SMALL-SIGNAL STABILITY, TRANSIENT STABILITY AND
VOLTAGE REGULATION ENHANCEMENT OF POWER SYSTEMS
WITH DISTRIBUTED RENEWABLE ENERGY RESOURCES
by
Adirak Kanchanaharuthai
Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy
Dissertation Advisor: Dr. Kenneth A. Loparo and Dr. Vira Chankong
Department of Electrical Engineering & Computer Science
CASE WESTERN RESERVE UNIVERSITY
January 2012
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the thesis/dissertation of
Adirak Kanchanaharuthai
______________________________________________________
Doctor of Philosophy
candidate for the ________________________________degree *.
Kenneth A. Loparo
(signed)_______________________________________________
(chair of the committee)
M. Cenk Cavusoglu
________________________________________________
Vira Chankong
________________________________________________
Marc Buchner
________________________________________________
________________________________________________
________________________________________________
11/21/2011
(date) _______________________
*We also certify that written approval has been obtained for any
proprietary material contained therein.
Table of Contents
Table of Contents
List of Tables . .
List of Figures . .
Acknowledgement
Abstract . . . . .
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iii
v
vi
ix
x
1 Introduction
1.1 Motivation and Literature Survey . . . . . . . . . . . . . . . . . . . .
1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . .
1
1
8
9
2 Power System Models
2.1 Synchronous Generators: SG . . . . . . . . . . . . . . . . . . . . . . .
2.2 Doubly-Fed Induction Generators: DFIG . . . . . . . . . . . . . . . .
2.3 STATCOM/Battery Energy Storage Models . . . . . . . . . . . . . .
12
12
13
18
3 Small-Signal Stability Enhancement
3.1 Introduction . . . . . . . . . . . . . . . . . . .
3.2 Problem Formulation . . . . . . . . . . . . . .
3.3 Transmitted Power using STATCOM/Battery
3.3.1 Single Machine Infinite Bus . . . . . .
3.3.2 Two-Machine Infinite Bus . . . . . . .
3.4 Linearized Power System Models including
STATCOM/Battery . . . . . . . . . . . . . .
3.5 Controller Design with D-stability . . . . . . .
3.6 Simulation Results . . . . . . . . . . . . . . .
3.6.1 Single Machine Infinite Bus . . . . . .
3.6.2 Two-Machine Infinite Bus . . . . . . .
3.7 Conclusion . . . . . . . . . . . . . . . . . . . .
iii
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20
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24
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29
35
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4 Transient Stability and Voltage Regulation Enhancement
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . .
4.3 IDA-PBC Controller Design . . . . . . . . . . . . . . . . . .
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Single Machine Infinite Bus . . . . . . . . . . . . . .
4.4.2 Two-Machine Infinite Bus . . . . . . . . . . . . . . .
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
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49
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5 Nonlinear Voltage Regulation
71
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 IDA-PBC Controller Design . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 Nonlinear Observer Design . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 Asymptotic stability of the combined IDA-PBC controller and nonlinear observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.6 Dynamic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.7.1 Single Machine Infinite Bus . . . . . . . . . . . . . . . . . . . 95
5.7.2 Two-Machine Infinite Bus . . . . . . . . . . . . . . . . . . . . 96
5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6 Conclusion
110
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Future Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A Appendix of Chapter 3
113
A.1 Fi (x) of the voltage dynamics . . . . . . . . . . . . . . . . . . . . . . 113
A.1.1 Fi (x) of the voltage dynamics in (3.25) . . . . . . . . . . . . . 113
A.1.2 Fi (x) of the voltage dynamics in (3.31) . . . . . . . . . . . . . 114
A.2 Aij and Buij in (3.27) . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.3 Aij and Buij in (3.30) . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.4 Aij and Buij in (3.33) . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.5 Parameters of the SMIB and TMIB . . . . . . . . . . . . . . . . . . 119
A.6 PSS/AVR parameters for SG case (PSS/AVR) . . . . . . . . . . . . 120
A.7 PI gains for DFIG case (PIC) . . . . . . . . . . . . . . . . . . . . . . 120
A.8 PSS/AVR paramenters and PI gains for SG-DFIG case (PSS/AVR/PIC)120
B Appendix of Chapter 5
121
B.1 Computing ϕ of SG-DFIG cases . . . . . . . . . . . . . . . . . . . . . 121
B.2 Initial conditions of SG, DFIG, and SG-DFIG cases . . . . . . . . . 122
iv
List of Tables
4.1 CCT of SG cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 CCT of DFIG cases . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 CCT of two-machine bus (SG-DFIG cases) . . . . . . . . . . . . . . .
63
63
63
B.1 Initial conditions of SG and DFIG cases . . . . . . . . . . . . . . . . 122
B.2 Initial conditions of SG-DFIG case . . . . . . . . . . . . . . . . . . . 122
v
List of Figures
1.1 Top 10 new installed and cumulative capacity in 2010 [25] . . . . . .
1.2 Wind power installation throughout the world at the end of 2010 [25]
1.3 Classification of power system stability [42] . . . . . . . . . . . . . . .
2
3
4
2.1 Equivalent circuit of DFIG [66] . . . . . . . . . . . . . . . . . . . . .
14
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
24
25
26
27
37
41
41
42
42
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phasor Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phasor diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A stable D region [15] . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotor speed (frequency) and voltage responses of SG case . . . . . . .
Control input responses of SG case . . . . . . . . . . . . . . . . . . .
Control input responses of SG case . . . . . . . . . . . . . . . . . . .
Unmeasured state responses of SG case . . . . . . . . . . . . . . . . .
Closed-loop pole location of SG case: “ ◦ ” without D-stability and
“ + ” with D-stability . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotor speed (frequency) and voltage responses of DFIG case . . . . .
Control input responses of DFIG case . . . . . . . . . . . . . . . . . .
Control input responses of DFIG case . . . . . . . . . . . . . . . . . .
Unmeasured state responses of DFIG case . . . . . . . . . . . . . . .
Closed-loop pole location of DFIG case: “ ◦ ” without D-stability and
“ + ” with D-stability . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotor speed (frequency) and voltage responses of SG and DFIG case
Control input responses of SG-DFIG case . . . . . . . . . . . . . . . .
Control input responses of SG-DFIG case . . . . . . . . . . . . . . . .
Unmeasured state responses of SG-DFIG case . . . . . . . . . . . . .
Unmeasured state responses of SG-DFIG case . . . . . . . . . . . . .
Closed-loop pole location of DFIG case: “ ◦ ” without D-stability and
“ + ” with D-stability . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
43
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44
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48
48
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
A single line diagram of SMIB . . . . . . . . . . . . .
A single line diagram of two-machine infinite bus . .
Phase portraits of SG case (D = 0) . . . . . . . . . .
Voltage and active power responses at tcr = 0.40 sec.
Phase portraits of SG case (D = 0.2) . . . . . . . . .
Voltage and active power responses at tcr = 1.56 sec.
Phase portraits of DFIG case . . . . . . . . . . . . .
Voltage and active power responses at tcr = 0.69 sec.
Phase portraits of SG in SG-DFIG case . . . . . . . .
Phase portraits of DFIG in SG-DFIG case . . . . . .
Voltage and active power responses at tcr = 0.425 sec.
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59
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5.1 A single line diagram of SMIB . . . . . . . . . . . . . . . . . . . . . .
5.2 A single line diagram of two-machine infinite bus . . . . . . . . . . .
5.3 Temporary fault in SG case: Time histories of rotor speed (frequency)
(ω − ωs ), terminal voltage (Vt ) and active and reactive currents of
GEC/STAT and GEC/STAT/BATT . . . . . . . . . . . . . . . . . .
5.4 Temporary fault in SG case: Time histories of the estimated and actual
states of power angles (δ̂ and δ) and transient voltages (Ê and E) of
GEC/STAT/BATT . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Temporary fault in DFIG case: Time histories of DFIG rotor slip (frequency) (1 − ω/ωs ), terminal voltage (Vt ) and active and reactive currents of DRVC/STAT and DRVC/STAT/BATT . . . . . . . . . . . .
5.6 Temporary fault in DFIG case: Time histories of the estimated and
actual states of power angles (δ̂ and δ) and transient voltages (Ê and
E) of DRVC/STAT/BATT . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Temporary fault in SG-DFIG case: Time histories of rotor speed (frequency) (ω − ωs ), DFIG rotor slip (frequency) (1 − ω/ωs ) and terminal
voltage (Vt ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Temporary fault in SG-DFIG case: Time histories of active and reactive currents of GEC/DRVC/STAT and GEC/DRVC/STAT/BATT .
5.9 Temporary fault in SG-DFIG case: Time histories of the estimated and
actual states of power angles (δ̂i and δi, i = 1, 3) and transient voltages
(Êi and Ei , i = 1, 3) of GEC/DRVC/STAT/BATT . . . . . . . . . . .
5.10 Permanent fault in SG case: Time histories of rotor speed (frequency)
(ω − ωs ) and terminal voltage (Vt ) . . . . . . . . . . . . . . . . . . . .
5.11 Permanent fault in SG case: Time histories of active and reactive currents of GEC/STAT and GEC/STAT/BATT . . . . . . . . . . . . . .
5.12 Permanent fault in SG case: Time histories of the estimated and actual
states of power angles (δ̂ and δ) and transient voltages (Ê and E) of
GEC/STAT/BATT . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
76
vii
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102
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106
5.13 Permanent fault in DFIG case: Time histories of DFIF rotor slip (frequency) (1 − ω/ωs ) and terminal voltage (Vt ) . . . . . . . . . . . . . .
5.14 Permanent fault in DFIG case: Time histories of active and reactive
currents of DRVC/STAT and DRVC/STAT/BATT . . . . . . . . . .
5.15 Permanent fault in DFIG case: Time histories of the estimated and
actual states of power angles (δ̂ and δ) and transient voltages (Ê and
E) of DRVC/STAT/BATT . . . . . . . . . . . . . . . . . . . . . . . .
5.16 Permanent fault in SG-DFIG case: Time histories of rotor speed (frequency) (ω − ωs ), DFIF rotor slip (frequency) (1 − ω/ωs ) and terminal
voltage (Vt ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.17 Permanent fault in SG-DFIG case: Time histories of active and reactive
currents of GEC/DRVC/STAT and GEC/DRVC/STATCOM/BATT
5.18 Permanent fault in SG-DFIG case: Time histories of the estimated
and actual states of power angles (δ̂i and δi, i = 1, 3) and transient
voltages (Êi and Ei , i = 1, 3) of GEC/DRVC/STAT/BATT . . . . .
viii
107
107
108
108
109
109
ACKNOWLEDGEMENTS
First of all, I would like to thank my research advisor, Professor Kenneth A.
Loparo for his constant guidance, encouragement, and support. I am also thankful for
numerous discussion, critical comment, constructive suggestion, and careful readings
of this dissertation.
I am indebted to Professor Vira Chankong for serving as my academic and research
co-advisor on his helps, the interesting comments, the orals and reading committees.
I am thankful to Professor Marc Buchner and Professor M. Cenk Cavusoglu for
being on my advisory committee and reviewing my dissertation. I am also grateful
to Professor Wei Lin for bringing my interest to the nonlinear control theory and its
application through his classes and research group seminars.
Many thanks are due to my friends who had given numerous helps while I studied at CWRU. In particular, I would like to thank Dr. Sakchai Rakkarn of Kasem
Bundit University for his generous help during my arrival. I also thank Dr. Arsit Boonyaprapasorn of Chulachomklao Royal Military Academy for numerous helps
during my study at CWRU and sharing his experience with me. His friendship and
gentleness are sincerely appreciated. In addition, many thanks go to Wanchat Theeranaew, Thanyaseth Sethaput, Sorn Srimatrang, and Anurak Thungtong who shared
numerous experiences and offered an enormous number of assistance.
I am grateful to the C.S. Draper Laboratory, Cambridge, MA. for providing a
University Research and Development (URAD) grant through which my work was
partially supported. My entire study and invaluable experiences at CWRU were
financially supported by Rangsit University, which is gratefully acknowledged.
Finally, I would like to express my gratitude to my mother and two sisters for their
constant support and encouragement. Besides, I would like to thank Professor Pinit
Ngamsom of Rangsit University for introducing me to study abroad in the Ph.D.
level. I deeply thank my close sister: Piraporn Konkhum for her invaluable helps.
ix
Small-Signal Stability, Transient Stability and Voltage Regulation Enhancement of
Power Systems with Distributed Renewable Energy Resources
Abstract
by
ADIRAK KANCHANAHARUTHAI
The environmental and economic impact of fossil fuel generating plants has resulted in the need to find more efficient and clean sources of electricity power generation. As an alternative, promising power generation options such as renewable energy
sources, especially wind energy, are receiving increased interest.
This dissertation addresses some important problems in power system operations
(stability and power quality) when renewable energy resources (generation and storage) are integrated into the conventional power system. In particular, the use of
energy storage (STATCOM/Battery) to enhance the small-signal stability, transient
stability, and voltage regulation of an electrical power system with renewable power
generation is investigated as follows.
• A linearized model of the nonlinear power system in the region of a steadystate operating point is used in conjunction with an LMI-based control design
method, including D-stability, to achieve small-signal angle and frequency sta-
bility, voltage regulation, when the system is perturbed by changes in mechanical power inputs to the synchronous generators in the system.
• A nonlinear model of the power systems with energy storage is used with an In-
terconnection and Damping Assignment-Passivity-Based Control design (IDAx
PBC) framework to achieve transient power angle stability along with frequency
and voltage regulation after the occurrence of a large disturbance (a symmetrical
three-phase short circuit transmission line fault).
• Using the combination of the IDA-PBC control law and a reduced-order non-
linear observer, voltage regulation and system stability enhancement are simultaneously accomplished following both temporary and permanent faults.
To illustrate the effectiveness of the STATCOM/Battery to enhance small-signal
and transient stability, the LMI-based controller design with D-stability and IDAPBC controller design are validated using the following simulation studies: (1) a
single machine infinite bus (SMIB) (2) a two-machine (synchronous generator and
doubly-fed induction generator) connected to an infinite bus. For the small-signal
case, simulation results show that the proposed controller can simultaneously achieve
frequency and voltage regulation along with improved transient performance. For
the large-signal case, the results show that the proposed controller provides improved
critical clearing times (CCTs) and an enlarged domain of attraction (DOA) along
with improved frequency and voltage regulation.
xi
Chapter 1
Introduction
1.1
Motivation and Literature Survey
Due to environmental and economic concerns, there are numerous attempts to find
more efficient and clean ways of generating electric power. As an alternative generation option, renewable energy is gaining increasing interest throughout the world. In
particular, wind energy is one of the fastest-growing and promising renewable energy
options because it is a clean and inexhaustible resource that is available throughout
the world.
The Global Wind Energy Council (GWEC)1 [25] recently reported (2010) the
global status of wind power in 2010. Figure 1.1 summarizes the status of global wind
energy installations, top 10 new installations and cumulative capacity, while wind
power installation statistics throughout the world at the end of 2010 are shown in
Figure 1.2 in luring the quantity of global cumulative installed and global annual
installed wind capacity from 1996 to 2010 along with the annual installed capacity
by region from 2003-2010. As shown in Figures 1.1 and 1.2, it is evident that there
is a 22.5% increase in global installed wind power capacity during the year, and total
installed capacity of 194.4 GW despite the decrease in annual installations. Asia and
Europe are the main markets driving growth, with 19 GW and 9.9 GW installed
1
Refer to www.gwec.net for more details
1
capacity, respectively, in 2010. In particular, Asia was the world’s largest regional
market for wind energy for the third year in a row. China was the world’s largest
market in 2010, adding a staggering 16.5 GW of new capacity over the USA to become
the world’s leading wind power country. For the USA, 5.1 GW of wind energy was
installed in 2010 for a total of 40.2 GW of wind power capacity, up from 35.1 GW
at the end of 2009. From the GWEC’s growth projection of the global wind market,
more than 40 GW of new wind power capacity will be added in 2011 and by 2015, the
global installed wind power capacity will more than double to 450 GW from 194.4
GW at the end of 2010. It is anticipated that such growth will continue into the
future. The increase in installed wind energy capacity and the need to rely more on
alternative and renewable sources of energy provides motivation for this thesis.
Figure 1.1: Top 10 new installed and cumulative capacity in 2010 [25]
2
Figure 1.2: Wind power installation throughout the world at the end of 2010 [25]
The effective integration of wind, and other distributed renewable energy resources, into the conventional power grid is a challenging task. As the diversity
of the generation mix increases and larger amounts of power are being obtained from
renewable sources, differences in the dynamic behavior between conventional and
alternative generators can play an increasingly important role in the stability and
security of the grid. Therefore, with larger amounts of installed wind power, both
conventional and alternative energy generation are generally faced with the difficult
task of maintaining power system stability and quality when small or large distur-
3
bances occur in the power system. Power system stability and security are important
issue for the reliable operation of power systems and it is essential to develop methods
that mitigate an adverse effects of intermittent renewable resources on the power grid.
As shown in Figure 1.3, power system stability [41, 42] can be mainly classified as
(1) Rotor angle stability (2) Voltage stability (3) Frequency stability. The first two
are subcategorized into small-signal and transient stability.
Figure 1.3: Classification of power system stability [42]
As usual, the power system stability needs to be maintained subject to the occurrence of small or large disturbances. Small disturbances, for example load changes or
fluctuations in mechanical power input to a generator, are constantly perturbing the
power system, especially when the influence of variable wind speed is considered in a
wind energy conversion system. Large disturbances, for instance loss of a generator or
a symmetrical three-phase short circuit transmission line fault, are closely associated
with the transient stability related to dynamic behavior of the system during a fault
before the fault is cleared from the system.
In order to enable us to effectively tackle the existing transmission facilities and
overcome a large number of constraints to build new transmission lines, there are
4
currently rapid developments in power electronic devices, thus leading to the development of reliable and high speed flexible AC transmission system (FACTS) devices.
FACTS can be mainly used to enhance the controllability of power flow and voltages
augmenting the utilization and stability and are common equipments in the power
industry. In addition, they have been used to replace a significant number of mechanical control devices [29, 62]. FACTS [58, 75] are currently composed of SVC (Static
VAR Compensator), STATCOM (Static Synchronous Compensator), TCSC (Thyristor Controlled Series Compensator), SSSC (Static Synchronous Series Compensator),
TCPAR (Thyristor Controlled Phase Angle Regulator), and UPFC (Unified Power
Flow Controller). Applications for FACTS are often used in interconnected and longdistance transmission systems to improve a lot of technical problems, namely load flow
control, voltage control, system oscillations, inter-area a oscillation, reactive power
control, steady state stability, and dynamic stability. Among the family of FACTS
devices, the Static Synchronous Compensator (STATCOM) is of particular interest in
this study due to its ability to increase the grid transfer capability through enhanced
voltage stability, to significantly provide smooth and rapid reactive power compensation for voltage support, and to slightly improve both damping oscillation [50] and
transient stability [16, 26, 27].
It is well-known that potential, kinetic or chemical energy can be converted to
another important form, especially electrical energy. As a result, electrical energy
storage technologies have currently a considerable number of types of energy storage
media such as batteries, fuel cells, flywheels, super conducting magnetic energy storages (SMEs), super capacitor, compressed air energy storage (CAES), pumped hydro,
and so on. In particular, the energy storage system (BESS) [10, 19, 43, 59] is essential
to expending the use of both conventional and renewable power energy. Integrating
wind and other renewable energy to the power grid is an ongoing issue for the utility
industry and is becoming even more important as the penetration of these renewable
resources increases. A benefit of using energy storage is to provide the opportunity to
improve power system reliability and power quality, especially frequency and power
5
stability, as reported in [3, 37, 45, 64]. For stability enhancement of wind energy conversion systems, battery energy storages has been used to improve frequency stability
and the regulation of active power levels. As reported in [51, 63, 65, 70], battery
energy storage can be used to (1) enable a shift of wind-generated energy from offpeak to on-peak availability, (2) reduce the need to compensate for the variability
and limited predictability of wind generation resources, (3) to allow the wind energy
storage system to be a viable peak-load resource, (4) reduce the power fluctuations
from large wind farms, (5) deliver large amounts of energy in a short time period
when needed, and (6) regulate active power output to improve the economics of wind
farms. Among a significant number types of storage media, of particular interest
in this work is the battery capable of improving several power qualities mentioned
above.
To further increase the capability of the existing FACTS device alone, the battery
energy storage system, as an effective choice mentioned previously, is a promising
and proven technology. There, however, are commercially very few available BESSFACTS devices due to these limited applications being expensive and bulky. It is
however possible that the suitable mix of battery energy storage system and one
of the family of FACTS is likely to be the battery and the STATCOM. Since both
are a common FACT device and widely used as storage devices for power system
application, of particular interest in this work is the integration of STATCOM and
battery to enhance the small-signal and transient stability along with frequency and
voltage regulation.
The STATCOM and (battery) energy storage so far have independently been
used to improve power system operations, and by integrating these devices using the
STATCOM to improve voltage stability and the battery to regulate active power and
improve frequency stability, provides an opportunity to improve overall small-signal
stability of the power system. Relatively little prior work has been devoted to the
integration of STATCOM and battery energy storage, Yang et al. [69] have shown
the integration of STATCOM and battery to enhance voltage stability and damping
6
oscillation capable of providing additional benefits beyond the STATCOM in conventional power systems through simulation results compared with the experimental
results. Cheng et al. [14] have shown that stability of a single generator system including STATCOM/Battery under 3-phase fault of different durations through simulation
results compared in terms of modularity, switching losses and transient stability impact of different STATCOM/Battery topologies experimentally. Baran et al. [8] have
shown the ability of the STATCOM and to smooth intermittent wind farm power
and compensate reactive power through simulation. In addition, Arulampalam et al.
[7] have shown that improvements in stability and power quality can be obtained
by a STATCOM and battery combination for a fixed speed wind farm application.
More recently, Castaneda et al. [12] have proposed a conbined STATCOM/Battery
model, both independently separated, different from [69] but there is the identical
ability to mitigate some power transmission problems arising in integration of the
wind farm and the power grid. To improve further performances beyond integrating STATCOM/Batteries, Muyeen et al. [52, 53] have proposed the combination
of STATCOM/Battery and hydrogen generation in the fixed-speed wind farm as a
positive solution in wind farm applications.
By integrating of batteries with other FACTS, Leung and Sutanto [44] have shown
that load leveling for energy management, power factor correction, harmonic filter and spinning reserve are improved through using battery and UPFC (SPFC or
UPFC/BESS). Zhan et al. [71] have shown that voltage dip on a distribution system
is mitigated through simulation presented for a one- and two-phase voltage sag. In
addition, Zhang et al. [72, 73, 74] have shown that integrating battery into FACTS
(STATCOM, SSSC and UPFC) can improve power flow control, oscillation damping
and voltage control and a comparison of the dynamic performances of FACTS/battery
and the traditional FACTS is provided.
