University of the Ryukyus JAPAN
Modeling-Based Design of Intelligent Control Paradigms for Modern Wind Generating Systems by
MUHANDO, Billy Endusa
A dissertation submitted to the Graduate School of Engineering and Science in partial fulfillment of the requirements for the degree of Doctor of Engineering in Interdisciplinary Intelligent Systems Engineering
March 2008
Copyright by MUHANDO, Billy Endusa March 2008
Preface
G
ERMAN physicist Albert Einstein (1879-1955), who fancied himself as a violinist, was rehearsing a Haydn string quartet. When he failed for the fourth time to get his entry in the
second movement, the cellist looked up and said, “The problem with you, Albert, is that you simply can’t count—what is your occupation anyhow?” to which he answered that he (Einstein) was an artist’s model, reflecting his feeling that he was constantly posing for sculptures and paintings. Einstein’s conundrum aside, modeling has been embraced by engineers of various persuasion in systems’ design, and forms the basis of control design for wind generating systems in this research. The dissertation has been submitted in partial fulfillment of the requirements for the degree of Doctor of Engineering in Interdisciplinary Intelligent Systems. It has been prepared in Japan at the Power Energy System Control (PESC) Laboratory of the Electrical & Electronics Department, Faculty of Engineering, University of the Ryukyus. The project has been carried out as a harmonization between the Laboratory’s core research areas (Power Systems, Power Electronics, and Control Systems) with Sustainable Energy. The dissertation is a condensed report based on investigation of generator torque control for optimal performance of a three-bladed, variable-speed wind generating system with active pitch regulation. It is based on articles published or submitted to peer-reviewed journals during the period of the PhD project. The problem definition, methodology and steps leading to solutions to the problems, and finally the results, are presented in a concise manner outlined in two parts. The first part of the dissertation is concerned with modeling the aerodynamic conversion system. As an introduction, a brief rundown is offered on recent trends in world energy demand, the integration of wind energy in the global energy mix, challenges to the wind industry, and the problems thereof. Modeling the various wind turbine subsystems — wind speed and power train system (comprising the electrical and mechanical parts) — aims to provide an argumentative framework for a prototype that can be independently evaluated for validation. The main contribution is the harmonization of the various state-space models with varying dynamics to facilitate multiobjective controller design. The second part deals with optimal controller design based on some defined control strategies. It discusses the potential of several advanced intelligent control paradigms for meeting the two contradictory control objectives: power conversion maximization and active attenuation of structuraldynamic load-oscillations as well as static loads of the drive-train. Computer simulations executed in C-programming and MATLAB /Simulink™ environments confirm the efficacy of the paradigms
(albeit often in amalgamated configurations) when applied to the developed performability models. i
ii In 1985 I came across A. F. Abbott’s book: Ordinary Level Physics, at my brother Clyde’s room. Unaware of the fact that I would not return it for two years and only then under threat of severe penalty, Clyde let me borrow it. I am glad he did. Around the same time I was enrolled in junior high at the Alliance High School. Physics at this level was an interesting and at the same time frustrating experience. Looking back, the book was elemental in arousing my thinking about machines and very deeply influenced me in pursuing a career in engineering. My time in studying and practising engineering has been intellectually stimulating, thought-provoking, challenging and above all, fun, mostly thanks to practising engineers I have met in the field including Kenya, Belgium, and Japan. Thanks are due first to Prof Tomonobu Senjyu — my doctoral studies supervisor — for his great insights, perspectives, and sense of humor. Prof Senjyu, as always, both challenged me and guided me throughout my thesis work, kept me abreast of current work in wind turbine research and helped put this work on a strong foundation by facilitating presentation of our research results in various colloquiums across the globe. My sincere thanks go to both Prof Hiroshi Kinjo and Prof Tetsuhiko Yamamoto (formally) of Mechanical Systems Engineering who dedicated their time in seeing me through the two-year masters course leading to an M.Eng degree. They particularly inducted me to life in Japan, showed me the essence of research in control engineering, and encouraged me to publish and make presentations at national and international symposia. Special thanks go to dissertation committe member, Prof Koji Kurata for his time in reviewing this manuscript. Sincere gratitude is also extended to members of the academia at the University of the Ryukyus who have influenced my work and made this educational process a success, notably Prof R. M. Alsharif, Dr N. Urasaki, Dr K. Nakazono, Ms A. Kelly (for Japanese language tutorship), and members of the PESC laboratory who offered a conducive environment for research. I have been humbled by the altruistic commitment of Messrs. S. Murata and H. Arizono who were instrumental in my expertise in C-language, GNU-Plot, and (typesetting in) LATEX 2ε —the scientific word processor that effortlessly couples magnificent layout with user-unfriendliness of varying degrees! Of course none of this would have been possible without support from the following. Firstly, my parents: my Mom and dad’s loving encouragement (may your souls rest in peace) and inspiration (dad’s) at a young age to be a scientist. Secondly, my partner Senta Judy Haron, whom I can never thank enough for her endless love, company and encouragement. Last but not least, the Japan Ministry for Education, Culture, Sports, Science and Technology ( 文部科学省) for advancing me the Monbukagakusho (MEXT) scholarship for my 5-year graduate (master and doctoral) studies.
Okinawa, Japan.
March 2008
Muhando, B. E.
Abstract
A
GAINST the backdrop of increasing awareness of the effects of global warming due to greenhouse gas emissions and with fossil-fuel prices on the rise and their supply increasingly un-
stable, the need for more environmentally benign electric power systems is a critical part of the new thrust of engineering for sustainability. To address security of supply and energy diversification, wind energy is regarded the most attractive vanguard of the world’s energy challenges as it is clean, fuelfree (produces no CO2 ), and a renewable source of power. Wind plants have benefited from steady
advances in technology, and much of the advance has been made in the components dealing with the utility interface, the electrical machine, the power electronic converter, and the control capability. Wind turbines have become the most cost-effective renewable energy systems available today and are now completely competitive with essentially all conventional generation systems. However, the major problem is the wind’s unpredictable nature that forces utility operators to think differently about power generation, with the main challenge being to provide governor functions and controlled rampdown during high wind speed events. Additionally, wind turbines present nonlinear dynamic behavior and lightly damped resonant modes. This thesis examines design of advanced control paradigms geared toward lessening the negative impact of wind stochasticity on modern MW-class wind energy conversion systems (WECS) during high turbulence. The main control design objectives are to maximize power conversion throughout the operating envelope for steady output power as well as to actively attenuate structural-dynamic load-oscillations of the drive-train. The proposed advanced paradigms include the linear quadratic Gaussian (LQG), artificial neural networks (ANNs) in form of neurocontrollers, the self-tuning regulator (STR), and a model-based predictive control (MBPC) scheme. These yield, singly or in combination, digital systems whereby control is exercised through regulation of generator torque. Their design is enhanced by modeling: the plant and its environment are structured as a system of interacting subsystems that constitute an equivalent model defined in state space. The disturbance (input) signal is the wind that is modelled as a stochastic process constituted by the seasonal mean wind speed and the instantaneous turbulence component, while drive-train components (turbine, gearing and generator subsystems) are represented as a series of inertias linked by ‘soft’ shafts without friction. Computer simulations conducted using the MATLAB /Simulink™ software, with the generator model as an interface between the mechanical and electrical characteristics of the WECS reveal that achieving the objectives of optimal operation for reliability by the proposed multiobjective schemes becomes more attractive vis-`a-vis the classical proportional-integral-derivative (PID) controller. iii
Glossary I. Acronyms and Abbreviations (A)NN
(artificial) neural network
ARMA
auto-regressive moving average
CF
capacity factor
COE
cost of energy
CSS
constrained stochastic simulation
DOIG
double output induction generator
FSIG
fixed speed induction generator
GHGs
greenhouse gas emissions
GSC
grid side converter
HAWT
horizontal axis wind turbine
LQ
linear quadratic
LQG
linear quadratic Gaussian
MBPC
model-based predictive control
MPPT
maximum power point tracking
NC
neurocontroller
OP
operational point
PI
proportinal-integral (controller)
PID
proportinal-integral-derivative
RSC
rotor side converter
Std
The IEC61400-1 Standard
STR
self-tuning regulator
VAWT
vertical axis wind turbine
VSIG
variable speed induction generator
WECS
wind energy conversion system
iv
v
II. Nomenclature Notation for various symbols is defined as they occur in the text, however, the following are the common ones encountered across chapters: α β Γ θ λ µw ξ ρ
stator firing angle rotor collective pitch torque torsional angular twist tip-speed ratio (TSR) seasonal mean wind speed Gaussian noise air density
σ τ ϕ Ψ ω ∆ ∆t Λ
standard deviation actuator time constant hidden neuron output flux rotor speed deviation from reference simulation time step area of rotor disk
c cP cT f0 fn ird,rq isd,sq k kω kvw kβ rr rs t u urd,rq usd,sq vr vt vw x y
Weibull scale parameter power coefficient torque coefficient mechanical eigenfrequency grid nominal frequency rotor d- and q-axis current stator d- and q-axis current Weibull shape parameter partial derivative of Γt w.r.t. rotor speed partial derivative of Γt w.r.t. wind speed partial derivative of Γt w.r.t. pitch angle rotor resistance stator resistance time control input rotor d- and q-axis voltage stator d- and q-axis voltage rated (design) wind speed for WECS perturbed wind disturbance free-stream wind speed state vector control (or measured) output
A B C C(s) Ds G J Jg Jt Ks Kp Ki P Pe Pm Pr Q R Vhub Xm Xr Xs
state matrix control input gain matrix relates plant output to states controller transfer function drive-train torsional damping coefficient gain in full state feedback law quadratic cost function generator mass moment of inertia rotor mass moment of inertia drive-train torsional spring stiffness classical controller proportional gain classical controller integral gain weighting on the states x WECS electrical power WECS mechanical power WECS rated power weighting on the input u rotor radius wind speed at hub height mutual reactance rotor reactance stator reactance
Notes: 1. The subscript OP is used to denote the operating point (value of respective quantity at control design point). 2. The superscripts x˙ and x¨ denote the first and second derivatives of x w.r.t. time, i.e. d/dt and d2 /dt2 respectively, while xˆ represents the estimated value of x, an arbitrary dynamic quantity.
Contents I Analytic Models for Wind Energy Conversion Systems
1
1
Introduction
2
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
WECSs Generation Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.1
WECS Siting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.2
WECS Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.3
WECS Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.4
Problem Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.5
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.6
Goals and Scope of Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.6.1
Aim of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.6.2
Scientific and Technological Contribution of this Work . . . . . . . . . . . .
11
1.6.3
Outline of Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2
Aerodynamic Conversion Modeling
18
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2
Theoretical Development for Aerodynamic Conversion . . . . . . . . . . . . . . . .
19
2.2.1
Energy Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2.2
Power Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.3
Electrical Output Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.4
Capacity Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.3
Turbine Linearization for Steady-state Analysis . . . . . . . . . . . . . . . . . . . .
24
2.4
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
vi
vii 3
Drive-train Modeling
29
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2
Power train Modeling Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.3
Mechanical State Space System . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.4
Drive-train Torque Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.4.1
Steady-state Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.4.2
Operation under High Turbulent Inflow . . . . . . . . . . . . . . . . . . . .
35
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.5
4
Electrical System Modeling
38
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
4.2
Detailed Model of DOIG Unit with Converters . . . . . . . . . . . . . . . . . . . .
39
4.2.1
Construction and Operation Principle . . . . . . . . . . . . . . . . . . . . .
39
4.2.2
DOIG: Electrical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.2.3
DOIG: a Mechanical Perspective . . . . . . . . . . . . . . . . . . . . . . . .
46
DOIG Operation under Steady-state and Fault Conditions . . . . . . . . . . . . . . .
47
4.3.1
Steady-state Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.3.2
Transient Response and Fault-ride-through Analysis . . . . . . . . . . . . .
48
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.3
4.4
5
Modeling Wind Field Dynamics
52
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
5.2
Determination of Mean Wind Speed, vm . . . . . . . . . . . . . . . . . . . . . . . .
53
5.3
CSS Model for Wind Turbulence, vt (t) . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.3.1
Formulating the Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.3.2
Setting the Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
5.4
Real-time Wind Speed Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
5.5
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
viii
II Control Strategies and Design for Wind Energy Conversion Systems
62
6
Control Philosophy
63
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
6.2
Control Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
6.2.1
Model Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
6.2.2
Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
6.3.1
Active Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
6.3.2
Power-train Torsional Load Alleviation . . . . . . . . . . . . . . . . . . . .
68
Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
6.4.1
Assigning the Control Tasks . . . . . . . . . . . . . . . . . . . . . . . . . .
69
6.4.2
Pitch Actuator and Blade Servo . . . . . . . . . . . . . . . . . . . . . . . .
70
6.4.3
Generator Torque Controller . . . . . . . . . . . . . . . . . . . . . . . . . .
72
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
6.3
6.4
6.5
7
Full-State Feedback Digital Control by LQG
78
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
7.2
State Development for the Power-train . . . . . . . . . . . . . . . . . . . . . . . . .
80
7.3
LQG Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
7.3.1
State Estimation and LQG Design . . . . . . . . . . . . . . . . . . . . . . .
82
7.3.2
Choice of Weighting Matrices for LQG Cost Function, J
. . . . . . . . . .
84
7.3.3
Solution of the Stochastic Linear Regulator Problem . . . . . . . . . . . . .
85
Hybrid Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
7.4.1
NC Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
7.4.2
NC Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
7.5.1
Tracking Performance by Proposed Technique . . . . . . . . . . . . . . . .
90
7.5.2
Optimization of Power Output . . . . . . . . . . . . . . . . . . . . . . . . .
92
7.5.3
Minimization of Shaft Torsional Torque . . . . . . . . . . . . . . . . . . . .
94
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
7.4
7.5
7.6
ix 8
Predictive Control I: STR
97
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
8.2
WECS Multi-objective Control Concept . . . . . . . . . . . . . . . . . . . . . . . .
99
8.3
STR Design and Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.4
8.5
8.3.1
Outer Loop: Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . 101
8.3.2
Inner Loop: Control Law, Γg,ref . . . . . . . . . . . . . . . . . . . . . . . . 104
Simulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.4.1
Control for Energy Extraction . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.4.2
Control for Load Alleviation . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9
Predictive Control II: MBPC
111
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9.2
Control Concept for Power Regulation . . . . . . . . . . . . . . . . . . . . . . . . . 113
9.3
Generator Torque Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.4
9.5
9.3.1
Γg,ref by MBPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.3.2
Γg,ref by PI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Simulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.4.1
Aerodynamic Power Production . . . . . . . . . . . . . . . . . . . . . . . . 121
9.4.2
Drive-train Torque Variation Minimization . . . . . . . . . . . . . . . . . . 122
9.4.3
Comparison: MBPC and Classical PID . . . . . . . . . . . . . . . . . . . . 123
9.4.4
Evolution of Electrical Parameters . . . . . . . . . . . . . . . . . . . . . . . 124
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
10 Analysis, Perspectives, and Conclusions
128
10.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10.2 Modeling: an Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 10.3 WECS Modeling: Assessment of Approach and Validation . . . . . . . . . . . . . . 133 10.4 Control: an Appraisal of Classical and Advanced Paradigms . . . . . . . . . . . . . 136 10.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Part I Analytic Models for Wind Energy Conversion Systems
Chapter 1 Introduction 1.1 Background Wind Energy: Basis for Investment
R
ESOLVING the world’s growing demand for energy, minimizing related impacts on the environment and reducing the potential geopolitical tensions associated with increased competition
for energy supplies represent some of the greatest technical and policy challenges of the next several decades. These global energy and environmental challenges require a multidisciplinary systems approach that integrates policy design and technology development. Fossil fuels supply more than 80 percent of the world’s primary energy [1] but they are finite resources and major contributors to global climate change. The ways and means for their ultimate replacement with clean, affordable and sustainable energy sources at the scale required to power the world are not yet fully obvious, readily available or, in many instances, technically feasible. Also, these alternative sources are not all benign and their impacts on the environment, particularly when deployed at scale, are not fully understood. Turning off the carbon spigot is the first step, and many of the solutions are familiar: windmills, solar panels, nuclear plants. All three technologies are part of the energy mix, although each has its issues, including noise from windmills and radioactive waste from nukes. Moreover, existing energy infrastructures around the world are complex and very large, represent enormous capital investment and have operational life spans of 50 years or more. Wholesale or even piecemeal replacement of these infrastructures will be costly, will take time and will be frequently resisted by entrenched interests. In addition, the local, regional and global impacts of climate change require unique understanding of the scientific and technical underpinnings of the problems in order to formulate informed and timely responses at unprecedented national and international levels.
CHAPTER 1. INTRODUCTION
3
(a) Early windmill design, Denmark
(b) Modern WECS integrate well in urban environments
Figure 1.1: Evolution of WECS through the decades: structure influenced by purpose.
Meeting dramatic increases in energy demand, particularly in the developing world, will compound these problems at the same time that it enables opportunities for enhanced national stability, economic development and improved quality of life. To meet the energy, environmental, and security imperatives of the 21st century, it is essential that energy policy, technology development, regulatory and diplomatic decisions and actions be coordinated and based on the strongest, most informed and integrated scientific, economic and social analyses to: ◦ avoid or minimize the stranding of assets, ◦ optimize the investment in research, ◦ minimize potential economic dislocation during the transition to a sustainable energy future, ◦ preserve fundamental drivers of free markets by internalizing environmental stewardship, and ◦ maximize the opportunities for successful transformation of global energy systems.
Wind Energy: Decades of Technological Development In windmills (a much older technology), wind energy is used to turn mechanical machinery to do physical work; historically, windmills were used traditionally for grinding grain or spices, pumping water, sawing wood or hammering seeds (Fig. 1.1(a)). The evolution of modern turbines is a remarkable success story of engineering and scientific skill, coupled with a strong entrepreneurial spirit. The progress of wind energy around the world in recent years has been consistently impressive, with the main engineering challenge to the wind industry being to design an efficient wind turbine to harness that energy and turn it into electricity. Fig. 1.1(b) shows a modern wind turbine — structural design has been influenced by need to be a good neighbor!
CHAPTER 1. INTRODUCTION
1980 Nominal power (kW)
30 Rotor diameter (m) 15 Hub height (m) 30 Annual energy yield (kWh) 35,000
4
1985
1990
1995
2000
80 20 40 95,000
250 30 50 400,000
600 46 78 1,250,000
1,500 74 100 3,500,000
2005 5,000 124 120 17,000,000
Figure 1.2: Upscaling: size increased 100×, and energy yield grew 500-fold in just 25 years.
In the last 25 years turbines have increased in power by a factor of 100, the cost of energy has reduced (from $0.80/kWh in 1980 to $0.03 − 0.06/kWh in 2005 (in 2005 dollars) [2]), and the industry has moved from an idealistic fringe activity to the edge of conventional power generation. The cumulative global wind power production capacity has expanded rapidly, with global installed capacity standing at over 74 GW of electricity generating wind turbines that are operating in over 50 countries by the end of 2006, almost 4.5 times greater than in 2000 [3]. Further, future prospects are very promising: it is envisaged the total wind power installed world-wide could rise to 160 GW by 2012 [4], due to a broadening of the global wind energy market to engage a spread of new countries across all continents. Fig. 1.2 illustrates the growth in size of commercial wind turbines since 1980. There are, however, several impediments to truly large-scale deployment, including intermittency [5]-[7], the location of high-quality wind resources far from large demand centers, and public opposition to siting of WECS facilities [8]-[10]. The major concerns can be summed up as follows: • It is commonly held that the introduction of intermittent sources of electricity such as wind energy into a utility network causes operational problems and necessitates the provision of energy storage. • Reliability and durability of the structural assembly, based on O&M costs. • Wind power sceptics have raised questions on the conceivable environmental aspects, considering both physical and biological receptors as well as socio-economic impacts. • First-hour critics argued that with continued upscaling, the huge dimensions would limit the number of suitable potential locations.
CHAPTER 1. INTRODUCTION
5
Regarding integration of the output into the grid, in practice, most utility networks are able to maintain grid stability with penetrations of wind energy above 10% without any change to their operating procedures. In typical grid systems there may be an adverse economic impact for penetration levels above 20%, but there is no overriding technical difficulty that would limit wind energy penetration to very low values. Advances in energy-storage technologies can address intermittency issues. The modernization of the power network and increased efficiency of the grid will enable the integration and transmission of wind energy over longer distances. The point that is overlooked is the fact that there are numerous uncertainties in the electricity supply and demand balance; the variability associated with wind energy only causes problems once wind energy raises the statistical error margin. Local reinforcement of grids and the ability of variable speed turbines to contribute to grid stability counteract concerns about variability of supply, mismatch with demand, and the need for storage. Concerning reliability, most manufacturers peg the lifetime of a wind turbine at 20-25 years [11], and technological advances in the control system coupled with pertinent materials for blade strength have ensured long maintenance-free operation times, and reduced overheads. The last two arguments focus not so much on technological challenges but on aesthetics (visual impact), landscape integration and transport logistics. Public opposition to facility siting can be addressed, in part, through development of novel wind power technologies. Mechanical noise has practically been eliminated and aerodynamic noise has been vastly reduced (a WECS installation at 350m emits a noise level of 35–40 dB, which is comparable to a quiet indoor room. Wind itself is noisy!). Careful siting can avoid potential interference with electromagnetic radiation for communication. Besides, there is evidence from independent studies suggesting wind farms do not have a significant adverse effect on AM radio, navigation systems, mobile phone transmission, and military radar operation, with the exception of low level air-defence radar. On the brighter side, there has been considerable potential created for employment in all aspects of the wind industry (manufacture, project design, installation, and O&M), though there are different ways of estimating the personnel employed in the wind energy sector. Overall, the trend towards lower costs for wind-generated electricity has driven manufacturers to less conservative, more optimized machine design at an increasingly large scale 1 . Now industry insiders are talking about next-generation offshore turbine giants of 7.5 to 12 MW with rotor diameters of up to 200 metres. But how realistic are these plans? Is bigger better, and are there limits to wind turbine upscaling? Regarding these, the jury is still out. 1
The pace of upscaling can only be described as breathtaking. The world’s first commercial 4.5 MW prototype was created in 2002 and two 5 MW prototypes developed in 2004, with one installed in the North Sea 15 miles off the East coast of Scotland near the Beatrice Oil Field (assembly/commissioning in 2006). Currently, the world’s most powerful wind turbine — the E112 manufactured by Enercon GmbH of German — delivers up to 6 MW, has an overall height of 186 m and a diameter of 114 m. Interestingly, this is not the world’s largest wind turbine, it just produces the most power!
CHAPTER 1. INTRODUCTION
6
1.2 WECSs Generation Technologies 1.2.1 WECS Siting WECSs have traditionally been installed on land, but recent trends favor offshore siting because wind speeds are higher (may be 25% higher than at the coast) and less turbulent than onshore winds [12], and there is reduced social imapact.
1.2.2 WECS Configurations I. HAWT/VAWT Turbine development over the years has experimented with both horizontal-axis wind turbine (HAWT) and verical-axis wind turbine (VAWT) types. Due to their expected advantages of omni-directionality and having gears and generating equipment at the tower base, vertical axis designs were considered. However, several disadvantages have caused the vertical axis design route to disappear from the mainstream commercial market, including: • reduced aerodynamic efficiency — much of the blade surface is close to the axis • albeit usually at ground level, it is not feasible to have the gearbox of large VAWT at ground level because of the weight and cost of the transmission shaft • invariably have a lot of structure per unit of capacity (catenary curve loaded only in tension). II. Variable-speed, Pitch-regulated Variable speed is facilitated by pitch regulation that involves turning the blades about their lengthwise axes (pitching the blades) to regulate the power extracted by the rotor. • Advantages: (i) ability to supply power at a constant voltage and frequency while the rotor speed varies (ii) control of the active and reactive power, thus enhancing grid integration [13] (iii) variable-speed capability allows the turbine to operate at ideal tip-speed ratios over a larger range of wind speeds; peculiarly, the most dramatic increase in performance is realized at lower wind speeds. • Disadvantages: (i) require some active protection system to keep the turbine connected to the network but also protected against any over-current in the case of short-term grid disturbances [14] (ii) the alternating current (ac) they produce has a variable frequency that cannot be safely delivered to existing power transmission grids without conditioning [15],[16].
CHAPTER 1. INTRODUCTION
7
III. Fixed-speed, Stall-regulated As wind speed increases, the blades become increasingly stalled to limit power to acceptable levels without any additional active control. The rotor speed is held essentially constant, achieved through the connection of the electric generator to the stiff grid frequency. • Advantage: simple and robust construction, hence lower capital cost. • Disdvantages: (i) do not have the capability of independent control of active and reactive power (ii) offer no inherent means of torque oscillation damping which places a greater load and cost on their gearbox. Industry has been shifting toward variable speed for reasons related to overall wind turbine performance: they take full advantage of variations in the incident wind speed, encounter lower mechanical stress and less power fluctuations, and provide 10–15% higher energy output compared with constant speed operation [17],[18]. They are routinely connected “indirectly” to the grid to allow for power conditioning to occur (at the wind farm). The majority of modern turbines include transmissions, clutches, and rotor shaft braking systems or aerodynamic stall features that act on the rotor blades to maintain the variations in a rotor shaft’s rotation within prescribed design limits. Such turbines are also equipped with SCADA2 systems [19],[20] that can adjust operating conditions (e.g., aerodynamic stall and blade pitch) to changing wind conditions.
1.2.3 WECS Control Recent research concentrates on improving the technological advantage of wind plants over existing conventional power generating systems. Such research has seen various proposals for the wind industry to embrace novel digital control systems geared toward low installation and maintenance costs while ensuring maximum energy extraction efficiency. The most common strategies incorporate the linear proportional-integral-derivative (PID) controller [21]-[23] that has been tested in the field environment. Recently, multivariable control paradigms have been gaining prominence as they are multiobjective hence several control goals can be met simultaneously. Such robust schemes include sensorless techniques [24]-[26], adaptive control that incorporates gain-scheduling by the linear quadratic Gaussian (LQG) controller [27]-[31], the self-tuning regulator (STR) [32]-[34], and fuzzy control systems [35]-[37] that may be considered an extension of maximum power-point tracking (MPPT) schemes and yield more flexible but quite context-dependent controllers. 2
Supervisory Control And Data Acquisition — a system that collects data from various sensors at a plant or in other remote locations and then transmits this data to a central computer that manages and controls the data.
CHAPTER 1. INTRODUCTION
8
1.3 Motivation Enviromental and technological considerations form the conceptual framework for the essence and the design of modern WECS that incorporate sophisticated control paradigms, a theme for this thesis. Motivation for this work stems from, firstly, the need for stabilization of greenhouse-gas (GHG) emissions, which requires that annual emissions be brought down to the level that balances the Earth’s natural capacity to remove such gases from the atmosphere. Though prediction of the extent of climate change with complete certainty has not been established, the risks can be envisaged. Mitigation — taking strong action to reduce emissions — must be viewed as an investment, a cost incurred now and in the coming few decades to avoid the risks of very severe consequences in the future. The stocks of hydrocarbons that are profitable to extract (under current policies) are more than enough to take the world to levels of GHGs concentrations well beyond 750ppm CO 2 e, with very dangerous consequences3 . GHGs emissions contributed by the power sector can be cut by switching to lowercarbon technologies for electricity, to be at least 60%, and perhaps as much as 75%, decarbonized by 2050 to stabilize at or below 550ppm CO 2 e. While a portfolio of technologies to achieve this already exist, the priority is low-cost abatement so that they are competitive with fossil-fuel alternatives under a carbon-pricing policy regime. Most countries have formulated policies to support the wind industry, which is a powerful motivation that has seen wind turbine innovation across the globe. Secondly, computer processing power and available memory have increased at a phenomenal rate over the 20 years that the modern wind turbine industry has existed. Coupled with the possibilities for extremely user-friendly software environments, sophisticated design calculations can be executed in a straightforward and convenient manner by the wind turbine designer, using standard desktop PC hardware. Simulation — the time domain approach to calculating the response of a system subject to some disturbance — forms the basis of all current, state-of-the-art wind turbine design calculations. Computer simulation is a most powerful tool to investigate the means and capabilities of different technologies for integrating WECSs to the power network. When incorporation of large amount of wind power into electric power systems takes place, a number of technical problems will be encountered that need innovative solutions. The approach relies on computer modeling and simulations to develop effective control schemes to ensure reliability of the WECS and smooth integration of wind power into the grid. In this report, calculations are run on Intel Celeron™ CPU, 128 MB RAM, Unix compiler (C–programming) and Windows 2000 OS (MATLAB /Simulink™ environment) to develop computational tools for modeling. 3
CO2 e designates CO2 -equivalent
CHAPTER 1. INTRODUCTION
9
1.4 Problem Identification There are two intrinsic issues common to wind power systems that are explored in this thesis, relating to the operating environment, and robustness of the installed control scheme, viz: I Stochastic Operating Environment Control design in this rersearch focuses mainly on what will happen when grid-connected wind power plants experience large amounts of highly-fluctuating wind. The issues relate to ensuring steady electrical power output, alleviation of cyclic (torsional) loads on the power train components, and maintaining transient voltage stability, specifically, to avoid voltage collapse in the power system. II Limitation of Linear Control Systems For a long period in the wind industry, controller design has centered mainly on simple, linear, proportional-integral-derivative (PID) controllers that are easily implemented in the field environment. Conventional PID controllers must be conservatively tuned in order to ensure closed–loop stability over the full range of operating conditions. Gain selection for these controllers has generally been a trial-and-error process relying on experience and intuition from the engineers. Unfortunately, this means that the plant can not operate at high efficiency, since the wind turbine is a highly nonlinear process [38], [39].
1.5 Methodology The aforementioned objectives and control problem stem from wind stochasticity that impacts on both power quality and drive-train fatigue life for a WECS, and the nonlinearity in the system respectively. The WECS under consideration is an onshore, HAWT, variable speed, pitch-regulated type. Its electrical part is comprised by the double output induction generator (DOIG) — a configuration that employs a wound-rotor induction machine and a rotor converter cascade consisting of a back-to-back double-bridge inverter configuration based on IGBTs. The approach, as detailed below, entails two fundamentals: modeling of the various WECS dynamic components, and design of advanced control paradigms to enhance optimal operation geared toward low cost of energy (COE). The generality of the developed models strongly depends on the modeling requirements, i.e. time scale (transient and 120-second simulations), and nature of the phenomena to be reproduced (power quality and power train loading), as follows:
CHAPTER 1. INTRODUCTION
10
I Modeling Aspects No general model can be introduced that would represent with sufficient accuracy the dynamic behavior of all variable speed WECS schemes. In this report dynamic models are presented for a variable speed WECS configuration that uses an induction generator and stator or rotor AC/DC/AC converter cascade, for representing the behavior of the output power in both relatively slow wind variations, and also for calculating its stability margin during above rated turbulent inflow. In the sequel the modeling equations for each subsystem that constitute the WECS are presented and the main assumptions outlined. The following main subsystems are modelled independently: • Rotor aerodynamics, (includes a wind speed model) • Power-train, i.e. the torsional subsystem of the axes, gearbox and elastic couplings linking the turbine rotor to the electrical generator • Electrical and control subsystem, consisting of the electrical generator, the power electronics converters, and the associated controls, and • Blade-pitch regulation system and speed controller. II Advanced Control Design Although industry has embraced the PID controller, researchers have begun to investigate the capabilities of more sophisticated control designs [40]-[43]. The fundamental concept common to these designs is that they are both adaptive and depend on state feedback (often with state estimation to render full-state feedback). This study proposes several control schemes and evaluates their design and performance, notably: • Linear Quadratic Gaussian (LQG) — this converts control system design problems to an optimization problem with quadratic time-domain performance criteria; disturbances and measurement noise are modelled as stochastic processes. A hybrid scheme is also mooted, based on augmenting the LQG with a neurocontroller (NC), whereby the control load is shared such that the LQG handles the linear part while the NC utilizes the intrinsic properties of NNs to handle the nonlinearities inherent in the WECS system, to execute generator torque control. • Self-Tuning Regulator (STR) — control is exercised through a self-tuning regulator, and incorporates a recursive least squares (RLS) algorithm to estimate plant parameters. • Model-Based Predictive Control (MBPC) — control algorithm based on solving an online optimal control problem via a receding horizon policy.
CHAPTER 1. INTRODUCTION
11
1.6 Goals and Scope of Present Work 1.6.1 Aim of the Work The objective of this work is to develop advanced control techniques for variable-speed, pitch-regulated WECS by a modeling approach and validate their performance. This study is focused mainly on: • An investigation of the capabilities of advanced control paradigms for wind-electrical energy conversion performance. • Reduction of power train fatigue loads by enhanced damping through generator torque control. • An investigation on the impact of detailed DOIG wind turbine modeling on the accuracy of electrical system performance analysis. • The transient and steady-state analysis of a wind-power DOIG operating under high turbulence wind inflow using the developed power-train and field-circuit simulator models.
1.6.2 Scientific and Technological Contribution of this Work 1. Applying the developed methodology of combining detailed wind turbine subsystems’ modeling with a Matlab-Simulink environment for the analysis of the whole electric drive system and electric part of a wind energy conversion system, and validation of the developed simulator. 2. Comparative study of different variable speed wind turbine control approaches from the point of view of transient simulation accuracy, gauged upon the classical PID. 3. Verification of the method for coupling the magnetic field and circuit equations of the electrical machine with the drive-train dynamic equations. Adverse climatic change and hence the need for ‘green’ energy in contemporary times aside, global energy demand is consistently exponential, and wind energy is becoming a significant player in the energy mix. This research focuses on the need to design control systems that properly account for the flexible modes of the turbine, and maintain the stable closed-loop behavior of the WECS. Overall, this research contributes to advancement in wind technology geared toward lower cost of energy (COE), by proposing advanced control paradigms whose stock-in-trade is robustness. In addition, they are easily implemented in a microprocessor matrix. Most of the research has been published in peer-reviewed journals, establishing permanent reference value.
CHAPTER 1. INTRODUCTION
12
WECS Background: Chapter 1
Future work
Electrical system
Grid & Converter
DOIG model
Chapter 4
(H2 /Hinf & Fuzzy control)
Mechanical system
Aero dynamics model
Drive train model
Wind speed model
Chapter 2 Chapter 3 Chapter 5
Control system
Control concept (Gen. torque & pitch angle)
Chapter 6
LQG control
STR control
Chapter 7 Chapter 8
MBPC control
Chapter 11
Analysis & conclusions
Chapter 9 Chapter 10
Figure 1.3: Outline of presentation of the work in this thesis.
1.6.3 Outline of Presentation This project is a multi-task work; it contains elements of electric machinery theory, shaft system representation, aerodynamic relations, control features and controller design, and the overall interaction of the wind energy conversion system (wind turbine and power system), as depicted in Fig. 1.3. The content of the work is separated into two parts: • Part I deals with development of WECS subsystems’ dynamic models. State-space representation of the mechanical and electrical subsystems are harmonized as a foundation for the control design in the sequel. The models are discretized to enable sampling during the simulations, and the various time constants associated with the subsystems are defined. • Part II analyzes design, for optimality and stability in operation, of several advanced controllers applied to the perfomability models developed in Part I. Furthermore, the thesis is divided into eleven chapters, including the Introduction (Chapter 1) that has given a background check on the development of wind power through the decades, status of wind power today, the challenges, future trends, and the various WECS configurations. Motivation for this thesis has been presented, as well as the statement of the problem, the objectives, and the methodology employed in addressing the problems. The rest of the work is detailed in the following fashion. Chapter 2 validates the economic viability of WECSs by a theoretical development of a model for the energy conversion as well as the concept of turbine linearization that is essential for the control formulation in the steady-state analysis. The research work presented in this chapter appears in IET Proc. Renewable Power Generation. (Accepted for publication, 2007).
CHAPTER 1. INTRODUCTION
13
Chapter 3 deals with the mechanical dynamics of the WECS with regard to torsional loading. A spring-mass-damper model for the mechanical construction of the drive-train as a series of elastically coupled frictionless shafts is developed, and analysis of the system reliability is tackled. This concept is developed in most of the published works by the author as detailed in Appendix C. Chapter 4 describes the DOIG as the interface between the wind turbine and grid, with the converter control mentioned with a generic scheme. The work is based on DOIG modeling as presented in the author’s work: IEEE Transactions on Energy Conversion, (Forthcoming). Chapter 5 presents in detail the model for generating a real-time wind speed profile for the simulations, with particular emphasis on modeling of gust events for the turbulent inflow. This work is published in Renewable Energy, vol. 32, no. 14, pp. 2407-2423, and Wind Energy, doi:10.1002/we.236. Chapter 6 elucidates on the control philosophy, and describes the global model detailing interaction of the WECS subsystems. Description of controller formulation is presented. The chapter examines maximum power-point tracking (MPPT) schemes as well as their demerits, and suggests the necessary shift in controller design: use of multivariable schemes for generator torque control. These are handled in subsequent chapters. This research is published in Renewable Energy, vol. 31, no. 11, pp. 1764-1775, and Int. J. of Emerging Electric Power Systems, vol. 8, no. 2, Art. 3, pp. 1-19. Chapter 7 presents the first of several advanced control paradigms — the LQG in combination with a nuerocontroller (NC). Investigations are carried out on the suitability of the proposed controllers in meeting the two objectives at above rated wind speeds: output power leveling and drivetrain load mitigation. This research is published in Renewable Energy, vol. 32, no. 14, pp. 2407-2423, and Renewable Energy, doi:10.1016/j.renene.2007.12.001. Chapter 8 proposes the self-tuning regulator (STR). This research work is published in IET Procs. Control Theory & Applications, vol. 1, no. 5, pp. 1431-1440. Chapter 9 develops the model-based predictive control (MBPC) as an alternative control scheme that relies on prediction to minimize errors in control design and performance. This work appears in the paper submitted to IET Procs. Renewable Power Generation. Chapter 10 renders the analyses and perspectives — a crispy discussion on the implications of advanced control paradigms to the wind industry from the dual techno-economic viewpoint, as well as a conclusion of the thesis. Chapter 11 gives directions for future research, based on an on-going study of several schemes: H2 /H∞ and neurofuzzy logic, as a foundation for viable alternatives, albeit only qualitatively. The Appendix serves to provide important features regarding the modeling, in respect of WECS parameters and mathematical derivations for supporting various concepts developed therein.
CHAPTER 1. INTRODUCTION
14
References [1] The European Wind Energy Association (EWEA), “The current status of the wind industry — Industry overview, market data, employment, policy,” Available online, http://www.ewea.org [2] E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “RLS-based self-tuning regulator for WTG dynamic performance enhancement under stochastic setting,” Proc. The International Conference on Electrical Engineering, ICEE 2007, Hong Kong, 8-12 July 2007. [3] American Wind Energy Association (AWEA), “The economics of wind energy,” Available online, http://www.awea.org/pubs/factsheets/EconomicsOfWind-Feb2005.pdf [4] International Energy Agency (IEA), “Key world energy statistics 2007,” Available online, http://www.iea.org/statistics [5] K. Stunz, and J. Nedrud, “Multilevel energy storage for intermittent wind power conversion: computer system analogies,” IEEE Power Engineering Society General Meeting, 12–16 June 2005, vol. 2, pp. 1950-1951. [6] R. Piwko, D. Osborn, R. Gramlich, G. Jordan, D. Hawkins, and K. Porter, “Wind energy delivery issues (transmission planning and competitive electricity market operation),” IEEE Power and Energy Magazine, vol. 3, no. 6, pp. 47-56, Nov.-Dec. 2005. [7] J. P. Barton, and D. G. Infield, “Energy storage and its use with intermittent renewable energy,” IEEE Trans. Energy Conversion, vol. 19, no. 2, pp. 441-448, June 2004. [8] T. Bookman, “Wind energy’s promise, offshore,” IEEE Technology and Society Magazine, vol. 24, no. 2, pp. 9-15, Summer 2005. doi:10.1109/MTAS.2005.1442376. [9] V. Colello, “Wind energy projects in New York: facing the siting issue,” Environmental Quality Management, vol. 15, no. 1, pp. 105-110, Sept. 2005. doi:10.1002/tqem.20073. [10] J. Everaert, and E. W. M. Stienen, “Impact of wind turbines on birds in Zeebrugge (Belgium): significant effect on breeding tern colony due to collisions,” Biodiversity and Conservation, vol. 16, no. 12, Nov. 2007. doi:10.1007/s10531-006-9082-1. [11] British Wind Energy Association (BWEA), “Onshore wind turbines,” Available online, http//:www.bwea.com [12] A. R. Henderson, C. Morgan, B. Smith, H. C. Sørensen, R. J. Barthelmie, and B. Boesmans, “Offshore wind energy in europe – a review of the state-of-the-art,” Wind Energy, vol. 6, no. 1, pp. 36-52, Feb. 2003. doi:10.1002/we.82.
