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Optimal Control of wind energy systems, Notas de estudo de Engenharia Elétrica

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2014

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Baixe Optimal Control of wind energy systems e outras Notas de estudo em PDF para Engenharia Elétrica, somente na Docsity! Advances in Industrial Control Other titles published in this series: Digital Controller Implementation and Fragility Robert S.H. Istepanian and James F. Whidborne (Eds.) Optimisation of Industrial Processes at Supervisory Level Doris Sáez, Aldo Cipriano and Andrzej W. Ordys Robust Control of Diesel Ship Propulsion Nikolaos Xiros Hydraulic Servo-systems Mohieddine Jelali and Andreas Kroll Model-based Fault Diagnosis in Dynamic Systems Using Identification Techniques Silvio Simani, Cesare Fantuzzi and Ron J. Patton Strategies for Feedback Linearisation Freddy Garces, Victor M. Becerra, Chandrasekhar Kambhampati and Kevin Warwick Robust Autonomous Guidance Alberto Isidori, Lorenzo Marconi and Andrea Serrani Dynamic Modelling of Gas Turbines Gennady G. Kulikov and Haydn A. Thompson (Eds.) Control of Fuel Cell Power Systems Jay T. Pukrushpan, Anna G. Stefanopoulou and Huei Peng Fuzzy Logic, Identification and Predictive Control Jairo Espinosa, Joos Vandewalle and Vincent Wertz Optimal Real-time Control of Sewer Networks Magdalene Marinaki and Markos Papageorgiou Process Modelling for Control Benoît Codrons Computational Intelligence in Time Series Forecasting Ajoy K. Palit and Dobrivoje Popovic Modelling and Control of Mini-Flying Machines Pedro Castillo, Rogelio Lozano and Alejandro Dzul Ship Motion Control Tristan Perez Hard Disk Drive Servo Systems (2nd Ed.) Ben M. Chen, Tong H. Lee, Kemao Peng and Venkatakrishnan Venkataramanan Measurement, Control, and Communication Using IEEE 1588 John C. Eidson Piezoelectric Transducers for Vibration Control and Damping S.O. Reza Moheimani and Andrew J. Fleming Manufacturing Systems Control Design Stjepan Bogdan, Frank L. Lewis, Zdenko Kova i and José Mireles Jr. Windup in Control Peter Hippe Nonlinear H2/H Constrained Feedback Control Murad Abu-Khalaf, Jie Huang and Frank L. Lewis Practical Grey-box Process Identification Torsten Bohlin Control of Traffic Systems in Buildings Sandor Markon, Hajime Kita, Hiroshi Kise and Thomas Bartz-Beielstein Wind Turbine Control Systems Fernando D. Bianchi, Hernán De Battista and Ricardo J. Mantz Advanced Fuzzy Logic Technologies in Industrial Applications Ying Bai, Hanqi Zhuang and Dali Wang (Eds.) Practical PID Control Antonio Visioli (continued after Index) Advances in Industrial Control Series Editors Professor Michael J. Grimble, Professor of Industrial Systems and Director Professor Michael A. Johnson, Professor (Emeritus) of Control Systems and Deputy Director Industrial Control Centre Department of Electronic and Electrical Engineering University of Strathclyde Graham Hills Building 50 George Street Glasgow G1 1QE United Kingdom Series Advisory Board Professor E.F. Camacho Escuela Superior de Ingenieros Universidad de Sevilla Camino de los Descubrimientos s/n 41092 Sevilla Spain Professor S. Engell Lehrstuhl für Anlagensteuerungstechnik Fachbereich Chemietechnik Universität Dortmund 44221 Dortmund Germany Professor G. Goodwin Department of Electrical and Computer Engineering The University of Newcastle Callaghan NSW 2308 Australia Professor T.J. Harris Department of Chemical Engineering Queen’s University Kingston, Ontario K7L 3N6 Canada Professor T.H. Lee Department of Electrical Engineering National University of Singapore 4 Engineering Drive 3 Singapore 117576 Professor Emeritus O.P. Malik Department of Electrical and Computer Engineering University of Calgary 2500, University Drive, NW Calgary Alberta T2N 1N4 Canada Professor K.-F. Man Electronic Engineering Department City University of Hong Kong Tat Chee Avenue Kowloon Hong Kong Professor G. Olsson Department of Industrial Electrical Engineering and Automation Lund Institute of Technology Box 118 S-221 00 Lund Sweden Professor A. Ray Pennsylvania State University Department of Mechanical Engineering 0329 Reber Building University Park PA 16802 USA Professor D.E. Seborg Chemical Engineering 3335 Engineering II University of California Santa Barbara Santa Barbara CA 93106 USA Doctor K.K. Tan Department of Electrical Engineering National University of Singapore 4 Engineering Drive 3 Singapore 117576 Professor Ikuo Yamamoto The University of Kitakyushu Department of Mechanical Systems and Environmental Engineering Faculty of Environmental Engineering 1-1, Hibikino,Wakamatsu-ku, Kitakyushu, Fukuoka, 808-0135 Japan To our families Preface Actual strategies for sustainable energy development have as prior objective the gradual replacement of fossil-fuel-based energy sources by renewable energy ones. Among the clean energy sources, wind energy conversion systems currently carry significant weight in many developed countries. Following continual efforts of the international research community, a mature wind energy conversion technology is now available to sustain the rapid dynamics of concerned investment programs. The main problem regarding wind power systems is the major discrepancy between the irregular character of the primary source (wind speed is a random, strongly non-stationary process, with turbulence and extreme variations) and the exigent demands regarding the electrical energy quality: reactive power, harmonics, flicker, etc. Thus, wind energy conversion within the parameters imposed by the energy market and by technical standards is not possible without the essential contribution of automatic control. The stochastic nature of the primary energy source represents a risk factor for the viability of the mechanical structure. The literature concerned emphasises the importance of the reliability criterion, sometimes more important than energy conversion efficiency (e.g., in the case of off-shore farms), in assessing global economic efficiency. This aspect must be taken into account in control strategies. Many research works deal with wind power systems control, aiming at optimising the energetic conversion, interfacing wind turbines to the grid and reducing the fatigue load of the mechanical structure. Meanwhile, the gap between the development of advanced control algorithms and their effective use in most of the practical engineering domain is widely recognized. Much work has been and continues to be done, especially by the research community, in order to bridge this gap and ease the technology transfer in control engineering. This book is aimed at presenting a point of view on the wind power generation optimal control issues, covering a large segment of industrial wind power applications. Its main idea is to propose the use of a set of optimization criteria which comply with a comprehensive set of requirements, including the energy conversion efficiency, mechanical reliability, as well as quality of the energy provided. This idea opens the perspective toward a multi-purpose global control approach. Prefacexii A series of control techniques are analyzed, assessed and compared, starting from the classical ones, like PI control, maximum power point strategies, LQG optimal control techniques, and continuing with some modern ones: sliding-mode techniques, feedback linearization control and robust control. The discussion is aimed at identifying the benefits of dynamic optimization approaches to wind power systems. The main results are presented along with illustration by case studies and MATLAB®/Simulink® simulation assessment. The corresponding software programmes and block diagrams are included on the back-of-book software material. For some of the case studies presented real-time simulation results are also available. The discourse of this book concludes by stressing the point on the possibility of designing WECS control laws based upon the frequency separation principle. The idea behind this is simple. First, one must define the set of quality demands the control law must comply with. Then one seeks to split this set into contradictory pairs, for each of them a component of the control law being separately synthesized. Finally, these components are summed to yield the total control input. This approach is possible because the different WECS dynamic properties usually involved in the imposed quality requirements are exhibited in disjointed frequency ranges. Offering a thorough description of wind energy conversion systems – principles, functionality, operation modes, control goals and modelling – this book is mainly addressed to researchers with a control background wishing either to approach or to go deeper in their study of wind energy systems. It is also intended to be a guide for control engineers, researchers and graduate students working in the field in learning and applying systematic optimization procedures to wind power systems. The book is organised in seven chapters preceded by a glossary and followed by a concluding chapter, three appendices, a list of pertinent references and an index. Chapter 1 realises an introduction about the wind energy resource and systems. Chapter 2 presents a systemic analysis of the main parts of a wind energy conversion system and introduces the associated control objectives. The modelling development needed for control purposes is presented in the Chapter 3. Chapter 4 is dedicated to explaining the fundamentals of the wind turbine control systems. In Chapter 5 some powerful control methods for energy conversion maximization are presented, each of which is illustrated by a case study. Chapter 6 deals with mixed optimization criteria and introduces the frequency separation principle in the optimal control of the wind energy systems, whose effectiveness is suggested by two case studies. Chapter 7 is focused on using the hardware-in-the-loop simulation philosophy for building development systems that experimentally validate the wind energy systems control laws. A case study is presented to illustrate the proposed methodology. Chapter 8 discusses general conclusions and suggestions for future development of WECS control laws. Appendix A offers detailed information about the features of systems used in the case studies. Appendix B resumes the main theoretical results supporting the sliding-mode, feedback linearization and QFT robust control methods. Finally, Preface xiii reported case studies. We would like to acknowledge the Romanian National Authority for Scientific Research (ANCS – CEEX Research Programme) and the Romanian National University Research Council (CNCSIS) for their partial financial support during the period in which this manuscript was written. Gala i and Roskilde, Iulian Munteanu August 2007 Antoneta Iuliana Bratcu Nicolaos-Antonio Cutululis Emil Ceang Appendix C presents some illustrations accompanying the implementation of the Contents xvii 5.2.3 Case Study (2): Classical MPPT vs. MPPT with Wind Turbulence as Searching Signal..................................................................... 124 5.2.4 Conclusion .................................................................................. 128 5.3 PI Control ............................................................................................... 129 5.3.1 Problem Statement...................................................................... 129 5.3.2 Controller Design........................................................................ 130 5.3.3 Case Study (3): 2 MW WECS Optimal Control by PI Speed Control ........................................................................................ 132 5.3.4 Case Study (4): 6 kW WECS Optimal Control by PI Power Control ........................................................................................ 134 5.4 On–Off Control ...................................................................................... 135 5.4.1 Controller Design........................................................................ 135 5.4.2 Case Study (5)............................................................................. 140 5.5 Sliding-mode Control............................................................................. 142 5.5.1 Modelling.................................................................................... 143 5.5.2 Energy Optimization with Mechanical Loads Alleviation ......... 143 5.5.3 Case Study (6)............................................................................. 146 5.5.4 Real-time Simulation Results ..................................................... 147 5.5.5 Conclusion .................................................................................. 150 5.6 Feedback Linearization Control............................................................. 150 5.6.1 WECS Modelling........................................................................ 151 5.6.2 Controller Design........................................................................ 152 5.6.3 Case Study (7)............................................................................. 