For other types of energy storage media integrating with STATCOM, Molina et
al. [47] have shown with incorporation of super conducting magnetic energy storages (SMES) and STATCOM the transient stability and dynamic stability of power
7
systems are augmented.
The purpose of this dissertation are as follows.
• to investigate how a STATCOM and battery energy storage system can be used
to improve the small-signal stability, transient stability, and voltage regulation
of a conventional power system and a wind energy conversion system interconnected to the grid
• to simultaneously enhance some significant problems of power system stability,
namely voltage, frequency, and power regulation
• to design a dynamic output feedback controller to achieve such stabilities
1.2
Contributions
The major contributions of this dissertation can be summarized as follows:
• An analysis of transmitted power with the STATCOM/battery configuration
using simplified models consisting of conventional (SG), wind power generators
(DFIG), and the combination of both kinds of generators is investigated.
• Based on an LMI approach, the dynamic output feedback controller design with
D-stability in linearized power system models including STATCOM/Battery is
presented, thereby providing the proposed controller achieving the small-signal
stability and voltage and frequency regulation along with improved transient
performances.
• Under the assumption of full state feedback, the IDA-PBC design technique to
design a state feedback control law for conventional and wind power systems
including STATCOM/Battery is applied, thus providing the nonlinear controller
achieving transient stability, voltage and frequency regulation in terms of the
increased critical clearing time (CCT) and the enlarged domain of attraction
(ROA).
8
• A reduced-order nonlinear observer for conventional and wind power systems
including STATCOM/Battery is proposed.
• The incorporation of the IDA-PBC state feedback controller and the reduced-
order nonlinear observer to achieve transient stability and voltage and frequency
regulation is proposed.
1.3
Outline of the Dissertation
The rest of the dissertation is organized as follows.
• Chapter 2: Simplified Power System Models
The simplified power system models consisting of Synchronous generators, Doublyfed induction generators, and STATCOM/Battery energy storage systems are
given.
• Chapter 3: Small-Signal Stability Enhancement
We start with a power system analysis with STATCOM/Battery configuration
providing for two operating scenarios: (1) Single machine infinite bus with
either a synchronous generator or a DFIG connected to the infinite bus, (2) a
two-machine, both and DFIG, connected to the infinite bus. Linearized power
system models including STATCOM/Battery along with voltage dynamics and
integral action to achieve voltage and frequency regulation are described. In
order to achieve the desired performance requirements and further improve the
transient performances, the existence conditions of the LMI-based controller
with D-stability are derived.
• Chapter 4: Transient Stability and Voltage Regulation Enhancement
Based on the power system analysis with STATCOM/Battery mentioned previously, energy storage used to enhance the transient stability of an electrical
power system with renewable power generation is investigated. The passivitybased control design method (IDA-PBC) is used to achieve power angle stability
9
and frequency and voltage regulation after the occurrence of a large disturbance
(a symmetrical three-phase short circuit transmission line fault). The performance of the proposed controller is illustrated in simulation. The results show
that the proposed controller provides improved critical clearing times (CCTs)
and an enlarged domain of attraction (DOA) with improved frequency and
voltage regulation.
• Chapter 5: Nonlinear Voltage Regulation
The problem of nonlinear output feedback voltage regulation of power systems
with renewable distributed energy resources is investigated. The problem we address is to design a state feedback controller to achieve simultaneously not only
system stability enhancement but also voltage regulation. Using an interconnection and damping assignment passivity-based control (IDA-PBC) technique,
the existence of the desired nonlinear controller is derived. Moreover, dynamic
extensions are used in the proposed controller in absence of the knowledge of
the equilibrium values. Due to impossibility to measure the full states to be
used in the control law, a reduced-order nonlinear observer design is used to
estimate the unmeasured states. The proposed controller is validated using the
simulation studies. The results show that the proposed controller can simultaneously achieve the system stability enhancement and voltage regulation after
the occurrence of a large disturbance (a symmetrical three-phase short circuit
transmission line fault)
• Chapter 6: Summary and Conclusions
we summarize the results of this dissertation, and suggest future research directions.
• Appendix A: Appendix of Chapter 3
Matrices from linearized power systems including STATCOM/Battery of Chap-
ter 3 are given.
10
• Appendix B: Appendix of Chapter 5
Computing ϕ in SG-DFIG case along with initial conditions are given.
11
Chapter 2
Power System Models
In this Chapter, simplified dynamic models of synchronous and doubly-fed induction
generators, STATCOMs, and batteries are briefly developed for use in LMI-based and
nonlinear control design presented in subsequent chapters of this thesis.
2.1
Synchronous Generators: SG
Consider a power system consisting of generators interconnected through a transmission network to a set of loads. For the propose of controller design, we consider a
one-axis model for each generator based on an integral manifold [60] of a two-axis
0
model with a sufficiently small rotor circuit time constant Tq0
. A dynamic model for
a synchronous generator (SG) is obtained by representing the machine by a transient
voltage source, E, behind a transient reactance, Xs0 as follows:
δ̇ = ω − ωs
ωs
ω̇ =
(Pm − PE )
2H
Xs E
(Xs − Xs0 )V cos(δ − θ) uf
Ė = − 0 0 +
+ 0
Xs T0
Xs0 T00
T0
(2.1)
where δ is the power angle of the generator, ω denotes the relative speed of the
generator, Pm is the mechanical input power, PE =
12
EV sin(δ−θ)
Xs0
is the electrical power
delivered by the generator to the terminal voltage V of the SG, ωs is the synchronous
machine speed, ωs = 2πf, H represents the per unit inertial constant, f is the system
frequency, and θ is the angle of the SG terminal voltage. uf denotes a control input.
Refer to [6, 45, 46] for more details.
2.2
Doubly-Fed Induction Generators: DFIG
It has been common to use Induction generators (IG), Doubly-Fed Induction Generators (DFIG), and Permanent Magnet Generators (PMG) in wind energy applications.
Recently, the trend seems to be use DFIG in variable-speed wind power system installations to achieve improved dynamic performance and optimized power generation
profiles.
The principle of the Doubly Fed Induction Generator is to directly connect the
rotor windings to the grid through slip rings and a back-to-back voltage source converter that can simultaneously control rotor and grid currents. As a result, rotor
frequency is different from grid frequency and by controlling rotor currents through
the converter, it is possible to adjust the active and reactive power fed to the grid
from the stator.
If the drive train of a wind energy conversion system is connected to a DFIG,
a simplified model includes a one degree of freedom mechanical model of the drive
train, and a second order DFIG model where the electromagnetic transients of the
stator are neglected.
The equivalent circuit of the DFIG is shown in Figure 2.1 which is similar to a
synchronous generator. The voltage equations of the DFIG can be transformed to a
synchronous rotating reference frame (the d − q coordinates) as follows:
13
ids + jiqs
Rs + jXs
Rr + jXr
idr + jiqr
Rr (1−s)
s
jXm
Vds + jVqs
+
−
(Vdr +jVqr )
s
Figure 2.1: Equivalent circuit of DFIG [66]
• Stator voltages:
vds = Rs ids − ϕqs +
1 dϕds
ωs dt
(2.2)
vqs = Rs iqs + ϕds +
1 dϕqs
ωs dt
(2.3)
vdr = Rr idr − sϕqr +
1 dϕdr
ωs dt
(2.4)
vqr = Rr iqr + sϕdr +
1 dϕqr
ωs dt
(2.5)
• Rotor voltages
where all quantities except the the synchronous speed ωs are in pu. The DFIG
rotor slip s = 1 − ω/ωs , the terminal voltage is Vs = vds + jvqs and the stator
current is is = ids + jiqs .
The rotor parameters including the rotor current ir = idr + jiqr are all in the stator
windings. In addition, the relationship between the flux linkages and the current is
14
given by:
ϕds = Lss ids + Lm idr
(2.6)
ϕqs = Lss iqs + Lm iqr
(2.7)
ϕdr = Lrr idr + Lm ids
(2.8)
ϕqs = Lrr idr + Lm iqs
(2.9)
where Lss is the reactance of the stator circuit, Lrr is the reactance of the rotor circuit,
and Lm represents the magnetizing reactance.
From the expressions above, it is easy to get
idr =
ϕdr − Lm ids
Lrr
(2.10)
iqr =
ϕqr − Lm iqs
Lrr
(2.11)
Substituting the two expressions in (2.10)-(2.11) into (2.2)-(2.3), we have
where
Xs0
= ωs Lss −
L2m
Lrr
ϕds = Xs0 ids +
Lm
ϕdr
Lrr
(2.12)
ϕqs = Xs0 iqs +
Lm
ϕqr
Lrr
(2.13)
.
Let us define Ed = − ωLs Lrrm ϕqr and Eq =
ωs Lm
ϕdr
Lrr
be a voltage behind a transient
model for DFIG. Substituting Ed and Eq into (2.12)-(2.13), we obtain
ϕds = Xs0 ids + Eq
ϕqs = Xs0 iqs − Ed
Substituting the two expression above into (2.2)-(2.3) along with (2.10)-(2.11) into
15
(2.4)-(2.5), we get
vds = Rs ids − Xs0 iqs + Ed +
1 dEq
Xs0 dids
+
ωs dt
ωs dt
vqs = Rs iqs − Xs0 ids + Eq +
1 dEd
Xs0 diqs
−
ωs dt
ωs dt
dEq
dt
= −
Eq − (Xs − Xs0 )ids
Lm
− sωs Ed + ωs
vdr
0
T0
Lrr
Ed + (Xs − Xs0 )iqs
Lm
dEd
= −
+
sω
E
−
ω
vqr
s
q
s
dt
T00
Lrr
where T00 =
Lrr
.
Rr
The per unit electromagnetic torque (positive for a motor) is computed as follows:
TE = ϕds iqs − ϕqs ids
= vds ids + vqs iqs
= Ed ids + Eq iqs
Power system transient studies generally require increased numbers of dynamic
variables and small integration time steps for numerical simulation, causing increased
computational time. Model simplification by reducing the order of the generator,
for example by neglecting the stator electric transients [21, 39], can decrease the
computational burden.
In what follows, the DFIG is represented as an induction generator with a transient
voltage source, E = Ed + jEq , with nonzero rotor voltage. The DFIG system is
represented as a third-order model as follows:
Dynamic Equations:
2Htol
ds
= PE − Pm
dt
dEq
dt
= −sωs Ed + ωs
Lm
Eq − (Xs − Xs0 )ids
Vdr −
Lrr
T00
dEd
Lm
Ed + (Xs − Xs0 )iqs
= sωs Eq − ωs
Vqr −
dt
Lrr
T00
16
Electrical Equations:
PE = −Ed ids − Eq iqs , QE = Ed iqs − Eq ids
Ed = −Rs ids + Xs0 iqs + vds , Eq = −Rs iqs − Xs0 ids + vqs
The DFIG dynamic equations have two rotor inputs, Vdr and Vqr , in the d − q
reference frame. By neglecting the stator resistance Rs and transforming to polar
q
Ed2 + Eq2 and δ = tan−1 Eq /Ed , an equivalent DFIG dynamic
coordinates E =
model is:
δ̇ = ω − ωs −
ω̇ =
Xs − Xs0 V sin(δ − θ) ωs Vr cos(δ − θr )
+
Xs0
ET00
E
ωs
(Pm − PE )
2H
Ė = −
Xs E
Xs − Xs0
+
V cos(δ − θ) + ωs Vr sin(δ − θr )
Xs0 T00
Xs0 T00
where θ represents the angle of the DFIG terminal voltage V , Vr ∠θr = Vdr + jVqr =
Vr ejθr = Vr cos θr + jVr sin θr denotes the control input of the rotor side of DFIG,
and PE =
EV sin(δ−θ)
Xs0
is the active electrical power delivered by the generator to the
terminal voltage V (identical to SG case).
Remark 1. This form of the DFIG model is very similar to a one-axis SG model
with excitation current, however E in the DFIG model does not come from an external excitation current. Further, δ in the DFIG model is the angle of the rotor flux
magnitude and is not the rotating angle of the SG rotor shaft. Refer to [2, 4, 20] for
more details.
Remark 2. It is easy to obtain the dynamic model of Induction Generator (IG) by
setting Vr equal to zero in the DFIG model.
17
2.3
STATCOM/Battery Energy Storage Models
The STATCOM represents a three-phase voltage source inverter that includes capacitance on the DC link for connection to the grid. The STATCOM can be used to
inject and absorb compensating reactive power to maintain a desired terminal voltage
level. Battery energy storage can be used to improve power and frequency regulation,
and can also be used to maintain constant DC capacitor voltage, e.g. in STATCOM
applications.
Roughly speaking, STATCOMs and Batteries can be used to support electrical
power networks that have poor voltage, frequency, and power stability (both smallsignal and large-signal (transient)),[29, 59, 62] and references therein. The STATCOM/Battery dynamic models used in this work are described next.
In the d − q coordinates, the dynamic model of a STATCOM and Battery are as
follows:
Rs ωs
ωs Vt cos β
I˙d = −
Id + ωIq + hk cos(γ + β) −
Ls
Ls
ωs Vt sin β
Rs ωs
Iq + hk sin(γ + β) −
I˙q = −ωId −
Ls
Ls
1
Rb Rdc
ωs Vb ωs Id
ωs Iq
Vdc +
−
k cos(γ + β) −
k sin(γ + β)
V̇dc = −
C Rdc + Rb
Rb C
C
C
where Id and Iq are the injected or absorbed per unit d − q STATCOM/Battery
currents. Vdc is the per unit voltage across the dc capacitor C. Rs and Ls are used
to model the STATCOM/battery transformer losses. Vb denotes the per unit battery
voltage. Vt ∠β represents the per unit system side (AC) bus voltage and h =
ωs Vdc
.
Ls
Rb
and Rdc model the battery and the switching losses, respectively. k and γ represent the
PWM modulation gain and firing angle, respectively, which are also control variables
to be designed. Vdc is treated as a constant. See [69] for more details.
18
Therefore, one can rewrite the STATCOM/Battery dynamic model as follows:
ωs Vt cos β
Rs ωs
I˙d = −
Id + ωIq −
+ (h cos β) · uM − (h sin β) · uγ
Ls
Ls
Rs ωs
ωs Vt sin β
I˙q = −ωId −
Iq −
+ (h sin β) · uM + (h cos β) · uγ
Ls
Ls
where uM = k cos γ, uγ = k sin γ, k =
and sin β are given in Section 3.4.
q
(2.14)
u2M + u2γ , and γ = arctan uγ /uM . Vt , cos β,
19
Chapter 3
Small-Signal Stability
Enhancement
3.1
Introduction
Due to the environmental impact of fossil fuels (KYOTO Protocol, European Directive on renewable energy, etc.), renewable energy is receiving increased interest as an
alternative power generation option. Recently there have been efforts to integrate
wind power into the conventional power system, e.g. [2, 28]. As the diversity of
the generation mix increases and larger amounts of power are being obtained from
renewable sources, differences in the dynamic behavior between conventional and
alternative generators can play an increasingly important role in the stability and
security of the grid.
In general, both conventional and alternative energy generation are faced with
the difficult task of maintaining stability when small or large disturbances occur in
the power system. Small disturbances, for example fluctuations in mechanical power
input to a generator are constantly perturbing a power system, especially when the
influence of variable wind speed is considered in a wind energy conversion system.
Our objective is to investigate small-signal stability in power systems that include
the integration of renewable energy sources (generation and storage) with conventional generation. Flexible AC Transmission System (FACTS) devices can be used
20
to enhance power transfer capability and augment the stability of a power system
[29, 62], and of particular interest in this work is the Static Synchronous Compensator (STATCOM) that is capable of providing smooth and rapid reactive power
compensation for voltage support and to improve damping and transient stability
[26, 27].
It is evident that energy storage technologies [59], are important for dealing with
the intermittency of many alternative energy sources, and also provide the opportunity to improve power quality, especially frequency and power stability, as reported
in [45]. For stability enhancement of wind energy conversion systems, battery energy
storages has been used to improve frequency stability through the regulation of active
power levels. As reported in [63, 65], battery energy storage can be used to (1) reduce
the power fluctuations from large wind farms, (2) deliver large amounts of energy in
a short time period when needed, and (3) regulate active power output to improve
the economics of wind farms.
The STATCOM and (battery) energy storage have independently been used to improve power system operations, and by integrating these devices using the STATCOM
to improve voltage stability and the battery to regulate active power and improve frequency stability, provides an opportunity to improve overall small-signal stability of
the power system. Relatively little prior work has been devoted to the integration of
STATCOM and battery energy storage; in [14, 69] the authors have shown that the
integration of a STATCOM and battery can provide additional benefits beyond the
STATCOM in conventional power systems, in [8] the authors have shown the ability
of the STATCOM to smooth intermittent wind farm power and compensate reactive
power in a simulation study, and in [7] the authors have shown that improvements in
power quality can be obtained by a STATCOM and battery combination for a wind
farm application.
This chapter continues this line of investigation and examines how a STATCOM
and battery system can be used to improve the small-signal stability of a wind energy
conversion system connected to the grid through a Doubly Fed Induction Generator
21
(DFIG). An LMI-based controller is designed to achieve both frequency and voltage
regulation. Simulation results are provided for two operating scenarios: (1) Single
machine infinite bus with either a Synchronous Generator (SG) or DFIG connected
to the infinite bus, (2) a two-machine (SG and DFIG) connected to the infinite bus.
This chapter is organized as follows. The problem formulation is provided in
Section 3.2. Power analysis for the STATCOM and battery system is investigated in
Section 3.3. Linearized models of the various operating configurations are derived in
Section 3.4. An LMI-based controller is given in Section 3.5. Simulation results are
given in Section 3.6. Conclusions are given in Section 3.7.
3.2
Problem Formulation
In this Chapter we are interested in studying the small-signal (local) stability of a
nonlinear power system in the region of a steady-state operating point. The nonlinear
system is linearized at an operating point to obtain the (stabilizable and detectable)
linear state space model given by:
ẋ(t) = Ax(t) + Bw w(t) + Bu u(t)
(3.1)
y(t) = Cy x(t)
(3.2)
where x(t) ∈ <n is the state variables, u(t) ∈ <nu is the control input, y(t) ∈
<ny is the measured output, and w(t) ∈ <nw is the disturbance (small mechanical
power fluctuations, ∆Pm ) and the system matrices A, Bw , Bu and Cy have appropriate
dimensions.
The problem of interest can be formulated as a dynamic output feedback control
design problem where xK (t) is the state of the controller and:
ẋK (t) = AK xK (t) + BK y(t)
(3.3)
u(t) = CK xK (t) + DK y(t)
(3.4)
with AK , BK , CK , and DK matrices of appropriate dimensions to be determined so
that the closed-loop system satisfies:
22
1. All closed-loop poles are located in a stable D region of the open left-half plane
(Re(λ(Acl )) ∈ D ⊆ C − ) to achieve the desired transient performance.
2. Frequency and voltage regulation is simultaneously achieved.
The closed-loop system is as follows;
ẋcl (t) = Acl xcl (t) + Bcl w(t)
y(t) = Ccl xcl (t)
(3.5)
(3.6)
where
Acl
Bcl
A + Bu DK Cy BCK
x(t)
=
, xcl (t) =
,
BK Cy
AK
xK (t)
Bw
=
, Ccl = Cy 0 .
0
In the next section, we develop power analysis for the STATCOM/Batery and
use this along with the power system models presented in the previous chapter to
develop a linearized model of the system for the design of a dynamic output feedback
controller that meets the requirements 1) and 2) given above.
3.3
Transmitted Power using STATCOM/Battery
In this section, we study the transmitted power characteristics of the STATCOM/Battery
system using two simplified models consisting of conventional and wind power generators, especially SG and DFIG. We assume that any losses in the STATCOM/Battery
are negligible, and we model the STATCOM/Battery system as a parallel current
source that can inject or absorb both active and reactive power at the same time. We
focus on the transmitted power in single-machine (SMIB) and two-machine infinite
bus systems with STATCOM/Battery.
23
Vt ∠β
E∠δ
SG/DF IG
V∞ ∠0
jX1
jX2
Ī1
Ī2
I¯SB = Id + jIq
Figure 3.1: Network
3.3.1
Single Machine Infinite Bus
Consider the network and phasor diagram shown in Figures 3.1 and 3.2, respectively,
where X1 denotes the transformer and transient reactance of the SG or DFIG. X2
denotes the transmission line reactance between the bus terminal voltage Vt and the
infinite bus voltage V∞ . Id and Iq are the active and reactive current components,
respectively. E is the transient voltage of either the SG or DFIG.
Using Kirchoff’s voltage and current laws and extending the results from [62] to
calculate I¯1 and I¯2 , we have:
Vt ∠β = E∠δ − j I¯1X1
Vt ∠β − V∞ ∠0
I¯2 =
= I¯1 − I¯SB
jX2
(3.7)
(3.8)
From the above equations, I¯1 can be easily obtained as
E∠δ − V∞ ∠0 ¯
I¯1 =
+ ISB M̃
j(X1 + X2 )
(3.9)
where M̃ = X2 /(X1 + X2 ). The terminal STATCOM/Battery voltage Vt in (3.7) is
24
given by:
Vt ∠β = E∠δ −
(E1 ∠δ − V∞ ∠0)X1
− j I¯SB M̃
(X1 + X2 )
= Vs ∠α − j(Id + jIq )M̃
Vs ∠α
|Vs |
(3.10)
where (X1 + X2 )Vs ∠α = X2 E∠δ + X1 V∞ ∠0. |Vs | represents the unit vector of Vs .
Using the phasor diagram shown in Figure 3.2, the terminal STATCOM/Battery
voltage (Vt ) can be written as follows:
M̃
M̃
Vt ∠β = Vs ∠α 1 + Iq
− jId
|Vs |
|Vs |
(3.11)
β = α − θ̄
(3.12)
X1 V∞ ∠0
Iq M̃ V|Vs∠α
s|
jId M̃ V|Vs∠α
s|
(X1 + X2 )Vs ∠α
X2 E∠δ
Vt ∠β
θ̄
δ
X2 E sin δ
α
X2 E cos δ
X1 V∞ ∠0
Figure 3.2: Phasor Diagram
Then, the power, PE , transmitted from bus 1 to the infinite bus is:
EVt
EVt
sin(δ − β) =
sin(δ − α + θ̄)
X1
X1
EV∞ sin δ
Iq X1 X2
EX2 Id
V∞ X1 cos δ + EX2
=
1+
+
·
(X1 + X2 )
∆
(X1 + X2 )
∆
PE =
25
(3.13)
where ∆ =
p
(EX2 )2 + (V∞ X1 )2 + 2X1 X2 EV∞ cos δ.
Remark 3. The transmitted power between the generator and the infinite bus contains three terms (3.13): the first is the active power from the generator, the second
is the active power from the STATCOM [62], and the third is the active power from
the Battery.
3.3.2
Two-Machine Infinite Bus
Vt ∠β
E1 ∠δ1
SG
V∞ ∠0
jX1
jX2
Ī1
Ī2
Ī3
jX3
I¯SB = Id + jIq
E3 ∠δ3
DF IG
Figure 3.3: Network
Consider the network and phasor diagrams shown in Figures 3.3 and 3.4, respectively, incorporating a DFIG wind power generator. Here, X3 denotes the transient
reactance of the DFIG and transformer and using Kirchoff’s voltage and current laws,
we obtain:
Vt ∠β = E1 ∠δ1 − j I¯1X1 = E3 ∠δ3 − jX3 I¯3
(3.14)
Vt ∠β − V∞ ∠0
= I¯1 + I¯3 − I¯SB
I¯2 =
jX2
(3.15)
26
Then, I¯1 is given by:
I¯1 =
where K = 1 +
X2
.
X3
KE1 ∠δ1 − V∞ ∠0 −
X2
E ∠δ3
X3 3
j(KX1 + X2 )
+
I¯SB X2
KX1 + X2
(3.16)
The STATCOM/Battery terminal voltage (3.14) is
Vt ∠β = E1 ∠δ1 −
KE1 ∠δ1 − V∞ ∠0 −
X2
E ∠δ3
X3 3
KX1 + X2
= Vs ∠α − j(Id + jIq )M̃
−j
I¯SB X1 X2
KX1 + X2
Vs ∠α
|Vs |
(3.17)
where ΛVs ∠α = X2 X3 E1∠δ1 + X1 X3 V∞ ∠0 + X1 X2 E3 ∠δ3, M̃ = X1 X2 X3 /Λ, Λ =
X1 X2 + X2 X3 + X1 X3 , Id and Iq denote the Battery and STATCOM currents, respectively, and |Vs | represents the unit vector of Vs .
X1 X3 V∞ ∠0
X2 X3 E1 ∠δ1
X1 X2 E3 ∠δ3
δ3
ΛVs ∠α
θ̄
Iq M̃ V|Vs∠α
s|
Vt ∠β
jId M̃ V|Vs∠α
s|
X2 X3 E1 sin δ1 + X1 X2 E3 sin δ3
δ1 α
X2 X3 E1 cos δ1
X1 X3 V∞ ∠0 X1 X2 E3 cos δ3
Figure 3.4: Phasor diagram
The STATCOM/Battery terminal voltage can be rewritten as follows:
M̃
M̃
Vt ∠β = Vs ∠α 1 + Iq
− jId
|Vs |
|Vs |
β = α − θ̄
(3.18)
(3.19)
and the transmitted power, P1E , from bus 1 to the infinite bus is:
P1E =
E1 Vt
E1 Vt
sin(δ1 − β) =
sin(δ1 − α + θ̄)
X1
X1
27
(3.20)
From the phasor diagrams we obtain the following:
X1 X3 V∞ sin δ1 + X1 X2 E3 sin(δ1 − δ3)
,
Vs Λ
sin(δ1 − α) =
X2 X3 E1 + X1 X3 cos δ1 + X1 X2 E3 cos(δ1 − δ3 )
,
Vs Λ
Vs Id M̃
Vs
M̃
sin θ̄ =
, cos θ̄ =
1 + Iq
.
Vt |Vs |
Vt
|Vs |
cos(δ1 − α) =
Substituting the expressions above into (3.20), yields
P1E = (E1 V∞ |Y12 | sin δ1 + E1E3 |Y13 | sin(δ1 − δ3)) · (1 + M)
X2 X3 E12
+
+ E1V∞ |Y12 | cos δ1 + E1 E3 |Y13| cos(δ1 − δ3) · N
X1 Λ
where M =
Iq X1 X2 X3
,
Π
N =
Id X1 X2 X3
,
Π
Π=
√
(3.21)
A2 + B 2, A = X2 X3 E1 cos δ1 +X1 X3 V∞ +
X1 X2 E3 cos δ3, B = X1 X2 E3 sin δ3 + X2 X3 E1 sin δ1, |Y12 | = X3 /Λ, and |Y13| = |Y31| =
X2 /Λ.
Likewise, the transmitted power, P3E , from bus 3 to the infinite bus is:
P3E =
E3 Vt
E3 Vt
sin(δ3 − β) =
sin(δ3 − α + θ̄)
X3
X3
(3.22)
Similarly, we have:
sin(δ3 − α) =
X1 X3 V∞ sin δ3 + X2 X3 E1 sin(δ3 − δ1)
,
Vs Λ
cos(δ3 − α) =
X1 X2 E3 + X1 X3 cos δ3 + X2 X3 E1 cos(δ3 − δ1 )
.
Vs Λ
Substituting the two expressions above into (3.22), yields
P3E = (E3 V∞ |Y32 | sin δ3 + E1E3 |Y31 | sin(δ3 − δ1)) · (1 + M)
X1 X2 E32
+
+ E3V∞ |Y32 | cos δ3 + E3 E1 |Y31| cos(δ3 − δ1) · N
X3 Λ
where |Y32| = X1 /Λ.
28
(3.23)
Remark 4. Similar to the SMIB case, there are three terms in (3.21) and (3.23),
including the active power of the generator and the active powers of the STATCOM
and Battery.
3.4
Linearized Power System Models including
STATCOM/Battery
Power system models are highly nonlinear, and for the study of small-signal stability
we can use linearize models that include the dynamics of the terminal voltage Vt for
the single machine infinite bus case, to address the problem of voltage regulation.
Define ∆Vt = Vt − Vref as a voltage deviation where Vref represents the reference
voltage and
Vt
∆
=
(X1 + X2 )
s
Iq X1 X2
1+
∆
2
+
Id X1 X2
∆
2
where ∆ is given in the previous section.
The objective is to drive ∆Vt to zero at steady state, so we incorporate an additional state variable ξ (integral action) as follows
ξ̇(t) = ∆Vt = Vt − Vref
(3.24)
The voltage dynamics for the single machine infinite bus system are then given as
follows:
∆V̇t = F1 (x)δ̇ + F2 (x)Ė + F3 (x)I˙d + F4 (x)I˙q
where x =
δ ω E Id Iq ∆Vt ξ
T
(3.25)
are the state variables, Fi , i ∈ {1, ..., 4} are
given in Appendix A.1.1, and M T is the transpose of matrix M.
In order to illustrate the benefits of the STACOM/Battery to both conventional
and wind power systems, consider the SMIB case as shown in Figure 3.1. There are
two SMIB cases of interest, SG and DFIG.
29
Case 1: SG Dynamic models of the SG, STATCOM/Battery, the integral action
(3.24), and voltage dynamics (3.25) are described as follows:
δ̇ = ω − ωs
EV∞ sin δ
Iq X1 X2
EX2 Id
V∞ X1 cos δ + EX2
ωs
ω̇ =
Pm −
1+
−
·
2H
(X1 + X2 )
∆
(X1 + X2 )
∆
Ė = −aE + b cos δ +
uf
T00
Rs ωs
ωs Vt cos β
I˙d = −
Id + ωIq −
+ (h cos β) · uM − (h sin β) · uγ
Ls
Ls
Rs ωs
ωs Vt sin β
I˙q = −ωId −
Iq −
+ (h sin β) · uM + (h cos β) · uγ
Ls
Ls
∆V̇t = F1 (x)δ̇ + F2(x)Ė + F3 (x)I˙d + F4 (x)I˙q
ξ̇(t) = ∆Vt
where a =
(3.26)
(Xs +Xtr +X2 )
(X1 +X2 )T00
and b =
(Xs −Xs0 )V∞
.
(X1 +X2 )T00
cos β and sin β are as follows:
1
Iq X1 X2
Id X1 X2
cos β =
(X2 E cos δ + X1 V∞ ) 1 +
+ X2 E sin δ ·
Vt (X1 + X2 )
∆
∆
Iq X1 X2
IdX1 X2
1
X2 E sin δ · 1 +
− (X2 E cos δ + X1 V∞ )
sin β =
Vt (X1 + X2 )
∆
∆
Substituting the dynamics of δ̇, Ė, I˙d and I˙q into ∆V̇t and linearizing the model
given above, we obtain:
∆ẋ = AS ∆x + BSw ∆Pm + BSu ∆u
(3.27)
where

