Supervisory Voltage Control Scheme for
Grid-Connected Wind Farms
by
Hee-Sang Ko
B.S., Cheju National University, Republic of Korea, 1996
M.S., Pennsylvania State University, USA, 2000
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in
THE FACULTY OF GRADUATE STUDIES
(ELECTRICAL AND COMPUTER ENGINEERING)
THE UNIVERSITY OF BRITISH COLUMBIA
December 2006
© Hee-Sang Ko, 2006
Abstract
Modern variable speed wind turbines utilize power electronic converters for the grid
connection requirement and to improve performance. Most commonly used converters
enable the wind turbines to maintain the required power factor (power factor control) or
voltage (local voltage control) at the terminals. However, in many wind farm applications
there is a need to control voltage at a specified remote location, which may require the
installation of additional compensating devices (transformer tap changers, switchedcapacitors, SVCs, etc.) to meet local power quality conditions. This thesis proposes a
supervisory control scheme that uses the individual wind turbines to regulate voltage at the
required location, i.e., point of common coupling. The proposed approach considers that each
turbine may have somewhat different instantaneous wind speeds and real power outputs, and
therefore different amounts of reactive power available for achieving the main control
objectives. The operating limits of each turbine are also taken into account to ensure that all
power electronic converters operate in the allowable region. Since the proposed supervisory
scheme is general and can work with different controllers, we investigate several controllers
in this thesis. The problem of control design is formulated as a linear matrix inequality. An
innovative cost-guaranteed linear-quadratic-regulator-based controller with an observer is
proposed and tuned for a range of operating conditions.
In this thesis, we apply the proposed supervisory voltage control methodology to a candidate
wind farm site on Vancouver Island, BC, Canada, made available through collaboration with
Powertech Labs Inc. We have developed a detailed model of the system, using three 3.6 MW
wind turbines, to carry out the simulation studies. The proposed control solution is compared
with traditional approaches and shown to be very effective during load disturbances and
faults. The proposed methodology is also flexible and readily applicable to larger wind farms
of different configurations.
Ill
Table of Contents
Abstract
ii
Table of Contents
iii
List of Tables
vii
List of Figures
viii
List of Symbols
xii
List of Abbreviations
^
Acknowledgments
xi
xviii
Chapter
1. Introduction
1
1.1
Wind Power Status
1
1.2
Wind Turbine Technologies
4
1.2.1
Stand-along wind turbine grid connection
5
1.2.2
Wind farm grid connection
5
1.2.3
Fault-ride-through capability
6
1.3
Voltage Control in Power Systems
7
1.4
Voltage Control Using Existing Wind Turbine Technologies
8
2. Impact of Wind Energy on Power Systems
10
2.1
Local Impacts
11
2.2
System-Wide Impacts
12
2.3
Voltage Control in Power Systems with Wind Turbines
2.3.1
Impact of wind power at the distribution level
•
14
••••
15
2.3.2
2.4
2.5
2.6
Impact of wind power at the transmission level
16
Voltage Control at Remote Locations
Research Objectives and Approaches
16
;
19
2.5.1
Problem statement
19
2.5.2
Research objectives
20
2.5.3
Proposed approach
20
Contributions
22
3. Model Description of Grid-Connected Wind Farm System
3.1
Study System
3.2
System Model Components
27
3.2.1
Mechanical components
27
3.2.2
Electrical components
31
Voltage Source Converter Controller
41
3.3.1
Rotor-side converter controller
42
3.3.2
Grid-side converter controller
47
3.3.3
DC-link controller
49
3.3
3.4
•.
23
•••••
Conventional Voltage Control of Wind Turbine
24
50
4. Wind Farm Voltage Control
52
4.1
Common Practice
••••
52
4.2
Available Reactive Power in a Multi-Turbine System
53
4.3
Supervisory Voltage Control Scheme
55
4.4
Plant Model and Conventional Controllers
56
4.4.1
Linearized and reduced-order model
56
4.4.2
PID supervisory controller design
61
4.4.3
Evaluation of conventional controllers
65
5. Advanced Voltage Control Schemes
67
5.1
Observer-Based Framework
67
5.2
State Observer Design
69
5.3
Linear Quadratic Regulator Approach
71
5.3.1
Formulation of LQR
72
5.3.2
Conventional approach
74
5.3.3
Cost-guaranteed approach
77
5.3.4
Evaluation of controllers
78
Advanced LMI Representation of LQR
79
5.4
5.5
5.4.1
Taking into account cross-product terms
in the conventional approach
^
5.4.2
Taking into account cross-product terms
in the cost-guaranteed approach
^
5.4.3
Evaluation of controllers
81
Summary of Control Gains
83
6. Simulation Studies
6.1
6.2
86
Small Disturbances
,•
86
6.1.1
Wind speed variations
86
6.1.2
Load variations
91
6.1.3
Summary
96
Large Disturbances
97
6.2.1
Three-phase fault
97
6.2.2
Summary
7. Conclusion and Future Work
7.1
Conclusion
7.2
Future Work
102
103
103
•••
104
Bibliography
106
Appendix A
System Parameters and Operating Conditions
112
Appendix B
Voltage Source Converter Controller Design
119
Appendix C
Local Voltage Controller Design
Appendix D
Matlab Script Files
134
Appendix E
Proof of Theorem
140
Appendix F
Effect of Cross-Product Terms
145
••••
128
List of Tables
Chapter 3
Table 3.1
Switching Operations
34
Chapter 4
Table 4 1
Eigenvalues, Damping ratio, and Frequency
of the 5 th -order Reduced Model
Table 4.2
Gains of the PID-Supervisory Controller
^
62
Chapter 5
Table 5.1
Eigenvalues, Damping ratio, and Frequency of
^
the Closed-Loop Observer
Table 5.2
Design Parameters and Control Gains
84
Table 5.3
Eigenvalues, Damping ratio, and Frequency
85
Table 6.1
Voltage Magnitude Deviations
97
Table 6.2
Comparisons of the Voltage Control Performance (pu)
Chapter 6
102
Appendix A
Table A. 1
Wind Power Model Parameters Model Parameters
112
Table A.2
Turbine Controller Parameters
Table A.3
Two-Mass Rotor Model Parameters
113
Table A.4
DFIG and DC link Parameters
114
Table A.5
Maximum Operating Limit of VSC
114
Table A.6
PI Controller Gains of VSC
114
Table A.7
Line Parameters
114
Table A.8
Thyristor Excitation System
115
Table A.9
Synchronous Generator Parameters
Table A. 10
Operating Conditions
.•••
••••
113
115
116
Vlll
List of Figures
Chapter 1
Figure 1 1
Installed wind power capacity in the Europe, US, and the World
Figure 1.2
Cost of wind energy (Year 2000 US$) and cumulative
• • ••
2
3
domestic capacity (US)
Figure 1.3
Voltage sag magnitude for 132kV Fault
8
Figure 1.4
Wind-energy system utilizing constant speed wind turbine
9
Figure 1.5
Wind-energy system utilizing variable speed wind turbine
••
9
Chapter 2
Figure 2.1
Voltage regulation based on the X/R
Figure 2.2
Diagram
ratio
•••
17
depicting the wind farm interconnection impedance
18
Chapter 3
Figure 3.1
Wind power system considered for dynamic studies
25
Figure 3.2
Block
Figure 3.3
Variable speed wind turbine with DFIG
Figure 3.4
Block diagram of the pitch control
28
Figure 3.5
Simplified block diagram of the two-mass rotor
29
Figure 3.6
Block diagram of the two-mass drive train rotor model
30
Figure 3.7
Schematic representation of the voltage source converter
33
Figure 3.8
Representation of the switching function on the grid-side converter
Figure 3.9
Transmission line lumped-parameter gd-model
36
Figure 3.10
Exciter model block diagram
39
Figure 3.11
RL load model representation in the ^-synchronous
diagram showing subsystem input-output variables
25
•••
26
••
33
4 Q
reference frame
Figure 3.12
Block diagram of the voltage source controller modules
Figure 3.13
Turbine
power versus speed tracking characteristic
41
42
Figure 3.14
Block diagram of the rotor-side converter controller
43
Figure 3.15
Block diagram of the grid-side converter controller
Figure 3.16
Block diagram of the dc-link model and its controller, PI7
50
Figure 3.17
Overall control block diagram of the voltage source converter
51
••
48
Chapter 4
Figure 4.1
Real and reactive power operating limits of
^
voltage source converter
Figure 4.2
Block diagram of the supervisory voltage control scheme
55
Figure 4.3
Bode diagrams of the full-order model (104 t h )
58
Figure 4.4
Figure 4.5
Bode diagrams of the reduced-order model (4 t h ) • Step response of the open-loop and the closed-loop
reduced-order system
58
^
Figure 4.6
Implementation of the PID controller
with distributed anti-windup loop
^
Figure 4.7
Voltage transient at the PCC resulted
from a three-phase fault
^
Chapter 5
Figure 5.1
Block diagram of the supervisory voltage control
with observer
gg
Figure 5.2
Comparison of PID, LQRS, and LQRCG controllers
79
Figure 5.3
Comparison of LQRCG and ALQRS controllers
82
Figure 5.4
Comparison of PID, ALQRS, and ALQRCG controllers
83
Figure 6.1
Wind speed (m/sec)
87
Figure 6.2
Real power set-point for each WT
due to wind speed variation
gg
Figure 6.3
Real power output from each WT
due to wind speed variation
gg
Chapter 6
Figure 6.4
Real power output from the wind farm
due to wind speed variation
^
Figure 6.5
Reactive power output from each WT
due to wind speed variation
g^
Figure 6.6
Real power output from the wind farm
due to wind speed variation
pQ
Figure 6.7
Voltage fluctuations due to wind variation, as observed
at the WT terminals
^
Figure 6.8
Voltage fluctuations due to wind variation, as observed
at the PCC
^
Figure 6.9
Voltage transient observed at the PCC
due to load impedance changes
Figure 6.10
Voltage transient observed at the PCC due to load impedance
changes: Detailed view of the PID-supervisory and ALQRCG
controllers
••••••
93
Figure 6.11
Voltage transient observed at the WT terminals
93
Figure 6.12
Real power output from each WT
Figure 6.13
Real power output from the wind farm
94
Figure 6.14
Reactive power output from each WT
95
Figure 6.15
Reactive power output from the wind farm
95
Figure 6.16
Reactive power set-point and maximum at each WT
96
Figure 6.17
Voltage transient observed at the PCC due to the fault
98
Figure 6.18
Real power output from each WT due to the fault
99
Figure 6.19
Real power output from the wind farm due to the fault
99
Figure 6.20
Reactive power output from each WT due to the fault
100
Figure 6.21
Voltage transient observed at the terminal of each WT
jqq
.•••••
94
due to the fault
Figure 6.22
Reactive power output from the wind farm due to the fault
Figure 6.23
Reactive power set-point and maximum at each WT
due to the fault
101
^
Appendix B
Figure B. 1
Bode diagrams of the transfer function of the open-loop
and closed-loop system
Figure B.2
Comparison of the step response of the open-loop
and closed-loop system
Figure B.3
Bode diagrams of the transfer function of the open-loop
and closed-loop system
Figure B.4
Step response of the closed-loop system
Figure B.5
Bode diagrams of the transfer function of the open-loop
and closed-loop system
Figure B.6
Step response of the closed-loop system
Figure B.7
Bode diagrams of the transfer function of the open-loop
and closed-loop system
Figure B.8
Step response of the closed-loop system
^q
^
123
125
_
127
Appendix C
Figure C.l
Bode diagram of the full-order model (23 t h ) and the
reduced-order model (3 r d )
Figure C.2
Nyquist plot of a compensated loop transfer function
Figure C.3
Bode diagrams of the transfer function of the open-loop
and closed-loop system
Figure C.4
Step response of the closed-loop system
Figure C.5
Block diagram of the PI controller
with distributed anti-windup scheme
130
132
Appendix F
Figure F.l
Optimal and neighbouring optimal paths
146
List of Symbols
The unit is based on the per unit (pu) if there is no specification.
Matrices
A
B
C, F
system matrix
D,H
feedforward matrix
A
system matrix of the reduced-order model
B
input matrix of the reduced-order model
C
output matrix of the reduced-order model
D
feedforward matrix of the reduced-order model
G
randomly chosen matrix
input matrix
output matrix
observer gain matrix
Ke
P
S
positive-definite Lyapunov matrix
Y
change variable
Z
slack variable
positive-definite matrix
Matrices - Greek Letters
A
decision matrix
Vectors
k
LQR gain vector
u
input vector
w
stator noise signal vector
x
state vector
x(0)
stationary random initial state vector
x
estimated stator vector
z
measured system output vector
z
observer output vector
y
system output vector
y
controlled system output vector
Vectors - Greek Letters
V
flux vector
output noise signal vector
Scalars
AR
area swept by the rotor (m 2 )
C P (A, 6)
power coefficient
C
capacitance
dc field voltage
P
real power
Q
reactive power
R
resistance
Rt
rotor ratio (m)
S
apparent power
T
torque
v,i
voltage and current
V,I
steady state voltage and current
vw
wind speed (m/s)
Z
impedance
Scalars - Greek Letters
coe
stator angular speed
cor
rotor angular speed
slip angular speed
<os
mechanical rotor angular speed
cob
base angular speed (rad/sec)
tyjiase
base mechanical angular speed (rad/sec)
e
pitch angle (degree)
A
ratio of the rotor blade tip speed and wind speed
P
air density (kg/m 3 )
c
damping ratio
K
phase margin (degree)
Q/7' R W
observer design parameter
¥
flux
Mathematical Symbols
a <0
matrix a is negative definite
a>0
matrix a is positive definite
aeA
a is an element of the set A
a
first
order time derivative of a
a
second order time derivative of a
a
third order time derivative of a
d/dt
d B
first
order time derivative
201og 1 0 |G|
E
Expectation
G
transfer function
51
field of real number
w-dimensional real vector space
s
Laplace operator
tr
trace of matrix
A
error
Superscripts
ref
reference
set
set-point
T
transpose
-1
inverse
max
min
maximum
minimum
Subscripts
abc
a
phase abc
c
voltage source converter
ca
cable
cl
closed-loop
d
J-axes of reference frame
dc
direct current
f
field winding of synchronous generator
filter
gc
j
armature of synchronous generator
filter connected to the grid-side converter
gain crossover frequency
sub systems
k
damper winding of synchronous generator
load
load
max
maximum
mech
mechanical
min
minimum
m
magnetizing
o
steady state
<1
q-axes of reference frame
r
s
rotor of generator
TL
transmission line
t
turbine rotor
tr
transformer
stator of generator
List of Abbreviations
List is given in Alphabetical order.
Acronyms
ac
alternating current
dc
direct current
DFIG
doubly fed induction generator
FACTS
flexible ac transmission system
IPPs
independent power producers
LMI
linear matrix inequality
LQR
linear quadratic regulator
LVC
local voltage control
PCC
point of common coupling
PFC
power factor control
PI
proportional-plus-integral
PID
proportional-integral-derivative
pu
per unit
PWM
pulse width modulation
SCR
short circuit ratio
SG
synchronous generator
SMES
superconducting magnetic energy storage
STATCOM
static compensator
SVC
static VAR compensator
TR
transformer
TL
transmission line
K
voltage at the connection point
V GRI D
VSC
V pcc
voltage at the grid
voltage source converter
voltage at the point of common coupling
WF
wind farm
WT / wt
wind turbine
ZF ARM
equivalent impedance of the wind farm
Z GRI (I
equivalent impedance of the grid
LQRS
LQR supervisory
LQRCG
LQR cost-guaranteed
ALQRS
Advanced LQR supervisory
ALQRCG
Advanced LQR cost-guaranteed
XVlll
Acknowledgments
I would like to acknowledge the essential role of both my supervisors, Dr. Guy Dumont
and Dr. Juri Jatskevich. I wish to express my deepest gratitude for their guidance,
assistance, encouragement and advice in this project. They were always ready to discuss
the intricate details of this research and share their expertise in control theory and power
systems which made this project possible.
I also would like to thank Dr. Prabha Kundur and Dr. Ali Moshref of Powertech Labs
Inc., for providing financial support as well as valuable information for the model that
made this research project practical and more relevant to the industry. I will never forget
my visits to Powertech Labs and the many discussions that I had with them.
I would like to thank my parents, who raised me and supported me in every possible way.
They have enabled me to study at UBC and develop my career. I will always remain
grateful to them. I am also endlessly grateful to my wife for sharing my life and
encouraging me during the times when research was not going smoothly, and to my little
son Kevin, who is very busy learning to walk and talk and makes our life happy and
worthwhile.
UBC, December 2006
Hee-Sang Ko
Chapter 1
Introduction
1.1
Wind Power Status
The advantages of conventional thermal, nuclear, and hydro power generation include a
relatively low price, as well as complete control of the generated power. Renewable power
generation, however, poses less severe environmental consequences, but relies on available
primary energy sources, such as sunlight and/or wind, that are not controllable in the same
sense as the traditional energy sources.
Renewable wind energy technology uses wind turbines to convert the energy contained in the
wind into electrical energy. Wind is an inexhaustible primary energy source. Furthermore,
the environmental impact of harnessing wind power is small. Although wind turbines affect
the visual scenery and emit some noise, the overall consequences appear to be small with no
significant impact on the ecosystem. Moreover, when the wind turbines are installed at
remote locations on the ground or offshore, the visual effect and noise are no longer a
concern. Compared with other renewable energy sources, such as photovoltaic (PV), ocean
waves, and tidal power generation, wind power appears to be less expensive and gives higher
returns per affected (required) area. That is why many countries including Germany,
Denmark, Spain, etc., demonstrate strong growth in the wind energy sector. Figure 1.1 shows
the growth of wind power in Europe, the US, and worldwide [1]. As can be seen in Figure
1.1, the installed wind power capacity shows a steady growth; during the last five years,
annual growth has been higher than 30%.
Total Installed Wind Capacity
Rest of
World
Europe
o
CO
Q.
(0
O
United
States
1982
1985
1988
1991
1994
1997
2000
2003
Year
Figure 1.1: Installed wind power capacity in Europe, the US, and worldwide [1].
Worldwide, many countries value the advantages of renewable power generation and support
the expansion of its capacity in various ways. However, the installation/equipment cost
involved and lack of direct control remain concerns, especially when the penetration levels
arehigh[2].
The cost disadvantage of wind power is reduced in many, cases by some form of subsidy. For
example, power companies may be forced to buy power from renewable energy providers at
a guaranteed price that is not based on the actual value of the power, but is calculated such
that the renewable energy project becomes profitable for the developer. Unless the power
companies are able to sell this power as "green power" at a premium price, such subsidies
will lead to a general increase in the electricity price, whereas all consumers would end up
paying for the additional cost of electricity generated from renewable sources. Alternative
subsidies may include a direct support given to the developers of renewable energy projects,
which spreads the cost burden associated with renewable energy over all taxpayers. Using a
variety of incentives, the cost disadvantages associated with developing renewable energy
sources continue to diminish. For example, Figure 1.2 shows the changes in renewable
electricity cost and installed capacity growth over the last decade in the US [1], which is tied
economically to Canada. Current wind energy in Canada produces a very small portion of the
electricity supply. Canada's total installed capacity of 444 MW satisfies only 0.2% of the
nation's energy demand. However, with new projects totalling about 2000 MW coming online in the near future, and more planned, it is expected that wind energy will cover up to 3%
or more of Canadian energy needs by the year 2012 [3]. As seen in Figure 1.2, the present
renewable electricity cost is reaching below 10 cents/kWh and becomes directly competitive
with the traditional energy production.
Cost of Wind Energy (cents/kWh)
— • — Capacity (MW)
i2
cd)
E>
0)
c
LL1
o
TO
CL
CO
--
to
O
o
1980
1984
1988
1992
1996
2000
1000
O
2004
Year
Figure 1.2: Cost of wind energy (Year 2000 US$) and cumulative domestic capacity [1].
The present practice requires the independent power producers (IPPs) and/or generators who
want to connect to the grid to meet the so-called connection requirements of the local electric
utility (the grid company). These requirements may also include the steady-state and
dynamic interaction between the generator and the grid.
In order to maintain the power generation and consumption balance, necessary for stable
functioning of the power system, the traditional power plants always exert necessary control
actions. However, renewable energy sources are presently exempted from such control
functions. This, in turn, simplifies the requirements for the renewable energy source
interconnection as well as the project developer, allowing connection to the system without
having to take part in the overall stabilization effort.
1.2
Wind Turbine Technologies
Although the fundamental principle of a wind turbine is straightforward, modern wind
turbines are very complex systems. The design and optimization of the wind turbine's blades,
drive train, and tower require extensive knowledge of aerodynamics, mechanical and
structural engineering, control and protection of electrical subsystems, etc.
Two major technologies are prevalent in the wind power energy sector today. First, a
substantial scaling up has taken place to further reduce the cost of wind power and the
individual wind turbines. For modern wind turbines of the multi-MW class, both the nacelle
height and rotor diameter are in the order of 100 meters. Thus, at the vertical position, the
blade tip can reach heights of up to 150 meters. The largest wind turbine presently developed
is a 5MW unit [4] that is based on a new design concept involving a carbon-fiber material
type blade, and gearless and permanent synchronous generator technology developed
especially for offshore wind power generation. Enercon is also presently upgrading their E112 turbine technology and advertising up to 6MW of output power [5]. Second, most of the
presently developed large wind turbines are based on variable-speed operation rather than
fixed-speed technology, which was used initially and is simpler. The fixed-speed wind
turbines would typically include an induction or synchronous generator that is directly
connected to the grid; hence, the rotor speed remains essentially constant, or varies very
slightly with the speed of the wind. This simple design entails lower manufacturing costs.