156 5.7 QFT Robust Control............................................................................... 158 5.7.1 WECS Modelling........................................................................ 158 5.7.2 QFT-based Control Design......................................................... 158 5.7.3 Case Study (8)............................................................................. 160 5.8 Conclusion ............................................................................................. 166 6 WECS Optimal Control with Mixed Criteria ............................................ 169 6.1 Introduction............................................................................................ 169 6.2 LQ Control of WECS............................................................................. 170 6.2.1 Problem Statement...................................................................... 170 6.2.2 Input–Output Approach .............................................................. 170 6.2.3 Case Study (9): LQ Control of WECS with Flexibly-coupled Generator Using R-S-T Controller ............................................. 173 6.3 Frequency Separation Principle in the Optimal Control of WECS........ 176 6.3.1 Frequency Separation of the WECS Dynamics.......................... 176 6.3.2 Optimal Control Structure and Design Procedure (2LFSP) ....... 177 6.3.3 Filtering and Prediction Algorithms for Wind Speed Estimation180 6.4 2LFSP Applied to WECS with Rigidly-coupled Generator .................. 182 6.4.1 Modelling.................................................................................... 182 6.4.2 Steady-state Optimization Within the Low-frequency Loop...... 185 6.4.3 LQG Dynamic Optimization Within the High-frequency Loop. 185 6.4.4 LQ Dynamic Optimization Within the High-frequency Loop.... 187 6.4.5 Case Study (10)........................................................................... 190 6.4.6 Global Real-time Simulation Results ......................................... 193 xviii Contents 6.5 2LFSP Applied to WECS with Flexibly-coupled Generator ................. 197 6.5.1 Modelling.................................................................................... 197 6.5.2 Steady-state Optimization Within the Low-frequency Loop...... 199 6.5.3 Dynamic Optimization Within the High-frequency Loop.......... 199 6.5.4 Case Study (11)........................................................................... 201 6.6 Concluding Remarks on the Effectiveness of 2LFSP............................ 204 6.7 Towards a Multi-purpose Global Control Approach ............................. 205 6.7.1 Control Objectives in Large Wind Power Plants........................ 205 6.7.2 Global Optimization vs. Frequency Separation Principle for a Multi-objective Control...................................................... 206 6.7.3 Frequency-domain Models of WECS......................................... 208 6.7.4 Spectral Characteristics of the Wind Speed Fluctuations........... 209 6.7.5 Open-loop Bandwidth Limitations of WECS Control Systems . 211 6.7.6 Frequency Separation Control of WECS.................................... 214 7 Development Systems for Experimental Investigation of WECS Control Structures ....................................................................... 219 7.1 Introduction............................................................................................ 219 7.2 Electromechanical Simulators for WECS.............................................. 220 7.2.1 Principles of Hardware-in-the-loop (HIL) Systems.................... 220 7.2.2 Systematic Procedure of Designing HIL Systems...................... 223 7.2.3 Building of Physical Simulators for WECS ............................... 223 7.2.4 Error Assessment in WECS HIL Simulators.............................. 225 7.3 Case Study (12): Building of a HIL Simulator for a DFIG-based WECS .................................................................................................... 229 7.3.1 Requirements Imposed to the WECS Simulator......................... 230 7.3.2 Building of the Real-time Physical Simulator (RTPS)............... 230 7.3.3 Building of the Investigated Physical System (IPS) and Electrical Generator Control....................................................... 233 7.3.4 Global Operation of the Simulated WECS................................. 236 7.4 Conclusion ............................................................................................. 237 8 General Conclusion ....................................................................................... 239 A Features of WECS Used in Case Studies .................................................... 243 B Elements of Theoretical Background and Development ........................... 247 B.1 Sliding-mode Control............................................................................. 247 B.2 Feedback Linearization Control............................................................. 249 B.3 QFT Robust Control............................................................................... 255 C Photos, Diagrams and Real-time Captures................................................. 261 References............................................................................................................ 269 Index..................................................................................................................... 281 Notation Wind Power System Aerodynamic Subsystem and Drive Train v , sv , tv Total, steady-state and turbulence wind speed [m/s] w Relative wind speed to the blades [m/s] Air density [kg/m3] tI , tL Turbulence intensity [–] and length [m] l , h Rotational speed of a wind turbine rotor (low- speed shaft) and of the high-speed shaft respectively [rad/s] R Blade length of a wind turbine [m] Pitch angle bN Number of blades of a wind turbine 2A R Area swept by the rotor blades [m2] P Prandtl’s coefficient [–] , opt Tip speed ratio of a wind turbine and its optimal value [–] Sv , nv , Mv Cut-in, rated and respectively cut-out wind speed of a wind turbine [m/s] airP Total power of a delimited moving mass of air [W] P , emP Generated active power and generator mechanical power respectively [W] wtP , nP Harvested and respectively rated power of a wind turbine [W] xxii Notation Acronyms and Abbreviations 2LFSP Two-loop control structure based on the frequency separation principle AS Aerodynamic subsystem BET Blade element theory BPS Basic physical system CS Control subsystem DFIG Doubly-fed induction generator DFT Discrete Fourier Transform DT Drive train EFT Effector (part of the HIL simulator) EMS Electromagnetic subsystem EPS Emulated physical system EPSM Model of the EPS FFT Fast Fourier Transform H/V AWT Horizontal-/vertical-axis wind turbine HFL High-frequency loop HIL Hardware-in-the-loop HILS Hardware-in-the-loop simulation HPF High-pass filter HSS High-speed shaft IPS Investigated physical system LFL Low-frequency loop LPF Low-pass filter LSS Low-speed shaft OP Operating point OOP Optimal operating point ORC Optimal regimes characteristic PMSG Permanent-magnet synchronous generator PWM Pulse-width modulation RTPS Real-time physical simulator RTSS Real-time software simulator SCIG Squirrel-cage induction generator TSC Tip speed controller TSR Tip speed ratio (V/C S) WECS / WPS (Variable-/constant-speed) wind energy conversion system / wind power system WRIG Wound-rotor induction generator WRSG Wound-rotor synchronous generator 1 Wind Energy 1.1 Introduction The use of the wind has a history of thousands of years. Since ancient times wind power has been used for different purposes, varying from agricultural activities, like grain milling and water pumping to, nowadays, electricity production. Since the early 1970s oil crisis, wind power technology has experienced an important development, moving – in just two decades – from a low level, experimental technology used mainly for batteries charging to a mainstream power technology. Today, wind power is by far the fastest-growing renewable energy source. Wind power is free, clean and endless. Furthermore, the cost of the electricity produced by wind turbines is fixed once the plant has been built (EWEA 2005) and it has already reached the point where the cost of the electricity produced by wind is comparable with that of electricity produced by some of the conventional, fossil- based power plants (Parfit and Leen 2005). The power produced by wind worldwide reached, at the end of 2004, 48 GW, representing 0.57% of the total world electricity supply. The figure might not seem impressive, but when compared to other renewable energy technologies, it becomes clear that wind power is the most promising one. As an example, wind power is still a small electricity player on the European market, producing 2.4% of its total electricity production. This will change as the European Union has decided to make wind power a major electricity source, with a 12% market share in 2020 and 20% in 2030 (EWEA 2005). 1.2 State of the Art and Trends in Wind Energy Conversion Systems Wind energy conversion systems (WECS) constitute a mainstream power technology that is largely underexploited. Wind technology has made major progression from the prototypes of just 25 years ago. Two decades of technological progress has resulted in today’s wind turbines looking and being much more like 1 Wind Energy 2 power stations, in addition to being modular and rapid to install. A single wind turbine can produce 200 times more power than its equivalent two decades ago (EWEA 2005). The low-power WECS have not however lost their importance, being nowadays of great interest in islanding generation, hybrid microgrid systems, distributed energy production, etc. Today, WECS represent a mature technology still with important development potential. 1.2.1 Issues in WECS Technology The development of various wind turbine concepts in the last decade has been very dynamic. The main differences in wind turbine concepts are in the electrical design and control. Thus, WECS can be classified according to speed control and power control ability, leading to wind turbine classes differentiated by the generating system (speed control) and the method employed for limiting the aerodynamic efficiency above the rated power (power control). The speed-control criterion leads to two types of WECS: fixed-speed and variable-speed wind turbines, while the power control ability divides WECS into three categories: stall-controlled, pitch-controlled and active-stall-controlled wind turbines. Fixed-speed WECS Fixed-speed wind turbines are the pioneers of the wind turbine industry. They are simple, reliable and use low-cost electrical parts. They use induction generators and they are connected directly to the grid, giving them an almost constant rotor speed stuck to the grid frequency, regardless of the wind speed. Variable-speed WECS Variable-speed wind turbines are currently the most used WECS. Their advantages, compared to fixed-speed wind turbines, are numerous. First of all and most important, the decoupling between the generating system and the grid frequency makes them more flexible in terms of control and optimal operation. Of course, this comes at a price, namely the use of power electronic converters, which are the interfaces between the electrical generator and the grid and thus they actually make the variable-speed operation possible. But still, the high controllability offered by the variable-speed operation is a powerful advantage in achieving higher and higher wind energy penetration levels (Sørensen et al. 2005; Hansen and Hansen 2007). The variable-speed operation allows the rotational speed of the wind turbine to be continuously adapted (accelerated or decelerated) in such a manner that the wind turbine operates constantly at its highest level of aerodynamic efficiency. While fixed-speed wind turbines are designed to achieve maximum aerodynamic efficiency at one wind speed, variable-speed wind turbines achieve maximum aerodynamic efficiency over a wide range of wind speeds. Furthermore, variable- speed operation allows the use of advanced control methods, with different objectives: reduced mechanical stress, reduced acoustical noise, increased power capture, etc. (Ackermann 2005; Burton et al. 2001). 