AS




= 




A11
A21
A31
A41
A51
A61
0
A12
A22
A32
A42
A52
A62
0
A13
A23
A33
A43
A53
A63
0
A14
A24
A34
A44
A54
A64
0
A15
A25
A35
A45
A55
A65
0
A16
A26
A36
A46
A56
A66
1
30
0
0
0
0
0
0
0










 , ∆x = 








∆δ
∆ω
∆E
∆Id
∆Iq
∆Vt
∆ξ










 , BSw = 








0
1
0
0
0
0
0





,





BSu
0
0
0
0
0
0
0
0

 1
 T0
 0
= 
 0 Bu42 Bu43
 0 Bu52 Bu53

 Bu61 Bu62 Bu63
0
0
0






∆uf

 , ∆u =  ∆uM  ,


∆uγ


where Aij and Buij are given in Appendix A.2. ∆Pm denotes a small mechanical
power perturbation as the disturbance.
Case 2: DFIG Dynamic models of the DFIG, STATCOM/Battery, integral
action (3.24), and voltage dynamics (3.25) are described as follows:
b sin δ ωs Vr cos(δ − θr )
+
E
E
ωs
EV∞ sin δ
Iq X1 X2
EX2 Id
V∞ X1 cos δ + EX2
Pm −
1+
−
·
ω̇ =
2H
(X1 + X2 )
∆
(X1 + X2 )
∆
δ̇ = ω − ωs −
Ė = −aE + b cos δ + ωs Vr sin(δ − θr )
Rs ωs
ωs Vt cos β
I˙d = −
Id + ωIq −
+ (h cos β) · uM − (h sin β) · uγ
Ls
Ls
ωs Vt sin β
Rs ωs
Iq −
+ (h sin β) · uM + (h cos β) · uγ
I˙q = −ωId −
Ls
Ls
∆V̇t = F1 (x)δ̇ + F2(x)Ė + F3 (x)I˙d + F4 (x)I˙q
ξ̇(t) = ∆Vt
(3.28)
where a, b, Vt, cos β, and sin β are identical to the SG case.
For the DFIG case, to achieve frequency regulation (ω → (1 − s0 )ωs ) where
s0 = 1 − ω0 /ωs is the DFIG slip at steady state, we incorporate an additional state
variable ξω (integral control) as follows:
ξ˙ω = ω − (1 − s0 )ωs
(3.29)
Substituting the dynamics of δ̇, Ė, I˙d and I˙q into ∆V̇t and linearizing the overall
31
dynamic model we obtain the following:
∆ẋ = AD ∆x + BDw ∆Pm + BDu ∆u
(3.30)
where

AD





= 






BDw





= 





A11 A12 A13 A14 A15 A16 0
A21 A22 A23 A24 A25 A26 0
A31 A32 A33 A34 A35 A36 0
A41 A42 A43 A44 A45 A46 0
A51 A52 A53 A54 A55 A56 0
A61 A62 A63 A64 A65 A66 0
0
0
0
0
0
1 0
0
1
0
0
0
0 0


0

1 





∆V
0 
dr






∆Vqr 
0 


,
∆u
=
,
B
=
Du

 ∆uM 
0 




∆uγ
0 


0 
0
0
0
0
0
0
0
0
0












 , ∆x = 










∆δ
∆ω
∆E
∆Id
∆Iq
∆Vt
∆ξ
∆ξω






,





Bu11 Bu12
0
0
0
0
0
0
Bu31 Bu32
0
0
0
0 Bu43 Bu44
0
0 Bu53 Bu54
Bu61 Bu62 Bu63 Bu64
0
0
0
0
0
0
0
0












where Aij and Bij are given in Appendix A.3. uM , uγ and ∆Pm are identical to SG
case.
For the two machine infinite bus case (SG-DFIG) shown in Figure 3.3, let ∆Vt =
Vt − Vref where
Vt =
Πp
(1 + M)2 + N 2
Λ
The integral action ξ in (3.24) is intended to drive ∆Vt to zero at steady state,
and the voltage dynamics for the two machine infinite bus case are as follows:
∆V̇t = F1(x)δ̇1 + F2 (x)Ė1 + F3 (x)δ̇3 + F4(x)Ė3 + F5(x)I˙d + F6(x)I˙q
where x =
δ1 ω1 E1 δ3 ω3 E3 Id Iq ∆Vt ξ
{1, ..., 6} are given in Appendix A.1.2.
32
T
(3.31)
are state variables and Fi , i ∈
Case 3: SG-DFIG Dynamic models of the two machine infinite bus system
consisting of SG, DFIG, STATCOM/Battery, integral action (3.24), and voltage dynamics (3.25) are described as follows:
δ̇1 = ω1 − ωs
ωs
ω̇1 =
(Pm1 − P1E )
2H1
Ė1 = −a1E1 + a2 cos δ1 + a3E3 cos(δ1 − δ3) +
δ̇3 = ω3 − ωs −
ω̇3 =
uf
,
0
T10
b1E1 sin(δ3 − δ1) + b2 sin δ3 ωs Vr cos(δ − θr )
+
E3
E3
ωs
(Pm3 − P3E )
2H3
Ė3 = −c1 E3 + c2 cos δ3 + c3E1 cos(δ3 − δ1) + ωs Vr sin(δ − θr )
ωs Vt cos β
Rs ωs
Id + ωIq −
+ (h cos β) · uM − (h sin β) · uγ
I˙d = −
Ls
Ls
Rs ωs
ωs Vt sin β
I˙q = −ωId −
Iq −
+ (h sin β) · uM + (h cos β) · uγ
Ls
Ls
∆V̇t = F1 (x)δ̇1 + F2(x)Ė1 + F3(x)δ̇3 + F4(x)Ė3 + F5(x)I˙d + F6(x)I˙q
˙
ξ(t)
= ∆Vt
ξ̇ω = ω3 − (1 − s30)ωs ,
where a1 =
Xm3 |Y32 |V∞
,
0
T30
(1+Xm1 |Y11 |)
,
0
T10
c1 =
a2 =
(1+Xm3 |Y33 |)
,
0
T30
c2 =
(3.32)
Xm1 |Y12 |V∞
,
0
T10
Xm3 |Y32 |V∞
,
0
T30
Xm1 |Y13 |
,
0
T10
a3 =
c3 =
Xm3 |Y31 |
,
0
T30
b1 =
Xm3 |Y31 |
,
0
T30
b2 =
0
Xm1 = Xs1 − Xs1
, and
0
Xm3 = Xs3 − Xs3
. P1E and P3E represent the power transmitted from bus 1 and bus
3, respectively, to the STATCOM/Battery terminal in (3.21) and (3.23). For this
case, cos β and sin β are given below.
cos β =
A(1 + M) + BN
,
Vt Λ
sin β =
B(1 + M) + AN
.
Vt Λ
with A, B, M, N , Vt and Λ are given in the previous section.
33
Substituting the dynamics of δ̇1, Ė1 , δ̇3, Ė3 , I˙d and I˙q into ∆V̇t and linearizing the
overall dynamic model we obtain the following:
∆ẋ = ASD ∆x + BSDw ∆Pm + BSDu ∆u
(3.33)
where
ASD
BSDw

A11
A21
A31
A41
A51
A61
A71
A81
A91
0
0

0
1
0
0
0
0
0
0
0
0
0








= 
















= 









0
0
0
0
1
0
0
0
0
0
0
∆uf
 ∆Vdr

∆u = 
 ∆Vqr
 ∆uM
∆uγ
A12
A22
A32
A42
A52
A62
A72
A82
A92
0
0

A13
A23
A33
A43
A53
A63
A73
A84
A94
0
0
A14
A24
A34
A44
A54
A64
A74
A84
A94
0
0

A15
A25
A35
A45
A55
A65
A75
A85
A95
0
1
A16
A26
A36
A46
A56
A66
A76
A86
A96
0
0
0
0
0
0
0
A17
A27
A37
A47
A57
A67
A77
A87
A97
0
0
A18
A28
A38
A48
A58
A68
A58
A88
A98
0
0
0
0
0
0
0
0
0
0
0
A19
A29
A39
A49
A59
A69
A79
A89
A99
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0


 1

 T0

 10

 0 Bu41 Bu42



 0

0
0


 , BSDu =  0 Bu61 Bu62


 0

0
0 Bu74 Bu75


 0

0
0 Bu84 Bu85


 B

B
B
u92
u93 Bu94 Bu95
 u91

 0

0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0










,

























 , ∆x = 
















∆δ1
∆ω1
∆E1
∆δ3
∆ω3
∆E3
∆Id
∆Iq
∆Vt
∆ξ
∆ξω









,










 , ∆Pm = ∆Pm1

∆Pm3

with Aij and Buij are given in Appendix A.4.
Remark 5. From the matrices BSu , BDu , and BSDu in (3.27), (3.30), and (3.33), the
generator excitation controller (∆uf ), the DFIG rotor controller (∆Vdr and ∆Vqr ),
34
and STATCOM/Battery controller (∆uM and ∆uγ ) are considered as decentralized
controllers while such controllers perform together in the dynamics of ∆Vt .
The linearized power system models for the three cases of interest are given in
(3.27), (3.30), and (3.33). We assume that full state information is not available, the
T
output variables that are directly measurable are: y = ∆ω ∆Vt
= Cy ∆x.
The linearized models of the conventional and wind power systems including a
STATCOM/Battery are used in the next section to design an output feedback controller to achieve small-signal stability and voltage regulation under small mechanical
power perturbations to the system.
3.5
Controller Design with D-stability
The linearized power system models derived in the previous section are used to achieve
the performance requirements given in the Problem Statement by using a full order output feedback controller, (3.3)-(3.4). We use an LMI-based controller design
methodology, see for example [15] and [61], to achieve the following.
Theorem 1. For the linearized system, the closed-loop system can accomplish the
expected performance requirements: All closed-loop poles are located in a stable D
region (the intersection of α-stability regions, disks, and conic sectors) in the open
left-half plane with frequency and voltage regulation simultaneously achieved, if and
only if there exist symmetric matrices X, Y and matrices Ā, B̄, C̄, and D̄ such that
the following LMIs are simultaneously satisfied.

