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Supervisory voltage control scheme for grid-connected wind farms Ko, Hee-Sang 2006

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Supervisory Voltage Control Scheme for Grid-Connected Wind Farms by Hee-Sang Ko B.S., Cheju National University, Republic of  Korea, 1996 M.S., Pennsylvania State University, USA, 2000 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (ELECTRICAL AND COMPUTER ENGINEERING) THE UNIVERSITY OF BRITISH COLUMBIA December 2006 © Hee-Sang Ko, 2006 Abstract Modern variable speed wind turbines utilize power electronic converters for  the grid connection requirement and to improve performance.  Most commonly used converters enable the wind turbines to maintain the required power factor  (power factor  control) or voltage (local voltage control) at the terminals. However, in many wind farm  applications there is a need to control voltage at a specified  remote location, which may require the installation of  additional compensating devices (transformer  tap changers, switched-capacitors, SVCs, etc.) to meet local power quality conditions. This thesis proposes a supervisory control scheme that uses the individual wind turbines to regulate voltage at the required location, i.e., point of  common coupling. The proposed approach considers that each turbine may have somewhat different  instantaneous wind speeds and real power outputs, and therefore  different  amounts of  reactive power available for  achieving the main control objectives. The operating limits of  each turbine are also taken into account to ensure that all power electronic converters operate in the allowable region. Since the proposed supervisory scheme is general and can work with different  controllers, we investigate several controllers in this thesis. The problem of  control design is formulated  as a linear matrix inequality. An innovative cost-guaranteed linear-quadratic-regulator-based controller with an observer is proposed and tuned for  a range of  operating conditions. In this thesis, we apply the proposed supervisory voltage control methodology to a candidate wind farm  site on Vancouver Island, BC, Canada, made available through collaboration with Powertech Labs Inc. We have developed a detailed model of  the system, using three 3.6 MW wind turbines, to carry out the simulation studies. The proposed control solution is compared with traditional approaches and shown to be very effective  during load disturbances and faults.  The proposed methodology is also flexible  and readily applicable to larger wind farms of  different  configurations. Ill Table of Contents Abstract ii Table of  Contents iii List of  Tables vii List of  Figures viii List of  Symbols xii List of  Abbreviations ^ xi Acknowledgments xviii Chapter 1. Introduction 1 1.1 Wind Power Status 1 1.2 Wind Turbine Technologies 4 1.2.1 Stand-along wind turbine grid connection 5 1.2.2 Wind farm  grid connection 5 1.2.3 Fault-ride-through capability 6 1.3 Voltage Control in Power Systems 7 1.4 Voltage Control Using Existing Wind Turbine Technologies 8 2. Impact of  Wind Energy on Power Systems 10 2.1 Local Impacts 11 2.2 System-Wide Impacts 12 2.3 Voltage Control in Power Systems with Wind Turbines • 14 2.3.1 Impact of  wind power at the distribution level •••• 15 2.3.2 Impact of  wind power at the transmission level 16 2.4 Voltage Control at Remote Locations 16 2.5 Research Objectives and Approaches ; 19 2.5.1 Problem statement 19 2.5.2 Research objectives 20 2.5.3 Proposed approach 20 2.6 Contributions 22 3. Model Description of  Grid-Connected Wind Farm System 23 3.1 Study System •. ••••• 24 3.2 System Model Components 27 3.2.1 Mechanical components 27 3.2.2 Electrical components 31 3.3 Voltage Source Converter Controller 41 3.3.1 Rotor-side converter controller 42 3.3.2 Grid-side converter controller 47 3.3.3 DC-link controller 49 3.4 Conventional Voltage Control of  Wind Turbine 50 4. Wind Farm Voltage Control 52 4.1 Common Practice •••• 52 4.2 Available Reactive Power in a Multi-Turbine System 53 4.3 Supervisory Voltage Control Scheme 55 4.4 Plant Model and Conventional Controllers 56 4.4.1 Linearized and reduced-order model 56 4.4.2 PID supervisory controller design 61 4.4.3 Evaluation of  conventional controllers 65 5. Advanced Voltage Control Schemes 67 5.1 Observer-Based Framework 67 5.2 State Observer Design 69 5.3 Linear Quadratic Regulator Approach 71 5.3.1 Formulation of  LQR 72 5.3.2 Conventional approach 74 5.3.3 Cost-guaranteed approach 77 5.3.4 Evaluation of  controllers 78 5.4 Advanced LMI Representation of  LQR 79 5.4.1 Taking into account cross-product terms ^ in the conventional approach 5.4.2 Taking into account cross-product terms ^ in the cost-guaranteed approach 5.4.3 Evaluation of  controllers 81 5.5 Summary of  Control Gains 83 6. Simulation Studies 86 6.1 Small Disturbances ,• 86 6.1.1 Wind speed variations 86 6.1.2 Load variations 91 6.1.3 Summary 96 6.2 Large Disturbances 97 6.2.1 Three-phase fault  97 6.2.2 Summary 102 7. Conclusion and Future Work 103 7.1 Conclusion 103 7.2 Future Work ••• 104 Bibliography 106 Appendix A System Parameters and Operating Conditions 112 Appendix B Voltage Source Converter Controller Design 119 Appendix C Local Voltage Controller Design • • • • 128 Appendix D Matlab Script Files 134 Appendix E Proof  of  Theorem 140 Appendix F Effect  of  Cross-Product Terms 145 List of Tables Chapter 3 Table 3.1 Switching Operations 34 Chapter 4 Table 4 1 Eigenvalues, Damping ratio, and Frequency ^ of  the 5 th-order Reduced Model Table 4.2 Gains of  the PID-Supervisory Controller 62 Chapter 5 Table 5.1 Eigenvalues, Damping ratio, and Frequency of  ^ the Closed-Loop Observer Table 5.2 Design Parameters and Control Gains 84 Table 5.3 Eigenvalues, Damping ratio, and Frequency 85 Chapter 6 Table 6.1 Voltage Magnitude Deviations 97 Table 6.2 Comparisons of  the Voltage Control Performance  (pu) 102 Appendix A Table A. 1 Wind Power Model Parameters Model Parameters 112 Table A.2 Turbine Controller Parameters .••• 113 Table A.3 Two-Mass Rotor Model Parameters 113 Table A.4 DFIG and DC link Parameters 114 Table A.5 Maximum Operating Limit of  VSC 114 Table A.6 PI Controller Gains of  VSC 114 Table A.7 Line Parameters 114 Table A.8 Thyristor Excitation System 115 Table A.9 Synchronous Generator Parameters •••• 115 Table A. 10 Operating Conditions 116 Vlll List of Figures Chapter 1 Figure 1 1 Installed wind power capacity in the Europe, US, and the World • • • • 2 Figure 1.2 Cost of  wind energy (Year 2000 US$) and cumulative 3 domestic capacity (US) Figure 1.3 Voltage sag magnitude for  132kV Fault 8 Figure 1.4 Wind-energy system utilizing constant speed wind turbine 9 Figure 1.5 Wind-energy system utilizing variable speed wind turbine • • 9 Chapter 2 Figure 2.1 Voltage regulation based on the X/R  ratio • • • 17 Figure 2.2 D i a g r a m depicting the wind farm  interconnection impedance 18 Chapter 3 Figure 3.1 Wind power system considered for  dynamic studies 25 Figure 3.2 B l o c k diagram showing subsystem input-output variables 25 Figure 3.3 Variable speed wind turbine with DFIG • • • 26 Figure 3.4 Block diagram of  the pitch control 28 Figure 3.5 Simplified  block diagram of  the two-mass rotor 29 Figure 3.6 Block diagram of  the two-mass drive train rotor model 30 Figure 3.7 Schematic representation of  the voltage source converter 33 Figure 3.8 Representation of  the switching function  on the grid-side converter • • 33 Figure 3.9 Transmission line lumped-parameter gd-model 36 Figure 3.10 Exciter model block diagram 39 Figure 3.11 RL load model representation in the ^-synchronous 4 Q reference  frame Figure 3.12 Block diagram of  the voltage source controller modules 41 Figure 3.13 T u r b i n e power versus speed tracking characteristic 42 Figure 3.14 Block diagram of  the rotor-side converter controller 43 Figure 3.15 Block diagram of  the grid-side converter controller • • 48 Figure 3.16 Block diagram of  the dc-link model and its controller, PI7 50 Figure 3.17 Overall control block diagram of  the voltage source converter 51 Chapter 4 Figure 4.1 Real and reactive power operating limits of  ^ voltage source converter Figure 4.2 Block diagram of  the supervisory voltage control scheme 55 Figure 4.3 Bode diagrams of  the full-order  model (104 th) 58 Figure 4.4 Bode diagrams of  the reduced-order model (4 t h) • - 58 Figure 4.5 Step response of  the open-loop and the closed-loop ^ reduced-order system Figure 4.6 Implementation of  the PID controller ^ with distributed anti-windup loop Figure 4.7 Voltage transient at the PCC resulted ^ from  a three-phase fault Chapter 5 Figure 5.1 Block diagram of  the supervisory voltage control gg with observer Figure 5.2 Comparison of  PID, LQRS, and LQRCG controllers 79 Figure 5.3 Comparison of  LQRCG and ALQRS controllers 82 Figure 5.4 Comparison of  PID, ALQRS, and ALQRCG controllers 83 Chapter 6 Figure 6.1 Wind speed (m/sec) 87 Figure 6.2 Real power set-point for  each WT gg due to wind speed variation Figure 6.3 Real power output from  each WT gg due to wind speed variation Figure 6.4 Real power output from  the wind farm  ^ due to wind speed variation Figure 6.5 Reactive power output from  each WT g^ due to wind speed variation Figure 6.6 Real power output from  the wind farm  pQ due to wind speed variation Figure 6.7 Voltage fluctuations  due to wind variation, as observed ^ at the WT terminals Figure 6.8 Voltage fluctuations  due to wind variation, as observed ^ at the PCC Figure 6.9 Voltage transient observed at the PCC due to load impedance changes Figure 6.10 Voltage transient observed at the PCC due to load impedance changes: Detailed view of  the PID-supervisory and ALQRCG •••••• 93 controllers Figure 6.11 Voltage transient observed at the WT terminals 93 Figure 6.12 Real power output from  each WT .••••• 94 Figure 6.13 Real power output from  the wind farm  94 Figure 6.14 Reactive power output from  each WT 95 Figure 6.15 Reactive power output from  the wind farm  95 Figure 6.16 Reactive power set-point and maximum at each WT 96 Figure 6.17 Voltage transient observed at the PCC due to the fault  98 Figure 6.18 Real power output from  each WT due to the fault  99 Figure 6.19 Real power output from  the wind farm  due to the fault  99 Figure 6.20 Reactive power output from  each WT due to the fault  100 Figure 6.21 Voltage transient observed at the terminal of  each WT jqq due to the fault Figure 6.22 Reactive power output from  the wind farm  due to the fault  101 Figure 6.23 Reactive power set-point and maximum at each WT ^ due to the fault Appendix B Figure B. 1 Bode diagrams of  the transfer  function  of  the open-loop ^ q and closed-loop system Figure B.2 Comparison of  the step response of  the open-loop ^ and closed-loop system Figure B.3 Bode diagrams of  the transfer  function  of  the open-loop and closed-loop system Figure B.4 Step response of  the closed-loop system 123 Figure B.5 Bode diagrams of  the transfer  function  of  the open-loop and closed-loop system Figure B.6 Step response of  the closed-loop system 125 Figure B.7 Bode diagrams of  the transfer  function  of  the open-loop _ and closed-loop system Figure B.8 Step response of  the closed-loop system 127 Appendix C Figure C.l Bode diagram of  the full-order  model (23 th) and the reduced-order model (3 r d) Figure C.2 Nyquist plot of  a compensated loop transfer  function  130 Figure C.3 Bode diagrams of  the transfer  function  of  the open-loop and closed-loop system Figure C.4 Step response of  the closed-loop system 132 Figure C.5 Block diagram of  the PI controller with distributed anti-windup scheme Appendix F Figure F.l Optimal and neighbouring optimal paths 146 List of Symbols The unit is based on the per unit (pu) if  there is no specification. Matrices system matrix input matrix output matrix feedforward  matrix system matrix of  the reduced-order model input matrix of  the reduced-order model output matrix of  the reduced-order model feedforward  matrix of  the reduced-order model randomly chosen matrix observer gain matrix positive-definite  Lyapunov matrix positive-definite  matrix change variable slack variable Matrices - Greek Letters A decision matrix Vectors k LQR gain vector u input vector w stator noise signal vector x state vector x(0) stationary random initial state vector A B C, F D,H A B C D G K e P S Y Z x estimated stator vector z measured system output vector z observer output vector y system output vector y controlled system output vector Vectors - Greek Letters V flux  vector output noise signal vector Scalars AR area swept by the rotor (m2) CP (A,  6) power coefficient C capacitance dc field  voltage P real power Q reactive power R resistance Rt rotor ratio (m) S  apparent power T  torque v,i voltage and current V,I  steady state voltage and current vw wind speed (m/s) Z impedance Scalars - Greek Letters coe cor stator angular speed rotor angular speed slip angular speed mechanical rotor angular speed base angular speed (rad/sec) base mechanical angular speed (rad/sec) pitch angle (degree) ratio of  the rotor blade tip speed and wind speed air density (kg/m3) damping ratio phase margin (degree) observer design parameter flux Mathematical Symbols a < 0 matrix a is negative definite a > 0 matrix a is positive definite a e A a is an element of  the set A a first  order time derivative of  a a second order time derivative of  a a third order time derivative of  a d/dt  first  order time derivative d B 201og1 0 |G| E Expectation G transfer  function 51 field  of  real number w-dimensional real vector space s Laplace operator tr trace of  matrix A error <os cob tyjiase e A P c K Q/7'RW ¥ Superscripts ref set T -1 max min Subscripts abc a c ca cl d dc f filter gc j k load max mech min m o <1 r s TL t tr reference set-point transpose inverse maximum minimum phase abc armature of  synchronous generator voltage source converter cable closed-loop J-axes of  reference  frame direct current field  winding of  synchronous generator filter  connected to the grid-side converter