysteresis based DTC and DPC. Therefore, for further considerations a shorter name
will be used for common control method of full controlled AC/DC/AC converter:
Direct power and torque control with space vector modulation – DPTC-SVM.
Tab. 3. 4. Comparison of control techniques for AC/DC/AC converter – fed IM drives
Feature
IFOC/V-FOC
DTC/VF-DPC
DPC-SVM/DTC-SVM
Constant switching freq.
Yes (5 kHz)
No
Yes (5 kHz)
SVM blocks
Yes
No
Yes
Coordinates transform.
Yes
No
Yes (only one)
Direct control of
(VSI side)
Stator currents
Torque, Stator Flux
Torque, Stator Flux
Estimation of
(VSI side)
Rotor flux angle
Torque, Stator Flux
Torque, Stator Flux
Rotor flux
Stator flux
Stator flux
Line currents
Line powers
Line powers
Estimation of
(VSR side)
Virtual flux
Powers, Virtual flux
Powers, Virtual flux
Coordinates orientation
(VSR side)
Virtual flux
Virtual flux
Virtual flux
Line voltage sensorless
Yes
Yes
Yes
Sampling frequency
5 kHz
50kHz
5kHz
Independence from
rotor parameters;
universal for IM and
permanent magnet
synchronous motor –
PMSM
No
Yes
Yes
Coordinates orientation
(VSI side)
Direct control of
(VSR side)
68
Chapter 4
4. Direct Power and Torque Control with Space Vector Modulation
– DPTC-SVM
4.1. Introduction
As mentioned in the Chapter 3 DPC-SVM and DTC-SVM seems to be most
attractive for control of the AC/DC/AC converter. When both methods are joined for
control of the AC/DC/AC converter direct power and torque control with space
vector modulation is obtained – DPTC-SVM.
In this Chapter DPTC-SVM scheme of AC/DC/AC converter-fed IM drive will be
considered and power flow between VSR and VSI side will be also analyzed. Some
techniques for reduction of the DC-link capacitor will be described. When the active
rectifier DC-link current I dc is equal to the DC-link inverter current I load in the
AC/DC/AC converter, no current will flow through the DC-link capacitor. As a
result, DC-link voltage will be constant.
However, in spite of very good dynamics behaviors of DPTC-SVM scheme, the
control of the DC-link voltage can be improved [23], [37], [81]. Therefore, active
power feedforward – PF from inverter side to rectifier side is introduced. The PF
deliver information about motor states directly to active power control loop of the
VSR. Thanks to faster control of power flow between VSR and VSI, the fluctuation
of the DC-link voltages will decrease. So, the size of the DC-link capacitor can be
significantly reduced.
4.2. Model of the AC/DC/AC Converter-Fed Induction Motor Drive with
Active power feedforward
In Fig.4. 1 is shown simplified diagram of the AC/DC/AC converter which consist
of VSR-fed DC-link and VSI-fed IM. Both VSR and VSI are IGBT bridge converter.
4. Direct Power and Torque Control with Space Vector Modulation – DPTC-SVM
VSR
UL
I dc
I load
U dc
Ic
VSI
US
3~
VM
IM
Fig.4. 1. Configuration of the AC/DC/AC converter-fed induction motor drive with PF loop
The mathematical models of the VSR and VSI are given in Chapter 2 and
description of the DPC-SVM, and DTC-SVM have been presented in Chapter 3.
Here the whole system with PF will be discussed and studied.
Note again that, the coordinates system for control of the VSR is oriented with VF
vector. Therefore, I Lxc is set to zero to meet the unity power factor – UPF condition.
With this assumption the VSR input power can be calculated as:
PVSR =
3
(I LxU px + I LyU py ) = 3 I LyU py
2
2
(4. 1)
Under steady states operation I Ly = const . and, with assumption that resistance of the
input choke is R = 0 , the following equation can be written:
PVSR =
3
I LyU Ly
2
(4. 2)
On the other hand the power consumed/produced by the VSI-fed IM is defined by:
PVSI =
3
(I SxU Sx + I SyU Sy )
2
(4. 3)
Another form of the above equations can be derived based on power Eq. (3.51) in
Subsection 3.3.2 where is clearly seen that the active power of the VSR is
proportional to the virtual torque - VT. Therefore, based on Subsection 3.3.2, the Eq.
(4.1) can be written that:
3
3
PVSR = ωL (Ψ Lx I Ly −Ψ Ly I Lx ) = ω LΨ Lx I Ly
2
2
(4. 4)
On the VSI-fed IM side electromagnetic power of the motor is defined by:
Pe = M e Ω m
(4. 5)
Taking into account Eq. (2.39) yields:
70
Pe = pb
mS
Ω mΨ Sx I Sy
2
(4. 6)
Moreover, it can be assumed (neglecting power losses) that electromagnetic power of
the IM is equal to an active power delivered to the motor Pe = PVSI , hence:
PVSI = pb
mS
Ω mΨ Sx I Sy
2
(4. 7)
But this is not sufficient assumption because of power losses Plosses in the real
system, so it should be written:
PVSI = pb
mS
Ω mΨ Sx I Sy + Plosses
2
(4. 8)
Further, please consider a situation at stand still (Ω m = 0) when nominal torque is
applied. In such a case the electromagnetic power will be zero but the IM power PVSI
will have a significant value. Estimation of this power is quite difficult, because the
parameters of the IM and power switches are needed. Hence, for simplicity of the
control structure a power estimator based on command stator voltage U Sc and actual
current I S will be taken into consideration:
PVSI =
3
(I SxU Sxc + I SyU Syc )
2
(4. 9)
4.2.1. Analysis of the Power Response Time Constant
Based on Eq. (3.68) the time constant delay of the VSR response TIT is
determined. With assumption that power losses of the converters can be neglected,
power tracking performance can be expressed by:
PVSR (s ) =
1
PVSRc
1 + sTIT
(4. 10)
Similarly for the VSI can be written:
PVSI (s ) =
1
PVSIc
1 + sTIF
(4. 11)
Where, TIF is the equivalent time constant of the VSI step response.
4.2.2. Energy of the DC-link Capacitor
The DC-link voltage can be described as (for more detail see Section 2.5.2):
71
dU dc 1
= (I dc − I load ),
dt
C
(4. 12)
So:
U dc =
1
(I dc − I load )dt ,
C∫
(4. 13)
Assuming the initial condition as in steady state, hence, the actual DC-link voltage
U dc is equal to commanded DC-link voltage U dcc . Therefore, Eq. (4.13) can be
rewritten:
U dc =
1
CU dcc
∫ (U
I − U dcc I load )dt =
dcc dc
1
CU dcc
∫ (P
dc
− Pload )dt ,
(4. 14)
Where, Pdc − Pload = Pcap , therefore is obtained:
U dc =
1
Pcap dt ,
CU dcc ∫
(4. 15)
If the power losses of the VSR and VSI are neglected (for simplicity), the energy
storage variation of the DC-link capacitor will be the integral of the difference
between the input power PVSR and the output power PVSI . Therefore, it can be written:
PVSR = Pcap + PVSI ,
(4. 16)
From this equation it can be concluded that for proper (accurate) control of the
VSR power PVSR the command power PVSRc should be as follows:
PVSRc = Pcapc + PVSIc ,
(4. 17)
Where, Pcapc = Pc denotes power of the DC-link voltage feedback control loop (see
Subsection 3.3.3.2), and PVSIc denotes the instantaneous active power feedforward
signal – PF (estimated by two different estimators see Subsections 4.2.2.2 or 4.2.2.3).
The command output power can be estimated based on different methods which
provide additional time constant T2 [37], [41], [56], [81], [137], hence:
PVSIc 2 (s ) =
1
PVSIc .
1 + sT2
(4. 18)
Moreover, it should be stressed here that the first order filter with time constant TU
should be added to DC-link voltage feedback which strongly delays the signal Pc
(see Subsection 3.3.3.2):
72
1
U dc
1 + sTU
U dcf (s ) =
(4. 19)
This delay is taken into account in DC-link voltage controller design. Hence:
Pc (s ) =
K PU (sTIU + 1)
eU dcf U dcc
sTIU
(4. 20)
Where, eU dcf = U dcc − U dcf . Therefore Eq. (4.17) can be rewritten:
PVSRc (s ) = Pc + PVSIc 2 ,
(4. 21)
Substituting Eqs. (4.18) and (4.20) into Eq. (4.21) yields:
K PU (sTIU + 1)
1
eU dcf U dcc +
PVSIc ,
sTIU
1 + sT2
PVSRc (s ) =
(4. 22)
From Eqs. (4.10) and (4.11), the open loop transfer function of the input power (of
the VSR) and output power (of the VSI) can be written as:
GVSRo (s ) =
PVSR
1
=
PVSRc 1 + sTIT
(4. 23)
GVSIo (s ) =
PVSI
1
=
PVSIc 1 + sTIF
(4. 24)
Based on this equations the analytic model of the AC/DC/AC converter-fed IM
drive with active power feedforward – PF can be defined as in Fig. 4. 2. Such a
system can be described by open loop transfer function as:
G Ao (s ) =
U dc
M ec
(4. 25)
DC-link Voltage Feedback
PI
K PU (sTIU + 1)
U dcc
sTIU
Active Power
Feedforward - PF
Ωm
M ec
PVSIc
1
sT2 + 1
PVSIc2
+
Pc
+
PVSRc
U dcc
+ U dcf
−
eU dcf
VSR
PVSR Pcap
1
1
+
sTIT + 1
− sCUdcc
PVSI
VSI
1
sTIF + 1
1
sTU + 1
U dc
Fig. 4. 2. Block diagram of the AC/DC/AC converter-fed IM with active power feedforward
73
Assuming initial steady state operation, Ω m = Ω mc = const . and U dc = U dcc = const .
the transfer function of the AC/DC/AC converter-fed IM drive can be described.
Therefore, in following Subsections different variants of the system presented in Fig.
4. 2 will be discussed.
4.2.2.1. Transfer Function of the AC/DC/AC Converter-Fed IM Drive with DClink Voltage Feedback only – PF0
In this case, only the DC-link voltage feedback is used – PF0 . Therefore, the
model takes a form as shown in Fig. 4. 3.
K PU (sTIU + 1)
U dcc
sTIU
M ec
Ωm
PVSIc
PVSIc2 = 0
+
Pc
+
PVSRc
U dcc
+ U dcf
−
eU dcf
1
sTU + 1
PVSR Pcap
1
+
sTIT + 1
− sCUdcc
PVSI
U dc
1
1
sTIF + 1
Fig. 4. 3. Block diagram of the AC/DC/AC converter-fed IM drive with DC-link voltage
feedback only – PF0
For such defined circuit the transfer function can be derived as:
G Ao1 (s ) = −
Ω m sTIU (1 + sTIT )(1 + sTU )
(1 + sTIF )M ( s )
(4. 26)
Where,
M ( s ) = s 4CU dccTIU TIT TU + s 3CU dccTIU (TIT + TU ) +
+ s 2CU dccTIU − K PU (sTIU + 1)U dc
(4. 27)
If the delay provided by VSR and VSI are equal i.e., TIT = TIF then Eq. (4.26)
thanks to pole-zero cancellation can be simplified to:
74
G Ao1 (s ) = −
Ω m sTIU (1 + sTU )
(4. 28)
M( s )
4.2.2.2. Transfer Function of the AC/DC/AC Converter-Fed IM Drive with DClink Voltage Feedback and Active Power Feedforward Calculated Based on
Mechanical Speed, Commanded Torque, and Power Losses – PFΩ
This case takes into consideration PF control loop which uses power of the VSI
calculated based on mechanical speed, commanded torque, and power losses as
follows:
PPFΩ = M ec Ω m + Plosses = M ec Ω m +
3
(RFVSI + RS + Rr )(I Sx2 + I Sy2 )
2
(4. 29)
There is a time delay between calculated power and actual power of the VSI,
defined by Eq. (4.18). Such defined system has been shown in Fig. 4. 2. The transfer
function of the system can be defined as:
G Ao 2 (s ) =
Ω mTIU (1 + sTU )s 2 (− T2 − TIT + TIF − sT2TIT )
(1 + sTIF )(1 + sT2 )M ( s )
(4. 30)
If the delay provided by VSR and VSI are equal i.e., TIT = TIF yields:
G Ao 2 (s ) =
Ω mTIU (1 + sTU )s 2 (− T2 − sT2TIT )
(1 + sTIF )(1 + sT2 )M ( s )
(4. 31)
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 - PFUI
The active power of the VSI can be calculated from commanded values of the
stator voltages and actual stator current – PFUI . In such a case there is smaller delay
between PVSIc and PVSIc 2 . For simplification it can be assumed that T2 = 0 . The
modified model for this case is shown in Fig. 4. 4. Hence, the transfer function can
be defined as:
G Ao 3 (s ) =
Ω mTIU (1 + sTU )s 2 (− TIT + TIF )
(1 + sTIF )M ( s )
(4. 32)
Assuming the case, when TIT = TIF . In such situation right hand side of Eq. (4.32)
should become 0. Hence, theoretically the DC-link voltage should not be affected by
the change of the load PINV power. However, in the real system the unbalanced
power difference that makes the fluctuations of the DC-link voltage occurs mainly
75
due to: the power estimation error, moment of inertia of the rotor and low pass filter
in speed control loop, and the saturation of the commanded control variable.
K PU (sTIU + 1)
U dcc
sTIU
M ec
Ωm
PVSIc = PVSIc2
+
Pc
+
PVSRc
U dcc
+ U dcf
−
eU dcf
1
sTU + 1
PVSR Pcap
1
+
sTIT + 1
− sCUdcc
PVSI
1
U dc
1
sTIF + 1
Fig. 4. 4. Block diagram of the AC/DC/AC converter-fed IM with active power feedforward –
PFUI
4.3. Simulation Study
Investigation of the active power feedforward - PF effectiveness has been carried
out in Matlab Simulink in continuous model as well as in SABER discrete model. In
this Subsection discrete simulation of the AC/DC/AC converter-fed IM drive will be
presented for two modes of operations:
•
for drives operated with closed torque control loop,
•
for drives operated with closed speed control loop.
4.3.1. Steady State Performances
For verification of the PF impact on the input current steady states operation
during motoring and regenerating modes were investigated. The results of this study
are presented in following tables (Tab. 4. 1 - Tab. 4. 4). In Tab. 4. 1 THD of the line
current under the steady state (at zero speed and zero torque) is presented.
76
Tab. 4. 1. THD of the line current under the steady state operation
(at zero speed and zero torque)
PF0
THD I L [%]
K PU = 0.0056
63
PFΩ
PFUI
THD I L [%]
THD I L [%]
53
53
Case
K PU = 0.056
64
60
50
In Tab. 4. 2 THD of the line current under the steady state (at 71 % of nominal
speed and 75% of nominal torque) is presented.
(at 71 % of nominal speed and 75% of nominal torque)
PF0
THD I L [%]
K PU = 0.0056
-
PFΩ
PFUI
THD I L [%]
THD I L [%]
4.3
4.3
Case
K PU = 0.056
4.9
5.2
In Tab. 4. 3 THD of the line current under the steady state (at 71 % of nominal
speed and -75% of nominal torque) is presented. In this case is a regenerating mode
and current flows from the AC/DC/AC converter-fed IM drive to the supply line.
(at 71 % of nominal speed and -75% of nominal torque)
PF0
THD I L [%]
K PU = 0.0056
9.9
PFΩ
PFUI
THD I L [%]
THD I L [%]
9.8
9.8
Case
K PU = 0.056
12
12.6
13
In Tab. 4. 4 THD of the line current under the steady state (100 % of nominal
speed and - 100% of nominal torque) is presented. In this case is also regenerating
mode.
