integration of small hydro distributed generation …
TRANSCRIPT
INTEGRATION OF SMALL HYDRO DISTRIBUTED
GENERATION INTO DISTRIBUTION NETWORKS: A
PUMPED HYDRO-STORAGE TOPOLOGY
by
James Odhiambo Owuor
Submitted in partial fulfillment of the requirements for the degree
DOCTOR TECHNOLOGIAE: ELECTRICAL ENGINEERING
in the
Department of Electrical Engineering
FACULTY OF ENGINEERING AND THE BUILT ENVIRONMENT
TSHWANE UNIVERSITY OF TECHNOLOGY
Supervisor : Prof. Josiah L. MundaCo-Supervisor : Prof. Adisa A. Jimoh
January 2014
DECLARATION
I hereby declare that this thesis submitted for the degree DTech: Electrical Engi-
neering, at the Tshwane University of Technology, is my own original work and has
not previously been submitted to any other institution of higher education. I further
declare that all sources cited or quoted are indicated and acknowledged by means of a
comprehensive list of references.
J.O. Owuor
Copyright c© Tshwane University of Technology 2014
ii
DEDICATION
This thesis is dedicated to my late parents, Henry Sibudi Owuor and Perez Adhiambo
Owuor.
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ACKNOWLEDGEMENTS
”If I have seen further it is only by standing on the shoulders of giants” - Isaac Newton.
I most sincerely thank my supervisors, Prof Josiah Munda, and Prof Adisa Jimoh,
for their valuable guidance and patience, without which none of this work would have
been possible. A special thank you to Prof Fred Otieno, former Executive Dean, Faculty
of Engineering and the Built Environment, TUT, and to Prof Ben van Wyk, the cur-
rent Executive Dean, for the enabling post graduate research environment. The South
African government and People, through the Electrical Engineering Department at TUT
accorded me the opportunity to conduct my research, the basic framework from which
my studies were possible, I thank you.
To aunt Diana and uncle John, your untiring support and encouragement in all
matters made a big difference...May God bless you abundantly!
Dr Adedayo Yusuff and Dr Lawrence Letting, thank you for always lending me a
patient ear whenever I posed my numerous questions, this often involved late night calls
too...I shall forever be grateful.
My wife, and soon to be Dr Grace Abera, best wishes in your aspirations. Your
presence, kindness and support all through the time I have known you has been second
to none! You brought me much needed balance...thank you.
My TUT Electrical Engineering Department colleagues and friends, you made the
whole experience wholesome and fulfilling.
Last and most important, Thank You God for making everything possible!
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ABSTRACT
An important solution to challenges posed by the intermittent nature of renewable
energy sources in electrical power generation lies in energy storage. One way of ensuring
a permanent and continuous source of power generation in this respect is pumped storage
small hydro power. Old and disused hydro power stations could with some modifications
be put back into use, existing hydro power stations could with some modification, be
adapted to incorporate pumped storage and existing pumped storage plants could be
adapted so that both pumping and generating modes run concurrently. Suitable loca-
tions on coastlines could also be identified and seawater pump-generator schemes could
be established. Of key significance in such schemes are; the overall plant layout, the
pump-generator set topology and its attributes.
This work is motivated by the need for innovative ways of exploiting small hydro
resources for power generation, and connecting remote areas that are usually supplied
by weak distribution systems. The objective of this study is to develop an embedded
generator-pump set topology using a wound rotor induction machine using the doubly
fed induction generator concept, and a synchronous machine electrically and mechani-
cally coupled to it, powering its magnetisation circuit. An adjustable pitch pump is also
coupled to the generating set on the same shaft to provide an embedded generating-
pumping solution that can provide co-incident generating and pumping functions . The
induction machine is in this case the main power machine, while the synchronous gen-
erator is used to excite the induction machine rotor circuit. The system speed can vary
over a small margin over and below the induction machines synchronous speed. The
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plant layout is similar to a conventional pumped storage layout with the exceptions
that; the pump blades can be adjusted for variable output as a means of slip power
recovery and, a separate penstock channel is used for pumping back to the reservoir.
The electrical interface between the two machines is implemented via a power converter
with a decoupling DC bus. The generating set topology is an adaptation from both
the symbiotic squirrel cage induction generator-synchronous condenser topology, and a
synchronous generator supplying a D.C load adapted from previous works on the two
machines as separate entities.
A mathematical model of the generator-pump system is developed from the equiv-
alent physical machine topology. Once controllability is ascertained, an open loop sta-
bility analysis is carried out to give insight into the stability issues likely to arise from
the proposed topology, and accompanying participation factors. Insight into physical
stability phenomena for the proposed topology is gained. An average model of the sys-
tem, which includes a developed adjustable pitch pump model, is then simulated in
Matlab/Simulink. Control is implemented using standard vector control schemes with
the generating-pumping set connected to the end of a 30 km distribution line. The
performance is evaluated against stability issues identified in the mathematical analy-
sis. Results are used to highlight key factors that need special consideration for design,
implementation, and operation of such schemes.
Keywords: pumped storage hydro power, cascaded electrical machines, stability of
rotating electrical machines, adjustable pitch pump, distributed generation.
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LIST OF TABLES
2.1 Technologies for renewable distributed generation . . . . . . . . . . 21
2.2 Voltage space vectors used in DTC . . . . . . . . . . . . . . . . . . 42
3.1 Eigenvalues at synchronous speed and start/stall conditions. . . . . 84
3.2 Eigenvalues at minimum and maximum rated speed conditions. . . . 84
3.3 Eigenvalues at intermediate sub-synchronous and super-synchronous
speeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
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LIST OF FIGURES
Figure 2.1 Typical layout of pumped storage plant (reprinted from (Jog, 1989)). 14
Figure 2.2 Reversible pump turbine (courtesy (Avellan, 2012)). . . . . . . . . . 16
Figure 2.3 Voltage variation on a radial feeder. . . . . . . . . . . . . . . . . . . 20
Figure 2.4 Synchronous machine with full scale thyristor converter. . . . . . . . 25
Figure 2.5 Synchronous machine with diode rectifier. . . . . . . . . . . . . . . . 25
Figure 2.6 Permanent magnet synchronous machine with diode rectifier. . . . . 25
Figure 2.7 Synchronous machine with cycloconverter. . . . . . . . . . . . . . . 26
Figure 2.8 DFIG with cycloconverter. . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 2.9 DFIG with back-to-back converter. . . . . . . . . . . . . . . . . . . 27
Figure 2.10 Induction machine equivalent circuit. . . . . . . . . . . . . . . . . . 28
Figure 2.11 Synchronous machine equivalent circuit. . . . . . . . . . . . . . . . . 30
Figure 2.12 Simplified synchronous machine phasor diagram. . . . . . . . . . . . 31
Figure 2.13 Block diagram of a distributed generation system with a power elec-
tronics interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 2.14 PMSG with diode rectifier and grid side compensator. . . . . . . . . 33
Figure 2.15 PMSG with diode rectifier and VSI grid side converter. . . . . . . . 34
Figure 2.16 PMSG with diode rectifier, VSI and intermediate DC chopper. . . . 34
Figure 2.17 Electrically excited synchronous machine with diode rectifier, VSI
and intermediate DC chopper. . . . . . . . . . . . . . . . . . . . . . 35
Figure 2.18 DFIG with static Kramer drive. . . . . . . . . . . . . . . . . . . . . 36
Figure 2.19 DFIG with back to back converters (static Scherbius drive). . . . . . 37
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Figure 2.20 Relationship between stationary and rotating space vector q − d axes. 38
Figure 2.21 Indirect field oriented control. . . . . . . . . . . . . . . . . . . . . . 40
Figure 2.22 DTC torque and flux control. . . . . . . . . . . . . . . . . . . . . . . 43
Figure 2.23 Ooi concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 2.24 Ansell concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Figure 2.25 Wound rotor induction machine steady state equivalent circuit. . . . 53
Figure 3.1 Proposed system topology. . . . . . . . . . . . . . . . . . . . . . . . 56
Figure 3.2 Variable pitch concept. . . . . . . . . . . . . . . . . . . . . . . . . . 57
Figure 3.3 Traditional pumped storage plant layout. . . . . . . . . . . . . . . . 58
Figure 3.4 Proposed pumped storage concept plant layout. . . . . . . . . . . . 58
Figure 3.5 Plant layout (Courtesy: Hitachi review, vol 47). . . . . . . . . . . . 59
Figure 3.6 Proposed plant d and q axes equivalent circuits. . . . . . . . . . . . 61
Figure 3.7 Interconnections in block model form. . . . . . . . . . . . . . . . . . 65
Figure 3.8 <λ1 and =λ1 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . 76
Figure 3.9 <λ2 and =λ2 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . 76
Figure 3.10 <λ3 and =λ3 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . 77
Figure 3.11 <λ4 and =λ4 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . 77
Figure 3.12 <λ5 and =λ5 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . 78
Figure 3.13 <λ6 and =λ6 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . 78
Figure 3.14 <λ7 and =λ7 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . 79
Figure 3.15 <λ8 and =λ8 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . 80
Figure 3.16 <λ9 and =λ9 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . 80
Figure 3.17 <λ10 and =λ10 trajectories. . . . . . . . . . . . . . . . . . . . . . . . 81
Figure 3.18 <λ11 and =λ11 trajectories. . . . . . . . . . . . . . . . . . . . . . . . 82
Figure 3.19 <λ12 and =λ12 trajectories. . . . . . . . . . . . . . . . . . . . . . . . 82
Figure 4.1 Stator flux orientation . . . . . . . . . . . . . . . . . . . . . . . . . 95
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Figure 4.2 DFIG Rotor side converter control scheme . . . . . . . . . . . . . . 96
Figure 4.3 Dc bus feedback loop. . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Figure 4.4 Synchronous machine rectifier space Vector Diagram . . . . . . . . . 98
Figure 4.5 Cp versus blade angles β. . . . . . . . . . . . . . . . . . . . . . . . . 101
Figure 4.6 Ppump versus water velocity vout. . . . . . . . . . . . . . . . . . . . . 102
Figure 4.7 Ppump versus vout for various blade angles β. . . . . . . . . . . . . . . 102
Figure 4.8 Physical layout for typical pumped storage hydro power station. . . 103
Figure 4.9 Variable pitch concept. . . . . . . . . . . . . . . . . . . . . . . . . . 105
Figure 4.10 Pump pitch angle control. . . . . . . . . . . . . . . . . . . . . . . . 107
Figure 4.11 Pitch control servo. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Figure 4.12 Functional system block diagram. . . . . . . . . . . . . . . . . . . . 108
Figure 4.13 Pitch angle and speed, steady state. . . . . . . . . . . . . . . . . . . 111
Figure 4.14 Synchronous machine state variables irqs, irds, i
rkq1, i
rfd, i
rkd, δ
0, undamped
DFIG stator mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Figure 4.15 DFIG state variables ieqs, ieds, i
eqr, i
edr, undamped DFIG stator mode . 113
Figure 4.16 DFIG ieqs stator currents spectral density analysis at 20, 40 and 65
sec, undamped DFIG stator mode . . . . . . . . . . . . . . . . . . . 114
Figure 4.17 DFIG ieqr rotor currents spectral density analysis at 20, 40 and 65
sec, undamped DFIG stator mode . . . . . . . . . . . . . . . . . . . 114
Figure 4.18 DFIG stator and rotor voltages veqs, veds, v
eqr, v
edr, undamped DFIG
stator mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Figure 4.19 Induction machine series resonance. . . . . . . . . . . . . . . . . . . 117
Figure 4.20 Synchronous machine stator voltages vrqs, vrds, undamped DFIG sta-
tor mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Figure 4.21 System responses of β, Vdc, Vf , to a step decrease in load . . . . . . 118
Figure 4.22 Real and reactive power following a step reduction in load . . . . . . 119
Figure 4.23 Response to a close 3 Phase short circuit (β, Vdc, Vfd). . . . . . . . . 120
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Figure 4.24 Speed response to a close 3 Phase short circuit. . . . . . . . . . . . . 120
Figure 4.25 Real and reactive power during a close 3 Phase short circuit P Q. . 121
Figure 4.26 Responses of β, DC bus voltage and Synchronous machine excitation
Vf to a step increase in load . . . . . . . . . . . . . . . . . . . . . . 122
Figure 4.27 Real and reactive power during a close 3 Phase short circuit P Q. . 123
Figure 4.28 Responses of β, Vdc and Vf to a distant fault on the distribution
network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Figure 4.29 Real and reactive power during a distant fault. . . . . . . . . . . . 125
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GLOSSARY
DC Direct Current
DFIG Doubly Fed Induction Generator
DG Distributed Generation
PCC Point of Common Coupling
PEI Power Electronic Interface
PMSG Permanent Magnet Synchronous Generator
PV Photovoltaic
PSHP Pumped Storage Hydro Power
UPF Unity Power Factor
PWM Pulse Width Modulation
VC Vector Control
FOC Field Oriented Control
DTC Direct Torque Control
PED Power Electronic Device
LVRT Low Voltage Ride Through
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TABLE OF CONTENTS
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENTS iv
ABSTRACT v
LIST OF TABLES vii
LIST OF FIGURES viii
GLOSSARY xii
CHAPTER 1. INTRODUCTION 1
1.1 Background and motivation of study 3
1.2 Problem Statement 5
1.3 Research Objectives 6
1.4 Methodology 6
1.5 Outline of main contributions 8
1.6 Delineations and Limitations 9
1.7 Thesis Chapter Overview 9
1.8 Publications 10
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CHAPTER 2. SMALL PUMPED HYDRO STORAGE ELECTRIC
POWER IN DISTRIBUTED GENERATION 11
2.1 Small hydro power and energy storage 11
2.1.1 Power available 12
2.1.2 Types of hydro plant 12
2.2 Hydraulic turbines and pumps for variable speed pumped storage applications 13
2.2.1 Turbine-pumps for hydro power applications 15
2.2.2 Pump characteristics 17
2.3 Technical considerations in integration of DG into distribution networks 18
2.3.1 Balancing of active power 18
2.3.2 Balancing of reactive power 19
2.3.3 Voltage profile 19
2.3.4 Power quality 19
2.3.5 Protection 20
2.3.6 Stability 20
2.3.7 Losses 21
2.3.8 Control and monitoring 21
2.3.9 Attributes of applied power conversion technologies used in pumped storage
small hydro power 22
2.4 Generators in small hydro power schemes 22
2.4.1 Synchronous generators 23
2.4.2 Induction generators 23
2.5 Power electronics interfacing for synchronous and induction generators 24
2.5.1 Topological solutions for synchronous machines 24
2.5.2 Topological solutions for DFIG machines 25
2.5.3 DFIG: Principle of operation and reactive power requirement 27
2.5.4 Synchronous machine: principle of operation 29
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2.6 Power electronics interfacing topologies 32
2.6.1 Topologies for synchronous machines 33
2.6.2 Topologies for the DFIG 35
2.7 Control of power electronics interfaces 37
2.7.1 Space vectors and transformations 37
2.7.2 Field oriented control (FOC) 39
2.7.3 Direct torque control (DTC) 41
2.8 Cascaded machines 42
2.9 Stability considerations 46
2.10 Small signal stability 47
2.10.1 Small signal stability of the synchronous generator 49
2.10.2 Small signal stability of the doubly fed induction generator 51
2.11 Magnetising reactive power supply for the DFIG 52
2.12 Summary 53
CHAPTER 3. DESCRIPTION OF PROPOSED SCHEME AND
PRIMITIVE MATHEMATICAL MODEL 55
3.0.1 Electrical sub-system 55
3.0.2 Mechanical sub-system 56
3.0.3 Plant layout 57
3.1 System mathematical model 60
3.1.1 DFIG mathematical model 60
3.1.2 Synchronous machine mathematical model 62
3.1.3 Combined inertia and electromagnetic torque equations 63
3.1.4 Combined electromechanical plant mathematical model 63
3.2 Linearisation of Machine equations 65
3.2.1 Linearised DFIG state space equations 68
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3.2.2 Linearised synchronous machine state space equations 69
3.3 Small signal stability analysis of proposed scheme 70
3.3.1 Small Signal Stability Analysis Model 70
3.3.2 Formulation of system Jacobian 71
3.4 Eigenvalue analysis of the system 72
3.4.1 Eigenvalue trajectories with original and modified parameters 74
3.4.2 Participation factors 85
3.4.3 Controllability 89
3.5 Summary 91
CHAPTER 4. COMPLETE SYSTEM MODEL 92
4.1 Model of generators 92
4.1.1 Model of DFIG 93
4.1.2 Model of synchronous machine 93
4.1.3 DFIG Rotor Side Converter Model 94
4.2 Synchronous Machine Stator Side Converter Model 97
4.3 DC link model 99
4.4 Model of adjustable pitch pump 100
4.4.1 Hydraulic Pump Model 100
4.4.2 Pump dynamic model 103
4.5 Time domain steady state simulation under no-load, (natural system re-
sponse) 110
4.6 Time domain simulation, dynamic system response 117
4.6.1 Time domain response, step reduction in load at DFIG stator terminals 117
4.6.2 Time domain response to a three phase short circuit at DFIG stator terminals119
4.6.3 Time domain response to a step increase in load at DFIG stator terminals 121
4.6.4 Time domain response to a distant fault 121
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4.7 Summary 123
CHAPTER 5. CONCLUSION 126
5.1 Future Research 129
REFERENCES 131
APPENDIX 152
xvii
CHAPTER 1. INTRODUCTION
Methods by which electrical energy is generated and distributed are changing. Security
of energy supply and environment continue to be key concerns in both industrialised and
industrialising nations worldwide. Central to the reduction of dependency on dwindling,
increasingly expensive and imported fossil fuel resources and fulfilling environmental
policy objectives, lies the attractive option of renewable energy. In addition, the liber-
alisation of national power sectors worldwide is shifting the operational and regulatory
paradigms into those of injection of electrical energy at distribution network levels. The
introduction of distributed generation (DG) is also leading to a fundamental change
in how distribution networks are utilised and viewed. Distribution networks are now
increasingly used as a means to connect geographically dispersed energy sources to the
electricity system, thereby converting what were originally energy delivery networks, to
networks used both for the delivery and harvesting of energy.
Renewable energy is derived from natural processes that are replenished constantly
(Agency, 2007) in forms useful to humanity on a sustainable basis. Some renewable
electricity sources, such as bioenergy, hydroelectricity, and geothermal power, are dis-
patchable, their output can, within limits, be made available quickly in response to
demand. Weather dependent renewable energy sources, unless they incorporate storage
such as those based on wind, wave and solar power are variable. This means that they
are not so readily dispatchable (Boyle, 2010).
Siting of renewable energy facilities for DG power injection into distribution networks
remains a broad and complex problem, for which solutions are not obvious or well
1
understood (Vajjhala, 2006). The most common sources of siting problems fall into
three main categories: (a) environmental constraints (b) public opposition, (c) regulatory
roadblocks, (Vajjhala, 2007). The proliferation of DG requires new methodologies for
planning and operating the power system, these involve issues like system dynamics and
issues of longer term significance such as generation mix (Jenkins et al., 2008; Doherty &
O’Malley, 2005). DG can either take the form of small scale conventional or renewable
generation.
Industry generally remains hesitant to changes in the paradigm of reliable electric
power supply that has been in place for most of the last century. This is mainly because
the new paradigm mainly involves renewable energy, which is mainly captured from
naturally intermittent renewable sources. For instance, solar electricity only functions
at night from battery storage systems, and wind being stochastic in nature means power
generation from it is only possible when the wind is blowing. This raises the challenge of
intermittent generation integration issues. One solution to the intermittent integration
challenge is electrical energy storage.
In many developing African countries, there are many potential, as yet untapped
sites for electrical energy storage in the form of small hydroelectric power stations that
could contribute significantly to electricity supply capacity generated through the local
distribution networks in the form of DG. Africa remains the region with the lowest
ratio of deployment-to-potential, and the opportunities for growth are very large. South
Africa is no exception and the perception that its potential for hydropower is very
low is often overstated. A baseline Study on Hydropower in South Africa has shown
that a significant potential for development of all categories of hydropower in the short
and medium-term in specific areas of the country actually exist (Ballance et al., 2000;
Winkler, 2005; Altinbilek et al., 2005). Indeed, such potential exists in plants such as
Drakensberg, Palmiet, Ingula and Tubatse schemes. One of the main objectives of this
thesis is the development of DG for small - hydro pumped storage.
2
Current key technical interconnection requirements for DG are stringent and have
been identified and published by the IEEE PES working group in accordance with IEEE
1547 series standards. Details of these requirements can be found in (Camm et al.,
2012). Relevant to the particular scope of this doctoral research project is the ability
to electrically excite the DFIG rotor so as to minimise, or even eliminate reactive power
import from the network.
‘
1.1 Background and motivation of study
Recent attention has mostly focussed on electrical power generation from wind power
and other sources, while considerably less attention has been paid to the use of small
hydro, which also happens to be an abundant resource not only in Africa, but worldwide.
As it happens, pumped storage hydropower (PSH) is the only conventional, mature and
commercial grid-scale electricity storage option available today. PSH can also provide
regulation of changing loads that would otherwise require modulation of output from
other generating assets, with concomitant wear and tear and expense (Dunn et al., 2011;
Dell & Rand, 2001).
Early hydroelectric pumped storage systems used separate pump and turbine units
(turbine, 2013). Three possible configurations for pumped storage plants in use today
are (Levine, 2007; Kuan, 1989; Brown, 2006):
• Four units: A pump coupled to a motor and a turbine coupled to a generator.
This configuration occupies a large amount of space and is no longer used.
• Three units: A pump and turbine are both coupled to a single reversible motor-
generator. The efficiencies of the pump and turbine can be optimized and multi-
stage pumps can be used for very high heads.
3
• Two units: A reversible pump-turbine is coupled to a reversible motor-generator.
This configuration takes up a smaller space compared to the other two and has a
lower installation cost. The disadvantage is a decrease in the efficiency.
More than 95 % of pumped storage power plants are of the type in item three
above (AIT, 2011), and the traditional concept of pumped storage systems consists
of two water reservoirs and a reversible pump-turbine (Mansoor, 2000; Yang, 2010).
Reversible turbine-generator sets act either as a pump or as a turbine, either in pumping
or generating mode. An important advance in the last decades has been the development
of variable speed PHS systems to allow for controllable power in pumping mode (Suul
et al., 2008). At the same time, the idea of small hydro has mainly been associated
with inland sources like rivers and streams, yet there is an additional potential pumped
storage hydro power resource in the form of long coastlines surrounding most countries.
Although variable speed operation of hydropower stations implemented using the
traditional synchronous machine is simple (Naidu & Mathur, 1989; Arrillaga et al.,
1992), it is based on the use of a full-scale converter and this is considered a main
drawback with respect to cost and losses for pumped storage systems with high output
ratings. Since for most pumped storage applications a limited controllable speed range
may be needed during normal operation (Bendl et al., 1999; Bocquel & Janning, 2005;
Mori et al., 1995), the DFIG with a power electronic converter with reduced rating may
provided an attractive option Janning & Schwery (2009). The DFIG would however
require a source of magnetisation for its rotor circuit and this could be a drawback if
this source were obtained from the network.
Pumped storage hydro power plants in operation utilising reversible turbine pumps
can operate only in one mode; either pumping or generating mode. The DFIG as widely
documented has increasingly become popular in variable speed constant frequency appli-
cations. This has been possible because of advances in power electronics and control. Its
4
major drawback is that it requires an independent source of magnetising power, usually
sourced from elsewhere in the system. The synchronous generator on the other hand,
if used in variable speed applications, would require full scale converters at its output.
This would significantly limit the size of the machine as it would entail converters of
large rating. At the same time, most potential small hydro power resources are located
at significantly long distances from the main power grid. This usually means that loads
thereto are connected at the end of long and electrically weak distribution systems.
The background given above justifies the need to find an integrated solution whereby
a pumped storage plant could operate in pumping and generation mode simultaneously,
while employing the DFIG as the main source of power and providing it with suitable
excitation. Seawater could be pumped to a reservoir during off-peak hours and released
for power generation to offer a load balancing solution for peak hours. Existing conven-
tional hydro power plants could be modified to have pumped storage capability. New
hydro power plants could be designed to have pumped storage as a standard functional
capability. Existing pumped hydro storage facilities using reversible pump turbines could
be adapted so that they can pump and generate at the same time which would provide
a form of load balancing for frequency support. In the South African context, inland
plants with such potential include the Drakensberg and Palmiet schemes (Wall, 2008)
and Ingula and Tubatse schemes (Louwinger, 2008), for example.
1.2 Problem Statement
The main problem can be broken down as follows:
• identify a suitable overall plant physical topology.
• identify a suitable pump/generator configuration.
• establish controllability of the proposed plant.
5
• explore means of controlling the plant.
• isolate control problems.
1.3 Research Objectives
The research objectives are as follows:
• to develop an overall plant topology.
• to identify plant attributes necessary for proper functionality of the proposed plant.
• to identify a pumping/generation topology that meets the required electro-mechanical
and overall topological layout attribute requirements.
• to develop a primitive mathematical model of the plant that provides insight into
fundamental physical behaviour of the plant.
• to investigate the stability issues arising from the electromechanical coupling of
the two machines used.
• to establish controllability of the proposed configuration.
• to identify influencing factors on the stable operation of the proposed plant.
• to develop an overall system model for simulation. This also entails developing a
suitable mathematical model for the variable pitch pump.
• to simulate the system steady state and dynamic behaviour.
1.4 Methodology
The approach adopted in this work is based on machine models that are congruent with
experimentally validated models found in literature such as (Krause et al., 2002; Kundur
6
et al., 1994; Krishnan, 2001; Slemon, 1989; Anderson & Fouad, 2003; Rogers & Shir-
mohammadi, 1987; Park, 1929). Operation and control of the DFIG and synchronous
machines as individual entities are well established in these publications. Numerous tech-
niques have hitherto also been applied in the rotor circuit control of the DFIG (Pena
et al., 1996; Blaschke, 1972; Bonwick & Jones, 1972; Casadei et al., 2002). The Mat-
lab/Simulink blockset library models used are based on the above mentioned literature.
A necessary condition for the successful operation of such controls is a stable DC supply.