gain crossover frequency sub systems damper winding of  synchronous generator load maximum mechanical minimum magnetizing steady state q-axes of  reference  frame rotor of  generator stator of  generator transmission line turbine rotor transformer List of Abbreviations List is given in Alphabetical order. Acronyms ac alternating current dc direct current DFIG doubly fed  induction generator FACTS flexible  ac transmission system IPPs independent power producers LMI linear matrix inequality LQR linear quadratic regulator LVC local voltage control PCC point of  common coupling PFC power factor  control PI proportional-plus-integral PID proportional-integral-derivative pu per unit PWM pulse width modulation SCR short circuit ratio SG synchronous generator SMES superconducting magnetic energy storage STATCOM static compensator SVC static VAR compensator TR transformer TL transmission line K voltage at the connection point V GRI D voltage at the grid VSC voltage source converter V pcc voltage at the point of  common coupling WF wind farm WT / wt wind turbine ZF ARM  equivalent impedance of  the wind farm Z GRI (I  equivalent impedance of  the grid LQRS LQR supervisory LQRCG LQR cost-guaranteed ALQRS Advanced LQR supervisory ALQRCG Advanced LQR cost-guaranteed Acknowledgments XVlll I would like to acknowledge the essential role of  both my supervisors, Dr. Guy Dumont and Dr. Juri Jatskevich. I wish to express my deepest gratitude for  their guidance, assistance, encouragement and advice in this project. They were always ready to discuss the intricate details of  this research and share their expertise in control theory and power systems which made this project possible. I also would like to thank Dr. Prabha Kundur and Dr. Ali Moshref  of  Powertech Labs Inc., for  providing financial  support as well as valuable information  for  the model that made this research project practical and more relevant to the industry. I will never forget my visits to Powertech Labs and the many discussions that I had with them. I would like to thank my parents, who raised me and supported me in every possible way. They have enabled me to study at UBC and develop my career. I will always remain grateful  to them. I am also endlessly grateful  to my wife  for  sharing my life  and encouraging me during the times when research was not going smoothly, and to my little son Kevin, who is very busy learning to walk and talk and makes our life  happy and worthwhile. UBC, December 2006 Hee-Sang Ko Chapter 1 Introduction 1.1 Wind Power Status The advantages of  conventional thermal, nuclear, and hydro power generation include a relatively low price, as well as complete control of  the generated power. Renewable power generation, however, poses less severe environmental consequences, but relies on available primary energy sources, such as sunlight and/or wind, that are not controllable in the same sense as the traditional energy sources. Renewable wind energy technology uses wind turbines to convert the energy contained in the wind into electrical energy. Wind is an inexhaustible primary energy source. Furthermore, the environmental impact of  harnessing wind power is small. Although wind turbines affect the visual scenery and emit some noise, the overall consequences appear to be small with no significant  impact on the ecosystem. Moreover, when the wind turbines are installed at remote locations on the ground or offshore,  the visual effect  and noise are no longer a concern. Compared with other renewable energy sources, such as photovoltaic (PV), ocean waves, and tidal power generation, wind power appears to be less expensive and gives higher returns per affected  (required) area. That is why many countries including Germany, Denmark, Spain, etc., demonstrate strong growth in the wind energy sector. Figure 1.1 shows the growth of  wind power in Europe, the US, and worldwide [1]. As can be seen in Figure 1.1, the installed wind power capacity shows a steady growth; during the last five  years, annual growth has been higher than 30%. Total Installed Wind Capacity o CO Q. (0 O 1982 1985 1988 1991 1994 Year 1997 2000 2003 Rest of World Europe United States Figure 1.1: Installed wind power capacity in Europe, the US, and worldwide [1]. Worldwide, many countries value the advantages of  renewable power generation and support the expansion of  its capacity in various ways. However, the installation/equipment cost involved and lack of  direct control remain concerns, especially when the penetration levels arehigh[2]. The cost disadvantage of  wind power is reduced in many, cases by some form  of  subsidy. For example, power companies may be forced  to buy power from  renewable energy providers at a guaranteed price that is not based on the actual value of  the power, but is calculated such that the renewable energy project becomes profitable  for  the developer. Unless the power companies are able to sell this power as "green power" at a premium price, such subsidies will lead to a general increase in the electricity price, whereas all consumers would end up paying for  the additional cost of  electricity generated from  renewable sources. Alternative subsidies may include a direct support given to the developers of  renewable energy projects, which spreads the cost burden associated with renewable energy over all taxpayers. Using a variety of  incentives, the cost disadvantages associated with developing renewable energy sources continue to diminish. For example, Figure 1.2 shows the changes in renewable electricity cost and installed capacity growth over the last decade in the US [1], which is tied economically to Canada. Current wind energy in Canada produces a very small portion of  the electricity supply. Canada's total installed capacity of  444 MW satisfies  only 0.2% of  the nation's energy demand. However, with new projects totalling about 2000 MW coming on-line in the near future,  and more planned, it is expected that wind energy will cover up to 3% or more of  Canadian energy needs by the year 2012 [3]. As seen in Figure 1.2, the present renewable electricity cost is reaching below 10 cents/kWh and becomes directly competitive with the traditional energy production. Cost of Wind Energy (cents/kWh) — • — Capacity (MW) i2 c d) E> 0) c LL1 to O o - - 1000 1980 1984 1988 1992 1996 Year 2000 2004 o TO CL CO O Figure 1.2: Cost of  wind energy (Year 2000 US$) and cumulative domestic capacity [1]. The present practice requires the independent power producers (IPPs) and/or generators who want to connect to the grid to meet the so-called connection requirements  of  the local electric utility (the grid company). These requirements may also include the steady-state and dynamic interaction between the generator and the grid. In order to maintain the power generation and consumption balance, necessary for  stable functioning  of  the power system, the traditional power plants always exert necessary control actions. However, renewable energy sources are presently exempted from  such control functions.  This, in turn, simplifies  the requirements for  the renewable energy source interconnection as well as the project developer, allowing connection to the system without having to take part in the overall stabilization effort. 1.2 Wind Turbine Technologies Although the fundamental  principle of  a wind turbine is straightforward,  modern wind turbines are very complex systems. The design and optimization of  the wind turbine's blades, drive train, and tower require extensive knowledge of  aerodynamics, mechanical and structural engineering, control and protection of  electrical subsystems, etc. Two major technologies are prevalent in the wind power energy sector today. First, a substantial scaling up has taken place to further  reduce the cost of  wind power and the individual wind turbines. For modern wind turbines of  the multi-MW class, both the nacelle height and rotor diameter are in the order of  100 meters. Thus, at the vertical position, the blade tip can reach heights of  up to 150 meters. The largest wind turbine presently developed is a 5MW unit [4] that is based on a new design concept involving a carbon-fiber  material type blade, and gearless and permanent synchronous generator technology developed especially for  offshore  wind power generation. Enercon is also presently upgrading their E-112 turbine technology and advertising up to 6MW of  output power [5]. Second, most of  the presently developed large wind turbines are based on variable-speed operation rather than fixed-speed  technology, which was used initially and is simpler. The fixed-speed  wind turbines would typically include an induction or synchronous generator that is directly connected to the grid; hence, the rotor speed remains essentially constant, or varies very slightly with the speed of  the wind. This simple design entails lower manufacturing  costs. The variable-speed wind turbine is technically more advanced. A typical variable-speed wind turbine consists of  more components and needs additional control system(s), and is therefore more expensive. However, it has various advantages over constant-speed wind turbines, such as increased energy yield, reduced noise emission, the ability to withstand higher mechanical operating limits, and additional controllability of  active and reactive power. 1.2.1 Stand-alone wind turbine grid connection In the majority of  installations, the wind turbines are connected to the grid. The grid connection of  solitary wind turbine is relatively straightforward.  The voltage at the turbine's generator terminal is typically low (690V is common); therefore,  a step-up transformer  is used to bring the voltage to the grid level at the point of  connection. Furthermore, some switchgear is necessary so that the wind turbine can be disconnected in the case of  a short circuit or in islanding [6]. 1.2.2 Wind farm grid connection The wind farm  represents an aggregation of  several or many tightly interconnected wind turbines that are then interconnected with the power grid. Although the individual wind farms may represent a large contribution to the local power pool that is comparable in size to the conventional medium-size power plants, their effect  on the power system is very different from  that of  conventional synchronous generators. The difference  is especially pronounced in terms of  response to disturbances in the terminal voltage, frequency,  and power, depending on the type of  wind turbines used. In the case of  fixed-speed  wind turbines based on induction generator technology, an installation of  additional capacitor banks is often  required to support the reactive power demand as well as to control the voltage. In the case of variable-speed wind turbines, the wind-farm  response and dynamic interaction are primarily determined by the wind turbines' internal power electronic converters and the respective controllers [7]. 1.2.3 Fault ride-through capability The wind turbine manufacturers  presently offer  a number of  practical solutions and control approaches to improve the reliability and stability of  power systems with wind turbines in the event of  large disturbances such as faults.  In particular, the fault-ride-through  capability of the wind turbine envisions that the wind turbine remains connected to the grid during the transient, and enables faster  recovery and more reliable operation of  the overall network after the source of  the disturbance is removed (fault  is being cleared). Option 1: Crowbar Protection The crowbar protection scheme may be used with wind turbines that are based on the doubly-fed  induction generator (DFIG) technology. This protection redirects the current from  the rotor-side converter by short-circuiting the rotor windings and thus blocking the rotor-side converter. Therefore,  the rotor current goes through the crowbar and does not damage the converter. This measure makes the DFIG resemble a conventional squirrel-cage induction generator during the transient, including the contribution to the short-circuit current [8]. Crowbar protection is usually activated when the peak value of  the rotor current exceeds approximately 2 times the normal rotor current. The crowbar is deactivated again when the ac voltage reaches 80% of  the predefined  voltage level and the rotor current is below that current for  activating the crowbar. Because crowbar protection makes the DFIG operate similar to the squirrel-cage induction machine and consume reactive power during the large transients (which basically disables the controls), it also has an undesirable effect  on voltage stability. Alternatively, the two control schemes described below allow the control actions of reactive power during the fault. Option 2: Power Factor Control The power factor  control (PFC) scheme [8]-[10] relies on the rotor- and grid-side converters to ensure the specified  power factor  (usually unity) at the wind turbine terminals. Under this scheme, reduction of  the generator terminal voltage leads to an increase in the real power injected by the wind turbine. When this happens, the rotor will decelerate while the power from  the wind is lower than the real power taken from  the generator. At this point, the rotor blade pitch controller is activated to avoid the wind turbine operation in the under-speed region. Option 3: Local Voltage Control Local voltage control (LVC) scheme [8], [9], [11], [12] utilizes the reactive power by controlling the rotor current to regulate the voltage at the wind turbine terminals. When this scheme is used, the wind turbine is more likely to remain connected to the grid during the fault  and the control operation may help to restore the voltage after  the disturbance. 