(at 100 % of nominal speed and -100% of nominal torque)
PF0
THD I L [%]
K PU = 0.0056
4.0
PFΩ
PFUI
THD I L [%]
THD I L [%]
4.0
4.1
Case
77
K PU = 0.056
4.0
4.0
4.1
Based on above considerations it can be concluded that introduced active power
feedforward has no negative impact on the line current. This can be derived
analytically. Please consider rotating reference frame concurrently with virtual flux
(Subsection 2.5.4). With assumption that the system is decoupled, and meets the
unity power factor condition i.e. I Lx = 0 the following equations can be derived:
L
C
L
L
dI Ly
dt
= U Ly − U py − RI Ly
(4.79)
U
dU dc
= I Ly py − I load
U dc
dt
dI Ly
dt
dI Lyc
dt
(4.80)
=
K Pi1
(I Lyc − I Ly )
TIi1
(4.81)
=
K PU
(U dcc − U dc )
TIU
(4.82)
I Lyc = K PU (U dcc − U dc )
(4.83)
Then, VSR voltage:
[
U py = K Pi1 K PU (U dcc − U dc ) + I PF − I y
]
(4.84)
Where, I PF – current proportional to signal of the active power feedforward – PF
Hence, during steady state operation above equations yields:
[
]
0 = U Ly − K Pi1 K PU (U dcc − U dc ) + I PF − I y − RI Ly
0 = I Ly
0=
[
]
1
{K Pi1 K PU (U dcc − U dc ) + I PF − I y }− Iload
U dc
K Pi1
(K PU (U dcc − U dc ) − I Ly ) = (K PU (U dcc − U dc ) − I Ly )
TIi1
(4.85)
(4.86)
(4.87)
Substituting Eq.(4.93) into Eqs. (4.91) and (4.92):
0 = U Ly − K Pi1I PF − RI Ly
0 = I Ly
(4.88)
I U
1
K Pi1I PF − I load ⇒ K Pi1I PF = load dcc
U dcc
I Ly
(4.89)
Based on Eq.(4.95) the current from active power feedforward can be eliminated:
78
0 = U Ly −
I loadU dcc
− RI Ly
I Ly
(4.90)
From Eq. (4.96) it can be seen that steady states do not depends on active power
feedforward. Further, based on Eq. (4.96) a stationary error elimination condition can
be defined as:
2
RI Ly
− U Ly I Ly − I loadU dcc = 0
(4.91)
If the solutions are real the stationary error is eliminated. It takes place for:
I Ly1 / 2 =
2
U Ly ± U Ly
− 4 RI loadU dcc
(4.92)
2R
When:
I loadU dcc <
2
U Ly
(4.93)
4R
4.3.2. AC/DC/AC Converter-Fed IM Drive Operated with Closed Torque Control
Loop
In Fig. 4. 5 oscillograms illustrate operation with torque control loop. Torque
reversal is applied when the mechanical speed reach +/- 71 % of nominal speed.
From Fig. 4. 5 can be seen very good stabilization of DC-link voltage through
PFΩ and PFUI . Moreover, for 10 time reduced DC-link capacitor C=47uF as in Fig.
4. 6 that qualifications are also true. Moreover, as it can be seen in Fig. 4. 7 higher
gain of DC-link voltage controller (i.e. with proportional gain as for C=470uF)
improves significantly the quality of the DC-link voltage stabilization in all cases.
79
Fig. 4. 5. Simulated torque tracking performance. Torque reversal from -75 % to 75% of nominal
torque; a) Only DC-link voltage feedback PF0 ; b) With active power feedforward PFΩ ,
c) With active power feedforward
PFUI .
From the top: line current, DC-link voltage, active and reactive power, stator current, mechanical speed of
IM, command and estimated torque. DC-link capacitor C= 470 uF
80
torque; a) Only DC-link voltage feedback PF0 ; b) With active power feedforward PFΩ ,
PFUI .
81
torque. The proportional gain K PU of DC-link voltage controller is 10 times higher than
PF0 ; b) With active power feedforward PFΩ ,
c) With active power feedforward PFUI .
calculated. a) Only DC-link voltage feedback
At this point it has to be stressed that in all previous consideration (Fig. 4. 5- Fig.
4. 7) the commanded torque have been generated not by speed controller but directly
82
by command value (open speed loop operation). The question is: Is this enough
accurate simplification for the studies? The answer is: yes, and no. Yes for a real case
when the system operates with open speed loop as in the tram, electrical vehicles
(train or in the subway). And finally “no” for the system which works with closed
speed loop (e.g. elevators, servo drives and so on).
For easier comparison integral squared error – ISE criterion for cases presented
in Fig. 4. 6 and Fig. 4. 7 is shown in Fig. 4. 8. From this figure can be seen that both
active power feedforwards have the same errors in steady states but in transients
PFUI gives higher fluctuations.
Fig. 4. 8. Integral squared error – ISE criterion for test like in: a) Fig. 4. 6 and b) Fig. 4. 7;
PFΩ , DC-link voltage error with
PFΩ , and (2) - ISE for the PFUI
From the top: DC-link voltage error with active power feedforward
active power feedforward
PFUI , (1) - ISE for the
4.3.3. AC/DC/AC Converter-Fed IM Drive Operated with Closed Speed Control
Loop
In Fig. 4. 9 it can be seen that very good stabilization of the DC-link voltage are
provided by PFΩ and PFUI while circuit with only PF0 is unstable. Moreover, for
the higher proportional gain of DC-link voltage controller (i.e. like for C=470uF)
from Fig. 4. 10 it can be concluded that it improves the control quality for both PF.
However, for the case PF0 behavior of the system is still unstable. Further, it can be
seen that DC-link voltage in case PFUI is stiff even during transients – this is very
good especially for life time of the electrolytic capacitor.
83
Fig. 4. 9. Simulated, torque tracking performance at constant speed (at 71% of nominal speed).
Torque load changes from 0% to 75 % to -75% of nominal torque;
a) Only DC-link voltage feedback PF0 , b) With active power feedforward PFΩ , c) With active
power feedforward
PFUI .
IM, command and estimated torque, DC-link capacitor C=47uF
84
Fig. 4. 10. Simulated torque tracking performance at constant speed (71% of nominal speed).
Torque load change from 0% to 75 % to -75% of nominal torque. The proportional gain
K PU of DC-link voltage controller is 10 times higher than calculated; a) Only DC-link
voltage feedback
PF0 , b) With active power feedforward PFΩ , c) With active power
feedforward PFUI .
IM, command and estimated torque. DC-link capacitor C=47uF
Let consider an ISE criterion for cases presented in Fig. 4. 9 and Fig. 4. 10 shown
in Fig. 4. 11.
85
Fig. 4. 11. Integral squared error – ISE criterion for test like in: a) Fig. 4. 9 and b) Fig. 4. 10.
PFΩ , DC-link voltage error with active
PFΩ , and (2) - ISE for the PFUI .
power feedforward
PFUI ; (1) - ISE for the
It is worth to note that in steady state both active power feedforwards have similar
error while in transient PFUI gives smaller fluctuation of the DC-link voltage.
Further, let check the quality of the active power feedforward loop during the
hardest conditions, it means speed reversal. From Fig. 4. 12 it is clearly seen that
PFUI and PFΩ gives very good stabilization of DC-link voltage, while PF0 is
unstable. However, form Fig. 4. 13 (when proportional gain of DC-link voltage
controller is higher) it seems that the fluctuation of the DC-link voltage for PFUI is
slightly higher then in case PFΩ ( PF0 is out of comparison). As has been reported in
[29], [30] the duration of the fluctuation in DC-link voltage has also impact for life
time of the capacitor (when duration is shorter it is better for life time of the
capacitor).
86
Fig. 4. 12. Simulated speed reversal from 71% to -71% of nominal speed; a) Only DC-link
voltage feedback PF0 , b) With active power feedforward PFΩ , c) With active power
feedforward
PFUI .
87
Fig. 4. 13. Simulated speed reversal from 71% to -71% of nominal speed. The proportional gain
K PU of DC-link voltage controller is 10 times higher than calculated; a) Only DC-link
voltage feedback
PF0 , b) With active power feedforward PFΩ , c) With active power
feedforward PFUI .
An ISE criterion for cases presented in Fig. 4. 12, and Fig. 4. 13 is shown in
Fig. 4. 14.
88
Fig. 4. 14. Integral squared error – ISE criterion for test like in: a) Fig. 4. 12 and b) Fig. 4. 13.
PFΩ , DC-link voltage error with
PFΩ , and (2) - ISE for the PFUI
PFUI ; (1) - ISE for the
From Fig. 4. 14b it can be seen that for exact parameters of the IM in the active
power feedforward estimator PFΩ gives slightly better results then PFUI . However,
as can be seen in experimental results (see Chapter 6) it is not easy to choose the
parameters of IM exactly. Moreover, in practice the proportional gain of DC-link
voltage controller can not be 10 times higher because of unstable reaction of the real
system.
Also interesting situation is in stand still (when mechanical speed of IM is zero
rpm) and the torque is applied (see Fig. 4. 15).
89
Fig. 4. 15. Simulated torque tracking performance at constant speed (zero speed). Torque load
change from 0% to 75 % to -75% of nominal torque. The proportional gain K PU of DC-link
voltage controller is 10 times higher than calculated; a) Only DC-link voltage feedback
b) With active power feedforward PFΩ , c) With active power feedforward
PF0 ,
PFUI
90
4.4. Conclusion
Analysis and synthesis of the AC/DC/AC converter-fed IM controlled by DPTCSVM were performed in this Chapter. To meet the simple, robust, fast and cost
effective power flow control needs, two novel active power feedforwards – PFs were
elaborated and introduced to DPTC-SVM control structure. Three variants of the
control structure i.e.:
•
DPTC-SVM without active power feedforward – PF0 ,
•
DPTC-SVM with active power feedforward calculated based on mechanical
speed, commanded torque, and power losses – PFΩ ,
•
DPTC-SVM with active power feedforward calculated from commanded
stator voltages and actual stator current – PFUI ,
were investigated under different operation conditions. Based on simulation
results and integral squared error – ISE index evaluation of the three variants has
been made. The scheme with closed torque control loop and scheme with closed
speed control loop were investigated during steady states as well as in transients.
It might be concluded that:
For drives operated with closed torque control loop:
•
case PF0 (only DC-link voltage feedback) is unstable or gives high
overshoot of DC-link voltage,
•
case PFΩ (DC-link voltage feedback with active power feedforward
calculated based on mechanical speed, commanded torque, and power
losses) provide superior results in transients then case PFUI . However, it
should be stressed that results are obtained for exact parameters used for
calculation the power losses Plosses .
For drives operated with closed speed control loop:
•
case PF0 (only DC-link voltage feedback) is unstable or gives significantly
higher overshoot of DC-link voltage,
91
•
case PFΩ (DC-link voltage feedback with active power feedforward
calculated based on mechanical speed, commanded torque, and power
losses) provide generally inferior results in transient then PFUI .
Several cases of the steady states have been studied during motoring mode and
regenerating mode. The conclusions are as follows: introduced PFs have no negative
impact on the system performance in steady states. Moreover, they can improve
slightly the line current THD under no load operation (as shown in Tab. 4. 1).
92
Chapter 5
5. Passive Components Design – DC-link Capacitor
5.1. Introduction
In this Chapter methods of passive components for AC/DC/AC converter design
will be discussed. There are input filter (L or LCL) and the DC-link capacitor which
have strong effect on size, weight and final price of AC/DC/AC converter.
To minimize the reactive filter requirements of a VSR circuit, certain terminal
constraints are needed: input and output voltages, the total harmonics distortion –
THD of the input and output current ( I L , I dc respectively). The filter requirements
are usually defined in terms of filter cost and/or size. Here, the selection of DC-link
capacitor will be discussed while L and LCL filter is out of the scope of the thesis.
However, the design of L as well as LCL filter is important and it has been well
discussed in many publications [17], [82], [83], [84], [85], [87], [130], [136].
The advantages of aluminum – Al electrolytic capacitors that have led to their
wide application range, are their high volumetric efficiency (i.e. capacitance per unit
volume), which enables the production of capacitors with up to one Farad
capacitance, and the fact that an Al electrolytic capacitor provides a high ripple
current capability together with a high reliability and an excellent price/performance
ratio [29], [30]. However, optimization and minimization of the DC-link capacitor
volume is very important for the cost, power density and reliability of the product.
Therefore, advanced control strategies have been investigated for better control the
DC-link voltage (see Chapter 4).
5.2. Selection of Filter Components
5.2.1. Nominal Voltage of the DC-link Capacitor
Relationship between the leakage current and the voltage applied under
temperature conditions shown in Fig. 5. 1. It can be seen that stability of the DC-link
voltage at desired level is very important for a capacitor life time.
5. Passive Components Design
[A]
0
U
URet
USu
UF
[V]
Fig. 5. 1. Current-voltage characteristic of an aluminum – Al electrolytic capacitor [30]
In Fig. 5. 1 URet (rated voltage) – is the DC-link voltage value for which the
capacitor has been designed; USu (surge voltage) – is the maximum voltage which
can be applied to the capacitor for short periods of time, i.e. up to 5 times for 1
minute per hour. The norm IEC 60384-4 specifies the above voltages as follows:
•
for: URet ≤ 315 V:
USu = 1,15URet,
•
for URet > 315 V:
USu = 1,10URet,
and finally UF is a forming voltage.
When the forming voltage UF is exceeded, the forming process starts a new and
large amounts of gas and heat are generated. The same effect, yet on smaller scale,
can already be observed in the knee of the curve. In order to achieve a high degree of
operating safety of the capacitor, the rated voltage U Re t is defined as being on the
quasi-linear part of the curve. As the capacitor is subjected to surge voltages U Su for
short periods only, this range lies between the rated voltage and the forming voltage.
The difference between forming voltage and operating voltage, the so-called overanodization, thus has a substantial effect on the operating reliability of the capacitor.
High overanodization offers the possibility of producing especially reliable
capacitors designed as long-life grade “LL” capacitors in accordance with IEC
60384-1 [29], [30].
94
5.2.2. Ripple Current Consideration
The term ripple current is used for the RMS value of the AC current that flows
through the device as a result of any pulsating or ripple voltage. In Fig. 5. 2 ripple
current waveforms for different cases are shown i.e.:
•
AC/DC/AC converter with diode rectifier,
•
AC/DC/AC converter with VSR without active power feedforward,
•
AC/DC/AC converter with VSR and active power feedforward.
The maximum permissible ripple current value depends on the ambient
temperature, the surface area of the capacitor (i.e. heat dissipation area), the
dissipation factor (or equivalent series resistance – ESR) and on the AC frequency.
Thermal stress has a strong affect on the capacitor’s life. Therefore, the
dissipation heat generated by the ripple current is an important factor affecting the
useful life. Thermal considerations imply that, under certain circumstances, it may be
necessary to select a capacitor with a higher voltage or capacitance rating then would
be required. The ripple current is a function of the frequency.
Useful life - service life, operational life, or life time is defined as the life achieved
by the capacitor without exceeding a specified failure rate. This time can be
prolongated by operating the capacitor at loads below the rating value (e.g. lower
operating voltage, current or temperature) and by appropriate cooling measures. The
general capacitor life time can be defined as a function:
LH = LB f ( TM − TC ) f1 (U dc )
(5. 1)
Where: LH is the life estimate in hours, LB is the base life elevated maximum
temperature TM, TC is the actual core temperature and Udc is the applied DC-link
voltage. The voltage multiplier f1 at higher stress level may reduce the life of the
capacitor [112]. Therefore, the stabilization of the DC-link voltage at the required
level is very important for useful life.
95
a)
b)
c)
a)
Fig. 5. 2. Ripple current in DC-link capacitors. a) AC/DC/AC converter with diode rectifier, b)
AC/DC/AC converter with VSR without PF, c) AC/DC/AC converter with VSR, with PF.
DC-link capacitor C=470uF.
From the top: capacitors current, DC-link voltages and zoom of them
The ripple in the DC-link voltage is a function of the ripple current in the DC-link.
The capacitor current I c is difference between the current I dc (output current of the
VSR) and I load (input current of the VSI) as shown in Eq. (5. 2). An approximate
minimum occurs when the DC-link currents pulses from the both converters overlap,
especially when the pulses are centered on the same time instant [4].
A modulation with symmetric zero vector placement well meets mentioned
condition [47]. PWM pulses are always centered on the half period of the switching
state. Fig. 5. 3 shows simulated DC-link voltage, DC-link currents I dc , I load , and
capacitors current I c over sampling period. It should be note that there are a lot of
modulation techniques. The problem of modulation and its impact on the controlled
system performance (switching losses, current or voltage shapes, noise level, and so
on) are presented in many publications [7], [12], [14], [44], [45], [47], [72], [76],
[138].
96
a)
c)
b)
Fig. 5. 3. Ripple in DC-link currents. a) AC/DC/AC converter with diode rectifier, b) AC/DC/AC
converter with VSR without PF, c) AC/DC/AC converter with VSR, with PF; C =470uF.