In the case of a synchronous machine, the point in case here is that of a synchronous
machine supplying an active DC load. In electrically and mechanically coupling the two
machines, an additional degree of complexity is introduced and it is useful to examine
the underlying basic physical interactions of the resulting common electrical circuit and
effect of the mechanical coupling. The method adopted in this work is implemented in
two stages.
At the first stage, an assumption is made that the power electronics interfacing the
two machines and their controls operate ideally. It is also assumed that the synchronous
generator excitation control operates ideally. It is then possible to abstract the interfaces
and controls away and form an equivalent circuit and primitive mathematical model of
the coupling in state variable form. The equivalent circuit and model then provide insight
into the basic interaction of the coupled machines, where each machine sees the other
as an external disturbance in the common circuit. The adopted process is decomposed
as follows:
• Assume ideal interface and control components and abstract them away.
• Formulate d−q primitive electromechanical mathematical model of combined plant
and express it in state variable form.
• Express the DFIG rotor circuit as a disturbance to the synchronous machine stator
circuit and vice-versa.
7
• Linearise the model.
• Formulate combined system Jacobian.
• Compute eigenvalues, establish their trajectories in the speed ranges of interest
and modal characteristics of the primitive model.
• Compute participation factors and establish key contributions to oscillatory modes.
• Determine controllability of the plant in the speed range of interest.
Tools used at this stage are the following Matlab toolboxes; symbolic math toolbox,
partial differential equation toolbox and signal processing toolbox.
In the second stage, a complete model and requisite controls are introduced. In
addition, a model for the variable pitch pump is developed and integrated with the
pump-generator set. The process is decomposed as follows:
• Adapt the primitive mathematical model to incorporate de-coupling DC link and
converters.
• Implement vector control of DFIG rotor circuit and close feedback loop between
DC link and synchronous machine exciter.
• Develop variable pitch pump model and its control; incorporate into overall model.
• Simulate the complete system and verify modal behaviour of the primitive model.
The modelling and simulations are implemented using the Matlab/Simulink/ Simpow-
ersystems, Control system and Simscape standard library block-sets.
1.5 Outline of main contributions
The main contributions of this work can be summarised as follows:
8
• An integrated pumped hydro storage solution layout that can deliver electrical
power while pumping water.
• Development of an integrated pump-generator set that can be extended to other
applications.
• Development and analysis of a primitive electromechanical coupling model of two
machines.
• Development of a solution to provide magnetising power to the rotor of a DFIG.
• Development of a variable pitch pump concept.
1.6 Delineations and Limitations
In the system under study, mechanical power inputs are assumed to be non-intermittent
and step changes are not considered. The distribution line to which the plant is connected
is lumped and parameters are generated from a network reduction. As such, simulated
system faults are only those at the immediate plant outputs or at the other end of
the line. The simulated short circuit faults are only three phase to ground and distant
faults are simulated as voltage dips. The reactive power problem is limited to adequate
magnetisation of the DFIG rotor circuit. An economic analysis is also not considered.
1.7 Thesis Chapter Overview
Chapter 1 provides a general introduction to the thesis, and provides a background and
justification for the study. Chapter 2 gives insight into current practise and technolo-
gies used in renewable energy in distribution networks and, to small hydro power and
energy storage technology. An overview of the necessary attributes of power conversion
plants used in renewable energy generation and their effects on the local power network
9
or load to which they are connected is also provided. Overall, a background of small
hydro power conversion technologies, their topologies and principles of control, and a
background of pumping technologies for variable speed pumped storage applications
is provided. Chapter 3 introduces the proposed system, mathematically describing a
primitive interconnection of the two machines that reveals their basic interaction. A
mathematical model is developed in state variable form and a stability assessment is
carried out using eigenvalue analysis to give insight into factors that may cause insta-
bility and their relative influence. In chapter 4, the full system model is developed and
simulated. Chapter 5 provides the conclusion and outlines scope for further research.
1.8 Publications
1. Owuor, J.O., Munda, J.L., Jimoh, A.A., “Analysis of Doubly Fed Induction Gen-
erator and Synchronous Machine Cascade for Small Hydropower Applications”.
Electric Power Components and Systems, Volume 41, Issue 7, pp. 669-692, 2013.
2. Owuor, J.O., Munda., “ Effects of embedded generation operating modes on power
system performance”. 9th International Conference on Power System Operation
and Planning (ICPSOP), Nairobi, Kenya, 2012.
3. Owuor, J.O., Munda, J.L., Jimoh, A.A., “The ieee 34 node radial test feeder
as a simulation testbench for Distributed Generation”. IEEE, AFRICON 2011,
Victoria Falls, Livingstone, Zambia.
10
CHAPTER 2. SMALL PUMPED HYDRO STORAGE
ELECTRIC POWER IN DISTRIBUTED GENERATION
2.1 Small hydro power and energy storage
A considerable proportion of the sun’s energy that reaches the earth’s surface causes
evaporation from lakes, seas and other water bodies. This evaporation causes vapour to
rise into the atmosphere against gravitational pull of the earth. The vapour condenses
and it either rains or snows. The rain or snow in hilly areas still has some of the solar
energy input stored in it in the form of stored gravitational energy.
Compared to wind and photovoltaic power, small hydropower is often undervalued.
The worldwide demand for energy could theoretically be covered by hydropower. But
this is not economically practical, least of all because of the uneven distribution of
worldwide water resources. Since hydropower doesn’t depend on the natural rhythm of
the sun or on the strength of wind, it can go a long way in ensuring security of energy
supply, and ensuring a permanent and continuous source of power generation.
Hydropower generation is considered a mature renewable technology. Numerous
publications have covered the subject in great detail. Therefore only a brief review is
presented here.
11
2.1.1 Power available
The output power of a hydro-turbine is given by (Jenkins et al., 2008)
P = QHηρg (2.1)
where:
P = output power (W )
Q = flow rate (m3/s)
H = effective head (m)
η = overall efficiency
ρ = water density (1000 kg/m3)
g = gravitational acceleration
2.1.2 Types of hydro plant
These may be: run of river, dammed or pumped storage. Run of river schemes are usually
established on swift flowing bodies of water and may incorporate a small diversion weir
or pondage. A drawback of run of river schemes is that during times of high inflow excess
water has to be spilled (Taylor & Schuman, 2012). This represents a lost opportunity for
generation. Dammed schemes have traditionally been based on storage of large amounts
of water and extensive civil works. While these schemes have the capability to store
large amounts of energy, its drawbacks include significant ecosystem and environmental
damage, loss of land and other attendant well documented problems arising from scale-
of-site. Pumped storage hydro power plants are used to pump water from a river or
lower reservoir up to a higher reservoir to allow its release during peak times (Taylor
12
& Schuman, 2012). Despite the losses of the pumping-to-storage-process, they can be
very useful in providing large-scale energy storage. Hydro power in general is the only
large-scale, low-cost electricity storage option available today. Their usefulness can be
extended to facilities, where the environmental impact of large hydro projects cannot be
undone, by modification of existing plants. They also provide an interesting prospect
for any further development of the worlds unexploited small and large hydro resources.
Energy storage is vital when integrating large amounts of intermittent and nondis-
patchable renewable energy sources into electric power systems and can be used as a
means of increasing penetration of renewable energy while maintaining a high quality
and reliable power supply (Barton & Infield, 2004; Schainker, 2004; Leonhard & Grobe,
2004). Flexible control is needed to maintain instantaneous active and reactive power
balance. Energy storage suitably interfaced to the electricity grid from hydropower is a
necessary and important factor towards achieving sustainable energy systems based on
renewable energy sources (Dell & Rand, 2001).
2.2 Hydraulic turbines and pumps for variable speed pumped
storage applications
Pumped storage hydro power plants are today the storage technology with the lowest
specific investment costs, the widest application range and the longest service life (Roth,
2012). In order for them to be competitive they need to have high operational flexibility,
this means that new designs and technologies must be continuously researched and
developed.
Figure 2.1 shows a typical layout of a pumped hydro storage system. In such systems,
water is pumped from a lower reservoir to an upper one to store energy. For generation,
water is released from the upper reservoir via a turbine into the lower reservoir. A
typical pumped storage plant is very similar to a conventional hydro power station
13
Figure 2.1: Typical layout of pumped storage plant (reprinted from (Jog, 1989)).
and has many of the same components. Apart from the difference in reservoir layout,
additional machinery is usually required and these may take the following configurations
(Brown, 2006).
• A pump-motor set and a turbine-generator set. Such a configuration occupies a
large amount of space and is rarely used anymore.
• A pump-turbine set mechanically coupled to a reversible motor-generator. The
efficiencies of the pump and turbine can be optimized and multi-stage pumps can
be used for very high heads. (Requires clutch mechanism for pump)
• A reversible pump-turbine coupled to a reversible motor-generator. This set-up
requires less space compared to the other two and has a lower installation cost.
The disadvantage is a decrease in the efficiency. Majority of pumped storage power
plants are of this type (Coleman et al., 1976).
Motivating factors for the drive towards variable speed operation of pumped hydro
systems have mainly been: increased efficiency (Kerkman et al., 1980a), possibility for
power control in pumping mode (Taguchi et al., 1991), increase in response time for
power control (Erlich & Bachmann, 2002).
14
While the reversible pump-turbine coupled to a reversible motor-generator scheme
is attractive for frequency regulation and high capacity energy storage, the station can
obviously not deliver electrical power while in pumping mode because of the reverse
turbine operation. A similar scheme is presented in for example, (Lung et al., 2007).
In this scheme, a hybrid wind-hydro-pumped storage power station is proposed, where
wind power is used to pump water for storage, while a conventional hydro-power station
is employed for power generation. Most such schemes in operation use conventional
synchronous generator -reverse turbine sets (Mansoor, 2000), or use doubly fed induction
generators as reported in (Lung et al., 2007).
A novel generator-pump topology that can deliver electrical power even while in
pumping mode forms a key contribution of this work.
2.2.1 Turbine-pumps for hydro power applications
Most pumped storage hydro power systems use the vertical axis Francis turbine (Dixon,
2005). The reverse pump turbine is effectively a Francis turbine whose runner geometry
is a compromise between the optimum for pumping and generation (Janning & Schw-
ery, 2009). Figure 2.2 shows a vertical axis Francis turbine, this is basically a reaction
turbine that generates energy as water pressure changes when passing through the tur-
bine shaft. Details of various types of turbine used in small hydro power applications
can be found in numerous publications. These are essentially fixed blade-pitch turbines.
Variable blade-pitch hydro-kinetic and other turbo machinery in use are usually found
in generator turbine, and aircraft and marine propulsion systems. The novel pump
generator proposed in this thesis involves a generator-pump set, with the pump having
adjustable-pitch blades.
Meanwhile, research efforts on the concept of variable pitch hydro-kinetic power
generation hydraulic machinery has mostly focussed on variable pitch turbines, while
there is a notable scarcity of research information on variable pitch pumps. Literature
15
(a) Francis turbine (b) Scale model of reverse turbine
Figure 2.2: Reversible pump turbine (courtesy (Avellan, 2012)).
available includes the patent (Patin, 1984) for the bladed turbine–pump with adjustable
guide vanes and (Ramsay, 1998) for the pump impeller with adjustable blades, in which
the pump has a retractable impeller that is used to vary the pumping rate. Some recent
research literature has focussed on the variable pitch operation of Darrieus water tur-
bines. (Kirke & Lazauskas, 2008) presented an adaptation of the Darrieus wind turbine
concept to hydro applications, and summarised recent developments in the adaptation.
(Schonborn & Chantzidakis, 2007) developed and described a novel hydraulic control
mechanism designed for cyclic vertical axis Darrieus concept marine turbines, while
(Paillard et al., 2013) described a new method for simulating a crossflow darrieus tur-
bine with active pitch variation. At the same time, (Cooper, 2003) described a fully
variable geometry pump derived from the original Darrieus pump as being of particular
interest in terms of efficiency. He names it as a prime candidate for typical impeller
pumps (such as centrifugal and Hydraulic Power Recovery Turbines HPRT) going main-
16
stream. He also predicts that creative mechanical development of reliable, durable, and
cost effective innovations in variable geometry pumps will make such pumps capable of
running virtually at shutoff - both smoothly and at very low power levels. Such capabil-
ity will obviously provide enormous flexibility in the operation of pumped hydro storage
plants, such as that proposed in this thesis.
2.2.2 Pump characteristics
While the focal interest in this research is the behaviour of the electric system, it is
recognised that the attached variable-pitch mechanical pumping system will have an
effect on overall system behaviour. It is therefore useful to provide some background on
pump theory in context.
Pump characteristics usually differ considerably and detailed data is only usually
provided by manufacturers. A prediction of pump characteristics is only possible using
empirical (or numerical) methods and empirical methods are based on statistical evalu-
ations of tested pumps (Glich, 2010). This makes a generic mathematical description of
such a unit very difficult, particularly when the runners of such units permit the blade
angle to be varied on the run depending on operating conditions as described in (Gorla
& Khan, 2003; Brezovec et al., 2006). In short, generic exact mathematical description
of complex pump characteristics using analytical expressions is usually not possible.
Variable pump characteristics can be obtained using lifting line propeller design theory,
details of which can be found in (Betz & Helmbold, 1932; Prandtl & Betz, 1927; Nicolet,
2007; Kimball & Epps, 2010; Khan et al., 2006; Epps, 2010).
17
2.3 Technical considerations in integration of DG into distri-
bution networks
In the process of managing renewable and generally difficult to control resources of elec-
trical energy, the existing distribution power systems are evolving from passive to active
systems. The proliferation of DG also leads to changes in the technical characteristics
of distribution networks mainly in the form of bi-directional active and reactive power
flows, which presents technical challenges of balancing the active and reactive power
(Lund, 2007) between source, load and system losses. Active power balancing on short
time scale is usually performed using spinning inertia while the long term scale is regu-
lated by the power and spot markets. Reactive capabilities of DG, with the aid of power
electronic converters can be used to minimize reactive power flows between the grid and
distribution networks (Morren, 2006). Systems are as a result usually pushed close to
their limits of safe and reliable operation in this new paradigm. The main technical
issues are treated in detail in (Jenkins et al., 2008) and a brief overview is presented
below.
2.3.1 Balancing of active power
Since storage of electricity is difficult and expensive, the load has to balance generation
at all times. This is normally carried out in two time scales; a short time scale where the
spinning inertia in the system is used to balance small demand variations and, long term
scale where scheduling is done and influenced by both the regulating and spot power
markets.
In the short time scale, most renewable sources do not contribute to system reserves
and to the total system inertia (Tielens & Hertem, 2012). System inertia is vital in
determining the immediate frequency response with respect to inequalities in the overall
power balance. Small scale renewable power generation based on rotating machinery,
18
especially with the aid of power electronics can sometimes provide this inertial response
using the so called synthetic inertia (Seyedi & Bollen, 2013).
2.3.2 Balancing of reactive power
DG units can be used to balance reactive power in distribution networks. They are how-
ever traditionally not used for active voltage control. Many grid codes worldwide such
as (Gmbh, 2006) and (Transmission, 2009) currently require DG to maintain reactive
power support to the power system and be able to ride through system voltage dips
without interruption.
Static reactive power balance can be achieved within reason by using suitably placed
capacitors. This is however fraught with technical difficulties when a large number of
capacitors are required. Dynamic reactive power balance is necessary especially during
and after system disturbances. The rule of thumb is generally that dynamic reactive
power quantity available should be equal to the static demand. The motivation for this
research project is in part, fortified by the need for reactive power supply to an induction
machine employed in small scale distributed generation.
2.3.3 Voltage profile
Because of the relatively low X/R ratio of distribution lines, the consumption of active
power always causes a drop in bus voltage. Conversely, an injection of active power
causes a voltage rise. Problems associated with voltage rise in distribution systems are
discussed in (Masters, 2002; Viawan, 2008) and Fig. 2.3. illustrates a typical voltage
profile on a radial feeder.
2.3.4 Power quality
DG can either improve or degrade power quality received by the consumer in the distri-
bution network. Quality of power is a measure of how close the system voltage waveform
19
G
MV feeder LV feeder
V
1 p.u.
Min load
Max load
Permissible voltage
rise for DG
Max permissible
voltage variation
Figure 2.3: Voltage variation on a radial feeder.
is to sinusoidal with rated voltage magnitude and frequency.
2.3.5 Protection
The active power flow in distribution networks has traditionally been from higher to
lower voltage levels. Connection of DG in MV and LV networks therefore leads to:
• increased fault levels.
• False tripping of healthy radial circuits.
• Protection blinding.
• Unintended islanding.
2.3.6 Stability
A single fault in the HV system may cause a voltage dips in several distribution laterals.
This can lead to voltage collapse as a result of many DG tripping out. Large enough
penetration of DG in the MV and LV networks can also influence transient stability
limits of the transmission system (Slootweg & Kling, 2002).
20
Table 2.1: Technologies for renewable distributed generation
Technology Typical module size
1. Small Hydro 1− 100 MW2. Micro Hydro 25 kW − 1 MW3. Wind Turbine 200 W − 7 MW4. Photovoltaic Arrays 20 W − 100 plus MW3. Solar Thermal, Central Receiver 1− 10 MW
2.3.7 Losses
DG can supply local loads thus reducing power flows across the network hence reducing
system losses. The converse may however be true when there is a low coincidence between
generation and local load.
2.3.8 Control and monitoring
Control and monitoring are usually centralized operations that are provided using super-
visory control and data acquisition (SCADA) systems in the transmission networks. A
lot of research is currently underway to set common communication standards (Buchholz
& Styczynski, 2006). This will substantially lower flexibility and costs of implementing
the same at distribution level.
One of the key influencing factors on the interaction between DG and the system at
large is the technology utilized in the DG and the mode of its control and operation.
Table 2.1 provides a brief summary of some commonly used technologies and typical
module sizes based on current trends gleaned from literature. The table is just a guide
as global definitions of distributed generation still vary widely (Pepermans et al., 2005;
van Werven & Scheepers, 2005; Lopes et al., 2007).
21
2.3.9 Attributes of applied power conversion technologies used in pumped
storage small hydro power
Attributes of the power conversion technology used in any renewable energy conversion
system affect how they interact with the local electrical power system or isolated load.
Details of these attributes can be found in (Saint & Friedman, 2002). A flexible and
effective implementation of such technology would allow variable speed operation within
allowable limits, in such a way that enables continuous control of input power during
pumping operation (Suul, 2006). Variable speed operation within allowable limits can be
even more advantageous for pumped storage units in case of islanding , since fluctuating
production-load balances within an islanded area introduces additional challenges to the
operation of the system.
2.4 Generators in small hydro power schemes
The most common power conversion technologies used in small hydro applications are
synchronous and induction generators (Jenkins et al., 2008). The possibility of variable
speed operation of a pumped hydro plant can be achieved by asynchronously tying the
hydro site and the main or local ac grid (Kerkman et al., 1980a; Kroposki et al., 2006a),
this provides new flexibility in machine design and configuration and also means that
it is no longer necessary to tie the generator to the grid frequency. Synchronizing the
conversion technology means that its final electrical output variables of voltage magni-
tude, frequency, phase rotation, and phase angle must be the same as that of the a.c.
electrical system to which it is connected within an acceptable range. This is usually
achieved using controlled power electronics equipment connected in various topological
configurations. By taking advantage of innovations in power electronics and interfaces,
much can be achieved in exploiting the individual advantages of the synchronous and
induction machine capabilities by exploring their complementary cascaded operation.
22
System optimisation and integration factors associated with distributed generation sys-
tems and the benefits of asynchronous ties between source and load are examined in
(Kroposki et al., 2006b).
2.4.1 Synchronous generators
This is the most common generator in service today (Krause et al., 2002). It is quite a
well understood machine and there have been numerous publications about it over the
years. The synchronous generator can either be of the permanent magnet or electrically
excited type. The rotor of the electrically excited synchronous generator is equipped with
a main field winding and one or more damper windings. The rotor windings all have
different electrical characteristics, and the main rotor winding carries a d.c. excitation
current. The rotor may be of the salient pole or cylindrical type. The permanent
magnet machine needs no additional power supply for rotor excitation and is simpler
in construction. It however has the drawbacks of; high cost of the permanent magnet,
difficulty in manufacturing, and demagnetization of the permanent magnet over time.
2.4.2 Induction generators
This is an asynchronous machine that has an armature or stator winding very similar
to that of the synchronous machine and a rotor that may either be of the squirrel cage
type or wound rotor. The induction generator however requires an external source of
magnetizing current (reactive power)in order for it to be able to establish the magnetic
field across it’s air gap. The doubly fed induction generator (DFIG) is often used in
windpower generation (Wegener et al., 2006). The stator is connected directly to a
local load or grid while the rotor is fed by an electronic power converter which enables
variable speed operation. The DFIG is essentially a wound rotor induction machine and
can equally find application in small hydro power generation (Ansel et al., 2006; Okafor
& Hofmann, 2004).
23
2.5 Power electronics interfacing for synchronous and induc-
tion generators
Power electronic converters may be used to interface a rotating generator to the network
by electrically decoupling it, thereby potentially increasing efficiency and flexibility of
prime mover operation. There is a wide variety of topological schemes in use and several
more being researched on. Most of these modern power electronic converters synthesise
a waveform from a voltage source. The basic operation is premised on the two equations
(2.2) and (2.3). The real power P , injected by the converter is mainly controlled by
manipulating the phase angle δ between that of the synthesised waveform and that of
the network. The reactive power Q is controlled by manipulating the relative magnitudes
of the synthesised waveform and that of the network. Synthesization based on current
source inverters have also been investigated and are currently in use.
A brief review of some common topological solutions currently in use is provided in
the following subsections for synchronous and DFIG machines.
P =EV
Xsin δ (2.2)
Q =EV
Xcos δ − V 2
X(2.3)
2.5.1 Topological solutions for synchronous machines
Some of the earlier uses of variable speed solutions for synchronous machines were based
on full scale thyristor inverters using current source converters as shown in figure 2.4
(Kerkman et al., 1980a,b). Other solutions include the synchronous machine with a
diode bridge rectifier, or with a cycloconverter. Figure 2.5 shows an electrically excited
synchronous generator, detailed descriptions and analyses of such a scheme can be found
in (Bonwick & Jones, 1972; Jadric, 1998), while figure 2.6 shows a permanent magnet
24
SM
DC field
supply
Thyristor based
current source converter
Figure 2.4: Synchronous machine with full scale thyristor converter.
SM
DC field
supply
Diode rectifier Thyristor inverterDC Link
Figure 2.5: Synchronous machine with diode rectifier.
synchronous machine with the same configuration. Figure 2.7 features an electrically
excited synchronous machines connected via a cycloconverter, detailed analyses and
descriptions of similar schemes can be found in (Nakano et al., 1984; Hasse, 1977).
Because of the manner of connection of the schemes of figures 2.5, 2.6 and 2.7, the losses
and cost of the full scale converter in each case imposes limits on its applicability. It
is evident that synchronous machines used for variable speed operation must have full
scale rated converters, this is because the power output can only be extracted from the
stator windings. Similar converters with lower ratings can be applied innovatively, in
different topologies, one of which forms part of the contribution of this research work.
2.5.2 Topological solutions for DFIG machines
Much research and publication has focussed on the DFIG in variable speed applica-
tions. The DFIG has mostly been used in wind power applications and a comprehensive
overview of developments of DFIG over 25 years is given in (Tazil et al., 2010). The
PMSM
Diode rectifierPermanent magnet
synchronous generatorDC Bus
inverter
Figure 2.6: Permanent magnet synchronous machine with diode rectifier.
25
SM
CyclonverterSynchronous
generator
Figure 2.7: Synchronous machine with cycloconverter.
result is that there are many low cost and innovative schemes that have been proposed,
researched on and implemented. Since most pumped hydro storage systems (and indeed
small hydro power systems) only need a limited, controllable speed range under normal
operating conditions (Freris & Infield, 2008), the DFIG is easily very suitable as a power
source, (Schreier et al., 2000; Bonnet et al., 2007; Bendl et al., 1999). The power elec-
tronics converter used in most DFIG systems is connected via the rotor circuit and is
rated at 20-30 % of the base machine rating (Pena et al., 1996).
While active power is usually considered to be the more important factor in power
generation, magnetising reactive power is equally important for a DFIG’s proper oper-
ation. The DFIG, requires a source of reactive power from an external source. This is
usually provided either from external capacitors, the grid or, in the case of an isolated
system, some other source such as batteries interfaced with a suitable converter for its
operation (Carattozzolo et al., 2000).
Cycloconverter
~~
DFIG
Figure 2.8: DFIG with cycloconverter.
26
DFIG
Voltage source converter
Figure 2.9: DFIG with back-to-back converter.
Figure 2.8 shows a basic configuration with a cyclonconverter feeding the rotor wind-
ings. This is one of the earlier solutions that were used during the advent of large scale
implementations of high power variable speed drives (Taguchi et al., 1991), while this
full scale converter concept provides a large degree of flexibility, the cost of converter
and associated losses and harmonics injected into the grid remain an obstacle. Figure
2.9 shows the most widely used scheme in variable speed wind energy applications, with
a back-to-back converter connected in the rotor circuit (Pena et al., 1996; Iwanski &
Koczara, 2008; Lipo, 2009). With this concept, control of active and reactive power are
easily decoupled (Li et al., 2009; Luna et al., 2008; Dendouga et al., 2007). Another
key advantage compared to the full scale converter solution is that the maximum power
flow through it under normal conditions is typically 20-30% of rated system power. A
drawback of the concept is that it cannot provide reactive power to the grid during
and right after a severe voltage dip. This drawback stems from the fact that the DFIG
constantly needs an external source of reactive power to function properly. In line with
the generation topology researched on and presented in this thesis, the following sections
focuses on reactive power supply to the DFIG.
2.5.3 DFIG: Principle of operation and reactive power requirement
The induction machine has been in use for many years, its theory and operational charac-
teristics are quite well understood and numerous publications attest to this. The theory
27
~
+ - + -
+
-
+ - + -
2
m
R
2j L 1j L1R
LRj M
2I1I
mImV LI tV
Figure 2.10: Induction machine equivalent circuit.
of the induction machine as a generator, its excitation and reactive power requirements
are extensively reported in (Barkle & Ferguson, 1954; Bansal, 2005; Elder et al., 1983).