The variable-speed wind turbine is technically more advanced. A typical variable-speed wind
turbine consists of more components and needs additional control system(s), and is therefore
more expensive. However, it has various advantages over constant-speed wind turbines, such
as increased energy yield, reduced noise emission, the ability to withstand higher mechanical
operating limits, and additional controllability of active and reactive power.
1.2.1 Stand-alone wind turbine grid connection
In the majority of installations, the wind turbines are connected to the grid. The grid
connection of solitary wind turbine is relatively straightforward. The voltage at the turbine's
generator terminal is typically low (690V is common); therefore, a step-up transformer is
used to bring the voltage to the grid level at the point of connection. Furthermore, some
switchgear is necessary so that the wind turbine can be disconnected in the case of a short
circuit or in islanding [6].
1.2.2 Wind farm grid connection
The wind farm represents an aggregation of several or many tightly interconnected wind
turbines that are then interconnected with the power grid. Although the individual wind farms
may represent a large contribution to the local power pool that is comparable in size to the
conventional medium-size power plants, their effect on the power system is very different
from that of conventional synchronous generators. The difference is especially pronounced in
terms of response to disturbances in the terminal voltage, frequency, and power, depending
on the type of wind turbines used. In the case of fixed-speed wind turbines based on
induction generator technology, an installation of additional capacitor banks is often required
to support the reactive power demand as well as to control the voltage. In the case of
variable-speed wind turbines, the wind-farm response and dynamic interaction are primarily
determined by the wind turbines' internal power electronic converters and the respective
controllers [7].
1.2.3 Fault ride-through capability
The wind turbine manufacturers presently offer a number of practical solutions and control
approaches to improve the reliability and stability of power systems with wind turbines in the
event of large disturbances such as faults. In particular, the fault-ride-through capability of
the wind turbine envisions that the wind turbine remains connected to the grid during the
transient, and enables faster recovery and more reliable operation of the overall network after
the source of the disturbance is removed (fault is being cleared).
Option 1: Crowbar Protection
The crowbar protection scheme may be used with wind turbines that are based on the doublyfed induction generator (DFIG) technology. This protection redirects the current from the
rotor-side converter by short-circuiting the rotor windings and thus blocking the rotor-side
converter. Therefore, the rotor current goes through the crowbar and does not damage the
converter. This measure makes the DFIG resemble a conventional squirrel-cage induction
generator during the transient, including the contribution to the short-circuit current [8].
Crowbar protection is usually activated when the peak value of the rotor current exceeds
approximately 2 times the normal rotor current. The crowbar is deactivated again when the
ac voltage reaches 80% of the predefined voltage level and the rotor current is below that
current for activating the crowbar. Because crowbar protection makes the DFIG operate
similar to the squirrel-cage induction machine and consume reactive power during the large
transients (which basically disables the controls), it also has an undesirable effect on voltage
stability. Alternatively, the two control schemes described below allow the control actions of
reactive power during the fault.
Option 2: Power Factor Control
The power factor control (PFC) scheme [8]-[10] relies on the rotor- and grid-side converters
to ensure the specified power factor (usually unity) at the wind turbine terminals. Under this
scheme, reduction of the generator terminal voltage leads to an increase in the real power
injected by the wind turbine. When this happens, the rotor will decelerate while the power
from the wind is lower than the real power taken from the generator. At this point, the rotor
blade pitch controller is activated to avoid the wind turbine operation in the under-speed
region.
Option 3: Local Voltage Control
Local voltage control (LVC) scheme [8], [9], [11], [12] utilizes the reactive power by
controlling the rotor current to regulate the voltage at the wind turbine terminals. When this
scheme is used, the wind turbine is more likely to remain connected to the grid during the
fault and the control operation may help to restore the voltage after the disturbance.
1.3
Voltage Control in Power Systems
Because transmission lines, cables, and transformers, etc. have impedance, voltage control is
necessary to maintain the bus voltages within the allowable range required for the safe and
reliable operation of all equipment. Appropriate measures must be taken to prevent and/or
reduce voltage deviations. It is important to stress that bus voltage is a local quantity, as
opposed to frequency, which is more often associated with the system (global) level. It is
therefore not possible to control the voltage at a certain bus from an arbitrary point in the
system without affecting the voltages at other buses, however, the voltage can be effectively
controlled locally.
Short-duration reductions in voltage are often referred to as a voltage sags and have been
associated with voltage instability in power systems [13], [14], The voltage sags due to
motor-starting transients are typically longer. The relatively short voltage sags are often
caused by faults in the power system, and are often more severe in magnitude and are
responsible for the majority of equipment trips. To get an idea of how the sag magnitude
propagates in a radial system, the voltage sag due to a fault on a 132kV transmission line is
shown in Figure 1.3 [14]. As shown, the voltage sag at Bus A is less severe as the distance
from the fault increases. Based on this observation, voltage sags from a distant fault can be
more easily mitigated and is less likely to trip local equipment than a sag due to a nearby
fault.
Set-point level
a,
Monitored at
Bus A
-M
132kV
BusA
TL
132kV line
Load
TR
7
fault
33kV
0
20
40
60
80
100
Distance to the fault in kilometers
TL
Load
TL: transmission line, TR: transformer
Figure 1.3: Voltage sag magnitude for 132kV fault [14].
1.4
Voltage Control Using Existing Wind Turbine Technologies
In this section, conventional voltage control of the grid with wind turbine/farm is reviewed
with respect to the existing wind turbine technologies. Figure 1.4 shows a power network
with a constant-speed wind turbine. This type of wind turbine consumes reactive power. To
achieve a power factor close to unity at the point of grid connection, an additional source of
reactive power such as static VAR compensator (SVC) or capacitor banks, etc., is always
needed, and is often placed close to the connection point, as shown in Figure 1.4. In addition,
real power generation fluctuates quite significantly with wind speed changes. Therefore,
regulating the voltage at the remote location, or point of common coupling (PCC), in terms
of critical load, often requires having another additional compensating device, which
increases the costs and complicates operation. Due to these disadvantages, this type of wind
turbine is not usually used when there is high penetration of wind power in the grid.
Figure 1.4: Wind-energy system utilizing constant-speed wind turbine.
A wind energy system with a grid-connected variable-speed wind turbine is shown in Figure
1.5. In this type of wind turbine, a voltage source converter (VSC) is used, which may be
used as a source of reactive power if the converter ratings and operating conditions permit.
Therefore, the reactive power available from VSC, if any, can be utilized for voltage control
purposes. Conventional control schemes include PFC and LVC. In PFC mode, the reactive
power QG is controlled to be zero, and the additional device at the wind turbine terminal is
not necessary. The LVC mode uses available reactive power from the VSC to regulate
voltage at the wind turbine terminal. Both of these control strategies are local with respect to
the wind turbine terminal and do not consider the voltages further away in the system. To
regulate the voltage at a remote PCC, additional reactive power compensating devices are
often still required, which entails undesirable costs.
Vwt
PCC
Figure 1.5: Wind-energy system utilizing variable-speed wind turbine.
Chapter 2
Impact of Wind Energy on Power Systems
The impact of wind energy on a power system is associated with its inherently fluctuating
unpredictable output power. The response of wind farms is also determined by the
technology and/or controls used in the individual wind turbines. For instance, when a
constant-speed wind turbine is used, controlling reactive power is made possible by using
additional compensating devices only. At the same time, when a variable-speed wind turbine
is used, controlling reactive power is possible at the wind turbine terminal by utilizing the
respective inverters [15].
Small wind farms and individual wind turbines by themselves are relatively weak power
sources. Because wind farms are often installed at remote locations and have a weak
connection with the grid, additional measures to ensure voltage control in the grid are
required, especially when the portion of the wind power in the grid is substantial [15], [16].
However, the exact measures that are necessary for achieving the desired voltages throughout
the whole system depend highly on the location and characteristics of the wind farm, the
network layout, the capabilities of the remaining conventional synchronous generators,
spinning reserves, etc. [17]. Depending on the extent to which the wind farms affect the grid,
their impacts may be broadly categorized as local or system-wide.
2.1
Local Impacts
The impacts observable in the close vicinity of the wind power interconnection include:
.
Change of fault currents, protection scheme settings, and switchgear ratings
Change of power flow in local distribution network
Change of voltages at nearby buses
•
Flicker
•
Harmonics
The first two impacts must be investigated whenever a new generation capacity, wind or
otherwise, is being considered for interconnection. The way in which wind farms affect
voltages at nearby buses depends on the type of wind turbine (variable- or fixed-speed) used
and their controls [15]. The contribution of wind farms to the fault current also depends on
the type of wind turbine used [15], [18], For instance, a constant-speed wind turbine based on
a squirrel-cage induction generator directly connected to the grid contributes to the fault
current and relies on conventional protection schemes (over-current, over-speed, over- and
under-voltage, over- and under-frequency). At the same time, a variable-speed wind turbine
also changes the fault current. However, due to the faster control action of power electronic
converters in variable-speed wind turbines, the fault current may be actively controlled to
enable the fault-ride-through capability.
Flicker is typical with constant-speed wind turbines [18], wherein the fluctuating wind speed
is directly translated into fluctuations of output power. Depending on the strength of the grid,
the resulting power fluctuations will result in voltage fluctuations propagating in the network.
These voltage fluctuations may lead to undesired fluctuations in the light brightness of
commercial and residential buildings and cause annoyance and irritation. The power quality
problem that results in light fluctuation is referred to asflicker. However, flicker problems
are not generally associated with variable-speed wind turbines because the wind speed
fluctuations are not directly translated into output-power fluctuations. With the rotor inertia
acting as a low-pass filter and the additional action of the power electronic converters, it is
possible to smooth out the effect of wind speed and power fluctuations.
Harmonics are mainly associated with variable-speed wind turbines [19] and their use of
switching power electronic converters. However, modern variable-speed wind turbines
utilize converters that operate at high switching frequencies and employ advanced control
algorithms and filtering techniques to minimize harmonics propagation [15], [18].
2.2
System-Wide Impacts
In addition to local impacts, wind power also introduces large-scale effects that become more
noticeable as the penetration level of wind power in the grid increases. In particular, high
penetration of wind energy has an impact on the following:
Power system dynamics and stability
•
Reactive power generation and network voltage control
System operation/balancing and dispatch of the remaining conventional units
Frequency control
The impact on the dynamics and stability of power systems is mainly due to the fact that
wind turbine generating systems [20] do not provide an inertial response similar to
conventional generators and do not participate in stabilizing control actions. Instead, the
voltage and frequency response of wind turbines is determined by the underlying technology,
interconnection inverters, and the corresponding internal controllers.
High penetration of wind energy in power systems has been noticed to affect reactive power
generation and voltage control in the system [15], [21], [22], First, not all wind farms are
capable of varying their reactive power output. This is, however, only one aspect of the
impact of wind power on voltage control in a power system. Apart from this, wind power
plants cannot be installed at arbitrary locations and must be erected at places with good wind
resources [23], The locations with good wind conditions are not necessarily favorable from
the perspective of grid voltage control. In choosing a location for a conventional power plant,
it is generally easier to take into account the voltage control aspect.
The impact of wind power on system balancing, i.e. the dispatch of remaining conventional
units and frequency control, is also due to the fact that wind turbine output is not traditionally
controlled. In general, the power generation from wind farms is uncontrolled as well, and
wind power does not contribute to the primary frequency regulation. Although this would be
technically possible, it would require a reduction in energy yield and financial loss for the
wind farm operators. Therefore, as long as the wind farms are not participating in power
system control and as long as there are cheaper means to keep the system balanced, wind
farms are not likely to contribute to system balancing.
However, the impact of wind power on system balancing should be given special
consideration in the case of higher wind power penetration,, wherein the numbers of
conventional generator units and the spinning reserve are decreased. Longer term wind
variations (often from 15 minutes to several hours) tend to complicate the dispatch of the
remaining conventional generators used to supply the load. The resulting demand profile that
is formed by the load minus the generated wind power now has to be met by remaining
conventional power generation. Due to the stochastic nature of wind, this resulting demand
profile is usually less smooth than that produced without a wind power contribution.
Therefore, faster dispatch action of the conventional generation and reserve units is required,
which is altogether more difficult to accommodate. Thus, imbalance between the generation
and the load may occur more often and affect the system frequency. To mitigate/reduce this
imbalance, it may be possible and/or necessary to incorporate a forecast of the wind speed
into the real-time dispatching of conventional generation.
2.3
Voltage Control in Power Systems with Wind Turbines
Traditionally, voltage control for transmission grids and distribution grids is achieved
differently. At the transmission level, large-scale centralized power plants keep the bus
voltages within the allowable range. At the distribution level, dedicated equipment such as
tap changers, switched capacitors and/or reactors, etc., are often utilized for voltage control at
a particular location. Overall, in a traditional power system, the bus voltages are regulated by
combining the action of large-scale power plants at the transmission level with the use of
additional devices at various levels and locations [13].
A number of recent developments in energy production have complicated the traditional
approach to voltage control. In particular, the increased use of wind turbines for generating
electricity makes voltage control more challenging, due to the unpredictable nature of wind
conditions. When individual wind turbines or small-sized wind farms are connected at the
distribution level, the action of the auxiliary compensating devices and/or tap-changers must
be coordinated with the operation of the wind turbines to ensure the required voltage
regulation at the affected buses. The problem of harmonized integration becomes even more
challenging as the level of wind power penetration increases and large-scale wind farms are
connected at the transmission level. Not only are the voltages at various locations affected,
but also the power flow, power system dynamic, transient stability, and reliability [24], [25],
The common practice in wind turbine operation is to disconnect them from the grid
immediately when a fault occurs somewhere in the system. However, research trends and
some applications suggest that wind turbines may be required to stay connected longer and
ride through part of or the entire fault transient(s) to enhance system stability [3], [15], [16],
[18], [21], [22], [26]. In this regard, in many countries with high levels of wind energy
penetration in power systems, the wind turbine grid connection standards are being revised in
terms of their impact on transient voltage stability. For example, some presently proposed
grid-connection requirements for allowable voltage levels at the connection point with the
transmission grid during operation are as follows [18]: DEFU in Denmark (<1%); VDEW
and E.ON in Germany (<2%); AMP in Sweden (<2.5%); ESBNG in the Republic of Ireland
(<2.5% for llOkV level and <1.6% for between 220kV to 400kV). To achieve these high
standards and to make grid integration easier and more reliable, active control of individual
wind turbines and wind farms is becoming increasingly important.
Maintaining the voltage at various locations becomes more of a concern where there is a high
level of wind power penetration in power systems. Thus, it is necessary to examine how the
operation of conventional power systems and voltage control at the distribution and
transmission levels are affected by wind power.
2.3.1 Impact of wind power at the distribution level
Traditionally, voltage control in distribution grids includes the tap-changing transformers
(i.e., transformers in which the turns ratio can be changed) and devices that can generate or
consume reactive power (i.e., shunt capacitors or reactors) [13], [18]. The use of tapchanging transformers is a rather cumbersome way of controlling bus voltages. Assuming a
radial network, rather than affecting the voltage at one bus and/or its direct vicinity, the
whole voltage profile of the distribution branch is shifted up or down, depending on whether
the transformer turns ratio is decreased or increased. Switched capacitors and reactors
perform better in this respect and have a more localized effect. In combination with installing
auxiliary voltage regulation devices, the converters of modern variable-speed wind turbines
may also be utilized for voltage control. However, the sensitivity of the bus voltage to
changes in reactive power often requires relatively large capacitors and reactors [13].
One might argue that with an increasing number of wind turbines connected to the
distribution grid, the voltage control possibilities might increase as well. However, in many
cases, the opposite is true, for following reasons:
Depending on the design type, wind turbines are not always (if ever) able to vary
reactive power generation in the required range.
•
It may be very costly to equip the wind turbines with additional voltage control
capabilities.
•
Adding the voltage control capabilities could increase the risk of islanding.
When there are many wind turbines, it may be difficult to coordinate the control
action(s), considering the varying network topology and operation.
2.3.2 Impact of wind power at the transmission level
At the transmission level, in addition to traditional large-scale power plants and synchronous
generators, dedicated equipment such as capacitor banks and flexible ac transmission systems
(FACTS) have also been used for voltage control [13], [18]. However, due to industry
deregulation, voltage control has become a more complicated task in the planning and
dispatch of power plants [14], Additionally, when large wind farms are installed at remote
locations or offshore [18], achieving the desired voltage control at some remote and weakly
connected locations may be difficult. Therefore, the voltage control capabilities of various
wind turbine types are expected to become increasingly important.
2.4
Voltage Control at Remote Locations
As mentioned in the previous section, to achieve easier grid integration and reliable voltage
control, voltage control of wind turbines is essential [15], [16], [18], [21], [22], [26].
However, in many wind farm installations, there is a need to control the voltage at a specified
remote location, or point of common coupling (PCC), which becomes more difficult due to
the fluctuating nature of wind power. Voltage control at remote PCCs may become even
more difficult in places with high penetration of wind energy and weak ties to strong
subsystems. In these cases, additional compensation devices are sometimes used.
Voltage fluctuation also depends on the effective or equivalent impedance of the grid. '
Broadly speaking, injecting power into a weak grid causes large voltage fluctuations
compared to in a strong grid. It is well known that lower grid impedance results in a higher
short circuit ratio [27]. In practice, a short circuit ratio of greater than 20 is considered to
indicate a grid that is strong [27],
The composition of the equivalent impedance of the grid, which is often expressed as the
X/R ratio [28], also has a pronounced effect on voltage control. To clarify the role of line
impedance in voltage regulation, a simplified phasor diagram of a grid-connected wind
turbine is shown in Figure 2.1. Here, V cp represents the voltage at the wind turbine
connection point, and V gricj represents a strong utility grid. The effectiveness of voltage
regulation by adjusting the reactive power depends significantly on the XjR ratio of the
connecting tie, represented here by an equivalent impedance Z.
Assuming certain fixed values of the grid voltage V grid
and the injected reactive current / ,
voltage V cp can be determined using the phasor relations depicted in Figure 2.1 (b). When
the XjR is high (diagram on the left), the voltage drop across the impedance Z is closer in
phase to the grid voltage V grid,
hand, when the X/R
which results in a significant increase in V cp. On the other
ratio is low (diagram on the right), the voltage drop across the
impedance Z is closer in phase to the current I, which results in a smaller increase in V r„.
i.p
Based on this observation, it can be concluded that when the X/R ratio is high, the voltage
at the connection point can be effectively controlled by injecting a reactive current.
connection point
^
R+jX
Z
I
(a)
V
gr,d
jXI
higher X/R ratio
V
(b)
I
Zrid
lower X/R ratio
Figure 2.1: Voltage regulation based on the X/R ratio.
jXI
Voltage regulation becomes more complicated if instead of regulating the voltage at the
connection point it is necessary to regulate it at an intermediate PCC, as shown in a
simplified diagram in Figure 2.2. In particular, when the wind turbine operates in the LVC
mode, the equivalent impedance Z is composed of the impedance to the wind farm
and the grid impedance Z GRI D
ZJ
ARM
combined, which reduces the short circuit ratio and makes the
grid interconnection appear weaker. However, if the wind turbine is controlled to regulate the
voltage at the PCC, then the effective value of impedance Z becomes smaller by the amount
of ZF ARM. This, in turn, increases the effective short circuit ratio and makes the grid appear
stronger.
V WT
PCC
Grid
Figure 2.2: Diagram depicting wind farm interconnection impedance.
Based on this observation, the wind farms or wind turbines may be used as very effective
voltage regulation tools and should no longer exempted from contributing to reliable
operation of the grid, especially where there is high wind power penetration.
2.5
Research Objectives and Approaches
2.5.1 Problem statement
As the present tendency of incorporating wind turbines into large wind farms continues, new'
possibilities for integrated design of individual turbines, the infrastructure within the wind
farm, and the grid-connection interface open up [21]. Furthermore, wind farms that generate
substantial amounts of electrical power may be connected at higher voltage levels and greater
distance [18]. The local impacts of wind power have been studied extensively in the literature
[15], [18], [19], [29], [30], The system-wide impacts of wind power are of special interest at
higher levels of wind power penetration [21], [22], [25], [31]—[33], and is expected continue
with the present rapid growth of wind power.
Modern variable-speed wind turbines utilize power electronic converters for the grid
connection and improved performance. By appropriately controlling the converters, it
becomes possible to locally maintain the power factor (power factor control mode, PFC) or
the voltage (local voltage control mode, LVC). Furthermore, in modern wind farm
applications, the wind farms have to contribute voltage control within a specified allowable
voltage level at the PCC. As wind power penetration increases, the PFC and LVC modes
(Options 2 and 3 in Section 1.2.3, respectively) are frequently not sufficient to achieve the
desired voltage control, especially during events such as faults [16], [21], [26], and may still
require installation of additional devices to meet the power quality specifications. However,
there are always costs associated with the installation and operation of supplementary
devices, which makes this option less attractable. Therefore, to achieve easier grid integration
and reliable voltage control, alternative active voltage control of wind turbines is required.
2.5.2 Research objectives
The research objective of this thesis is to investigate the control options that can be used
concurrently with existing wind turbine technologies to improve voltage regulation in the
system. In particular, the performance of traditional control schemes such as the PFC and the
LVC subject to small transients and large events like faults is investigated. Alternative design
and/or control solutions are proposed to improve the voltage control at required locations of
PCCs.