−rX −rI ΘT11 ΘT21
 −rI −rY ΘT
Θ222 
12


 Θ11 Θ12 −rX −rI  < 0,
Θ21 Θ22 −rI −rY
X I
Ω11 ΩT21
< 0,
>0
I Y
Ω21 Ω22


Σ11
Σ12
Σ13
T
 ΣT
Σ
−Σ
22
12
14

 −Σ13 −Σ14 Σ11
ΣT14 −Σ24 ΣT12

Σ14
Σ24 
 < 0, (3.34)
Σ12 
Σ22
(3.35)
35
where
Θ11 = AX + Bu C̄,
Θ12 = A,
Θ21 = Ā,
Θ22 = Y A + B̄Cy ,
Σ12 = (ĀT + (A + Bu D̄Cy )) sin θ,
Σ22 = (AT Y + Y A + B̄Cy + (B̄Cy )T ) sin θ,
Σ13 = (AX − XAT + Bu C̄ − (Bu C̄)T ) cos θ,
Σ14 = (A + Bu D̄Cy − ĀT ) cos θ,
Σ24 = (Y A − AT Y + B̄Cy − (B̄Cy )T ) cos θ,
Ω11 = AX + XAT + Bu C̄ + (Bu C̄)T + 2αX,
Ω21 = Ā + (A + Bu D̄Cy )T + 2αI,
Ω22 = AT Y + Y A + B̄Cy + (B̄Cy )T + 2αY,
Σ11 = (AX + XAT + Bu C̄ + (Bu C̄)T ) sin θ
and for X and Y symmetric matrices, X > Y means X − Y is positive definite.
A dynamic output feedback controller can be constructed as follows:
AK := (N T )−1 (Ā − NBK Cy X − Y Bu CK M T − Y (A + Bu DK Cy )X)(M T )−1 ,
BK := (N T )−1 (B̄ − Y Bu DK ),
CK := (C̄ − DK Cy X)(M T )−1 ,
DK := D̄.
where X and Y are arbitrary nonsingular matrices satisfying MN T = I − XY .
Proof: See [15] and [61].
Remark 6. Theorem 1 requires solving three LMIs to confine the closed-loop poles to
the intersection of α-stability regions, disks, and conic sectors, respectively, as shown
36
in Figure 3.5 to meet certain transient response requirements: a minimum decay α:
Re(s) ≤ −α, a minimum damping ratio (conic sectors): ζ = cos θ, and a maximum
p
undamped natural frequency (disks): ωn 1 − ζ 2 = r sin θ. These LMI conditions can
be used to meet transient performance metrics such as maximum overshoot, frequency
of oscillation, rise time, and settling time. Stability, closed-loop poles in the open left
half plane (α = 0, θ = 90◦ , and r → ∞) are selected which will be used to compare
with the controller design with D-stability in the next section.
Im(s)
Re(s)
θ
α
r
Figure 3.5: A stable D region [15]
Remark 7. There are a number of linear controller design methods using the LMI
approach that can be used to improve the results of this paper, for example assigning
the closed-loop poles in a stable D region while simultaneously satisfying a performance
index given by H2 or H∞ performance. Robust control design techniques can also be
applied to reduce the sensitivity of the system outputs to uncertainties obtained from
the linearization.
37
3.6
Simulation Results
Simulation results for SMIB and two machine infinite bus power systems are used to
illustrate the the effectiveness of the STATCOM/Battery in improving small-signal
stability.
3.6.1
Single Machine Infinite Bus
Consider the single line diagram given in Figure 3.1 with either SG or DFIG connected
through a transmission line to an infinite-bus. The parameters for this power system
model are given in Appendix A.5. The generators deliver 1.0 pu. power for SG
and 1.03 pu. power for DFIG, respectively, while the reference terminal voltages
are 0.9897 pu. for SG, and 0.9877 pu. for DFIG, respectively, and the infinite-bus
voltage is 1.0 pu. In these two cases, we confine the closed-loop poles in a stable
D region (α = 0, θ = 60◦ , r = 100). However, when there is a small step increase
of mechanical power (∆Pm = 0.05Pm ), this leads to rotor acceleration and voltage
deviation. The STATCOM/Battery can absorb the excess active power (improving
angle and frequency stability) and simultaneously inject reactive power to regulate
voltage and improve voltage stability.
We consider two cases mentioned in the previous section and investigate the effectiveness of the STATCOM/Battery to improve small-signal stability applied to
conventional and wind power systems through voltage and frequency regulation according to the expected performance requirements.
Case 1 SG: Figures 3.6–3.10 show rotor speed (frequency) and voltage responses,
unmeasured state responses, and closed-loop pole location of conventional power systems, respectively. It is obvious that the proposed controller with and without D-
stability can drive frequency and voltage deviations (∆ω and ∆Vt ) to zero at steady
state; system frequency ω goes to ωs (ω = ωs + ∆ω → ωs ) and the terminal voltage
goes to the reference voltage Vref (Vt = Vref +∆Vt → 0.9897 pu.). However, the voltage
response Vt with D-stability can decrease the overshoot from +7.1% to +1.5% along
38
with improving the settling time and reducing oscillations. Although the rotor speed
ω/ωs with D-stability may increase but clearly the oscillation is quickly damped out.
Case 2 DFIG: Figures 3.11–3.15 show rotor speed (frequency) and voltage re-
sponses, unmeasured state responses, and the location of closed-loop poles of wind
power systems, respectively. From the rotor speed and voltage responses without
D-stability, it is clear that the voltage deviation converges to zero. The STATCOM
is able to compensate reactive power to track the reference voltage following a small
change in mechanical power input. Similarly, the Battery is able to help manage
frequency deviations at steady state, and the DFIG frequency eventually recovers
(ω = 1.03ωs with slip = −0.03) at steady state. Nevertheless, the overshoot of the
voltage response without D-stability is around +34%, which is practically undesir-
able, while the voltage response with D-stability has no overshoot but a very small
voltage sag (−1.2%) and quickly returns to the reference voltage. Also, the DFIG
rotor speed responses are similar to the voltage responses, e.g. the overshoot, settling
time, and damped oscillation.
For the unmeasured states in both cases with and without D-stability above, it is
obvious that there is very little change in ∆δ, ∆E, ∆Id, and ∆Iq but their responses
are very slow compared to D-stability. Also, the closed-loop poles for both cases are
located in the desired D region according to the requirement (1). It is obvious in
Figures 3.10 and 3.15 that with D-stability the closed-loop poles are located in the
desired region where “ + ” and “ ◦ ” represent the closed-loop poles with and without
the pole placement constraints, respectively.
3.6.2
Two-Machine Infinite Bus
We consider two generators (SG-DFIG case) connected to an infinite bus and including a STATCOM/Battery. The parameters of this power system model are given in
Appendix A.5.
Consider the single line diagram given in Figure 3.3 consisting of a conventional
SG and a DFIG representing a wind power system. Both are connected through a
39
transmission line to an infinite bus. The SG and DFIG generators deliver 1.0 and
1.03 pu. power, respectively, while the reference terminal voltage is 0.9780 pu. and
the infinite-bus voltage is 1.0 pu. Identical to SMIB case, we would like to confine the
closed-loop poles to a stable D region (α = 0, θ = 60◦ , r = 100). A small step increase
in mechanical power (∆Pm3 = 0.05Pm3 ) of the SG causes a power imbalance and
T
voltage deviation. The measured output signals are y = ∆ω1 , ∆ω3 , ∆Vt
=
Cy ∆x.
Case 3: SG-DFIG Figures 3.16–3.21 show rotor speed (frequency) and voltage
responses, unmeasured state responses, and closed-loop pole locations for the SG and
DFIG power systems, voltage and frequency regulation are identical to the SMIB
cases, namely voltage and frequency deviations converge to zero at steady state (ω1 =
ωs + ∆ω1 → ωs , ω3 = 1.03ωs + ∆ω3 → 1.03ωs , and Vt = Vref + ∆Vt → 0.9780 pu.). In
addition, the transient responses are further improved in terms of reduced overshoot,
shorter settling time, and damped oscillations when place placement constraints are
included. In particular, the overshoot of voltage responses decreases from 6.85% to
0.72% and both SG and DFIG rotor speeds (frequency) have shorter settling times
and no oscillations.
3.7
Conclusion
In this chapter, we have shown the combination of a STATCOM and Battery energy
storage system can be used to enhance the small-signal stability of SG and DFIG generators. An analysis of transmitted power with the STATCOM/Battery configuration
is investigated. Simulation results have demonstrated that voltage and frequency regulation are achievable for small disturbances using a linearized system model and an
LMI-based control design methodology.
40
Output feedback
Output feedback with D stability
∆ω (rad/s)
1.01
1.005
1
0.995
0.99
0
1
2
3
4
5
6
7
8
9
10
∆Vt (pu.)
1.075
Output feedback
Output feedback with D stability
1.0175
0.96
0
1
2
3
4
5
6
7
8
9
10
Time (sec.)
Figure 3.6: Rotor speed (frequency) and voltage responses of SG case
∆uf
0.5
Output feedback
Output feedback with D−stability
0
−0.5
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
Time
(sec.)6
7
8
9
10
−3
∆uM
4
x 10
2
0
−2
0
−3
x 10
∆uγ
1
0.5
0
−0.5
−1
0
Figure 3.7: Control input responses of SG case
41
−3
4
x 10
Output feedback
Output feedback with D−stability
k
3
2
1
0
0
1
2
3
0
1
2
3
4
5
6
7
8
9
10
4
5
Time
(sec.)6
7
8
9
10
1.5
1
0.5
γ
0
−0.5
−1
−1.5
∆Iq (pu.) ∆I (pu.) ∆E (rad)
d
∆δ (rad)
Figure 3.8: Control input responses of SG case
−3
4
x 10
2
0
−2
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
0.01
0
−0.01
0
−3
5
x 10
0
Output feedback
Output feedback with D stability
−5
−10
0
1
2
3
0
1
2
3
4
5
6
7
8
9
10
4
5
Time
(sec.)6
7
8
9
10
0.02
0
−0.02
−0.04
Figure 3.9: Unmeasured state responses of SG case
42
(α = 0, θ = 60◦ , r = 100)
400
300
200
Im(s)
100
0
−100
−200
−300
−400
−100
−90
−80
−70
−60
−50
Re(s)
−40
−30
−20
−10
0
Figure 3.10: Closed-loop pole location of SG case: “ ◦ ” without D-stability and
“ + ” with D-stability
∆ω (rad/s)
1.06
Output feedback
Output feedback with D−stability
1.05
1.04
1.03
1.02
0
1
2
3
4
5
6
8
9
10
Output feedback
Output feedback with D−stability
1.3
∆Vt (pu.)
7
1.2
1.1
1
0
1
2
3
4
5
6
Time (sec.)
7
8
9
10
Figure 3.11: Rotor speed (frequency) and voltage responses of DFIG case
43
∆Vdr
8
6
4
2
0
∆Vqr
0
1
2
3
4
5
6
7
8
9
10
3
2
Output feedback
Output feedback with D−stability
1
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
Time
(sec.)6
7
8
9
10
∆uM
∆uγ
0.1
0
−0.1
−0.2
0.04
0.02
0
Figure 3.12: Control input responses of DFIG case
0.25
0.2
k
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
1
Output feedback
Output feedback with D−stability
0.5
γ
0
−0.5
−1
0
1
2
3
4
5
Time
(sec.)6
7
8
9
Figure 3.13: Control input responses of DFIG case
44
10
∆Iq (pu.) ∆Id (pu.) ∆E (pu.) ∆δ (rad)
0.2
0.1
0
−0.1
Output feedback
Output feedback with D−stability
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
Time
(sec.)6
7
8
9
10
0
−0.2
−0.4
0.02
0
−0.02
0.5
0
−0.5
Figure 3.14: Unmeasured state responses of DFIG case
(α = 0, θ = 60◦ , r = 100)
400
300
200
Im(s)
100
0
−100
−200
−300
−400
−100
−90
−80
−70
−60
−50
Re(s)
−40
−30
−20
−10
0
Figure 3.15: Closed-loop pole location of DFIG case: “ ◦ ” without D-stability and
“ + ” with D-stability
45
Output feedback
Output feedback with D−stability
1
0.995
0
1
2
3
4
5
6
7
8
9
10
1.032
1.03
1.028
Output feedback
Output feedback with D−stability
1.026
1.024
0
1
2
3
4
5
6
7
8
9
10
1.05
∆Vt (pu.)
∆ω3 (rad/s) ∆ω1 (rad/s)
1.01
1.005
Output feedback
Output feedback with D−stability
1
0
1
2
3
4
5
6
Time (sec.)
7
8
9
10
Figure 3.16: Rotor speed (frequency) and voltage responses of SG and DFIG case
Output feedback
Output feedback with D−stability
∆uf
0.2
0
−0.2
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
Time
(sec.)6
7
8
9
10
∆Vdr
0.1
0
−0.1
−0.2
∆Vqr
0.1
0.05
0
−0.05
−0.1
Figure 3.17: Control input responses of SG-DFIG case
46
∆uγ
0.1
0
−0.1
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
∆uM
0.02
0
−0.02
k
0.06
Output feedback
Output feedback with D−stability
0.04
0.02
0
0
1
2
3
0
1
2
3
4
5
6
7
8
9
10
4
5
Time
(sec.)6
7
8
9
10
γ
2
0
−2
Figure 3.18: Control input responses of SG-DFIG case
∆δ1 (rad)
−3
6
x 10
4
2
0
−2
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
∆E1 (pu.)
−3
x 10
15
10
5
0
−5
0
∆δ3 (rad)
−3
10
x 10
5
Output feedback
Output feedback with D−stability
0
0
1
2
3
4
5
Time
(sec.)6
7
8
9
10
Figure 3.19: Unmeasured state responses of SG-DFIG case
47
x 10
Output feedback
Output feedback with D stability
0
−5
−10
−15
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
time
6
7
8
9
10
0.005
0
−0.005
−0.01
−0.015
∆Iq (pu.)
∆Id (pu.)
∆E3 (pu.)
−3
5
0
−0.05
−0.1
Figure 3.20: Unmeasured state responses of SG-DFIG case
(α = 0, θ = 60◦ , r = 100)
600
400
Im(s)
200
0
−200
−400
−600
−100
−90
−80
−70
−60
−50
Re(s)
−40
−30
−20
−10
0
Figure 3.21: Closed-loop pole location of DFIG case: “ ◦ ” without D-stability and
“ + ” with D-stability
48
Chapter 4
Transient Stability and Voltage
Regulation Enhancement
4.1
Introduction
Distributed Renewable Energy Resources (DRERs) are receiving increasing interest
in power system applications [11] with efforts to integrate wind power into the conventional power grid, e.g. [2, 28]. As the diversity of the generation mix increases
and larger amounts of power are being obtained from renewable generating sources,
differences in the dynamic behavior between conventional and alternative generators
can play an increasingly important role in the stability and security of the grid.
The objective of this Chapter is to investigate transient stability and voltage regulation in power systems that in include DRERs (generation and storage). Transient
stability is related to dynamic behavior of the system during a fault and before the
fault is cleared from the system. Stability criteria are often given in terms of the
critical clearing time of fault, that is the maximum allowable time the fault can persist on the system before the trajectory exits the domain of attraction of the post
fault equilibrium state. Increasing the critical clearing time, increases the transient
stability margin of the system and improves overall security of the system.
Challenges with integrating renewable generation sources in a power system are
related to the potential for intermittent power availability as well as the differences
49
in their dynamic response characteristics as compared to conventional synchronous
machines. It is apparent that energy storage technologies [59] are important for
dealing with the intermittency of many alternative energy sources and can also to
provide the opportunity to improve power quality, especially frequency and power
stability, as reported in [45]. For stability enhancement of wind energy conversion
systems, battery energy storages has been used to improve frequency stability through
the regulation of active power levels. As reported in [13, 63], battery energy storage
can be used to (1) reduce the power fluctuations from large wind farms, (2) deliver
large amounts of energy in a short time period when needed, and (3) regulate active
power output to improve the economics of wind farms.
Flexible AC Transmission System (FACTS) devices continue receiving considerable attention in applications to further enhance power transfer capability and augment both small-signal and transient stability in power systems [29, 62]. Among the
family of FACTS devices, the Static Synchronous Compensator (STATCOM) is of
particular interest in this study because of its ability to provide smooth and rapid
reactive power compensation for voltage support and slightly improve damping of
oscillations and transient stability [26, 27].
Until now STATCOM and (battery) energy storage systems have independently
been used to improve power system operations, and integrating these devices provides
an opportunity to improve overall small-signal and transient stability of the power
system. Relatively little prior work has been devoted to the integration of STATCOM and battery energy storage, Yang et al. [69] has shown that the integration
of STATCOM and battery can provide additional benefits beyond the STATCOM
in conventional power systems and Baran et al. [8] have shown the ability of the
STATCOM to smooth intermittent wind farm power and compensate reactive power
in simulation studies. In addition, Arumlampalam et al. [7] has shown that improvements in power quality can be obtained by a STATCOM and battery combination for
a wind farm application and recently, Abedini and Nikkhajoei [1] has shown the use of
energy storage as an auxiliary source to not only reduce the wind power fluctuations
50
but also for load following using a PM synchronous generator in combination with
energy storage.
This chapter continues this line of investigation and examines how a STATCOM
and battery system can be used to improve the transient stability of a wind energy
conversion system connected to the grid through a Doubly Fed Induction Generator
(DFIG). An IDA-PBC control design approach is used to achieve angle stability
and provide frequency and voltage regulation. Simulation results are provided for
two operating scenarios: (1) Single machine infinite bus with either a Synchronous
Generator (SG) or DFIG connected to the infinite bus, and (2) a two-machine (SG
and DFIG) system connected to an infinite bus.
The rest of this chapter is organized as follows. The problem formulation is
provided in Section 4.2; IDA-PBC controller design is given in Section 4.3; simulation
results are given in Section 4.4; and conclusions are given in Section 4.5.
4.2
Problem Formulation
In this Chapter we are interested in studying the transient stability of a nonlinear
power system including a STATCOM/Battery. The nonlinear system considered can
be written in the general form:
ẋ(t) = f(x) + g(x)u
(4.1)
where x ∈ <n is the state vector, u ∈ <m , m < n is the control action. f(x) : <n → <n
is continuously differentiable on the domain D ⊂ <n and satisfies a local Lipschitz
condition to ensure the existence of a unique solution on [0, +∞). g(x) : <n → <n×m
is assumed to be a smooth, full rank function.
The problem is: Given a stable equilibrium point xe and under the assumption of
full state feedback, find a feedback control law u so that for the closed-loop system:
1. The equilibrium point xe is asymptotically stable.
2. The domain of attraction is enlarged.
51
3. Power angle stability and voltage and frequency regulation are simultaneously
achieved.
4.3
IDA-PBC Controller Design
In recent years, a number of nonlinear control design techniques [33, 36, 40, 48] have
been proposed to stabilize and control dynamic models of power systems. In this work,
the Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC),
a formulation of Passivity-Based Control (PBC), is chosen as the design methodology.
Refer to the survey and tutorial paper [56, 57] for more details, and in particular,
[24, 55] for transient stability of power systems with synchronous generators alone. In
this chapter, the IDA-PBC control method is used to design a state-feedback control
law to achieve the desired closed-loop system performance requirements.
In order to illustrate the benefits of the STACOM/Battery to both conventional
and wind power systems, consider the SMIB and two-machine infinite bus systems
as shown in Figures 3.1 and 3.3. There are two SMIB cases and one two machine
infinite bus case of interest: (1) SG, (2) DFIG, and (3) SG-DFIG.
However, the design of the IDA-PBC state feedback control laws for these three
cases will be provided in Chapter 5.
Case 1: SG Dynamic models of the SG and STATCOM/Battery are written in
the general form (5.1) as follows:




f(x) = 





g(x) = 




ω − ωs
δ
1
(Pm − PE − D(ω − ωs )) 
 ω − ωs
M


s +Xtr +X2 )E
Xm V∞ cos δ 
− (X(X
+
0
0 , x = 
+X
)T
(X
+X
)T
1
2 0
1
2
 E
0
ωs Vt cos β 
R s ωs
 Id
− Ls Id + ωIq − Ls

ωs Vt sin β
R s ωs
Iq
−ωId − Ls Iq − Ls

0
0
0
 uf 

0
0
0
T00

 , u(x) =  huM 
1
0
0

0 cos β − sin β 
huγ
0 sin β cos β
52






where PE represents the power transmitted from bus 1 to the STATCOM/Battery
terminal in (3.13).
The closed-loop SMIB system (SG) can accomplish the expected performance
requirements (1)-(3) with the following control law u(x):
Xm V∞ (cos δ − cos δe )
uf
= −J23(δ, E)(ω − ωs ) −
0
T0
(X1 + X2 )T00
huM = M cos β + N sin β
huγ = −M sin β + N cos β
(4.2)
where
J23 (δ, E) = −
V∞ sin δ
γ1 (X1 + X2 )
J24 (δ, E) = −
Ee X2 (V∞ X1 cos δ + Ee X2 )
∆e (X1 + X2 )γ2 Ls
J25 (δ, E) = −
Ee X1 X2 V∞ sin δ
∆e (X1 + X2 )γ3Ls
M = −J24(δ, E)(ω − ωs ) − r2Ls γ2 (Id − Ide ) +
Rs ωs
ωs Vt cos β
Id − ωIq +
Ls
Ls
N = −J25(δ, E)(ω − ωs ) − r3Ls γ3 (Iq − Iqe ) + ωId +
and ∆e =
p
ωs Vt sin β
Rs ωs
Iq +
Ls
Ls
(Ee X2 )2 + (V∞ X1 )2 + 2X1 X2 Ee V∞ cos δ. γi and ri , i = 1, 2, 3 are arbi-
trary positive constants. Further, the equilibrium point xe = (δe , ωs , Ee , 0, 0)T of the
closed-loop system is asymptotically stable with energy function:
M
γ1
γ2 Ls
γ3 Ls
(ω − ωs )2 + (E − Ee )2 + ϕ(δ) +
(Id − Ide )2 +
(Iq − Iqe )2
2
2
2
2
Z δ
Ee X2 Ide (V∞ X1 cos δ + Ee X2 )
ϕ(δ) = −Pm (δ − δe ) +
·
dδ
∆e
δe (X1 + X2 )
Z δ
Ee V∞ sin δ
Iqe X1 X2
+
1+
dδ
∆e
δe (X1 + X2 )
Hd (x) =
53
A domain of attraction is given by the largest bounded level set {x ∈ <5 |Hd (x) ≤ c}
that contains the desired equilibrium point xe . Further, the desired interconnection
matrix Jd (x) and the damping matrix Rd (x) are selected as follows:


1
0
0
0
0
M
 − 1 −D J23 J24 J25 

 M
Jd (x) − Rd (x) = 
0 

 0 −J23 −r1 0
 0 −J24 0 −r2 0 
0 −J25 0
0 −r3
Remark 8. For the control law in (5.5), there are two coordinated controllers: a
Generator Excitation Controller (GEC), uf /T00 for conventional generation power
systems, and a STATCOM/Battery controller (STAT/BATT), (huM and huγ ).
Case 2: DFIG Dynamic models of the DFIG and STATCOM/Battery are written in the general form (5.1) as follows:




ω − ωs − XXm0 V∞ETsin0 δ
δ
s
0
1


 ω − Sωs 
(Pm − PE )


M


 − (Xs+Xtr +X2 )E + Xm V∞ cos δ 


E
f(x) = 
(X1 +X2 )T00
(X1 +X2 )T00  , , x = 



ω
V
cos
β


R
ω
s
t
Id

 − Ls s Id + ωIq − L
s
s
β
Iq
−ωId − RLsωs s Iq − ωs VLt sin
s

 cos δ
sin δ


0
0
E
E
ω
V
s
dr
 0

0
0
0




 , u(x) =  ωs Vqr 

0
0
g(x) =  sin δ − cos δ

 huM 
 0
0
cos β − sin β 
huγ
0
0
sin β cos β
where Vdr = Vr cos θr and Vqr = Vr sin θr . PE is identical to SG case.
A state feedback controller u(x) for SMIB (DFIG) system that achieves the performance requirements (1)-(3) is given by:
ωs Vdr = X cos δ + Y sin δ
ωs Vqr = X sin δ − Y cos δ
huM = M cos β + N sin β
huγ = −M sin β + N cos β
54
(4.3)
where
X = s0 ωs E +
Xm V∞ sin δ
(X1 + X2 )T00
Y = −J23(δ, E)(ω − Sωs ) −
Xm V∞ cos δ
(X1 + X2 )T00
M = −J24(δ, E)(ω − Sωs ) − r2 Ls γ2 (Id − Ide ) +
Rs ωs
ωs Vt cos β
Id − ωIq +
Ls
Ls
N = −J25(δ, E)(ω − Sωs ) − r3 Ls γ3 (Iq − Iqe ) + ωId +
Rs ωs
ωs Vt sin β
Iq +
Ls
Ls
S = 1 − s0 = 1 − ωe /ωs
and J2i(δ, E), i = 3, 4, 5 are identical to SG case. γi and ri , i = 1, 2, 3 are arbitrary
positive constants. The closed-loop system equilibrium point xe = (δe , Sωs , Ee , 0, 0)T
is asymptotically stable with the energy function Hd (x), which is identical to the SG
case except that the first term is changed to
M
(ω
2
− Sωs )2 .
Remark 9. The control law in (5.12) is similar to the SG case; there are two coordinated controllers: a DFIG rotor voltage controller (DRVC), (ωs Vdr and ωs Vqr ) for
alternative (wind) generation power systems, and a STATCOM/Battery controller
(STAT/BATT), (huM and huγ ). In addition, under disturbances the rotor voltage
Vdr and Vqr may exceed the voltage capacity of the feedback converter; therefore, Vdr
and Vqr should be limited.
Case 3: SG-DFIG Dynamic models of the SG, DFIG and STATCOM/Battery
are written in the general form (5.1) as follows:




δ1
f1 (x)
 ω1 − ωs 
 f2 (x) 






 f3 (x) 
E1






 f4 (x) 
δ
3

,x = 
f(x) = 
 ω3 − Sωs  ,
 f5 (x) 






 f6 (x) 
E3






 f7 (x) 
Id
Iq
f8 (x)
55






g(x) = 





0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
sin δ3
cos δ3
0
0
0 E3
E3
0
0
0
0
0
0 sin δ3 − cos δ3
0
0
0
0
0
cos β − sin β
0
0
0
sin β cos β









 , u(x) = 







uf
0
T10
ωs Vdr
ωs Vqr
huM
huγ






where
f1 (x) = ω1 − ωs ,
f2 (x) =
1
(Pm1 − P1E − D(ω1 − ωs )) ,
M1
f3 (x) = −
(1 + Xm1 |Y11 |)E1 Xm1 |Y12|V∞ cos δ1 Xm1 |Y13|E3 cos(δ1 − δ3 )
+
+
,
0
0
0
T10
T10
T10
f4 (x) = ω3 − ωs −
f5 (x) =
Xm3 |Y31|E1 sin(δ3 − δ1) Xm3 |Y32|V∞ sin δ3
−
,
0
0
T30
E3
T30
E3
1
(Pm3 − P3E ) ,
M3
f6 (x) = −
(1 + Xm3 |Y33 |)E3 Xm3 |Y32|V∞ cos δ3 Xm3 |Y31|E1 cos(δ3 − δ1 )
+
+
,
0
0
0
T30
T30
T30
f7 (x) = −
Rs ωs
ωs Vt cos β
Id + ωIq −
,
Ls
Ls
f8 (x) = −ωId −
Rs ωs
ωs Vt sin β
Iq −
,
Ls
Ls
0
0
Xm1 = Xs1 − Xs1
, Xm3 = Xs3 − Xs3
,
and P1E and P3E represent the power transmitted from bus 1 and bus 3, respectively,
to the STATCOM/Battery terminal in (3.21) and (3.23).
A state feedback controller u(x) that achieves the expected performance require-
56
ments (1)-(3) for the two machine infinite bus (SG-DFIG) system is given as follows:
Xm |Y12|V∞ (cos δ1 − cos δ1e )
uf
= −J23(δ, E)(ω1 − ωs ) + J35(δ, E)(ω3 − Sωs ) −
0
0
T10
T10
−
Xm |Y13|E1 E3 (cos(δ1 − δ3) − cos(δ1e − δ3e ))
0
T10
ωs Vdr = X cos δ3 + Y sin δ3
ωs Vqr = X sin δ3 − Y cos δ3
huM = M cos β + N sin β
huγ = −M sin β + N cos β
(4.4)
where
J23 (δ, E) = −
V∞ |Y12| sin δ1 + E3e |Y13| sin(δ1 − δ3)
γ1
E1 |Y13| sin(δ1 − δ3 )
γ2
2
X2 X3 E1e
X1 X2 X3
+ E1e V∞ |Y12 | cos δ1 + E1e E3e |Y13 | cos(δ1 − δ3 )
J27 (δ, E) = −
X1 Λ
Πe γ3 Ls
J26 (δ, E) = −
J28 (δ, E) = −(E1e V∞ |Y12| sin δ1 + E1e E3e |Y13 | sin(δ1 − δ3))
J35 (δ, E) =
X1 X2 X3
Πe γ4 Ls
E3 |Y31| sin(δ3 − δ1)
γ1
V∞ |Y32| sin δ3 + E1e |Y31| sin(δ3 − δ1)
γ2
2
X1 X2 E3e
X1 X2 X3
J57 (δ, E) = −
+ E3e V∞ |Y32 | cos δ3 + E1e E3e |Y31 | cos(δ3 − δ1 )
,
X3 Λ
Πe γ3 Ls
X1 X2 X3
J58 (δ, E) = − E3e V∞ |Y32 | sin δ3 + E1e E3e |Y31 | sin(δ3 − δ1)
,
Πe γ4 Ls
J56 (δ, E) = −
X = s0 ωs E3 +
Xm3 |Y31|E1 sin(δ3 − δ1 ) Xm3 |Y12 |V∞ sin δ3
+
,
0
0
T30
T30
57
Y = −J26(δ, E)(ω1 − ωs ) −
−
Xm3 V∞ cos δ3
− J56 (δ, E)(ω3 − Sωs )
0
T30
Xm |Y31 |E1 cos(δ3 − δ1 )
0
T30
M = −J27(δ, E)(ω1 − ωs ) − r3 Ls γ3 (Id − Ide ) − J57(δ, E)(ω3 − Sωs ) − f7 (x)
N = −J28(δ, E)(ω1 − ωs ) − r4 Ls γ4 (Iq − Iqe ) − J58(δ, E)(ω3 − Sωs ) − f8 (x)
S = 1 − s0 = 1 − ω3e /ωs
and Πe = Π|E1 =E1e ,E3 =E3e . γi , i = 1, 2, 3, 4 and ri , i = 2, 3 are arbitrary positive
constants. The equilibrium point xe = (δ1e, ωs , E1e , δ3e , Sωs , E3e , 0, 0)T of the closedloop system is asymptotically stable with the energy function:
Hd (x) =
M1
γ1
M3
(ω1 − ωs )2 + (E1 − E1e )2 + ϕ(δ) +
(ω3 − Sωs )2
2
2
2
γ2
γ3 Ls
γ4 Ls
(E3 − E3e )2 +
(Id − Ide )2 +
(Iq − Iqe )2
2
2
2
Z δ1
Z δ3
ϕ(δ) = −Pm1 (δ1 − δ1e) − Pm3 (δ3 − δ3e) +
P1Ee dδ1 +
P3Ee dδ3
+
δ1e
δ3e
with P1Ee and P3Ee representing P1E and P3E evaluated at E1 = E1e and E3e . A domain of attraction is given by the largest bounded level set {x ∈ <8 |Hd (x) ≤ c} that
contains the desired equilibrium point xe . Furthermore, the desired interconnection
matrix Jd (x) and the damping matrix Rd (x) are selected as given below.
Jd (δ, E) − Rd (x) =

1
0
0
0
0
0
0
0
M1
 − M1
D
J23(δ, E)
0
0
J26 (δ, E) J27(δ, E) J28 (δ, E)
1

 0
−J
(δ,
E)
−r
0
J
(δ,
E)
0
0
0
23
1
35

1
 0
0
0
0
0
0
0
M3

1
 0
0
−J35(δ, E) − M3
0
J56 (δ, E) J57(δ, E) J58 (δ, E)