1.3 Voltage Control in Power Systems Because transmission lines, cables, and transformers,  etc. have impedance, voltage control is necessary to maintain the bus voltages within the allowable range required for  the safe  and reliable operation of  all equipment. Appropriate measures must be taken to prevent and/or reduce voltage deviations. It is important to stress that bus voltage is a local quantity, as opposed to frequency,  which is more often  associated with the system (global) level. It is therefore  not possible to control the voltage at a certain bus from  an arbitrary point in the system without affecting  the voltages at other buses, however, the voltage can be effectively controlled locally. Short-duration reductions in voltage are often  referred  to as a voltage sags and have been associated with voltage instability in power systems [13], [14], The voltage sags due to motor-starting transients are typically longer. The relatively short voltage sags are often caused by faults  in the power system, and are often  more severe in magnitude and are responsible for  the majority of  equipment trips. To get an idea of  how the sag magnitude propagates in a radial system, the voltage sag due to a fault  on a 132kV transmission line is shown in Figure 1.3 [14]. As shown, the voltage sag at Bus A is less severe as the distance from  the fault  increases. Based on this observation, voltage sags from  a distant fault  can be more easily mitigated and is less likely to trip local equipment than a sag due to a nearby fault. a, Set-point level Monitored at Bus A -M 132kV T L BusA 132kV line Load TR 7 fault 33kV TL 0 20 40 60 80 100 Load Distance to the fault  in kilometers TL: transmission line, TR: transformer Figure 1.3: Voltage sag magnitude for  132kV fault  [14]. 1.4 Voltage Control Using Existing Wind Turbine Technologies In this section, conventional voltage control of  the grid with wind turbine/farm  is reviewed with respect to the existing wind turbine technologies. Figure 1.4 shows a power network with a constant-speed wind turbine. This type of  wind turbine consumes reactive power. To achieve a power factor  close to unity at the point of  grid connection, an additional source of reactive power such as static VAR compensator (SVC) or capacitor banks, etc., is always needed, and is often  placed close to the connection point, as shown in Figure 1.4. In addition, real power generation fluctuates  quite significantly  with wind speed changes. Therefore, regulating the voltage at the remote location, or point of  common coupling (PCC), in terms of  critical load, often  requires having another additional compensating device, which increases the costs and complicates operation. Due to these disadvantages, this type of  wind turbine is not usually used when there is high penetration of  wind power in the grid. Figure 1.4: Wind-energy system utilizing constant-speed wind turbine. A wind energy system with a grid-connected variable-speed wind turbine is shown in Figure 1.5. In this type of  wind turbine, a voltage source converter (VSC) is used, which may be used as a source of  reactive power if  the converter ratings and operating conditions permit. Therefore,  the reactive power available from  VSC, if  any, can be utilized for  voltage control purposes. Conventional control schemes include PFC and LVC. In PFC mode, the reactive power QG is controlled to be zero, and the additional device at the wind turbine terminal is not necessary. The LVC mode uses available reactive power from  the VSC to regulate voltage at the wind turbine terminal. Both of  these control strategies are local with respect to the wind turbine terminal and do not consider the voltages further  away in the system. To regulate the voltage at a remote PCC, additional reactive power compensating devices are often  still required, which entails undesirable costs. Vwt PCC Figure 1.5: Wind-energy system utilizing variable-speed wind turbine. Chapter 2 Impact of Wind Energy on Power Systems The impact of  wind energy on a power system is associated with its inherently fluctuating unpredictable output power. The response of  wind farms  is also determined by the technology and/or controls used in the individual wind turbines. For instance, when a constant-speed wind turbine is used, controlling reactive power is made possible by using additional compensating devices only. At the same time, when a variable-speed wind turbine is used, controlling reactive power is possible at the wind turbine terminal by utilizing the respective inverters [15]. Small wind farms  and individual wind turbines by themselves are relatively weak power sources. Because wind farms  are often  installed at remote locations and have a weak connection with the grid, additional measures to ensure voltage control in the grid are required, especially when the portion of  the wind power in the grid is substantial [15], [16]. However, the exact measures that are necessary for  achieving the desired voltages throughout the whole system depend highly on the location and characteristics of  the wind farm,  the network layout, the capabilities of  the remaining conventional synchronous generators, spinning reserves, etc. [17]. Depending on the extent to which the wind farms  affect  the grid, their impacts may be broadly categorized as local or system-wide. 2.1 Local Impacts The impacts observable in the close vicinity of  the wind power interconnection include: . Change of  fault  currents, protection scheme settings, and switchgear ratings Change of  power flow  in local distribution network Change of  voltages at nearby buses • Flicker • Harmonics The first  two impacts must be investigated whenever a new generation capacity, wind or otherwise, is being considered for  interconnection. The way in which wind farms  affect voltages at nearby buses depends on the type of  wind turbine (variable- or fixed-speed)  used and their controls [15]. The contribution of  wind farms  to the fault  current also depends on the type of  wind turbine used [15], [18], For instance, a constant-speed wind turbine based on a squirrel-cage induction generator directly connected to the grid contributes to the fault current and relies on conventional protection schemes (over-current, over-speed, over- and under-voltage, over- and under-frequency).  At the same time, a variable-speed wind turbine also changes the fault  current. However, due to the faster  control action of  power electronic converters in variable-speed wind turbines, the fault  current may be actively controlled to enable the fault-ride-through  capability. Flicker is typical with constant-speed wind turbines [18], wherein the fluctuating  wind speed is directly translated into fluctuations  of  output power. Depending on the strength of  the grid, the resulting power fluctuations  will result in voltage fluctuations  propagating in the network. These voltage fluctuations  may lead to undesired fluctuations  in the light brightness of commercial and residential buildings and cause annoyance and irritation. The power quality problem that results in light fluctuation  is referred  to as flicker.  However, flicker  problems are not generally associated with variable-speed wind turbines because the wind speed fluctuations  are not directly translated into output-power fluctuations.  With the rotor inertia acting as a low-pass filter  and the additional action of  the power electronic converters, it is possible to smooth out the effect  of  wind speed and power fluctuations. Harmonics are mainly associated with variable-speed wind turbines [19] and their use of switching power electronic converters. However, modern variable-speed wind turbines utilize converters that operate at high switching frequencies  and employ advanced control algorithms and filtering  techniques to minimize harmonics propagation [15], [18]. 2.2 System-Wide Impacts In addition to local impacts, wind power also introduces large-scale effects  that become more noticeable as the penetration level of  wind power in the grid increases. In particular, high penetration of  wind energy has an impact on the following: Power system dynamics and stability • Reactive power generation and network voltage control System operation/balancing and dispatch of  the remaining conventional units Frequency control The impact on the dynamics and stability of  power systems is mainly due to the fact  that wind turbine generating systems [20] do not provide an inertial response similar to conventional generators and do not participate in stabilizing control actions. Instead, the voltage and frequency  response of  wind turbines is determined by the underlying technology, interconnection inverters, and the corresponding internal controllers. High penetration of  wind energy in power systems has been noticed to affect  reactive power generation and voltage control in the system [15], [21], [22], First, not all wind farms  are capable of  varying their reactive power output. This is, however, only one aspect of  the impact of  wind power on voltage control in a power system. Apart from  this, wind power plants cannot be installed at arbitrary locations and must be erected at places with good wind resources [23], The locations with good wind conditions are not necessarily favorable  from the perspective of  grid voltage control. In choosing a location for  a conventional power plant, it is generally easier to take into account the voltage control aspect. The impact of  wind power on system balancing, i.e. the dispatch of  remaining conventional units and frequency  control, is also due to the fact  that wind turbine output is not traditionally controlled. In general, the power generation from  wind farms  is uncontrolled as well, and wind power does not contribute to the primary frequency  regulation. Although this would be technically possible, it would require a reduction in energy yield and financial  loss for  the wind farm  operators. Therefore,  as long as the wind farms  are not participating in power system control and as long as there are cheaper means to keep the system balanced, wind farms  are not likely to contribute to system balancing. However, the impact of  wind power on system balancing should be given special consideration in the case of  higher wind power penetration,, wherein the numbers of conventional generator units and the spinning reserve are decreased. Longer term wind variations (often  from  15 minutes to several hours) tend to complicate the dispatch of  the remaining conventional generators used to supply the load. The resulting demand profile  that is formed  by the load minus the generated wind power now has to be met by remaining conventional power generation. Due to the stochastic nature of  wind, this resulting demand profile  is usually less smooth than that produced without a wind power contribution. Therefore,  faster  dispatch action of  the conventional generation and reserve units is required, which is altogether more difficult  to accommodate. Thus, imbalance between the generation and the load may occur more often  and affect  the system frequency.  To mitigate/reduce this imbalance, it may be possible and/or necessary to incorporate a forecast  of  the wind speed into the real-time dispatching of  conventional generation. 2.3 Voltage Control in Power Systems with Wind Turbines Traditionally, voltage control for  transmission grids and distribution grids is achieved differently.  At the transmission level, large-scale centralized power plants keep the bus voltages within the allowable range. At the distribution level, dedicated equipment such as tap changers, switched capacitors and/or reactors, etc., are often  utilized for  voltage control at a particular location. Overall, in a traditional power system, the bus voltages are regulated by combining the action of  large-scale power plants at the transmission level with the use of additional devices at various levels and locations [13]. A number of  recent developments in energy production have complicated the traditional approach to voltage control. In particular, the increased use of  wind turbines for  generating electricity makes voltage control more challenging, due to the unpredictable nature of  wind conditions. When individual wind turbines or small-sized wind farms  are connected at the distribution level, the action of  the auxiliary compensating devices and/or tap-changers must be coordinated with the operation of  the wind turbines to ensure the required voltage regulation at the affected  buses. The problem of  harmonized integration becomes even more challenging as the level of  wind power penetration increases and large-scale wind farms  are connected at the transmission level. Not only are the voltages at various locations affected, but also the power flow,  power system dynamic, transient stability, and reliability [24], [25], The common practice in wind turbine operation is to disconnect them from  the grid immediately when a fault  occurs somewhere in the system. However, research trends and some applications suggest that wind turbines may be required to stay connected longer and ride through part of  or the entire fault  transient(s) to enhance system stability [3], [15], [16], [18], [21], [22], [26]. In this regard, in many countries with high levels of  wind energy penetration in power systems, the wind turbine grid connection standards are being revised in terms of  their impact on transient voltage stability. For example, some presently proposed grid-connection requirements for  allowable voltage levels at the connection point with the transmission grid during operation are as follows  [18]: DEFU in Denmark (<1%); VDEW and E.ON in Germany (<2%); AMP in Sweden (<2.5%); ESBNG in the Republic of  Ireland (<2.5% for  llOkV level and <1.6% for  between 220kV to 400kV). To achieve these high standards and to make grid integration easier and more reliable, active control of  individual wind turbines and wind farms  is becoming increasingly important. Maintaining the voltage at various locations becomes more of  a concern where there is a high level of  wind power penetration in power systems. Thus, it is necessary to examine how the operation of  conventional power systems and voltage control at the distribution and transmission levels are affected  by wind power. 