2.5.3 Proposed approach
The system and modifications considered in this thesis are based on an industrial site located
on Vancouver Island, Canada, that is presently being investigated by Powertech Labs Inc.,
for a possible wind farm installation [34]. The wind farm is assumed to be connected at the
transmission level and provide a significant portion of the local power demand (20 to 50%).
Although aggregate wind farm models have traditionally been used in the analysis of wind
power generation systems, the multiple wind turbines farm model is more appropriate for this
purpose as it enables us to portray possible interactions among the individual turbines due to
disturbances and variation in wind speed, seen by each wind turbine unit. Thus, the approach
taken here relies on developing models of various power system components, including the
wind turbines and wind farm, to study and predict the behaviour of the wind farm and its
interaction with the grid. The utility grid is represented by a large synchronous generator to
capture the possible grid dynamics and its influence on voltage control performance. Our
reasons for considering the multiple wind turbine model instead of a simple aggregate model
include the following:
Voltage control is often achieved by appropriately regulating the reactive power.
However, in a realistic wind farm, each wind turbine has somewhat different
instantaneous wind speed and real power output. Consequently, the availability of
reactive power generation produced by each wind turbine is also different.
•
The controllers of individual wind turbines may interact with each other, and their
action may affect the grid dynamics.
For these reasons, it is not appropriate to represent the wind farm using an aggregate model.
Instead, each wind turbine is included as a separate module in the overall model of the
system.
Since the overall system is nonlinear and the system state depends on the operating/loading
conditions, a robust controller should be designed to account for the variations in the system.
The linear matrix inequality (LMI) techniques have been previously considered for robust
tuning of controllers [35]—[45]. The use of the LMI-based approach provides readily
applicable and robust tuning considering multiple linear systems. In this thesis, a linear
quadratic regulator (LQR) tuning is achieved using the LMI-based technique. The LQRLMI-based approach is considered because it guarantees closed-loop stability for varying
operating conditions.
To design a controller that is valid for a range of operating conditions of interest, the overall
system is linearized at the three operating conditions - nominal load, 50% increase, and 50%
decrease of the local load impedance. This choice is based on expected daily local load
deviations. Since the order of each linearized model is very high, we use the balanced modelorder reduction technique to find the lower-order transfer functions that are more suitable for
the purpose of a controller design.
Finally, we carry out the simulation studies and analyses of system impacts due to small
disturbances (such as wind-speed variations and changes in load impedance) and a large
disturbance (three-phase symmetrical fault).
2.6
Contributions
To improve the voltage control under nominal wind variations as well as during the
disturbances (load variations, fault and/or line tripping), we propose an innovative
supervisory voltage control scheme in this thesis. We compare the proposed control to the
traditional methods and show that it improves the transient performance and fault-ridethrough capability of the considered wind farm system. Overall, the contributions of this
thesis include the following:
1. The proposed supervisory control scheme achieves a direct voltage control at a
remote location, PCC, and does not require installation of additional compensating
devices to meet the grid-connection requirements.
2. The proposed scheme takes into account the power output and limits of individual
wind turbines and is readily applicable to larger wind farms of different
configurations. Voltage control is achieved by appropriately regulating the reactive
power injected by each wind turbine, whereas the control of real power may be
allowed and/or included whenever the limits have been reached.
3. An innovative cost-guaranteed LQR-LMI formulation of the controller design is also
proposed. The final controller is tuned for a range of operating conditions using the
proposed cost-guaranteed LQR-LMI approach, and is shown to improve the system's
dynamic performance over that of traditional control solutions.
Chapter 3
Model Description of Grid-Connected Wind Farm System
Traditionally, studies of wind power generation systems have been carried out using a socalled aggregate representation of wind farms. Even though such studies have limited
accuracy and application [46], using an aggregate wind farm model is sometimes acceptable
especially at the distribution level or in cases when the interaction among the individual wind
turbines is not likely to be of importance. However, as the size of modern wind farms and the
individual wind turbines continues to increase, it is important to develop a more general
model, wherein each wind turbine is represented as a subsystem. Such dynamic models
would be of great help in more accurately evaluating a wind-power generation systems
performance during normal operation as well as during disturbances.
Although personal computers are becoming increasingly faster, computational speed is still
one of the limiting factors in dynamic simulation of power systems [47], [48]. Electrical
transients have very small time constants that require small integration time steps and result
in long computation time. To keep the simulation speed reasonable, special attention should
be given to model development. In particular, in this thesis, to increase the simulation speed
of various electrical components, these components are modeled in the qd - synchronous
reference frame [49]. For the same reason, the power electronic converters are represented
using average-value models that are also expressed in appropriate qd - reference frames.
This chapter describes a model of the system considered in this thesis, including mechanical
and electrical components of each wind turbine. The system considered herein corresponds to
a candidate industrial site on Vancouver Island, Canada [34], The system parameters are
summarized in Appendix A. All model measurements are expressed in per unit, such that the
values of all variables are in pu., except time t. To preserve the units of time t in sec., the
respective state equations are normalized by the base frequency
3.1
a .
Study System
A simplified diagram of the system considered herein is shown in Figure 3.1. Without loss of
generality, only three wind turbines are included here to represent a possible dynamic
interaction among individual wind turbines. The model itself, the control methodology, and
conclusions are readily extendable to larger systems. In the system considered, each wind
turbine is equipped with a step-up 0.69/34.5kV transformer. The wind turbines are connected
in a chain using cables (9km). The details of the individual wind turbine considered in the
model are shown in Figure 3.2. In this thesis, the GE 3.6MW wind turbine [8] is considered.
The wind farm is connected to the grid through a 34.5/132kV transformer and a 132kV
double-transmission-line (100km). A large synchronous generator (SG) represents the grid.
At the 34.5kV level, the wind farm feeds a local load connected at the PCC. The block
diagram of the overall model showing the individual components and the respective inputoutput variables is depicted in Figure 3.2. Since the overall model includes the individual
wind turbines, each turbine can have an independent wind speed, denoted by v w t ] through
v
wt3 •
WT2
WT3
0.69/ Q
34.5kVCj
.6
0.69/
34.5kV
1,..., 9 - bus number
WT1,... - wind turbines
T1, T2,... - transformers
C1, C2, C3-cables
TL - transmission line
PCC - point of common
coupling
WT1
0.69/
34.5kV
C1
9km
9km
100km
Local I
• -i X
Load T
T
'
' T4 "
Utility Grid
(Syn. Gen.)
TL
(PCC)
Figure 3.1: Wind power system considered for dynamic studies.
Wind speed input(s)
v
g,Wtl
14 -<
(3.17)
t4
TL
(3.15)
t5
SG
(3.18)
sg
Figure 3.2: Block diagram showing subsystem input-output variables.
A more detailed diagram of the individual wind turbine is shown in Figure 3.3. The wind
turbine consists of the following major components: a three-blade rotor with the
corresponding pitch controller; a mechanical gearbox; a doubly-fed induction generator
(DFIG) with two voltage source converters (sometimes known as the back-to-back voltage
source converter, VSC); a dc-link capacitor; and a grid filter. Mechanical power comes
through the three-blade rotor and the gearbox to the shaft of the DFIG, which has rotor speed
denoted by cor. The power is then taken from the DFIG through the stator side Ps and the
rotor side Pr. The stator side is directly coupled to the 0.69/34.5kV transformer, which
operates at the grid frequency. Variable-speed operation is achieved by appropriately
controlling the two converters. In particular, the rotor-side converter provides the real and the
reactive power necessary to attain the control objectives for either the PFC or the LVC
modules. The grid-side converter is connected through the filter, and its main objective is to
maintain the dc-link capacitor voltage by exchanging the real power with the grid. The
mechanical and electrical components of the wind turbine are described in more detail in the
following section.
W i n
d
speed
Mechanical
power
v qd,r
Terminal of WT
P
Rotor-side
converter
DC-link Grid-side
converter
VSC Controller
Figure 3.3: Variable-speed wind turbine with DFIG.
g
Generally, the absolute value of slip cos is much lower than 1; consequently, the real power
of the rotor Pr is a fraction of the real power of the stator Ps as Pr ~ cosPs. The grid-side
converter is used to generate or absorb the power Pf,i
constant. In steady-state for a lossless converter, Pfii
depends on the power
3.2
ter
ter
in order to keep the dc-link voltage
is equal to Pr and the rotor speed cor
absorbed or generated by the rotor-side converter.
System Model Components
3.2.1 Mechanical components
In most applications, the wind turbine is operated to extract as much power from the
available wind as possible without exceeding the ratings of the equipment. The mechanical
components of the wind turbine include a pitch control, a gearbox, and a three-blade rotor.
A. Pitch control
The block diagram of the pitch control [8] is shown in Figure 3.4. The mechanical power
generated from the wind can be calculated using a well-known relationship,
Pmech=^A rvlc p(A,d)
where Pmech
(3.1)
is the mechanical power in W; p is the air density in kg/rr? ; Ar is the area
swept by the rotor blades in m2; vw is the wind speed in mjsec; C p(X,6)
is the power-
conversion function, which is commonly defined in terms of the ratio of the rotor blade tip
P CO
speed and the wind speed here denoted by Z=
1
1
vw-
; Rt is the rotor radius in meters; cot is
the turbine rotor speed in rad/sec; and 6 is the blade pitch angle in degrees. The function
C p (A, 6) is often obtained as a numerical lookup table for a given type of turbine. The GE
wind turbine parameters for the energy conversion function C P (A;0) are given in Appendix
A in per unit on a 4MW base.
The pitch control attempts to keep the value of the turbine rotor speed constant by providing
the set-point to the pitch-angle actuator. The response of pitch control is relatively slow
compared to other controllers such as the torque control and pitch compensation. Thus, the
turbine control results in an auxiliary control signal into the pitch actuator for faster damping.
When the available wind power is higher than the rated power of the wind turbine, the blades
are pitched out to reduce the mechanical power delivered to the shaft PME CH
such that it does
not exceed the power rating P m a x . When the available power is less than PMA X,
are set at minimum pitch to maximize the mechanical power PME CH•
the blades
The variable PGBT is the
set-point value of the output of the wind turbine.
pset
rg
Figure 3.4: Block diagram of the pitch control.
B. Two-mass rotor model
A block diagram of a two-mass rotor model of a wind turbine with separate masses for the
turbine and generator is presented schematically in Figure 3.5. The aerodynamic model
describes the energy conversion from kinetic energy of the wind to the mechanical energy on
the wind turbine rotor. The inputs to the aerodynamic model are wind speed vw,
and the
blade-pitch angle 6. The mechanical rotor speed cot depends on the mechanical torque
T mecfj acting on the drive train. The drive-train model receives the mechanical torque
T MEC H
and electrical torque T S and computes the electrical rotor speed (Or. Here, H T and Hq are
the turbine rotor and gearbox inertias, respectively, and H R is the generator rotor inertia. The
coefficient DT S represents the shaft damping, and K T S is shaft stiffness.
Aerodynamic
Turbine rotor
Shaft
Gear
Generator
Drive train
Figure 3.5: Simplified block diagram of the two-mass rotor.
The block diagram shown in Figure 3.6 represents the rotor model, drive train, and generator
model, all expressed in per unit [8], In this representation, since the gear inertia is very small
compared with that of the wind-turbine blade rotor and generator rotor, the shaft and the gear
are represented by a common damping coefficient Dtg
and the stiffness K tg coefficient,
respectively. Since the GE energy conversion function C p (X, 0) is given on the 4MW base
but the overall model here is developed in per unit on the 100MW base, the corresponding
coefficients must be multiplied by a constant K pu = 4/100, which is derived from the
following relationship
rp
_ rpOld rpOld _ rpYieW rpfieW
actual ~1 base ' 1 pu ~ base ' 1 pu
(3.2)
1
The calculation of T mecij and X with the Pmecf,
^mech
mech = Kpu
V °>t J
1T
and Rt are then rewritten as follows:
Rtcot
X =-
and
(3-3)
+A
1
2H t
1F +
1mech
D,'tg
1
s
°h,base
-o
1
CM K,tg
+ a
Ts — K J
•
2H„
tyfiase
Figure 3.6: Block diagram of the two-mass drive-train rotor model.
3.2.2 Electrical components
The electrical components of the wind turbine include the DFIG and the voltage source
converter. The remaining electrical components of the overall system include the
transmission line, transformers, cables, and the load. The corresponding subsystem modules
are described below.
A. Doubly-fed induction generator
The DFIG is represented in the qd- synchronous reference frame. The corresponding
equations in per unit [49] are
R n
l
• .
qs= s qs +COe¥d
v
S
+
-
1
dX
fqs
dt
~
1 dVds
ds = Rslds - <»eVqs +
dt
v
v
qr= Rr iqr+®sVdr
(3.4)
1 dWqr
+
dt
1 dWdr
dr = Rr'dr ~ ®s¥qr +
cob dt
v
with the flux linkages expressed as
Wqs
=
(As + An )iqs + Aw V»
Wds
=
(As + An )*ds
+
An^dr
(3-5)
Wqr ~ (A- + Lm)iqr + Lmiqs,
y/ d r — (L r + Lm)icjr
+ Lmid s
where vqs and vd s are the stator voltage; vqr and vd r are the rotor voltage; iqs and id s are
the stator current; iqr and ijr are the rotor current; Rs and Rr are the stator and the rotor
resistance, respectively; Lm is the mutual inductance; Ls and Lr are the stator and the rotor
leakage inductance, respectively; co^ = Info is the base angular speed (rad/sec) with /q at
60Hz; coe and cor are the stator and the rotor electrical angular speed, respectively;
0)s-caeand yd r
cor is slip electrical angular speed; y/ qs and y/^ are the stator flux linkage; y/ qr
are the rotor flux linkage; the subscripts q and d indicate the quadrature and the
direct axis components as expressed in the reference frame; and the subscripts s and r
indicate the stator and the rotor quantities, respectively.
The stator voltages can be compactly expressed as the vector Vqd>s =
, vds
. These
voltages are the input to the DFIG model and are obtained from the low voltage side of the
0.69/34.5kV transformer model as the voltage vector V QCJJ
R
• The electrical torque T S, the
stator real power PS , and the stator reactive power QS delivered by the generator are
calculated as
T s = Vdr'qs -Vqrids
(3-6)
(3.7)
Qs=vqsids- vds iqs
(3-8)
B. Voltage source converter
The variable-speed operation of the DFIG is achieved by means of two converters linked via
a capacitor as shown in Figure 3.7. A detailed description of the pulse-width-modulation
(PWM) switching scheme of the converter can be found in [49], [50]. The rotor-side
converter feeds the DFIG rotor with the reactive power and takes out the real power as
necessary to attain its control objectives. These objectives usually consist of maintaining
turbine speed and either controlling the stator power factor (PFC) or terminal voltage (LVC).
Real power requirements for the rotor-side converter are provided by drawing current from
or supplying current to the dc-link capacitor. The grid-side converter is connected to the grid
through the filter. The main objective of the grid-side converter is to maintain the voltage
level on the dc-link capacitor by exchanging real power with the grid.
Rotor side
DC-link
Grid side
R
t j -HCj -HtJ
l
al
T-W.
-w.—
v
dc
eft
anl
f'lter
ia2
™
'62
'41
v
L
filter
>c2
-•EJ
v
bn\ vcnl
v
cn2 vbn2van2
V
cn2 vbn2 van2
Figure 3.7: Schematic representation of the voltage source converter.
Grid side
DC-link
T1
v
T3
£
'a 2
l
C
dc
T6
T4
Js}
4
>
b2
A
duty cycle
d c(t)
triangle signal AAAAM
g c(
0
0N
•OFF
A
v
v
cnl
comparator
>
bn2
g c ( 0 = 4 : (OS
T=0_
Figure 3.8: Representation of the switching function on the grid-side converter.
an2
The output of the comparator is the pulse train g c{t) depicted in Figure 3.8, which shows the
c- phase only. In particular, the high-frequency triangle waveform is compared with the
sinusoidally varying duty-ratio function d c(t) [49]. When the magnitude of the triangle
waveform is greater than that of the duty-ratio waveform, the switch is in ON mode. The
overall switch operations in the 6-pulse converter are listed in Table 3.1 along with the
corresponding on/off status of the switch, i.e.,
TI/T4
. For example, State 3 indicates that the
switches T3, T4, and T5 are ON, and the switches Tl, T2, and T6 are OFF.
TABLE 3 . 1
SWITCH OPERATIONS
State
1
2
3
4
5 .
6
Tl/T4
1
0
0
1
1
0
T2/T5
0
1
0
1
0
1
T3/T6
0
0
1
0
1
1
( l : O N , OiOFF)
The duty cycle for switching each phase can be specified as follows:
d a=d cos(6>c)
f
df,=d cos
d c=d cos
3
(3.9)
/
3 ,
where 6 C = 6 e + 6 . The variable 6 is the phase shift between the synchronous reference of
the system and the converter, and 6 e is the synchronous angular displacement.
The switching action and harmonics of the converters may be ignored and replaced in the
model with the appropriate average-value relationships [49]. By assuming that the frequency
of the triangle wave is much higher than the frequency of the desired waveform, the average
t
magnitude terminal voltages van2,
i
t
Vbnh
ar)
d vc„2 of the grid-side converter are described as
v
an2 = da vdc
»
(3.10)
v
bn2 = db vdc
v
cn2 = dc vdc
A change of variables to the qd- synchronous reference frame [49] is then applied such that
v
qd
where
(3.11)
=T qd(8 e)yabc
(vf
= vqI
T
t
v
d and ( «6c) = van2
v
v
bn2
v
cn2
and the qd - transformation
matrix is defined as
cos(0 e )
cos 0o-
e
Tqd( e) = -Z
sin(0 c )
In
T
cos
In
sin
sin Ge +
In
(3.12)
The terminal voltage of the converter in the qd - synchronous reference frame is then
vq=dcos{6)v
d c,
vd=-dsm{6)v
dc
(3.13)
By defining the control signals as vqu =c/cos(0) and vd u = - d s i n ( 0 ) , the magnitude of
the duty cycle is d = yjvqu +vd u
and the angle displacement is 9 = - t a n - 1 [v d ujvqu
j .
Hence, using the control signals vqu and vd u, the average value of voltages of the grid-side
converter can be expressed as
v
q=vquvdc>
v
d =
(3-14)
v
du vdc
C. Transmission line, transformer, and cable models
Since the transmission line considered here has medium length (80km~240km), we
considered it appropriate to represent this line using an equivalent lumped-parameter n model [28], Such a model can be expressed in the ^ - s y n c h r o n o u s reference frame
depicted in Figure 3.9 using the equivalent R , L , and C elements [49]. The cables and
transformers are also represented using the model structure similar to that shown in Figure
3.9, wherein the appropriate R , L , and C parameters are used. For the formulation of
transformer models, the capacitors at the sending and receiving ends are not used. The
detailed parameters for each component model are summarized in Appendix A. The
respective equations of the line segment depicted in Figure 3.9 are as follows
/it T.i
1
R
»
T ^
d\
l
dc\
>
0)Li,
e
-Idl
qi
<
13
l
di
l
dc2
+
i
v
d\
C-'
Ve C Vq2
•? c
Figure 3.9: Transmission line lumped-parameter qd - model.
A v
q = Vq\ ~ vq2 = RTl}ql
Av
=vd\
d
~vd2
_C TL d v<l
,
+ — 0 )
(Ofo
= RTL'dl +
r
L j
L r l
(Ob
l . „
e
ut
dt
L
i
d
l
-OeLniql
v
;7T + ^e^TL vdl
(Of,
at
_ C T L dv d x
dc\ ~
a L
e TL vql
T
COfo
at
_CTLdvq2
l
T~ + °>e LTL vd2
qc2 ~
*
;
z
<£2
A v
dt
_ C T L dv d 2
(fy
a> L
e TL vq2
T
at
q = vq\ - vq2 = Rcalql
Av
d
iqcl =
(3.15)
L
+
(Ofr
,, +
_ C ca
Vcl - —
T
(% dt
;
Av
at
q\ ~ ^qcl
v
d\ =
R
oidcl
at
^A^rf/
+ L c a d l f - coeLcai qi
cojj dt
(3.16)
„ ^ „
^e c ca v ol
y
Ltr d lql + 0) L i i
e tr d
(% dt
+
Avd = vd l - vd 2 = Rtrid l
(% dt
v
I +
4QaV£/l
=Rtriql
<7 =
di qi
°
CO/,
= vd\ ~ vd 2 =
C ca dv q\
C
• CO eLtri qi
(3.17)
Note: For the notation used here to match with that used in Figure 3.2, the voltage equations in
(3.15)—(3.17) need to be correlated to the bus number indicated in Figure 3.1. The current
equations for the transmission line, cable, and transformer need to be referred to as
TL, ca, and tr as well.
D. Utility grid
A large synchronous generator is considered to represent the utility grid. The model of the
synchronous generator including the excitation system is taken from [13]. The generator
equations are as follows
^r =0)b( vs
at
-Ra*a +kl<°eV a)
d¥f
=
dt
®b(Efd-
(3.18)
R
f if)
dVk
dt
where
[» a
h
'/]7'=L1[Vfl
V*
Wfl
with
k1
"0
=
Vk =
- 1 "
1
sq
»
i
«
=
jsd _
Vsq
¥sd
V
l
'Wkq
}fkd_
and
0
Rkq
0
0 R•kd
, k2 =
»
v
5 =
sq
ySd.