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[13] B. Rabelo, and W. Hoffmann, “Optimal active and reactive power control with the doubly-fed induction generator in the MW-class wind turbines,” Procs. 4th IEEE International Conference on Power Electronics and Drive Systems, 22-25 Oct. 2001, vol. 1, pp. 53-58. [14] L. Holdsworth, X. G. Wu, J. B. Ekanayake, and N. Jenkins, “Comparison of fixed speed and doubly-fed induction wind turbines during power system disturbances,” IEE Procs. Generation, Transmission and Distribution, vol. 150, no. 3, pp. 343-352, May 2003. doi:10.1049/ipgtd:20030251. [15] R. C. Portillo, M. M. Prats, J. I. Leon, J. A. Sanchez, J. M. Carrasco, E. Galvan, and L. G. Franquelo, “Modeling strategy for back-to-back three-level converters applied to high-power wind turbines,” IEEE Trans. Industrial Electronics, vol. 53, no. 5, pp. 1483-1491, Oct. 2006. doi:10.1109/TIE.2006.882025. [16] A. H. Ghorashi, S. S. Murthy, B. P. Singh, and B. Singh, “Analysis of wind driven grid connected induction generators under unbalanced grid conditions,” IEEE Trans. Energy Conversion, vol. 9, no. 2, pp. 217-223, June 1994. doi:10.1109/60.300156. [17] Q. Wang, and L. Chang, “An intelligent maximum power extraction algorithm for inverter-based variable speed wind turbine systems,” IEEE Trans. Power Electronics, vol. 19, no. 5, pp. 12421249, Sept. 2004. [18] K. Tan, and S. Islam, “Optimum control strategies in energy conversion of PMSG wind turbine system without mechanical sensors,” IEEE Trans. Energy Conversion, vol. 19, no. 2, pp. 392399, June 2004. [19] G. J. Smith, “SCADA in wind farms,” IEE Colloquium on Instrumentation in the Electrical Supply Industry, 29 June 1993, pp. 11/1-11/2. [20] O. Anaya-Lara, N. Jenkins, and J. R. McDonald, “Communications requirements and technology for wind farm operation and maintenance,” First International Conference on Industrial and Information Systems, 8-11 Aug. 2006, pp. 173-178. doi:10.1109/ICIIS.2006.365659. [21] A. Tapia, G. Tapia, J. X. Ostolaza, and J. R. Saenz, “Modeling and control of a wind turbine driven doubly fed induction generator,” IEEE Trans. Energy Conversion, vol. 18, no. 2, pp. 194-204, 2003. [22] B. Boukhezzar, L. Lupu, H. Siguerdidjane, and M. Hand, “Multivariable control strategy for variable speed, variable pitch wind turbines,” Renewable Energy, vol. 32, pp. 1273-1287, 2007.
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[23] M. Sedighzadeh, D. Arzaghi-Harris, and M. Kalantar, “Adaptive PID control of wind energy conversion systems using RASP1 mother wavelet basis function,” IEEE Region 10 Conference TENCON 2004, 21-24 Nov. 2004, vol. 3, pp. 524-527. doi:10.1109/TENCON.2004.1414823. [24] T. Senjyu, E. B. Muhando, A. Yona, N. Urasaki, H. Kinjo, and T. Funabashi, “Maximum wind power capture by sensorless rotor position and wind velocity estimation from flux linkage and sliding observer,” Int. Journal of Emerging Electric Power Systems, vol. 8, no. 2 Art. 3, pp. 1-19, 2007. [25] T. Senjyu, S. Tamaki, E. B. Muhando, N. Urasaki, H. Kinjo, T. Funabashi, H. Fujita, and H. Sekine, “Wind velocity and rotor position sensorless maximum power point tracking control for wind generation system,” Renewable Energy, vol. 31, no. 11, pp. 1764-1775, 2006. [26] R. Cardenas, R. Pena, G. Asher, and J. Cilia, “Sensorless control of induction machines for wind energy applications,” IEEE 33rd Annual Power Electronics Specialists Conference PESC 02.2002, vol. 1, pp. 265-270. doi:10.1109/PESC.2002.1023880. [27] E. B. Muhando, T. Senjyu, N. Urasaki, A. Yona, H. Kinjo, and T. Funabashi, “Gain scheduling control of variable speed WTG under widely varying turbulence loading,” Renewable Energy, vol. 32, no. 14, pp. 2407-2423, 2007. [28] P. Novak, T. Ekelund, I. Jovilk, and B. Schimidtbauer, “Modeling and control of variable speed wind turbine drive systems dynamics,” IEEE Control Systems Magazine, vol. 15, no. 4, pp. 28-38, 1995. [29] E. S. Abdin, and W. Xu, “Control design and dynamic performance analysis of a wind turbineinduction generator unit,” IEEE Trans. Energy Conversion, vol. 15, no. 1, pp. 91-96, 2000. [30] W. E. Leithead, S. de La Salle, and D. Reardon, “Role and objectives of control of wind turbines,” IEEE Proceedings, vol. 138 Pt. C, pp. 135-148, 1991. [31] I. Munteanu, N. A. Cutululis, A. I. Bratcu, and E. Ce˘ anga, “Optimization of variable speed wind power systems based on a LQG approach,” Control Engineering Practice, vol. 13, pp. 903-912, 2005. [32] E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “Disturbance rejection by dual pitch control and self-tuning regulator for WTG parametric uncertainty compensation,” IET Proc. Control Theory and Applications, vol. 1, no. 5, pp. 1431-1440, Sept. 2007. [33] W. Ren, and P. R. Kumar, “Stochastic adaptive prediction and model reference control,” IEEE Trans. Automatic Control, AC-30, pp. 2047-2060, 1994.
CHAPTER 1. INTRODUCTION
17
[34] T. Senjyu, R. Sakamoto, N. Urasaki, H. Higa, K. Uezato, and T. Funabashi, “Output power control of wind turbine generator by pitch angle control using minimum variance control,” IEEJ Trans. Power and Energy, vol. 124B, no. 12, pp. 1455-1463, 2004. [35] A. M. Mohamed, M. N. Eskander, and F. A. Ghali, “Fuzzy logic control based maximum power tracking of a wind energy system,” Renewable Energy, vol. 23, pp. 235-245, 2001. [36] M. G. Simoes, B. K. Bose, and R. J. Spiegel, “Fuzzy logic based intelligent control of a variable speed cage machine wind generation system,” IEEE Trans. Power Electronics, vol. 12, no. 1, pp. 87-95, 1997. [37] R. G. de Almeida, J. A. P. Lopes, and J. A. L. Barreiros, “Improving power system dynamic behavior through doubly fed induction machines controlled by static converter using fuzzy control,” IEEE Trans. Power Systems, vol. 19, no. 4, pp. 1942-1950, Nov. 2004. [38] E. B. Muhando, T. Senjyu, N. Urasaki, H. Kinjo, and T. Funabashi, “Online WTG dynamic performance and transient stability enhancement by evolutionary LQG,” IEEE Power Engineering Society General Meeting, 24-28 June 2007, pp. 1-8. doi:10.1109/PES.2007.385499. [39] C. W. Kung, R. K. Joseph, and N. K. Thupili, “Evaluation of classical and fuzzy logic controllers for wind turbine yaw control,” Procs. The First IEEE Regional Conference on Aerospace Control Systems, 25-27 May 1993, pp. 254-258. [40] E. B. Muhando, T. Senjyu, O. Z. Siagi, and T. Funabashi, “Intelligent optimal control of wind power generating system by a complemented linear quadratic Gaussian approach,” Procs. IEEE Power Engineering Society Conference and Exhibition, PowerAfrica 2007, 16-20 July 2007. [41] T. Ekelund, “Speed control of wind turbines in the stall region,” Procs. 3rd IEEE Conference on Control Applications, 24-26 Aug. 1994, vol. 1, pp. 227-232, 2007. doi:10.1109/CCA.1994.381194. [42] I. Kraan, and P. M. M. Bongers, “Control of a wind turbine using several linear robust controllers,” Procs. 32nd IEEE Conference on Decision and Control, 15-17 Dec. 1993, vol. 2, pp. 1928-1929. doi:10.1109/CDC.1993.325530. [43] K. Stol and M. Balas, “Full-state feedback control of a variable-speed wind turbine: a comparison of periodic and constant gains,” ASME Journal of Solar Energy Engineering, vol. 123, no. 4, pp. 319-326, Nov. 2001.
Chapter 2 Aerodynamic Conversion Modeling 2.1 Introduction
A
LL the successful megawatt-class wind technology developments to date are results of rather conventional evolutionary design efforts whose basis is the premise that control can signifi-
cantly improve energy capture and reduce dynamic loads in a WECS [1],[2]. As wind turbines grow in size, their components will be subjected to additional wind loading associated with complex environments of their installation. Indeed, rotor structural dynamics significantly influence the wind turbine response during electrical faults. To provide industry with the support it needs to develop technologies capable of cost-effective operation in stochastic wind speed resource areas, it is important for researchers to understand drive train design for effective power conversion through advanced power electronic components. Because blades and rotor comprise up to 25% of the WECS’s total capital cost, and the rotor captures 100% of the energy, technology improvements in these areas can provide as much as 50% of the cost reduction. Typically, when power rating goes up, rotor diameter increases too. Maintaining an optimum ratio between rated power and rotor swept area is essential, but the optimal value depends to a large extent on average wind speed at hub height. One implication of increasing rotor diameters is increased aerodynamic noise, hence rotor speed, as a rule, has to come down to curb these emissions. Beginning with simple calculations based largely on engineering intuition, the approach to wind turbine design has been transformed to the point where sophisticated computer-based analysis is now performed routinely throughout the industry. This chapter develops both aerodynamic and electrical models for power production based on empirical formulations. These assist in visualizing the concept of control design as a philosophy geared toward achieving near-ideal performance within the technological constraints that abound in the wind industry.
CHAPTER 2. AERODYNAMIC CONVERSION MODELING WINDMILL
ELECTRICAL SYSTEM
DRIVE TRAIN
AC generator
vw
Torque control
19 Γref
AC Gearbox Γt
AC Γg
Grid
Figure 2.1: Generalized block diagram of the WECS’s main subsystems.
2.2 Theoretical Development for Aerodynamic Conversion Fig. 2.1 depicts the interconnection of the main drive train components. The windmill comprises the blades and the hub. The working principle of the WECS encompasses two conversion processes that are executed by its principle subsystems. The wind turbine generates torque from the wind pressure, which is transmitted via the shaft and gearing to the generator rotor. The generator converts this torque into electric power. The control system serves to regulate the rotor speed and damp out torque fluctuations at the shaft by pitch and torque controllers respectively, as explained in Part II.
2.2.1 Energy Extraction With regard to energy production, the wind power, P w , available from the turbine blades’ rotation is a derivative of the kinetic energy, Ew , of the wind with respect to time Pw =
∂Ew 1 ∂(mvw2 ) 1 ∂(ρΛvw3 t) 1 = = = ρΛvw3 ∂t 2 ∂t 2 ∂t 2
(2.1)
where m is the mass of the air (Kg) in the area swept by the blades, vw is the wind speed at the centre of the rotor (m/s), ρ is the air density (kg/m 3 ), and Λ = πR2 is the frontal area of the wind turbine (m2 ), R being the rotor radius (m). The portion of the extracted wind power converted to mechanical power by the rotor can be simulated by the static relation obtained according to the Rankine-Froude theory [3] of propellers in incompressible fluids Pm = cP (λ, β)
ρΛ 3 v 2 w
(2.2)
where Pm is the mechanical power (W), and cP (λ, β) denotes the performance coefficient of the turbine, determined by the pitch angle, β, of the blades and the tip-speed ratio (TSR), λ.
CHAPTER 2. AERODYNAMIC CONVERSION MODELING cp(λ,β)
0.6
β=−2 β=0 β=3 β=5 β=7 β=10
0.5 0.4 cP
0.6 0.5 0.4 0.3 0.2 0.1 00
2
4
6
8 10 12 14 16 18 β 20
0
2
4
6
8
10
12
14
16
λ
20
0.3 0.2 0.1 0 0
2
4
6
8
10
12
14
λ
(a) cP (λ, β) according to (2.3)
(b) Variation of c P with λ and β
Figure 2.2: Performance curves for a 3-bladed WECS. Negative cP values have been set to zero.
2.2.2 Power Curve Research in advanced control for development of efficient production tools is in line with the framework that aims to improve performance of WECS to get the best benefit from the wind energy source. With the primary objective of maintaining steady electrical power, reducing rotor speed fluctuations, and minimizing control actuating loads, controller design requires a formulation of the power curve of the WECS. One common way to control the active power of a wind turbine is by regulating the cP value. Information on the power coefficient for commercial wind turbines is not readily given by turbine manufacturers [4]. Several numerical approximations have been developed to compute c P [5]-[8]. This chapter analyzes an achetype for modeling cP , approximated using a nonlinear function based on the turbine characteristics, according to [9]. It is modelled as
−21 116 − 0.4β − 5 e λi + 0.0068λ cP (λ, β) = 0.5176 λi
(2.3)
where the TSR is computed from the blade tip-speed and wind speed λ=
ωt R vw
(2.4)
with ωt designating the rotor angular velocity, in rad/sec. Further, the value λ i is determined from 1 1 0.035 = − 3 . λi λ + 0.08β β + 1
(2.5)
For practical purposes cP may be determined using a graphical method; the power coefficient is illustrated as a three-dimensional mesh surface in Fig. 2.2(a). The profile in Fig. 2.2(b) shows that cP = 0.48 for β = 0◦ and λ = 8.1. This tip-speed value is assigned as the optimum tip-speed
CHAPTER 2. AERODYNAMIC CONVERSION MODELING 30 25 20 15 10 5 0 -5
21
ωt λ β
5
10
15 Wind speed, vw [m/s]
20
25
(a) Rotor speed (rpm), TSR, and pitch angle (deg.) Power coefficient, cP
0.5 0.4 0.3 0.2 0.1 0 5
10
15 Wind speed, vw [m/s]
20
25
Torque coefficient, cT
(b) Performance coefficient c P (λ, β) 0.07 0.06 0.05 0.04 0.03 0.02 0.01 5
10
15
20
25
Wind speed, vw [m/s]
(c) Torque coefficient c T (λ, β)
Figure 2.3: Steady power curve calculations: performance coefficients variation with wind speed.
ratio, λopt , and the optimum turbine speed curve at any given wind speed can be obtained based on this value. This curve is then used as a reference in the active power control. c P can be thought of as a correction factor, introduced into the above power equation to reflect the reality that the rotor’s power-capturing efficiency is less than perfect. It is noteworthy that WECSs are now highly efficient with less than 10% thermal losses in the system transmission. The aerodynamic efficiency of turbines has gradually risen from the early 1980s with cP rising from 0.44 to about 0.50 for state-of-the-art technology, which is near the theoretical maximum value of 16/27≈0.593, called the Lancaster-Betz limit [3]. This value is based upon the physical reality that even the most aerodynamically efficient turbine blade disrupts the airflow of incident wind before the wind front reaches the rotating blade. In actuality, the air molecules within the cross-sectional area swept by the rotor slow down as they approach rotating turbine blades and thus lose kinetic energy proportional to the cube of that velocity loss. Note that the maximum theoretical c P value in Fig. 2.2(b) from the empirical formulation in (2.3) is about 0.59, corresponding to β = −2 ◦ and λ = 14. Fig. 2.3(a) shows variation in ωt , λ, and β while Figs. 2.3(b) and (c) show, respectively, the desired variations in cP (λ, β) and torque coefficient cT (λ, β) for the WECS over a range of wind speeds, for optimum power production.
Power, [pu]
CHAPTER 2. AERODYNAMIC CONVERSION MODELING 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
22
cP=1.0 (Eqn 2.1) Actual (Eqn 2.2) Linearized (Eqn 2.6)
0
5
10 15 uw(t), [m/s]
20
25
Figure 2.4: Ideal and actual shaft power, and linearized electrical output power.
2.2.3 Electrical Output Model The electrical power output is a function of various parameters including wind speed, rotor speed, efficiencies of the drive train components, type of turbine, system inertias, and the gustiness of the wind. By defining a model for the output electrical power P e , more accurate energy estimates can be attained. A closed form expression [10] for energy production is obtained by assuming that P e varies as vwk between cut-in, vc , and rated, vr , wind speeds, where k is the Weibull shape parameter: Pe = 0
for (vw < vc )
Pe = a + bvwk for (vc ≤ vw ≤ vr ) Pe = Pr
(2.6)
for (vw > vr )
where Pr is the rated electrical power, and the coefficients a and b are given by a=
Pr vck vck − vrk
and
b=
vrk
Pr . − vck
Fig. 2.4 shows the power curve for the variable speed, pitch-regulated WECS. The first curve (cP =1.0) gives the maximum wind power available, while the second one (‘Actual’) is the mechanical power (including all the generator and transmission losses) for production of useful electrical power. For the linearized plot of Pe versus vw in (2.6) for k = 2, the electrical power varies as vw2 between the cut-in and rated wind speeds, rising above zero at a wind speed of 5.6 m/s and then assumes a constant value at and above rated wind speeds. It is noteworthy that other turbines, transmissions, and generators will produce somewhat different curves with approximately the same shape.
CHAPTER 2. AERODYNAMIC CONVERSION MODELING
23
2.2.4 Capacity Factor Average power output, Pavg , of a turbine is a much better economical indicator of the total energy production as compared to the rated power Pr , as the latter is chosen by the manufacturer with less accurate regard to wind speed at a site. Pavg may be computed as a product of the power produced at each wind speed and the fraction of the time that wind speed is experienced, integrated over all possible wind speeds
Pavg =
∞ 0
Pe f (vw )dvw
(2.7)
where f (vw ) is the Weibull probability density function k vw k−1 vw k exp − f (vw ) = c c c
(2.8)
with c as the scale parameter. Generally, c is about 12% larger than the mean wind speed, and since most good wind regimes will have the shape parameter k in the range 1.5 ≤ k ≤ 3.0, the estimate c = 1.12µw suffices, where µw designates the long term mean wind speed at the site. Thus the optimum design for energy production is a rated speed of about twice the mean speed. Substituting (2.6) and (2.8) into Eq. (2.7) yields
vr
Pavg =
(a +
bvwk )f (vw )dvw
∞
+ Pr
vc
f (vw )dvw .
(2.9)
vr
The Rayleigh distribution [11] is a χ 2 density function with 2 DOF, a subset of the Weibull distribution when k = 2 and is sufficiently accurate for analysis of wind power systems when statistics at a given site are unknown. Substituting the limits of integration in (2.9) and neglecting small terms Pavg = Pr
exp[−(vc /c)2 ] − exp[−(vr /c)2 ] (vr /c)2 − (vc /c)2
.
(2.10)
The quantity inside the brackets in (2.10) is the Capacity Factor (CF), thus Pavg = Pr (CF).
(2.11)
The CF may be envisaged as a correction factor that reflects the turbine’s technical availability. CFs of at least 25% are considered minimally necessary for a site to be considered economically viable [12]. In practice, the most efficient wind farms exhibit individual turbine CFs of 30 to 35% [13],[14]. However, values as high as 45% have been observed [12],[13],[15].
CHAPTER 2. AERODYNAMIC CONVERSION MODELING
24
2.3 Turbine Linearization for Steady-state Analysis Linear controller design requires that the nonlinear turbine dynamics be linearized about a specified operating point (OP). The linearization process, carried out by numerical simulation, determines an optimal OP that yields maximum energy extraction. Once stability is attained, observation of the system response to step inputs provides direction in choosing gain values that provide adequate performance [16]. The assumption is that the plant dynamics are adequately described by a set of ordinary differential equations in state-variable form. For small-signal approximations, stationarity is assumed i.e. variables do not change significantly from their initial values at the operating point [4]. The wind turbine is driven by a rotor torque, Γ t , extracted from the wind, and delivered through a gearbox to the DOIG, expressed as ρΛvw2 cT (λ, β) . Γt = R 2
(2.12)
This continuous function, Γ t = f (ωt , vw , β), possesses nonlinearity, being a function of the third power of wind speed. At the OP, Γt |OP = Γg |OP , and the turbine may be linearized along the optimal trajectory by considering a small signal value, ∆Γ t Γt = Γt,OP + ∆Γt
(2.13)
that may be expanded as a Taylor series with respective values ω t,OP , vw,OP and βOP at the OP: 1 ∂2f ∂f ∂f ∂f ∂2f 2 ∆ωt + ∆vw + ∆β + = f (ωt,OP , vw,OP , βOP ) + (∆ω ) + (∆vw )2 t ∂ω ∂v ∂β 2! ∂ω 2 ∂v 2 ∂2f ∂2f ∂2f ∂2f 2 + (∆β) + 2 (2.14) ∆ωt ∆vw + 2 ∆vw ∆β + 2 ∆ωt ∆β + hots ∂β 2 ∂ω∂v ∂v∂β ∂ω∂β
Γt
where partial differentials are computed around the OP. ∆ indicates instantaneous change, while ∆ω t , ∆vw and ∆β designate deviations from the chosen OP i.e. (ω t − ωt,OP ), (vw − vw,OP ) and (β − βOP ), respectively, and “hots” refers to “higher order terms”, which are neglected. Thus Γt − Γt,OP =
∂f ∂f ∂f ∆ωt + ∆vw + ∆β . ∂ω ∂v ∂β
(2.15)
Adopting a local convention and denoting the respective slopes as ∂f
∂f
∂f
kω =
, kv =
, and kβ =
∂ω OP ∂v OP ∂β OP
(2.16)
∂Γt/∂ωt
CHAPTER 2. AERODYNAMIC CONVERSION MODELING
25
0 -500 -1000 -1500 -2000 -2500 -3000 -3500 5
10
15 Wind speed, vw [m/s]
20
25
(a) Partial derivative of aerodynamic torque w.r.t. rotor speed 350000
∂Γt/∂vw
300000 250000 200000 150000 100000 50000 5
10
15
20
25
Wind speed, vw [m/s]
(b) Partial derivative of aerodynamic torque w.r.t. wind speed
∂Γt/∂β
0 -5000 -10000 -15000 5
10
15 Wind speed, vw [m/s]
20
25
(c) Partial derivative of aerodynamic torque w.r.t. pitch angle
Figure 2.5: Variation in linearization coefficients with wind speed.
then (2.15) becomes ∆Γt = kω ∆ωt + kv ∆vw + kβ ∆β
(2.17)
and this linearizes the turbine torque around the OP. Since the torque coefficient c T (λ, β) cP (λ, β)/λ the aerodynamic torque in (2.12) may be determined from the modelled c P (λ, β) in (2.3) to yield the respective aerodynamic (linearization) coefficients in (2.16), as follows [17]
kω kv kβ
∂Γt
ρΛ 2 ∂c
P
= v v + ω R −c
=
P w,OP t
∂ωt
2ωt2 w,OP ∂λ
OP
OP
OP
∂Γt
ρΛ ∂cP
= vw,OP 3cP vw,OP − ωt R
=
∂vw
2ωt ∂λ
OP
OP
OP
∂cP
∂Γt
ρΛ 3 = vw,OP
=
. ∂β
2ωt ∂β
OP
(2.18) (2.19) (2.20)
OP
Figs. 2.5(a)–(c) show the respective partial derivatives of aerodynamic torque with rotor angular frequency, wind speed, and pitch angle, representing the linearization coefficients in (2.18)–(2.20), and are computed from the cP (λ, β) surface according to the OP loci.
CHAPTER 2. AERODYNAMIC CONVERSION MODELING
26
2.4 Remarks In rotor design, an operating speed range is normally selected first, having regard to issues such as acoustic noise emission. With the speed chosen it then follows that there is an optimum total blade area for maximum rotor efficiency. Energy capture improves with increasing turbine size, and it is often asserted that component mass and costs increase less than cubically with scale. However, the underlying physics is often confused with the effects of technology development and the influence of volume on production cost. Most modern WECSs have three blades. The two-bladed rotor design is technically a little less efficient aerodynamically than the established three-bladed design, though both are at par in the overall cost benefit. In general, there are some small benefits from increasing blade number, relating to minimizing losses that take place at the blade tips. In aggregate, these losses are less for a larger number of narrow blade tips than for fewer, wider ones. Two-bladed rotors generally run at a much higher tip speed than three-bladed rotors so most historic designs would consequently have noise problems. There is, however, no fundamental reason for the higher tip speed, and this should be discounted in an objective technical comparison of the design merits of two versus three blades. Thus, the one-bladed rotor is, perhaps, more problematic technically whilst the two-bladed rotor is basically acceptable technically. The decisive factor in eliminating the one-blade rotor design from the commercial market and in almost eliminating the two-bladed design has been visual impact. Although the power curve is an accurate measure of the turbine’s ability to generate electricity from incident wind, it does not adequately describe expectations of real-world power production. For a more realistic analysis, the average power, Pavg , that is dependent on both the Rayleigh probability density function as well as the Capacity factor, is utilized. Since the Rayleigh density function is dependent only on the mean wind speed, all its statistics to describe a measurement site are immediately available without massive amounts of additional computation. Indeed the Rayleigh is very easy to use and will yield quality, acceptable results in most cases, as confirmed by various studies [18]-[20]. Capacity factors are normally represented as annualized values to account for seasonal variations in wind regimes, thus are considered as the most realistic and reliable predictors of the energy yield for a given candidate site. Because it is rooted in the real world, the capacity factor becomes a much more valuable tool for supporting decisions about wind farm development than the turbine’s power curve alone. CFs are dimensionless, expressed as a ratio in which the WECS’s annual predicted energy production is divided by the energy it would produce if it operated at its nameplate rating continuously.
CHAPTER 2. AERODYNAMIC CONVERSION MODELING
27
References [1] E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “RLS-based self-tuning regulator for WTG dynamic performance enhancement under stochastic setting,” Proc. The International Conference on Electrical Engineering, ICEE 2007, 8-12 July 2007, pp. 1-8. [2] A. D. Wright, and M. J. Balas, “Design of state-space-based control algorithms for wind turbine speed regulation,” ASME Journal of Solar Energy Engineering, vol. 125, no. 4, pp. 386-395, June 2003. [3] T. Burton, D. Sharpe, N. Jenkins, and E. Bossanyi, Wind Energy Handbook, New York: Wiley, 2001. ISBN:978-0471489979. [4] E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “Disturbance rejection by dual pitch control and self-tuning regulator for wind turbine generator parametric uncertainty compensation,” IET Procs. Control Theory and Applications, vol. 1, no. 5, pp. 1431-1440, Sept. 2007. doi:10.1049/iet-cta:20060448. [5] H. Akagi, Y. Kanazawa, and A. Nabae, “Instantaneous reactive power compensators comprising switching devices without energy storage components,” IEEE Trans. Industrial Applications, vol. IA-20, no. 3, pp. 625-630, May/June 1984. [6] N. W. Miller, W. W. Price, and J. J. Sanchez-Gasca, “Dynamic modeling of GE 1.5 and 3.6 wind turbine-generators,” GE Power Systems Energy Consulting, GE WTG Modeling-v3.0.doc, October 27, 2003. [7] O. Wasynczuk, D. T. Man, and J. P. Sullivan, “Dynamic behavior of a class of wind turbine generators during random wind fluctuations,” IEEE Trans. Power App. Syst., vol. 100, pp. 28372845, 1981. [8] P. M. Anderson, and A. Bose, “Stability simulation of wind turbine systems,” IEEE Trans. Power App. Syst., vol. 102, pp. 3791-3795, 1983. [9] S. Heir, Grid Integration of Wind Energy Conversion Systems, John Wiley & Sons Ltd, 1998, ISBN 0-471-97143-X. [10] W. R. Powell, “An analytical expression for the average output power of a wind machine,” Solar Energy, vol. 26, no. 1, pp. 77-80, 1981.
CHAPTER 2. AERODYNAMIC CONVERSION MODELING
28
[11] E. B. Muhando, T. Senjyu, N. Urasaki, A. Yona, H. Kinjo, and T. Funabashi, “Gain scheduling control of variable speed WTG under widely varying turbulence loading,” Renewable Energy, vol. 32, no. 14, pp. 2407-2423, 2007. [12] J. G. McGowan, and S. Connors, “Windpower: A turn of the century review,” Annual Review of Energy and the Environment, vol. 25, pp. 147-197, 2000. [13] EPRI (Electric Power Research Institute), “Big Spring Wind Power Project, Second-Year Operating Experience: 2000-2001,” Final Report, DOE-EPRI Wind Turbine Verification Program, Dec. 2001. [14] DOE/TVA/EPRI (DOE, Tennessee Valley Authority, and Electric Power Research Institute), “Tennessee Valley Authority Buffalo Mountain Wind Power Project, First- and Second-Year Operating Experience: 2001-2003,” DOE-EPRI Wind Turbine Verification Program, Dec. 2003. [15] J. F. Manwell, A. Rogers, and J. G. McGowan, Wind Energy Explained: Theory, Design, and Application, Chichester, United Kingdom: John Wiley & Sons Ltd., 2002. ISBN:978-0471499725. [16] E. B. Muhando, T. Senjyu, N. Urasaki, H. Kinjo, and T. Funabashi, “Online WTG dynamic performance and transient stability enhancement by evolutionary LQG,” Proc. IEEE Power Engineering Society General Meeting, 24-28 June 2007, pp. 1-8. doi:10.1109/PES.2007.385499. [17] E. B. Muhando, T. Senjyu, O. Z. Siagi, and T. Funabashi, “Intelligent optimal control of wind power generating system by a complemented linear quadratic Gaussian approach,” Proc. IEEE Power Engineering Society Conference and Exhibition, PowerAfrica 2007, 16-20 July 2007, pp. 1-8. [18] R. B. Corotis, “Stochastic modeling of site wind characteristics,” ERDA Report, September 1977, RLO/2342-77/2. [19] R. B. Corotis, A. B. Sigl, and J. Klein, “Probability models of wind velocity magnitude and persistence,” Solar Energy, vol. 20, no. 6, pp. 483-493, 1978. [20] J. Asmussen, D. Manner, G. L. Park, and E. L. Harder, “An analytical expression for the specific output of wind turbine generators,” Proceedings of the IEEE, vol. 66, no. 10, pp. 1295-1298, Oct. 1978. ISSN: 0018-9219.
Chapter 3 Drive-train Modeling 3.1 Introduction
V
ARIABLE speed wind turbine systems provide better dynamic performance characteristics than fixed speed configurations. The power train components are subject to highly irregular
loading input from turbulent wind conditions, and the number of fatigue cycles experienced by the major structural components can be orders of magnitude greater than for other rotating machines. A modern wind turbine operates for about 13 years in a design life of 20 and is almost always unattended [1],[2]. Thus, considering challenges posed by the severity of the fatigue environment, wind technology has a unique technical identity and R&D demands, and repeated loadings need to be taken into account in wind generating system design. In the sequel, the relevance of detailed representations of the structural dynamics of variable speed WECS on transient stability studies is assessed. Most of the DOIG wind turbine models used in dynamic stability studies include a drive-train model. Due to the increased compliance of the drive-train of almost every wind turbine (usually achieved by “soft” axes or special elastic couplings), suitable multimass equivalents must be employed in order to represent the low frequency torsional modes that dominate the dynamic behavior of the wind turbine. Such multimass equivalents for modeling the drive-train fall in either of two categories: the so-called two-mass model [3]–[6], or the frequently used lumped model approach, which assumes that all the rotating masses can be treated as one concentrated mass [7]–[9]. However, simplification of the drive-train model may have a negative impact on the accuracy of wind-generator modeling [10],[11]. Particularly, the lumped model approach may be insufficient in the case of transient analysis. In this thesis the model of the wind turbine drive-train is represented by means of a three-mass model considering an equivalent system with an equivalent stiffness and damping factor on the wind turbine rotor side [12].
CHAPTER 3. DRIVE-TRAIN MODELING
30
Pitch control
Torque control
Converter AC
CONTROL SYSTEM
AC
Wind Gearbox Ratio 1:n
Low-speed shaft
High-speed shaft
Grid
Generator
Wind turbine
(a) Main components of the WECS Jt J1
ωt
Dt
ω1 Kgr
Γt
Turbine rotor
Kt
Γ1
J2
ω2
Dg
ωg
Γ2
Kg
Γg
Jg
Generator
(b) 3-inertia system
Figure 3.1: Dynamic drive-train equivalenced by a 3-inertia system interlinked by a flexible shaft.
3.2 Power train Modeling Concept The physical diagram of the variable speed WECS system is shown in Fig. 3.1(a). The mechanical part of the wind turbine consists of a shaft system and the rotor of the wind turbine itself. The torque induced by the aerodynamics on the rotor disk is transmitted to the generator by a series of turbine structures: blades, hub, low-speed shaft, gearbox, and high-speed shaft. The torque applied to the generator shaft is not the same as the aerodynamic torque in the blades because of the flexibilities of these rotor structures. During a transient event torque oscillations may be introduced in addition to the aerodynamic torque due to the rotor structural dynamics. These torque oscillations are associated with the torsional flexibility of the shafts. When the shafts are assumed rigid, the drive-train can be represented by a single-mass model. If the torsional flexibility of the shafts is included, then the drive-train is represented as a multimass model where the blade bending dynamics are neglected. Variable speed WECS show large inertias and low shaft stiffness, and the interaction between the wind turbine and electrical generator could give rise to low frequency oscillations that can limit the transient stability of the system. Representing the mechanical system as a lumped mass may give optimistic results especially for fixed speed induction generators (FSIGs), but an elaborate model is necessary for a variable speed WECS, as the mechanical and electrical frequencies are decoupled.
CHAPTER 3. DRIVE-TRAIN MODELING
31
To enhance design of a suitable controller for damping the torsional oscillations, a multimass model is adopted in this study to formulate the state space for analyzing system response to disturbances at steady state. Fig. 3.1(b) illustrates the mechanical equivalent 3 rd -order model of the WECS drive-train, consisting of rotating masses (rotor with the asynchronous generator) elastically coupled to each other by a linear torsional spring and a linear torsional damper. The nomenclature is explained as follows. Jt , Jg are the wind turbine and generator moments of inertia, J 1,2 represents the inertia of the gearwheels, ωt , ω1,2 , ωg are the wind turbine, gearbox wheels, and generator mechanical speeds, Kt , Kg are the spring constants indicating the torsional stiffness of the shaft on wind turbine and generator parts, and Dt , Dg are respective damping constants on turbine and generator sides (K represents the elastic properties of the shaft element while D models internal viscous friction). The wind turbine is driven by a rotor torque Γ t extracted from the wind, which is delivered, through a gearbox with gear ratio Kgr , to the generator that yields a generator torque Γg . As a consequence, the gearbox experiences a torsional torque, Γd . The shaft system gives a soft coupling between the heavy turbine and the light generator rotor, thus the effective shaft stiffness, K s , is reduced by the 2 ratio 1/Kgr . In modern turbines Kgr is normally in the range 50 – 70, thereby rendering the shafts
extremely soft, with Ks typically in the range 0.15 – 0.40 pu. This may be compared with shafts in conventional power plants incorporating synchronous generators, where 20 ≤ K s ≤ 80 pu [13]. Considering the model with a single dominant resonant mode, the dynamic response of the rotor driven at a speed ωt by the aerodynamic torque Γt written on the generator side has the expression Γt = Jt
dωt + Dt ωt + Kt (θt − θ1 ). dt
(3.1)
Similarly, the generator is driven by the high speed shaft torque Γ 2 and braked by Γg −Γg = Jg
dωg + Dg ωg + Kg (θg − θ2 ). dt
(3.2)
The torsional torque experienced by the low speed shaft is comprised by the torques developed at the gearbox, resulting from the torsional effects due to the difference between θt and θg : dω1 + Dt ω1 + Kt (θ1 − θt ) dt dω2 + Dg ω2 + Kg (θ2 − θg ) = J2 dt
Γ1 = J1
(3.3)
Γ2
(3.4)
where θt , θg are the angular positions of the shaft at the rotor and generator sides, Γ 1 is torque that goes in the gearbox, Γ2 (= Γ1 /Kgr ) is torque out from the gearbox, and ω2 = Kgr ω1 .
CHAPTER 3. DRIVE-TRAIN MODELING ω t ,θ t Γt
ωgb ,θ gb
Khgb
Kgbg
Hg
H gb Dhgb
dgb
ωg ,θ g Γg
Γd
Ht
dt
32
Dgbg
dg
Figure 3.2: Schematic representation of the drive-train as a series of elastically coupled inertias.
3.3 Mechanical State Space System Fig. 3.2 illustrates the mechanical equivalent. The interconnecting axes, disc brakes etc, are incorporated in the lumped inertias of the model. The elasticity and damping elements between adjacent inertias correspond to the low and high speed shaft elasticities and internal friction, whereas the external damping elements represent the torque losses. Adopting the per unit (pu) system (see Appendix B.1), a Hamiltonian matrix [14],[15] may be generated by the state equations for the drive train mechanical equivalent, obtained using the inertias’ angular positions and velocities as state variables θ d ··· dt ω
.. .
[I]3×3 [0]3×3 = ........................... . −[2H]−1 [K] .. −[2H]−1 [D]
θ · · · ω
[0]3×3 + ...... [2H]−1
Γ
(3.5)
where [0]3×3 and [I]3×3 are the zero and identity 3 × 3 matrices, respectively. Further, [2H] is the diagonal 3 × 3 inertia matrix of turbine, gearbox and generator inertias, [K] is the 3 × 3 stiffness matrix, where Khgb and Kgbg are the hub to gearbox and gearbox to generator stiffness coefficients, while [D] is the 3×3 damping matrix, where D hgb and Dgbg are relative dampings of elastic couplings, and dt , dgb , dg are the external damping coefficients. These are expressed as follows:
=
diag(2Ht , 2Hgb , 2Hg ) Khgb −Khgb 0 = −Khgb Khgb + Kgbg −Kgbg K 0 −Kgbg Kgbg d + Dhgb −Dhgb t = −Dhgb dgb + Dhgb + Dgbg D 0 −Dgbg 2H
0 −Dgbg dg + Dgbg
CHAPTER 3. DRIVE-TRAIN MODELING
33
Additionally, θ T and ω T are the vectors of the angular positions and angular velocities of the rotor, gearbox and generator respectively, while Γ T is the vector of the external torques acting on the turbine rotor and on the generator rotor, conventionally accelerating, viz. θT = [θt , θgb , θg ],
ω T = [ωt , ωgb , ωg ],
and
Γ T = [Γt , Γd , Γg ].
In the steady state condition, the input aerodynamic torque Γ t applied on the turbine rotor should be counter-balanced by an opposing electromagnetic torque developed inside the induction machine. Due to high turbine inertia relative to J g , and low shaft stiffness, this subjects the elastic shaft element to a torsional twist, causing a point on the circumference of one end of the shaft to shift by a large electrical twist angle θtg , in electrical radian, from the corresponding point on the other end of the shaft. The angle generated per unit applied torque is computed as dθtg = ωb (ωt − ωg ) dt
(3.6)
where ωb = 2πfn is the base angular frequency and fn is nominal grid frequency (Hz). The resonance lies in the most flexible part of the rotational system. Neglecting damping, the natural frequency of vibration of the three mass model is given as [16],[17]
f1 f2
√ 1 b2 − 4c 2 b − − 2 2 √ 1 b2 − 4c 2 1 b = − + 2π 2 2 1 = 2π
(3.7) (3.8)
where 1 1 1 1 + + + Kg b = − Kt Jt Jgb Jgb Jg
and
c = Kt K g
Jt + Jgb + Jg . Jt Jgb Jg
The first-mode mechanical frequency of a typical wind turbine is in the 0 to 10 Hz range [18], which is also the range for electromechanical oscillations. Consequently, the mechanical vibrations of the WECS interact with the electromechanical dynamics. Therefore, in order to create an accurate model of a wind generator for transient stability analysis, the first-mode mechanical turbine dynamics must be accurately represented. By conducting a spectral analysis of the low-speed shaft torque for the 2 MW WECS (wind turbine data is given in Appendix A), the frequencies of vibration of the rotor structure are: f1 = 2.7 Hz, and f2 = 11 Hz.
CHAPTER 3. DRIVE-TRAIN MODELING
34
1.2
Γt [MNm]
1 0.8 0.6 0.4 0.2 0 5
10
15 Wind speed, vw [m/s]
20
25
20
25
20
25
(a) Aerodynamic torque 1.2
Γd [MNm]
1 0.8 0.6 0.4 0.2 0 5
10
15 Wind speed, vw [m/s]
Γg [kNm]
(b) Gearbox torque 14 12 10 8 6 4 2 0 5
10
15 Wind speed, vw [m/s]
(c) Generator torque
Figure 3.3: Steady state variation in aerodynamic, gearbox and generator torques with wind speed.