From the top: period of sampling, DC-link voltage and its potentials (positive and negative), capacitors
current, DC-link currents
I dc , I load
In Fig. 5. 3 one sampling period during load step condition (from 0 to 85% of the
M N ) are presented. In spite of short time period the average value of the ripple
currents can be valuated. It can clearly be seen that the highest current flows from the
capacitor the case presented in Fig. 5. 3a. The average value is negative. It means the
energy is transferred from the capacitor to the load. In Fig. 5. 3b the capacitor current
is significantly reduced. Further reduction of the current I c is observed in Fig. 5. 3c,
where active power feedforward is adopted.
5.2.3. Ratings of the DC-link Capacitor
Beside variety design criteria [20], [103], [143] of the DC-link capacitor, the
minimum capacitance value is designed to limit the DC-link voltage ripple at a
specified level, typically ∆U dc is 1% or 2% of U dc . Therefore, peak to peak voltage
ripple in DC-link is adopted as the design criterion for the DC-link capacitor sizing.
With assumption of a balanced tree-phase line and neglecting the power losses in
the power switches, the VSR's DC-link part can be described as Eq. (5.2).
97
C
C
dU dc
P
= I dc − I load = ∑ I Lk S k − I load ≈ I LA S A + I LB S B + I LC SC − load
dt
U dc
k=A
(5. 2)
For given allowable peak ripple voltage and switching frequency, the minimum
capacitor for the converter in Fig.4. 1 can be found from [24]:
3U LL
U dc
,
2 3∆U dc f sU LL
2+
Cmin_ VSR1 = Pload _ max
(5. 3)
Where: U LL - is a line to line voltage, Pload _ max - maximal load power, ∆U dc specified peak to peak voltage ripple in DC-link during steady states.
Another approach of the DC-link capacitors design, takes into account following
considerations:
•
the voltage ripple, due to the high-frequency components of the modulated
DC- link currents of both converters (i.e. VSR and VSI), have to remain
within desired limits,
•
when all switches of VSR are off, the inductors energy flows into the
capacitor, increasing its voltage,
•
the capacitor energy has to sustain the output power demand in a period of
the time delay of the DC-link voltage control loop.
The first and the second point of considerations are less important, while the third
one practically determines the capacitor value. Assuming time delay of the DC-link
voltage control loop TUT and variation of the maximal load power ∆Pload _ max , the
energy exchanged by the DC-link capacitor can be estimated as:
∆Wdc = ∆Pload _ maxTUT
(5. 4)
Where, TUT is defined in Section 3.3.3.2.
From this equation the maximal DC-link voltage variation during transient is
expressed by:
∆U dc _ max =
∆Wdc
(5. 5)
Cmin_ VSR 2U dc
Considering the maximal voltage variation during transient ∆U dc _ max and
rearrange an Eq. (5.5) the minimal capacitance can be calculated as [90]:
98
Cmin_ VSR 2 =
TUT ∆Pload _ max
(5. 6)
U dc ∆U dc _ max
Other expression for minimum capacitance can be derived, given the maximum
2
power step modeled by maximal load power in square Pload
_ max and the maximum
allowable DC-link voltage variation ∆U dc _ max [4].
Cmin_ VSR 3 =
2
2 LPload
_ max
(5. 7)
U dc ∆U dc _ maxU L2
2
Because of simplification ( Pload
_ max , TUT and TS are not taken into account) used in
Eq. (5. 7) calculated capacitance is slightly too large.
For a comparison, the DC-link capacitor value of AC/DC/AC converter with
diode rectifier (presented in Fig.1. 1a), is given by [24]:
Cdi = Pload _ max
π2
54 2∆U dc f LU LL
(5. 8)
In Fig. 5. 4 are shown values of the DC-link capacitor in AC/DC/AC converter
with diode rectifier and with VSR. It can be seen that difference between
capacitances raised proportional to power rating. Therefore, only for higher power
the installation of the VSR is more cost-effective.
The power ratio of the AC/DC/AC converter considered in the thesis is 3 kW.
Therefore, for better illustration in Fig. 5. 5 are presented values of the DC-link
capacitor up to 6 kW.
99
0.015
C
di
C [F]
0.01
Cmin_VSR3
0.005
Cmin_VSR2
Cmin_VSR1
0
0
0.2
0.4
0.6
0.8
1
1.2
Pload_max
1.4
1.6
1.8
2
x 10 4
Fig. 5. 4. Value of the DC-capacitor versus rated load power up to 20 kW:
Cdi - Capacitance defined by Eq. (5. 8), Cmin_VSR1 - Capacitance defined by Eq. (5. 3),
Cmin_VSR 2 - Capacitance defined by Eq. (5. 6), Cmin_VSR 3 - capacitance defined by Eq. (5. 7).
4.5
x 10
-3
C value
C
di
4
3.5
3
2.5
2
1.5
1
0.5
0
0
1000
2000
3000
PLoad
4000
5000
Cmin_VSR3
C min_VSR 2
Cmin_VSR1
6000
Fig. 5. 5. Value of the DC-capacitor versus rated load power up to 6 kW:
Cdi - Capacitance defined by Eq. (5. 8), Cmin_VSR1 - Capacitance defined by Eq. (5. 3),
Cmin_VSR 2 - Capacitance defined by Eq. (5. 6), Cmin_VSR 3 - capacitance defined by Eq. (5. 7).
100
Therefore, it can be concluded that for given nominal power, the DC-link
capacitor value depends only on the switching pattern and quality of the applied
control methods. In considered case the switching frequency is equal to the sampling
frequency. For higher TS the DC-link capacitor can be smaller because the DC-link
voltage error is significantly reduced, as shown in Fig. 5. 6. However, the switching
frequency is limited by the switching loses of the devices used in the VSC. Hence, to
further reduction of the DC-link filter the device loses should be made independent
of switching frequency. It can be realized by the use of soft-switched VSC such as
resonant DC-link converter [41], [59], [117].
800
700
600
2
1
2
3
4
|eUdc| [V]
500
400
300
no PF C=470uF
no PF C=47uF
with PF C=470uF
with PF C=47uF
200
100
1
3
4
0.2
0.1
0.05
0.02
0.01
Ts [ms]
Fig. 5. 6. Module of the DC-link voltage fluctuation eU dc as a function of sampling time
during load change from 0 to 3kW with command DC-link voltage
TS ,
U dcc = 560V .
Simulated results in Saber model of DPTC-SVM.
From Fig. 5. 6 can be concluded that for ideal control methods, with sufficiently high
sampling time, the DC-link voltage fluctuation during transient for reduced capacitor
can be even smaller (case with active power feedforward (PF C=47uF)) then for
higher value of the capacitance (case without active power feedforward (no PF
C=470uF)). Moreover, taking into consideration only the cases with active power
feedforward – PF it can be seen that the fluctuations of DC-link voltage during
transient for C= 470uF and for C= 47uF are almost equal for higher sampling time
(Fig. 5. 7). Base on this the postulate can be formed as follows: stabilization of the
DC-link voltage in AC/DC/AC converter with transistor bridge rectifier (VSR)
101
mainly depends on quality of the used DC-link voltage and input/output power flow
control methods.
80
70
60
|eUdc| [V]
50
3
4
40
with PF C=470uF
with PF C=47uF
30
20
10
4
3
0.2
0.1
0.05
0.02
0.01
Ts [ms]
Fig. 5. 7. Module of the DC-link voltage fluctuation eU dc as a function of sampling time
during load change from 0 to 3kW with command DC-link voltage
TS ,
U dcc = 560 [V ] for a
case when in control structure the active power feedforward – PF is implemented. Simulated
results in Saber model of DPTC-SVM.
5.2.3.1. Consideration of Operation with Reduced DC-link Capacitor
The minimum value of the DC-link capacitor is limited by a maximal acceptable
switching ripple level and fluctuation during load power steps. In case of small DClink capacitor that phenomena can not be reduced by the reactive component (DClink capacitor), but have to be compensated by the other way.
One alternative is to use a load itself e.g. induction motor. The VSI-fed IM can be
controlled in such a way that, all energy which intent to drown or especially inject to
the DC-link can be dissipated in the IM itself. It is necessary because in a typical
motor drive, the mechanical energy is several times higher then electrical energy
stored in the DC-link capacitor of the AC/DC/AC converter-fed IM. In cost effective
case the diode rectifier with reduced DC-link capacitor can be used. However, in
such application line current is still distorted (Fig. 5. 8a). Moreover, the fluctuation
of the DC-link voltage (Fig. 5. 8b) is significant for life time of the capacitor.
Regenerative energy has to be wasted in IM (this increase the temperature of the
102
motor too) or in DC-link chopper. While the energy during braking is not dissipated
Mag(V)
Mag(A)
(V)
(A)
in the motor (or chopper) the DC-link voltage grown fast (see Fig. 5. 9a).
Fig. 5. 8. Simulated AC/DC/AC converter with diode rectifier and reduced DC-link capacitor.
Steady state at nominal load; a) line current and its harmonics spectrum, b) DC-link voltage
and its harmonics spectrum
Fig. 5. 9. Simulated AC/DC/AC converter with diode rectifier and reduced DC-link capacitor.
Transient to: a) start up, loading (20Nm), and speed reversal +/- 71% of nominal speed, b)
load torque +/- 100% of nominal torque at zero speed.
From the top: DC-link voltage [V], mechanical speed of IM [rpm], torque [Nm]
Another solution is just transfer all fluctuations from the load power to the supply
line power. It can be realized by active power feedforward – PF (see Chapter 4).
However, in the case when the DC-link capacitor has a minimized value its capacity
103
is not enough for storing the energy from the motor. Hence, this approach places
very high demands for the control structure which has to be very robust and accurate
[20], [68], [78], [81], [89], [90], [91], [115], [116], [128], [137], [142], [143], [153].
If the control of the VSR will fail, the energy stored in DC-link may cause raise of
the DC-link voltage above acceptable level e.g. like in the Fig. 5. 9a (which can
damaged the VSR and VSI semiconductors). Therefore, to limit the DC-link voltage
at safe level the brake chopper can be needed. However, its average power rating can
be significantly smaller then in AC/DC/AC converter with diode rectifier. This
chopper should be independent from VSR control. As has been mentioned before to
transfer the load power fluctuations to supply line power the active power
feedforward is needed. Therefore, it is necessary to investigate its impact on the line
current quality during steady states (see Subsection 4.3.1).
5.3. Conclusion
In this Chapter the problem of the DC-link capacitor ratings have been discussed.
It can be concluded that DC-link capacitor value strongly depends on quality of the
instantaneous power flow control strategy between input AC/DC converter and
output DC/AC converter (VSI). Moreover, it was shown that when the active power
feedforward – PF is applied, the DC-link voltage fluctuation during transient can be
significantly reduced even for smaller capacitance of the DC-link capacitor. It should
be stressed that for minimized value of the DC-link capacitor very high demands for
the control structure robustness and accuracy are placed. Moreover, for protection the
power modules against no foresee faults in VSR control the chopper is needed. The
critical circumstances are in case of fault in VSR control during regenerating
operation.
104
Chapter 6
6. Simulation an Experimental Results
6.1. Introduction
In this Chapter selected experimental results measured on the laboratory set-up
with 5 kVA AC/DC/AC converter-fed 3 kW induction motor will be shown. The
parameters of the laboratory setup are given in Appendix A. 4. 3.
6.2. Steady States Operation
The steady states are first step to investigation the features of the power electronic
circuit. Especially, for voltage source rectifiers – VSR, it is very interesting to see
the waveform of line current in different modes of operation. In Fig. 6. 1 and in
Fig. 6. 2 are presented operation in rectifying mode of the diode rectifier and VSR
with DPC-SVM control methods, respectively.
From Fig. 6. 1 steady states of diode rectifier shows that line current is strongly
distorted. Therefore, the estimated active power is distorted too. Moreover, value of
the reactive power is very high.
Fig. 6. 1. Experimental steady state oscillograms of diode rectifier; a) resistive load 3.0 kW,
b) resistive load 1.0 kW.
From the top: 1) line voltage 500V/div, 2) line current 10A/div, 3) line active power 1kW/div and
4) line reactive power 1kVar/div. DC-link capacitor C=470uF
6. Simulation and Experimental Results
From Fig. 6. 2 can be seen that steady states of the VSR controlled by DPC-SVM
provide sinusoidal like line current. Moreover, the estimated active power is less
distorted then in previous case. Reactive power has a value close to zero.
Fig. 6. 2. Experimental steady state oscillograms of VSR controlled by DPC-SVM, resistive load
3.0 kW, b) resistive load 1.0 kW.
Operation with induction motor (active load) is very interesting and more critical
then operation with resistive load. Regenerating mode can be observed and analyzed.
Therefore in, Fig. 6. 3 and Fig. 6. 4 some cases of operation with active load are
presented.
Fig. 6. 3. Experimental steady state oscillograms of VSR controlled by DPC-SVM in AC/DC/AC
converter-fed IM with reduced DC-link capacitor from C=470uF to C=47uF, active load;
a) 75% of nominal load torque, b) -75% of nominal load torque.
The comparison of the total harmonics distortion – THD factor is shown in Tab.
6. 1. All methods have been compared in the same supply conditions. However, it
can be seen the difference in THDU L for diode rectifier and for VSR. This difference
106
is caused by negative impact of the distorted line current on the line voltage
waveform.
Tab. 6. 1 THD of line current for steady states (resistive load)
Control Method
Diode rectifier
Diode rectifier
VSR with DPC-SVM
VSR with DPC-SVM
THDU L [%]
THD I L [%]
4.6
4.6
3.4
3.4
28.3
25.9
2.8
1.8
a)
b)
c)
d)
Load Power kW
1.6
3.0
1.6
3.0
Fig. 6. 4. Experimental steady state operation of VSR controlled by DPC-SVM in AC/DC/AC
converter-fed IM. a) load torque 80% of nominal torque , mechanical speed 50% of nominal
speed, b) load torque -80% of nominal torque , mechanical speed 50% nominal speed, c) load
torque -80% of nominal torque , mechanical speed 100% of nominal speed,
d) load torque -80% of nominal torque , mechanical speed 110% of nominal speed.
From the top: 1) line voltage 100V/div, 2) line current 10A/div, 3) DC-link voltage 200V/div and
4) line active power 1kW/div
It is interesting to observe that the VSR takes different AC side line current levels
to provide the same torque level on the IM shaft for motoring and regenerating
modes. This is because of power losses in IM and AC/DC/AC converter. Moreover,
the shape of the motoring and regenerating current is different.
107
6.3. Active and Reactive Power Controllers
Controller design verification has been done for the same condition as in
simulation presented in Section 3.3.3.1. However, it should be note that during
command reactive line power step command active line power is set to 1kW. The
active and reactive power controller syntheses as well as analysis were done
according to symmetry optimum – SO.
In Fig. 6. 5 is presented active power steps response at different sampling
frequency. From this figure can be seen similarities with simulated result shown in
Fig. 3. 29 and Fig. 3. 30.
Fig. 6. 5. Experimental active and reactive power tracking performance at different sampling
frequency: f s = 2.5kHz a) active power step, and b) reactive power step;
f s = 5Hz c) active power step, and d) reactive power step.
From the top: command and actual active power 0.5kW/div, command and actual reactive power
0.5kVar/div
108
Fig. 6. 6. Experimental active and reactive power tracking performance at different sampling
frequency: f s = 10kHz a) active power step, and b) reactive power step;
f s = 20kHz c) active power step, and d) reactive power step.
From the top: command and actual active power 0.5kW/div, command and actual reactive power
0.5kVar/div
6.4. AC/DC/AC Converter-Fed IM Drive Operated with Closed Torque
Control Loop
In Fig. 6. 7 and Fig. 6. 8 operation with closed torque control loop with C=470uF
and C=47uF is presented. Torque reversal is applied when the mechanical speed
reaches +/- 71% of nominal speed. Based on this results can be concluded that better
results can obtained with PFUI . This is different than in simulations. The difference
is caused by errors in determinations of the power estimator parameters (Eq. 4.29)
used in active power feedforward – PFΩ (calculated based on command torque,
mechanical speed and power losses).
Remark:
For reduced DC-link capacitor the case with DC-link voltage feedback only –
PF0 was not investigated for safety of the laboratory setup.
109
Fig. 6. 7. Experimental torque tracking performance. Step change from – 75 % to 75% of nominal
torque; a) Only DC-link voltage feedback PF0 , b) With active power feedforward PFΩ ,
PFUI .
From the top: DC-link voltage 100V/div, line current 10A/div, active 2.5kW/div and reactive 2.5kVar/div
power, mechanical speed of IM 1000rpm/div, stator current 10A/div, command and estimated torque 20
Nm/div. DC-link capacitor C= 470 uF
110
a)
b)
c)
d)
Fig. 6. 8. Experimental torque tracking performance. Step change from – 75 % to 75% of nominal
torque; a) With active power feedforward PFΩ , b) With active power feedforward PFUI .