2.3.1 Impact of wind power at the distribution level Traditionally, voltage control in distribution grids includes the tap-changing transformers (i.e., transformers  in which the turns ratio can be changed) and devices that can generate or consume reactive power (i.e., shunt capacitors or reactors) [13], [18]. The use of  tap-changing transformers  is a rather cumbersome way of  controlling bus voltages. Assuming a radial network, rather than affecting  the voltage at one bus and/or its direct vicinity, the whole voltage profile  of  the distribution branch is shifted  up or down, depending on whether the transformer  turns ratio is decreased or increased. Switched capacitors and reactors perform  better in this respect and have a more localized effect.  In combination with installing auxiliary voltage regulation devices, the converters of  modern variable-speed wind turbines may also be utilized for  voltage control. However, the sensitivity of  the bus voltage to changes in reactive power often  requires relatively large capacitors and reactors [13]. One might argue that with an increasing number of  wind turbines connected to the distribution grid, the voltage control possibilities might increase as well. However, in many cases, the opposite is true, for  following  reasons: Depending on the design type, wind turbines are not always (if  ever) able to vary reactive power generation in the required range. • It may be very costly to equip the wind turbines with additional voltage control capabilities. • Adding the voltage control capabilities could increase the risk of  islanding. When there are many wind turbines, it may be difficult  to coordinate the control action(s), considering the varying network topology and operation. 2.3.2 Impact of wind power at the transmission level At the transmission level, in addition to traditional large-scale power plants and synchronous generators, dedicated equipment such as capacitor banks and flexible  ac transmission systems (FACTS) have also been used for  voltage control [13], [18]. However, due to industry deregulation, voltage control has become a more complicated task in the planning and dispatch of  power plants [14], Additionally, when large wind farms  are installed at remote locations or offshore  [18], achieving the desired voltage control at some remote and weakly connected locations may be difficult.  Therefore,  the voltage control capabilities of  various wind turbine types are expected to become increasingly important. 2.4 Voltage Control at Remote Locations As mentioned in the previous section, to achieve easier grid integration and reliable voltage control, voltage control of  wind turbines is essential [15], [16], [18], [21], [22], [26]. However, in many wind farm  installations, there is a need to control the voltage at a specified remote location, or point of  common coupling (PCC), which becomes more difficult  due to the fluctuating  nature of  wind power. Voltage control at remote PCCs may become even more difficult  in places with high penetration of  wind energy and weak ties to strong subsystems. In these cases, additional compensation devices are sometimes used. Voltage fluctuation  also depends on the effective  or equivalent impedance of  the grid. ' Broadly speaking, injecting power into a weak grid causes large voltage fluctuations compared to in a strong grid. It is well known that lower grid impedance results in a higher short circuit ratio [27]. In practice, a short circuit ratio of  greater than 20 is considered to indicate a grid that is strong [27], The composition of  the equivalent impedance of  the grid, which is often  expressed as the X/R  ratio [28], also has a pronounced effect  on voltage control. To clarify  the role of  line impedance in voltage regulation, a simplified  phasor diagram of  a grid-connected wind turbine is shown in Figure 2.1. Here, V cp represents the voltage at the wind turbine connection point, and V gricj represents a strong utility grid. The effectiveness  of  voltage regulation by adjusting the reactive power depends significantly  on the XjR  ratio of  the connecting tie, represented here by an equivalent impedance Z. Assuming certain fixed  values of  the grid voltage V grid  and the injected reactive current / , voltage V cp can be determined using the phasor relations depicted in Figure 2.1 (b). When the XjR  is high (diagram on the left),  the voltage drop across the impedance Z is closer in phase to the grid voltage V grid,  which results in a significant  increase in V cp. On the other hand, when the X/R  ratio is low (diagram on the right), the voltage drop across the impedance Z is closer in phase to the current I,  which results in a smaller increase in V r„. i.p Based on this observation, it can be concluded that when the X/R  ratio is high, the voltage at the connection point can be effectively  controlled by injecting a reactive current. connection point ^ R+ jX Z I (a) Vgr,d  jXI V Zrid  jXI I higher X/R ratio (b) lower X/R ratio Figure 2.1: Voltage regulation based on the X/R  ratio. Voltage regulation becomes more complicated if  instead of  regulating the voltage at the connection point it is necessary to regulate it at an intermediate PCC, as shown in a simplified  diagram in Figure 2.2. In particular, when the wind turbine operates in the LVC mode, the equivalent impedance Z is composed of  the impedance to the wind farm  ZJ ARM and the grid impedance Z GRI D combined, which reduces the short circuit ratio and makes the grid interconnection appear weaker. However, if  the wind turbine is controlled to regulate the voltage at the PCC, then the effective  value of  impedance Z becomes smaller by the amount of  ZF ARM.  This, in turn, increases the effective  short circuit ratio and makes the grid appear stronger. V W T  PCC Grid Figure 2.2: Diagram depicting wind farm  interconnection impedance. Based on this observation, the wind farms  or wind turbines may be used as very effective voltage regulation tools and should no longer exempted from  contributing to reliable operation of  the grid, especially where there is high wind power penetration. 2.5 Research Objectives and Approaches 2.5.1 Problem statement As the present tendency of  incorporating wind turbines into large wind farms  continues, new' possibilities for  integrated design of  individual turbines, the infrastructure  within the wind farm,  and the grid-connection interface  open up [21]. Furthermore, wind farms  that generate substantial amounts of  electrical power may be connected at higher voltage levels and greater distance [18]. The local impacts of  wind power have been studied extensively in the literature [15], [18], [19], [29], [30], The system-wide impacts of  wind power are of  special interest at higher levels of  wind power penetration [21], [22], [25], [31]—[33], and is expected continue with the present rapid growth of  wind power. Modern variable-speed wind turbines utilize power electronic converters for  the grid connection and improved performance.  By appropriately controlling the converters, it becomes possible to locally maintain the power factor  (power factor  control mode, PFC) or the voltage (local voltage control mode, LVC). Furthermore, in modern wind farm applications, the wind farms  have to contribute voltage control within a specified  allowable voltage level at the PCC. As wind power penetration increases, the PFC and LVC modes (Options 2 and 3 in Section 1.2.3, respectively) are frequently  not sufficient  to achieve the desired voltage control, especially during events such as faults  [16], [21], [26], and may still require installation of  additional devices to meet the power quality specifications.  However, there are always costs associated with the installation and operation of  supplementary devices, which makes this option less attractable. Therefore,  to achieve easier grid integration and reliable voltage control, alternative active voltage control of  wind turbines is required. 2.5.2 Research objectives The research objective of  this thesis is to investigate the control options that can be used concurrently with existing wind turbine technologies to improve voltage regulation in the system. In particular, the performance  of  traditional control schemes such as the PFC and the LVC subject to small transients and large events like faults  is investigated. Alternative design and/or control solutions are proposed to improve the voltage control at required locations of PCCs. 2.5.3 Proposed approach The system and modifications  considered in this thesis are based on an industrial site located on Vancouver Island, Canada, that is presently being investigated by Powertech Labs Inc., for  a possible wind farm  installation [34]. The wind farm  is assumed to be connected at the transmission level and provide a significant  portion of  the local power demand (20 to 50%). Although aggregate wind farm  models have traditionally been used in the analysis of  wind power generation systems, the multiple wind turbines farm  model is more appropriate for  this purpose as it enables us to portray possible interactions among the individual turbines due to disturbances and variation in wind speed, seen by each wind turbine unit. Thus, the approach taken here relies on developing models of  various power system components, including the wind turbines and wind farm,  to study and predict the behaviour of  the wind farm  and its interaction with the grid. The utility grid is represented by a large synchronous generator to capture the possible grid dynamics and its influence  on voltage control performance.  Our reasons for  considering the multiple wind turbine model instead of  a simple aggregate model include the following: Voltage control is often  achieved by appropriately regulating the reactive power. However, in a realistic wind farm,  each wind turbine has somewhat different instantaneous wind speed and real power output. Consequently, the availability of reactive power generation produced by each wind turbine is also different. • The controllers of  individual wind turbines may interact with each other, and their action may affect  the grid dynamics. For these reasons, it is not appropriate to represent the wind farm  using an aggregate model. Instead, each wind turbine is included as a separate module in the overall model of  the system. Since the overall system is nonlinear and the system state depends on the operating/loading conditions, a robust controller should be designed to account for  the variations in the system. The linear matrix inequality (LMI) techniques have been previously considered for  robust tuning of  controllers [35]—[45]. The use of  the LMI-based approach provides readily applicable and robust tuning considering multiple linear systems. In this thesis, a linear quadratic regulator (LQR) tuning is achieved using the LMI-based technique. The LQR-LMI-based approach is considered because it guarantees closed-loop stability for  varying operating conditions. To design a controller that is valid for  a range of  operating conditions of  interest, the overall system is linearized at the three operating conditions - nominal load, 50% increase, and 50% decrease of  the local load impedance. This choice is based on expected daily local load deviations. Since the order of  each linearized model is very high, we use the balanced model-order reduction technique to find  the lower-order transfer  functions  that are more suitable for the purpose of  a controller design. Finally, we carry out the simulation studies and analyses of  system impacts due to small disturbances (such as wind-speed variations and changes in load impedance) and a large disturbance (three-phase symmetrical fault). 2.6 Contributions To improve the voltage control under nominal wind variations as well as during the disturbances (load variations, fault  and/or line tripping), we propose an innovative supervisory voltage control scheme in this thesis. We compare the proposed control to the traditional methods and show that it improves the transient performance  and fault-ride-through capability of  the considered wind farm  system. Overall, the contributions of  this thesis include the following: 1. The proposed supervisory control scheme achieves a direct voltage control at a remote location, PCC, and does not require installation of  additional compensating devices to meet the grid-connection requirements. 2. The proposed scheme takes into account the power output and limits of  individual wind turbines and is readily applicable to larger wind farms  of  different configurations.  Voltage control is achieved by appropriately regulating the reactive power injected by each wind turbine, whereas the control of  real power may be allowed and/or included whenever the limits have been reached. 3. An innovative cost-guaranteed LQR-LMI formulation  of  the controller design is also proposed. The final  controller is tuned for  a range of  operating conditions using the proposed cost-guaranteed LQR-LMI approach, and is shown to improve the system's dynamic performance  over that of  traditional control solutions. Chapter 3 Model Description of Grid-Connected Wind Farm System Traditionally, studies of  wind power generation systems have been carried out using a so-called aggregate  representation of  wind farms.  