3.4 Drive-train Torque Dynamics 3.4.1 Steady-state Operation The WECS system considered in this thesis employs a frequency converter to decouple the generator from the fixed frequency of the grid, and uses pitch control to limit the power above rated wind speed. The steady-state operating curve can be described with reference to the torque-speed characteristic: • below rated the operating curve resembles a stall-regulated variable speed case • above rated, blade pitch is adjusted to maintain the chosen OP. Figs. 3.3(a)–(c) show the steady-state calculation results for aerodynamic, low-speed shaft, and generator torques, respectively. Fig. 3.3(a) shows how the aerodynamic torque increases with wind speed. At high wind speeds (above rated) changing the pitch alters the trajectory of constant wind speed, constraining it to the OP locus. Fig. 3.3(b) represents development of the low-speed shaft torque — the shaft should experience reduced fluctuations to avoid cyclic fatigue stresses. From Fig. 3.3(c) it can be observed that the torque demand is kept constant at rated value for all higher wind speeds (to actively damp shaft torsional oscillations). Pitch control then regulates rotor speed.
CHAPTER 3. DRIVE-TRAIN MODELING
35
3.4.2 Operation under High Turbulent Inflow Controller design in this thesis deals particularly with operation under high turbulence. Focus is on the need to design control systems that properly account for the flexible modes of the turbine, and maintain the stable closed-loop behavior of the WECS, mainly because • under turbulent wind conditions, the power train components of a WECS are subject to highly irregular loading input, and the number of fatigue cycles experienced by the major structural components can be orders of magnitude greater than for other rotating machines; • control that optimizes energy capture in medium to high wind speed regimes can also cause undesirable torque fluctuations that result from the inertia of the rotor as the torque control attempts to follow the wind. The torque applied to the generator shaft is not equivalent to the aerodynamic torque due to flexibilities of the drive train structures. During a transient event torque oscillations associated with the torsional flexibility of the shafts are introduced in addition to the aerodynamic torque. A twisted shaft contains potential energy; when a wind gust strikes the turbine, part of the extra power goes into shaft potential energy rather than instantly appearing in the electrical output. This stored energy will then go from the shaft into the electrical system during a wind lull. Thus a shaft helps to smooth out power fluctuations.
3.5 Remarks An appropriate model of system behavior is the heart of control design. It is appreciated, however, that a challenge is introduced in defining the level of detail required for each study (modeling, analysis, and control design). In some situations representation of certain details of structural dynamics may not be necessary if they have no impact on the electrical performance during the time frames of interest in a particular study. At present, there are models that accurately represent the aerodynamic, mechanical and electrical systems of WECS [19]-[22]. However, these models are normally developed in different simulation platforms and the availability of reliable studies that investigate the dynamic interaction that exists between the electrical and structural systems is limited. In this study the importance of a detailed representation of the power train is assessed. Although a nonlinear model is required for the simulation, a simple linear model is preferred for control design purposes. Controller design is enhanced by the 3 rd −order model developed in this Chapter, by devising effective control algorithms that reflect the plant dynamic characteristics as well as the anticipated working environment.
CHAPTER 3. DRIVE-TRAIN MODELING
36
References [1] W. D. Jones, “I’ve Got the Power,” IEEE Spectrum Magazine, vol. 43, no. 10 (INT), October 2006. Available online, http://www.spectrum.ieee.org/oct06/4661. [2] W. D. Kellogg, M. H. Nehrir, G. Venkataramanan, and V. Gerez, “Generation unit sizing and cost analysis for stand-alone wind, photovoltaic, and hybrid wind/PV systems,” IEEE Trans. Energy Conversion, vol. 13, no. 1, pp. 70-75, Mar. 1998. DOI: 10.1109/60.658206. [3] E. B. Muhando, T. Senjyu, H. Kinjo, and T. Funabashi, “Augmented LQG controller for enhancement of online dynamic performance for WTG system,” Renewable Energy, doi:10.1016/j.renene.2007.12.001. [4] P. Ledesma, and J. Usaola, “Minimum voltage protection in variable speed wind farms,” Proceedings IEEE Porto Power Tech 2001, 10-13 Sept. 2001, vol. 4, pp. 1-6. [5] E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “Regulation of WTG dynamic response to parameter variations of analytic wind stochasticity,” Wind Energy, DOI:10.1002/we.236. [6] ——, “Disturbance rejection by dual pitch control and self-tuning regulator for wind turbine generator parametric uncertainty compensation,” IET Procs. Control Theory & Applications, vol. 1, no. 5, pp. 1431-1440, Sept. 2007. DOI: 10.1049/iet-cta:20060448. [7] L. Holdsworth, X. G. Wu, J. B. Ekanayake, and N. Jenkins, “Direct solution method for initializing doubly-fed induction wind turbines in power system dynamic models,” IEE Procs. Generation, Transmission and Distribution, vol. 150, no. 3, pp. 334-342, May 2003. [8] ——, “Comparison of fixed speed and doubly fed induction wind turbines during power system disturbances,” IEE Procs. Generation, Transmission and Distribution, vol. 150, no. 3, pp. 343352, May 2003. [9] J. G. Slootweg, S. W. H. de Haan, H. Polinder, and W. L. Kling, “General model for representing variable speed wind turbines in power systems dynamics simulations,” IEEE Trans. Power Systems, vol. 18, no. 1, pp. 144-151, Feb. 2003. doi:10.1109/TPWRS.2002.807113. [10] S. K. Salman, and A. L. J. Teo, “Windmill modeling consideration and factors influencing the stability of a grid-connected wind power-based embedded generator,” IEEE Trans. Power Systems, vol. 18, no. 2, pp. 793-802, May 2003.
CHAPTER 3. DRIVE-TRAIN MODELING
37
[11] E. B. Muhando, T. Senjyu, H. Kinjo, Z. Siagi, and T. Funabashi, “Intelligent optimal control of nonlinear wind generating system by a modeling-based approach,” IET Procs. Renewable Power Generation, (Accepted for publication). [12] E. B. Muhando, T. Senjyu, H. Kinjo, and T. Funabashi, “Extending the modeling framework for wind generation systems: RLS-based paradigm for performance under high turbulence inflow,” IEEE Trans. Energy Conversion, 2007. (Accepted for publication). [13] E. N. Henrichsen, and P. J. Nolan, “Dynamics and stability of wind turbine generators,” IEEE Trans. Power App. Syst., vol. PAS-101, no. 8, 1982. [14] K. R. Meyer, and G. R. Hall, Introduction to Hamiltonian Dynamical Systems and the ‘N’-body Problem, Springer, pp. 34-35, 1991. ISBN 0-387-97637-X. [15] S. Skogestad, and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, John Wiley & Sons, Inc., 2/Ed., 2005. ISBN 978-0-470-01167-6. [16] C. M. Harris, Shock and Vibration Handbook, 4 th edition, McGraw Hill, pp. 38.1–38.14, 1996. ISBN: 0 07 026920 3. [17] W. T. Thomson, Theory of Vibration with Applications, 4 th edition, Chapman & Hall, pp. 131– 145, 268–337, 1993. ISBN: 0 412 78390 8. [18] D. J. Trudnowski, A. Gentile, J. M. Khan, and E. M. Petriz, “Fixed-speed wind-generator and wind-park modeling for transient stability studies,” IEEE Trans. Power Systems, vol. 19, no. 4, pp. 1911-1917, Nov. 2004. doi:10.1109/TPWRS.2004.836204. [19] A. Petersson, T. Thiringer, L. Harnefors, and T. Petru, “Modeling and experimental verification of grid interaction of a DFIG wind turbine,” IEEE Trans. Energy Conversion, vol. 20, no. 4, pp. 878-886, Dec. 2005. doi:10.1109/TEC.2005.853750. [20] P. Novak, T. Ekelund, I. Jovik, and B. Schmidbauer, “Modeling and control of variable-speed wind-turbine drive-system dynamics,” IEEE Control Systems Magazine, vol. 15, no. 4, pp. 28-38, Aug. 1995. doi:10.1109/37.408463. [21] L. Yazhou, A. Mullane, G. Lightbody, and R. Yacamini, “Modeling of the wind turbine with a doubly fed induction generator for grid integration studies,” IEEE Trans. Energy Conversion, vol. 21, no. 1, pp. 257-264, Mar. 2006. doi:10.1109/TEC.2005.847958. [22] T. Petru, and T. Thiringer, “Modeling of wind turbines for power system studies,” IEEE Trans. Power Systems, vol. 17, no. 4, pp. 1132-1139, Nov. 2002. doi:10.1109/TPWRS.2002.805017.
Chapter 4 Electrical System Modeling 4.1 Introduction
T
HE increasing integration of the double-output induction generator (DOIG) systems controlled by static converters for wind generation into power grids is currently a generalized tendency
in numerous countries. This fact is directly related with the control flexibility offered by static converters that enhance maintaining the terminal voltage at a constant value when the IG operates with variable speed as well as to allow independent active and reactive power control exchanged between the machine and the grid with better use of the available wind energy. In addition to constant voltage, the grid-connected DOIG provides several attractive features during variable speed operation, for instance: constant frequency, generation above the machine rated power, and relatively cheaper and smaller converter as compared with squirrel cage or synchronous machines [1]–[3]. Nowadays one of the most widely used generator types for units above 1 MW (installed either offshore or onshore), both for reasons of network compatibility and reduction in mechanical loads, is the DOIG for effective variable speed operation [4]. Several studies undertaken on the DOIG as a mainstream configuration for large wind turbines have shown that it is possible for the wind turbines to remain grid-connected during grid faults so that they can contribute to the stability of the power transmission system [5],[6]. The main advantage of the DOIG concept is that only a percentage of the power produced in the generator has to pass through the power converter. Typically this is only 20%–30% compared with full power (100%) for a synchronous generator-based wind turbine concept, and thus it has a substantial cost advantage compared with the conversion of full power [7],[8]. The control performance of the DOIG is excellent under normal grid conditions, allowing active and reactive power changes in the range of few milliseconds owing to the presence of power electronics.
CHAPTER 4. ELECTRICAL SYSTEM MODELING
39
Turbine
ps ,q s Drive train gearbox
DOIG Rotor side converter
~
ωt Pitch angle control
Grid side converter
=
~
= p r,q r
Pref Crowbar Fault detection Torque control
Q ref
kc pr
Power converter control (voltage or PF)
Control mode selection: - normal operation - fault operation
Figure 4.1: General schematic of the WECS: DOIG, converters and controllers.
4.2 Detailed Model of DOIG Unit with Converters 4.2.1 Construction and Operation Principle Fig. 4.1 illustrates the wind turbine coupled to a grid connected 2 MW asynchronous DOIG. The DOIG is a brushless wound-rotor electric machine incorporating the most optimum electromagnetic core structure of any electric machine, but without the traditional Achilles’ Heel of the wound-rotor doubly-fed electric machine (DIFG), which is the multiphase slip ring assembly with potential control instability [9],[10]. By eliminating the multiphase slip ring assembly and guaranteeing stability at any speed, the theoretical attributes of the wound-rotor DOIG are acquired: upto 50% reduction in system cost, system electrical loss, and system physical size. Nothing approaches the brushless wound-rotor DOIG machine, if cost, efficiency, and power density, combined, are the deciding factors. The rotor in the generator has three pole pairs while the three phase stator winding is connected directly to the grid synchronous frequency, ω 0 . Since the simulation of the fundamental power system dynamic behavior does not require a detailed modeling of power electronics, the converters are modelled as voltage source and/or current source. The rotor side converter (RSC) is assumed as a voltage source injected into the rotor, whereas the grid side converter (GSC) is assumed to be a controlled current source. As the RSC can provide reactive power control, the GSC may offer additional voltage support capabilities in conditions of excessive speed ranges or in transient operations. Crowbar protection is included: in the event of excessive rotor current, this disconnects the converter and connects the rotor circuit to a crowbar resistor instead. When the current drops back below a set value, the crowbar disengages and the converter is reconnected.
CHAPTER 4. ELECTRICAL SYSTEM MODELING
40
I dc DOIG
Generator
U dc
f ref
Power system
Figure 4.2: Main components of the frequency converter.
Fig. 4.2 shows the main components of the frequency converter. The DOIG connects to the grid with a back-to-back voltage source converter that controls the excitation system. The main components are an AC/DC converter, a DC-link and a DC/AC converter. When power is flowing from the generator, the AC/DC converter acts like a rectifier, and the DC/AC converter acts like an inverter. The DC-link can be used to attenuate voltage fluctuations. Control of the converter firing angle makes it possible to control the electrical torque in the generator, allowing the turbine to be run at variable speed. The frequency converter is used to transform the constant frequency and constant voltage of the grid to variable frequency and voltage on the generator side, thereby maintaining the frequency out of the generator on a stable level independent of the generator’s angular speed. The stator active and reactive power (ps and qs ) are fed directly to the network, while the rotor active and reactive power (pr and qr ) pass through the power converter. The converter efficiency, kc , results in a loss of active power. The converter is controlled by two main control loops: • a torque control loop that works by injecting a quadrature-axis voltage into the rotor circuit, and • a voltage or power factor control loop, which works by injecting a direct-axis voltage. The stator is directly coupled to the electrical power supply network, thus the generator stator voltage always equals the grid voltage. By utilizing the converter, the network frequency is decoupled from the mechanical speed of the machine and variable speed operation is possible, permitting maximum absorption of wind power. A great advantage of the DOIG wind turbine is that it has the capability to independently control active and reactive power. Moreover, the mechanical stresses on a DOIG wind turbine are reduced in comparison to a fixed speed induction generator (FSIG). Due to the decoupling between mechanical speed and electrical frequency that results from DOIG operation, the rotor can act as an energy storage system, absorbing torque pulsations caused by wind gusts. Other advantages of the DOIG include reduced flicker and acoustic noise in comparison to FSIGs. The main disadvantage of DOIG wind turbines is their increased capital cost. The fundamental dynamics of the frequency converter are very complex and nonlinear, albeit considerably faster than the fundamental drive train dynamics and therefore can be neglected in the modelling. This means that the generator torque will be equal to its reference value, Γ g = Γg,ref .
CHAPTER 4. ELECTRICAL SYSTEM MODELING Step-down η = U s T Ui transformer
IT
Grid Us Is
Turbine
41
Rd Ir
Ld Ii
Id
Ur
Ui U d1
Gearbox
U d2
Generator Diode rectifier
α
Line-commutated inverter
Figure 4.3: Simplified schematic of the electrical system. For purposes of formulating the system equations, the diode rectifier represents the RSC while the line-commutated inverter models the GSC.
4.2.2 DOIG: Electrical Model Fig. 4.3 is the functional scheme of the WECS with DOIG, detailing the frequency converter system with dc–link. Finding an operational point of the DOIG in steady-state operation corresponds to initialization [11]. Initialization of the DOIG model is essential prior to starting dynamic simulations, and the following considerations are taken into account: the DOIG consists of a wound rotor IG with a converter feeding into the rotor circuit, and it has a symmetrical three-phased winding distributed around the uniform air-gap. Additionally, the voltage in the stator, U s , is applied from the grid while the voltage in the rotor, Ur , is induced by the converter. At initialization, the electric power operation point is defined by the incoming wind. The reactive power initialized is in accordance with the control strategy chosen. In this research, the electric power and the reactive power are initialized independently, and the generalized reduced order DOIG model is developed based on the following conditions and assumptions: 1. The stator current is positive when flowing toward the machine. 2. The equations are derived in the synchronous reference frame fixed to the stator flux, using direct (d) and quadrature (q) axis representation [12]. 3. The q-axis is 90◦ ahead of the d-axis in the direction of rotation. 4. The q component of the stator voltage is chosen to be equal to the real part of the generator busbar voltage obtained from the load flow solution that is used to initialize the model. 5. The dc component of the stator transient current is ignored, permitting representation of only fundamental frequency components. Similarly, the higher order harmonic components in the rotor injected voltages are neglected.
CHAPTER 4. ELECTRICAL SYSTEM MODELING
42
The DOIG can be simulated by the standard 4th order dq model, described by the following equations against an arbitrary reference frame [13],[14] 1 d . Ψsd ωb dt 1 d = −rs isq + ωΨsd + . Ψsq ωb dt 1 d = −rr ird − (ω − ωt )Ψrq + . Ψrd ωb dt 1 d = −rr irq + (ω − ωt )Ψrd + . Ψrq ωb dt
usd = −rs isd − ωΨsq +
(4.1)
usq
(4.2)
urd urq
(4.3) (4.4)
where usd , usq are the stator voltage d and q components, u rd , urq are the rotor voltage d and q components, isd , isq , ird , irq are the stator and rotor d and q windings currents, and rs , rr are the stator and rotor windings resistances. The inputs u sd and usq of the model are directly available from the known stator voltage, while the rotor voltages u rd and urq are computed from the converters and dc filter equations. Additionally, ω is the arbitrary dq frame electrical angular speed, ω b = 2πfn is the base angular frequency, and fn is the nominal grid frequency in Hz. The flux equations are obtained as Ψsd = −Xs isd + Xm ird
(4.5)
Ψsq = −Xs isq + Xm irq
(4.6)
Ψrd = −Xm isd + Xr ird
(4.7)
Ψrq = −Xm isq + Xr irq
(4.8)
where Xs , Xr are the stator and rotor windings reactance, and Xm is the magnetizing reactance. The state-space modelling of the induction machine considers the voltage equations (4.1)–(4.4) and flux equations (4.5)–(4.8) in the arbitrary d–q synchronous reference frame. The space model for the flux can be written using fluxes as state variables [4], as Ψ sd Ψ d sq dt Ψrd Ψrq
=−1 B
rX −ωB −rs Xm 0 s r ωB r s Xr 0 −rs Xm −rr Xm 0 r r Xs −(ω − ωt )B 0 −rr Xm (ω − ωt )B r r Xs
Ψ sd Ψsq Ψrd Ψrq
u sd usq + urd urq
(4.9)
2 and ω = pω0 , with p being the number of pole pairs of the machine, and where B = Xs Xr − Xm
ω0 being the synchronous mechanical speed obtained as ω 0 = 2πf0 , where f0 is the mechanical drive train eigenfrequency (Hz).
CHAPTER 4. ELECTRICAL SYSTEM MODELING
43
For convenience, (4.9) is rewritten with the currents as inputs to the system model, achieved by substituting (4.5)–(4.8) in (4.1)–(4.4) and solving for the derivatives of the currents. This yields the state equations with the currents as state variables, expressed in the arbitrary dq reference frame:
isd
i d sq dt ird irq
−rs Xr
2 (ωB + ωt Xm )
2 −(ωB + ωt Xm ) −rs Xr = ωb B −rs Xm ω t Xs Xm −rs Xm −ωt Xs Xm 0 Xm 0 −Xr u sd 0 Xm usq 0 −Xr ωb + B −Xm 0 Xs 0 urd urq 0 −Xm 0 Xs
−rr Xm
−ωt Xr Xm
isd
isq ω t Xr Xm −rr Xm −rr Xs (ωB − ωt Xs Xr ) ird −(ωB − ωt Xs Xr ) −rr Xs irq .
(4.10)
The inputs to the model u sd , usq are directly available from the known stator voltage, while the rotor voltages urd , urq are computed from the converters and dc filter equations. The generator electromagnetic torque, Γg , can be expressed in terms of stator and rotor fluxes as Γg = Ψsd isq − Ψsq isd
≡
Ψrd irq − Ψrq ird
(4.11)
which may be set in terms of the reactances and currents as Γg = Xm (isq ird − isd irq )
(4.12)
and from (3.2), assuming generator convention
whence
dωg = − Kg (θ2 − θg ) + Dg ωg + Γg −Jg dt
(4.13)
dωg 1 Γd − Xm (isq ird − isd irq ) . = dt Jg
(4.14)
By controlling the firing angle of the converter, it is possible to control the electrical torque in the generator. The torque control using the frequency converter allows the wind turbine to run at variable speed and thereby makes possible a reduction of the stress on the drive train and the gearbox [15].
CHAPTER 4. ELECTRICAL SYSTEM MODELING
44
The following relations hold for the diode rectifier (RSC) and thyristor inverter (GSC) in Fig. 4.3: Ud1 = Ir = Ud2 = Ii =
√ 3 3 Ur π √ 2 3 Id π √ √ 3 3 1 3 3 · Ui cosα ≡ Us cosα π ηT π √ 2 3 Id π
(4.15) (4.16) (4.17) (4.18)
where Ud1 , Ud2 are the rectifier and inverter dc voltages, Id the dc current, Ur , Ir the peak phase rotor voltage and current, Ui , Ii the peak phase values of inverter output voltage and current, α the inverter firing angle, Us the bus voltage peak phase value, and ηT = Us /Ui is the rotor transformer ratio. The dc link RL filter differential equation is obtained from Ud1 − Ud2 = Rd Id + Ld
dId dt
(4.19)
where Rd and Ld are the choke resistance and inductance, and all quantities are in absolute values (i.e. V, A, Ω, and H). Expressing (4.19) in p.u., eliminating U d1 and Ud2 by (4.15) and (4.17) respectively, and substituting I r for the dc current Id from (4.16): Ur + Uc = Rd Ir +
Xd dIr ω0 dt
(4.20)
where Uc is the voltage of the inverter cosine firing angle controller: Uc =
ηM Us cosα ηT
(4.21)
while Rd and Xd are the dc filter resistance and reactance, referred to the stator of the DOIG: 2 Rd = ηM
π 2 Rd 18 ZBs
and
2 Xd = ηM
π 2 ω0 Ld 18 ZBs
(4.22)
with ZBs the stator base resistance and ηM the equivalent stator/rotor turn ratio of the DOIG. The rotor voltage and current, Ur and Ir , are related to the respective d and q components by: u2rd + u2rq = i2rd + i2rq .
Ur =
(4.23)
Ir
(4.24)
CHAPTER 4. ELECTRICAL SYSTEM MODELING
45
q
~ Ir
i rq urd
d i rd
~
urq
Ur
Figure 4.4: Relative position of rotor fundamental voltage and current phasors.
Ignoring the harmonics and the commutation phenomena of the diode rectifier, its reactive power consumption is zero and therefore the rotor voltage and current are displaced by 180 ◦ as shown in Fig. 4.4 (the rotor current conventionally enters the rotor terminals). Hence Ir ird irq Ur = − Ur ⇔ urd = Ur , urq = Ur . Ir Ir Ir
(4.25)
ird didtrd + irq didtrq Re{ ddtIr Ir∗ } dIr = = dt Ir Ir
(4.26)
Differentiating (4.24)
e
where the complex representation F of a dq quantity f (voltage, current or flux) and its derivatives are defined respectively as F = fd + jfq
and
dfd dfq dF = +j dt dt dt
(4.27)
and its complex conjugate is denoted by the superscript “∗”. Substituting the derivatives of the rotor current from (4.11) in (4.26), the following expression is obtained ωb dIr = − Xm P1 + rr Xs Ir2 − Xs (urd ird + urq irq ) dt BIr ωb = − Xm P1 + rr Xs Ir2 − Xs Ur Ir BIr
(4.28)
where the quantity P1 is given by P1 = Re Us + (rs + jωr Xs )Is Ir∗ .
(4.29)
Combining (4.28) with (4.20) and solving for the rotor voltage, U r , yields the following relation Ur =
(rr Xs Xd − BRd )Ir + BUc + Xm Xd P1 /Ir . B + Xs Xd
(4.30)
CHAPTER 4. ELECTRICAL SYSTEM MODELING
46
4.2.3 DOIG: a Mechanical Perspective The asynchronous DOIG has mechanical properties that render it very suitable for WECS applications, including good overload handling and ability to accommodate changes in the torque applied by the wind turbine’s rotor shaft (via the transmission), thereby reducing overall mechanical wear and tear over the generator’s service life. A modern variable speed drive is capable of accepting a torque demand and responding to this within a very short time to give the desired torque at the generator air-gap, irrespective of the generator speed (as long as it is within specified limits). A first order lag model is provided for this response Γg =
Γg,ref 1 + τe s
(4.31)
where Γg,ref is the demanded torque, Γg is the air-gap torque, and τe is the time constant of the first order lag. Note that the use of a small time constant may result in slower simulations, without much effect on accuracy. A variable speed WECS requires a controller to generate an appropriate torque demand, such that the turbine speed is regulated appropriately. Additionally, the minimum and maximum generator torque must be specified; motoring may occur if a negative minimum is specified. The phase angle between current and voltage, and hence the power factor, is specified on the assumption that, in effect, both active and reactive power flows into the network are being controlled with the same time constant as the torque, and that the frequency converter controller is programmed to maintain constant power factor. An option for drive-train damping feedback is provided. This represents additional fuctionality that may be available in the frequency converter controller, which adds a term derived from measured generator speed onto the incoming torque demand. This term is defined as a transfer function acting on the measured speed. The transfer function is supplied as a ratio of polynomials in the Laplace operator, s. Thus the equation for the air-gap torque becomes Γg =
Num(s) Γg,ref + ωg 1 + τe s Den(s)
(4.32)
where, in this study, Num(s) and Den(s) are the following polynomials 15.123s Num(s) = . Den(s) 0.002643s2 + 0.0257s + 1.0
(4.33)
The transfer function represents a tuned bandpass filter designed to provide additional damping for the drive-train torsional vibrations, which in the case of variable speed operation may otherwise be very lightly damped, sometimes causing severe gearbox loads.
Generator speed, [rpm]
CHAPTER 4. ELECTRICAL SYSTEM MODELING
47
1600 1500 1400 1300 1200 1100 1000 900 800 5
10
15 Wind speed, vw [m/s]
20
25
20
25
20
25
Γg [kNm]
(a) Generator speed 14 12 10 8 6 4 2 0 5
10
15 Wind speed, vw [m/s]
(b) Generator torque Power, [MW]
2.5
Pm Pe
2 1.5 1 0.5 0 5
10
15 Wind speed, vw [m/s]
(c) Shaft power, P m , and electrical output power, P e
Figure 4.5: Steady-state generator parameters’ variation with wind speed.
4.3 DOIG Operation under Steady-state and Fault Conditions 4.3.1 Steady-state Analysis Below rated wind speeds, the steady-state (Γg —ωg ) operating curve is determined by the target of maximizing energy capture by following a constant TSR load line that corresponds to operation at the maximum cP . Pitch control is used to limit the power above rated wind speed. Blade pitch is adjusted to maintain the chosen OP by altering the lines of constant wind speed and constraining the WECS to the OP locus. Once rated torque is reached, the torque demand is kept constant for all higher wind speeds, and pitch control regulates the rotor speed. The parameters needed to specify the steady state operating curve are: the minimum speed, ω g,min , the maximum speed in constant TSR mode, ω g,max , the maximum steady-state operating speed, and above-rated torque set-point. Figs. 4.5(a)–(c) show results of steady state calculations for variation in generator speed and generated powers with wind speed. Electric power, Pe , is generated when ωt > ω0 , where ω0 denotes the system synchronous speed.
Speed, [pu]
CHAPTER 4. ELECTRICAL SYSTEM MODELING
48
1.16 1.15 1.14 1.13 1.12 1.11 1.1 1.09 1.08 1.07 1.06 0
0.2
0.4
0.6
0.8
1
t, [s]
(a) Generator speed Rotor current, Ir [pu]
3
d-Axis q-Axis
2 1 0 -1 -2 0
0.2
0.4
0.6
0.8
1
0.8
1
t, [s]
(b) Rotor current: d-axis and q-axis 1.2
Power, [pu]
1 0.8 0.6 0.4 0.2 0
0.2
0.4
0.6 t, [s]
(c) Turbine power, P m
Figure 4.6: DOIG single phase fault.
4.3.2 Transient Response and Fault-ride-through Analysis Analysis of voltage restoration capability with various controllers is dealt with in Part II of the thesis. However, Fig. 4.6 serves to illustrate the relatively fast recovery of the DOIG when subjected to a single phase fault. Time t = 0 is the time immediately after a fault. It is seen in Fig. 4.6(a) that the fault causes the speed to rise from 1.08 pu to a high of 1.14 pu at 0.14 seconds. More importantly, severe fluctuations in both d- and q-axis currents are attenuated within 0.4 seconds after the fault, as observed in Fig. 4.6(b). In this case, the fault is not significant to trigger the over-current protection, and thus the DOIG is able to ride through the incurred voltage dip. The implication is that approximately 60 ms after the fault is cleared the terminal voltage is recovered (back to the steady state value) and the currents resume their respective variation. Fig. 4.6(c) shows the generated power. The fast voltage recovery is a plus with respect to the DOIG’s capability to control the reactive power. For a serious fault, the current flowing through the power converter may be too high, which may cause damage to the RSC. Thus the DOIG is equipped with an over-current protection — in case the rotor current magnitude reaches the setting value of the protection relay, the converter is subsequently blocked. The setting point of the protection relay is set at 1.5 pu.
CHAPTER 4. ELECTRICAL SYSTEM MODELING
49
4.4 Remarks Variable speed WECS utilize the available wind resource more efficiently especially during light wind conditions. The effect of wind power integration in the grid depends on both the power system design to which the WECS is connected and the turbine control ability to fulfil the grid requirements. This fact has challenged different wind turbine manufacturers regarding the ability of different wind turbine concepts to comply with high-power system operator requirements [16]-[18]. Model simulation studies to understand the impact of system disturbances on wind turbines and consequently on the power system itself abound [19]-[21]. The presence of power electronics inside modern WECS provides large potential for control capability and provides a versatile electronic interface for the grid connection. The doubly outage induction machine is a wound-rotor type and is directly connected to the grid with little additional conditioning. Due to the relatively constant operating conditions, the DOIG has several advantages over conventional induction machines in wind power applications: 1) Ability to control reactive power — since the rotor voltage is controlled by a power electronics converter, the DOIG is able to both import and export reactive power; this has important consequences for power system stability and allows the machine to remain connected to the system during severe voltage disturbances. 2) Ability to control the rotor voltage — this enables the induction machine to remain ‘synchronized’ with the grid while the wind turbine varies in speed. 3) Decoupling of the electric and the reactive power control with independent control of torque and rotor excitation current. By decoupling (item 3), the DOIG can be excited from the rotor circuit by the rotor converter, but not necessarily from the power grid. Thus there exist two distinct principal situations: • When connected to a strong power system where the voltage is (or about) 1 pu, the DOIG will be excited from the rotor circuit by the rotor converter; however, the DOIG does not exchange reactive power with the power system (i.e., the DOIG will produce electric power and be reactive-neutral with the power network). • When connected to a weak power system characterized by fluctuating voltages, the DOIG can be ordered to produce or absorb an amount of reactive power to control voltage; the DOIG will produce electric power and exchange some reactive power with the grid to reach a desired voltage in the vicinity of the connection point.
CHAPTER 4. ELECTRICAL SYSTEM MODELING
50
References [1] H. de Battista, P. F. Puleston, R. J. Mantz, and C. F. Christiansen, “Sliding mode control of wind energy systems with DOIG – power efficiency and torsional dynamics optimization,” IEEE Trans. Power Systems, vol. 15, no. 2, pp. 728-734, May 2000. doi:10.1109/59.867166. [2] M. Ermis, H. Ertan, M. Demirekler, B. M. Saribatir, Y. Uctug, M. E. Sezer, and I. Cardici, “Various induction generator schemes for wind-electricity generation,” Electric Power Systems Research, vol. 23, pp. 71-83, 1992. [3] Z. M. Salameh, and L. F. Kazda, “Analysis of the steady-state performance of the double output induction generator,” IEEE Trans. Energy Conversion, vol. 1, no. 1, pp. 26-32, 1986. [4] S. Muller, M. Deicke, and R. W. de Doncker, “Doubly fed induction generator systems for wind turbines,” IEEE Industrial Applications Magazine, pp. 26-33, May/June 2002. [5] I. Cadirei, and M. Ermis, “Performance evaluation of a wind driven DOIG using a hybrid model,” IEEE Trans. Energy Conversion, vol. 13, no. 2, pp. 148-155, June 1998. doi:10.1109/60.678978. [6] M. Y. Uctug, I. Eskandarzadeh, and H. Ince, “Modeling and output power optimization of a wind turbine driven double output induction generator,” IEE Procs. Electric Power Applications, vol. 141, no. 2, pp. 33-38, March 1994. [7] Z. M. Salameh, and L. F. Kazda, “Commutation angle analysis of a double-output induction generator using a detailed d-q model,” IEEE Trans. Power Apparatus and Systems, vol. PAS-104, no. 3, pp. 512-518, March 1985. doi:10.1109/TPAS.1985.318966. [8] Z. M. Salameh, and S. Wang, “Microprocessor control of double output induction generator I: inverter firing circuit,” IEEE Trans. Energy Conversion, vol. 4, no. 2, pp. 172-176, June 1989. [9] Z. Fengge, T. Ningze, H. Wang, W. Li, and W Fengxiang, “Modeling and simulation of variable speed constant frequency wind power generation system with doubly fed brushless machine,” Int. Conf. Power System Technology, PowerCon 2004, 21-24 Nov. 2004, vol. 1, pp. 801-805. [10] R. Krishnan, and G. H. Rim, “Modeling, simulation, and analysis of variable-speed constant frequency power conversion scheme with a permanent magnet brushless DC generator,” IEEE Trans. Industrial Electronics, vol. 37, no. 4, pp. 291-296, Aug. 1990. June 1989. [11] E. B. Muhando, T. Senjyu, H. Kinjo, Z. O. Siagi, and T. Funabashi, “Intelligent optimal control of nonlinear wind generating system by a modeling-based approach,” IET Proc. Renewable Power Generation, (Forthcoming).
CHAPTER 4. ELECTRICAL SYSTEM MODELING
51
[12] R. Pena, J. C. Clare, and G. M. Asher, “Doubly fed induction generator using back-to-back PWM converters and its applications to variable speed wind-energy generation,” IEE Proc. Electric Power Applications, vol. 143, no. 3, pp. 231-241, May 1996. [13] J. B. Ekanayake, L. Holdsworth, W. XueGuang, and N. Jenkins, “Dynamic modeling of doubly fed induction generator wind turbines,” IEEE Trans. Power Systems, vol. 18, no. 2, pp. 803-809, May 2003. doi:10.1109/TPWRS.2003.811178. [14] E. B. Muhando, T. Senjyu, and H. Kinjo, “Disturbance rejection by stochastic inequality constrained closed-loop model-based predictive control of MW-class wind generating system,” in Proceedings of the Joint IEEJ and IEICE Conference, 19 Dec. 2007, pp. 91-99. [15] E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “Disturbance rejection by dual pitch control and self-tuning regulator for wind turbine generator parametric uncertainty compensation,” IET Procs. Control Theory and Applications, vol. 1, no. 5, pp. 1431-1440, Sept. 2007. doi10.1049/iet-cta:20060448. [16] L. Yazhou, A. Mullane, G. Lightbody, and R. Yacamini, “Modeling of the wind turbine with a doubly fed induction generator for grid integration studies,” IEEE Trans. Energy Conversion, vol. 21, no. 1, pp. 257-264, Mar. 2006. doi.10.1109/TEC.2005.847958. [17] J. L. Rodriguez-Amenedo, S. Arnalte, and J. C. Burgos, “Automatic generation control of a wind farm with variable speed wind turbines,” IEEE Trans. Energy Conversion, vol. 17, no. 2, pp. 279-284, June 2002. doi:10.1109/TEC.2002.1009481. [18] J. Kabouris, and C. D. Vournas, “Application of interruptible contracts to increase wind-power penetration in congested areas,” IEEE Trans. Power Systems, vol. 19, no. 3, pp. 1642-1649, Aug. 2004. doi:10.1109/TPWRS.2004.831702. [19] V. Akhmatov, and P. B. Eriksen, “A large wind power system in almost island operation – a Danish case study,” IEEE Trans. Power Systems, vol. 22, no. 3, pp. 937-943, Aug. 2007. doi:10.1109/TPWRS.2007.901283. [20] J. G. Slootweg, H. Polinder, and W. L. Kling, “Representing wind turbine electrical generating systems in fundamental frequency simulations,” IEEE Trans. Energy Conversion, vol. 18, no. 4, pp. 516-524, Dec. 2003. doi:10.1109/TEC.2003.816593. [21] E. Muljadi, C. P. Butterfield, B. Parsons, and A. Ellis, “Effect of variable speed wind turbine generator on stability of a weak grid,” IEEE Trans. Energy Conversion, vol. 22, no. 1, pp. 29-36, Mar. 2007. doi:10.1109/TEC.2006.889602.
Chapter 5 Modeling Wind Field Dynamics 5.1 Introduction
W
INDS come about as a consequence of the differential heating that powers a global atmospheric convection system reaching from the Earth’s surface to the stratosphere that acts as
a virtual ceiling, leading to global circulation patterns. Globally, the wind energy resource is plentiful, renewable, widely distributed, clean, and reduces toxic atmospheric and greenhouse gas emissions if used to replace fossil-fuel-derived electricity. However, wind speed — certainly the most significant wind energy parameter — is considered as one of the most difficult meteorological phenomena due to its non-predictability [1]. Though the intermittency of wind seldom creates problems when using wind power at low to moderate penetration levels, such intermittency has reportedly caused problems for grid stability in areas where penetration is greatest. In an effort to eliminate the need for measured data acquired over long periods of time, the IEC 61400-1 Standard [2] allows the use of statistical methods to generate turbulent wind fields. Current design standards and certification rules accept the use of standard spectral models of turbulence such as von Karman [3],[4] and Kaimal [5]. The importance of turbulent loading is now universally recognized, and it is now common practice to base load calculations on a model of the three turbulent velocity components [6]. A considerable body of research has been undertaken for reliable prediction and/or simulation of real-time wind speeds for analyzing WECS response to wind gusts [7]–[11]. Several models have been proposed, including the point source Box-Muller algorithm [12], the autoregressive moving average (ARMA) model [13],[14], among others. All these methods use, as a starting point, auto-spectral and coherence descriptions of the turbulence. In this thesis, a constrained stochastic simulation (CSS) approach [15],[16] is adopted. The method can be applied to generate wind gusts from time series around events defined by means of a linear condition (constraint).
CHAPTER 5. MODELING WIND FIELD DYNAMICS
53 k=1.0 k=2.0 k=3.0 k=4.0 k=5.0
Rayleigh function
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0
5
10
15
20
25
30
35
40
Wind speed [m/s]
Figure 5.1: Rayleigh distribution for annual wind speed. k is the shape parameter.
5.2 Determination of Mean Wind Speed, v m In the absence of manufacturer specifications with regard to turbine rated wind speed, the Rayleigh distribution [17],[18] may be utilized in determining the average annual wind speed, µ w . The Rayleigh is a probability density function that describes the annual wind speed distribution and is used for estimating the energy recovery from a wind turbine. The annualized energy output, E, for the WECS is obtained as (see Appendix A.2.4)
vf
E = 8760
P f (vw )dvw
(5.1)
vc
where 8760 is the number of hours in a 365-day year and P is the non-zero power captured corresponding to wind speed vw , in the range from cut-in speed (vc ) to furling-speed (vf ). The Rayleigh function f (vw ) is derived from the general Weibull function in (2.8) for k = 2. It has the form 2 vw vw f (vw |α) = 2 exp − √ α 2α
(5.2)
where α is the mode of the distribution, and v w is the instantaneous wind speed. This expression is √ analogous to (2.8), with α related to the scale parameter c as α = c/ 2. Fig. 5.1 shows annual Rayleigh distribution curves for a range of shape parameters; the area under each curve is unity, with a standard deviation σ = α 4−π . Due to physical reasons, the 2 seasonal wind speed, µw , cannot change abruptly, but instead only continuously. Once the Rayleigh distribution is established, the turbulence level, σ ∗ , may be obtained and utilized in determining the
CHAPTER 5. MODELING WIND FIELD DYNAMICS
54
hub height average wind speed for simulations. Since σ 2 σ µw ∗
(5.3)
then using the the IEC 61400-1 Standard (ed. 3) for the representative turbulence intensity as detailed in Appendix A.2.3
∗
σ = Iref
(15m/s + 3Vhub ) + 1.28 × 1.44m/s 3
(5.4)
the average wind speed, Vhub , at hub height is empirically determined, which, to all intents and purposes, is the effective mean wind speed vm . In this study, the seasonal mean wind speed at site, µw = 7 m/s, and vm ≡ Vhub = 12.205 m/s.
5.3 CSS Model for Wind Turbulence, v t(t) CSS imposes a set of linear constraints on a turbulent wind field — the speed increases with a certain amount over a certain period in time and space. The simulated Gaussian field is the sum of a deterministic and a stochastic part. The deterministic part is time series given the specified constraints, formulated as a variational problem where the constraints are introduced by means of Lagrange multipliers. The starting point for the stochastic part is an ordinary Gaussian, homogeneous simulation that is subsequently projected onto the orthogonal complement of the subspace spanned by the constraints. For operation under turbulent wind, the following assumptions are made in the modeling: a) at t < 0 the power system is under steady-state conditions (almost constant wind speed) so that the load flow algorithm can be used to evaluate the initial conditions; b) no wind shear, either vertical or horizontal, is taken into account; c) variations in the horizontal direction of wind speed are not considered thus ensuring perfect tracking in the yaw direction (in practice this is not possible and causes 1–2% energy loss and additional stress on components [16]). In the model the turbulent wind field is represented by an expression for the temporal and spatial cross correlation of wind speed fluctuations which is transformed to a frame of reference moving with the rotor blades. In order to compute the dynamic response and loading of the wind turbine, this cross correlation function is integrated with the linear model of the rotor aerodynamics model.