Figure 2.10 shows the equivalent circuit of an induction machine that is a general
case of all induction machines whether squirrel cage or wound rotor (DFIG) (Ooi &
David, 1979a). In figure 2.10 R1, RL represent the stator resistance and load impedance
respectively, whereas
R2ω
ω − ωm(2.4)
represents a fictitious slip - dependent equivalent rotor resistance. With the assumption
that the machine’s ferromagnetic non-linearity is negligible, M describes the machine’s
constant airgap magnetisation inductance. The system electrical angular frequency is
given by ω = 2πf whereas the airgap synchronous speed is given by 2ωPf
. Assuming a
purely resistive load and neglecting losses in the rotor circuit as represented in figure
2.10, for a DFIG, when the mechanical speed given by
R2ωmPf
(2.5)
exceeds the airgap synchronous speed, the equivalent resistance of (2.4) becomes negative
and both the rotor and stator windings supply power to the load. On the other hand,
when the mechanical speed is less than the air-gap speed (2.4) becomes positive, only the
stator current supplies power to the load while the rotor current draws reactive power.
28
The stator current is given by
I1 = − VtZ
(2.6)
and I1 is the phasor sum of the rotor current I2 and magnetisation current Im, where
Vt is the terminal voltage and Z is the equivalent circuit impedance. It is Im flowing
through the mutual inductance M that is responsible for the machine excitation (Barkle
& Ferguson, 1954). With known values of slip s and magnetising reactance xm, the
value of Z (equivalent circuit impedance) can be calculated from the equivalent circuit
of figure 2.10 as
Z = RS + jXs +
(Rss
+ jxr)
(jxm)Rss
+ j (xr + xm)(2.7)
and the magnetisation current is given by
Im = −j VmωM
(2.8)
where Vm is the air-gap voltage and Im must be supplied by overexcited synchronous
capacity, shunt connected capacitors on the system, or some other source. The maximum
(or limiting) real and reactive power output of the DFIG are given by
Pg = − V 2t Rs
R2s + (xm + xs) (x′)
(2.9)
Qg = − V 2t [xm + xs + x′]
2 [R2s + (xm + xs) (x′)]
(2.10)
where x′ = xs + xmxrxm+xr
.
2.5.4 Synchronous machine: principle of operation
The synchronous generator being currently the world’s largest source of electricity gen-
eration, has well established theory of operation and practical applications. It consists
29
+
-
sj LsR
LR
sI
LI
tV~
fI
2
P
gE
Figure 2.11: Synchronous machine equivalent circuit.
of a d.c. field winding on the rotor and an a.c. armature winding on the stator. The
machine may function as a motor or generator. Figure 2.11 shows a simplified single
phase equivalent circuit of the synchronous machine. In the figure, Rs is the stator
resistance, ωLs is the synchronous reactance and Eg is the internal generated voltage.
The stator winding inductance stores energy. Since the inductance of the stator
winding depends on rotor position, this energy changes with rotor position. In the case
of a generator, this energy flows through the stator and into the electrical network. The
magnitude of the internally generated voltage Eg has the following non-linear relation-
ship with the field current If .
Eg = ωf(If
)(2.11)
The field current can therefore be used to control the magnitude of internal generated
voltage (or the excitation voltage). Taking Kirchoff’s voltage law around the circuit
gives us Eg in terms of the terminal voltage and voltage drops.
Eg = Vt +RsIS + jωLS Is (2.12)
30
gE
tV
singE s AjX I
AI
Figure 2.12: Simplified synchronous machine phasor diagram.
The basic relationship between power, torque and speed is
P = Tω (2.13)
Average torque is produced only when the machine rotates at synchronous speed
Ns =120f
P(2.14)
Figure 2.12 shows the relationship between the excitation voltage Eg, terminal voltage
Vt and stator current I. For active power to flow out of the machine, Eg must lead Vt
by angle δ and the active power transfer equation is given by.
P =EgVtXeq
sin δ (2.15)
Equation (2.15) represents an idealized synchronous machine model where the active
power varies as a pure sinusoidal function of δ.
31
Equation (2.16) gives the reactive power delivered by a synchronous generator
Q =EgVtXeq
cos δ − V 2t
Xeq
(2.16)
From equation (2.16), it is evident that the synchronous machine is able to supply
reactive power to any system it is connected to, and this depends on the square of the
magnitude of the machine’s terminal voltage. The configuration proposed in this work
entails the synchronous generator charging a d.c. link, the generator essentially needs
to operate in unity power factor (UPF) mode and supply only active power at specified
voltage to charge the d.c. link. This means the excitation system needs to be operated
in voltage supporting mode. A functional description of the voltage supporting mode
operation of synchronous machines is provided in (Hurley et al., 1999).
2.6 Power electronics interfacing topologies
Any type of distributed generation source can be connected to an electric power system
or load via a power electronics interface. Converters may consist of both an inverter and
rectifier or just an inverter. Figure 2.13 shows a block diagram of distributed generation
system interfaced to an electric power system.
Five general categories of power electronics systems used for distributed genera-
tion are: (i) AC-DC rectifiers (controlled and uncontrolled); (ii) DC-AC inverters; (iii)
DC-DC switched mode converters; (iv) solid state breakers; (v) AC-AC converters (cy-
clonconverters). The back-to-back topology involving (i), (ii) and (iii) and methods of
control are discussed further because of their relevance to this research work.
Prime
moverGenerator Rectifier Inverter
Load/
system
Figure 2.13: Block diagram of a distributed generation system with a power electronicsinterface.
32
PM
Synchronous
machine
Compensator
Turbine
SCR
inverter
Diode
rectifier
Local
loads/
grid
DC link
Grid side
controller
Figure 2.14: PMSG with diode rectifier and grid side compensator.
Converter topologies
The categories of power electronics systems referred to in section 2.6 above are usually
arranged in topologies best suited to particular applications with appropriate control
schemes. A comprehensive review of such topologies is provided in (Baroudi et al.,
2005). A brief discussion of some common topologies follows.
2.6.1 Topologies for synchronous machines
Figure 2.14 shows a permanent magnet synchronous generator charging a D.C. link via
an uncontrolled diode rectifier (Chen et al., 2003). A grid side thyristor based converter
regulates the turbine speed through the D.C. link voltage. While this scheme has a lower
cost than a hard switched inverter system, it employs an active compensator for reactive
power supply and for smoothing out harmonic distortion.
Figure 2.15 shows a permanent magnet synchronous machine (PMSG) connected
to a diode rectifier, and a self commutated current controlled pulse-width-modulated
(PWM) inverter connects the system to the load to the grid (Tan & Islam, 2004). While
this system is capable of supplying power from a low voltage to a stiff voltage system,
it nevertheless has the drawback of relatively low efficiency (Mohr & Fuchs, 2005). The
33
PM
Synchronous
machine
Turbine
VSI
Diode
rectifier
Local
loads/
grid
DC link
Grid side
controller
Figure 2.15: PMSG with diode rectifier and VSI grid side converter.
PM
Synchronous
machine
Turbine
VSIDiode
rectifier
Local
loads/
grid
DC link
Grid side
controller
DC
chopper
Generator
controller
Figure 2.16: PMSG with diode rectifier, VSI and intermediate DC chopper.
34
Turbine
VSIDiode
rectifier
Local
loads/
grid
DC link
Grid side
controller
DC
chopper
Generator
controller
~=
SM
Figure 2.17: Electrically excited synchronous machine with diode rectifier, VSI andintermediate DC chopper.
configuration of figure 2.16 has a voltage source inverter with an intermediate DC–
DC converter (Chen & Spooner, 2001, 1998a), which is an improvement to figure 2.15.
Its advantages over the scheme of figure 2.15 include possibility of: controlling the
generator side DC voltage, maintaining appropriate grid side inverter voltage, reduced
losses, greater flexibility of control (Chen & Spooner, 1998b). The system of figure
2.17 provides a satisfactory compromise between those of figures 2.15, 2.16 and 2.15. A
similar system is designed and analysed in (Jadric, 1998). The main generator output
is rectified by a diode bridge, in order to form a dc-link that feeds an inverter. Since
the diode rectifier used here provides no means of regulation, constant dc-link voltage
can be achieved only by adjusting the synchronous machine’s exciters field voltage vfd
using a DC link voltage feedback loop. Because of the advantages and relative simplicity
of the configuration shown in figure 2.17, it is adopted with some modification in this
research work to charge the DC link that provides an input to the inverter connected to
the DFIG rotor.
2.6.2 Topologies for the DFIG
In the modern DFIG configurations control is implemented through the rotor circuit and
the converter power rating is reduced since it is connected to the rotor. Two common
35
TurbineDiode
rectifier
SCR
inverter
Local
loads/
grid
DC link
Grid side
controller
DFIG
Figure 2.18: DFIG with static Kramer drive.
DFIG concepts are shown in figures 2.18 and 2.19
The static Kramer drive of figure 2.18 connects through a diode rectifier to the DFIG
rotor and a line commutated inverter to the grid or local load. The major drawback of
this arrangement is that the DFIG rotor circuit can only provide power under super-
synchronous operation but cannot receive power under sub-synchronous operation. A
solution to this drawback is the modified Kramer drive presented in (Uctug et al., 1994),
where the diode rectifier is replaced with a controlled thyristor rectifier (SCR). This so-
lution allows power delivery at both sub and super synchronous speeds, but is hampered
by firing and commutation problems with the rotor side converter. Harmonic distortion
generated by the grid side thyristor poses an additional problem.
The advent of IGBT and BJT power semiconductor devices gave rise to the modified
Scherbius drive (back-to-back rectifier-inverter pair) scheme of figure 2.19 (Pena et al.,
1996). This scheme consists of two conventional pulse-width-modulated (PWM) voltage
source converters connected across a capacitor on a common DC link. The grid side
converter is used to keep the DC voltage constant while the rotor side converter can be
used to control the torque and speed including the stator terminal power factor.
36
Turbine
Rectifier Inverter
Local
loads/
grid
DC link
Grid side
controller
DFIG
Rotor side
controller
Figure 2.19: DFIG with back to back converters (static Scherbius drive).
2.7 Control of power electronics interfaces
The control of power electronics interfacing electrical machines is a very widely published
discipline. The de-facto industry standards are based on two techniques (Casadei et al.,
2002; Le-Huy, 1999): Vector control (VC) or field oriented control (FOC), and direct
torque control (DTC). The two techniques are based on space vectors.
2.7.1 Space vectors and transformations
The idea of space vectors is a very useful tool in the analysis and control of electrical
machines. It can be used to describe any three phase system using an orthogonal set of
axes.
If we have a resultant vector quantity derived from a balanced three phase set of e.g.
currents i rotating at some speed ωe given by
i =√
2 iaejωet (2.17)
where ia is the rms time phasor of the phase a current. An observer moving at the same
37
a s
b s
c s
d s
q s
(a) three phase and stationary qd
d s
q s
q
d
0
(b) Stationary and rotating qd
Figure 2.20: Relationship between stationary and rotating space vector q − d axes.
speed ωe will see the current space vector i , as a constant spatial distribution. This is
equivalent to mathematically resolving any variables in the system that we want to see
onto an orthogonal rotating reference frame moving at the same speed as the observer.
Figure 2.20 (a) illustrates the geometrical relationship between the three phase a−
b− c axis and the stationary qsds axis while 2.20 (b) shows the relationship between the
stationary qsds and rotating q − d axes. The resolution of the space current vector i of
the balanced set of a − b − c currents in terms of its stationary qs − ds components on
the new rotating q − s axes is given by:
iqid
=
cos θ − sin θ
sin θ cos θ
isqisd
(2.18)
The angle θ is a function of the angular speed, ωt, of the rotating q − d axes the initial
38
angle δ(0)
θ(t) =
t∫0
ω(t)dt+ θ(0) (2.19)
where θ(0) is the initial angle at time t = 0. The inverse transformation is
isqisd
=
cos θ sin θ
− sin θ cos θ
iqid
(2.20)
The space vectors of other machine quantities (voltages, currents, magnetic fluxes, etc.)
can be defined in the same way as the stator current space vector.
2.7.2 Field oriented control (FOC)
Field oriented control is implemented in one of two ways; either by measuring specific
values of flux (direct vector control) or by calculating them (indirect vector control)
(Blaschke, 1972; Ogasawara et al., 1988). Figure 2.21 (a) illustrates the implementation
of an indirect field control system for the rotor circuit of a DFIG, while (b) illustrates
how the d − q reference frame is aligned to the rotor flux vector which rotates at the
stator frequency ωe. With this alignment, the variables are decoupled so flux and torque
can be independently controlled by the stator d- axis current ids and q-axis current iqs
respectively.
The reference value of stator current, i∗qs, is generated from the torque input command
as
i∗qr = − 3
P· LsLm· λd,s · T ∗e (2.21)
where
λq,s = 0 (2.22)
39
,abc ri
,abc sV
Current
regulator
DC-link
Grid
Encoder
Rotor side
converter
Gen
dq
abc
Compu
tation
*
qri
a
b
c
S
S
S
* * *, ,as bs csi i i
dq
abc
dq
abc
,abc si
Stator
flux
computation
Stator
flux angle
computation
,sV
,si
0( )sje
*
eT+
sje
,s
,ss
0
ddt
*
dri
*
r
*
,ri
r
+
-
-+
,d s
,abc ri
P
(a) FOC block diagram
q
d
SI
dsi
qsi
r
rI
e
a
Rotor axis
Rotor flux axis
sle
r
(b) FOC space vectors
Figure 2.21: Indirect field oriented control.
40
The stator α, β values of current are calculated as follows
λαs =
∫(Vα,s − iα,sRS) dt (2.23)
λαs =
∫(Vβ,s − iβ,sRS) dt (2.24)
The stator flux angle or position is used for coordinate transformation and it is generated
as follows
θs = arctan
(λβ,sλα,s
)(2.25)
and the torque command, T ∗e , is generated by passing the speed error through a PI
controller.
The generated current references i∗qr and i∗dr are then converted to a−b−c coordinates
i∗ar, i∗br and i∗cr which form the command to the current regulator. The commands are
then processed in the regulator to form the Sa, Sb, Sc switching pulses for the inverter .
2.7.3 Direct torque control (DTC)
The idea of direct torque control (DTC) is a departure from coordinate transformation
and decoupling control as applied in FOC. The method was introduced and published
in (Takahashi & Noguchi, 1986) and (Depenbrock, 1988) almost concurrently. The basic
concept is premised on the independent direct control of electromagnetic torque and flux
using voltage space vectors found in lookup tables. It is seen from equation (2.26) that
electromagnetic torque, Te, is a function of rotor and stator flux linkages λ and the angle
γ between them.
Te ∝LmLsLr
· λr · λs · sin γ (2.26)
If the rotor flux magnitude is kept constant, the generator torque can be controlled using
angle γ, (assuming a grid connection where the stator flux magnitude and angular speed
41
Table 2.2: Voltage space vectors used in DTC
Sector Sector Sector Sector Sector Sectorθ(1) θ(2) θ(3) θ(4) θ(5) θ(6)
∆Ψ = 1 ∆T = 1 Vr2 Vr3 Vr4 Vr5 Vr6 Vr1∆T = 0 Vr6 Vr1 Vr2 Vr3 Vr4 Vr5
∆Ψ = 0 ∆T = 1 Vr3 Vr4 Vr5 Vr6 Vr1 Vr2∆T = 0 Vr5 Vr6 Vr1 Vr2 Vr3 Vr4
is constant).
The lookup table sets an inverter switching sequence that depends on the outputs of
flux and hysteresis comparators as input. Upon establishment of a switching sequence,
a voltage phasor applied to the rotor makes the rotor flux change. The tangential
component of this applied voltage is used to control (increase or decrease) the torque
angle whereas the radial component is used to control flux magnitude.
Figure 2.22 (a) shows a block diagram of the DTC control method while (b) illustrates
the principle of implementation. In 2.22 (b), stator flux has constant magnitude and
rotates at slip speed, ωslip, in the rotor reference frame. Any of the six active voltage
vectors (~VR1 − ~VR6) and two zero vectors can be used to control rotor flux. An optimal
switching pattern is used to define the voltage vector that corresponds to any torque
and flux controller logical outputs and current sector. This is illustrated in table 2.2
2.8 Cascaded machines
Over the past several years, there has been a revival of interest in new ways of connecting
generators at distribution network level. Novel configurations can expand the means by
which electrical power can be generated and improve the efficiency with which electricity
is generated and transmitted. The induction machine as a generator has recently been
favoured as a means of variable speed constant frequency electricity generation owing
42
,abc si
DC-link
Grid
Rotor side
converter
Gen
Switching
table
a
b
c
S
S
S
*
r
rEstimator
eT
,abc sV
ddt
( )t
eT*
eT
(a) DTC block diagram
a
b
s
1RV
3RV 2RV
4RV
5RV6RV
R
3 0*R RV T
0R
slip
1
23
4
5 6
(b) DTC space vectors
Figure 2.22: DTC torque and flux control.
43
to it’s versatility and a number of publications have looked at ways of connecting these
machines in cascade. Indeed a significant proportion of suggested constant-frequency
variable-speed a-c generators employ some form of induction machine coupled with aux-
iliary a-c or commutator machines (Riaz, 1959) which in many cases, form an integral
part of the induction machine and a selection of the more relevant ones are given here.
It is well known that an induction generator requires a separate source of leading
reactive power to function (Murthy et al., 1982; Barkle & Ferguson, 1954) and this is
usually obtained from synchronous plant elsewhere in the electrical system (Anaya-Lara
et al., 2005). Some early works that sparked interest in the possibility of cascaded op-
eration of electrical machines include the publications (Burbidge et al., 1967; Broadway
et al., 1974; Wallace et al., 1993, 1990; Williamson et al., 1997).
(Ooi & David, 1979b) proposed a system comprising of a mechanically and electri-
cally coupled induction generator and synchronous condenser set, with the synchronous
condenser supplying the induction generator with magnetising reactive current and real
power to overcome its mechanical and electrical losses. This was essentially a DFIG
with a static scherbius drive as illustrated in figure 2.23. In the Ooi concept, the sta-
tors of the two machines are directly connected together while the rotor of the DFIG
is asynchronously connected to the stator of the synchronous machine through a power
electronics interface and the synchronous machine only operates in motoring mode.
Meanwhile (Shibata & Taka, 1992) proposed an improvement to the Ooi concept
in which the induction machine is replaced by two cage rotor induction machines with
rotors connected in reverse phase to form a cascaded mechanical and electrical coupling
with the synchronous machine. The induction and synchronous machine systems are
again asynchronously interfaced with a frequency converter drive and the synchronous
machine still operates in motoring mode as in the Ooi scheme. Although this provides
an alternative solution to the magnetising reactive power needs of the induction machine
cascade, use of standard “off- the-shelf” machines is not possible and the self cascaded
44
DC-link
DFIG
mSC
Mechanical
coupling
DC field
supply
Load
Firing signal
Scherbius
drive
Figure 2.23: Ooi concept.
induction machine must be designed and built. There is also considerable complexity in
the system control. Similar configurations were considered in (Patin et al., 2009; Zhang
& Zhu, 2011) and the notable complexity was highlighted. Other publications that have
dwelt on cascaded induction machines with similar schemes include (Burbidge et al.,
1967; Ferreira et al., 2009; Hopfensperger et al., 1999).
(Ansel & Robyns, 2006) proposed an autonomous variable speed micro hydropower
station with a doubly fed induction generator coupled mechanically and electrically
via a static Kramer drive to a permanent magnet synchronous machine. The model
was based on an energetic macroscopic representation EMR approach and is depicted
in figure 2.24. In this scheme, the permanent magnet machine provides the reactive
power needs of the DFIG. Using permanent magnet synchronous machines eliminates
copper losses and introduces a degree of simplicity since they are generally smaller in
size and more efficient than their electrically excited counterparts. However, magnetic
characteristics of permanent magnets change with time and they become prohibitively
expensive for higher power applications. In addition, synchronous machines are generally
45
DC-link
DFIG
mPMSG
Mechanical
coupling
Load
Kraemer
drive
Kaplan
turbine
Figure 2.24: Ansell concept.
lightly damped (Lipo & Krause, 1968), and since permanent magnet machine rotors are
usually not equipped with damper windings (Ong, 1998), this is a potential source of
oscillatory instability (Kundur et al., 1994).
2.9 Stability considerations
The system under study, like any dynamical system, can be modelled by dynamic integro-
differential equations. Stability of such a system is defined as its ability to re-establish a
satisfactory steady state after a perturbation from its initial operating mode. Stability
of such systems is an important qualitative characteristic and is a major scientific and
technological factor in the design and operation of electrical systems and machines.
The stability problem is broadly classified under: steady state stability (small signal
stability), or transient stability (Kundur et al., 1994; Anderson & Fouad, 2003).
46
2.10 Small signal stability
There is ample literature available on small signal stability analysis for electrical systems
and machines and power systems in general. These include the works (Kundur et al.,
1994; Anderson & Fouad, 2003; Anderson et al., 1990; Lipo & Krause, 1968; Stephenson
& Ula, 1978; Mei & Pal, 2007) among others. The general small signal stability problem
for any dynamic electrical system may be formulated in a standard mathematical and
engineering form as a scalar eigenvalue, and vector eigenvector problem (Molzahn, 2010).
The system may be expressed by the set of nonlinear first order differential equations
(Kundur et al., 1994)
x = fi (x1, x2, · · · , xn; u1, u2, · · · , ur; t) i = 1, 2, · · · , n (2.27)
where n is the system order, r is the number of inputs, and the system can be written
in state variable format for an autonomous system as follows:
x = f (x,u) (2.28)
The system outputs may be expressed in terms of the state variables and inputs as
follows
y = g (x,u) (2.29)
Because it is difficult to analyse the dynamic relationship between the inputs and outputs
of the system due to its nonlinear nature, it is necessary to linearize the state space
representation. The mathematical representation then takes the form
∆x = A∆x+B∆u
∆y = C∆x+D∆u (2.30)
47
In (2.30), x represents the system states, u represents the system inputs, while A is a
linearized Jacobian matrix that contains information of all the system elements. Stability
implications for the system can be seen through eigenvalues of the system matrix [A],
and these can be solved as follows (Vittal et al., 2009)
det (A− λI) = 0 for λ1, λ2, · · · , λn (2.31)
The eigenvectors of A represent the change activity effects of A’s eigenvalues. The right
eigenvector is calculated as follows
Aφi = λiφi for i = 1, 2, · · · , n (2.32)
The left eigenvector is found by solving
ψA = λiψi for i = 1, 2, · · · , n (2.33)
Differentiating (2.32), with respect to akj, and appropriate substitution, we obtain
∂λi∂akk
= ψikφki = Pkk (2.34)
which is the linear product of the left and right eigenvectors. ψki is a measure of the
activity of the ith mode in relation to element k and gives the mode shape of that
particular state variable, known as the participation factor. The participation factor
indicates the net participation of a mode with respect to a particular system element.
A large participation factor indicates a significant impact on the system while a low one
indicates a negligible impact on the system. Contribution of an element ψik is a measure
of how much a mode contributes to the system stability, it weighs the contribution of
the eigenvalue’s activity to the mode.
48
The mode shape, together with the mode shape for a given sensitivity provides
information as to the system stability. The mode shape comes as a complex conjugate
pair for a given time instant
(a+ jb) e(σ−jω)t + (a− jb) e(σ+jω)t (2.35)
In (2.35), the real part of the complex conjugate determines the speed of system oscilla-
tion damping, while the imaginary part b determines the specific frequency ω at which
oscillations occur. Premised on (2.35), the proposed system model can be subjected to
various contingency events, and stability can be assessed for each situation. In addition,
the relative contribution of different components in the complex system to the system
instability can be readily identified.
For a complex eigenvalue set λ = σ± jω, the oscillation frequency in Hz is given by
f =ω
2π(2.36)
the damping ratio and amplitude of decay time constant respectively are given by (2.37)
and (2.38)
ζ =−σ√
[σ2 + ω2](2.37)
τ =−σ√|σ2 + ω2|
(2.38)
2.10.1 Small signal stability of the synchronous generator
The inherent instability of the synchronous machine was recognised as early as in the
1900s. The work of (Prescott & Richardson, 1934) established among other things, that
any normal alternator has a tendency towards instability because the armature reactance
49
is significantly larger than the armature resistance. The result is that a so called negative
damping appears, contributing to inherent stability of the machine. As an individual en-
tity, synchronous machine stability issues have received, and continue to receive a great
deal of attention. (Stephenson & Ula, 1978) established an effective method of stability
related computation using a complete model of the machine for the simple case of a sin-
gle machine infinite busbar (SMIB). This was based on well known methods published
by (Laughton, 1966), and showed it’s usefulness for more complex multimachine studies.
Meanwhile (Undrill, 1968) had earlier described an efficient method for extracting the
[A] system matrix and its application to a multi machine system, which provided im-
proved computational speed to the Laughton method. Relevant to present day demands
of flexibility in the form of variable speed operation of synchronous machines is an early
publication of (Lipo & Krause, 1968), in which a stability analysis for variable frequency
operation of synchronous machines is studied. Using the Nyquist stability criterion (Dorf
& Bishop, 2005), Lipo and Krause established that in some cases, synchronous machine
instability is prevalent at low speeds. An important contribution of (Lipo & Krause,
1968) relevant to the model used in this thesis is the expression of external machine dis-
turbances in the system matrix. Stability issues affecting the synchronous machine were
further comprehensively investigated in (Cornell & Novotny, 1972), in which the effects
of frequency, load torque, rotor inertia, stator resistance, and other machine parameters
and constraints are illustrated, and suggestions for improved stability performance are
given. A study on the synchronous machine connected via a power electronics interface
is treated in depth in (Krause & Lipo, 1969). (Colby et al., 1985) performed a state
space analysis on a line commutated inverter (LCI) fed synchronous machine drive, while
a more recent study with an average model of the machine with an electronic converter
was conducted in (Jadric, 1998). Comprehensive and detailed texts on methodologies for
the modelling and analysis of synchronous and induction machines as separate entities
are published in (Krause et al., 2002; Kundur et al., 1994; Anderson & Fouad, 2003).