.
kq
h = }kd _
Lamq
0
Lmq
0
0
0
Lamd
0
Lmd
Lmd
0
T-'kqmq
0
0
Lm d
0
Lkdmd
Lmd
0
L = Lmq
0
0
Lmd
L
Lmd
fmd
where v and i are the voltage and current vectors, respectively; coe is the angular speed; y/
is the flux linkage; R and L are the resistance and inductance, respectively; Ej d is the dc
field voltage; and the subscripts a,k,f
denote the armature, damper, and field quantities,
respectively.
The above-described generator model was considered together with the excitation system
[51] that regulates generator terminal voltage by controlling the field-winding voltage. The
block diagram of the exciter model considered here is depicted in Figure 3.10.
Exciter
Filter
1
sTp+Ti
s
1 + ST r
1
l +
sT
ex
+ ^
E,'id
Figure 3.10: Exciter model block diagram.
Here E i s
the dc field voltage and E i s
the initial value; and V s and V rsef are the
measured magnitude of the generator terminal voltage and the reference voltage, respectively.
The time constants necessary for filtering the rectified terminal voltage waveform are
reduced to a single time constant
. The exciter gain is represented by the parameter T ex.
Automatic voltage regulator (AVR) gains are given as T p and 7}. The thyristor-controlled
rectifier is represented by a scalar K t [51].
E. Load
The dynamic RL load model represented in the qd - synchronous reference frame is depicted
in Figure 3.11.
coLi.,
co Li,
Figure 3.11: RL load model represented in the qd - synchronous reference frame.
The corresponding state equations are
di
koad
gl
CQb dt
L
load
- vq2 - Rload lql
di
dl
_v
d2 ~ Rload ldl
t d t
v
q2 ~
R l
o qc2'
v
d2 =
~
^ekoad^l
(3.19)
+ Mekoadiql
R
oidc2
1
where Rq is 10 . The output voltage vector
v
b7,pcc
as
2
defines the voltage vector at the PCC
shown in Figure 3.2. The current vector i q ( j2 is equal to the sum of the current
vector i t 4 and i c a b 3 .
3.3
Voltage Source Converter Controller
An important part of the wind turbine is the voltage source converter controller that controls
the voltage of the rotor-side converter, the dc-link, and the grid-side converter. Different
schemes arid detailed information can be found in [10], [52], [53], This chapter presents
transfer functions, which are used to tune internal controllers. Figure 3.12 shows a block
diagram of the voltage source converter controller modules and the respective input-output
variables.
v ,
qdjilter
qa,r
set
Q'filter
. set
\dfilter
8
^g
Figure 3.12: Block diagram of the voltage source converter controller modules.
In Figure 3.12, PS GET
and QS GET
are the set values for the real and reactive power for the
terminal of the wind turbine. When the unity power factor control mode is applied, QS„ ET is
o
set to zero and all reactive power to the DFIG is provided via the rotor-side converter. When
the local voltage control mode is used, QS GET is adjusted by the local controller to maintain the
voltage at the wind turbine terminal. Figure 3.13 shows the maximum tracking power output
via the turbine power characteristic with respect to the rotational speed of the rotor. This
characteristic curve is obtained based on the maximum power tracking curve given in [54]
and is calculated for the GE 3.6MW turbines considered here on the 100MW base. The value
of PGET
is determined by this wind turbine energy harvesting tracking characteristic, which is
represented here as a look-up table PGET (CO R). The turbine power characteristics PME CH are
obtained at different wind speeds. The actual speed of the turbine cor is measured and the
corresponding mechanical power of the tracking characteristic is used as the set-point real
power PGET
for the real-power control loop.
3
-3
I*
i>
£
o
a,
"3
cx
3
O
T3
o
'3
C3
-C
ou
s
H
3
<D
0.6
C
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Rotational speed of the rotor {a>r, pu)
Figure 3.13: Turbine power versus speed tracking characteristic.
3.3.1 Rotor-side converter controller
Figure 3.14 shows a block diagram of the rotor-side converter controller module, which
includes four internal conventional proportional-plus-integral (PI) controllers, PI1 through
PI4. The controller is implemented as two branches, one for reactive power (PI1 and PI2) and
one for real power (PI3 and PI4), with the corresponding de-coupling terms between the qand d-axes, respectively. The actual electrical output power Pg, measured at the terminal of
the wind turbine, is compared with the set-point power obtained from the tracking
characteristic Pget(a> r).
The PI3 regulator is used to drive the error in power to zero. The
output of this regulator is the rotor current set-point
that must be injected into the rotor
by the rotor-side converter. The actual current id r is compared to i'^f and the error is driven
to zero by the PI4. The output of the PI4 is the voltage vd r
generated by the rotor-side
converter. Similarly, the PI1 regulates the set value of the reactive power at the terminal of
the wind turbine, and its output is the value of the g-axis rotor current. The PI2 regulates the
<?-axis rotor current set value. The output from PI2 is used to compute the command value
for the rotor voltage vqr . As can be seen in Figure 3.14, the two speed-voltage terms (O s\j/ d r
and cosy/ qr are compensating the coupling terms in (3.4). For consistency, a brief discussion
of the controller design is given below.
A. PI2 and PI4 controller design
The rotor voltage equations can be written with the stator flux dynamics neglected [53], as
follows: .
V
= R l
r qr
+
~1T + ®Wdr
(% dt
v
dr = Rr ldr +
(Ob
A
at
- Q)syf
(3.20)
'
• set
set
Ol
PI1
V
—*' rO
+~H
-
a
V
V
qr
+h
qr
+
V,
H p - H PI3 [ — » Q - H pi4 h > Q
I
I
g•
dr
+
V
qr
-f
v
dc
V,
dr
+
_
+ O — •
VSVDR
qr
: set
dr
. set
PI2
dr
"dr
>®—>
T
CO IIf
s t qr
Figure 3.14: Block diagram of the rotor-side converter controller.
By introducing the new variables
v
qr
Lr digr
R l
r qr
COfr
dt
(3.21)
Lty•
+
v
'dr ~ Rrldr
(%dt
(3.20) can be simplified to the following:
v
*qr=vqr+cos¥dr
(3.22)
V
dr
=V
'dr-^sWqr
To design PI2 and PI4, (3.22) can be rewritten in the Laplace domain to obtain the transfer
function from the rotor current to the rotor voltage in the q- and d-axes, as follows:
_ Iqris) _
1
Vqr
(3.23)
(.S)
Q ( :.)=Idr( s)
4 K J
V d r(s)
Rr+S^LrlcOb)
=
1
Rr+s{L r/03 b)
Based on (3.23), PI2 and PI4 can be designed with the gains summarized in Appendix A. In
this thesis, a pole-placement design technique [55], [56] is used, as summarized in Appendix
B.
B. PI1 and PI3 controller design
B.l
PI1 controller
To tune PI1, a transfer function from the #-axis rotor current to the reactive power of the
rotor-side converter should be considered. Because the g-axis of the selected synchronous
reference frame is tightly aligned with the stator voltage vector and the angle between the
stator and rotor voltages is relatively small, the reactive power is approximated as described
in [10], [48], [57]
Qr ~ v'qr ldr -vdr lqr
=
(3.24)
v
qr ldr
By using the approximate relationship Q' r ~ (O sQ' s and Q' g ~ (a> e - cos)Q' s where 0)e =1, the
chosen reference frame allows (3.24) to be rewritten as
1 -G>s
(3.25)
v
qr ldr
V % V
Assuming that slip frequency cos is small and that the controller is being tuned for the
nominal operating condition,
=1 on its local base, the term
c l1- 0 ) A
s
V
here as
f l *-cos
^
v <°s )
. Hence, (3.25) is rewritten as
f 1
f 1
N
V
y
I-6>s
v ^
ijf is approximated
j
qr
-
V
Lr di qr
Rfiqr +
dt
J
x
(3.26)
j
As a result, for the purpose of controller design, the transfer function from the rotor current in
the g-axis to the reactive power at the terminal of the wind turbine can be approximated as
Q'
G3(s) = -^- = Rr+sLr
qr
(3.27)
where Rr - {^>Rr
B.2
r
and
T
\
PI3 controller
The design of PI3 is similar to that of PI1. In particular, to tune PI3, a transfer function from
the c/-axis rotor current to the real power of the rotor-side converter should be considered.
The real power of the rotor-side converter can be approximated as
K ~ Vqriqr + vd^dr
(3.28)
v
dr ldr
~
By using the relationship
= asP' s and P' g ~ (co e -a>s )P' s where a>e-1 , the chosen
reference frame allows (3.28) to be rewritten as
rP'
g =
f 1
(3.29)
v
v °>s J
dr ldr
As it was previously assumed that slip frequency cos is small and id r ~ 1 on its local base
close to nominal operating conditions, (3.29) is then approximated as
P'8 -
r1
f 1
>
dr
v ^
j
\f
I -a>s
v
r
\
L
Rrid r + r
J
di dr
dt
(3.30)
As a result, the transfer function from the rotor current in the J-axis to the real power at the
terminal of the wind turbine can be approximated as
P'As)
GA(s) = -2— = Rr+sLr
!
dr( s)
(3.31)
Hence, PIl and PI3 can be designed using the transfer function G^s) or G ^ s ) . As before,
the pole-placement technique [55], [56] has been used to find the proportional and integral
gains, as summarized in Appendix B.
3.3.2 Grid-side converter controller
As shown in Figure 3.3, the grid-side converter is connected to the grid through the filter.
The voltage equations for the filter in the (^-synchronous reference frame are
D
v
q, filter ~ Kfilter
1
.
q, filter +
^filter
dig,filter
.
+ meLfilter
1
d, filter
(3.32)
v
d,filter
D
~ Kfilter
1
.
L
filter
filter
d,
^d, filter
~J t
T
co L
e filter'q,
filter
Equations (3.32) are coupled by the corresponding speed-voltage terms that must be
considered in the design of controller. By introducing the new variables
L
q, filter ~ Rfilter
D
K
v
d, filter ~ filter
1
q, filter +"
1
.
filter
dig,filter
(Ojj
dt
L
filter
d, filter
(3.33)
did, filter
the modified command voltages of (3.32) can be expressed as
V
hfilter
~ V'q,filter
+ ®eL filter
'd, filter ~ Vd, filter " ®eLfilter
1
1
d, filter
(3.34)
q, filter
The corresponding transfer function can be expressed as
G5(S) = I«/'^S) V
q, filter 0 )
Rf
Ute r
1
+ S (l FLHE R
j )
(3.35)
„ ,
&
N
1
d,filter
s
( )
6(f)=7T
Yd, filter 0 )
1
R filter
+ S
[
L
filter /<°b )
which is considered for tuning the controllers PI5 and PI6.
Figure 3.15 shows the grid-side converter controller block diagram. The input current setvalues are calculated by the real and reactive power commands P0ter
•set
q,filter
•S€t
_ld, filter
v
q,tr
v
~vd,tr
V
l
Here \ q < j > t r = \ y q > t r
transformer; Pfi/
ter
The value for Qflj
ter
d,tr
vdtr J
ter
ter
as
-1 tpset
q,tr_
and Qfif
and Qfi[
,filter
Qfiiter
is the voltage at the low-voltage-side wind turbine
are the set-point of the real and reactive power commands.
is set to zero if the unity power factor control is used; however, Pff/
ler
is provided by the dc-link controller, which determines the flow of real power and regulates
the dc-link voltage by driving it to a constant reference value. As depicted in Figure 3.15, the
compensation terms a>eLfiit erid,
filter
^e^filter'q,
filter
decouple the speed-voltage terms
in (3.32).
Figure 3.15: Block diagram of the grid-side converter controller.
3.3.3 DC-link controller
The capacitor in the dc-link is an energy storage device. Neglecting losses, the time
derivative of the energy in this capacitor depends on the difference of the power delivered to
the grid filter, PFII
TE R
, and the power provided by the rotor circuit of the DFIG, PR, which
can be expressed as
2 cob
J
dt
p
(3
3 7 )
K
Note that since all variables are in pu, except time t, the state equation (3.37) is normalized
by cob to get time in sec. Based on (3.37) the dc-link voltage will vary according to the
following state equation:
~~~vdc
- j f - = Pfilter
(0^
~Pr=
^cjllte^dc
~ [dc,r vdc
(
3
-38)
at
which can be rewritten as
dv
^ dc
Qc
where i d c
l
dc, filter
and idc,filter
r
(3.39)
l
dc,r
are
^ e current in the rotor-side converter and the grid-side
converter, respectively. The transfer function from the current id c, filter
t0
the dc voltage vdc
is therefore as follows:
c
W
. - S * « ! L - = - a dc, filter v*^)
(3.40)
sL
dc
The dc-link model with its controller is shown in Figure 3.16. The rotor-side converter
current i d c
r
is obtained by idc,r~ Pr/ vdc
obtained by Pf{
ter
= vd cis/ c[ fllte
r
.
The real power set-point Ps^l ter
in (3.36) is
• Pfllter
+r\
w
a
b
• set
l
dcfilter
>-
v
dc
ldc,r
Figure 3.16: Block diagram of the dc-link model and its controller, PI7.
The voltage source converter controllers PI1 through PI7 are tuned using the respective
transfer functions of G\ (s) through G 7 (5) . In this thesis, a conventional pole-placement
technique [55], [56] was used, as described in Appendix B, and the respective control gains
are summarized in Appendix A.
3.4
Conventional Voltage Control of Wind Turbine
Figure 3.17 shows the combination of the voltage source converter controller with respect to
the rotor-side converter, the dc-link, and the grid-side converter from the top to the bottom.
The conventional control modes PFC and LVC are also described in [10], [52],
The objective of PFC is to achieve zero reactive power consumption by the wind turbine Qg .
To implement this mode, the reactive power set-point Qsget in Figure 3.17 is simply set to
zero. To achieve the LVC, the reactive power set-point Q g e t is made available to another
control loop that drives the voltage at the wind turbine terminal to a specified value. The
design of this additional controller is summarized in Appendix C. Thereafter, these two
conventional controls schemes are used for comparison purposes.
Figure 3.17: Overall control block diagram of the voltage source converter.
Chapter 4
Wind Farm Voltage Control
4.1
Common Practice
Traditionally, additional compensating devices such as static VAR compensator (SVC), static
compensator (STATCOM), transformer with tap changer, etc., have been considered to
improve voltage regulation associated with the variable nature of wind energy [9]-[12], [31],
[32], [58]—[60]. A coordinated voltage control strategy for the DFIG and the on-load tap
changer has been proposed in [60], wherein a single DFIG was considered. However, there
are always costs associated with the installation and operation of any supplementary devices,
which makes this option less attractive. Moreover, increasing wind power penetration has
been noted to influence overall power system operation in terms of power quality, stability,
voltage control, and security [17], [20], [21], [23], [25]. Consequently, in many modern wind
farm applications, the voltage regulation at a specified remote PCC where a load of particular
concern is connected may need to be addressed as a requirement for the grid connection. To
achieve easier grid integration and reliable voltage control in the system, alternative voltage
control schemes for wind turbines become very important.
This chapter describes an innovative supervisory voltage control scheme that does not require
installation of additional compensating devices and is applicable to wind farms of different
configurations. The key to the supervisory control scheme is to regulate the voltage at a
specified remote PCC by adjusting the reactive power produced by the individual wind
turbines while taking into account its operating limits.
4.2
Available Reactive Power in a Multi-Turbine System
On a practical wind farm site covering a sufficiently large area, each wind turbine may have
somewhat different instantaneous wind speed and output of real power. Consequently, the
capacity for reactive power generation of each wind turbine is also different. Due to these
differences, it is not appropriate to represent a wind farm using the aggregate models.
Instead, we consider each wind turbine as a separate module in the overall model of the
system, as described in Chapter 3.
When controlling multiple turbines, it is important that the operating limits of each individual
turbine (in particular the internal voltage source converter) are not exceeded. Assuming a
proportional distribution, the portion of reactive power required from an individual
jth
voltage source converter can be computed as
set
2 j,g
where
=
m
m
QJT
max
Qj,c '
X
•>max
Qlc +Q2T+-
max
+ Qnl
c
AGpcc
(4.1)
is the maximum reactive power (limit) that the y' th voltage source converter
can provide, and AQpcc
is the total reactive power required to support the voltage at the
PCC.
The quantity Qj
c
may be different for each wind turbine and is dependent on operating
conditions. To understand how this quantity can be calculated, it is instructive to consider a
single voltage source converter first. Figure 4.1 shows the real and the reactive power
operating limits, wherein it is assumed that a converter (the rotor-side or the grid-side) should
not exceed its apparent power limit depicted by the half-circle. Suppose that at a given time
the converter is delivering real power, denoted herein by Pc, that is changing depending on
the wind condition. Then, in addition to the real power, the converter can supply or absorb a
maximum of 2 c
m a x
of the reactive power. So, the reactive power available from a single
converter lies within the limits [ - 0 C ; 1 T i a x ;
+ 0 C ; m a x ] , which are dependent on operating
conditions.
+P
*
Absorb
reactive power
0max
^(pu)
Qc,max
Supply
reactive power
Figure 4.1: Real and reactive power operating limits of the voltage source converter.
Since the same real power Pc must pass through both the rotor-side and the grid-side
converter, and each of them is limited as depicted in Figure 4.1,. the maximum available
reactive power from the voltage source converter can be expressed as
(4.2)
where it is assumed that reactive power, which has been supplied to the DFIG, is Qj r , and
the nominal apparent power of the converter is S ™ x , which is defined as 1/3 of the wind
turbine rating [8], Based on Figure 4.1, it also follows that -Sf
ax
< Pj
c
< Sf
ax
.
4.3
Supervisory Voltage Control Scheme
Equation (4.1) represents the basis for the proposed supervisory voltage control scheme
depicted in Figure 4.2. It should be noted that this controller will require information from all
wind turbines. However, since the overall control objective is to regulate the voltage at a
single remote PCC, this centralized scheme appears to be reasonable as well as justifiable. In
this scheme, the supervisory voltage controller provides the total reactive power Agp CC to
regulate the voltage at the PCC. Then, this total reactive power is distributed to each wind
turbine according to (4.1).
Since the system is nonlinear and the voltage at the PCC also changes due to load variations,
grid conditions, wind speed, etc., the supervisory controller in Figure 4.2 should be robust to
ensure stable and adequate dynamic performance in a wide range of conditions. In this thesis,
it is assumed that the system operating conditions are mainly determined by the load
variations, which are in the range of ±50% during the day. To accommodate this range, one
may consider designing several controllers and then switching them (or their gains),
depending on the system's condition. Alternatively, to design the supervisory controller for
robust operation, it is possible to view the entire model of the system as a collection of linear
systems (spanned by the range of operating conditions) for which a common controller is
designed and tuned. The latter approach is taken in this thesis.
A control-design technique known as the linear quadratic regulator (LQR) may be
conveniently utilized for multi-input multi-output systems. The LQR approach can be used
for tuning the controllers with some specified properties, such as phase margin (-60°,60°)
and gain margin ( - 6 dB, inf dB) [61], The design procedure consists of finding the solution
to the Riccati equation that satisfies a certain cost function. However, to make this approach
applicable simultaneously to several linear systems, the LQR problem can be formulated as a
linear matrix inequality (LMI) solution for which a common Lyapunov function for the set of
considered linear systems is found if it exists. The controller designed utilizing this common
Lyapunov function guarantees system stability (in the Lyapunov sense) in the range of the
considered region. This thesis adopts the LQR design approach formulated as a system of
LMIs.
4.4
Plant Model and Conventional Controllers
4.4.1 Linearized and reduced-order model
The first step in controller design considered here consists of finding a linearized plant model
that captures the relationship between the input and output with regard to the control
objectives. With respect to Figure 4.2, we need to find a transfer function from the reactive
power injected by the wind farm to the voltage at the PCC. It should be pointed out that
although state equations of all the model components are available as presented in Chapter 3,
it is not practical to derive the required linearized model analytically. Instead, in this thesis,
the respective linearized models are obtained using numerical linearization (a feature
available in Matlab/Simulink) of the overall model about a specified operating point. To
cover the operating range of interest, three operating points determined by the local load
impedance of -50%, nominal, and +50% were considered. These operating conditions
correspond to the expected average daily peak load variations and are summarized in Table
A. 10 in Appendix A. In this thesis, these three operating points are assumed sufficient to
represent the desired operating range and therefore are considered for control design
purposes. The corresponding numerically obtained transfer functions from the reactive power
to voltage at the PCC were found to have 104th-order. For a system of such high order,
analytical derivations would not have been possible and/or practical. The corresponding
magnitude and phase plots are shown in Figure 4.3. Note that even though the load
deviations are large, the deviations of the linearized models in the frequency domain are not
very significant, which suggests that a linear controller may work adequately for the given
system.
The original 104th-order linearized model for control design might be possible, but is not
desirable, as it would require significant computational resources. However, as the visual
inspection of Figure 4.3 and 4.4 reveals, these transfer functions may well be approximated
in the frequency range of interest by a system of much lower order. In this thesis, a balanced
realization model-order reduction technique [62], [63] is used to find the lower-order
approximate transfer functions that are more suitable for purpose of control design. This
technique is based on considering the dominant states (modes) in the input-output behaviour
of the system. The method uses Hankel singular values of the system, which are the common
eigenvalues of controllability and observability Gramians. The reduced-order model is
obtained by neglecting the appropriate number of smallest Hankel singular values.