CHAPTER 5. MODELING WIND FIELD DYNAMICS White noise
ξ(t)
Signal shaping filter
55 Colored noise
Gw Noise generator
Mean wind speed
τw
vt (t) σw k σw
Simulated wind speed
+ + vm
vw (t)
vm
Rayleigh distribution
Figure 5.2: Model for simulating wind speed behavior with CSS.
5.3.1 Formulating the Turbulence In the sequel a concise outline is given of CSS as a probabilistic method to determine a suitable wind speed profile as input for a wind turbine simulation tool. The design tool is for analyzing the extreme response as well as determining the internal loads of the WECS as a function of time. CSS presents a comprehensive method that may be applied for any event that can be expressed as a linear function of the involved random variables. A basic assumption in applying CSS for this purpose is that the extreme response is driven by wind turbulence and that the turbulence is Gaussian. Fig. 5.2 shows the model for executing CSS in simulating wind speed behavior. The driving force of the wind is normally distributed white noise produced by a random number generator. The discrete signal produced has mean value zero and unit variance. The sequential signal values with the sample time, T , are thus independent of each other. The stochastic component of the wind field is modeled as follows. The linear model of the turbulence component (wind gust), vt (t), is comprised by a first order filter disturbed by Gaussian noise v˙ t (t) = −
1 vt (t) + ξ(t) τw
(5.5)
where ξ(t) is white noise from the noise generator. The white noise is smoothed by a signal shaping filter with transfer function Gw (jω) and time constant τw , thereby transforming it into colored noise. The kσw block serves to standardize the colored noise by the standard deviation, σ w , of vm (obtained statistically from the Rayleigh distribution) to yield v t (t), which is ideally the summation of independent harmonics with random phases φ k uniformly distributed over [0, 2π] that follow from K 12 √ 2Gw (ωk )∆ω cos(ωk t + φk ) vt (t) = 2
(5.6)
k=0
where ∆ω = ωmax /K, and ωk = k∆ω, while ωmax is an upper cut-off of the noise spectrum. ωk a set of K equidistant frequencies.
CHAPTER 5. MODELING WIND FIELD DYNAMICS
56
The filter takes the form Gw (jω) =
Ξw 5
(1 + jωτw ) 6
(5.7)
where Ξw is the amplification factor. Selection of filter parameters depends on the long term mean wind speed, µw , and the characteristic turbulence length scale L that corresponds to the site roughness. These parameters are obtained as Ξw ≈
2π τw L and τw = 1 1 µw Γ( 2 , 3 ) T
(5.8)
where Γ designates the beta function. The turbulence component (5.6) may be rewritten as
vt (t) =
K
Ak cos(ωk t + φk )
(5.9)
k=0
where the amplitude, Ak , of each discrete frequency component represents the power in a specific frequency band
Ak (ωk ) ≈ 2
∞ ω0 2
Sk (ωk ) dω .
(5.10)
The integral in (5.10) may be discretely approximated thus:
Ak (ωk ) =
√
2
[Sk (ωk ) + Sk (ωk+1 )][ωk+1 − ωk ] 2
(5.11)
where the frequencies ωk are chosen to be logarithmically spaced to adequately represent the frequency content. Ak is based on the area under a density function S — the power spectral density of the turbulence — represented by the filter. Substituting (5.8) in (5.7) yields the von Karman distribution [19]
0.475σw2 vLm Sk (ωk ) = 56 1 + ωk µLw
and thus (5.9) becomes vt (t) =
K k=0
2Sk cos(ωk t + φk ). τw
(5.12)
(5.13)
where t is the discretized time. It is noteworthy that the development of v w (t) assumes normal distribution. Non-Gaussianity of wind turbulence and how to incorporate it in constrained simulation may be addressed by variational calculus, as suggested by Nielsen et al. [20].
CHAPTER 5. MODELING WIND FIELD DYNAMICS
57
5.3.2 Setting the Constraints Applying the Fourier transform to the wind gust component in (5.9) yields the series of the form:
vt (t) =
K
ak cosωk t + bk sinωk t
(5.14)
k=1
where, for normally distributed wind speed fluctuations, the Fourier coefficients a k and bk will also be normal. Their means are zero, they are mutually uncorrelated, and their variances are 2Sk /τw = 1. Selecting gusts with amplitude A at time t = t 0 corresponds to applying the following constraints: vw (t0 ) = A
(5.15)
v˙ w (t0 ) = 0
(5.16)
where the constraint (5.16) ensues from the fact that the ‘reference trajectory’ for the mean wind speed, vm , is a constant. The desired gusts are automatically selected by a combination of (5.14) and (5.15), leading to: Gc = a
(5.17)
with G
=
cosω1 t0
···
cosω2 t0
cosωK t0
sinω1 t0
···
sinωK t0
c
=
−ω1 sinω1 t0 −ω2 sinω1 t0 · · · −ωK sinωK t0 ω1 cosω1 t0 · · · ωK cosωK t0 T a1 a2 · · · aK b1 b2 · · · bK , and
a
=
(A 0)T .
,
To obtain the desired wind gust, the Fourier coefficients a k , bk , which are normally distributed, should satisfy the above conditions. The covariance matrix M of c is the diagonal with elements 2S k /Tw : 2 T M = E[cc ] = τw
S1
0
0
0
0
0
0
0
S2
0
0
0
0
0
0
0
···
0
0
0
0
0
0
0
SK
0
0
0
0
0
0
0
S1
0
0
0
0
0
0
0
···
0
0
0
0
0
0
0
SK
.
(5.18)
CHAPTER 5. MODELING WIND FIELD DYNAMICS
58
The constraint is that there is a peak of given height A at time t 0 , expressed in (5.17). The constraint may be conveniently expressed in terms of the unconstrained simulation time function: A − Gc =
A − vw (t0 ) −v˙ w (t0 )
(5.19)
and thus the constrained Fourier coefficients (5.14) are obtained as Sk cosωk t0 Sk ωk sinωk t0 % (A − vw (t0 )) + % 2 v˙ w (t0 ) Sk ω k Sk
(5.20)
Sk sinωk t0 Sk ωk cosωk t0 % (A − vw (t0 )) − % 2 v˙ w (t0 ) . Sk ω k Sk
(5.21)
ak,c = ak + and bk,c = bk +
Thus having made an unconstrained simulation of the wind velocity, (5.20) and (5.21) determine the Fourier coefficients that satisfy the gust constraints in (5.15) and (5.16).
5.4 Real-time Wind Speed Profile For analysis of wind turbine loading, it is appreciated that the rotor interacts with a complex spatially and temporally varying wind-field. However, the wind field may be represented by an effective wind speed, vw (t), over the rotor disk. This wind speed is modelled as a stochastic process with two components: the seasonal, slowly variable component, µ w , and the rapidly variable turbulence component, vt (t). Over short periods the wind speed can be approximated as the superposition of the mean wind speed and the instantaneous turbulence component vw (t) = vm + vt (t)
(5.22)
where vm Vhub is obtained from (5.4) that is based on µ w determined by the Rayleigh distribution while vt (t) is computed via CSS. Gaussian white noise and typical wind speed profiles are shown in Figs. 5.3(a) and (b), respectively. It should be noted that the spectral characteristic of this effective wind speed is very different from that of a point source. In order to obtain the distribution of the extreme loading caused by a gust with arbitrary amplitude (for a given v m ), the different distributions should be convoluted (weighed) with the occurrence probability of the individual gusts. Furthermore, using the load distribution and resistance distribution of the structure the probability of failure can be estimated. Together they constitute the tools leading to a more efficient and reliable WECS design.
White noise signal, Z
CHAPTER 5. MODELING WIND FIELD DYNAMICS
59
2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0
5
10
15
20
25
30
35
40
t, [s]
(a) Gaussian white noise signal of zero mean, unit variance. 24
Wind speed, v(t) [m/s]
22 20 18 16 14 12 10 vm=12,It=16% vm=12,It=18% vm=16,It=16%
8 6 0
20
40
60
80
100
t, [s]
(b) Wind speed profiles. Red line is v m = vr at a turbulence intensity of 16%, green line represents v m = vr at a turbulence intensity of 18%, and blue line is profile at v m = 16 m/s and turbulence intensity of 16%.
Figure 5.3: White noise and typical generated wind speed profiles by CSS at various mean wind speeds and turbulence intensities.
5.5 Remarks Time domain simulations of wind gust events are of practical interest for wind turbine design calculations. Until relatively recently, calculations of the loading and behavior of wind turbines were based on grossly simplified models of the wind: a steady wind speed, constant power or logarithmic law model of wind shear, a constant flow inclination, and a dominant longitudinal component of turbulence. Although such input enables a satisfactory calculation of the periodic loading, it provides no basis for evaluating the random loads due to turbulence. The current IEC-Standard considers extreme wind events as extreme load conditions that must be considered as ultimate load cases when designing a wind turbine. Within the framework of the IEC 61400-1 Std (ed. 3) [2],[21], these load situations are defined in terms of two independent site variables — a reference mean wind speed and a characteristic turbulence intensity. In this research CSS generates a spatial turbulent wind field at fixed points at the rotor disc, based on a Class A turbulence site. For the seasonal mean wind speed of 7 m/s, cut-in wind speed of 4.0 m/s, and operation at rated wind speed of the turbine equipment (12.205 m/s), the prevailing turbulence intensities (longitudinal, lateral and vertical) are obtained as 16.0108%, 12.5465%, and 8.92472%, respectively (see Appendix A.2.2, A.2.3).
CHAPTER 5. MODELING WIND FIELD DYNAMICS
60
References [1] E. B. Muhando, T. Senjyu, N. Urasaki, H. Kinjo, and T. Funabashi, “Online WTG dynamic performance and transient stability enhancement by evolutionary LQG,” IEEE Power Engineering Society General Meeting, 24-28 June 2007, pp. 1-8. doi:10.1109/PES.2007.385499. [2] International Electrotechnical Commission. IEC 61400-1: Wind Turbines Part 1: Design Requirements. IEC 2005-08, 3rd edition, 2005. Available online, http://www.iec.ch. [3] B. G. Rawn, P. W. Lehn, and M. Maggiore, “A control methodology to mitigate the grid impact of wind turbines,” IEEE Trans. Energy Conversion, vol. 22, no. 2, pp. 431-438, 2007. [4] W. E. Leithead, S. de la Salle, and D. Reardon, “Role and objectives of control for wind turbines,” IEE Procs. Generation, Transmission and Distribution, vol. 138, no. 2, pp. 135-148, March 1991. [5] T. Ekelund, “Speed control of wind turbines in the stall region,” Procs. 3rd IEEE Conference on Control Applications, 24-26 Aug. 1994, vol. 1, pp. 227-232. doi:10.1109/CCA.1994.381194. [6] W. Bierbooms, “A gust model for wind turbine design,” JSME International Journal, Series B, Vol. 47, No. 2, pp. 378-386, 2004. [7] F. Iov, F. Blaabjergg, A. D. Hansen, and Z. Chen, “Comparative study of different implementations for induction machine model in Matlab/Simulink for wind turbine simulations,” Procs. IEEE Workshop on Computers in Power Electronics, 3-4 June 2002, pp. 58-63. doi:10.1109/CIPE.2002.1196716. [8] R. A. Schlueter, G. L. Park, R. Bouwmeester, L. Shu, M. Lotfalian, P. Rastgoufard, and A. Shayanfar, “Simulation and assessment of wind array power variations based on simultaneous wind speed measurements,” IEEE Trans. Power Apparatus and Systems, vol. PAS-103, no. 5, pp. 1008-1016, 1984. doi:10.1109/TPAS.1984.318705. [9] R. Karki, P. Hu, and R. Billinton, “Reliability evaluation of a wind power delivery system using an approximate wind model,” IEEE Procs. 41st International Universities Power Engineering Conference, UPEC ’06, 6-8 Sept. 2006, vol. 1, pp. 113-117. doi:10.1109/UPEC.2006.367726. [10] P. Flores, A. Tapia, and G. Tapia, “Application of a control algorithm for wind speed prediction and active power generation,” Renewable Energy, vol. 30, pp. 523-536, 2005. [11] G. N. Kariniotakis, G. S. Stavrakakis, and E. F. Nogaret, “Wind power forecasting using advanced neural networks models,” IEEE Trans. Energy Conversion, vol. 11, no. 4, pp. 762-767, 1996.
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[12] G. E. P. Box, and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat., vol. 29, pp. 610-611, 1958. [13] E. B. Muhando, T. Senjyu, N. Urasaki, A. Yona, and T. Funabashi, “Robust predictive control of variable-speed wind turbine generator by self-tuning regulator,” IEEE Power Engineering Society General Meeting, 24-28 June 2007, pp. 1-8. doi:10.1109/PES.2007.385885. [14] E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “RLS-based self-tuning regulator for WTG dynamic performance enhancement under stochastic setting,” Proc. The International Conference on Electrical Engineering, ICEE 2007, 8-12 July 2007, pp. 1-6. [15] E. B. Muhando, T. Senjyu, Z. O. Siagi, and T. Funabashi, “Intelligent optimal control of wind power generating system by a complemented linear quadratic Gaussian approach,” IEEE Power Engineering Society Conference and Exposition, PowerAfrica 2007, 16-20 July 2007, pp. 1-8. [16] E. B. Muhando, T. Senjyu, N. Urasaki, A. Yona, H. Kinjo, and T. Funabashi, “Gain scheduling control of variable speed WTG under widely varying turbulence loading,” Renewable Energy, vol. 32, no. 14, pp. 2407-2423, 2007. [17] R. B. Corotis, “Stochastic modeling of site wind characteristics,” ERDA Report, RLO/234277/2, September 1977. [18] R. B. Corotis, A. B. Sigl, and J. Klein, “Probability models of wind velocity magnitude and persistence,” Solar Energy, vol. 20, no. 6, pp. 483-493, 1978. [19] C. Nichita, D. Luca, B. Dakyo, and E. Ce˘anga, “Large band simulation of the wind speed for real time wind turbine simulators,” IEEE Trans. Energy Conversion, vol. 17, no. 4, pp. 523-529, Dec. 2002. [20] M. Nielsen, G. C. Larsen, J. Mann, S. Ott, K. S. Hansen, and B. J. Pedersen, “Wind simulation for extreme and fatigue loads,” Risø-R-1437(EN), 2003. [21] IEC 61400-1 Standard:
Wind Turbine Safety and Design Ed 3. Available online,
http://www.awea.org/standards
Part II Control Strategies and Design for Wind Energy Conversion Systems
Chapter 6 Control Philosophy 6.1 Introduction
C
ONTROL can significantly improve the energy capture by a wind turbine. Part II of this thesis reviews techniques for the control of wind turbines during power production. Particularly, as
turbines become larger and more flexible, there is increasing interest in designing controllers with load reduction as part of the primary objective, to mitigate loads as far as possible. Terms can be introduced into the controller to help damp resonances, such as drive train torsion in variable-speed turbines. Classical methods based on proportional-integral (PI) and proportional-integral-derivative (PID) algorithms are a good starting point for many aspects of closed-loop controller design for fixedand variable-speed turbines. With regard to energy extraction efficiency of WECs, controller design has centered mainly on simple, linear, PID controllers [1]-[6] that are easily implemented in the field environment. Although industry has embraced the PID controller, researchers have begun to investigate the capabilities of more sophisticated control designs for ensuring efficient power conversion, especially multivariable, multiobjective paradigms. Several advanced controllers are proposed and analyzed in this research, including the linear quadratic Gaussian (LQG) [7]-[10] that has been shown to effectively optimize power conversion for a wind power system across a whole range of operating regions [11], the selftuning regulator (STR) scheme [12]-[15], and model-based predictive control (MBPC) [16],[17]. These have an advantage over the PID since they can incorporate multiple inputs and multiple outputs. However, in order to convince industry to invest in more complicated controllers it is necessary to show that they are potentially able to guarantee long service life of the WECS, ensure low maintenance and above all, maintain a high level of energy conversion. It is important to be able to quantify the benefits of any new controller with particular regard to the variability of the real wind.
CHAPTER 6. CONTROL PHILOSOPHY
64
Advanced controller design methods can offer an explicit mathematical formulation for the design of controllers with multiple objectives, including load reduction. Such controllers have been used on commercial turbines to a limited extent. For variable speed turbines, attention to detail in the interaction of pitch and torque controllers can significantly improve energy capture without any compromise on loads. Individual pitch control has potential for very significant load reduction but is not yet commercially proven. The design of the control algorithms is clearly of prime importance. Additional sensors such as accelerometers and load sensors can also help the controller achieve its objectives more effectively. As the size of wind turbines increases, and as cost reduction targets encourage lighter and hence more flexible and dynamic structures, these aspects of controller design become increasingly important. A very basic controller might consist of a classical PI or PID algorithm acting on a single measured signal (generator speed or power output) to generate a pitch demand. For variable-speed turbines a torque demand is generated independently from a speed–torque look-up table. This basic scheme can be greatly improved in a number of ways. This thesis covers the following possibilities: • joint control of pitch and torque to improve the trade-off between energy and loads; • using torque control to damp out torsional resonances, especially in the drive train. These strategies are now routinely used in the industry. Blade pitch control is primarily used to limit the aerodynamic power in above-rated wind speeds in order to keep the turbine within its design limits, but it also has an important effect on structural loads. Some optimization of energy capture below rated is also possible. Generator torque control in variable-speed turbines is used primarily to maximize energy capture below rated wind speed by controlling rotor speed, and to limit the transmission torque above rated, but it can also be used to reduce certain loads. The algorithms used for controlling pitch and torque need careful design. In addition to their effectiveness in meeting these primary objectives, the control algorithms can also have a major influence on the loads experienced by the wind turbine. Clearly the algorithms must be designed so as to prevent excessive loading, but it is possible to go further by designing them with load reduction as an explicit objective — the main theme for Part II of this report. This thesis illustrates the finding that, with careful design, more robust multiobjective and adaptive controllers can be developed that can achieve better performance levels relative to classical linear controllers and are much more likely to be adopted in practice. Finally, the importance of modeling for controller design is stressed. Although field trials are useful, computer simulations are also vital, and are utilized to evaluate performance.
CHAPTER 6. CONTROL PHILOSOPHY
65
5
ωt vw
Optimal control
Pref
βcmd
Q ref β
vw Wind speed model 6
ωt
2 Wind Γ t turbine rotor
Γt
Drive-train Γg dynamics
ωg
Stator i dq_s 4 voltage i dq_r Active/ udq Pref reactive udq_r power Q ref (P&Q) udq_s control ωg
3
udq_s i dq_s udq_r i dq_r Γg
ωg DOIG
ωt
1
Figure 6.1: Relational schematic of the WECS with DOIG, converters and controllers.
6.2 Control Concept 6.2.1 Model Overview The most significant dynamics of the wind turbine have been modelled in Part I with emphasis on control design. An entire nonlinear simulation model of the wind turbine can then be derived by connecting the individual sub-models. Fig. 6.1 is the block diagram of the dynamic WECS model that is applied to investigate power output performance, power train reliability, and transient voltage stability in the sequel. The interconnections between the different dynamic components are depicted as the respective blocks: 1) The aerodynamic model of the turbine rotor 2) The shaft system model — represents possible torsional oscillations in the shaft system 3) The electric generator model — it is a transient model 4) The converter and its control 5) The blade-angle (pitch) control and the servo model, and generator torque reference. 5) The wind model1. The work presented in Chapters 2–5 has been to develop subsystem models as part of a simulation platform project in which the main idea is to extend the ability of the existing wind turbine design tools to simulate the dynamic behavior of the wind turbines and the wind turbine–grid interaction. One of the main targets is to improve the generator models used in advanced aeroelastic tools and to add the electrical part of the wind turbine. In these aeroelastic tools the focus is on the frequency scale between 0 and 20 Hz, because the main contribution to fatigue loads is in this frequency range. 1
The wind model, strictly speaking, is not a component of the wind turbine model, but the output power calculation for the WECS requires the knowledge of instantaneous wind speed. The instantaneous wind speed v w (t) is described by (5.1) in Chapter 5.
CHAPTER 6. CONTROL PHILOSOPHY
Inputs u vt βcmd ω g,ref Γg,ref u ds u qs
66
States x ωt ωg β θtg i ds i qs i dr i qr
Outputs y
P
Q
Figure 6.2: Complete dynamic model of inputs, state variables, and outputs.
Fig. 6.2 illustrates the complete WECS dynamic model, characterized by six inputs u¯, two outputs y¯, and eight state variables. The inputs include pitch angle reference, β cmd , wind speed disturbance vt , and adjustable control variables of the converter. State variables comprise 8 nonlinear equations: (3.5) for the drive-train states from Chapter 3, and (4.11) for the induction machine (Chapter 4). The active (P), and reactive (Q) powers injected into the grid are taken as model outputs. Based on the state-space form of the induction machine dynamic model, the complete linearized model is obtained, and then the reduced-order model that neglects the stator transients, and the steadystate model are easily extracted. After that, the required transfer functions between the desired input– output pair can be obtained. These transfer functions are useful in the linear design of the control loops as well as in the analysis of the stability and response of the system under different operating conditions. Based on the steady-state model, an analysis of the control variables is performed in order to obtain the operational points of the DOIG. The developed models are used in controlling the DOIG with a power electronic converter in the WECS. The focus is on analysis of the state-space modeling of a 2 MW DOIG used in WECS applications, and the mechanical and aeroelastical aspects must be considered to visualize the dynamic behavior. Before the models can be used with confidence, they should have been validated by comparing model results to measurements. To highlight the importance that an accurate representation of the structural dynamics has for purposes of model validation of IG wind turbines, this study compares the performance of the proposed modeling and control to the actual prototype values detailed in Appendix A. The real advantage of the method is visible in the chosen highly turbulent wind environment, presenting a noisy signal to the system. The simulation model is implemented in a MATLAB/Simulink environment with the control target of ensuring response geared toward optimum power conversion and minimizing shaft torsional torque variations without additional filtering.
CHAPTER 6. CONTROL PHILOSOPHY
67
TURBULENCE
vt Wind speed
Proposed multivariable controller
Γprp
DYNAMIC OPTIMIZATION
ω
Γg,ref
Γbc
Σ
∆ω
Electric power
WECS
STEADY STATE OPTIMIZATION
Baseline controller
vm
Key:
Γprp
Generator command signal by proposed controller
Γbc
Generator command signal by baseline controller
SEASONAL
Figure 6.3: Control strategy by the frequency separation principle. Γ g,ref = .
' %& Γbc + Γprp
6.2.2 Control Objectives In this study the control problem is conveniently divided into two time scales corresponding to slow mean wind speed changes and rapid turbulent wind speed variations. The mean speeds are treated as steady state operating points. Fig. 6.3 illustrates the frequency separation principle utilized in analyzing the system: steady state optimization assumes operation at the optimal wind speed while with dynamic optimization the OP is bound to shift hence the need for an adaptive controller to regulate the aerodynamic effects on the system. The WECS can be started at the wind speed of 4 m/s and operated in the wind area up to 25 m/s. The control design objectives are: ◦ To optimize power production in low to medium wind speeds, and to regulate turbine speed in the above-rated region thereby maintaining rated power. ◦ To specify the demanded generator torque to maintain stable closed-loop behavior over the entire turbine operating envelope, which includes enhancing the damping to the drive train torsion and mitigating the effects of wind speed disturbances. The overall objective of the controller is to maximize energy production, whilst working within the operational limits of the turbine, and minimizing the peak loadings experienced. While the wind is highly stochastic, initial insight into this requirement can be gained by considering the situation when the wind is steady and the turbine is in equilibrium. Three operating modes can be identified: 1. Energy capture limited by available wind energy 2. Energy capture limited by rotor speed constraints 3. Energy capture limited by generator rating Overall, effectiveness of each proposed control scheme is evaluated based on the objectives, and subject to operating constraints. A further, deterministic, extreme gust is employed to confirm the ability of the controllers to maintain operation within the allowed rotor speed limits.
CHAPTER 6. CONTROL PHILOSOPHY
68
6.3 Control Strategy The devised strategy is twofold: active power control for optimal conversion throughout the WECS operating envelope, and generator torque control for alleviation of torsional loads on the power train.
6.3.1 Active Power Control The control objectives of the active power control loop are achieved by speed control, based on the following control strategies: (a) Power optimization strategy — utilized for below rated wind speed, where the energy capture is maximized by tracking the maximum power coefficient. • The power reference is the wind turbine available power • The speed reference is the optimal speed. The turbine has to produce the optimum power corresponding to the maximum tracking power point look-up table. The difference between the generator speed and its reference value is negative and, therefore, the generator torque controller’s output, Γg,ref , is increased systematically thereby driving the TSR to its optimal value by varying the rotational speed. In this operational regime, the pitch angle is kept constant at the lower limit (optimal value). (b) Power limitation strategy — for above rated wind speed; power is limited to rated power, P r . • The power reference is the rated power • The speed reference is the rated speed. Speed controller keeps the generator speed limited to its rated value by acting on the pitch angle. The difference between the generator speed and its rated value is positive, thus the pitch controller kicks in and drives the pitch angle to positive values until the rated generator speed is reached. The WECS has to produce less than it is capable of at a given wind speed. This action implies both a larger dynamical pitch activity and a larger steady-state pitch angle.
6.3.2 Power-train Torsional Load Alleviation In above rated wind regimes, generator torque control is utilized exclusively for overload prevention, whereas blade pitch control is used for power limitation. The inverter controller holds the electrical power constant at rated power, thus the turbine is prevented from following the c P,opt trajectory and constrained to operate at lower values of TSR and cP . The power that the inverter injects into the grid is completely independent of both the grid frequency and the DOIG speed. Γ g,ref is for damping only.
CHAPTER 6. CONTROL PHILOSOPHY
69 Speed controller
ref
Optimal speed
Wind speed
ωgen
∆ω
+
meas
ωgen
β ref
PI
_
+ _
K pi Gain scheduling
τ
Rate limiter
Angle limiter
β
Turbine rotor Γaero
β
Transmission system Γmech
Power controller Active power reference
ACTIVE POWER CONTROL LOOP
P ref grid
+
P
ACTIVE CURRENT CONTROL
i qref
PI
_
Reactive power reference
Rotor current controller P mq PI
Control signals
i qmeas
REACTIVE POWER CONTROL LOOP
Q ref grid
+ _
meas grid
+
meas
ωgen
REACTIVE CURRENT CONTROL
i dref
PI
_
Generator frequency converter
+ _
Q meas grid
P md PI
i dmeas
Available power
MPTP
P MPTP el P el
ω
Figure 6.4: WECS control level.
6.4 Controller Design 6.4.1 Assigning the Control Tasks As illustrated in Fig. 6.4, the WECS’s power capability is expressed in terms of instantaneous (shortterm) available power. This is based on the maximum power tracking point (MPTP) as a function of the optimal speed. The wind turbine control level contains • a slow control level (speed controller and a power controller), and • a fast control level (frequency converter-rotor current controller). In implementation, the converter controls the power of the WECS through two controllers in cascade: 1. The power controller (the external controller in the cascade controllers) provides a reference rotor current to the rotor current controller (the internal controller in the cascade controllers), which further controls the generator current and thus the generator torque. 2. The speed controller — controls generator speed to its reference value by acting on pitch angle. The power controller ensures the power reference by acting on the current reference of the rotor current controller and thus on the generator current/torque; this is achieved via two control loops: (i) the active power control is achieved by controlling the q-axis component of the rotor current (in a stator flux dq reference frame), while (ii) the reactive power control is achieved by controlling the d-axis component of the rotor current (the magnetizing current) collinear with the stator flux. Note: rotor current controller generates rotor voltage components as control variables of the converter.
CHAPTER 6. CONTROL PHILOSOPHY
70
PI Pref +
K pp _
Pe
+ K ip
βcmd
s
+
K PI
1
β
s
_
β
Gain scheduling
max
min
Figure 6.5: Pitch control system.
6.4.2 Pitch Actuator and Blade Servo The actuator dynamics and implementation of the pitch control are depicted in Fig. 6.5. The pitch angle controller is only active during high wind speeds. By varying the pitch angle β, the aerodynamic torque input to the rotor is altered and hence the output power. Because the inertia of the blades is large and the actuator should not consume a great deal of power, the actuator has limited capabilities. The goals of pitch control include: ◦ Total active power, Pe , as high as possible subject to the condition P e < Pr ; this implies holding the pitch angle at a mechanical limit: β = −2 ◦ . ◦ Pe remains at WECS rating, Pr , in the region of higher wind speeds; thus β has to be modified between −2◦ and 30◦ to reduce cP (λ, β). Controller dynamics are nonlinear with saturation limits on both pitch angle and pitch rate. When the pitch angle and pitch rate are less than the saturation limits, the pitch dynamics exhibit linear behavior, thus the dynamics of the servo with the blades may be described by a first order transfer function with a time constant τ β Kβ 1 βcmd . β˙ = − β + τβ τβ
(6.1)
The desired pitch command, βcmd , is the output of the pitch controller, and is fed to the pitch actuator to regulate the pitch angle of the turbine blades. The desired pitch angle is selected so that the generated power, Pe , follows Pref . The command βcmd is the integral sum of the small changes of pitch command (∆βcmd ) over the sampling intervals, and can be represented as βcmd =
−sgn{∆P }|∆βcmd |
(6.2)
where ∆P = Pe − Pref , and ∆βcmd is derived by gain scheduling in the PI block. Output power P e is smoothed by a hydraulic servo system that drives the blades around their lengthwise axes.
CHAPTER 6. CONTROL PHILOSOPHY
71
Pitch angle, β [deg]
25 20 15 10 5 0 -5 5
10
15
20
25
Wind speed, vw [m/s]
Figure 6.6: Variation of pitch angle with wind speed in steady state conditions. Fig. 6.6 is a typical variation in pitch angle with wind speed at steady state conditions for the WECS in this study. Gain scheduling serves to compensate for the large changes in the sensitivity of aerodynamic torque to pitch angle over the operating range, since the WECS aerodynamic characteristics vary according to the OP, and hence vw . Thus the proportional and integral gains are scaled by the gain scheduling constant, K P I , in order to ensure suitable control loop characteristics are attained at all wind speeds. Additionally, the rate limiter is applied to the output with instantaneous integrator desaturation to prevent wind-up. The transfer function C(s) between the power error and β cmd is: C(s) =
βcmd (s) sKpp + Kip = . ∆Pe (s) s
(6.3)
Selection of Kpp , Kip , and KP I is by trial and error, based on minimizing deviations from the setpoint without excessive control action and without causing any instabilities. The proportional and integral constants are respectively Kpp = 0.0246 s and Kip = 0.01025. KP I is given as follows: KP I =
1, β 15
for −2◦ < β ≤ 0◦
+ 1, for 0◦ < β ≤ 30◦ 3,
(6.4)
for β > 30◦
The servomotor, modeled as a first order system with time constant τ β = 0.05 s can operate very fast, but allowance has been made for servo system delay, and possibility of other delays e.g. communication delay, computational delay and conditional delay (to overcome Coulomb friction). Thus the response of the pitch actuation system is not instantaneous. The pitch rate commanded by the actuator is physically limited to maximum ±8 ◦ /s while the saturation level of the pitch angle is from –2◦ to 90◦ . These limits should not be reached during the normal operation in order to avoid not only the fatigue damage and wear of the pitch actuator, but also the loss of performance. It should be mentioned that in power control mode lower values of pitch rate are desirable, however, for speed control mode the larger pitch rate value shows better transient performance.
CHAPTER 6. CONTROL PHILOSOPHY
72
6.4.3 Generator Torque Controller 6.4.3.1 Baseline Controller The choice of generator torque as a control input is motivated by the fact that when connecting the generator to the grid via the frequency converter, the generator rotational speed, ωg , will be independent of the grid frequency. This decoupling enables variable speed operation, and a control strategy based on wind speed regime may be formulated: I. At low and moderate wind speeds generator speed, ωg , is controlled to maximize energy capture by operating continuously at the TSR that results in the maximum power coefficient. The target is to track the OP locus (λopt , cP,opt) by regulating the generator torque to yield the optimum power conversion, Pm,opt :
R 3 1 ωt3 . Pm,opt = ρΛcP,opt 2 λopt
(6.5)
A standard baseline controller is then designed to keep the turbine operating at the peak of its cP -TSR-pitch surface, executed in accord with the expression Γref =
KT ωt2
R 3 1 KT = ρΛcP,opt 2 λopt
where
(6.6)
Γref being the reference torque signal and KT the torque control gain. The gain algorithm is derived from the non-adaptive case presented in (6.6)
Γref =
0,
for ω < 0
ρΓ ∗ ω 2 , for ω ≥ 0
(6.7)
where Γ ∗ incorporates all the non-adaptive gain (KT ) parameters apart from air density ρ that is time-varying and thus uncontrollable. Most turbines have separate control mechanisms to prevent reverse operation; in this study the control law (6.6) assumes positive regions of ω. II. When the wind speed exceeds its nominal value, the control objective shifts from maximizing power capture to regulating power to the WECS’s rated output while reducing rotor speed fluctuations and minimizing both control actuating loads and shaft torsional moments. Hence in above rated wind regimes, both generator torque control and blade pitch control are used for overload prevention and power limitation. The generator torque controller utilizes only the local generator speed to produce appropriate control signals for meeting the control objectives.
CHAPTER 6. CONTROL PHILOSOPHY Optimum characteristic
Pref
73
PI
ωref +
ωg
K pt+ K it _
Γ g,ref
s
Torque to current translation
i rq,ref + _ i rq
K pv+ K iv
s
Rotor injected voltage
u rq
Figure 6.7: Generator torque and speed control. 6.4.3.2 PI Controller for Γg,ref The PI controller consists of a cascade speed and torque control-loop. The inner loop is the torque control that compares the electric torque and the output signal from the speed proportional plus integral controller, shown in Fig. 6.7. The speed controller compares the actual rotor speed and the reference rotor speed. The output signal from the cascade controllers is the q-axis rotor current. I. Current Control The stator current is regulated through control of the rotor current, and, by applying both d- and q-components, the reference values for the rotor current are calculated as irq,ref =
Us Xs − isq,ref · . Xs Xm Xm
(6.8)
where the reference stator current is calculated with the reference values for torque, Γg,ref isq,ref = Γg,ref ·
ω0 . Us
(6.9)
The rotor currents are controlled with a PI controller, equipped with anti-windup and decoupling terms to optimize the dynamic behavior [18]. II. Speed Control The speed controller in Fig. 6.7 is a PI regulator that gives the relationship between the input, ∆ωg , and the output, Γg,ref Γg,ref = Kpt ∆ωg + Kit
t 0
∆ωg dt
(6.10)
where the proportional and integral constants are Kpt = 500 Nms/rad and Kit = 250 Nm/rad, respectively. The output is Γg,ref , that is used to define isq,ref in (6.9). The reference generator speed is a function of wind speed: below rated wind speed the reference generator speed is proportional to the wind speed, above, it is constant at rated value.
CHAPTER 6. CONTROL PHILOSOPHY
74
6.4.3.3 Multiobjective Controllers for Γg,ref Two significant problems abound with the standard control in (6.7): 1. inaccuracies in determining Γ ∗ due to changing blade aerodynamics over time; 2. wind speed fluctuations force the WECS to operate off the peak of its power curve much of the time, resulting in less energy capture. Besides, control with PI (6.10) that optimizes energy can also cause undesirable torque fluctuations that result from the inertia of the rotor as the torque control attempts to follow the wind. Indeed, tracking of T SR = λopt at high frequency is not desirable because it would induce sudden variations of turbine rotational speed and thus high mechanical loads on the drive train. Moreover, the converter in the variable speed turbine neither adds inherent damping to the power system, nor is its speed inherently damped by the power system. To address these issues, adaptive controllers are proposed that reduce the negative effects of both the uncertainty regarding Γ ∗ and the change in optimal OP due to turbulence. These paradigms include: LQG, STR and MBPC, as detailed in Chapters 7–9. These controllers seek the gain that maximizes power capture regardless of whether this gain corresponds to the maximum of the power curve for the WECS. Mechanical stress and strain reduction are met by reducing the vibrations between the rotating parts. The several multivariable, multiobjective schemes that are proposed generate the appropriate generator torque signals respectively to compensate for the above contigencies as well as add damping to the drive-train. Although it may be possible to provide some damping mechanically, for example by means of appropriately designed rubber mounts or couplings, there is a cost associated with this. Another solution, which has been successfully adopted on many turbines, is to modify the generator torque control to provide some damping. Instead of demanding a constant generator torque above rated, a small ripple at the drive train frequency is added on, with its phase adjusted to counteract the effect of the resonance and effectively increase the damping. A highpass or bandpass filter of the form G
2ζωs(1 + sτ ) s2 + 2ζωs + ω 2
(6.11)
acting on the measured generator speed can be used to generate this additional ripple. The frequency ω must be close to the resonant frequency which is to be damped. The time constant τ can sometimes be used to compensate for time lags in the system, or to adjust the phase of the response.
CHAPTER 6. CONTROL PHILOSOPHY
75
6.5 Conclusions The overall objective of the controller is to maximize energy production, whilst working within actuator operational limits and minimizing the extreme loads and associated fatigue damage on the turbine structure and drive-train — a disturbance rejection task. Within this framework the designed controllers have the following measurements available (i) the instantaneous power, Pe = Γg · ωg , and (ii) the generator speed, ωg . The sensor dynamics can be assumed negligible, as is measurement noise. Note that the effective wind speed, vw , cannot be measured. Additionally, the controllers are able to adjust these manipulable variables: (i) the blade pitch angle, and (ii) the generator reaction torque. In the case of grid faults the controllability of the WECS embraces both the control for preventing rotor overspeed, and the control and protection of the power converter during and after the grid faults. Several advanced controllers are proposed in the sequel whose commonality is full-state feedback with state estimation and/or prediction. The greatest advantage of these paradigms over PID control is the fact that they are multiobjective, hence can incorporate multiple inputs and multiple outputs. Issues such as reducing shaft fatigue could be easily included in the control objectives. However, in order to convince industry to shift toward more sophisticated controllers, it is necessary to compare their functionality with PID controllers. Against the limitations of PID control, the aim is to establish, through systematic design methods, that these elaborate controllers offer greater benefit in form of robustness, efficiency, and eventual reduction in cost of energy. Of the analyzed multiobjective, multivariable controllers, no scheme is clearly favored against the others; the various paradigms are being tested and evaluated with respect to the classical PID controller. The design and development of the various multiobjective control paradigms is undertaken in the time domain, based on modeling the WECS components as discrete systems. Although the frequency domain approach has the advantage that it provides for a very rapid analysis of wind turbine loading, it suffers from the disadvantage that it cannot take account of system non-linearities associated, for example, with the rotor aerodynamics, structural dynamics and/or control system dynamics. For this reason in particular, the frequency domain approach is generally not utilized as the basis of final, detailed wind turbine design calculations. The method is, nevertheless, of some value in the very early stages of wind turbine design for optimization studies.
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References [1] A. Tapia, G. Tapia, J. X. Ostolaza, and J. R. Saenz, “Modeling and control of a wind turbine driven doubly fed induction generator,” IEEE Trans. Energy Conversion, vol. 18, no. 2, pp. 194204, 2003. [2] B. Boukhezzar, L. Lupu, H. Siguerdidjane, and M. Hand, “Multivariable control strategy for variable speed, variable pitch wind turbines,” Renewable Energy, vol. 32, pp. 1273-1287, 2007. [3] S. R. Dominguez, and M. D. McCulloch, “Control method comparison of doubly fed wind generators connected to the grid by assymetric transmission lines,” IEEE Trans. Industry Applications, vol. 36, no. 4, pp. 986-991, Jul/Aug. 2000. doi:10.1109/28.855951. [4] R. Teodorescu, and F. Blaabjerg, “Flexible control of small wind turbines with grid failure detection operating in stand-alone and grid-connected mode,” IEEE Trans. Power Electronics, vol. 19, no. 5, pp. 1323-1332, Sept. 2004. doi:10.1109/TPEL.2004.833452. [5] R. Cardenas, and R. Pena, “Sensorless vector control of induction machines for variable-speed wind energy applications,” IEEE Trans. Energy Conversion, vol. 19, no. 1, pp. 196-205, Mar. 2004. doi:10.1109/TEC.2003.821863 [6] H. S. Ko, S. Bruey, J. Jatskevich, G. Dumont, and A. Moshref, “A PI control of DIFG-based wind farm for voltage regulation at remote location,” IEEE Power Engineering Society General Meeting 2007, 24-28 June 2007, pp.1-6. doi:10.1109/PES.2007.385496. [7] E. B. Muhando, T. Senjyu, N. Urasaki, A. Yona, H. Kinjo, and T. Funabashi, “Gain scheduling control of variable speed WTG under widely varying turbulence loading,” Renewable Energy, vol. 32, no. 14, pp. 2407-2423, 2007. [8] W. E. Leithead, S. de La Salle, and O. Reardon, “Role and objectives of control of wind turbines,” Proceedings of the IEEE, vol. 138C, pp. 135-148, 1991. [9] P. Novak, T. Ekelund, I. Jovilk, and B. Schimidtbauer, “Modeling and control of variable speed wind turbine drive systems dynamics,” IEEE Control Systems Magazine, vol. 15, no. 4, pp. 28-38, 1995. [10] E. S. Abdin, and W. Xu, “Control design and dynamic performance analysis of a wind turbineinduction generator unit,” IEEE Trans. Energy Conversion, vol. 15, no. 1, pp. 91-96, 2000.