50
2.10.2 Small signal stability of the doubly fed induction generator
Early interest in the induction machine was sparked by its robust construction and sim-
plicity, it appears that limitations to its usefulness mainly stemmed from modelling and
control related difficulties. Recent trends in power generation from renewable resources,
particularly from wind powered systems, has spurred intense renewed interest in the
induction generator as a source of electric power generation. Advances in the areas of
power electronics, enhanced computing capabilities and control engineering, including
environmental concerns, have fostered further interest in it as a main source of electric
power generation.
Early works of the past century concentrated mostly on ways of modelling, analysing,
and providing the induction generator with suitable excitation and ensuring its stable
operation. In the publication of (Stanley, 1938), the need for analysis of the induction
machine under different loading conditions was evaluated. Theories on the process of
excitation of induction machines were postulated in (Murthy et al., 1982; Elder et al.,
1983), other publications that emphasized the feasibility of using the induction generator
as a low cost option for power generation include (Gadenz et al., 1968; De Mello &
Harmnett, 1981; Elder et al., 1984). In the development of solutions for US military
airborne applications, (Riaz, 1959) presented an approach for using the wound rotor
induction machine as an integrated variable speed, constant frequency power generator.
Riaz’s publication examined the characteristics of the induction machine in different
modes of operation.
Stability analysis of the induction machine has been addressed in several papers.
Some early works include (Rogers, 1965), in which a linearised analysis of the induc-
tion machine was presented and the root locus method was used for stability analysis,
while (Jackson & Phillips, 1968), used eigenvalues of the linearised machine equations
for a variable-speed doubly fed induction to establish steady-state stability boundaries.
51
Recent works on the stability of wound rotor induction machines have mainly centred
on wind driven generators. (Alakula et al., 1992) describes the induction machine as a
resonant system and explores the sources of oscillations. Other recent studies on small
signal stability of the DFIG based on eigenvalue/participation factor analysis include
works by (Ostadi et al., 2009; Mei & Pal, 2007; Mishra et al., 2009; Wu et al., 2007). A
similar study that includes Hopf bifurcation analysis is published in (Yang et al., 2011).
While the Hopf bifurcation analysis readily lends itself to the establishment of stability
boundaries and can help in the selection of practical parameters for maintaining stable
operation, it is not applied in this research thesis. Its usefulness and application are
however noted and will be explored in subsequent works.
Stable operation is key for any dynamical system. Since the proposed topology
is a novel and complex dynamic electrical system, a stability study is essential. It
is important to determine occurrences of instability, identify the contributing factors
and their relative participations. The modelling method chosen for this thesis for the
proposed machine topology is a natural extension of the method reported in (Laughton,
1966). The stability analysis is based on eigenvalue and modal methods.
2.11 Magnetising reactive power supply for the DFIG
The induction machine equivalent circuit shown in figure 2.25 can be used to describe
the relation between stator and rotor currents, and the required magnetisation current
for operation of the machine. The desired stator current is dictated by the complex
power, and the corresponding rotor current can easily be determined. Current in the
magnetising branch as seen in figure 2.25 is given by
iqm = iqs + iqr (2.39)
52
00qsV 0qr
r
V
s
sR qsjX
mV mjX
qrjXrR
s
qsiqri
mi
Figure 2.25: Wound rotor induction machine steady state equivalent circuit.
From (2.39), it is evident that the reactive power can be controlled through the d − q
components using appropriate control schemes that can be found in (Pena et al., 1996;
Silva & Lyra, 1993; Neris et al., 1999), among others. The magnetising current iqm,
is in quadrature to the stator voltage Vs. For a grid connected system, iqs can either
be supplied from the grid or grid side converter, while iqr is supplied via the rotor side
converter. Reactive power needs of the DFIG can therefore be supplied from either or
both of these two sources.
2.12 Summary
Many remote communities are located near sites with the potential for small hydro
generation, these areas often have no connection to the grid, or have long distribution
lines that are often weak. The presence of an exploitable small hydro resource in itself
provides a natural source of energy storage. This can solve the twin problem of supply
reliability and service continuity that are often a challenge when dealing with other
intermittent renewable energy sources.
Literature and current practice have shown that the doubly fed induction machine
is a robust, durable machine that is easily capable of operating over a wide speed range
and its performance is largely determined by the features and technical limits of its
controlled power electronic devices (PED). A major technical feature that characterises
modern DFIG power electronic control is very fast response. A drawback for these PEDs
53
are tight thermal constraints, and limited overload capacity. With a connection of the
PEDs in the rotor circuit, the converter sizes are greatly reduced. With a weak grid
connection or an isolated system (either intentional or unintentional island), the DFIG
may be constrained in terms of a reliable source of power both for its initial, and steady
state excitation.
While the synchronous machine offers excellent flexibility in terms of reactive power
support, an additional degree of control through field excitation, and overload capacity,
its dynamics are slower and its use as a main generator interfaced to power electronic
devices considerably increases converter size. This is because it can only operate with
a full scale converter, having at minimum the generators full load rating. This research
work aims to exploit advantages of the two machines in the form of an integrated power
generation topology that can be used in variable speed pumped storage applications.
54
CHAPTER 3. DESCRIPTION OF PROPOSED SCHEME
AND PRIMITIVE MATHEMATICAL MODEL
This chapter introduces the proposed integrated plant topology and describes its oper-
ation. A primitive mathematical model of the system is developed in d− q coordinates
based on an equivalent system circuit. An analysis carried out on this open loop model
gives insight into possible stability and controllability issues, including the state variables
and machine circuits that significantly influence system dynamics.
The derived model is cast in state variable form. A linearisation procedure is then
carried out to establish steady state operating points from which modal analysis is
possible. Participation factors are computed to determine key factors influencing modes.
From the model, rank sufficiency of the controllability matrix is established for the speed
range of interest.
3.0.1 Electrical sub-system
The proposed system is illustrated in figure 3.1, and comprises of a doubly fed induction
generator (DFIG) with a synchronous machine (SM) exciter mechanically and electrically
coupled to it. An electronic power converter electrically couples the DFIG rotor and SM
exciter stator. The electronic converter consists of a rotor side converter (RSC), a PWM
thyristor inverter connected across the terminals of the DFIG rotor and a diode rectifier
connected across the synchronous machine stator terminals. The capacitor connected
in the DC link decouples operation of the two converters. A fast responding DC-DC
55
DFIG
Control
Lo
ca
l
Lo
ad
s
Synchronous
machine pilot
exciter
Grid
Turbine
RSC ESC
Variable pitch
bladed pump
d/dt
Rotor angle
position encoder
r
r
SM
Figure 3.1: Proposed system topology.
IGBT-based PWM buck-boost converter is also connected in the DC link to damp short
term power oscillations and hence improve low voltage ride through (LVRT) capability
(Abbey & Joos, 2007). The converter has no direct connection to the load or connected
distribution network, switching harmonics are thus eliminated. The arrangement is such
that the synchronous machine exciter takes care of the slower electro-dynamics while the
power electronics converter takes care of the faster electro-dynamics associated with the
system.
3.0.2 Mechanical sub-system
The DFIG, synchronous machine and variable pitch pump are mechanically coupled
through a common drive-train. This type of coupling ensures that the overall plant
has unrestrained flexibility of operation; it can be operated in a purely electrical power
generation mode when the pump blades are fully retracted, it can be operated as a pure
pumping station with no power generation while avoiding the traditional reverse pump-
56
Radial piston-poppet
mechanism
Pitch at
1v
2v
1
2
090
Outflow
Inflow
Figure 3.2: Variable pitch concept.
turbine mode, and it can operate in partial power-pump mode depending on the pump
blade pitch angle β. In effect, slip energy is used to pump water back into the reservoir.
The pump is fitted with pitched blades and the pitch angle β depends on the electrical
loading of the DFIG. At light loads, the blades are protracted to increase the pumping
rate and at heavy loads, retracted to reduce the pumping rate. Figure 3.2 illustrates the
variable pitch concept. The only pumps known to the author with similar technology
are manufactured by Hitachi plant technologies (Hitachi Plant Technologies, 2008).
3.0.3 Plant layout
Traditional pumped hydro storage systems transfer water between reservoirs at two
different elevations. In the energy storage cycle, water is pumped from a lower reservoir
to an upper reservoir. In the energy recovery cycle, water released from the upper
reservoir is used to power a turbine after which it flows back to the lower reservoir. Such
a traditional pumped storage plant layout is shown in figure 3.3.
The proposed plant layout is illustrated in figure 3.4 and could be applied to schemes
for sea water pumped hydro power projects like that of the Okinawa project shown in
57
Reversible turbine-pump
Figure 3.3: Traditional pumped storage plant layout.
Main reservoir
Turbine
Tail race
Variable pitch pump
Synchronous
generator
DFIG
To natural
waterway
Discharge
to Storage
Drive shaft
Figure 3.4: Proposed pumped storage concept plant layout.
figure 3.5. The proposed scheme would only utilize a single reservoir but require a
separate channel for transferring water back into the reservoir.
58
(a) Okinawa sea water pumped storage hydro power station.
(b) Profile of pumped hydro power Okinawa station.
Figure 3.5: Plant layout (Courtesy: Hitachi review, vol 47).
59
3.1 System mathematical model
As a first step in the analysis of the proposed system, it is necessary to formulate a math-
ematical model of the plant, it is further necessary to express the system state variables
as explicit functions of the system parameters. This helps clarify various factors that
may influence stable operation of the system and their relative influences on stability. A
framework is therefore provided, from which a linearized model of the plant is formed.
The linearised model is then used to determine the small signal characteristics which
help in the determination of suitable control actions.
Figures 3.6 (a) and (b) show the equivalent q and d axes circuits of the topology
of figure 3.1 from which the electrical system equations can be derived. The converter
switching dynamics are ignored as they are not relevant at this stage. The current
control for the d.c. link is assumed to be adequately fast and accurate, therefore the
d.c. link capacitor dynamic model can be excluded at this stage (Mei & Pal, 2007). The
ensuing equations will then be cast in a form suitable for depicting the combined plant
dynamics.
3.1.1 DFIG mathematical model
The balanced condition Park transformed DFIG voltage equations in the synchronously
rotating reference frame with currents as state variables derived from the voltage and
flux linkage equations are as follows (Krause et al., 2002)
veqs
veds
v′eqr
v′edr
=
Rs + pωbLs
ωeωbLs
pωbLm
ωeωbLm
−ωeωbLs Rs + p
ωbLs −ωe
ωbLm
pωbLm
pωbLm sωe
ωbLm R
′r + p
ωbL′r sωe
ωbL′r
−sωeωbLm
pωbLm −sωe
ωbL′r R
′r + p
ωbL′r
ieqs
ieds
i′eqr
i′edr
(3.1)
60
DFIGSynchronous Machine
sm
s ds sm
qsL
sm
mqL
'
1
sm
l k qL
'
1
sm
k qR'
2
sm
k qR
'
2
sm
l k qL
sm
sR
sm
qse
+
-
sm
qsi+++
---
'
2
sm
kqi
'sm
kqi
'
1
sm
kqi
sm
mqi( )e im
qse
( )im
sR( )im
s d s ( )im
q sL( )im
qrL ( )im
s r dr ( )im
rR
( )im
mi
( )im
mL
( )im
qri( )e im
qsi
+
-
( )e im
qrV
(a) q-axis equivalent circuit.
sm
mqL
'
1
sm
l k dL
'
1
sm
fdR'
2
sm
k qR
'
2
sm
l k qL
sm
sR
+
-
+++
---
'
2
sm
kqi '
1
sm
kqi
sm
mqi
DFIGSynchronous Machine
( )im
de
( )im
sR( )im
dsL ( )im
drL ( )im
s r dr ( )im
rR
( )im
mi
( )im
mL
( )im
dri( )e im
dsi
( )e im
drV( )sm
de
+
-
sm
s qs sm
dsLsm
dsi'sm
fdi ' 'sm
fd kdi i
am
s qs
+
-'sm
fdV
(b) d-axis equivalent circuit.
Figure 3.6: Proposed plant d and q axes equivalent circuits.
61
where s is the slip defined by
s =ωe − ωrωe
(3.2)
The electromagnetic torque expressed in terms of the state variables is
Te = Lm
(ieqsi
′edr − iedsi
′eqr
)(3.3)
and the dynamic torque equation is
Te = 2Hpωrωb
+ Tm (3.4)
3.1.2 Synchronous machine mathematical model
The balanced Park transformed synchronous machine voltage equations in the rotor
reference frame are represented in the same manner as in subsection 3.1.1.
vrqsvrds
v′rkq1
v′rkq2
v′rfd
v′rkd
=
−Rs− pωbLq −ωr
ωbLd
pωbLmq
pωbLmq
ωrωbLmd
ωrωbLmd
ωrωbLq −Rs− p
ωbLd −ωr
ωbLmq −ωr
ωbLmq
pωbLmd
− pωbLmq 0 r
′kq1+ p
ωbL′kq1
pωbLmq 0 0
− pωbLmq 0 p
ωbLmq r
′kq2+ p
ωbL′kq2 0 0
0 −LmdR′fd
(pωbLmd
)0 0
Lmd
R′fd
(R′fd+ p
ωbLfd
)Lmd
R′fd
(pωbLmd
)0 − p
ωbLfd 0 0 p
ωbLmd R
′kd+ p
ωbL′kd
irqsirds
i′rkq1
i′rkq2
i′rfd
i′rkd
(3.5)
The electromagnetic torque is
Te = Lmd
(−irds + i
′rfd + i
′rkd
)irqs − Lmq
(−irqs + i
′rkq1 + i
′rkq2
)irds (3.6)
The dynamic torque equation is
Te = −2Hpωrωb
+ Tm (3.7)
62
whereas the rotor angle is
δ =ωbp
(ωr − ωeωb
)(3.8)
3.1.3 Combined inertia and electromagnetic torque equations
The total mechanical input torque is the sum of the two machines’ electromagnetic and
inertial torques. Using each machine’s individual electrical frequency base ωb1 and ωb2,
the inertias H1 and H2 for the DFIG and synchronous machine respectively can be
combined into a single equivalent inertia (Anderson & Fouad, 2003; Milano, 2010) which
results in the combined inertia
Heq =H1H2
(H1 +H2)(3.9)
and the combined plant dynamic equation given by
Tm1 + Tm2 = 2(Heq)pωrωb− Te2 + Te1 (3.10)
3.1.4 Combined electromechanical plant mathematical model
The proposed scheme involves complex electromechanical interactions of the two electro-
mechanically coupled machines. The voltages of the common electrical circuit shared by
the stator of the synchronous machine and rotor of the DFIG, plus the common shaft
dynamics form mutual constraints in the derived system equations. The independent
driving force that constitutes the rotor input voltages of the DFIG are the stator output
voltage of the synchronous machine, while the external disturbance to the synchronous
machine stator circuit are the DFIG rotor voltages. Assuming that both sources are
balanced, we must relate these voltages to the appropriate machine circuits through a
transformation matrix. This is because the individual machine equations are expressed
in different reference frames.
63
Equation (3.11) performs the transformation of the synchronous machine stator out-
put voltages to the DFIG, while (3.12) performs a similar transformation to the syn-
chronous machine.
f eqds = T · f rqds (3.11)
f rqds = T−1 · f eqds (3.12)
In (3.11) and (3.12), T and T−1 are the transformations (Krause et al., 2002) given by
(3.13) and (3.14)
T =
cos(θr) sin(θr)
− sin(θr) cos(θr)
(3.13)
T−1 =
cos(δ) − sin(δ)
sin(δ) cos(δ)
(3.14)
The electromechanical model of the plant with the transformations of (3.11) and
(3.12) is given in (3.15) and combines the full mathematical description of the plant and
its dynamics.
ve(im)qs
ve(im)ds
cos θrvrqs′+sin θrvrds′− sin θrvrqs′+cos θrvrds′
cos δveqr−sin δvedrsin δveqr+cos δvedr
v′r(sm)kq1
v′r(sm)kq2
v′r(sm)xfd
v′r(sm)kd
0Tm
=
χ11 ··· ··· ··· χ112χ21 ··· ··· ··· χ212χ31 ··· ··· ··· χ312
......
......
...
χ51
......
... χ512
......
......
...
χ91
......
... χ912
......
......
...
χ111
......
... χ1112χ121 ··· ··· ··· χ1212
ie(im)qs
ie(im)ds
cos θrirqs′+sin θrirds′− sin θrirqs′+cos θrirds′
cos δieqr−sin δiedrsin δieqr+cos δiedr
i′r(sm)kq1
i′r(sm)kq2
i′r(sm)fd
i′r(sm)kdδsmωrωb
(3.15)
The elements χij of matrix equation (3.15) are standard system matrix elements of
64
DFIG machine
equations.
.
.
.
r rdtω θ=∫imT
r
qdsVr
qdsi
T1T −
e
qdsV e
q d si
Synchronous
machine
equations.
.
.
.
δsmT
e
qdsVe
qdsi1−TT
r
qdsV r
q d si
MTMech
input
rθ
rθ
δ δ
Figure 3.7: Interconnections in block model form.
voltage equations for induction and synchronous machines and can be found in any
good electrical machines textbook such as (Krause et al., 2002) so they are not given
here. Elements 3 − 6 of the left-hand vector depict the cross-coupling in the machine
voltage equations, while those of the right hand vector depict the cross coupling of
the machine currents, and figure (3.7) depicts the interconnection with the machine
constraint dynamics in block-model form.
3.2 Linearisation of Machine equations
The behaviour of a non-linear dynamic system at a critical point x0, is similar to the
behaviour of the same system linearised at x0. Any autonomous dynamic system may
be represented by a set of first order differential equations of the form
x = f(x, u) (3.16)
65
The set of equations depicted by (3.16) linearised about a point x0 satisfies (3.17),
(Tomim, 2005).
∆x = J(x0, u0)∆x (3.17)
J(x0, u0), is the Jacobian of the system at x0. Using Lyapunov’s first method, stabil-
ity of the system represented by (3.16) can be assessed by examining the eigenvalues.
Linearisation of the system machine equations is accomplished by applying Taylor’s ex-
pansion about an operating point. After expressing each machine in its own reference
frame, all the machine variables are perturbed from their equilibrium points by letting:
x = x0 + ∆x and u = u0 + ∆u (3.18)
Hence:
x = x0 + ∆x (3.19)
= f [(x0 + ∆x), (u0 + ∆u)] (3.20)
The linearised form of (3.16) takes the form
∆x = A∆ x+B∆ u (3.21)
Equation (3.21) depicts the linearised small signal stability analysis model. The Jacobian
matrix J(x0, u0) of the dynamic system at x0 is evaluated by solving f(x, u) = 0.
Linearising the transformations of (3.11) and (3.12) yields respectively
∆f rqds = T∆f eqds + F r∆δ (3.22)
66
∆f eqds = T−1∆f rqds + F e∆δ (3.23)
where F r and F e are steady state values of the state variables
F r =
−f rds0f rqs0
; F e =
f rds0
−f rqs0
(3.24)
and the T transforms now take the respective forms
T =
cos(θo) sin(θo)
− sin(θo) cos(θo)
(3.25)
T−1 =
cos(δo) − sin(δo)
sin(δo) + cos(δo)
(3.26)
The subscripts ′0′ denote steady state values while δo - is the steady state angle by which
the synchronous machine terminal voltage angle δ leads/lags the DFIG rotor angle θo in
electrical radians at any given point in time.
Equation (3.27), provides a linearised mathematical description of the plant, the
transformations T and T−1 are defined in 3.25 and 3.26.
∆ve(im)qs
∆ve(im)ds
T−1·∆ve(im)qr
T−1·∆ve(im)dr
T·vr(sm)qs
T·vr(sm)ds
∆v′r(sm)kq1
∆v′r(sm)kq2
∆e′r(sm)xfd
∆v′r(sm)kd0
∆Tm
=
z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112z11 z12 z13 z14 z15 z16 z17 z18 z19 z110 z111 z112
∆ie(im)qs
∆ie(im)ds
T−1·ieqrT−1·iedrT·irqsT·irds
∆i′r(sm)kq1
∆i′r(sm)kq2
∆i′r(sm)fd
∆i′r(sm)kd
∆δsm∆ωrωb
(3.27)
In the seventh order model used in this thesis for the synchronous machine, the voltage
67
dynamics of the dampers can and will be neglected henceforth. Although the approach
described in this thesis can also be used with the inclusion of these damper voltages,
studies have shown that the damper circuit parameters can not be estimated accurately
using small perturbation of the field voltage (Karrari & Malik, 2003). Therefore ∆v′r(sm)kq1 ,
∆v′r(sm)kq2 and ∆v
′r(sm)kd in (3.27) will be set to zero in the affine linearized state space
model.
It can be noticed that additional non-linearities are introduced in the leftmost and
rightmost vectors of (3.15) and also appear in (3.27). These non-linearities must be de-
coupled from the variables in order to cast them in standard state variable form. This
allows compact formulation of the Jacobian, which in turn eases integro-differential cal-
culations. After decoupling it is now possible to partition and apply the transformations.
The complete equations are expressed as follows
∆ve(im)qs
∆ve(im)ds
∆ve(im)qr
∆ve(im)dr
∆vr(sm)qs
∆vr(sm)ds
∆v′r(sm)kq1
∆v′r(sm)kq2
∆e′r(sm)xfd
∆v′r(sm)kd0
∆Tm
=
a11 a12 a13 a14 a15 a16 a17 a18 a19 a110 a111 a112a21 a22 a23 a24 a25 a26 a27 a28 a29 a210 a211 a212a31 a32 a33 a34 a35 a36 a37 a38 a39 a310 a311 a312a41 a42 a43 a44 a45 a46 a47 a48 a49 a410 a411 a412a51 a52 a53 a54 a55 a56 a57 a58 a59 a510 a511 a512a61 a62 a63 a64 a65 a66 a67 a68 a69 a610 a611 a612a71 a72 a73 a74 a75 a76 a77 a78 a79 a710 a711 a712a81 a82 a83 a84 a85 a86 a87 a88 a89 a810 a811 a812a91 a92 a93 a94 a95 a96 a97 a98 a99 a910 a911 a912a101 a102 a103 a104 a105 a106 a107 a108 a109 a1010 a1011 a1012a111 a112 a113 a14 a15 a16 a17 a18 a19 a110 a111 a112a11 a12 a13 a14 a15 a16 a17 a18 a19 a110 a111 a112
∆ie(im)qs
∆ie(im)ds
∆ieqr∆iedr∆irqs∆irds
∆i′r(sm)kq1
∆i′r(sm)kq2
∆i′r(sm)fd
∆i′r(sm)kd
∆δsm∆ωrωb
(3.28)
The elements of a12,12 in (3.28) and the decoupling multiplications are shown in the
appendix.
3.2.1 Linearised DFIG state space equations
In order to express the DFIG equations in linearised state variable form, the derivative
terms of the DFIG portion of (3.15) are separated out and state variable representation
68
takes the form
Epp∆x = Fk∆x + ∆u (3.29)
where p is the differential operator. The vector of state variables is ∆x = [∆ids,∆iqs,∆idr,∆iqr,∆ωrωb
]T
and ∆u is the input vector [∆vds,∆vdr,∆vqs,∆vqr,∆TL]T . Elements Ep contain deriva-
tive terms while Fk contain the remaining variables and they are given respectively as
follows
Ep =1
ωb
[ Ls 0 Lm 0 00 Ls 0 Lm 0Lm 0 Lr 0 00 Lm 0 Lr 00 0 0 0 −2H1ωb
](3.30)
Fp =
Rs
ωeωbLs 0 ωe
ωbLm 0
−ωeωbLs Rs −ωe
ωbLm 0 0
0 S0ωeωbLm Rr S0
ωeωbLr −Lmieds0−Lri
edr0
−S0ωeωbLm 0 −S0
ωeωbLr Rr Lmieqs0−Lrieqr0
Lmiedr0 −Lmieqr0 −Lmieds0 Lmieqs0 0
(3.31)
Then
p∆x = Ep−1Fk∆x + Ep
−1∆u (3.32)
which is the standard state variable form and
A = Ep−1Fk (3.33)
B = Ep−1 (3.34)
3.2.2 Linearised synchronous machine state space equations
The linearised synchronous machine equations in state variable form are derived and
arranged in a similar manner to the DFIG ones and derivatives of the synchronous
machine portion of (3.15) are also separated and written in the same form.