1.2 State of the Art and Trends in Wind Energy Conversion Systems 5 1.2.3 Low-power WECS Interest has also grown in low-power wind turbines, due to their application in insulated grids and distributed energy production, from which the microgrid concept has emerged (Kanellos and Hatziargyriou 2002). Low-power WECS are being incorporated both in stand-alone generation systems, as well as elements of hybrid power systems. Related to the latter, some typical applications are the hybrid wind-photovoltaic generation systems or wind turbines in conjunction with fuel-cell/diesel, all of them using accumulator batteries for energy storage. Because of the very high penetration level, the control problems are here somehow different from those related to wind farms, being strongly dependent on the current application. For example, for water pumping or house heating the control objectives are obviously different from ensuring power quality standards of an insulated utility grid. Therefore, the main problems in insulated grids relate to the wind energy sources scheduling depending on the instantaneous consumption and on the power reserve from other generators (taking account of energy storage). Besides the captured power maximisation and the reliability-related issues, control focuses on the local power system stability and the delivered power conditioning (fluctuations, harmonics, etc.). In some cases the generators contained in hybrid systems (e.g., wind- photovoltaic-accumulator) feed a common DC-bus. Here, the problem is the global control of the system in order to ensure continuity of power supply, while complying with operation requirements. The latter can relate to the life-time of the system components which must not be affected by the control action (e.g., regularity of accumulator charge/discharge cycles, the diesel-generator on/off regimes, etc.). 1.2.4 Issues in WECS Control The challenge in WECS control is to ensure good quality electrical energy delivery from a profoundly irregular primary source, the wind. Modern wind generation systems are equipped with control and supervision subsystems implementing the supervisory control and data acquisition (SCADA) concept. Generally, there are three low-level control systems, which are briefly reviewed in the following. Aerodynamic power control acting on the blades is based upon well-established and widely-used techniques. Industrial applications already benefit from the classical PI or optimal control structure. As regards generator control ensuring variable-speed operation, the literature offers a multitude of control techniques waiting for field testing; however, none of them has become classical such as to be widely used by wind turbine integrators. A unitary variable-speed strategy has not yet been established and the real-world applications actually implement only the basic control laws. Finally, grid interface control and output power conditioning are intensively researched because the grid connection standards are continuously changing. The control objectives, problem formulations and their methods of 1 Wind Energy 6 solution depend greatly on the current generation structure, local utility grid, operating regime (i.e., islanding or grid-connected), etc. Many research works deal with WECS control, aiming at optimising the energy conversion, interfacing wind turbines to the grid and even reducing the fatigue load of the mechanical structure. The idea of building unitary approaches based on optimization criteria, complying with a comprehensive set of requirements that depend on the actual application, opens the perspective toward a multi-criteria global control approach. 1.3 Outline of the Book The book is organised in eight chapters preceded by a glossary and followed by three appendices, a list of references and an index. After this first, introductory chapter, in the second chapter the wind energy resource is presented and the main parts of a wind energy conversion system are analysed from a functional point of view: the turbine (rotor), the drive train and the electrical subsystem. The associated control objectives are stated at the end of this chapter. The modelling development needed for control purposes is presented in the third chapter. The analysis starts with the exogenous variable, namely the wind, and provides fixed-point wind speed models and also models of the wind speed experienced by the turbine rotor. Then the models of the subsystems described in the previous chapter are detailed. This chapter ends with a case study illustrating the dynamic properties analysis of a class of wind power systems. The fourth chapter is dedicated to explaining the fundamentals of the wind turbine control systems. Here are included the closed-loop systems to fulfil the so- called primary objectives – stall and pitch control – as well as more advanced control systems generally derived from mixed optimization criteria. Controllers for the reactive power and for the energy quality when operating under grid conditions are also presented. In the fifth chapter some powerful control methods for energy conversion maximization in the partial-load regime are presented, which can be classified depending on how rich the knowledge is that they use about the system. Each such method is illustrated by a case study in order to allow the assessment of their performances and drawbacks. The conclusion of this chapter suggests the idea of expressing the various WECS control requirements by mixed criteria. When mixed optimization criteria are formulated, for example, if, apart the energy conversion maximization, a mechanical reliability constraint is imposed, then more complex control structures are needed. In the sixth chapter the frequency separation principle in the optimal control of the wind energy systems is formulated, which is fundamental for the intended design methodology. Two case studies are presented here to illustrate the application of this principle to rigidly- and flexibly-coupled-generator-based wind power systems. The seventh chapter deals with development systems used for experimentally validating the control laws associated with wind power systems. These experimental simulators are based on the hardware-in-the-loop (HIL) philosophy, 1.3 Outline of the Book 7 consisting of closed-loop connecting hardware and software elements, in order to replicate the real-world systems and their operating conditions. A case study is presented to illustrate the closed-loop optimised functioning of an induction- generator-based variable-speed wind energy conversion system. The last chapter of the book presents some general conclusions and suggests future directions in developing WECS control laws. Appendix A provides extensive information about the parameters of WECS used in the case studies. Both low- and high-power, rigid- and flexible-drive-train, induction- or permanent-magnet-synchronous-generator-based WECS have been chosen as illustrative examples. Appendix B resumes the main theoretical results supporting the sliding-mode, feedback linearization and QFT robust control methods. Appendix C groups together some photos, diagrams and real-time captures that accompany the implementation of the reported case studies. 2 Wind Energy Conversion Systems 10 where c is the Weibull scale parameter, with units equal to the wind speed units, k is the unitless Weibull shape parameter, v is the wind speed, vi is a particular wind speed, dv is the wind speed increment, diP v v v v is the probability that the wind speed is between v and v + dv and 0P v is the probability that the wind speed exceeds zero. The cumulative distribution function is given by 0 1 exp k i i v P v v P v c (2.2) The two Weibull parameters and the average wind speed are related by 11v c k , (2.3) where v is the average wind speed and ( ) is the complete gamma function. A special case is when 2k , the Weibull distribution becoming a Rayleigh distribution. In this case, the factor 1 1 k has the value 2 0.8862 . The influence of the k parameter on the probability density function is presented in Figure 2.1, with the scale factor kept constant. Simply speaking, the variation of the hourly mean speed around the annual mean is small as k is higher, as depicted in Figure 2.1. 0 5 10 15 20 25 0 0.05 0.1 0.15 0.2 Wind speed [m/s] P ro ba bi lit y de ns ity k=1.5 k=2 k=2.5 k=3 k=3.5 Figure 2.1. Weibull distributions as a function of k (constant c) The scale factor c shows how “windy” a location is or, in other words, how high the annual mean speed is. The influence of the scale factor on the probability density function is presented in Figure 2.2, with the shape factor kept constant. The estimation of the Weibull distribution parameters – c and k – is usually done with two methods. One method for calculating parameters c and k is starting from Equation 2.2 and taking the natural logarithm of both sides: 2.1 Wind Energy Resource 11 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 Wind speed [m/s] P ro ba bi lit y de ns ity c=2 c=3 c=4 c=5 c=6 Figure 2.2. Weibull distributions as a function of c (constant k) ln ln 1 ln lni iP v k v c (2.4) Using the notation 0ln ln 1 ; ln ; lni iy P v u v k c , (2.5) Equation 2.4 becomes: 0y ku k , (2.6) where k and k0 are calculated using linear regression of the cumulative distribution function. Finally, the scale factor c is calculated: 0 exp kc k (2.7) The second method, the maximum likelihood method (Seguro and Lambert 2000), uses the time-series wind data instead of the cumulative distribution function. The two parameters are calculated respectively with 1 1 1 1 ln ln n nk i i i i i n k i i v v v k n v , (2.8) 1 1 1 kn k i i c v n , (2.9) 2 Wind Energy Conversion Systems 12 where vi is the wind speed in time step i and n is the number of nonzero wind speed data points. Since Equation 2.8 must be solved using an iterative procedure, it is suitable to start with the initial guess 2k . Knowing the annual variation of the wind on a given site is important, but it is not sufficient for assessing the economic viability of the wind turbine installation. For that purpose, the level of wind resource is often defined in terms of the wind- power-density value, expressed in watts per square meter (W/m2). This value incorporates the combined effects of the wind speed frequency distribution and the dependence on the air density and the cube of the wind speed. The power of the wind over an area A is given by 31 2t P Av , (2.10) where is the air density. Thus, the mean wind power density, over an area A, can be calculated with 31 2 meantP v A , (2.11) where the mean value of the cube of the wind speed is 3 3 0 dv v p v v , (2.12) with p v being the probability density function. After integrating Equation 2.12 and using the Weibull function 3 3 3 1 3 1 1 k v v k , (2.13) where 3 1 3 1 1 k e k k (2.14) is called energy pattern factor (EFP – Jamil 1990). Using Equations 2.13 and 2.14, Equation 2.11 becomes 31 2 meantP e k v A (2.15) Thus, the mean wind power density is proportional to the EFP and the cube of the wind speed. The evolution of the mean wind power density as a function of the Weibull distribution function parameters, c and k, is presented in Figure 2.3. 2.3 Wind Turbine Aerodynamics 15 All wind turbines have a mechanism that moves the nacelle such that the blades are perpendicular to the wind direction. This mechanism could be a tail vane (small wind turbines) or an electric yaw device (medium and large wind turbines). Concerning the power conversion chain, it involves naturally some loss of power. Because of the nonzero wind velocity behind the wind turbine rotor one can easily understand that its efficiency is less than unity. Also, depending on the operating regime, both the motion transmission and the electrical power generation involve losses by friction and by Joule effect respectively. Being directly coupled one with the other, the energy conversion chain elements dynamically interact, mutually influencing their operation. 2.3 Wind Turbine Aerodynamics The wind turbine rotor interacts with the wind stream, resulting in a behaviour named aerodynamics, which greatly depends on the blade profile. 2.3.1 Actuator Disc Concept The analysis of the aerodynamic behaviour of a wind turbine can be done, in a generic manner, by considering the extraction process (Burton et al. 2001). Consider an actuator disc (Figure 2.5) and an air mass passing across, creating a stream-tube. vu v0 vw pu pw 0p 0p Figure 2.5. Energy extracting actuator disc The conditions (velocity and pressure) in front of the actuator disc are denoted with subscript u, the ones at the disc are denoted with 0 and, finally, the conditions behind the disc are denoted with w. The momentum u wH m v v transmitted to the disc by the air mass m passing through the disc with cross-section A produces a force, expressed as 0 0 u w u w u w m v v Av t v vHT Av v v t t t (2.20) or 0 0T A p p (2.21) 2 Wind Energy Conversion Systems 16 Using Bernoulli’s equation, the pressure difference is 2 2 0 0 1 2 u w p p v v (2.22) and, replacing Equation 2.12 in Equation 2.21, results 2 21 2 u w T A v v (2.23) From Equations 2.20 and 2.23 one gets 0 0 1 2 2 u w u w u v v v v v v v (2.24) The kinetic energy of an air mass travelling with a speed v is 21 2k E mv , (2.25) where m is the air mass that passes the disc in a unit length of time, e.g., 0m Av ; then the power extracted by the disc is 2 2 0 1 2 u w P Av v v (2.26) or 231 4 1 2 P Av a a , (2.27) with 01 ua v v . The power coefficient, denoting the power extraction efficiency, is defined as 23 3 0.5 4 1 0.5 p t Av a aPC P Av (2.28) Therefore 24 1pC a a (2.29) The maximum value of pC occurs for 1 3a and is max 0.59pC , known as the Betz limit (Betz 1926) and represents the maximum power extraction efficiency of a wind turbine. 