Brushless Doubly Fed Induction Generator based
Wind Turbine Drivetrain under Grid Fault
Conditions
Udai Shipurkar
Master of Science Thesis
Supervisor:
Dr.ir. H. Polinder
Ir. T.D. Strous
Electrical Power Processing
Department of Electrical Sustainable Energy
Faculty of Electrical Engineering, Mathematics and Computer Science
Delft University of Technology
July 8, 2014
Abstract
With growing interest in sustainable forms of energy, the wind industry is growing rapidly.
The Doubly Fed Induction Generator is the most popular choice for the drivetrain because
it is cost effective. However, it suffers from reliability and maintenance issues due to the slip
rings and brushes it requires. The Brushless Doubly Fed Induction Generator (B-DFIG)
aims to address these drawbacks.
With increased wind power penetration, tripping of wind turbines during grid disturbances is no longer acceptable for the power system. Therefore, it is important to study the
performance of such wind turbine drivetrain under low voltage events. This thesis studies
the Low Voltage Ride Through (LVRT) characteristics of the B-DFIG for its application in
wind turbines.
This thesis first looks at the modelling and control of the Brushless Doubly Fed Induction
Generator (B-DFIG) to be used in a wind turbine drivetrain. It develops the steady state
model using circuit theory to study the steady state characteristics of the machine. The
dynamic model is developed to form the basis of the study of the machine during low voltage
events. Further, the controller is developed based on vector control.
The second part of the thesis looks at the performance of a B-DFIG based wind turbine
under symmetric low voltage dips. It compares the performance of this generator with
that of the Permanent Magnet Synchronous Machine (PMSM) and Doubly Fed Induction
Generator (DFIG) - two of the most prevalent generators used in wind turbine drivetrains
today. The thesis also looks at protection methods for these generators. The issue with
LVRT performance of the PMSM is the rise in the DC link voltage which is due to the
mismatch in power generated by the machine and the power transferred to the grid. It has
been found that apart from offering better reliability through the exclusion of slip rings and
brushes, the B-DFIG also has an improved LVRT performance when compared with the
DFIG. The protection is simpler and can be built into the control algorithm of the machine
controller.
iii
iv
Acknowledgements
First, I would like to thank Henk Polinder for his help and guidance throughout the duration
of this thesis. Discussions with him have always been enlightening and thought provoking.
I would also like to thank Tim Strous who has been a great support in carrying out this
thesis work. He has always been ready to discuss my questions and doubts and has been
very helpful when I was writing this document.
I would also like to thank the rest of the B-DFIG team - Nils van der Blij, Einar Vilmarsson and Xuezhou Wang - discussions with whom have helped build my understanding
of this machine.
Thanks also to my fellow master students of the student room - Foivos, Nikolas, Joost,
Didier, Einar, Ralino and JK - who have made these months fun-filled and informative.
Thanks also to Didier for permission to use his photograph in the cover and Siddhartha for
his help in designing it.
v
Contents
Abstract
iii
Acknowledgements
v
Contents
vii
List of Symbols
1 Introduction
1.1 Background . . .
1.2 Thesis Objective
1.3 Methodology . .
1.4 Contribution . .
ix
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1
1
2
2
4
2 B-DFIG Steady State Modelling
2.1 Machine Description . . . . . . . . . . . . . .
2.1.1 Machine Operation . . . . . . . . . . .
2.2 Equivalent Circuit . . . . . . . . . . . . . . .
2.2.1 Development of the Equivalent Circuit
2.2.2 Power Balance Equations . . . . . . .
2.3 Simplified Equivalent Circuit . . . . . . . . .
2.4 The Γ-Equivalent Circuit . . . . . . . . . . .
2.5 Steady State Characteristics . . . . . . . . . .
2.6 Discussion . . . . . . . . . . . . . . . . . . . .
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5
5
6
7
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16
17
18
19
21
3 B-DFIG Dynamic Modelling
3.1 The Dynamic Equations of the Machine . .
3.2 Dynamic Model in the Block Diagram Form
3.3 Dynamic Behaviour . . . . . . . . . . . . .
3.4 Discussion . . . . . . . . . . . . . . . . . . .
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23
23
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4 B-DFIG Control
4.1 Wind Turbine Drivetrain System . . . . . . . .
4.2 Reference Signal Generation . . . . . . . . . . .
4.2.1 Modelling for Optimal Power Extraction
4.2.2 B-DFIG Reference Signal . . . . . . . .
4.3 Active and Reactive Power Control . . . . . . .
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31
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34
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37
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72
6 Conclusion
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
73
74
Bibliography
75
Appendices
79
A Additional LVRT Performance Simulations
81
B B-DFIG Analytical Parameter Calculation
89
C B-DFIG Simple Case Study Machine
93
D PMSM 3.2MW Case Study Machine
97
E DFIG 3.2MW Case Study Machine
99
4.4
4.5
4.6
4.3.1 Active Power Control .
4.3.2 Reactive Power Control
PI Tuning . . . . . . . . . . . .
Control Behaviour . . . . . . .
Discussion . . . . . . . . . . . .
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5 Performance Under Grid Events
5.1 Grid Code Requirements . . . . . . . . . . . . . . . . .
5.2 Simulation of the LVRT Performance for the PMSM .
5.2.1 LVRT Performance without Protection . . . . .
5.2.2 LVRT Performance with Protection . . . . . .
5.3 Simulation of the LVRT Performance of the DFIG . .
5.3.1 Performance without Protection . . . . . . . .
5.3.2 Performance with Protection . . . . . . . . . .
5.4 Simulation of the LVRT Performance for the B-DFIG
5.4.1 Performance without Protection . . . . . . . .
5.4.2 Performance with Protection . . . . . . . . . .
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
F B-DFIG 3.2MW Case Study Machine
viii
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103
List of Symbols
A
bp
bc
Cp
ip
ic
ir,l
ix,d−ref
p
ipx,q−ref
ix,d−ref
c
icx,q−ref
Lp,σ
Lc,σ
Lp,m
Lc,m
Lr,lσ
Lr,lm
Mp,nl
Mc,nl
Nn
Nl
pp
pc
Pp
Pc
Pp,ag
Pc,ag
Pp,cu
Pc,cu
Pr,cu
PF e
Ppref
Qref
p
R
Rp
Rc
swept area of wind turbine rotor
Power Winding air-gap
Control Winding air-gap
power coecient
Power Winding current
Control Winding current
rotor loop - l current
reference Power Winding d-axis current in the ‘x’ reference frame
reference Power Winding q-axis current in the ‘x’ reference frame
reference Control Winding d-axis current in the ‘x’ reference frame
reference Control Winding q-axis current in the ‘x’ reference frame
Power Winding leakage inductance
Control Winding leakage inductance
Power Winding main inductance
Control Winding main inductance
rotor loop -l leakage inductance
rotor loop - l main inductance
mutual inductance between Power Winding and rotor loop - l
mutual inductance between Control Winding and rotor loop - l
number of rotor nests
number of loops in each rotor nest
Power Winding pole number
Control Winding pole number
Power Winding real power
Control Winding real power
Power Winding air-gap power
Control Winding air-gap power
Power Winding copper loss
Control Winding copper loss
rotor copper loss
machine iron loss
reference Power Winding active power
reference Power Winding reactive power
wind turbine rotor radius
Power Winding resistance
Control Winding resistance
ix
Rr,l
s
sp
Sp
Sc
Tref
up
uc
ūxp
ūxc
rotor loop - l resistance
Control Winding circuit slip
rotor circuit slip
Power Winding apparent power
Control Winding apparent power
reference torque
Power Winding voltage
Control Winding voltage
voltage vector of Power Winding in ‘x’ reference frame
voltage vector of Control Winding in ‘x’ reference frame
reference Control Winding d-axis voltage in the ‘x’ reference frame
reference Control Winding q-axis voltage in the ‘x’ reference frame
DC link voltage
ωp
ωc
ωm
φp,c
λ̄xp
λ̄xc
λ̄xr
ρ
Power Winding excitation frequency
Control Winding excitation frequency
rotor mechanical angular velocity
phase angle between Power and Control Winding voltages
Power Winding flux linkage vector in ‘x’ reference frame
Control Winding flux linkage vector in ‘x’ reference frame
rotor flux linkage vector in ‘x’ reference frame
density of air
ux,d−ref
c
ux,q−ref
c
udc
x
Chapter 1
Introduction
The Brushless Doubly Fed Induction Machine (B-DFIM) traces its origin to the beginning of
the twentieth century. It has recently been explored for use as generators in wind turbines.
This chapter serves as an introduction to the work in this thesis.
Section 1.1 looks at the historical developments of the B-DFIM. Section 1.2 describes
the objectives of this thesis work. Section 1.3 describes the approach taken in this document
and Section 1.4 details the contribution made by this thesis.
This document deals with the Brushless Doubly Fed Induction Machine used in a wind
turbine. Although this machine can operate both as a motor and as a generator, for the
wind turbine application its operation is restricted to use as a generator. Therefore, the
term Brushless Doubly Fed Induction Generator (B-DFIG) is used interchangeably with
Brushless Doubly Fed Induction Machine (B-DFIM).
1.1
Background
The development of the Brushless Doubly Fed Induction Machine can be traced back to
the early twentieth century. Before the use of power electronics became prevalent for the
control of electric machines, the use of two cascade connected slip ring induction machines
for speed control was common. In 1907, Hunt [1] proposed a machine that incorporated two
machine windings in the stator and a special rotor design. This machine achieved speed
control through resistors connected to one of the stator windings, thus doing away with slip
rings. The design of the B-DFIM was further developed by Creedy [2] in the early 1920s.
Further advancements for the B-DFIM came in 1970 when Broadway et al. [3] proposed
a caged rotor (nested loop) design. Modern B-DFIMs still used rotors based on this design.
They also developed the equivalent circuit for the machine and analysed the performance
in steady state and also noted the operation of the machine in the ’synchronous mode’.
Kusko and Somuah [4] studied the operation of the single-frame brushless induction motor
with a rectifier-inverter to control speed by slip-power pump back to the line. Till this
point all stator designs, including those of Hunt and Creedy, used a single stator winding
which produced both fields of different pole numbers. However, Rochelle et al. in 1990
compared the alternatives for stator winding design and concluded that electrically isolated
windings were more advantageous [5]. Subsequently all BDFIM’s have used this stator
winding design.
1
CHAPTER 1. INTRODUCTION
Wallace et al. at Oregan State University developed a dynamic model1 of the machine
[6] and used simulation models to investigate the behaviour [7] in the mid-1980s. Li et
al. developed a two-axis model suitable for dynamic studies [8] and presented results of
dynamic simulations [9]. Further developments in the modelling of the Brushless Doubly
Fed Machine took place at Cambridge University. Williamson et al. presented a generalised
mathematical model for the machine operating in the synchronous mode [10] [11].
Zhou et al. presented the first field-oriented control algorithm for the machine [12]
and followed that with a simplified version [13]. However, these algorithms were heavily
dependant on machine parameters. In 2002, Poza et al. presented a vector control algorithm
[14] based on the Power Winding flux.
There has been little research on the Low Voltage Ride Through (LVRT) performance of
the B-DFIG. Shao et al. studied the dynamic behaviour of the machine during symmetrical
voltage dips [15]. However, this study was limited as it did not consider the subsequent
voltage rise of the grid and it did not propose any methods to improve the performance. In
2011 they proposed a control scheme that gives the B-DFIG the capability to ride through
low voltage faults [16] and this was extended for asymmetric low voltage faults [17].
This thesis work has been carried out under the aegis of the Project Windrive - ‘Industrialization of a 3MW Medium-Speed Brushless DFIG Drivetrain for Wind Turbine
Applications’ . One aspect of this project, that this thesis attempts to address, is the
investigation of the effects of grid events (LVRT) on the electrical systems of the Brushless Doubly Fed Induction Generators, Doubly-Fed Induction Generators and Permanent
Magnet Synchronous Generators. The investigation covers the protection of the power
electronics during voltage events and the reaction of the generator in these situations.
1.2
Thesis Objective
The primary objective of this thesis is the study of the Low Voltage Ride Through (LVRT)
characteristics of the Brushless Doubly Fed Induction Generator (B-DFIG) for its application in wind turbines. In order to achieve this goal, the objective has been broken down
into a number of subsidiary objectives. These are describes below;
1. Develop a dynamic model for the B-DFIG.
2. Develop a controller based on the use of the B-DFIG as a wind turbine generator.
3. Determine the LVRT performance of the B-DFIG.
4. Compare the LVRT performance of the B-DFIG with those of other prevalent wind
turbine generator technologies such as Permanent Magnet Synchronous Machines
(PMSM) and Doubly Fed Induction Generators (DFIG).
1.3
Methodology
This section describes the approach taken in this thesis. The steps described here also form
the layout of this document.
1
Previously published analyses assumed the system to be equivalent to two separate motors with rotors
coupled mechanically and electrically.
2
CHAPTER 1. INTRODUCTION
Steady State Model
The first step taken as part of this thesis is the development of the steady state model of the
B-DFIG. This is done by forming phasor equations from the time varying voltage equations
of the machine. The machine parameters are derived from the geometry of the machine.
Apart from the steady state model forming the basis of the development of the dynamic
model, the steady state model becomes a way to check the results of the dynamic model.
It also forms a preliminary tool to understand the behaviour expected from the machine.
The development of the steady state model has been described in Chapter 2 of this
thesis.
Dynamic Model
The dynamic model of the B-DFIG is developed with a view to study the transient behaviour
of the machine. This model is created in the state-space form and implemented as a
MATLAB Simulink block model. When the LVRT performance of the machine is to be
studied the dynamic model becomes the tool used to study the transition period between
two steady states.
The development of the dynamic model is described in Chapter 3.
Controller for the Machine
The B-DFIG operates in the synchronous mode under conditions described in Section 2.1.1.
Therefore, for the machine to be used in the intended application (i.e. as a wind turbine
generator) it is required to be controlled by the power electronic converter connected to
the Control Winding. The control developed is a form of vector control in the stator flux
reference frame.
The controller developed for use with the B-DFIG as a wind turbine generator is described in Chapter 4.
LVRT Performance of the B-DFIG
Using the dynamic model and controller developed for the B-DFIG the LVRT performance
of the machine under constraints laid down by grid codes is investigated. Methods to
improve this performance are also studied.
This study is detailed in Chapter 5 of this document.
Comparison with other Generators
To compare the LVRT performance of the B-DFIG, the LVRT performance of the PMSM
and DFIG are simulated. All the three case study generators used are developed for a
3.2MW wind turbine drivetrain. Therefore, they provide a good basis on which a comparison can be made. The dynamic model and controller for all three types of machines have
been developed for this thesis, however, only the model and controller for the B-DFIG has
been covered in detail.
The details of the LVRT performance for the PMSM and DFIG are given in Chapter 5.
3
CHAPTER 1. INTRODUCTION
1.4
Contribution
This thesis focuses on the Brushless Doubly Fed Induction Generator. It looks at the
modelling, control and performance during low voltage events of a wind turbine based on
this machine. Some of the contributions this thesis makes are listed below.
This thesis develops the dynamic model and controller for the B-DFIG to be used in
a wind turbine. For the control of the machine, a method to generate a reference (Ppref )
signal for optimal operation is developed. This has not been widely covered in literature.
It also develops and implements the concept of ‘Cross-Coupling Compensation’. The thesis
also compares the LVRT performance of case study B-DFIG, PMSM and DFIG generator
based wind turbine drivetrains.
4
Chapter 2
B-DFIG Steady State Modelling
Steady state models are an important tool for the study of electrical machines. This chapter
explores the development of the steady state model and the steady state characteristics for
the Brushless Doubly Fed Induction Machine (B-DFIM).
Section 2.1 describes the machine and looks at its operation. Section 2.2 develops the
mathematical model of the machine which is used to study the machine in the steady state.
Section 2.3 extends this model into a simplified one that helps develop greater insight into
the behaviour of the machine. Section 2.4 further used the Γ-transformation to form another
simplified model of the machine. Finally, Section 2.5 studies the steady state behaviour of
the machine.
2.1
Machine Description
The B-DFIM has two sets of 3-phase windings with different pole numbers. One of these
windings is termed the ‘Power Winding’ while the other ‘Control Winding’. For machine
operation the Power Winding is connected directly to the supply while the Control Winding
is connected through a power electronic converter. This is shown in Figure 2.1.
B − DF IM
Partially Rated
P.E. Converter
Figure 2.1
Schematic Description of the Brushless Doubly Fed Induction Machine
5
CHAPTER 2. B-DFIG STEADY STATE MODELLING
2.1.1
Machine Operation
As can be expected from a machine with two different stator winding configurations the
machine can be run in the induction machine mode where either one of the windings are
used to run the machine. The machine can also be run as a cascaded induction machine by
short circuiting one of the stator windings. However, the most promising operation method
of the machine is the synchronous operation.
The synchronous mode of operation occurs due to the coupling of the two stator windings
(with different pole numbers) through the rotor [10]. In this arrangement, the Power
Winding is connected to the supply while the Control Winding is supplied with a voltage
of variable frequency as shown in Figure 2.1.
In this condition the fundamental airgap fields produced by these windings are given
by,
bp (θ, t) = B̂p cos(ωp t − pp θ + αp )
bc (θ, t) = B̂c cos(ωc t − pc θ + αc )
(2.1)
(2.2)
where bp and bc describe the fields produced by the Power Winding and Control Winding
respectively with pole pairs pp and pc . ωp and ωc are the excitation frequencies and αp
and αc are the phase angles. When the rotor rotates with an angular velocity of ωm the
expressions for airgap fields may be written in a reference frame rotating with the rotor as
expressed in Equation 2.3 and Equation 2.4.
bp (θ0 , t) = B̂p cos((ωp − pp ωm )t − pp θ0 + αp )
bc (θ0 , t) = B̂c cos((ωc − pc ωm )t − pc θ0 + αc )
(2.3)
(2.4)
The two stator windings will be coupled through the rotor when the frequency and
distribution of the currents induced by these two fields in the rotor are equal [10]. For
induced frequencies to be equal Equation 2.5 needs to be satisfied.
ωp − pp ωm = ωc − pc ωm
which gives,
ωm =
ωp − ωc
pp − pc
(2.5)
(2.6)
For the induced currents to have an equal distribution [10],
pp
2π
2π
= pc
+ 2qπ
Nn
Nn
(2.7)
where Nn are the number of rotor bars. Which gives rise to the condition,
pp − pc = qNn
(2.8)
Nn = pp − pc
(2.9)
if q=1, this gives,
However, as cos(θ) = cos(−θ), Equation 2.4 is equivalent to,
bc (θ0 , t) = B̂c cos(−(ωc − pc ωm )t + pc θ0 − αc )
6
(2.10)
CHAPTER 2. B-DFIG STEADY STATE MODELLING
Again equating frequencies and phase differences, as in Equation 2.5 and Equation 2.7,
gives,
ωp − pp ωm = −(ωc − pc ωm )
or,
ωm =
(2.11)
ωp + ωc
pp + pc
(2.12)
For the induced currents to have an equal distribution [10],
pp
2π
2π
= −pc
+ 2qπ
Nn
Nn
(2.13)
which gives Equation 2.14 if it is assumed that q=1,
Nn = pp + pc
(2.14)
It seems it is better to use Equation 2.14 in selecting the number of rotor bars. This
number, however, is still small for practical machine pole numbers. This results in a high
rotor leakage inductance [3]. This is combated by using rotor Nest structures. Here, Nn is
also taken to be the total number of nests in the rotor structure. When these conditions
mentioned above are met, the machine operates in the synchronous mode of operation and
performs as a synchronous machine [18].
2.2
Equivalent Circuit
The equivalent circuit for the B-DFIM is shown in Figure 2.2. Here, the equivalent circuit
and the voltage equations are shown, these will be derived later in this section. The voltage
Rr,1
Rp
Lp,σ
Up
•
Lp,m
Rc
s
Lc,σ
Uc
s
•
•
Lr,1p
Rr,2
Lr,2p
•
Lc,m
Lr,1c
Lr,2c
•
•
Figure 2.2
Lr,1σ
Single Phase Equivalent Circuit.
7
Lr,2σ
CHAPTER 2. B-DFIG STEADY STATE MODELLING
equations are given by,
U p =Rp I p + ωp Lp,σ I p + ωp Lp,m I p +
U c Rc
= I c + ωp Lc,σ I c + ωp Lc,m I c −
s
s
Nl
X
l=1
Nl
X
ωp Mp,nl I 0l
(2.15)
ωp Mc,nl I 0l
(2.16)
l=1
Rr,l
I + ωp Lr,lσ I r,l + ωp Lr,lm I r,l + ωp Mp,nl I p − ωp Mc,nl I c
0=
sp r,l
(2.17)
where the quantity X represents a complex phasor quantity. All parameters, such as s and
sp , are discussed and explained later in the section.
2.2.1
Development of the Equivalent Circuit
The remaining part of this section looks at the development of this equivalent circuit.
To derive the steady state characteristics of the Brushless Doubly Fed Induction Machine
(B-DFIM) a number of assumptions have been made. These are,
• The influence of saturation, hysterisis and eddy currents have been neglected.
• The rotor is cylindrical, the air gap is uniform and the surfaces of the rotor and stator
are smooth neglecting slotting effects.
• The phase conductors distributions are perfectly sinusoidal and the currents through
the conductors are balanced and sinusoidal.
• There is no coupling between the power and control circuit windings. The mutual
inductance between the power and control winding is zero. This has been shown in
Appendix B.
Further,
• Each loop is identified by the indices nl where n represents the Nest number and l
represents the Loop number. The total number of loops is Nl and the total number
of nests is Nn .