In case c), and d) the proportional gain of DC-link voltage controller
K PU is two times
higher then calculated.
Nm/div. DC-link capacitor C= 47 uF
111
6.5. AC/DC/AC Converter-Fed IM Drive Operated with Closed Speed
Control Loop
From Fig. 6. 9 and Fig. 6. 10 it can be seen that better stabilization of the DC-link
voltage provide PFUI (with active power feedforward from command stator voltages
and actual stator current).
a)
b)
Fig. 6. 9. Experimental torque tracking performance at constant speed (71%). Step change from
0% to 75 % to 0% of nominal torque; a) Only DC-link voltage feedback PF0 , b) With active
power feedforward PFΩ , c) With active power feedforward
PFUI .
power, mechanical speed of IM 1000rpm/div, stator current 10A/div, command and estimated torque
20Nm/div. DC-link capacitor C=470uF
112
a)
b)
c)
d)
0% to 75 % to 0% of nominal torque; a) With active power feedforward PFΩ , b) With
PFUI . In case c), and d) the proportional gain of the DC-link
voltage controller K PU is two times higher then calculated.
113
a)
b)
t=680 ms
0% to 75 % to 0% of nominal torque. The proportional gain of the DC-link voltage controller
K PU is 5.5 times higher then calculated; a) With active power feedforward PFΩ , b) With
PFUI .
In Fig. 6. 10c,d and Fig. 6. 11 the results with higher proportional gain of the DClink voltage controller K PU are shown. Note that the gain is up to 5.5 times higher
than calculated while in simulation is 10 times higher than calculated. The reason is
in instable operation for higher then 5.5 gain in experiment. In Fig. 6. 11a an unstable
operation of VSR with active power feedforward – PFΩ is shown. In time t=680 ms
the over current protection switched off the control of the VSR and the system
operates with diode rectifier as an AC/DC converter. While, in Fig. 6. 11b very good
operation of the system with active power feedforward – PFUI is presented. It should
be stressed that in this case of transients the stiff DC-link voltage is obtained (see
Fig. 6. 11b).
114
a)
b)
t=TA
t=TA
0% to 75 % to 120% of nominal torque. The proportional gain of the DC-link voltage
controller K PU is 5.5 times higher then calculated and with active power feedforward
PFUI ; a) load fault and control of VSR is off, b) load fault and control of VSR is on.
20Nm/div, C=47uF
In case presented in Fig. 6. 12 operation of the system with active power
feedforward PFUI is shown. In time t=TA the load fault is applied (120% of nominal
torque). In Fig. 6. 12a over current protection switched off the VSR control and
system operates with diode rectifier as an AC/DC converter. While, in Fig. 6. 12b
VSR controller mitigated the fault and keep the DC-link voltage at desired level with
reactive power equal to zero.
Note that even under distorted load conditions, the scheme with active power
feedforward operates properly (satisfactory).
115
a)
b)
c)
Fig. 6. 13. Experimental torque tracking performance at constant speed (71% of nominal speed).
Step change from 0% to 75 % to -75% of nominal torque. a) Only DC-link voltage feedback
PF0 , b) With active power feedforward PFΩ , c) With active power feedforward PFUI .
From the top DC-link voltage 100V/div, DC motor current 10A/div, active 2.5kW/div and reactive
2.5kVar/div power, mechanical speed of IM 1000rpm/div, stator current 10A/div, command and
estimated torque 20Nm/div. DC-link capacitor C=470uF
116
DC motor without load
Fig. 6. 14. Experimental DC motor operates without load (i.e. control of IM is off).
From the top DC-link voltage 100V/div, DC motor current 10A/div, active 2.5kW/div and reactive
2.5kVar/div power
A waveform of the DC motor current (DC motor works as an active load for the
IM) is presented in Fig. 6. 14. In this figure DC motor operates without load i.e.
control of IM is off.
In Fig. 6. 13 are shown the waveforms during active load step and reversal. Note
that in spite of distorted load condition the DC-link voltage is well stabilized at
desired level when active power feedforwards are applied. Better stabilization gives
the PFUI .
In further investigation the most difficult test will be applied.
So, let check the quality of the active power feedforward during the hardest
conditions - speed reversal. From Fig. 6. 15 it can be seen that PFUI (with active
power feedforward calculated from command stator voltages and actual stator
current) is superior in respect to PF0 (with DC-link voltage feedback only) and PFΩ
(with active power feedforward calculated form actual speed and commanded
torque).
In Fig. 6. 16 and Fig. 6. 17 (AC/DC/AC converter with reduced DC-link capacitor
C=47uF) results only for control with active power feedforwards are presented.
117
a)
b)
c)
Fig. 6. 15. Experimental speed reversal from 71% to -71% of nominal speed; a) Only DC-link
voltage feedback PF0 ; b) With active power feedforward PFΩ , c) With active power
feedforward
PFUI .
Nm/div. DC-link capacitor C=470uF
118
a)
b)
c)
Fig. 6. 16. Experimental speed reversal from 53% to -53% of nominal speed; a) With active
power feedforward PFΩ , b) With active power feedforward PFUI , c) speed reversal from
71% to -71% of nominal speed with active power feedforward
PFUI .
119
Fig. 6. 17. Experimental speed reversal from 71% to -71% of nominal speed. The proportional
gain K PU of DC-link voltage controller is 3.5 times higher then calculated; a) with active
power feedforward PFΩ , b) with active power feedforward
PFUI .
In Fig. 6. 16a,b waveforms for case with active power feedforwards PFΩ and
PFUI are presented. Shown speed reversal are from -53% to 53% of nominal speed
because DC-link voltage fluctuation for case PFΩ under transient +/- 71% of
nominal speed is too high. Only for higher proportional gain of DC-link controller
K PU equals to 3.5 times of the calculated value (based on Eq. (3.73)) the DC-link
voltage fluctuations are acceptable (see Fig. 6. 17). From Fig. 6. 17 can also be
concluded that PFUI gives superior results then PFΩ .
Further, for stressing the contrast between AC/DC/AC converter with VSR results
presented in Fig. 6. 15, Fig. 6. 16, and Fig. 6. 17 can be compared with results of
AC/DC/AC converter with diode rectifier shown in Fig. 6. 18.
120
a)
b)
c)
Fig. 6. 18. AC/DC/AC converter with diode rectifier. Experimental speed reversal; a) from 71%
to -71% of nominal speed. DC-link capacitor C=470uF, b) from 35% to -35% of nominal
speed. DC-link capacitor C=47uF, c) from 53% to -53% of nominal speed. DC-link capacitor
C=47uF.
20 Nm/div
121
6.6. Conclusion
Experimental results presented in this Chapter confirm proper operation of the
AC/DC/AC control system. Introduced second transistor bridge (VSR) on the line
side instead of the diode rectifier eliminates the main drawbacks of diode rectifiers,
such as:
•
distorted line current,
•
reactive power consumption,
•
unidirectional power flow.
Moreover, assured a number of advantages like minimized energy storage in DClink capacitor, controllable and stable DC-link voltage as well as active filtering and
reactive power compensation capabilities [170]. The VSI operates on the boosted
DC-link voltage; thanks to this the output power of a connected motor can be
increased over its rated power. Further, a difficult to overestimate is an advantage
that the breaking energy can be feedback to the line instead wasting it in a DC-link
circuit (or in the IM itself).
The active power feedforward – PF loop from the motor side to the VSR
improved control dynamics of the DC-link voltage. It allows better fulfilling power
matching conditions during transient for VSR-fed DC-link and VSI-fed IM drive.
Hence, it gives a possibility of fast control of the power flow. Thanks to fast control
of the power flow to the line, the DC-link capacitor can be reduced without affecting
VSI performances. The value of the capacitor can be small enough that gives
possibility for implementation with film capacitor. Moreover, very low THD of the
line current at steady states is kept. However, it should be stressed that power
estimator used in active power feedforward – PF has to be sufficiently fast, accurate,
and robust. Otherwise errors in provided by PF can cause a malfunction in control
system, unstable operation, or in worst case damage of the AC/DC/AC converter.
122
Chapter 7
7. Summary and Conclusion
In the thesis a line power friendly AC/DC/AC voltage source converter-fed
induction motor drives has been studied and investigated under aspect of DSP-based
control and industrial serial manufacturing. Such a control system should meet
following requirements: operation with unity power factor – UPF, low total
harmonics distortion – THD of the line current, very good stabilization of the DClink voltage, four-quadrant operation, and be universal for different types of drive
application (open loop control, sensorless control and control with motion sensor).
Also, for serial manufacturing is important to have simple reliable and low cost
solution with repeatable parameters of drive.
After literature studies two control strategies were chosen: direct torque control
with space vector modulation – DTC-SVM for the motor side DC/AC converter
(VSI) and direct power control with space vector modulation DPC-SVM for the line
side AC/DC converter (VSR). As results the novel scheme for control of AC/DC/AC
converter-fed induction motor – IM called Direct Power and Torque Control with
Space Vector Modulation – DPTC-SVM (see Fig. 3. 10) has been proposed.
After simulation study it has been shown that when control structure of the
voltage source rectifier – VSR-fed DC-link operates independently from control of
the VSI-fed IM, the DC-link voltage stabilization is not sufficiently fast to balancing
instantaneous DC-link power flow. Therefore, a high value of DC-link capacitor is
required. However, it could be significantly reduced if a new elaborated active power
feedforward – PF estimator has been introduced to whole control structure. As shown
in Chapter 4 of the thesis, thanks to PF loop the value of the DC-link capacitor have
been reduced ten times while good stabilization of DC-link voltage in both dynamic
and stationary states as well as low line current THD are kept (Fig. 6. 3). However, it
should be stressed that the system has to be equipped with fast PF loop and/or DClink chopper (see Fig. 6. 18).
The detailed investigation has shown that PF loop with power estimation based on
motor voltage and current sensing gives better results as in the case of estimation
based on torque and speed sensing (see Chapter 4.2.2).
For proper operation of the closed control loops in DPTC-SVM scheme, the
digital PI controllers design is very important. Therefore, in Chapter 3 synthesis of
active and reactive power as well as DC-link voltage controller has been given. This
design is based on continuous transfer function approach (symmetric optimum – SO),
because fast sampling used in the system (dominated time constant – line inductor
TRL - is much higher as sampling time TS ), and has been confirmed by both
simulated and experimental results. The flux and torque control design has been
implemented similarly (for more details see [152]).
The main features and advantages of developed DPTC-SVM scheme (Fig. 3. 10)
for AC/DC/AC converter-fed IM can be summarized as follows:
•
four-quadrant operation (bi-directional power flow) (Fig. 6. 4),
•
fast flux and torque response (Fig. 6. 17),
•
available maximum output torque in wide range of speed operation region,
•
constant switching frequency and uni-polar voltage pulses (thanks to use
SVM),
•
low flux and torque ripple (Fig. 6. 12b),
•
robustness to rotor parameter variations,
•
motion sensorless operation,
•
nearly sinusoidal line current (low THD typically below 5%) (Fig. 6. 3),
•
direct adjustable reactive power (including unity power factor – UPF),
•
controllable DC-link voltage (well stabilized at desired level),
•
AC side voltage sensors are eliminated (only DC-link voltage sensor is
required),
•
wide speed range operation of the IM connected to the output of the VSI,
•
reduction of DC-link capacitor, thanks to active power feedforward – PF
(gives possibility to use a foil capacitor),
124
•
easy operation in open or closed loop fashion, because (separately SVM
blocks are used).
In author opinion the results of the thesis could be easily expand for design of
AC/DC/AC converter-fed permanent magnet synchronous motor – PMSM drives.
Finally, it should be stressed that the thesis is a continuation of PhD-projects
developed at Institute of Control and Industrial Electronics [24], [93], [152] which
have brought into serial production DTC-SVM inverters with diode rectifiers, and
also will be introduced into a new family of AC/DC/AC converter-fed drives
produced by Polish company Power Electronics Manufacture TWERD, Toruń.
125
References
[1] ABB ACS 800 Single Drive Manuals, ACS800-01/U1 Hardware Manual 0.55 to 110 kW.
[2] F. Abrahamsen, and A. David, “Adjustable Speed Drive with Active Filtering Capability for
Harmonic Current Compensation”, in Proceedings of the PESC 1995 Conf., 1995, pp.1137-1143.
[3] H. Akagi, “New trends in active filters for power conditioning”, IEEE Trans. on Ind. Appl., Vol.
32, 1996, pp. 1312–1332.
[4] M. Alakula, and J. E. Persson, “Vector Controlled ac/ac converters with a minimum of energy
storage,” in Proceedings of the IEEE Conf., 1994, pp. 1130–1134.
[5] M. Bagher, B. Sharifian, and J. Faiz, ”Different switching patterns in DTC of IM,” Power
Electronics and Motion Control Conference, 2004. IPEMC 2004. The 4th International Vol. 1,
2004 Page(s):384 - 388 Vol.1.
[6] M. Bagher, B. Sharifian, and J. Faiz, ”Implementation of snsorless DTC techique for speed control
of induction motor using a novel switching pattern,” pp. 379–383.
[7] M. M. Bech, F. Blaabjerg, and J.-K. Pedersen, ”Random Modulation Techniques with Fixed
Switching Frequency for Three-Phase Power Converters,” IEEE Trans. on Power Electronics,
vol. 15, No. 4, July 2000, pp. 753–761.
[8] R. Barlik, M. Nowak, ”Jakość energii elektrycznej – stan obecny i perspektywy,” Przegląd
Elektrotechniczny, R. 81, No. 7-8, 2005, pp. 1–12, (in Polish).
[9] F. Blaabjerg, and J. K. Pedersen, “An integrated high power factor three-phase AC-DC-AC
converter of AC-machines implemented in one microcontroller,” Power Electronics Specialists
Conference, 1993. PESC '93 Record., 24th Annual IEEE, 20-24 June 1993, pp. 285–292.
[10] F. Blaabjerg, J. K. Pedersen, and U. Jaeger, “Evaluation of modern IGBT-modules for hardswitched AC/DC/AC converters,” Industry Applications Conference, 1995. Thirtieth IAS Annual
Meeting, IAS '95., Conference Record of the 1995 IEEE, Vol. 2, 8-12 Oct. 1995, pp. 997–1005.
[11] F. Blaschke, "A New method for the structural decoupling of A.C. induction machines," in
Proceedings of the Conf. Rec. IFAC, Duesseldorf, Germany, Oct. 1971, pp. 1-15.
[12] V. Blasko, ”Analysis of a Hybrid PWM Based on Modified Space-Vector and TriangleComparison Methods,” IEEE Trans. on Industrial Applications, vol. 33, No. 3, May/June 1997,
pp. 756–764.
[13] V. Blasko, V. Kaura, “A new mathematical model and control of a tree-phase AC-DC voltage
source converter,” IEEE Trans. on Power Electronics, vol. 12, No. 1, January 1997, pp. 116–123.
[14] A. Boglietti, G. Griva, M. Pasterelli, F. Profumo, and T. Adam, "Different PWM Modulation
Techniques Indexes Performance Evaluation,” in Proceedings of the IEEE Conf., 1993, pp.193 –
199.
[15] B. K. Bose, Modern Power Electronics and AC Drives, Prentice-Hall, 2002.
[16] J. Brzózka, Regulatory cyfrowe w automatyce , Mikom, 2002, p.258.
[17] E. J. Bueno, F. Espinosa, F. J. Rodriguez, J. Urena, and S. Cobreces, ”Current Control of
Voltage Source Converters connected to the grid through an LCL-filter,” in Proceedings of the
PESC 2004, IEEE Conf., 2004, pp. 68–73.
References
[18] G. Buja, M. P. Kazmierkowski, ”Direct Torque Control of PWM Inverter Fed AC Motors- A
Survey”, IEEE Transaction on Ind. Electronics, vol. 51, No. 4, 2004, pp. 744–755.
[19] R. P. Burgos, E. P. Wiechmann, and J. Holtz, ”Complex State Variables Modeling and Nonlinear
Control of PWM Voltage- and Current-Source Rectifiers,” IECON 02, Industrial Electronics
Society, IEEE 2002 28th Annual Conference, Volume 1, 5-8 Nov. 2002 Page(s):187 - 192.
[20] A. Carlsson, “The back-to-back converter control and design,” Lund Institute of Technology,
Ph.D. Thesis, Lund 1998.