Ep1p∆x = Fk1∆x + ∆u1 (3.35)
69
The state variables are[∆i′eqs′ ,∆i
′eds′ ,∆i
′rkq1,∆i
′rkq2,∆i
′rfd,∆i
′rkd,
∆ωrωb,∆δ
]T, the input vector
is[∆v
′eqs′ ,∆v
′eds′ , 0, 0,∆v
′rfd, 0,∆TL, 0
]T. Separating out derivative terms from the rest
yields
Ep1 =1
ωb
Lq 0 −Lmq Lmq 0 0 0 00 −Ld 0 0 Lmd Lmd 0 0
−Lmq 0 Lkq1 Lmq 0 0 0 0−Lmq 0 −Lmq Lkq2 0 0 0 0
0 −L2mdRf
0 0LmdLfRf
L2mdRf
0 0
0 −Lmd 0 0 Lmd Lkd 0 00 0 0 0 0 0 2H2ωb 00 0 0 0 0 0 0 ωb
(3.36)
and
Fk1 =1
ωb
−Rs −ωeωbLd 0 0 ωe
ωbLmd
ωeωbLmd −Ldirds0+Lmdi
rf0 0
ωeωbLq −Rs −ωe
ωbLmq 0 0 Lmd Lqirqs0 0
0 0 Rkq1 0 0 0 0 00 0 0 Rkq2 0 0 0 00 0 0 0 Lmd 0 0 00 0 0 0 0 Lkd 0 0
Lmqirds0− Lmdirqs0+ −Lmqirds0 −Lmqi
rds0 −Lmdi
rqs0 −Lmdirqs0 0 0
Lmd(irds0−irf0) Lmqirqs00 0 0 0 0 0 −ωb 0
(3.37)
Then
p∆x = Ep1−1Fk1∆x + Ep1
−1∆u (3.38)
which is the standard state variable form and
A = Ep1−1Fk1 (3.39)
B = Ep1−1 (3.40)
3.3 Small signal stability analysis of proposed scheme
3.3.1 Small Signal Stability Analysis Model
The state matrix is of dimension 12×12. The state variables as defined in the preceding
subsections are arranged as follows
x = [∆ieqs, ∆ieds, ∆i′rqr, ∆i
′rdr, ∆ieqs′ ∆i
eds′ , ∆i
′rkq1, ∆i
′rkq2,∆i
′rfd, ∆i
′rkd, ∆δ,
∆ωrωb
] (3.41)
70
and the complete state space equation is
p
∆ieqs
∆ieds
∆i′rqr
∆i′rdr
∆ieqs′
∆ieds′
∆i′rkq1
∆i′rkq2
∆i′rfd
∆i′rkd
∆δ
∆ωrωb
=
D1A11 · · · D1A14 0 · · · · · · 0 D1A112
D1A21 · · · D1A24 0 · · · · · · 0 D1A212
D1A31 · · · D1A34 0 · · · · · · 0 D1A312
D1A41 · · · D1A44 0 · · · · · · 0 D1A412
0 · · · 0 D2A55 · · · D2A510 D2A511 D2A512
0 · · · 0 D3A65 · · · D3A610 D3A611 D3A612
0 · · · 0 D2A75 · · · D2A710 D2A711 D2A712
0 · · · 0 D2A85 · · · D2A810 D2A811 D2A812
0 · · · 0 D3A95 · · · D3A910 D3A911 D3A912
0 · · · 0 D3A105 · · · D3A1010 D3A1011 D3A1012
0 · · · · · · · · · · · · 0 A1111 0
A1201 · · · · · · · · · · · · A1210 A1211 0
∆ieqs
∆ieds
∆i′rqr
∆i′rdr
∆ieqs′
∆ieds′
∆i′rkq1
∆i′rkq2
∆i′rfd
∆i′rkd
∆δ
∆ωrωb
+
D1b11 · · · D1b13 · · · · · · · · · · · · · · · · · · · · ·
· · · D1b22 · · · D1b24 · · · · · · · · · · · · · · · · · ·
D1b31 · · · D1b33 · · · · · · · · · · · · · · · · · · · · ·
· · · D1b42 · · · D1b44 · · · · · · · · · · · · · · · · · ·
· · · · · · · · · · · · D2b55 · · · · · · · · · · · · · · ·
· · · · · · · · · · · · · · · D3b66 · · · D3b69 · · · · · ·
· · · · · · · · · · · · D2b75 · · · · · · · · · · · · · · ·
· · · · · · · · · · · · D2b85 · · · · · · · · · · · · · · ·
· · · · · · · · · · · · · · · D3b96 · · · · · ·D3b99 · · · · · ·
· · · · · · · · · · · · · · · D3b106 · · · · · · D3b109 · · · · · ·
· · · · · · · · · · · · · · · · · · · · · · · · b1111 · · ·
· · · · · · · · · · · · · · · · · · · · · · · · · · · b1212
∆veqs
∆veds
∆v′rqr
∆v′rdr
∆veqs′
∆veds′
0
0
e′rxfd
0
0
∆Tm
(3.42)
where, D1, is a multiplier for the DFIG A and b matrix equations whereas D2, D3, are
multipliers for the synchronous machine A and b matrix equations. All the · · · entries
in the A and b matrices imply zeros. The elements and multipliers can all be found in
the Appendix .
3.3.2 Formulation of system Jacobian
The equilibrium point x0 is calculated by solving (3.17) at the equilibrium point. The
resulting Jacobian is given in (3.43) below, and the elements are provided in the Ap-
pendix.
71
J(X0) = A =∂f
∂x
∣∣∣∣x=x0
=
J11 J12 J13 J14 J15 J16 J17 J18 J19 J110 J111 J112
J21 J22 J23 J24 J25 J26 J27 J28 J29 J210 J211 J212
J31 J32 J33 J34 J35 J36 J37 J38 J39 J310 J311 J312
J41 J42 J43 J44 J45 J46 J47 J48 J49 J410 J411 J412
J51 J52 J53 J54 J55 J56 J57 J58 J59 J510 J511 J512
J61 J62 J63 J64 J65 J66 J67 J68 J69 J610 J611 J612
J71 J72 J73 J74 J75 J76 J77 J78 J79 J710 J711 J712
J81 J82 J83 J84 J85 J86 J87 J88 J89 J810 J811 J812
J91 J92 J93 J94 J95 J96 J97 J98 J99 J910 J911 J912
J101 J102 J103 J104 J105 J106 J107 J108 J109 J1010 J1011 J1012
J111 J112 J113 J114 J115 J116 J117 J118 J119 J1110 J1111 J1112
J121 J122 J123 J124 J125 J126 J127 J128 J129 J1210 J1211 J1212
(3.43)
In equation (3.43), Jij (i = 1, 2, 3, 4, and j = 1, 2, ....4 and 12) represents the linearised
dynamics of the DFIG; Jij (i = 5, .., 10, and j = 5, ...10 and 12) represents the
linearised dynamics of the synchronous machine;Jij (i = 11, and j = 12) represents
the linearised dynamics of the power transfer between the two machines; Jij (i = 12,
and j = 1, 2, ....11) represents the combined linearised dynamics of the equations of
motion.
3.4 Eigenvalue analysis of the system
In this section, an eigenvalue analysis of the system is carried out using the Jacobian
matrix of equation (3.43). The system parameters used are provided in the Appendix.
For the DFIG, the standard system parameters are from the Matlab/simulink DFIG
blockset for a 1.5 MW machine. The synchronous machine is chosen based on a typical
machine rating of approximately 1/3rd the rating of the DFIG, which is in turn based
on conventional converter ratings used for back-to-back sets.
Eigenvalues associated with speed variations are analysed with the assumption that
the angular mechanical speed of the two machines is the same. The common (or natural)
speed of the cascade is derived from the speed/poles relationship of cascaded machines
(Li et al., 1994; Gish et al., 1981; Cook & Smith, 1979; Williamson et al., 1997). The
72
DFIG steady-state rotor electrical frequency ωr0, and the synchronous machine electrical
speed ωrs, as functions of their number of poles P1 and P2, are cast as functions of the
DFIG slip s as follows:
ωr0 = ωe(1− s) (3.44)
ωr(sm) =P2
P1
(ωr0)
In the ensuing stability calculations, the parameters of intuitive interest regarding
stability are; the system speed and, the circuit parameters of the individual machines.
Operation of the two machines and issues regarding stability of the two machines as
individual entities are quite well understood and literature covering the pertinent issues
has been pointed out in section 2.9.
The system speed variations are expressed as a functional relation of the DFIG slip
s as shown in (3.44) for different values of synchronous machine stator, and DFIG rotor
circuit parameters. Simulation runs are carried out and the key modal characteristics
are extracted. Analysis is carried out with variation in rotor speeds with the assumption
that both machines have equal number of poles. The speed is in this section linked to
the synchronous machine steady state power angle δ0. To accomplish this, the speed
variations are expressed as a function of the DFIG slip s. Therefore it is actually the slip
that is varied in each calculation. The slip in (3.44) is varied between 100% and −30%.
For each value of slip, ωr0 and ωe2 are calculated, from which the associated power angle
is also calculated from (3.45) (Krause et al., 2002). The values are substituted into the
elements of (3.43) to obtain eigenvalue trajectories.
δ =ωbp
(ωr0 − ωe2
ωb
)(3.45)
73
3.4.1 Eigenvalue trajectories with original and modified parameters
Eigenvalues for both machines are highly dependent on machine parameters, and it is
quite difficult to relate a change in an eigenvalue with a change in a specific parameter
through analytic expressions. The following figures show variations of speed, with cor-
responding variations of slip between 1 to −1 pu in figures 3.8-3.19 for original machine
parameters in figures 3.8 (a)-3.19 (a), and modified parameters in figures 3.8 (b)-3.19
(b). Although numerous calculations were performed based on parameters with crite-
ria based on findings from (Cook & Smith, 1983; Cornell & Novotny, 1972; Nickle &
Pierce, 1930) with the relation between steady state speed ωr0 and slip given in section
3.4 maintained, only one specific change in parameter, that of the synchronous machine
resistance rs′ is shown as an example.
The synchronous machine stator resistance rs′ was varied in the range 0 < rs′ <
0.0095 p.u. with all other base parameters held constant while the DFIG stator resistance
was varied in the range 0 < rs < 0.0085 with all other base parameters constant. The
base parameters can be found in the Appendix.
It was observed that generally, the stability boundaries were enhanced by reduction
of the synchronous machine rs′ and increase in DFIG rs. frequency of the eigenvalues
into the right hand plane give a graphic indicator of stability boundaries as shown in
figures 3.8-3.19 as an example. It was also observed that the same excursions were
less with values near the critical limiting angle for stability given in (3.46) which was
observed in (Nickle & Pierce, 1930)
rs′ =< xq tan δ′
(3.46)
In (3.46), rs′ is the synchronous machine stator resistance, xq is the quadrature axis
synchronous reactance, and δ′
is the steady state power angle.
The physical significance is that in general, lowering synchronous machine stator
74
resistance while increasing DFIG stator resistance enhances stability boundaries. How-
ever, parameters used here are from standard off-the-shelf machines and this information
is useful in formulating control strategies needed for stable operation. A much better
option would be for optimal-parameter designed machines where analysis such as that
carried out here would prove very useful but that is beyond the scope of this work.
In figure 3.8 with original parameters, real λ1 is in the right plane for all speed ranges.
With modified parameters, it is a simple pole that moves to the origin at synchronous
speed.
In figure 3.9 with original parameters, real λ1 is a simple pole in the left plane for all
speed ranges. With modified parameters it is located at the origin at all speed ranges
but moves away leftwards from the imaginary axis at synchronous speed.
In figure 3.10 with original parameters, real λ1 makes several excursions between the
left and right planes. With modified parameters it is a simple pole through all speed
ranges but moves leftwards, away from the imaginary axis at synchronous speed.
In figure 3.11 with original parameters, between s = −1 to s = 1, the real part of
λ4 makes several excursions between left and right planes. At s = 0 and s = 1, λ4 is a
simple pole. With modified parameters, the real part is in the right plane. At s = 0, it
is a simple pole in the left plane, while at s = 1 it is in the right plane.
In figure 3.12 with original parameters, between s = −1 to s = 0.2, the real part
of λ5 makes several excursions between left and right planes. At s = 0, it is located in
the right plane and, at s = 1, λ5 is located on the imaginary axis with an oscillatory
frequency equal to the base frequency. With modified parameters, the real part is a
simple pole upto s = −1. At s = 0, it makes an excursion to the right plane, while
at s = 1 it is located on the imaginary axis with a frequency equal to the base system
frequency.
In figure 3.13 with original parameters, between s = −1 to s = 0.2, the real part of
λ6 makes several excursions between left and right planes. At s = 0, it is located on the
75
2762.7
2762.8
2762.9
2763
2763.1
2763.2
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
Real λ1
Imaginary λ1
Slip
(pu
)
(a) <λ1 and =λ1 transitions (original parame-ters).
−6−5
−4−3
−2−1
0
x 107
−1
−0.5
0
0.5
1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ1
Imaginary λ1
Slip
(pu
)
(b) <λ1 and =λ1 transitions(modified param-eters).
Figure 3.8: <λ1 and =λ1 trajectories.
−1134.4
−1134.3
−1134.2
−1134.1
−1134
−1133.9
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
Real λ2Imaginary λ
2
Slip
(pu
)
(a) <λ2 and =λ2 transitions (original parame-ters).
−6−5
−4−3
−2−1
0
x 107
−1
−0.5
0
0.5
1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ2
Imaginary λ2
Slip
(pu
)
(b) <λ2 and =λ2 transitions(modified param-eters).
Figure 3.9: <λ2 and =λ2 trajectories.
76
−400
−200
0
200
400
600
−1
−0.5
0
0.5
1−1
−0.5
0
0.5
1
Real λ3
Imaginary λ3
Slip
(pu
)
(a) <λ3 and =λ3 transitions (original parame-ters).
−3.5
−3
−2.5
−2
−1.5
x 104
−1
−0.5
0
0.5
1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ3
Imaginary λ3
Slip
(pu
)
(b) <λ3 and =λ3 transitions(modified param-eters).
Figure 3.10: <λ3 and =λ3 trajectories.
−400
−200
0
200
400
600
−1
−0.5
0
0.5
1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ4
Imaginary λ4
Slip
(pu
)
(a) <λ4 and =λ4 transitions (original parame-ters).
−5000
0
5000
10000
15000
−1
−0.5
0
0.5
1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ4
Imaginary λ4
Slip
(pu
)
(b) <λ4 and =λ4 transitions(modified param-eters).
Figure 3.11: <λ4 and =λ4 trajectories.
77
−500
0
500
050
100150
200250
300350
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ5Imaginary λ
5
Slip
(pu
)
(a) <λ5 and =λ5 transitions (original parame-ters).
−600
−400
−200
0
200
400
0
50
100
150
200
250
300
350−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ5Imaginary λ
5
Slip
(pu
)
(b) <λ5 and =λ5 transitions(modified param-eters).
Figure 3.12: <λ5 and =λ5 trajectories.
−400
−200
0
200
400
−400−300−200−1000100200300400
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ6
Imaginary λ6
Slip
(pu
)
(a) <λ6 and =λ6 transitions (original parame-ters).
−1000
100200
300400
500
−300
−200
−100
0−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ6
Imaginary λ6
Slip
(pu
)
(b) <λ6 and =λ6 transitions(modified param-eters).
Figure 3.13: <λ6 and =λ6 trajectories.
78
−200
0
200
400
600
−400−200
0200
400
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ7
Imaginary λ7
Slip
(pu
)
(a) <λ7 and =λ7 transitions (original parame-ters).
−150
−100
−50
0
50
0
10
20
30
40
50
60
70−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ7
Imaginary λ7
Slip
(pu
)
(b) <λ7 and =λ7 transitions(modified param-eters).
Figure 3.14: <λ7 and =λ7 trajectories.
pure imaginary axis with an undamped oscillatory frequency equal to base frequency. At
s = 1, λ6 is again located on the imaginary axis with an undamped oscillatory frequency
equal to system base frequency. With modified parameters, the real part is in the right
plane between s = −1 and s = −0.1. At s = 0, it is a simple pole in the left plane,
while at s = 1 it is located at the origin.
In figure 3.14 with original parameters, between s = −1 to s = −0.4, λ7 is mostly on
the imaginary plane. plane with oscillatory frequency equal to base system value, this
is also the case at s = 0. At s = 1, it is located in the right hand plane. With modified
parameters, λ7 at the origin between s = −1 and s = −0.2. Between s = −0.2 and
s = 0.2, it is in the right plane, while at s = 1 it has an undamped low frequency.
In figure 3.15 with original parameters, at s = 0, λ8 is in the right hand plane. At
s = 1, it is a simple pole. With modified parameters, λ8 at the origin between s = −1
and s = −0.2. Between s = −0.2 and s = 0.2, it is in the right plane, while at s = 1 it
has an undamped low frequency.
79
−200
−100
0
100
200
300
−400−300
−200−100
0100−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ8
Imaginary λ8
Slip
(pu
)
(a) <λ8 and =λ8 transitions (original parame-ters).
−50
0
50
100
150
−80
−60
−40
−20
0
20
40
60
−1
−0.5
0
0.5
1
Real λ8
Imaginary λ8
Slip
(pu
)
(b) <λ8 and =λ8 transitions(modified param-eters).
Figure 3.15: <λ8 and =λ8 trajectories.
−50
0
50
100
−5
0
5−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ9
Imaginary λ9
Slip
(pu
)
(a) <λ9 and =λ9 transitions (originalparame-ters).
−25−20
−15−10
−50
5
−100
0
100
200
300
400
−1
−0.5
0
0.5
1
Real λ9
Imaginary λ9
Slip
(pu
)
(b) <λ9 and =λ9 transitions(modified param-eters).
Figure 3.16: <λ9 and =λ9 trajectories.
80
−100
0
100
200
300
400
500
−5
0
5−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ10
Imaginary λ10
Slip
(pu
)
(a) <λ10 and =λ10 transitions (original param-eters)..
−20
0
20
40
60
80
−350
−300
−250
−200
−150
−100
−50
0−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ10
Imaginary λ10
Slip
(pu
)
(b) <λ10 and =λ10 transitions(modified pa-rameters).
Figure 3.17: <λ10 and =λ10 trajectories.
In figure 3.16 with original parameters, at s = 0, λ9 is in the right hand plane. At
s = 1, it is also in the right plane. It makes several excursions between left and right
planes. With modified parameters, λ9 has undamped natural base system frequency,
while at s = 1 it is located at the origin.
In figure 3.17 with original parameters, at s = 0, λ10 is in the right hand plane. At
s = 1, it is also in the right plane. It makes several excursions between left and right
planes. With modified parameters, λ10 is a simple pole, while at s = 1 it is located at
the origin.
In figure 3.18 with original parameters, at s = 0, λ11 is in the right hand plane. At
s = 1, it is at the origin. With modified parameters at s = 0, the real part is on the
right hand plane, while at s = 1 it is located at the origin.
In figure 3.19 with original the real part of λ12 makes several excursions between left
and right planes. parameters, at s = 0, λ12 is a simple pole. At s = 1, it is located at
origin. With modified parameters, λ12 is a simple pole, while at s = 1 it is located at
the origin.
81
020
4060
80100
−100
0
100
200
300
400−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ11
Imaginary λ11
Slip
(pu
)
(a) <λ11 and =λ11 transitions (original param-eters).
−5000
0
5000
10000
15000
050100150200250300350
−1
−0.5
0
0.5
1
Real λ11
Imaginary λ11
Slip
(pu
)(b) <λ11 and =λ11 transitions(modified pa-rameters).
Figure 3.18: <λ11 and =λ11 trajectories.
−50
0
50
100
−300
−200
−100
0−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ12
Imaginary λ12
Slip
(pu
)
(a) <λ12 and =λ12 transitions (original param-eters).
−20000
−15000
−10000
−5000
0
5000
−350
−300
−250
−200
−150
−100
−50
0−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Real λ12
Imaginary λ12
Slip
(pu
)
(b) <λ12 and =λ12 transitions(modified pa-rameters).
Figure 3.19: <λ12 and =λ12 trajectories.
82
Numerous calculations were carried out for different machine parameters whilst main-
taining the critical limiting criterion reported in (Nickle & Pierce, 1930) and given in
(3.47), for the synchronous machine to varying degrees of x/R ratio. It was noted that
increasing x/R ratios resulted in less frequent excursions of all the eigenvalue real parts
into the right hand plane.
Rs < x tan δ′
(3.47)
Where Rs is the synchronous machine stator resistance, x is the equivalent synchronous
machine reactance, and δ′
is the steady state power angle of the synchronous machine.
At all speeds within the rated speed range of the proposed scheme, and with all
conditions of varied parameters of (3.47), two distinct oscillatory modes were consistently
observed to appear: a mode near 50 Hz and a sub-synchronous torsional mode (IEEE
et al., 1985) near 40 Hz. The DFIG 50 Hz stator mode is consistently lightly damped,
and either stable, or unstable. When it is unstable, the real parts are always very
close to the origin. The sub-synchronous mode is consistently well damped and always
stable. The same oscillatory modes appear under the parameter variations with the only
difference between the original parameters and modified parameters being manifested
in the frequency of real λ excursions between left and right planes. The remaining
analysis in this chapter is therefore limited to the situation with original parameters
as no new information is provided with modified parameters at this stage. Eigenvalues
for the base case at synchronous speed s = 0 pu, stall s = 1 pu, and two intermediate
speeds at s = 0.1 pu and s = −0.1 pu, plus minimum and maximum DFIG speeds at
slip s = −0.3 pu and s = 0.3 pu are given in tables 3.1, 3.2 and 3.3. The lightly damped
DFIG stator mode is a manifestation of the 50Hz rated DFIG stator frequency, hence it
is termed the stator eigenvalue mode. The well damped sub-synchronous mode involves
electromechanical torsional dynamics, and is termed the torsional mode.
83
Table 3.1: Eigenvalues at synchronous speed and start/stall condi-tions.
s = 0 f1σ± jω Hz Nature of mode
λ2 −172.26 + j264.68 42 Torsionalλ3 −172.26− j264.68 42 Torsionalλ11 0.315 + j314.15 49.99 Unstable (DFIG stator)λ12 0.315− j314.15 49.99 Unstable (DFIG stator)
s = 1 σ± jω
λ3 −0.000062424502 + j314.15 49.99 DFIG statorλ4 −0.000062424502− j314.15 49.99 DFIG statorλ5 −179.36 + j268.51 42.73 Torsionalλ6 −179.36− j268.51 42.73 Torsional
Table 3.2: Eigenvalues at minimum and maximum rated speedconditions.
s = 0.3 f1σ± jω Hz Nature of mode
λ3 0.2276 + j315.66 50.23 (Unstable)DFIG statorλ4 0.2276− j315.66 50.23 (Unstabele)DFIG statorλ5 −174.62 + j265.63 42.27 Torsionalλ6 −174.62− j265.63 42.27 Torsional
s = −0.3 σ± jω
λ4 0.382 + j311.62 49.59 (Unstable)DFIG statorλ5 0.382− j311.62 49.59 (Unstable)DFIG statorλ6 −169.69 + j263.92 42 Torsionalλ7 −169.69− j263.92 42 Torsional
Table 3.3: Eigenvalues at intermediate sub-synchronous andsuper-synchronous speeds.
s = 0.1 f1σ± jω Hz Nature of mode
λ4 0.2885 + j314.79 50.10 (Unstable)DFIG statorλ5 0.2885− j314.79 50.10 (Unstable)DFIG statorλ6 −173.07 + j264.97 42.17 Torsionalλ7 −173.07− j264.97 42.17 Torsional
s = −0.1 σ± jω
λ4 0.340 + j313.40 49.87 (Unstable)DFIG statorλ5 0.340− j313.40 49.87 (Unstable)DFIG statorλ6 −171.42 + j264.40 42.08 Torsionalλ7 −171.42− j264.40 42.08 Torsional
84
3.4.2 Participation factors
In this subsection, the influence of the state variables on the modes identified in sub-
section 3.4.1 are assessed using participation factors computed as outlined in section
2.9. The left and right eigenvectors Φ and Ψ were computed for the values of speed
considered in subsection 3.4.1. The right eigenvectors were computed using
(A− λI)Φ = 0 (3.48)
and the left eigenvectors by
Ψ = adj (Φ) (3.49)
The elements of the participation matrices for each speed were then calculated by com-
bining the left and right modal matrices as
Pki = Φki ·Ψik (3.50)
In the matrices (3.51)–(3.56), each Pij entry indicates the sensitivity of a particular
eigenvalue λi to the diagonal elements of Aij ∀, i = j of the state matrix of equation
(3.42). From this, we can readily ascertain which circuits participate most in different
oscillatory modes. The state variables are shown on the left of the matrix for clarity and
the eigenvalues are shown at the top. It is to be noted the participation matrix must
be of same dimension as the state matrix. The computed participation factors are thus
85
presented.