• The resistance of the stator Power Winding is given by Rp while that of the stator
Control Winding is Rc .
• The self inductance of the stator Power Winding is Lp while the self inductance of
the stator Control Winding is given by Lc . The mutual inductance between phases
of the Power Winding is Mp while for the Control Winding this is given by Mc .
• The mutual inductance between the stator Power winding and the nl loop is given
by M̂p,nl cos(pp ωm t + βnl + pp βp ). Here, βnl is the initial angle between the nl rotor
loop and the reference axis while βp is the angle between the stator Power Winding
and the reference axis. Similarly, the mutual inductance between the stator Control
Winding and the nl loop is given by M̂c,nl cos(pp ωm t + βnl + pc βc ). Again, βc is the
angle between the stator control winding and the reference axis. This is detailed in
Appendix B and Figure 2.3 shows some of these details.
8
CHAPTER 2. B-DFIG STEADY STATE MODELLING
Power Winding Axis
Rotor Loop - nl Axis
βp
βnl
a
Reference Axis
a’
ωm
Figure 2.3
Winding Details
Given the resistances and inductances of the windings and the mutual inductances
between windings, the voltage equations for each winding in the time domain is given by,
N
up,a
N
n
l X
X
d
= Rp ip,a + (Lp ip,a + Mp ip,b + Mp ip,c +
M̂p,nl cos(pp ωm t + βnl )inl )
dt
up,b = Rp ip,b +
up,c = Rp ip,c +
d
(Lp ip,b + Mp ip,a + Mp ip,c +
dt
d
(Lp ip,c + Mp ip,b + Mp ip,a +
dt
l=1 n=1
Nl X
Nn
X
l=1 n=1
Nl X
Nn
X
(2.18)
M̂p,nl cos(pp ωm t + βnl )inl )
(2.19)
M̂p,nl cos(pp ωm t + βnl )inl )
(2.20)
l=1 n=1
Similarly, the voltage equations for the control winding are,
N
uc,a = Rc ic,a +
uc,b = Rc ic,b +
uc,c = Rc ic,c +
N
l X
n
X
d
(Lc ic,a + Mc ic,b + Mc ic,c +
M̂c,nl cos(pc ωm t + βnl )inl )
dt
d
(Lc ic,b + Mc ic,a + Mc ic,c +
dt
d
(Lc ic,c + Mc ic,b + Mc ic,a +
dt
l=1 n=1
Nl X
Nn
X
l=1 n=1
Nl X
Nn
X
(2.21)
M̂c,nl cos(pc ωm t + βnl )inl )
(2.22)
M̂c,nl cos(pc ωm t + βnl )inl )
(2.23)
l=1 n=1
where, up,a , up,b and up,c are the power winding voltage in the three phases. uc,a , uc,b and
uc,c are the control winding voltage in the three phases. Similarly, ip,a and ic,a are the
currents in the power and control winding in the respective phases. These equations can
9
CHAPTER 2. B-DFIG STEADY STATE MODELLING
be written as,
N
up,a
N
l X
n
X
dip,a
d
d
M̂p,nl (inl cos(pp ωm t + βnl )
=Rp ip,a + Lp
+ Mp (ip,b + ip,c ) +
dt
dt
dt
l=1 n=1
d
+ cos(pp ωm t + βnl ) inl )
dt
(2.24)
N
uc,a
N
l X
n
X
dic,a
d
d
=Rc ic,a + Lc
+ Mc (ic,b + ic,c ) +
M̂c,nl (inl cos(pc ωm t + βnl )
dt
dt
dt
l=1 n=1
d
+ cos(pc ωm t + βnl ) inl )
dt
(2.25)
A few aspects of the rotor are now discussed. They help in the next step of the development of the steady state model. It has been seen that the mutual inductance between
the stator and rotor depends on the angle between the loop and the reference axis as shown
2π
, where Nn is the
in Figure 2.3. This value for consecutive rotor nests will differ by
Nn
number of rotor nests. Assuming that a rotor nest is aligned with the reference axis, the
value of the mutual inductance can therefore be written as,
2π
)
Nn
2π
= M̂c,nl cos(−pc ωm t + pc βc − (n − 1)
)
Nn
Mp,nl = M̂p,nl cos(pp ωm t − pp βp + (n − 1)
(2.26)
Mc,nl
(2.27)
The machine convention used in this derivation is that the Control Winding field is
positive when it rotates opposite to the Power Winding field [19]. This causes the difference
between the expression for Mc,nl and Mp,nl , i.e. the expression for Mp,nl has a pp ωm t term
2π
electrical degrees,
while Mc,nl has a −pc ωm t term. Since the rotor loops are displaced by
Nn
the current will also be displaced by this quantity. The frequency of the currents in these
loops is given by Equation 2.3. It can be seen that the magnitude of the currents of the lth
loop in all the nests will be equal. The rotor loop currents are given by,
2π
)
Nn
2π
= Iˆl sin((ωc + pc ωm )t + φnl − (n − 1)
)
Nn
inl = Iˆl sin((ωp − pp ωm )t + φnl − (n − 1)
(2.28)
(2.29)
where φnl is the phase shift due to the impedence of the loops.
Using Equations 2.24, 2.26 and 2.28 the voltage equation can be developed into,
up,a
Nl X
Nn
X
M̂p,nl Iˆl d
dip,a
d
=Rp ip,a + Lp
+ Mp (ip,b + ip,c ) +
(sin(ωp t − pp βp + φnl )
dt
dt
2 dt
l=1 n=1
+ sin((ωp − 2pp ωm ) + φnl + pp βp − (n − 1)
4π
)
Nn
(2.30)
Here the summation of the second sine term over l = 1 to Nl will be zero due to the
2π
(n−1)
term. The assumption of balanced currents will also be used and ip,b +ip,c = −ip,a .
Nn
10
CHAPTER 2. B-DFIG STEADY STATE MODELLING
The Equation 2.30 is therefore simplified to,
up,a
Nl
ωp M̂p,nl Nn Iˆl
dip,a X
= Rp ip,a + (Lp − Mp )
+
cos(ωp t − pp βp + φ0 )
dt
2
(2.31)
n=1
A similar analysis on the voltage equation of the stator control winding gives us the following
equation,
uc,a
Nl
ωc M̂c,nl Nn Iˆl
dic,a X
−
cos(ωc t − pc βc + φ0 )
= Rc ic,a + (Lc − Mc )
dt
2
(2.32)
n=1
Now, the complete voltage equations for the two stator circuits are reproduced below.
N
up,a = Rp ip,a + (Lp − Mp )
l
ωp M̂p,nl Nn Iˆl
dip,a X
+
cos(ωp t − pp βp + φ0 )
dt
2
up,b = Rp ip,b + (Lp − Mp )
dip,b
+
dt
up,c = Rp ip,c + (Lp − Mp )
dip,c
+
dt
uc,a = Rc ic,a + (Lc − Mc )
dic,a
−
dt
uc,b = Rc ic,b + (Lc − Mc )
dic,b
−
dt
uc,c = Rc ic,c + (Lc − Mc )
dic,c
−
dt
n=1
Nl
X
n=1
Nl
X
n=1
ωp M̂p,nl Nn Iˆl
cos(ωp t − pp βp + φ0 )
2
(2.34)
ωp M̂p,nl Nn Iˆl
cos(ωp t − pp βp + φ0 )
2
(2.35)
Nl
X
ωc M̂c,nl Nn Iˆl
n=1
Nl
X
n=1
Nl
X
n=1
(2.33)
cos(ωc t − pc βc + φ0 )
(2.36)
ωc M̂c,nl Nn Iˆl
cos(ωc t − pc βc + φ0 )
2
(2.37)
ωc M̂c,nl Nn Iˆl
cos(ωc t − pc βc + φ0 )
2
(2.38)
2
For the rotor, the voltage equation of the nlth loop is given by,
0 =Rr inl +
Nl X
Nn
X
l=1 n=1
Mnlnl
d
d
2π
inl + (M̂p,nl cos(pp ωm t − pp βp + (n − 1)
)ip,a
dt
dt
Nn
2π
2π
−
)ip,b
Nn
3
2π
4π
+ M̂p,nl cos(pp ωm t − pp βp + (n − 1)
−
)ip,c )
Nn
3
d
2π
)ic,a
+ (M̂c,nl cos(−pc ωm t + pc βc − (n − 1)
dt
Nn
2π
2π
+ M̂c,nl cos(−pc ωm t + pc βc − (n − 1)
−
)ic,b
Nn
3
4π
2π
+ M̂c,nl cos(−pc ωm t + pc βc − (n − 1)
−
)ic,c )
Nn
3
+ M̂p,nl cos(pp ωm t − pp βp + (n − 1)
11
(2.39)
CHAPTER 2. B-DFIG STEADY STATE MODELLING
where Mnlnl is the mutual inductance between the loops. The current in the stator circuits
is given by,
ip,a = Iˆp sin(ωp t + φp )
2π
= Iˆp sin(ωp t + φp −
)
3
4π
)
= Iˆp sin(ωp t + φp −
3
ip,b
ip,c
(2.40)
(2.41)
(2.42)
and
ic,a = Iˆc sin(ωc t + φc )
2π
)
3
4π
)
= Iˆc sin(ωc t + φc +
3
(2.43)
ic,b = Iˆc sin(ωc t + φc +
(2.44)
ic,c
(2.45)
Here, it can be seen that the phase sequence of the stator control circuit currents is taken
to be opposite to that of the stator power circuits. The Equation 2.39 is simplified to,
0 =Rr inl +
Nl X
Nn
X
d 3
2π
d
inl + ( M̂p,nl Iˆp sin((ωp − pp ωm )t + pp βp + φp − (n − 1)
)
dt
dt 2
Nn
Mnlnl
l=1 n=1
2π
3
))
− M̂c,nl Iˆc sin((ωc + pc ωm )t − pc βc + φc + (n − 1)
2
Nn
(2.46)
or,
0 =Rr inl +
Nl X
Nn
X
2π
(ωp − pp ωm )Mnlnl Iˆnl cos((ωp − pp ωm )t + φnl − (n − 1)
)
Nn
l=1 n=1
3
2π
+ (ωp − pp ωm )M̂p,nl Iˆp cos((ωp − pp ωm )t + pp βp + φp − (n − 1)
)
2
Nn
3
2π
− (ωc + pc ωc )M̂c,nl Iˆc cos((ωc + pc ωm )t − pc βc + φc + (n − 1)
)
2
Nn
(2.47)
This defines the complete set of voltage equations for the machine. They are now
transformed into phasor equations from which the equivalent circuit can be derived. The
voltage equation of the stator power winding for a single phase reproduced below for surveyability.
N
up,a
l
ωp M̂p,nl Nn Iˆl
dip,a X
= Rp ip,a + (Lp − Mp )
+
cos(ωp t + pp βp + φ0 )
dt
2
(2.48)
n=1
Nn
A transformation for the rotor current such that, Il0 =
Il is used. The transform Mp,nl =
3
3
M̂p,nl is also used. Therefore,
2
N
up,a = Rp ip,a + (Lp − Mp )
l
dip,a X
+
ωp Mp,nl Iˆl0 cos(ωp t + pp βp + φ0 )
dt
n=1
12
(2.49)
CHAPTER 2. B-DFIG STEADY STATE MODELLING
Taking up,a = Ûp sin(ωp t), Equation 2.49 can be extended to all three phase equations as,
Im(Up eωp t ) =Im(Rp Ip eφp eωp t ) + Im(ωp (Lp − Mp )Ip eφp eωp t )
+
Nl
X
Im(ωp Mp,nl In e(pp βp +φnl ) eωp t )
Nl
X
Im(ωp Mp,nl In e(pp βp +φnl ) eωp t )
Nl
X
Im(ωp Mp,nl In e(pp βp +φnl ) eωp t )
0
(2.50)
n=1
Im(Up e−
2π
3
eωp t ) =Im(Rp Ip e(φp −
+
2π
)
3
eωp t ) + Im(ωp (Lp − Mp )Ip e(φp −
2π
)
3
eωp t )
0
(2.51)
n=1
− 4π
ωp t
3
Im(Up e
e
) =Im(Rp Ip e(φp −
+
4π
)
3
eωp t ) + Im(ωp (Lp − Mp )Ip e(φp −
4π
)
3
eωp t )
0
(2.52)
n=1
Removing the imaginary function and using phasor notation to represent the equations,
U p = Rp I p + ωp (Lp − Mp )I p +
Nl
X
ωp Mp,nl I 0l
(2.53)
l=1
A similar exercise is now done on the voltage equations of the stator control winding
circuit.
N
uc,a = Rc ic,a + (Lc − Mc )
l
dic,a X
−
ωc Mc,nl Iˆl cos(ωc t + pc βc + φ0 )
dt
(2.54)
n=1
First, the slips for the machine are defined, these are sp and s which are given by,
ωp − pp ωm
ωc + pc ωm
=
ωp
ωc
ωc
s=
ωp
sp =
(2.55)
(2.56)
Also, uc,a = Ûc sin(ωc t + θc ) and,
Im(Uc eθc eωc t ) =Im(Rc Ic eφc eωc t ) + Im(ωc (Lc − Mc )Ic eφc eωc t )
+
Nl
X
0
Im(ωc Mcn In e(pc βc +φnl ) eωc t )
n=1
13
(2.57)
CHAPTER 2. B-DFIG STEADY STATE MODELLING
which, using Equation 2.56 can be written for all phases as,
Im(Uc eθc esωp t ) =Im(Rc Ic eφc esωp t ) + Im(sωp (Lc − Mc )Ic eφc esωp t )
−
Im(Uc e(θc −
2π
)
3
Im(sωp Mcn In e(pc βc +φnl ) esωp t )
Nl
X
Im(sωp Mcn In e(pc βc +φnl ) esωp t )
Nl
X
Im(sωp Mcn In e(pc βc +φnl ) esωp t )
0
(θc − 4π
) sωp t
3
e
2π
)
3
esωp t ) + Im(sωp (Lc − Mc )Ic e(φc −
2π
)
3
esωp t )
0
(2.59)
n=1
) =Im(Rc Ic e(φc −
−
(2.58)
n=1
esωp t ) =Im(Rc Ic e(φc −
−
Im(Uc e
Nl
X
4π
)
3
esωp t ) + Im(sωp (Lc − Mc )Ic e(φc −
4π
)
3
esωp t )
0
(2.60)
n=1
Using phasor notation, the equation can be expressed as,
N
l
X
Uc
Rc
=
I c + ωp (Lp − Mp )I c −
ωp Mp,nl I 0l
s
s
(2.61)
l=1
The rotor voltage equations are now reproduced below.
0 =Rr inl +
Nl X
Nn
X
(ωp − pp ωm )Mnlnl Iˆnl cos((ωp − pp ωm )t + φnl − (n − 1)
l=1 n=1
2π
)
Nn
2π
+ (ωp − pp ωm )Mp,nl Iˆp cos((ωp − pp ωm )t − pp βp + φp − (n − 1)
)
Nn
2π
− (ωc + pc ωc )Mc,nl Iˆc cos((ωc + pc ωm )t − pc βc + φc + (n − 1)
)
Nn
(2.62)
or,
Nl
X
0 =Rr inl +
(ωp − pp ωm )Mnlnl Iˆl0 cos((ωp − pp ωm )t + φnl )
l=1
2π
)
+ (ωp − pp ωm )Mp,nl Iˆp cos((ωp − pp ωm )t − pp βp + φp − (n − 1)
Nn
2π
− (ωc + pc ωc )Mc,nl Iˆc cos((ωc + pc ωm )t − pc βc + φc + (n − 1)
)
Nn
(2.63)
The procedure followed for the stator voltage equations are repeated, and the rotor voltage
equation becomes,
Nl
X
Rr,l 0 φnl sp ωp t
0 =Im(
Ie e
)+
Im(ωp Mnlnl Il0 eφnl esp ωp t )
sp l
m=1
2π
(−pp βp +φp −(n−1) N
) sp ωp t
n
+ Im(ωp Mp,nl Ip e
− Im(sp ωp Mc,nl Ic e
e
)
2π
) sp ωp t
(−pc βc +φc+(n−1) N
n
e
14
)
(2.64)
CHAPTER 2. B-DFIG STEADY STATE MODELLING
which is expressed in the phasor form for each rotor loop as,
0=
Rr,l 0
I + ωp Lr,l I 0l + ωp Mp,nl I p − ωp Mc,nl I c
sp l
(2.65)
The voltage equations are now reproduced in the phasor form, with inductances split
into the leakage and main part, below,
U p =Rp I p + ωp Lp,σ I p + ωp Lp,m I p +
U c Rc
= I c + ωp Lc,σ I c + ωp Lc,m I c −
s
s
Nl
X
l=1
Nl
X
ωp Mp,nl I 0l
(2.66)
ωp Mp,nl I 0l
(2.67)
l=1
Rr,l
0=
I + ωp Lr,lσ I r,l + ωp Lr,lm I r,l + ωp Mp,nl I p − ωp Mc,nl I c
sp r,l
(2.68)
On the basis of these equations, i.e. Equations 2.66, 2.67 and 2.68 the equivalent circuit
for the BDFM shown in Figure 2.2 is formed.
It is also important to look at what happens to the equivalent circuit under hypernatural operation of the machine. As per the convention used, the voltage for the stator
control circuit would be given by,
uc,a =Ûc sin(ωc t + θc )
4π
)
3
2π
=Ûc sin(ωc t + θc −
)
3
(2.69)
uc,b =Ûc sin(ωc t + θc −
(2.70)
uc,c
(2.71)
As the phase sequence of this is different to that of the power and rotor circuits, it needs
to be transformed so as to include it in the equivalent circuit. This is done as follows,
uc,a = − Ûc sin(−ωc t − θc )
(2.72)
uc,b
(2.73)
uc,c
4π
= − Ûc sin(−ωc t − θc +
)
3
2π
= − Ûc sin(−ωc t − θc +
)
3
(2.74)
The phase sequence now is identical for all the circuits and the control circuit can now
be added to the equivalent circuit. Also, in the convention a negative value for ωc during
operation at hyper-natural speeds has been considered. Therefore we have,
uc,a =Ûc sin(ωc t + (−θc + π))
4π
)
3
2π
=Ûc sin(ωc t + (−θc + π) +
)
3
(2.75)
uc,b =Ûc sin(ωc t + (−θc + π) +
(2.76)
uc,c
(2.77)
Therefore, during hyper-natural operation, a phase difference of θc for the stator control
circuit voltage must be taken as (−θc + π) in the steady state equivalent circuit.
15
CHAPTER 2. B-DFIG STEADY STATE MODELLING
2.2.2
Power Balance Equations
The apparent power input to the machine is given by,
Sp = Up Ip∗
(2.78)
Sc = Uc Ic∗
(2.79)
This gives the expressions for the Active Power input to the machine as,
Ip2Rp
Ir2Rr
Pm
Ic2Rc
Pp
Pp,ag
Pc,ag
Pc
(a)
P OW ER
W IN DIN G
ROT OR
CON T ROL
W IN DIN G
Ip2Rp
Ir2Rr
Pm
Pp
Ic2Rc
Pp,ag
Pc,ag
Pc
(b)
Figure 2.4
Power Flow in Motoring Mode neglecting Iron losses (a) for Sub-Natural Speeds and
(b) for Hyper-Natural Speeds.
Pp = <(Up Ip∗ )
Pc = <(Uc Ic∗ )
(2.80)
(2.81)
A part of this power is lost in the form of copper losses in the stator windings. Thus the
air-gap power transferred to the rotor may be given by,
Pp,ag = Pp − Pp,cu
Pc,ag = Pc − Pc,cu
(2.82)
(2.83)
A part of this air-gap power is lost as copper losses and iron losses in the rotor. The remaining power is available as mechanical power to the shaft. This is shown in Equation 2.84.
Pm = Pp,ag + Pc,ag − Pr,cu − PF e
(2.84)
Figure 2.4 gives a schematic representation of this while neglecting the iron losses in the
machine.
16
CHAPTER 2. B-DFIG STEADY STATE MODELLING
2.3
Simplified Equivalent Circuit
A simplification to this equivalent circuit shown in Figure 2.2 can be made by transforming
the stator and rotor circuits onto a single circuit. This can be done by first transforming
the control winding circuit onto the rotor circuit. Then transforming this onto the power
winding circuit. This has been shown below for a single rotor loop, but can be extended to
include multiple rotor loops.
The first step is the transformation of the control winding to the rotor. The transformation ratio is given by,
Lc,m
= a2
(2.85)
Lr,c
This transform can be seen in Figure 2.5. The transformed parameters are given by,
Rp
Up
Uc0
s
•
Lp,σ
•
Figure 2.5
Lr,σ
Lr,p
Lp,m
Rc0
s
Rr
sp
L0c,σ
L0c,m
First Step in the Simplified Equivalent Circuit Formation.
1
Rc
a2
1
= 2 Lc,σ
a
= Lr,c
Rc0 =
L0c,σ
L0c,m
(2.86)
(2.87)
(2.88)
The next step is to transform this circuit onto the power winding circuit. This transformation ratio is given by
Lp,m
= a02
(2.89)
Lr,p
The transformed parameters are given by,
a02
Rc
a2
a02
= a02 L0c,σ = 2 Lc,σ
a
02 0
= a Lc,m = a02 Lr,c
Rc00 = a02 Rc0 =
L00c,σ
L00c,m
Rr0
L0r,σ
(2.90)
(2.91)
(2.92)
02
= a Rr
(2.93)
= a02 Lr,σ
(2.94)
17
CHAPTER 2. B-DFIG STEADY STATE MODELLING
The thus simplified equivalent circuit is shown in Figure 2.6.
Rp
Rr0
sp
Lp,σ
Up
L0r,σ
Rc00
s
Uc00
s
L0c,m
Lp,m
Figure 2.6
L00c,σ
Simplified Equivalent Circuit.
This can be extended to machine with multiple rotor loops to form the equivalent circuit
shown in Figure 2.7.
Rp
Up
Lp,σ
L0r,2σ
0
Rr,1
sp
L0r,1σ
L00c,σ
L0c,m
Lp,m
Figure 2.7
2.4
0
Rr,2
sp
Rc00
s
Uc00
s
Simplified Equivalent Circuit.
The Γ-Equivalent Circuit
The Γ-equivalent is another simplified model. This equivalent circuit can be seen in Figure 2.8. This has been shown for a circuit with a single rotor loop but can be extended to
include multiple loops as well. The parameters [20] are given by,
where,
L0c,m−γ = αL0c,m
(2.95)
Lp,m−γ = α0 Lp,m
Lr,σ−γ = α02 α2 L0r,σ + αL00c,σ + α0 Lp,σ
(2.96)
L00c,σ + L0c,m
L0c,m
Lp,σ + Lp,m
α0 =
Lp,m
α=
18
(2.97)
(2.98)
(2.99)
CHAPTER 2. B-DFIG STEADY STATE MODELLING
Up
L0rσ−γ
Rr0
sp
Rp
Rc00
s
Uc
s
L0cm−γ
Lpm−γ
Figure 2.8
Γ-Equivalent Circuit.
As described by McMahon et al. [18], the equivalent circuit for the inner core of the
machine (i.e. the circuit obtained by omiting the magnetising reactances and the stator
and rotor resistances) resembles the circuit for a synchronous machine. This has been
reproduced in Figure 2.9,
ωp L0rσ−γ
Uc00
s
Up
Figure 2.9
Equivalent Circuit for Core.
This leads us to believe that the performance of the BDFM must be similar to that of
the synchronous machine with a synchronous speed given by,
ωm,syn =
2.5
ωp − ωc
Nn
(2.100)
Steady State Characteristics
The steady state characteristics are shown in Figure 2.10. The figure follows the convention
that power flow into the machine is positive and power flow out of the machine is negative.
It can be seen that at speeds below the natural speeds (i.e. sub-natural speeds) the control
circuit acts in the motoring mode while the power winding acts in the generating mode.
While at speeds above the natural speed (i.e. hyper-natural speeds), the control and power
winding circuits, both act in the generating mode. This is also highlighted in Figure 2.4.
The machine parameters used for the simulations are based on the case study machine
detailed in Appendix C.
The Power Winding circuit and the Control Winding circuit are fed by two independent
voltage sources. As the power circuit is connected directly to the grid, we assume a constant
voltage and frequency for this circuit. For the results shown in this section, the power circuit
frequency ωp was taken to be fixed at 2 × π × 50 rad/s. The frequency of the control circuit
is fixed as per the Equation 2.12 for the rotor speeds taken in each case. As the control
19
CHAPTER 2. B-DFIG STEADY STATE MODELLING
200
Net Power
Power Circuit Power
Control Circuit Power
0
Power (kW)
−200
−400
−600
−800
−1000
0
5
10
Figure 2.10
15
20
25
Speed of Rotation (rad/s)
30
35
40
45
50
Power-Speed Characteristics of B-DFIG.
circuit is fed through a power electronic converter it is possible to control its phase with
respect to Up which is taken as the reference vector.
Figure 2.11 explores the operation of the machine when ωc = −25% of ωp and the
ωp
. The phase difference between Uc and the
rotational speed is given by, ωm = 0.75
Nn
reference Up is varied between −π and +π. The effect of this on the circuit is shown.
450
Power Circuit Power
Control Circuit Power
Net Power
300
350
200
300
100
0
250
200
−100
150
−200
100
−300
50
−400
0
−100
0
100
Uc angle with Up (degrees)
Figure 2.11
Ip
Ic
Ir
400
Current (kA)
Power (MW)
400
Ip angle
Ic angle
Ir angle
150
Angle of Current with Up (degrees)
500
100
50
0
−50
−100
−150
−100
0
100
Uc angle with Up (degrees)
−100
0
100
Uc angle with Up (degrees)
ωp
with constant magnitude input voltages and
Nn
variation of phase angle between them.
Characteristics for ωm = 0.75
Figure 2.12 explores the operation of the machine when ωc = +25% of ωp and the
ωp
rotational speed is given by, ωm = 1.25
. The circuit used for these simulations is the
Nn
equivalent circuit as in Figure 2.2.
20
CHAPTER 2. B-DFIG STEADY STATE MODELLING
450
Power Circuit Power
Control Circuit Power
Net Power
Ip
Ic
Ir
400
400
350
300
Current (kA)
Power (MW)
200
0
−200
250
200
150
100
−400
50
−600
−100
0
0
100
Uc angle with Up (degrees)
Figure 2.12
2.6
Ip angle
Ic angle
Ir angle
150
Angle of Current with Up (degrees)
600
100
50
0
−50
−100
−150
−100
0
100
Uc angle with Up (degrees)
−100
0
100
Uc angle with Up (degrees)
ωp
with constant magnitude input voltages and
Nn
variation of phase angle between them.
Characteristics for ωm = 1.25
Discussion
From the characteristics of Section 2.5, a number of observations can be made. First, the
power flow for the conditions of sub-natural and hyper-natural speeds of operation can
be clearly observed from Figure 2.11 and Figure 2.12. During sub-natural operation, the
power flow in the power and control winding are in the opposite direction, while in the
hyper-natural mode of operation, the power flow in both the stator circuits flow in the
same direction (i.e. if power flows into the stator power winding circuit then power also
flows into the stator control winding circuit and vice versa).
Second, it can be observed that at two separate operating points, i.e. at φp,c = 0 and
φp,c = ±π (where, φp,c is the phase angle difference between Up and Uc ), we have zero power
output. It is interesting to note that at these points the values of power winding, control
winding and rotor current are different. This fact can be explained by looking at the stator
flux wave and rotor position. Figure 2.13 describes the condition for φp,c = ±π. The flux
linked with the rotor loop is maximum and hence, the rotor currents are maximum. The
torque in this position on both conductors that make up the loop will be opposite and equal,
causing the net torque to be zero. Figure 2.14 decribes the condition for φp,c = 0. The flux
linked with the rotor loop will be zero and therefore, rotor currents will be minimum. Due
to this zero flux linkage, the torque produced will be zero.
Third, for the condition φp,c = ±π, we see that a small perturbation in the positive
x direction will increase the torque in that direction will reducing the opposing torque,
causing the rotor to run-off in the positive x direction. Similarly, a perturbation in the
negative x direction will cause a run-off in the negative x direction. This operating point
is therefore, unstable. In contrast, the condition given by φp,c = 0 is stable. This will be
further studied during the dynamic simulation of the machine.
21
CHAPTER 2. B-DFIG STEADY STATE MODELLING
Stator Flux Wave
Rotor Loop
Stator Flux Wave and Rotor Loop at φp,c = ±π.
Figure 2.13
Stator Flux Wave
Rotor Loop
Figure 2.14
Stator Flux Wave and Rotor Loop at φp,c = 0.
500
Power Circuit Power
Control Circuit Power
Net Power
Stable Region
Unstable Region
400
Power (MW)
300
Unstable Region
200
100
0
−100
−200
−300
−400
−150
−100
−50
0
50
100
150
Uc angle with Up (degrees)
Figure 2.15
Stable and Unstable Operating Ranges for ωm = 0.75
22
ωp
.
Nn
Chapter 3
B-DFIG Dynamic Modelling
To study the behaviour of the machine during grid events, it is necessary to develop a
dynamic model of the B-DFIG based wind turbine drivetrain. The development of the
dynamic model is discussed in this chapter.
Section 3.1 describes the dynamic model using differential equations. Section 3.2 represents the dynamic model graphically sing block diagrams and finally, Section 3.3 describes
the response of the developed model.
3.1
The Dynamic Equations of the Machine
In this section the dynamic voltage equations of the B-DFIG are developed. First, the
convention used in this chapter and the rest of the document is described. As an example,
ūxa , gives the voltage vector ū for the a circuit in the x reference frame. Similarly, ux,d
a ,
gives the voltage u for the a circuit in the d-axis of the x reference frame. The vector ūxa
can be represented as,
x,d u
x
ūa = ax,q
(3.1)
ua
Now, the voltage equations for the B-DFIG are given by,
dλ̄sp
dt
dλ̄s
ūsc = Rc i¯sc + c
dt
dλ¯r
0 = Rr i¯rr + r
dt
ūsp = Rp i¯sp +
(3.2)
where, the flux linkages are given by,
λ¯sp = Lp i¯sp + Mpr i¯sr
λ¯sc = Lc i¯sc − Mcr i¯sr
λ¯rr = Lr i¯rr + Mpr i¯rp − Mcr i¯rc
(3.3)
(3.4)
(3.5)
These equations come from the equivalent circuit developed in the previous chapter. Here,
the vectors (such as λ̄rr ,īsp etc.) are all in different reference frames. To combine these
23
CHAPTER 3. B-DFIG DYNAMIC MODELLING
equations to form the dynamic model of the machine, they are transformed into a common
reference frame. First, the transformation of the quantities in the abc domain to the stator
reference frame is looked at. This is done through the Clarks’ transform shown below.