Numerical linearization and model reduction have been carried out using the Control System
Toolbox [64]. Based on Figure 4.3 and 4.4, it was considered sufficient to approximate the
respective transfer function with 4 th -order. The corresponding reduced-order transfer
functions magnitude and phase are plotted in Figure 4.4. As can be concluded by comparing
Figure 4.3 and 4.4, the 4 th -order transfer functions approximate the original 104th-order
system very well in the frequency range of interest. Hereafter, these reduced-order models
are considered to represent the plant.
Frequency (rad/sec)
Figure 4.3: Bode diagrams of the full-order model (104 th ).
Frequency (rad/sec)
Figure 4.4: Bode diagrams of the reduced-order model (4 th )
The 4 th -order reduced model may be realized in terms of the state variables that are related to
the voltage at the PCC as
i =
[Avpcc
A
Av
Vc
(4.3)
Avpcc]7
pcc
which contains proportional and derivative states. To guarantee zero steady-state error in
tracking the set-point voltage Vpf c, an integral action is needed from the controller [65], This
integral action can be expressed by adding an auxiliary state j A v ^ to (4.3) as
x ( 0 = rjAv-<
where |Avp CC =
x
(4.4)
(v^c - v p c c
. This state clearly delivers the integral action for the
difference between the set-point voltage
and the system output voltage v p c c . The
combined state-space equations for three systems with the auxiliary state can then be
expressed as
x(0 = A,-x(0 + Bju(t)
where A B
/ ;
"o
0
(4.5)
z (t) = Cj\(t) + D,-w(f)
C h Dj are the augmented system matrices
-Ci
B
i=
-V
»7
c
i =
I
"1 0 0 0 0"
0 10 0 0
'
D
Ii =
"0"
oj'
(4.6)
Here i = o,l,h and the subscript "o" denotes the nominal operating condition; "/" denotes a
50% decrease and "h" denotes a 50% increase of local load impedance, respectively;
Ae
D G Si mXP
is the state matrix; B e 3 i n X p is the input matrix; C e 3 i m X n is the output matrix;
is the direct feed-through matrix; x e
is the state vector; u e
is the input
signal vector; ze 3l m is the output vector; and A,-, B ; , C,-, D ; are the system matrices of the
4 l -order reduced model. Overall, this system has one input (p = 1), five states (n = 5 ), and
two outputs (m = 2), respectively. The corresponding state-space matrices are
"0
6.89150
10.2209
-1.7963
6.09420"
"0.02820"
0 -10.408
-54.899
6.36430
-17.017
-6.8915
A 0 = 0 54.8990
0 -6.3643
-52.306
13.7505
13.7505 -76.688 >
-4.7933 72.8850
0 -17.017
76.6880
-72.885
-99.634
-6.0942
"0 6.40620
9.55480
1.70940
4.54920"
0.00390"
0 -10.593
-51.313
-6.5215 -14.052
51.3130
-61.339
-16.300 -71.572
6.52150
0 -14.052
-16.300
-5.6663 -64.272
71.5720
64.2720
-75:609_
1.70940
-4.5492
7.08100
10.4324
-1.8855
6.79060"
"0.04200"
0 -10.294
-55.954
6.46570
-18.208
-7.0810
0
55.9540
-48.792
13.2378
-77.843
Bh - 10.4324
0 -6.4657
13.2378
-4.7963
77.4240
-1.8855
0 -18.208
77.8430
-77.424
-112.34
-6.7906
A,= 0
0
"0
Ah
=
C 0-C t-C h-
1
0
0
0 10
0
0'
0 0
(4.7)
B 0 - 10.2209
-1.7963
-6.4062
>
» /
=
9.55480
=D; =Di =
(4.8)
(4.9)
(4.10)
and the output and the state vector are
=[K"pcc
~\T
AVpCC
= [ jAvpcc
Avpcc
Avpcc
Avpcc
& v p c c J(4.11)
The output vector z is assumed to be directly available (measurable) for feedback purposes.
Table 4.1 summarizes the eigenvalues and damping ratios, as well as the corresponding
frequencies, of the 5 th -order reduced model.
TABLE 4 . 1
EIGENVALUES, DAMPING RATIO, AND FREQUENCY OF THE 5™-ORDER REDUCED MODEL
Damping ratio
Eigenvalues
0
-22.6±j20.8
-60.9±jl06
-
0.786
0.5
Frequency (rad/sec)
0
30.8
122
4.4.2 PID supervisory controller design
Before considering advanced controllers, it is prudent to investigate available traditional
approaches to see if satisfactory dynamic performance could be obtained using them. A
proportional-integral-derivative (PID) controller is commonly used in power industries. In
the remainder of this chapter, a PID-supervisory controller is designed based on the transfer
function corresponding to the nominal condition, and system performance is evaluated
subject to the three-phase symmetrical fault. More detail on the simulation studies is
presented in Chapter 5.
The PID gains are tuned to meet the specifications of less than 10% overshoot and greater
than 60 degrees phase margin. The gain crossover frequency 0)gc , which corresponds to this
phase margin, is 65 rad/sec, as can be found from the Bode diagram in Figure 4.4 (see the
line that corresponds to the nominal condition). The integral gain k t was first chosen as
2.2347, which corresponds to the dc-gain. By using the values of (Ogc and kj , the
proportional gain kp and the derivative gain kj can be computed using the following
equation [56]
k
V
or
p
+jo)
g c
k
h
d
+
ico,
-
G r{jCD
J
gc)
= Xe j6{ Wz c\
(4.12)
eM°>gc)
P
s
G r(jco gc)
where Gr(ja> gc)
jk
.
(4.13)
(o gc
is the transfer function of the 4' -order reduced model obtained for the
nominal operating condition, as follows:
Gr{s) =
-0.028s 4 - 2 7 . 7 8 S 3 - 7 3 5 4 s 2 +15618025 + 31501951
(4.14)
s 4 + 167.14s 3 + 21340s 2 + 788654s+ 14079701
By solving (4.12) with the gain crossover frequency (Ogc = 65 rad/sec, the gains of the PID
supervisory controller may be expressed as k p =a, k d =bja)gc
, and k t = d c g a j n . To speed
up the settling time, the integral gain k t is increased to 6.7122. The final computed gains are
summarized in Table 4.2.
To limit the effect of high-frequency noise, the derivative control branch is implemented as a
r
/
filter with an equivalent gain given as
k d-k dt
\
\
1s +1 . This is usually done to
kd
Vv y
"
avoid large transients in the control signal resulting from sudden changes in the set-point
[55]. The typical range of values for N d is from 2 to 20 (higher value implies stronger
derivative action); N d - 20 was used herein. The corresponding step responses of the
reduced-order system (4.14) in the open-loop and the closed-loop are plotted in Figure 4.5,
which shows that the overshoot of the closed-loop system is less than 10%, which satisfied
the design specification.
TABLE 4.2
GAINS OF THE PID-SUPERVISORY CONTROLLER
States
jAvpcc
Gains
6.7122
Av
pcc
0.4635
Avpcc .
0.0009
time(sec)
Figure 4.5:
Step response of the open-loop and the closed-loop reduced-order system.
To make this controller practical, the output control signal should be limited by the amount
of currently available reactive power. At the same time, limiting the control action should be
implemented together with the integrator-anti-windup scheme that would stop integrating the
error when the limit is being reached. To take into account the individual wind turbines, a
distributed anti-windup scheme that takes into account the limits of each turbine has been
considered. A combined diagram of the PID controller with the proposed anti-windup
scheme is depicted in Figure 4.6. The anti-windup scheme requires the currently available
reactive power limits Q™* defined in (4.2). As shown in Figure 4.6, the output of the
controller is distributed among the wind turbines according to (4.1), wherein each output is
compared to the respective limit Q™*. When none of the limits are reached, the overall antiwindup loop is inactive and the integral control branch operates in a normal way. However,
when one or more limits are being reached, the difference between the actual output(s) before
and after the limiters will be non-zero, which in turn will make the anti-windup loop active
and reduce the integral action. The anti-windup loop gain is determined by \/k t [55], where
k t is the integral reset time constant, calculated as k t - ^k^k, .
1
s
Figure 4.6:
k
i
Implementation of the PID controller with distributed anti-windup loop.
4.4.3 Evaluation of conventional controllers
To evaluate the dynamic performance of the PID-supervisory controller designed in this
section, the controller was put back into the original detailed full-order model of the overall
system. A symmetrical three-phase fault was implemented on one of the transmission lines
(see TL in Figure 3.1). For comparison purposes, the same fault study was also implemented
using the PFC and LVC modes. The corresponding transient responses of the voltage at the
PCC produced by models with different controllers are superimposed in Figure 4.7 for better
comparison. As can be seen in Figure 4.7, the system initially operates in a steady state such
that each control scheme results in the same bus voltage at the PCC. At t = 1.0s, a fault is
being applied and is then cleared after t = 1.15s . The fault results in voltage sag observed at
the PCC, which is different for the three control techniques considered. As can be seen in
Figure 4.7, the PFC does not provide voltage support during the fault. At the same time, the
LVC and the PID resulted in 0.26/s and 0.44/s of the voltage recovery rate. When the fault is
cleared by opening the faulted line, the PFC and LVC resulted in 1.32% and 0.25%
deviations from the pre-fault value, respectively. Although the PID controller enabled a much
faster voltage recovery during the fault, it still shows an undesirable oscillatory behaviour
during and after the fault.
Overall, it is desirable to achieve a faster damping with less oscillatory behaviour during the
fault, as well as when the fault is cleared. In addition, to make the proposed controller
practical, the noise of the measured signals (voltage at the PCC) should also be taken into
consideration. This conclusion motivates investigating further options in designing advanced
controllers.
Figure 4.7:
Voltage transient at the PCC resulting from a three-phase fault.
Chapter 5
Advanced Voltage Control Schemes
5.1
Observer-Based Framework
To make the overall control scheme applicable for realistic cases, one should consider from
the beginning that noise and signal distortions are unavoidable in measuring voltage at the
PCC. For improving dynamic performance beyond what was demonstrated in the previous
section with the PID controller, it is desirable to make use of the entire state vector in (4.11)
instead of just the output voltage. However, in general, measuring the high-order derivatives
is even more problematic in the presence of noise.
To address the above-mentioned considerations, an observer-based controller design
framework is taken in this thesis. A block diagram of the overall proposed supervisory
voltage control scheme, with observer, is depicted in Figure 5.1. Here, for the purposes of
controller design, the plant denoting the wind farm and electric grid combined is represented
by the following collection of reduced-order linear systems:
x = Aj-x + BjU + Gw
y = CjX + D,u + n
(5.1)
,
(5.2)
where, as before i — o,l,h and the subscript "o" denotes the nominal operating condition; "/"
denotes a 50% decrease and "h" denotes a 50% increase of local load impedance,
respectively; and G e
is the randomly chosen real matrix. In Figure 5.1,
w = A2 P cc = _ k x is the output of the supervisory controller, which is the total reactive power
required to control the voltage at the PCC. The variable x is the observer state vector. The
noise signals in the system state and the measured output are denoted by w and r\,
respectively. The variable z represents the measured system output vector
; = [f
Av
pcc
Av
(5.3)
pcc + •
where A v p c c = Vpf c - v p c c , and Vpf c is the predefined value of the voltage at the PCC.
The presence of an observer allows using observer states for feedback control instead of the
system states as given in (4.11), which in practice are not measured directly. In this thesis,
the Separation Principle [61], [66] is used and the state-feedback controller gain k and the
observer gain K e are designed independent from each other.
W
Figure 5.1: Block diagram of the supervisory voltage control with observer.
5.2
State Observer Design
The necessary condition in the design of an observer is observability. The concept of
observability is dual to that of controllability, which is the necessary condition for statefeedback controller design. Roughly speaking, controllability refers to the ability to steer the
state from the input; observability refers to the possibility of estimating the state from the
output signal. In this section, a standard Kalman filter is designed to deal with noise signals
[66], wherein finding the observer gain is achieved through solution of the following
algebraic Riccati equation:
A0S
+ S T\L
+GR WG T
-S TCIQ~ 1C 0S
= 0
(5.4)
T
where S = S is the positive-definite solution matrix and A 0 and C 0 are the state matrices
corresponding to the nominal operating condition. The noise covariance matrices denoted
here by R w e Sl nX n and Q„ e y( mxm
signal as R w = E wwT
and Qn=E
are
J
defined using the expectation E of each noise
, respectively [61], [67]. The noisy signals are
assumed to be white, Gaussian, and to have zero-mean such that £[w] = 0 and £[11] = 0.
Here, the subscripts w and rj relate the matrices to the state and output noise signals,
respectively. Finally, the observer gain is calculated using the solution to (5.4) as
T* 1
K e = SC 0 Q^ . The observer can be expressed as
x = A 0 x + B 0 w + K e ( z - z)
(5.5)
A
A
z = C0x
where z is the measured system output vector, and z is the observer output vector.
Choosing Q^ to be very small compared to R w implies that the measurement noise TJ is
also small. An optimal state observer then interprets a large deviation of the observer output
z from z as an indication that the estimate x is bad and needs to be corrected. In practice,
this lead to large matrices of the observer gain K e and corresponding fast poles in
(A 0 -K EC 0).
Alternatively, choosing Q^ to be very large implies that the measurement
noise q is large too. An optimal state observer is then much more conservative regarding
deviations of z from z . This generally leads to small matrices for the observer gain K e and
consequently slow poles in ( A 0 - K e C 0 ) .
For the work presented here, we chose to have a faster observer. We chose the covariance
matrices as Q^ = diag (0.0006, 0.0006) and R w = 0.5, and the randomly chosen constant
G = [1.9515, 2.3081, 2.0722, 1.5768, 0.4277] 7 .
matrix
Applying
these
design
parameters and using the Control System Toolbox [64], the observer gain is obtained as
follows:
K
e -SCoQj
61.5334
20.6683
44.7403
48.6022
1.2318
20.6683
26.3330
25.4970
17.7305
5.7823
T
(5.6)
Table 5.1 shows the eigenvalues, damping ratio, and frequencies of the closed-loop observer
that correspond to ( A 0 - K e C 0 ) . Note that since the observer is designed based on the 5 t h order reduced model, the covariance matrices are chosen to place the eigenvalues in the
complex domain close to the frequencies of the reduced-order model, specifically at 30.8
rad/sec and 122 rad/sec. With R w = 0.5, the covariance matrix Q^ was increased from
0.0001 to 0.0006, where the maximum damping ratio of the closed-loop observer was
reached at 0.503. This is close to the maximum damping ratio of the reduced-order model,
0.5, and at the same time the eigenvalues are placed close to the frequency of the reduced-
order model at 30.8 rad/sec. As a design observation, if the damping ratio is less than 0.5 at
frequency 122 rad/sec, or the eigenvalues of the observer are significantly different than
those of the reduced-order model at a frequency of around 30.8 rad/sec, system performance
becomes sluggish or oscillatory.
TABLE 5.1
EIGENVALUES, DAMPING RATIO, AND FREQUENCY OF THE CLOSED-LOOP OBSERVER
5.3
Eigenvalues
Damping ratio
Frequency (rad/sec)
-22.0
-31.4
-78.5
-61.5±jl06
1
1
1
0.503
22.0
31.4
78.5
122
Linear Quadratic Regulator Approach
We chose the LQR approach as a framework for tuning the controller gains in this thesis as
this methodology is general and flexible, and can be formulated in terms of a performancebased optimization problem, for which the numerical solution techniques and software tools
are widely available [64], [68]. At the same time, the cost function (function to be
minimized) may be defined in a number of ways that can simultaneously include several
performance-based criteria. Another important advantage of using the LQR is that it can be
formulated for the case when the overall plant is described by a set of linear systems that
span a particular range of operating conditions. This is accomplished by representing the
underlying control optimization problem in terms of a system of LMI constraints and matrix
equations that are simultaneously solved. The solution of LMI equations involves a form of
quadratic Lyapunov function that; if it exists, not only gives the stability property of the
controlled system but can also be used for achieving certain performance specifications.
5.3.1 Formulation of LQR
The LQR control design problem looks for a feedback controller gain k e 9 \ p X n for the
system
x = Ax + Bu,
u =-kx
(5.7)
that minimizes the cost function [69]
J = mm J ^ ( x r Q x + urRu)flfr
(5.8)
where Q and R are design parameters, Q e Si" X n is a symmetric non-negative definite
matrix and R e 3 l p X p is a symmetric positive definite matrix. The final control gain k
should satisfy the following Lyapunov equation:
( A - B k ) r P + P(A-Bk) + Q + k rRk = 0
where P e %nXn
(5.9)
> 0, known as the Lyapunov matrix.
The LQR controller minimizes the quadratic function of the state xTQx
and that of the
control signal u Ru . These quadratic functions are often associated with the energy in the
system's state and control signal. Matrices Q and R are the relative weights of the state
dynamics and the control action. For example, choosing Q large and R small will result in a
control gain that will attempt to reduce the deviations of the state at the expense of a very
strong control action. On the other hand, choosing Q small and R large will result in a
control gain that will attempt to reduce the control action at the expense of allowing large
state variations. Therefore, these matrices are chosen to achieve some balance between the
desired performance and the required control action.
Since (5.9) is nonlinear and difficult to solve, the solution to the LQR is found by the wellknown algebraic Riccati equation
^ rP = 0
A r P + PA + Q - P B R _l 1r B
(5.10)
which is linear in variable P and is readily solved numerically using software [64], The
controller gain is then computed as k = R B P.
Since in our case all signals are expected to have some noise, the cost function (5.8) can be
rewritten in terms of expectation as .
(5.11)
Moreover, since we have a set of linear systems representing a range of operating conditions,
the feedback gain k should now satisfy a number of Lyapunov equations, as follows:
(Aj - B j k f P + P ( A 7 - B y k) + Q + k r R k = 0
(5.12)
The Lyapunov equations (5.12) are nonlinear and therefore difficult to solve. The problem is
even more complicated by the fact, that these equations for multiple systems have to be
solved simultaneously for a common Lyapunov matrix P > 0.
Instead of trying to solve (5.11) and (5.12) directly, in the following Subsections we reformulate this problem as a set of LMIs, which are then solved for a common matrix P . In
doing so, we provide two LMI formulations. The first conventional LMI formulation is based
on minimizing the quadratic cost function [70]. The second method presents an LMI
formulation based on minimizing the upper bound of the cost function [71], which is a
preferred approach to dealing with uncertain signals/variables.
5.3.2 Conventional approach
Using the H 2 representation of the LQR problem [61], [67], we would like to find the statefeedback gain matrix k that minimizes the following cost function in terms of output y as
(5.13)
J = minjiiTy^yT]
(k) 1 L
K
subject to
(Ay-Byk) rP + P(A/-B/k) +Q+krRk<0
V
and
J
(5.14)
P>0
We first formulate an LMI for the cost function (5.13), using the output y as given in [67]
>1/2
y=
0
(5.15)
Rl/2
then substitute (5.15) into (5.13) with u = -kx to obtain
.T~ = E tr
yy
r
r
= E x Qx + u Ru
/ r
• tr
QV2
2
-RV k
(5.16)
'
xx
2 T
\QV )
1
2
-k^R / )
7
where the function tr stands for trace, which is the sum of all its diagonal entries.
By utilizing the identity tr{ABC) = tr(CBA) and the state covariance matrix Y = E xx
the H 2 representation of the LQR problem (5.13) can be expressed as
yy
( '
'
7
= fr(QY) + fr R 1 / 2 k Y k r ( R 1 / 2 )
(5.17)
where it is assumed that E
= R w > 0 and Y = E xx
WW
. We then utilize the change
of variables [67] such that
X>R1/2kYkr(R1/2)7
(5.18)
Let us define a change of variables as used in [67] such that
k = AY"
1
k G3IPXN,
and
A E
(5.19)
P = Y"
and
P e Si nXn
(5.20)
Y=Y
and
Ye5R nxn
(5.21)
Using the well-chosen variable (5.19) and the Schur complement, (5.18) can be put into a
matrix inequality as follows:
X-jR^AjY-'j/R
R1/2A
X
A V
2
1
/
2
^
(5.22)
>0
Y
Finally, the cost function (5.13) is formulated as follows:
Minimize:
J = min {fr(QY) + /r(X)l
(Y,X)1
J
subject to LMI constraint (5.22).
(5.23)
To complete the LMI formulation of the H 2 LQR problem, we still need to obtain an LMI
for (5.14). Using the change of variables (5.19)—(5.21), Equation.(5.14) can be re-formulated
as follows:
I
(Ay-Byk) P + P(Ay-Byk) + Q + krRk
=Y
-1
Y T Ay - ArBy + AyY - By A + YQY + ArRA
Y
<0
(5.24)
= Y 7 Ay - ArB y + Ay Y - By A + YQY + ArRA < 0
Then, utilizing the Schur complement, (5.24) can be put into an inequality as follows:
Ay Y + YrAy - ByA - A^By ) AT
Yr
-R1
0
0 -Q"
A
Y
<0
(5.25)
with A > 0 and Y > 0 .
Finally, LMI formulation of the H 2 LQR problem based on the conventional approach with
cost function (5.23) is as follows:
Minimize:
J = min {;r(QY) + /r(X)|
(Y,X)1
1
subject to LMI constraints (5.22) and (5.25).
5.3.3 Cost-guaranteed approach
As defined in [67], the cost-guaranteed approach is to replace the cost function (5.11) with a
certain upper bound when the system is subject to noise. Thus, if we write the Lyapunov
equations (5.12) as a matrix inequality, the solution of this inequality will be an upper bound.