 0
−J26(δ, E)
0
0
−J56(δ, E)
−r2
0
0

 0
−J27(δ, E)
0
0
−J57(δ, E)
0
−r3
0
0
−J28(δ, E)
0
0
−J58(δ, E)
0
0
−r4
58












Remark 10. The control law in (5.16) resembles the SG and DFIG cases and the coordinated controllers include the combination of generator excitation controller (GEC)
0
for conventional power systems, uf /T10
, DFIG rotor voltage controller (DRVC), (ωs Vdr
and ωs Vqr ) for wind power systems, and STATCOM/Battery controller (STAT/BATT),
(huM and huγ ).
Remark 11. The energy functions Hd for all three cases can be split into two groups,
namely kinetic energy and potential energy. The former stems from
synchronous generators and
M
(ω − Sω2 )2
2
M
(ω
2
− ω2 )2 of
of the DFIG. The latter stems from SG and
DFIG (independent of Id and Iq ) and the STATCOM/Battery (dependent on Id and
Iq ), respectively.
4.4
Simulation Results
In this section, simulation results for the integration of STATCOM/Battery in SMIB
and two-machine infinite bus systems are presented. System performance enhancements are evaluated according to (1) the critical clearing time (CCT), (2) the DOA,
and (3) power angle stability including voltage and frequency regulation.
Vt∠β
E∠δ
j2X2
jX1
j2X2
P
F
SG/DFIG
I¯SB = Id + jIq
Figure 4.1: A single line diagram of SMIB
59
V∞∠0
4.4.1
Single Machine Infinite Bus
Consider the single line diagram as shown in Fig. 5.1 with either SG or DFIG connected through parallel transmission line to the infinite bus. The generators deliver
1.0 pu. power for the SG and 1.03 pu. power for the DFIG, respectively, with terminal voltages 0.9897 pu. for the SG and 0.9877 pu. for DFIG, respectively, and the
infinite bus voltage is 1.0 pu. A three-phase fault occurs at P , the midpoint of one
of the transmission lines, which leads to rotor acceleration and voltage sag. During
the fault circuit breakers on the affected line are opened and re-closed after the fault
is cleared.
There are four basic stages associated with transient stability studies of a power
system:
• Stage 1: Pre-fault steady state operation.
• Stage 2: Fault initiation at time t0 = 0.
• Stage 3: Fault isolation through the opening of circuit breakers on the affected
line at time tcr , with critical clearing time CCT = tcr − t0 = tcr .
• Stage 4: Transmission line is restored without the fault at t = tr sec, and the
system is in a post-fault state at t = tf sec.
Remark 12. Such stages can be regarded as a temporary disconnection since the
transmission line is reconnected after the fault is cleared, thus it is reasonable to
assume that in the pre-fault and post-fault states, Ide and Iqe in M and N are equal
to zero. If the transmission line is permanently disconnected, Ide and Iqe will be
unknown and cannot be chosen as zero.
We consider two cases (SG and DFIG) and investigate the effectiveness of the
STATCOM/Battery and nonlinear control system to improve transient stability (power
angle stability) along with voltage, frequency, and power regulation.
60
The benefits of integrating STATCOM/Battery system in terms of improving the
critical clearing time (CCT) for both SG and DFIG cases are shown in Table 4.1 and
4.2 where the CCT with GEC/STAT/BATT for SG case (DRVC/STAT/BATT for
DFIG case), is compared to the cases with GEC/STAT for SG case (DRVC/STAT
for DFIG case), and only GEC for SG case (only DRVC for DFIG case) along with
the performance of the system with a conventional PSS/AVR generator excitation
controller [41] for the SG case and a PI Controller [18]: PIC for the DFIG case.
The parameters for the state feedback controllers in both cases are set as γ1 r1 =
(Xs +Xtr +X2 )
, γ2
(X1 +X2 )T00
= γ3 = 1 and r2 = r3 = 3 for SG case and γ1 r1 =
(Xs +Xtr +X2 )
, γ2
(X1 +X2 )T00
=
γ3 = 10 and r1 = r2 = 1 for DFIG case.
Case 1: SG Figures 4.3 and 4.5 show phase portraits between power angle δ and
SG rotor speed (frequency) ω − ωs with and without damping constant. Voltage and
active power responses are shown in Figures 4.4 and 4.6 with and without damping.
From this case, it can easily be seen that power angles (δ) return to the pre-fault
value (δ0). The domain of attraction is clearly enlarged and CCT is increased when
STATCOM/Battery are integrated into the system. Further, the domain of attraction
for GEC/STAT/BATT is larger than that for the GEC/STAT, GEC, and PSS/AVR.
Likewise, the percentage increase of the GEC/STAT/BATT is significantly larger
than that of GEC/STAT and GEC when compared to the case with PSS/AVR (0%)
(e.g. approximately +19% and +116% increase in CCT for GEC/STAT/BATT and
+11% and +95% increase in CCT for GEC/STAT along with +8% and +88% increase in CCT for GEC). For voltage and power regulation at CCT = 0.4 sec., the
GEC/STAT/BATT system responses are further improved over the cases with GEC,
GEC/STAT while the terminal voltage and active power responses from PSS/AVR
cannot return to the desired reference voltages and mechanical power, respectively,
and become eventually unstable as shown in Figures 4.4 and 4.6. Simulation results
for this case indicate that when using the STATCOM/Battery and IDA-PBC control design, transient (power angle) stability is improved (increased critical clearing
time and enlarged domain of attraction) along with simultaneous enhancements to
61
frequency, power, and voltage regulation. Moreover, the use of STATCOM/Battery
and IDA-PBC design can clearly provide additional benefits beyond the PSS/AVR
in conventional power systems.
Case 2: DFIG Figures 4.7 show phase portraits between power angle δ and
DFIG slip s = 1 − ω/ωs . Voltage and active power responses are shown in Figure 4.8.
From this case, it is apparent that the power angle (δ) converges to the pre-
fault values (δ0) and DFIG rotor slip (s = 1 − ω/ωs → s0 = −0.03) returns to its
pre-fault value. The domain of attraction and CCT are obviously enhanced when
STATCOM/Battery are integrated into the system. Moreover, the domain of attrac-
tion for DRVC/STAT/BATT is larger than that for the DRVC/STAT and DRVC
but for PIC, there are different dynamic behaviors that lead to a different domain
of attraction and a new equilibrium point. Similarly, there is a distinct percentage
increase in CCT for the DRVC/STAT/BATT that is significantly larger than that for
the DRVC/STAT and DRVC, as compared to the case with PIC (0%), (e.g. +182%
increase in CCT for DRVC/STAT/BATT, +154% increase in CCT for DRVC/STAT,
+146% increase in CCT for DRVC). Identical to the SG case, voltage and power regulation transient responses for the DRVC/STAT/BATT are further improved over the
cases with DRVC/STAT, DRVC, or PIC. In particular at CCT = 0.69 sec., there are
oscillations on terminal voltage and active power responses of PIC, tending to eventual instability while time responses of the proposed controllers are able to regulate
both terminal voltages and active power to the desired pre-fault steady state values
quickly following a fault. Simulation results illustrate that transient (power angle)
stability, in terms of increasing critical clearing time and enlarging domain of attraction, is improved and frequency, power, and voltage regulation are simultaneously
enhanced via STATCOM/Battery integration and IDA-PBC design.
4.4.2
Two-Machine Infinite Bus
Next we consider two generators (SG-DFIG) connected to an infinite bus, including
a STATCOM/Battery.
62
Table 4.1: CCT of SG cases
PSS/AVR [41]
GEC
GEC/STAT
GEC/STAT/BATT
0.83
1.56
1.62
1.79
SG (CCT (sec.))
D = 0.2
D=0
(No damping)
(0.00%) 0.37
(0.0%)
(+87.95%) 0.40 (+8.1%)
(+95.18%) 0.41 (+10.8%)
(+115.66%) 0.44 (+18.9%)
Table 4.2: CCT of DFIG cases
DFIG (CCT (sec.))
PIC [18]
0.28
(0.00%)
DRVC
0.69
(+146.4%)
DRVC/STAT
0.71
(+153.6%)
DRVC/STAT/BATT 0.79
(+182.1%)
Table 4.3: CCT of two-machine bus (SG-DFIG cases)
SG-DFIG (CCT (sec.))
PSS/AVR/PIC
0.316
(0.00%)
GEC/DRVC
0.425
(+34.49%)
GEC/DRVC/STAT
0.430
(+36.07%)
(+55.75%)
GEC/DRVC/STAT/BATT 0.489
Consider the single line diagram given in Figure 4.2 consisting of a conventional
SG and a DFIG representing a wind power system. Both are connected through a
transmission line to a infinite bus. The SG and DFIG generators deliver 1.0 and 1.03.
pu. power, respectively, while the terminal voltage is 0.9780 pu and the infinite-bus
voltage is 1.0 pu. A three-phase fault occurring at P causes a power imbalance and
voltage sag.
Table 4.3 presents the critical clearing times (CCT) in percentage for the twomachine (SG-DFIG) infinite bus system including GEC/DRVC, GEC/DRVC/STAT,
and GEC/DRVC/STAT/BATT cases when compared to PSS/AVR/PIC.
63
Vt∠β
E1 ∠δ
V∞∠0
j2X2
jX1
j2X2
F
SG
P
I¯SB = Id + jIq
jX3
E3 ∠δ3
DFIG
Figure 4.2: A single line diagram of two-machine infinite bus
Case 3: SG-DFIG Figures 4.9 and 4.10 show phase plane between a power angle
and frequency. Voltage and active power responses are shown in Figure 4.11.
The parameters of the state feedback controllers in this case are r1 γ1 =
and r2γ2 =
(1+Xm3 |Y33 |)
,
0
T30
(1+Xm1 |Y11 |)
0
T10
γ3 = γ4 = 1 and r3 = r4 = 10. From the simulation re-
sults it is clear that GEC/DRVC/STAT/BATT can be used to enhance the transient
stability margin through improved CCT, in particular +55.75% increase in CCT,
enlarged domain of attraction, and improved voltage and frequency regulation compared with GEC/DRVC (+34.49%) , with GEC/DRVC/STAT (+36.07%), or with
PSS/AVR/PIC (0%). Similar to last two cases, the use of GEC/DRVC/STAT/BATT
provides better performances than the GEC/DRVC/STAT, GEC/DRVC and PSS/
AVR/PIC in terms of improving (power angle stability) both CCT and DOA, along
with voltage and frequency regulation whereas the combination of conventional controllers (PSS/AVR/PIC) does not.
64
Remark 13. The PSS/AVR parameters and PI gains of the conventional and alternative (wind) power system controllers in three cases (PSS/AVR, PIC, and PSS/AVR/
PIC) are chosen to simultaneously achieve rotor speed and terminal voltage control
using a trial and error procedure via simulations. The values obtained from this tuning process are presented in Appendix A.6-A.8. Also, in order to further increase the
transient stability margin and accomplish both rotor speed and terminal voltage regulation, the parameters and gains are designed to minimize power system oscillations
and regulate the performance after the occurrence of a fault through an optimization
method.
4.5
Conclusions
In this Chapter, we have shown that the combination of a STATCOM and Battery
energy storage can be used to enhance the transient stability of SG and DFIG generators. Simulation results have demonstrated that power angle stability, along with
voltage and frequency regulation are achievable for large (transient) disturbances using the IDA-PBC nonlinear model-based control design methodology. In particular,
using the STATCOM/Battery and IDA-PBC control design (GEC/STAT/BATT,
DRVC/STAT/BATT, GEC/DRVC/STAT/BATT), there is a large increase in CCT
and enlarged DOA for SG, DFIG, and SG-DFIG cases when compared to operation
with GEC (DRVC), GEC/STAT (DRVC/STAT), GEC/DRVC/STAT or conventional
and alternative (wind) power system controllers (PSS/AVR, PIC, and PSS/AVR/PIC),
thereby leading to a significant improvement in power system stability and robustness.
65
6
GEC/STAT/Batt: tcr = 0.44 s.
GEC/STAT: tcr = 0.41 s.
GEC: tcr = 0.40 s.
PSS/AVR: tcr = 0.37
5
4
t0 = 0 s.
ω − ωs (rad/s)
3
tcr
2
1
0
−1
−2
tr = 5 s.
tf = 20 s.
−3
−4
0
0.5
δ (rad) 1.5
1
2
2.5
Figure 4.3: Phase portraits of SG case (D = 0)
Vt (pu.)
1
0.8
0.6
0.4
0.2
0
2
4
6
8
10
12
14
16
18
20
PE (pu.)
1.5
1
GEC
GEC/STAT/Batt
GEC/STAT
PSS/AVR
0.5
0
0
2
4
6
8
10
Time
(sec.) 12
14
16
18
20
Figure 4.4: Voltage and active power responses at tcr = 0.40 sec.
66
2.5
GEC/STAT/BATT: tcr: 1.79 s.
GEC/STAT: tcr = 1.62 s.
GEC: tcr = 1.56 s.
PSS/AVR: tcr = 0.83 s.
2
ω − ωs (rad/s)
1.5
t0 = 0 s.
1
tr = 5 s.
0.5
tcr
0
−0.5
−1
tf = 20 s.
−1.5
0.4
0.6
0.8
1
1.2
δ 1.4
(rad) 1.6
1.8
2
2.2
2.4
Figure 4.5: Phase portraits of SG case (D = 0.2)
1.2
Vt (pu.)
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
16
18
20
PE (pu.)
1.6
GEC
GEC/STAT/Batt
GEC/STAT
PSS/AVR
1.4
1.2
1
0.8
0.6
0
2
4
6
8
10
Time
(sec.) 12
14
16
18
20
Figure 4.6: Voltage and active power responses at tcr = 1.56 sec.
67
tf = 20 s.
tr = 5 s.
−0.025
s = 1 − ω/ωs
DRVC/STAT/Batt: tcr = 0.79 s.
DRVC/STAT: tcr = 0.71 s.
DRVC: tcr = 0.69 s.
PIC: tcr = 0.28 s.
−0.03
−0.035
t0 = 0 s.
−0.04
tcr
−0.045
0
0.5
1
δ (rad) 1.5
2
2.5
Figure 4.7: Phase portraits of DFIG case
Vt (pu.)
1
0.8
0.6
0.4
0.2
PE (pu.)
0
0
1
2
3
4
5
6
7
8
9
10
DRVC
DRVC/STAT/Batt
DRVC/STAT
PIC
−0.5
−1
−1.5
0
1
2
3
4
5
Time (sec.)
6
7
8
9
10
Figure 4.8: Voltage and active power responses at tcr = 0.69 sec.
68
GEC/DRVC/STAT/Batt: tcr = 0.489 s
GEC/DRVC/STAT: tcr = 0.430 s.
GEC/DRVC: tcr = 0.425 s.
PSS/AVR/PIC: tcr = 0.316 s.
4
ω1 − ωs (rad/s)
3
tr = 2 s.
2
tcr
t0 = 0 s.
1
0
tf = 10 s.
−1
0
0.5
1
δ1 (rad)
1.5
2
2.5
Figure 4.9: Phase portraits of SG in SG-DFIG case
−0.022
−0.024
GEC/DRVC/STATBatt: tcr = 0.489 s.
GEC/DRVC/STAT: tcr = 0.430 s.
GEC/DRVC: tcr = 0.425 s.
PSS/AVR/PIC: tcr = 0.316 s.
tf = 10 s.
−0.026
s = 1 − ω3 /ωs
−0.028
−0.03
−0.032
t0 = 0 s.
−0.034
−0.036
tcr
−0.038
tr = 2 s.
−0.04
−0.042
0
0.5
1
1.5
δ3 (rad)
2
2.5
Figure 4.10: Phase portraits of DFIG in SG-DFIG case
69
3
Vt (pu.)
1.5
1
0.5
0
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
P1E (pu.)
2
1
0
−1
P3E (pu.)
0
GEC/DRVC/STAT/BATT
GEC/DRVC/STAT
GEC/DRVC
PSS/AVR/PIC
−0.5
−1
−1.5
0
1
2
3
4
Time (sec.)
5
6
7
Figure 4.11: Voltage and active power responses at tcr = 0.425 sec.
70
Chapter 5
Nonlinear Voltage Regulation
5.1
Introduction
Both conventional and alternative energy generation sources are faced with difficult
operating challenges, including load following and disturbance rejection (small- and
large-signal), while achieving desired power quality including frequency and voltage
regulation. Voltage regulation requires maintaining generator terminal voltages and
bus voltages close to the desired reference voltages under normal operating conditions
and to regulate voltage to desired pre-fault steady state values quickly after the
occurrence of a fault.
Conventional control designs methods include linear control [6], robust control [23,
76], and nonlinear control [5, 22, 49, 54]. To further enhance the problem of voltage
regulation, there have been efforts to incorporate Flexible AC Transmission System
(FACTS) devices to enhance power transfer capability and augment the stability of
the power system [29, 62] using a Static Var Compensator (SVC) [17, 68] and in
particular, a Static Synchronous Compensator (STATCOM) [34]. For the controller
design for alternative power generations, particularly wind power generators, there
have been efforts to enhance voltage regulation via robust control [31, 32].
It is evident that energy storage technologies [59], are important for dealing with
the intermittency of many alternative energy sources, and also provide the opportu71
nity to improve power quality, especially frequency and power stability. For stability
enhancement of wind energy conversion systems, battery energy storage has been
used to improve frequency stability through the regulation of active power levels.
STATCOM and (battery) energy storage systems have been independently used
to improve power system operations, and integrating these devices provides an opportunity to simultaneously improve overall voltage regulation and system stability
of power systems. Relatively little prior work has been devoted to the integration
of STATCOM and battery energy storage, Yang et at. [69] have shown that the
integration of STATCOM and battery can provide additional benefits beyond the
STATCOM in conventional power systems and Arulampalam et at. [7] have shown
that improvements in power quality can be obtained by a STATCOM and battery
combination for a wind farm application. Recently, Hossain et at. [30] have shown
that improvements in voltage regulation and transient stability via robust control
techniques can be obtained by a combined STATCOM and energy storage system for
a wind farm application.
Our objective is to design a dynamic output feedback controller with an IDA-PBC
nonlinear state feedback control law and a reduced-order nonlinear observer in order
to achieve the problem of voltage regulation and system stability of power systems
that include the integration of renewable energy sources (generation and storage) with
conventional generation.
This Chapter continues the work reported in Chapter 4 by examining how a
STATCOM and battery energy storage system can be used to improve the voltage
regulation and system stability of a wind energy conversion system interconnected
to the grid through a Doubly Fed Induction Generator (DFIG). The wind power
system considered here is represented using a DFIG. Simulation results are provided
for two operating scenarios: (1) Single machine infinite bus with either a Synchronous
Generator (SG) or DFIG connected to the infinite bus, (2) a two-machine, both SG
and DFIG, system connected to an infinite bus.
The approach can be summarized as follows:
72
• With complete state feedback, the existing (IDA-PBC) control design methodology [55] is further extended to achieve system stability and voltage regulation
with and without the knowledge of the equilibrium states of power systems that
include STATCOM/Battery.
• A reduced-order observer is designed based on available power system measurements to estimate power angles and transient terminal voltages.
• An observer-based controller is proposed using a Lyapunov-based approach to
guarantee asymptotic stability of the closed-loop system.
The Chapter is organized as follows. The problem formulation is provided in
Section 5.2. IDA-PBC controller design is given in Section 5.3. A reduced-order
nonlinear observer design method is presented in Section 5.4 and the asymptotic
stability with IDA-PBC controller and nonlinear observer is demonstrated in Section
5.5. A dynamic extension for the unknown post-fault equilibrium point is provided
in Section 5.6. Simulation results are given in Section 5.7. Conclusions are given in
Section 5.8.
5.2
Problem Formulation
In this Chapter we are interested in studying the system stability and voltage regulation of a nonlinear power system including STATCOM/Battery. The nonlinear
system considered can be written in the nonlinear affine form as follows:
ẋ = f(x) + g(x)uI (x)
y = h(x)
(5.1)
where x ∈ <n is the state vector, uI (x) ∈ <m , m < n is the control action, y ∈ <p
are the measured output. f(x) : <n → <n is continuously differentiable on the
domain D ⊂ <n and satisfies a local Lipschitz condition to ensure the existence of
73
solution on [0, +∞). g(x) : <n → <n×m is assumed to be smooth, full rank function.
h(x) : <n → <p is continuous.
The problem is: Given a stable equilibrium point xe , find a dynamic output
feedback controller in the following form:
x̂˙ = ν(x̂, y),
x̂ ∈ <n−p
uI = uI (x̂, y)
(5.2)
such that the closed-loop system satisfies:
1. The desired equilibrium point xe is asymptotically stable,
2. System stability and voltage regulation are simultaneously achieved.
In the next two sections, we develop the output feedback controller that meets
the expected performance requirements (1)-(2) given above.
5.3
IDA-PBC Controller Design
In this section, IDA-PBC control method, proposed in [56], is extended to design a
static state feedback control law with an integral action to achieve the performance
requirements given in the problem formulation.
Proposition 1. [56] Consider the nonlinear systems in (5.1) and assume that the
desired closed-loop system is in the following form:
ẋ = (Jd (x) − Rd (x))∇xHd
where the notation ∇x =
∂
.
∂x
(5.3)
Matrices Jd (x) = −Jd (x)T and Rd (x) = Rd (x)T ≥ 0
are the desired interconnection and damping matrices, respectively, and both can be
chosen by the designer. The Hamiltonian function Hd : <n → < denotes the desired
total stored energy function satisfying xe = arg min Hd such that the PDE below is
solved.
g ⊥ (x)f(x) = g ⊥ (x)[Jd(x) − Rd (x)]∇xHd
74
(5.4)
where g ⊥ (x)g(x) = 0. Then the closed-loop system in (5.1) with a feedback control
law:
u(x) = [g T (x)g(x)]−1g T (x){(Jd (x) − Rd (x))∇xHd − f(x)}
will be a port-controlled Hamiltonian (PCH) system with dissipation of the form (5.3)
with a stable equilibrium xe and Lyapunov function Hd (x). Further, the system will
be asymptotically stable if xe is in the largest invariant set for the closed-loop system
(5.3) contained in
T
n
x ∈ < (∇xHd ) Rd (x)∇xHd = 0
A domain of attraction is given by the largest bounded level set {x ∈ <n |Hd (x) ≤ c}
that contains the desired equilibrium point xe .
The foregoing Proposition ensures the asymptotic stability of the nonlinear systems. The problem of voltage regulation is addressed by defining ∆Vt = Vt − Vref
where Vref represents the reference voltage and Vt denotes the terminal voltage shown
in Figures 5.1 and 5.2, respectively. The objective is to drive ∆Vt to zero at steady
state, so we incorporate an additional state ξ (integral action) as follows:
˙ = −KI ∆Vt = −KI (Vt − Vref )
ξ(t)
Therefore, an extension to Proposition 1 to simultaneously achieve voltage regulation is as follows:
Proposition 2. [56] (Integral action) Consider the system of Proposition 1 in closedloop with uI (x) = u(x) + ξ, where
ξ˙ =
−KI gIT (x)∇xHd
⇒ξ=
Z
t
0
−KI gIT (x)∇xHd dt
with KI = KIT > 0 and a gIT (x) matrix given by the designer. Then all stability
properties of xe mentioned in Proposition 1 are preserved.
75
In order to show the benefits of the STATCOM/Battery to both conventional and
wind power systems, consider the SMIB case and a two-machine infinite bus system
as shown in Figures 5.1 and 5.2. There are two SMIB cases and one two-machine
infinite bus case that are of interest, (1) SG (2) DFIG and (3) SG-DFIG.
Vt∠β
E∠δ
V∞∠0
j2X2
jX1
j2X2
P
F
SG/DFIG
I¯SB = Id + jIq
Figure 5.1: A single line diagram of SMIB
Vt∠β
E1 ∠δ
V∞∠0
j2X2
jX1
j2X2
F
SG
P
I¯SB = Id + jIq
jX3
E3 ∠δ3
DFIG
Figure 5.2: A single line diagram of two-machine infinite bus
76
Case 1: SG Dynamic models of the SG and STATCOM/Battery are written in
the general form (5.1) as follows:




f(x) = 





g(x) = 




ω − ωs
δ
1

(P
−
P
−
D(ω
−
ω
))

m
E
s
M

 ω − ωs
Xm V∞ cos δ 
s +Xtr +X2 )E
 E
+
− (X(X
0
0
,
x
=
(X1 +X2 )T0 
1 +X2 )T0

ωs Vt cos β 
R s ωs
 Id

− Ls Id + ωIq − Ls
β
Iq
−ωId − RLsωs s Iq − ωs VLt sin
s

0
0
0
 uf 

0
0
0
T00



1
0
0
huM 
 , u(x) =
0 cos β − sin β 
huγ
0 sin β cos β



,


where PE represents the power transmitted from bus 1 to the STATCOM/Battery
bus as follows:
EV∞ sin δ
Iq X1 X2
EX2 Id
V∞ X1 cos δ + E1 X2
PE =
1+
+
·
(X1 + X2 )
∆
(X1 + X2 )
∆
s
2 2
Iq X1 X2
∆
Id X1 X2
1+
+
Vt =
(X1 + X2 )
∆
∆
p
(EX2 )2 + (V∞ X1 )2 + 2X1 X2 EV∞ cos δ
∆ =
and cos β and sin β given in Chapter 3. Due to physical consideration, the behaviors
of the system used in SG and DFIG cases are restricted within the closed set as
follows:
D = {(δ, ω, E, Id, Iq )| 0 ≤ δ <
π
, E > 0}
2
A state feedback controller can be constructed from the following proposition.
77
Proposition 3. The closed-loop system of SMIB (SG case) can accomplish the expected performance requirements (1)-(2) with the control law uI (x):
Xm V∞ (cos δ − cos δe )
uf
= −J23(δ, E)(ω − ωs ) −
0
T0
(X1 + X2 )T00
huM = M cos β + N sin β
huγ = −M sin β + N cos β
uI (x) = u(x) + ξ
(5.5)
where
J23 (δ, E) = −
V∞ sin δ
γ1 (X1 + X2 )
J24 (δ, E) = −
Ee X2 (V∞ X1 cos δ + Ee X2 )
∆e (X1 + X2 )γ2 Ls
J25 (δ, E) = −
Ee X1 X2 V∞ sin δ
∆e (X1 + X2 )γ3Ls
M = −J24(δ, E)(ω − ωs ) − r2Ls γ2 (Id − Ide ) +
Rs ωs
ωs Vt cos β
Id − ωIq +
Ls
Ls
N = −J25(δ, E)(ω − ωs ) − r3Ls γ3 (Iq − Iqe ) + ωId +
Rs ωs
ωs Vt sin β
Iq +
Ls
Ls
(5.6)
and ∆e = ∆|E=Ee , γi and ri , i = 1, 2, 3 are arbitrary positive constants. The closedloop system is asymptotically stable at the desired equilibrium point xe = (δe , ωs , Ee , Ide , Iqe)T
with the Lyapunov and energy functions as follows:
Hd (x) =
M
γ1
γ2 Ls
γ3 Ls
(ω − ωs )2 + (E − Ee )2 + ϕ(δ) +
(Id − Ide )2 +
(Iq − Iqe )2
2
2
2
2
(5.7)
1
W(x, ξ) = Hd (x) + ξ T KI−1 ξ
2
(5.8)
78
+
Z
δ
δe
δ
Ee X2 Ide (V∞ X1 cos δ + Ee X2 )
·
dδ
∆e
δe (X1 + X2 )
Ee V∞ sin δ
Iqe X1 X2
1+
dδ
(X1 + X2 )
∆e
ϕ(δ) = −Pm (δ − δe ) +
Z
(5.9)
A domain of attraction is given by the largest bounded level set {x ∈ <5 |Hd (x) ≤ c}
that contains the desired equilibrium point xe . Further, the desired interconnection
matrix Jd (x) and the damping matrix Rd (x) are selected as follows:



Jd (x) − Rd (x) = 


1
M
0
− M1
0
0
0
−D
−J23
−J24
−J25
0
0
0
J23 J24 J25
−r1 0
0
0 −r2 0
0
0 −r3






Proof. From Proposition 1, the PDE in (5.4) for this case is
Pm − PE − D(ω − ωs )
1
= − ∇δ Hd − D∇ω Hd + J23(δ, E)∇E Hd
M
M
+J24 (δ, E)∇Id Hd + J25(δ, E)∇Iq Hd
(5.10)
Substituting (5.7) into the expression above and evaluating at the equilibrium
point (E = Ee , Id = Ide , Iq = Iqe ), we have:
Ee V∞ sin δ
Iqe X1 X2
Ee X2 Ide (V∞ X1 cos δ + Ee X2 )
∇ϕ = −Pm +
+
·
1+
(X1 + X2 )
∆e
(X1 + X2 )
∆e
Plugging this expression back in (5.10) and using some lengthy, but straightforward,
calculation, we obtain J2j (δ, E) in (5.6).
According to Proposition 1, the state feedback control law can be computed by
choosing the free parameter as r1 γ1 =
(Xs +Xtr +X2 )
(X1 +X2 )T00
as given in (5.5) as well as using
the pre-fault equilibrium equations.
From Proposition 1, the asymptotic stability is proved by using the Lyapunov
function obtained by integrating ∇ϕ and then substituting back into (5.7). Asymp-
totic stability is then established through LaSalle’s invariance principle that requires
79
(1) the energy function Hd is positive definite and proper, (2) all trajectories are
d T
d
) Rd (x) ∂H
=
bounded and remain inside the set where Ḣd ≤ 0, and (3) Ḣd = ( ∂H
∂x
∂x
0 ⇒ ω = ωs , E = Ee , Id = Ide , Iq = Iqe , this implies δ = δe . The domain of attrac-
tion can be determined explicitly using the invariance principle, namely Ḣd ≤ 0 and
Hd ≤ c.
To achieve voltage regulation, we combine Proposition 1 and 2, with all properties
preserved, to get Proposition 3 that enables the design of a control law that meets
the expected performance requirements. Therefore, the control law in (5.5) can be
replaced by uI (x) = u(x) + ξ. The asymptotic stability with the integral action is
completed using the energy function W(x, ξ) along with the invariance principle to
conclude that Ẇ(x, ξ) = 0 ⇒ x = xe and ξ = 0 ⇒ Vt = Vref .
Case 2: DFIG Dynamic models of the DFIG and STATCOM/Battery are written in the general form (5.1) as follows:




f(x) = 





g(x) = 


ω − ωs − XXm0 V∞ETsin0 δ
s
0
1
(P
−
P
)
m
E
M
s +Xtr +X2 )E
m V∞ cos δ
− (X(X
+X
0
(X1 +X2 )T00
1 +X2 )T0
β
− RLsωs s Id + ωIq − ωs VLt cos
s
−ωId − RLsωs s Iq − ωs VLt ssin β
cos δ
E
sin δ
E







,x = 




0
0
0
0
0
0
sin δ − cos δ
0
0
0
0
cos β − sin β
0
0
sin β cos β

δ
ω − Sωs
E
Id
Iq



,



ωs Vdr


 ωs Vqr
 , u(x) = 

 huM

huγ


,

(5.11)
where Vdr = Vr cos θr and Vqr = Vr sin θr . PE , ∆, cos β, sin β, and Vt are identical
to SG case.
80
A state feedback controller can be constructed from the following proposition.
Proposition 4. The closed-loop system of SMIB (DFIG case) can accomplish the
performance requirements (1)-(2) with the control law uI (x):
ωs Vdr = X cos δ + Y sin δ
ωs Vqr = X sin δ − Y cos δ
huM = M cos β + N sin β
huγ = −M sin β + N cos β
uI (x) = u(x) + ξ
(5.12)
where
X = s0 ωs E +
Xm V∞ sin δ
(X1 + X2 )T00
Y = −J23(δ, E)(ω − Sωs ) −
Xm V∞ cos δ
(X1 + X2 )T00
M = −J24(δ, E)(ω − Sωs ) − r1 Ls γ2 (Id − Ide ) +
Rs ωs
ωs Vt cos β
Id − ωIq +
Ls
Ls
N = −J25(δ, E)(ω − Sωs ) − r2 Ls γ3 (Iq − Iqe ) + ωId +
ωs Vt sin β
Rs ωs
Iq +
Ls
Ls
S = 1 − s0 = ωe /ωs
and J2i (δ, E), i = 3, 4, 5 are identical to SG case. γi and ri , i = 1, 2, 3 are arbitrary
positive constants. Further, the closed-loop system is asymptotically stable at a desired
equilibrium point xe = (δe , Sωs , Ee , Ide , Iqe)T with the Lyapunov and energy functions
Hd (x), W(x, ξ) and ϕ(δ) that are almost identical to SG case except for replacing
M
(ω
2
− Sωs )2 in Hd (x). A domain of attraction is given by the largest bounded level
set {x ∈ <5|Hd (x) ≤ c} that contains the desired equilibrium point xe . Further,
the desired interconnection matrix Jd (x) and the damping matrix Rd (x) are selected
identical to the SG case.
81
Proof. The proof is almost identical to the SG case except that there are slight differences in the control law, particularly DFIG control law, but this is straightforward.
Similar to the SG case the integral action is combined with the results of Propositions
1 and 2 to replace u(x) in (5.12) by uI (x) = u(x) + ξ to achieve voltage regulation.
The asymptotic stability with and without the integral action and the domain of
attraction are established identical to SG case.
Case 3: SG-DFIG Dynamic models of the SG, DFIG and STATCOM/Battery
are written in the general form (5.1) as follows:

f1(x)
f2(x)
f3(x)
f4(x)
f5(x)
f6(x)
f7(x)
f8(x)

0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
cos δ3
sin δ3
0 E3
0
0
E3
0
0
0
0
0
0 sin δ3 − cos δ3
0
0
0
0
0
cos β − sin β
0
0
0
sin β cos β





f(x) = 










g(x) = 

















,x = 










δ1
ω1 − ωs
E1
δ3
ω3 − Sωs
E3
Id
Iq






,














 , u(x) = 







uf
0
T10
ωs Vdr
ωs Vqr
huM
huγ



 . (5.13)


where
f1 (x) = ω1 − ωs ,
f2 (x) =
1
(Pm1 − P1E − D(ω1 − ωs )) ,
M1
f3 (x) = −
(1 + Xm1 |Y11 |)E1 Xm1 |Y12|V∞ cos δ1 Xm1 |Y13|E3 cos(δ1 − δ3 )
+
+
,
0
0
0
T10
T10
T10
f4 (x) = ω3 − ωs −
Xm3 |Y31|E1 sin(δ3 − δ1) Xm3 |Y32|V∞ sin δ3
−
,
0
0
T30
E3
T30
E3
82
f5 (x) =
1
(Pm3 − P3E ) ,
M3
f6 (x) = −
(1 + Xm3 |Y33 |)E3 Xm3 |Y32|V∞ cos δ3 Xm3 |Y31|E1 cos(δ3 − δ1 )
+
+
,
0
0
0
T30
T30
T30
f7 (x) = −
ωs Vt cos β
Rs ωs
Id + ωIq −
,
Ls
Ls
f8 (x) = −ωId −
Rs ωs
ωs Vt sin β
Iq −
,
Ls
Ls
0
0
Xm1 = Xs1 − Xs1
, Xm3 = Xs3 − Xs3
,
and P1E and P3E represent the power transmitted from bus 1 and bus 3, respectively,
to the STATCOM/Battery bus as follows:
P1E
P3E
Vt
Iq X1 X2 X3
= (E1 V∞ |Y12 | sin δ1 + E1 E3 |Y13| sin(δ1 − δ3)) 1 +
Π
X2 X3 E12
Id X1 X2 X3
+ E1 V∞ |Y12| · cos δ1 + E1 E3 |Y13| cos(δ1 − δ3) ·
+
X1 Λ
Π
Iq X1 X2 X3
= (E3 V∞ |Y32 | sin δ3 + E1 E3 |Y31| sin(δ3 − δ1)) · 1 +
Π
Id X1 X2 X3
X1 X2 E32
+ E3 V∞ |Y32| · cos δ3 + E3 E1 |Y31| cos(δ3 − δ1) ·
+
X3 Λ
Π
s
2 2
Π
Iq X1 X2 X3
Id X1 X2 X3
=
1+
+
Λ
Π
Π
and Π =
√
A2 + B 2 , A = X2 X3 E1 cos δ1 +X1 X3 V∞ +X1 X2 E3 cos δ3, B = X1 X2 E3 sin δ3+
X2 X3 E1 sin δ1. In this case, cos β and sin β are given in Chapter 3. Similar to SG and
DFIG cases, the operation of the system is physically restricted within the closed set
as follows:
D = {(δi , ωi , Ei , Id, Iq )| 0 ≤ δi <
π
, Ei > 0, i = 1, 3.}
2
A state feedback controller can be constructed from the following Proposition.
83
Proposition 5. The closed-loop system of the two-machine infinite bus (SG-DFIG
case) can accomplish the expected performance requirements (1)-(2) with stable operation given the following assumptions:
Assumption 1.
|δ1e − δ3e | ≤
π
2
(5.14)
Assumption 2.
Z δ3 2
∂P1Ee
∂ P3Ee
+
dδ3 > 0
∂δ1
∂ 2 δ1
δ3e
2
∂ 2P1Ee
∂P1Ee ∂P3Ee
dδ1 −
+
>0
∂ 2 δ3
∂δ3
∂δ1
∂P1Ee
+
∂δ1
Z
δ3
δ3e
∂ 2 P3Ee
dδ3
∂ 2 δ1
∂P3Ee
+
∂δ3
Z
δ1
δ1e
(5.15)
and the control law uI (x):
uf
Xm |Y12|V∞ (cos δ1 − cos δ1e )
= −J23(δ, E)(ω1 − ωs ) + J35(δ, E)(ω3 − Sωs ) −
0
0
T0
T10
−
Xm |Y13|E1 E3 (cos(δ1 − δ3) − cos(δ1e − δ3e ))
0
T10
ωs Vdr = X cos δ3 + Y sin δ3
ωs Vqr = X sin δ3 − Y cos δ3
huM = M cos β + N sin β
huγ = −M sin β + N cos β
uI (x) = u(x) + ξ
(5.16)
where
J23(δ, E) = −
V∞ |Y12 | sin δ1 + E3e |Y13| sin(δ1 − δ3 )
γ1
J26(δ, E) = −
E1|Y13 | sin(δ1 − δ3)
γ3
84
2
X2 X3 E1e
X1 X2 X3
J27 (δ, E) = −
+ E1e V∞ |Y12 | cos δ1 + E1e E3e |Y13 | cos(δ1 − δ3 )
X1 Λ
Πe γ3 Ls
J28 (δ, E) = −(E1e V∞ |Y12| sin δ1 + E1e E3e |Y13 | sin(δ1 − δ3))
J35 (δ, E) =
X1 X2 X3
Πe γ4 Ls
E3 |Y31| sin(δ3 − δ1)
γ1
V∞ |Y32| sin δ3 + E1e |Y31| sin(δ3 − δ1)
γ3
2
X1 X2 X3
X1 X2 E3e
J57 (δ, E) = −
+ E3e V∞ |Y32 | cos δ3 + E1e E3e |Y31 | cos(δ3 − δ1 )
,
X3 Λ
Πe γ3 Ls
X1 X2 X3
J58 (δ, E) = − E3e V∞ |Y32 | sin δ3 + E1e E3e |Y31 | sin(δ3 − δ1)
,
Πe γ4 Ls
J56 (δ, E) = −
X = s0 ωs E3 +
Xm3 |Y31|E1 sin(δ3 − δ1 ) Xm3 |Y12 |V∞ sin δ3
+
0
0
T30
T30
Y = −J26(δ, E)(ω1 − ωs ) −
−
Xm3 V∞ cos δ3
− J56(δ, E)(ω3 − Sωs )
0
T30
Xm |Y31|E1 cos(δ3 − δ1 )
0
T30
M = −J27(δ, E)(ω1 − ωs ) − r3 Ls γ3 (Id − Ide ) − J57(δ, E)(ω3 − Sωs ) − f7 (x)
N = −J28(δ, E)(ω1 − ωs ) − r4 Ls γ4 (Iq − Iqe ) − J58(δ, E)(ω3 − Sωs ) − f8 (x)
S = 1 − s0 = ω3e /ωs
(5.17)
and Πe = Π|E1 =E1e ,E3 =E3e . γi and ri , i = 1, 2, 3, 4 are arbitrary positive constants.
Further, the closed-loop system is asymptotically stable at the desired equilibrium point
xe = (δ1e, ωs , E1e , δ3e , Sωs , E3e , Ide , Iqe )T with the Lyapunov and energy functions as
follows:
Hd (x) =
M1
γ1
M3
(ω1 − ωs )2 + (E1 − E1e )2 + ϕ(δ) +
(ω3 − Sωs )2
2
2
2
γ2
γ3 Ls
γ4 Ls
(E3 − E3e )2 +
(Id − Ide )2 +
(Iq − Iqe )2
2
2
2
1
W(x, ξ) = Hd (x) + ξ T KI−1 ξ.
2
+
85
(5.18)
(5.19)
ϕ(δ) = −Pm1 (δ1 − δ1e ) − Pm3 (δ3 − δ3e ) +
Z
δ1
P1Ee dδ1 +
δ1e
Z
δ3
P3Ee dδ3 (5.20)
δ3e
A domain of attraction is given by the largest bounded level set {x ∈ <8 |Hd (x) ≤ c}
that contains the desired equilibrium point xe . Furthermore, the desired interconnection matrix Jd (x) and the damping matrix Rd (x) are selected as follows.
Jd (δ, E) − Rd (x) =

1
0
0
0
0
0
0
0
M1
1
 −
D
J23(δ, E)
0
0
J26 (δ, E) J27(δ, E) J28 (δ, E)
 M1
 0
−J
(δ,
E)
−r
0
J
(δ,
E)
0
0
0
23
1
35

1
 0
0
0
0
0
0
0
M3

1
 0
0
−J35(δ, E) − M3
0
J56 (δ, E) J57(δ, E) J58 (δ, E)