2.3.2 Wind Turbine Performance A wind turbine is a power extracting device. Thus, the performance of a wind turbine is primarily characterized by the manner in which the main indicator – power – varies with wind speed. Besides that, other indicators like torque and 2.3 Wind Turbine Aerodynamics 17 thrust are important when the performances of a wind turbine are assessed. The generally accepted way to characterize the performances of a wind turbine is by expressing them by means of non-dimensional characteristic performance curves (Burton et al. 2001). The tip speed ratio of a wind turbine is a variable expressing the ratio between the peripheral blade speed and the wind speed. It is denoted by and computed as l R v , (2.30) where R is the blade length, l is the rotor speed (the low-speed shaft rotational speed) and v is the wind speed. The tip speed ratio is a key variable in wind turbine control and will be extensively used in the rest of the book. It characterizes the power conversion efficiency and it is also used to define the acoustic noise levels. The power coefficient, pC , describes the power extraction efficiency of a wind turbine. The aerodynamic performance of a wind turbine is usually characterized by the variation of the non-dimensional pC vs. curve. Based upon Equation 2.28, the power extracted by a wind turbine whose blade length is R is expressed as 2 31 2wt p P R v C (2.31) Therefore, the pC performance curve gives information about the power efficiency of a wind turbine. Figure 2.6 presents this curve for a typical two-bladed wind turbine. One can see that the conversion efficiency is lower than the Betz limit (0.59), which is normal since the Betz limit assumes perfect blade design. The theoretical reasons for such an allure of the pC curve lie in the aerodynamic blade theory; some justifications are given in Chapter 3. 0 0 0.1 0.2 0.3 0.4 0.5 pC opt 15 Figure 2.6. Cp( ) performance curve For control purposes, useful information arising from the pC performance curve is the fact that the power conversion efficiency has a well determined maximum for a specific tip speed ratio, denoted by opt . 2 Wind Energy Conversion Systems 20 2.5.1 Fixed-speed WECS Fixed-speed WECS operate at constant speed. That means that, regardless of the wind speed, the wind turbine rotor speed is fixed and determined by the grid frequency. Fixed-speed WECS are typically equipped with squirrel-cage induction generators (SCIG), softstarter and capacitor bank and they are connected directly to the grid, as shown in Figure 2.8. This WECS configuration is also known as the “Danish concept” because it was developed and widely used in Denmark (Hansen and Hansen 2007). Figure 2.8. General structure of a fixed-speed WECS Initially, the induction machine is connected in motoring regime such that it generates electromagnetic torque in the same direction as the wind torque. In steady-state, the rotational speed exceeds the synchronous speed and the electromagnetic torque is negative. This corresponds to the squirrel-cage induction machine operation in generation mode (or in the over-synchronous regime – Bose 2001). As it is directly connected to the grid, the SCIG works on its natural mechanical characteristic having an accentuated slope (corresponding to a small slip) given by the rotor resistance. Therefore, the SCIG rotational speed is very close to the synchronous speed imposed by the grid frequency. Furthermore, the wind velocity variations will induce only small variations in the generator speed. As the power varies proportionally with the wind speed cubed, the associated electromagnetic variations are important. SCIG are preferred because they are mechanically simple, have high efficiency and low maintenance cost. Furthermore, they are very robust and stable. One of the major drawbacks of the SCIG is the fact that there is a unique relation between active power, reactive power, terminal voltage and rotor speed (Ackermann 2005). That means that an increase in the active power production is possible only with an increase in the reactive power consumption, leading to a relatively low full-load power factor. In order to limit the reactive power absorption from the grid, SCIG- based WECS are equipped with capacitor banks. The softstarter’s role is to smooth the inrush currents during the grid connection (Iov 2003). SCIG-based WECS are designed to achieve maximum power efficiency at a unique wind speed. In order to increase the power efficiency, the generator of some 2.5 Power Generation System 21 fixed-speed WECS has two winding sets, and thus two speeds. The first set is used at low wind speed (typically eight poles) and the other at medium and large wind speeds (typically four to six poles). Fixed-speed WECS have the advantage of being simple, robust and reliable, with simple and inexpensive electric systems and well proven operation. On the other hand, due to the fixed-speed operation, the mechanical stress is important. All fluctuations in wind speed are transmitted into the mechanical torque and further, as electrical fluctuations, into the grid. Furthermore, fixed-speed WECS have very limited controllability (in terms of rotational speed), since the rotor speed is fixed, almost constant, stuck to the grid frequency. An evolution of the fixed-speed SCIG-based WECS are the limited variable- speed WECS. They are equipped with a wound-rotor induction generator (WRIG) with variable external rotor resistance; see Figure 2.9. The unique feature of this WECS is that it has a variable additional rotor resistance, controlled by power electronics. Thus, the total (internal plus external) rotor resistance is adjustable, further controlling the slip of the generator and therefore the slope of the mechanical characteristic. Obviously, the range of the dynamic speed control is determined by how big the additional resistance is. Usually the control range is up to 10% over the synchronous speed. Figure 2.9. General structure of a limited variable-speed WECS 2.5.2 Variable-speed WECS Variable-speed wind turbines are currently the most used WECS. The variable- speed operation is possible due to the power electronic converters interface, allowing a full (or partial) decoupling from the grid. The doubly-fed-induction-generator (DFIG)-based WECS (Figure 2.10), also known as improved variable-speed WECS, is presently the most used by the wind turbine industry. 2 Wind Energy Conversion Systems 22 Figure 2.10. General structure of an improved variable-speed WECS The DFIG is a WRIG with the stator windings connected directly to the three- phase, constant-frequency grid and the rotor windings connected to a back-to-back (AC–AC) voltage source converter (Akhmatov 2003; Ackermann 2005). Thus, the term “doubly-fed” comes from the fact that the stator voltage is applied from the grid and the rotor voltage is impressed by the power converter. This system allows variable-speed operation over a large, but still restricted, range, with the generator behaviour being governed by the power electronics converter and its controllers. The power electronics converter comprises of two IGBT converters, namely the rotor side and the grid side converter, connected with a direct current (DC) link. Without going into details about the converters, the main idea is that the rotor side converter controls the generator in terms of active and reactive power, while the grid side converter controls the DC-link voltage and ensures operation at a large power factor. The stator outputs power into the grid all the time. The rotor, depending on the operation point, is feeding power into the grid when the slip is negative (over- synchronous operation) and it absorbs power from the grid when the slip is positive (sub-synchronous operation). In both cases, the power flow in the rotor is approximately proportional to the slip (Lund et al. 2007). The size of the converter is not related to the total generator power but to the selected speed variation range. Typically a range of 40% around the synchronous speed is used (Akhmatov 2003). DFIG-based WECS are highly controllable, allowing maximum power extraction over a large range of wind speeds. Furthermore, the active and reactive power control is fully decoupled by independently controlling the rotor currents. Finally, the DFIG-based WECS can either inject or absorb power from the grid, hence actively participating at voltage control. Full variable-speed WECS are very flexible in terms of which type of generator is used. As presented in Figure 2.11, it can be equipped with either an induction (SCIG) or a synchronous generator. The synchronous generator can be either a wound-rotor synchronous generator (WRSG) or a permanent-magnet synchronous generator (PMSG), the latter being the one mostly used by the wind turbine industry. The back-to-back power inverter is rated to the generator power and its operation is similar to that in DFIG-based WECS. Its rotor-side ensures the 2.7 Control Objectives 25 energy. Depending on the operating regime the turbine can be controlled either for maximum power point tracking or for power limiting. The reliability requirements are important in these structures. The control of hybrid generation structure envisages the entire system; its management and supervision are beyond the scope of this book. Figure 2.13. DC-coupled hybrid generation system 2.7 Control Objectives Control plays an ever increasing role in modern WECS. There are numerous research articles dedicated to WECS control, all of them having starting from the idea that control can and does significantly improve all aspects of WECS. In any process, control has two main objectives: protection and optimization of operation. Furthermore, when applied to WECS, control becomes more important, in all aspects, as the main characteristic of WECS is that they have to cope with the highly variable, intermittent and unpredictable nature of the wind. To this end, as previously mentioned, all WECS have some sort of power control. The passive-stall wind turbines manage to limit the aerodynamic power, for protection reasons, without any active controllers. This approach is simple and offers hardware robustness, but can lead to unacceptable levels of mechanical loads 2 Wind Energy Conversion Systems 26 (Burton et al. 2001). Thus, control in that sense has as its only objective the protection of wind turbines. Active stall implies that WECS are equipped with several additional hardware components: electromechanical or hydraulic actuators used to move the blades (or parts of them), sensors and controllers. All of these add complexity and increase the operation and maintenance costs but they also allow one to extend the control objectives to increase the power capture, thus optimizing the WECS operation. Fixed-speed WECS, with either passive or active stall, dominated the wind power industry for a long time. Their main drawback is their rigidity, as the fixed generator speed does not offer any control flexibility. This disappears with the use of DFIG-based WECS and, later, with the use of full scale power converter WECS. Variable-speed operation became possible by incorporating power electronics converters. Variable-speed WECS control system generally includes three main control subsystems: aerodynamic power control, through pitch control; variable-speed operation and energy capture maximization, by means of generator control; grid power transfer control, through the power electronics converter. Furthermore, the specific objectives of each control subsystem vary in accordance to the operating regime (see Figure 2.7). When the wind speed is between the cut-in and the rated speed (partial load regime), the pitch control system is typically inactive, with two exceptions: when the pitch system is used to assist the start-up process, as the two- or three-bladed wind turbines have a relatively low starting torque, and when the rotational speed is limited by pitch control as the wind speed approaches the rated value. The pitch control system is active when the wind speed exceeds the rated wind speed. Its objective is to limit the aerodynamic power to the rated one and, when the wind speed reaches the cut-out value, to stop the wind turbine. Thus, the pitch control system deals mainly with alleviating the mechanical loads on the wind turbine structure. During the partial load regime, the generator control is the only active control and aims at maximizing the energy captured from the wind and/or at limiting the rotational speed at rated. This is possible by continuously accelerating or decelerating the generator speed in such a way that the optimum tip speed ratio is tracked. At rated wind speed, the generator control limits the generator speed. Thus, the generator control deals mainly with the power conversion efficiency optimization. Sometimes this means that the generator torque varies along with the wind speed and, in some conditions, can induce supplementary mechanical stress to the drive train. Consequently, maximizing the power conversion