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[11] I. Munteanu, N. A. Cutululis, A. I. Bratcu, and E. Ce˘anga, “Optimization of variable speed wind power systems based on a LQG approach,” Control Engineering Practice, vol. 13, pp. 903-912, 2005. [12] E. B. Muhando, T. Senjyu, N. Urasaki, A. Yona, and T. Funabashi, “Robust predictive control of variable speed wind turbine generator by self-tuning regulator,” 2007 IEEE Power Engineering Society General Meeting, 24-28 June, 2007, pp. 1-8.doi:10.1109/PES.2007.385885. [13] E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “Disturbance rejection by dual pitch control and self-tuning regulator for wind turbine generator parametric uncertainty compensation,” IET Procs. Control Theory and Applications, vol. 1, no. 5, pp. 1431-1440, 2007. doi:10.1049/iet-cta:20060448. [14] R. Sakamoto, T. Senjyu, T. Kinjo, N. Urasaki, and T. Funabashi, “Output power leveling of wind turbine generator by pitch angle control using adaptive control method,” IEEE Int. Conf. on Power Systems Technology, PowerCon 2004, 21-24 Nov. 2004, vol. 1, pp. 834-839. doi:10.1109/ICPST.2004.1460109. [15] M.-u.-D. Mufti, R. Balasubramanian, and S. C. Tripathy, “Self tuning control of wind-diesel power systems,” Procs. of the 1996 Int. Conf. on Power Electronics, Drives and Energy Systems for Industrial Growth, 8-11 Jan. 1996, vol. 1, pp. 258-264. doi:10.1109/PEDES.1996.539550. [16] E. B. Muhando, T. Senjyu, and H. Kinjo, “Disturbance rejection by stochastic inequality constrained closed-loop model-based predictive control of MW-class wind generating system,” Procs. Joint IEEJ-IEICE Conference, 19 Dec. 2007, pp. 91-99. [17] E. Gallestey, A. Stothert, M. Antoine, and S. Morton, “Model predictive control and the optimization of power plant load while considering lifetime consumption,” IEEE Trans. Power Systems, vol. 17, no. 1, pp. 186-191, Feb. 2002. doi:10.1109/59.982212. [18] J. Soens, J. Driesen, and R. Belmans, “A comprehensive model of a doubly fed induction generator for dynamic simulations and power system studies,” Procs. Int. Conference on Renewable Energies and Power Quality, ICREPQ ’03, Vigo, Spain, April 9–12, 2003.
Chapter 7 Full-State Feedback Digital Control by LQG 7.1 Introduction Meeting the world’s growing demand for energy is a challenge that requires heavy investment in power sources that minimize related impacts on the environment. Renewables, particularly wind power, have an important part to play in widening the diversity of the energy mix. However, with high wind penetration levels, the need for grid operators to quickly assess the impacts of the wind generation on system stability has become critical. With regard to power production, industry has been shifting toward variable speed WECSs as they encounter lower mechanical stress, less power fluctuations, and provide 10–15% higher energy output compared with constant speed operation [1],[2]. However, variable speed WECs present nonlinear dynamic behavior and lightly damped resonant modes. When the frequency range of the disturbances matches one of the resonant modes, the life of the turbine components is reduced, and the generated power quality is deteriorated [3]. A sophisticated control strategy incorporating a standard baseline controller and the LQG — a multi-objective, full state feedback with state estimation scheme — for generator torque control is proposed to meet the following objectives: 1. Ensure operation geared toward optimal power conversion 2. Ensure system reliability by enhancing reduction of stresses on the drive-train; achieved by regulating large torque variations at the shaft to avoid damage to mechanical subsystems. In the above rated wind speeds, the LQG’s main purpose is to add damping to the drive-train, thereby minimizing cyclic fatigue, while a pitch control mechanism prevents rotor overspeed thus ensuring the maximum power constraint is respected. However, the power generated may change rapidly due to continuous fluctuation of wind speed and direction: the baseline controller tracks wind speed variations with the target of optimizing aerodynamic efficiency during below rated wind speed events.
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The firt part of this chapter explores the LQG [4],[5] as a control scheme for WECS. The acronym refers to Linear Quadratic Guassian — Linear systems with Quadratic performance criteria that include Gaussian white noise in the LQ paradigm. By including Gaussian white noise in the LQ paradigm linear optimal feedback systems based on output feedback rather than state feedback may be found. LQG design methods convert control system design problems to an optimization problem with quadratic time-domain performance criteria. Disturbances and measurement noise are modeled as stochastic processes. MIMO problems can be handled almost as easily as SISO problems. Several studies have shown the efficacy of LQG in WECS control [6]-[10]. A practical implementation is reported by Lescher et al. [11], where the LQG is incorporated in intelligent micro-sensors placed on the wind turbine blades and tower to monitor fatigue loads during above rated wind speed operation. The second part of this chapter proposes a hybrid control paradigm, as developed in [12],[13], to ensure maximum power capture and regulation of shaft load variations via generator torque control. The neurocontroller (NC) is introduced to work in tandem with the LQG since the turbine system is dynamically nonlinear. The scheme takes advantage of the qualities of the NC, made up of an artificial neural network (ANN). The basis for including the NC is influenced by two properties of ANNs: 1. Computational speed, and 2. Ability to learn and generalize even in cases where full information for the problem at hand is absent [14]. With either control strategy, the main control objective is the regulation of turbine speed. Other objectives include maintaining stable closed-loop behavior as well as enhancement of damping in various flexible modes of the turbine. Overall, the designed control scheme should achieve a trade-off between two contradictory demands: • maximization of energy capture from the wind by operating at the optimum power coefficient • alleviation of mechanical dynamical loads due to very lightly damped resonant modes of the system [15]. The controller utilizes feedback from just one output variable, generator speed, to achieve stability, performance, and robustness. State estimation is employed in modeling the unknown states to attain full-state feedback, a process undertaken by a Kalman filter [16]-[18]. The WECS is dynamically nonlinear; to aid in the design synthesis of the controllers and gain insight into approximate behavior of the WECS, the system has to be linearized as explained in Section 2.3. Once this is established, the generator torque line can be controlled by the LQG (or hybrid), and the generator loading of the WECS made to follow the desired optimum shaft power locus.
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Jt
θtg
J1
ωt
Dt
ω1
ωt
Kgr Γt
Turbine rotor
Kt
ω2
Γ1
J
2
Γ2
Dg
ωg
Γg
Kg
D
Jg
N gr Jt
Generator
(a) 3-inertia model
ωg Ke
Jg
(b) 2-inertia model
Figure 7.1: Dynamic drive-train equivalent system: rotating masses interlinked by a flexible shaft.
7.2 State Development for the Power-train Fig. 7.1 illustrates the multimass model of the drive-train, simplified to a spring-mass-damper mechanical representation. The moments of inertia of the shafts and the gearbox wheels can be neglected when assumed to be small compared with either J t or Jg . Thus there is justification for model reduction prior to realizing a simpler LQG controller design: the McMillan degree 1 should be minimal for practical implementation to avoid complex control laws. Further, external damping is assumed negligible, and the moments of inertia of the shafts and the gearbox wheels can be neglected because they are small compared with that of the wind turbine or generator. Therefore the resultant model is essentially a two-mass system connected by a flexible shaft of equivalent stiffness and damping factor (Fig. 7.1(b)). Only the gearbox ratio has influence on the new equivalent system. Generally, the drive-train modifies the dynamics of the system because they include torsional modes that relate to the aerodynamic rotor mass swinging with the induction generator mass through the flexible transmission shaft. In the event that a strong gust is experienced, the system would be subjected to an instantaneous speed change, ∆ω t . The dynamics of the drive train are dωt = Γt − Γd dt dωg Jg = Γd − Γg . dt Jt
(7.1) (7.2)
The low speed shaft torque, Γd , acts as braking torque on the rotor; it results from the torsion and friction effects due to the difference between ωt and ωg and may be modelled to represent the torsional moments that relate to the cyclic twist of the shaft during operation Γd = Ke (θt − θg ) + D 1
(ω˙ t − ω˙ g )
This is the model order, and refers to the dimension of the state vector.
(7.3)
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where D represents the damping index and Ke is the equivalent shaft compliance, given by 1 1 1 = + 2 Ke Kt /Ngr Kg
(7.4)
and further, from (7.3) and Fig. 7.1(b), θtg = (θt − θg ),
dθt = ωt , dt
and
dθg = ωg dt
(7.5)
where θt , θg are the angular positions of the shaft at the rotor and generator sides. In the analysis, Γ d is the torsional torque experienced by the flexible shaft that couples the two rotating inertias. The linearized model locally valid around the OP may be developed on an equivalent mathematical state-space representation of the form x˙ = Ai ∆x + B∆u + Bw ξ y = Cx
(7.6)
where x ∈ N is a vector consisting of the system states, u ∈ M represents the command signals, ξ(≡ mw ) ∈ O is the disturbance input vector, A ∈ N ×N is the system matrix while the inputs affect the state dynamics through the control input gain distribution matrix B ∈ N ×M , and Bw ∈ N ×O is the disturbance input matrix. The output variable y ∈ P , which is the measured output (generator speed), is constructed from the states and the inputs through matrix C ∈ P ×N . Model orders are defined in {M, N, O, P }. Note that friction of the shaft at the rotor and generator sides is implied in D, since the elasticity and damping elements between the adjacent inertias correspond to the low- and high-speed shaft elasticities and internal friction, respectively. The vector x ∈ N in (7.6) consists of the system states defined respectively as follows: x1 is the perturbed turbine rotor speed, ∆ωt x2 is the perturbed generator speed, ∆ωg x3 is the perturbed shaft torsional torque, ∆Γ d x4 is the perturbed actuator pitch rate, ∆β, and x5 is the wind disturbance over the rotor disk, ∆v w . The states x1 –x5 are obtained from (7.1), (7.2), (7.3), (6.1), and (5.5) respectively. For each t ≥ 0 the state x(t) and input u(t) are dimensional vectors. The output y(t) is the controlled output. The signal ξ models the wind disturbances on the plant, and is a vector-valued Gaussian white noise process.
CHAPTER 7. FULL-STATE FEEDBACK DIGITAL CONTROL BY LQG
x(k-1)
x(k)
x’(k) Turbine dynamics
u(k-1)
Optimal state feedback
y’(k-1) y(k-1)
82
u(k)
Cost function
Correction
J=xT Px + uT Qu
State estimator
Figure 7.2: Schematic of the proposed LQG controller with state estimator.
7.3 LQG Controller Design 7.3.1 State Estimation and LQG Design Fig. 7.2 is a schematic of the proposed control paradigm. Unknown states are determined from just one measured variable — generator speed. In the figure, y represents the predicted measurements and x the predicted states. By estimating the aerodynamic torque, Γ t , from vw , ωref and β by the relation in (2.17), the state estimator makes a one-step-ahead prediction of the states, and a correction updates the state estimates, taking into account the prediction error. Thus xk+1 = xk+1 + M(yk − yk )
(7.7)
where, assuming the stochastic disturbances acting on the system are Gaussian, the matrix M is computed from the system dynamics and the disturbances, subject to minimization of the expected sum of squares of the prediction error, (yk − yk ). The initial state x(0) is assumed to be a random vector. At any time t the entire past measurement signal y(s), s ≤ t, is assumed to be available for feedback. The system states are generated using the estimated aerodynamic torque, Γt in (2.17) with respective aerodynamic coefficients kω , kv , kβ in (2.19)–(2.21). With generator speed being the only measurement, C = (0 1 0 0 0) in (7.6), thus the state-space mathematical equivalent becomes
ω˙ t
kω Jt
0
− J1t
kβ Jt
1 0 0 0 Jg Dkβ Dkω D D Γ˙ d = Ke + Jt −Ke −( Jt + Jg ) Jt 0 0 0 − τ1β β˙ ω˙ g
v˙ w
0
0
0
0
kv Jt
0 Dkv Jt
0 − τ1w
ωt
0
0
ωg − J1g 0 Γd + Ke 0 Jg β 0 − τ1β vw
0
0
Γg,ref + β cmd
0 0 0 0 − τ1w
ξ
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So, why the LQG? Generally, control that optimizes energy by tracking λ opt (as formulated in (6.6) and (6.7)) at high frequency can be undesirable a) torque fluctuations that result from the inertia of the rotor as the torque control attempts to follow the wind would induce sudden variations of rotor speed and thus high mechanical loads on the drive train b) the converter in the variable speed turbine neither adds inherent damping to the power system, nor is the turbine speed inherently damped by the power system. To address these issues, the proposed LQG seeks the gain that maximizes power capture regardless of whether this gain corresponds to the maximum of the power curve for the WECS. The LQG, designed using the state-space model, takes into account stochastic properties of the system disturbances. The problem of controlling the system is the stochastic linear regulator problem, and the target is to control the WECS plant from any initial state x(0) such that the output y(≡ ω g ) is regulated to the desired value as quickly as possible without making the input u(≡ Γ g,ref ) unduly large. To this end, the system is discretized and a performance index J introduced, with the following formulation: J =
∞ 0
∞ T E[x Qx(t) + u (t)Ru(t)]dt ≈ (t)RΓg,ref (t) x(t)T Qx(t) + Γg,ref T
T
(7.8)
t=0
where Q and R are symmetric weighting matrices, that is, Q = Q T and R = RT . The LQG is synthesized for each linearization point Si (xi ; ui), composed by the state estimator for linear system state vector estimation ∆x = (x − xi ) and by state feedback ∆u = G∆x . Static state feedback G is calculated in order to minimize the quadratic function J depending on control objectives, which are dependent on operating zone: I. when (vw ≤ vr ), the system has to operate at λ = λopt to extract the maximum of energy J =
∞
¯ 2 + q2 ∆Γ¯d (t)2 + r∆Γg,ref (t)2 q1 ∆λ(t)
(7.9)
t=0
where ∆λ(t) = λ(t) − λopt and ∆Γd (t) = Γd (t) − Γd,i . II. when (vw > vr ), the produced electric power has to be regulated to its nominal value J =
∞ q1 ∆P¯e (t)2 + q2 ∆Γ¯d (t)2 + r1 ∆Γg,ref (t)2 + r2 ∆βcmd (t)2 . t=0
(7.10)
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7.3.2 Choice of Weighting Matrices for LQG Cost Function, J Often it is adequate to let the two matrices simply be diagonal. The two terms, x T (t)Qx(t) and T (t)RΓg,ref (t), are quadratic forms in the components of the output, ω g , and the input signal, Γg,ref
Γg,ref , respectively. The first term in the integral criterion (7.8) measures the accumulated deviation of the states from their references. The second term measures the accumulated amplitude of the control input. It is most sensible to choose the weighting matrices Q and R such that the two terms are nonnegative, that is, to take Q and R nonnegative-definite 2 . If the matrices are diagonal then this means that their diagonal entries should be nonnegative. The cost function, J , has no physical significance; it provides a means to trade-off opposing objectives: state regulation versus control usage. The choice of the weighting matrices Q and R is a trade-off between control performance (Q large) and low input energy (R large). Increasing both Q and R by the same factor leaves the optimal solution invariant. Thus, only relative values are relevant. The Q and R parameters generally need to be tuned until satisfactory behavior is obtained. An initial guess is to choose both Q and R diagonal
Q1
0
0
0
0
0 Q2 0 0 0 = 0 0 0 Q3 0 Q 0 0 0 Q4 0 0 0 0 0 Q5 R1 0 = R 0 R2
(7.11)
(7.12)
where Q and R have positive diagonal entries such that
Qi =
1
, i = 1, 2, · · ·, m ; max
yi
Ri =
1 umax i
, i = 1, 2, · · ·, k
(7.13)
where the number yimax denotes the maximally acceptable deviation value for the ith component of has a similar meaning for the ith component of the input u. the output y. The other quantity u max i Starting with this initial guess the values of the diagonal entries of Q and R may be adjusted by systematic trial and error.
An n × n symmetric matrix R is nonnegative-definite if x T Rx ≥ 0 for every n-dimensional vector x. R is positivedefinite if xT Rx > 0 for all nonzero x. 2
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85
7.3.3 Solution of the Stochastic Linear Regulator Problem The stochastic linear regulator problem consists of minimizing J for the system in (7.6). For the case when there is no state noise (ξ = 0) and the state x(t) may be directly and accurately accessed for measurement, then for T → ∞ the performance index is minimized by the state feedback law Γg,ref (t) = −Gx(t)
(7.14)
with the G being the k × n state feedback gain matrix (k=1, n=5), given by G = R−1 B T X
(7.15)
and the matrix X is the nonnegative-definite solution of the algebraic Riccati equation (ARE) [19],[20] AT X + XA + D T QD − XBR−1 B T X = 0.
(7.16)
However, for the WECS under consideration, white noise disturbance ξ is present, but some of the states cannot be accessed for measurement, but may be optimally estimated with the help of the Kalman filter. Then the solution of the stochastic linear regulator problem with output feedback (rather than state feedback) is to replace the state x(t) in the state feedback law (7.14) with the estimated state xˆ(t). Thus, the optimal controller is given by , xˆ˙ = Aˆ x(t) + BΓg,ref (t) + K ωg (t) − C xˆ(t) x(t). Γg,ref (t) = −Gˆ
(7.17)
The controller minimizes the steady-state mean square error , T (t)RΓg,ref (t) limT →∞ E ωgT (t)Qωg (t) + Γg,ref
(7.18)
under output feedback. The signal xˆ is meant to be an estimate of the state x(t). It satisfies the , state differential equation of the system (7.6) with an additional input term K ωg (t) − C xˆ on the right-hand side. K is the observer gain matrix that needs to be suitably chosen. The observation error ˆ g (t) = C xˆ(t) ωg (t)−C xˆ(t) is the difference between the actual measured output ωg (t) and the output ω , as reconstructed from the estimated state xˆ(t). The extra input term K ωg (t) − C xˆ(t) on the righthand side of (7.17) provides a correction that is active as soon as the observation error is nonzero.
CHAPTER 7. FULL-STATE FEEDBACK DIGITAL CONTROL BY LQG
Γg,ref
_
x Gain
86
ωg
WECS Plant
Observer
Figure 7.3: Observer based feedback control. Figure 7.3 shows the arrangement of the closed-loop system. By certainty equivalence (using the estimated state as if it were the actual state), state estimation is divorced from control input selection, which in effect is the separation principle. The closed-loop system that results from interconnecting the plant (7.6) with the compensator (7.17) is stable, and may be recognized as follows. By connecting the observer xˆ˙ = Aˆ x(t) + BΓg,ref (t) + K[ωg (t) − C xˆ(t)], t ∈
(7.19)
to the noisy system in (7.6), then differentiation of e(t) = xˆ(t) − x(t) leads to the error differential equation e(t) ˙ = (A − KC)e(t) − Bw ξ(t), t ∈ .
(7.20)
Substitution of u(t) = −Gˆ x(t) into x(t) ˙ = Ax(t)+Bu(t)+B w ξ(t) yields with the further substitution xˆ(t) = x(t) + e(t) x˙ = (A − BG)x(t) − BGe(t) + Bw ξ(t).
(7.21)
Together with (7.20) then
x(t) ˙ e(t) ˙
=
A − BG
−BG
0
A − KC
x(t) e(t)
+
Bw ξ(t) −Bw ξ(t)
(7.22)
The eigenvalues of this system are the eigenvalues of the closed-loop system. Inspection shows that these eigenvalues consist of the eigenvalues of A − BG (the regulator poles) together with the eigenvalues of A − KC (the observer poles). If the plant (7.6) has order n then the compensator also has order n. Hence, there are 2n closed-loop poles.
CHAPTER 7. FULL-STATE FEEDBACK DIGITAL CONTROL BY LQG Augmented system
X
ref
LQG
+
GA
NC
uNC
Plant
Drive-train damper
uLQ
Hidden layer, H j Input layer, I i
DT loads
Γg, ref
Drive-train dynamics
+
∆ωt ∆ωg
ωg
Wind speed model
Output layer, O l
∆Γd Γt
Pitch controller
87
WECS aerodynamics
uNC
∆β
wi j
∆vw
Other loads
β cmd
wj i
(a) Simulation block diagram for the hybrid control scheme
(b) Feedforward ANN architecture
Figure 7.4: Hybrid control scheme illustrating the augmented LQG with NC.
7.4 Hybrid Controller Design The main goals of the control system are to control the power interchange within the WECS system, and accommodating the fluctuations in wind speed for reliability, by controlling large torque variations at the shaft to avoid damage to mechanical subsystems. Instabilities would be obtained at high wind speeds if the only controller utilized is a linear one. By introducing a hybrid control system comprised by a LQG and a neurocontroller (NC) acting in tandem, the nonlinearities in the system are handled by the latter. Fig. 7.4(a) shows the simulation block diagram. To ensure optimal operating conditions, the hybrid controller effects minimization of errors between actual and reference states, and outputs the generator torque command signal Γg,ref = uLQ + uN C
(7.23)
where uLQ is the control contribution by the LQG and u N C is the NC control component. GA denotes the genetic algorithm procedure that serves to train the NC.
7.4.1 NC Architecture In its formulation, the NC is constructed from artificial neural network (ANN) units — a radial-basis feedforward neural network whose hidden layer is nonlinear whereas the output layer is linear. A relatively compact design having a 4:5:1 configuration is employed, and its architecture is shown in Fig. 7.4(b). Defining ui (k) and IiI (k) as input and output of the ith input neuron at time k, H jI (k) as output of the jth neuron of the hidden layer at time k, O lI (k) as output of the lth neuron of the (1)
(2)
(3)
output layer at time k, f (x) as the activation function, and w ji , wj , wlj as the connection weights
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88
from the input layer to the hidden layer, between hidden layers and from hidden layer to output layer respectively, the structure of the 3-layered recurrent NN has the following mathematical description: (7.24) IiI (K) = ui(K), i = 1, 2, ..., nII I nI (1) (2) I wji ui (k) + wj HjI (k − 1) , j = 1, 2, ..., nIH (7.25) Hidden layer : Hj (k) = f Input layer :
i=1 nIH
Output layer : OlI (k) =
(3)
wlj HjI (k), l = 1, 2, ..., nIO
(7.26)
j=1
First Layer Consists of input nodes. Each neuron model receives 5 inputs: ∆ω t , ∆ωg , ∆Γd , ∆β, and ∆vw . Associated with each input are scalar weights wi (i = 1, 2, ..., n) that multiply the inputs, x i . Hidden layer Composed of the kernel nodes whose effective range is determined by their center and width. The argument of the activation function of each hidden unit computes the Euclidean distance between the input vector and the center of that unit. The combined inputs from the first layer are fed into an activation function of the second layer that produces the output, y, of the neuron: y=k
n
wi xi + b
(7.27)
i=1
where k is a logistic logarithmic function with sigmoidal nonlinearity, defined by fj (xj (n)) =
a , 1 + e−bxj (n)
{for − ∞ < xj (n) < ∞, and b > 0}
(7.28)
where xj (n) is the weighted sum of all synaptic inputs of neuron j, f j is the output of the neuron, with the gain a set to 1.0 and b = 1.0, both chosen by trial and error. Such an asymmetric activation function typically learns faster [14], and is differentiable everywhere. Third layer Consists of the output node that simply computes the weighted sum of the hidden node outputs. It is a linear mapping un = φxi where φ = 0.1 and x is the input vector.
(7.29)
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7.4.2 NC Training Training of the NC by the GA is undertaken during preprocessing. The weight vectors along the interconnection paths between layers are determined with an algorithm so that the signals are scaled down to the range of [0,1]. The real-coded GA ensures fast training with good representational accuracy, thereby yielding the desired input-output mapping. The evaluation function, E, of the NC represents the mean square errors between the WECS output and the reference values E=
2 Qi x(t) − x .(t)
(7.30)
i
where the factor Qi = 1.0, chosen by trial and error, denotes the weight associated with the squared error function and adjusts the importance of the control variables, x .(t) is the actual output and x(t) is the desired state variable. After the identifier neural network is trained, outputs of the nonlinear system are same as those of the NN when the plant is controlled. Adjustment of the connection weights for training the NN is as follows: w(k + 1) = w(k) − η(k)
∂E I (k) + α∆w(k) ∂w
(7.31)
nI
O 1 E I (k) = (x1 (k) − xˆl (k))2 2 l=1
∂E I (k) (3) ∂wlj
∂E I (k) (2) ∂wj
∂E I (k) (1) ∂wji
(7.32)
I
= −
nO
(xl (k) − xˆl (k))HjI (k)
(7.33)
l=1 I
= −
nO
(3)
(xl (k) − xˆl (k))wlj δj (k)
(7.34)
l=1 I
= −
nO
(3)
(xl (k) − xˆl (k))wlj βji (k)
(7.35)
l=1
(7.36) where w(k) and ∆w(k) are connection weights and change of connection weights at time k respectively, η and α are learning rate and momentum factor. Further, δj (0) = 0, and βji(0) = 0, where I nl (1) (2) (2) wji ui(k) + wj HjI (k − 1) (HjI (k − 1) + wj δj (k − 1)) δj (k) = f
(7.37)
i=1
I nl (1) (2) (2) wji ui(k) + wj HjI (k − 1) (ui (k) + wj βji (k − 1)). βji (k) = f i=1
(7.38)
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22
vw(t) [m/s]
20 18 16 14 12 10 0
5
10
15
20 t [s]
25
30
35
40
(a) Simulated wind speed 2
Γt [pu]
1.5 1 0.5
Γˆ t Γt
0 0
5
10
15
20 25 30 35 40 t [s] ˆt (b) Bold line shows the actual aerodynamic torque Γ t while the dotted line represents the estimated value Γ
Figure 7.5: Aerodynamic torque tracking with the proposed hybrid scheme.
7.5 Simulation Results 7.5.1 Tracking Performance by Proposed Technique Fig. 7.5(a) shows a wind profile generated for a 42-second simulation. The prevailing mean wind speed is 12.205 m/s under gusty conditions, with turbulence intensity of 19%. For the most part vw > vr and the target for the LQG controller is to mitigate against torsional loading on the drive train. Pitch control assures rated power. The philosophy of LQG control is ability to estimate plant states so as to generate the command signal necessary to compensate for parameter variations. This is the essence of turbine linearization about an OP. Fig. 7.5(b) shows good tracking performance of the aerodynamic torque as estimated by the Kalman filter. The apparent deviations from actual values may be explained as follows. The controller is designed using one set of gains appropriate for a particular wind speed and blade pitch angle. As the OP deviates from the design point of the WECS, the nonlinear aerodynamics of the turbine cause the estimator to get less accurate. However, overall, the estimation of Γ t is achieved with appreciably significant precision.
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22
vw(t) [m/s]
20 18 16 14 12 10 0
5
10
15
20 t [s]
25
30
35
40
(a) Simulated wind speed 25
λ β
TSR, Pitch angle
20 15 10 5 0 0
5
10
15
20 t [s]
25
30
35
40
35
40
(b) TSR (bold line) and pitch angle (dotted line) 0.5 0.4 cP
0.3 0.2 0.1 0 0
5
10
15
20 t [s]
25
30
(c) Coefficient of performance, c P (λ, β)
Power, [pu]
2 1.5 1 0.5 IG real power Aerodynamic power
0 0
5
10
15
20
25
30
35
40
t [s]
(d) IG real power (bold line) and aerodynamic power (dotted line) with the proposed method
Figure 7.6: Evolution of plant parameters for power conversion.
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22
vw(t) [m/s]
20 18 16 14 12 10 0
5
10
15
20 25 t [s] (a) Simulated wind speed
1.1
30
35
40
Proposed method LQG
1.08 Ve, [pu]
1.06 1.04 1.02 1 0.98 0.96 0
5
10
15
20 25 30 35 40 t [s] (b) Bold line shows Ve with the proposed hybrid controller while dotted line represents V e with LQG
Figure 7.7: Variation in phase voltage Ve with either controller for the 42 s simulation.
7.5.2 Optimization of Power Output It is observed from Fig. 7.6 that wind turbulence considerably affects the evolution of the various power parameters. With the progression of the wind speed beyond the rated value, power conversion has to be checked to avoid damage to mechanical subsystems, by systematically decrementing c P . From Fig. 7.6(b) it is seen that pitch angle β rises in direct relation to the wind speed, since the demanded pitch signal, βcmd , is large. This results in pitching the blades to regulate aerodynamic conversion. Further, increase in wind speed results in a decrease in the TSR (λ ∝ 1/vw ). Since both the TSR and β determine the value of cP , the overall effect is that as wind speed increases, the power coefficient is lowered appropriately, thereby limiting harvested aerodynamic power (Fig. 7.6(c)). The IG real power is maintained at a steady output (rated) value in wind speed regimes beyond nominal, as seen in Fig. 7.6(d), by the action of the pitch controller. It is noteworthy that above nominal wind speed, an initial mean wind speed value has to be given for the simulations since there is no unique relation between wind speed and generated power. This relates to the initialization procedure for above rated wind speed analysis.
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22
vw(t) [m/s]
20 18 16 14 12 10 0
5
10
15
20 t [s]
25
30
35
40
(a) Simulated wind speed 1
Γg,cmd, [pu]
0.5 0 -0.5 Proposed method LQG
-1 0
5
10
15
20 25 30 35 t [s] (b) Demanded generator torque command signal, Γ g,ref
40
1.5
Γd [pu]
1 0.5 0 -0.5 -1
Proposed method LQG
-1.5 0
5
10
15
20 25 t [s] (c) Drive-train torque, Γ d .
30
35
40
Figure 7.8: Variation in torque parameters with either controller for the 42 s simulation. Bold and dotted lines represent quantities under the proposed hybrid controller and LQG, respectively.
The phase voltage response is shown in Fig. 7.7(b). Though both controllers achieve steady V e over the simulation period, it is seen that in instances when v w < vr e.g. at 3.2 s from beginning of simulation, the system is destabilized somewhat, and the LQG takes longer to regain stability (large Ve fluctuations for longer). This is an instance of voltage recovery after a transient fault. As a whole it is seen that the proposed hybrid controller can enhance voltage transient stability of the wind turbine generator during high turbulence when wind fluctuations around v m are severe, and maintain the output voltage and power at rated levels when wind speed is over the rated speed.
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7.5.3 Minimization of Shaft Torsional Torque At high wind speeds the generator torque serves only to add damping to the drive train thus should be maintained at a fairly constant value to ensure rated power output. Fig. 7.8(b) shows that Γ g,ref is more steady with the proposed method relative to the LQG. This is attributable to the learning and generalization-ability of the neural network in the NC system. By use of Γ g,ref both the speed set-point and damping of excess aerodynamic torque are effected, alleviating undue loads on the shaft. Fig. 7.8(c) depicts the torsional torque variations experienced by the flexible shaft, with either controller. It can be seen that the objective of ensuring the drive train is cushioned from severe torque fluctuations is attained more readily with the proposed hybrid controller.
7.6 Conclusion The control objective aims to robustly stabilize the system while maintaining good disturbance attenuation and small tracking error despite actuator saturation. More specifically, the requirement is to design a controller to trade-off minimizing the control usage due to the penalty thereof, defined in the cost function, while also reducing the deviations from reference input ω g,ref i.e., tracking error e (tracking performance). This involves disturbance attenuation to guarantee robust stability. In this chapter a sophisticated control strategy is presented to compensate for the complicated effects of a stochastic operating environment and nonlinearities inherent in WECS dynamics that cause parametric uncertainties. To meet the objectives, the approach involves designing an adaptive controller and applying it to a performability model. The essence of the NC is to handle the nonlinearities in the system and alleviate part of the control load on the LQG. Pitch control determines the power coefficient, while the generator torque command is used to compensate for variations in parameters. Influence of the torsional dynamics on the grid through delivered active power is eliminated by using the generator torque command to achieve damping. The energy required to regulate these variations is exchanged with the turbine hub rather than the grid. Comparisons are made between the proposed controller and the LQG with regard to robustness to the evolution of plant parameters with changing operating conditions. By utilizing either control scheme, a trade-off is imposed between the contradictory objectives of maximizing energy capture from the wind and minimizing both the stress on the mechanical parts of the WECS and power fluctuations in the grid. Though both the LQG and the proposed hybrid controller show good conversion performance and robustness, simulation results validate the effectiveness of the latter scheme in satisfying both objectives relative to the former.
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References [1] Q. Wang, and L. Chang, “An intelligent maximum power extraction algorithm for inverter-based variable speed wind turbine systems,” IEEE Trans. Power Electronics, vol. 19, no. 5, pp. 12421249, 2004. [2] K. Tan, and S. Islam, “Optimum control strategies in energy conversion of PMSG wind turbine system without mechanical sensors,” IEEE Trans. Energy Conversion, vol. 19, no. 2, pp. 392-399, June 2004. [3] P. Novak, T. Ekelund, J. Jovik and B. Schmidtbauer, “Modeling and control of variable-speed wind turbine drive system dynamics,” IEEE Control Systems Magazine, vol. 15, no. 4, pp. 28-38, 1995. [4] J. M. Maciejowski, Multivariable Feedback Design, Reading, MA: Addison-Wesley Publishers Limited, 1990. [5] J. C. Doyle, and G. Stein, “Multivariable feedback design: concepts for a classical/modern synthesis,” IEEE Trans. Automatic Control, vol. AC-26, pp. 4-16, Feb. 1981. [6] E. B. Muhando, T. Senjyu, N. Urasaki, A. Yona, H. Kinjo, and T. Funabashi, “Gain scheduling control of variable speed WTG under widely varying turbulence loading,” Renewable Energy, vol. 32, no. 14, pp. 2407-2423. 2007. doi:10.1016/j.renene.2006.12.011. [7] A. Ben-Abdennour, K. Y. Lee, and R. M. Edwards, “Multivariable robust control of a power plant generator,” IEEE Trans. Energy Conversion, vol. 8, no. 1, pp. 123-129, 1993. [8] E. S. Abdin, and W. Xu, “Control design and dynamic performance analysis of a wind turbine induction generator unit,” IEEE Trans. Energy Conversion, vol. 15, no. 1, pp. 91-96, 2000. [9] C. Y. Kuo, C. L. Yang, and C. Margolin, “Optimal controller design for nonlinear systems,” IEE Proc. Control Theory and Applications, vol. 145, no. 1, pp. 97-105, 1998. [10] I. Munteanu, N. A. Cutululis, A. I. Bratcu, and E. Ceanga, “Optimization of variable speed wind power systems based on a LQG approach,” Control Engineering Practice, vol. 13, pp. 903-912, 2005. [11] F. Lescher, H. Camblong, R. Briand, and R. O. Curea, “Alleviation of wind turbines loads with a LQG controller associated to intelligent micro sensors,” Presented at the IEEE International Conference on Industrial Technology, 15-17 Dec. 2006. doi:10.1109/ICIT.2006.372245.
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[12] E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “Regulation of WTG dynamic response to parameter variations of analytic wind stochasticity,” Wind Energy, doi:10.1002/we.236. 2007. [13] E. B. Muhando, T. Senjyu, H. Kinjo, and T. Funabashi, “Augmented LQG controller for enhancement of online dynamic performance for WTG system,” Renewable Energy, doi:10.1016/j.renene.2007.12.001. [14] S. Haykin, Neural Networks: A Comprehensive Foundation, 2nd ed. Prentice Hall, 1999. ISBN:978-0132733502. [15] T. Petru, and T. Thiringer, “Modeling of wind turbines for power system studies,” IEEE Trans. Power Syst., vol. 17, no. 4, pp. 1132-1139, 2002. [16] R. E. Kalman, “A new approach to linear filtering and prediction problems,” ASME Trans. Journal of Basic Engineering, vol. 82, pp. 35-45, 1960. [17] Z. Xin-Fang, X. Da-Ping, and L. Yi-Bing, “Adaptive optimal fuzzy control for variable speed pitch wind turbines,” 5th World Congress on Intelligent Control and Automation, WCICA 2004, 15-19 June 2004, vol. 3, pp. 2481-2485. doi:10.1109/WCICA.2004.1342041. [18] L. Shuhui, D. C. Wunsch, E. O’Hair, and M. G. Giesselmann, “Wind turbine power estimation by neural networks with Kalman filter training on a SIMD parallel machine,” Int. Joint Conf. on Neural Networks, IJCNN ’99, 10-16 July 1999, vol. 5, pp. 3430-3434. doi:10.1109/IJCNN.1999.836215. [19] T. E. Duncan, L. Guo, and B. Pasik-Duncan, “Adaptive continuous-time linear quadratic Gaussian control,” IEEE Trans. Automatic Control, vol. 44, no. 9, pp. 1653-1662, Sept. 1999.doi:10.1109/9788532. [20] S. Boyd, and C. Barratt, Linear Controller Design: Limits of Performance, Prentice-Hall, 1991. ISBN-13:978-0135386873.
Chapter 8 Predictive Control I: STR 8.1 Introduction
R
ENEWABLE energy systems that take advantage of energy sources that will not diminish over time and are independent of fluctuations in price and availability are playing an ever-increasing
role in modern power systems. Low cost, plentiful, clean, and, in all other respects, “green” — these words describe wind power in a nutshell. To ensure smooth integration of the wind power into the grid, modern control techniques for WECS have become a prerequisite, often centred around various types of self-tuning control. Such adaptive control interfaces include Minimum Variance Control (MVC) [1]-[4], Generalized Minimum Variance (GMV) [3],[5]-[8], Pole Assignment (PA) [5],[9],[10], and optimal predictors [11]. MVC generally gives very lively control and can be highly sensitive to nonminimum phase plants. GMV, albeit more robust and generalized, is vulnerable to unknown or varying plant dead time and can have difficulty with dc offsets. PA aims to locate the closed-loop poles of the system at pre-specified locations leading to ‘smooth’ controllers, but the algorithm can show numerical sensitivity when the plant model is overparameterized. In this Chapter a self-tuning algorithm based on principles of Generalized Predictive Control (GPC) [12] has been selected. One of the most commonly used and well studied adaptive controllers is the Self-tuning regulator (STR) [13]–[15]. STRs consist of two parts: an estimator and a control law, which are usually invoked at every sample period. The most commonly used estimator in STR is recursive least-squares (RLS) [16],[17]. The purpose of this estimator is to dynamically estimate a model of the system relating the measured metrics with the actuation. The control law will then, based on this model, set the actuators such that the desired performance is achieved. The ability of the controller to achieve the performance goals is explicitly tied to how well the model represents the system at that instant. Use of STR for the adaptive control of WECSs has been shown to offer considerable promise [18]–[20].
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The development of a control system involves many tasks such as modeling, design of the control law, implementation, and validation. The STR attempts to automate several of these tasks. In the proposed paradigm, control is exercised through the STR that incorporates a recursive least squares (RLS) algorithm to predict the process parameters and update the states. The STR design is carried out as a nonlinear stochastic problem and is incorporated into the dynamical system, where the structure comes from tracking error. Variations in parameters are identified by a Kalman filter and their influence is compensated by generating a control signal to minimize output error. A gradient-based tuning algorithm guarantees the boundedness of all the closed-loop system signals. Motivation for the choice of STR is attributable to advances in microcomputer technology that has made more sophisticated algorithms feasible. The STR has been selected for control of generator torque of the WECS for these practical reasons: 1) Capability: can control difficult systems such as wind turbines without special adjustments. 2) Multi-step prediction: control signal can be influenced by future system output bounds, giving robustness. 3) Tuning knobs: enable customized performance and give flexibility. 4) Future target reference: adjustable setting to enhance control during scheduled changes. 5) Versatility: simple or complex controller structure as necessary. The RLS is one of the most widely used estimation algorithms in adaptive controllers due to its robustness against noise, and its proven convergence speed — factors elemental in effecting stability of the whole control loop. In its implementation, on-line recursive parameter estimation is employed to evaluate the time-varying or unknown parameters of a discrete time model of the WECS. Changes in the system dynamics are slow and the estimator should be and is able to track parameter variations well. Any concerns over the STR controller ranging too far can be met in the software by imposing limits or ‘jacketing’ the control. Below rated wind speeds, operation is executed on a trajectory that guarantees optimal energy conversion, by reference tracking. To prevent large torque and power peaks during high wind speeds, the turbine speed is regulated by the action of a pitch controller. In this case the output of the STR, Γg,ref , is used to add damping to the drive train torsional modes. In performing generator torque control, the scheme dictates the reference generator speed, which is a function of wind speed: below rated wind speed the reference generator speed is proportional to the wind speed, above, it is constant at rated value.