s=0 λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 λ12
ieqs 0 0 0 0 0 0 0 0 0 0 0 0
ieds 0 0 0 0 0 0 0 0 0 0 0 0
ieqr 0 0 0 0 0 0 0 0 0 0 0 0
iedr 0 0 0 0 0 0 0 0 0 0 0 0
irqs 0.159e−3 0.16e−3 0.029e−3 0.181e−3 0 0 0 0 0 0 0 0.911
irds 0 0 0.322e−3 0.076e−3 0 0 0 0 0 0 0.0021 0.184
irkq1 1.46e−3 1.067e−3 0.002e−3 0.010e−3 0 0 0 0 0 0 0.331e−3 0.0839
irkq2 0.002 0.007 0.0177e−3 0.009e−3 0 0 0 0 0 0 0.1772e−3 0.0708
irfd 0 0 0.626e−3 0.00112 0 0 0 0 0 0 0.89e−3 0.32
irkd 0 0 0.88125e−3 0.5513e−3 0.002 0 0 0 0 0 0.402e−3 0.1434
δ 0 0 0 0 0 0 0 0 0 0 0 0
ωr/ωb 0 0 0.507e−3 0 0 0 0 0 0 0 0.0017 0.417
(3.51)
s=1 λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 λ12
ieqs 0 0 0 0 0 0 0 0 0 0 0 0
ieds 0 0 0 0 0 0 0 0 0 0 0 0
ieqr 0 0 0 0 0 0 0 0 0 0 0 0
iedr 0 0 0 0 0 0 0 0 0 0 0 0
irqs 0.19e−3 0.16e−3 0.017e−3 0.21e−3 0 0 0 0 0 0 0.0231 0.833
irds 0.001e−3 0.008e−3 0.338e−3 0.043e−3 0.01e−3 0.103e−3 0 0 0 0 0.0061 0.552
irkq1 0.705e−3 0.433e−3 0 0 0 0 0 0 0 0 0 0.081e−3
irkq2 0.221 0.599e−3 0.011e−3 0.151e−3 0 0 0 0 0 0 0 0.081e−3
irfd 0.09e−3 0.04e−3 0.85e−3 0.75e−3 0 0 0 0 0 0 0.02e−3 0.006
irkd 0.003 0.004 0.032 0.749e−3 0.002 0 0 0 0 0 0 0
δ 0 0 0 0 0 0 0 0 0 0 0 0
ωr/ωb 0 0 0 0 0 0 0 0 0 0 0 0
(3.52)
86
s=0.3 λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 λ12
ieqs 0 0 0 0 0 0 0 0 0 0 0 0
ieds 0 0 0 0 0 0 0 0 0 0 0 0
ieqr 0 0 0 0 0 0 0 0 0 0 0 0
iedr 0 0 0 0 0 0 0 0 0 0 0 0
irqs 0.168e−3 0.165e−3 0.015e−3 0.108e−3 0 0 0 0 0 0 0.021 0.749
irds 0 0 0.327e−3 0.0248e−3 0 0 0 0 0 0 0.0576 0.661
irkq1 1.04e−3 0.08e−3 0.001e−3 0.02e−3 0 0 0 0 0 0 0.139e−3 0.02
irkq2 0.498e−3 0.0017 0 0.0003 0 0 0 0 0 0 0.066 0.020
irfd 0.001e−3 0 0.007e−3 0.65e−3 0.55e−3 0 0 0 0 0 0.0016 0.002
irkd 0.001 0.005 0.923 0.232 0.002 0 0 0 0 0 0.442 0.102
δ 0.0183e−3 0.0016e−3 0.447e−3 0.0013 0 0 0 0 0 0 0 0.917
ωr/ωb 0.0091 0.007 0.0166 0.4759 0 0 0 0 0 0 0.4679 0.08361
(3.53)
s=-0.3 λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 λ12
ieqs 0 0 0 0 0 0 0 0 0 0 0 0
ieds 0 0 0 0 0 0 0 0 0 0 0 0
ieqr 0 0 0 0 0 0 0 0 0 0 0 0
iedr 0 0 0 0 0 0 0 0 0 0 0 0
irqs 0.151e−3 0.165e−3 0.09e−3 0.197e−3 0 0 0 0 0 0 0.0229 0.661
irds 0 0 0 0.01e−3 0 0 0 0 0 0 0.03 0.75
irkq1 0.002 0.01 0.28e−3 0 0 0 0 0 0 0 0.251e−3 0.017e−3
irkq2 0.917e−3 0.002 0 0.001 0.129e−3 0 0 0 0 0 0.119e−3 0.017e−3
irfd 0 0 0 0 0 0 0 0 0 0 0.00127 0.002
irkd 0.0012 0.0033 0.0197 0.1434 0.002 0 0 0 0 0 0.6394 0.1078
δ 0.0183e−3 0.0016e−3 0.447e−3 0.0013 0 0 0 0 0 0 0 0.917
ωr/ωb 0 0 0 0.0072 0 0 0 0 0 0 0 0.9154
(3.54)
87
s=0.1 λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 λ12
ieqs 0 0 0 0 0 0 0 0 0 0 0 0
ieds 0 0 0 0 0 0 0 0 0 0 0 0
ieqr 0 0 0 0 0 0 0 0 0 0 0 0
iedr 0 0 0 0 0 0 0 0 0 0 0 0
irqs 0.162e−3 0.164e−3 0.08e−3 0.03e−3 0 0 0 0 0 0 0.0255 0.912
irds 0 0 0.03e−3 0.182e−3 0 0 0 0 0 0 0.002 0.183
irkq1 0.0012 0.9e−3 0.274e−3 0 0 0 0 0 0 0 0.33e−3 0.084
irkq2 0.001 0.003 0.001 0.001 0.129e−3 0 0 0 0 0 0.176e−3 0.071
irfd 0 0 0 0.0012 0 0 0 0 0 0 0.397e−3 0.318
irkd 0 0 0 0.0017 0.002 0 0 0 0 0 0.397e−3 0.142
δ 0 0 0.123e−3 0.0013 0 0 0 0 0 0 0 0.132
ωr/ωb 0 0 0 0 0 0 0 0 0 0 0.0017 0.422
(3.55)
s=-0.1 λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 λ12
ieqs 0 0 0 0 0 0 0 0 0 0 0 0
ieds 0 0 0 0 0 0 0 0 0 0 0 0
ieqr 0 0 0 0 0 0 0 0 0 0 0 0
iedr 0 0 0 0 0 0 0 0 0 0 0 0
irqs 0.156e−3 0.164e−3 0.08e−3 0.201e−3 0 0 0 0 0 0 0.0255 0.911
irds 0 0 0.03e−3 0.182e−3 0 0 0 0 0 0 0.002 0.184
irkq1 0.0017 0.0012 0.277e−3 0 0 0 0 0 0 0 0.33e−3 0.083
irkq2 0.015 0.049 0.001 0.001 0.32e−3 0 0 0 0 0 0.17e−3 0.067
irfd 0 0 0.124e−3 0.6e−3 0 0 0 0 0 0 0.903e−3 0.320
irkd 0 0 0 0..197e−3 0 0 0 0 0 0 0.403e−3 0.143
δ 0 0 0.103e−3 0.124e−3 0 0 0 0 0 0 0 0.129
ωr/ωb 0 0 0 0 0 0 0 0 0 0 0.0017 0.409
(3.56)
In the following observations referring to matrices (3.51) – (3.56), only participation
factors affecting the oscillatory modes are considered.
It is observed that at s = 0 (pu) which corresponds to synchronous speed, ωr0 has
the highest participation in the unstable DFIG stator mode corresponding to λ12 which
88
is associated with the system speed. At s = 1 (pu) (start or stall condition), ωr0 has
highest participation on the torsional modes corresponding to λ5,6 which are associated
with the synchronous machine stator d−q circuits. At s = 0.3 (pu) which corresponds to
minimum rated sub-synchronous speed, ωr0 has highest participation on the torsional
modes corresponding to λ5,6 which are associated with the synchronous machine stator
d − q circuits. At s = −0.3 (pu), ωr0 has highest participation on the unstable DFIG
stator mode corresponding to λ5 associated with the synchronous machine stator q axis
circuit, and the stable torsional mode corresponding to λ6, which is associated with
the synchronous machine stator d axis circuit. Meanwhile irkq1 has highest participation
on the torsional mode corresponding to λ7, which is associated with the synchronous
machine rotor kq1 damper circuit. The participations at s = 0.1 (pu) and s = −0.1 (pu)
which correspond to intermediate speeds around synchronous speed are similar. ωr0
has highest participation on the unstable DFIG stator mode corresponding to λ5 and
on torsional mode corresponding to λ6,7, which are associated with the synchronous
machine stator d axis circuit, and the synchronous machine rotor kq1 damper circuit
respectively.
3.4.3 Controllability
The plant in its primitive form is inherently unstable. The coupling of the two ma-
chine shafts and the inherent complex electromechanical interactions introduce various
eigen-modes. Machine parameters affect eigenvalues but it is difficult to link changes in
machine parameters and eigenvalues through analytical expressions. The fact that it is
possible to identify associations between eigenvalues and machine state variables makes
it possible to control these variables to ensure stability. This is however subject to the
plant being controllable and this is the topic of this sub-section.
Controllability of the proposed plant is now evaluated for the condition of vary-
ing speed examined in preceding sections. The controllability is calculated for ∀ s ∈
89
(−0.5 · · · 1) pu. The selection of the range of s is based on practical operating speeds of
30%, through to −30% of synchronous speed for the DFIG (Pena et al., 1996), and the
start-up condition when s = 1.
A control system is said to be (completely) controllable if, for all initial times t0 and
all initial states x(t0), there exists some input function u(t) that drives the state vector
x(t) to any final state at some finite time t0 ≤ t ≤ T (Franklin et al., 2005). Given a
system defined by the linear state equation:
x(t) = Ax(t) + Bu(t), x(t0) = x0 (3.57)
the controllability matrix is defined as:
P(t) =[B AB A2B · · · An−1B
](3.58)
and a system is controllable if and only if
rank(P) = n (3.59)
For values of slip (speed) ∀ s ∈ (−0.5 · · · 1) pu, the system was checked for condition
(3.59). The rank of P(t) in (3.58) was found to vary between 8, 9, 10 and 12 with speed,
there was no rank deficiency in the speed range of interest. Because of ill-conditioning
of the system matrices (due to change of base of synchronous machine variables, and
possible divisions by zero), the controllability staircase form (Edelman et al., 1999)
was used to calculate the controllable/uncontrollable subspaces and locate the possible
uncontrollable modes.
90
3.5 Summary
Analysis of the coupled primitive mathematical model derived from the electrical d− q
equivalent circuit exhibits the synchronous machines greater tendency towards instabil-
ity than the DFIG in general, and this is seen through an examination of the eigenvalue
trajectories. High synchronous machine stator resistance was also seen to result into
more frequent excursions of the eigenvalue real parts into the positive plane. A modal
analysis revealed two oscillatory modes. It was observed that a lightly damped DFIG
stator mode (near system 50 Hz frequency), and a well damped sub-synchronous tor-
sional mode (near 40 Hz) appeared under all circumstances. The system was found to
be sensitive to speed variations and this was seen through a participatory analysis. It
can be inferred that the system stability is greatly dependent on speed changes and due
to its lightly damped nature, the DFIG stator mode is prone to sustained oscillations.
Sudden speed changes are therefore likely to impact negatively on the system dynamics.
The controllability matrix was also found to be rank deficient below a certain speed.
This is due to the frequency dependence of machine inductive reactances on speed. At
low speeds, some rows of the controllability matrix attain values close to zero resulting
in a rank deficit.
91
CHAPTER 4. COMPLETE SYSTEM MODEL
In this chapter, a complete model for the system is presented and includes a model
developed for the adjustable pitch pump. A standard vector control scheme is then
implemented on the DFIG rotor side converter while the loop between the synchronous
generator excitation and d.c. link is closed. The system is simulated and both steady
state and dynamic response results are evaluated against those obtained in chapter 3.
The machine models used in the simulation model are the standard Matlab/Simulink
Simpowersystems block-sets for a sixth order induction machine and eighth order syn-
chronous machine models respectively, both with machine currents as state variables.
Since the block-set models are exactly consistent with the models used in this work both
in terms of order and state variables, they can be used as they are.
4.1 Model of generators
Both generator models were already presented in chapter 3 but are included here for
brevity.
92
4.1.1 Model of DFIG
The DFIG equations in a synchronously rotating reference frame, rotating at the angular
velocity of the DFIG airgap are given below
p
ieqs
ieds
ieqr
iedr
= A1
ieqs
ieds
ieqr
iedr
+B1
veqs
veds
veqr
vedr
(4.1)
With currents as state variables, the dynamic torque equation is
dωr
dt=
1
2H1
ωb (Te − Tm) (4.2)
and the electromagnetic torque expressed in terms of the state variables is
Te = Lm
(ieqsi
′edr − iedsi
′eqr
)(4.3)
4.1.2 Model of synchronous machine
The synchronous machine equations in a reference frame rotating at the angular velocity
of the rotor are given by
p
irqs
irds
irkq1
irkq2
irfd
irkd
= R
ieqs
irds
irkq1
irkq2
irfd
irkd
+ L
d
dt
vrqs
vrds
0
0
erxfd
0
(4.4)
93
The dynamic torque equation is
dωr
dt=
1
2H2
ωb (Tm − Te) (4.5)
the electromagnetic torque is
Te = Lmd
(−irds + i
′rfd + i
′rkd
)irqs − Lmq
(−irqs + i
′rkq1 + i
′rkq2
)irds (4.6)
whereas the rotor angle is
dδ
dt= ωb (ωr −ωe) (4.7)
4.1.3 DFIG Rotor Side Converter Model
The objective of the DFIG rotor control is to control the rotor currents such as to
maintain constant stator current frequency, and to control stator power factor. Under
steady state conditions, the rotor flux must be synchronized with the rotating stator
magnetic field. Under these conditions, equation (4.8) relates the rotor frequency with
the stator frequency and mechanical speed
fr = fs ±nrP
2π(4.8)
Where P is the DFIG pole pairs, fr, fs are the DFIG rotor and stator current frequencies
respectively and nr is the system mechanical speed. It is evident from equation (4.8) that
when the rotor speed changes, an adjustment of rotor current frequency will maintain a
constant DFIG stator current frequency. Therefore the action needed is to adjust rotor
current frequency to keep stator current frequency constant.
Assuming constant stator voltage amplitude, frequency, and negligible stator resis-
tance, if we align the rotor d-axis with the stator flux vector as shown in figure 4.1,
where the stator voltage vector is: us = uds + juqs, then uds = 0, uqs = us = ψsω1 and
94
d axis
q axis
su
s
1
Figure 4.1: Stator flux orientation
the rotor voltages can be expressed in the stator flux reference frame as follows
uϕdr = Rri
ϕdr − σLriϕqr (ωs − ωr) + σLr
ωb
diϕdrdt
uϕqr = Rriϕqr +
(σLri
ϕdr + Lm
LsΨs
)(ωs − ωr) + σLr
ωb
diϕqrdt
(4.9)
Where Ψs is the stator flux, Ls, Lr, Lm are stator, rotor and mutual inductances re-
spectively and σ = 1 − (L2m/LsLr) is the leakage factor. The ϕ superscript indicates
variables in the stator flux frame.
Equation (4.9) can be re-cast as follows
uϕ′
dr = Rriϕdr + σLr
ωb
diϕdrdt
uϕ′qr = Rri
ϕqr + σLr
ωb
diϕqrdt
∆uϕ′
dr = − (ωs − ωr)σLriϕqr
∆uϕ′qr =
(σLri
ϕdr + Lm
LsΨs
)(ωs − ωr)
(4.10)
where uϕ′
dr and uϕ′qr are the components that can be used in decoupled control of rotor
voltages and currents. ∆uϕ′
dr and ∆uϕ′qr are compensating components in the control
strategy. Stator flux oriented DFIG rotor side converter control can be implemented
95
DFIG
abc
GRID
sv3
2
sai
sbi
scisi
si
Sje
qsi
dsi
d
dt
r
abc
rai
rbi
rciri
ri j re
Rectangular
to polar+
+
++
+
-
Decoupling
voltages
abcPWM( )sj r
e
++
++
PI
PI
+
+
PI
PI
s
smi
+
+-
-
Sje
-
-
r
r dri qri smi
s r
s
m
L
L
s
m
L
L
sPsQ
*
s refP
*
s refQ
*
qri
*
dri
qri
dri*
drv
*
qrv
'
drv'
qrv
*
qrv
*
drv
rav
rbv
rcv
e
Figure 4.2: DFIG Rotor side converter control scheme
using equations (4.10). As shown in figure 4.2, the system comprises of an inner rotor
current control loop that eliminates cross coupling of rotor voltage and current so that
the voltage commands uϕ∗dr and uϕ∗qr have a linear relationship with the corresponding
current component. The voltage compensating ∆uϕ′
dr and ∆uϕ′
dr are superimposed on the
current error through the PI controller.
Independent control of the DFIG stator active and reactive powers is achieved by the
following linear relationships between P,Q and the appropriate rotor current components
as shown in (4.11) (Tapia et al., 2003).
Ps = −3
2|~vs|LmLs i
ϕqr
Qs = 32|~vs|LmLs
(|~im| − iϕdr
) (4.11)
where iϕqr and iϕqr are the DFIG quadrature and direct axis rotor currents respectively,
96
expressed in the stator flux oriented reference frame, |~im| is the modulus of the DFIG
stator magnetising current space vector and |~vs| is the modulus of the DFIG stator volt-
age space vector, whereas Lm and Ls are the magnetising and stator leakage inductances
respectively.
The outer active power loop is for rotor frequency control, implemented through ac-
tive power fed back. The hydraulic turbine power set-point is used to form the reference
value P ∗s ref . The error between it and actual power is then fed to a standard PI con-
troller to generate i∗qr. The error between i∗qr and measured iqr is fed into another PI
controller to form u∗qr. The compensating term of ∆uqr is added to this command value
and transformed to α/β coordinates to generate the commands for the space vector
PWM. The PWM then sends the duty signals to the rotor side converter.
The angle between the d axis of the synchronous reference frame and stator is ϕ =
arctan (ψqs/ψds), from which the rotor voltage in the DFIG reference frame can be
derived as in (4.12) udruqr
=
cosϕ −sinϕ
sinϕ cosϕ
uϕdruϕqr
(4.12)
4.2 Synchronous Machine Stator Side Converter Model
The synchronous machine stator converter is a non-controlled rectifier, and the averaging
modelling approach used here is similar to the one used in (Krause & Lipo, 1969) and
(Jadric, 1998). Since the diode rectifier is uncontrolled, constant dc-link voltage is
achieved by adjusting the field voltage vfd of the synchronous machine. This is achieved
via a closed loop dc-link feedback as shown in Figure. 4.3.
The average model used here accurately represents the fundamental harmonic com-
ponents of the system’s ac variables and the (varying) average value of the dc variables,
while accounting for the effects of the non-ideal diodes. Figure. 4.4 shows the rela-
tion between the average dc link voltages and currents with those of the synchronous
97
SM
Gain
filter
Exciterfdv
+
-refdcv
KPI5
dcv
Sensor
reffdv maxfv
minfv
Figure 4.3: Dc bus feedback loop.
d axis
q axis
dc
v
v
k
qi
di
qv
dv
dc
i
i
k
Figure 4.4: Synchronous machine rectifier space Vector Diagram
machine.
The corresponding equations are
vdc = kv(vd sin δ + vq cos δ) (4.13)
id =idcki
sin(δ + φ) (4.14)
iq =idcki
cos(δ + φ) (4.15)
δ = arctanvdvq
(4.16)
98
where kv, ki, φ are parameters that represent non-ideal diode rectification. φ represents
the angle between the fundamental harmonic components of the synchronous machine
ac voltage and current while δ is the synchronous machine power angle. Details of the
calculation procedure for parameters kv, ki, φ can be found in (Jadric, 1998). The power
fed by the synchronous machine to the diode rectifier is
Psm = vdid + vqiq =cosφ
kvkivdcidc (4.17)
Assuming an ideal diode rectifier, kv = 3√
2/π and ki = π/3√
2. All the above values
are average values expressed in the synchronous machine rotor reference frame.
4.3 DC link model
The DC link capacitor’s stored energy is given by
Wdc =1
2Cdcv
2dc =
1
2Cdc [kv(vd sin δ + vq cos δ)]2 (4.18)
Assuming minimum converter losses, DC link capacitor energy relates to the power
supplied by the synchronous machine and the power consumed by the DFIG rotor as
follows
dWdc
dt=
1
2Cdc
d
dt[kv(vd sin δ + vq cos δ)]2 = −Psm − Pr (4.19)
The DC link voltage therefore varies according to the power balance in the link as follows
Cdcvdcdvdcdt
= −Psm − Pr (4.20)
Cdc [kv(vd sin δ + vq cos δ)]d
dt[kv(vd sin δ + vq cos δ)] = −Psm − Pr (4.21)
99
where
Pr =3
2(udaidL + uqaidL) (4.22)
and
Psm =cosφ
kvkivdcidc (see section 4.2) (4.23)
Pr is the power consumed by DFIG rotor and Psm is the power delivered by the syn-
chronous machine. Which means that the condition Psm = −Pr must be maintained
for stability of the DC link. For a particular operating condition, kv, ki are constant.
For the same particular operating condition with the synchronous machine field voltage
vfd held constant, the power balance is a function of δ (Psm = f(δ) or ∆P = f(∆δ)).
Considering (4.31), it is evident that δ = f(β), the power balance can be maintained by
adjusting the pump turbine blades.
4.4 Model of adjustable pitch pump
While the key interest in this research is the dynamic behaviour of the electric system,
the scarcity of research information and literature on variable pitch pumps compelled
the author to develop a suitable model that exhibits the appropriate hydro-dynamics.
A combination of hydro-dynamic, blade element analysis and turbo-machinery theories
were used to develop the mathematical model, and the MIT open-source tool Openprop
(Epps, 2010), used to design a turbine with appropriate characteristics.
4.4.1 Hydraulic Pump Model
Pump characteristics usually differ considerably and detailed data is only usually pro-
vided by manufacturers. A prediction of pump characteristics is only possible using
empirical (or numerical) methods, and empirical methods are based on statistical evalu-
ations of tested pumps (Glich, 2010). This makes a generic mathematical description of
100
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Pitch Angle β0
Power Coefficient Cp
Figure 4.5: Cp versus blade angles β.
such a unit difficult, particularly when the runners of such units permit the blade angle
to be varied on the run depending on operating conditions (Gorla & Khan, 2003; Voros
et al., 2000; Brezovec et al., 2006). In short, generic exact mathematical description of
complex pump characteristics using analytical expressions is usually not possible. The
variable pitch pump characteristics used here were obtained using lifting line propeller
design theory, details of which can be found in (Betz, 1944; Nicolet, 2007; Kimball &
Epps, 2010; Khan et al., 2006; Epps, 2010; Anagnostopoulos & Papantonis, 2007).
Power coefficient Cp, versus blade pitch angle β, and pump power Ppump, versus
water output velocity vout curves were then generated as shown in figures 4.5, and 4.6
from equations (4.29)-(4.31). Ppump versus water output velocity , vout characteristics for
various blade pitch angles were also generated as shown in figure 4.7. The curves thus
obtained were then used to calculate the blade angles for different pumping rates for
different load conditions. Load balancing can be achieved by using slip power recovery
to pump water; the blades ’pitch-up’ to increase pumping rate during times of light load,
and they ’pitch-down’ to reduce pumping rate during times of heavy load. A simplified
scheme of the layout was shown in figure 3.4, while the conceptual runner is illustrated
in figure 4.9.
101
Figure 4.6: Ppump versus water velocity vout.
0 50 100 150 200 250−0.5
0
0.5
1
1.5
2
2.5
3Pump power versus water velocity
vout
(m/s)
P (
MW
)
β=00
β=50
β=100
β=150
β=200
β=250
β=300
β=350
β=400
β=450
β=500
β=550
β=600
Figure 4.7: Ppump versus vout for various blade angles β.
102
Effective area =a
Area =A
Velocity=vP1
P2
Head=h
Length=L
Lower reservoir
Upper
reservoir
Figure 4.8: Physical layout for typical pumped storage hydro power station.
4.4.2 Pump dynamic model
Consider the system of figure 4.8. Bernoulli’s equation for a trajectory between points
P1 and P2 is ∫ P2
P1
∂ v
∂t· dr +
1
2
(v2
2 − v21
)+ ω2 − ω1 +
∫ P2
P1
1
ρdp = 0 (4.24)
If the following assumptions are made in (4.24)
• v2=0, because water level doesn’t change in the time scale of interest.
• the water is incompressible (ρ = constant).
• Pressure at P1 and P2 are equal. (p1 = p2)
• ω2 − ω1 = −gh
With the above assumptions, considering the output channel length as L and with
vout = v1 the dynamic equation derived from (4.24) is
Ldv
dt− 1
2v2out − gh = 0 (4.25)
103
and from the equation of energy transfer dynamics of an impeller (Glich, 2010)
a vout = Av (4.26)
v is the water velocity in pipe L imparted by the impellers, a is the effective area of the
pump outlet opening, and the effective area of the pipe inlet is A. If we then express
the effective impeller pitch angle β = f(a), by applying dimensional similitude principle
(Manwaring, 2008; Nicolet, 2007), then
vout =A
β(a)v (4.27)
Equation (4.25) can therefore be re-written as
dv
dt=
1
Lgh+
1
2L
(A
β(a)v
)2
(4.28)
The power consumed by the pump as a function of water velocity and pump param-
eters is given as
Ppump =1
2ρCp(λ, β)a(β)v3
out =1
2ρCp
A3v3out
a(β)2(4.29)
In (4.29), ρ is the water density. The effective area swept by the pump blades, a, is
cast as a function of pitch angle β (a = f(β)), vout is the water exit velocity and Cp
is a dimensionless power coefficient that expresses the pump efficiency as a function of
pump pitch angle β (see figure 4.5), and tip-speed ratio λ. Figure 4.9 illustrates two
pitch angles β1 and β2. Then,
Cp = f(β, λ) (4.30)
Therefore (4.29) is modified as
Ppump =1
2ρCp(λβ)βv3
out =1
2ρCp
A3v3out
(β)2(4.31)
104
Radial piston-poppet
mechanism
Pitch at
1v
2v
1
2
090
Outflow
Inflow
Figure 4.9: Variable pitch concept.
then if we let x1 = v
u1 = β(a)A
y1 = Ppump
(4.32)
where x1 is the state variable, u1 is the input and y1 is the output (The subscripts ′1′
are used here to differentiate x, u, y from other intermediate variables found elsewhere
in the text). The standard state variable form of (4.28) is therefore
dvoutdt
= ghL
+ x21
12Lu2 ,
Ppump = ρAv3out
2u21
(4.33)
The conceptual parametrized pump characteristics used in the simulations have been
attached in Appendix 5.1.
105
4.4.2.1 Pump Pitch Angle Control
Pump pitch angle is controlled to vary pumping rate. A block diagram of the pump
pitch angle control is illustrated in figure 4.10. The error e between P ∗ref and Ps (see
figure 4.2) is used in the pitch angle control system to generate the pitch angle command
βcmd. βcmd is then used in the pump servo mechanism to adjust the pump’s blade angles
β0. Values of vout, the water velocity for each blade angle were generated from the pump
characteristics. The generalised mathematical description of Ppump follows:
Ppump = d1(β) + d2(β)v2out
d1(β) = α11 + α12β + α13β2 + α14β
3
d2(β) = α21 + α22β + α23β2 + α24β
3
(4.34)
where α11 − α24 are derived from the pump characteristics. The variable pump system
block diagram used to construct the simulation model is shown in figure 4.12. In the
actual simulation model as shown in the diagram, the variable pump output Ppump
is subtracted from the hydro turbine output to form the main synchronous generator
mechanical input. The synchronous generator speed output is then used as an input
to the DFIG. This is implemented in this way for the simulation because the inputs
available in the block-set models for the synchronous and induction machines allow
mechanical coupling only in this way. The dependence of Cp on β is obtained using the
curve of figure 4.5.
The pump pitch angle is controlled according to the power characteristic of figure
106
X ++ cmdβe
00
090
outv β
β∆P∆
( )G β
2D
Lookup
PD
Figure 4.10: Pump pitch angle control.
4.5, and the pitch angle control quantity is given by
G(β) =∆β
∆P=
1
A1 + A2v2out
A1(β) = α12 + 2α13β + 3α14β2
A2(β) = α22 + 2α23β + 3α24β2
(4.35)
where ∆P and ∆β are linearized small signal state variables of Ppump and β. Figure
4.10 shows how the pitch angle command βcmd is generated. The power output power
error e is fed into a PD controller, the pump pitch control quantity G(β) of eq. (4.35)
is multiplied by pump output power signal ∆P of the PD controller. The pitch angle
command is then generated by adding ∆β to β. The pitch angle control quantity is
obtained via a 2D look-up table calculated from pump characteristics. A block diagram
of the pitch servo is shown in Figure. 4.11 where the pitch angle command is limited to
0 − 600. While this is a very non-linear system, it can be represented by a first order
lag system as shown in the figure (Matsuzaka & Tsuchiya, 1997). A functional block
diagram of the overall scheme as implemented in Simulnik is shown in Figure. 4.12.
107
cmdβ 0β
00
060 1
1servo
sT+
Figure 4.11: Pitch control servo.