1
1
−  
  r  1 −
√2  xa
√2
xα

3
3  
→
−
−1
xβ  = 2 
x abc
(3.6)
 xb = Cαβ,abc
 0
−

3
2
2
x0
x

 1
c
1
1
√
√
√
2
2
2
For the ease of control of the machine the selection of a reference frame will be made
later in this document. Therefore, it is desirable to create a model in an arbitrary reference
frame. Figure 3.1 show the positioning of the two stator reference frames and the rotor
reference frames related to the pp and pc pole pair distributions. Here, the Power Winding
αp βp reference frame is taken to be the overall static reference frame.
βp
βrc
αrp, αrc
βrp
θm
αp, αc
π
2pc
βc
Figure 3.1
B-DFIG Reference Frames with mechanical angles
First, the quantities in the Control Winding αc βc reference frame are transformed to
the overall static reference frame, i.e. the αp βp . For the operation of the machine it has
been shown that Equation 2.11 and Equation 2.14 are to be met. Therefore, a vector x̄ in
the αrp βrp and the αrc βrc are related by Equation 3.7.
x̄αrc βrc = x̄αrp βrp
(3.7)
x̄αc βc = Crot (−pc θm )x̄αrc βrc
(3.8)
Using Figure 3.1,
x̄
αp βp
= Crot (pp θm )x̄
αrp βrp
(3.9)
where,
cos θ − sin θ
Crot (θ) =
sin θ cos θ
24
(3.10)
CHAPTER 3. B-DFIG DYNAMIC MODELLING
Using Equation 3.7 - Equation 3.9 the transformation of a quantity from the Control Winding αc βc reference frame to the Power Winding αp βp reference frame is given by,
x̄αp βp = Crot ((pp + pc )θm )x̄αc βc
(3.11)
Therefore, if an arbitrary reference frame rotating with an angular speed of ωk with
respect to the Power Winding static reference frame is chosen, the voltage equation can be
written as,
ūkp = Rp īkp + Crot (−θk )
d
Crot (θk )λ¯kp
dt
(3.12)
ūkc = Rc īkc + Crot (−(θk − (pp + pc )θm ))
0 = Rr īkr + Crot (−(θk − pp θm ))
d
Crot
dt
d
Crot ((θk − (pp + pc )θm ))λ¯kc
dt
((θ − p θ ))λ¯k
k
p m
r
(3.13)
(3.14)
where,
dθk
= ωk
dt
dθm
= ωm
dt
(3.15)
(3.16)
This simplifies to,
k
0 −1 ¯k dλ̄p
λp +
=
+ ωk
1 0
dt
0 −1 ¯k dλ̄kc
λc +
ūkc = Rc īkc + (ωk − (pp + pc )ωm )
1 0
dt
0 −1 ¯k dλ̄kr
λp +
0 = Rr īkr + (ωk − pp ωm )
1 0
dt
ūkp
Rp īkp
The equations of the dynamic model, rewritten in state-space form, are given by,
dλkp
0 −1 ¯k
k
k
λp
= ūp − Rp īp − ωk
1 0
dt
dλkc
0 −1 ¯k
k
k
= ūc − Rc īc − (ωk − (pp + pc )ωm )
λc
1 0
dt
dλkr
0 −1 ¯k
k
= −Rr īr − (ωk − pp ωm )
λr
1 0
dt
 k

  k 
īp
λ̄p
Lp
0
Mpr
īkc  = inv  0



Lc
−Mcr
λ̄kc 
k
Mpr −Mcr
Lr
īr
λ̄kr
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
A further look at these equations will be taken when the control for the machine is discussed.
To complete the dynamic model, the expression for power is computed. The total electrical
power at the terminals of a stator winding is given by,
ps = ūTs,abc īs,abc
25
(3.24)
CHAPTER 3. B-DFIG DYNAMIC MODELLING
using the Clark’s transform defined in Equation 3.6,
−1
−1
ps =(Cαβ,abc
ūs,αβ )T (Cαβ,abc
īs,αβ )
(3.25)
s,α
ps =us,α
s is
s
=īsT
s ūs
(3.26)
+
s,β
us,β
s is
(3.27)
The voltage equation of the stator in the stator reference frame is reproduced below,
ūss = Rs īss +
dλ̄ss
dt
(3.28)
the flux linkage λ̄ss may be considered to be made up of two parts, the leakage flux linkage
λ̄ss,σ and the main flux linkage λ̄ss,m . Therefore, the equation may be expressed as,
dλ̄ss,σ
dλ̄ss,m
+
dt
dt
s
d
ī
dīs
dīs
=Rs īss + Ls,σ s + Ls,m s + Msr r
dt
dt
dt
The differential of the currents in the stator reference frame is given by,
s,β dθ
d is,α
dθ
0 −1 is,α
−is
s
s
p
=
s,β p
s,β =
s,α
1 0
is
is
dt is
dt
dt
ūss =Rs īss +
The voltage equation can therefore be given by,
0 −1 s dθ
0 −1 s dθ
s
s
ī p + Msr
ūs =Rs īs + (Ls,σ + Ls,m )
ī p
1 0 s dt
1 0 r dt
From this voltage equation the power in the winding may be expressed as,
sT s
sT 0 −1 s dθ
sT 0 −1 s dθ
ps =Rs īs īs + (Ls,σ + Ls,m )īs
ī p + Msr īs
ī p
1 0 s dt
1 0 r dt
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
It can be seen that the first term represents the resistance loss in the winding, the
second and third terms will be equal to zero and the remaining term represents the power
converted into mechanical power. The torque is given by,
dθ
dt
Therefore, the electromagnetic torque can be given by,
sT 0 −1 s
Te =pMsr īs
ī
1 0 r
pm = Te
s,α
s,α s,β
=pMsr (is,β
s ir − is ir )
(3.34)
(3.35)
(3.36)
This expression is extended to form the torque expression for the B-DFIG by taking into
account the two stator windings. This is expressed as,
s,α
s,α s,β
s,β s,α
s,α s,β
Te = pp Mpr (is,β
p ir − ip ir ) + pc Mcr (ic ir − ic ir )
(3.37)
If the Park’s transform is used, the equation in the stator reference frame may be converted
to a rotating reference frame,
k,d
k,d k,q
k,q k,d
k,d k,q
Te = pp Mpr (ik,q
p ir − ip ir ) + pc Mcr (ic ir − ic ir )
26
(3.38)
CHAPTER 3. B-DFIG DYNAMIC MODELLING
3.2
Dynamic Model in the Block Diagram Form
From the voltage equations derived in the previous section (Equation 3.20-Equation 3.23
and Equation 3.38) the dynamic model of the B-DFIG in an arbitrary reference frame is
presented in block diagram form in Figure 3.2.
ūkp
−
0 −1
1 0
1
s
−
X
ωk
īkr
λ̄kp
Rp
ūkc
−
1
s
X
λ̄kc
−
1
s
Mux
X
Figure 3.2
De-Mux
+ Te
+
λ̄kr
īkr
0 −1
1 0
īkr
M
ωrk
Rr
īkp
īkc
0 −1
1 0
−
pp Mpr
ωck
Rc
.
īkp
0 −1
1 0
−
0 −1
1 0
.
pc Mcr
īkc
Dynamic Model of the B-DFIG in the arbitrary reference frame ‘k’
27
CHAPTER 3. B-DFIG DYNAMIC MODELLING
where the matrix M is given by,


Lp,σ + Lp,m
0
Mpr
0
Lc,σ + Lc,m
−Mcr 
M = inv 
Mpr
−Mcr
Lr,σ + Lr,m
(3.39)
and,
ωck =ωk − (pp + pc )ωm
ωrk =ωk − pp ωm
28
(3.40)
(3.41)
CHAPTER 3. B-DFIG DYNAMIC MODELLING
3.3
Dynamic Behaviour
In this section the dynamic behaviour of the B-DFIG is discussed. The operation in the
stable state for sub-natural speeds as described in Figure 2.15 is shown.
Figure 3.3a shows the torque output of the machine, operating in steady state in the stable region, with a 10% step increase in the load torque at t = 15s with a constant frequency
on the control circuit. Figure 3.3b shows the speed response for the condition. When the
5600
15.716
Generator Torque
Load Torque
5500
15.714
5400
15.712
Speed (rad/s)
Torque (kNm)
5300
5200
5100
5000
15.71
15.708
15.706
4900
15.704
4800
15.702
4700
4600
0
5
10
15
20
25
30
35
40
45
15.7
0
50
5
10
15
20
Time (s)
(a)
Figure 3.3
25
30
35
40
45
50
Time (s)
(b)
Step Response in Stable Region with a 10% Step increase in Load Torque (a) Torque
Response (b) Speed Response.
step change in load torque is applied the machine torque also increases and oscillates around
the load torque value. The speed also oscillates, however, the oscillations remains around
the stable speed before the change in load. As the Control Winding frequency has been
kept constant, the machine acts as a synchronous machine. This is the reason the speed of
the machine oscillates around the original speed.
The response for the machine in the unstable region is also shown, in Figure 3.4. Figure 3.4a shows the torque output of the machine, operating in steady state in the unstable
region, with a 10% step increase in the load torque at t = 15s with a constant frequency
on the control circuit. Figure 3.4b shows the speed response for this condition. In this case
the machine is unable to take an increase in load torque. The machine torque oscillates
around zero and the machine speed reduces.
8000
15.8
Generator Torque
Load Torque
6000
15.6
15.4
4000
Speed (rad/s)
Torque (kNm)
15.2
2000
0
−2000
15
14.8
14.6
14.4
−4000
14.2
−6000
−8000
0
14
5
10
15
20
25
30
35
40
45
13.8
0
50
Time (s)
10
15
20
25
30
35
40
45
50
Time (s)
(a)
Figure 3.4
5
(b)
Step Response in Stable Region with a 10% Step increase in Load Torque (a) Torque
Response (b) Speed Response.
These characteristics lead to a number of observations. First, they reinforce the discussion on the stable and unstable regions of operation in the previous chapter. It can be seen
29
CHAPTER 3. B-DFIG DYNAMIC MODELLING
that a perturbation in the unstable region causes the rotor to run off as can be seen in the
speed characteristics in Figure 3.4b.
Second, the characteristics can be seen to be similar to those of a synchronous machine.
This has already been seen in the previous chapter, refer Figure 2.9, the equivalent circuit
for the B-DFIG can be seen as a synchronous machine. This is confirmed by the response
of the B-DFIG dynamic model.
3.4
Discussion
This chapter has focussed on the creation of the dynamic model. Section 3.1 has developed
the dynamic model equations. This has been done in the arbitrary reference frame such
that a suitable choice of reference frame could be made when the control of the machine
is investigated. Section 3.2 has developed these equations into a block form which can be
implemented in Simulink. Finally, Section 3.3 discussed aspects of the dynamic behaviour
of the machine. The results confirmed the observations made in Chapter 2.
30
Chapter 4
B-DFIG Control
This chapter aims at the development of the control system for the operation of the B-DFIG.
A system level view of the B-DFIG in a wind turbine application is given in Figure 4.1.
The control developed in this thesis is based on the vector control method and employs two
cascaded current loops.
Figure 4.1
System Level B-DFIG Schematic
This chapter is organised in the following manner, Section 4.1 develops the wind turbine
drivetrain for the study. Section 4.2 develops the method to generate reference signals for
optimal power extraction. Section 4.3 looks at the control for Active and Reactive Power.
It looks at the rationale behind the choice of the controlled quantity, i.e. Power Winding
Active Power Pp in this case, and develops the control scheme for the machine. Section 4.4
looks at the tuning of the PI controllers used. Section 4.5 describes the control behaviour
of the machine and Section 4.6 discusses the results.
31
CHAPTER 4. B-DFIG CONTROL
4.1
Wind Turbine Drivetrain System
This section looks at the components of the wind turbine drivetrain. This is discussed for
the three generator types used in this study - the B-DFIG, the DFIG and the PMSM.
Wind Turbine Rotor
The wind turbine rotor is responsible for converting the kinetic energy in the wind to rotational energy that is used in the generator. This thesis does not focus in the characteristics
of the rotor. However, a simple model that calculates the torque on the rotor based on the
wind speed is discussed.
The power in the wind airflow is given by [21],
1
Pwind = ρAv 3
2
(4.1)
where, ρ is the air density, A is the swept area of the rotor and v is the upwind free wind
speed. The power transferred from the wind through the wind turbine rotor to the shaft
connected to the gearbox is given by,
Pturbine = Cp Pwind
(4.2)
where Cp is the power coefficient. The wind model is important for two reasons, first, it
is used to model the wind as a load for the turbine and second, it is used to generate the
reference signal for the control of the machine.
The power transferred to the turbine by the wind is given by Equation 4.2. It has been
ωm R
shown that Cp varies with the tip speed ratio (λt =
) and the pitch angle (θ), where
v
R is the radius of the wind turbine rotor. Here, the following numerical approximation [22]
is used for the estimation of Cp ,
1
−C 1
− C3 λi θ − C4 θCx − C5 )e 6 λi
λi
1
1
0.035
=
−
λi λt + 0.08θ θ3 + 1
Cp (λi , θ) =C1 (C2
(4.3)
(4.4)
The region of pitch control is not considered here. Therefore the pitch angle is assumed to
be constant, i.e. θ = 0. The coefficients used are, C1 = 0.5, C2 = 218, C3 = 0.4, C4 = 0,
C5 = 10.5 and C6 = 25. The variation of Cp with the tip speed ratio is shown in Figure 4.2.
For wind speeds above vrated the pitch of the blades is controlled to maintain constant load
torque. The Load Torque can therefore be defined as,

1
v3


 ρACp (λt )
if v < vrated
ωm
Tload = 2
(4.5)
3
v
1


 ρACp (λt , θ) rated if v ≥ vrated
2
ωm
32
CHAPTER 4. B-DFIG CONTROL
0.5
0.45
0.4
0.35
Cp
0.3
0.25
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
12
Tip Speed Ratio
Figure 4.2
Variation of Cp with λt
Gearbox
The gearbox transforms the speed of the rotor into one that can be better utilised by the
generator. In this thesis the gearbox has been considered to be lossless.
Generator
The generator converts the rotational kinetic energy of the wind turbine rotor into electrical
energy. Three generators have been considered here - the PMSM, the DFIG and the BDFIG. Figure 4.3 shows the drivetrains for the wind turbine with each of these generators.
(a)
(b)
(c)
Figure 4.3
Drivetrain Schematic (a) with B-DFIG (b) with DFIG (c) with PMSM
The generators in this study are based on a 3.2MW generator design and are detailed
in Appendix D, E and Appendix F. The dynamic model for the B-DFIG is developed in
33
CHAPTER 4. B-DFIG CONTROL
Chapter 3 while the equations describing the dynamic behaviour of the PMSM and DFIG
machines are in Appendix D and Appendix E respectively.
Power Electronic Converter and Controller
The power electronic converter for the B-DFIG and DFIG based wind turbine drivetrains
are partially rated while the PMSM employs a fully rated converter.
The design of the controller for this machine is detailed in this chapter. The control is
based on a vector control algorithm and an overview is shown in Figure 4.4.
idp
Qref
p
i¯c
i¯p
ωm
Ref. Gen.
−
id−ref
p
idc
icd−ref
PI
−
−
PI
ud,ref
c
+
dq Cross-Coupling Compensator
Ref. Gen.
iq−ref
p
−
q−ref
PI
+ ic
−
iqp
q,ref
PI
− uc
iqc
Figure 4.4
B-DFIG Control Scheme
The controller for the DFIG is also based on the vector control algorithm, however, it
is simpler and does not require cascaded current control loops. A number of studies have
described this and it is therefore not detailed in this document. Similarly, the controller for
the PMSM is based on vector control, but not detailed here.
4.2
Reference Signal Generation
This section deals with the first step of controlling the generator for a wind turbine, the
generation of a reference signal. It looks at obtaining this signal such that the generator
operates at the optimal operating point. Section 4.2.1 looks at the requirement to extract
maximum power from the wind while Section 4.2.2 looks at converting this reference signal
into one that can be used to control the machine.
4.2.1
Modelling for Optimal Power Extraction
The control of a wind turbine consists of two parts; the mechanical control of the pitch of the
rotor blades and the electrical control of the generator. For this thesis only electrical control
is considered. The aim of the control scheme is to maximise the power output of the wind
turbine. A typical wind turbine characteristic, with the optimal power extraction-speed
34
CHAPTER 4. B-DFIG CONTROL
curve and its intersection with the Cp,max for all wind speeds [23] is shown in Figure 4.5.
As Popt is the curve with Cp,max it is evident that if the turbine is controlled and kept on this
curve, the turbine will generate the maximum energy. This is followed for all speeds below
rated. For speeds above rated, rated power Prated is maintained through pitch control. This
is described in Equation 4.8.
Maximum Power Curve (Popt)
Generator Power
Rated Power
Generator Speed
Figure 4.5 Control Strategy for Optimal Power Extraction. The plot shows the generator
output power vs. speed curve for different wind speeds. The Popt curve connects all the points of
maximum power forming the curve for optimal power extraction.
Pref

 1 ρAC
3
p,max v
= 2
P
rated
if v < vrated
(4.6)
if v ≥ vrated
This equation can be modified to represent a reference torque by using the expression
ωm R
v=
. This is shown in Equation 4.7.
λt

3

 1 ρAR Cp,max ω 2 if ω < ω
m
rated
m
λ3t,max
Tref = 2
(4.7)