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PI Optimum characteristic
Pref
K pt+ K it
s
ωref + _
Γ g,ref
STR
ωg
Torque to current translation
i rq,ref + _ i rq
K pv+ K iv
s
Rotor injected voltage
u rq
Figure 8.1: Choice of STR or PI for generator torque control.
8.2 WECS Multi-objective Control Concept To achieve rotor speed and current control, the rotor side converter operates in a stator-flux oriented reference frame and executes control via the generator torque controller consisting of a cascade speed and torque control-loop, as shown in Fig. 8.1. In the simulations, generator torque control command, Γg,ref , is generated by the STR controller. Its performance in meeting the objectives of optimizing power conversion and alleviating power train loads is gauged against that of a PI set up in the same fashion. The inner loop is the torque control that compares the electric torque and the output signal from the speed controller. The speed controller compares the actual rotor speed and the reference rotor speed. The output signal from the cascade controllers is the q-axis rotor current. The d-,q-axis rotor currents are transformed to 3-phase currents prior to being applied to the rotor side converter. Current Control The reference stator current is calculated from the reference torque, Γg,ref , and is regulated by applying both d- and q-components; the reference values for the rotor current are calculated as irq,ref = Γg,ref ·
ω0 . Us
(8.1)
Generator Torque Control The output of the generator torque controller, Γg,ref , is used in determining the current in (8.1). This is accomplished by either control: I. PI controller — the PI regulator in Fig. 8.1 gives the relationship between the input, ∆ω g , and the output, Γg,ref
Γg,ref = Kpt ∆ωg + Kit
0
t
∆ωg dt.
(8.2)
where the proportional and integral gains are Kpt = 500 Nms/rad, and Kit = 250 Nm/rad. II. STR — this is explained in Section 8.3.
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Plant parameters, θ(k)
Optimal tracking controller design
RLS model Estimator
Controller parameters Target/Reference setting
+
_
Control signal Control law
Γg,ref (k)
WGS plant
System response g (k)
ω
SELF-TUNING REGULATOR
Figure 8.2: Self-tuning regulator block diagram.
8.3 STR Design and Implementation Fig. 8.1 shows the STR — a type of adaptive control system composed of two parts: an estimator and a control law. These constitute two loops that are executed to yield a generator torque command signal for stabilizing the WECS during operation under high turbulent inflow, viz. • an outer loop composed of a recursive parameter estimator and design calculations that adjusts the parameters of the controller, and • an inner loop that consists of the WECS plant and an ordinary linear feedback controller. An indirect adaptive algorithm is utilized for the overall execution of the WECS control in two steps: 1. estimate plant model parameters 2. update controller parameters as if estimates were correct (The Certainty Equivalence Principle)1 . Out of the several possible parameter estimation techniques, the RLS algorithm is selected to perform the above tasks; additionally, of the several possible controller design methods, a LQ tracking optimal control using state space models, is adopted. For the STR control, LQ tracking optimal control design employs the RLS algorithm based on an equivalent non-minimal state space realization of the WECS model as prior developed in Part I. Simulations assume the complete dynamic model is set on an equivalent mathematical state-space representation as x(k) ˙ = Ax(k) + Bu(k) + Bv w(k) and ωg (k) = Cx(k). Note that the state vector at time k is simply formed using past values of the input variables. No state observer is required. In contrast, if minimal state space realizations are used, then a state observer is usually required. Development and execution of the RLS algorithm and the control law are presented in Sections 8.3.1 and 8.3.2, respectively. 1
The outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more “uncertain” that characteristic is for the system — The Heisenberg Uncertainty Principle (HUP) in quantum physics.
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8.3.1 Outer Loop: Parameter Estimation The following notation is used in Fig. 8.1 for the least-squares estimation: Γ g,ref (k) is the vector of the M th actuator setting during sampling interval k, where Γg,ref (k) = [Γg,ref 1 (k), Γg,ref 2 (k) · ··, Γg,ref N ]T
(8.3)
and ωg (k) is the vector of the performance measurements of the N workloads, measured at the beginning of interval k ωg (k) = [ωg,1 (k), ωg,2 (k), · · ·, ωg,M ]T .
(8.4)
The relationship between ωg (k) and Γg,ref can be described by the following MIMO model:
ωg (k) =
N
Ai ωg (k − 1) +
i=1
N
Bj Γg,ref (k − 1).
(8.5)
i=1
Note that Ai ∈ N ×N , Bj ∈ N ×M , 0 < i ≤ n, 0 < j ≤ n, and {n = 8} ∈ N is the order of the model. This linear model is chosen for tractability since the relationship will indeed, in all but the most trivial cases, be nonlinear. However, it is a good local approximation of the nonlinear function and ample enough for the controller since it only makes small changes to the actuator settings. The plant model (8.5) can be written explicitly as ωg (k) = −a1 ωg (k − 1) − a2 ωg (k − 2) − · · · − an ωg (k − n) + b0 Γg,ref (k − d0 ) + · · · + Γg,ref (k − d0 − m).
(8.6)
where {m = 8} ∈ N is a system model order and d0 = 1 is the dead time. For notational convenience, the system model may take the following form, noting that this process is linear in the plant parameters: ωg (k) = ϕT (k − 1)θ(k − 1)
(8.7)
where T
ϕ (k − 1) = θT
− ωg (k − 1), · · ·, − ωg (k − n), Γg,ref (k − n), · · ·, Γg,ref (k − d0 − m) , = a1 , a2 , · · ·, an , b0 , · · ·, bm
with ϕ(k) being the regression vector and θ(k) the parameter matrix.
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The estimation block that utilizes the RLS algorithm is the heart of the STR. It recursively estimates the unknown process parameters for each measurement based on minimization of the leastsquare error. The whole RLS algorithm involves the following matrix computations: 1) new data ωg (k) and ϕ(k) are acquired, and the prediction error vector, ε(k), is computed from the old estimated parameter ˆ − 1) ε(k) = ωg (k) − ϕ(k)T θ(k
(8.8)
ˆ is calculated 2) new parameter θ(k) ˆ = θ(k ˆ − 1) + θ(k)
ϕ(k)T P (k − 1)ε(k) λ + ϕ(k)T P (k − 1)ϕ(k)
(8.9)
3) data in the covariance matrix, P (k), is updated for the next sample P (k − 1) ϕ(k)T P (k − 1)ϕ(k)P (k − 1) − P (k) = λ T λ 1 + ϕ(k) P (k − 1)ϕ(k)
(8.10)
ˆ = [ˆ ˆ2 , · · ·, ˆan, ˆb0 , · · ·, ˆbm] is the estimated process parameter vector, ε(k) ∈ N ×1 is the where θ(k) a1 , a ˆ error in predicting the signal ωg (k) one step ahead based on the estimate θ(k), P (k) ∈ N M n×N M n is the error covariance, k is an integer discrete time index, and λ is the forgetting factor: (0 < λ ≤ 1). A high forgetting factor means that RLS remembers a lot of old data when it computes the new model. Conversely, a low forgetting factor means that it largely ignores previous models and only focuses on producing a model from the last few samples. The intuition behind these equations is quite simple. (8.8) computes the error between the latest performance measurements and the performance prediction (8.7) of the model. This is the RLS error, ε(k). The model parameters are then adjusted in (8.9) according to the RLS error and another factor dependent on the covariance matrix P computed in (8.10). P contains the covariances between all the measurements and the actuators. The model θˆ is then used by the control law described in the next section to set the actuators correctly. The RLS algorithm enables the adaptive filter to find the filter coefficients that relate to producing, recursively, the least squares of the difference between the desired and actual signal. The benefits of using the RLS algorithm is that there is no need to invert extremely large matrices, thereby saving computation time, and that through it some intuition behind such results as the Kalman filter is gained. Note that the recursion for P follows a Riccati equation and thus draws parallels to the Kalman filter.
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In practice, implementation of plain RLS algorithms may lead to numerical problems, and a factorization of the covariance matrix should be considered to overcome this drawback. In calculating the parameter estimates, the stability of the RLS method is improved by means of LD decomposition (L = lower triangular matrix; D = diag(d1 , ..., dn )), while adaptation is supported by directional forgetting [21]. Since the task of recursive identification consists of searching for a parameter estimate vector θT (k) that minimizes the given criterion in (8.5), then the vector of the parameter estimates is computed according to the square root version (LD decomposition) of the recursive relations ˆ = θ(k ˆ − 1) + P (k − 1)ϕ(k − 1) εˆ(k − 1) θ(k) 1 + ζ(k − 1)
(8.11)
where ζ is an auxiliary scalar in step k such that ζ(k − 1) = ϕT (k − 1)P (k − 1)ϕ(k − 1). Case I:
If ζ(k−1) > 0, a rectangular covariance matrix is computed by the recurrent algorithm P (k) = P (k − 1) −
C(k − 1)ϕ(k − 1)ϕT (k − 1)P (k − 1) δ −1 (k − 1) + ζ(k − 1)
(8.12)
where δ(k − 1) = λ(k) −
1 − λ(k) ζ(k − 1)
with the adaptive directional forgetting factor, λ(k), computed at each sampling period. Case II:
When ζ(k − 1) = 0, then P (k) = P (k − 1).
(8.13)
The most complex situation is in the continuous-time domain since the straight differentiation of data must be avoided. The approach entails descretizing the continuous system to enable sampling and digital control analysis, thus the k th sampling (or at time kT ) represents a sample at the instant t in the continuous system. Owing to the unified approach of the control synthesis, the same set of control calculations can offer four types of control costs and two types of identified models. Each self-tuner of a given structure has two main “knobs” — the first is for model option (RLS applied to a regression (ARX) model), the second selects the performance criterion (LQ in this case).
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8.3.2 Inner Loop: Control Law, Γg,ref The concept behind the RLS filter is to minimize a weighted least squares error function in the relation between input and output {Γg,ref (k), ωg (k)} for k = 1...N based on the prediction error ε(k); this error function may be conceptualized as a performance index, P I(·) N N 2 1 1 2 |ε(k)| = ωg (k) − ϕT (k)θ . P I(θ) = N N k=1
(8.14)
k=1
From (8.7), for θ0 = (a1 , · · ·, an , b0 , · · ·, bm )T then P I(θ0 ) = 0,
∀ input, Γg,ref (k).
(8.15)
To stay within the adaptive filter terminology, this performance index is the Cost Function, J (k), that evaluates the performance of the control unit, and the task of the recursive identification consists of searching for a parameter estimate vector θˆT (k) that minimizes the criterion T J (θ) = ωg (k)T Qωg (k) + Γg,ref (k) − W Γg,ref (k − 1) R Γg,ref (k) − W Γg,ref (k − 1) (8.16) where Q is the weighting on the output, and R, W are the weights associated with the control signal. To minimize J (θ) along all possible trajectories of the system, a trade-off between control performance (Q large) and low input energy (R large) is desirable. The weight parameters are tuned until satisfactory behavior is obtained. As an initial guess, the two terms are chosen to be nonnegativedefinite, and their values adjusted by systematic trial and error. The proposed performance index also takes into account the prevention of excessive control. Due to the fact that the cost function is square in the parameter θ, then J (θ) ≥ 0 with only one minimum. Because J (θ0 ) = 0, this minimum is θ = θ 0 . The implication is that for suitable excitation of the system these parameters may be found by solution to the equations ∂ J (θ) = 0 ∂θ
(8.17)
with respect to θ. This defines the control law that governs the plant: Γg,ref (k) = − R+BiT QBi BiT Q NAi ωg (k+1−i)+ NBi (k+1−i) −RW Γg,ref (k−1) . i=1
i=2
(8.18)
Wind speed, Rotor speed
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22 20 18 16 14 12 10 8
Wind speed (m/s) Rotor speed (rpm)
0
10
20
30
40
50
60
Time, t [s]
(a) Nominal wind speed at hub position and rotor speed 15
Γg,ref [kNm]
14 13 12 11 10 9 0
10
20
30 Time, t, [s]
40
50
60
50
60
Pm [MW]
(b) Demanded generator torque, Γ g,ref 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 0
10
20
30 Time, t, [s]
40
(c) Shaft mechanical power
Figure 8.3: Evolution of control and controlled parameters.
8.4 Simulation Analysis 8.4.1 Control for Energy Extraction Fig. 8.3(a) shows the generated wind speed signal, as well as variation in rotor speed for the 1-minute simulation. Fig. 8.3(b) demonstrates, as expected, how the demanded generator torque is kept very nearly constant at above rated wind speeds so as to provide damping to the drive train. For above rated wind speeds the turbine operates at full load and the output electric power has to be regulated at nominal generator power. The inverter controller holds the electrical power constant, thus the turbine is prevented from following the cP,opt trajectory and constrained to operate at lower values of λ and cP . The turbine rotational speed is maintained around nominal generator speed and β is controlled in order to reduce cP (λ, β). Control is thus multivariable in this zone, because it acts on both generator torque and pitch angle. It is observed from Fig. 8.3(c) that the mechanical power extracted from the wind is successfully kept steady in cases of wind speeds above rated to guarantee the electric power output is kept within the allowable ±5% of the WECS’s rating.
Wind speed, vw [m/s]
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18 16 14 12 10 8 0
10
20
30
40
50
60
Time, t, [s]
Voltage [kV]
(a) Nominal wind speed at hub position, v m =12.205 m/s 11.002 11.0019 11.0018 11.0017 11.0016 11.0015 11.0014 11.0013 11.0012 0
10
20
30 Time, t, [s]
40
50
60
(b) Line voltage (rms)
Current [A]
120 110 100 90 80 70 0
10
20
30 Time, t, [s]
40
50
60
40
50
60
Reactive power, [VAr]
(c) WECS current 400 350 300 250 200 150 0
10
20
30 Time, t, [s]
(d) WECS reactive power
Figure 8.4: Electrical parameters.
With regard to network compliance, Figs. 8.4(b)–(d) depict variations in electrical parameters: the grid voltage, current and reactive power, respectively. It is observed from Fig. 8.4(b) that the network voltage is virtually undisturbed by variations in operating conditions. Fig. 8.4(c) shows variation in the rotor current; for the current loop, the active power control is achieved by controlling the q-axis component of the rotor current (in a stator flux dq reference frame). The assumption that the dc link voltage remains constant is valid if the dc link capacitor and converters are designed to enable continued operation of the DOIG with low generator busbar voltages caused by close-up faults. The available reactive power, shown in Fig. 8.4(d), depends on the active power. The fast-acting reactive power control (applied through either converter) improves the stability of the generator.
Wind speed, vw [m/s]
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18 16 14 12 10 8 0
10
20
30 Time, t, [s]
40
50
60
40
50
60
40
50
60
50
60
(a) Nominal wind speed at hub position, v w 1.4
Γt [MNm]
1.3 1.2 1.1 1 0.9 0.8 0
10
20
30 Time, t, [s]
(b) Aerodynamic torque, Γ t 1.4
Γd [MNm]
1.3 1.2 1.1 1 0.9 0.8 0
10
20
30 Time, t, [s]
(c) Low speed shaft torque, Γ d 15
Γg [kNm]
14 13 12 11 10 9 0
10
20
30 Time, t, [s]
40
(d) Generator torque, Γ g
Figure 8.5: Variation in various WECS torques.
8.4.2 Control for Load Alleviation Figs. 8.5(b)–(d) show evolution of various torques during the 1-minute simulation. Fig. 8.5(b) gives the corresponding aerodynamic torque developed by the turbine that is a function of v w . Fig. 8.5(c) depicts the low speed shaft torque, Γd , that acts as a braking torque on the rotor. It results from the torsional and frictional effects due to the difference between ωt and ωg and is modelled to represent the torsional moments that relate to the cyclic twist of the shaft during operation. It is seen that at above rated wind speeds, severe shaft torsional moments that may cause mechanical stress and strain are prevented by reducing vibrations between the rotating parts. The control is very effective in maintaining generator torque control — it is observed from Fig. 8.5(d) that the generator torque does not exceed Γg,max despite the turbulence that occassionally drives the wind speed above rated.
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8.5 Conclusion The two core problems that face wind energy conversion systems today include grid integration issues and reliability of the turbine structure, both attributable to the stochasticity of the wind. This study is set against the background of need for modern digital controls to ensure optimum power conversion in all operating ranges as well as alleviation of drive train loads that occur due to highly turbulent wind environments that cause cyclic fatigue on the mechanical components. A self-tuning regulator is proposed for the coordinated active power control and shaft torsional moments reduction for a variable speed WECS that is incorporated into the grid. In implementing the control topology, the considered constraints imposed on the control input signal are the rate, amplitude and energy types. The control philosophy of the proposed paradigm relies on feedback, including state estimation to approximate unmeasured plant states using the single turbine parameter, ω g , to significantly enhance dynamic compensation and response of the closed loop system. For the nonlinear WECS system, the fundamental concept of feedback is tremendously compelling as it enhances stability, improves the steady-state error characteristics, enables state estimation for unmodelled states, and provides disturbance rejection due to a stochastic wind. The appeal of the proposed STR is that the RLS algorithm is easy to implement and does not require massive processing power. Once designed, execution of the STR is reduced to a set of difference equations connecting the measured outputs, ω g , to the new control signals, Γg,ref . The power of the mathematical model lies in the fact that it can be simulated in hypothetical situations, be subject to highly stochastic states due to large turbulence that would be dangerous in reality, and it can be used as a basis for synthesizing controllers. With regard to drive-train load mitigation capability, the STR performs well in attenuating drivetrain vibrational magnitudes at the rated wind speed. Effectiveness of the generator torque control is gauged on capacity to improve damping for suppression of torsional vibration. Relative to the classical PI controller, the STR control scheme shows considerable improvement in achieving the dual objectives of maximization of energy capture and regulation of torsional dynamics under turbulent wind conditions, and also guarantees that uncertainties in the WECS and wind models are explicitly taken into account, resulting in a reduction in pitch activity.
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References [1] T. Senjyu, R. Sakamoto, N. Urasaki, T. Funabashi, H. Fujita, and H. Sekine, “Output power leveling of wind turbine generator for all operating regions by pitch angle control,” IEEE Trans. Energy Conversion, vol. 21, no. 2, pp. 467-475, 2006. doi:10.1109/TEC.2006.874253. [2] R. Sakamoto, T. Senjyu, T. Kinjo, N. Urasaki, and T. Funabashi, “Output power leveling of wind turbine generator by pitch angle control using adaptive control method,” Int. Conf. on Power System Technology, PowerCon 2004, 21-24 Nov. 2004, vol. 1, pp. 834-839. doi:10.1109/ICPST.2004.1460109. [3] H. S. Ko, T. Niimura, and K. Y. Lee, “An intelligent controller for a remote wind-diesel power system – design and dynamic performance analysis,” IEEE Power Engineering Society General Meeting, 13-17 July 2003, vol. 4. doi:10.1109/PES.2003.1270948. [4] F. Jurado, and J. R. Saenz, “Adaptive control for biomass-based diesel-wind system,” 11th IEEE Mediterranean Electrotechnical Conf., MELECON 2002, 7-9 May 2002, pp. 55-60. doi:10.1109/MELECON.2002.1014529. [5] G. Fusco, and M. Russo, “Generalized minimum variance implicit self-tuning nodal voltage regulation in power systems with pole-assignment technique,” 9th IEEE Int. Conf. on Control, Automation, Robotics and Vision, ICARCV ’06, 5-8 Dec 2006, pp. 1-6. doi:10.1109/ICARCV.2006.345452. [6] W. T. Chung, and A. K. David, “Digital and laboratory implementation of a generalized minimum variance controller for an HVDC link,” IEE Procs. Generation, Transmission and Distribution, vol. 146, no. 2, pp. 181-185, Mar 1999. doi:10.1049/ip-gtd.19990039. [7] W. Gu, and K. E. Bollinger, “A self-tuning power system stabilizer for wide range synchronous generator operation,” IEEE Trans. Power Systems, vol. 4, no. 3, pp. 1191-1199, Aug. 1989. doi:10.1109/59.32617. [8] J. Y. Fan, T. H. Ortmeyer, and R. Mukundan, “Power system stability improvement with multivariable self-tuning control,” IEEE Trans. Power Systems, vol. 5, no. 1, pp. 227-234, Feb. 1990. doi:10.1109/59.49110. [9] H. Camblong, G. Tapia, and M. Rodriguez, “Robust digital control of a wind turbine for ratedspeed and variable-power operation regime,” IEE Procs. Control Theory and Applications, vol. 153, no. 1, pp. 81-91, Jan. 2006. doi:10.1049/ip-cta:20045190.
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[10] C. Woei-Luen, and H. Yuan-Yih, “Controller design for an induction generator driven by a variable-speed wind turbine,” IEEE Trans. Energy Conversion, vol. 21, no. 3, pp. 625-635, Sept. 2006. doi:10.1109/TEC.2006.875478. [11] A. Chandra, K. K. Wong, O. P. Malik, and G. S. Hope, “Implementation and test results of a generalized self-tuning excitation controller,” IEEE Trans. Energy Conversion, vol. 6, no. 1, pp. 186-192, Mar 1991. doi:10.1109/60.73806. [12] D. W. Clarke, C. Mohtadi, and P. S. Tuffs, “Generalized predictive control, the basic algorithm, and extensions and interpretations,” Automatica, vol. 23, no. 2, pp. 137-160, 1987. ˚ om, and B. Wittenmark, Adaptive Control. Electrical Engineering: Control Engineer[13] K. J. Astr¨ ing, 2 ed., Reading, MA: Addison-Wesley, 1995. ISBN 0-201-55866-1. [14] R. Bitmead, M. Gevers, and V. Wertz, Adaptive Optimal Control. The Thinking Man’s GPC, Englewood Cliffs, NJ: Prentice-Hall, 1990. [15] P. E. Welstead, and M. B. Zarrop, Self-Tuning Systems: Control and Signal Processing, Chichester: Wiley, 1991. [16] M. Honig, and D. Messerschmitt, Adaptive Filters: Structures, Algorithms, and Applications, Hingham, MA: Kluwer Academic Publishers, 1984. ISBN0-898-38163-0. [17] E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “Disturbance rejection by dual pitch control and self-tuning regulator for wind turbine generator parametric uncertainty compensation,” IET Procs. Control Theory and Applications, vol. 1, no. 5, pp. 1431-1440, 2007. doi:10.1049/iet-cta:20060448. [18] E. B. Muhando, T. Senjyu, N. Urasaki, A. Yona, and T. Funabashi, “Robust predictive control of variable-speed wind turbine generator by self-tuning regulator,” IEEE Power Engineering Society General Meeting, 24-28 June, 2007, pp. 1-8. doi:10.1109/PES.2007.385885. [19] W. Ren, and P. R. Kumar, “Stochastic adaptive prediction and model reference control,” IEEE Trans. Automatic Control, vol. AC-30, pp. 2047-2060, 1994. [20] M.-u.-D. Mufti, R. Balasubramanian, and S. C. Tripathy, “Self tuning control of wind-diesel power systems,” Procs. of the 1996 Int. Conf. on Powewr Electronics, Drives and Energy Systems for Industrial Growth, 8-11 Jan. 1996, vol. 1, pp. 258-264. doi:10.1109/PEDES.1996.539550. [21] R. Kulharv´ y, “Restricted exponential forgetting in real time identification,” Automatica, vol. 23, no. 5, pp. 586-600, 1987.
Chapter 9 Predictive Control II: MBPC 9.1 Introduction
I
F the Earth is choking on greenhouse gases, it is not hard to see why. Global carbon dioxide output approached a staggering 32 billion tons in 2006. Turning off the carbon spigot is the
first step, and many of the solutions are familiar: windmills, solar panels, nuclear plants. All three technologies are part of the energy mix, although each has its issues, including noise from windmills and radioactive waste from nukes. Greenhouse-gas-induced global-warming worries are not the only reasons to consider a power-grid shift to wind power. With fossil-fuel prices on the rise and their supply increasingly unstable, the need for more environmentally benign electric power systems is a critical part of the new thrust of engineering for sustainability. Wind turbines have become the most cost-effective renewable energy systems available today and are now completely competitive with essentially all conventional generation systems. Wind plants have benefited from steady advances in technology, and much of the advance has been made in the components dealing with the utility interface, the electrical machine, the power electronic converter, and the control capability [1]. However, the wind’s unpredictable nature forces utility operators to think differently about power generation, and the main challenge is to provide governor functions and controlled ramp-down during high wind speed events. Wind stochasticity results in fluctuations in output power as well as undesirable dynamic loading of the drive-train during high turbulence. The only advanced control methodology that has made a significant impact on industrial control engineering is Model Predictive Control (MPC) [2]-[6]. The main reasons for this success in applications include: • ability to handle multivariable control problems naturally, • capacity to account for actuator limitations, and • allows operation closer to constraints that frequently lead to more profitable operation.
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Model predictive control, in the form of Generalized Predictive Control (GPC), was first proposed by Clark et al. [7],[8] and the properties of GPC are further presented for a set of continuous linear systems [9] and nonlinear problems [10]. Several researchers have reported the potential of GPC for WECS control [11]-[13]. Nonetheless, these references apply predictive control for energy extraction maximization, but do not consider the cyclic loading impact on the drive train due to high turbulence. This chapter develops the Model-Based Predictive Control (MBPC) strategy [14]-[17] for current and speed control of the field oriented induction machine drive as well as regulation of drive train shaft torsional moments reduction through generator torque control. The idea of MBPC is based on computing a control function for the future time in order to force the controlled system response to attain the reference value. An optimization process generates the control sequence, and the system response is based on future control action, model parameters, and the actual system state. A criterion to regulate the costing horizon of the MBPC is defined in the form of minimizing a quadratic cost function. In its implementation, the plant is dynamically decoupled from the stiff grid frequency since the mechanical dynamics are slower than the electrical ones. To execute the control, MBPC requires an equivalent model defined in state space for online estimation and prediction of future states, including disturbances. The proposed controller associates the predictive control action and ensures the smooth transition of control from region to region. MBPC provides a systematic procedure for dealing with constraints (both input and state) in MIMO control problems, and is widely used in industry. Remarkable properties of the method include global asymptotic stability provided certain conditions are satisfied (e.g. appropriate weighting on the final state). A remarkable property of MBPC is that stability of the resultant feedback system (at least with full state information) can be established. This is made possible by the fact that the value function of the optimal control problem acts as a Lyapunov function for the closed loop system. The key elements in the design of the MBPC digital system for WECS control are: • state space (or equivalent) model (developed in Part I), • on-line state estimation (including disturbances), • prediction of future states (including disturbances), • implementation of first step of control sequence, and • on-line optimization of future trajectory subject to constraints using Quadratic Programming. Simulations are conducted using the MATLAB/Simulink software to validate the MBPC technique vis-`a-vis the classical PI controller. Computer simulations reveal that achieving the two objectives of maximizing energy extraction and load reduction by the proposed control paradigm becomes more attractive relative to the classical linear controller.
CHAPTER 9. PREDICTIVE CONTROL II: MBPC Optimum characteristic
ωg
ωref
Pref
_
113
Above rated
PI Pitch controller MBPC
+ Below rated
PI
βcmd Γref Γref
Figure 9.1: Pitch regulated variable speed WECS speed control loops.
9.2 Control Concept for Power Regulation Fig. 9.1 shows the overall control loops for the WECS. Control action to achieve both objectives of conversion performance as well as drive-train load mitigation throughout the operating envelope is undertaken by two controllers: generator torque control and pitch angle control, as follows: • At low and moderate wind speeds, the rotor speed is controlled to maximize energy capture by operating continuously at the TSR that results in the maximum power coefficient. This is achieved by regulating the generator torque via the torque reference, Γg,ref , given by the proposed MBPC. The target is to track the trajectory with cP,opt. • When vw > vr , the rotor collective pitch controller kicks in and generates a pitch signal, β cmd , for adjusting blade pitch to regulate c P (λ, β), thereby ensuring rated power output: Pe = Pr . In this region generator torque control serves only to add damping to the drive train. The proposed control scheme first derives Pref throughout the operating region, and then the corresponding generator torque control, Γg,ref , and pitch command, βcmd , are computed to follow the actual power along the Pref trajectory. Such regional control aims to maintain the desired power command, Pref , at various wind speeds in different wind regimes by the control expression P if P1 ≤ Pr 1 Pref =
& (vw < vc )
P2 if P2 ≤ Pr & (vc ≤ vw ≤ vr ) P if P > P & (v > v ) r 1,2 r w r
(9.1)
where Pr is rated power, vc , vr are cut-in and rated wind speeds, P1 is the maximum power command calculated for the impressed wind speed in region 1 (safe operating region), and P 2 is the optimum power command in region 2 of operation, at which priority is given to power system stability rather than to producing maximum wind energy conversion. When v w > vr , the target is rated power output.
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114
@time=t k
Take process measurements
WECS plant model = Current & future
Objectives
Future plant outputs Constraints
Control actions Disturbances
Solve above optimization problem Best current and future control actions Implement best current control action @time=t k+1
Figure 9.2: MBPC scheme.
9.3 Generator Torque Control 9.3.1 Γg,ref by MBPC The flow-chart in Fig. 9.2 illustrates the concept of MBPC applied to the WECS model. It utilizes a control algorithm founded on solving an online optimal control problem, with the objective of determining the control function for the future time in order to constrain the WECS response to attain the reference values, which are known. A receding horizon approach is used, which involves the following control algorithm: (i) At time k and for the current states xi (k), solve, online, an open-loop control problem over some future interval taking into account the current and future constraints. (ii) Apply the first step in the optimal control sequence. (iii) Repeat the procedure at time (k + 1) using the current states, xi (k + 1). The solution is converted into a closed loop strategy by using the measured value of x i (k) as the current state. In this case only the generator speed is measurable; the rest of the states are obtained by an observer (state estimation), and the closed loop policy is obtained by replacing the respective states by the estimates.
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x - predicted state _ x - set-point x - actual future state
x - past state
Prediction horizon
Measured past k
Unpredicted future k+N
Figure 9.3: Receding horizon control principle.
MBPC is based on iterative, finite horizon optimization of a plant model. At time k the current plant state is sampled and a cost minimizing control strategy is computed (via a numerical minimization algorithm) for a relatively short time horizon in the future: [k, k + N]. Specifically, an online calculation is used to explore state trajectories that emanate from the current state and find (via the solution of Euler-Lagrange equations) a cost-minimizing control strategy until time k, k + N. Only the first step of the control strategy is implemented, then the plant state is sampled again and the calculations are repeated starting from the now current state, yielding a new control and new predicted state path. The prediction horizon keeps being shifted forward and for this reason MPC is thus called receding horizon control. Fig. 9.3 demonstrates the receding horizon control principle, where the reference values are assumed to be a constant sequence while the system response is based on future control action, model parameters, and the actual system state. MBPC algorithms have a greater “prediction horizon”, characterized by an explicit model of the controlled system, which can be identified separately. The model is used for precalculating the future behavior of the controlled system as well as for the selection of optimal control values. With respect to the wide prediction horizon MBPC algorithms need more calculation power relative to standard linear controls. The calculation performance of microcontrollers, however, has been increasing steadily. Although this approach is not optimal, in practice it has given very good results. Fig. 9.4 defines the fundamental principles of the proposed control scheme. MBPC is a multivariable control algorithm that uses: ◦ an internal dynamic model of the plant ◦ a history of past control moves, and ◦ an optimization cost function J over the prediction horizon, to calculate the optimum control moves.
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Cost function Constraints
Future error
w
+ _
Future reference value
Total response
Optimizer dJ(Γg,ref,w,...) dΓg,ref
WECS plant
ωg(=y)
Γg,ref (=u)
Free response
ωg (k-i k) Γg,ref (k-i k)
Model Predictor
ωg +
Model
Forced response
Figure 9.4: Structure of MBPC as applied to the WECS.
For a traditional MBPC formulation, consider a SISO plant with input u(≡ Γ g,ref ), and output y(≡ ωg ). The MBPC online optimization problem is developed as follows. At time k find
min u[k|k],···,u[k+p−1|k]
p
y[k + i|k] − y
i=1
SP
+
m
ri ∆u[k + i − 1|k]2
(9.2)
i=1
subject to umax ≥ u[k + i − 1|k] ≥ umin , i = 1, · · ·, m
(9.3)
umax ≥ u[k + i − 1|k] ≥ − ∆umax , i = 1, · · ·, m
(9.4)
ymax ≥ y[k + i − 1|k] ≥ ymin , i = 1, · · ·, p
(9.5)
where p and m < p are the lengths of the plant output prediction and manipulated plant input horizons respectively, and u[k + i − 1|k], for i = 1, · · ·, p, is the set of future plant input values with respect to which the optimization will be performed, where u[k + i − 1|k] = u[k + m − 1|k], i = m, · · ·, p − 1
(9.6)
and y SP is the set-point, while ∆ is the backward difference operator, i.e. ∆u[k + i − 1|k] ≈ u[k + i − 1|k] − u[k + m − 2|k].
(9.7)
In typical MPC fashion, the above optimization problem is solved at time k, and the optimal input u[k] = uopt [k|k] is applied to the plant. This procedure is repeated at subsequent times k+1, k+2, etc. It is clear that the above problem formulation necessitates the prediction of future outputs y[k + i|k]. This, in turn, makes necessary the use of a model for the plant and external disturbances. Assuming
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the following finite-impulse-response (FIR) model describes the dynamics of the controlled plant:
y[k] =
n
hi u[k − j] + d[k]
(9.8)
j=1
where hi are the model coefficients (convolution kernel) and d is the disturbance. Then y[k + i|k] =
n
hj u[k + i − j|k] + d[k + i|k]
(9.9)
j=1
where u[k + i − j|k] = u[k + i − j], i − j < 0.
(9.10)
The prediction of the future disturbance d[k + i|k] clearly can be neither certain nor exact. An approximation or simplification has to be employed, such as d[k + i|k] = d[k|k] = y[k] −
n
hj u[k − j]
(9.11)
j=1
where y[k] is the measured value of the plant output y at sampling point k and u[k − j] are past values of the process input u. Substitution of (9.9) to (9.11) into (9.2) to (9.5) yields p
min u[k|k],···,u[k+p−1|k]
i=1
n j=1
hj u[k + i − j|k] −
n
2 hj u[k − j] + y[k] − y
SP
j=1
+
m
ri ∆u[k + i − 1|k]2
i=1
(9.12) subject to umax ≥ u[k + i − 1|k] ≥ umin , i = 1, · · ·, m
(9.13)
∆umax ≥ ∆u[k + i − 1|k] ≥ − ∆umax , i = 1, · · ·, m (9.14) n n hj u[k + i − j|k] − hj u[k − j] + y[k] ≥ ymin , i = 1, · · ·, p (9.15) ymax ≥ j=1
j=1
The above optimization problem is a quadratic programming problem that can easily be solved at each time k. The elaborate models developed for the drive train and electrical system in Part I are appropriate to ensure the system behavior is calculable. For the WECS plant model, x( + 1) = f (x(), Γg,ref ()), x(k) = x
(9.16)
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the MBPC at event (x, k) is computed by solving the constrained optimal control problem: PN (x) : VNo (x) = min VN (x, U)
(9.17)
U = {Γg,ref (k), Γg,ref (k + 1), ..., Γg,ref (k + N − 1)} k+N −1 L x(), Γg,ref () + F (x(k + N)) VN (x, U) =
(9.18)
U ∈uN
where
(9.19)
=k
and UN is the set of U that satisfy the constraints over the entire interval [k, k + N − 1], i.e. Γg,ref () ∈ U x() ∈ X
= k, k + 1, ..., k + N + 1
(9.20)
= k, k + 1, ..., k + N
(9.21)
together with the terminal constraint x(k + N) ∈ W.
(9.22)
Here, U ⊂ m is convex and compact, X ⊂ n is convex and closed, and the set W is appropriately selected to achieve stability. With the constraint (9.17), the model and cost function are time invariant, thus a time-invariant feedback control law is obtained by setting U = {Γg,ref (0), Γg,ref (1), ..., Γg,ref (N − 1)} N −1 & ' VN (x, U) = L x(), Γg,ref () + F x(N)
(9.23) (9.24)
=0
Uxo = {uox (0), uox (1), ..., uox (N − 1)}
(9.25)
then, the actual control applied at time k is the first element of this sequence, i.e. u = uox (0)
where u Γg,ref .
(9.26)
The predictive control law in (9.26) generates a control sequence that forces the future system response to be equal to the reference values. Expression (9.26) is a necessary condition for optimality: Theorem 1 (Optimality Principle Bellman) For the above problem if {u(t) = u o(t), t ∈ [to , tf ]} is the optimal solution, then u o (t) is also the optimal solution over the (sub)interval [t o + ∆t, tf ], where to < to + ∆t < tf . Proof: See [18].
2
The essence is that any part of an optimal trajectory is necessarily optimal in its own right [19],[20].
CHAPTER 9. PREDICTIVE CONTROL II: MBPC Optimum characteristic
PI
ωref
Pref
119
+
ωg
K pt+ K it
Γ g,ref
s
_
Torque to current translation
i rq,ref
Figure 9.5: Rotor speed and active power control by PI.
9.3.2 Γg,ref by PI In the steady state the generator torque is set to be proportional to ω g2, and thus the required generator torque demand, Γg,ref , is obtained thus Γg,ref = Kopt ωg2
(9.27)
where Kopt is the optimal mode gain Kopt = πρR5
cP (λ, β) 3 Ngr . 2λ3
(9.28)
Thus below and around rated wind speeds, the variable speed WECS tries to stay at the desired TSR wherever possible by tracking wind disturbances, achieved by using generator torque control for regulating rotor speed in proportion to the wind speed. This maximizes c P (λ, β) and hence the aerodynamic power available. Note that in steady state conditions, energy output may not necessarily be maximized by maximizing aerodynamic efficiency (tracking optimum TSR, λ opt ) since the energy losses may also vary with the operating point (OP). It is therefore better to track a slightly different TSR that yields cP,opt (λ, β), by computing Kopt . The generator torque controller in Fig. 9.5 (shown as dotted block in Fig. 9.1) is a PI regulator that gives the relationship between the input, ∆ω g , and the output, Γg,ref via the transfer function C(s) C(s) =
Kit Γg,ref = Kpt + ∆ωg s
(9.29)
where the proportional and integral gains of the PI controller are respectively: K pt = 500 Nms/rad, and Kit = 250 Nm/rad. PI tuning involves the long process of carefully adjusting the gains through several simulations by trial and error in order to obtain minimum variations for the controlled variables. The reference generator speed is a function of wind speed: below rated wind speed it is proportional to the wind speed, above, it is constant at rated value.
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Wind speed [m/s]
20 18 16 14 12 10 8 0
20
40
60
80
100
120
$\beta$ [deg], $\dot\beta$ [deg/s]
Time, t, [s]
(a) Simulated wind speed, v m = 12.205 m/s 8
Pitch angle Pitch rate
6 4 2 0 -2 -4 -6 0
20
40
60
80
100
120
Time, t, [s]
(b) Variation in pitch angle control variables, β ref and β˙ Generator speed [rpm]
1560 1540 1520 1500 1480 1460 1440 0
20
40
60
80
100
120
100
120
Time, t, [s]
(c) Generator rotor speed
Power [MW]
2.2 2 1.8 1.6 1.4
Electrical Power Mechanical Power
1.2 0
20
40
60 Time, t, [s]
80
(d) Aerodynamical power, P m , and electrical power, P e
Figure 9.6: Evolution of power parameters at rated speed (vr = 12.205 m/s).
9.4 Simulation Analysis In the sequel the behavior of the output variables of the WECS system i.e. active and reactive powers injected into the utility grid, and low speed shaft torque variations, in response to variations of the input variables, is presented. In executing the MBPC, gain-scheduling is carried out to compensate for non-linearities of the WECS characteristics. The effectiveness of the paradigm is validated against two criteria: comparison in performance with a classical PI controller, and manufacturer’s data that has been tested in the field environment. The wind model provides a wind profile for the rated wind speed of 12.205 m/s with prevailing turbulence of 16%, as shown in Fig. 9.6(a).