Turbine
Pitch angle
control
system
Pump servo ,pC ,pump out pP v C
maxP
minP
e 0cmdpC pumpP
Lookup
0,pumpP
,r outv
outv
DFIG SYNCH GEN
Exciter
Compensator
turbinePr mP
r
eP
eP refP
ref refPe 0eP
wd
d cV
dc refV
DFIG ROTOR
SIDE CONTROL
+-
-+
+-
Figure 4.12: Functional system block diagram.
108
A series of computer simulations using Matlab/Simulink are presented here, and are
based on the model of the system presented in chapter 3. The block-sets and components
used for the two machines are the standard ones available in the Matlab/Simulink/
SimPowerSystems library. The block-set machine models are exactly the same as those
used in the theoretical study, with the same state variables. The plant is connected at
the end of a 3-phase 30 km 25 kV radial distribution line, R,L and C parameters of
which are given in Appendix 5.1.
The DFIG rotor is vector controlled, while the loop between the dc bus and syn-
chronous machine exciter is closed via a compensator as illustrated in figure 4.12. The
pump pitch angle control tracks the electrical power output at the DFIG stator terminals
and adjusts the blade angle β accordingly, with the inverse relationship:
β u K1
Pe(4.36)
Focus is on qualitative analysis, and the system response is checked against findings in
the theoretical study.
At start-up, a battery is connected in the DC bus that sets its initial voltage to
1150V , and the blade angle is set to an initial position of β = 00. The idea is to run
the system to steady state, then use the steady state variables to initialise the system
for further simulations. The time domain simulations are carried out in two stages. In
the first stage, an attempt is made to run the system from standstill to steady state
under no-load. In the second stage, the system response to a variety of disturbances is
investigated.
109
4.5 Time domain steady state simulation under no-load, (nat-
ural system response)
Figure 4.13 (a) shows the interaction of the DC bus voltage with blade angle β from
t = 0 to t = 70 s. It is observed that shortly after t = 0, the blade angle increases
rapidly, trying to attain maximum position. At the same time, the DC bus voltage
dips. The blade angle reaches a peak of about 70 and then falls again, trying to track
the DC bus voltage. This trend continues all through the 70 seconds and the angle
levels out at a maximum of approximately 490 when the DC bus transients die out.
There is a noticeable time lag between the action of the blade angle adjustment and the
fast transients occurring in the DC bus as expected because of the relatively large time
constant of the blade adjustment system. A faster response is observed in the action of
the synchronous machine excitation system tracking the DC bus voltage transients as
shown in figure 4.13 (b). It is however noted that the blade angle does not attain the
theoretical design maximum of approximately 570 shown in 4.6.
In figure 4.14, the synchronous machine stator currents settle to steady state after
the initial transients. The action of the synchronous machine kq1 and kd damper circuits
is evident in (c) and (e), the kq2 damper circuit trace is not shown here as it is zero.
The field winding current ifd is seen to have an average value of approximately 1.65 pu.
The power angle δ settles to a steady state value of approximately 350.
An inspection of figure 4.15 shows that the DFIG currents oscillate with monoton-
ically increasing amplitude. The nature of these oscillations points to unstable focus
singularity (Kundur et al., 1994). This means that there is at least one dominant com-
plex conjugate pair of eigenvalues with the real part located to the right of the imaginary
axis. In chapter 3, a complex conjugate pair was identified which either appeared as a
lightly damped eigenvalue with a frequency close to the 50Hz base system frequency or
as a complex conjugate pair with positive real parts, still with frequency close to 50Hz.
110
0 10 20 30 40 50 60 70850
900
950
1000
1050
1100
1150
DC
bus
vol
ts
t (sec)
0 10 20 30 40 50 60 700
5
10
15
20
25
30
35
40
45
50
β0
DC bus voltage
β0
(a)
0 10 20 30 40 50 60 70850
900
950
1000
1050
1100
1150
DC
bus
vol
ts
t (sec)
0 10 20 30 40 50 60 700
0.5
1
1.5
2
2.5
3
SM
exc
itatio
n, V
olts
(pu
)DC bus voltageSM excitation volts (pu)
(b)
Figure 4.13: Pitch angle and speed, steady state.
111
0 10 20 30 40 50 60 70−0.2
0
0.2
0.4
0.6
0.8
1
1.2
t (s)
i qsr (
pu)
iqsr
(a)
0 10 20 30 40 50 60 70−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
t (s)
i dsr (
pu)
idsr
(b)
0 10 20 30 40 50 60 70−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
t (s)
i kq1
r (
pu)
ikq1r
(c)
0 10 20 30 40 50 60 700.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
t (s)
i fdr (
pu)
ifdr
(d)
0 10 20 30 40 50 60 70−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
t (s)
i kdr (
pu)
ikdr
(e)
0 10 20 30 40 50 60 70−100
−80
−60
−40
−20
0
20
40
60
t (s)
δ0
δ 0
(f)
Figure 4.14: Synchronous machine state variables irqs, irds, i
rkq1, i
rfd, i
rkd, δ
0, undampedDFIG stator mode
112
0 10 20 30 40 50 60 70−100
−80
−60
−40
−20
0
20
40
60
80
100
t (s)
i qse (
pu)
iqse
(a)
0 10 20 30 40 50 60 70−100
−80
−60
−40
−20
0
20
40
60
80
100
t (s)
i dse (
pu)
idse
(b)
0 10 20 30 40 50 60 70−100
−80
−60
−40
−20
0
20
40
60
80
100
t (s)
i qre (
pu)
iqre
(c)
0 10 20 30 40 50 60 70−100
−80
−60
−40
−20
0
20
40
60
80
100
t (s)
i dre (
pu)
idre
(d)
Figure 4.15: DFIG state variables ieqs, ieds, i
eqr, i
edr, undamped DFIG stator mode
This mode was called the DFIG stator mode.
A Fast Fourier Transform carried out on the unstable DFIG stator and rotor cur-
rents reveals high spectral densities of frequencies close to the base 50Hz frequency.
This corroborates the finding of the existence of the oscillatory DFIG stator mode iden-
tified in the theoretical analysis of chapter 3. Since the Fourier analyses yielded similar
information, only results for ieqs and ieqr are presented here.
An inspection of the spectral densities in figures 4.16 and 4.17 reveals that at lower
speeds, there is relatively heavy spectral density of low frequency oscillations. As the
system speeds up, the spectral density of these lower frequencies reduce, which indicates
that they are damped. This is readily observed by the evolving forms of the shapes of
figures 4.16 and 4.17 with advancement of time. It is particularly noticeable that while
the spectral densities of the low frequency components reduce with increase in system
speed, those of components slightly below and above base system frequency remain
constant throughout. An closer inspection of the actual waveforms in the observed time
windows (in red, above the spectral distributions of figures 4.16 and 4.17) also confirms
this. In figure 4.18 the DFIG stator and rotor circuit voltages are also observed to
113
20 20.02 20.04 20.06 20.08 20.1 20.12 20.14 20.16 20.18
−4
−2
0
2
4
FFT window: 10 of 3500 cycles of selected signal
Time (s)
0 20 40 60 80 100 120 140 160 180 2000
2
4
6
8
10
12
Frequency (Hz)
Fundamental (50Hz) = 3.133 , THD= 0.96%
Mag
(%
of F
unda
men
tal)
(a)
40 40.02 40.04 40.06 40.08 40.1 40.12 40.14 40.16 40.18
−5
0
5
Time (s)
0 20 40 60 80 100 120 140 160 180 2000
2
4
6
8
10
12
Frequency (Hz)
Fundamental (50Hz) = 6.987 , THD= 0.65%
Mag
(%
of F
unda
men
tal)
(b)
65 65.02 65.04 65.06 65.08 65.1 65.12 65.14 65.16 65.18
−50
0
50
Time (s)
0 20 40 60 80 100 120 140 160 180 2000
2
4
6
8
10
Frequency (Hz)
Fundamental (50Hz) = 57.1 , THD= 2.11%
Mag
(%
of F
unda
men
tal)
(c)
Figure 4.16: DFIG ieqs stator currents spectral density analysis at 20, 40 and 65 sec,undamped DFIG stator mode
20 20.02 20.04 20.06 20.08 20.1 20.12 20.14 20.16 20.18
−4
−2
0
2
4
Time (s)
0 20 40 60 80 100 120 140 160 180 2000
2
4
6
8
10
12
14
Frequency (Hz)
Fundamental (50Hz) = 3.009 , THD= 0.80%
Mag
(%
of F
unda
men
tal)
(a)
40 40.02 40.04 40.06 40.08 40.1 40.12 40.14 40.16 40.18
−5
0
5
Time (s)
0 20 40 60 80 100 120 140 160 180 2000
2
4
6
8
10
12
Frequency (Hz)
Fundamental (50Hz) = 6.987 , THD= 0.65%
Mag
(%
of F
unda
men
tal)
(b)
65 65.02 65.04 65.06 65.08 65.1 65.12 65.14 65.16 65.18−50
0
50
Time (s)
0 20 40 60 80 100 120 140 160 180 2000
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
Fundamental (50Hz) = 53.58 , THD= 0.98%
Mag
(%
of F
unda
men
tal)
(c)
Figure 4.17: DFIG ieqr rotor currents spectral density analysis at 20, 40 and 65 sec,undamped DFIG stator mode
114
0 10 20 30 40 50 60 70−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t (s)
v qse (
pu)
vqse
(a)
0 10 20 30 40 50 60 70−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t (s)
v dse (
pu)
vdse
(b)
0 10 20 30 40 50 60 70−5
−4
−3
−2
−1
0
1
2
3
4
5
t (s)
v qre (
pu)
vqre
(c)
0 10 20 30 40 50 60 70−5
−4
−3
−2
−1
0
1
2
3
4
5
t (s)
v dre (
pu)
vdre
(d)
Figure 4.18: DFIG stator and rotor voltages veqs, veds, v
eqr, v
edr, undamped DFIG stator
mode
be unstable. It is instructive to note that the mode of oscillations are electrical, as
established by the participation factor analysis in chapter 3.
Meanwhile, the synchronous machine stator voltages are observed to stably settle to
steady state values as shown in figure 4.20. This is as a result of the strong electrome-
chanical coupling of the two machines, and the stabilizing effect of the hydraulic turbine
and governor driving the system. The relatively weak coupling between the DFIG stator
circuit and the distribution network however, is what gives rise to the electrical oscilla-
tions appearing in the output. The underlying physics of the process and its mitigation
follows.
The induction machine circuits are known to have two resonant frequencies as de-
scribed in (Peterson & Valis, 1991; Peterson, 1991; Alakula et al., 1992). Of particular
interest given the machine size in this thesis is the series resonance, where the stator
and rotor fluxes oscillate in the tangential direction. This means that the angle δ will
vary with oscillations. This can be visualised in figure 4.19. If we think of the rotor
and stator flux vectors as being connected by a ’spring’ representing total leakage in-
115
ductance Ll as shown in figure 4.19, then if we stretch the spring, energy is increased
and when we compress it, energy is reduced. L−L determines resonant frequency while
Rr and Rs, which are the rotor and stator resistances respectively, provide damping for
this oscillating energy. This is a classic second order series resonant system with the
characteristic equation
s2 + 2Ωζ + Ω2 = 0 (4.37)
where the resonant frequency Ωs and damping ζs are given as follows
Ωs =Ψ√JLL
(4.38)
ζs =Rr +Rs
2Ψ
√J
LL(4.39)
where Ψ is the flux amplitude and J is the moment of inertia. This proves that global
asymptotic stability of the DFIG is very sensitive to the machine parameters and there-
fore the design of such a system must take this into account. However, since machine
design is beyond the scope of this work, the DFIG damping had to be improved for
the simulated system to achieve steady state. This was done by changing the stator
and rotor resistances by first computing the analytical Lyapunov condition derived in
(Oteafy & Chiasson, 2010) as follows:
RsRr −(LmPωR0
2
)> 0 (4.40)
In (4.40), Lm is the mutual inductance, P is the number of pole pairs and ωR0 is the
steady state speed.
The adjustment of the DFIG stator and rotor resistances in the manner prescribed
in (4.40) successfully damped out the series resonance and the simulation was able to
116
δ
sψ
rψ
LL
Figure 4.19: Induction machine series resonance.
0 10 20 30 40 50 60 700
0.2
0.4
0.6
0.8
1
1.2
1.4
t (s)
v qsr (
pu)
vqsr
(a)
0 10 20 30 40 50 60 70−0.5
0
0.5
1
t (s)
v dsr (
pu)
vdsr
(b)
Figure 4.20: Synchronous machine stator voltages vrqs, vrds, undamped DFIG stator mode
reach steady state. It can be inferred that the relatively weak coupling between the
DFIG stator circuit and the (weak) distribution network is what gives rise to the elec-
trical oscillations appearing in the output. This manifested itself through the equivalent
distribution network and DFIG impedance.
4.6 Time domain simulation, dynamic system response
In this section, the system is subjected to various disturbances and the performance is
evaluated.
4.6.1 Time domain response, step reduction in load at DFIG stator termi-
nals
The system is initially running with a load connected to the stator terminals. At time
t = 1 s, the load is disconnected. In figure 4.21 (a), this change in load causes the blades
117
0 1 2 3 4 5 6 7 8 9 1040
42
44
46
48
50
52
54
56
t (s)
βo
βo
(a)
0 1 2 3 4 5 6 7 8 9 101117.5
1118
1118.5
1119
1119.5
1120
1120.5
1121
1121.5
t (s)
VD
C (
volts
)
DC bus voltage
(b)
0 1 2 3 4 5 6 7 8 9 101.5
2
2.5
3
3.5
4
4.5
5
5.5
t (s)
Vf
SM excitation volts
(c)
Figure 4.21: System responses of β, Vdc, Vf , to a step decrease in load
to pitch accordingly. The blades eventually settle to a larger new angle with a higher
pumping rate. The disturbance is also picked by the hydro-turbine speed regulator
which brings the system speed back to 1 pu. In figure (b), the DC bus voltage initially
dips in response. This is because of the change in power balance in the DC bus, the
transition period before the pumping action takes up the slack to divert the slip energy
to pumping action. The DC bus voltage eventually levels out to a steady state value,
slightly higher than the pre-disturbance value. Vf . In figure (c), the response of the
synchronous machine excitation system to the DC bus voltage initial dip is observed,
it rises with the drop in DC bus voltage then after a transient period, settles back to
a value close to the pre-disturbance level. Meanwhile, the active and reactive powers
demanded from the DFIG reduces with the change in load as can be seen in figure 4.22.
118
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 530
32
34
36
38
40
42
44
46
48
50
t (s)
P ⋅
10−
3 (M
W)
Active power, P
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
t (s)
Q ⋅
10−
3 (Mva
r)
Reactive power, Q
(b)
Figure 4.22: Real and reactive power following a step reduction in load
4.6.2 Time domain response to a three phase short circuit at DFIG stator
terminals
The system is initially running unloaded, when a 3-phase to ground short circuit is
simulated at the DFIG stator terminals from t = 1 to t = 1.3. During the period no
power is available to the load. The pitching system interprets this as a reduction of
Pe and sends a pitch-up command, and this happens in an oscillatory manner as seen
in figure 4.23 (a). An analysis of the frequencies shows them to have spectral density
very similar to those reported in section 4.5. The oscillations are however damped as is
evident. The DC bus oscillations are similar in nature. The reason the DC bus voltage
rises in the circumstances arises from the combined action of the synchronous machine
excitation attempting to maintain DC bus voltage and the hydraulic turbine governor
accelerating the system see figure 4.24. The speed has greater influence on this rise of
the DC bus voltage than the synchronous machine excitation. A careful examination of
the peaks and troughs of the oscillations of figures 4.23 and 4.24 (1st cycle), reveals the
nature of the interactions. Figure 4.25, shows the net active and reactive powers feeding
the fault.
119
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.551.82
51.84
51.86
51.88
51.9
51.92
51.94
t (s)βo
βo
(a)
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5900
1000
1100
1200
1300
1400
1500
1600
t (s)
VD
C (
volts
)
VDC
DC bus
(b)
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.52.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
t (s)
Vfdr
(pu
)
Vfdr
(c)
Figure 4.23: Response to a close 3 Phase short circuit (β, Vdc, Vfd).
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.99
0.9905
0.991
0.9915
0.992
0.9925
0.993
0.9935
t (s)
ωr
Speed
(a)
Figure 4.24: Speed response to a close 3 Phase short circuit.
120
0.9 1 1.1 1.2 1.3 1.4 1.5 1.648
48.1
48.2
48.3
48.4
48.5
48.6
48.7
48.8
48.9
t (s)
P ⋅
10−
3 (M
W)
Active power, P
(a)
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t (s)
Q ⋅
10−
3 (Mva
r)
Reactive power, Q
(b)
Figure 4.25: Real and reactive power during a close 3 Phase short circuit P Q.
4.6.3 Time domain response to a step increase in load at DFIG stator ter-
minals
Figures 4.26, 4.27 show the system responses to a step load demand. β reduces to
settle at a lower value while a temporary the dip in DC bus voltage is accompanied by
corrective action of the synchronous machine excitation. Corresponding changes in the
active and reactive powers are also observed.
4.6.4 Time domain response to a distant fault
A fault on the far end of the network is simulated as a voltage dip in the distribution
network. As a result of this, the net electrical power demand on the plant increases as
seen in figure 4.29. The increase in electrical power demand is interpreted by the pump-
ing system as a pitch-down command and the β decreases. This is also accompanied
121
0 1 2 3 4 5 6 7 8 9 1051.65
51.7
51.75
51.8
51.85
51.9
t (s)
βo
βo
(a)
0 1 2 3 4 5 6 7 8 9 101000
1020
1040
1060
1080
1100
1120
1140
t (s)
VD
C (
volts
)
VDC
DC bus
(b)
0 1 2 3 4 5 6 7 8 9 102.3
2.35
2.4
2.45
t (s)
Vf (
pu)
Vf SM
(c)
Figure 4.26: Responses of β, DC bus voltage and Synchronous machine excitation Vf toa step increase in load
122
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 548
49
50
51
52
53
54
55
56
t (s)
P ⋅
10−
3 (M
W)
Active power, P
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.35
0.4
0.45
0.5
0.55
t (s)
Q ⋅
10−
3 (Mva
r)
Reactive power, Q
(b)
Figure 4.27: Real and reactive power during a close 3 Phase short circuit P Q.
by an increased reactive power demand. The result is also a dip in the DC bus voltage
accompanied by the regulating action of the synchronous machine excitation.
4.7 Summary
In this chapter, the system model was developed for simulation and both the steady
state and dynamic behaviour were evaluated through simulations. The objective was to
control the DFIG rotor currents such as to maintain stator frequency, control torque and
to control the reactive power. Another objective was to power the DC link and control
it using the synchronous machines field voltage Vfd. In addition, a model was developed
for the adjustable pitch pump.
Theoretical analysis of the primitive model developed in chapter 3 established sensi-
tivity of the system to sudden changes in speed and this was confirmed in this chapter.
123
0 1 2 3 4 5 6 7 8 9 1051.832
51.834
51.836
51.838
51.84
51.842
51.844
51.846
51.848
t (s)
βo
βo
(a)
0 1 2 3 4 5 6 7 8 9 101000
1020
1040
1060
1080
1100
1120
1140
t (s)
Vdc
DC bus voltage
(b)
0 1 2 3 4 5 6 7 8 9 102.3
2.35
2.4
2.45
t (s)
Vf (
pu)
Vf SM
(c)
Figure 4.28: Responses of β, Vdc and Vf to a distant fault on the distribution network.
124
0 1 2 3 4 5 6 7 8 9 1048
49
50
51
52
53
54
55
56
t (s)
P ⋅
10−
3 (M
W)
Active power, P
(a)
0 1 2 3 4 5 6 7 8 9 100.34
0.36
0.38
0.4
0.42
0.44
0.46
t (s)
Q ⋅
10−
3 (Mva
r)
Reactive power, Q
(b)
Figure 4.29: Real and reactive power during a distant fault.
The most drastic speed change observed here was that resulting from the simulated
three phase short circuit near the terminals of the DFIG. The oscillations observed were
analysed and found to have components close to the oscillatory frequencies encountered
in the theoretical analysis. The tortional mode, was however found to vanish in the
complete system model presented here, and this can be attributed to damping provided
by the common system shaft through the system’s hydraulic turbine speed regulator.
The other was found to persist and in some cases cause instability (see subsection 4.5)
In general, the system responds to control commands and it can be inferred that
design parameters, particularly the DFIG rotor and stator circuit resistances, including
pump parameters and well coordinated control actions significantly influence perfor-
mance.
125
CHAPTER 5. CONCLUSION
An extensive survey of literature shows that with the upsurge in usage of electrical en-
ergy, there is an urgent need both to diversify means of generation and increase the
amount of electrical power that is generated. This requires innovation, both in the use
of existing technology and, providing new means of power generation. The method
proposed in this work focussed on pumped storage hydro power generation. Exist-
ing conventional hydro power plants could be modified to incorporate pumped storage
capability. Existing pumped storage facilities could be modified in such a way that
pumping and generating functions don’t have to be mutually exclusive. The design of
new hydro power plants could be done such that pumped storage using the proposed
generating scheme and plant layout are mainstream. This would result in better use of
water resources for power generation and greater flexibility in operation of such facili-
ties. Continued advancements in power electronics and control provide ever increasing
opportunities for control of such schemes.
The objective of this study was to develop an integrated solution for pumped storage
hydro power generation that can operate with the flexibility of mutually inclusive modes
of pumping and generating power at the same time. The method used after identifying
a suitable topology of overall plant, and technology layout, was to develop a primitive
mathematical model from which insights were gleaned into the fundamental physical
interactions of the basic components. From this, potential stability issues were exposed.
Thereafter it was then established that the plant is controllable and an overall model
including control components was developed and simulated. Steady state and dynamic
126
simulation results substantially agreed with the mathematical stability analysis which is
based on well established methods, and with available literature on expected behaviour
of the components as individual units.
The key contributions lie in; the development of an integrated solution layout in-
cluding a coupled generator -pump set, the mathematical analysis method applied to
mechanically and electrically coupled machines, and development of a variable pitch
pump concept
It was concluded that small hydro power is grossly undervalued and can provide
permanent solutions to electrical energy supply where the resource is available. It was
also concluded that attributes of power conversion plants used in renewable energy
generation affect their interaction with the local power network or load to which they
are connected.
The induction generator can be connected in cascade with the synchronous generator
in such a way that the synchronous generator provides magnetisation power for the
induction generator. In this way, the induction machine would be the main power
machine while the synchronous generator would be a smaller powered exciting machine.
Successful operation is subject to controllability and stability. It was also highlighted
that reversible turbine pumps in pumped storage applications in current use are of fixed
pitch type, and that such pumped storage plants in operation only function either in
generating mode or pumping mode. The need for a power conversion topology utilising
advantages of both the machines together was identified.
With the primitive mathematical model developed in state variable form and eigen-
value analysis carried out, it was concluded that stability of the scheme has significant
dependence on machine parameters and that operational speed was likely to influence
stable operation of the scheme. Dependence of stability on machine parameters was
shown through the frequency of eigenvalue excursions into the right hand plane. It was
also shown that two dominant modes of oscillation were expected to influence the system
127
dynamics. One sub-synchronous torsional mode (IEEE et al., 1985) near 40 Hz, and
the other was a lightly damped electrical one, near the base system frequency of 50Hz.
A participation factor analysis was then carried out to determine the influence of the
state variables on the identified modes. The inherently unstable nature of the plant and
the most likely sources of instability were thus made apparent. Finally it was considered
that even if inherently unstable, there was a possibility to control the scheme if it met
certain analytical controllability conditions. As a result, a controllability analysis was
carried out and it was established that the system was controllable.
The dearth of literature on variable pitch pumps warranted the development of a
model that would exhibit variable pitch operation. This was achieved using hydraulic
and turbo-machinery theory that is widely available in literature. Open source soft-
ware was then used to design and parametrise the pump and pitch angle control was
implemented. The DFIG rotor control was implemented using standard vector control
methods while the loop between the DC bus and synchronous machine was closed for
control of the DC bus. The torque control command for the DFIG rotor was generated
from the hydraulic turbines speed set-point. Global asymptotic instability was shown to
emanate mainly from the DFIG circuits with the given coupling, resulting in unstable
output and the cause was shown to emanate from poorly damped poles. Adjustment
of circuit parameters corrected this condition. The system in general, was found to be
responsive to control commands and of the disturbances simulated, the three phase short
circuit at the DFIG stator terminals was found to have greatest impact on oscillations.
An analysis of the oscillations again revealed a high spectral density of frequencies close
to system frequency. The simulation of the system also revealed undesirable coupling
between the active and reactive power responses, but this was relegated to be a subject
of subsequent research work. It is also recognised that further work is needed on con-
trolling the DC bus voltage response and enhanced coordination of the different control
elements to cover all contingencies.
128
In summary, there are both positive and negative implications of mechanically and
electrical coupling the induction and synchronous generators. While the coupling is to
some extent restrictive in terms of efficiently controlling intermittently varying energy
input sources, it provides an efficient way of utilising pumped storage hydro energy,
since it enables mutually inclusive generation and pumping functionality. In addition,
it was found that the shaft provides damping of eigenvalues associated with torsional
oscillations emanating from the synchronous machine. This is particularly so for a
system that has a regulated mechanical input such as that provided by a hydro-turbine
governor. The system as it is therefore, is more suited to controlled mechanical input
such as hydro turbines than erratic input such as that from wind turbines. It can
therefore comfortably fit into dispatch oriented or load balancing applications.
It is evident that for different combinations of machine circuit parameters and speed,
there are several different accompanying eigenvalue locations, some of which give satis-
factory operating stability . It is however a challenge to establish a universal analytical
link between machine design parameters and eigenvalues which would satisfy the whole
spectrum of possibilities. The spectrum of possibilities for the expected operating speeds
indicates that it is possible to obtain a fairly universal compromise and indications are
that there is merit in exploring these universal links in further research. Most of the
findings here largely agree in relevant aspects with those found in available literature,
and knowledge of the individual machines.
5.1 Future Research
Areas for future research are identified as follows
• Extension of the reactive power problem to include reactive power support for local
loads under steady state and contingency conditions.