Therefore, the H j LQR problem based on the cost-guaranteed approach can be defined as
follows:
Minimize:
(5.26)
subject to (5.14), which is not an LMI.
This optimization problem provides a necessary and sufficient condition to guarantee the
system asymptotic stability. The proof for these properties is given in Appendix E. Using the
change of variables (5.20), the bound on the cost function /r (P) in (5.26) may be changed to
tr (Y
1
) . We further introduce a slack matrix variable Z as used in [67] such that
Z> Y_ 1
(5.27)
which is used to write the following matrix inequality:
Z-IY
--1
1
Z
I
I
>0,
Y
and
Y>0
where I is an nxn identity matrix, and Z = Z T e 3inX n
matrix.
(5.28)
is the symmetric positive definite
Thereafter, the H 2 LQR problem based on the cost-guaranteed approach is formulated as
follows:
Minimize:
V = minjfr-(Z))
1
J
(Z)
(5.29)
subject to (5.25) and (5.28).
5.3.4 Evaluation of controllers
The LQR formulations presented in Subsections 5.3.2 and 5.3.3 have been used for tuning
the controller gains. The numerical solutions were carried out using the LMI Control System
Toolbox [68] with the input script files as given in Appendix D. All tuning parameters and
LQR gains are summarized in Table 5.2 in Section 5.5 for consistency and further
comparison. The two controllers resulting from Subsections 5.3.2 and 5.3.3 are hereafter
referred to as the LQR supervisory (LQRS) and the LQR cost-guaranteed (LQRCG),
respectively.
The same symmetrical fault study as described in Section 4.4.3 was used here to compare the
system's response with different controllers. In particular, the voltage transient observed at
the PCC for the system with PID supervisory (Section 4.4.2), LQRS (Section 5.3.2), and
LQRCG (Section 5.3.3) controllers is shown in Figure 5.2. As can be seen, the two LQR
controllers perform much better than the PID controller, which has lower order. During the
fault, the PID-supervisory controller resulted in 0.44/sec of the voltage recovery rate,
whereas the LQRS and LQRCG controllers resulted in 0.875/sec of the voltage recovery rate.
While performing better than the PID controller during the fault, both the LQRS and LQRCG
controllers showed very similar behaviour, with some undesirable oscillations after the fault.
The following section presents some control modifications to reduce the oscillatory
behaviour and further improve the transient response of the system.
0.92
1.2
time(sec)
Figure 5.2:
5.4
Comparison of PID, LQRS, and LQRCG controllers.
Advanced LMI Representation of LQR
As an attempt to further improve controller performance, in this section we consider taking
into account the cross-product of the state and control signals [61]. This is accomplished by
including these cross-product terms into the LQR cost function in addition to the quadratic
functions of the state and control signal, as done in Section 5.3.
5.4.1 Taking into account cross-product terms in the conventional approach
Adding the cross-product terms, the cost function for the
LQR problem (5.16) can be
expressed as
J = mini? x^Qx + u r R u + x r N u + u r N r x
(5.30)
00
where N e S l n X p satisfies the condition Q - N r R
]
N>0.
We further proceed by substituting u = -kx into (5.30) to obtain
m
rr< m
rp
T"
rp rrt
J = E xl Qx + x k R k x - x N k x - x ^ k ' N 7 x
= ^ ( Q + k R k r - N k - k r N r \}tr{E
t r \ E xxT
T ^
= /r(QY) + fr R ^ K Y k ^ R 1 / 2 )
(5.31)
)
-^(NkY + k r N r Y r )
Hence, the conventional H 2 LQR problem with the cross-product terms becomes
J = min tr QY + R ! / 2 k Y k r ( R 1 / 2 ) - NkY - k r N r Y
T
(5.32)
(k,Y)
subject to (5.25).
Note that the second term was already presented in (5.23) as X . However, (5.32) includes
kY and therefore cannot be easily solved. By using the change of variables shown in (5.19),
we obtain
?r(-NkY-krNrYr) = /r(-NkY-YrkrNr) =^(-NA-ArNr)
(5.33)
Hence, the advanced H 2 LQR problem based on the conventional approach can be
formulated as follows:
Minimize:
J-
min
fr(QY + x)-/r(NA + A r N 3
(A,Y,X)
subject to (5.22) and (5.25).
(5.34)
5.4.2 Taking into account cross-product terms in the cost-guaranteed
approach
Utilizing (5.29) and (5.34), the H 2 LQR problem based on the cost-guaranteed approach can
be described as following:
J = min\E |"x r Qx + u r R u + x r N u + u ^ N 7 * ] ]
(k) I L
JJ
= min ^ r ( Q Y + x ) - / r ( N A + A r N r )
< min
(Z,A)
(5.35)
?r(Z)-/r(NA + A r N r )
Hence, the advanced H 2 LQR problem based on the cost-guaranteed approach can be
formulated as follows:
Minimize'.
V = min
(Z,A)
(5.36)
subject to (5.25) and the slack matrix variable constraint (5.28).
5.4.3 Evaluation of controllers
The formulations presented in Subsections 5.4.1 and 5.4.2 have been used for tuning the
controller gains. The numerical solutions were carried out using the LMI Control System
Toolbox [68] with the input script files as shown in Appendix D. All tuning parameters and
gains are summarized in Section 5.5 for consistency and further comparison. The two
controllers resulting from Subsections 5.4.1 and 5.4.2 are hereafter referred to as the
advanced LQR supervisory (ALQRS) and the advanced LQR cost-guaranteed (ALQRCG),
respectively.
The same symmetrical fault study as described in Section 4.4.3 was used here to compare the
system's response with different controllers. In particular, the voltage transient observed at
the PCC for the system with different controllers is shown in Figures 5.3 and 5.4. To get an
idea of what has been gained by taking into account the cross-product terms, Figure 5.3 first
compares the system's response with the ALQRS (see Section 5.4.1) versus the LQRCG (see
Section 5.3.3). As can be observed in Figure 5.3, the ALQRS does improve performance and
reduces oscillatory behaviour, especially after the fault has been cleared. This achievement
already well justifies the extra effort involved in formulating and tuning this controller.
Performance of the ALQRCG (see Section 5.4.2) is depicted in Figure 5.4, wherein this
controller is further compared with the PID supervisory and ALQRS controllers. An
interesting observation can be made here. In particular, the ALQRCG controller provides
even further improvement and damping of the post-fault oscillations over the ALQRS. The
formulation and design of the ALQRCG has paid off with the best transient performance of
the system, which was the goal of this Chapter. The following section summarizes controller
gains. The computer studies presented in Chapter 6 compare the proposed ALQRCG with
traditional control solutions such as PFC, LVC, and PID-supervisory controller.
Figure 5.3:
Comparison of LQRCG and ALQRS controllers.
Figure 5.4:
5.5
Comparison of PID, ALQRS and ALQRCG controllers.
Summary of Controller Gains
The tuning parameters and gains of the LQR-based controllers are summarized in Table 5.2.
In selecting the design parameters, we begin with R = 10 and Q = diag{\, 1, 1, 1, 1). Since
these design parameters showed very slow settling time, we increased the entry Q y and
Q2 2
speed up the integral action, and decreased R. Table 5.3 summarizes the eigenvalues
and damping ratios, along with corresponding frequencies, of the closed-loop 5 th -order
reduced model ( A 0 - B 0 k ) .
An important observation can be made regarding the data in Table 5.2, namely, that
considering the cross-product terms results in noticeably higher derivative and integral gains.
This is especially pronounced in the third derivative, where the ALQRCG has the highest
gain of 3.3509. The integral gain of ALQRCG also increases to 2.7747. Thus, better damping
and faster performance can be expected.
TABLE 5.2
DESIGN PARAMETERS AND CONTROL GAINS
Conventional Approach
Design
parameter
LQRS
R
1.2
LQRCG
2.2 .
ALQRS
ALQRCG
0.955
4
diag{ 70, 10, 1, 1, 1)
Q
N
Advanced Approach
[0, 0, 0, 0, o f
[0.55, 0.55, 0.55, 0.55, 0.55f
Gains
State
LQRS
LQRCG
ALQRS
ALQRCG
fAvp CC
-5.5182
-5.5099
-5.6543
-5.7190
Av
pcc
-2.2902
-2.2342
-2.1965
-1.3626
Av
pcc
0.8600
0.8917
1.5886
2.7747
vpcc
0.0660
0.0106
0.1861
0.1217
-0.1852
-1.8105
0.1958
3.3509
A
Avpcc
TABLE 5.3
EIGENVALUES, DAMPING RATIO, AND FREQUENCY
LQRS
Eigenvalues
-9.10
-28.3±j20.6
-63.4±jl07
Damping ratio
1
0.808
0.508
Frequency (rad/sec)
9.10
25.0
125
LQRCG
Eigenvalues
-9.88
-35.3±jll.6
-61.0±j94.9
Damping ratio
1
0.905
0.541
Frequency (rad/sec)
9.88
37.1
113
ALQRS
Eigenvalues.
-10.6
-25.0±jl9.4
-68. l±j 110
Damping ratio
1
0.790
0.526
Frequency (rad/sec)
10.6
31.7
130
ALQRCG
Eigenvalues
-11.2
-12.7±j23.6
-73.7±jl30
Damping ratio
1
0.475
0.493
Frequency (rad/sec)
11.2
26.8
149
Chapter 6
Simulation Studies
We modelled the system depicted in Figure 3.1 and described in Chapter 3 together with the
various controllers described in Chapters 4 and 5. We implemented the overall detailed
model of the system using Matlab/Simulink software [72] was implemented, and carried out
computer studies to study the impact of wind speed variations, load variations at the PCC,
and the remote three-phase symmetrical fault. In the simulation studies presented in this
chapter, the proposed advanced LQR-cost-guaranteed controller (ALQRCG) is compared
with the PID-supervisory controller, the conventional power factor control (PFC) and the
local voltage control (LVC).
6.1
Small Disturbances
6.1.1 Wind speed variations
To study the effect of wind speed variations, different wind speeds were assumed for the
three wind turbines (WTs), and are shown in Figure 6.1. These wind variations are assumed
to represent a realistic wind gust that is unavoidable, especially if the WTs are located apart
from each other, as is in the case of the system depicted in Figure 3.1. We performed the
computer simulations of the system with different controllers and plotted the corresponding
results in Figures 6.2 through 6.8.
The real power output from each wind turbine is controlled by the maximum power tracking
curve (see Figure 3.13),. which determines the real power set-point in each turbine, as shown
in Figure 6.2. As can be observed, the internal controllers in each turbine track the
instantaneous power command very well. The actual real power output plotted in Figure 6.3
follows that of Figure 6.2, which altogether corresponds to the wind speed trends. The
combined real power from the wind farm that is injected into the grid is shown in Figure 6.4.
As can be seen, the variations are relatively small, which is due to the pitch control action.
The reactive power outputs at the terminal of each wind turbines are shown in Figure 6.5 for
the system with different controllers. When the WTs operate in PFC, they output no
additional reactive power to the grid to maintain a unity power factor. When the LVC is used,
the reactive power injected by each WT is somewhat different, due to the different wind
speeds. However, when any of the supervisory controllers are used, the commanded reactive
power is evenly distributed among the participating WTs, because they all operate somewhat
below the limit. In this case, the WTs contribute evenly to the overall reactive power injected
by the wind farm, as depicted in Figure 6.6. The voltage fluctuations observed at the WT
terminals and at the PCC are shown in Figures 6.7 and 6.8, respectively. Overall, these
variations are small and therefore do not represent a concern for the power quality in the
system. Voltage fluctuation is minimized by the variable-speed wind turbine technology and
the action of the internal controllers.
Figure 6.1: Wind speed (m/sec).
0.027
- a y ,
0.0265
LVC
3Q.
0.027
JP'yC-
C
o 0.0265
CD
o
Q.
PID - supervisory
0.027 -
"S
<D
CC
0.0265 ALQRCG
0.027
0.0265
0
I
10
20
i
30
I
40
I
I
50
time(sec)
60
I
70
i
i
80
-
90
100
Figure 6.2: Real power set-point for each WT due to wind speed variation.
(
WT1 — 1 WT2
WT3)
PFC
0.027
/••,.. X.
0.0265
3Q.
0.027
D
Q.
0.0265
LVC
''/py^X/
PID - supervisory
0.027
0
<5
1
CL
"5 0.0265
a>
CC
fKoC
ALQRCG
0.027
\
0.0265 10
20
-A^rv
, J Jy\ .A"' / / vV... f\/""'"\.
I
30
I
40
I
50
time(sec)
I
60
i
70
i
80
i
90
Figure 6.3: Real power output from each WT due to wind speed variation.
(—WT1 — W T 2
WT3)
-
100
PFC
0.078
0.0775
0.077
3Q.
3Q.
LVC
0.078
0.0775
0.077
PID - supervisory
0.078
O
Q
g) 0.0775
O
Q.
"(5 0.077
i_
CD
cc
ALQRCG
0.078
0.0775
0.077.
0
10
20
30
40
50
time(sec)
60
70
80
90
100
Figure 6.4: Real power output from the wind farm due to wind speed variation.
PFC
0.00005
-0.00005
0.00005
Q.
1Q.3
-0.00005
"5
o 0.00005
<D
PID - supervisory
1
—
o
Q.
a>
>
13
to -0.00005
CD
DC
v V A A A A V N / V X ^
ALQRCG
0.00005
V W A a A A A A / V \
-0.00005.
10
20
30
40
50
time(sec)
60
70
80
90
Figure 6.5: Reactive power output from each WT due to wind speed variation.
(—WT1 — W T 2
WT3)
100
0.0031
0.003
0.0029
3Q.
0.0028
0.0031
LVC
0.003
0.0029
0.0028
30
1
<D
3o
Q.
<
}
>1
O
TO
CD
tr
0.0031
. 0.003
0.0029
0.0028
0.0031
0.003
0.0029
0.0028,
40
50
time(sec)
60
100
Figure 6.6: Reactive power output from the wind farm due to wind speed variation.
3Q.
C
D
•5
CO
O
>
3
CD
0
10
20
30
40
50
60
time(sec)
70
80
90
100
Figure 6.7: Voltage fluctuations due to wind variation, as observed at the WT terminals.
0.905 -
0.9049.
Figure 6.8: Voltage fluctuations due to wind variation, as observed at the PCC.
6.1.2 Load variations
In the following simulation study, a sequence of step-changes in the load impedance is
implemented, wherein the load is first increased by 20%, then decreased by 20%, and finally
decreased by further 20%. For test clarity, the wind speed here is assumed constant for all
WTs. The corresponding transient responses of the system with different controllers are
provided in Figures 6.9 through 6.16.
The voltage transients observed at the PCC due to load changes are plotted in Figures 6.9 and
6.10. The performance of the four controllers is compared in Figure 6.9. As can be seen in
this figure, the PFC results in the largest voltage changes. This result is expected for this
control mode, wherein the voltage deviations are somewhat proportional to load changes.
The voltages at the WT terminals are plotted in Figure 6.11. When the LVC is used, the
changes in the load cause a small transient, but overall the voltages at the WT terminals
return to the same set values, which is not the case for the PFC. However, due to the
impedance of the cables and transformers connecting the WTs to the PCC, the voltage at the
PCC changes with the load variations, as can be seen in Figure 6.9. The real power produced
by each WT is depicted in Figure 6.12 and the combined real power injected by the wind
farm is shown in Figure 6.13. As can be observed in these two figures, there is a very small
transient there due to the load changes, but the level of injected real power returns to the
same level which is determined by the wind speed. This behaviour is the same for all the
controllers shown in Figures 6.12 and 6.13, as they all operate using the reactive power only.
Reactive power output for each WT and the farm are shown in Figures 6.14 and 6.15,
respectively. As can be seen from the figures, the amount of injected reactive power depends
on the control scheme. The reactive power set values are plotted in Figure 6.16. As can be
seen from these studies, the PID-supervisory and ALQRCG controllers perform very
similarly and better than the PFC or LVC. However, as shown in Figure 6.10, the ALQRCG
responds somewhat faster, which is especially noticeable in the beginning of each transient.
0.92
PFC
LVC
0.915
3
PID-supervisory
ALQRCG
Q.
in
3
0.885
0
2
4
6
8
10
12 .
14
time(sec)
Figure 6.9: Voltage transient observed at the PCC due to load impedance changes.
16
0.912
PID-supervisory
0.91
ALQRCG
0.908
S
0.906
O
O
0.904
oto> 0.902
C
D
CT)
to
o
0.9
>
v>
m
0.898
0.896
0.894.
0
8
12
10
14
time(sec)
16
Figure 6.10: Voltage transient observed at the PCC due to load impedance changes:
Detailed view of the PID-supervisory and ALQRCG controllers.
PFC
LVC
3Q. 0.9?
-
0.9
-
0)
•C
_
v..
F
-t—i
to
& 0.92
_
O)
to
§
0.9
C
O
3
CQ
0.92
- j
R
PID - supervisory
-
L
f"
ALQRCG
'
0.9
i
i
6
i
8
time(sec)
i
10
1
12
1
14
Figure 6.11: Voltage transient observed at the WT terminals.
16
0.027
••• WT1
1
\
0.0265
D
O.
H
§
0.027
WT2
WT3
14
16
J
LVC
"o 0.0265
"3
^a.
PID - supervisory
0.027
o
a)
0a.
"ro 0.0265
<D
01
ALQRCG
0.027
i
i
I
I
p-s/-I
0.0265.
0
I
6
8
time(sec)
10
12'
Figure 6.12: Real power output from each WT.
PFC
0.08
i
/
y
0.075
LVC
0.08
t
3
CL
0.075
3CL
0.08
o
L_
<D
o
a.
0.075
lr
r
PID - supervisory
-t—1
ro
<D
a:
0.08
0.075.
0
r
ALQRCG
I
|
8
10
time(sec)
I
I
12
14
16
6
8
time(sec)
10
12
Figure 6.14: Reactive power output from each WT.
PFC
0.02
-0.02
LVC
2
4
6
8
time(sec)
10
12
14
16
0.02
E
3
E
'x
(0
E
0
Q.
1
<D
O
)
w
d)
g
o
Q.
d)
>
o(0
(U
OL
8
time(sec)
Figure 6.16: Reactive power set-point and maximum at each WT.
(WT1
WT2
WT3, j=1,2,3)
6.1.3 Summary
The level of voltage deviation observed at the PCC depends on the control scheme being
used. When the wind turbine operates in the PFC mode, the load changes result in the most
noticeable deviations in the voltage level at the PCC. When the LVC mode is used, the
voltage fluctuations are significantly reduced, because the PCC is relatively close to the wind
farm (9km cable). However, the proposed ALQRCG controller shows best performance. A
summary of steady state values is given in Table 6.1, where it is shown that only the
supervisory controllers provide a required voltage tracking at the remote PCC.
TABLE 6.1
MAGNITUDE OF VOLTAGE DEVIATIONS
Steady State
Value
High (pu)
Low (pu)
Deviation (%)
6.2
PFC
LVC
PID- Supervisory & ALQRCG
0.9154
0.8892
2.62
0.9067
0.9021
0.46
0.90497
0.90497
0
Large Disturbances
From the point of view of power system stability, it is desirable to keep the wind farm
operational and connected to the grid for as long as possible, even during large system-wide
disturbances. This can be achieved by actively controlling the WTs during the disturbances
and attempting to keep the voltage as close as possible to the pre-disturbance level (within
the current limits), as well as suppressing the voltage swings that may activate the protection
circuitry and prematurely trip the turbine.
6.2.1 Three-phase fault
To study system response to a large disturbance, the same symmetrical fault study as was
used in Chapter 5 is presented here, with more detail. In this study, at t -1 s a fault is
assumed upstream in one of the transmission lines. This fault is cleared after 0.15s by
disconnecting the faulted line. The transient responses of the system with different
controllers are plotted in Figures 6.17 through 6.23.
We analyzed the performance of various controllers and their ability to regulate voltage at the
PCC in Chapters 4 and 5. For consistency, the voltage transient observed at the PCC is
shown again in Figure 6.17. This figure shows that the final proposed ALQRCG controller
outperforms the PFC and LVC basic schemes, as well as providing a superior transient
response, compared to the standard PID controller employed in the proposed supervisory
scheme.
For evaluating the performance of different controllers, two time intervals are of importance,
one during the fault and another after the fault has been cleared. During the fault, the system
is stressed by a disturbance, which is also evident from the transients observed in the real
power output of each individual WT, as shown in Figure 6.18, as well as of the wind farm, as
shown in Figure 6.19. However, the major differences among the controllers are found in
terms of the reactive power provided during and after the fault, shown in Figures 6.20
through 6.23. As can be seen in these figures, the proposed control provides the most reactive
power support during the fault, as well as better damping after the fault.
0.92
1.2
time(sec)
Figure 6.17: Voltage transient observed at the PCC due to the fault.
....... W
0.03H
T 1
WT2---WT3
0.025
0.02
LVC
0.03
0.025
0.02
1.2
time(sec)
Figure 6.18: Real power output at each WT due to the fault.
PFC
°'S.9
1
1.1
1.2
time(sec)
1.3
1.4
1.5
0.015r
• WT1
0.01 •
- WT2
WT3
0.005-
0•
-0.005:
0.015r
LVC
—vw—
"Www 3
S
fK
?s
m
m
m
t
n
t
M
A
w
0.01
0.005
0
-0.005
1.2
time(sec)
Figure 6.20: Reactive power output at each WT due to the fault.