Even though such studies have limited accuracy and application [46], using an aggregate wind farm  model is sometimes acceptable especially at the distribution level or in cases when the interaction among the individual wind turbines is not likely to be of  importance. However, as the size of  modern wind farms  and the individual wind turbines continues to increase, it is important to develop a more general model, wherein each wind turbine is represented as a subsystem. Such dynamic models would be of  great help in more accurately evaluating a wind-power generation systems performance  during normal operation as well as during disturbances. Although personal computers are becoming increasingly faster,  computational speed is still one of  the limiting factors  in dynamic simulation of  power systems [47], [48]. Electrical transients have very small time constants that require small integration time steps and result in long computation time. To keep the simulation speed reasonable, special attention should be given to model development. In particular, in this thesis, to increase the simulation speed of  various electrical components, these components are modeled in the qd  - synchronous reference  frame  [49]. For the same reason, the power electronic converters are represented using average-value models that are also expressed in appropriate qd  - reference  frames. This chapter describes a model of  the system considered in this thesis, including mechanical and electrical components of  each wind turbine. The system considered herein corresponds to a candidate industrial site on Vancouver Island, Canada [34], The system parameters are summarized in Appendix A. All model measurements are expressed in per unit, such that the values of  all variables are in pu., except time t. To preserve the units of  time t in sec., the respective state equations are normalized by the base frequency  a . 3.1 Study System A simplified  diagram of  the system considered herein is shown in Figure 3.1. Without loss of generality, only three wind turbines are included here to represent a possible dynamic interaction among individual wind turbines. The model itself,  the control methodology, and conclusions are readily extendable to larger systems. In the system considered, each wind turbine is equipped with a step-up 0.69/34.5kV transformer.  The wind turbines are connected in a chain using cables (9km). The details of  the individual wind turbine considered in the model are shown in Figure 3.2. In this thesis, the GE 3.6MW wind turbine [8] is considered. The wind farm  is connected to the grid through a 34.5/132kV transformer  and a 132kV double-transmission-line (100km). A large synchronous generator (SG) represents the grid. At the 34.5kV level, the wind farm  feeds  a local load connected at the PCC. The block diagram of  the overall model showing the individual components and the respective input-output variables is depicted in Figure 3.2. Since the overall model includes the individual wind turbines, each turbine can have an independent wind speed, denoted by v w t ] through vwt3 • WT3 0.69/ Q 0.69/ 34.5kVCj 34.5kV .6 9km WT2 0.69/ 34.5kV C1 9km 100km WT1 Local I ' T4 • -i X ' " Load T T (PCC) TL 1,..., 9 - bus number WT1,... - wind turbines T1, T2,... - transformers C1, C2, C3-cables TL - transmission line PCC - point of common coupling Utility Grid (Syn. Gen.) Figure 3.1: Wind power system considered for  dynamic studies. Wind speed input(s) v g,Wtl 14 (3.17) -< TL (3.15) SG (3.18) t4 t5 sg Figure 3.2: Block diagram showing subsystem input-output variables. A more detailed diagram of  the individual wind turbine is shown in Figure 3.3. The wind turbine consists of  the following  major components: a three-blade rotor with the corresponding pitch controller; a mechanical gearbox; a doubly-fed  induction generator (DFIG) with two voltage source converters (sometimes known as the back-to-back voltage source converter, VSC); a dc-link capacitor; and a grid filter.  Mechanical power comes through the three-blade rotor and the gearbox to the shaft  of  the DFIG, which has rotor speed denoted by cor. The power is then taken from  the DFIG through the stator side Ps and the rotor side Pr. The stator side is directly coupled to the 0.69/34.5kV transformer,  which operates at the grid frequency.  Variable-speed operation is achieved by appropriately controlling the two converters. In particular, the rotor-side converter provides the real and the reactive power necessary to attain the control objectives for  either the PFC or the LVC modules. The grid-side converter is connected through the filter,  and its main objective is to maintain the dc-link capacitor voltage by exchanging the real power with the grid. The mechanical and electrical components of  the wind turbine are described in more detail in the following  section. W i n d Mechanical speed power Terminal of WT P g Rotor-side DC-link Grid-side v converter converter qd,r VSC Controller Figure 3.3: Variable-speed wind turbine with DFIG. Generally, the absolute value of  slip cos is much lower than 1; consequently, the real power of  the rotor Pr is a fraction  of  the real power of  the stator Ps as Pr ~ cosPs. The grid-side converter is used to generate or absorb the power Pf,i ter in order to keep the dc-link voltage constant. In steady-state for  a lossless converter, Pfii ter is equal to Pr and the rotor speed cor depends on the power absorbed or generated by the rotor-side converter. 3.2 System Model Components 3.2.1 Mechanical components In most applications, the wind turbine is operated to extract as much power from  the available wind as possible without exceeding the ratings of  the equipment. The mechanical components of  the wind turbine include a pitch control, a gearbox, and a three-blade rotor. A. Pitch control The block diagram of  the pitch control [8] is shown in Figure 3.4. The mechanical power generated from  the wind can be calculated using a well-known relationship, Pmech=^Arvlcp(A,d)  (3.1) where Pmech is the mechanical power in W; p is the air density in kg/rr?  ; Ar is the area swept by the rotor blades in m2; vw is the wind speed in mjsec; Cp(X,6)  is the power-conversion function,  which is commonly defined  in terms of  the ratio of  the rotor blade tip P CO speed and the wind speed here denoted by Z= 1 1 ; Rt is the rotor radius in meters; cot is v w -the turbine rotor speed in rad/sec; and 6 is the blade pitch angle in degrees. The function Cp (A,  6) is often  obtained as a numerical lookup table for  a given type of  turbine. The GE wind turbine parameters for  the energy conversion function  CP (A;0)  are given in Appendix A in per unit on a 4MW base. The pitch control attempts to keep the value of  the turbine rotor speed constant by providing the set-point to the pitch-angle actuator. The response of  pitch control is relatively slow compared to other controllers such as the torque control and pitch compensation. Thus, the turbine control results in an auxiliary control signal into the pitch actuator for  faster  damping. When the available wind power is higher than the rated power of  the wind turbine, the blades are pitched out to reduce the mechanical power delivered to the shaft  PME CH  such that it does not exceed the power rating P m a x . When the available power is less than PMA X,  the blades are set at minimum pitch to maximize the mechanical power PME CH•  The variable PGBT  is the set-point value of  the output of  the wind turbine. pset r g Figure 3.4: Block diagram of  the pitch control. B. Two-mass rotor model A block diagram of  a two-mass rotor model of  a wind turbine with separate masses for  the turbine and generator is presented schematically in Figure 3.5. The aerodynamic model describes the energy conversion from  kinetic energy of  the wind to the mechanical energy on the wind turbine rotor. The inputs to the aerodynamic model are wind speed vw, and the blade-pitch angle 6. The mechanical rotor speed cot depends on the mechanical torque T mecfj  acting on the drive train. The drive-train model receives the mechanical torque T MEC H and electrical torque T S and computes the electrical rotor speed (Or. Here, H T  and Hq  are the turbine rotor and gearbox inertias, respectively, and H R is the generator rotor inertia. The coefficient  DT S represents the shaft  damping, and K T S is shaft  stiffness. Aerodynamic Turbine rotor Shaft Gear Generator Drive train Figure 3.5: Simplified  block diagram of  the two-mass rotor. The block diagram shown in Figure 3.6 represents the rotor model, drive train, and generator model, all expressed in per unit [8], In this representation, since the gear inertia is very small compared with that of  the wind-turbine blade rotor and generator rotor, the shaft  and the gear are represented by a common damping coefficient  Dtg and the stiffness  K tg coefficient, respectively. Since the GE energy conversion function  Cp (X,  0) is given on the 4MW base but the overall model here is developed in per unit on the 100MW base, the corresponding coefficients  must be multiplied by a constant K pu = 4/100, which is derived from  the following  relationship rp _ rpOld  rpOld  _ rpYieW  rpfieW 1 actual ~1 base ' 1 pu ~ base ' 1 pu (3.2) The calculation of  T mecij and X  with the Pmecf,  and Rt are then rewritten as follows: Rtcot T  = K 1 mech ^mech pu V °>t J and X  = - (3-3) 1F + 1mech +A D, 'tg -o Ts — K J • 2H„ tyfiase 1 2H t 1 s °h,base 1 CM + a K, tg Figure 3.6: Block diagram of  the two-mass drive-train rotor model. 3.2.2 Electrical components The electrical components of  the wind turbine include the DFIG and the voltage source converter. The remaining electrical components of  the overall system include the transmission line, transformers,  cables, and the load. The corresponding subsystem modules are described below. A. Doubly-fed  induction generator The DFIG is represented in the qd-  synchronous reference  frame.  The corresponding equations in per unit [49] are n • . 1 d Xfqs vqs=Rslqs+COe¥d S+- ~ vds  = Rslds  - <»eVqs  + vqr=Rriqr+®sVdr  + vdr  = Rr'dr  ~ ®s¥qr  + dt 1 dVds dt 1 dWqr dt 1 dWdr (3.4) cob dt with the flux  linkages expressed as Wqs  = (As + An )iqs + Aw V»  Wds  = (As + An )*ds  + An^ dr (3-5) Wqr  ~ (A- + Lm)iqr + Lmiqs, y/d r — (L r + Lm)icjr + Lmid s where vqs and vd s are the stator voltage; vqr and vd r are the rotor voltage; iqs and id s are the stator current; iqr and ijr are the rotor current; Rs and Rr are the stator and the rotor resistance, respectively; Lm is the mutual inductance; Ls and Lr are the stator and the rotor leakage inductance, respectively; co^  = Info  is the base angular speed (rad/sec) with /q at 60Hz; coe and cor are the stator and the rotor electrical angular speed, respectively; 0)s-cae- cor is slip electrical angular speed; y/ qs and y/^  are the stator flux  linkage; y/ qr and yd r are the rotor flux  linkage; the subscripts q and d  indicate the quadrature  and the direct  axis components as expressed in the reference  frame;  and the subscripts s and r indicate the stator and the rotor quantities, respectively. The stator voltages can be compactly expressed as the vector Vqd>s = , v d s . These voltages are the input to the DFIG model and are obtained from  the low voltage side of  the 0.69/34.5kV transformer  model as the voltage vector V QCJJ R • The electrical torque T S, the stator real power PS , and the stator reactive power QS delivered by the generator are calculated as T s = Vdr'qs  -Vqrids  (3-6) (3.7) Qs=vqsids- vds iqs (3-8) B. Voltage source converter The variable-speed operation of  the DFIG is achieved by means of  two converters linked via a capacitor as shown in Figure 3.7. A detailed description of  the pulse-width-modulation (PWM) switching scheme of  the converter can be found  in [49], [50]. The rotor-side converter feeds  the DFIG rotor with the reactive power and takes out the real power as necessary to attain its control objectives. These objectives usually consist of  maintaining turbine speed and either controlling the stator power factor  (PFC) or terminal voltage (LVC). Real power requirements for  the rotor-side converter are provided by drawing current from or supplying current to the dc-link capacitor. The grid-side converter is connected to the grid through the filter.  The main objective of  the grid-side converter is to maintain the voltage level on the dc-link capacitor by exchanging real power with the grid. Rotor side DC-link Grid side lal '41 vdc eft  -•EJ t j -HCj  -HtJ Rfilter Lf'lter  ia2 T-W. ™ '62 -w.— >c2 vanl vbn\ vcnl Vcn2  vbn2 van2 vcn2 vbn2van2 Figure 3.7: Schematic representation of  the voltage source converter. DC-link Grid side T1 vdc T3 C T4 Js} comparator gc(  0 duty cycle d c(t) triangle signal AAAAM 0 N g c ( 0 = 4 : (OS •OFF £ T6 'a 2 lb2 > A T = 0 _ 4 > A cnl vbn2 van2 Figure 3.8: Representation of  the switching function  on the grid-side converter. The output of  the comparator is the pulse train gc{t)  depicted in Figure 3.8, which shows the c- phase only. In particular, the high-frequency  triangle waveform  is compared with the sinusoidally varying duty-ratio function  d c(t)  [49]. When the magnitude of  the triangle waveform  is greater than that of  the duty-ratio waveform,  the switch is in ON mode. The overall switch operations in the 6-pulse converter are listed in Table 3.1 along with the corresponding on/off  status of  the switch, i.e., TI/T4  . For example, State 3 indicates that the switches T3, T4, and T5 are ON, and the switches Tl, T2, and T6 are OFF. TABLE 3 .1 SWITCH OPERATIONS State Tl/T4 T2/T5 T3/T6 1 1 0 0 2 0 1 0 3 0 0 1 4 1 1 0 5 . 