efficiency through generator control should be done, bearing in mind the possibility that supplementary loads are induced to the mechanical structure. Finally, the power electronics converter control ensures that the strict power quality standards (frequency, power factor, harmonics, flicker, etc.) are met. Recently, the increasing requirements for WECS to remain connected and to provide active grid support have added control objectives for the power electronics converters. In the case of a grid fault, the WECS should remain connected; thus 2.7 Control Objectives 27 they should cope with sudden and important loads, and even assist the grid in voltage or frequency control (Akhmatov 2003; Ackermann 2005; Sørensen et al. 2005). Thus, the power electronics converter control deals mainly with power quality standards. The role and objectives of WECS control, as presented above, can be summarized as follows (De La Salle et al. 1990; Leithead et al. 1991): starting on the WECS at the cut-in speed, stopping it at the cut-out speed and switching controllers corresponding to the specific operating conditions; controlling the aerodynamic power and the rotational speed above rated wind speed; maximising the wind harvested power in partial load zone, with respect to the speed and captured power constraints; alleviating the variable loads, in order to guarantee a certain level of resilience of the mechanical parts, in all operating regimes; guaranteeing a desired response to isolated wind gusts; transferring the electrical power to the grid at an imposed level, for wide range of wind velocities; meeting strict power quality standards (power factor, harmonics, flicker, etc.); protecting the WECS and, at the same time, offer active grid support during grid faults. The list is not exhaustive; several other control objectives, deriving from those listed above, can be formulated. Variable-speed WECS is a highly nonlinear time- variant system excited by stochastic inputs which significantly affect its reliability and leads to non-negligible variations in the dynamic behaviour of the system over its operating range. This is the reason why the control of variable-speed wind turbines is still in the phase of searching technical solutions suitable to be widely implemented in the wind turbine industry. 3 WECS Modelling 30 3 3~ AC/DC/AC Drive train PWM PWM Electric grid Generator: SCIG / PMSG (gears) Wind Pitchable rotor HSS LSS 3 3 3~ AC/DC/AC Drive train PWM PWM Electric grid Induction generator (gears) Pitchable rotor HSS LSS DFIG S1 S3 S2 S4 a) b) Wind Figure 3.1. General configuration of a variable-speed-controlled WECS from a system point of view: a SCIG/PMSG-based; b DFIG-based The WECS subsystems are usually treated individually; a global WECS model suitable for control structure design is obtained by adding models of their interactions. The modelling of WECS components, presented in the following, relies upon some assumptions, which are specified when they appear. These modelling assumptions depend on the desired level of detail, the operating regime and on the control goal. Thus, since the variable-speed operation regime is effective in partial load, some wind turbine simplified models – e.g., fixed aerodynamic characteristics – can be used for generator control purposes, irrespective of the turbine size. Furthermore, since the issue of getting accurate measure information is beyond the scope here, all necessary measurements, such as the wind velocity at the hub level, the blades’ position, the HSS rotational speed and all necessary electrical variables, are assumed available for control/supervision. 3.2 Wind Turbine Aerodynamics Modelling 3.2.1 Fixed-point Wind Speed Model From a system point of view, the wind speed represents the main exogenous signal applied to the WECS and determines its behaviour. Its erratic variation, highly dependent on the given site and on the atmospheric conditions, makes the wind 3.2 Wind Turbine Aerodynamics Modelling 31 speed quite difficult to model. Usually the thermic equilibrium of the atmosphere nearby Earth is assumed (neutral atmosphere – Burton et al. 2001). Therefore, turbulence results mainly from the friction between air and ground, due to the ground roughness. When designing WECS, the history of the wind speed extreme values (gusts) is considered for the mechanical structure design and also for control purposes. Wind near the Earth’s surface is generally modelled by a spatial (3D) speed distribution. Assuming that the turbine is equipped with a vane (or yawing equipment) and that changes in wind direction are sufficiently slow, then the turbine rotor is maintained normal to the wind and WECS analysis requires only the longitudinal wind speed being synthesized/modelled. Thus, in the present book only scalar (1D) wind speed models will be used. As the interest here is focused on WECS behaviour in normal operating regimes, the developed models will not include extreme operating conditions like wind gusts. Wind dynamics result from combining meteorological conditions with particular features of a given site. Thus, wind speed is modelled in the literature as a non-stationary random process, yielded by superposing two components (Burton et al. 2001; Nichita et al. 2002; Vihriälä 2002; Bianchi et al. 2006): s tv t v t v t , (3.1) where ( )sv t is the low-frequency component (describing long term, low-frequency variations) and ( )tv t is the turbulence component (corresponding to fast, high- frequency variations). These components can be identified in Van der Hoven’s large band (six decades) model (Figure 3.2). The spectral gap of around 0.5 mHz suggests that the turbulence component can be modelled as a zero average random process (there is little energy in the spectral range between 2 h and 10 min). sv is considered constant (equal to the average wind speed) when viewed at the turbulence time scale. Averaging is usually performed on a 10-min time window (Burton et al. 2001). vvf S f log f Figure 3.2. Van der Hoven’s spectral model of the wind speed 3 WECS Modelling 32 The low-frequency component corresponds to the very slow wind speed variations and characterizes the site from the energy viewpoint. It can be modelled as a Weibull’s distribution (see Chapter 2) or a Rayleigh’s distribution (Leithead et al. 1991): 21 2e avsv a v , where v is the wind speed’s hourly average and a is a parameter depending on the wind speed’s very long term average. The value of this component influences the turbulence amplitude, but its evolution is not crucial for short and medium term dynamic behaviour of WECS. Fast wind speed variations (typically occurring within 10 min) are modelled by the turbulence component. This is mathematically described as a zero average normal distribution, whose standard deviation, , depends on the current value of the hourly average, sv . The turbulence intensity is a measure of the global level of turbulence, depends on the ground surface roughness and is defined as t s I v (3.2) The mathematical description of the turbulence’s dynamical properties, tv t , can be obtained by using two kinds of spectra: von Karman’s and Kaimal’s respectively. According to Burton et al. (2001), Kaimal’s spectrum reflects better the correspondence to experimental data, when turbulence is present. But von Karman’s spectrum is more consistently theoretically founded (an analytical connection with the correlation function is provided) and allows a realistic representation of turbulence data in wind tunnels. The von Karman’s model for the longitudinal component of the turbulence is 2 5 / 62 4 / 1 70.8 / vv t s t s f S f f L v f L v , (3.3) where vvS f is the power spectral density, tL is the length of turbulence, specific to the site (ground roughness), and f is the frequency in Hz. Kaimal’s spectral model has the form 2 5 / 3 4 / 1 6 / vv t s t s f S f f L v f L v (3.4) One can note that in both models the power spectral density is influenced by the turbulence intensity, It, which determines the turbulence “level” (i.e., its variance, 2 ) and the turbulence length, tL , which impresses the turbulence dynamic properties (the spectral function bandwidth). Both these parameters are adopted according to various standards. For example, in the Danish standard (DS 742 2007), the following relations are used to compute these parameters: 3.2 Wind Turbine Aerodynamics Modelling 35 5 / 61 F t F K H j j T , (3.8) where parameters KF and TF depend on the low-frequency wind speed, sv . Welfonder et al. (1997) propose a procedure for obtaining the turbulence component, which is based upon experimentally identifying the shaping filter’s parameters from Equation 3.8. The time constant results from F t sT L v , with tL being found empirically. The filter is fed with a normally distributed white noise having one-unit variance, whose sampling time, Ts, can be configured. By computing the static gain, KF: 2 1 2,1 3 F F s T K B T , (3.9) where B is the beta function, one gets a one-unit-variance coloured noise at the filter’s output. In order to obtain variance 2 corresponding to the average wind speed, sv , the coloured noise is multiplied by the product sI v , with the turbulence intensity, I, being empirically determined. The procedure proposed by Welfonder et al. (1997) is adapted by Nichita et al. (2002) for obtaining the non-stationary wind speed, within a large time window, by using the block diagram from Figure 3.6. Here, the time scales of the two components, sv t and tv t , are different. Usually the sampling time are Tss = 10 min for sv and Tst = 1 s for the turbulence component. Time series generator, sv t White noise generator, e t Shaping filter, tH tv t ,F FT K sI v tv t v tsv t Figure 3.6. Nonstationary wind speed generation The low-frequency component is obtained either based upon a model fitted to measured data, or by using a generic model, e.g., the van der Hoven’s spectrum. In the latter case, the low-frequency component must be sampled. Let i, 1,i N , be the discrete angular frequency and ( ) s sv v iS the corresponding values of the power spectral density. The i harmonic has the amplitude 1 1 2 1 ( ) 2 s s s si v v i v v i i i A S S (3.10) and component sv is thus computed: 3 WECS Modelling 36 0 cos t N s i i i i v t A , (3.11) where phase i is generated randomly in the [– , ] range. For 0=0 it is set 0=0 and 0A v , where v is the average wind speed, calculated on a time horizon greater than the largest period in Van der Hoven’s characteristic (i.e., 12T ). For each new computed value of sv , the current time constant of the shaping filter, ( ) ( )F t sT t L v t , is computed. The static gain KF is computed based upon Equation 3.9, ensuring unitary variance at the filter’s output, and then the turbulence’s variance is adjusted by means of factor sI v , as shown in Figure 3.6. Parameters TF, KF and turbulence’s variance remain constant along the time interval Tss as long as sv is constant; they are re-calculated as soon as a new sv value is obtained. The turbulence component of the wind speed is simulated at Tst sampling time and involves numerical difficulties if the non-integer order filter at Equation 3.8 is employed. This filter can be approximated by a two pole and one zero transfer function (Nichita et al. 2002): 1 2 1 1 1 F t F F F m T s H s K T s m T s , 1 0.4m , 2 0.25m (3.12) The non-stationary wind speed, computed with the procedure presented above, can be seen in Figure 3.7. It covers a 5-h time range. The evolution of the low- frequency wind speed component is highlighted. Figure 3.8 presents details of the wind speed profile when sv varies in two distinct ranges. 0 0.5 1 1.5 2 2.5 3 x 104 0 2 4 6 8 10 12 14 16 18 20 st m/sv Figure 3.7. Non-stationary wind speed: total wind speed, ( )v t (black), and its low- frequency component, ( )sv t (white) (z=50 m, z0=0.005 m, Danish standard DS472) 3.2 Wind Turbine Aerodynamics Modelling 37 1.5 1.51 1.52 1.53 1.54 1.55 1.56 0 5 10 15 20 410 st m/sv 600 700 800 900 1000 1100 1200 0 2 4 6 8 10 12 14 16 18 20 st m/sv b)a) Figure 3.8. Details from Figure 3.7: a sv varies around 12 m/s; b sv varies around 7 m/s 3.2.2 Wind Turbine Characteristics Variable-pitch Case The main modelling purpose is to provide the wind torque developed by the turbine rotor in the form , ,wt wt lv (3.13) as the wind velocity, v, rotational speed, l , and blade pitch, , are given. For this purpose, the blade element theory (BET) is used. Some very interesting aspects concerning the turbine operation and interaction between the rotor and the air stream can also be revealed. According to BET, the blade is divided into a number of transversal elements, placed along the blade. A blade element j (Figure 3.9) is obtained by sectioning the blade with two parallel planes, situated at distances r and r+dr from the hub, and normally disposed to the blade. 