129
• Faster dynamic absorption of slip energy that appears in DC bus during transient
conditions.
• Islanded operation of the plant.
• Analysis of unsymmetrical operating conditions and faults.
• Detailed modal analysis of complete system model.
• A more thorough development of a unified damping solution for DFIG stator flux
dynamics.
130
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APPENDIX
Elements of equation 3.28
Elements 0i,j are zero matrices, elements with subscripts ′0′ denote steady state values,
S0 is DFIG steady state slip and the multipliers Tand T−1 are given in (3.22) and (3.23).a11 a12
a21 a22
=
Rs + pωbLs
ωeωbLs
−ωeωbLs Rs + p
ωbLs
;
a13 a14
a23 a24
=
pωbLm
ωeωbLm
−ωeωbLm
pωbLm
·T−1;
a15 · · · a112
a25 · · · a212
=
[02,8
];
a31 a32
a41 a42
= T ·
pωbLm S0
ωeωbLm
−S0ωeωbLm
pωbLm
;
a33 a34
a43 a44
= T ·
Rr + pωbLm S0
ωeωbLr
−S0ωeωbLr Rr + p
ωbLr
·T−1;
a35 · · · a310
a45 · · · a410
=
[02,6
];
a311 a312
a411 a412
= T ·
0 Lmdieds0 − Lriedr0
0 Lmqieqs0 + Lri
eqr0
;
a51 · · · a54
a101 · · · a104
=
[06,4
];
a55 a56
a65 a66
= T−1 ·
−Rs − pωbLq −ωe
ωbLd
ωeωbLq −Rs − p
ωbLd
·T;
a57 · · · a510
a67 · · · a610
= T−1 ·
pωbLmq
pωbLmq
ωeωbLmd
ωeωbLmd
−ωeωbLmq −ωe
ωbLmq
pωbLmd
pωbLmd
;
a511 · · · a512
a611 · · · a612
=
[04
];
152
a75 a76
a85 a86
=
− pωbLmq 0
− pωbLmq 0
·T;
a77 · · · a712
a87 · · · a812
=
Rkq1 + pωbLkq1
pωbLq 0 0 0 0
pωbLmq Rkq2 + p
ωbLkq2 0 0 0 0
;
a95 a96
a105 a106
=
0 −LmdRf
(pωbLmd
)0 − p
ωbLmd
·T;
a97 · · · a912
a107 · · · a1012
=
0 0 0 0 LmdRf
(Rf + p
ωbLf
)LmdRf
(pωbLmd
)0 0 0 0
0 0 0 0 pωbLmd Rkd + p
ωbLkd 0 0 0 0
;
a111 a112
a121 a122
=
0 0
Lmiedr0 −Lmieqr0
;
a113 a114
a123 a124
=
0 0
−Lmieds0 LmieqS0
·T−1;
a115 a116
a125 a126
=
0 0
Lmqirds0 − Lmd
(irds0 − irf0
)−Lmdirqs0 + Lmqi
rqs0
·T;
a117 a118
a127 a128
=
0 0
−Lmqirds0 −Lmqirds0
;
a119 · · · a1112
a129 · · · a1212
=
0 0 p −ωb
Lmdirqs0 Lmdi
rqs0 0 2Heqp
;
Elements of equation 3.42
D1 =1(
Lm2 − L′r Ls
)D2 =
1
L′kq1 Lmq
2 − L′kq2 Lmq2 + Lmq
2 Lq − L′kq1 L
′kq2 Lq
D3 =1
Ld L′kd
2 − 2Lmd3 + Ld Lmd
2
A11 = ωbL′
rRs; A12 = ωe
(S0Lm
2 − L′rLs)
; A13 = −ωbLmR′
r;
153
A14 = ωe (S0 − 1)LmLr; A112 = Lm
(ieds0 Lm + i
′edr0 Lr
)ωb
A21 = −ωe(S0 Lm
2 − Lr Ls)
; A22 = Lr Rs ωb; A23 = −Lm Lr ωe (S0 − 1) ;
A24 = −LmRr ωb; A212 = −Lm(ieqs0 Lm + i
′eqr0 Lr
)ωb;
A31 = −LmRs ωb; A32 = −Lm Ls ωe (S0 − 1) ; A33 = LsRr ωb;
A34 = ωe(Lm
2 − S0 Lr Ls)
; A312 = −Ls(ieds0 Lm + i
′edr0 Lr
)ωb;
A41 = Lm Ls ωe (S0 − 1) ; A42 = −LmRs ωb; A43 = ωe(S0 Lr Ls − Lm2
);
A44 = LsRr ωb; A412 = Ls
(ieqs0 Lm + i
′eqr0 Lr
)ωb;
A55 = Rss
(L′
kq1 L′
kq2 − Lmq2)ωb; A56 = Ld
(L′
kq1 L′
kq2 − Lmq2)ωe;
A57 = A58 = −R′kq1 Lmq
(L′
kq2 + Lmq
)ωb; A59 = Lmd
(Lmq
2 − L′kq1 L′
kq2
)ωe
A510 = Lmd
(Lmq
2 − L′kq1 L′
kq2
)ωe; A511 = 0
A512 = (irds0 Ld − irf0 Lmd)(L′
kq1 L′
kq2 − Lmq2)ωb;
A65 = Lq
(L′
kd
2+ Lmd
2)ωe; A66 = −Rss
(L′
kd
2+ Lmd
2)ωb;
A67 = −Lmq
(L′
kd
2+ Lmd
2)ωe; A68 = −Lmq
(L′
kd
2+ Lmd
2)ωe;
A69 = r′
kd Lmd
(L′
kd − Lmd
)ωb; A610 = −R′kd Lmd
(L′
kd + Lmd
)ωb;
A611 = 0 A612 = 0
A75 = Rss Lmq
(L′
kq2 − Lmq
)ωb; A76 = Ld Lmq
(L′
kq2 − Lmq
)ωe;
A77 = R′
kq1
(L′
kq2 Lq − Lmq2)ωb; A78 = R
′
kq2 Lmq (Lmq − Lq) ωb;
A79 = Lmd Lmq
(Lmq − L
′
kq2
)ωe; A710 = Lmd Lmq
(Lmq − L
′
kq2
)ωe;
A711 = 0
A712 = (irds0 Ld − irf0 Lmd) Lmq
(L′
kq2 − Lmq
)ωb;
154
A85 = Rss Lmq
(L′
kq1 − Lmq
)ωb; A86 = Ld Lmq
(L′
kq1 − Lmq
)ωe;
A87 = −R′kq1 Lmq (Lmq + Lq) ωb; A88 = R′
kq2
(Lmq
2 + L′
kq1 Lq
)ωb;
A89 = Lmd Lmq
(Lmq − L
′
kq1
)ωe; A810 = Lmd Lmq
(Lmq − L
′
kq1
)ωe;
A811 = 0
A812 = (irds0 Ld − irf0 Lmd) Lmq
(L′
kq1 − Lmq
)ωb;
A95 = Lq Lmd
(Lmd − L
′
kd
)ωe; A96 = Rss Lmd
(L′
kd − Lmd
)ωb;
A97 = Lmq Lmd
(L′
kd − Lmd
)ωe; A98 = Lmq Lmd
(L′
kd − Lmd
)ωe;
A99 = r′
kd
(L′
kd Ld − Lmd2)ωb; A910 = −R′kd Lmd (Ld − Lmd) ωb;
A911 = 0 A912 = 0;
A105 = Lq Lmd
(L′
kd + Lmd
)ωe; A106 = −Rss Lmd
(L′
kd + Lmd
)ωb;
A107 = −Lmq Lmd
(L′
kd + Lmd
)ωe; A108 = −Lmq Lmd
(L′
kd + Lmd
)ωe;
A109 = −r′kd Lmd (Ld − Lmd) ωb; A1010 = −R′kd
(Lmd
2 + L′
kd Ld
)ωb;
A1011 = 0 A1012 = 0;
A115 = 0; A116 = 0; A117 = 0; A118 = 0; A119 = 0; A1110 = 0; A1111 = ωb; A1112 = 0;
b11 = −Lr ωb; b12 = 0; b13 = Lm ωb; b14 = 0;
b21 = 0; b22 = −Lr ωb; b23 = 0; b24 = Lm ωb;
b31 = Lm ωb; b32 = 0; b33 = −Ls ωb; b34 = 0;
b41 = 0; b42 = Lm ωb; b43 = 0; b44 = −Ls ωb;
b55 =1
2H1
155
b55 =(L′
kq1 L′
kq2 − Lmq2)ωb; b56 = 0;
b57 = Lmq
(L′
kq2 + Lmq
)ωb; b58 = −Lmq
(L′
kq1 + Lmq
)ωb;
b59 = 0; b510 = 0;
b511 = 0;
b65 = 0; b66 = −(L′
kd
2+ Lmd
2)ωb;
b67 = 0; b68 = 0;
b69 = r′
kd
(Lmd − L
′
kd
)ωb; b610 = Lmd
(L′
kd + Lmd
)ωb;
b611 = 0;
b75 = Lmq
(L′
kq2 − Lmq
)ωb; b76 = 0;
b77 =(Lmq
2 − L′kq2 Lq
)ωb; b78 = −Lmq (Lmq − Lq) ωb;
b79 = 0; b710 = 0;
b711 = 0;
b85 = Lmq
(L′
kq1 − Lmq
)ωb; b86 = 0;
b87 = Lmq (Lmq + Lq) ωb; b88 = −(Lmq
2 + L′
kq1 Lq
)ωb;
b89 = 0; b810 = 0;
b811 = 0;
b95 = 0; b96 = Lmd
(L′
kd − Lmd
)ωb;
b97 = 0; b98 = 0;
b99 = r′
kd
(Lmd
2 − L′kd Ld
)ωb; b910 = Lmd (Ld − Lmd) ωb;
b911 = 0;
b105 = 0; b106 = −Lmd
(L′
kd + Lmd
)ωb;
b107 = 0; b108 = 0;
b109 = r′
kd (Ld − Lmd) ωb; b1010 =(Lmd
2 + L′
kd Ld
)ωb;
156
b1011 = 0;
b115 = 0; b116 = −Lmd
(L′
kd + Lmd
)ωb;
b117 = 0; b118 = 0;
b119 = r′
kd (Ld − Lmd) ωb; b1110 =(Lmd
2 + L′
kd Ld
)ωb;
b1111 = 0;
Elements of Jacobian, eq 3.43
J11 = D1ωb
[L′
rrs + Lmωb∂∆veqr∂∆ieds
];
J12 = D1
[ωe(S0 Lm
2 − Lr Ls)
+ Lmωb∂∆veqr∂∆ieqs
]J13 = D1
[−r′rLmωb + Lmωb
∂∆veqr∂∆ieqr
];
J14 = D1
[Lm Lr ωe (S0 − 1) + Lmωb
∂∆veqr∂∆iedr
];
J112 = D1ωb
[Lm
(ieds0 Lm + i
′edr0 Lr
)+ Lmωb
∂∆veqr
∂∆ωrωe
];
J21 = D1
[−ωe
(S0 Lm
2 − Lr Ls)
+ ωbLm∂∆vedr∂∆ieqs
];
J22 = D1ωb
[L′
rrs + Lm∂∆vedr∂∆ieds
];
J23 = D1ωb
[−Lm Lr ωe (S0 − 1) + Lm
∂∆vedr∂∆ieqr
];
J24 = D1ωb
[−r′rLmωb + Lm
∂∆vedr∂∆iedr
];
J212 = D1ωb
[−Lm
(ieqs0 Lm + i
′eqr0 Lr
)ωb + Lm
∂∆vedr∂∆ωr
ωe
];
157
J31 = D1ωb
[−L′rrsωb − Ls
∂∆v′eqr
∂∆ieqs
];
J32 = D1ωb
[−Lm Ls ωe (S0 − 1)− Ls
∂∆veqr∂∆ieds
];
J33 = D1ωb
[−r′rLs − Ls
∂∆veqr∂∆ieqr
];
J34 = D1
[ωe(Lm
2 − S0 Lr Ls)− ωbLs
∂∆veqr∂∆iedr
];
J312 = D1ωb
[−Ls
(ieds0 Lm + i
′edr0 Lr
)− ωbLs
∂∆veqr
∂∆ωrωe
];
J41 = D1
[Lm Ls ωe (S0 − 1)− ωbLs
∂∆vedr∂∆ieqs
];
J42 = D1ωb
[−Lmrsωb − ωbLs
∂∆vedr∂∆ieds
];
J43 = D1
[ωe(S0 Lr Ls − Lm2
)− ωbLs
∂∆vedr∂∆ieqr
];
J44 = D1ωb
[r′
rLsωb − Ls∂∆vedr∂∆iedr
];
J412 = D1ωb
[Ls
(ieqs0 Lm + i
′eqr0 Lr
)− ωbLs
∂∆vedr∂∆ωr
ωe
];
J55 = D2ωb
(L′
kq1 L′
kq2 − Lmq2)[
rs′ωb +∂∆v
′rqs′
∂∆i′rqs′
];
J56 = D2ωe
(L′
kq1 L′
kq2 − Lmq2)[
Ldωe +∂∆v
′rqs′
∂∆i′rds′
];
J57 = D2ωb
[−r′kq1 Lmq
(L′
kq2 + Lmq
)ωb +
(L′
kq1 L′
kq2 − Lmq2) ∂∆v
′rqs′
∂∆i′rkq1
];
J58 = D2
[R′
kq2 Lmq
(L′
kq1 + Lmq
)ωb +
(L′
kq1 L′
kq2 − Lmq2) ∂∆v
′rqs′
∂∆i′rkq2
];
J59 = D2ωe
(L′
kq1 L′
kq2 − Lmq2)[−Lmdωe +
∂∆v′rqs′
∂∆i′rfd
ωb
];
158
J510 = D2ωe
(L′
kq1 L′
kq2 − Lmq2)[−Lmd +
∂∆v′rqs′
∂∆i′rkd
ωb
];
J11 = 0.
J512 = −D2ωb
(L′
kq1 L′
kq2 − Lmq2)[
(irds0 Ld − irf0 Lmd) +∂∆v
′rqs′
∂∆ωrωe
];
J65 = D3
[−Lq
(L′
fdL′
kd − Lmd2)ωe + ωb
(L′
fdL′
kd − L2md
) ∂∆vrds′
∂∆i′rqs
−r′fdωb(L′
kd − Lmd) ∂∆e
′rxfd
∂∆i′rqs
];
J66 = D3ωb
[rs′(L′
fdL′
kd − L2md
)+ ωb
(L′
fdL′
kd − L2md
) ∂∆vrds′
∂∆i′rkq1
−r′fdωb(L′
kd − Lmd) ∂∆e
′rxfd
∂∆i′rkq1
];
J67 = D3
[Lmq
(L′
fdL′
kd − Lmd2)ωe + ωb
(L′
fdL′
kd − L2md
) ∂∆vrds′
∂∆i′rkq2
−r′fdωb(L′
kd − Lmd) ∂∆e
′rxfd
∂∆i′rkq2
];
J68 = D3
[Lmq
(L′
fdL′
kd − Lmd2)ωe + ωb
(L′
fdL′
kd − L2md
) ∂∆vrds′
∂∆i′rkq2
−r′fdωb(L′
kd − Lmd) ∂∆e
′rxfd
∂∆i′rkq2
];
J69 = D3ωb
[r′
fdLmd
(L′
kd − Lmd)
+ ωb
(L′
fdL′
kd − L2md
) ∂∆vrds′
∂∆i′rfd
−r′fdωb(L′
kd − Lmd) ∂∆e
′rxfd
∂∆i′rfd
];
J610 = D3ωb
[−r′kd
(Lmd
2 − L′fdLmd)ωb + ωb
(L′
fdL′
kd − L2md
) ∂∆vrds′
∂∆i′rkd
159
−r′fdωb(L′
kd − Lmd) ∂∆e
′rxfd
∂∆i′rkd
];
J611 = D3ωb
[(L′
fdL′
kd − Lmd2) (vrqs0ωb + ∂∆irqs0rsωb + ∂∆irds0Lqωe
)+ωb
(L′
fdL′
kd − L2md
) ∂∆vrds′
∆δ− r′fdωb
(L′
kd − Lmd) ∂∆e
′rxfd
∆δ
];
J612 = 0;
J75 = D2ωb
[−rs′ +
∂∆v′rqs′
∂∆irqs′
]; J76 = D2ωb
[−Ldωe +
∂∆v′rqs′
∂∆irds′
];
J77 = D2ωb
[−r′kq1ωb +
∂∆v′rqs′
∂∆irkq1′
]; J78 = D2
[ωbr
′
kq2
∂∆v′rqs′
∂∆irkq2′
];
J79 = D2ωb
[Lmd +
∂∆v′rqs′
∂∆irfd′
]; J710 = D2
[Lmd +
∂∆v′rqs′
∂∆irkd′
];
J711 = D2ωb
[vrds0ωb + irds0rsωb − irqs0Ldωe +
∂∆v′rqs′
∂∆δ
];
J712 = D2
[(irds0Lmq
2Lq − irds0Lmq2Lqωb
−L′kq2irds0LmqLq + L′
kq2irds0LmqLqωb) + ωb
∂∆v′rqs′
∂∆ωrωe
];
J85 = D2ωb
[−rs′ −
∂∆vrqs′
∂∆irqs′
]; J86 = D2
[−Ldωe − ωb
∂∆vrqs′
∂∆irds′
];
J87 = D2ωb
[r′
kq1 − ωb∂∆vrqs′
∂∆irkq1′
]; J88 = D2ωb
[−r′kq2 −
∂∆vrqs′
∂∆irkq2′
];
160
J89 = D2
[Lmdωe − ωb
∂∆vrqs′
∂∆irfd′
]; J810 = D2
[Lmdωe − ωb
∂∆vrqs′
∂∆irkd′
];
J811 = D2ωb
[(vrds0 + irds0rsωb − irqs0Ldωe
)−∂∆vrqs′
∂∆δ
];
J812 = D2
[(irds0Lmq
2Lq − irds0Lmq2Lqωb−
−L′kq2irds0LmqLq + L′
kq2irds0LmqLqωb) + ωb
∂∆v′rqs′
∂∆ωrωe
];
J95 = D3
[Lqωe + ωb
r′
fd
Lmd
∂∆e′rxfd
∆i′rqs′− ωb
∂∆vrqs′
∂∆i′rqs′
];
J96 = D3ωb
[−rs′ +
r′
fd
Lmd
∂∆e′rxfd
∆i′rds′−∂∆vrqs′
∂∆i′rds′
];
J97 = D3
[−Lmqωe + ωb
r′
fd
Lmd
∂∆e′rxfd
∆i′rkq1′− ωb
∂∆vrqs′
∂∆i′rkq1′
];
J98 = D3
[−Lmqωe + ωb
r′
fd
Lmd
∂∆e′rxfd
∆i′rkq2′− ωb
∂∆vrqs′
∂∆i′rkq2′
];
J99 = D3ωb
[−r′fd +
r′
fd
Lmd
∂∆e′rxfd
∆i′rfd′−∂∆vrqs′
∂∆i′rfd′
];
J910 = D3ωb
[r′
kd +r′
fd
Lmd
∂∆e′rxfd
∆i′rkd′−∂∆vrqs′
∂∆i′rkd′
];
J911 = D3
[(−vrqs0ωb − irqs0rsωb − irds0Lqωe
)+
r′
fd
Lmd
∂∆e′rxfd
∂∆δ−∂∆vrqs′
∂∆δ
];
J912 = 0;
J105 = D3
[ωeLq
(Lq
(Lmd
2 − L′fdLmd))
161
+r′
fdωb (Ld − Lmd)∂∆e
′rxfd
∂∆i′rqs′− ωb
(L2md − L
′
fdLmd
) ∂∆vrds′
∂∆i′rqs′
];
J106 = D3ωb
[rs′(Lq
(Lmd
2 − L′fdLmd))
+r′
fd (Ld − Lmd)∂∆e
′rxfd
∂∆i′rds′−(Lmd
2 − L′fdLmd) ∂∆vrds′
∂∆i′rds′
];
J107 = D3
[−ωeLmd
(Lmd
2 − L′fdLmd)
+r′
fdωb (Ld − Lmd)∂∆e
′rxfd
∂∆i′rkq1′− ωb
(Lmd
2 − L′fdLmd) ∂∆vrds′
∂∆i′rkq1′
];
J108 = D3
[−ωeLmq
(Lmd
2 − L′fdLmd)
+r′
fdωb (Ld − Lmd)∂∆e
′rxfd
∂∆i′rkq2′− ωb
(Lmd
2 − L′fdLmd) ∂∆vrds′
∂∆i′rkq2′
];
J109 = D3ωb
[−r′fdLmd (Ld − Lmd) + r
′
fd (Ld − Lmd)∂∆e
′rxfd
∂∆i′rfd′−(Lmd
2 − L′fdLmd) ∂∆vrds′
∂∆i′rfd′
]
J1010 = D3
[ωeLq
(Lmd
2 − L′fdLmd)
+ r′
fdωb (Ld − Lmd)∂∆e
′rxfd
∂∆i′rkd′
−ωb(Lmd
2 − L′fdLmd) ∂∆vrds′
∂∆i′rkd′
];
J1011 = D3ωb
[−(Lmd
2 − L′fdLmd) (vrqs0 + irqs0rs + irds0Lqωe
)+r
′
fd (Ld − Lmd)∂∆e
′rxfd
∂∆δ−(Lmd
2 − L′fdLmd) ∂∆vrds′
∂∆δ
];
J1012 = D3ωb
[+r
′
fd (Ld − Lmd)∂∆e
′rxfd
∂∆ωrωe
−(Lmd
2 − L′fdLmd) ∂∆vrds′
∂∆ωrωe
]
J1112 = ωb2;
J121 =
[i′edr0Lm2H1
−(
1
2(H1 −H2)
∂∆Tm∂∆i′eqs
)];
J122 =
[−i′eqr0Lm
2H1
−(
1
2(H1 −H2)
∂∆Tm∂∆i
′eds
)];
162
J123 =
[−i
eds0Lm2H1
−(
1
2(H1 −H2)
∂∆Tm∂∆i′eqr
)];
J124 =
[−ieqs0Lm
2H1
−(
1
2(H1 −H2)
∂∆Tm∂∆i
′edr
)];
J125 =
[(Lmd
(irds0 − i
′rfd0
)− irds0Lmq
)2H2ωb
−
(1
2(H1 −H2)
∂∆Tm∂∆i
′rqs′
)];
J126 =
[(irqs0Lmd − irqs0Lmq
)2H2ωb
−(
1
2(H1 −H2)
∂∆Tm∂∆i
′rds′
)];
J127 =
[irds0Lmq
2H2
−
(1
2(H1 −H2)
∂∆Tm∂∆i
′rkq1′
)];
J128 =
[irds0Lmq
2H2
−
(1
2(H1 −H2)
∂∆Tm∂∆i
′rkq2′
)];
J129 =
[−irqs0Lmd
2H2
−
(1
2(H1 −H2)
∂∆Tm∂∆i
′rfd′
)];
J1210 =
[−irqs0Lmd
2H2
−(
1
2(H1 −H2)
∂∆Tm∂∆i
′rkd′
)];
J1211 =
[irqs0(irqs0Lmd − irqs0Lmq
)+ irds0
(Lmd
(irfd0 − irds0
)+ irds0Lmq
)2H2
−(
1
2(H1 −H2)
∂∆Tm∂∆δ
)];
J1212 = 0
T and T−1 transformation multiplications into Jacobian ele-
ments of equation 3.43J13 J14
J23 J24
·T; T−1 ·
J31 J32
J41 J42
; T−1 ·
J33 J34
J43 J44
·T; T−1 ·
J311 J312
J411 J412
;
T ·
J55 J56
J65 J66
·T−1; T ·
J57 · · · J510
J67 · · · J610
;
J75 J76
J85 J86
·T−1;
J95 J96
J105 J106
·T−1;
J113 J114
J123 J124
·T−1;
J115 J116
J125 J126
·T−1;
163
Pump and blade design geometry and features
Diameter, D = 1.5m
Revolutions per minute, rpm = 1200 /min
Fluid output velocity, v = 40m/s
Number of blades , B = 30
Blade solidity σ = 0.8537
Efficiency η = 57.667%
Power, P = 100 kW
loading medium Pitch, H = 3.11 m
DFIG parameters
Rs = 0.0076 p.u.
Rr = 0.005 p.u.
Ls = 0.0076 p.u.
Lr = 0.156 p.u.
Lm = 3.5 p.u.
pole pairs = 2
Hg = 0.5 s
Prated = 1.5 MW
Vrated = 575 V
Synchronous machine parameters
Parameters are adopted from the General Electric Marelli 3 phase generators MJB 315
MA 4 series Vrated = 440 V
Prated = 328 kW
164
Short circuit ratio = 0.46
Rs = 0.085 p.u. at 200C
xd = 2.75 p.u.
xq = 1.45 p.u.
x′
d = 0.244 p.u.
x′′
d = 0.109 p.u.
x′′q = 0.129 p.u.
T′
do = 1.6
T′′
d = 0.014
T′
d = 0.145
Ta = 0.0018
Distribution line R, L C parameters
Resistance matrix (ohm/km):
0.0890 0.0790 0.0773
0.0790 0.0915 0.0790
0.0773 0.0790 0.0890
(.1)
Inductance matrix (H/km):
1.6100e− 03 7.8539e− 04 6.4938e− 04
7.8539e− 04 1.6053e− 03 7.8539e− 04
6.4938e− 04 7.8539e− 04 1.6100e− 03
(.2)
165
Capacitance matrix (F/km):
1.1661e− 08 −2.1268e− 09 −5.8362e− 10
−2.1268e− 09 1.2117e− 08 −2.1268e− 09
−5.8362e− 10 −2.1268e− 09 1.1661e− 08
(.3)
Positive- & zero- sequence resistance [R1 R0] (ohm/km):
[0.0114 0.2466
](.4)
Positive- & zero- sequence inductance [L1 L0] (H/km):
[8.6839e− 04 3.0886e− 03
](.5)
Positive- & zero- sequence capacitance [C1 C0] (F/km):
[1.3426e− 08 8.5885e− 09
](.6)
166