 0
−J26(δ, E)
0
0
−J56(δ, E)
−r2
0
0

 0
−J27(δ, E)
0
0
−J57(δ, E)
0
−r3
0
0
−J28(δ, E)
0
0
−J58(δ, E)
0
0
−r4












Proof. From Proposition 1, the PDE in (5.4) for this case is
Pm1 − P1E − D(ω1 − ωs )
1
= −
∇δ Hd − D∇ω Hd + J23 (δ, E)∇E1 Hd
M1
M1 1
+J26(δ, E)∇E3 Hd + J27(δ, E)∇Id Hd
+J28(δ, E)∇Iq Hd
Pm3 − P3E
M3
= −
(5.21)
1
∇δ Hd − J35 (δ, E)∇E1 Hd + J56 (δ, E)∇E3 Hd
M3 3
+J57(δ, E)∇Id Hd + J58(δ, E)∇Iq Hd
(5.22)
Substituting (5.18) into the expressions above, using the Leibniz integral rule1,
and then evaluating at the equilibrium point (E1 = E1e , E3 = E3e , Id = Ide , Iq = Iqe ),
we have:
(Ee,Ie )
(δ, E),
(Ee,Ie )
(δ, E)
∇δ1 ϕ = −Pm1 + P1Ee + P1 (δ1, δ3) = −F1
∇δ3 ϕ = −Pm3 + P3Ee + P3 (δ1, δ3) = −F3
(5.23)
R b(y)
R b(y)
d
The derivative of the integral of the form a(y) f(x, y)dy is dy
f(x, y)dx = f(b(y), y) ∂b(y)
∂y −
a(y)
R b(y) ∂f(x,y)
∂a(y)
f(a(y), y) ∂y + a(y) ∂y dx if both f and ∂f/∂x are continuous.
1
86
R δ3
where P1 (δ1, δ3) =
δ3e
∂P3Ee (δ1 ,δ3)
dδ3
∂δ1
and P3(δ1, δ3) =
P1E |(E1e,E3e ,Ide ,Iqe ) and P3Ee = P3E |(E1e ,E3e,Ide ,Iqe ) .
R δ1
δ1e
∂P1Ee (δ1,δ3 )
dδ1.
∂δ3
P1Ee =
Substituting these expressions back into (5.21) and (5.22) and using some lengthy,
but straightforward, calculations, we obtain Jij (δ, E) given in (5.17).
According to Proposition 1, the control law can be computed by selecting the free
parameter as r1γ1 =
(1+Xm1 |Y11 |)
0
T10
and r2 γ2 =
(1+Xm3 |Y33 |)
0
T30
and using the pre-fault equilib-
rium equations as given in (5.16). The Lyapunov function can be obtained, satisfying
(Ee ,Ie )
Poincare’s Lemma in [56] that requires ∇δ1 F3
(Ee,Ie )
= ∇δ3 F1
=
∂P1Ee
∂δ3
+ ∂P∂δ3Ee
, from
1
integrating ∇ϕ and then substituting back into (5.18) as shown in Appendix B.1. Fur-
thermore, to ensure the minimum condition (xe = arg min Hd and ∇ϕ(xe ) = 0) the
Hessian of ϕ(δ) evaluated at the desired equilibrium is shown to be positive semidefinite as follows:
∇2ϕ(δe ) =
Rδ 2
∂P1Ee
+ δ3e3 ∂ ∂P2 3Ee
dδ3
∂δ1
δ1
∂P3Ee
∂P1Ee
+ ∂δ1
∂δ3
∂P1Ee
∂δ3 R
∂P3Ee
∂δ3
+
+
δ1
δ1e
∂P3Ee
∂δ1
∂ 2 P1Ee
dδ1
∂ 2 δ3
!
|(δ1 ,δ3 )=(δ1e ,δ3e )
≥0
which is equivalent to (5.15).
Analogous to SG and DFIG cases, the asymptotic stability with and without
the integral action and the domain of attraction are established using the invariance
principle and voltage regulation is achieved by the control law uI (x) = u(x) + ξ in
(5.16).
5.4
Nonlinear Observer Design
In this section, a reduced-order observer is combined with the IDA-PBC state feedback control law with either relative speed (ω−ωs ) for SG or DFIG rotor slip (1−ω/ωs )
along with active and reactive currents (Id and Iq ) as the measured outputs. Unmeasured states include the generator power angle (δ) and transient voltage (E) for the
SG and the power angle (δ) and transient voltage (E) obtained from δ = tan−1 Eq /Ed
q
and E = Ed2 + Eq2 for DFIG.
87
Remark 14. In recent years, the power (rotor) angle can be monitored using Phasor
Measurement Units (PMUs) [67] suitable for only synchronous generators. PMUs
cannot be used for DFIG because the power angle δ is not a stroboscopic angle of
the shaft but rather the virtual angle of the rotor flux magnitude transformed from
transient voltage Ed and Eq .
Consider the nonlinear system (5.1) represented in the form
η̇ = f˜1 (η, y, u),
ẏ = f˜2 (η, y, u),
η ∈ <n−p
(5.24)
(5.25)
T
is the system state, y = ω − ωs , Id , Iq
is the measurable
T
part of the system state and η = δ, E
is the unmeasured part of the system
where x =
η, y
T
state. The proposed observer is obtained by including the power angle and tran-
sient voltage dynamics along with suitably chosen correction terms. A reduced order
observer is proposed in the following Proposition:
Proposition 6. Consider the nonlinear system in (5.24)-(5.25) with the measured
Pp
output y and define the new variable z := η + L i=1 (y − yie ) where L is a constant
(n − p) × 1 matrix to be designed and yie is the equilibrium state of yi . If there exist
an (n − p) × (n − p) matrix P = P T > 0, a positive constant , and the matrix L such
that
˜
T
˜
∂ f1
∂ f˜2
∂ f1
∂ f˜2
−L
P +P
−L
≤ −I
∂η
∂η
∂η
∂η
(5.26)
then
ż = f˜1 (η̂, y, u(y, η̂)) − L
η̂ = z + L
p
X
i=1
(yi − yie )
88
p
X
i=1
ŷ˙i ,
z ∈ <n−p
(5.27)
(5.28)
is a globally asymptotic observer and the observer error e(t) := η − η̂ satisfies
lim (η − η̂) = 0.
t→+∞
Proof. To begin, the error dynamics can be computed using the difference between
(5.24) and the time derivative of (5.28); we have
ė = η̇ − η̂˙
X
p
p
X
= f˜1(η, y, u) − f˜1 (η̂, y, û) − L
ẏ −
ŷ˙
i=1
i=1
where û = u(y, η̂) and ŷ˙ = f˜2 (η̂, y, û). Based on the fundamental theorem of integral
calculus for a vector-valued function of several variables [38] we can rewrite ė as
ė =
Z
0
1
∂ f˜1 (ζ(s), y)
ds − L
∂ζ
= Ae
Z
1
0
∂ f˜2 (ζ(s), y)
ds
∂ζ
e
(5.29)
where ζ = sη + (1 − s)η̂ and s ∈ [0, 1].
Introduce a Lyapunov function candidate V (e) = eT P e, with P a positive definite
symmetric matrix, then V (e) > 0 when e 6= 0 and V (0) = 0. It is easy to compute
the time derivative V̇ = ėT P e + eT P ė ≤ −kek2, leading to the inequality (5.26).
From condition (5.26), we easily obtain V̇ ≤ − λmax (P ) V with λmax (P ) denoting the
maximum eigenvalue of P , resulting in the global exponential convergence of e to zero
(e = η − η̂ → 0 as t → +∞).
Remark 15. The observer gain L is chosen to guarantee the asymptotic stability of
the observer error dynamics (e(t) → 0 as t → ∞) of (δ(t), E(t)), t ≥ 0 for every e(0).
The observer convergence relies on a positive constant that can also appropriately be
selected to achieve the closed-loop stability of the IDA-PBC control law and nonlinear
observer.
89
From the foregoing Proposition, the reduced order observers and error dynamics
of three cases are given in the following.
Case 1: SG The nonlinear observer and error dynamics are as follows:
˙
˙
ż1 = ω − ωs − L1 (ω̂˙ + Iˆd + Iˆq )
ż2 = −aÊ + b cos δ̂ +
˙
ėδ = δ̇ − δ̂,
ûf
˙
˙
− L2 (ω̂˙ + Iˆd + Iˆq )
0
T0
˙
ėE = Ė − Ê
δ̂ = z1 + L1 (ω − ωs + Id − Ide + Iq − Iqe )
Ê = z2 + L2 (ω − ωs + Id − Ide + Iq − Iqe )
where ûf = uf (y, δ̂, Ê) and L =
L1 , L2
T
(5.30)
∈ <2 .
Case 2: DFIG The nonlinear observer and error dynamics are as follows:
ωs V̂dr cos δ̂ ωs V̂qr sin δ̂
˙
˙
− L1 (ω̂˙ + Iˆd + Iˆq ) +
−
Ê
Ê
Ê
˙
˙
= −aÊ + b cos δ̂ − L2 (ω̂˙ + Iˆd + Iˆq ) + ωs V̂dr sin δ̂ − ωs V̂qr cos δ̂
ż1 = ω1 − ωs −
ż2
˙
ėδ = δ̇ − δ̂,
b sin δ̂
˙
ėE = Ė − Ê
δ̂1 = z1 + L1 (ω − Sωs + Id − Ide + Iq − Iqe )
Ê = z2 + L2 (ω − Sωs + Id − Ide + Iq − Iqe )
where V̂dr = Vdr (y, δ̂, Ê), V̂qr = Vqr (y, δ̂, Ê) and L =
(5.31)
T
L1 , L2
∈ <2 .
Case 3: DFIG The nonlinear observer and error dynamics are as follows:
ż1 = ω − ωs − L1 P
0
ż2 = −a1Ê1 + b1 cos δ̂1 + c1 Ê3 cos(δˆ1 − δˆ3) + ûf /T10
− L2 P
ż3 = ω3 − ωs −
c3 Ê1 sin(δ̂3 − δ̂1)
Ê3
−
b3 sin δ̂3
Ê3
90
+
ωs V̂dr cos δ̂3
Ê3
−
ωs V̂qr sin δ̂3
Ê3
− L3 P
ż4 = −a3Ê3 + b3 cos δ̂3 + c3 Ê1 cos(δ̂3 − δ̂1) + ωs V̂dr cos δ̂3 − ωs V̂qr sin δ̂3 − L4 P
˙
ėδ1 = δ̇1 − δ̂1,
˙
ėE1 = Ė1 − Ê1
˙
ėδ3 = δ̇3 − δ̂3,
˙
ėE3 = Ė3 − Ê3
δ̂1 = z1 + L1 Q,
Ê1 = z2 + L2 Q
δ̂3 = z3 + L3 Q,
Ê3 = z4 + L4 Q
(5.32)
˙
˙
where P = ω̂˙ 1 + ω̂˙ 3 + Iˆd + Iˆq , ûf = uf (y, δ̂1, δ̂3, Ê1 , Ê3 ), V̂dr = Vdr (y, δ̂1, δ̂3, Ê1 , Ê3),
T
V̂qr = Vqr (y, δ̂1, δ̂3, Ê1 , Ê3), L = L1 , L2 , L3 , L4
∈ <4 and Q = ω1 + ω3 − (1 +
S)ωs + Id − Ide + Iq − Iqe .
From the proposed observers in the three cases above, it is easy to compute the
nonlinear observer through zi and żi in (5.30)-(5.32) to obtain δ̂ and Ê.
5.5
Asymptotic stability of the combined IDA-PBC
controller and nonlinear observer
Contrary to linear systems, the separation principle does not hold for general nonlinear systems; for instance, the combination of a state feedback control law using state
estimates from an observer may lead to instability and finite escape time. In this
section, we show that the combined state feedback control law and observer preserve
stability of the closed-loop system along with all trajectories bounded for the SG,
DFIG, and SG-DFIG cases.
The closed-loop system for the three cases is written as a cascade interconnection
of the error dynamics (5.29) and the desired full state-feedback dynamics (5.3) in the
following form:
ẋ = (Jd (η, y) − Rd (η, y))∇xHd + ∆f(η, η̂, y)
(5.33)
ė = Ae
(5.34)
where ∆f(η, η̂, y) = F (η̂, y) − F (η, y), F (η̂, y) = f(η̂, y) + g(η̂, y)u(η̂, y) and F (η, y) =
f(η, y) + g(η, y)u(η, y).
91
The first term in (5.33) can be viewed as the closed-loop system with only the
IDA-PBC full-state control law while the second term is regarded as a perturbation
from the error dynamics resulting in a difference of full-state feedback control laws
u(η, y) and u(η̂, y).
It is noted in (5.33) the equilibrium xe of the system with ∆f(η, η̂, y) = 0 is
asymptotically stable in particular limt→∞ ∆f(η, η̂, y) = ∆f(η, η − e, y) = 0 or
limt→∞ e(t) = 0. In order to check that closed-loop stability is preserved, boundedness
of the trajectories (y, η, e) can be established. Furthermore, checking boundedness of
all trajectories requires establishing a suitable bound of ∆f(η, η − e, y).
Consider the Lyapunov function candidate: V(η, y, e) = Hd (η, y) + eT P e. The
time derivative of V is:
V̇ = −∇xHTd Rd (x)∇xHd + ∇xHTd ∆f(η, η̂, y) − kek2
It is clear that the nonlinear vector function ∆f(η, η̂, y) given by (5.33) verifies the
local Lipschitz condition in (y, η) coordinates. There exists, therefore, a positive
constant k, such that
kF (η̂, y) − F (η, y)k ≤ kkη̂ − ηk = kkek
Thus, V̇ is bounded above by
V̇(η, y, e) ≤ −∇xHTd Rd (x)∇xHd + k∇xHTd ∆f(η, η̂, y)k − kek2
≤ −∇xHTd Rd (x)∇xHd + kk∇xHTd kkek − kek2
T k∇x Hd k
k∇xHd k
R
≤ −
kek
kek
where R =
r̃ − k2
− k2 with r̃ = λmin (Rd (x)) > 0.
It is obvious that V̇(η, y, e) can be made negative semi-definite by choosing R > 0,
in particular for and r̃ sufficiently large. Therefore, according to Lyapunov stability
theory, the closed-loop stability of the power systems including STATCOM/Battery
92
for the three cases with IDA-PBC control law (5.5), (5.12), (5.16) and observers
(5.30)-(5.32) are asymptotically stable. Further, there exists a positive real k̃ such
that
V̇(η, y, e) ≤ −k̃ T 2
.
∇xHd , e
From V(η, y, e) and its time derivative, it immediately follows that all trajectories
(k∇xHd k and kek) are bounded, t ≥ 0.
Using LaSalle’s invariance principle, we have that V̇(η, y, e) = 0 ⇒ ω = ωs for
SG case (or ω = Sωs for DFIG case), E = Ee , Id = Ide , Iq = Iqe , e = 0, and this
implies δ = δe ; therefore, the closed-loop system at the desired equilibrium (xe , e) =
(δe , ωs , Ee , Ide , Iqe , 0) for SG and (xe , e) = (δe , Sωs , Ee , Ide , Iqe, 0) for DFIG cases are
asymptotically stable. For the SG-DFIG case, we follow the same manner to check the
boundedness and to show (xe , e) = (δ1e, ωs , E1e , δ3e , Sωs , E3e , Ide , Iqe , 0) as the desired
equilibrium of the closed-loop system.
Remark 16. From the IDA-PBC controller design methodology above, we follow the
idea from [55, 56] but the main differences are as follows:
1. Extension to the coordinated control of generator excitation, STATCOM and
Battery energy storage systems to further enhance the transient stability providing an increased critical clearing time and an enlarged domain of attraction,
See [35] or Chapter 4
2. Application to conventional and wind power system generation, and the combination of both,
3. Extension to the problem of voltage regulation subject to the two fault sequences
below.
In this chapter, the fault of interest is a symmetrical three-phase fault occurring on
one of the transmission lines as shown in Figures 5.1 and 5.2. There are two different
fault sequences of interest:
93
1. Temporary fault: A fault occurs at t = 0, the fault is isolated by opening
breakers of the faulted line at t = tc sec., the transmission line is recovered
free-fault at t = tr sec. Afterward the system is in a post-fault state.
2. Permanent fault: A fault occurs at t = 0, the fault is isolated by permanently
opening breakers of the faulted line at t = tc sec. The system is eventually in a
post-fault state.
Remark 17. From the fault sequences above, it is clear that the temporary fault keeps
the network structure (X2 ) of the pre-fault and post-fault state unchanged whereas
the permanent fault does not. Obviously, the STATCOM/Battery bus voltage is a
nonlinear function of δ, E, Id, Iq and the system structure, so any change in network
structure will cause a change in the STATCOM/Battery bus voltage, Vt , that may
reach either the same post-fault equilibrium, or a new post-fault equilibrium. In addition, the control design in the previous section can be considered as a coordinated
controller of the generator excitation control, DFIG rotor voltage control and STATCOM/Battery energy storage system to keep Vt close to the reference voltage Vref given
fault sequences.
Remark 18. The proposed control laws in (5.5), (5.12) and (5.16) are nonlinear
functions of δ, E, Id, Iq , the system structure and especially the post-fault equilibrium
point of Ide and Iqe . To meet the performance requirements simultaneously, the proposed controller needs to address the following two situations. For a temporary fault,
the pre-fault and post-fault network structure are identical and thus the currents (Id
and Iq ) should return to zero at the post-fault steady state. In contrast, for a permanent fault both currents need to adjust to keep Vt close to Vref so the post-fault
equilibrium for both currents is unknown.
94
5.6
Dynamic Extensions
In this section, we consider the situation without knowledge of the post-fault equilibrium point, especially Ide and Iqe , in the case of a permanent fault. In the case of
a temporary fault, it is reasonable to assume Ide and Iqe are equal to zero at steady
state.
With the help of a dynamic extension reported in [9], the unknown post-fault
equilibrium point of Ide and Iqe can be treated as unknown parameters in the control
law (5.5), (5.12) and (5.16) so we can employ a certainly equivalence adaptive control
methodology by including Id − Iˆde and Iq − Iˆqe in the control law. In addition, the
following dynamics are incorporated
˙
Iˆde = αd (Id − Iˆde )
˙
Iˆqe = αq (Iq − Iˆqe )
with αd > 0, αq > 0 are free parameters selected by designer. For the proof of
asymptotic stability and selecting a Lyapunov function candidate, refer to [9] for
more details.
5.7
Simulation Results
In this section, simulation results are given for the following: (1) SMIB, and (2) the
two-machine infinite bus system mentioned in previously.
Two operating scenarios (for SG, DFIG, and SG-DFIG) are tested on the single
line diagram of SMIB and two-machine infinite bus systems, as shown in Figures 5.1
and 5.2 respectively.
5.7.1
Single Machine Infinite Bus
Consider the single line diagram as shown in Figure 5.1 with either SG or DFIG
connected through parallel transmission line to an infinite-bus. The generators deliver
95
1.0 pu. power for the SG and 1.03 pu. power for the DFIG, respectively, with terminal
voltages 0.9897 pu. for the SG and 0.9877 pu. for DFIG, respectively, and the infinitebus voltage is 1.0 pu. A three-phase fault occurs at P , the midpoint of one of the
transmission lines, this leads to rotor acceleration and voltage sag. To this end, the
output feedback controller from generators (SG or DFIG) and STATCOM/Battery
is designed to enhance the voltage regulation and system stability.
5.7.2
Two-Machine Infinite Bus
Consider the single line diagram of two generators (SG-DFIG) connected to an infinite
bus, including a STATCOM/Battery shown in Figure 5.2 consisting of a conventional
SG and a DFIG representing a wind power system. Both are connected through a
transmission line to a infinite bus. The SG and DFIG generators deliver 1.0 and 1.03.
pu. power, respectively, while the terminal voltage is 0.9780 pu and the infinite-bus
voltage is 1.0 pu. A three-phase fault occurring at P causes a power imbalance and
voltage sag. Identical to SMIB scenario, the incorporation of two generators (SG
and DFIG) and STATCOM/Battery is used to accomplish the expected performance
requirements in Section 5.2.
We consider two operating scenarios with three cases (SG, DFIG, and SG-DFIG)
and investigate the effectiveness of the coordinated controllers we propose (generator excitation controller (GEC), DFIG rotor voltage controller (DRVC) and STATCOM/Battery (STAT/BATT) controllers) along with a comparison with conventional
controllers (PSS/AVR [41] and PIC [18]) for conventional and alternative (wind)
power systems, respectively, through the two fault sequences mentioned previously:
Temporary fault:
• Case 1: SG Figures 5.3-5.4 show time histories of rotor speed (ω − ωs ), STATCOM/Battery bus voltage (Vt ), and active and reactive currents (Id and Iq )
along with estimates and actual power angle (δ̂ and δ) and transient voltage (Ê
and E).
96
• Case 2: DFIG Figures 5.5-5.6 shows time histories of DFIG rotor slip (1 −
ω/ωs ), STATCOM/Battery bus voltage (Vt ), and active and reactive currents
(Id and Iq ) along with of the estimated and actual power angle (δ̂ and δ) and
transient voltage (Ê and E).
• Case 3: SG-DFIG Figures 5.7-5.9 show time histories of rotor speed (ω1 −ωs ),
DFIG rotor slip (1 − ω3 /ωs ), STATCOM/Battery bus voltage (Vt ), active and
reactive currents (Id and Iq ), estimated and actual responses of power angle (δ̂i
and δi, i = 1, 3) and transient voltage (Êi and Ei , i = 1, 3), respectively.
The parameters for the IDA-PBC control law and the observer gain are set as
r1 γ1 = a, γ2 = γ3 = 10, r2 = r2 = 1, and Li = 1, i = 1, 2 for the SG case,
r1 γ1 = a, γ2 = γ3 = 10, r2 = r3 = 1, and Li = 1, i = 1, 2 for the DFIG case,
r1 γ1 = a1 and r2 γ2 = a3, γ3 = γ4 = 0.01, r3 = r4 = 20, and Li = 1, i = 1, 2, 3, 4
for the SG-DFIG case. In these cases, KI = αd = αq = 0 when the pre-fault and
post-fault network structures are the same. Initial conditions for the three cases are
given in Appendix B.2.
For this fault sequence, the fault is cleared by opening the breaker at tc = 0.2 sec.
and then fault free operation is restored at tr = 2 sec. for the SG, DFIG, and
SG-DFIG cases. From the simulation results given in Figures 5.3, 5.5, 5.7, and
5.8, it can be seen that the proposed controller without the energy storage systems (GEC, DRVC, GEC/DRVC) can perform well but when the energy storage
systems (STATCOM/Battery) are incorporated, rotor speed (ω1 − ωs → 0), DFIG
rotor slip 1 − ω3 /ωs → s0 = −0.03) and STATCOM/Battery bus voltage (Vt ) are
more quickly stabilized and eventually return to the pre-fault state with shorter settling time and smaller overshoot. However, the time histories from the conventional
controllers (PSS/AVR, DRVC, PSS/AVR/DRVC) provide worse transient response
performance than those of the proposed controller with and without the energy storage systems such as higher overshoot and slow reduction of oscillations (e.g. DFIG
rotor slip).
97
For the voltage regulation problem, it is clear that in each case of the proposed
controllers, Vt returns to the reference (pre-fault) voltage value Vref (0.9897 pu. for SG,
0.9877 pu. for DFIG, and 0.9780 pu. for SG-DFIG, respectively). Further, transient
responses from GEC/STAT/BATT, DRVC/STAT/BATT, and GEC/DRVC/STAT/
BATT provide reduced overshoot, shorter settling time, and reduced oscillations when
compared to other proposed and conventional controllers. Although time histories of
SG and DFIG rotor speed (frequency) from conventional controllers eventually return
to the pre-fault values following a fault, achieving the expected requirements, their
terminal voltage responses at steady state cannot go to the desired reference voltages
Vref and there is a large voltage sag before the fault can be cleared, which may be
practically undesirable, particularly in DFIG and SG-DFIG cases.
For active and reactive currents (Id and Iq ), both currents expectedly tend to
zero after the fault is cleared and the system is in the post-fault state. In addition,
the integral action may not be necessary to regulate each case of terminal voltages
because the network structure for the pre-fault and post-fault states are the same.
For estimated and actual states of power angle δ and transient voltage E as shown in
Figures 5.4, 5.6, and 5.9, it can be seen that the estimated states δ̂ and Ê converge
asymptotically to the actual states δ and E, respectively, after the fault has been
cleared.
Permanent fault:
• Case 1: SG Figures 5.10-5.12 show time histories of rotor speed (ω − ωs ),
STATCOM/Battery bus voltage (Vt ), and active and reactive currents (Id and
Iq ) along with estimated and actual power angle (δ̂ and δ) and transient voltage
(Ê and E).
• Case 2: DFIG Figures 5.13-5.15 show time histories of DFIG rotor slip (1 −
ω/ωs ), STATCOM/Battery bus voltage (Vt ), and active and reactive currents
(Id and Iq ) along with estimated and actual power angle (δ̂ and δ) and transient
voltage (Ê and E).
98
• Case 3: SG-DFIG Figures 5.16-5.18 show time histories of rotor speed (ω −
ωs ), DFIG rotor slip (1 − ω/ωs ), STATCOM/Battery bus voltage (Vt ), active
and reactive currents (Id and Iq ), estimated and actual responses of power angle
(δ̂i and δi, i = 1, 3) and transient voltage (Êi and Ei , i = 1, 3), respectively.
The parameters for the IDA-PBC control law and the observer gain are identical
to those of the temporary fault but KI = 5 for SG and DFIG cases and KI = 25
for SG-DFIG case along with αd = αq = 1 for SG, DFIG, SG-DFIG cases. However,
selecting different KI , αd and αq makes the time responses of Vt , Iˆde , and Iˆde , approach