0i Reference chord d aF dF dL d tF Rotation plane dD i j-th blade element Direction of blade movement 1l r b 1 a v l r 0w w av v lb r Wind velocity direction Figure 3.9. Aerodynamic loads along the blade profile 3 WECS Modelling 40 Each blade element develops an elementary torque: d ( ) ( ) d ( )tj r j F j (3.20) By integrating Equation 3.20 along the blade length and by using Equation 3.18, one obtains the total torque developed by the wind turbine rotor: d ( ) ( ) d ( )tj r j F j (3.21) This result is obtained assuming that the rotor has an infinite number of blades. As in fact there is a finite number of blades, BN , Equation 3.21 is amended by a correction factor, called Prandtl’s coefficient: 2 0.931 ( ) 0.445 P B rN j (3.22) In conclusion, the wind turbine torque computation procedure can be synthesized, using the expressions presented above, as in Algorithm 3.1 (Diop et al. 1999; Nichita et al. 2006). Algorithm 3.1. Computation of wind torque according to blade element theory #0. Input (constant) data: number of blades, blade length, air density, number of finite elements, chord variation along the blade, pitch variation along the blade, aerodynamic characteristics zC and xC depending on the incidence angle. #1. Input variables: wind velocity, rotational speed, pitch angle (at the hub). #2. For each element j, compute: – elementary tip speed ratio ( )r j ; – elementary distance to the hub r(j), total pitch corresponding to the element, ( )j ; – elementary incidence angle, i(j); – elementary axial and tangential interference factors, a(j) and b(j); – relative wind speed, w(j); – lift and drag coefficients, ( )zC i and ( )xC i , and also their report 1 ( )i ; – elementary torque (function of many above computed variables). #3. Apply the Prandtl’s coefficient. #4. Numerically integrate the elementary torque equation (using a suitably chosen method). Fixed-pitch Case The aerodynamic subsystem model describes (by averaging) the interaction of the turbine rotor with the air masses (wind); this subsystem is modelled by the mechanical torque provided by the rotor motion (Wilkie et al. 1990; Miller et al. 2003). Following the assumptions listed above, for a fixed-pitch ( ) wind turbine, this torque depends on the low-speed shaft rotational speed and on the wind speed: constant,wt wt l v (3.23) 3.2 Wind Turbine Aerodynamics Modelling 41 2 31 ( ) 2 wt wt l P v R C , (3.24) where pC C is the torque coefficient (introduced by Equation 2.32). More detailed aerodynamic models can be developed, emphasizing the rotational sampling, spatial filtering or induction lag (Rodriguez-Amenedo et al. 1998; Vihriälä 2002; Molenaar 2003), leading to a more complex expression of the developed aerodynamic torque, wt . Furthermore, the power coefficient characteristic should take into account the effects due to Reynolds number and air density variations. For low-power wind turbines, these effects, together with the structural dynamics, can be neglected, and simplified models are preferred (Wilkie et al. 1990). Passive stall Active stall C 0 2 4 6 8 10 12 14 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 a) b) pC 0 2 4 6 8 10 12 14 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure 3.11. Typical HAWT C (a) and pC (b) curves The torque coefficient can be described by a polynomial function of the tip speed ratio, (Figure 3.11a – Nichita 1995; Miller et al. 2003): 6 5 4 3 26 5 4 3 2 1 0( )C a a a a a a a (3.25) Parameters , 0...6ia i , are usually determined by fitting the look-up table representing an experimental torque characteristic in a least squares sense. Typical static C and pC variations with respect to tip speed ratio for two bladed HAWT are given in Figure 3.11. Different blade profiles imply different aerodynamic characteristic shapes; for example, in Figure 3.11 are depicted C and pC curves belonging to a passive-stall regulated rotor (dotted line) and to an active- stall regulated rotor (continuous line). The corresponding torque and power characteristics with respect to the LSS speed (obtained using Equations 3.24 and 2.31 respectively), parameterised by the wind speed, are shown in Figure 3.12. According to Equation 2.31, one obtains 3 WECS Modelling 42 a) v increases b) Nmwt rad/sl WwtP v increases rad/sl Figure 3.12. Typical HAWT torque (a) and power (b) characteristics (LSS) 3.2.3 Wind Torque Computation Based on the Wind Speed Experienced by the Rotor The fixed-point wind speed model, as experienced by the turbine’s hub, has been presented in Section 3.2.1. It represents only the initial information for determining the effective wind speed model, as experienced across the turbine’s blades. The wind speed’s effective properties are also influenced by a series of effects due to the rotor motion. Two of these effects induce deterministic variations of the wind torque, namely: the tower shadow effect, which takes place when one of the blades passes the tower while moving and results in a decreasing fluctuation of the wind torque; the wind shear effect, due to the wind speed periodic variation with the height to ground, producing periodic wind torque variations. The two effects cumulatively yield periodic wind torque variations with a frequency that is an integer multiple of the blades’ rotational speed. The interaction between wind and turbine requires that the spectral properties of the fixed-point wind speed variations be strongly modified, as compared with those of the fixed-point wind turbulence model. These changes are significant, especially for large wind turbines. Rodriguez-Amenedo et al. (1998) present a model for obtaining the wind torque, wt (Figure 3.13), which allows a simplified description of the wind speed fluctuations due to wind-turbine interaction. In the following, the main dynamic subsystems appearing in the block diagram in Figure 3.13 are described. The spatial filter performs the averaging of the wind speed’s variations across the area swept by the rotor. The fixed-point spectrum is modified in such a way that a spectral representation of the average wind speed across the rotor is obtained. The filter’s transfer function is (Wilkie et al. 1990; Rodriguez-Amenedo et al. 1998) 2 2 1 sf sf sf sf sf sf b s H s b b a s s a , (3.26) 3.2 Wind Turbine Aerodynamics Modelling 45 10-2 10-1 100 101 -30 -20 -10 0 10 20 10-3 10-2 10-1 10-2 100 102 gain dB rad/s a) b) ( )vvS f Hzf Figure 3.15. a Bode characteristics of shaping filters: fixed-point (solid line), rotational sampling (dashed line). b Rotationally-sampled power spectral density 1170 1175 1180 1185 1190 1195 1200 7 8 9 10 11 12 13 st m/sv 1000 1100 1200 1300 1400 1500 1600 0 2 4 6 8 10 12 m/sv 1000 1100 1200 1300 1400 1500 1600 0 2 4 6 8 10 12 m/sv stst a) b) c) zoom Figure 3.16. a Fixed-point wind speed. b Wind speed with the effect of rotational sampling. c Zoom on the rotationally-sampled wind speed Let us now remark that the effects described above are in fact important only for large scale WECS. For the low/medium-power WECS, the rotor diameter is sufficiently small for one to neglect the tower effect and structural dynamics of the turbine, especially if it is equipped with a teeter hub or some other equipment ensuring the damping of these undesired dynamic effects. Also, the wind speed can be considered constant for the entire area swept by the rotor. This means that the wind shear effect is negligible too. The rotational sampling effect is reduced; 3 WECS Modelling 46 therefore a fixed-point wind speed model can be employed, usually neglecting the blade torsional dynamics and induction lag. These remarks suggest that the aerodynamic model of a high-power wind turbine is far more complicated compared to that employed in low-power WECS. 3.3 Electrical Generator Modelling Electrical generators are systems whose power regime is generally controlled by means of power electronics converters. From this viewpoint, irrespective of their particular topologies, controlled electrical generators are systems whose inputs are stator and rotor voltages, having as state variables the stator and rotor currents or fluxes (Leonhard 2001). They are composed of an electromagnetic subsystem, which outputs the electromagnetic torque, further referred to as G , and the electromechanical subsystem, through which the generator experiences a mechanical interaction. Figure 3.17 illustrates the modelling principle for the SCIG case. The necessity of using (d,q) models comes from vector control implementation, which has the advantage of ensuring torque variation minimization and thus better motion control. 3~ AC/DC PWM HSS , ,a b cu 2 32 3 h G du qu,Sd q i 2 3 ,Rd qi , ,Ra b ci , ,Sa b ci SCIG DC-link … … 3 Park Transforms Figure 3.17. Generator modelling: identifying inputs, outputs and states – SCIG case Below, the generator modelling is focused on capturing the evolution of the electromagnetic subsystem into a mathematical form. Thus, a set of equations involving the generator’s electrical variables – voltages, fluxes and currents – results. In WECS the generator interacts with the drive train; hence, to this set of equations is usually added the high-speed shaft (HSS) motion equation in the form d d h mec GJ t , (3.30) where the static and viscous frictions have been neglected, J is the equivalent inertia rendered to the HSS, mec is the mechanical torque, h is the HSS rotational speed and G is the electromagnetic torque resulting from the interaction between the stator and rotor fluxes and depending upon each particular configuration, as listed below. The modelling has assumed that the influence of the generator constructive features on its dynamics (e.g., higher harmonics, asymmetries, etc.) is neglected and its parameters are constant. 3.3 Electrical Generator Modelling 47 3.3.1 Induction Generators Doubly-fed Induction Generator (DFIG) The doubly-fed induction generator’s electromagnetic torque is expressed in (d,q) frame as (Leonhard 2001; Bose 2001): 3 2G m Sq Rd Rq SdpL i i i i , (3.31) with p being the pole pairs number, Lm the stator-rotor mutual inductance, Sdi , Sqi , Rdi and Rqi are the stator, respectively rotor current (d,q) components, obtained by integrating the following differential equations: d d ( ) d d d d ( ) d d d d ( ) ( ) d d d d ( ) ( d d Sd Sd S m Rd m Sd S Sq Rq S S S S Sq Sq RqS m m Sq S Sd Rd S S S S Rd Rd m Sd mR Rd S Rq Sq R R R R Rq Rq SqmR Rq S R R R i V R L i L i i i t L L L t L i V iR L L i i i t L L L t L i V L i LR i i i t L L L t L i V iLR i i t L L L t )mRd Sd R L i L , (3.32) where hp is the speed in electrical radians per second (where h is the generator rotational speed), d dS S t (rad/s) is the stator field frequency, RS, RR are the stator and rotor resistances, LS, LR are the stator and rotor inductances; SdV , RdV , 0Rd RqV V are the stator, respectively rotor voltage (d,q) components; Rd R RdL i , Rq R RqL i are the rotor flux (d,q) components. By adopting the notation 1 2 3 4( ) ( ) ( ) ( ) TT Sd Sq Rd Rq T Sd Sq Rd Rq x t x t x t x t i i i i V V V V x u (3.33) for the state and input vector respectively, the DFIG state model can be presented as a fourth-order model: 2 3 1 4 3 2 h m G pL x x x x x A x B u y i , (3.34) where 21 m S RL L L and 3 WECS Modelling 50 where 2 1 1 S S SL R . Stator fluxes are then obtained from purely algebraic equations: 2 2 2 2 2 2 S Sm S m SS S SdR S S SqS m m S SS SS R SS R RL R L L VL L L VR L L R R L L LL S R (3.40) Finally, currents are also computed based on an algebraic dependence: 10 0 0 0 0 0 0 0 Sd S m Sq S m m RRd m RRq i L L i L L L Li L Li (3.41) Starting from the fourth-order DFIG model given in Equation 3.34, one can obtain a reduced-order model by employing a second method. For this purpose, the singular perturbation method is applied, which takes into account that the stator currents influence the dynamic of the rotor currents by their steady-state values. Consequently, the state vector x from Equation 3.33 is partitioned: T T S Sd Sq R Rd Rqi i i ii i , (3.42) such that the state equation from Equation 3.34 to be put into the form 11 12 1 21 22 2 | | S S R R A A B i i u i A A Bi i i , (3.43) where 2 11 122 21 22 S h m m R h mS S S R S R S h m m Rh m S S S S RS R S m S h m hR S S R R R h m S m h R S R S R R R p L L R p L L L L L L L p L L Rp L R L L LL L L L R p L pR L L L L p L R L p R L L L L A A A A 3.3 Electrical Generator Modelling 51 1 2 1 10 0 0 0 1 10 0 0 0 m m S S R S R R m m S S R S R R L L L L L L L L L L L L L L L L B B To obtain the steady-state values one must zero the derivatives. By zeroing Si i , Equation 3.43 becomes 11 12 1 21 22 2 0S R S R R A i A i B u A i A i B u i i (3.44) Steady-state stator currents result from the first relation of Equation 3.44 as 1 11 12 1S Ri A A i B u , (3.45) where the inverse of matrix 11A is 2 1 11 22 22 1 S h m S S S R h m SS h m SS S R SS S R R p L L L L p L RR p L L L LL L L A , (3.46) and are replaced in the second relation of Equation 3.44 to provide the DFIG reduced-order (second-order) dynamic, concerning only the d and q rotor currents: 1 1 22 21 11 12 2 11 1R Ri A A A A i B A B u i (3.47) 3.3.2 Synchronous Generators Permanent-magnet Synchronous Generator (PMSG) The PMSG is modelled under the following simplifying assumptions: sinusoidal distribution of stator winding, electric and magnetic symmetry, negligible iron losses and unsaturated