T
if ωm ≥ ωrated
rated
or,
Pref

3

 1 ρAR Cp,max ω 3
m
λ3t,max
= 2

P
rated
if ωm < ωrated
(4.8)
if ωm ≥ ωrated
where λt,max is the tip speed ratio for Cp,max . This equation is valid till the speed of the
generator equals the rated speed ωrated . At this speed the generator is generating rated
torque and rated power. Once this speed is crossed, the power output is held constant by
varying the pitch angle of the rotor blades [21].
This method of generating the Tref is used in the control of the DFIG and the PMSM[23]
[24] [25]. However, for the B-DFIG, control using torque is complex and hence the Power
Winding active power is used. The rationale behind this choice is explained in the following
sections.
35
CHAPTER 4. B-DFIG CONTROL
4.2.2
B-DFIG Reference Signal
Section 4.2.1 discusses the generation of a reference signal for optimal power extraction
from wind turbines. This signal is the net torque or power from the machine. Section 4.3.1
discusses the control of the machine, this control is based on the Active Power of the Power
Winding alone. Therefore, it is required to generate the control signal (Ppref ) from the rotor
speed (ωm ).
The steady state characteristics, as defined in Chapter 2, are used to generate a function
for the relation between ωm and Ppref . The characteristics depend on two variables, i.e.
magnitude of the Control Winding voltage and the phase angle between Control Winding
and Power Winding voltages. Therefore, given a value of ωm and P ref there are a number
of operating points possible. To select an optimal operating point the efficiency of the
machine is used as a selection criteria.
The curve of the shaft power with rotor speed, as per Equation 4.8, is shown in Figure 4.6. Solving the system for maximum efficiency, the curve for the Power Winding
Power corresponding to each point of the net power curve shown in Figure 4.6 is shown in
Figure 4.7. These curves have been drawn for the B-DFIG 3.2MW Case Study Machine
detailed in Appendix F.
−0.5
Active Power (MW)
−1
−1.5
−2
−2.5
−3
−3.5
24
26
28
30
32
34
36
38
Rotor Speed (rad/s)
Figure 4.6
Power Curve for optimal power generation in the motor convention
Curve fitting on the Power Winding Power curve gives the polynomial equation of the
relation between Ppref and ωm . This is given in Equation 4.9.
3
2
Ppref = 1.4185 × ωm
− 2071.7218 × ωm
+ 7406.5342 × ωm − 1.3713 × 105
36
(4.9)
CHAPTER 4. B-DFIG CONTROL
−0.5
Net Power
Power Winding Power
Active Power (MW)
−1
−1.5
−2
−2.5
−3
−3.5
24
26
28
30
32
34
36
38
Rotor Speed (rad/s)
Figure 4.7
4.3
Curve for net Shaft Power and Power Winding Active Power.
Active and Reactive Power Control
A number of control strategies for the B-DFIG have been proposed in literature. They can
be classified into the following categories amongst others,
• Scalar Voltage Control [26][27]
• Vector Control [28][14][29] [30]
• Direct Torque Control [31] [32][33].
The control of the B-DFIG in this thesis is based on vector control (also used for DFIG
based systems[34]). This choice has been made because the speed and accuracy of the
response of the system with a vector control strategy is adequate for the study taken up in
this thesis.
4.3.1
Active Power Control
The Power Winding power in the arbitrary reference frame ‘k’ is given by Equation 4.10.
k,d
k,q k,q
Pp = uk,d
p ip + up ip
(4.10)
Here, the voltage equations provided in Chapter 3 Equation 3.17 are reproduced below,
uk,d
p
uk,q
p
dλk,d
p
=
−
+
dt
dλk,q
p
k,d
= Rp ik,q
+
ω
λ
+
k
p
p
dt
Rp ik,d
p
ωk λk,q
p
Using Equation 4.11 and Equation 4.12 and substituting in Equation 4.10 gives,
37
(4.11)
(4.12)
CHAPTER 4. B-DFIG CONTROL
k,d
Pp = Rp (ik,d2
+ ik,q2
p
p ) + ip
k,q
dλk,d
p
k,q
k,q dλp
k,d
− ωk ik,d
λ
+
i
+ ωk ik,q
p
p
p
p λp
dt
dt
(4.13)
Choosing a reference frame rotating with the Power Winding flux results in Equation 4.14 and Equation 4.15 for the d and q components of Power Winding flux.
λPp W,d =|λp |
λPp W,q
(4.14)
=0
(4.15)
This reference frame is referred to as the PW reference frame in the rest of this document.
The equation for Power Winding power in this reference frame is,
Pp = Rp (iPp W,d iPp W,d + iPp W,q iPp W,q ) + iPp W,d
d|λp |
+ ωk iPp W,q |λp |
dt
(4.16)
Here, if the assumption is made that Up is constant and Rp is small enough to be neglected,
the flux |λp | will be constant. This gives,
Rp (iPp W,d iPp W,d + iPp W,q iPp W,q ) ≈ 0
d|λp |
≈0
dt
(4.17)
(4.18)
Therefore, Pp simplifies to,
Pp ≈ ωP W |λp |iPp W,q
(4.19)
This is the first step in the control of the machine, i.e. the generation of a reference iPp W,q
value from the Ppref value. This can be visualised in the control scheme shown in Figure 4.8.
Ppref
÷
|λp |
×
ωP W
Figure 4.8
iPp W,q−ref
W,q
Reference iP
Generation
p
For this machine, only the Control Winding circuit is controllable through the power
electronic converter. Therefore, the next step would be to obtain a reference iPc W,q current.
Consider the flux equations,
PW
PW
PW
λ¯p
= Lp i¯p
+ Mpr i¯r
λ¯c
PW
λ¯r
PW
=
PW
Lc i¯c
=
PW
Lr i¯r
−
+
(4.20)
PW
Mcr i¯r
PW
Mpr i¯p
38
(4.21)
−
PW
Mcr i¯c
(4.22)
CHAPTER 4. B-DFIG CONTROL
From these equations the relation between the Power Winding and Control Winding currents is given by,
1 P W,d
λ
−
Lc c
1
= λPc W,q −
Lc
iPc W,d =
iPc W,q
Mcr Lp P W,d
Mcr
ip
+
|λp |
Mpr Lc
Mpr Lc
Mcr Lp P W,q
i
Mpr Lc p
(4.23)
(4.24)
From Equation 4.24 it is seen that iPc W,q depends on iPp W,q and λPc W,q . λPc W,q is weakly
dependant on iPp W,d and iPc W,d through Equation 3.21 reproduced below in the PW reference
frame.
dλPc W,q
= uPc W,q − Rc iPc W,q − (ωP W − (pp + pc )ωm )λPc W,d
dt
(4.25)
This influence of d−axis terms on q−axis quantities and vice versa is termed ‘CrossCoupling’. For accurate control it is required that the d and q axis terms be completely
de-coupled such that the control of both parameters is independent of the other. This is
done through the addition of a compensation term, calculated using Equation 4.24, shown
in Equation 4.26.
iPc W,q = f (iPp W,q , |λp |) +
1 P W,q
λ
Lc c
| {z }
(4.26)
Cross-Coupling Term
It is also seen that iPc W,q varies with −iPp W,q . The control scheme for this step is shown in
Figure 4.9.
ipP W,q−ref−
+
PI
+
iqcomp
iPp W,q
Figure 4.9
iPc W,q−ref
W,q
Reference iP
Generation
c
where,
iqcomp =
1 P W,q
λ
Lc c
(4.27)
The power electronic converter can be controlled by the duty ratio for the switches which
can be calculated from the reference Control Winding voltage and the DC bus voltage.
For the simulations here, the reference voltage uc is used as input to the machine. The
dependence of uqc on iqc can be calculated from the Control Winding voltage equation in
Equation 4.28.
uPc W,q = Rc iPc W,q + (ωP W − (pp + pc )ωm )λPc W,d +
39
dλPc W,q
dt
(4.28)
CHAPTER 4. B-DFIG CONTROL
A similar cross-coupling term, due to λPc W,d , as seen in Equation 4.26 is seen in the equation
above. This can also be expressed as in Equation 4.29.
Lp Mcr P W,d
uPc W,q = f (iPc W,q , λPc W,q ) − (ωP W − (pp + pc )ωm )(
i
+ Lc iPc W,d )
Mpr p
|
{z
}
(4.29)
Cross-Coupling Term
The control scheme for this is shown in Figure 4.10.
iPc W,q−ref
−
uPc W,q−ref
PI
−
uqcomp
iPc W,q
Figure 4.10
W,q
Generation
Reference uP
c
where,
uqcomp = (ωP W − (pp + pc )ωm )(
Lp Mcr P W,d
i
+ Lc iPc W,d )
Mpr p
(4.30)
The complete control scheme for Active Power control of the B-DFIG is shown in Figure 4.4. The next section will look at the control of Reactive Power for the machine.
4.3.2
Reactive Power Control
The Reactive Power of the Power Winding for the B-DFIG in the arbitrary reference frame
‘k’ is given by,
k,d
k,d k,q
Qp = uk,q
p ip − up ip
(4.31)
Substituting Equation 4.11 and Equation 4.12 in Equation 4.31 gives,
Qp = ik,d
p
k,d
dλk,q
p
k,d
k,q dλp
k,q
+ ωk ik,d
λ
−
i
+ ωk ik,q
p
p
p
p λp
dt
dt
(4.32)
Again, choosing the PW reference frame results in Equation 4.33 and Equation 4.34 for
the d and q components of Power Winding flux.
λPp W,d =|λp |
λPp W,q =0
(4.33)
(4.34)
Using the assumption that Up is constant and Rp is small enough to be neglected, the flux
|λp | will be constant. This gives,
d|λp |
≈0
dt
(4.35)
The expression for Qp can therefore be expressed as,
Qp ≈ ωP W |λdp |iPp W,d
40
(4.36)
CHAPTER 4. B-DFIG CONTROL
Qref
p
|λp |
ωP W
Figure 4.11
iPp W,d−ref
÷
×
W,d
Reference iP
Generation
p
This is the first step in the control of the reactive power of the machine, i.e. the generation
of a reference iPp W,d value from the Qref
value. This can be visualised in the control scheme
p
shown in Figure 4.11. From Equation 4.23 it is seen that iPc W,d depends on iPp W,d and λPc W,d .
λPc W,d is weakly dependant on iPp W,q and iPc W,q through Equation 3.21 reproduced below in
the PW reference frame.
dλPc W,d
= uPc W,d − Rc iPc W,d + (ωP W − (pp + pc )ωm )λPc W,q
(4.37)
dt
Again there is a dq cross-coupling term which must be compensated for. This compensation
term can be calculated using Equation 4.23, as seen in Equation 4.38.
iPc W,d = f (iPp W,d , |λp |) −
1 P W,d
λ
Lc c
| {z }
(4.38)
Cross-Coupling Term
This part of the control scheme is shown in Figure 4.12.
id,ref
p
−
+
iPc W,d−ref
PI
iPp W,d
−
idcomp
Figure 4.12
W,d
Reference iP
Generation
c
where,
idcomp =
1 P W,d
λ
Lc c
(4.39)
The dependence of uPc W,d on iPc W,d can be calculated from the Control Winding voltage
equation.
uPc W,d = Rc iPc W,d − (ωP W − (pp + pc )ωm )λPc W,q +
dλPc W,d
dt
(4.40)
Again, we see a dq cross-coupling term due to λPc W,q . This can be seen in Equation 4.41.
Lp Mcr q
uPc W,d = f (iPc W,d , λPc W,d ) − (ωP W − (pp + pc )ωm )(
i
+ Lc iPc W,q )
Mpr P W,p
|
{z
}
Cross-Coupling Term
41
(4.41)
CHAPTER 4. B-DFIG CONTROL
The control scheme for this is shown in Figure 4.13.
iPc W,d−ref
−
PI
−
udcomp
iPc W,d
Figure 4.13
uPc W,d−ref
Reference udc Generation
Here, it is important to discuss current limits. In the control scheme described both
P W,ref
P W,ref
¯
ip
and i¯c
must be maintained within the operating limits of the machine for
safe operation. This is described by the equations,
q
(iPp W,d−ref )2 + (iPp W,q−ref )2 ≤ Ip,limit
(4.42)
q
(iPc W,d−ref )2 + (iPc W,q−ref )2 ≤ Ic,limit
(4.43)
In the event that the reference d − q currents are larger than that allowed by the current
limiters. The current d component is given preference over the q component. This is
because the d current component is responsible for the creation of the stator flux |λp |. For
operation it is important to maintain the flux of the machine and therefore the d component
is important to maintain. The complete control scheme is shown in Figure 4.14.
42
abc
PW
i¯c
abc
PW
λ¯c
uc
ic
abc
up
|λp |
ωp
PW
Estimator
u¯p
measurement
B-DFIG
PW
abc
ip
i¯p
ωm
PW
λ¯p
PW
×
P.E. Controller
iPp W,d
Qref
p
43
÷
−
iPp W,d−ref
iPc W,d
iPc W,d−ref
PI
−
−
PI
idcomp
iqcomp
Ppref
|λp |
ωp
÷
iPp W,q−ref
−
PI
−
udcomp
PW
i¯c
PW
dq Cross-Coupling Compensator
i¯p
ωm
iPp W,q
iPc W,q
×
Figure 4.14
uPc W,q
uqcomp
P W,q−ref
+ ic
−
B-DFIG Controller
measurement
uPc W,d
PI
−
CHAPTER 4. B-DFIG CONTROL
u¯c
CHAPTER 4. B-DFIG CONTROL
4.4
PI Tuning
The PI controller used in this thesis work is described by,
K(1 +
1
)
τs
(4.44)
The controller is designed using the internal mode control method [35] where the pole of
the open-loop system is compensated with the zero of the PI controller.
Inner Current Loop
The open loop transfer function for the inner current control loop, taking cross-coupling
compensation into account is given by,
ic
1
=
uc
Rc + sLc
G1 =
(4.45)
The PI parameters are given by [35],
K =α1 Lc
Lc
τ=
Rc
(4.46)
(4.47)
where, α1 is the required closed loop bandwidth. The value of α is limited by the equations
[35],
ωs ≥ 10α
ωsw ≥ 5α
(4.48)
(4.49)
where, ωs is the angular sampling frequency of the system and ωsw is the angular switching
frequency. α is also related to the rise-time tr as,
α=
ln9
tr
(4.50)
Selecting the rise time for the inner loop as 1ms we have,
α1 =
ln9
= 2197.224 rad/s
0.001
(4.51)
(4.52)
Outer Current Loop
A simplified model for the plant of the outer current control loop is shown in Figure 4.15.
This transfer function takes cross-coupling compensation and the inner current control loop
into account. The open loop transfer function is,
G2 = −
Lc Mpr
s
Mcr Rp + sLp
44
(4.53)
CHAPTER 4. B-DFIG CONTROL
iref
p
iref
c
PI
−
−Lc Mpr
s
Mcr Rp + sLp
ic Control
ip
Reference idc Generation
Figure 4.15
The PI parameters are given by,
K =α2 Lp
Lp
τ=
Rp
(4.54)
(4.55)
where, α2 is the closed loop bandwidth. Here, we choose a rise time tr of 2.5ms and can
calculate α2 as,
α2 =
4.5
ln9
= 878.89 rad/s
0.0025
(4.56)
Control Behaviour
This section describes the behaviour of the B-DFIG with the control scheme detailed in
the previous sections under the condition of a step change applied to the load torque. A
step increase of 10% in load torque is simulated to observe the response of the system. The
machine used in the simulations is detailed in Appendix C.
The simulated torque response is shown in Figure 4.16a. This figure shows both the
Load Torque generated by the wind and the Electrical Torque produced by the B-DFIG.
The rotor angular speed under this condition is shown in Figure 4.16b. The Power Winding
and Control Winding currents in the PW reference frame are shown in Figure 4.17a and
Figure 4.17b respectively. In these simulations Qref
has been maintained at a constant
p
value.
−520
26
Load Torque
B−DFIG Torque
25.8
−530
Rotational Speed (rad/s)
25.6
Torque (kNm)
−540
−550
−560
−570
25.4
25.2
25
24.8
24.6
24.4
−580
24.2
−590
800
900
1000
1100
1200
1300
1400
1500
1600
1700
24
800
1800
Time (s)
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
Time (s)
(a)
(b)
Figure 4.16 Response of the B-DFIG Control Scheme with a 10% Load Torque Step at t=1000s
(a) B-DFIG and Load Torque Response (b) Rotational Speed of the B-DFIG
45
CHAPTER 4. B-DFIG CONTROL
1900
−2000
d−axis Current
q−axis Current
1700
1600
1500
1400
1300
1200
1100
d−axis Current
q−axis Current
−3000
Control Winding d−q Current (A)
Power Winding d−q Current (A)
1800
−4000
−5000
−6000
−7000
−8000
−9000
−10000
1000
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
−11000
800
Time (s)
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
Time (s)
(a)
(b)
Figure 4.17 Response of the B-DFIG Control Scheme with a 10% Load Torque Step at t=1000s
(a) Power Winding Currents in the PW Reference Frame (b) Control Winding Currents in the
PW Reference Frame
It can be seen that the response of the control scheme is acceptable, however, the speed
of response is limited by the inertia of the turbine.
4.6
Discussion
This chapter has focussed on the development of a control method for the B-DFIG in a
wind turbine. The control has been based on the vector control method and is implemented
in the Power Winding flux reference frame.
Section 4.2 describes the generation of the reference signal to control the machine such
that it operates at an optimal operating point. This step is a challenge for a B-DFIG as
the control is based on the Power Winding Active Power.
Section 4.3 discussed the control methodology for the machine. This includes the concept of ‘Cross-Coupling Compensation’ to decouple the d and q axis control. The d axis
is used to control the reactive power supplied to the system while the q axis component
controls the active power.
The behaviour of the control has been shown in Section 4.5. The response shown is
for a step change in wind speed. The response of the B-DFIG in this case is found to be
limited by the inertia of the turbine.
46
Chapter 5
Performance Under Grid Events
With increased wind energy penetration, wind turbine generators are required to have Fault
Ride Through (FRT) or Low Voltage Ride Through (LVRT) capabilities. This chapter
looks at this capability of the B-DFIG and compares this with those of other wind turbine
generating technologies such as permanent magnet generators and doubly fed induction
generators. This comparison is made for case study machines developed for a 3.2MW wind
turbine which gives a good basis on which a comparison can be made.
Section 5.1 looks at the grid code requirements for LVRT capabilities. It defines the test
cases under which these capabilities will be tested. Section 5.2 discusses the LVRT response
of the Permanent Magnet Synchronous Machine while Section 5.3 looks at the LVRT response of the Doubly Fed Induction Generator. Section 5.4 describes the performance of
the B-DFIG and Section 5.5 compares the response of all three generators.
5.1
Grid Code Requirements
The European Network of Transmission System Operators for Electricity (ENTSO-E) represents the electric Transmission System Operators (TSO) in the European Union. The
ENTSO-E released its Network Code on Requirement of Generators (NC RfG) [36] [37]
in March 2013. The purpose of the NC RfG is to develop a set of coherent requirements
to create harmonised solutions and products for the pan-European market. The NC RfG
gives the requirements for the Fault Ride Through (FRT) or Low Voltage Ride Through
(LVRT) capabilities for generators connected to the grid. These are divided into conditions
for connections below 110kV and above 110kV.
The objective is to limit the potential loss of generation after a fault on the distribution
or transmission system in order to avoid more severe disturbances, i.e. frequency collapse
in a synchronous area causing demand tripping and unexpected power flows resulting in
overloads both on internal transmission lines and tie lines with neighbouring systems possibly
leading to cascading tripping, system splitting, load shedding, major faults, brown outs and
even black outs. In the case of a fault on the transmission system level a voltage drop will
propagate across large geographical areas around the point of the fault during the period
of the fault. The increased levels of distributed generation will need to be tolerant to such
faults, especially where the total installed volume of embedded generation possibly affected by
a transmission system fault exceeds the maximum designed generation loss. It is also possible
that a large generator may have been tripped depending upon the exact fault location. Unless
47
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
this capability is established, the total loss would then be the sum of this large generator plus
any lower level distributed generation tripping.[36]
1.0
Urec2
Urec1
Uclear
Uret
tclear
Figure 5.1
trec1
trec2
trec3
Voltage Profile for LVRT capability of generators connected to the grid.
The NC RfG specifies a profile that represents the worst voltage variation during a
fault and after its clearance. Power Generating Modules are required to stay connected
to the grid and continue stable operation for voltages above the worst case conditions.
The specification of the profile comprises of a set of parameters for times and voltages as
shown in Figure 5.1. The parameters are described in Table 5.1. TSOs specify a voltageVoltage Parameters (pu)
Uret
0.05-0.15
Uclear
Uret -0.15
Urec1
Uclear
Urec2
0.85
Table 5.1
Time Parameters (s)
tclear
0.14-0.25
trec1
tclear
trec2
trec1
trec3
1.5-3.0
Parameters for generators connected below 110kV as per NC RfG
against-time profile for LVRT capabilities based on the parametrised curve in Figure 5.1.
Therefore, for the purpose of this study, two voltage profiles cases which are the worst
case scenarios within the constraints defined in the NC RfG will be considered. These are
shown in Figure 5.2 and Figure 5.3. Case A has been selected as the steep drop and rise of
voltage, at t = 0 s and t = 0.25 s respectively, are expected to cause the highest transients
current magnitudes in the case of the DFIG and B-DFIG. Here, a 95% symmetric voltage
dip is considered. Case B has been selected as can be expected to cause transient currents
48
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
for longer periods of time. It can also be expected to cause larger DC-link Voltage rise for
PMSM machines as it involves power mismatch for longer periods.
V oltage (pu)
1.0
0.25
Figure 5.2
T ime (s)
Voltage profile for Case A of the LVRT response study
V oltage (pu)
1.0
0.25
Figure 5.3
T ime (s)
3.0
Voltage profile for Case B of the LVRT response study
In the following sections, the LVRT response of the B-DFIG and the two most prevalent
generators used in wind turbines, the Permanent Magnet Synchronous Machine (PMSM)
and the Doubly Fed Induction Generator (DFIG), will be investigated. In this chapter
the response of these generators for the ‘Case A’ voltage profile described in Figure 5.2 is
discussed in detail. The response for the ‘Case B’ voltage profile is discussed in Appendix A.
49
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
5.2
Simulation of the LVRT Performance for the PMSM
The schematic layout of the PMSM based wind turbine system is shown in Figure 5.4.
Figure 5.4
Schematic of the Permanent Magnet Synchronous Machine (PMSM) as a Wind
Turbine
The PMSM is completely isolated from the grid through an AC-DC-AC converter.
Therefore, disturbances on the grid do not directly affect the machine. However, in the
occurance of an LVRT event the ability of the grid side inverter to transfer power to the
grid is greatly reduced. This results in the rise of the DC link voltage which is a cause for
concern to the DC link capacitors. Section 5.2.1 looks at the performance of the PMSM
under an LVRT event. Section 5.2.2 discusses methods for protecting the DC link during
such events and shows the response of the system under LVRT conditions. The machine
used in these simulation is the PMSM Case Study Machine detailed in Appendix D.
The operating point of the generator at the instant of the event is the limit of maximum
output i.e. Pout = 3.2MW. As the PMSM is completely isolated from the grid the Reactive