1 0 1 6 0 1 1 ( l : O N , OiOFF) The duty cycle for  switching each phase can be specified  as follows: d a=d  cos(6>c) f df,=d  cos d c=d  cos 3 / 3 , (3 .9) where 6C = 6 e + 6 . The variable 6 is the phase shift  between the synchronous reference  of the system and the converter, and 6 e is the synchronous angular displacement. The switching action and harmonics of  the converters may be ignored and replaced in the model with the appropriate average-value relationships [49]. By assuming that the frequency of  the triangle wave is much higher than the frequency  of  the desired waveform,  the average t i t magnitude terminal voltages van2, Vbnh a r ) d vc„2 of  the grid-side converter are described as van2 = da vdc » vbn2 = db vdc vcn2 = dc vdc (3.10) A change of  variables to the qd-  synchronous reference  frame  [49] is then applied such that vqd  =T qd(8 e)yabc (3.11) where ( v f = I t vq vd and T (v«6c) = van2 vbn2 vcn2 and the qd - transformation matrix is defined  as Tqd( ee) = -Z cos(0e) cos 0o-sin(0c) sin In T In cos sin Ge + In (3.12) The terminal voltage of  the converter in the qd  - synchronous reference  frame  is then vq=dcos{6)v d c, vd=-dsm{6)v d c (3.13) By defining  the control signals as vqu =c/cos(0) and vd u = - d s i n ( 0 ) , the magnitude of the duty cycle is d  = yjvqu +vd u and the angle displacement is 9 = - t an - 1 [v d ujvqu j . Hence, using the control signals vqu and vd u, the average value of  voltages of  the grid-side converter can be expressed as vq=vquvdc> vd = vdu vdc (3-14) C. Transmission line, transformer,  and cable models Since the transmission line considered here has medium length (80km~240km), we considered it appropriate to represent this line using an equivalent lumped-parameter n -model [28], Such a model can be expressed in the ^ - synch ronous reference  frame depicted in Figure 3.9 using the equivalent R , L , and C elements [49]. The cables and transformers  are also represented using the model structure similar to that shown in Figure 3.9, wherein the appropriate R , L , and C parameters are used. For the formulation  of transformer  models, the capacitors at the sending and receiving ends are not used. The detailed parameters for  each component model are summarized in Appendix A. The respective equations of  the line segment depicted in Figure 3.9 are as follows /it  T.i vd\ 1d\ R 0)Li, e qi -I dl » T ^ 1 3 < + ldc\  > ldi ldc2  i C-' Ve CVq2 ? • c Figure 3.9: Transmission line lumped-parameter qd  - model. A v q = Vq\  ~ vq2 = RTl}ql  + — 0 ) e L j L i d l (Ofo  ut Avd =vd\  ~vd2 = RTL'dl  + L r l -OeLniql (Ob  dt , _CTL dv<l l . „ r v ;7T + ^e^TL vdl (Of,  at _ CT L dv d x dc\  ~ T aeLTL vql COfo  at _ C T L d v q 2 lqc2 ~ T~  + °>eLTL vd2 * dt ; _ CT L dv d  2 z<£2 T a>eLTL vq2 (fy  at (3.15) L di qi A v q = vq\ - vq2 = Rcalql  + C° I  + ^A^rf/ CO/,  at Avd = vd\ ~ vd2 = + L c a d l f  - coeLcaiqi cojj dt Cca dv q\ iqcl  = ,, + 4QaV£/l (Ofr  at ; _ Cca „ ^ „ Vcl - — T ^e ccavol (%  dt y (3.16) Av<7 = =Rtriql  + Ltr d lql (%  dt + 0)eLtridi Avd  = vd l - vd 2 = Rtrid l (%  dt • CO eLtriqi (3.17) vq\ ~ ^qcl vd\  = Roidcl Note: For the notation used here to match with that used in Figure 3.2, the voltage equations in (3.15)—(3.17) need to be correlated to the bus number indicated in Figure 3.1. The current equations for  the transmission line, cable, and transformer  need to be referred  to as TL,  ca, and tr as well. D. Utility grid A large synchronous generator is considered to represent the utility grid. The model of  the synchronous generator including the excitation system is taken from  [13]. The generator equations are as follows ^r =0)b( vs -Ra*a +kl<°eV a) at d¥f dt dVk dt = ®b(Efd- Rf if) (3.18) where [»a h ' / ] 7 ' = L 1 [ V f l  V* Wfl with " 0 - 1 " k1 = 1 0 , k 2 = Rkq  0 0 R •kd Vsq ¥sd Vk  = 'Wkq » i « = lsq » v 5 = Vsq . h = kq }fkd_ jsd  _ ySd. }kd  _ and L = Lamq 0 Lmq 0 0 0 Lamd 0 Lmd Lmd Lmq 0 T-'kqmq 0 0 0 Lm d 0 Lkdmd Lmd 0 Lmd 0 Lmd Lfmd where v and i are the voltage and current vectors, respectively; coe is the angular speed; y/ is the flux  linkage; R and L are the resistance and inductance, respectively; Ejd  is the dc field  voltage; and the subscripts a,k,f  denote the armature, damper, and field  quantities, respectively. The above-described generator model was considered together with the excitation system [51] that regulates generator terminal voltage by controlling the field-winding  voltage. The block diagram of  the exciter model considered here is depicted in Figure 3.10. Filter Exciter 1 1 + ST r sTp+Ti 1 s l  + sTex + ^ E, 'id Figure 3.10: Exciter model block diagram. Here E i s the dc field  voltage and E i s the initial value; and V s and V rsef  are the measured magnitude of  the generator terminal voltage and the reference  voltage, respectively. The time constants necessary for  filtering  the rectified  terminal voltage waveform  are reduced to a single time constant . The exciter gain is represented by the parameter T ex. Automatic voltage regulator (AVR) gains are given as T p and 7}. The thyristor-controlled rectifier  is represented by a scalar K t [51]. E. Load The dynamic RL load model represented in the qd  - synchronous reference  frame  is depicted in Figure 3.11. coLi., co Li, Figure 3.11: RL load model represented in the qd  - synchronous reference  frame. The corresponding state equations are koad d igl CQb dt - vq2 - Rload lql  ~ ^ekoad^l Lload d id l _ t d t vd2 ~ Rload ldl  + Mekoadiql (3.19) vq2 ~ Rolqc2' vd2 = Roidc2 1 where Rq is 10 . The output voltage vector 2 defines  the voltage vector at the PCC vb7,pcc a s shown in Figure 3.2. The current vector iq ( j2 is equal to the sum of  the current vector i t 4 and i c a b 3 . 3.3 Voltage Source Converter Controller An important part of  the wind turbine is the voltage source converter controller that controls the voltage of  the rotor-side converter, the dc-link, and the grid-side converter. Different schemes arid detailed information  can be found  in [10], [52], [53], This chapter presents transfer  functions,  which are used to tune internal controllers. Figure 3.12 shows a block diagram of  the voltage source converter controller modules and the respective input-output variables. v , qa,r qdjilter Q set 'filter . set \dfilter 8 ^g Figure 3.12: Block diagram of  the voltage source converter controller modules. In Figure 3.12, PSGET  and QSGET  are the set values for  the real and reactive power for  the terminal of  the wind turbine. When the unity power factor  control mode is applied, QS„ET  is o set to zero and all reactive power to the DFIG is provided via the rotor-side converter. When the local voltage control mode is used, QSGET  is adjusted by the local controller to maintain the voltage at the wind turbine terminal. Figure 3.13 shows the maximum tracking power output via the turbine power characteristic with respect to the rotational speed of  the rotor. This characteristic curve is obtained based on the maximum power tracking curve given in [54] and is calculated for  the GE 3.6MW turbines considered here on the 100MW base. The value of  PGET  is determined by this wind turbine energy harvesting tracking characteristic, which is represented here as a look-up table PGET  (CO R). The turbine power characteristics PME CH  are obtained at different  wind speeds. The actual speed of  the turbine cor is measured and the corresponding mechanical power of  the tracking characteristic is used as the set-point real power PGET  for  the real-power control loop. 3 -3 I* i> £ o a, "3 cx 3 O T3 o '3 C3 -C o u s <D C 3 H 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Rotational speed of  the rotor {a>r, pu) Figure 3.13: Turbine power versus speed tracking characteristic. 3.3.1 Rotor-side converter controller Figure 3.14 shows a block diagram of  the rotor-side converter controller module, which includes four  internal conventional proportional-plus-integral (PI) controllers, PI1 through PI4. The controller is implemented as two branches, one for  reactive power (PI1 and PI2) and one for  real power (PI3 and PI4), with the corresponding de-coupling terms between the q-and d-axes,  respectively. The actual electrical output power Pg, measured at the terminal of the wind turbine, is compared with the set-point power obtained from  the tracking characteristic Pget(a> r). The PI3 regulator is used to drive the error in power to zero. The output of  this regulator is the rotor current set-point that must be injected into the rotor by the rotor-side converter. The actual current id r is compared to i'^f  and the error is driven to zero by the PI4. The output of  the PI4 is the voltage vdr generated by the rotor-side converter. Similarly, the PI1 regulates the set value of  the reactive power at the terminal of the wind turbine, and its output is the value of  the g-axis rotor current. The PI2 regulates the <?-axis rotor current set value. The output from  PI2 is used to compute the command value for  the rotor voltage vqr . As can be seen in Figure 3.14, the two speed-voltage terms (O s\j/ d r and cosy/ qr are compensating the coupling terms in (3.4). For consistency, a brief  discussion of  the controller design is given below. A. PI2 and PI4 controller design The rotor voltage equations can be written with the stator flux  dynamics neglected [53], as follows:  . (%  dt V = Rrlqr + ~1T + ®Wdr vdr  = Rrldr  + - Q)syf (Ob  at ' (3.20) A set O l • set V PI1 V V V qr qr _ qr —'*rO~H P I2 h + O — • + - + + - f a qr VSVDR . set : set dr V , V , dr  dr vdc "dr H p - H PI3 [ — » Q - H pi4 h > Q >®—> + I I + T g • dr CO IIf s t qr Figure 3.14: Block diagram of  the rotor-side converter controller. By introducing the new variables vqr Rrlqr v'dr  ~ Rrldr  + Lr digr COfr  dt (3.21) Lty• (%dt (3.20) can be simplified  to the following: v*qr=vqr+cos¥dr (3.22) Vdr =V'dr-^sWqr To design PI2 and PI4, (3.22) can be rewritten in the Laplace domain to obtain the transfer function  from  the rotor current to the rotor voltage in the q- and d-axes,  as follows: _ Iqris)  _ 1 Vqr (.S) Rr+S^LrlcOb) Q ( :.)=Idr( s) = 1 4 K J  V d r(s)  Rr+s{L r/03 b) (3.23) Based on (3.23), PI2 and PI4 can be designed with the gains summarized in Appendix A. In this thesis, a pole-placement design technique [55], [56] is used, as summarized in Appendix B. B. PI1 and PI3 controller design B.l PI1 controller To tune PI1, a transfer  function  from  the #-axis rotor current to the reactive power of  the rotor-side converter should be considered. Because the g-axis of  the selected synchronous reference  frame  is tightly aligned with the stator voltage vector and the angle between the stator and rotor voltages is relatively small, the reactive power is approximated as described in [10], [48], [57] Qr ~ v'qr ldr  -vdr lqr = vqrldr (3.24) By using the approximate relationship Q' r ~ (O sQ' s and Q'g ~ (a> e - cos)Q' s where 0)e =1, the chosen reference  frame  allows (3.24) to be rewritten as 1 -G>s V % V vqrldr (3.25) Assuming that slip frequency  cos is small and that the controller is being tuned for  the ijf  is approximated nominal operating condition, =1 on its local base, the term . Hence, (3.25) is rewritten as c 1 A l - 0 ) s here as f  * ^ l -cos v <°s ) f  1 N I-6>s v ^ j V  -yqr f  1 V J Rfiqr  + Lr di qr x dt V j (3.26) As a result, for  the purpose of  controller design, the transfer  function  from  the rotor current in the g-axis to the reactive power at the terminal of  the wind turbine can be approximated as Q' G3(s)  = -^- = Rr+sLr (3.27) qr where Rr -{^>Rr and r T \ B.2 PI3 controller The design of  PI3 is similar to that of  PI1. In particular, to tune PI3, a transfer  function  from the c/-axis rotor current to the real power of  the rotor-side converter should be considered. The real power of  the rotor-side converter can be approximated as K  ~ Vqriqr  + vd^dr  ~ vdr ldr (3.28) By using the relationship = asP' s and P' g ~ (co e -a>s )P' s where a>e-1 , the chosen reference  frame  allows (3.28) to be rewritten as P'  = rg f  1 v °>s J vdr ldr (3.29) As it was previously assumed that slip frequency  cos is small and id r ~ 1 on its local base close to nominal operating conditions, (3.29) is then approximated as P'  -r8 r 1 > v ^ j vdr f  1 \f I  -a>s \ J Rrid r + Lr di dr dt (3.30) As a result, the transfer  function  from  the rotor current in the J-axis to the real power at the terminal of  the wind turbine can be approximated as P'As) GA(s)  = -2— = Rr+sLr !dr( s) (3.31) Hence, PIl and PI3 can be designed using the transfer  function  G^s) or G ^ s ) . As before, the pole-placement technique [55], [56] has been used to find  the proportional and integral gains, as summarized in Appendix B. 3.3.2 Grid-side converter controller As shown in Figure 3.3, the grid-side converter is connected to the grid through the filter. The voltage equations for  the filter  in the (^-synchronous reference  frame  are D . ^filter  dig,filter  . vq, filter  ~ Kfilter 1 q, filter  + + meLfilter 1 d,  filter (3.32) D . Lfilter  ^d,  filter T  vd,filter  ~ Kfilter 1 d,  filter  ~J t coeLfilter'q,  filter Equations (3.32) are coupled by the corresponding speed-voltage terms that must be considered in the design of  controller. By introducing the new variables q, filter  ~ Rfilter 1 q, filter  +" Lfilter  dig,filter (Ojj  dt D . Lfilter  did,  filter vd,  filter  ~ Kfilter 1 d,  filter the modified  command voltages of  (3.32) can be expressed as (3.33) V hfilter  ~ V'q,filter  + ®eL filter 1 d,  filter 'd,  filter  ~ Vd,  filter  " ®eLfilter 1 q, filter (3.34) The corresponding transfer  function  can be expressed as G5(S) = I«/'^S) -  1 Vq,  filter  0 ) R f Ute r + S (l FLHE R j ) „ , N 1d,filter  ( s ) 1 & 6 ( f ) = 7 T (3.35) Yd,  filter  0 ) R filter + S [ L filter  /<°b  ) which is considered for  tuning the controllers PI5 and PI6. Figure 3.15 shows the grid-side converter controller block diagram. The input current set-values are calculated by the real and reactive power commands P0ter and Qfi[ ter as •set lq,filter vq,tr vd,tr -1 pset t,filter •S€t _ld,  filter ~vd,tr Vq,tr_ Qfiiter Here \ q < j > t r = \ y q > t r v d t r J is the voltage at the low-voltage-side wind turbine transformer;  Pfi/ ter and Qfif ter are the set-point of  the real and reactive power commands. The value for  Qflj ter is set to zero if  the unity power factor  control is used; however, Pff/ ler is provided by the dc-link controller, which determines the flow  of  real power and regulates the dc-link voltage by driving it to a constant reference  value. As depicted in Figure 3.15, the compensation terms a>eLfiit erid,  filter  ^e^filter'q,  filter  decouple the speed-voltage terms in (3.32). Figure 3.15: Block diagram of  the grid-side converter controller. 