[21] D. Casadei, G. Grandi, C. Rossi, A. Trentin, and L. Zarri, ”Comparison between Back-to-Back
and Matrix Converters Based on Thermal Stess of the Switches,” in Proceedings of the IEEE
ISIE 2004 Conf., 2004, pp. 1081–1086.
[22] L. Chen, “Field vector strategy for an induction motor system using PWM rectifier/inverter link,”
Aalborg University, Ph.D. Thesis, Denmark 1995.
[23] W.-S. Chien, and Y.-Y. Tzou, ”Analysis and Design on the Reduction of DC-link Electrolytic
Capacitor for AC/DC/AC Converter Applied to AC Motor Drives,” Power Electronics Specialists
Conference, 1998. PESC 98 Record. 29th Annual IEEE Volume 1, 17-22 May 1998 Page(s):275 –
279.
[24] M. Cichowlas, PWM Rectifier with Active Filtering, Warsaw University of Technology, Ph.D.
Thesis, Warsaw, Poland, 2004.
[25] Danfoss, VLT 5000 FLUX Operating Instructions Software version: 5.4x, MG.55.A5.02, pg.160.
[26] R. W. De Doncker, and D. W. Novotny, ”The Universal Field Oriented Controller,” IEEE Trans.
on Industrial Applications, vol. 30, No. 1, January/February 1994, pp. 92–100.
[27] dSPACE, Manulas for DS1103 PPC Controller Board, Relase 3.5, 2003, on CD.
[28] P. N. Enjeti, P. D. Ziogas, and M. Eshani, ”Unbalanced PWM converter analysis and corrective
measures,” IEEE, 1989, pp. 861–870.
[29] EPCOS, “Aluminium Electrolic Capacitors- Applications,” Data Sheet, www.epcos.com.
[30] EPCOS, “Aluminium Electrolic Capacitors- General Technical Information,” Data Sheet,
www.epcos.com.
[31] G. F. Franklin, J. David Powell, and A. Emami-Naeini , Feedback Control of Dynamic Systems,
Addison-Wesley Publishing Company, Third Edition, 1994, p. 778.
[32] M. E. Fraser, C. D. Manning, and B. M. Wells, ”Transformless four-wire PWM rectifier and its
application in AC-DC-AC converters,” IEE Trans. Proc.-Electr. Power Appl., vol. 142, No. 6,
November 1995, pp. 410–416.
[33] G. O. Garcia, R. M. Stephan, E. H. Watanabe ”Comparing the Indirect Field-Oriented Control
with a Scalar Method,” IEEE Trans. on Industrial Electronics, vol. 41, No. 2, April 1994, pp.
201–207.
[34] P. Z. Grabowski, “Direct Flux and Torque Neuro-Fuzzy Control of Inverter Fed Induction Motor
Drives,” Warsaw University of Technology, Ph.D. Thesis, Warszawa 1999.
[35] P. Z. Grabowski, M. P. Kazmierkowski, B. K. Bose, and F. Blaabjerg, ”A Simple Direct-Torque
Neuro-Fuzzy Control of PWM-Inverter-Fed Induction Motor Drive,” IEEE Trans. on Industrial
Electronics, vol. 47, No. 4, August 2000, pp. 863–870.
[36] M. Grotzbach, and R. Redmann, ”Line Current Harmonics of VSI-Fed Adjustable-Speed
Drives,” IEEE Trans. on Industrial Applications, vol. 36, No. 2, March/April 2000, pp. 683–690.
[37] B.-g. Gu, and K. Nam, “A Dc Link Capacitor Minimization Method through Direct Capacitor
Current Control,” Industry Applications Conference, 2002. 37th IAS Annual Meeting. Conference
Record of the Volume 2, 13-18 Oct. 2002 Page(s):811 - 817.
[38] T. G. Habatler, F. Profumo, M. Pastorelli, and L. M. Tolbert, ”Direct Torque Control of
Induction Machines Using Space Vector Modulation,” IEEE Trans. on Industrial Applications,
vol. 28, No. 5, September/October 1992, pp. 1045–1053.
127
References
[39] T.G. Habatler, ”A space vector-based rectifier regulator for AC/DC/AC converters,” IEEE
Trans. on Power Electronics, vol. 8, February 1993, pp. 30–36.
[40] T.G. Habatler, and D. M. Divan, ”Angle controlled current regulated rectifiers for AC/AC
converters,” IEEE Trans. on Power Electronics, vol. 6, No. 3, July 1991, pp. 463–469.
[41] T.G. Habatler, and D. M. Divan, ”Rectifier/Inverter Reactive Component Minimalization,” IEEE
Trans. on Industrial Applications, vol. 25, No. 2, March/April 1989, pp. 307–316.
[42] S. Hansen, P. N. Enjeti, J.-H. Hahn, and F. Blaabjerg, ”An Integrated Single-Switch Approach
to Improve Harmonic Performance of Standard PWM Adjustable-Speed Drives,” IEEE Trans. on
Industrial Applications, vol. 36, No. 4, July/August 2000, pp. 1189–1196.
[43] S. Hansen, P. Nielsen, and P. Thogersen, ”Harmonic Distortion and Reduction Techniques of
PWM Adjustable Speed Drives- A Cost-Benefit analysis,”, pp. 271–277.
[44] A. M. Hava, R. J. Karkman, T. A. Lipo, ”A High-Performance Generalized Discontinuous PWM
Algorithm,” IEEE Trans. on Industrial Applications, vol. 34, No. 5, September/October 1998, pp.
1059–1071.
[45] A. M. Hava, S.-K. Sul, R. J. Kerkman, ”Dynamic Overmodulation Characteristics of Triangle
Intersection PWM Methods,” IEEE Trans. on Industrial Applications, vol. 35, No. 4, July/August
1999, pp. 896–907.
[46] M. Hinkkanen, and J. Luomi, ”Modified integrator for Voltage Model Flux Estimation of
Induction Motors,” IEEE Trans. on Industrial Electronics, vol. 50, No. 4, August 2003, pp. 818–
820.
[47] D. G. Holmes, and T. A. Lipo, Pulse Width Modulation for Power Converters, Principles and
Practice, Wiley-Interscience and IEEE Press, 2003.
[48] J. Holtz, ”Sensorless Vector Control of Induction Motors at Very Low Speed using a Nonlinear
Inverter Model and Parameter Identification,” Industry Applications, IEEE Transactions on,
Volume 38, Issue 4, July-Aug. 2002 Page(s):1087 - 1095.
[49] J. Holtz, ”The Representation of AC Machine Dynamics by Complex Signal Flow Graphs,” IEEE
Trans. on Industrial Electronics, vol. 42, No. 3, June 1995, pp. 263–271.
[50] J. Holtz, ”Pulsewidth Modulation for Electronics Power Conversion,” in Proceedings of the
IEEE, vol. 82, no. 8, August 1994, pp.1194- 1214.
[51] J. Holtz, ”Speed Estimation and Sensorless Control of Ac Drives,” Industrial Electronics,
Control, and Instrumentation, 1993. Proceedings of the IECON '93., International Conference on,
15-19 Nov. 1993 Page(s):649 - 654.
[52] J. Holtz, and J. Quan, ”Drift- and Parameter-Compensated Flux Estimator for Presistent ZeroStator-Frequency Operation of Sensorless-Controlled Induction Motors,” IEEE Trans. on
Industrial Applications, vol. 39, No. 4, July/August 2003, pp. 1052–1060.
[53] H. Hur, J. Jung and K. Nam, ”A Fast Dynamics DC-link Power-Balancing Scheme for a PWM
Converter-Inverter System,” IEEE Trans. on Industrial Electronics, vol. 48, No. 4, August 2001,
pp. 794–803
[54] IEEE Std 519-1992, “IEEE Recommended Practices and Requirements for Harmonic Control in
Electrical Power Systems ,” The Institute of Electrical and Electronics Engineers, Inc., USA,
1993.
[55] G. Joos, N. R. Zargari, and P. D. Ziogas, ”A new class of current-controlled suppressed-link ac to
ac frequency changers,” Power Electronics Specialists Conference, 1991. PESC '91 Record.,
22nd Annual IEEE, 24-27 June 1991 Page(s):830 - 837.
[56] J. Jung, S. Lim, and K. Nam, ”A Feedback Linearizing Control Scheme for a PWM ConverterInverter Having a Very Small DC-link Capacitor,” IEEE Trans. on Industrial Applications, vol.
35, No. 5, September/October 1999, pp. 1124–1131.
[57] F. Kamran, and T. G. Habatler, ”An Improved Deadbeat Rectifier Regulator Using a Neural Net
Predictor,” IEEE Trans. on Power Electronics, vol. 10, No. 4, July 1995, pp. 504–510.
128
References
[58] J. G. Kassakian, M. F. Schlecht, and G. C. Verghese, Principles of Power Electronics, AddisonWesley Publishing Company, 1991, p. 738.
[59] M. P. Kazmierkowski, and M. A. Dzieniakowski, A. Kasprowicz, S. Kanoza, ”Control of DC
link resonant inverter-fed induction motor drive without speed sensor,” Industrial Electronics,
1995. ISIE '95., Proceedings of the IEEE International Symposium on, Volume 2, 10-14 July
1995 Page(s):628 - 632.
[60] M. P. Kazmierkowski, and A. Kasprowicz, ”Regulacja bezpośrednia momentu i stumienia silnika
klatkowego zasilanego z falownika napięcia,” Przgląd elektotechniczny, R. LXX Z 2/1994, 1994,
pp. 30–33, (in Polish).
[61] M. P. Kazmierkowski, and L. Malesani, ”Current Control Techniques for Three-Phase VoltageSource PWM Converters: A Survey,” IEEE Transaction on Ind. Electronics, vol.45, No. 5,
October 1998, pp. 691–703.
[62] M. P. Kazmierkowski, and W. Sulkowski, ”A Novel Vector Control Scheme for Transistor PWM
Inverter-Fed Induction Motor Drive,” IEEE Trans. on Industrial Electronics, vol. 38, No. 1,
February 1991, pp. 41–47.
[63] M. P. Kazmierkowski, H. Tunia, Automatic control of converter-fed drives, Elsevier,
Amsterdam-London-New York-Tokyo-PWN Warszawa, 1994, p.559.
[64] M. P. Kazmierkowski, M. Nowak, and A. Wojcik, Urządzenia elektroniki przemysłowej, WKŁ,
1982, p.211.
[65] M. P. Kazmierkowski, R. Krishnan, and F. Blaabjerg, Control in Power Electronics, Academic
Press, 2002, pg. 579.
[66] J. S. Kim and S. K. Sul, ”New control scheme for ac–dc–ac converter without dc link electrolytic
capacitor,” in Proceedings of the IEEE PESC’93, 1993, pp. 300–306.
[67] S. Kim, S.-K. Sul, and T. A. Lipo, ”AC/AC Power Conversion Based on Matrix Converter
Topology with Unidirectional Switches,” IEEE Trans. on Industrial Applications, vol. 36, No. 1,
January/February 2000, pp. 139–145.
[68] J. S. Kim, and S. K. Sul “New control scheme for ac-dc-ac converter without dc link electrolytic
capacitor,” Power Electronics Specialists Conference, 1993. PESC '93 Record., 24th Annual
IEEE, 20-24 June 1993, Page(s):300 - 306.
[69] H. Kim, H. Akagi, ”The Instantaneous Power Theory on the Rotating p-q-r Reference Frames,”
Power Electronics and Drive Systems, 1999. PEDS '99. Proceedings of the IEEE 1999
International Conference on, Volume 1, 27-29 July 1999 Page(s):422 - 427.
[70] C. Klumpner, and F. Blaabjerg, ”A Direct Power Electronic Conversion Topology for MultiDrive Applications,” EPE Jurnal, vol. 13, No. 1, February 2003, pp. 25–34.
[71] W. Koczara, ”Compensation in demand for drive power,” in Proceedings of the PEMC 1990
Conf., 1990, pp. 354–358.
[72] J. W. Kolar, H. Ertl, and F. C. Zach, ”Influence of the Modulation Method on the Conduction and
Switching Losses of a PWM Converter System,” EEE Trans. on Industrial Applications, vol. 27,
No. 6, November/December 1991, pp. 1063–1075.
[73] J. W. Kolar, H. Ertl, K. Edelmoser, and F. C. Zach, ”Analysis of the control behaviour of a
bidirectional three-phase PWM rectifier system,” in Proceedings of the EPE 1991 Conf., 1991,
pp. (2-095)–(2-100).
[74] C. Lascu, I. Boldea, and F. Blaabjerg, ”A Modified Direct Torque Control for Induction Motor
Sensorless Drive,” IEEE Trans. on Industrial Applications, vol. 36, No. 1, January/February
2000, pp. 122–130.
[75] C. Lascu, I. Boldea, and F. Blaabjerg, ”Variable-Structure Direct Torque Control- A Class of
Fast and Robust Controllers for Induction Machine Drives,” IEEE Trans. on Industrial
[76] D.-C. Lee, and G.-M. Lee, ”A Novel Overmodulation Technique for Space-Vector PWM
Inverters,” IEEE Trans. on Power Electronics, vol. 13, No. 6, November 1998, pp. 1144–1151.
129
References
[77] B.-K. Lee, and M. Ehsani, ”A Simplified Functional Simulation Model for Three-Phase VoltageSource Inverter Using Switching Function Concept,” IEEE Trans. on Industrial Electronics, vol.
48, No. 2, April 2001, pp. 209–321.
[78] D.-C. Lee, K.-D. Lee, and G- M. Lee, “Voltage control of PWM converters using Feedback
Linearization,” Industry Applications Conference, 1998. Thirty-Third IAS Annual Meeting. The
1998 IEEE, Volume 2, 12-15 Oct. 1998 Page(s):1491 - 1496.
[79] W.-C. Lee, T.-K. Lee, and D.-S. Hyun, ”Comparison of Single-Sensor Current Control in the DC
link for Three-Phase Voltage-Source PWM Converters,” IEEE Trans. on Industrial Electronics,
vol. 48, No. 3, June 2001, pp. 491–505.
[80] H. Le-Huy, ”Comparison of Field-Oriented Control and Direct Torque Control for Induction
Motor Drives,” in Proceedings of the IEEE Conf. , 1999, pp. 1245–1252.
[81] J.Ch. Liao, and S. N. Yen, ”A Novel Instantaneous Power Control Strategy and Analytic Model
for Integrated Rectifier/Inverter Systems,” IEEE Trans. on Power Electronics, vol. 15, No. 6,
November 2000, pp. 996–1006.
[82] M. Lindgren, and J. Svensson, ”Control of a Voltage-source Converter Connected to the Grid
through an LCL-filter-Application to Active Filtering,” IEEE , 1998, pp. 229–235.
[83] M. Liserre, “Innovative control techiques of power converters for industrial automation,”
Politecnico di Bari, Ph.D. Thesis, Italy 2001.
[84] M. Liserre, A. Dell'Aquila, and F. Blaabjerg, ”Genetic Algorithm-Based Design of the Active
Damping for an LCL-Filter Three-Phase Active Rectifier,” IEEE Trans. on Power Electronics,
vol. 19, No. 1, January 2004, pp. 76–86.
[85] M. Liserre, A. Dell'Aquila, and F. Blaabjerg, ”Stability Improvements of an LCL-filter Based
Three-phase Active Rectifier,” IEEE , 2002, pp. 1195–1201.
[86] M. Liserre, C. Klumpner, F. Blaabjerg, V. G. Monopoli, and A. Dell'Aquila, ”Evaluation of the
Ride-Through Capability of an Active-Front-End Adjustable Speed Drive under Real Grid
Conditions,” in Proceedings of the Annual Conf. of the IEEE-IES, 2004.
[87] M. Liserre, F. Blaabjerg, and S. Hansen, ”Design and Control of an LCL-filter based Three-phase
Active Rectifier,” Industry Applications, IEEE Transactions on, Volume 41, Issue 5, Sept.-Oct.
2005 Page(s):1281 - 1291.
[88] K.J.P. Macken, M. H. J. Bollen, and R. J. M. Belmans, ”Mitigation of Voltage Dips Through
Distributed Generation Systems,” IEEE Trans. on Industrial Applications, vol. 40, No. 6,
November/December 2004, pp. 1686–1693.
[89] L. Malesani, L. Rossetto, P. Tenti and P. Tomasin, ”AC/DC/AC Power Converter with Reduced
Energy Storage in the DC Link,” IEEE Trans. on Industrial Applications, vol. 31, No. 2,
March/April 1995, pp. 287–292.