Power transferred to the grid is generated at the grid side converter. Therefore, Qout of the
generator is taken to be zero.
For the power electronic converter, the shape of the DC link voltage depends on the
value of Capacitance used. Here, an arbitrary value of 1F has been chosen.
50
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
5.2.1
LVRT Performance without Protection
The response of the PMSM (without any protection circuit or algorithm) to a Low Voltage
event is shown in Figure 5.5.
0
32
−50
31
−100
30
−150
29
0.8
0.6
0.4
0.2
0
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−200
999
1001
999.2
999.4
999.6
999.8
Time (s)
1000.2
1000.4
1000.6
1000.8
28
1001
(b)
50
0.2
d−axis Current
q−axis Current
0
d−axis Current
q−axis Current
40
30
Grid d−q Currents (kA)
−0.2
−0.4
−0.6
−0.8
−1
−1.2
20
10
0
−10
−20
−30
−1.4
−40
−1.6
999
1000
Time (s)
(a)
Generator d−q Currents (kA)
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−50
999
1001
999.2
999.4
999.6
999.8
Time (s)
(c)
1000
Time (s)
1000.2
1000.4
1000.6
1000.8
1001
(d)
1.9
1.8
9
9
7
7
5
5
3
3
1
1
1.6
1.5
1.4
1.3
1.2
1.1
Reactive Power (MVAr)
Active Power (MW)
DC−link Voltage (pu)
1.7
1
0.9
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−1
999
1001
999.2
999.4
Time (s)
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−1
1001
Time (s)
(e)
(f )
Figure 5.5 LVRT Performance of PMSM Without Protection for a 95% Symmetric Voltage Dip
at t=1000s (a) Grid Voltage magnitude (b) Torque and Speed (c) Generator Currents in the PW
reference frame (d) Grid Currents in the grid reference frame (e) DC link voltage (f ) Active and
Reactive Power to the grid
The flow of energy through the Power Electronic Converter is described in Figure 5.6.
The difference between the input energy and the output energy is stored in the capacitor
(causing the voltage rise). This can be described using Equation 5.1.
Pin − Pout = udc C
dudc
dt
(5.1)
During a Low Voltage event, the grid voltage reduces and if the current output of the grid
side converter is kept within the current limits specified, the power delivered to the grid is
substantially reduced. This results in a rise in the DC link voltage seen in Figure 5.5e. As
51
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
Pin
Pout
C
Generator-Side Converter
Figure 5.6
Grid-Side Converter
Power Flow in the Converter for a PMSM
the generator is completely isolated from the grid, there is no effect of the Low Voltage event
on the generator currents and torque. Therefore, the issue with the LVRT performance of
a PMSM does not arise in the machine but the DC link due to a mismatch in power flow
capabilities.
5.2.2
LVRT Performance with Protection
The previous section highlighted the issue of DC link voltage rise during a low voltage
event. A number of methods have been devised to manage the mismatch in power flow
which causes this problem. Two such methods, Energy Discharge Circuits [38] and Power
Balancing [39] have been simulated and discussed, while others have been described in brief.
Energy Discharge Circuit
This method prevents the rise of the DC link voltage by using resistive elements to dissipate
a part of the power fed to the DC circuit through the generator side converter. A schematic
of this is shown in Figure 5.7. This minimises the difference between the power fed into
the DC circuit and the power taken out of the DC circuit, thus limiting the rise in DC link
voltage. Figure 5.8 shows the performance of the PMSM equipped with a Energy Discharge
Circuit in the DC link.
Pin
Pout
Energy Discharge Circuit
C
R
Generator-Side Converter
Figure 5.7
Grid-Side Converter
Schematic of the Energy Discharge Circuit
The results in Figure 5.8 show that with the Energy Discharge Circuit implemented it
is possible to control the rise of the DC link voltage. This method however has a number
of drawbacks,
• It requires additional power electronic switches and large resistor banks capable of
carrying large currents.
• The excess energy, however small, is dissipated and lost.
52
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
0
32
−50
31
−100
30
−150
29
0.8
0.6
0.4
0.2
0
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−200
999
1001
999.2
999.4
999.6
999.8
1000
Time (s)
1000.2
1000.4
1000.6
1000.8
28
1001
Time (s)
(a)
(b)
20
0.2
d−axis Current
q−axis Current
0
d−axis Current
q−axis Current
15
−0.2
Grid d−q Currents (kA)
Generator d−q Currents (kA)
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
−0.4
−0.6
−0.8
−1
−1.2
10
5
0
−5
−10
−1.4
−15
−1.6
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−20
999
1001
999.2
999.4
999.6
999.8
1000
Time (s)
Time (s)
(c)
1000.2
1000.4
1000.6
1000.8
1001
(d)
1.02
9
9
7
7
5
5
3
3
1
1
0.99
0.98
0.97
0.96
0.95
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−1
999
1001
Time (s)
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
Reactive Power (MVAr)
1
Active Power (MW)
DC−link Voltage (pu)
1.01
−1
1001
Time (s)
(e)
(f )
Braking Resistor Current (A)
2500
2000
1500
1000
500
0
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
1001
Time (s)
(g)
Figure 5.8 LVRT performance of PMSM with Energy Discharge Circuit for a 95% Symmetric
Voltage Dip at t=1000s (a) Grid Voltage magnitude (b) Torque and Speed (c) Generator
Currents in the PW reference frame (d) Grid Currents in the grid reference frame (e) DC link
voltage (f ) Active and Reactive Power to the grid (g) Energy Discharge Circuit - Resistor
Current
Power Balancing
This method controls the rise of the DC link voltage by minimising the difference between
the power transferred to the grid and the power generated by the PMSM. This is done
53
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
by forcing the generator side converter to follow the grid side converter during Low Voltage events. Figure 5.9 shows the performance of the PMSM when the Power Balancing
Algorithm is implemented.
0
32
−50
31
−100
30
−150
29
0.8
0.6
0.4
0.2
0
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−200
999
1001
999.2
999.4
999.6
999.8
Time (s)
1000
1000.2
1000.4
1000.6
1000.8
28
1001
Time (s)
(a)
(b)
10
0.2
d−axis Current
q−axis Current
0
d−axis Current
q−axis Current
5
−0.2
Grid d−q Currents (kA)
Generator d−q Currents (kA)
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
−0.4
−0.6
−0.8
−1
−1.2
0
−5
−10
−1.4
−1.6
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−15
999
1001
999.2
999.4
999.6
999.8
Time (s)
(c)
1000
Time (s)
1000.2
1000.4
1000.6
1000.8
1001
(d)
1.008
1.006
9
9
7
7
5
5
3
3
1
1
1
0.998
0.996
0.994
0.992
Reactive Power (MVAr)
1.002
Active Power (MW)
DC−link Voltage (pu)
1.004
0.99
0.988
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−1
999
1001
Time (s)
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−1
1001
Time (s)
(e)
(f )
Figure 5.9 LVRT performance of PMSM with Power Balancing for a 95% Symmetric Voltage
Dip at t=1000s (a) Grid Voltage magnitude (b) Torque and Speed (c) Generator Currents in the
PW reference frame (d) Grid Currents in the grid reference frame (e) DC link voltage (f ) Active
and Reactive Power to the grid
The results in Figure 5.9 show that with the Power Balancing Algorithm implemented
it is possible to control the rise of the DC link voltage. This method does not require any
additional components and does not dissipate the excess energy, instead it saves this excess
in the rotor in the form of kinetic energy. However, it has a number of drawbacks,
• It causes sharp torque ripples as can be seen in Figure 5.9b. This may be addressed
with a better choice of reference torque (for example, a ramped change in reference
in place of a step change) during the Low Voltage event.
• As the generated power, and hence the generated torque, reduces during the event it
may cause the rotor speed to increase beyond the limits of operation. This may not
54
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
be a major issue when the rotor has a large inertia and the low voltage event occurs
for a short period of time. This could also be limited by pitch control.
Other Methods
There are a number of other methods that have been proposed and used to address the
issues arising due to Low Voltage events. Pitching Control seems to be an obvious solution.
When a disturbance in the form of a Low Voltage event is sensed on the grid, the pitch of
the rotor blades is increased to reduce the torque on the generator. This will reduce the
imbalance of power in the DC circuit. However, the speed at which the blades can be pitched
is very important to determine the success of this method. Conroy et al. demonstrated
that even with a maximum pitching rate of 20◦ per second the DC link voltage rise is not
significantly mitigated [40].
Another method is to use Energy Storage Systems to manage and store the energy
during Low Voltage events [38]. In this case the excess energy is stored during the voltage
event and transferred to the grid after the voltage is restored. Wang et al. simulated the
use of Vanadium Redox Flow Battery (VRB) based Energy Storage System for the PMSM
[41]. The system could also be used to smoothen the power transferred to the grid apart
from improving LVRT response. The drawbacks of this system are its size and cost.
55
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
5.3
Simulation of the LVRT Performance of the DFIG
The schematic layout of the DFIG based wind turbine system is shown in Figure 5.10.
Figure 5.10
Schematic of the Doubly Fed Induction Generator (DFIG) as a Wind Turbine
Unlike the PMSM, the DFIG is not isolated from the grid. The stator is directly
connected to the grid and any disturbance in the form of a Low Voltage event can be
expected to have an effect on the stator and rotor currents. Under normal operating
conditions the flux space vectors of the rotor and stator rotate with synchronous speed.
When the grid voltage drops, the magnitude of the stator flux vector also reduces. The rotor
flux vector however retains its magnitude and speed, causing oscillatory stator currents. The
voltage dip also induces large oscillating currents in the rotor which could harm the power
electronics connected in the circuit.
Section 5.3.1 looks at the performance of the DFIG under LVRT without protection
while Section 5.3.2 looks at some methods for improving LVRT performance. The machine
used in these simulations is the DFIG Case Study Machine detailed in Appendix E. As with
the PMSM, the operating point of the DFIG used for the simulations is Pout = 3.2MW.
For the reactive power generation, it has been assumed that the complete reactive power
transferred to the grid by the machine is through the stator winding. This means that it
has been assumed that the power electronic converter does not have any reactive power
interaction with the grid. As per the NC RfG discussed in Section 5.1 the outer limits for
the reactive power capabilities of the generator are −0.5Pmax and 0.65Pmax . For this study,
the condition Qout = −0.5Pmax is used.
Considering that the power electronic converter for a DFIG is partially rated (say, 25%
of rated power) the capacitance value has been chosen such that the energy stored in it
1
( CV 2 ) is a quarter of the energy stored in the PMSM capacitor under standard operating
2
conditions.
56
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
5.3.1
Performance without Protection
The response of the DFIG without any means of protection angainst LVRT is shown in
Figure 5.11.
1000
0.8
0.6
0.4
500
30
0
0.2
0
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−500
999
1003
1000
1001
Time (s)
1003
1004
28
1005
(b)
20
25
d−axis Current
q−axis Current
15
d−axis Current
q−axis Current
20
Rotor d−q Currents (kA)
10
5
0
−5
−10
−15
15
10
5
0
−5
−10
−20
−15
−25
−30
999
1002
Time (s)
(a)
Stator d−q Currents (kA)
Generator Speed (rad/s)
32
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
1000
1001
1002
1003
1004
−20
999
1005
1000
1001
Time (s)
1002
1003
1004
1005
Time (s)
(c)
(d)
1.8
15
15
10
10
5
5
0
0
−5
−5
−10
−10
1.7
1.5
1.4
1.3
1.2
Reactive Power (MVAr)
Active Power (MW)
DC−link Voltage (pu)
1.6
1.1
1
0.9
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−15
999
1003
Time (s)
1000
1001
1002
1003
1004
−15
1005
Time (s)
(e)
(f )
Figure 5.11 LVRT performance of DFIG without protection for a 95% Symmetric Voltage Dip
at t=1000s (a) Grid Voltage magnitude (b) Torque and Speed (c) Stator Currents in the Stator
Flux reference frame (d) Rotor Currents in the Stator Flux reference frame (e) DC link voltage
(f ) Active and Reactive Power to the grid
A number of issues with the performance of the machine during a Low Voltage event are
seen. First, as with the case of the PMSM, there is a substantial rise in the DC link voltage.
This however, may be easily controlled using methods discussed in the Section 5.2.2. Also,
the voltage disturbance causes large oscillatory currents in the stator as seen in Figure 5.11c.
Such currents, due to the magnetic coupling between the stator and the rotor, also flow in
the rotor circuit as shown in Figure 5.11d. This is further highlighted in the plots of stator
and rotor phase currents shown in Figure 5.12. The transients occur at both instances
of voltage change (i.e. voltage drop when fault in the system occurs and the subsequent
rise in voltage when the fault is cleared). These transient currents in the rotor circuit are
57
20
20
15
15
Rotor Phase Currents (kA)
Stator Phase Currents (kA)
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
10
5
0
−5
−10
−15
10
5
0
−5
−10
−15
−20
−25
999.5
1000
1000.5
1001
−20
999.5
1001.5
1000
1000.5
Time (s)
(a)
Figure 5.12
1001
1001.5
Time (s)
(b)
LVRT performance of DFIG without protection for a 95% Symmetric Voltage Dip
at t=1000s (a) Stator Phase Currents (b) Rotor Phase Currents
especially dangerous to the power electronics and hence must be contained. It is seen that
the maximum current magnitude in the rotor circuit is approximately 2.5 times the current
at rated operation. This is when the voltage limits on the power electronic converter is 1.25
times that required for rated operation.
One way to control the rotor currents would be by applying a sufficiently large rotor
voltage to control them. Figure 5.13 shows the response of the DFIG under the assumption
that the voltage rating of the power electronic converter is no longer a constraint. Such a
study may be used as a tool to compare the LVRT response of different machines. It can
0
1
0.8
0.6
0.4
−100
−150
−200
30
−250
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
−50
0.2
−300
0
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−350
999
1003
999.5
1000
1000.5
Time (s)
1001
1001.5
1002
1002.5
1003
Time (s)
(a)
(b)
6
−4.86
d−axis Current
q−axis Current
d−axis Current
q−axis Current
−4.88
Rotor d−q Currents (kA)
Stator d−q Currents (kA)
4
2
0
−2
−4.9
−4.92
−4.94
−4.96
−4.98
−5
−4
−5.02
−6
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−5.04
999
1003
Time (s)
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(c)
(d)
Figure 5.13 LVRT performance of DFIG without constraints on Voltage of Converter for a 95%
Symmetric Voltage Dip at t=1000s (a) Grid Voltage magnitude (b) Torque and Speed (c) Stator
Currents in the Stator Flux reference frame (d) Rotor Currents in the Stator Flux reference
frame
be seen that the rotor currents are indeed controlled. This is further highlighted in the
phase currents shown in Figure 5.14b. The magnitude of the excess rotor voltage required
58
6
6
4
4
Rotor Phase Currents (kA)
Stator Phase Currents (kA)
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
2
0
−2
−4
−6
999.5
2
0
−2
−4
1000
1000.5
1001
−6
999.5
1001.5
1000
1000.5
Time (s)
1001
1001.5
Time (s)
(a)
(b)
2000
Rotor Phase Voltages (V)
1500
1000
500
0
−500
−1000
−1500
−2000
999.5
1000
1000.5
1001
Time (s)
(c)
Figure 5.14 LVRT performance of DFIG without constraints on Voltage of Converter for a 95%
Symmetric Voltage Dip at t=1000s (a) Stator Phase Currents (b) Rotor Phase Currents (c)
Rotor Phase Voltages
to control the rotor currents during LVRT can be seen in Figure 5.14c. It can be seen that
the Volt-Ampere (VA) rating required of the power electronic converter is approximately
9.3 times that required for rated operation. The next section discusses methods to control
the issues of rotor transient currents.
59
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
5.3.2
Performance with Protection
A common method that is used to improve the LVRT performance of DFIG based wind
turbines is the use of a Crowbar [38]. The performance of a DFIG with crowbar protection
has been simulated while other methods have been briefly discussed in the following sections.
Crowbar
In the event of large rotor currents being generated during a Low Voltage event, it may be
necessary to by-pass the power electronic converter for its protection. The crowbar circuit is
a set of switches and resistors that short circuit the rotor terminals through the resistors in
the case of the Low Voltage event and disconnect the generator side converter. Figure 5.15
and Figure 5.16 show the LVRT response of the DFIG with a crowbar circuit implemented
such that the crowbar is triggered in the event that the rotor current exceeds the rated
current.
The connection of the crowbar circuit serves a number of purposes. First, it bypasses the
power electronic converter when large transient rotor currents are present. This protects
the converter in the event of a Low Voltage event. Second, the added resistance in the rotor
circuit lowers the magnitude of the transient rotor currents. Third, the added resistance
Lr
in the crowbar circuit reduces the time constant of the rotor circuit (τr =
),
Rr + Rcrowbar
this means that the currents during the fault will decay faster.
There are however a number of issues with the use of the crowbar circuit as well. First,
when the crowbar circuit is activated, the DFIG resembles a induction motor with an
external rotor resistance. This would mean that the machine would absorb reactive power
from the grid which may further exasperate the condition of the grid. Second, the vector
control is lost during the action of the crowbar circuit and re-establishing control after the
crowbar is released is a challenge.
60
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
1000
0.8
0.6
0.4
500
30
0
0.2
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−500
999
1003
1000
1001
Time (s)
(a)
1004
1005
20
d−axis Current
q−axis Current
d−axis Current
q−axis Current
15
Rotor d−q Currents (kA)
10
Stator d−q Currents (kA)
1003
(b)
15
5
0
−5
−10
−15
−20
−25
999
1002
Time (s)
10
5
0
−5
−10
−15
1000
1001
1002
1003
1004
−20
999
1005
1000
1001
Time (s)
1002
1003
1004
1005
Time (s)
(c)
(d)
20
20
1.4
15
15
10
10
5
5
0
0
−5
−5
−10
−10
Active Power (MW)
1.5
1.3
1.2
1.1
1
0.9
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−15
999
1003
Time (s)
1000
1001
1002
1003
1004
Reactive Power (MVAr)
0
999
DC−link Voltage (pu)
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
−15
1005
Time (s)
(e)
(f )
Figure 5.15 LVRT performance of DFIG with Crowbar protection for a 95% Symmetric Voltage
Dip at t=1000s (a) Grid Voltage magnitude (b) Torque and Speed (c) Stator Currents in the
Stator Flux reference frame (d) Rotor Currents in the Stator Flux reference frame (e) DC link
voltage (f ) Active and Reactive Power to the grid
Other Methods
A number of methods have been proposed to protect the DFIG during low voltage events.
Liang et al. proposed an additional control [43], i.e. injection of additional feed-forward
transient compensation terms, for the DFIG with crowbar. This controll scheme results in
minimum transient rotor currents and minimum crowbar usage. However, apart from this
still requiring the crowbar circuit, it requires additional computation effort [42].
Another solution employs a parallel grid side converter with a series grid side converter
to provide robust voltage disturbance ride through [42]. However, this method also requires
additional devices.
Most proposed methods either require the crowbar circuit or other additional circuits
(like the parallel grid side converter). Therefore, an additional circuit is required for the
protection of the DFIG.
61
20
20
15
15
Rotor Phase Currents (kA)
Stator Phase Currents (kA)
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
10
5
0
−5
−10
−15
−20
999.5
10
5
0
−5
−10
−15
1000
1000.5
1001
−20
999.5
1001.5
1000
1000.5
Time (s)
1001
1001.5
Time (s)
(a)
(b)
20
Rotor Phase Currents (kA)
15
10
5
0
−5
−10
−15
−20
999.5
1000
1000.5
1001
1001.5
Time (s)
(c)
Figure 5.16 LVRT performance of DFIG with Crowbar protection for a 95% Symmetric Voltage
Dip at t=1000s (a) Stator Phase Currents in the Stator Reference Frame (b) Rotor Currents in
the Stator Reference Frame (c) Crowbar Resistance Phase Currents
62
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
5.4
Simulation of the LVRT Performance for the B-DFIG
The schematic layout of a B-DFIG based wind turbine system is shown in Figure 5.17.
Figure 5.17
System Level B-DFIG Schematic
The B-DFIG suffers from the same issues faced by the DFIG during Low Voltage events.
When a low voltage occurs the magnitude of the Power Winding flux reduces, in this event
the rotor flux induces transient currents in the Power Winding circuit. This sets up transient
currents in the Control Winding circuit through the rotor circuit. Control Winding transient
currents occur when the controller is unable to match the induced voltages in this winding
caused by the rotor currents.
Section 5.4.1 describes the LVRT performance of the B-DFIG without protection while
Section 5.4.2 looks at methods to improve this performance. The machine used in these