3.3.3 DC-link controller The capacitor in the dc-link is an energy storage device. Neglecting losses, the time derivative of  the energy in this capacitor depends on the difference  of  the power delivered to the grid filter,  PFII TE R , and the power provided by the rotor circuit of  the DFIG, PR, which can be expressed as p ( 3 3 7 ) 2 cob dt J K Note that since all variables are in pu, except time t, the state equation (3.37) is normalized by cob to get time in sec. Based on (3.37) the dc-link voltage will vary according to the following  state equation: ~~~vdc  - j f -  = Pfilter  ~Pr= ^cjllte^dc  ~ [dc,r vdc  ( 3 - 3 8 ) (0^  at which can be rewritten as Qc d vdc ^ ldc,  filter ldc,r (3.39) where i d c r and idc,filter a r e ^ e current in the rotor-side converter and the grid-side converter, respectively. The transfer  function  from  the current id c, filter t 0 the dc voltage vdc is therefore  as follows: c W . - S * « ! L - = - a - (3.40) dc,  filter  v*^)  sLdc The dc-link model with its controller is shown in Figure 3.16. The rotor-side converter current i d c r is obtained by idc,r~ Pr/ vdc  The real power set-point Ps^l ter in (3.36) is obtained by Pf{ ter = vd cis/ c[ fllte r . • Pfllter • set ldcfilter +r\ w ab >- vdc ldc,r Figure 3.16: Block diagram of  the dc-link model and its controller, PI7. The voltage source converter controllers PI1 through PI7 are tuned using the respective transfer  functions  of  G\ (s)  through G7 (5) . In this thesis, a conventional pole-placement technique [55], [56] was used, as described in Appendix B, and the respective control gains are summarized in Appendix A. 3.4 Conventional Voltage Control of Wind Turbine Figure 3.17 shows the combination of  the voltage source converter controller with respect to the rotor-side converter, the dc-link, and the grid-side converter from  the top to the bottom. The conventional control modes PFC and LVC are also described in [10], [52], The objective of  PFC is to achieve zero reactive power consumption by the wind turbine Qg . To implement this mode, the reactive power set-point Qsget in Figure 3.17 is simply set to zero. To achieve the LVC, the reactive power set-point Q g e t is made available to another control loop that drives the voltage at the wind turbine terminal to a specified  value. The design of  this additional controller is summarized in Appendix C. Thereafter,  these two conventional controls schemes are used for  comparison purposes. Figure 3.17: Overall control block diagram of  the voltage source converter. Chapter 4 Wind Farm Voltage Control 4.1 Common Practice Traditionally, additional compensating devices such as static VAR compensator (SVC), static compensator (STATCOM), transformer  with tap changer, etc., have been considered to improve voltage regulation associated with the variable nature of  wind energy [9]-[12], [31], [32], [58]—[60]. A coordinated voltage control strategy for  the DFIG and the on-load tap changer has been proposed in [60], wherein a single DFIG was considered. However, there are always costs associated with the installation and operation of  any supplementary devices, which makes this option less attractive. Moreover, increasing wind power penetration has been noted to influence  overall power system operation in terms of  power quality, stability, voltage control, and security [17], [20], [21], [23], [25]. Consequently, in many modern wind farm  applications, the voltage regulation at a specified  remote PCC where a load of  particular concern is connected may need to be addressed as a requirement for  the grid connection. To achieve easier grid integration and reliable voltage control in the system, alternative voltage control schemes for  wind turbines become very important. This chapter describes an innovative supervisory voltage control scheme that does not require installation of  additional compensating devices and is applicable to wind farms  of  different configurations.  The key to the supervisory control scheme is to regulate the voltage at a specified  remote PCC by adjusting the reactive power produced by the individual wind turbines while taking into account its operating limits. 4.2 Available Reactive Power in a Multi-Turbine System On a practical wind farm  site covering a sufficiently  large area, each wind turbine may have somewhat different  instantaneous wind speed and output of  real power. Consequently, the capacity for  reactive power generation of  each wind turbine is also different.  Due to these differences,  it is not appropriate to represent a wind farm  using the aggregate models. Instead, we consider each wind turbine as a separate module in the overall model of  the system, as described in Chapter 3. When controlling multiple turbines, it is important that the operating limits of  each individual turbine (in particular the internal voltage source converter) are not exceeded. Assuming a proportional distribution, the portion of  reactive power required from  an individual jth voltage source converter can be computed as 2 set j,g = m m Q max j,c ' QJT QlcX+Q2T+-  + Qnl •>max max c AG pcc (4.1) where is the maximum reactive power (limit) that the y' th voltage source converter can provide, and AQpcc is the total reactive power required to support the voltage at the PCC. The quantity Qj c may be different  for  each wind turbine and is dependent on operating conditions. To understand how this quantity can be calculated, it is instructive to consider a single voltage source converter first.  Figure 4.1 shows the real and the reactive power operating limits, wherein it is assumed that a converter (the rotor-side or the grid-side) should not exceed its apparent power limit depicted by the half-circle.  Suppose that at a given time the converter is delivering real power, denoted herein by Pc, that is changing depending on the wind condition. Then, in addition to the real power, the converter can supply or absorb a maximum of  2 c m a x of  the reactive power. So, the reactive power available from  a single converter lies within the limits [ - 0 C ; 1 T i a x ; + 0 C ; m a x ] , which are dependent on operating conditions. Absorb reactive power +P ^ ( p u ) * 0max Qc, max Supply reactive power Figure 4.1: Real and reactive power operating limits of  the voltage source converter. Since the same real power Pc must pass through both the rotor-side and the grid-side converter, and each of  them is limited as depicted in Figure 4.1,. the maximum available reactive power from  the voltage source converter can be expressed as (4.2) where it is assumed that reactive power, which has been supplied to the DFIG, is Qj r , and the nominal apparent power of  the converter is S ™ x , which is defined  as 1/3 of  the wind turbine rating [8], Based on Figure 4.1, it also follows  that -Sf ax < Pj c < Sf ax . 4.3 Supervisory Voltage Control Scheme Equation (4.1) represents the basis for  the proposed supervisory voltage control scheme depicted in Figure 4.2. It should be noted that this controller will require information  from  all wind turbines. However, since the overall control objective is to regulate the voltage at a single remote PCC, this centralized scheme appears to be reasonable as well as justifiable.  In this scheme, the supervisory voltage controller provides the total reactive power AgpCC to regulate the voltage at the PCC. Then, this total reactive power is distributed to each wind turbine according to (4.1). Since the system is nonlinear and the voltage at the PCC also changes due to load variations, grid conditions, wind speed, etc., the supervisory controller in Figure 4.2 should be robust to ensure stable and adequate dynamic performance  in a wide range of  conditions. In this thesis, it is assumed that the system operating conditions are mainly determined by the load variations, which are in the range of  ±50% during the day. To accommodate this range, one may consider designing several controllers and then switching them (or their gains), depending on the system's condition. Alternatively, to design the supervisory controller for robust operation, it is possible to view the entire model of  the system as a collection of  linear systems (spanned by the range of  operating conditions) for  which a common controller is designed and tuned. The latter approach is taken in this thesis. A control-design technique known as the linear quadratic regulator (LQR) may be conveniently utilized for  multi-input multi-output systems. The LQR approach can be used for  tuning the controllers with some specified  properties, such as phase margin (-60°,60°) and gain margin ( -6 dB, inf  dB) [61], The design procedure consists of  finding  the solution to the Riccati equation that satisfies  a certain cost function.  However, to make this approach applicable simultaneously to several linear systems, the LQR problem can be formulated  as a linear matrix inequality (LMI) solution for  which a common Lyapunov function  for  the set of considered linear systems is found  if  it exists. The controller designed utilizing this common Lyapunov function  guarantees system stability (in the Lyapunov sense) in the range of  the considered region. This thesis adopts the LQR design approach formulated  as a system of LMIs. 4.4 Plant Model and Conventional Controllers 4.4.1 Linearized and reduced-order model The first  step in controller design considered here consists of  finding  a linearized plant model that captures the relationship between the input and output with regard to the control objectives. With respect to Figure 4.2, we need to find  a transfer  function  from  the reactive power injected by the wind farm  to the voltage at the PCC. It should be pointed out that although state equations of  all the model components are available as presented in Chapter 3, it is not practical to derive the required linearized model analytically. Instead, in this thesis, the respective linearized models are obtained using numerical linearization (a feature available in Matlab/Simulink) of  the overall model about a specified  operating point. To cover the operating range of  interest, three operating points determined by the local load impedance of  -50%, nominal, and +50% were considered. These operating conditions correspond to the expected average daily peak load variations and are summarized in Table A. 10 in Appendix A. In this thesis, these three operating points are assumed sufficient  to represent the desired operating range and therefore  are considered for  control design purposes. The corresponding numerically obtained transfer  functions  from  the reactive power to voltage at the PCC were found  to have 104th-order. For a system of  such high order, analytical derivations would not have been possible and/or practical. The corresponding magnitude and phase plots are shown in Figure 4.3. Note that even though the load deviations are large, the deviations of  the linearized models in the frequency  domain are not very significant,  which suggests that a linear controller may work adequately for  the given system. The original 104th-order linearized model for  control design might be possible, but is not desirable, as it would require significant  computational resources. However, as the visual inspection of  Figure 4.3 and 4.4 reveals, these transfer  functions  may well be approximated in the frequency  range of  interest by a system of  much lower order. In this thesis, a balanced realization model-order reduction technique [62], [63] is used to find  the lower-order approximate transfer  functions  that are more suitable for  purpose of  control design. This technique is based on considering the dominant states (modes) in the input-output behaviour of  the system. The method uses Hankel  singular  values of  the system, which are the common eigenvalues of  controllability and observability Gramians. The reduced-order model is obtained by neglecting the appropriate number of  smallest Hankel  singular  values. Numerical linearization and model reduction have been carried out using the Control System Toolbox [64]. Based on Figure 4.3 and 4.4, it was considered sufficient  to approximate the respective transfer  function  with 4th-order. The corresponding reduced-order transfer functions  magnitude and phase are plotted in Figure 4.4. As can be concluded by comparing Figure 4.3 and 4.4, the 4th-order transfer  functions  approximate the original 104th-order system very well in the frequency  range of  interest. Hereafter,  these reduced-order models are considered to represent the plant. Frequency (rad/sec) Figure 4.3: Bode diagrams of  the full-order  model (104th). Frequency (rad/sec) Figure 4.4: Bode diagrams of  the reduced-order model (4th) The 4th-order reduced model may be realized in terms of  the state variables that are related to the voltage at the PCC as i = [ A v pcc A V c Avpcc A v p c c ] 7 (4.3) which contains proportional and derivative states. To guarantee zero steady-state error in tracking the set-point voltage Vpf c, an integral action is needed from  the controller [65], This integral action can be expressed by adding an auxiliary state j A v ^ to (4.3) as x ( 0 = rjAv-< x (4.4) where |AvpCC = (v^c - v p c c . This state clearly delivers the integral action for  the difference  between the set-point voltage and the system output voltage v p c c . The combined state-space equations for  three systems with the auxiliary state can then be expressed as x(0 = A,-x(0 + Bju(t)  z (t)  = Cj\(t)  + D,-w(f) where A B / ; Ch Dj are the augmented system matrices (4.5) "o -Ci - V "1 0 0 0 0" "0" Bi = » c i = Di = 0 7 I 0 1 0 0 0 ' I oj' (4.6) Here i = o,l,h and the subscript "o" denotes the nominal operating condition; "/" denotes a 50% decrease and "h" denotes a 50% increase of  local load impedance, respectively; A e is the state matrix; B e 3 i n X p is the input matrix; Ce 3 i m X n is the output matrix; D G SimXP  is the direct feed-through  matrix; x e is the state vector; u e is the input signal vector; ze 3lm is the output vector; and A,-, B ; , C,-, D ; are the system matrices of  the 4 l -order reduced model. Overall, this system has one input (p  = 1), five  states (n  = 5 ), and two outputs (m = 2), respectively. The corresponding state-space matrices are "0 6.89150 10.2209 -1.7963 6.09420" "0.02820" 0 -10.408 -54.899 6.36430 -17.017 -6.8915 A 0 = 0 54.8990 -52.306 13.7505 -76.688 > B 0 - 10.2209 0 -6.3643 13.7505 -4.7933 72.8850 -1.7963 0 -17.017 76.6880 -72.885 -99.634 -6.0942 "0 6.40620 9.55480 1.70940 4.54920" 0.00390" 0 -10.593 -51.313 -6.5215 -14.052 -6.4062 A , = 0 51.3130 -61.339 -16.300 -71.572 > » / = 9.55480 0 6.52150 -16.300 -5.6663 -64.272 1.70940 0 -14.052 71.5720 64.2720 -75:609_ -4.5492 "0 7.08100 10.4324 -1.8855 6.79060" "0.04200" 0 -10.294 -55.954 6.46570 -18.208 -7.0810 A h = 0 55.9540 -48.792 13.2378 -77.843 B h - 10.4324 0 -6.4657 13.2378 -4.7963 77.4240 -1.8855 0 -18.208 77.8430 -77.424 -112.34 -6.7906 (4.7) (4.8) (4.9) C0-C t-Ch-1 0 0 0 0' 0 1 0 0 0 =D; =Di = (4.10) and the output and the state vector are =[K ~\T "pcc AVpCC = [ jAv p c c Av p c c Av p c c Av p c c & v p c c J (4.11) The output vector z is assumed to be directly available (measurable) for  feedback  purposes. Table 4.1 summarizes the eigenvalues and damping ratios, as well as the corresponding frequencies,  of  the 5th-order reduced model. TABLE 4 .1 EIGENVALUES, DAMPING RATIO, AND FREQUENCY OF THE 5™-ORDER REDUCED MODEL Eigenvalues Damping ratio Frequency (rad/sec) 0 - 0 -22.6±j20.8 0.786 30.8 -60.9±jl06 0.5 122 4.4.2 PID supervisory controller design Before  considering advanced controllers, it is prudent to investigate available traditional approaches to see if  satisfactory  dynamic performance  could be obtained using them. A proportional-integral-derivative (PID) controller is commonly used in power industries. In the remainder of  this chapter, a PID-supervisory controller is designed based on the transfer function  corresponding to the nominal condition, and system performance  is evaluated subject to the three-phase symmetrical fault.  