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9.4.1 Aerodynamic Power Production The proposed MBPC controller associates the predictive control action and ensures the smooth transition of control from region to region. In low to medium speed winds the controller regulates the TSR for optimal power extraction. At above rated wind speeds, the predictive controller first derives the desired power command, and the corresponding pitch command is computed to follow the actual power along the trajectory of the desired power command. 9.4.1.1 Power Optimization Strategy (vw ≤ vr ) The energy capture is maximized by tracking the maximum power coefficient: ◦ the power reference is the wind turbine available power; ◦ the speed reference is the optimal speed. The turbine has to produce the optimum power corresponding to the maximum tracking power point. Thus the speed controller keeps the pitch angle constant to its optimal value, while the TSR is driven to its optimal value by varying the rotational speed. Hence the demanded pitch angle command signal, βcmd , is kept at –2◦ as seen in Fig. 9.6(b). The mechanical power, Pm , extracted from the wind and the corresponding output electrical power, Pe , are shown in Fig. 9.6(d). 9.4.1.2 Power Limitation Strategy (vw > vr ) The controllers limit Pe and speed to the rated values of the WECS, thus ◦ the power reference is the rated power; ◦ the speed reference is the rated speed. The speed controller keeps the generator speed, ωg , limited to its rated value, ω r , by acting the pitch angle — β is driven to positive values so as to keep the generator speed around the rated value of 1500 rpm, as shown in Fig. 9.6(c). During instances when v w > vr , the WECS has to produce less than it is capable of at a given wind speed. This action implies both a larger dynamical pitch activity and a larger steady-state pitch angle of the wind turbine. This is observed in Fig. 9.6(b), around the 75 th to the 95th seconds of simulation. A reduction of the power conversion when the mean wind speed is over rated speed implies an increase in the demanded pitch angle. Fig. 9.6(d) illustrates the steady state power curves at high wind speeds for the 2 MW wind turbine. It is easily noticed that at any given moment when vw > vr , the deviation in power production from the optimum is satisfactorily small, |Pe − Pr | < 5%. It is noteworthy that pitch rates are kept within ±8 ◦ /s, thereby lessening pitch activity despite meeting control objectives (Fig. 9.6(b)), and that ω g is kept within ±3% of rated rpm of 1500 (Fig. 9.6(c)).
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Wind speed [m/s]
20 18 16 14 12 10 8 0
20
40
60
80
100
120
Time, t, [s]
Aerodynamic torque [MNm]
(a) Simulated wind speed at v m = 12.205 m/s 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0
20
40
60
80
100
120
100
120
100
120
Time, t, [s]
(b) Aerodynamic torque, Γ t Generator torque [kNm]
15 14 13 12 11 10
Γg,ref Γg
9 8 0
20
40
60
80
Time, t, [s]
Low speed shaft torque [MNm]
(c) Generator torques, Γ g,ref and Γg 1.3 1.2 1.1 1 0.9 0.8 0.7 0
20
40
60 Time, t, [s]
80
(d) Variation in low-speed shaft torque, Γ d
Figure 9.7: Development of torques during 120 s simulation.
9.4.2 Drive-train Torque Variation Minimization Fig. 9.7(a) shows the simulated turbulent wind speed at a mean of 12.205 m/s under a prevailing turbulence intensity of 16%. Despite the high fluctuations in the generated aerodynamic torque as shown in Fig. 9.7(b), MBPC yields a good tracking of the reference generator torque, Γg,ref , as seen in Fig. 9.7(c). More importantly, Fig. 9.7(d) confirms that variations in the drive-train torque, Γ d , are kept to a minimum to ensure undue cyclic loads are not experienced. Essentially, by implementing the MBPC scheme, generator torque control and current control are used to limit shaft moments thereby put in check the cyclic fatigue stresses that may ensue thereof.
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123
Wind speed [m/s]
20 18 16 14 12 10 8 0
20
40
60
80
100
120
Time, t, [s]
(a) Simulated wind speed at v m = 12.205 m/s Drive train torque [MNm]
1.3 1.2 1.1 1 0.9 0.8 MBPC PID 0.7 0
20
40
60
80
100
120
80
100
120
Time, t, [s]
Generator power loss [kW]
(b) Drive train torque 120 110 100 90 80 70
MBPC PID 0
20
40
60 Time, t, [s]
(c) Generator power loss
Figure 9.8: Comparison of performance in power conversion and alleviation of drive train loads by MBPC (red line) and classical PID (green line).
9.4.3 Comparison: MBPC and Classical PID The main objective of the generator torque controller in the above rated region is to enhance damping in the first drive train torsional mode. This ensures a smooth transition when gusts are experienced, to avoid exciting flexible turbine modes that increase dynamic loads. Fig. 9.8 serves to validate the choice of the proposed MBPC when a comparison is made between the MBPC scheme and the PID with regard to performance in meeting the objectives: • During the two-minute simulation it is seen that the PID shows relatively higher fluctuations in the drive train torque as opposed to the MBPC (Fig. 9.8(b)). This is attributable to the fact that the PID is a linear controller and is unable to handle the nonlinear WECS dynamics fully, especially in regions of high wind speeds. • The generated power is compromised by using the PID. It can be observed from Fig. 9.8(c) that the generator power loss is higher with the linear PID than the MBPC, especially at critical instances (vw < vr ), when the objective is energy conversion maximization.
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Wind speed [m/s]
20 18 16 14 12 10 8 0
20
40
60
80
100
120
Time, t, [s]
Voltage [kV]
(a) Simulated wind speed at v m = 12.205 m/s 11.002 11.0019 11.0018 11.0017 11.0016 11.0015 11.0014 11.0013 11.0012 0
20
40
60
80
100
120
Time, t, [s]
(b) Voltage at connection point (line, rms)
Current [A]
110 100 90 80 70 0
20
40
60
80
100
120
80
100
120
Time, t, [s]
(c) Current Reactive power [VAr]
400 350 300 250 200 150 0
20
40
60 Time, t, [s]
(d) Reactive power
Figure 9.9: Variation in electrical parameters.
9.4.4 Evolution of Electrical Parameters Figs. 9.9(b)–(d) show, respectively, the stator voltage, current, and reactive power for simulation at the mean wind speed of 12.205 m/s. The power controller ensures the power reference by acting on the current reference of the rotor current controller and thus on the generator current/torque. This is achieved via two control loops: one for the active power control and the other for reactive power control. The active power control is achieved by controlling the q-axis component of the rotor current (in a stator flux dq reference frame), while the reactive power control is achieved by controlling the d-axis component of the rotor current (the magnetizing current) collinear with the stator flux. The rotor current controller generates rotor voltage components as control variables of the converter.
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9.5 Conclusion MBPC is a powerful methodology for solving challenging control problems, particularly the ones where prescribed point-wise-in-time input and/or state constraints have to be satisfied. It relies on a feedback-control methodology suitable to enforce efficiently hard constraints on the variables of the controlled WECS system. It is shown that the method hinges upon a constrained open-loop optimal control problem along with the adoption of the so-called receding-horizon control strategy. Being a predictive control paradigm based on iterative, finite horizon optimization whose key elements include a state space (or equivalent) model and online state estimation and prediction of future states (including disturbances), MBPC is utilized successfully in meeting the control objectives of energy maximization as well as regulation of drive-train torsional moments. The control sequence is computed by solving online, over a finite control horizon, an open-loop optimal control problem, given the WECS plant dynamical model and current state. Though this computation relies upon an open-loop control problem, MBPC yields a feedback-control action. Indeed, in a discrete-time setting only the first control of the open-loop control sequence is applied to the plant, and, according to the receding horizon control policy, the whole optimization cycle is repeated at the subsequent time-instant based on the new plant-state. Because it involves a control horizon made up by only a finite number of time-steps, MBPC can be often calculated online by existing optimization routines so as to minimize a performance index in the presence of hard constraints on the time evolutions of input and/or state. MBPC’s ability of handling constraints is of paramount importance whenever constraints are part of the control design specifications. In fact, constraints are typically present in WECS dynamics, as they stem from actuators’ saturations and/or physical, safety or economical requirements. The main reason for the interest of control engineers in MBPC is therefore its ability to systematically and effectively handle hard constraints. The nonlinear WECS model may be linearized to derive a Kalman filter or specify a model for linear MBPC. The time derivatives may be set to zero (steady state) for applications of real-time optimization or data reconciliation. Alternatively, the nonlinear model may be used directly in nonlinear MBPC and nonlinear estimation (e.g. moving horizon estimation). Though MBPC has been applied effectively in the chemical and process industries, there is great potential for its integration in the control modules of modern wind turbines.
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References [1] J. C. Smith, “Winds of Change: Issues in utility wind integration,” IEEE Power & Energy Magazine, vol. 3, no. 6, pp. 20-25, Nov-Dec. 2005. [2] J. M. Maciejowski, Predictive Control with Constraints, Harlow, England: Prentice Hall, 2002. [3] D. Q. Mayne, and H. Michalska, “Receding horizon control of nonlinear systems,” IEEE Trans. Automatic Control, vol. 35, pp. 631-643, 1990. [4] C. E. Garcia, D. M. Prett, and M. Morari, “Model predictive control: theory and practice –a survey,” Automatica, vol. 25, pp. 335-348, 1989. [5] W. H. Kwon, A. M. Bruckstein, and T. Kailath, “Stabilizing state feedback design via the moving horizon method,” Int. Journal of Control, vol. 37, pp. 631-643, 1983. [6] J. Qin, and T. Badgwell, “An overview of industrial model predictive control technology,” 5th Int. Conf. on Chemical Process Control, AIChE Symposium Series, J. C. Kantor, C. E. Garcia, and B. Carnahan (Editors), vol. 93, pp. 232-256, 1997. [7] D. W. Clark, C. Mohtadi, and P. S.Tuffs, “Generalized predictive control I: the basic algorithm”, Automatica, vol. 23, no. 2, pp. 137-148, 1987. [8] ——, “Generalized predictive control II: extensions and interpretations”, Automatica, vol. 23, no. 2, pp. 149-160, 1987. [9] A. W. Pike, M. J. Grimble, M. A. Johnson, A. W. Ordys, and S. Shakoor, “Predictive control,” Chapter 51, pp. 805-814 in The Control Handbook, W. S. Levine (ed.), CRC and IEEE Press, 1996. [10] J. B. Rawlings, “Tutorial overview of model predictive control,” IEEE Control Systems Magazine, pp. 35-52, June 2000. [11] L. Lavoie, and P. Lautier, “Nonlinear predictive power controller with constraint for a wind turbine system,” IEEE Int. Symposium on Industrial Electronics, 9-13 July 2006, vol. 1, pp. 124-129. [12] T. Senjyu, R. Sakamoto, N. Urasaki, T. Funabashi, H. Fujita, and H. Sekine, “Output power leveling of wind turbine generator for all operating regions by pitch angle control,” IEEE Trans. Energy Conversion, vol. 21, no. 2, pp. 467-475, June 2006.
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[13] X. Zhang, D. Xu, and Y. Liu, “Predictive functional control of a doubly fed induction generator for variable speed wind turbines,” Proc. IEEE-WCICA 2004 Fifth World Congress on Intelligent Control and Automation, 15-19 June 2004, vol. 4, pp. 3315-3319. [14] E. B. Muhando, T. Senjyu, and H. Kinjo, “Disturbance rejection by stochastic inequality constrained closed-loop model-based predictive control of MW-class wind generating system,” Proceedings of the Joint IEEJ-IEICE Conference, 19 Dec. 2007, pp. 91-99. [15] E. Camponogara, D. Jia, B. H. Krogh, and S. Talukdar, “Distributed model predictive control,” IEEE Control Systems Magazine, vol. 22, no. 1, pp. 44-52, Feb. 2002. doi:10.1109/37.980246. [16] E. Gallestey, A. Stothert, M. Antoine, and S. Morton, “Model predictive control and the optimization of power plant load while considering lifetime consumption,” IEEE Trans. Power Systems, vol. 17, no. 1, pp. 186-191, Feb. 2002. doi:10.1109/59.982212. [17] M. Larsson, and D. Karlsson, “Coordinated system protection scheme against voltage collapse using heuristic search and predictive control,” IEEE Trans. Power Systems, vol. 18, no. 3, pp. 1001-1006, Aug. 2003. doi:10.1109/TPWRS.2003.814852. [18] R. Bellman, “On the theory of dynamic programming,” Proc. National Academy of Sciences of the United States of America, vol. 38, pp. 716-719, 1952. [19] C. D. Johnson, “Beyond Bellman’s principle of optimality: the principle of ‘real-time optimality’ RTO,” Proc. 37th Southeastern Symposium on System Theory, SSST’05, 20-22 March, 2005. pp. 326-335, DOI: 10.1109/SSST.2005.1460931. [20] M. Sniedovich, “A new look at Bellman’s principle of optimality,” Journal of Optimization Theory and Applications, vol. 49, no. 1, pp. 161-176, 1986. DOI: 10.1007/BF00939252.
Chapter 10 Analysis, Perspectives, and Conclusions 10.1 Preamble
T
HE debate on whether climate change is real is virtually over — the facts are in: the Earth is heating up. Building on action taken by signatories of the Kyoto Protocol 1 , the Intergov-
ernmental Panel on Climate Change (IPCC) estimates that this century, global temperatures will rise between 1.8◦ C and 4◦ C, while some predictions for the next century are sobering, projecting this figure to as much as 6.4◦ C. Evidence of climate change is already here. A warmer and less stable climate has potential for massive ecological and economic challenges. Unpredictable weather and natural disasters — drought, floods, hurricanes and heat waves — are becoming more common. The year 2007’s World Environment Day highlighted the consequences of the melting of the polar ice caps, in an attempt to give a human face to environmental degradation. Commitment to the Kyoto Protocol by some of the greatest GHGs emitters has been lukewarm. The US is a party to the UN Framework Convention on Climate Change, whose Kyoto Protocol — a European Union-led effort to reduce greenhouse gas emissions, primarily carbon dioxide associated with global warming — has been reluctant to ratify the Kyoto Protocol, because it would unfairly hurt the American economy 2 . The stalemate notwithstanding, most countries have turned to renewables to meet electricity de1
According to a press release from the United Nations Environment Programme: “The Kyoto Protocol is an agreement under which industrialized countries will reduce their collective emissions of greenhouse gases by 5.2% compared to the year 1990” (but note that, compared to the emissions levels that would be expected by 2010 without the Protocol, this limitation represents a 29% cut). The goal is to lower overall emissions of six greenhouse gases - carbon dioxide, methane, nitrous oxide, sulfur hexafluoride, hydrofluorocarbons, and perfluorocarbons - averaged over the period of 2008-2012. National limitations range from 8% reductions for the European Union and some others to 7% for the US, 6% for Japan, 0% for Russia, and permitted increases of 8% for Australia and 10% for Iceland. 2 An independent analysis showed that compliance with Kyoto would dramatically increase energy costs, substantially reduce GDP growth and force energy-intensive industries overseas, taking with them 4 million jobs. As a result, Kyoto would lead to an increase in associated emissions in countries without obligations under the treaty, such as China and India, and therefore produce no real environmental benefit. Remarks atributed to Mr Michael Ranneberger, US ambassador in Nairobi, and reported in the Kenyan newspaper, Daily Nation, of 26 May 2007.
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mand, and wind power is among the preferred sources of ‘green’ energy. Strong growth figures prove that wind is now a mainstream option for new power generation. Wherever the wind speed exceeds approximately 6 m/s there are possibilities for exploiting it economically, depending on the costs of competing power sources. Numerous utility studies have shown that a unit of wind energy saves a unit of energy generated from coal, gas or oil — depending on the utility’s plant — thereby saving emissions of greenhouse gases, pollutants and waste products. The exact amount of emissions saved depends on which fossil plants are displaced by wind energy. Energy has since been established as a fundamental ingredient of socio-economic development and economic growth. Renewable energy sources like wind energy are indigenous and can help in reducing the dependency on fossil fuels. It has been estimated that roughly 10 million MW of energy are continuously available in the earth’s wind. Wind energy provides a variable and environmental friendly option and national energy security at a time when decreasing global reserves of fossil fuels threatens the long-term sustainability of the global economy. The contribution of wind energy in the global energy mix has been steadily increasing, thanks in part to remarkable advances in the wind power design that has been achieved due to modern technological developments. The wind turbine technology has evolved into a unique technical identity to meet unique demands in terms of the methods used for design. Growth in size and the optimization of WECSs has enabled wind energy to become increasingly competitive with conventional energy sources. However, these developments raise a number of challenges. The penetration of wind energy in the grid raises questions about the compatibility of the wind turbine power production with the grid. In particular, the contribution to grid stability, power quality and behavior during fault situations plays therefore as important a role as the reliability. Regarding installation and O&M costs, it is claimed that a wind turbine used for electricity generation will repay the energy used in its manufacture within 6–9 months of its operation. Further, a modern wind turbine operates for about 13 years in a design life of 20 and is almost always unattended. Development of advanced power electronic components is integral to providing industry with the support it needs to develop technologies capable of cost-effective operation. It is against this backdrop that the research presented in this thesis explores advanced schemes for control of wind power plants with regard to: 1. optimized power output, and 2. reliability assurance. The methodology entailed modeling the various subsystems to derive mathematical state-space representations for multiobjective controller design.
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10.2 Modeling: an Overview I. Wind Speed Model for Simulations Being the stock-in-trade for the WECS, real-time wind speed has been simulated by a reliable CSS method, taking into consideration the annual average as well as the turbulence spectrum. The mean wind speed is obtained from an annual Rayleigh probability density curve and the turbulence component modelled as an instantaneous variation. The problem is complicated and further work will undoubtedly be required before it will be possible to formulate reliable guidelines to assist the wind turbine designer. Turbulence models of the form described above are now widely used for the calculation of fatigue loads for design purposes. For calculation of extreme loads, however, it is standard practice to base calculations on deterministic descriptions of extreme wind conditions. Current design standards and certification rules specify extreme events in terms of discrete gusts, wind direction changes and wind shear transients. The form, amplitude and time period specified for these discrete events remain rather arbitrary and largely unvalidated. The development of more reliable methods for the evaluation of extreme design loads, based possibly on the use of probabilistic analysis, requires considerable effort but is crucially important in the context of refining wind turbine design analysis.
II. Drive-train Model With larger or more flexible wind turbine structures, other issues of concern during variable-speed operation are drive train dynamics and avoiding operation at system resonant frequencies. Moreover, system losses must be dealt with. These can be incorporated with modification to the control methodology. Blade pitch can also be used to improve energy capture when the turbine is operating at large errors in power output. The improvement in energy capture from these methods depends on the turbine and operating environment. However, use of variable-speed control increases the fluctuation of output power and somewhat increases the shaft fatigue cycles. These issues must be weighed against the increase in power output obtained from use. Drive train dynamics, system losses, and avoiding resonant frequencies can be incorporated using proper control system implementation, by modifying the reference value for the aerodynamic torque near the resonant rotor speed. The proposed control schemes (LQG, LQG/NC, STR, and MBPC) consistently show lower fluctuations in shaft torsional torque when compared with the classical PI(D). These advanced paradigms are a better alternative for the wind turbine industry.
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Phase Voltage, [pu]
1.5 1 0.5 0 -0.5 -1 -1.5 0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
0.6
0.8
1
0.6
0.8
1
t, [s]
(a) Phase voltage Phase current, [pu]
1.5 1 0.5 0 -0.5 -1 -1.5 0
0.2
0.4 t, [s]
Speed, [pu]
(b) Phase current 1.16 1.15 1.14 1.13 1.12 1.11 1.1 1.09 1.08 1.07 1.06 0
0.2
0.4 t, [s]
(c) Rotor speed 1.2
Power, [pu]
1 0.8 0.6 0.4 0.2 0
0.2
0.4 t, [s]
(d) Power output
Figure 10.1: DOIG single phase fault: results of the WECS system.
III. DOIG Model: Fault Current Contribution and Post-Fault Behavior Fig. 10.1 shows the DOIG response to a single phase fault introduced at t = 0 s. The stator current, speed and power output during the fault and after the fault is cleared is shown in Figs. 10.1(b)–(d). The crowbar is designed to be triggered by the high rotor current when the fault is cleared. However, normal operation of the machine is maintained once the fault clears, as seen in Fig. 10.1(b). All the parameters (phase current, rotor speed and power) stabilize within ample time, meaning that the power grid will still maintain transient voltage stability, attesting to the effectiveness of the control in re-establishment after a short circuit fault. The assumption is that both converters continue to operate normally during and after the fault. MBPC was utilized in generator torque control.
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1.2 Voltage, [pu]
1 0.8 0.6 0.4 0.2 0 -0.2 0
0.5
1 t, [s]
1.5
2
(a) Phase voltage 5 Current, [pu]
4 3 2 1 0 0
0.5
1 t, [s]
1.5
2
(b) Current phasor 1.25
Speed, [pu]
1.2 1.15 1.1 1.05 1 0
0.5
1
1.5
2
1.5
2
1.5
2
t, [s]
PQ control, P
(c) Generator speed 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0
0.5
1 t, [s]
PQ control, Q
(d) PQ control: active power, P 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 0
0.5
1 t, [s]
(e) PQ control: reactive power, Q
Figure 10.2: 3P fault: RMS simulation.
Similar results are obtained for a 3-phase fault, in Figs. 10.2(a)–(d). When the fault is applied, the DOIG model shows a high current peak but the decay is rapid. When the fault clears the over-current protection operates the crowbar circuit. During the fault, the speed of the generator is maintained close to its prefault value and returns to normal operation. The simulation results demonstrate the importance of the control system in limiting the generator current perturbations during a fault.
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10.3 WECS Modeling: Assessment of Approach and Validation Limitations of Presented Modeling Concepts Even using the medley of subcomponent models as presented in Part I, there are bound to be significant differences with expected results, suggesting unresolved deficiencies in the models, inconsistencies in empirical input parameters to the models, and coupling issues between the subcomponent models cannot be discounted. Other shortcomings include: 1. Use of lower fidelity (reduced order) models – in a model-based control approach, increased model accuracy implies reduced uncertainty. High-fidelity physical models for WECS are usually developed during the design process of these components, but their dimensionality is excessive for current control architectures. As a consequence, model-based controls typically recourse to lower fidelity models that penalize the achievable performance. A systematic methodology for obtaining reduced order models directly from the design models as presented in this thesis reduce the development cycle for high performance model-based controllers. 2. Neglecting tower shadow modeling – changes in pitch also have a major effect on the thrust load, which in turn drives the fore-aft motion of the tower. This is turn affects the relative wind speed seen by the blades, which then feeds back into the pitch control via the aerodynamic torque — a strong feedback which has a major effect on the stability of the pitch control system. While this has been neglected in the analysis, the tower shadow involves an unsteady wake structure, and in a time-averaged sense it can be represented as a velocity deficit in the flow behind the support tower. Therefore, to a first level of approximation, it can be modelled as a spatial variation in the flow velocity normal to the chord of the blade section. The emphasis is on two key areas that need continued serious consideration in WECS modeling: • the representation of the rotor wake using dynamic inflow and vortex methods, and • the representation of the unsteady aerodynamics of the blade sections. Despite their limitations, the presented dynamic inflow models have attractive mathematical forms and good computational efficiency that will always be appealing for certain types of rotor analyses. However, it is in the area of vortex wake modelling and the incorporation of these models into wind turbine analyses that many future challenges lie for the wind turbine analyst. Vortex wake methods are attractive because of their appealing physical nature and flexibility to handle a broad range of steady and transient operating conditions.
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Model Validation Many strides have been made in the understanding and modeling of wind turbine aerodynamics, as evidenced by several publications [1]–[4]. Like all knowledge, however, this understanding of aerodynamics is not absolute and can be viewed as tentative, approximate and always subject to revision. The presented models and control codes are yet to be validated: development of tools for wind turbine analysis need verification and validation before they can be tested in the field environment [5]–[7]: ◦ Verification: concerned with building the model right. It is utilized in the comparison of the conceptual model to the computer representation that implements that conception. It asks the questions: Is the model implemented correctly in the computer? Are the input parameters and logical structure of the model correctly represented? ◦ Validation: concerned with building the right model. It is utilized to determine that a model is an accurate representation of the real system. Validation is usually achieved through the calibration of the model, an iterative process of comparing the model to actual system behavior and using the discrepancies between the two, and the insights gained, to improve the model. This process is repeated until model accuracy is judged to be acceptable. Model verification and validation are essential parts of the model development process if models are to be accepted and used to support decision making. Both work together for model credibility — to establish an argument that the model produces sound insights and comparable results to data from the real system after a wide range of tests and criteria, to remove barriers and objections to model use. Verification is done to ensure that the model is programmed correctly, the algorithms have been implemented properly, and that the model does not contain errors, oversights, or bugs. Verification does not, however, ensure the model solves an important problem, meets a specified set of model requirements, or, correctly reflects the workings of a real wind turbine. Practical verification recognizes that no computational model will ever be fully verified, guaranteeing 100% error-free implementation. In principle, the end result is technically not a verified model, but rather one that has passed a properly structured testing program that increases the level of statistical certainty to acceptable levels! Practical validation exercises amount to a series of attempts to invalidate a model — explicitly formulate a series of mathematical tests designed to “break the model”. Presumably, once a model is shown to be invalid, the model is salvageable with further work and results in a model having a higher degree of credibility and confidence. The end result of validation is, technically, not a validated model, but rather a model that has passed all the validation tests — offers a better understanding of the model’s capabilities, limitations, and appropriateness for addressing a range of important questions.
CHAPTER 10. ANALYSIS, PERSPECTIVES, AND CONCLUSIONS Compare model to actual
Compare revised model to actual
Actual System
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Initial model Revise First revision of model Revise
Compare 2nd revised model to actual
Second revision of model Revise
Figure 10.3: Iterative process of calibrating the WECS model.
The constantly growing size of WECS and wind parks is today’s most challenging aspect in power system analysis. It is thus imperative to compare developed codes with those that have been tested on flexible control structures of actual prototypes and whose long term simulation capability offer confidence to allow for an integrated analysis of fault response, control principles, blade and tower dynamics, and stochastic wind model impact. As an aid in the validation process, Naylor and Finger [8] formulated a three-step approach which has been widely followed: 1. Build a model that has high face validity. 2. Validate model assumptions. 3. Compare model input-output transformations to corresponding input-output transformations for the real (‘Actual’) system (or prototype). Fig. 10.3 illustrates this concept in the validation of the simulation models developed in this thesis. Two of the commercially available software for validation purposes include: 1. FAST (Fatigue, Aerodynamics, Structures, and Turbulence) Code [9] — a comprehensive aeroelastic simulator capable of predicting both the extreme and fatigue loads of two- and threebladed horizontal-axis wind turbines, and 2. DIgSILENT PowerFactory [10] — incorporates extensive modelling capabilities with advanced solution algorithms, to provide the analyst with tools to carry out the most complex power system studies. Both find typical applications in wind park design studies, verification of connection conditions, generator control design, harmonic penetration analysis, voltage stability analysis, fault recovery studies, as well as integrated wind park modeling. Especially in wind power applications, both offer standard tools, as all required models and simulation algorithms provide unmet accuracy and performance.
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10.4 Control: an Appraisal of Classical and Advanced Paradigms Evolutionary Basis for Intelligent Control Design for Modern WECS Several attempts have been made to integrate the inspiration, philosophy, history, mathematics, actualizations, and perspectives of evolutionary computation [9]–[16]. Intelligence and evolution are intimately connected. Intelligence is a natural part of life. It is also, however, a mechanical process that can be simulated and emulated. Intelligence is not a property that can be limited philosophically solely to biological structures. It must be equally applicable to machines. For the process of intelligence to be understood, methods for its generation should converge functionally and become fundamentally identical, relying on the same physics whether the intelligence occurs in a living system or machine. Intelligence is defined as the capability of a system to adapt its behavior to meet its goals in a range of environments. The form of the intelligent system is irrelevant, for its functionality is the same whether intelligence occurs within an evolving species, an individual, or a social group. If intelligent decision making is viewed as a problem of optimally allocating available resources in light of diverse criteria (environmental demands and goals), then machine intelligence can be achieved by simulating evolution to effectively design controllers for modern WECSs. The process of adaptation is one of minimizing surprise to the adaptive organism, and requires three basic elements: (i) Prediction — operates as a mechanistic mapping from a set of observed environmental symbols to another set of symbols that represents the expected new circumstance. The mapping is essentially a model that relates previous experiences to future outcomes. Prediction is an essential ingredient of intelligence, for if a system cannot predict future events, every environmental occurrence comes as a surprise and adaptation is impossible. (ii) Control — an intelligent system must not simply predict its environment but must use its predictions to affect its decision making to be able to allocate its resources (i.e., control its behavior) with regard to the anticipated consequences of those actions to avoid relegating its future to nothing but pure luck. (iii) Feedback — the adaptive mechanism must act on the error in prediction and the associated cost of inappropriate behaviors to improve the quality of its forecasting. Environmental adaptability for the intelligent system relies on future event prediction, control of its actions in light of those predictions, and revising its bases for making predictions based on feedback on the degree to which it is achieving its goals.
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Linear Control Schemes All complex power production applications nowadays, from gas and steam turbines, to wind turbines, to integrated gasification combined cycle, require some degree of closed-loop control, for stability and performance. Classical designs (PI and PID) controllers are widely used throughout industry and are a good starting point for many wind turbine control applications. A PID controller can be written in terms of the Laplace variable s (similar to a differentiation operator) as y=
Kd s Ki + Kp + s 1 + sτ
x
(10.1)
where x is the input error signal to be corrected, y is the control action, and the time constant τ prevents the derivative term from becoming large at high frequency, where it could respond excessively to signal noise. The tuning parameters are Ki : Integral Gain – larger Ki implies steady state errors are eliminated quicker. The trade-off is larger overshoot: any negative error integrated during transient response must be integrated away by positive error before a steady state is reached. Kp : Proportional Gain – larger Kp typically means faster response since the larger the error, the larger the feedback to compensate. An excessively large proportional gain will lead to plant instability. Kd : Derivative Gain – larger Kd decreases overshoot, but slows down transient response and may lead to instability (K d is zero in a PI controller). For a variable speed pitch-regulated turbine, x is the difference between the measured generator rotational speed and the demanded or rated speed and y is the demanded pitch angle. Above rated, the pitch is used to regulate the rotor speed to the desired value, while the generator torque or power is held constant. Below rated the pitch is forced to the fine pitch limit, but the generator torque is varied in order to control the generator speed. A measure of ingenuity in selection of tuning parameters in (10.1) with respect to pitch control design is a prerequisite. For example, a pitch controlled machine crossing rated wind speed may take too long before the pitch starts acting. On the other hand, a very short time constant may result in slower simulations. If the PI controller being modelled is actually implemented in discrete form, as is usual, then the desaturation time constant should be chosen to be somewhat smaller than the discrete controller timestep. Alternatively, specify a zero time constant for instantaneous desaturation.
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Typically, a well-tuned PID was sufficient for the first generation of control solutions where, generally, the problems in complex plants were related to performance improvement of different local loops. However, nonlinear limitations imposed by the actuators (magnitude, rate, duty cycle) limit the achievable performance of the controllers, which is also coupled with the design of the different components. The major limitations of PI(D) control for WECS applications include: 1. PID controllers, when used alone, can give poor performance when the PID loop gains must be reduced so that the control system does not overshoot, oscillate or “hunt” about the control setpoint value. The control system performance thus needs to be improved by combining the PID controller functionality with that of a feed-forward control output. Since the feed-forward output is not a function of the plant feedback, it can never cause the control system to oscillate, thus improving the system response and stability. 2. PID controllers are linear, hence their performance in WECS systems that are non-linear is variable. Thus practical application issues can arise from instrumentation connected to the controller, such as need for high sampling rate, measurement precision, and measurement accuracy. Often PID controllers need to be enhanced through methods such as gain scheduling or fuzzy logic. 3. Integral windup during implementation — refers to the situation where the integral, or reset action continues to integrate (ramp) indefinitely. This usually occurs when the controller’s output can no longer affect the controlled variable, which in turn can be caused by controller saturation (the output being limited at the top or bottom of its scale), or if the controller is part of a selection scheme and it is not the selected controller. This needs to be addressed by: a) Initializing the controller integral to a desired value, commonly the process present value for startup problems b) Disabling the integral function until the plant has entered the controllable region c) Limiting the time period over which the integral error is calculated d) Preventing the integral term from accumulating above or below pre-determined bounds. 4. Due to the differential term in the PID, small amounts of measurement or process noise can cause large amounts of change in the output. This requires the additional use of a low-pass filter to filter the measurements. However, low-pass filtering and derivative control cancel each other out, so reducing noise by instrumentation means is a much better choice. Alternatively, the differential band can be turned off in most systems with little loss of control — equivalent to using the PID controller as a PI controller.
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Advanced Multiobjective Controllers Starting from inability of classical PI or PID algorithms that act on a single input signal (to generate a pitch demand), advanced controllers proposed in this thesis rely on a synthesis of both pitch regulation and a torque demand to address the following objectives: • controlling pitch and torque together to improve the trade-off between energy capture, actuator duty, and loads; • the use of higher-order controllers to tackle particular problems in the turbine dynamics; • using the control algorithm to provide damping for lightly damped resonant responses; • algorithm design using optimal feedback or other techniques in which the trade-off between different design objectives can be included explicitly in the design. None of these ideas is new and all of them have been explored to some extent for wind turbines. In this thesis the above are handled by the proposed control schemes, including the LQG, STR, and MBPC, and all rely on an elaborate model of the WECS system. Problem formulation is normally the most difficult part of the process. It is the selection of design variables, constraints, objectives, and models of the disciplines. A further consideration is the strength and breadth of the interdisciplinary coupling in the problem. Once the design variables, constraints, objectives, and the relationships between them have been chosen, the problem can be expressed in the following standard format: find x that minimizes J (x) subject to g(x) ≤ 0, h(x) = 0 and xlb ≤ x ≤ xub
(10.2)
where J is an objective, x is a vector of design variables, g is a vector of inequality constraints, h is a vector of equality constraints, and xlb and xub are vectors of lower and upper bounds on the design variables. Maximization problems can be converted to minimization problems by multiplying the objective by -1. Constraints can be reversed in a similar manner. Equality constraints can be replaced by two inequality constraints. This leads to the optimization cost function J generally used to calculate the optimum control law, and is given by: J =
wxi (ri − xi )2 +
wui ∆u2i
(10.3)
without violating constraints (low/high limits). x i is the i-th control variable, ri is the i-th reference variable, ui is the i-th manipulated variable, w xi is weighting coefficient reflecting the relative importance of xi , and wui is weighting coefficient penalizing relative big changes in u i . This is the basis of adaptive multiobjective control design that relies on state-estimation for full-state feedback.
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While moving towards larger wind turbine installations, more stringent requirements and systematic design of the plants to meet the specifications while minimizing cost, model-based multiobjective control design is becoming more frequently the design method of choice. In power generation applications, model-based control design methods have to address typical problems associated with complex applications such as large order models and actuator nonlinearities, but also specific issues – dynamic nonlinearities, mode coupling or limitations due to conflicting control objectives. The performance obtained by these controllers is crucial in wind power generation for which feedback control is a vital component of the overall operation. The appeal of the proposed advanced schemes is that they are multiobjective, and make use of the following concepts, singly or in combination: • Observers – utilize a subset of the known dynamics to make estimates of a particular variable. In this case, the estimated wind speed can then be used to define the appropriate pitch angle. • State estimators – using a full model of the dynamics, a Kalman filter can be used to estimate all the system states from the prediction errors. Thus it is possible explicitly to take account of the stochastic nature of the wind input by formulating a wind model driven by a Gaussian input. • Optimal feedback – the cost function approach means that the trade-off between partially competing objectives is explicitly defined by selecting suitable weights for the terms of J . It is noteworthy that efficiency of a WECS does not scale simply by physical dimensions and control plays a significant role in increasing the size of the machine while decreasing the structural loads and improving the rotor efficiency. Challenges in implementing advanced control paradigms include: 1. Most of these techniques require large numbers of evaluations of the objectives and the constraints. The disciplinary models are often very complex and can take significant amounts of time for a single evaluation. The solution can therefore be extremely time-consuming. Fortunately, many of the optimization techniques are adaptable to parallel computing, and much current research is focused on methods of decreasing the required time. 2. A vital aspect of the development of new control algorithms for the novel schemes is the assessment of their effectiveness. This is difficult because of the variability of the wind input. Suitable approaches to the evaluation of controller performance is thus limited to simulations. 3. Field trials – despite the power and reliability of some of the simulation models now available, there is no substitute for field trials in real wind conditions. The variability of the wind makes it particularly difficult to carry out field trials repeatably and reliably, particularly if the effectiveness of two or more alternative controllers is to be compared (or their cost benefit!).
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10.5 Conclusions In Part I, various concepts are brought forth as a basis for modelling the various components of the drive train in order to formulate the control objectives for controller design. Chapter 1 gives an overview of wind energy as well as motivation for this study. The control objectives are determined as twofold: the optimization of energy conversion and mitigation of shaft torque torsional moments to check cyclic-stress-induced fatigue damage to mechanical subsystems. Chapter 2 details the essential concepts in aerodynamic conversion modeling, and formulates a model for the expected output of the WECS. The importance of turbine linearization in controller design is emphasized and developed. Reliability of wind turbine system is based on the performance of its components under assigned environment, manufacturing process, handling, and the stress and aging process. As part of the design process, a wind turbine must be analyzed for aerodynamic loads, gravitational loads, inertia loads and operational loads it will experience during its design life. Chapter 3 develops a mathematical model for the mechanical drive train as a multi-inertial system coupled by elastic linkages. The main idea is to examine stresses on the drive shaft as well as the gearbox — a source of failures and defects in many wind turbines. The electrical system of the wind turbine includes all components for converting mechanical energy into electrical power. A brief review of the generator has been illustrated in Chapter 4. Chapter 5 analyzes the simulation of a real-time wind speed (which is, invariably, the stock-intrade for the WECS) for a generic site. The mean wind speed is obtained from an annual Rayleigh probability density curve and the turbulence component modelled as an instantaneous variation, obtained via a constrained stochastic simulation scheme. The generated gusts have the desired properties and are used as input for wind turbine design tools in order to assess the extreme loading. This is then used to formulate a real-time wind profile from Gaussian noise thus enabling the determination of the response of the WECS under highly stochastic environmental conditions. Part II develops a control strategy for WECS control based on harmonization between pitch angle control and generator torque control. One of the main goals of control is to increase power production and reduce loads with a minimum number of control inputs required for turbine measurement. Chapter 6 describes the control philosophy and expounds on the classical linear PI(D) controllers to regulate power. Several control configurations, whose common denominator is multiobjectivity, are mooted for control of a 2-MW class WECS, based on subsystems’ modeling. The design of the control schemes attempts to maximize performance (e.g. efficiency, throughput, specific energy consumption) while maintaining stability and physical integrity under both wind and load disturbances.
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Chapter 7 reviews the LQG controller for generator torque control. For the nonlinear WECS system, the basic idea of feedback is tremendously compelling as it enhances stability, improves the steady-state error characteristics, and provides disturbance rejection due to a stochastic wind. The LQG control objective for the WECS multivariable system has been to obtain a desirable behavior of several output variables by simultaneously manipulating several input channels. A hybrid controller is also mooted, based on the idea of augmenting the LQG with a neurocontroller, the latter to cater for the nonlinearities in the system. The hybrid shows remarkable improvement in control. Chapter 8 introduces the STR that consists of two parts: an estimator and a control law, that are invoked at every sample period. The purpose of the RLS algorithm (estimator) is to dynamically estimate the model of the WECS system relating the measured metrics with the actuation. For the control law, LQ tracking optimal control design employs the RLS algorithm based on an equivalent non-minimal state space realization of the WECS model. No state observer is required. MBPC is conceptually a natural method for generating feedback control actions for linear and nonlinear plants subject to pointwise-in-time input and/or state-related constraints. Chapter 9 proposes the MBPC scheme for generator torque control. An important observation is made: in contrast to MBPC, in feedback-control systems of more traditional type, e.g., LQG or H ∞ control, constraints are indirectly enforced, by imposing, whenever possible, a conservative behavior at a performancedegradation expense. Other instances where MBPC can be advantageously used comprise unconstrained plants for which off-line computation of a control law is a difficult task as compared with on-line computations via receding-horizon control. Chapter 10 gives an overview of the modeling approach employed in the thesis, as well as an evaluation of the merits and demerits of the various control paradigms proposed herein.
Remark Classical methods based on PI(D) algorithms are a good starting point for many aspects of closed loop controller design for variable speed turbines. However, as turbines become larger and more flexible, it is increasingly important not only to consider the effect that the controller has on component loads, but even to design the controller with load reduction as part of the primary objective. Of the proposed control techniques, clearly, no paradigm is extremely superior to the others: all have inherent capabilities as well as shortcomings. The appeal is that they are able to capture the nonlinearities in the turbine and devise the pertinent control signals for effective energy conversion and mitigation of drive train loads. However, most commercial wind turbines still use fairly basic control techniques, leaving a wide scope for improvement in the coming years.