magnetic circuit. Under these assumptions, the generator model in the so-called steady-state (or stator) coordinates is first obtained (Leonhard 2001). Another simpler model can be obtained in (d,q) rotor coordinates; conversion between (a,b,c) and (d,q) coordinates can be realized by means of the Park Transform (Leonhard 2001). Then, after neglecting the homopolar voltage, 0u , by virtue of symmetry, the (d,q) PMSG model becomes d d d d q S q q q q d S u Ri L i u Ri L i i i , (3.48) 3 WECS Modelling 52 where R is the stator resistance, ,d qu u are d and q stator voltages, ,d qL L are d and q inductances and S is the stator (or else electric) pulsation, d d d mL i (3.49) q q qL i (3.50) are d and q fluxes and m is the flux that is constant due to permanent magnets. Thus, the model at Equation 3.48 becomes d d d d q q S q q q q d d m S u Ri L i L i u Ri L i L i i i (3.51) The electromagnetic torque is obtained as G d q q d m q d q d qp i i p i L L i i , (3.52) where p is the number of pole pairs. If the permanent magnets are mounted on the rotor surface, then d qL L and the electromagnetic torque becomes G m qp i (3.53) When the machine operates as a grid connected generator, Equation 3.51 becomes d d d d q q S q q q q d d m S u Ri L i L i u Ri L i L i i i (3.54) The stator frequency, S , is proportional to the shaft rotational speed, S hp , which depends on how the electrical generator interacts mechanically. The state and input vector are identified respectively as 1 2 T TT d q d qx t x t i t i t u ux u (3.55) Hence, the grid connected PMSG state model is obtained in the form 1 2 1 1 2 1 0 10 q h dd d d m h qd q G m LR x p x LL L L xR x p LL L p x x u y i (3.56) 3.4 Drive Train Modelling 55 transmission has a constant efficiency for the whole speed range; the influence of the constructive features (e.g., vibrations, gear type, gear backlash, etc.) on its behaviour is considered parasitic and will be neglected. Furthermore, the mechanical and electrical systems are considered to be perfectly balanced, any fault is taken as a special operating regime. 3.4.1 Rigid Drive Train The main element of a rigid drive train is the single-stage rigidly-coupled speed multiplier, of (fixed) ratio i and efficiency (Figure 3.19). In this case, the model consists of a first-order motion equation, rendered either at the low-speed or at the high-speed shaft. G wt , i Rotor wtJ gJ Multiplier Generator HSS LSS lJ hJ axA axB Figure 3.19. Rigid drive train Due to the speed multiplier, the generator experiences an i times reduced torque and an i times increased speed, h li ; the system equivalent inertia is reduced when rendered at the high-speed shaft, hJ (Munteanu 1997). Under the previously stated modelling assumptions, the WECS dynamic may be expressed rendered at either the HSS or at the LSS in two equivalent forms (Nichita 1995; Welfonder et al. 1997): d ( , ) ( , ) d h h wt l G hJ v ct i , (3.64) d ( , ) ( , ) d l l wt l G h iJ v c t , (3.65) where wt( l,v) is the aerodynamic torque, parameterized by the wind speed, v; G( h,c) is the electromagnetic torque, parameterized by a generically called load variable, denoted by c; hJ , lJ are inertias rendered at HSS and LSS respectively, being computed as 1 22h wt gJ J J J Ji , (3.66) 2 1 2l wt g iJ J J J J , (3.67) 3 WECS Modelling 56 where J1, J2 are inertias of the multiplier gearings; wtJ , gJ are inertias of turbine rotor and of electrical generator respectively. As the drive train is rigid, the dynamical torque is given only by the first-order linear rotational speed variation. Figure 3.20a contains a Simulink® implementation of the motion equation rendered at the high-speed shaft (Equation 3.66). This equation is usually referred to as being the single-mass WECS model. 2 1J2 J1 1 s -K- -K- -K- -K- em em 2 1 Steady-state wt G h Drive train modelling HSS equivalent inertia 1 i Wind torque Generator torque 2 1J HSS LSS h lHSS movement equation a) b) Torques i 1 i OP Figure 3.20. a Motion equation rendered at the high-speed shaft – Simulink® implementation; b steady-state regime of WECS Figure 3.20b presents the steady-state regime of the interaction between the turbine rotor and the induction generator by means of the high-speed shaft turning at h . The steady-state operating point of WECS is thus determined as the cross- point of the rotor mechanical characteristic with that of the generator. 3.4.2 Flexible Drive Train Elements of a flexible drive train are depicted in Figure 3.21. li h G wt , i Rotor wtJ gJ Gearbox Generator High-speed shaft Low-speed shaft ,s sK B Spring axA axC axB l DT Figure 3.21. Schematics of a flexible drive train Different from the rigid coupling, the two parts of the high-speed shaft, axB and axC in Figure 3.21, are now turning at different speeds, i l and h respectively, where i is the transmission ratio of the gearbox. The elastic energy variations yield 3.5 Power Electronics Converters and Grid Modelling 57 a new state variable, the internal torque, . Denoting by Jg the inertia of axC and by JB the inertia of axB inertia, it holds that 2 B wtJ i J , where is the transmission efficiency and Jwt is the low-speed shaft inertia. The flexible drive train model is composed of axB and axC motion equations and the dynamic of the internal torque (De Battista and Mantz 1998; Akhmatov 2003): 1 1 1 l wt wt wt h g g G l hs l h s J i J J J K i B i i i i i i Finally, a third-order linear model results, having Tl hx as states, T wt Gu as inputs and T l hy as outputs: 11 00 0 1 10 0 0 1 1 1 0 0 0 1 0 wtB g g s s s s s wt gB g Ji J J J i B B i K K B J JJ J x x u y x i , (3.68) where Ks and Bs are respectively the stiffness and the damping coefficients of the spring. 3.5 Power Electronics Converters and Grid Modelling Power Electronics Interface The power electronics converter realizing the interface between WECS and the electrical grid has some key roles in the wind turbine variable-speed operation. It realizes a certain decoupling between the two above elements and allows the effective power flow control. The interest here will be focused on one of the most popular converter structures, namely the back-to-back AC–AC converter, employing two PWM- controlled voltage-source inverters (VSI), as shown in Figure 3.22. The voltages and frequencies differ on one side of the converter vs. the other. In fact, this 3 WECS Modelling 60 0 d 0 q 0 0 0 d q 0 d d 2 d d 2 d 3 d 2 invd gr m gr invd gr invq invq gr gr invd gr invq invd invq rec I V L V R I L I t I V L R I L I t V V C I I I t R (3.71) Model at Equation 3.71, describing the interaction between the power inverter and the grid, is useful especially for active/reactive power control. Other models can also be used, depending on a well-defined control goal (Lubosny 2003). For the medium/high-power case, the controlled generator and the power converters can be considered without any dynamic when used in a global modelling approach. 3.6 Linearization and Eigenvalue Analysis The aim of this subsection is to present linearized models of different WECS configurations resulted from combining the above presented types of WECS parts (aerodynamics, drive train, electrical generator). The wind speed is an input variable. For grid-connected WECS, the other inputs are the stator (or also the rotor) voltages, whereas for autonomous WECS the load is an input. 3.6.1 Induction-generator-based WECS In the following, the linearization procedure and eigenvalue analysis are illustrated in the case of a variable-speed fixed-pitch rigid-drive-train induction-generator- based WECS. The global nonlinear model results from adding to the DFIG model (Equation 3.34) the motion equation in the form of the rigid drive train equation rendered at the high-speed shaft (Equation 3.64), where the wind torque is given by Equation 3.24. Consequently, the state and input vectors are 1 2 3 4 5 TT Sd Sq Rd Rq h T Sd Sq Rd Rq x x x x x i i i i V V V V v x u (3.72) Linearized Model An operating point is characterized by a sextuple , , , , , ,Sd Sq Rd Rq hi i i i v vx . The following notations are adopted to represent variations of variables around such an operating point: T Sd Sq Rd Rq h T Sd Sq Rd Rq i i i i V V V V v x u (3.73) 3.6 Linearization and Eigenvalue Analysis 61 Linearization begins with obtaining a linearized model of the wind torque. Points of WECS usual operation are placed on the falling part of the wind torque characteristic, ,wt l v , where this one crosses the load characteristic, l , rendered at the high-speed shaft; Figure 3.23 depicts the case of a linear (or else called static) load characteristic. Generally, the load characteristic is controllable; let c be the control variable, therefore ,l l h c . In the case of a linear load characteristic, c is either its slope, lK , or its initial abscissa, 0l . Figure 3.23. WECS operating point as cross point of the aerodynamic characteristic and (linear) load characteristic In the following, for a generic functional variable x associated to the static operating point the following notation will be adopted: static operating pointx x ; x x x ; xx x (3.74) Being defined in a static operating point, x is called steady-state value. Curves ,wt l v and ,l h c can be linearized around such an operating point, ,l wt , by considering the first two terms of their Taylor’s series (Ekelund 1997; Munteanu et al. 2005; Munteanu 2006). Using well-known linearization procedures leads to results showing that the linearized model exhibits a first-order dynamic like suggested in Figure 3.24. In this figure the gains are 2wt l wt v l l K i v 2l l wtlc l l K i c The time constant of the linearized model is 2 l wt h l l T i J i , v3>v2 0 v1 [m/s] v2>v1 Nm rad/s wt l OP wt ,l hi c wtK 2 li K 0l 3 WECS Modelling 62 vK lcK + – 1 1Ts l v c Figure 3.24. Linearized first-order dynamic of a rigid-drive-train WECS with static load where Jh is given by Equation 3.66 and i is the drive train’s ratio. If the load characteristic is a linear one (see Figure 3.23), then, by noting wt wt lK and l l lK , one obtains the time constant in the form 2 h wt lT i J K i K . Therefore, the linearized dynamic depends on the system inertia and on the slopes of the two torque curves, wtK and lK . When the load characteristic is specified as being the torque controlled characteristic of the generator, supposing that the electromagnetic torque can be instantaneously obtained, then l G and 0lK . Thus, the simplest linearized model of WECS results in the form of the transfer function from the electromagnetic torque, G , to the low-speed shaft rotational speed, l : 1 2 ( )lin K H s s K , (3.75) where 1 1 hK J and 2 wt hK K i J . Therefore, supposing that G is instantaneously obtained (its dynamic is hundreds of times smaller than the time constant of the linearized system, 21 K ), then the rigid-drive-train WECS slow dynamic can be approximated as being of first-order (Figure 3.25). ( )v t wt G Speed multiplier S1 Jt S2 Drive train R o t o r l 1 2 2 1 1 K K s K lG v Output Control input Figure 3.25. Simplest linearized model of rigid-drive-train torque-controlled WECS (aerodynamic subsystem and drive train): a first-order dynamic 3.6 Linearization and Eigenvalue Analysis 65 -50 -40 -30 -20 -10 0 -400 -300 -200 -100 0 100 200 300 400 -50 -48 -46 -44 -42 -40 -10 -5 0 5 10 -24.5 -24 -23.5 -23 41 42 43 44 45 46 a) b) c) v increases v increases Figure 3.26. Migration of poles for a fifth-order uncontrolled induction-generator-based WECS model as the operating point (wind speed) varies: a stator, rotor and motion poles. b Zoom on rotor complex poles’ distribution; c zoom on the motion real pole migration -2 -1.5 -1 -0.5 0 -10 -5 0 5 10 -50 -40 -30 -20 -10 0 -400 -200 0 200 400 b) v increases U f increases a) Figure 3.27. a Migration of poles for a fifth-order controlled induction-generator-based WECS through a U/f=constant strategy at constant wind speed as the U/f ratio varies. b Migration of the dominant pole through constant torque control on the ORC The above remarks allow us to draw a conclusion to be further exploited for control purposes. Namely, the global dynamics of a controlled WECS depend on the shape of the generator characteristic, which further depends on the chosen control structure. More precisely, the relative position of the two mechanical characteristics – that of the turbine and that of the generator – determines the WECS response time. Thus, for example, dynamics of a constant torque control (when the load characteristic has zero slope in the ,h G plane) can be up to 10 times slower than those of the case where a U/f = constant control (when the same characteristic has a large slope) is adopted. Therefore, the possibility of controlling the slope of the load characteristic is a key issue in WECS control. 