[90] L. Malesani, L. Rossetto, P. Tenti, and P. Tomasin, “AC/DC/AC PWM converter with minimum
energy storage in the dc link,” Applied Power Electronics Conference and Exposition, 1993.
APEC '93. Conference Proceedings 1993., Eighth Annual, 7-11 March 1993 Page(s):306-311.
[91] L. Malesani, L. Rossetto, P. Tenti, P. Tomasin, and A. Zuccato, “Performance optimalization of
quasi-direct converters,” Power Conversion Conference, 1993. Yokohama 1993., Conference
Record of the, 19-21 April 1993 Page(s):99 - 104.
[92] M. Malinowski, "Adaptive Space Vector Modulation for three-phase two-level PWM
rectifiers/inverters," Archives of Electrical Engineering, vol. L1, No. 3, pp. 281-295
[93] M. Malinowski, “Sensorless Control Strategies for Three - Phase PWM Rectifiers,” Warsaw
University of Technology, Ph.D. Thesis, Warszawa 2001.
[94] M. Malinowski, M. P. Kazmierkowski, and A. M. Trzynadlowski, ”A Comparative Study of
Control Techiques for PWM Rectifiers in AC Adjustable Speed Drives,” IEEE Trans. on Power
Electronics, vol. 18, No. 6, November 2003, pp. 1390–1396.
[95] V. Manninen, ”Application of direct torque control modulation technology to a line converter,”
in Proceedings of the EPE 1995 Conf., 1995, pp. 1.292–1.296.
130
References
[96] H. Mao, D. Boroyevich, and F. C. Y. Lee, ”Novel Reduced-Order Small-Signal Model of a
Three-Phase PWM Rectifier and Its Application in Control Design and System Analysis,” IEEE
Trans. on Power Electronics, vol. 13, No. 3, May 1998, pp. 511–521.
[97] G. D. Marques, and J. F. Silva, ”Direct Voltage Control of a PWM AC/DC Voltage Converter,”
in Proceedings of the EPE 1997 Conf., 1997, pp. (3.222)–(3.227).
[98] S. Mazumder, “DSP based implementation of a PWM ac/dc/ac converter using space vector
modulation with primary emphasis on the analysis of the practical problems involved,” pplied
Power Electronics Conference and Exposition, 1997. APEC '97 Conference Proceedings 1997.,
Twelfth Annual Vol. 1, 23-27 Feb. 1997, pp. 306–312.
[99] K. Mikołajuk, Podstawy analizy obwodów energoelektronicznych, Wydawnictwo Naukowe
PWN, 1998, p. 525.
[100] N. Mohan, T. M. Undeland, and W. P. Robbins, Power Electronics: Converters, Applications,
and Design, John Wiley & Sons, 1989, p. 667.
[101] E. Monmasson, and J. P. Louis, ”Presentation of a control law for IM drive based on the
dynamic reconfiguration of a DTC algorithm and a SVM-DTC algorithm,” in Proceedings of the
Electrimacs 2002 Conf., August 2002.
[102] J. Monteiro, G. D. Maraques, and J. Palma, ”A Simplified Stator Fux Vector Control Method in
dq Co-ordinates for Induction Machines,” Industrial Electronics, 2000. ISIE 2000. Proceedings of
the 2000 IEEE International Symposium on, Volume 1, 4-8 Dec. 2000 Page(s):271 - 276.
[103] L. Moran, P. D. Ziogas, and G. Joos, ”Design Aspects of Synchronous PWM Rectifier-Inverter
Systems under Unbalanced Input Voltage Conditions,” IEEE Trans. on Industrial Applications,
vol. 28, No. 6, November/December 1992, pp. 1286–1293.
[104] J. M. Moreno-Eguilaz, and J. Peracula, ”Ral-Time Power Measuring and Monitoring for an
Efficient Vector-Controlled Induction Motor Drive,” in Proceedings of the IECON'98 Conf.,
1998, pp. 871–876.
[105] T. Noguchi, H. Tomiki, S. Kondo, I. Takahashi, ”Direct Power Control of PWM converter
without power-source voltage sensors,” IEEE Trans. on Industrial Applications. Vol.34, No.3,
1998, pp. 473-479.
[106] M. Nowak, R. Barlik, Poradnik inżyniera energoelektronika, Wydawnictwo NaukowoTechniczne WT, 1998, p. 700.
[107] T. Ohnishi, ”Three-phase PWM converter/inverter by means of instantaneous active and
reactive power control,” in Proceedings of the IEEE-IECON Conf., 1991, pp. 819-824.
[108] J. A. Oliver, R. Lawrence, and B. B. Banerjee, ”How to Specify Power-Quality-Tolerant
Process Equipment,” IEEE Industrial Applications Magazine, September/October 2002, pp. 21–
30.
[109] B. T. Ooi, J. W. Dixon, A. B. Kulkarni, and M. Nishimoto, ”An integrated AC drive system
using a controlled-current PWM Rectifier/Inverter link,” IEEE Trans. on Power Electronics, vol.
3, No. 1, January 1988, pp. 64–71.
[110] S. Osowski, Modelowanie układów dynamicznych z zastosowaniem języka Simulink, Oficyna
Wydawnicza Politechniki Warszawskiej, 1997, p.186.
[111] R. Ottersten, and J. Svensson, ”Vector Current Controlled Voltage Source Converter-Deadbeat
Control and Saturation Strategies,” IEEE Trans. on Power Electronics, vol. 17, No. 2, March
2002, pp. 279–285.
[112] S. G. Perler, ”Deriving Life Multipliers for Electrolytic Capacitors,” IEEE PES Newsletter,
First Quarter 2004, pp. 11-12.
[113] A. R. Prasad, P. D. Ziogas, and S. Manias, ”An Active Power Factor Correction Technique for
Three-Phase Diode Rectifiers,” IEEE Trans. on Power Electronics, vol. 6, No. 1, January 1991,
pp. 83–92.
[114] A. R. Prasad, P. D. Ziogas, and S. Manias, ”Passive input current waveshaping method for
three-phase diode rectifiers,” IEE Trans. Proc.-B., vol. 139, No. 6, November 1992, pp. 512–520.
131
References
[115] H. Raffel, and B. Orlik, “High Dynamic Quasi-Direct Converter with Very Low DC-link
Capacity,” Industry Applications Conference, 2000. Conference Record of the 2000 IEEE,
Volume 4, 8-12 Oct. 2000 Page(s):2323 - 2330.
[116] J. Sander-Larsen, K. Jespersen, M. R. Pedersen, F. Blaabjerg, and J. K. Pedersen, “Control of
complete digital-based component-minimized single-phase to three-phase AC/DC/AC converter,”
Industrial Electronics Society, 1998. IECON '98. Proceedings of the 24th Annual Conference of
the IEEE, Volume 2, 31 Aug.-4 Sept. 1998 Page(s):618 - 625.
[117] A. Sikorski, „Problemy dotyczące minimalizacji strat łączeniowych w przekształtniku
AC/DC/AC- PWM zasilającym maszynę indukcyjną,” Politechnika Białostocka, Rozprawy
Naukowe nr. 58, Białystok 1998, pg. 217. (In Polish).
[118] A. Sikorski, „Warunki regulacji prądu przekształtnika AC/DC,” Prądy Niesinusoidalne
EPN'2000, Zielona Góra, 2000, (In Polish).
[119] J. F. Silva, ”Modelling and control of a-b space-vector current-regulated unity-power-factor
three-phase boost rectifiers,”, pp. (II-111)–(II-116).
[120] B. N. Singh, B. Singh, and B. P. Singh, “Fuzzy control of integrated current controlled
converter-inverter fed cage induction motor drive,” Industry Applications, IEEE Transactions on,
Volume 35, Issue 2, March-April 1999 Page(s):405 - 412.
[121] D. L. Sobczuk, “Application of ANN for Control of PWM Inverter Fed Induction Motor Drives,”
Warsaw University of Technology, Ph.D. Thesis, Warszawa 1999.
[122] K. Stockman, M. Didden, F. D'Hulster, and R. Belmans, ”Embaded solutions to protect
TEXTILE PROCESSES agains VOLTAGE SAGS,” IEEE Industrial Applications Magazine,
September/October 2004, pp. 59–65.
[123] R. Strzelecki, and H. Supronowicz, Współczynnik mocy w systemach zasilania prądu
przemiennego i metody jego popraw, OWPW, 2000, p. 451.
[124] D. Swierczynski, and M. P. Kazmierkowski, ”Direct Torque Control of Permanent Magnet
Synchronous Motor (PMSM) Using Space Vector Modulation (DTC-SVM)– Simulation and
Experimental Results,” in Proceedings of the IECON 2002, Sevilla, Spain, on-CD.
[125] H. Tajima, and Y. Hori, ”Speed Sensorless Field-Oriented Control of the Induction Machine,”
IEEE Trans. on Industrial Applications, vol. 29, No. 1, 1993, pp. 175–180.
[126] I. Takahashi, and T. Noguchi, ”A New Quick-Response and High Efficiency Control Strategy
of an Induction Machine” IEEE Trans. on Industrial Applications, vol. IA-22, No. 5,
September/October 1986, pp. 820–827.
[127] I. Takahashi, and T. Noguchi, ”Take a Look Back upon the Past Decade of Direct Torque
Control,” Industrial Electronics, Control and Instrumentation, 1997. IECON 97. 23rd
International Conference on, Volume 2, 9-14 Nov. 1997 Page(s):546 - 551.
[128] R. M. Tallam, and R. Naik, ”Practical Issues in the Design of Active Rectifiers for AC Drives
with Reduced DC-link Capacitance,” Industry Applications Conference, 2003. 38th IAS Annual
Meeting. Conference Record of the, Volume 3, 12-16 Oct. 2003 Page(s):1538 - 1545.
[129] D. Telford, M. W. Dunigan, and B. W. Williams, ”A comparison of Vector Control and Direct
Torque Control of an Induction Machine,” Power Electronics Specialists Conference, 2000.
PESC 00. 2000 IEEE 31st Annual, Volume 1, 18-23 June 2000 Page(s):421 - 426.
[130] R. Teodorescu, F. Blaabjerg, M. Liserre, and A. Dell'Aquila, ”A stable three-phase LCL-filter
based active rectifier without damping,” Industry Applications Conference, 2003. 38th IAS
Annual Meeting. Conference Record of the, Volume 3, 12-16 Oct. 2003 Page(s):1552 - 1557.
[131] Texas Instruments Manuals, TMS320C2000 DSP Platform Code composer Studio, DSK v2.21
IDE, eZdsp F2812, 2002, on CD.
[132] A. Tripathi, A. M. Khambadkone, and S. K. Panda, ”Predictive Stator flux control with
overmodulation and dynamic torque control at constant switching frequency in AC drives,” in
Proceedings of the IEEE Conf., 2002, pp. 2080–2085.
132
References
[133] A. Tripathi, A. M. Khambadkone, and S. K. Panda, ”Space-vector based, Constant Frequency,
Direct Torque Control and dead beat Stator Flux Control of AC machines,” in Proceedings of the
Annual Conf. of the IEEE-IES, 2001, pp. 1219–1224.
[134] H. Tunia, and M. P. Kazmierkowski, Automatyka napędu przekształtnikowego, Państwowe
Wydawnictwo Naukowe, 1987, p.572.
[135] H. Tunia, and R. Barlik, Teoria przekształtników, WPW - Oficyna Wydawnicza Politechniki
Warszawskiej, 2003, p.304.
[136] E. Twining, and D. G. Holmes, ”Grid Current Regulation of a Three-Phase Voltage Source
Inverter With LCL Input Filter,” IEEE Trans. on Power Electronics, vol. 18, No. 3, May 2003,
pp. 888–895.
[137] R. Uhrin, and F. Profumo, “Performance comparison of output power estimators used in
AC/DC/AC converters,” Industrial Electronics, Control and Instrumentation, 1994. IECON '94.,
20th International Conference on, Volume 1, 5-9 Sept. 1994 Page(s):344 - 348.
[138] H. W. Van Der Broeck, and H. C. Skudelny ”Analysis and Realization of a Pulswidth
Modulator Based on Voltage Space Vectors,” IEEE Trans. on Industrial Applications, vol. 24,
No. 1, January/February 1988, pp. 142–150.
[139] A. van Zyl, R. Spee, A. Faveluke, and S. Bhowmik, ”Voltage Sag Ride-Through for
Adjustable-Speed Drives With Active Rectifiers,” IEEE Trans. on Industrial Applications, vol.
34, No. 6, November/December 1998, pp. 1270–1276.
[140] P. Vas, Sensorless Vector and Direct Torque Control, Oxford University Press, 1998, pg. 729.
[141] E. P. Weichmann, P. D. Ziogas, and V. R. Stefanovic, ”Generalized Functional Model for
Three-Phase PWM Inverter/Rectifier Converters,” IEEE Trans. on Industrial Applications, vol.
IA-23, No. 2, March/April 1987, pp. 236–246.
[142] N. J. Wheeler, H. Zhang, and D. A. Grant, “Minimalisation of reactive component values in
back-to-back converters”, in Proceedings of the PEVSD1994 Conf., 1994, pp.240-245.
[143] M. Winkelnkemper, and S. Bernet, ”Design and Optimalization of the DC-link Capacitor of
PWM Voltage Source Inverter with Active Frontend for Low-Voltage Drives,” in Proceedings of
the EPE 2003 Conf., 2003.
[144] T. M. Wolbank, A. Moucka, and J. L. Machl, ”A comparative study of field-oriented and directtorque control of induction motors reference to shaft-sensorless control at low and zero-speed,”
in Proceedings of the ISIC 2002 IEEE Conf. , 2002, pp. 391–396.
[145] R. Wu, S. B. Dewan, and G. L. Slemon, ”Analysis of a PWM ac to dc Voltage Source Converter
under the Predicted Current Control with Fixed Switching Frequency,” IEEE Trans. on Industrial
Applications, vol. 27, No. 4, 1991, pp. 756–764.
[146] R. Wu, S. B. Dewan, and G. R. Slemon, ”A PWM AC-to-DC Converter with Fixed Switching
Frequency,” IEEE Trans. on Industrial Applications, vol. 26, No. 5, September/October 1990, pp.
880–885.
[147] R. Wu, S. B. Dewan, and G. R. Slemon, ”Analysis of an ac-to-dc Voltage Source Converter
Using PWM with Phase and Amplitude Control,” IEEE Trans. on Industrial Applications, vol. 27,
No. 2, March/April 1991, pp. 355–364.
[148] K. Xing, F. C. Lee, J. S. Lai, Y. Gurjit, and D. Borojevic, ”Adjustable Speed Drive Neutral
Voltage Shift and Grounding Issues in a DC Distributed System,” in Proceedings of the Annual
Meeting of the IEEE-IAS, 1997, pp. 517–524.
[149] Y. Xue, X. Xu, T. G. Habatler, and D. M. Divan, ”A Low Cost Stator Flux Oriented Voltage
Source Variable Speed Drive,” Industry Applications Society Annual Meeting, 1990., Conference
Record of the 1990 IEEE, 7-12 Oct. 1990 Page(s):410 - 415.
[150] N. R. Zargari, and G. Joós, ”Performance Investigation of a Current-Controlled VoltageRegulated PWM Rectifier in Rotating and Stationary Frames,” IEEE Trans. on Industrial
[151] M. Zelechowski, M. Malinowski, P. Kaczynski, W. Kołomyjski, M. Twerd, J. Zaleski, "DSP
Based Sensorless Direct Torque Control - Space Vector Modulated (DTC-SVM) for Inverter Fed
133
References
Induction Motor Drives," Problems of Automated Electro drives Theory and Practice, September
15-20, 2003, Crimea, Ukraine, pp. 90-92
[152] M. Zelechowski, Space Vector Modulated-Direct Torque Controlled (DTC-SVM) Inverter-Fed
Induction Motor Drive, Warsaw University of Technology, Ph.D. Thesis, Warsaw, Poland, 2005.
[153] H. Zhang, N. J. Wheeler, and D. A. Grant “Switching harmonics in the dc link current in a
PWM ac-dc-ac converter,” Industry Applications Conference, 1995. Thirtieth IAS Annual
Meeting, IAS '95., Conference Record of the 1995 IEEE, Volume 3, 8-12 Oct. 1995 Page(s):2649
- 2655.
[154] D. Zhou, and D. Rouaud, ”Regulation and Design Issues of A PWM Three-Phase Rectifier,” in
Proceedings of the 25th Annu. Conf. IECON’99 , vol.1., pp.485 – 489.