simulations is the B-DFIG 3.2MW Case Study Machine detailed in Appendix F.
The operating conditions of the B-DFIG used for the simulations here are identical to
that of the DFIG. Therefore, Pout = 3.2MW and Qout = −0.5Pmax . As with the DFIG,
this operating point has been chosen as it defines the boundary of operation as defined in
the NC RfG. The capacitance value has been selected using the method described for the
DFIG.
63
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
5.4.1
Performance without Protection
The response of the B-DFIG without any protection against an Low Voltage event is shown
in Figure 5.18.
1000
0.8
0.6
0.4
500
39
0
38
0.2
0
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−500
999
1003
999.5
1000
1000.5
Time (s)
1001
1001.5
1002
37
1003
(b)
4
15
d−axis Current
q−axis Current
Control Winding d−q Currents (kA)
d−axis Current
q−axis Current
2
0
−2
−4
−6
−8
999
1002.5
Time (s)
(a)
Power Winding d−q Currents (kA)
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
999.5
1000
1000.5
1001
1001.5
1002
1002.5
10
5
0
−5
−10
−15
−20
999
1003
999.5
1000
1000.5
Time (s)
1001
1001.5
1002
1002.5
1003
Time (s)
(c)
(d)
1.1
1.08
20
20
10
10
0
0
1.04
1.02
1
0.98
−10
−10
Reactive Power (MVAr)
Active Power (MW)
DC−link Voltage (pu)
1.06
0.96
0.94
0.92
999
−20
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
999
Time (s)
−20
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(e)
(f )
Figure 5.18 LVRT performance of B-DFIG without protection for a 95% Symmetric Voltage
Dip at t=1000s (a) Grid Voltage magnitude (b) Torque and Speed (c) Power Winding Currents
in the PW reference frame (d) Control Winding Currents in the PW reference frame (e) DC link
voltage (f ) Active and Reactive Power to the grid
Large transient currents are seen in the Power Winding and Control Winding. These
are further highlighted in Figure 5.19. These transients occur at both the instant of voltage
dip (i.e. when fault occurs) and the subsequent voltage rise (i.e. when the fault is cleared).
The mechanism of transient current generation in the B-DFIG is similar to that of the
DFIG, however, there is an additional circuit (i.e. the control winding circuit). Therefore,
it can be expected that the Control Winding transients for the B-DFIG are lower than that
for the rotor in the DFIG. Another reason is the typically larger leakage inductances of the
B-DFIG. It is seen that the maximum current magnitude in the Control Winding circuit is
approximately 1.5 times the current required for rated operation when the voltage limits
64
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
15
Control Winding Phase Currents (kA)
Power Winding Phase Currents (kA)
6
4
2
0
−2
−4
−6
−8
999.5
1000
1000.5
1001
1001.5
1002
1002.5
10
5
0
−5
−10
−15
999.5
1003
1000
1000.5
1001
Time (s)
1001.5
1002
1002.5
1003
Time (s)
(a)
(b)
Figure 5.19 LVRT performance of B-DFIG without protection for a 95% Symmetric Voltage
Dip at t=1000s (a) Power Winding Phase Currents (b) Control Winding Phase Currents
on the power electronic converter is 1.25 times the voltage for rated operation.
As with the DFIG, the response of the B-DFIG under the assumption that the voltage on
the power electronic converter is not a constraint, is shown in Figure 5.20. The magnitude of
the Control Winding voltage required to control the currents is seen in Figure 5.20c. It can
be seen that the Volt-Ampere (VA) rating required for the converter is approximately 5.85
times that required for rated operation which is lower than that required for the DFIG. The
next section discusses methods to control the transient currents in the Control Winding.
15
Control Winding Phase Currents (kA)
Power Winding Phase Currents (kA)
6
4
2
0
−2
−4
−6
−8
999.5
1000
1000.5
1001
1001.5
1002
1002.5
10
5
0
−5
−10
−15
999.5
1003
1000
1000.5
1001
Time (s)
1001.5
1002
1002.5
1003
Time (s)
(a)
(b)
5000
Rotor Phase Voltages (V)
4000
3000
2000
1000
0
−1000
−2000
−3000
−4000
−5000
999.5
999.6
999.7
999.8
999.9
1000
1000.1
1000.2
1000.3
1000.4
1000.5
Time (s)
(c)
Figure 5.20 LVRT performance of B-DFIG without constraints on Voltage of Converter for a
95% Symmetric Voltage Dip at t=1000s (a) Power Winding Phase Currents (b) Control Winding
Phase Currents (c) Control Winding Phase Voltages
65
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
5.4.2
Performance with Protection
The issue of transient currents in the Control Winding circuit, which is dangerous for the
power electronic converter, has been highlighted in the previous section. This section looks
at some methods to combat this.
Crowbar
As with the DFIG, a crowbar circuit may be used to by-pass the power electronic converter
in the event of large transient rotor currents. Figure 5.21 shows the response of the B-DFIG
when crowbar protection is used.
1000
0.8
0.6
0.4
500
39
0
38
0.2
0
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−500
999
1003
999.5
1000
1000.5
Time (s)
1001
1001.5
1002
37
1003
(b)
4
15
d−axis Current
q−axis Current
Control Winding d−q Currents (kA)
d−axis Current
q−axis Current
2
0
−2
−4
−6
−8
999
1002.5
Time (s)
(a)
Power Winding d−q Currents (kA)
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
999.5
1000
1000.5
1001
1001.5
1002
1002.5
10
5
0
−5
−10
−15
−20
999
1003
999.5
1000
1000.5
Time (s)
1001
1001.5
1002
1002.5
1003
Time (s)
(c)
(d)
1.05
1.04
20
20
10
10
0
0
1.01
1
0.99
0.98
−10
−10
Reactive Power (MVAr)
1.02
Active Power (MW)
DC−link Voltage (pu)
1.03
0.97
0.96
0.95
999
−20
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
999
Time (s)
−20
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(e)
(f )
Figure 5.21 LVRT performance of B-DFIG with Crowbar protection for a 95% Symmetric
Voltage Dip at t=1000s (a) Grid Voltage magnitude (b) Torque and Speed (c) Stator Currents in
the Stator Flux reference frame (d) Rotor Currents in the Stator Flux reference frame (e) DC
link voltage (f ) Active and Reactive Power to the grid
66
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
Crowbarless Protection
The lower transient currents in the B-DFIG indicate the ability of controlling the Control
Winding currents without the use of external circuits such as the crowbar. This is done by
appropriately setting reference currents [44]. Transient currents in the control winding are
observed for the duration of disturbance in the Power Winding flux. Therefore, one option
would be to set these reference currents to zero for the duration of this disturbance. The
results with such a control strategy are shown in Figure 5.22 and Figure 5.23.
1000
0.8
0.6
0.4
500
39
0
38
0.2
0
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−500
999
1003
999.5
1000
1000.5
Time (s)
1001
1001.5
1002
37
1003
(b)
4
15
d−axis Current
q−axis Current
Control Winding d−q Currents (kA)
d−axis Current
q−axis Current
2
0
−2
−4
−6
−8
−10
999
1002.5
Time (s)
(a)
Power Winding d−q Currents (kA)
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
999.5
1000
1000.5
1001
1001.5
1002
1002.5
10
5
0
−5
−10
−15
−20
999
1003
999.5
1000
1000.5
Time (s)
1001
1001.5
1002
1002.5
1003
Time (s)
(c)
(d)
1.03
Active Power (MW)
DC−link Voltage (pu)
1.04
1.02
1.01
1
20
20
0
0
Reactive Power (MVAr)
1.05
0.99
−20
0.98
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
999
Time (s)
−20
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(e)
(f )
Figure 5.22 LVRT performance of B-DFIG with Crowbarless protection for a 95% Symmetric
Voltage Dip at t=1000s (a) Grid Voltage magnitude (b) Torque and Speed (c) Stator Currents in
the Stator Flux reference frame (d) Rotor Currents in the Stator Flux reference frame (e) DC
link voltage (f ) Active and Reactive Power to the grid
67
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
15
Control Winding Phase Currents (kA)
Power Winding Phase Currents (kA)
6
4
2
0
−2
−4
−6
−8
999.5
1000
1000.5
1001
1001.5
1002
1002.5
10
5
0
−5
−10
−15
999.5
1003
Time (s)
1000
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(a)
(b)
Figure 5.23 LVRT performance of B-DFIG with Crowbarless protection for a 95% Symmetric
Voltage Dip at t=1000s (a) Power Winding Phase Currents (b) Control Winding Phase Currents
It is seen that such a control strategy is successful in controlling the Control Winding
currents in a Low Voltage event and also controls torque oscillations in the machine. However, this strategy limits the ability of the machine to generate power immediately after the
clearance of the fault (at t = 1000.25 s in this case). This is not healthy for the grid.
68
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
Therefore, it is advantageous to control the machine such that the reference currents
are set to zero only for the duration of the Low Voltage event. The response in this case is
shown in Figure 5.24 and Figure 5.25. In this case the Control Winding current is limited
to approximately 1.1 times the current at rated operation.
1000
0.8
0.6
0.4
500
39
0
38
0.2
0
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−500
999
1003
999.5
1000
1000.5
Time (s)
1001
1001.5
1002
37
1003
(b)
4
15
d−axis Current
q−axis Current
Control Winding d−q Currents (kA)
d−axis Current
q−axis Current
2
0
−2
−4
−6
−8
−10
999
1002.5
Time (s)
(a)
Power Winding d−q Currents (kA)
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
999.5
1000
1000.5
1001
1001.5
1002
1002.5
10
5
0
−5
−10
−15
−20
999
1003
999.5
1000
1000.5
Time (s)
1001
1001.5
1002
1002.5
1003
Time (s)
(c)
(d)
1.05
Active Power (MW)
DC−link Voltage (pu)
1.03
1.02
1.01
1
10
0
0
−10
Reactive Power (MVAr)
20
1.04
0.99
−20
0.98
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
999
Time (s)
−20
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(e)
(f )
Figure 5.24 LVRT performance of B-DFIG with Crowbarless protection for a 95% Symmetric
Voltage Dip at t=1000s (a) Grid Voltage magnitude (b) Torque and Speed (c) Stator Currents in
the Stator Flux reference frame (d) Rotor Currents in the Stator Flux reference frame (e) DC
link voltage (f ) Active and Reactive Power to the grid
Chapter 4 has discussed the control of the B-DFIG. It is seen in Section 4.3 that the
reference signal id−ref
and iq−ref
are calculated on the basis of the d-axis Power Winding
p
p
flux (see Equation 4.19 and Equation 4.36). Therefore in the case of a Low Voltage event,
it is required to maintain the pre-fault reference values until oscillations in the flux are
minimised. The control during the Low Voltage event may therefore be expressed as the
following,
1. Detect Low Voltage event.
2. Set inner loop reference currents (i.e. id−ref
and iq−ref
) to zero.
c
c
69
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
15
Control Winding Phase Currents (kA)
Power Winding Phase Currents (kA)
6
4
2
0
−2
−4
−6
−8
999.5
1000
1000.5
1001
1001.5
1002
1002.5
10
5
0
−5
−10
−15
999.5
1003
1000
1000.5
1001
Time (s)
1001.5
1002
1002.5
1003
Time (s)
(a)
(b)
Figure 5.25 LVRT performance of B-DFIG with Crowbarless protection for a 95% Symmetric
Voltage Dip at t=1000s (a) Power Winding Phase Currents (b) Control Winding Phase Currents
3. Store the pre-fault current reference values and set post-fault reference to these values
until flux disturbance is minimised.
An advantage of such a control algorithm is the ability to inject reactive current to the
grid during Low Voltage events. This is done by setting the id−ref
to the rated value and
p
q−ref
ic
to zero. This is shown in Figure 5.26.
Ipd−rated
Qref
p
Ref. Gen.
idp
−
id−ref
p
idp
idc
−
PI
PI
icd−ref
−
idc
−
PI
PI
ud,ref
c
+
dq Cross-Coupling Compensator
ωm
Ref. Gen.
iq−ref
p
−
q−ref
PI
+ ic
−q
ic
i,q
p
0
−q
ic
Figure 5.26
q,ref
PI
− uc
PI
B-DFIG Control Scheme with LVRT Protection
The performance of the B-DFIG with such a control algorithm is shown in Figure 5.27.
Figure 5.28 shows the Power and Control Winding phase currents.
70
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
750
39
1
0.8
0.6
0.4
0.2
0
999
250
38
0
−250
37
−500
999.5
1000
1000.5
1001
1001.5
1002
1002.5
−750
999
1003
999.5
1000
1000.5
1001
Time (s)
1001.5
1002
36
1003
(b)
4
15
d−axis Current
q−axis Current
Control Winding d−q Currents (kA)
d−axis Current
q−axis Current
Power Winding d−q Currents (kA)
1002.5
Time (s)
(a)
2
0
−2
−4
−6
−8
−10
999
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
500
999.5
1000
1000.5
1001
1001.5
1002
1002.5
10
5
0
−5
−10
−15
−20
999
1003
999.5
1000
1000.5
1001
Time (s)
1001.5
1002
1002.5
1003
Time (s)
(c)
(d)
1.03
Active Power (MW)
DC−link Voltage (pu)
1.04
1.02
1.01
1
20
20
0
0
Reactive Power (MVAr)
1.05
0.99
−20
0.98
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
999
−20
999.5
1000
Time (s)
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(e)
(f )
Figure 5.27 LVRT performance of B-DFIG with Crowbarless protection and Reactive Current
Injection for a 95% Symmetric Voltage Dip at t=1000s (a) Grid Voltage magnitude (b) Torque
and Speed (c) Stator Currents in the Stator Flux reference frame (d) Rotor Currents in the
Stator Flux reference frame (e) DC link voltage (f ) Active and Reactive Power to the grid
15
Control Winding Phase Currents (kA)
Power Winding Phase Currents (kA)
6
4
2
0
−2
−4
−6
−8
999.5
1000
1000.5
1001
1001.5
1002
1002.5
10
5
0
−5
−10
−15
999.5
1003
Time (s)
1000
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(a)
(b)
Figure 5.28 LVRT performance of B-DFIG with Crowbarless protection and Reactive Current
Injection for a 95% Symmetric Voltage Dip at t=1000s (a) Power Winding Phase Currents (b)
Control Winding Phase Currents
71
CHAPTER 5. PERFORMANCE UNDER GRID EVENTS
5.5
Discussion
This chapter has discussed and compared the LVRT performance of the PMSM, the DFIG
and the B-DFIG based wind turbines.
The PMSM is not affected by a Low Voltage event because of the power electronic
converter that isolates the machine from the grid. The issue for such a wind turbine lies in
the voltage rise of the DC link due to a mismatch in the power generated and the power
transferred to the grid. A number of methods have been discussed to combat this issue
which give an adequate protection against the voltage rise. This wind turbine system has
the further advantage that it does not suffer from torque disturbances during voltage dips.
When the LVRT performance of the DFIG and the B-DFIG are compared, it is seen that
the performance of the B-DFIG is better, i.e. it has lower transient current magnitude as
well as lower torque disturbances. This is attributed to the higher leakage inductance of the
B-DFIG (in this case the leakage inductance of the B-DFIG is an order of magnitude higher
than that of the DFIG) as well as an additional Control Winding circuit. A comparison
between the DFIG and B-DFIG is shown in Table 5.2.
Table 5.2 Comparison of Maximum Current (without protection) for Winding Circuit
connected to Power Electronic Converter during a 95% Symmetric Low Voltage Dip
Machine
PMSM
DFIG
B-DFIG
without Protection
Crowbar
without Protection
Crowbar-less Protection
Maximum Current
Israted−operation
2.5 × Irrated−operation
2.2 × Irrated−operation
1.5 × Icrated−operation
1.1 × Icrated−operation
Requirement of Extra
Circuits for Protection
–
––
Crowbar based Protection
+
Algorithm based Protection
Here a (-) sign in the last column signifies that the machine has a disadvantage for
this criteria, i.e. an extra circuit is required for protection, while a (+) sign signifies that
the machine has an advantage for this criteria and does not require an extra circuit for
protection.
When protection is used, both the DFIG and the B-DFIG can handle the currents
generated in the winding connected to the power electronic converter. The DFIG does
this by bypassing the converter with the help of a crowbar circuit. The B-DFIG, however,
can achieve this without the use of an external circuit. The torque oscillations for the BDFIG are also lower than that in the DFIG. From the results obtained it can be concluded
that Crowbarless operation of the B-DFIG is possible during Low Voltage events. The
advantages of the LVRT performance of the B-DFIG over the DFIG is,
• The currents during the low voltage event are lower in the B-DFIG.
• The protection for the B-DFIG can be achieved with a control algorithm and does
not require an external protection circuit.
• The B-DFIG controller retains control of the machine through the low voltage event.
For the DFIG the controller is disconnected when the low voltage event is disconnected.
72
Chapter 6
Conclusion
6.1
Conclusions
This thesis work has looked at the modelling and control of the Brushless Doubly Fed
Induction Generator (B-DFIG). The controller developed for the machine is based on vector
control methods and has two current loops. It has been found that using the Power Winding
Active Power (Pp ) is a good way to control the output of the machine and a method
to generate a reference (Ppref ) signal for optimal operation is developed. This has been
done using the steady state characteristics and calculating the point at which the machine
has the highest efficiency for a given machine torque. The concept of ‘Cross-Coupling
Compensation’ for control has also been developed and implemented.
Further this thesis focussed on the performance of the machine under symmetric low
voltage events. This performance is compared with that of two other generators - the Permanent Magnet Synchronous Machine (PMSM) and the Doubly Fed Induction Generator
(DFIG). The concept of protection for these generators is also looked at. All the three
case study generators used are developed for a 3.2MW wind turbine drivetrain. Therefore,
they provide a good basis on which a comparison can be made. The dynamic model and
controller for all three types of machines have been developed for this thesis, however, only
the model and controller for the B-DFIG has been covered in detail.
It has been found that for the PMSM based wind turbine, the issue with Low Voltage
Ride Through (LVRT) is the rise in the DC link voltage . This is due to the mismatch in
power generated by the machine and the power transferred to the grid (which is limited
by the reduced voltage at the grid side converter terminals). Methods for controlling this
DC voltage rise, such as the use of an energy discharge circuit, have also been investigated.
Another method described is Power Balancing, where the power generated by the machine
is controlled to match the power transferred to the grid in the event of a low voltage event.
This method has been seen to control the issue of DC link voltage rise as well, however, it
introduces torque disturbances in the machine.
For the DFIG it has been found that voltage dips cause large transient currents in the
stator and rotor circuit. The over-currents in the rotor circuit are a cause for concern
as they may lead to adverse effects on the power electronic converter connected to the
circuit. A possible method to overcome this issue - the use of a Crowbar circuit - has been
investigated. The crowbar circuit manages the problem by bypassing the power electronic
converter in the event of current rise. Additionally, the crowbar resistors reduce the time
73
CHAPTER 6. CONCLUSION
constant of the rotor circuit, allowing for a faster decay of transient currents.
In the case of the B-DFIG, it has been found that again transient currents are set up
in the power and control windings in the event of a voltage dip. However, the magnitude
of these currents is lower when compared to that in a DFIG. This is attributed to the
higher leakage inductance of the B-DFIG. A Crowbarless method has also been studied
which is successful in controlling the high currents generated in the control winding. This
Crowbarless control method also reduces the torque oscillations observed in the B-DFIG
for a low voltage event.
In conclusion, apart from offering better reliability through the exclusion of slip ring
and brushes, the B-DFIG also has an improved LVRT performance when compared with
the DFIG. The protection is also simpler and does not require an external circuit, like the
crowbar, but can be built into the control algorithm of the machine controller.
6.2
Future Work
Over the course of this thesis a number of possible avenues of further investigation have
been identified. First, the response of the machine with the controller developed for normal
operation as well as under low voltage events need to be practically validated.
Second, this study has been conducted on a machine whose design has not been optimised. Although, it is not expected that there would be a large variation in the nature of
the results presented, it would be beneficial if this study is extended to such an optimised
machine.
Third, in this study the complete reactive power interaction is considered from the power
winding of the machine. In reality, there is a possibility to use the grid side converter for
the transfer of reactive power to and from the grid as well. This aspect and its effect on
the machine response should be looked into.
Finally, this study has focussed only on symmetric voltage dips. It would be interesting
to study the effect of asymmetric dips (such as those caused by single line or double line
faults) on the machine.
74
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78
Appendices
79
Appendix A
Additional LVRT Performance
Simulations
Chapter 5 has investigated the performance of the PMSM, DFIG and B-DFIG based machines for the ‘Case A’ Symmetric Voltage Dip. In this section performance of these
machines far a symmetric voltage dip as shown in Figure A.1 is discussed.
V oltage (pu)
1.0
0.25
Figure A.1
A.1
T ime (s)
3.0
Voltage profile for Case B of the LVRT response study
LVRT Performance of the PMSM
In this section the performance of the PMSM machine with and without protection as