3 WECS Modelling 66 3.6.2 Synchronous-generator-based WECS The case of a variable-speed fixed-pitch rigid-drive-train synchronous-generator- based WECS is discussed here, where its autonomous operation on a symmetric tri-phased resistive load, lR , is considered. Linearized Model In order to obtain the state model of an autonomous synchronous-generator-based WECS, one uses the model of a permanent-magnet synchronous generator supplying an isolated resistive load, lR (Equation 3.58), together with the motion equation in the form of Equation 3.30, where the wind torque, wt , plays the role of the mechanical torque, mec . wt is a polynomial function of the tip speed (Equation 3.25 replaced in Equation 3.24). For optimal control purposes, it is not necessary to use such a complicated expression of the torque coefficient, C , as that of Equation 3.25, which captures all operating regimes, including the starting one. Instead, one can rather employ a second-order polynomial approximation of C vs. the tip speed, : 2 0 1 2( )C a a a , (3.82) which satisfactorily describes the appearance of C near optimal regimes. Taking into account the tip speed given by Equation 2.30 and wind torque variation upon the torque coefficient (Equation 3.24), Equation 3.82 corresponds to the following expression of the wind torque depending on the wind speed, v, and on the generator’s rotational speed, h : 2 2 1 2 3wt h hd v d v d , (3.83) where 3 4 5 1 0 2 1 3 21 2 , 1 2 , 1 2d R a d R a d R a , (3.84) with R being the blade length and being the air density. In relation to the model at Equation 3.58, a new input is added, namely the wind speed, and also a new state variable, the rotational speed, h , which is also chosen as output variable. Thus, with the new notation: 1 2 3 1 2 TT d q h T T l h x t x t x t i t i t u t u t R v x u y , (3.85) the WECS model has the form 3.6 Linearization and Eigenvalue Analysis 67 1 2 3 1 1 2 1 3 3 2 1 2 2 3 3 2 1 2 2 2 3 1 1 1 1 1 1 0 0 1 q s d s d a d s m q s q s m Rx p L L x x x u L L L L Rx p L L x x p x x u L L L L d x p x d u d u x J J x y x i (3.86) The third-order nonlinear model (Equation 3.86) is linearized around an arbitrarily chosen operating point. Letting 1 2 3 Tx x xx and TlR vu be the variations of the state variables and of inputs around this point leads to 1 2 3 3 3 2 2 1 1 3 2 3 1 1 4 3 2 4 2 3 3 1 3 3 0 0 0 0 0 0 1 l l a a R a x a x a x b x b b R b x b b x c c v c x c v c x x x u y x i , (3.87) where 1 2 3 1 2 3 4 31 2 1 2 3 4 1; ; ; 1; ; ; ; 22 ; ; ; q s d s d s d s d s m q s q s q s q s m L LRa a a p L L L L L L L L pRb p b b b L L L L L L L L dd d c c c c p J J J J (3.88) Eigenvalue Analysis The linearized model given by Equations 3.87 and 3.88 has been used for studying the zero-pole distribution of the PMSG-based WECS in various operating points, defined by the resistance load, as well as by the wind speed. Three cases have been analyzed, described by the way the operating point variation takes place, namely: 1. variable wind speed and constant load; 2. variable load and constant wind speed; 3. variable wind speed and variable load, provided that the lv R pair ensures the optimal operation, i.e., optR v . Case 1. Wind speed takes the values 4 m/s, 7 m/s and 10 m/s and the load resistance is constant (80 ). By analyzing the zero-pole distribution presented in Figure 3.28, the following remarks can be made: 3 WECS Modelling 70 –7.7106 + j 313.9884 –7.7106 – j 313.9884 –4.5486 + j 27.0832 –4.5486 – j 27.0832 –9.0671 –7.7101 + j 313.9882 –7.7101 – j 313.9882 –4.5489 + j 27.08313 –4.5489 – j 27.08313 –9.0676 11.8 12 12.2 12.4 12.6 12.8 13 -5000 -3000 -1000 0 1000 3000 5000 11.8 12 12.2 12.4 12.6 12.8 13 -5000 -3000 -1000 0 1000 3000 5000 11.8 12 12.2 12.4 12.6 12.8 13 -5000 -3000 -1000 0 1000 3000 5000 11.8 12 12.2 12.4 12.6 12.8 13 -40 -30 -20 -10 0 5 15 [ ]t s [ ]t s [ ]t s 11.8 12 12.2 12.4 12.6 12.8 13 -5000 -3000 -1000 0 1000 3000 5000 [ ]t s 156.5 157 157.5 158 11.8 12 12.2 12.4 12.6 12.8 13 [ ]t s [ ]t s [ ]Sdi A [ ]Sqi A [ ]Rdi A [ ]Rqi A [rad/s]h [KNm]G Figure 3.31. Transients of state variables of a 2-MW SCIG-based WECS in response to voltage supply ( SqV ) step changes: fifth-order (thin line) vs. third-order (thick line) model Details about the MATLAB®/Simulink® implementation of this case study can be found in the folder case_study_1 from the software material. 4 Basics of the Wind Turbine Control Systems 4.1 Control Objectives Taking into account the ideas presented in the previous chapters, one can highlight the objectives of the WECS control (see Section 2.7). The list bellow selects the most important: controlling the wind captured power for speeds larger than the rated; maximising the wind harvested power in partial load zone as long as constraints on speed and captured power are met; alleviating the variable loads, in order to guarantee a certain level of resilience of the mechanical parts; meeting strict power quality standards (power factor, harmonics, flicker, etc.); transferring the electrical power to the grid at an imposed level, for wide range of wind velocities; There can be three main control subsystems (see Figure 4.1). Aerodynamics Electromagnetic subsystem Grid connection subsystemDrive train Wind stream Electric grid Pitch control Variable speed control Output power conditioning WECS Control system Figure 4.1. Main control subsystems of a WECS The first control subsystem affects the pitch angle following aerodynamic power limiting targets. The second implements the generator control, in order to 4 Basics of the Wind Turbine Control Systems 72 obtain the variable-speed regime and the third controls the transfer of the full (or a fraction) of electric power to the electric grid, with effects on WECS output power quality. The control structures result from defining one or more of the above goals stated in relation to a certain mathematical model of WECS. The controller determines the desired global dynamic behaviour of the system, such that ensuring power regulation, energy maximization in partial load, mechanical loads alleviation and reduction of active power fluctuations. 4.2 Physical Fundamentals of Primary Control Objectives Consider that the turbine operates in partial load at fixed pitch – often named “fine pitch” – that gives good aerodynamic performance and which can be considered pitch reference. When the wind velocity exceeds the rated, the turbine is operating in what is called full-load regime and the captured power – which potentially can vary with the wind speed cubed – must be aerodynamically limited (controlled). This is the formulation of the primary objective of the WECS control. There are several techniques usually used in order to fulfil this objective, which are reviewed next (Burton et al. 2001). The wind turbine aerodynamic behaviour fundamentals can be analysed in Figures 3.9 and 3.10. Some elements also result from the associated analysis using blade element theory (see Algorithm 3.1). One can remark that the key variable in aerodynamics behaviour, the incidence angle, increases with wind velocity and decreases with increase of rotational speed and pitch angle. Consequently, the aerodynamic efficiency ( ) ( )z xC i C i , will be affected by the incidence angle evolution as presented in Figure 4.2. 0 5 10 15 20 25 0 10 20 30 40 50 z x C C iopti Stall effect Feathering Aerodynamic efficiency Incidence angle Maximum capture Small ,l , Large v Small Large ,l , Small v Large Figure 4.2. Feathering and stall effects on the aerodynamic efficiency curve When the turbine experiences high winds, the aerodynamic power can be reduced by controlling the incidence angle through the rotational speed and/or pitch angle. The appearance of the aerodynamic efficiency curve in Figure 4.2 suggests two courses of action. Decrease of the incidence angle, which corresponds 4.3 Principles of WECS Optimal Control 75 0 Ratedwt P v RatedCut-in Cut-out Pitch regulated Passive stall Figure 4.4. Comparison between passive-stall and active-pitch control features 4.3 Principles of WECS Optimal Control This section is dedicated to the basics of WECS energy conversion optimization in the partial load regime. 4.3.1 Case of Variable-speed Fixed-pitch WECS Control of variable-speed fixed-pitch WECS in the partial load regime generally aims at regulating the power harvested from wind by modifying the electrical generator speed; in particular, the control goal can be to capture the maximum power available from the wind. For each wind speed, there is a certain rotational speed at which the power curve of a given wind turbine has a maximum ( pC reaches its maximum value). 0 ORC a)wtP l 0 b) ORC wt l Figure 4.5. Optimal regimes characteristic, ORC: a in the l wtP plane; b in the l wt plane All these maxima compose what is known in the literature as the optimal regimes characteristic, ORC (see Figure 4.5a – Nichita 1995). In the l wt plane, the ORC is placed at the right of the torque maxima locus (Figure 4.5b). By keeping the static operating point of the turbine around the ORC one ensures an optimal steady-state regime, that is, the captured power is the maximal 4 Basics of the Wind Turbine Control Systems 76 one available from the wind. This is equivalent to maintaining the tip speed ratio at its optimal value, opt (Figure 4.6) and can be achieved by operating the turbine at variable speed, corresponding to the wind speed (Connor and Leithead 1993). pC opt Figure 4.6. The unimodal power coefficient curve, expressing the aerodynamic efficiency Basically, the control approaches encountered in the WECS control field vary in accordance with some assumptions concerning the known models/parameters, the measurable variables, the control method employed and the version of WECS model used. Depending on how rich the information is about the WECS model, especially about its torque characteristic, the optimal control of variable-speed fixed-pitch WECS is based upon the following approaches. Maximum Power Point Tracking (MPPT) This approach is adequate when parameters opt and p max p optC C are not known. The reference of the rotational speed control loop is adjusted such that the turbine operates around maximum power for the current wind speed value. In order to establish whether this reference must be either increased or decreased, it is necessary to estimate the current position of the operating point in relation to the maximum of wt lP curve. This can be done in two ways: the speed reference is modified by a variation l , the corresponding change in the active power, P, being determined in order to estimate the value wt lP . The sign of this value indicates the position of the operating point in relation to the maximum of characteristic wt lP . If the speed reference is adjusted in ramp with a slope proportional to this derivative, then the system evolves to optimum, where wt lP = 0; a probing signal is added to the current speed reference; this signal is a slowly variable sinusoid; its amplitude does not significantly affect the system operation, but still produces a detectable response in the active power evolution. In order to obtain the position of the operating point in relation to the maximum, one compares the phase lag of the probing sinusoid and that of the sinusoidal component of active power. If the phase lag is zero/ , then the current operating point is placed on the ascending/descending part of wt lP , therefore, the slope of the speed reference must increase/decrease. Around the maximum, the probing signal does not produce any detectable response and the speed reference does not have to change. 4.3 Principles of WECS Optimal Control 77 In this simplified presentation of MPPT techniques, factors like the influence of wind turbulence and system dynamics that distort the information concerning the operating point position have been neglected. A more detailed description and analysis of performances can be found in Sections 5.1.1 and 5.2. Shaft Rotational Speed Optimal Control Using a Setpoint from the Wind Speed Information This solution can be applied if the optimal value of the tip speed ratio, opt , is known. The turbine operates on the ORC if ( ) optt , (4.1) which supposes that the shaft rotational speed is closed-loop controlled such that to reach its optimal value: ( ) ( ) opt opt l t v tR (4.2) This approach has some drawbacks related to the wind speed being measured by an anemometer mounted on the nacelle, which offers information on the fixed- point wind speed. But this information differs from the wind speed experienced by the blade (see Section 3.2.3), mainly because of the time lag between the two signals. Active Power Optimal Control Using a Setpoint from the Shaft Rotational Speed Information This method is used when both opt and maxp p optC C are known. From the expression of the power extracted by a turbine (Equation 2.31), it follows that 2 3 3 3 ( )1 1( ) 2 2 p wt p l C P C R v R (4.3) By replacing ( ) optt and p p optC C , one obtains the power reference for the second region of the power–wind speed curve: 3 opt optwt ref lP P K , (4.4) where 5 3 1 2 p opt opt C K R (4.5) This approach supposes an active power control loop being used, whose reference is determined based upon Equation 4.5. This method is widely employed, especially for medium and high-power WECS. In both methods an ORC tracking control loop is used. The wind turbulence
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