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References [1] Y. Coughlan, P. Smith, A. Mullane, and M. O’Malley, “Wind turbine modeling for power system stability analysis – a system operator perspective,” IEEE Trans. Power Systems, vol. 22, no. 3, pp. 929-936, Aug. 2007. doi: 10.1109/TPWRS.2007.901649. [2] D. J. Trudnowski, A. Gentile, J. M. Khan, and E. M. Petritz, “Fixed-speed wind-generator and wind-park modeling for transient stability studies,” IEEE Trans. Power Systems, vol. 19, no. 4, pp. 1911-1917, Nov. 2004. doi: 10.1109/TPWRS.2004.836204. [3] A. Rauh, and J. Peinke, “A phenomenological model for the dynamic response of wind turbines to turbulent wind,” J. Wind Engineering and Industrial Dynamics, vol.92, pp. 159-183, 2004. [4] V. Akhmatov, and H. Knudsen, “An aggregate model of a grid-connected, large-scale, offshore wind farm for power stability investigations – importance of windmill mechanical system,” Electrical Power and Energy Systems, vol. 24, pp. 709-717, 2002. [5] M. Martins, A. Perdana, P. Ledesma, E. Agneholm, and O. Carlson, “Validation of fixed speed wind turbine dynamic models with measured data,” Renewable Energy, vol. 32, pp. 1301-1316, 2007. [6] C. Eisenhut, F. Krug, C. Schram, and B. Klockl, “Wind turbine model for system simulations near cut-in wind speed,” IEEE Trans. Energy Conversion, vol. 22, no. 2, pp. 414-420, June 2007. doi: 10.1009/TEC.2006.875473. [7] P. N. Finlay, and J. M. Wilson, “The paucity of model validation in operational research projects,” Journal of the Operational Research Society, vol. 38, no. 4, pp. 303-308, April 1987. doi: 10.2307/2582053. [8] T. H. Naylor, and J. M. Finger, “Verification of computer simulation models,” Management Science, vol. 14, no. 1, pp. 92-101, Oct. 1967. [9] J. M. Jonkman, and M. L. Buhl Jr, FAST User’s Guide, Technical Report, NREL/EL-500-38230. A publication of the National Renewable Energy Laboratory (A national laboratory of the U.S. Department of Energy, Office of Energy Efficiency & Renewable Energy). Aug. 2005. Available online: http://wind.nrel.gov/designcodes/simulators/fast/ [10] DIgSILENT PowerFactory. An Integrated Power System Analysis Software for Wind Power Applications, from DIgSILENT GmbH. Homepage: http://www.digsilent.de
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[11] C. Darwin, On the Origin of Species by Means of Natural Selection or the Preservations of Favored Races in the Struggle for Life, London: John Murray, 1859. [12] J. H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence, Cambridge, MA: MIT Press, 1995. [13] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, MA: Addison-Wesley 1989. [14] A. Hoffman, Arguments on Evolution: A Paleontologist’s Perspective, New York: Oxford University Press, 1989. [15] B. D. Fogel, Evolutionary Computation: Toward a New Philosophy of Machine Intelligence, NY: IEEE Press, 1995. [16] K. A. De Jong, An Analysis of the Behaviour of a Class of Genetic Adaptive Systems, Doctoral Dissertation, Dept. of Computer and Communications Sciences, University of Michigan, Ann Arbor, 1975. [17] F. Rosenblatt, Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms, Washington, DC: Spartan Books, 1962. [18] R. Axelrod, “The Evolution of Strategies in the Iterated Prisoner’s Dilemma,” In Genetic Algorithms and Simulated Annealing, edited by L. Davis, London: Pitman, pp. 32-41, 1987.
Chapter 11 Future Work 11.1 Introduction
T
HERE are two approaches to allow for uncertainty in system models and disturbances: adaptive versus robust control. The first approach is to use an adaptive controller, which estimates
parameters and calculates the control accordingly. Self-tuning devices have been very successful, but they involve online design computations and are therefore not as simple as a fixed controller to implement. The second approach is to allow for uncertainty in the design of the fixed controller, thus producing a robust control scheme — one which is insensitive to parameter variations or disturbances. Future research is in the direction of control based on the following two concepts: • H2 /H∞ — these formulations eliminate the stochastic element and permit a frequency domain view by allowing the introduction of frequency dependent weighting functions; and • (Neuro)Fuzzy controllers — rely on fuzzy logic to model imprecise concepts and evolve context-dependent controllers via optimization. The H∞ design approach can be combined with self-tuning action to obtain a robust adaptive
controller [1]-[3]. The H∞ concept is particularly appropriate when improving robustness, in the face of WECS plant perturbations due to high wind turbulence, or parameter uncertainty. Fuzzy controllers are implemented using fuzzy rules, which can reduce the number of computations in conventional controllers. It is also claimed that they can be implemented more easily than conventional controllers. The most popular kind of fuzzy systems are based on either the Mamdami fuzzy model, Takagi-Sugeno-Kang (TSK) fuzzy model, Tsukamoto fuzzy model or Singleton fuzzy model. To define a fuzzy logic controller it is necessary to introduce IF-THEN rules to establish how probable the process variable is. To evaluate the rules, the definition of fuzzy operations is also needed. The application of the rules defines fuzzy set values of fuzzy output sets.
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11.2 H∞ -Optimization for WECS Design by H∞ -optimization as a design tool for linear multivariable WECS control involves the minimization of the peak magnitude of a suitable closed-loop system function. It is very well suited to frequency response shaping. Moreover, robustness against plant uncertainty is handled more directly. Introducing µ-synthesis in the design aims at reducing the peak value of the structured singular value. It accomplishes joint robustness and performance optimization. H ∞ -optimization amounts to the minimization of the ∞-norm of the relevant frequency response function. The name 1 derives from the fact that mathematically the problem may be set in the space H ∞ , which consists of all bounded functions that are analytic in the right-half complex plane. An important aspect of H ∞ optimization is that it allows to include robustness constraints explicitly in the criterion.
Properties of H∞ Robust Control Design There are several advantages of the H∞ control design approach. The technique can be easily computerised and formalized design procedures can be introduced. Design issues can be considered in the frequency domain and classical design intuition can be employed. However, the most important advantage is that stability margins can be guaranteed and performance requirements can also be satisfied, in a unified design framework. The H∞ design approach is distinguished by the following features and properties: • It is a design procedure developed specifically to allow for the modeling errors, which are inevitable and limit high-performance control systems design. • There is a rigorous mathematical basis for the design algorithms, which enables stability and robustness properties to be predicted with some certainty. • There are close similarities between state-space versions of H∞ controllers and the well-known Kalman filtering or H2 /LQG control structures. • If the uncertainty lies within the class considered, stability properties can be guaranteed and safe reliable systems can be assured. Note that the design procedures cannot be used blindly, since poor information can still lead to controllers with poor performance properties. • The trade-offs between good stability properties and good performance are easier to make in a H∞ context than with many of the competing designs. 1
Named after the British mathematician G. H. Hardy.
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• The approach can be interpreted in terms of the stochastic nature of the system, but if disturbances and noise are important H2 /LQG may still be the preferred solution. • The H∞ design technique is easy to use, since the algorithms are available in commercial software. The main disadvantage is that methods of handling parametric uncertainty are not handled so naturally in the H∞ framework. A high-performance robust design would take account of this structure, but the basic H∞ approach does not account for this type of information. However, there are several ways of modifying the method to allow for parametric uncertainties, including µ-synthesis and H ∞ adaptive control. The H∞ design approach is a strong contender to provide a general purpose control design procedure, which can account for uncertainties and is simple to use with computer-aided design tools.
Comparison of H∞ and H2 /LQG Controllers The similarities and differences between the H∞ and the H2 /LQG approaches are detailed below: 1. Similarities (a) Both H2 and H∞ optimal controllers are based on the minimization of a cost index. (b) Some of the closed-loop poles of the LQG solution will be the same as those of the H ∞ solution in certain limiting cases. (c) The dynamic cost weights have a similar effect in both types of cost function, e.g. integral action can be introduced via an integrator in the error weighting term in both cases. (d) Closed-loop stability can be guaranteed, whether the plant be non-minimum phase, or unstable (neglecting for the moment uncertainty and assuming controllers are implemented in full). 2. Differences (a) The basic conceptual idea behind H∞ design involves the minimization of the magnitudes of a transfer function, which is quite different from the H2 /LQG requirement to minimize a complex domain integral representing error and control signal power spectra. (b) The H∞ design approach is closer to that of classical frequency response design in that the frequency-response shaping of desired transfer functions is attempted. (c) The calculation of H∞ controllers is more complicated than the equivalent H 2 /LQG controllers, whether this be in the time or frequency domains.
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11.3 Fuzzy Logic Control The Fuzzy Logic System (FLS) employs a set of N fuzzy linguistic rules. These rules may be provided by experts or can be extracted from numerical data. In either case, engineering rules in FLS are expressed as a collection of IF – THEN statements. Therefore a fuzzy rule base R containing N fuzzy rules can be expressed as: R = [Rule1 , Rule2 , ..., Rulei , ..., RuleN ]
(11.1)
Rulei : IF z(k) is A˜ THEN u(k) is βi
(11.2)
where the i − th rule is:
where k refers to the variable values at time t = k∆t. Moreover, the vector z(k) = [z1 (k), ..., zl (k)]T
(11.3)
represents all the l fuzzy inputs to the FLS. On the other hand, u(k) represents the fuzzy output of the FLS. In the antecedent of the i − th rule, the term A˜ = [A˜1i , ..., A˜li ]T
(11.4)
represents the vector of the fuzzy sets referring to the input fuzzy vector z(k). The membership functions of both the input vector z(k) and the vector A˜ of the fuzzy sets are Gaussian, and assume the following expressions: µzj (k) = e−1/2[(zj (k)−ˆzj )/σzj ]
2
–2 » ˆi )/σ ˜i −1/2 (zj (k)−A j A
µA˜ij (k) = e
j
(11.5) (11.6)
where zˆj and σzj are the mean value and the variance of the Gaussian membership function of the j − th input, zj (k). Likewise, Aˆij and σA˜i are the mean value and the variance of the Gaussian j
membership function of the j − th fuzzy set referring to the i − th fuzzy rule, Aˆij . The terms zˆj and σzj are known constants, while Aˆij and σA˜ij represent the unknown parameters of the FLS. These parameters will be adapted to the controlled wind system by minimizing an appropriate cost function.
CHAPTER 11. FUTURE WORK
149 z -1
y(k-1) r(k)
z(k)
Fuzzy controller
u(k)
WECS Plant
y(k)
u(k-1) z -1
Figure 11.1: Layout of the fuzzy control system.
The output of the fuzzy controller, u(k), assumes the following expression [4] %N u(k) =
/l i=1 βi j=1 µQij [zj,max (k)] %N /l i=1 j=1 µQij [zj,max (k)]
(11.7)
where µQij [zj (k)] = µzj (k)µA˜ij (k). Moreover, zj,max (k) =
2 zˆj σzj + Aˆij σA2˜i
j
2 σzj
+
σA2˜i j
(11.8)
(11.9)
is the value of the j − th input that maximizes (11.8). The maximization of Eq. (11.8) represents the supremum operation in the sup-star composition of the i − th rule [4]. This fuzzy controller appears to be parameterized by 0 1 θ(k) = Aˆij (K), σA˜ij (k), βi (k)i = 1, 2, ..., N; j = 1, 2, ..., l .
(11.10)
In the next section, a procedure that allows an on-line adaptation of the parameters θ(k) to the controlled wind system will be introduced. The fuzzy logic control system adopted is represented in Fig. 11.1. The fuzzy input vector is defined as: z(k) = [y(k − 1), r(k), u(k − 1)]T
(11.11)
where y(k) is the output of the plant (controlled variable), u(k) is the control variable (output of the fuzzy controller), and r(k) represents a reference signal for y(k).
CHAPTER 11. FUTURE WORK
150 z -1
y(k-1) r(k)
z(k)
u(k)
Fuzzy controller
WECS Plant
y(k)
u(k-1) z -1 LSA parameter estimator
+ r(k+1)
_ y(k+1)
Figure 11.2: Fuzzy control system with the parameter estimator.
11.3.1 Estimator-Based Adaptive Fuzzy Logic In general, an Adaptive Fuzzy Logic (AFL) control starts from an initially assumed set of parameters θ(0), whose only requirement is to stabilize the plant. Then, at each time step, the AFL control adapts the set of parameters θ(k), in order to minimize the cost function: 1 J (k) = e2y (k) 2
(11.12)
where ey (k) is the control error defined as: ey (k) = r(k) − y(k).
(11.13)
The control error ey (k) can be determined only if a deterministic model of the controlled system is available. In this case it is supposed that no a priori deterministic model of the controlled system is available. The Estimator-based Adaptive Fuzzy Logic (EAFL) control here suggested allows to solve this class of problems. Indeed, instead of deriving the appropriate change in each internal parameter from the control error ey (k), the EAFL refers to an approximate estimation of the control error eˆy (k) = r(k) − yˆ(k)
(11.14)
CHAPTER 11. FUTURE WORK
151
and to the corresponding cost function: 1 Jˆ(k) = eˆ2y (k). 2
(11.15)
In Eq. (11.14), yˆ(k) represents the estimated value of the output at the time k, to be evaluated. As stated in [5], the present system can be expressed as follows: y(k) = ak y(k − 1) + bk u(k − 1)
(11.16)
where ak and bk represent the time-varying coefficients of model (11.16). If the controlled plant is observable, then (11.16) represents its model in state space notation. In such a case, the model coefficients ak and bk are unknown. These coefficients can be on-line estimated by applying the Least Square Algorithm (LSA) in recursive form [5],[6]. As a consequence, the basic scheme of the fuzzy control system has to be modified as shown in Fig. 11.2, where the LSA estimator evaluates the coefficients a ˆk and ˆbk . Assuming that such coefficients do not change from the time k to the time k + 1, the estimated model of the controlled system one-step-ahead, i.e., at time k + 1, assumes the following expression: yˆ(k + 1) = a ˆk y(k) + ˆbk u(k)
(11.17)
which is the output of the LSA parameter estimator (Fig. 2). The signal yˆ(k + 1) is compared to the reference signal r(k + 1) and the difference determines the modification of the fuzzy controller parameters θ(k). This is implemented by rewriting the cost function Jˆ at time k + 1 as: , -12 10 r(k + 1) − a ˆk y(k) + ˆbk u(k) . Jˆ(k + 1) = 2
(11.18)
The minimization of the cost function Jˆ(k + 1) can be easily accomplished by using the gradient descent algorithm as follows: θ(k) = θ(k − 1) − η
∂ Jˆ(k + 1) ∂θ
(11.19)
CHAPTER 11. FUTURE WORK
152
where the sensitivity derivatives of Jˆ(k + 1) with respect to θ (refer to (11.10)) are given by: /l ∂ Jˆ(k + 1) j=1 wij (k) = −ˆbk eˆy (k + 1) %N /l ∂βi i=1 j=1 wij(k) %N / −vij (k) i=1 cij lj=1 wij (k)[βi − u(k)] ∂ Jˆ(k + 1) = −ˆbk eˆy (k + 1) %N /l ∂ Aˆij i=1 j=1 wij(k) / % l σA˜ij vij2 (k) N i=1 cij j=1 wij (k)[βi − u(k)] ∂ Jˆ(k + 1) = −ˆbk eˆy (k + 1) %N /l ∂σA˜ij i=1 j=1 wij(k)
(11.20) (11.21)
(11.22)
where: vij =
Aˆij − zˆj 2 σA2˜i + σzj
(11.23)
j
2 2 2 ˆi −ˆ −1/2(A j zj ) / σ ˜i +σzj
wij (k) = e
!
A j
(11.24)
The coefficient η is the rate of descent which can be chosen arbitrarily. Moreover, cij is equal to 1 if the i-th rule is dependent on the j-th input, otherwise it is equal to 0.
11.4 Remarks After several years of efforts in design of control schemes for wind turbines, the wind industry is slowly succumbing to advanced controllers that offer a series of advantages over the classical linear systems. Research has been undertaken and various configurations proposed, for WECS control involving H∞ controllers [7]–[9] and Fuzzy logic schemes [10],[11], though successful practical implementation of these paradigms is not (yet) documented. From the pedagogical overview of some of the most promising and recent developments in advanced control for WECS discussed in this report, future research work is motivated by two issues: ◦ capability of the novel multiobjective controllers for effective energy conversion and drive-load reduction for MW-class WECSs, and ◦ flexibility on the part of wind turbine manufacturers to embrace a shift from the classical PID. Furthermore, rapid improvements in computer hardware, combined with stiff competition in the wind industry as well as various governments’ regulations are largely responsible for research in advanced control. Future research aims to address the frequently expressed improvement sought — to decrease control response time (including model development, computation, programming, communications, user interface), not just the cost benefit — as these advanced paradigms mature to a commodity status.
CHAPTER 11. FUTURE WORK
153
References [1] M. J. Grimble, “H∞ robust controller for self-tuning applications, Part 1: Controller design,” Int. Journal of Control, vol. 46, no. 4, pp. 1429-1444, 1987. [2] M. J. Grimble, “H∞ robust controller for self-tuning applications, Part 2: Self-tuning and robustness,” Int. Journal of Control, vol. 46, no. 5, pp. 1819-1840, 1987. [3] N. A. Fairbairn, and M. J. Grimble, “H∞ robust controller for self-tuning applications, Part 3: Self-tuning controller implementation,” Int. Journal of Control, vol. 52, no. 1, pp. 15-36, 1990. [4] J. M. Mendel, “Fuzzy logic systems for engineering: a tutorial,” Proceedings of the IEEE, vol. 83, no. 3, 1995. [5] G. C. Goodwin, and K. S. Sin, Adaptive, filtering, predfiction and control Prentice-Hall Inc., Englewood Cliff, New Jersey, 1984. [6] A. L. Dadone, L. D’Ambrosio, and B. Fortunato, “One-step-ahead adaptive technique for wind systems,” Energy Conversion and Management, vol. 39, no. 5/6, pp. 399, 1998. [7] T. Senjyu, E. Omine, D. Hayashi, E. B. Muhando, A. Yona, and T. Funabashi, “Balancing control for dispersed generators considering torsional torque suppression and AVR performance for synchronous generators,” IEEJ Trans. Power and Energy, vol. 128, no. 1, pp. 75-83, 2008 (in Japanese). [8] B. Connor, S. N. Iyer, W. E. Leithead, and M. J. Grimble, “Control of a horizontal axis wind turbine using H infinity control,” First IEEE Conference on Control Applications, 13-16 Sept. 1992, vol. 1, pp. 117-122. doi: 10.1109/CCA.1992.269889. [9] Dengying, Y. Shiming, W. Xiangming, L. Sun, and L. Jiangjing, “Researches on a controller for reducing load of driving chain in wind turbine based on H ∞ control,” IEEE Int. Conf. on Automation and Logistics, 18-21 Aug. 2007, pp. 1-4. doi: 10.1109/ICAL.2007.4338573. [10] M. G. Simoes, B. K. Bose, and R. J. Spiegel, “Fuzzy logic based intelligent control of a variable speed cage machine wind generation system,” IEEE Trans. Power Electronics, vol. 12, no. 1, Jan. 1977. [11] I. G. Damousis, M. C. Alexiadis, J. B. Theocharis, and P. S. Dokopoulos, “A fuzzy model for wind speed prediction and power generation in wind parks using spatial correlation,” IEEE Trans. Energy Conversion, vol. 19, no. 2, pp. 352-361, June 2004. doi: 10.1109/TEC.2003.821865.
Appendix A Parameters Setting A.1 WECS Model Details Table A.1: WECS parameters and baseline safety operational limits PARAMETER VALUE Wind turbine and rotor blade radius, R 35 m number of blades 3 hub height 61.5 m rated wind speed, vr 12.205 m/s cut-in/cut-out wind speed 4/25 m/s gearbox ratio, Kgr 83.33 6.029E+06 kgm2 turbine inertia, Jt low speed shaft torsional stiffness, K s 1.6E+08 Nm/rad low speed shaft torsional damping, D s 1.0E+07 Nms/rad Generator and grid network 2 MW rated capacity, Pr optimal mode maximum generator speed 1500 rpm generator inertia, Jg 60 kgm2 max/min generator torque, Γg,max /min 14.4/0 kNm generator torque set-point 13.4 kN max/min generator speed 1800/850 rpm generator stator resistance 0.01 Ω generator rotor resistance 0.01 Ω stator leakage inductance 95.5E-06 H rotor leakage inductance 95.5E-06 H generator magnetizing (mutual) inductance 0.0955 H stator rated voltage, Ve 690 V 50 Hz stator rated (electrical) frequency, fn rotor rated magnetizing current 1700 A Pitch controller max/min pitch angle, βmax/min 90/-2 deg ˙ max/min pitch rate, βmax/min 8/-8 deg/s
APPENDIX A. PARAMETERS SETTING
155
A.2 Aerodynamics Information A.2.1 Steady-state Operation Point Parameters Table A.2: Performance coefficients calculation rated wind speed 12.205 m/s minimum tip-speed ratio 2 maximum tip-speed ratio 20 tip-speed ratio step 0.1 pitch angle -2 deg rotor speed 20 rpm
A.2.2 Wind Speed Simulation Parameters Table A.3: Simulated time-dependent wind field parameters at hub height mean wind speed for simulation 12.205 m/s flow inclination 8 deg interpolation scheme cubic sampling period, ∆tw 0.1 s turbulence intensity: longitudinal 16.0108 % lateral 12.5465 % vertical 8.92472 % turbulence charactertistics: spectrum type von Karman width of turbulent wind field 100 m height of turbulent wind field 100 m length of turbulent wind field 1804.8m step-size of turbulent wind field 0.88125 m
Table A.4: Physical constants air density, ρ 1.225 kg/m3 air viscosity, ν 1.82E-05 kg/ms gravitational acceleration, g 9.81 m/s2 density of water, ρw 1027 kg/m3
Notes: 1. For Table A.3, the rated speed is taken as the mean wind speed for simulation. 2. Details on the determination of the various values for 3-D turbulence intensity are obtained from the IEC 61400-1 Standard, as explained in Section A.2.3.
APPENDIX A. PARAMETERS SETTING
156
A.2.3 The IEC61400-1 Standard for Turbulence Model The extreme wind events experienced by the WECS are included in the currently available draft of the IEC-Standard as extreme load conditions that must be considered as ultimate load cases when designing a wind turbine. Within the framework of the IEC 61400-1 Std these load situations are defined in terms of two independent site variables — a reference mean wind speed and a characteristic turbulence intensity, TI. Turbulence in random ten-minute periods has more scatter at low wind speeds. This is both because the uncertainty depends on the ratio of the time scale and sample duration, and because deviation from neutral atmospheric stability is more pronounced at low wind speeds. These effects are accounted for by an empirical formula in edition 3 of the IEC61400-1 Standard: The Standard defines the representative turbulence intensity, σ1 , as the mean + 1.28 times the standard deviation of random ten-min measurements. Load cases are defined by the reference turbulence intensity, Iref , which is equal to the mean turbulence intensity at 15 m/s σ1 = Iref
(15m/s + aVhub ) + 1.28 × 1.44m/s . (1 + a)
(A.1)
Note that in the formula the variability is added as an extra term because I ref refers to mean TI. The representative TI may be defined by the actual edition 3 formula that is equal to the more complex formulation when a = 3 as follows: σ1 = Iref (0.75Vhub + 5.6m/s).
(A.2)
The values in Table A.5 are specified for the turbulence model: Table A.5: IEC 61400-1 Ed. 3 parameter assignments Class A B C S Iref 0.16 0.14 0.12 Designer specifies
In this thesis, a Class A turbulence site is considered. For the seasonal mean wind speed of 7 m/s, cut-in wind speed of 4.0 m/s, and operation at rated wind speed of the turbine equipment (12 m/s), the prevailing turbulence intensities (longitudinal, lateral and vertical) are obtained as (16.0108%, 12.5465%, and 8.92472%), respectively, as given in Table A.3. In almost all circumstances the horizontal component of the wind is much larger than the vertical — the exception being violent convection.
APPENDIX A. PARAMETERS SETTING
157
A.2.4 Annual Energy Yield The annual energy yield is calculated by integrating the power curve for the turbine together with a Weibull distribution of hourly mean wind speeds. The power curve is defined at a number of discrete wind speeds, and a linear variation between these points is assumed. The Weibull distribution is defined by: vw
F (vw ) = 1 − e−( c¯vw )
k
(A.3)
where F is the cumulative distribution of wind speed. Thus the probability density f (v w ) is given by f (vw ) = −k
vwk−1 −( c¯vvw )k w e (c¯ vw )k
(A.4)
with k as the Weibull shape parameter and c as the scale factor. For a true Weibull distribution, these two parameters are related by the gamma function: 1 . c = 1/Γ 1 + k
(A.5)
Unless the user supplies the value for c, its value is calculated as above. Note that if a different value is supplied, the resulting distribution will have a mean value that is different from v¯ w . The annual energy yield is calculated as cut−out
P (vw )f (vw )dvw
E=Y
(A.6)
cut−in
where P (vw ) = power curve, i.e., electrical power as a function of wind speed, given in (2.6) Y = length of a year, taken as 365 days. The result is further multiplied by the availability of the turbine, which is assumed for this purpose to be uncorrelated with wind speed (see Section 2.2.4: Capacity Factor). The aforegoing is the basis of mean wind speed calculation for a given site as mentioned in Section 5.2: Determination of Mean Wind Speed, vm .
Appendix B Supporting Concepts B.1 Per Unit System for the WECS Model In electrical engineering in the field of power transmission a per-unit system is the expression of system quantities as fractions of a defined base unit quantity. A per-unit (pu) system provides units for power, voltage, current, impedance, and admittance. Pu system—widely used in the power industry in power flow studies to express values of quantities — is adopted in the dynamic analysis of the drivetrain as well as the electrical system in order to simplify calculations by expressing the parameters on a common power base. These are based on the following formulation: base value in pu =
quantity expressed in SI units . base value
(B.1)
B.1.1 DQ Base Values The dq system base voltage and current are taken equal to the respective abc instantaneous base values: Vb,dq =
√
2Vb,abc
Ib,dq =
√
2Ib,abc .
(B.2)
Using these definitions, the base power Sb is given by 3 Sb = Vb,dq Ib,dq 2
(B.3)
whereas the base resistance is equal to the respective abc value Zb,dq =
Vb,dq = Zb,abc Ib,dq
(B.4)
APPENDIX B. SUPPORTING CONCEPTS
159
B.1.2 Mechanical System If Sb is the base power (VA), ω0 the base electrical angular velocity (rad/sec) and p the number of poles of the generator, then the base values at the high speed side (generator-side) of the drive train are defined as follows: ω0 p/2 Sb Γb = ωb Sb Γb Jb = = 0.5ωb2 0.5ωb Γb Sb Kb = = 2 ωb ωb Γb Sb Db = = 2 ωb ωb ωb =
the base mechanical speed, in mechanical rad/sec
(B.5)
the base torque, in Nm
(B.6)
the base inertia, in Nm/(rad/sec)
(B.7)
the base stiffness coefficient, in Nm/(rad/sec)
(B.8)
the base damping coefficient, in Nm/(rad/sec)
(B.9)
The low speed side (rotor-side) base quantities are calculated from the above quantities using the gearbox ratio Ngr as follows: ωB = Ngr ωB ,
Γb = Ngr Γb ,
2 Jb = Ngr Jb ,
2 Kb = Ngr Kb
and
2 Db = Ngr Db .
(B.10)
where primed and double-primed respectively are the low and high speed side base quantities. Generally, the pu inertia values relate to the mass moments as follows: Ht =
Jt ωb2 2Sb Ngr p2
and
Hg =
Jg ωb2 . 2Sb Ngr p2
(B.11)
Further, the shaft stiffness is obtained from KS =
2ω02 Ht ωb
(B.12)
while the electrical twist angle of the shaft, θtg , is given by dθtg = ωb (ωt − ωg ). dt
(B.13)
In the aforegoing, ω0 = 2πfn , where fn is nominal grid frequency (Hz), and ωb = 2πf0 , with f0 being the mechanical drive train eigenfrequency (Hz).
APPENDIX B. SUPPORTING CONCEPTS
160
B.2 Pole-placement For the second-order system
and a second-order controller
B(z) b1 z + b2 = 2 A(z) z + a1 z + a2
(B.14)
S(z) s0 z 2 + s1 z + s2 = 2 R(z) z + r1 z + r2
(B.15)
the polynomial A(z)R(z) + B(z)S(z) becomes (z 2 + a1 z + a2 )(z 2 + r1 z + r2 ) + (b1 z + b2 )(s0 z 2 + s1 z + s2 ) = z 4 + (a1 + r1 + b1 s0 )z 3 + r2 (a2 + a1 r1 + r2 + b1 s1 + b2 s0 )z 2 + (a2 r1 + a1 r2 + b1 s2 + b2 s1 )z + (a2 r2 + b2 s2 ) .
(B.16)
If the control coefficients (r1 , r2 , s0 , s1 , s2 ) are known the coefficients in the polynomial A(z)R(z) + B(z)S(z) = P (z) = z 4 + p1 z 3 + p2 z 2 + p3 z + p4 become a1 + r1 + b1 s0 = p1 a2 + a1 r1 + r2 + b1 s1 + b2 s0 = p2 a2 r1 + a1 r2 + b1 s2 + b2 s1 = p3 a2 r2 + b2 s2 = p4 (B.17) and the closed loop poles are found from P (z) = 0. If the poles are specified in advance these equations may be solved with respect to the unknown control coefficients (r 1 , r2 , s0 , s1 , s2 ) and the above expression in matrix form becomes b 0 0 1 0 1 b2 b1 0 a1 1 0 b2 b1 a2 a1 0 0 b2 0 a2
p1 − a1 s1 p −a 2 2 s2 = p 3 r1 p4 r2
s0
(B.18)
APPENDIX B. SUPPORTING CONCEPTS
161
Having 5 unknown controller parameters and 4 equations means that an extra equation in the controller parameters may be fulfilled. If integral action of the controller is specified i.e. the DC-gain of the controller is infinite, then the following extra equation is obtained: R(z = 1) = 0
(B.19)
1 + r1 + r2 = 0.
(B.20)
or
The combined pole-placement controller with integral action then becomes the solution to
b1
0
0
1
s0
0
p1 − a1
b2 b1 0 a1 1 s1 p2 − a2 0 b2 b1 a2 a1 s2 = p3 0 0 b2 0 a2 r1 p4 0 0 0 1 1 r2 −1
.
(B.21)
For the DC-gain, consider the system: Y (z) =
z2
b1 z + b2 + a1 z + a2
(B.22)
that has the discrete time realization y(k + 2) + a1 y(k + 1) + a2 y(k) = b1 u(k + 1) + b2 u(k).
(B.23)
If the system is assumed stable then a constant input u(k) = u 0 will after a while lead to a constant output y(k) = y0 satisfying the equation y0 + a1 y0 + a2 y0 = b1 u0 + b2 u0
(B.24)
or y0 =
b1 + b2 u0 1 + a1 + a2
and the DC-gain is then seen to be the value of the transfer function for z = 1.
(B.25)
Appendix C List of Publications C.1 Journal Publications 1. Endusa Billy Muhando, Tomonobu Senjyu, Atsushi Yona, Hiroshi Kinjo, and Toshihisa Funabashi, 2007. “Disturbance Rejection by Dual Pitch Angle and Self-tuning Regulator for WTG Parametric Uncertainty Compensation,” IET - Control Theory and Applications, Vol. 1, No. 5, pp. 1431-1440, Sept. 2007. DOI:10.1049/iet-cta:20060448. 2. Endusa Billy Muhando, Tomonobu Senjyu, Naomitsu Urasaki, Atsushi Yona, Hiroshi Kinjo, and Toshihisa Funabashi, “Gain Scheduling Control of Variable Speed WTG Under Widely Varying Turbulence Loading,” Renewable Energy, Vol. 32, No. 14, pp. 2407-2423, 2007. DOI:10.1016/j.renene.2006.12.011. 3. Endusa Billy Muhando, Tomonobu Senjyu, Atsushi Yona, Hiroshi Kinjo, and Toshihisa Funabashi, “Regulation of WTG Dynamic Response to Parameter Variations of Analytic Wind Stochasticity,” Wind Energy, (In Press), DOI:10:1002/we.236. 4. Endusa Billy Muhando, Tomonobu Senjyu, Hiroshi Kinjo, and Toshihisa Funabashi, “Augmented LQG Controller for Enhancement of Online Dynamic Performance for WTG System,” Renewable Energy, (In Press), DOI:10.1016/j.renene.2007.12.001. 5. Endusa Billy Muhando, Tomonobu Senjyu, Eitaro Omine, Hiroshi Kinjo, and Toshihisa Funabashi, “Model Development for Nonlinear Dynamic Energy Conversion System: an Advanced Intelligent Control Paradigm for Optimality and Reliability,” IEEJ Trans. Power and Energy, 2007. (Accepted for publication).
APPENDIX C. LIST OF PUBLICATIONS
163
6. Endusa Billy Muhando, Tomonobu Senjyu, Hiroshi Kinjo, and Toshihisa Funabashi, “Extending the Modeling Framework for Wind Generation Systems: RLS-Based Paradigm for Performance under High Turbulence Inflow,” IEEE Trans. Energy Conversion, 2007. (Forthcoming). 7. Endusa Billy Muhando, Tomonobu Senjyu, Hiroshi Kinjo, Zachary Otara Siagi, and Toshihisa Funabashi, “Intelligent Optimal Control of Nonlinear Wind Generating System by a ModelingBased Approach,” IET - Renewable Power Generation, 2007. (Accepted for publication). 8. Tomonobu Senjyu, Endusa Billy Muhando, Atsushi Yona, Naomitsu Urasaki, Hiroshi Kinjo, and Toshihisa Funabashi, “Maximum Wind Power Capture by Sensorless Rotor Position and Wind Velocity Estimation from Flux Linkage and Sliding Observer,” Int. Journal of Emerging Electric Power Systems, Vol. 8, No. 2, Art. 3, pp. 1- 9, 2007. 9. Tomonobu Senjyu, Satoshi Tamaki, Endusa Billy Muhando, Naomitsu Urasaki, Hiroshi Kinjo, Toshihisa Funabashi, Hideki Fujita, and Hideomi Sekine, “Wind Velocity and Rotor Position Sensorless Maximum Power Point Tracking Control for WGS,” Renewable Energy, Vol. 31, No. 11, pp. 1764-1775, 2006. DOI:10.1016/j.renene.2005.09.020. 10. Tomonobu Senjyu, Eitaro Omine, Daisuke Hayashi, Endusa Billy Muhando, Atsushi Yona, and Toshihisa Funabashi, “Balancing Control for Dispersed Generators Considering Torsional Torque Suppression and AVR Performance for Synchronous Generators,” IEEJ Trans. Power and Energy, vol. 128, no. 1, pp. 75-83, 2008. (in Japanese). —————————————————————————————————————– 11*. Endusa Billy Muhando, Hiroshi Kinjo, Eiho Uezato, Tomonobu Senjyu, and Tetsuhiko Yamamoto, “Online Neurocontroller Design Optimized by a Genetic Algorithm for a Multi-trailer System,” Journal of the Society of Instrument and Control Engineers (SICE), Vol. 42, No. 9, pp. 1017-1026, 2006. 12*. Endusa Billy Muhando, Hiroshi Kinjo, and Tetsuhiko Yamamoto, “Enhanced Performance for Multivariable Optimization Problems by Use of Genetic Algorithms with Recessive Gene Structure,” Artificial Life & Robotics - Springer Japan, Vol. 10, No. 1, pp. 11-17, 2006. DOI:10.1007/s10015-005-0355-7. * Not related to PhD research work presented in the Thesis.
APPENDIX C. LIST OF PUBLICATIONS
164
C.2 Journal Papers under Peer Review 1. Endusa Billy Muhando, Tomonobu Senjyu, Hiroshi Kinjo, and Toshihisa Funabashi, “Stochastic Inequality Constrained Closed-loop Model-based Predictive Control of MW-Class Wind Generating System in the Electric Power Supply,” IET Procs. Renewable Power Generation 2. Endusa Billy Muhando, Tomonobu Senjyu, Hiroshi Kinjo, and Toshihisa Funabashi, “Model Fidelity Prerequisites for Individual Blade Pitch Regulation of Wind Generating System with State-Feedback Control,” IEEE Trans. Energy Conversion —————————————————————————————————————– 3*. Endusa Billy Muhando, Hiroshi Kinjo, Tomonobu Senjyu, Tetsuhiko Yamamoto, and Toshihisa Funabashi, “Multi-trailer Back-up Conundrum Revisited: LQR for Control Load Mitigation on Neurocontroller,” Automatica * Not related to PhD research work presented in the Thesis.
APPENDIX C. LIST OF PUBLICATIONS
165
C.3 Conference Papers: Presented 1. Endusa Billy Muhando, Tomonobu Senjyu, and Hiroshi Kinjo, “Disturbance Rejection by Stochastic Inequality Constrained Closed-loop Model-Based Predictive Control of MW-Class Wind Generating System,” Presented at the IEEJ-IEICE Joint Conference 2007), University of the Ryukyus, Okinawa, Japan, 19 Dec. 2007. 2. Endusa Billy Muhando, Tomonobu Senjyu, Otara Zachary Siagi, and Toshihisa Funabashi, “Intelligent Optimal Control of Wind Power Generating System by a Complemented LQG Approach,” Presented at the IEEE-Power Engineering Society Conference & Exhibition (Powerafrica2007), Johannesburg, South Africa, 16-20 July 2007. 3. Endusa Billy Muhando, Tomonobu Senjyu, Atsushi Yona, Hiroshi Kinjo, and Toshihisa Funabashi, “RLS-Based Self-Tuning Regulator for WTG Dynamic Performance Enhancement Under Stochastic Setting,” Presented at the International Conference on Electrical Engineering (ICEE 2007), Hong Kong, 8-12 July 2007. 4. Endusa Billy Muhando, Tomonobu Senjyu, Atsushi Yona, and Hiroshi Kinjo, “Evolutionary Intelligent Control of Wind Turbines for Optimized Performance and Reliability,” Presented at the IEEE-Power Engineering Society General Meeting 2007, Tampa, FL. USA, 24-28 June 2007. 5. Endusa Billy Muhando, Tomonobu Senjyu, Naomitsu Urasaki, Atsushi Yona, and Toshihisa Funabashi, “Robust Predictive Control of Variable-Speed Wind Turbine Generator by SelfTuning Regulator,” Presented at the IEEE-Power Engineering Society General Meeting 2007, Tampa, FL. USA, 24-28 June 2007. 6. Endusa Billy Muhando, Tomonobu Senjyu, Naomitsu Urasaki, Hiroshi Kinjo, and Toshihisa Funabashi, “Online WTG Dynamic Performance and Transient Stability Enhancement by Evolutionary LQG,” Presented at the IEEE-Power Engineering Society General Meeting 2007, Tampa, FL. USA, 24-28 June 2007. 7. Tomonobu Senjyu, Yasutaka Ochi, Endusa Billy Muhando, Naomitsu Urasaki, and Hideomi Sekine, “Speed and Position Sensorless Maximum Power Point Tracking Control for WGS with Squirrel Cage Induction Generator,” Presented at the IEEE-Power Engineering Society Power Systems Conference & Exposition (PSCE’06), Atlanta, GA. USA, 29 Oct.–01 Nov. 2006.
APPENDIX C. LIST OF PUBLICATIONS
166
C.4 Conference Papers: Scheduled 1. Endusa Billy Muhando, Tomonobu Senjyu, Eitaro Omine, Yuri Yonaha, and Toshihisa Funabashi, “Steady-state and Transient Dynamic Response of Grid-Connected WECS with Asynchronous DOIG by Predictive Control under Turbulent Inflow,” To be presented at the IEEEPower Engineering Society General Meeting, Pittsburg, Pennsylvania, USA, 20–24 July 2008. 2. Endusa Billy Muhando, Tomonobu Senjyu, Eitaro Omine, and Toshihisa Funabashi, “Full State Feedback Digital Control of WECS with State Estimation by Stochastic Modeling Design,” To be presented at the IEEE-Power Engineering Society General Meeting, Pittsburg, Pennsylvania, USA, 20–24 July 2008. 3. Endusa Billy Muhando, Tomonobu Senjyu, and Toshihisa Funabashi, “Model Fidelity Prerequisites for Variable Speed Pitch-Regulated WECS with State-Feedback Control,” To be presented at the IEEE International Symposium on Industrial Electronics (ISIE 2008), Cambridge, UK, 30 June–02 July 2008. 4. Endusa Billy Muhando, Tomonobu Senjyu, Hideomi Sekine, and Toshihisa Funabashi, “Individual Blade Pitch Regulation for Variable Speed Wind Energy Conversion System with State-Feedback Control,” To be presented at the IEEJ Power Engineering Society Conference (PES’08), Hiroshima, Japan, 24–26 September, 2008.
C.5 Conference Papers: Other 1. Hiroshi Kinjo, Endusa Billy Muhando, Kunihiko Nakazono, Eiho Uezato, and Tetsuhiko Yamamoto, “Real-time Design and Control of Multi-trailer System Using Neurocontroller Optimized by a Genetic Algorithm,” Presented at the 9th International Conference on Mechatronics Technology (ICMT 2005), Kuala Lampur, Malaysia, 5-8 Dec. 2005