discussed in Chapter 5 is shown.
81
APPENDIX A. ADDITIONAL LVRT PERFORMANCE SIMULATIONS
A.1.1
Performance without Protection
Figure A.2 shows the performance of the PMSM machine when faced with a 95% symmetric
voltage dip as shown in Figure A.1.
0
32
−50
31
−100
30
−150
29
0.8
0.6
0.4
0.2
0
999
1000
1001
1002
1003
1004
−200
999
1005
1000
1001
1002
Time (s)
1004
28
1005
(b)
50
0.2
d−axis Current
q−axis Current
0
d−axis Current
q−axis Current
40
30
Grid d−q Currents (kA)
−0.2
−0.4
−0.6
−0.8
−1
−1.2
20
10
0
−10
−20
−30
−1.4
−40
−1.6
999
1003
Time (s)
(a)
Generator d−q Currents (kA)
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
1000
1001
1002
1003
1004
−50
999
1005
1000
1001
1002
Time (s)
Time (s)
(c)
1003
1004
1005
(d)
3.5
DC−link Voltage (pu)
3
2.5
2
1.5
1
0.5
999
1000
1001
1002
1003
1004
1005
Time (s)
(e)
Figure A.2 LVRT performance of PMSM without Protection (a) Grid Voltage magnitude (b)
Torque and Speed (c) Generator Currents in the PW reference frame (d) Grid Currents in the
grid reference frame (e) DC link voltage
A.1.2
Performance with Protection
In this section the performance of the PMSM with protection against LVRT is shown. The
protection methods used in Chapter 5 are covered.
82
APPENDIX A. ADDITIONAL LVRT PERFORMANCE SIMULATIONS
Energy Discharge Circuit
The LVRT performance of the PMSM with an Energy Discharge circuit is shown in Figure A.3.
0
32
−50
31
−100
30
−150
29
0.8
0.6
0.4
0.2
0
999
1000
1001
1002
1003
1004
−200
999
1005
1000
1001
Time (s)
1002
1003
1004
28
1005
Time (s)
(a)
(b)
20
0.2
d−axis Current
q−axis Current
0
d−axis Current
q−axis Current
15
−0.2
Grid d−q Currents (kA)
Generator d−q Currents (kA)
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
−0.4
−0.6
−0.8
−1
−1.2
10
5
0
−5
−10
−1.4
−15
−1.6
999
1000
1001
1002
1003
1004
−20
999
1005
1000
1001
Time (s)
(c)
1004
1005
1003
1004
1005
2.5
Braking Resistor Current (kA)
1.01
DC−link Voltage (pu)
1003
(d)
1.015
1.005
1
0.995
0.99
0.985
999
1002
Time (s)
1000
1001
1002
1003
1004
2
1.5
1
0.5
0
999
1005
Time (s)
1000
1001
1002
Time (s)
(e)
(f )
Figure A.3 LVRT performance of PMSM with Energy Discharge Circuit (a) Grid Voltage
magnitude (b) Torque and Speed (c) Generator Currents in the PW reference frame (d) Grid
Currents in the grid reference frame (e) DC link voltage (f ) Energy Discharge Circuit - Resistor
Current
Power Balancing
Figure A.4 shows the performance of the PMSM when protectied with the Power Balancing
Algorithm.
83
APPENDIX A. ADDITIONAL LVRT PERFORMANCE SIMULATIONS
0
32
−50
31
−100
30
−150
29
0.8
0.6
0.4
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
0.2
0
999
1000
1001
1002
1003
1004
−200
999
1005
1000
1001
1002
Time (s)
1003
1004
28
1005
Time (s)
(a)
(b)
10
d−axis Current
q−axis Current
d−axis Current
q−axis Current
5
Grid d−q Currents (kA)
Generator d−q Currents (kA)
0
−0.5
−1
0
−5
−10
−1.5
−15
−2
999
1000
1001
1002
1003
1004
−20
999
1005
1000
1001
1002
Time (s)
Time (s)
(c)
1003
1004
1005
(d)
1.04
1.03
DC−link Voltage (pu)
1.02
1.01
1
0.99
0.98
0.97
0.96
0.95
999
1000
1001
1002
1003
1004
1005
Time (s)
(e)
Figure A.4 LVRT performance of PMSM with Power Balancing (a) Grid Voltage magnitude (b)
Torque and Speed (c) Generator Currents in the PW reference frame (d) Grid Currents in the
grid reference frame (e) DC link voltage
A.2
LVRT Performance of the DFIG
This section shows the performance of the DFIG for a symmetric voltage drop as shown in
Figure A.1.
A.2.1
Performance without Protection
Figure A.6 shows the performance of the DFIG without protection.
84
APPENDIX A. ADDITIONAL LVRT PERFORMANCE SIMULATIONS
1000
32.5
0.8
0.6
0.4
500
30.5
29.5
0
28.5
0.2
0
999
1000
1001
1002
1003
1004
−500
999
1005
1000
1001
Time (s)
27.5
1005
d−axis Current
q−axis Current
15
Rotor d−q Currents (kA)
Stator d−q Currents (kA)
1004
20
d−axis Current
q−axis Current
10
5
0
−5
−10
−15
−20
10
5
0
−5
−10
−15
1000
1001
1002
1003
1004
−20
999
1005
1000
1001
Time (s)
1002
1003
1004
1005
Time (s)
(c)
(d)
15
20
10
15
Rotor Phase Currents (kA)
Stator Phase Currents (kA)
1003
(b)
15
5
0
−5
−10
−15
−20
999.5
1002
Time (s)
(a)
−25
999
Generator Speed (rad/s)
31.5
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
10
5
0
−5
−10
−15
1000
1000.5
1001
1001.5
−20
999.5
Time (s)
1000
1000.5
1001
1001.5
Time (s)
(e)
(f )
Figure A.5 LVRT performance of DFIG without Protection (a) Grid Voltage magnitude (b)
Torque and Speed (c) Stator Currents in the Stator Flux Reference Frame (d) Rotor Currents in
the Stator Flux Reference Frame (e) Stator Phase Currents (f ) Rotor Phase Currents
A.2.2
Performance with Crowbar Protection
Figure A.6 shows the performance of the DFIG when a Crowbar circuit as defined in
Section 5.3.2 is used.
85
APPENDIX A. ADDITIONAL LVRT PERFORMANCE SIMULATIONS
1000
32.5
0.8
0.6
0.4
500
30.5
29.5
0
28.5
0.2
0
999
1000
1001
1002
1003
1004
−500
999
1005
1000
1001
1002
Time (s)
1003
1004
(b)
15
10
d−axis Current
q−axis Current
d−axis Current
q−axis Current
10
5
Rotor d−q Currents (kA)
Stator d−q Currents (kA)
27.5
1005
Time (s)
(a)
5
0
−5
0
−5
−10
−10
−15
999
1000
1001
1002
1003
1004
−15
999
1005
1000
1001
1002
Time (s)
1003
1004
1005
Time (s)
(c)
(d)
15
15
10
10
Rotor Phase Currents (kA)
Stator Phase Currents (kA)
Generator Speed (rad/s)
31.5
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
1
5
0
−5
−10
5
0
−5
−10
−15
999.5
1000
1000.5
1001
−15
999.5
1001.5
1000
1000.5
Time (s)
1001
1001.5
Time (s)
(e)
(f )
15
Rotor Phase Currents (kA)
10
5
0
−5
−10
−15
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
1003.5
1004
1004.5
1005
Time (s)
(g)
Figure A.6 LVRT performance of DFIG without Protection (a) Grid Voltage magnitude (b)
Torque and Speed (c) Stator Currents in the Stator Flux Reference Frame (d) Rotor Currents in
the Stator Flux Reference Frame (e) Stator Phase Currents (f ) Rotor Phase Currents (g)
Crowbar Phase Currents
A.3
LVRT Performance of the B-DFIG
This section charts the performance of the B-DFIG with and without protection for voltage
dips as shown in Figure A.1
86
APPENDIX A. ADDITIONAL LVRT PERFORMANCE SIMULATIONS
A.3.1
Performance without Protection
Performance of the B-DFIG without protection is shown in Figure A.7.
750
39
1
0.6
0.4
0.2
250
38
0
−250
37
−500
0
999
1000
1001
1002
1003
1004
−750
999
1005
1000
1001
Time (s)
1002
1003
(b)
5
15
d−axis Current
q−axis Current
Control Winding d−q Currents (kA)
Power Winding d−q Currents (kA)
d−axis Current
q−axis Current
0
−5
−10
999
1000
1001
1002
1003
1004
10
5
0
−5
−10
−15
−20
999
1005
1000
1001
Time (s)
1002
1003
1004
1005
Time (s)
(c)
(d)
15
Control Winding Phase Currents (kA)
6
Power Winding Phase Currents (kA)
36
1005
1004
Time (s)
(a)
4
2
0
−2
−4
−6
−8
999.5
Generator Speed (rad/s)
Generator Torque (kNm)
Grid Voltage
in Grid Reference Frame (pu)
500
0.8
1000
1000.5
1001
1001.5
1002
1002.5
1003
1003.5
1004
1004.5
10
5
0
−5
−10
−15
999.5
1005
Time (s)
1000
1000.5
1001
1001.5
1002
1002.5
1003
1003.5
1004
1004.5
1005
Time (s)
(e)
(f )
Figure A.7 LVRT performance of B-DFIG without Protection (a) Grid Voltage magnitude (b)
Torque and Speed (c) Power Winding Currents in the PW Reference Frame (d) Control Winding
Currents in the PW Reference Frame (e) Stator Phase Currents (f ) Rotor Phase Currents
87
APPENDIX A. ADDITIONAL LVRT PERFORMANCE SIMULATIONS
88
Appendix B
B-DFIG Analytical Parameter
Calculation
This appendix deals with the calculation of some parameters of the Brushless Doubly Fed
Machine.
B.1
Mutual Inductance - Power and Control Windings
The flux linkage between the two windings is,
∞ X
∞ Z
X
λsaA =
2π
φpakp (αm )ZcAkc (αm )rdαm
(B.1)
kp =1 kc =1 0
where αm is the mechanical angle along the stator, φpakp is the expression for the kp
harmonic flux produced by a current in the 0 a0 phase of the Power Winding and ZcAkc is
the kc harmonic conductor distribution of the 0 a0 phase of the Control Winding. φpa and
ZcA are of the form,
lrB̂ pa
sin(kpp αm + pp βkpa )
kpp
Nsek
=
sin(kpc αm )
2r
φpak =
ZcAk
(B.2)
Therefore,
λsaA =
∞ X
∞ Z
X
kp =1 kc =1 0
2π
ls rNsekp B̂ pa
sin(kp pp αm + pp βkp pa ) sin(kc pc αm )
2kp pp
(B.3)
This will have a value of zero ∀kp pp 6= kc pc . Therefore, the two stator windings are not
mutually coupled except for the case of certain harmonic frequencies depending on the
number of poles of each winding. Also, the pole number of the two windings can be chosen
such that there is no coupling for these windings for any harmonic number.
89
APPENDIX B. B-DFIG ANALYTICAL PARAMETER CALCULATION
Figure B.1
B.2
Rotor Structure with Nn nests of 3 loops each
Mutual Inductance between Rotor loops
The rotor structure consists of a set of Nn nests each with a set of Nl loops. This is shown
in figure B.1. To calculate the mutual inductance between two rotor loops first we define
the conductor distribution of the rotor loops. This distribution can be represented in the
form of the dirac delta function. This distribution will be further broken down into a set
of sinusoidally distributed windings using Fourier Analysis.
Figure B.2 gives the distribution of the rotor conductors in space. This is used to
calculate the equivalent fourier distribution. Because the conductor distribution of the
Figure B.2
Rotor Loops in the Rotor Space
90
APPENDIX B. B-DFIG ANALYTICAL PARAMETER CALCULATION
rotor loops is odd, an of the fourier distribution will be zero. The value of bn is given by,
Z π
2Nn Nn
bn =
δ(α − γ) sin(nNn α)dα
π 0
Z π
2Nn Nn
1
(B.4)
bn =
lim sin(nNn α)dα
π 0 →0 Z γ+
1
2Nn
bn =
lim
sin(nNn α)dα
→0
π
γ
Therefore,
2Nn
sin(nNn γ)
π
The conductor distribution is now given as,
bn =
Zj (α) =
∞
X
2Nn Zc
sin(kNn γj ) sin(kNn α)
π
k=1
(B.5)
(B.6)
where Zc is the number of rotor conductors per slot. The MMF due to a current flowing
through the loop j is given by,
Fj (α) = ij
Z
α+ Nπ
n
Zj dα
α
Fj (α) =
Fj (α) =
∞
X
2Nn Zc
k=1
∞
X
k=1
π
sin(kNn γj )
Z
α+ Nπ
n
sin(kNn α)dα
α
(B.7)
4ij Zc
sin(kNn γj ) cos(kNn α)
kπ
From Ampere’s Law and equation B.7 we get,
Bj (α) =
∞
X
2µ0 ij Zc
k=1
gkπ
sin(kNn γj ) cos(kNn α)
(B.8)
where g is the effective air-gap length. The flux linkage between the loops j and m (i.e.
λjm ) is now given by,
λjm =
Z 2π
∞ X
∞
X
4µ0 Nn Zcj Zcm ij ls r
sin(k
N
γ
)
sin(k
N
γ
)
sin(kj Nn α) sin(km Nn α)dα
j n j
m n m
gkj π 2
0
kj =1 km =1
(B.9)
When the two loops are displaced from a common reference point by the mechanical angles
βj and βm , λjm can be written as,
λjm
∞ X
∞
X
4µ0 Nn Zcj Zcm ij ls r
=
sin(kj Nn γj ) sin(km Nn γm )
gkj π 2
kj =1 km =1
Z 2π
sin(kj Nn α − Nn βj ) sin(km Nn α − Nn βm )dα
0
91
(B.10)
APPENDIX B. B-DFIG ANALYTICAL PARAMETER CALCULATION
This will have a non-zero value only if kj = km which will be given by,
λjm =
∞
X
4µ0 Zcj Zcm ij ls r
k=1
gk 2 π
sin(kNn γj ) sin(kNn γm ) cos(Nn βj − Nn βm )
(B.11)
Therefore, the mutual inductance is given by,
Mjm =
∞
X
4µ0 Zcj Zcm ls r
k=1
B.3
gk 2 π
sin(kNn γj ) sin(kNn γm ) cos(Nn βj − Nn βm )
(B.12)
Mutual Inductance between Rotor and Stator
The flux due to a current in the rotor loop is given by,
φr (α) =
∞
X
2µ0 Zc ir ls r
k=1
gkn π
sin(kNn γr ) sin(kNn α − βr )
(B.13)
where βr is the angle between the rotor loop and an arbitrary frame of reference. The
conductor distribution for the stator coils is given by,
Zs (α) =
∞
X
Nse
k=1
2r
sin(kpα − pβs )
(B.14)
where Nse is the effective number of winding turns and βs is the angle between the stator
winding and the arbitrary frame of reference. The flux linkage is given by,
λsr
Z 2π
∞ X
∞
X
µ0 ir ls rNse Zc
=
sin(kr Nn γr )
sin(kr Nn α − pβr ) sin(ks pα − pβs )dα (B.15)
gkr π
0
kr =1 ks =1
which will have a non zero value only when kr Nn = ks p which is given by,
λsr =
∞
X
µ0 ir ls rNse Zc
k=1
where k = (2n − 1)
gk 2
sin(kNn γr ) cos(pβr − pβs )
(B.16)
Nn
where k, nN . Therefore, the mutual inductance is given by,
p
Msr =
∞
X
µ0 ls rNse Zc
k=1
gk 2
sin(kNn γr ) cos(pβr − pβs )
92
(B.17)
Appendix C
B-DFIG Simple Case Study
Machine
This appendix gives the overview of the case study machine that has been used in the
preliminary development of the machine and its control. It is based on a single loop rotor
design for simplicity and neglects all saturation. The electric circuit parameters are derived
from the discussion in Appendix B.
Table C.1
B-DFIG Design
Number of Pole Pairs (PW & CW)
Rated Electric Frequency (PW & CW)
Rated Generator Speed
Stator Outer Radius
Stack Length
Rotor Inner Radius
Air-Gap Length
Ratio Length-Diameter
Ratio Slot height-Slot width
Stator Yoke Width
Stator Slot Width
Table C.2
Hz
rpm
m
m
m
mm
mm
mm
4&6
50 & 20
275
0.83
1.57
0.58
1.5
0.96
2.35
80
29.7
Winding Parameters
Power winding pole-pairs
Power winding slots/pole/phase
Power winding no. of turns per coil
Power winding pole-pitch
Power winding coil-pitch
Control winding pole-pairs
Control winding slots/pole/phase
Control winding no. of turns per coil
Control winding pole-pitch
Control winding coil-pitch
93
pp
qp
np
ypp
ypc
pc
qc
nc
ycp
ycc
4
3
1
9
9
6
2
1
6
6
APPENDIX C. B-DFIG SIMPLE CASE STUDY MACHINE
94
APPENDIX C. B-DFIG SIMPLE CASE STUDY MACHINE
Table C.3
Electrical Circuit Parameters
Power winding resistance
Power winding main inductivity (main harmonic)
Power winding leakage inductivity
Control winding resistance
Control winding main inductivity (main harmonic)
Control winding leakage inductivity
Rotor nest loop-1 resistance
Rotor nest loop-1 (pp main harmonic) inductivity
Rotor nest loop-1 (pc main harmonic) inductivity
Rotor nest leakage inductivity
95
Rp
Lpm(p)
Lpσ
Rc
Lcm(c)
Lcσ
Rr
Lrm(p)
Lrm(c)
Lrσ
H
mH
H
mH
m
mH
mH
mH
0.0023
0.0116
0.1447
0.0021
0.0052
0.1447
0.0296
0.1
0.0843
0.0068
APPENDIX C. B-DFIG SIMPLE CASE STUDY MACHINE
96
Appendix D
PMSM 3.2MW Case Study
Machine
This appendix describes the details of the PMSM generator. The design of this generator
has been based on a number of optimisation variables, including stack length, stator inner
radius and stator outer radius, for the REpower 3.2M114 wind turbine. This design and
optimisation is not within the scope of this thesis.
D.1
Machine Details
Table D.1
PMSM Design
Number of Pole Pairs
Rated Electric Frequency
Rated Generator Speed
Stator Outer Radius
Stack Length
Rotor Inner Radius
Air-Gap Length
Ratio Length-Diameter
Ratio Slot height-Slot width
Stator Yoke Width
Stator Slot Width
Table D.2
Stator
Stator
Stator
Stator
winding
winding
winding
winding
Hz
rpm
m
m
m
mm
mm
mm
11
51.02
278.3
0.85
0.63
0.66
2.58
0.37
1.91
52.07
34.71
Winding Parameters
slots/pole/phase
number of turns per coil
pole-pitch
coil-pitch
97
q
n
yp
ypc
1
1
3
3
APPENDIX D. PMSM 3.2MW CASE STUDY MACHINE
Table D.3
Electrical Circuit Parameters
Stator voltage
Frequency
Stator nominal current
Stator resistance
Stator main inductivity (main harmonic)
Stator leakage inductivity
0
V
Hz
A
m
mH
mH
205
50
5969.9
0.32518
0.0081
1.9000E-03
−0.8475
Figure D.1
D.2
U snom
fe
Isnom
Rs
Lsm
Lsσ
Geometric Plot of the PMSM Case Study Machine
Machine Dynamic Equations in Rotor Flux Reference
Frame
ūrs = Rs īrs + pωm
r,d
λr,d
s = Ls is + ψr
λr,q
s
=
T =
0 −1 r dλ̄rs
λ̄s +
1 0
dt
Ls ir,q
s
pψr ir,q
s
(D.1)
(D.2)
(D.3)
(D.4)
98
Appendix E
DFIG 3.2MW Case Study Machine
E.1
Machine Details
This appendix describes the details of the DFIG generator. The design of this generator
has been based on a number of optimisation variables, including stack length, stator inner
radius and stator outer radius, for the REpower 3.2M114 wind turbine. This design and
optimisation is not within the scope of this thesis.
Table E.1
DFIG Design
Number of Pole Pairs
Rated Electric Frequency
Rated Generator Speed
Stator Outer Radius
Stack Length
Rotor Inner Radius
Air-Gap Length
Ratio Length-Diameter
Ratio Slot height-Slot width
Stator Yoke Width
Stator Slot Width
Table E.2
Stator
Stator
Stator
Stator
Hz
rpm
m
m
m
mm
mm
mm
13
50.30
278.30
0.85
0.97
0.66
2.58
0.57
1.15
46.52
31.01
Winding Parameters
slots/pole/phase
number of turns per coil
pole-pitch
coil-pitch
99
q
n
ysp
ysc
1
1
3
3
APPENDIX E. DFIG 3.2MW CASE STUDY MACHINE
Table E.3
Electrical Circuit Parameters
Stator voltage
Stator frequency
Stator nominal current
Rotor nominal current
Stator resistance
Stator main inductivity (main harmonic)
Stator leakage inductivity
Rotor resistance
Rotor main inductivity (main harmonic)
Rotor leakage inductivity
U snom
f se
Isnom
Irnom
Rs
Lsm
Lsσ
Rr
Lrm
Lrσ
V
Hz
A
A
mH
mH
m
mH
mH
333
50
2880.7
3256
0.0011
0.487
0.0495
0.911
0.4859
0.0588
0
−0.8522
Figure E.1
Geometric Plot of the DFIG Case Study Machine
100
APPENDIX E. DFIG 3.2MW CASE STUDY MACHINE
E.2
Machine Dynamic Equations in Arbitrary Reference Frame
0 −1 k dλ̄ks
=
+ (pωm + ωk )
λ̄s +
1 0
dt
k
0 −1 k dλ̄r
ūkr = Rr īkr + ωk
λ̄r +
1 0
dt
1
(Lr λ̄ks − Msr λ̄kr )
īks =
2
Ls Lr − Msr
1
īkr =
(−Msr λ̄ks + Ls λ̄kr )
2
Ls Lr − Msr
ūks
Rs īks
k,d
k,d k,q
T = pMsr (ik,q
s ir − is ir )
101
(E.1)
(E.2)
(E.3)
(E.4)
(E.5)
APPENDIX E. DFIG 3.2MW CASE STUDY MACHINE
102
Appendix F
B-DFIG 3.2MW Case Study
Machine
This appendix describes the details of the B-DFIG generator, based on a 4-rotor loop rotor
design. The design of this generator has been based on a number of optimisation variables,
including stack length, stator inner radius and stator outer radius, for the REpower 3.2M114
wind turbine. This design and optimisation is not within the scope of this thesis.
Table F.1
B-DFIG Design
Number of Pole Pairs (PW & CW)
Rated Electric Frequency (PW & CW)
Rated Generator Speed
Stator Outer Radius
Stack Length
Rotor Inner Radius
Air-Gap Length
Ratio Length-Diameter
Ratio Slot height-Slot width
Stator Yoke Width
Stator Slot Width
Table F.2
Hz
rpm
m
m
m
mm
mm
mm
4&6
50 & 20
275
0.83
1.57
0.58
2.58
0.95
2.35
80
29.7
Winding Parameters
Power winding slots/pole/phase
Power winding number of turns per coil
Power winding pole-pitch
Power winding coil-pitch
Control winding slots/pole/phase
Control winding number of turns per coil
Control windig pole-pitch
Control winding coil-pitch
103
qp
np
ypp
ypc
qc
nc
ycp
ycc
3
1
9
9
2
1
6
6
APPENDIX F. B-DFIG 3.2MW CASE STUDY MACHINE
Table F.3
Electrical Circuit Parameters
Power winding resistance
Power winding main inductivity (main harmonic)
Power winding leakage inductivity
Control winding resistance
Control winding main inductivity (main harmonic)
Control winding leakage inductivity
Rotor nest loop-1 resistance
Rotor nest loop-2 resistance
Rotor nest loop-3 resistance
Rotor nest loop-4 resistance
Rotor nest loop-1 (pp main harmonic) inductivity
Rotor nest loop-1 (pc main harmonic) inductivity
Rotor nest loop-2 (pp main harmonic) inductivity
Rotor nest loop-2 (pc main harmonic) inductivity
Rotor nest loop-3 (pp main harmonic) inductivity
Rotor nest loop-3 (pc main harmonic) inductivity
Rotor nest loop-4 (pp main harmonic) inductivity
Rotor nest loop-4 (pc main harmonic) inductivity
Rotor nest leakage inductivity
Rp
Lpm(p)
Lpσ
Rc
Lcm(c)
Lcσ
Rr1
Rr2
Rr3
Rr4
Lrm(p)1
Lrm(c)1
Lrm(p)2
Lrm(c)2
Lrm(p)3
Lrm(c)3
Lrm(p)4
Lrm(c)4
Lrσ
H
mH
H
mH
m
m
m
m
mH
mH
mH
mH
mH
mH
mH
mH
mH
0.0023
0.0073
0.1613
0.0021
0.0033
0.1613
0.1168
0.1108
0.1047
0.0986
0.4704
0.2482
0.3733
0.23
0.2397
0.1617
0.08259
0.0581
0.3865
0
−0.83
Figure F.1
Geometric Plot of the B-DFIG 3.2MW Case Study Machine
104
LVRT Performance of Brushless Doubly Fed
Induction Machines
U. Shipurkar, T. Strous, H. Polinder
Abstract—The Brushless Doubly Fed Induction Machine (BDFIM) shows promise for use in wind turbine drivetrains. This
paper discusses the performance of this machine under symmetric low voltage dips and compares this with the performance
of two other machines - the Permanent Magnet Synchronous
Machine (PMSM) and the Doubly-Fed Induction Generator
(DFIG). Attention is paid to the controller for the B-DFIM and
protection methods for improved Low Voltage Ride Through
(LVRT) performance are discussed.
Index Terms—Brushless Doubly-Fed Machine (BDFM),
DFIG, Cross coupling, LVRT.
dλ̄sc
dt
d
λ¯r
0 = Rr i¯rr + r
dt
where, the flux linkages are given by,
ūsc = Rc i¯sc +
(1)
λ¯sp = Lp i¯sp + Mpr i¯sr
λ¯sc = Lc i¯sc − Mcr i¯sr
λ¯r = Lr i¯r + Mpr i¯r − Mcr i¯r
r
r
p
c
(2)
(3)
(4)
These are transformed to a common arbitrary reference frame
rotating with an angular velocity ‘ωk ’ with respect to the
I. I NTRODUCTION
Power Winding stator reference frame. This is given by,
ITH growing interest in sustainable forms of energy,
k
the wind industry is growing rapidly. The DFIG is
0 −1 ¯k dλ̄p
k
k
ūp = Rp īp + ωk
λp +
(5)
a popular choice for the wind turbine drivetrain because it
1 0
dt
is cost effective. However, it suffers from reliability and
0 −1 ¯k dλ̄kc
maintenance issues due to the slip rings and brushes it
ūkc = Rc īkc + (ωk − (pp + pc )ωm )
λc +
1 0
dt
requires. The B-DFIM aims at addressing these drawbacks.
(6)
With increased wind power penetration, it is no longer
k
0 −1 ¯k dλ̄r
acceptable for wind turbines to trip during grid disturbances.
0 = Rr īkr + (ωk − pp ωm )
(7)
λp +
1 0
dt
Therefore, the study of the performance of wind turbine
drivetains under low voltage events is important. There has The equations that form the dynamic model, rewritten in
been little research on the LVRT performance of the B-DFIG. state-space form, are given by,
Shao et al. studied the dynamic behaviour of the machine
dλkp
0 −1 ¯k
during symmetrical voltage dips [1]. However, this study was
k
k
= ūp − Rp īp − ωk
λp
(8)
1 0
limited as it did not consider the subsequent voltage rise of
dt
the grid and it did not propose any methods to improve the
dλkc
0 −1 ¯k
= ūkc − Rc īkc − (ωk − (pp + pc )ωm )
λc
performance. In 2011 they proposed a control scheme that
1 0
dt
gives the B-DFIG the capability to ride through low voltage
(9)
faults [2] and this was extended for asymmetric low voltage
k
dλr
0 −1 ¯k
faults [3].
= −Rr īkr − (ωk − pp ωm )
λr
(10)
1 0
dt
This paper develops a controller for the B-DFIM as part