( - - - WT1 — WT2
PFC
0.95
0.9
0.85
0.95
0.9 :
0.85
0.95
WT3)
• WT1
"I
V
WT2
WT3
awwuwj t if.
"'•'f i/.'ivii'.^'.'i yjtti i v
LVC
: 11» UtUtKfitf£K&'MVi
f
H
W
W
W
U
1
MitttVlttl
PID - supervisory
A '
0.9-
:v
n
^
W
m
w
M
M
W
U
.
'U
.
's
: ii:u;mnu tuivt tr-.-'.'I'.'l'.T.U'.'.
1.2
time(sec)
Figure 6.21: Voltage transient observed at the terininal of each WT due to the fault.
L
V
i
.
' A
V
V
i
0.040.02-
0=
3
" V V ^
-0.02 -
LVC
Q.
3Q.
"3
0
0)
5o
Q.
a)
>
oto
0)
EC
1
1.2
time(sec)
. 1.5
Figure 6.22: Reactive power output from the wind farm due to the fault.
LVC
Q.
E'
E
'x
CO
PID - supervisory
/
E
C
03
0Q.
L_
\
/
T3
1
0)
U)
0>
\
0
o
\
I „
1 1
1
ALQRCG
Q.
<1)
> 0.01
O
C
CD
D 0.005
CC
8
1.1
1.2
time(sec)
1.3
1.4
Figure 6.23: Reactive power set-point and maximum at each WT due to the fault.
(---WT1
WT2
WT3, j=1,2,3)
1.5
6.2.2 Summary
Several observations can be made with regard to the performance of different controllers.
When the PFC mode was used, the voltage at the PCC depended on the reactive power
balance of the network. When the network impedance was changed, the reactive power level
was also changed. The PFC mode enabled zero reactive power consumption by the wind
farm at every event. However, after the fault, the voltage at the PCC settled down below the
pre-fault level, due to weaker transmission line impedance. When the LVC mode was in use,
the voltage response at the PCC was significantly improved because of its partial reactive
power contribution. A summary of the steady state voltages at the PCC after the fault is given
in Table 6.2, where it is shown that only the supervisory controllers can restore the voltage
after the fault with zero steady state error. During the transient, the LVC and PID-supervisory
controls showed voltage recovery rates of 0.26/sec and 0.44/sec, respectively, while the
ALQRCG controller demonstrated a voltage recovery rate of 0.875/sec.
An important observation can be made regarding the proposed ALQRCG controller. In
particular, although the proposed controller was tuned for a range of operating conditions
defined by normal variation of the load, the overall supervisory ALQRCG controller
demonstrated outstanding performance, even under a severe disturbance such as fault, with
significantly improved transient performance and faster damping of the voltage swings.
TABLE 6.2
COMPARISONS OF VOLTAGE CONTROL PERFORMANCE (PU)
Set-point
0.90497
Deviation (%)
PFC
0.8918
1.32
LVC
0.9025
0.25
PID- supervisory & ALQRCG
0.90497
0
(Deviation: deviation between the set-point and the steady-state value in percent)
Chapter 7
Conclusion and Future Work
7.1
Conclusion
This thesis addresses the operation of wind power generation systems and their contribution
to' voltage control in the network. Voltage control in the system becomes particularly
important when wind energy penetration is high. We developed a detailed model of a
candidate industrial site with multiple wind turbines and used it to perform simulation studies
and evaluate alternative control solutions. The goal of our investigation was to make use of
available wind turbine technology, namely the variable-speed doubly-fed induction generator
with power electronic converters, to actively participate in improving voltage control in the
system without using additional compensating devices. To ensure reliable operation of the
proposed control scheme, the operating-point-dependent reactive power limit of each wind
turbine was taken into account. The overall supervisory voltage control scheme and the
control design methodology developed in this thesis can be applied to larger wind farms and
network configurations.
In Chapter 3, we presented the model of the grid-connected wind farm to investigate the
impact of wind power on power system dynamics. The overall component modules were
represented in the ^^-synchronous reference frame. In Chapter 4, based on the models
presented in Chapter 3, the transfer functions to be used in the design of the VSC controllers
were presented.
In Chapter 5, we proposed an innovative supervisory control scheme that allows using a wind
farm for regulating voltage at a point of common coupling (PCC) that is remote and/or
different from the wind farm grid-connection point. The supervisory scheme acts as a
distributed controller and makes use of reactive power that is available from all participating
wind turbines, while taking into account time-varying operating conditions and the limits of
individual turbines to ensure their safe and reliable operation. Through numerous simulation
studies, this supervisory scheme, even with a generic PID controller, outperformed the
traditional control modes of the wind turbines, such as reactive power support and/or local
voltage support, in cases of both small and large disturbances.
As the next step in this research, we investigated several advanced control approaches that
would work together with the supervisory control scheme. To enable a linear and robust
control framework, the overall system was represented by a set of reduced-order linear
systems that cover an operating range of interest determined by variations of the load. To
make the control design applicable to realistic systems, with noise and disturbances in the
measured signals, we considered an observer. Several control solutions based on the linearquadratic-regulator design were investigated. The best controller was designed using the
linear-quadratic-regulator and linear-matrix-inequality approach, which takes into account
cross-coupling between the state and control inputs, and minimizes an upper bound on the
cost-guaranteed objective function.
In Chapter 6, we carried out detailed simulation studies that considered the impact of wind
speed variations, load variations, and faults in the network on the voltage at a point of
common coupling. For the case system considered in this thesis, small disturbances such as
wind speed and load variations were not likely to represent an objectionable voltage control
problem, even when the wind farm was equipped with traditional controllers. However, the
faults resulted in much larger disturbances that should be mitigated, if possible. The proposed
final controller was further compared with traditional control techniques and shown to
provide an improved transient response under small disturbances as well as faults. In
particular, the final controller achieved a faster response and better-damped behaviour during
and after the fault. The achieved response is less likely to trigger the protection circuitry and
therefore more likely to ride through the fault in a favourable way.
7.2
Future Work
The modelling work in this thesis was not validated against the hardware system, which
would be of definite value. Such validation and tests could be possible if an industrial partner
with an appropriate facility becomes involved in this research. However, the wind energy
facilities in British Columbia are only in the planning stage at present. Contacts with other
provinces may perhaps lead to potential industrial collaborators.
Our research goal is to encourage active use of wind turbines and wind farms in power
system operations. As an extension of the supervisory controller, more advanced linear
controllers, such as a gain scheduling and/or nonlinear controller, could be studied to further
improve the system's performance.
Beyond the voltage control problem, which was the primary focus .of this thesis, the overall
theme of making wind farms active participants in improving the operation of power systems
can be extended. An area of application of a similar supervisory control scheme would be to
use the wind farms for frequency control, similar to conventional generation stations, by
utilizing real power instead of (or in addition to) reactive power. This approach would be
particularly important for places with very large wind power penetration or islands with
relatively small total inertia of conventional generators. However, it may require not
extracting the maximum possible energy from the available wind at all times. However, the
benefit of improving the power system's operation (quality, stability, and reliability) may
well justify the cost.
Another application would be to look for control solutions that would enable automatic
operation of wind energy resources in stand-alone mode and continue supplying some islands
of local loads in cases of wide-area power outages and/or blackouts. This would require a
distributed control approach, wherein the voltages and frequencies over a possibly larger area
would have to be monitored and integrated into the wind farm control to enable continuous
and reliable operation in a given area.
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Appendix A
System Parameters and Operating Conditions
Table A. 1 Wind Power Model Parameters [8]
(all quantities are given in per unit on 4MVW base)
Fixed constant
Power coefficient
C p{9,X)
4
4
X
=X
i=0j=0
where 2<A<13
1/2 pAr
0.0145
Kb
69.5
i
4
4
4
4
4
3
3
3
3
3
2
2
2
2
2
1
1
1
1
1
0
0
0
0
0
j
4
3
2
1
0
4
3
2
1
0
4
3
2
1
0
4
3
2
1
0
4
3
2
1
0
a
ij
4.9686e-010
-7.1535e-008
1.6167e-006
-9.4839e-006
1.4787e-005
-8.9194e-008
5.9924e-006
-1.0479e-004
5.705 le-004
-8.6018e-004
2.7937e-006
-1.4855e-004
2.1495e-003
-1.0996e-002
1.5727e-002
-2.3895e-005
1.0638e-003
-1.3934e-002
6.0405e-002
-6.7606e-002
1.1524e-005
-1.3365e-004
-1.2406e-002
2.1808e-001
-4.1909e-001
Table A.2 Turbine Controller Parameters [8]
(all quantities are given in per unit on 4MVW base)
K
150
K
25
PP
Pitch controller gains
Actuator time constant
ip
T p (second)
0.01
Time constant
T
0.05
A
pc
K
Torque controller gains
pt
3
K it
0.6
K
3
pc
Pitch compensator gains
K
30
ic
de
27
^max( g)
Pitch angle limtation
0
0
mm ( d e g )
rpmax
t
rpmin
t
Power maximum and minimum
Cut-in wind speed (m/s)
vw
Cut-off wind speed (m/s)
1
0.1
,
3.5
25
Table A.3 Two-Mass Rotor Model Parameters [8]
(all quantities are given in per unit on 100MW base)
Rotor inetia constant
Ht
Hr
0.1716
Shaft stiffness
K
11.868
Shaft damping coefficient
D
tg
0.06
Reference rotor speed
oI tef
1.15
Base rotor speed
fyfrase
Generator inertia constant
tg
0.036
1.335
Table A.4 DFIG and DC Link Parameters [6]
(all quantities are given in per unit on 100MW base)
Stator resistance
Rs
0.1285
Rotor resistance
Rr
0.1519
Stator inductance
Ls
2.82
Rotor inductance
Lr
2.9535
Magnetizing inductance
Lm
110.54
DC link capacitance
C
0.01
dc
ref
V
dc
Reference dc voltage
1
Table A.5 Maximum Operating Limit of VSC
(all quantities are given in per unit on 100MW base)
and Qn
0.012
Table A.6 PI Controller Gains of VSC
PI1 and PI3
Rotor-side converter
PI2 and .PI4 l _
Grid-side converter
PI5 and PI6
DC link
PI7
Local voltage controller
Proportional gain
0.0426
2.8013
0.0018
0.03155
0.0137
Integral gain
59.853
39.376
0.6436
2.3873
6.7057
Table A.7 Line Parameters [34]
(all quantities are given in per unit on 100MW base, 132kV base)
Transformer (TR1,TR2,TR3)
Transfomer (TR4)
Transmission line
Cable
Grid-side filter
Resistance
0.0092
0.0216
0.3775
0.0378
0.000435
Inductance
0.233
0.539
0.6689
0.0669
0.002696
Capacitance
0.024
0.0016
Table A.8 Thyristor Excitation system [13]
Amplifier gain
Tt
0.05
T
0.2
10
Thyristor gain
±
Tex
Kt
Filter gain
TR
0
Exciter gain
10
Table A.9 Synchronous Generator Parameters [13]
(all quantities are given in per unit on 100MW base, 132kV base)
Stator resistance
R
a
0.000135
Stator d-axis inductance
Ld
0.0815
Stator ^-axis inductance
L
1
0.0793
J-axis magnetizing inductance
Lmd
0.0748
g-axis magnetizing inductance
Lmq
0.0725
g-axis mutual inductance
Lamq
0.0815
J-axis mutual inductance
^amd
0.0793
q-axis mutual inductance
Lkqmq
0.0825
^-axis mutual inductance
Lkdmd
0.1052
g-axis mutual inductance
L
0.0822
Field resistance
R
f
0.00027
Field inductance
L
f
0.0074
d-axis damper winding resistance
R
kd
0.0013
d-axis damper winding inductance
L
kd
0.0077
g-axis damper winding resistance
R
kq
0.000279
^r-axis damper winding inductance
L
kq
0.0327
Electrical angular speed
coe
fmq
1
Table A. 10 Operating Conditions
Case
Grid
Load
n o m i n a l local l o a d i m p e d a n c e
BUS
V(PU)
P(PU)
Q(PU)
9
0.94860
0.07338
0.001595
8
0.93425
0.07303
0.049410
TR
0.07284
0.044610
WF
0.07738
0.002935
Total
0.15022
0.047555
Resistance
Reactance
7
(PCC)
0.90497
Load
WF
R
(• load)
( Xload)
5
1.6
BUS
V(PU)
P(PU)
Q (PU)
1
0.90936
0.02671
0
2
0.90912
0.02588
-0.0001888
3
0.90866
0.02671
0
TR
0.02588
-0.0001888
Cable
0.02586
0.00110100
Total
0.05174
0.00091190
0.02671
0
TR
0.02588
-0.0001891
Cable
0.05166
0.00210200
Total
0.07755
0.00191200
4
5
6
0.90843
0.90728
0.90704
Case
Grid
Load
50% decrease of the local load impedance
BUS
V(PU)
P(PU)
Q (PU)
9
0.94860
0.18900
0.072600
8
0.90584
0.18330
0.110300
TR
0.18210
0.080190
WF
0.07766
0.002245
Total
0.25976
0.082435
Resistance
Reactance
7
(PCC)
0.84259
Load
WF
(• Rload)
. ( Xload
2.5
0.8
)
BUS
V(PU)
P (PU)
Q (PU)
1
0.84725
0.02671
0
2
0.84700
0.02599
-0.0002193
3
0.84651
0.02671
0
TR
0.02599
-0.0002197
Cable
0.02596
0.0008907
Total
0.05195
0.0006710
0.02671
0
TR
0.02599
-0.0002205
Cable
0.05187
0.00166500
Total
0.07785
0.00144450
4
5
6
0.84626
0.84504
0.84479
Case
Grid
Load
50% increase of the local load impedance
BUS
V(PU)
P(PU)
Q(PU)
9
0.94860
0.02793
-0.018460
8
0.94900
0.02782
0.0310500
TR
0.02778
0.0300000
WF
0.07728
0.0031640
Total
0.10506
0.0331640
Resistance
Reactance
7
(PCC)
0.92565
Load
WF
R
(• load)
( Xload)
7.5
2.4
BUS
V(PU)
P (PU)
Q(PU)
1
0.92996
0.02671
0
2
0.92972
0.02584
-0.0001800
3
0.92928
0.02671
0
TR
0.02584
-0.0001803
Cable
0.02582
0.00117200
Total
0.05166
0.00099170
0.02671
0
TR
0.02584
-0.0001808
Cable
0.05159
0.00224800
Total
0.07743
0.00206760
4
5
6
0.92904
0.92792
0.92768
Appendix B
Voltage Source Converter Controller Design
The idea of the pole-placement techniques is to make an open-loop system behave as the
desired closed-loop system.
Suppose that the desired closed-loop characteristic equation for a system with a proportionalplus-integral (PI) controller can be expressed [56] as
1 + GC(5)G/,(5) = 1+
-kp+-L
G p{s)
(B.L)
where G p (5) is the first-order transfer function of a system, and G c (s) is a PI controller.
The objective with the pole-placement technique is to make the closed-loop characteristic
equation (B.l) behave as the desired closed-loop characteristic equation such that
Acl(s) = s2 +2gco ns + co2
The gain coefficients of a PI controller can be obtained by comparing (B.l) with (B.2).
(B.2)
B.l PI controller design for PI2 and PI4
To illustrate the conclusions drawn in this section, the Bode diagrams of the transfer function
of the open-loop and closed-loop systems are shown in Figure B.l.
Bode diagram
10
10
Frequency (rad/sec)
Figure B.l: Bode diagrams of transfer function of the open-loop and closed-loop system.
The transfer function of (3.23) with the numerical values is
40.6339
G2(s) = Ga(S) =
5 + 6.1723
(B.3)
The closed-loop characteristic equation for (B.3) with a PI controller is described as
Ad(s) = s1 +(6A723 + 40.6339k p)s + 40.6339k i
(B.4)
The natural frequency con is chosen as 40 rad/sec, where the phase approaches a constant
value and corresponds to a 20 dB per decade change in the magnitude. The damping ratio
C, - 1 . 5 is selected by sweeping from 0.1, which starts to provide a positive proportional gain
k p . With this damping ratio, less than 5% overshoot to the step response of the closed-loop
system is obtained. Thus, the desired closed-loop characteristic equation is
Ac!(S)
= s2 +2£A> ns + A>2 = s 2 + 1 2 0 s + 4 0 2
(B.5)
By comparing (B.4) and (B.5), the gain coefficients k p and k t are obtained as 2.8013 and
39.3760, respectively. The Bode diagram in Figure B.l shows the improving bandwidth and
the improvement of phase margin for the closed-loop system. Due to the increased
bandwidth, the closed-loop system now features faster step response time as seen in Figure
B.2.
Step response of the open-loop
Step response of the closed-loop
0
0.1
0.2
0.3
0.4
0.5
time (sec)
0.6
0.7
0.8
Figure B.2: Comparison of the step response of the open-loop and the closed-loop system.
B.2
PI controller design for PI1 and PI3
The transfer function of (3.27) or (3.31) with the numerical values is
(B.6)
G{ (s) = G3 (S) = 1.1646 + 0.060 Is
Bode diagrams of the transfer function of the open-loop and closed-loop systems are shown
in Figure B.3 to illustrate the conclusions drawn in this section.
Bode diagram
60
m" 40
-o
CD
T3
3
open-loop
closed loop
20
C
n
>
CO
2
0
-20
90
Frequency (rad/sec)
Figure B.3: Bode diagrams of transfer function of the open-loop and closed-loop system.
The closed-loop characteristic equation of (B.6) with a PI controller can be expressed as
Ad{s) = s2 +
d
0.060life,
v
(1 + 0.0601*; +\A646k p)s
P 1
+^ ^ p nnfinifr
(B.7)
By choosing con as 165 rad/sec where the phase approaches a constant value and
corresponds to a 20 dB per decade change in the magnitude, the desired closed-loop
characteristic equation is
,4 c / O) = s 2 + 3 3 0 ^ y + 165 2
(B.8)
By comparing (B.7) and (B.8), the coefficients of the PI controller are computed as
k p =0.0426 and k t =59.853. The damping ratio £ = 5.5 is selected by sweeping from 5.0,
which yields a positive proportional gain. The Bode diagram plotted in Figure B.3 shows the
feature of pole-zero cancellation, and its closed-loop step response is shown in Figure B.4.
Step response of the closed-loop
Figure B.4: Step response of the closed-loop system.
B.3
PI controller design for PI5 and PI6
The transfer function of (3.35) with the numerical values is
139830
G5(s) = G6(s) =
5 + 60.8276
(B.9)
Figure B.5 shows a comparison of the Bode diagrams of the transfer function of the openloop and closed-loop systems.
Bode diagram
open-loop
closed loop
10
10
Frequency (rad/sec)
Figure B.5: Bode diagrams of transfer function of the open-loop and closed-loop system.
The closed-loop characteristic equation of (B.9) with a PI controller can be expressed as
Arf(s) = s 2 + (60.8276 + 139830^)^ + 139830^-
(B.10)
To have a desired closed-loop system, a>n is chosen as 300 rad/sec, where the phase becomes
constant and corresponds to a 20 dB per decade change in the magnitude. The damping ratio
is chosen as 2 by sweeping from 1. With this damping ratio, the overshoot in the closed-loop
step response is less than 5%, as can be seen in Figure B.6. With these parameters, the
desired closed-loop characteristic equation becomes
+ 300,2:
Ad{s) = s2+1200^5
(B.ll)
By comparing (B.10) and (B.l 1), the proportional and integral gains are computed as 0.0081
and 0.6436, respectively.
Step response of the open-loop
2500
]| 2000
-g 1500
=
1000
I
500
0
0
0.01
0.02
0.03
0.04
0.05
time (sec)
0.06
0.07
0.08
0.07
0.08
Step response of the closed-loop
1.15
3Q.
<u
3 0.5
T3
CL
<£
0
0
0.01
0.02
0.03
0.04
0.05
0.06
time (sec)
Figure B.6: Step response of the open-loop and closed-loop system.
B.4
PI controller design for PI7
The transfer function of (3.40) is
GJ(S) =
376.9911
(B.12)
Figure B.7 shows a comparison of the Bode diagrams of the transfer function of the openloop and closed-loop systems.
Bode diagram
Frequency (rad/sec)
Figure B.7: Bode diagrams of transfer function of the open-loop and closed-loop system.
The closed-loop characteristic equation of the nominal loop (B.12) with a PI controller is as
follows:
Ad (5) = s 2 +37699.11k p s + 37699.1 \k t
(B. 13)
Since the controllers for the grid-side filter in the previous section B.3 have been designed up
to a frequency range of 300 rad/sec, the natural frequency con is chosen as 300 rad/sec. With
this parameter, the desired closed-loop characteristic equation becomes
4,/O) = S 2 + 6 0 0 ^ + 300 2
The damping ratio
(B.14)
is chosen as 1.6, where the step response of the closed-loop system
features less than 5% overshoot, as shown in Figure B.8. With this damping ratio, the
proportional and integral gains are 0.03155 and 2.3873, respectively.
Step response of the closed-loop
/
0.8
3"
D.
J 0.6
Z3
Q.
|
0.4
0.2
0
0
0.01
0.02
0.03
0.04
0.05
time (sec)
0.06
0.07
Figure B.8: Step response of the closed-loop system.