the desired values at different convergence rates so these values should be suitably
selected.
In this fault sequence, the fault is cleared at t = tc identical to the temporary
fault. From the simulation results, it is obvious from Figures 5.10, 5.13, and 5.16
that even though the time histories with the proposed controller without the energy
storage systems (GEC, DRVC, GEC/DRVC) can stabilize the system, time histories
from the proposed controller including the energy storage systems (STATCOM and
STATCOM/Battery) can further improve the transient response performance while
satisfying the expected performance requirements: rotor speed (ω − ωs → 0), DFIG
rotor slip (1 − ω/ωs → s0 = −0.03) and STATCOM/Battery bus voltage Vt are stabi-
lized and eventually converge to the pre-fault state identical to the temporary fault.
Contrary to the proposed controllers, time histories of the conventional controllers
(PSS/AVR, PIC, and PSS/AVR/PIC), show higher overshoot and poor damping of
oscillations.
It is also clear that voltage regulation (Vt → Vref ) is achieved and Vt returns to
the reference (pre-fault) voltage Vref when the proposed controllers are used with
integral action and with without the energy storage in all three cases. Analogous
to the temporary fault case, without the incorporation of the energy storage, time
histories of the conventional controllers cannot drive voltage deviations to zero at
steady state. Therefore, the integration of the energy storage (STATCOM/Battery)
99
and IDA-PBC design are obviously able to provide additional benefits beyond the
conventional PSS/AVR and PIC controllers.
The new equilibrium are asymptotically stable, in particular power angle δe , active
and reactive currents (Ide and Iqe ), even with the change in network structure where
Ide and Iqe are unknown. Fortunately, the dynamic extension can be effectively used
to estimate the two new unknown equilibrium currents, (Iˆde → Ide and Iˆqe → Iqe ) at
the post-fault states, as shown in Figures 5.11, 5.14, and 5.17, respectively. Similar to
the temporary fault case, it is easy to see that the estimated states δ̂ and Ê converge
asymptotically to the actual states δ and E as shown in Figures 5.12, 5.15, and
5.18. Observe also that the transient voltages (E → Ee and Ê → Ee ) return to the
same pre-fault states while power angles (δ → δe and δ̂ → δe ) converge to the new
equilibrium states due to the permanent disconnection of the transmission line.
Remark 19. In order to simultaneously achieve rotor speed and terminal voltage
control, the PSS/AVR parameters and PI gains of the conventional and alternative
(wind) power system controllers (PSS/AVR, PIC, and PSS/AVR/PIC) are adjusted
using a trial and error tuning procedure via simulation. The parameters and gains
obtained from tuning are shown in Appendix A.6-A.8. Also, to further increase the
transient stability margin and achieve rotor speed and terminal voltage regulation,
optimization methods are used to tune the parameters and gains to minimize power
system oscillations and regulate the dynamic performance following a large disturbance.
5.8
Conclusions
In this Chapter, we have presented a nonlinear output feedback controller that combines an IDA-PBC state feedback control law and a nonlinear reduced order observer
for SG and DFIG generators, representing conventional and wind power systems.
The power systems incorporate STATCOM and Battery energy storage and after the
occurrence of three-phase short circuit transmission line faults (both temporary and
100
permanent), through coordinated control, achieve asymptotic stability at a desired
equilibrium point and achieve voltage regulation.
Simulation results have demonstrated that the proposed controller can achieve
asymptotic stability of the two operating scenarios (SMIB and Two-machine Infinite
Bus). Further, voltage regulation and system stability are simultaneously achieved
for large disturbances (temporary and permanent faults). Due to the absence of the
knowledge of the post-fault equilibrium states in the permanent fault case, dynamic
extensions are used to estimate the unknown post-fault active and reactive currents.
Moreover, the proposed observers for the three cases can effectively estimate unmeasured states in short time duration and the combined IDA-PBC control law and
nonlinear observer can preserve the asymptotic stability of the closed-loop systems
along with all trajectories bounded.
101
ω − ωs (rad/s)
2
GEC/STAT/BATT
GEC/STAT
GEC
PSS/AVR
1
0
−1
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
Vt (pu.)
1.2
1
0.8
0.6
Id&Iq (pu.)
0.4
1
Id: GEC/STAT/BATT
Iq: GEC/STAT/BATT
Id: GEC/STAT
Iq: GEC/STAT
0.5
0
−0.5
0
1
2
3
Time
(sec.)4
5
6
7
Figure 5.3: Temporary fault in SG case: Time histories of rotor speed (frequency)
(ω − ωs ), terminal voltage (Vt ) and active and reactive currents of GEC/STAT and
GEC/STAT/BATT
1
δ̂
δ & δ̂
0.8
0.6
0.4
δ
0.2
0
0.5
1
1.5
1
1.5
2
2.5
3
3.5
4
2.5
3
3.5
4
1.4
Ê
E & Ê
1.2
1
E
0.8
0.6
0.4
0
0.5
Time 2(sec.)
Figure 5.4: Temporary fault in SG case: Time histories of the estimated and
actual states of power angles (δ̂ and δ) and transient voltages (Ê and E) of
GEC/STAT/BATT
102
1 − ω/ωs
−0.02
DRVC
DRVC/STAT
DRVC/STAT/BATT
PIC
−0.03
Id&Iq (pu.)
Vt (pu.)
−0.04
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
1.2
1
0.8
0.6
0.4
0.6
Id: DRVC/STAT/BATT
Iq: DRVC/STAT/BATT
Id: DRVC/STAT
Iq: DRVC/STAT
0.4
0.2
0
−0.2
0
1
2
3
Time
(sec.)4
5
6
7
Figure 5.5: Temporary fault in DFIG case: Time histories of DFIG rotor slip
(frequency) (1 − ω/ωs ), terminal voltage (Vt ) and active and reactive currents of
DRVC/STAT and DRVC/STAT/BATT
1.2
δ & δ̂
1
δ̂
0.8
0.6
δ
0.4
0.2
0
0.5
1
1.5
2
1.5
2
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
1.8
1.6
Ê
E & Ê
1.4
1.2
1
E
0.8
0.6
0.4
0
0.5
1
Time 2.5(sec.)
Figure 5.6: Temporary fault in DFIG case: Time histories of the estimated and
actual states of power angles (δ̂ and δ) and transient voltages (Ê and E) of
DRVC/STAT/BATT
103
ω1 − ωs (rad/s)
2
GEC/DRVC/STAT/BATT
GEC/DRVC/STAT
GEC/DRVC
PSS/AVR/PIC
0
−2
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
2
4
5
6
1 − ω3 /ωs
0
−0.02
−0.03
Vt (pu.)
−0.04
1.5
1
0.5
3
Time (sec.)
Iq (pu.)
Id (pu.)
Iq (pu.)
Id (pu.)
Figure 5.7: Temporary fault in SG-DFIG case: Time histories of rotor speed (frequency) (ω − ωs ), DFIG rotor slip (frequency) (1 − ω/ωs ) and terminal voltage (Vt )
2
GEC/DRVC/STAT/BATT
1
0
−1
0
1
2
3
4
5
6
7
8
9
10
2
GEC/DRVC/STAT/BATT
1
0
−1
0
1
2
3
4
5
6
7
8
9
10
10
GEC/DRVC/STAT
5
0
−5
0
1
2
3
4
5
6
7
8
2
1
0
−1
9
10
GEC/DRVC/STAT
0
1
2
3
4
5
Time (sec.)
6
7
8
9
10
Figure 5.8: Temporary fault in SG-DFIG case: Time histories of active and reactive
currents of GEC/DRVC/STAT and GEC/DRVC/STAT/BATT
104
E1 & Ê1
δ1 & δ̂1
1
0.8
0.6
δ̂1
δ1
0.4
0
0.5
1
1.5
2
2
2.5
3
Ê1
1.5
E1
1
0.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
E3 & Ê3
δ3 & δ̂3
1.5
1
0.5
0
δ3
0
δ̂3
0.5
1
1.5
2
2.5
3
2
Ê3
E3
1.5
1
0.5
0
0.2
0.4
0.6
Time 1(sec.) 1.2
0.8
1.4
1.6
1.8
2
ω − ωs (rad/s)
Figure 5.9: Temporary fault in SG-DFIG case: Time histories of the estimated and
actual states of power angles (δ̂i and δi , i = 1, 3) and transient voltages (Êi and
Ei , i = 1, 3) of GEC/DRVC/STAT/BATT
GEC
GEC/STAT
GEC/STAT/BATT
PSS/AVR
2
1
0
−1
0
1
2
0
1
2
3
4
5
6
4
5
6
Vt (pu.)
1.2
1
0.8
0.6
0.4
3
Time (sec.)
Figure 5.10: Permanent fault in SG case: Time histories of rotor speed (frequency)
(ω − ωs ) and terminal voltage (Vt )
105
Id (pu.)
Iq (pu.)
Id: GEC/STAT/BATT
Ide: GEC/STAT/BATT
0.2
0.1
0
0
5
Id (pu.)
15
Iq: GEC/STAT/BATT
Iqe: GEC/STAT/BATT
0.4
0.2
0
0
Iq (pu.)
10
5
10
15
0.2
Id: GEC/STAT
Ide: GEC/STAT
0.1
0
0
5
10
15
Iq: GEC/STAT
Iqe: GEC/STAT
0.4
0.2
0
0
5
Time (sec.)
10
15
Figure 5.11: Permanent fault in SG case: Time histories of active and reactive currents
of GEC/STAT and GEC/STAT/BATT
1
δ̂
δ & δ̂
0.8
0.6
0.4
0.2
δ
0
0.5
1
1.5
2
2.5
3
2
2.5
3
1.4
Ê
E & Ê
1.2
1
E
0.8
0.6
0.4
0
0.5
1
Time 1.5(sec.)
Figure 5.12: Permanent fault in SG case: Time histories of the estimated and
actual states of power angles (δ̂ and δ) and transient voltages (Ê and E) of
GEC/STAT/BATT
106
−0.02
1 − ω/ωs
−0.025
−0.03
DRVC
DRVC/STAT
DRVC/STAT/BATT
PIC
−0.035
−0.04
0
5
0
5
10
15
10
15
1.4
Vt (pu.)
1.2
1
0.8
0.6
0.4
Time (sec.)
Figure 5.13: Permanent fault in DFIG case: Time histories of DFIF rotor slip (frequency) (1 − ω/ωs ) and terminal voltage (Vt )
Id: DRVC/STAT/BATT
Ide: DRVC/STAT/BATT
0.2
0
0
5
10
15
Iq: DRVC/STAT/BATT
Iqe: DRVC/STAT/BATT
0.5
0
Id (pu.)
−0.5
−0.1
Iq (pu.)
Iq (pu.)
Id (pu.)
0.4
0.8
0.6
0.4
0.2
0
−0.2
0
2
4
6
8
10
12
14
16
18
20
Id: DRVC/STAT
Ide: DRVC/STAT
0.2
0.1
0
0
2
4
6
8
10
12
14
16
18
20
Iq: DRVC/STAT
Iqe: DRVC/STAT
0
2
4
6
8
Time 10
(sec.)
12
14
16
18
20
Figure 5.14: Permanent fault in DFIG case: Time histories of active and reactive
currents of DRVC/STAT and DRVC/STAT/BATT
107
1.2
δ & δ̂
1
δ̂
0.8
δ
0.6
0.4
0.2
0
0.5
1
1.5
2
1
1.5
2
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
1.8
Ê
E
1.6
E & Ê
1.4
1.2
1
0.8
0.6
0.4
0
0.5
Time 2.5(sec.)
ω1 − ωs (rad/s)
Figure 5.15: Permanent fault in DFIG case: Time histories of the estimated and
actual states of power angles (δ̂ and δ) and transient voltages (Ê and E) of
DRVC/STAT/BATT
2
GEC/DRVC/STAT/BATT
GEC/DRVC/STAT
GEC/DRVC
PSS/AVR/PIC
0
−2
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
0
1
2
3
Time
(sec.)4
5
6
7
1 − ω3 /ωs
0
−0.02
−0.03
Vt (pu.)
−0.04
1.5
1
0.5
Figure 5.16: Permanent fault in SG-DFIG case: Time histories of rotor speed (frequency) (ω − ωs ), DFIF rotor slip (frequency) (1 − ω/ωs ) and terminal voltage (Vt )
108
Iq (pu.)
Id (pu.)
2
1
Id: GEC/DRVC/STAT/BATT
Ide: GEC/DRVC/STAT/BATT
0
0
1.5
1
0.5
0
−0.5
1
2
3
4
5
6
7
8
9
10
Iq: GEC/DRVC/STAT/BATT
Iqe: GEC/DRVC/STAT/BATT
0
1
2
3
4
5
6
7
8
9
10
Iq (pu.)
Id (pu.)
10
Id: GEC/DRVC/STAT
Ide: GEC/DRVC/STAT
5
0
0
1
2
3
4
5
6
7
8
9
10
2
1
Iq: GEC/DRVC/STAT
Iqe: GEC/DRVC/STAT
0
−1
0
1
2
3
4
5
Time (sec.)
6
7
8
9
10
Figure 5.17: Permanent fault in SG-DFIG case: Time histories of active and reactive
currents of GEC/DRVC/STAT and GEC/DRVC/STATCOM/BATT
δ1 & δ̂1
1
0.8
δ̂1
0.6
δ1
0.4
0
0.5
1
1.5
2
2.5
3
3.5
4
δ3 & δ̂3
E1 &Ê1
3
E1
2
Ê1
1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
1
1.5
2
2.5
3
3.5
4
2.5
3
3.5
4
1
0.5
0
δ3
0
δ̂3
0.5
E3 &Ê3
2
E3
1.5
Ê3
1
0.5
0
0.5
1
1.5
Time 2(sec.)
Figure 5.18: Permanent fault in SG-DFIG case: Time histories of the estimated and
actual states of power angles (δ̂i and δi , i = 1, 3) and transient voltages (Êi and
Ei , i = 1, 3) of GEC/DRVC/STAT/BATT
109
Chapter 6
Conclusion
6.1
Summary
In this dissertation, we have investigated the application of energy storage to enhance
power system stability and maintain power quality for both small- and large-signal
disturbances.
Small-signal stability enhancement has been extensively studied in the literature
to improve power system operations for a nonlinear power system operating in the
region of a steady-state operating point. However, there is a relatively little prior
work devoted to the integration of STATCOM and battery energy storage systems
as related to small-signal stability enhancement. In Chapter 3, we examined how
a STATCOM and battery can be used to enhance the small-signal (local) stability
of a wind energy conversion system interconnected to the grid via an LMI-based
controller design methodology with D-stability to achieve both frequency and voltage
regulation as well as improve the transient responses in terms of overshoot, damping
ratio, settling time, etc. Simulation results are used to demonstrate that voltage and
frequency regulation are achievable for small disturbances.
Transient stability enhancement is one of the most important problems investigated in the literature of power and control system engineering.
In Chapter 4, we address the transient stability problem in conventional power
110
systems with renewable energy sources (generation and storage) by using an IDAPBC state feedback control design approach. The objectives of controller design are
to increase the critical clearing time (CCT), to enlarge the domain of attraction, and
to regulate power angle, terminal voltage and frequency (relative rotor speed of SG
and rotor slip of DFIG). Using the IDA-PBC nonlinear model-based control design
methodology, the simulation results have shown that power angle stability, along with
voltage and frequency regulation can be achieved when large (transient) disturbances
occur in the system.
In Chapter 5, the IDA-PBC control methodology and nonlinear observer design
are combined to develop a dynamic output feedback controller that can achieve system stability and voltage regulation enhancement after the occurrence of large temporary and permanent disturbances. Besides, the output feedback controller can
preserve asymptotic stability of the closed-loop systems along with all trajectories
being bounded. Simulation results have illustrated that the proposed controller can
achieve asymptotic stability, voltage regulation, and system stability in the two operating scenarios, even without the knowledge of the post-fault equilibrium states in
the permanent fault case. The unmeasured states (power angles and transient voltages) can be quickly estimated via the proposed observers. In addition, the combined
IDA-PBC control law and nonlinear observers are validated, leading to the capability to preserve the asymptotic stability of the closed-loop systems and maintain all
trajectories bounded.
6.2
Future Development
The following list contains feasible development for future research:
• Extensions to multi-machine power systems (at least three machines) with conventional and wind power generation and energy storage systems.
• Extensions to structure preserving power system models represented by a set of
Differential-Algebraic Equations (DAEs).
111
• Investigation in the impact of dynamic loads on the system and the effectiveness
of integrating energy storage to improve both dynamic and transient stability
of power systems with a diverse mix of generation.
• Incorporation of actuator saturation (excitation, DFIG rotor voltages, and
STATCOM/Battery) in the controller design.
• Extensions to robust and adaptive control design in the presence of uncertain
or unknown parameters.
112
Appendix A
Appendix of Chapter 3
A.1
A.1.1
Fi (x) of the voltage dynamics
Fi(x) of the voltage dynamics in (3.25)
2 2
E1 V∞ X1 X2 sin δ
Id X1 X2 + Iq X1 X2 H∆ − G∆2
·
−G ,
F1 (x) =
(X1 + X1 )∆
∆2
(Id X1 X2 )2 Iq X1 X2 H
X2 L
F2 (x) =
G−
+
,
2
∆G
G∆
∆(X1 + X2 )
F3 (x) =
Id (X1 X2 )2
,
(X1 + X2 )∆G
F4 (x) =
X1 X2 H
.
(X1 + X2 )G
where G =
s
1+
Iq X1 X2
∆
2
+
Id X1 X2
∆
2
, L = E1 X2 + V∞ X1 cos δ, and H = 1 +
Iq X1 X2
.
∆
113
A.1.2
Fi(x) of the voltage dynamics in (3.31)
where G0 =
p
F1(x) =
G0 X2 X3 E1X1 X2 X3 E1 X1Z
−
,
ΛΠ
ΛG0 Π2
F2(x) =
G0 X2 X3 Y1 X2 X3 Y1 Z
−
,
ΛΠ
ΛG0 Π2
F3(x) =
G0 X1 X2 E3X3 X1 X2 E3 X3Z
−
,
ΛΠ
ΛG0 Π2
F4(x) =
G0 X1 X2 Y3 X1 X2 Y3
− 2 0 ,
ΛΠ
Π ΛG
F5(x) =
Id (X1 X2 X3 )2
,
ΛG0 Π
F6(x) =
X1 X2 X3 H 0
ΛG0
(1 + M)2 + N 2 , H 0 = (1 + M), A = V∞ X1 X2 + E1 X2 X3 cos δ1 +
E3 X1 X2 cos δ3 and B = E1 X2 X3 sin δ1 + E3 X1 X2 sin δ3. Xi = (B cos δi − A sin δi),
Yi = (B sin δi + A cos δi ), i = 1, 3 and Z =
(Id X1 X2 X3 )2
Π
114
+ Iq X1 X2 X3 H 0 .
A.2
Aij and Buij in (3.27)
Let us define the following:
f1S (x) = ω − ωs
EV∞ sin δ
Iq X1 X2
EX2 Id
V∞ X1 cos δ + EX2
ωs
f2S (x) =
Pm −
1+
−
·
2H
(X1 + X2 )
∆
(X1 + X2 )
∆
f3S (x) = −aE + b cos δ +
f4S (x) = −
uf
T00
ωs Vt cos β
Rs ωs
Id + ωIq −
+ (h cos β) · uM − (h sin β) · uγ
Ls
Ls
f5S (x) = −ωId −
Rs ωs
ωs Vt sin β
Iq −
+ (h sin β) · uM + (h cos β) · uγ
Ls
Ls
f6S (x) = F1(x)δ̇ + F2 (x)Ė + F3(x)I˙d + F4(x)I˙q
f7S (x) = ∆Vt
where x =
x1 x2 x3 x4 x5 x6 x7
T
=
δ ω E Id Iq ∆Vt ∆ξ
T
.
Using diff function in Matlab Symbolic Math Toolbox to figure out the
entries of matrices AS and BSu of SG case, we have:
The entries of matrix AS are as follows:
Aij =
∂fiS
= diff(fiS , xj ),
∂xj
i ∈ {1, ..., 6}, j ∈ {1, ..., 6}
The entries of matrix BSu are as follows:
Bu42 =
∂f4S
∂f4S
= diff(f4S , uM ), Bu43 =
= diff(f4S , uγ ),
∂uM
∂uγ
Bu52 =
∂f5S
∂f5S
= diff(f5S , uM ), Bu53 =
= diff(f5S , uγ ),
∂uM
∂uγ
Bu61 =
∂f6S
= diff(f6S , uf ),
∂uf
Bu62 =
∂f6S
∂f6S
= diff(f6S , uM ), Bu63 =
= diff(f6S , uγ )
∂uM
∂uγ
115
A.3
Aij and Buij in (3.30)
Let us define the following:
b sin δ ωs Vr cos(δ − θr )
+
E
E
EV∞ sin δ
Iq X1 X2
EX2 Id
V∞ X1 cos δ + EX2
ωs
f2D (x) =
Pm −
1+
−
·
2H
(X1 + X2 )
∆
(X1 + X2 )
∆
f1D (x) = ω − ωs −
f3D (x) = −aE + b cos δ + ωs Vr sin(δ − θr )
f4D (x) = −
ωs Vt cos β
Rs ωs
Id + ωIq −
+ (h cos β) · uM − (h sin β) · uγ
Ls
Ls
f5D (x) = −ωId −
Rs ωs
ωs Vt sin β
Iq −
+ (h sin β) · uM + (h cos β) · uγ
Ls
Ls
f6D (x) = F1 (x)δ̇ + F2(x)Ė + F3 (x)I˙d + F4 (x)I˙q
f7D (x) = ∆Vt
f8D (x) = ω − (1 − s0 )ωs ,
where x =
x1 x2 x3 x4 x5 x6 x7 x8
T
=
δ ω E Id Iq ∆Vt ∆ξ ∆ξω
Using diff function in Matlab Symbolic Math Toolbox to figure out the
entries of matrices AD and BDu of DFIG case, we have:
The entries of matrix AD are straightforwardly computed analogous to SG case
as follows:
Aij =
∂fiD
= diff(fiD , xj ),
∂xj
i ∈ {1, ..., 6}, j ∈ {1, ..., 6}
The entries of matrix BDu are the following:
Bu11 =
∂f1D
∂f1D
= diff(f1D , Vdr ), Bu12 =
= diff(f1D , Vqr ),
∂Vdr
∂Vqr
Bu31 =
∂f3D
∂f3D
= diff(f3D , Vdr ), Bu32 =
= diff(f3D , Vqr ),
∂Vdr
∂Vqr
Bu43 =
∂f4D
∂f4D
= diff(f4D , uM ), Bu44 =
= diff(f4D , uγ ),
∂uM
∂uγ
116
T
.
A.4
Bu53 =
∂f5D
∂f5D
= diff(f5D , uM ), Bu54 =
= diff(f5D , uγ ),
∂uM
∂uγ
Bu61 =
∂f6D
∂f6D
= diff(f6D , Vdr ), Bu62 =
= diff(f6D , Vqr ),
∂Vdr
∂Vqr
Bu63 =
∂f6D
∂f6D
= diff(f6D , uM ), Bu64 =
= diff(f6D , uγ )
∂uM
∂uγ
Aij and Buij in (3.33)
Let us define the following:
f1SD (x) = ω1 − ωs
f2SD (x) =
ωs
(Pm1 − P1E )
2H1
f3SD (x) = −a1E1 + a2 cos δ1 + a3E3 cos(δ1 − δ3) +
f4SD (x) = ω3 − ωs −
f5SD (x) =
uf
,
0
T10
b1 E1 sin(δ3 − δ1) + b2 sin δ3 ωs Vr cos(δ − θr )
+
E3
E3
ωs
(Pm3 − P3E )
2H3
f6SD (x) = −c1E3 + c2 cos δ3 + c3 E1 cos(δ3 − δ1 ) + ωs Vr sin(δ − θr )
f7SD (x) = −
Rs ωs
ωs Vt cos β
Id + ωIq −
+ (h cos β) · uM − (h sin β) · uγ
Ls
Ls
f8SD (x) = −ωId −
Rs ωs
ωs Vt sin β
Iq −
+ (h sin β) · uM + (h cos β) · uγ
Ls
Ls
f9SD (x) = F1(x)δ̇1 + F2 (x)Ė1 + F3 (x)δ̇3 + F4(x)Ė3 + F5(x)I˙d + F6(x)I˙q
f10SD (x) = ∆Vt
f11SD (x) = ω3 − (1 − s30)ωs ,
where
x =
=
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
T
δ1 ω1 E1 δ3 ω3 E Id Iq ∆Vt ∆ξ ∆ξω3
117
T
.
Using diff function in Matlab Symbolic Math Toolbox to figure out the
entries of matrices ASD and BSDu of SG-DFIG case, we have:
The entries of matrix ASD are straightforwardly computed identical to SG and
DFIG cases as follows:
Aij =
∂fiSD
= diff(fiSD , xj ),
∂xj
i ∈ {1, ..., 9}, j ∈ {1, ..., 9}
The entries of matrix BSDu are the following:
Bu42 =
∂f4SD
∂f4SD
= diff(f4SD , Vdr ), Bu43 =
= diff(f4SD , Vqr ),
∂Vdr
∂Vqr
Bu62 =
∂f6SD
∂f6SD
= diff(f6SD , Vdr ), Bu63 =
= diff(f6SD , Vqr ),
∂Vdr
∂Vqr
Bu74 =
∂f7SD
∂f7SD
= diff(f7SD , uM ), Bu75 =
= diff(f7SD , uγ ),
∂uM
∂uγ
Bu84 =
∂f8SD
∂f8SD
= diff(f8SD , uM ), Bu85 =
= diff(f8SD , uγ ),
∂uM
∂uγ
Bu91 =
∂f9SD
= diff(f9SD , uf )
∂uf
Bu92 =
∂f9SD
∂f9SD
= diff(f9SD , Vdr ), Bu93 =
= diff(f9SD , Vqr ),
∂Vdr
∂Vqr
Bu94 =
∂f9SD
∂f9SD
= diff(f9SD , uM ), Bu95 =
= diff(f9SD , uγ )
∂uM
∂uγ
118
A.5
Parameters of the SMIB and TMIB
δ0(rad.)
0.4964
V∞ (pu.)
1.0
δ0(rad.)
0.5196
T00 (pu.)
0.4
Case 1: SG Case
ωs (rad/s) E10 (pu.) Pm (pu)
120π
1.05
1.0
X1 (pu)
X2 (pu.)
H
0.3
0.2
5
ω0 (rad/s)
ωs
T00
4
Case 2: DFIG Case
ωs (rad/s) E10 (pu.) Pm (pu) Vdr (pu.)
120π
1.05
1.03
-6.942
0
Xs (pu)
Xtr (pu.) Xs (pu.)
ω0
1.1
0.1
0.2
1.03ωs
Vqr (pu)
-7.62
H
4
STATCOM/Battery Parameters
Vdc (pu.) Rs (pu) Ls (pu.) Rt (Ω) Ct (µF)
1
0.001
0.1
1000
1000
δ10(rad)
0.7181
δ30 (rad)
0.7265
X3 (pu.)
0.3
Y33 (pu.)
2.38
ωs (rad/s)
120π
ω30 (rad/s)
1.03ωs
Y12 (pu.)
1.43
Xs3 (pu.)
1.1
Case: SG-DFIG Case
E10 (pu.) E30 (pu) Pm1 (pu.) Pm3 (pu.)
1.074
1.073
1
1.03
Vdr (pu.) Vqr (pu.)
H1
H3
-5.73
-5.87
5
4
Y13 (pu.) Y32 (pu.)
0.95
1.43
0
Xs3 (pu.)
T00
0.2
0.4
119
A.6
PSS/AVR parameters for SG case (PSS/AVR)
SG case: PSS/AVR parameters
KP SS KA Te T1
T2
50
4.5 100 0.1 0.025
A.7
PI gains for DFIG case (PIC)
KP 1
0.2
KI1
0.2
A.8
DFIG case: PI gains
KP 2 KP 3 KP 4
Kv
1
0.1
0.1
1
KI2 KI3 KI4
Tv
0.15 1.2
1.2 0.001
Kt
1
Tt
0.001
PSS/AVR paramenters and PI gains for SGDFIG case (PSS/AVR/PIC)
SG-DFIG case: PSS/AVR parameters and PI gains
KP 1 KP 2 KP 3 KP 4
Kv
Kt
KP SS
KA
Te
0.2
1
10
10
1
1
50
4.5 100
KI1 KI2 KI3 KI4
Tv
Tt
T1
T2
10.2 1.15 1.2
1.2 0.001 0.001
0.1
0.025
120
Appendix B
Appendix of Chapter 5
B.1
Computing ϕ of SG-DFIG cases
According to (5.23), integrating with respect to δ1 yields
ϕ = −Pm1 (δ1 − δ10) +
+
Z
Z
δ1
P1Ee (δ1, δ3 )dδ1
δ10
δ3
δ30
P3Ee dδ3 + G(δ3)
(B.1)
Differentiating the above with respect to δ3 and recalling Leibniz integral rule, we
have
∇δ3 ϕ =
Z
δ1
δ10
∂P1Ee(δ1 , δ3)
∂G(δ3)
dδ1 + P3Ee +
∂δ3
∂δ3
Thus, we have
G(δ3) = −Pm3 (δ3 − δ30).
Substituting G(δ3) into (B.1), we get ϕ(δ) given in (5.20).
121
(B.2)
B.2
Initial conditions of SG, DFIG, and SG-DFIG
cases
Table B.1: Initial conditions of SG and DFIG cases
Synchronous Generator
δ0(rad.) ω0 (rad/s) E0 (pu.)
0.4964
ωs
1.05
δ0(rad.)
0.5196
DFIG
ω0 (rad/s)
1.03ωs
δ̂0(rad.)
0.2482
δ̂0(rad.)
0.2598
Ê0 (pu.)
0.525
Ê0 (pu.)
0.525
E0 (pu.)
1.05
Table B.2: Initial conditions of SG-DFIG case
δ10(rad.)
0.718
δ30 (rad.)
0.726
SG: Gen#1
ω10 (rad/s) E10 (pu.) δ̂10 (rad.)
ωs
1.074
0.359
DFIG: Gen#3
ω30 (rad/s) E30 (pu.) δ̂30 (rad.)
1.03ωs
1.073
0.363
122
Ê10 (pu.)
0.537
Ê30 (pu.)
0.5365
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