Papers written during work on this thesis
Journal Papers
[155] M. Malinowski, M.P. Kazmierkowski, and M. Jasiński, “Virtual Flux based Direct Power
Control of Tree-Phase PWM Rectifiers,” Electrical Power Quality and Utilization, Volume VII,
Sheet 1, Year 2001, pp. 129–138.
[156] M. Jasiński, M. Liserre, and M. P. Kaźmierkowski, "Regulator rozmyty w napięciowo
zorientowanym sterowaniu prostownikiem PWM," Przegląd Elektrotechniczny, 2’ 2003, pp. 101104, (in Polish).
[157] M. Malinowski, M. Jasiński, and M. P. Kazmierkowski, "Simple Direct Power Control of
three-Phase PWM Rectifier Using Space-Vector Modulation (DPC-SVM),” IEEE Transaction on
Industrial Electronics, vol. 51, No. 2, April 2004, pp. 447– 454.
[158] M. Jasinski, P. Antoniewicz, M. P. Kaźmierkowski, „Napędy indukcyjne zasilane w układzie
prostownik PWM /falownik PWM ze sterowaniem wektorowym”, Przegląd Elektrotechniczny, r.
81 Nr 6/2005, pp. 1-5.
[159] M. Jasinski, M. Cichowlas, M. P. Kaźmierkowski, „Direct control for AC/DC/AC converterfed induction motor with active filtering function”, Special Issue of COMPEL, (under
publication).
Conference Contribution
[160] M. Jasiński (Angielczyk), M. Malinowski, and D. Świerczyński, "Stanowisko laboratoryjne
oparte o kartę DSP do sterowania prostownikiem PWM oraz falownikiem zasilającym silnik
PMSM,” in Proceedings of the MiS’2000 Conf., 2000, pp. 480–487, (in Polish).
[161] M. Malinowski, and M. Jasiński, “DSP Implementation of Sensorless Control Strategies Based
on Virtual Flux for PWM Rectifier,” In. Proc. of the Energy Saving in Electrical Engineering,
May 2001, pp. 335–339.
[162] M. Jasiński, and M. Malinowski, “DSP Implementation of Control Strategies for PWM
Rectifiers in AC Adjustable Speed Drives,” in Proceedings of the I Summer Seminar on Nordick
Network for Multi Disciplinary Optimised Electric Drives 2001, pp. 81–85.
[163] M. Jasiński, M. Liserre, and M.P. Kaźmierkowski, "Regulator rozmyty w napięciowo
zorientowanym sterowaniu prostownikiem PWM,” in Proceedings of the MiS-2Conf., 2002, pp.
509–517.
[164] M. Malinowski, M. Jasiński, and M. P. Kazmierkowski, "Simple Direct Power Control of
Three-Phase PWM Rectifier Using Space Vector Modulation,” in Proceedings of the EPE-PEMC
2002, 2002, on CD-ROM T6-045.
[165] M. Jasiński, M. Liserre, F. Blaabjerg, and M. Cichowlas, "Fuzzy Logic Current Controller for
PWM Rectifiers,” in Proceedings of the IECON’02, 2002, on CD-ROM SF-006578.
[166] M. Cichowlas, M. Malinowski, M. Jasiński, and M. P. Kaźmierkowski, “DSP Based Direct
Power Control for three-phase PWM Rectifier with Active Filtering Function,” in Proceedings of
the ISIE’03, 2003, on CD-ROM BF-000011.
134
References
[167] M. Jasiński, D. Świerczyński, M. P. Kaźmierkowski, and M. Malinowski, “Direct Control of an
ac-dc-ac converter,” in Proceedings of the III Summer Seminar on Nordick Network for Multi
Disciplinary Optimised Electric Drives Conf., 2003, pp. 33-39.
[168] M. Jasiński, D. Świerczyński, M. P. Kaźmierkowski, and M. Malinowski, “Direct Torque and
Power Control of an ac-dc-ac converter,” in Proceedings of the V Power Electronics Seminar,
2003.
[169] M. P. Kaźmierkowski, M. Cichowlas, and M. Jasiński , “Artificial Intelligence Based
Controllers for Industrial PWM Power Converters,” in Proceedings of the INDIN’03 Conf., 2003,
on CD-ROM.
[170] M. Cichowlas, M. Malinowski, M. P. Kaźmierkowski, and M. Jasiński, “Novel active filtering
function for DPC based three-phase PWM rectifier,” in Proceedings of the EPE’03 Conf., 2003,
on CD-ROM 0642,.
[171] M. Jasiński, D. Świerczyński, M. P. Kazmierkowski, and M. Żelechowski, “Sensorless Direct
Power and Torque Control of Space Vector Modulated AC/DC/AC Converter- Fed Induction
Motor,” in Proceedings of the SENE’03 Conf., 2003, pp. 179- 185.
[172] M. Jasiński, D. Świerczyński, and M. P. Kaźmierkowski, "Novel Sensorless Direct Power and
Torque Control of Space Vector Modulated AC/DC/AC Converter,” in Proceedings of the ISIE
2004 Conf., 2004, pp. 1147- 1152.
[173] M. Jasiński, "Direct power flow control of a back-to-back converter fed induction motor,” in
Proceedings of the NorFa Summer Seminar 2004 Conf., 2004, pp.11- 17.
[174] M. Jasiński, M. Cichowlas, and M. P. Kazmierkowski, "Direct Control for AC/DC/AC
Converter- fed Induction Motor with Active Filtering Function,” in Proceedings of the EPNC
2004 Conf., 2004, pp. 103-105.
[175] M. Jasinski, M. P. Kaźmierkowski, and M. Żelechowski, "Unified Scheme of Direct Power and
Torque Control for Space Vector Modulated AC/DC/AC Converter- Fed Induction Motor,” in
Proceedings of the EPE- PEMC'04 Conf., 2004, on CD-ROM A31206.
[176] M. Jasiński, M. P. Kazmierkowski, and M. Żelechowski, "Direct Power and Torque Control
Scheme for Space Vector Modulated AC/DC/AC converter- Fed Induction Motor,” in
Proceedings of the ICEM Conf., 2004, on CD-ROM 901.
[177] M. Jasiński, and M. P. Kaźmierkowski, "Sensorless Direct Power and Torque Control of PWM
Back-To-Back Converter-Fed Induction Motor,” in Proceedings of the IECON Conf., 2004, on
CD-ROM P1608.
[178] M. Jasiński, and M. P. Kaźmierkowski, "Direct Constant Switching Frequency Control for
Back-to-Back Converter- fed Induction Motor,” in Proceedings of the ICIT Conf., 2004, on CDROM TF-003104.
[179] M. Jasinski, P. Antoniewicz, M. P. Kazmierkowski, „Vector Control of PWM Rectifier –
Inverter Fed Induction Machine – A Comparison”, in Proc. of the CPE Conf., 1- 3 June, Gdańsk,
Polska, (CPE2005/2.10 , on CD)
[180] P. Antoniewicz , M. Jasinski, „Przekształtnik AC/DC/AC ze zmniejszoną wartością
kondensatora w obwodzie pośredniczącym”, in Proc. of PES-2 Conf., 20 - 24 June, Zakopane,
Polska, (under publication).
[181] M. Jasinski, P. Antoniewicz, „Vector Control of PWM Rectifier – Inverter Fed Induction
Motor – A Comparison”, in Proc. of the NorFa Summer Seminar, 2- 4 September, Jurata, Polska,
pp. 97-101.
[182] M. Jasinski, „Direct Power and Torque Control for AC/DC/AC Converter with Reduced DClink Capacitor”, in Proc. of PELINCEC Conf., 17-19 October, Warszawa, Polska, (in progress).
[183] P. Antoniewicz, M. Jasinski, M. P. Kaźmierkowski, „AC/DC/AC Converter with Reduced DC
Side Capacitor Value”, in Proc. of
EUROCON Conf., 22-24, November, Belgrade,
Serbia&Montenegro, (in progress).
.
135
Symbols Employed
Main Symbols
a=e
C –
j
2π
3
1
3
=− + j
–
2
2
complex unit vector
capacitance, absolute value
C = C1 + C2 – DC-link capacitor capacitance,
Cmin_ VSR –
minimum capacitance value of the DC-link capacitor for
AC/DC/AC converter with VSR,
Cdi – capacitance of DC-link capacitor for AC/DC/AC converter with diode
rectifier,
D1 (DA , DB , DC ) –
duty cycle of the VSR,
D2 –
duty cycle of the VSI,
E –
internal voltage induced in the motor winding (EMF), absolute value,
E A , EB , EC –
instantaneous values of the voltage induced in the motor
windings,
e –
error of control variables for example eΩ m ,
fL –
line voltage frequency,
fs –
sampling frequency,
f sw – switching frequency,
I–
current, absolute value,
I A , I B , I C – instantaneous values of the phase currents generally,
Ic –
current in DC-link capacitor,
I dc –
current on the VSR output,
I load – current on the VSI input,
Symbols Employed
IL –
line current space vector,
I LA , I LB , I LC – instantaneous values of the line phase currents,
Ir –
rotor current space vector,
I ra , I rb , I rc –
IS –
instantaneous values of the rotor phase currents,
stator current space vector,
I SA , I SB , I SC – instantaneous values of the stator phase currents,
I Sd , I Sq –
stator current vector components of the induction motor in
field-oriented dq coordinates,
I Sx , I Sy –
field-oriented xy coordinates,
I Sα , I Sβ –
stationary αβ coordinates,
J–
moment of inertia,
k–
space vector generally,
L–
inductance, absolute value
L–
input (AC line side) choke phase inductance,
LM – main, magnetizing inductance,
Lr –
rotor winding self-inductance,
LS –
stator winding self-inductance,
LSσ – stator winding leakage inductance,
Lσ –
total leakage inductance of the induction motor,
M–
mutual inductance, absolute value,
M–
torque, absolute value,
M–
virtual torque –VT,
M e – electromagnetic torque,
M L – load torque,
M N – nominal torque,
ms –
number of phases,
P–
active power, absolute value,
Pe –
electromagnetic power,
137
Symbols Employed
PVSI – output (AC side) power of VSI,
PVSR – input (AC side) power of VSR,
pb –
number of pole pairs,
Q–
reactive power, absolute value,
R–
resistance, absolute value,
R–
resistance of the input (AC line side) choke,
RFVSI – forward resistance of the VSI,
Rload – resistive load of the VSR,
Rr = Rra = Rrb = Rrc – rotor phase windings resistance,
RS = RSA = RSB = RSC – stator phase windings resistance,
RSr = RS + Rr ,
s–
Laplace variable,
S1 (S A , S B , SC ) –
S2 –
switching pattern of the VSR,
switching pattern of the VSI,
S A , S B , SC –
switching states of the VSR for appropriate phase,
t–
time,
T–
time constant, absolute value,
TIT –
equivalent time constant of the closed control loop,
Tpf –
time constant of prefilter,
TPWM – time constant of VSC, statistical delay of PWM generation,
U–
TS –
time constant of S&H block, sampling period,
TU –
time constant of DC-link voltage filter,
voltage, absolute value,
U A ,U B ,U C – instantaneous values of the phase voltages generally,
U dc – DC-link voltage in AC/DC/AC converter circuit,
Ui –
voltage drop on the input (AC line side) choke space vector,
UL –
line voltage space vector,
U LA ,U LB ,U LC – instantaneous values of the line phase voltages,
U p – VSR input voltage space vector,
138
Symbols Employed
U pA ,U pB ,U pC – instantaneous values of the VSR input phase voltages,
Ur –
rotor voltage space vector,
U ra ,U rb ,U rc – instantaneous values of the rotor phase voltages,
US –
stator voltage space vector,
U SA ,U SB ,U SC – instantaneous values of the stator phase voltages,
γ I – angle between Ψ rd and I Sq ,
S
γΨ –
angle between Ψ S and Ψ r ,
σ = 1−
L2M
– total leakage factor,
LS Lr
ϕ–
angle between I L and U L , power angle,
Ψ–
flux linkage, absolute value,
Ψ L – space vector of the line virtual flux,
Ψ L 0 – initial value of the line virtual flux space vector,
Ψr –
space vector of the rotor flux linkage,
Ψ ra ,Ψ rb ,Ψ rc – flux linkages of the rotor phase windings,
Ψ S – space vector of the stator flux linkage,
Ψ SA ,Ψ SB ,Ψ SC – flux linkages of the stator phase windings,
Ω–
angular frequency, absolute value,
ΩK –
angular frequency of the coordinate system,
Ωm –
angular frequency of the motor shaft,
ΩN –
nominal angular frequency ( 2π 50 ),
Ωr –
rotor (slip) angular frequency,
ωL –
line voltage angular frequency,
Rectangular Coordinates System
αβ – stator-oriented (stationary) coordinates,
dq –
rotor flux (field)-oriented (rotated) coordinates,
xy –
stator flux (field)-oriented (rotated) coordinates,
xy –
line voltage or line virtual flux-oriented (rotated) coordinates on VSR side,
139
Symbols Employed
Indices
c
–
command value, reference value,
dq
–
… space vector in synchronous rotating dq coordinates,
m
–
maximum, peak value,
min
xy
–
–
minimum value,
… space vector in synchronous rotating xy coordinates,
Mathematical symbols
UL ⋅ IL –
UL × IL –
I*L –
{ }
vector product = Im{U I }= (U
scalar product = Re U L I L = (U Lα I Lα + U Lβ I Lβ ),
*
*
L L
conjugate complex number,
Abbreviations
EMI – electromagnetic interference,
IM – induction motor,
MO– modulus optimum,
PI –
proportional-integral controller,
PWM – pulse width modulation,
SO–
symmetry optimum,
UPF – unity power factor,
VM – virtual machine,
VSC – voltage source converter,
VSI – voltage source inverter,
VSR – voltage source rectifier,
SVM – space vector modulator,
140
I
Lβ Lα
− U Lα I Lβ ) ,
A. Appendices
A.1. Space vector in coordinate systems
An advantage of space vectors is possibility their representation in various
systems of rectangular coordinates. Therefore, space vectors are very convenient
mathematical tool described three phase systems.
Spatial projection of space vector k on the phase axes ( A, B ,C ) are expressed as
(see [63]):
k A ( t ) = Re[k ],
[ ]
k B ( t ) = Re a 2k ,
kC ( t ) = Re[ak ]
(A.1. 1)
Assuming special steady-state case (phase waveforms are sinusoidal with
frequency Ω = 2πf ) an important relation between space vector and positivesequence system can be described:
k = K 1 = K A = k m e jϕ e
jΩ t
(A.1. 2)
Where:
ϕ - angle of initial phase.
A.1.1. Fixed System of Coordinates - αβ
Introducing a fixed system of rectangular coordinates the following space vector
can be defined:
k = kα + jk β
(A.1. 3)
Where α is the real axis and β the imaginary one:
kα = Re[k ] , k β = Im[k ]
(A.1. 4)
A.1.2. Rotating System of Coordinates
Let consider that space vector in fixed system of coordinates αβ is known. It will
be transformed into a rotating system of coordinates with angular frequency
Ωw =
dγ w
, where γ w is an angle between real axis α of the fixed system of
dt
coordinates and real axis d m of rotating one. Then, taking into account rotating
system of coordinates the following expressions could be written:
k w = ke − jγ w , and opposite k = k w e jγ w
Where:
(A.1. 5)
Appendices
k - space vector described in fixed system of coordinates αβ , k = ke jγ ; γ - angle
between complex space vector and α axe of stationary system of coordinates (see
Fig. A. 1. 1).
k w - space vector described in rotating system of coordinates with angular
frequency Ω w , k w = ke j (γ −γ w ) .
It is convenient to define rotating system of coordinates in special condition when,
Ω w = Ω m is an angular speed of the rotor. This system of coordinates is known in
literature as dq (here is marked by dqm ) system:
k w = k dq m = k d m + jk q m
(A.1. 6)
Where d m is the real axis and coincides with the direction of rotor phase a , and
qm the imaginary axis:
[ ]
[ ]
kd m = Re k dq m , kq m = Im k dq m
(A.1. 7)
It is also convenient to define rotating system of coordinates with arbitrary
angular speed Ω K .
Let consider that space vector k dq m in rotating system of coordinates dqm is
known. It will be transformed into a rotating system of coordinates with angular
frequency Ω K =
dγ K
, where γ K is an angle between real axis α of the fixed system
dt
of coordinates and real axis of the rotating one with arbitrary angular speed.