0.08
Appendix C
Local Voltage Controller Design
To design a local voltage controller, there is a need to find to a transfer function from the
voltage to the reactive power at the wind turbine terminal. In this thesis, WT3 in Figure 3.1 is
selected as a candidate model for our design of a local voltage controller since its terminal
voltage is close to the terminal voltage of the wind farm. To have a representative model, the
rest of system seen from the WT3 is modelled as an equivalent RL load model. Thus, this
representative model has system input, which is reactive power, and system output, which is
the magnitude voltage from this equivalent load. By applying the balanced model-order
reduction technique, the 3 rd -order reduced transfer function is obtained as
^ , N -0.002786s 3 + 204.4s 2 + 1277000s + 228900000
Gp(s) =
=
=
s +10880s + 2784000s+ 163600000
(C.l)
Bode diagrams of the full-order model (23 th ) and the reduced-order model (3 rd ) are shown in
Figure C.l. The reduced-order transfer function approximates the full-order model in the
frequency range of interest. Thereafter, this reduced-order model is considered to represent
the plant.
Bode Diagram
• Reduced (3rd)
-Full(23th)
10
10
10
Frequency (rad/sec)
Figure C. 1: Bode diagrams of the full-order model (23t h ) and the reduced-order model (3 rd ).
In this thesis, we design a PI controller, since it is frequently used in industry. The
compensated loop transfer function can be expressed [56] as
G c(s)G p(s)
where G c(s)
=
kpS "t* kj
(C.2)
G p(s)
stands for the controller, G p(s)
proportional and integral gain, respectively.
is a nominal plant, and kp
and ki are
As in Figure C.2, it is assumed that the compensated Nyquist diagram is to pass through the
point 1Z(-180° +(/>m), for the frequency a\, to achieve the phase margin (f> m [56], Or,
Gc{jco [)GJj(D [) = \Z{-m a
+ 0m)
(C.3)
Gc(s)Gp(s)
Re
Figure C.2: Nyquist plot of a compensated loop transfer function.
If the angle of a controller Gc(jo\)
is denoted by 6, then from (C.3),
9 = Z.G C (M) = -180° + 4>m ~ ^G p
From Gc(ja\)
(M)
(C.4)
and (C.3),
Gc (.ja\ ) = k
p
k- j - ^ = \Gc <Ja\ )| (cos 9+j sin 9)
(C.5)
where, from (C.3),
\GC(M)\
(C.6)
=
Gp(M)
|
From (C.5), equating real part to real part yields
cos 6
k p
(C.7)
\GPim)\
and equating imaginary part to imaginary part yields
o\ sin#
(C.8)
Gp(M)\
With the chosen design specifications such that the settling time r s = 0.075 seconds and the
phase margin is 85° , the phase-margin frequency, or the gain-crossover frequency, is
g
calculated as o\=—
T
s
t a n
= 9.3321 rad/sec, to yield a specified settling time r,, . The
Qm
coefficients of k p and k t are then calculated as 0.0137 and 6.7057, respectively. The antikwindup gain can be computed as k a=—
[55],
kp
Figure C.3 shows the compensated system Bode diagram, and Figure C.4 shows the step
response of the open-loop and the closed-loop system. In this thesis, the LVC mode is
implemented as shown in Figure C.5.
Figure C.3: Bode diagrams of transfer function of the open-loop and closed-loop system.
Figure C.4: Step response of the closed-loop system.
set
WT,1
•
+
V,
WT,1
\
)
. set
WT,2"
1
i,
>
I
ki
s
]
+
V,
WT,2 '
1
>
set
V,
WT,3'
ki
s
+
W3K Z
±) ^
y)—
WT,3
1
*
s
k
a
ki
+
ka
+
ka
'
Figure C.5:
+
Block diagram of the PI controller with distributed anti-windup scheme.
Appendix D
Matlab Script Files
% Written by Hee-Sang Ko.
% UBC Power and Control Lab. in May 2006.
% In these codes, the integral state is added in the last row.
D.l
Matlab Code for LQRCG and ALQRCG
%
clear all
load systemA
load systemB
load systemC
A=aA; B=aB; C=aC;
[n,m]=size(A);
[nn,mm]=size(B);
%
% Need the modification for the integral action
Ata=zeros(n+1 ,n+1);
Atb=zeros(n+1 ,n+1);
Atc=zeros(n+1 ,n+1);
Ata((l:n),[l:n])=aA;
Atb((l:n),[l:n])=bA;
Atc((l :n),[l :n])=cA;
Ata(n+1,[ 1 :n])=-aC;
Atb(n+1,[ 1 :n])=-bC;
Atc(n+1,[ 1: n])=-cC;
Bta=[aB; -aD];
Btb=[bB; -bD];
Btc=[cB; -cD];
Cta=zeros(2,n+l);
Cta(l,l)=l; Cta(2,n+l)=l;
Ctb=Cta;
Ctc=Cta;
Dta=zeros(2,l);
%
%
nstate=size(Ata, 1);
ncon=size(Bta,l);
%
Q=eye(size(Ata,2))* 1;
Q(nstate,nstate)=70; Q(1,1)=10;
R=eye(size(Bta,2))*2.2; % For ALQRCG
Qhalf = sqrtm(Q);
Rhalf= sqrtm(R);
N = ones(nstate,l)*0.55;
4
% check for the condition of Q-N'RA-1N>=0
Posi=Q-N*inv(R)*N';
% Start LMI
% Define the problem variables and matrix inequality constraints
setlmis([])
% Define and describe the matrix variables
X = lmivar(2, [1 nstate]);
Y = lmivar(l, [nstate 1]);
Z = lmivar(l, [nstate 1]);
% Define the individual LMIs. See pp.8-11 (Mathworks).
LMI_Sys_l = newlmi;
lmiterm([LMI_Sys_ 1 1 1 Y], Ata, 1, 's');
lmiterm([LMI_Sys_ 1 1 1 X], Bta, -1, 's');
lmiterm([LMI_Sys_l 1 2 -Y], 1, Qhalf);
lmiterm([LMI_Sys_l 1 3 -X],l, Rhalf);
lmiterm([LMI_Sys_l 2 2 0], -1);
lmiterm([LMI_Sys_l 3 3 0], -1);
%(1,1) block: A1*Y + Y*A1'
%(1,1) block: -B1*X - X'*A1'
%(1,2) block: Y*Ql A (l/2)
%(1,3) block: X'*Rl A (l/2)
%(2,2) block: -eye(3,3)
%(3,3) block: -eye(l,l)
LMI_Sys_2 = newlmi;
lmiterm([LMI_Sys_2
lmiterm([LMI_Sys_2
lmiterm([LMI_Sys_2
lmiterm([LMI_Sys_2
lmiterm([LMI_Sys_2
lmiterm([LMI_Sys_2
1 1 Y], Atb, 1, 's');
1 1 X], Btb, -1, 's');
1 2 -Y], 1, Qhalf);
1 3 -X],l, Rhalf);
2 2 0], -1);
3 3 0], -1);
%(1,1) block: A1*Y + Y*A1'
%(1,1) block: -B1*X - X'*A1'
%(1,2) block: Y*Ql A (l/2)
%(1,3) block: X'*Rl A (l/2)
%(2,2)block: -eye(3,3)
%(3,3) block: -eye(l.l)
LMI_Sys_3 = newlmi;
lmiterm([LMI_Sys_3
lmiterm([LMI_Sys_3
lmiterm([LMI_Sys_3
lmiterm([LMI_Sys_3
lmiterm([LMI_Sys_3
lmiterm([LMI_Sys_3
1 1 Y], Ate, 1, 's'); %(1,1) block: A1*Y + Y*A1'
1 1 X], Btc, -1, 's'); %(1,1) block: -B1*X - X'*A1'
1 2 -Y], 1, Qhalf);
%(1,2) block: Y*Ql A (l/2)
1 3 -X],l, Rhalf);
%(1,3) block: X'*Rl A (l/2)
2 2 0], -1);
%(2,2) block: -eye(3,3)
3 3 0], -1);
%(3,3) block: -eye(l,l)
% Coupling constraint
LMI_Couple = newlmi;
lmiterm([-LMI_Couple 1 1 Z],l,l);
lmiterm([-LMI_Couple 1 2 0],1);
lmiterm([-LMI_Couple 2 2 Y],l,l);
% (1,1) block: [Z I; I Y] >0
% (1,2) block: [Z I; I Y] >0
% (2,2) block: [Z I; I Y] >0
% Positive definition constraint
LMI_YPos = newlmi;
lmiterm([-LMI_YPos 1 1 Y], 1, 1);
% Y > 0 from [Z I; I Y] >0
% Stroe the internal representation of the LMI system (pp. 8-6)
lmisys = getlmis;
% Solve
ns = decnbr(lmisys);
for j=l:ns
[Xj, Yj, Zj] = defcx(lmisys, j, X, Y, Z);
% c(j) = trace(Zj);
c(j) = trace(Zj)-trace(N*Xj)-trace(Xj'*N');
end
options = [le-5 0 0 0 0];
[copt,xopt] = mincx(lmisys,c,options);
dispC
')
disp(")
disp('The optimized variable matrix X is ...');
Xstar = dec2mat(lmisys,xopt,X)
disp('The optimized variable matrix Y is ...');
Ystar = dec2mat(lmisys,xopt,Y)
disp('The optimized variable matrix Z is ...');
Zstar = dec2mat(lmisys,xopt,Z)
disp(")
disp('The robust-optimal gain is ...');
Kstar = Xstar*inv(Ystar);
dispO
% No cross-product term (LQRCG)
% Cross-product term (ALQRCG)
D.2 Matlab Codes for LQRS and ALQRS
%—
clear all
load systemA
load systemB
load systemC
A=aA; B=aB; C=aC;
[n,m]=size(A);
[nn,mm]=size(B);
%
% Need the modification for the integral action
Ata=zeros(n+1 ,n+1);
Atb=zeros(n+l ,n+l);
Atc=zeros(n+1 ,n+1);
Ata((l :n),[l :n])=aA;
Atb((l:n),[l:n]j=bA;
Atc((l:n),[l:n])=cA;
Ata(n+l,[l:n])=-aC;
Atb(n+l,[l :n])=-bC;
Atc(n+1, [ 1: n])=-cC;
Bta=[aB; -aD];
Btb=[bB; -bD];
Btc=[cB; -cD];
Cta=zeros(2,n+l);
Cta(l,l)=l; Cta(2,n+l)=l;
Ctb=Cta;
Ctc=Cta;
Dta=zeros(2,l);
%
:
%
....
nstate=size(Ata,
1);
ncon=size(Bta,
1);
%
Q=eye(size(Ata,2))*l;
Q(nstate,nstate)=70, Q(l,l)=10;
R=eye(size(Bta,2))* 1.2; % For ALQRS -> 0.955
Qhalf = sqrtm(Q);
Rhalf = sqrtm(R);
N = ones(nstate,l)*0.55;
Q-N*inv(R)*N'
% Start LMI
% Define the problem variables and matrix inequality constraints
setlmis([])
% Define and describe the matrix variables
X = lmivar(2, [1 nstate]);
Y = lmivar(l, [nstate 1]);
M = lmivar(2, [1,1]);
% Define the individual LMIs. See pp.8-11 (Mathworks).
LMI_Sys_l = newlmi;
lmiterm([LMI_Sys_l
lmiterm([LMI_Sys_l
lmiterm([LMI_Sys_l
lmiterm([LMI_Sys_l
lmiterm([LMI_Sys_l
lmiterm([LMI_Sys_l
1 1 Y], Ata, 1, 's');
1 1 X], Bta, -1, 's');
1 2 -Y], 1, Qhalf);
1 3 -X],l, Rhalf);
2 2 0], -1);
3 3 0], -1);
%(1,1) block: A1*Y + Y*A1'
%(1,1) block: -B1*X-X'*A1'
%(1,2) block: Y*Ql A (l/2)
%(1,3) block: X'*Rl A (l/2)
%(2,2) block: -eye(3,3)
%(3,3)block: -eye(l,l)
LMI_Sys_2 = newlmi;
lmiterm([LMI_Sys_2
lmiterm([LMI_Sys_2
lmiterm([LMI_Sys_2
lmiterm([LMI_Sys_2
lmiterm([LMI_Sys_2
lmiterm([LMI_Sys_2
1 1 Y], Atb, 1, 's');
1 1 X], Btb, -1, 's');
1 2 -Y], 1, Qhalf);
1 3 -X],l, Rhalf);
2 2 0], -1);
3 3 0], -1);
%(1,1) block:
%(1,1) block:
%(1,2) block:
%(1,3) block:
%(2,2) block:
%(3,3)block:
A1*Y + Y*A1'
-B1*X - X'*A1'
Y*Ql A (l/2)
X'*Rl A (l/2)
-eye(3,3)
-eye(l,l)
LMI_Sys_3 = newlmi;
lmiterm([LMI_Sys_3
lmiterm([LMI_Sys_3
lmiterm([LMI_Sys_3
lmiterm([LMI_Sys_3
lmiterm([LMI_Sys_3
lmiterm([LMI_Sys_3
1 1 Y], Ate, 1, 's');
1 1 X], Btc, -1, 's');
1 2 -Y], 1, Qhalf);
1 3 -X],l, Rhalf);
2 2 0], -1);
3 3 0], -1);
%(1,1) block: A1*Y + Y*A1'
%(1,1) block: -B1*X - X'*A1'
%(1,2) block: Y*Ql A (l/2)
%(1,3) block: X'*Rl A (l/2)
%(2,2) block: -eye(3,3)
%(3,3) block: -eye(l,l)
% Subject function, LMI #4:
% [ M Sqrt(R)X ]
%[]>0
% [X'Sqrt(R) Y ]
LMI_Sys_4 = newlmi;
lmiterm([-LMI_Sys_4 1 1 M], 1, 1);
lmiterm([-LMI_Sys_4 1 2 X], Rhalf, 1);
lmiterm([-LMI_Sys_4 2 2 Y], 1, 1);
% LMI #4: M
% LMI #4: X'*sqrt(R)
% LMI #4: Y
% Stroe the internal representation of the LMI system (pp.8-6)
lmisys = getlmis;
% Solve
ns = decnbr(lmisys);
for j=l:ns
[Xj, Yj, Mj] = defcx(lmisys, j, X, Y, M);
c(j) = trace(Q*Yj) + trace(Mj);
%c(j) = trace(Q*Yj) + trace(Mj) - trace(N*Xj) - trace(Xj'*N');
end
options = [le-5 0 0 0 0];
[copt,xopt] = mincx(lmisys,c,options);
% No cross-product term (LQRS)
% Cross-product term (ALQRS)
disp('
')
disp(")
disp('The optimized variable matrix X is ...');
Xstar = dec2mat(lmisys,xopt,X)
disp('The optimized variable matrix Y is ...');
Ystar = dec2mat(lmisys,xopt,Y)
disp('The optimized variable matrix M is ...');
Mstar = dec2mat(lmisys,xopt,M)
disp(")
disp('The robust-optimal gain is ...');
Kstar = Xstar*inv(Ystar)
dispC)
Appendix E
Proof of Theorem
Theorem:
If common positive definite P exists that satisfies the Lyapunov inequality (5.14) as in (E.l),
(Ay- - B y k ) 7 , P + P ( A y - B y k ) + Q + k r R k < 0
then
the
cost
function
J
of
(5.11)
is
bounded
(E.l)
by
the
scalar
expression
£^x(0) r Px(0) = £ ^ r ( x ( 0 ) P x ( 0 ) r ) =<r(X 0 P) asin(E.2):
K = m m j ^ { J ° ( x r Q x + urRu)<fc J < £ [ / r ( x ( 0 ) P x ( 0 ) r ) ] = /r(XoP)
where j = l,...,N
(E.2)
where N is the total number of multiple systems. The variables Q and
R are design parameters; Q > 0 is positive semi-definite matrix, and R is positive definite
symmetric matrix. The variable P is the Lyapunov matrix and is positive definite such that
P > 0. The variable X 0 is the expectation of the covariance of the stationary random initial
vector such that £ ^ x ( 0 ) x ( 0 ) r ] = X 0 .
For proof of the theorem, the following Definition and Lemma are needed.
Definition:
Consider the system (5.1) and the cost function (5.11). If there exists a control law u and a
positive scalar JG such that the closed-loop is stable and the closed-loop value of the cost
function (5.11) satisfies J < JQ , then Jq is said to be a guaranteed cost and u is said to be a
guaranteed cost controller for the system (5.1).
Lemma [67]:
The closed-loop system
x = ( A - B k ) x + Gw
(E.3)
for given system matrices (A,B) is asymptotically stable if and only if there exists a positive
definite P > 0 satisfying
(A-Bkf P + P(A-Bk)<0
(E.4)
where the state noise signal has zero mean £ [ w ] = 0 and symmetric positive definite
covariance matrix E
WW
T
= RW>0.
This lemma can be extended to the multiple systems situation. If matrix Lyapunov inequality
(E.5) is satisfied by a common positive definite P for all the systems, the systems are
guaranteed to have asymptotic stability within the linear regions for which these multiple
systems are defined, i.e.,
(Ay--B7-k)rp + p(Ay-B7-k)<0
(E.5)
Proof
For convenience, the proof is carried out based on a single system.
The solution of the minimization of a cost function
V = min
( x r Q x + u r R u ) dt
(E.6)
can be found from utilizing the parameter optimization problem solved by the second
Lyapunov method [63], [65] such that
xrQx + urRu = E
(E.7)
dt
where P is the Lyapunov matrix.
Substitute u = - k x into (E.7) to obtain
E(x TQx + xTk TRkx\
=E- - ( x ' W
dt
= E ( - X
T
PX-X^PX)
(E.8)
Substitute the system (5.1) with u = -kx into the right-hand side of (E.8) to obtain
xT ^Q + k^Rkjx 7^ = E -xT ( ( ( A - B k ) r + G w r ) P - P ( ( A - B k ) + Gw))x (E.9)
Since i?[w] = 0, (E.9) can be rewritten as
xT ( ( A • - B k ) r P + P ( A - B k ) j x = -E x r E x < 0
where E is ^Q + k ^ R k j , which is a positive definite matrix.
(E.10)
Thus, there exists a positive definite P , and by Lemma, there exist stable ( A - B k ) as
t —»
Equation (E. 10) is then posed in a matrix Lyapunov inequality such that
(A-Bk) P + P(A-Bk) + E < 0
(E.ll)
Multiply each term in the Lyapunov inequality (E.ll) by the system state transition matrix
Bk) t f
e(A-Bk)
r
or
^ g ief t a n c j
e
(A
/p(A__Bk)e(A-Bk).
Bk
V for the right to obtain
+ g ( A - B k f / ( A _ B k ) p . ( A - B k ) . +e(A-Bk)%e(A-Bk),
(E.12)
Equation (E; 12) can be simplified to
dt
'
A-Bk) 7 "/ pg (A-Bk)/ <0
(E.13)
Take integral to (E.13) to obtain
•"o dt
dt<0
(E.14)
Multiply each term of (E.14) by the stationary random initial vector x(0) and x(0) and
utilize expectation to obtain
K x ( 0 f
e
(
A
-
B k
^
dt <0
(E.15)
< Q
Equation (E.l5) can be rewritten as
<E x(0) Px(0)
(E-l 6)
Since £ , j^x(0) 7, Px(0)J is scalar, (E.l6) can be rewritten as
f^Ex)*
<E x(0) Px(0) = E^tr (x(O)Px(O)7,
= tr ( X 0 P )
with utilizing trace operation property such that tr(ABC) = tr(CBA) = tr{CAB).
(E.17)
To avoid
the dependency of the cost function V on initial conditions, we assume the initial conditions
are random variables with zero mean and a covariance equal to the identity such that
£[x(0)x(0)r] = I
and
£[x(0)] = 0
(E.l 8)
Thus, (E.17) can be reduced
E\ J^(x rEx^
and hence, (E.2) holds.
=E\ 0xr(Q
+ k rRk)x)<#
(E.l 9)
Appendix F
Effect of the Cross-Product Terms
It is well known from the LQR theory that for the linear time-invariant system
x ( 0 =F x ( 0 +G u ( 0 ,
x ( 0 ) given
(F.l)
the determination of an optimal control with the associated optimal cost function
v = J ^ ° ^ x r Q x+ u r R u + x r N u + u r N r x d t
( F . 2 )
is reduced as follows:
u(t) = u0(t) + R~1l$ Tx(t)
where
(
F
.
3
)
u* (7) = -kx(7) = R - 1 G^Px(0, P is the positive definite matrix, which is the
of the Riccati equation in the LQR problem [61],
Now we show how linear quadratic problems with the cross-product terms arise when
dealing with linearized systems [61]. Suppose that beginning at x(? 0 ), the optimal control
•
*
u 0 ( 7 ) drives the state along the trajectory x ( t ) . However, for whatever reason, as shown in
$
*
Figure F.l the state at time t is not x (t) but is x ( 7 ) + £ x ( 7 ) with Sx(t) small. Intuitively,
additional optimal control can be expected related to Sx(t).
Thus, the expected optimal control with the additional optimal control S\(t)
can be
described as follows:
*
*
(F.4)
V (0 = v o ( 0 + *v(0
5x(0
x(0 +8x(0
Figure F. 1: Optimal and neighbouring optimal paths.
Hence, comparing (F.4) with (F.3), the control signal £ v ( 0 in (F.4) can be seen to be
equivalent to ^ R ^ N ^ x j in (F.3), which is related to the cross-product terms. Therefore, an
important application of the cross-product terms compounds to the case when an optimal
control is in place for a nonlinear system, but additional closed-loop regulation is required to
maintain, as closely as possible, the optimal trajectory in the presence of disturbances that
cause small perturbations from the trajectory.
END