  k 
 k
īp
λ̄p
Lp
0
Mpr
of a wind turbine drivetrain. This controller is based on
īkc  = inv  0
Lc
−Mcr  λ̄kc 
(11)
vector control and consists of cascaded current control loops
Mpr −Mcr
Lr
λ̄kr
īkr
- one for the power winding current and the inner loop for
the control winding current. The paper also investigates the The model is completed with the expression for electrical
performance of the B-DFIM under symmetric low voltage power at the terminals of a stator winding. This is given by,
events and compares this with the performance of PMSM
and DFIG based drivetrains.
ps = ūTs,abc īs,abc
(12)
Section II develops the dynamic model of the B-DFIG.
Section III develops the controller for the B-DFIG which using the Clark’s transform,
−1
−1
is based on vector control. Section IV discusses the LVRT
ps =(Cαβ,abc
ūs,αβ )T (Cαβ,abc
īs,αβ )
(13)
performance of the PMSM, the DFIG and the B-DFIG.
s,α s,α
s,β s,β
ps =us is + us is
(14)
W
II. B-DFIM DYNAMIC M ODEL
A. B-DFIM Dynamic Equations
The voltage equations for the B-DFIG, in vector form, are
given by,
ūsp = Rp i¯sp +
dλ̄sp
dt
s
=īsT
s ūs
(15)
Using Equation 8, Equation 9 and Equation 15 the power
may be expressed as,
sT s
sT 0 −1 s dθ
ps =Rs īs īs + (Ls,σ + Ls,m )īs
ī p
1 0 s dt
0 −1 s dθ
+ Msr īsT
ī p
(16)
s
1 0 r dt
It can be seen that the first term represents the resistance
loss in the winding, the second term will be equal to zero
and the remaining term represents the power converted into
mechanical power. The torque is given by,
pm = Te
dθ
dt
(17)
Therefore, the electromagnetic torque can be given by,
sT 0 −1 s
Te =pMsr īs
ī
(18)
1 0 r
s,α
s,α s,β
=pMsr (is,β
s ir − is ir )
(19)
This expression is extended to form the torque expression
for the B-DFIG by including both the stator windings. This
is expressed as,
s,α
s,α s,β
s,β s,α
s,α s,β
Te = pp Mpr (is,β
p ir − ip ir ) + pc Mcr (ic ir − ic ir )
(20)
From these equations the relation between the Power Winding
and Control Winding currents is given by,
1 P W,d
λ
−
Lc c
1
W,q
= λP
−
Lc c
W,d
iP
=
c
W,q
iP
c
Mcr Lp P W,d
Mcr
i
+
|λp |
Mpr Lc p
Mpr Lc
Mcr Lp P W,q
i
Mpr Lc p
(31)
(32)
W,q
W,q
From Equation 32 it is seen that iP
depends on iP
c
p
P W,q
P W,q
P W,d
P W,d
and λc
. λc
is weakly dependant on ip
and ic
.
This influence of d−axis terms on q−axis quantities and vice
versa is termed ‘Cross-Coupling’. For accurate control it is
required that the d and q axis terms be completely de-coupled
such that the control of both parameters is independent of the
other. This is done through the addition of a compensation
term, shown in Equation 33.
W,q
W,q
iP
= f (iP
, |λp |) +
c
p
1 P W,q
λ
L c
| c {z }
(33)
Cross-Coupling Term
If the Park’s transform is used, the equation in the stator
reference frame may be converted to a rotating reference
frame,
W,q
W,q
It is also seen that iP
varies with −iP
.
c
p
The power electronic converter can be controlled by the
duty
ratio for the switches which can be calculated from the
k,d
k,d k,q
k,q k,d
k,d k,q
Te = pp Mpr (ik,q
p ir − ip ir ) + pc Mcr (ic ir − ic ir ) reference Control Winding voltage and the DC bus voltage.
(21) For the simulations here, the reference voltage uc is used
as input to the machine. The dependence of uqc on iqc can
be calculated from the Control Winding voltage equation in
III. B-DFIM C ONTROL
Equation 34.
A. Active Power Control
W,q
dλP
c
W,q
P W,q
P W,d
A reference frame rotating with the Power Winding flux uP
=
R
i
+
(ω
−
(p
+
p
)ω
)λ
+
c
P
W
p
c
m
c
c
c
dt
is chosen and results in Equation 22 and Equation 23 for the
(34)
d and q components of Power Winding flux.
A similar cross-coupling term, due to λP W,d , as seen in
W,d
λP
=|λp |
(22) Equation 33 is seen in the equation above.c This can also be
p
W,q
λP
=0
(23) expressed as in Equation 35.
p
This reference frame is referred to as the PW reference frame
in the rest of this document. Here, if the assumption is made
that Up is constant and Rp is small enough to be neglected,
the flux |λp | will be constant. This gives,
W,d P W,d
W,q P W,q
Rp (iP
ip
+ iP
ip
)≈0
p
p
d|λp |
≈0
dt
(24)
W,q
W,q
W,q
uP
= f (iP
, λP
)
c
c
c
Lp Mcr P W,d
W,d
i
+ Lc iP
)
+ (ωP W − (pp + pc )ωm )(
c
Mpr p
|
{z
}
Cross-Coupling Term
(35)
(25)
The complete control scheme for Active Power control of
the B-DFIG is shown in Figure 3.
(26)
B. Reactive Power Control
Therefore, Pp simplifies to,
W,q
Pp ≈ ωk |λp |iP
p
Therefore, a reference power winding current signal can be
generated from a reference Pp signal using Equation 27.
W,q−ref
iP
p
Ppref
=
ωk |λp |
(27)
For this machine, only the Control Winding circuit is
controllable through the power electronic converter. Therefore,
W,q
the next step would be to obtain a reference iP
current.
c
Consider the flux equations,
PW
PW
PW
λ¯p
= Lp i¯p
+ Mpr i¯r
PW
PW
PW
λ¯c
= Lc i¯c
− Mcr i¯r
PW
PW
PW
PW
λ¯r
= Lr i¯r
+ Mpr i¯p
− Mcr i¯c
(28)
(29)
(30)
The Reactive Power of the Power Winding for the B-DFIG
in the arbitrary reference frame ‘k’ is given by,
k,d
k,d k,q
Qp = uk,q
p ip − up ip
(36)
Substituting Equation 8 and Equation 9 in Equation 36 gives,
Qp = ik,d
p
dλk,q
dλk,d
p
p
k,d
k,q
+ ωk ik,d
λ
−
i
p
p
p
dt
dt
k,q
+ωk ik,q
p λp
(37)
Again, the PW reference frame is chosen (Equation 22 and
Equation 23) and Up is assumed to be constant and Rp is
neglected. The expression for Qp can be expressed as,
W,d
Qp ≈ ωP W |λdp |iP
p
(38)
The reference power winding current signal can be generated
from the reference Qp signal using Equation 39.
Qref
p
ωP W |λp |
Rated Power
Generator Power
W,d−ref
iP
=
p
Maximum Power Curve (Popt)
(39)
W,d
W,d
From Equation 31 it is seen that iP
depends on iP
c
p
P W,d
P W,d
P W,q
P W,q
and λc
. λc
is weakly dependant on ip
and ic
.
Again there is a dq cross-coupling term which must be
compensated for. This compensation term can be calculated
as seen in Equation 40.
1 P W,d
λ
L c
| c {z }
(40)
Cross-Coupling Term
The dependence of
on
can be calculated
from the Control Winding voltage equation.
W,d
uP
c
W,d
iP
c
W,d
W,d
W,q
uP
= Rc iP
− (ωP W − (pp + pc )ωm )λP
c
c
c
+
W,d
dλP
c
dt
(41)
W,q
Again, we see a dq cross-coupling term due to λP
. This
c
can be seen in Equation 42.
W,d
W,d
W,d
uP
= f (iP
, λP
)
c
c
c
Lp Mcr q
W,q
− (ωP W − (pp + pc )ωm )(
i
+ Lc iP
)
c
Mpr P W,p
|
{z
}
Cross-Coupling Term
The previous two sections discussed the control of the
active and reactive power in a B-DFIM. When such a machine
is used in a wind turbine drivetrain, the reference power signal
is chosen so as to extract the maximum energy from the
blades. Therefore, this section describes the generation of
such a Ppref signal from the rotational angular velocity of
the machine rotor.
The aim of the control scheme is to maximise the
power output of the wind turbine. A typical wind turbine
characteristics with the optimal power extraction-speed curve
and its intersection with the Cp,max for all wind speeds [4]
is shown in Figure 1. As Popt is the curve with Cp,max it
is evident that if the turbine is controlled and kept on this
curve, the turbine will generate the maximum energy. This
is followed for all speeds below rated. For speeds above
rated, rated power Prated is maintained. This is described in
Equation 43.
Pref =
or,
Pref =
(
The steady state characteristics are used to generate
a function for the relation between ωm and Ppref . The
characteristics depend on two variables, i.e. magnitude of the
Control Winding voltage and the phase angle between Control
Winding and Power Winding voltages. Therefore, given a
value of ωm and P ref there are a number of operating points
possible. To select an optimal operating point the efficiency
of the machine is used as a selection criteria. Solving the
system for maximum efficiency, the curve for the Power
Winding Power corresponding to each point of the net power
curve is shown in Figure 2.
−0.5
Net Power
Power Winding Power
(42)
C. Reference Signal Generation
(
Fig. 1. Control Strategy for Optimal Power Extraction. The plot shows
the generator output power vs. speed curve for different wind speeds. The
Popt curve connects all the points of maximum power forming the curve
for optimal power extraction.
−1
Active Power (MW)
W,d
W,d
iP
= f (iP
, |λp |) −
c
p
Generator Speed
−1.5
−2
−2.5
−3
−3.5
24
26
28
30
Prated
if v < vrated
if v ≥ vrated
3
1 ρAR Cp,max 3
ωm
2
λ3t,max
if ωm < ωrated
Prated
if ωm ≥ ωrated
34
36
38
Fig. 2. Curve for net Shaft Power and Power Winding Active Power based
on maximising Efficiency.
idp
Qref
p
−
Ref. Gen.
idc
PI
ipd−ref
id−ref
c
−
−
PI
ud,ref
c
+
i¯c
i¯p
ωm
dq Cross-Coupling Compensator
Ref. Gen.
iq−ref
p
−
PI
iqp
1
3
2 ρACp,max v
32
Rotor Speed (rad/s)
q−ref
+ ic
−
PI
q,ref
− uc
iqc
(43)
Fig. 3. B-DFIG Control Scheme
(44)
where λt,max is the tip speed ratio for Cp,max . Section III-A
discusses the control of the machine, this control is based
on the Active Power of the Power Winding alone. Therefore,
it is required to generate the control signal (Ppref ) from the
rotor speed (ωm ).
IV. LVRT P ERFORMANCE
The symmetric voltage dip considered here is shown in
Figure 4. The details of the machines used in the simulations
in the rest of the section are given in the Appendix.
A. PMSM Performance
The PMSM is completely isolated from the grid through an
AC-DC-AC converter. Therefore, disturbances on the grid do
50
d−axis Current
q−axis Current
40
1.0
V oltage (pu)
Grid d−q Currents (kA)
30
20
10
0
−10
−20
−30
−40
−50
999
999.2
999.4
999.6
999.8
1000
Time (s)
1000.2
1000.4
1000.6
1000.8
1001
1000.2
1000.4
1000.6
1000.8
1001
(d)
T ime (s)
0.25
1.9
1.8
Fig. 4. Voltage Profile for LVRT Response Study
0.8
1.5
1.4
1.3
1.2
1
0.9
999
999.2
999.4
999.6
999.8
1000
Time (s)
(e)
9
9
7
7
5
5
3
3
1
1
0.6
Reactive Power (MVAr)
Grid Voltage
in Grid Reference Frame (pu)
1
1.6
1.1
Active Power (MW)
not directly affect the machine. However, in the occurrence
of an LVRT event, the ability of the grid side converter to
transfer power to the grid is greatly reduced. This results
in the rise in voltage of the DC link which is a cause for
concern for the DC link capacitors.
Figure 5 shows the response of the machine when it
encounters a 95% symmetric voltage dip as shown in
Figure 5a.
DC−link Voltage (pu)
1.7
0.4
−1
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
−1
1001
Time (s)
0.2
(f)
0
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
1001
Fig. 5. LVRT Performance of PMSM for a 95% Symmetric Voltage Dip
(a) Grid Voltage magnitude (b) Torque and Speed (c) Stator Currents in the
Stator Flux reference frame (d) Grid Currents in the Grid Reference Frame
(e) DC link voltage (f) Active and Reactive Power to the grid
Time (s)
0
32
−50
31
−100
30
−150
29
−200
999
999.2
999.4
999.6
999.8
1000
1000.2
1000.4
1000.6
1000.8
Generator Speed (rad/s)
Generator Torque (kNm)
(a)
events. Other methods include Power Balancing [6] which
follows the generator side converter to follow the grid side
converter during low voltage events. This reduces the power
mismatch during such events and hence reduces the DC link
voltage rise.
28
1001
Time (s)
B. DFIG Performance
(b)
Unlike the PMSM, the DFIG is not isolated from the grid.
The stator of the machine is connected directly to the grid
and any disturbance in the form of a low voltage event can
be expected to have an effect on the stator and rotor currents.
The response of the DFIG is shown in Figure 6. It is seen
that large transient currents are induced in the stator winding.
Such transient currents are also induced in the rotor circuit
due to the magnetic coupling of the stator and rotor circuits.
These induced transient rotor currents are dangerous to the
(c)
power electronic converter connected to the circuit.
Fig. 5. LVRT Performance of PMSM for a 95% Symmetric Voltage Dip
A widely used method for the protection of the DFIG under
(contd.)
low voltage events is the Crowbar circuit [5]. The crowbar
Figure 7a and Figure 7b show that there is little effect of circuit is a set of switches and resistors that short circuit
the low voltage event on the machine. However, Figure 7d the rotor terminals through the resistors in the case of the
low voltage event and by-pass the generator side converter.
shows that the DC voltage rise is indeed an issue.
A number of methods have been devised to address this The connection of the crowbar circuit serves a number of
issue. One such method is the use of Energy Discharge purposes. First, it bypasses the power electronic converter
Circuits [5]. This limits the rise of the DC link voltage by when large transient rotor currents are present. This protects
using resistive elements to dissipate a part of the power the converter in the event of a Low Voltage event. Second, the
fed to the DC circuit through the generator side converter. added resistance in the rotor circuit lowers the magnitude of
Another method used is the use of Energy Storage Systems the transient rotor currents. Third, the added resistance in the
[5] to manage and store the excess energy during low voltage crowbar circuit reduces the time constant of the rotor circuit
0.2
d−axis Current
q−axis Current
Generator d−q Currents (kA)
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−1.4
−1.6
999
999.2
999.4
999.6
999.8
1000
Time (s)
1000.2
1000.4
1000.6
1000.8
1001
challenge.
1000
500
30
0
−500
999
1000
1001
1002
1003
Generator Speed (rad/s)
Generator Torque (kNm)
32
28
1005
1004
C. B-DFIM Performance
The B-DFIM suffers from the same issues faced by
the DFIG under low voltage events. Figure 7 shows the
performance of the B-DFIM under the condition of a 95%
symmetric voltage dip.
Time (s)
(a)
Generator Torque (kNm)
20
d−axis Current
q−axis Current
15
Stator d−q Currents (kA)
10
5
0
−5
500
39
0
38
Generator Speed (rad/s)
1000
−10
−15
−500
999
−20
999.5
1000
1000.5
−30
999
1001
1001.5
1002
1002.5
37
1003
Time (s)
(a)
−25
1000
1001
1002
1003
1004
1005
Time (s)
(b)
4
Power Winding d−q Currents (kA)
d−axis Current
q−axis Current
25
d−axis Current
q−axis Current
20
Rotor d−q Currents (kA)
15
10
5
2
0
−2
−4
−6
0
−5
−8
999
999.5
1000
1000.5
1000
1001
1002
1003
1004
1002
1002.5
1003
1005
Time (s)
15
d−axis Current
q−axis Current
Control Winding d−q Currents (kA)
(c)
1.8
1.7
1.6
DC−link Voltage (pu)
1001.5
(b)
−15
−20
999
1001
Time (s)
−10
1.5
1.4
10
5
0
−5
−10
−15
1.3
−20
999
1.2
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
1001.5
1002
1002.5
1003
Time (s)
(c)
1.1
1
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
1.05
Time (s)
15
15
10
10
5
5
0
0
−5
−5
−10
−10
−15
999
1000
1001
1002
1003
1004
DC−link Voltage (pu)
1.04
Reactive Power (MVAr)
Active Power (MW)
(d)
1.03
1.02
1.01
1
0.99
0.98
999
999.5
1000
1000.5
1001
Time (s)
(d)
−15
1005
Time (s)
20
20
10
10
0
0
Fig. 6. LVRT Performance of DFIG for a 95% Symmetric Voltage Dip (a)
Torque and Speed (b) Stator Currents in the Stator Flux reference frame (c)
Rotor Currents in the Stator Flux reference frame (d) DC link voltage (e)
Active and Reactive Power to the grid
Active Power (MW)
(e)
−10
−10
−20
Lr
Rr +Rcrowbar ),
this means that the currents during the
(τr =
fault will decay faster.
There are however a number of issues with the use of the
crowbar circuit as well. First, when the crowbar circuit is
activated, the DFIG resembles a induction motor with an
external rotor resistance. This would mean that the machine
would absorb reactive power from the grid which may further
exasperate the condition of the grid. Second, the vector
control is lost during the action of the crowbar circuit and
re-establishing control after the crowbar is released is a
999
Reactive Power (MVAr)
0.9
999
−20
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(e)
Fig. 7. LVRT Performance of B-DFIM for a 95% Symmetric Voltage Dip
(a) Torque and Speed (b) Power Winding Currents in the PW Reference
Frame (c) Control Winding Currents in the PW Reference Frame (d) DC
link voltage (e) Active and Reactive Power to the grid
It can be seen that the magnitude of the transient currents
is lower in the B-DFIM than that in the DFIG. This can be
attributed to the larger leakage inductance of the B-DFIM.
Also, it can be seen that the current induced in the control
15
Control Winding Phase Currents (kA)
winding circuit has an additional circuit (power winding rotor - control winding) when compared to the DFIG (stator
winding - rotor winding).
Although the B-DFIM could also be protected against
the effects of a low voltage event by the use of a Crowbar
circuit, this machine can be protected without the use of an
external circuit and with the use of an addition in the control
algorithm. Figure 8 shows the performance of the B-DFIM
with Crowbarless protection [7].
10
5
0
−5
−10
−15
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(a)
15
39
0
38
Control Winding Phase Currents (kA)
500
Generator Speed (rad/s)
Generator Torque (kNm)
1000
10
5
0
−5
−10
−15
999.5
−500
999
999.5
1000
1000.5
1001
1001.5
1002
1002.5
37
1003
Time (s)
Power Winding d−q Currents (kA)
d−axis Current
q−axis Current
2
−2
−4
−6
−8
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(b)
15
Control Winding d−q Currents (kA)
d−axis Current
q−axis Current
10
1002.5
1003
V. C ONCLUSIONS
0
−5
−10
−15
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
1001.5
1002
1002.5
1003
Time (s)
(c)
1.05
1.04
DC−link Voltage (pu)
1002
5
−20
999
1.03
1.02
1.01
1
0.99
0.98
999
999.5
1000
1000.5
1001
Time (s)
(d)
10
0
0
−10
−20
999
Reactive Power (MVAr)
20
Active Power (MW)
1001.5
Time (s)
Figure 9 shows the control winding phase currents both,
without and with Crowbarless protection. Crowbarless protection is achieved by detecting a low voltage event and setting
the reference values of the inner control winding current
loops to zero.
Section III-A and Section III-B has discussed the calculation of the id−ref
and iq−ref
reference signal depends on the
p
p
d-axis power winding flux. Therefore, in the case of a low
voltage event it is required to maintain pre-fault reference
values.
0
1000
1001
Fig. 9. Control Winding Phase Currents (a) Without Protection (b) With
Crowbarless Protection
4
999.5
1000.5
(b)
(a)
−10
999
1000
−20
999.5
1000
1000.5
1001
1001.5
1002
1002.5
1003
Time (s)
(e)
Fig. 8. LVRT Performance of B-DFIM with Crowbarless Protection for
a 95% Symmetric Voltage Dip (a) Torque and Speed (b) Power Winding
Currents in the PW Reference Frame (c) Control Winding Currents in the
PW Reference Frame (d) DC link voltage (e) Active and Reactive Power to
the grid
For the PMSM based wind turbine, the issue with Low
Voltage Ride Through (LVRT) is the rise in the DC link
voltage . This is due to the mismatch in power generated by
the machine and the power transferred to the grid.
For the DFIG it has been found that voltage dips cause
large transient currents in the stator and rotor circuit. The
oscillations in the rotor circuit are cause for concern as
they may lead to adverse effects on the power electronic
converter connected to the circuit. A possible method to
overcome this issue is the use of a Crowbar circuit. The
crowbar circuit manages the problem by bypassing the power
electronic converter in the event of current rise. Additionally,
the crowbar resistors reduce the time constant of the rotor
circuit, allowing for a faster decay of transient currents.
In the case of the B-DFIG, it has been found that again
transient currents are set up in the power and control windings
in the event of a voltage dip. However, the magnitude of
these currents is lower when compared to that in a DFIG.
This is attributed to the higher leakage inductance of the
B-DFIG. A Crowbarless method has also been discussed
which is successful in controlling the high currents generated
in the control winding. This Crowbarless control method also
reduces the torque oscillations observed in the B-DFIG for
a low voltage event.
In conclusion, apart from offering better reliability through
the exclusion of slip ring and brushes, the B-DFIG also has
an improved LVRT performance when compared with the
DFIG. The protection does not require an external circuit,
like the crowbar, and can be built into the machine controller.
R EFERENCES
[1] S. Shao, E. Abdi, and R. McMahon, “Dynamic Analysis of the
Brushless Doubly-Fed Induction Generator during Symmetrical ThreePhase Voltage Dips,” International Conference on Power Electronics
and Drive Systems, pp. 464–469, 2009.
[2] S. Shao, T. Long, E. Abdi, R. McMahon, and Y. Wu, “Symmetrical Low
Voltage Ride-Through of the Brushless Doubly-Fed Induction Generator,”
IEEE Industrial Electronics Society Conference, pp. 3209–3214, 2011.
[3] T. Long, S. Shao, E. Abdi, R. a. McMahon, and S. Liu,
“Asymmetrical Low-Voltage Ride Through of Brushless
Doubly Fed Induction Generators for the Wind Power
Generation,” IEEE Transactions on Energy Conversion,
vol. 28, no. 3, pp. 502–511, Sep. 2013. [Online]. Available:
http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6544266
[4] R. Pena, J. Dlare, and G. Asher, “Doubly fed induction generator uising
back-to-back PWM converters and its application to variable- speed
wind-energy generation,” IEE Proc.-Electr. Power Appl., vol. 143, no. 3,
pp. 231–241, 1996.
[5] C. Abbey, W. Li, L. Owatta, and G. Joos, “Power Electronic Converter
Control Techniques for Improved Low Voltage Ride Through Performance in WTGs,” Power Electronics Specialists Conference, pp. 1–6,
2006.
[6] X.-P. Yang, X.-F. Duan, F. Feng, and L.-L. Tian, “Low Voltage
Ride-Through of Directly Driven Wind Turbine with Permanent
Magnet Synchronous Generator,” 2009 Asia-Pacific Power and
Energy Engineering Conference, pp. 1–5, Mar. 2009. [Online]. Available:
http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4918470
[7] T. Long, S. Shao, P. Malliband, E. Abdi, and R. McMahon, “Crowbarless
Fault Ride-Through of the Brushless Doubly Fed Induction Generator in
a Wind Turbine Under Symmetrical Voltage Dips,” IEEE Transactions
on Industrial Electronics, vol. 60, no. 7, pp. 2833–2841, 2013.
VI. A PPENDIX
TABLE I: PMSM Parameters
Stator Resistance
Stator Leakage Inductance
Stator Main Inductance
Rs
Ls,σ
Ls,m
3.25E-4
1.95E-5
8.105E-5
TABLE II: DFIG Parameters
Stator Resistance
Stator Leakage Inductance
Stator Main Inductance
Rotor Resistance
Rotor Leakage Inductance
Rotor Main Inductance
Rs
Ls,σ
Ls,m
Rr
Lr,σ
Lr,m
0.0011
4.95E-5
4.88E-4
9.11E-4
5.88E-5
4.86E-4
TABLE III: B-DFIM Parameters
Power Winding Resistance
Power Winding Inductance
Control Winding Resistance
Control Winding Inductance
Rotor Loop - 1 Resistance
Rotor Loop - 2 Resistance
Rotor Loop - 3 Resistance
Rotor Loop - 4 Resistance
Rotor Loop - 1 Inductance
Rotor Loop - 2 Inductance
Rotor Loop - 3 Inductance
Rotor Loop - 4 Inductance
Rp
Lp
Rc
Lc
Rr1
Rr2
Rr3
Rr4
Lr1
Lr2
Lr3
Lr4
0.0023
3.5E-3
0.0021
1.74E-3
1.17E-4
1.11E-4
1.05E-4
0.99E-4
2.35E-4
1.84E-4
1.41E-4
1.02E-4