Therefore, space vector k dq m can be described in rotating system of coordinates with
angular frequency Ω K with taking into account rotating of the both system:
k K = k dq me− j (γ K −γ m )
(A.1. 8)
And in respect to stationary αβ system:
k K = k dq m e − j (γ K −γ m ) = ke − jγ m e − j (γ K −γ m ) = ke j (γ −γ m )e − j (γ K −γ m ) = ke j (γ −γ K ) = ke − jγ K ,
(A.1. 9)
Where:
k K - is a space vector k described in rotating system of coordinates with arbitrary
angular frequency Ω K .
142
Appendices
In the special condition when Ω K = Ω S is a synchronous frequency the
synchronous rotating system of coordinates - xy is obtained. Such a system is related
to arbitrary space vector (voltage, current or flux linkage). This synchronous rotating
system of coordinates is known in literature as xy system (see Fig. A. 1. 1). The
space vector is defined as:
k xy = k x + jk y
(A.1. 10)
Where x is the real axis and coincides with the direction of the selected space
vector (voltage, current or flux linkage), and y the imaginary axis:
[ ]
[ ]
k x = Re k xy , k y = Im k xy
(A.1. 11)
ΩK = Ω s
β
qm
x
k
γK −γm
y
γK
γ
Fig. A. 1. 1. Transformation of the space vector
Ωw = Ωm
dm
γw
α
k from fixed system of coordinates αβ to
dqm with angular frequency Ω m , and to synchronous
rotating reference frame xy with arbitrary angular frequency Ω S
rotating system of coordinates
A.1.3. Model of the Induction Motor in Natural ABC Coordinates
Voltages equations for stator of the induction motor in natural coordinates:
dΨ SA
dt
dΨ SB
= I SB RS +
dt
dΨ SC
= I SC RS +
dt
U SA = I SA RS +
U SB
U SC
(A.1. 12)
From analogy for rotor voltages:
143
Appendices
dΨ ra
dt
dΨ rb
U rb = I rb Rr +
dt
dΨ rc
U rc = I rc Rr +
dt
U ra = I ra Rr +
(A.1. 13)
And flux-currents equations:
Ψ SA = LS I SA + LM I ra
Ψ SB = LS I SB + LM I rb
Ψ SC = LS I SC + LM I rc
(A.1. 14)
Ψ ra = Lr I ra + LM I SA
Ψ rb = Lr I rb + LM I SB
Ψ rc = Lr I rc + LM I SC
(A.1. 15)
Matrix form of above equations:
U SK = RS I SK +
dΨ SK
dt
(A.1. 16)
U rK = Rr I rK +
dΨ rK
dt
(A.1. 17)
Ψ SK = LS I SK + LM I RK
(A.1. 18)
Ψ rK = Lr I rK + LM I SK
(A.1. 19)
In order to reduce the set of equations (A.1.12)-(A.1.15) from 12 to 4 the complex
space vectors are used. Additionally, to eliminate the dependence of mutual
inductance on the angle of rotor rotation relative to the stator, transformation into a
common rotating coordinate system with and an arbitrary angular speed Ω K is
needed. Therefore, following equations can be written:
U SK = RS I SK +
dΨ SK
+ jΩ K Ψ SK
dt
(A.1. 20)
U rK = Rr I rK +
dΨ rK
+ j (Ω K − pb Ω m )Ψ rK
dt
(A.1. 21)
Ψ SK = LS I SK + MI rK
(A.1. 22)
Ψ rK = Lr I rK + MI rK
(A.1. 23)
Where the
d (LS I SK + MI rK )
represents the inner voltage (EMF) of the
dt
transformation, while jΩ K (LS I SK + MI rK ) is the inner voltage of the motor rotation
[63].
144
Appendices
A.2. Coordinate Transformation
A.2.1. Three-Phase to Two-Phase Conversion (ABC/ αβ )
⎡ 1
⎡kα ⎤ ⎢
⎢k ⎥ = ⎢ 1
⎣ β⎦ ⎢ 3
⎣
0 0 ⎤ ⎡k A ⎤
⎥ ⎢k ⎥
2
0⎥⎢ B ⎥
⎥⎦ ⎢⎣kC ⎥⎦
3
(A.2. 1)
A.2.2. Two-Phase to Three-Phase Conversion ( αβ /ABC)
⎡
⎢ 1
⎡k A ⎤ ⎢
⎢k ⎥ = ⎢− 1
⎢ B⎥ ⎢ 2
⎢⎣kC ⎥⎦ ⎢
⎢− 1
⎢⎣ 2
0
3
2
3
−
2
⎤
0⎥
⎥
⎡kα ⎤
0 ⎥⎥ ⎢ ⎥
k
⎥⎣ β ⎦
0 ⎥⎥
⎦
(A.2. 2)
A.2.3. Rectangular to Rectangular Coordinate Conversion ( αβ /xy) and (xy/ αβ )
With the assumption that the rectangular components kα , k β of space vector k in
stationary coordinates αβ are known. Knowing from space vector theory that:
k = ke jγ - space vector in stationary αβ coordinates,
and :
k K = ke j (γ −γ K ) = ke − jγ K - space vector in rotating xy coordinates with overall
angular speed Ω K (see Fig. A. 1. 1).
Therefore the components k x , k y of the xy rotating system with angular speed
Ω K can be written:
ke − jγ K = (kα + jk β )(cos γ K − j sin γ K ) =
= (kα cos γ K + kβ sin γ K + j (− kα sin γ K + k β cos γ K ))
(A.2. 3)
Rewritten into matrix form:
⎡k x ⎤ ⎡cosγ K sinγ K ⎤ ⎡kα ⎤
⎢k ⎥ = ⎢
⎥⎢ ⎥
⎣ y ⎦ ⎣- sinγ K cosγ K ⎦ ⎣k β ⎦
(A.2. 4)
In an analogous way the inverse transformation can be found:
k K e jγ K = (k x + jk y )(cos γ K + j sin γ K ) =
= (k x cos γ K − k y sin γ K + j (k x sin γ K + k y cos γ K ))
Rewritten into matrix form:
145
(A.2. 5)
Appendices
⎡kα ⎤ ⎡cosγ K - sinγ K ⎤ ⎡k x ⎤
⎢k ⎥ = ⎢
⎥⎢ ⎥
⎣ β ⎦ ⎣sinγ K cosγ K ⎦ ⎣k y ⎦
(A.2. 6)
A.3. Apparent, Active, and Reactive Power
A.3.1. Complex Representation of the Power
Assuming that voltage and current have a complex forms and its vector scheme is
presented in Fig. A. 3. 1 [99].
Fig. A. 3. 1. a) Voltage and current space vectors; b) apparent power, active power, reactive
power space vectors
U = Ue jϕU , I = Ie jϕ I - complex space vector representation. From Fig. A. 3. 1 can
be noted that:
ϕ = ϕU − ϕ I
(A.3. 1)
From Fig. A. 3. 1b) can be written equation for apparent power:
S = P + jQ = Se jϕ
(A.3. 2)
therefore:
Se jϕ = UIe j (ϕU −ϕ I ) = Ue jϕU Ie− jϕ I = UI*
(A.3. 2a)
Base on these equations can be written:
S = UI* -
apparent power,
(A.3. 3)
{ }
Q = Im{UI }- instantaneous reactive power,
P = Re UI* - instantaneous active power,
*
(A.3. 4)
(A.3. 5)
It can be seen that in inductance like receiver reactive power is positive.
In three phase system:
146
Appendices
S=
ms * 3 *
UI = UI - apparent power,
2
2
P=
3
Re UI* 2
Q=
3
Im UI* 2
(A.3. 6)
{ }
instantaneous active power,
(A.3. 7)
{ }
instantaneous reactive power,
(A.3. 8)
The correctness of the above relations can be tested by definition of the space
vector (Appendix A.1):
Knowing that:
a* = a −1 = a 2
(A.3. 9)
a*2 = a −2 = a
(A.3. 9a)
Re a = Re a − 2 = −
1
2
(A.3. 9b)
That gives:
S=
(
) (
)
ms * 3 * 3 ⎧ 2
2
⎫
UI = UI = ⎨ U A + aU B + a 2U C
I A + a 2 I B + aI C ⎬
2
2
2 ⎩3
3
⎭
(A.3. 10)
For simplicity S can be divided into P and Q :
Re{S} = P =
P=
(
) (
)
3 ⎧2
2
⎫
Re⎨ U A + aU B + a 2U C
I A + a 2 I B + aI C ⎬
2 ⎩3
3
⎭
(A.3. 11)
2⎧
1
⎫
⎨(U A I A + U B I B + U C I C ) − [U A (I B + I C ) + U B (I A + I C ) + U C (I A + I B )]⎬
3⎩
2
⎭
(A.3. 11a)
Based on assumption:
I A + I B + IC = 0
(A.3. 12)
can be written:
I B + IC = − I A , I A + IC = − I B , I A + I B = − IC
(A.3. 13)
Therefore instantaneous active power:
P = U A I A + U B I B + U C IC .
(A.3. 14)
And instantaneous reactive power:
147
Appendices
Im{S} = Q =
Q=
Q=
1
3
1
3
(
) (
)
3 ⎧2
2
⎫
Im⎨ U A + aU B + a 2U C
I A + a 2 I B + aI C ⎬
2 ⎩3
3
⎭
(A.3. 15)
[I A (U B − U C ) + I B (U C − U A ) + I C (U A − U B )]
(A.3.15a)
(I AU BC + I BU CA + I CU AB )
(A.3.15b)
A.4. Simulation Model and Laboratory setup
A.4.1. Saber Model
The control schemes of AC/DC/AC converter-fed induction motor drive were
implemented in Saber Synopsys Inc. v. 2.4. Saber gives possibilities to analysis
behavior of analog and digital control structures as well as electrical circuits.
Constructed model is presented in Fig. A. 4. 1 .The main data of the model are
shown in Tab. A. 4. 1-Tab. A. 4. 3.
Tab. A. 4. 1. Data of the Induction Motor in Saber Model
Nominal power
PN = 3 kW
Nominal voltage
U SN = 380 V
Nominal current
I SN = 6.9 A
Nominal frequency
f SN = 50 Hz
Base speed
Ω mN = 1415 rpm
Number of pole pairs pb = 2
J = 0.0154 kgm2
Moment of inertia
Ψ SN = 0.98 Wb
Nominal stator flux
M N = 20 Nm
Nominal torque
Stator resistance
Rotor resistance
Stator inductance
Rotor inductance
Main inductance
Phase number
RS = 1.84 Ω
Rr = 1.84 Ω
LS = 0.17 H
Lr = 0.17 H
LM = 0.16 H
mS = 3
Tab. A. 4. 2.Data of the Supply Line and AC/DC/AC Converter Model
Line voltage
Line frequency
DC-link voltage
U L = 141 V
f L = 50 Hz
U DC =560 V
Resistive load
DC-link capacitor
Rload = 100 or 150 Ω
C =470 µF or 47 µF
148
Appendices
Tab. A. 4. 3. Date of the LCL Input Filter in Saber Model
Line side inductance
Converter side ind.
Resistance of L1
Resistance of L
L1 = 590 µH
L = 10 mH
R1 = 0.1 Ω
R = 0.08 Ω
C F = 20 µF
Filter capacitors
Fig. A. 4. 1. Saber Model
Tab. A. 4. 4. Modeled distorted line voltage
Number of
Harmonics
5
7
11
13
1) % of 1 harmonic
of UA (THD 4 %)
4
0.0
0.0
0.0
2) % of 1 harmonic
of UA (THD ~4 %)
3.5
0.0
1.1
0.3
Remarks:
-simulated results taken only 5 or 5, 7, 11, and 13 harmonics for THD calculation,
whereas measurements on the laboratory setup using LEM D6000 analyzer taken till
49th harmonics (maximum range is up to 99th harmonics).
A.4.2. Matlab Simulink Power Toolbox Model
This model has been built for comparison with Saber simulations results.
Therefore, validation and verification of the Saber model as well as Matlab Simuling
Power Toolbox model has been obtained.
149
Appendices
Fig. A. 4. 2. Model in Matlab Simulink Power Toolbox
Fig. A. 4. 3. Block of the V-FOC in Matlab Simulink Power Systems
A.4.3. Laboratory setup
An experimental investigation was conducted on a laboratory setup. The scheme
of the laboratory setup is presented in Fig. A. 4. 4. The setup consists of: input LCL
filter, AC/DC/AC converter constructed based on two VSIs (VLT5005, serially
produced by Danfoss with replaced control interfaces [25]) controlled by dSPACE
DS1103 and 3 kW induction motor set. The computer is used for software
development and process visualization. Main parameters of the laboratory setup are
shown in Tab. A. 4. 5 - Tab. A. 4. 10.
150
LINE
Appendices
L1
L
CF
Fig. A. 4. 4. Scheme of the laboratory setup
Tab. A. 4. 5. Data of the Induction Motor on Laboratory setup
Nominal power
PN = 3 kW
Nominal voltage
U SN = 380 V
Nominal current
I SN = 6.9 A
Nominal frequency
f SN = 50 Hz
Base speed
Ω mN = 1415 rpm
Number of pole pairs
Moment of inertia
Nominal stator flux
pb = 2
cos ϕ = 0.89
J = 0.0154 kgm2
Ψ SN = 0.98 Wb
Nominal torque
M N = 20 Nm
Stator resistance
RS = 1.84 Ω
Rotor resistance
Stator inductance
Rr = 1.84 Ω
LS = 0.17 H
Rotor inductance
Main inductance
Lr = 0.17 H
LM = 0.16 H
mS = 3
Phase number
151
Appendices
Fig. A. 4. 5. Pictures of the laboratory setup, 1–isolating interface, 2- AC/DC/AC converter
constructed based on two VSIs (Danfoss VLT 5005 with replaced control boards), 3–
isolating transformer, 4– resistive load, 5– reversible rectifier, 6– motor set, 7– input choke.
152
Appendices
Tab. A. 4. 6. Data of the DC Motor on Laboratory setup
Nominal power
PN = 2.7 kW
Nominal voltage
U N = 420 V
Nominal current
I N = 8.4 A
Nominal frequency
f SN = 50 Hz
Base speed
Ω mN = 1450 rpm
Tab. A. 4. 7.Data of the Supply Line and AC/DC/AC Converter
Line voltage
Line frequency
DC-link voltage
U L = 141 V
f L = 50 Hz
U DC =560 V
Resistive load
R load = 100 or 150 Ω
C =470 µF / 47 µF
DC-link capacitor
Tab. A. 4. 8. Date of the LCL Input Filter in laboratory setup
Line side inductance
Converter side ind.
Resistance of L1
Resistance of L
Filter capacitors
L1 = 590 µH
L = 10 mH
R1 =0.093 Ω
R = 0.08 Ω
C F = 20 µF
Tab. A. 4. 9.Data of the inverter VLT 5005
1)
Nominal output power
SVLTN = 5.5 kW
Nominal power on the shaft of
the IM fed by VSI
Nominal voltage
PVLTN = 3 kW
U LN = 380 V
Nominal input (line) current
I LVLTN = 7 A
Nominal output (IM) current
I SVLTN = 7.2 A
Dead time
Efficiency
TD = 2 µs
0.961)
Measured using 30 m screened motor cables at rated load – 3 kW and rated frequency – 3kHz .
Two inverters VLT 5005 have been connected through DC-link connectors. The
arisen DC-link circuit has been modified as in Fig. A. 4. 6.
153
Appendices
C = 470 µF or 47 µF
RC = 274KΩ
RC
C
C
RC
RC
C
C
RC
Fig. A. 4. 6. Modified DC-link circuit
The researches on the laboratory setup were carried out under distorted line
voltage condition. Typical THDs of the line voltage are presented in Tab. A. 4. 10.
Tab. A. 4. 10. Typical distorted line voltage at the laboratory setup measured by Norma LEM.
Number of
Harmonics
5
7
11
13
% of 1 harmonic of
UA (THD 4 %) at
morning
2.9
2.7
1.1
0.3
% of 1 harmonic of
UA (THD 4 %)
At afternoon
2.1
2.9
0.9
0.4
A.4.4. List of Equipment
Tab. A. 4. 11. List of equipment
Power quality analyzer
Oscilloscope
Oscilloscope
Oscilloscope
Voltage differential probe
Current probe
Current probe
Norma LEM D6000
Tektronix TDS3034B
Tektronix TDS3014
Tektronix TDS5034B
Tektronix P5200
Tektronix TCP A300
LEM PR30
154
% of 1 harmonic
of UA (THD 3.4
%) At evening
2.2
2.4
0.4
0.1

Direct Power and Torque Control of AC/DC/AC Converter

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