More detail on the simulation studies is presented in Chapter 5. The PID gains are tuned to meet the specifications  of  less than 10% overshoot and greater than 60 degrees phase margin. The gain crossover frequency  0)gc , which corresponds to this phase margin, is 65 rad/sec, as can be found  from  the Bode diagram in Figure 4.4 (see the line that corresponds to the nominal condition). The integral gain kt was first  chosen as 2.2347, which corresponds to the dc-gain. By using the values of  (Ogc and kj , the proportional gain kp and the derivative gain kj can be computed using the following equation [56] Gr{jCD gc) = Xe j6{ Wz c\ (4.12) V - J k p + j o ) g c k d + h ico, or eM°>gc) j k . P s Gr(jco gc) (o gc (4.13) where Gr(ja> gc) is the transfer  function  of  the 4' -order reduced model obtained for  the nominal operating condition, as follows: Gr{s)  = -0.028s4 -27.78S 3 - 7 3 5 4 s 2 +15618025 + 31501951 s 4 + 167.14s3 + 21340s2 + 788654s+ 14079701 (4.14) By solving (4.12) with the gain crossover frequency  (Ogc = 65 rad/sec, the gains of  the PID supervisory controller may be expressed as kp =a, kd  =bja)gc , and k t = d c g a j n . To speed up the settling time, the integral gain kt is increased to 6.7122. The final  computed gains are summarized in Table 4.2. To limit the effect  of  high-frequency  noise, the derivative control branch is implemented as a filter  with an equivalent gain given as kd-k dt / r \ \ kd  1 s +1 Vv y " . This is usually done to avoid large transients in the control signal resulting from  sudden changes in the set-point [55]. The typical range of  values for  N d  is from  2 to 20 (higher value implies stronger derivative action); N d  - 20 was used herein. The corresponding step responses of  the reduced-order system (4.14) in the open-loop and the closed-loop are plotted in Figure 4.5, which shows that the overshoot of  the closed-loop system is less than 10%, which satisfied the design specification. TABLE 4.2 GAINS OF THE PID-SUPERVISORY CONTROLLER States j A v p c c A v p c c Avpcc . Gains 6.7122 0.4635 0.0009 time(sec) Figure 4.5: Step response of  the open-loop and the closed-loop reduced-order system. To make this controller practical, the output control signal should be limited by the amount of  currently available reactive power. At the same time, limiting the control action should be implemented together with the integrator-anti-windup scheme that would stop integrating the error when the limit is being reached. To take into account the individual wind turbines, a distributed anti-windup scheme that takes into account the limits of  each turbine has been considered. A combined diagram of  the PID controller with the proposed anti-windup scheme is depicted in Figure 4.6. The anti-windup scheme requires the currently available reactive power limits Q™* defined  in (4.2). As shown in Figure 4.6, the output of  the controller is distributed among the wind turbines according to (4.1), wherein each output is compared to the respective limit Q™*. When none of  the limits are reached, the overall anti-windup loop is inactive and the integral control branch operates in a normal way. However, when one or more limits are being reached, the difference  between the actual output(s) before and after  the limiters will be non-zero, which in turn will make the anti-windup loop active and reduce the integral action. The anti-windup loop gain is determined by \/k t [55], where k t is the integral reset time constant, calculated as k t - ^k^k,  . 1 s ki Figure 4.6: Implementation of  the PID controller with distributed anti-windup loop. 4.4.3 Evaluation of conventional controllers To evaluate the dynamic performance  of  the PID-supervisory controller designed in this section, the controller was put back into the original detailed full-order  model of  the overall system. A symmetrical three-phase fault  was implemented on one of  the transmission lines (see TL in Figure 3.1). For comparison purposes, the same fault  study was also implemented using the PFC and LVC modes. The corresponding transient responses of  the voltage at the PCC produced by models with different  controllers are superimposed in Figure 4.7 for  better comparison. As can be seen in Figure 4.7, the system initially operates in a steady state such that each control scheme results in the same bus voltage at the PCC. At t = 1.0s, a fault  is being applied and is then cleared after  t = 1.15s . The fault  results in voltage sag observed at the PCC, which is different  for  the three control techniques considered. As can be seen in Figure 4.7, the PFC does not provide voltage support during the fault.  At the same time, the LVC and the PID resulted in 0.26/s and 0.44/s of  the voltage recovery rate. When the fault  is cleared by opening the faulted  line, the PFC and LVC resulted in 1.32% and 0.25% deviations from  the pre-fault  value, respectively. Although the PID controller enabled a much faster  voltage recovery during the fault,  it still shows an undesirable oscillatory behaviour during and after  the fault. Overall, it is desirable to achieve a faster  damping with less oscillatory behaviour during the fault,  as well as when the fault  is cleared. In addition, to make the proposed controller practical, the noise of  the measured signals (voltage at the PCC) should also be taken into consideration. This conclusion motivates investigating further  options in designing advanced controllers. Figure 4.7: Voltage transient at the PCC resulting from  a three-phase fault. Chapter 5 Advanced Voltage Control Schemes 5.1 Observer-Based Framework To make the overall control scheme applicable for  realistic cases, one should consider from the beginning that noise and signal distortions are unavoidable in measuring voltage at the PCC. For improving dynamic performance  beyond what was demonstrated in the previous section with the PID controller, it is desirable to make use of  the entire state vector in (4.11) instead of  just the output voltage. However, in general, measuring the high-order derivatives is even more problematic in the presence of  noise. To address the above-mentioned considerations, an observer-based controller design framework  is taken in this thesis. A block diagram of  the overall proposed supervisory voltage control scheme, with observer, is depicted in Figure 5.1. Here, for  the purposes of controller design, the plant denoting the wind farm  and electric grid combined is represented by the following  collection of  reduced-order linear systems: x = Aj-x + BjU + Gw (5.1) y = CjX  + D,u + n , (5.2) where, as before  i — o,l,h  and the subscript "o" denotes the nominal operating condition; "/" denotes a 50% decrease and "h" denotes a 50% increase of  local load impedance, respectively; and G e is the randomly chosen real matrix. In Figure 5.1, w = A2Pcc = _ k x is the output of  the supervisory controller, which is the total reactive power required to control the voltage at the PCC. The variable x is the observer state vector. The noise signals in the system state and the measured output are denoted by w and r\, respectively. The variable z represents the measured system output vector ; = [ f A v pcc A vpcc + • (5.3) where Av p c c = Vpf c - v p c c , and Vpf c is the predefined  value of  the voltage at the PCC. The presence of  an observer allows using observer states for  feedback  control instead of  the system states as given in (4.11), which in practice are not measured directly. In this thesis, the Separation Principle [61], [66] is used and the state-feedback  controller gain k and the observer gain K e are designed independent from  each other. W Figure 5.1: Block diagram of  the supervisory voltage control with observer. 5.2 State Observer Design The necessary condition in the design of  an observer is observability. The concept of observability is dual to that of  controllability, which is the necessary condition for  state-feedback  controller design. Roughly speaking, controllability refers  to the ability to steer the state from  the input; observability refers  to the possibility of  estimating the state from  the output signal. In this section, a standard Kalman filter  is designed to deal with noise signals [66], wherein finding  the observer gain is achieved through solution of  the following algebraic Riccati equation: A0S + ST\L  +GRWG T  -STCIQ~ 1C 0S = 0 (5.4) T where S = S is the positive-definite  solution matrix and A 0 and C 0 are the state matrices corresponding to the nominal operating condition. The noise covariance matrices denoted here by R w e Sl nX n and Q„ e y( mxm a r e defined  using the expectation E of  each noise signal as R w = E wwT  and Qn=E , respectively [61], [67]. The noisy signals are assumed to be white, Gaussian, and to have zero-mean such that £[w] = 0 and £[11] = 0. Here, the subscripts w and rj relate the matrices to the state and output noise signals, respectively. Finally, the observer gain is calculated using the solution to (5.4) as T* 1 K e = SC0Q^ . The observer can be expressed as J x = A 0 x + B0w + K e ( z - z) (5.5) A A z = C 0 x where z is the measured system output vector, and z is the observer output vector. Choosing Q^ to be very small compared to R w implies that the measurement noise TJ is also small. An optimal state observer then interprets a large deviation of  the observer output z from  z as an indication that the estimate x is bad and needs to be corrected. In practice, this lead to large matrices of  the observer gain K e and corresponding fast  poles in (A 0 -K EC 0). Alternatively, choosing Q^ to be very large implies that the measurement noise q is large too. An optimal state observer is then much more conservative regarding deviations of  z from  z . This generally leads to small matrices for  the observer gain K e and consequently slow poles in (A 0 - K e C 0 ) . For the work presented here, we chose to have a faster  observer. We chose the covariance matrices as Q^ = diag  (0.0006, 0.0006) and R w = 0.5, and the randomly chosen constant matrix G = [1.9515, 2.3081, 2.0722, 1.5768, 0.4277]7. Applying these design parameters and using the Control System Toolbox [64], the observer gain is obtained as follows: Table 5.1 shows the eigenvalues, damping ratio, and frequencies  of  the closed-loop observer that correspond to ( A 0 - K e C 0 ) . Note that since the observer is designed based on the 5 th-order reduced model, the covariance matrices are chosen to place the eigenvalues in the complex domain close to the frequencies  of  the reduced-order model, specifically  at 30.8 rad/sec and 122 rad/sec. With R w = 0.5, the covariance matrix Q^ was increased from 0.0001 to 0.0006, where the maximum damping ratio of  the closed-loop observer was reached at 0.503. This is close to the maximum damping ratio of  the reduced-order model, 0.5, and at the same time the eigenvalues are placed close to the frequency  of  the reduced-61.5334 20.6683 44.7403 48.6022 1.2318 20.6683 26.3330 25.4970 17.7305 5.7823 T K e - S C o Q j (5.6) order model at 30.8 rad/sec. As a design observation, if  the damping ratio is less than 0.5 at frequency  122 rad/sec, or the eigenvalues of  the observer are significantly  different  than those of  the reduced-order model at a frequency  of  around 30.8 rad/sec, system performance becomes sluggish or oscillatory. TABLE 5.1 EIGENVALUES, DAMPING RATIO, AND FREQUENCY OF THE CLOSED-LOOP OBSERVER Eigenvalues Damping ratio Frequency (rad/sec) -22.0 1 22.0 -31.4 1 31.4 -78.5 1 78.5 -61.5±jl06 0.503 122 5.3 Linear Quadratic Regulator Approach We chose the LQR approach as a framework  for  tuning the controller gains in this thesis as this methodology is general and flexible,  and can be formulated  in terms of  a performance-based optimization problem, for  which the numerical solution techniques and software  tools are widely available [64], [68]. At the same time, the cost function  (function  to be minimized) may be defined  in a number of  ways that can simultaneously include several performance-based  criteria. Another important advantage of  using the LQR is that it can be formulated  for  the case when the overall plant is described by a set of  linear systems that span a particular range of  operating conditions. This is accomplished by representing the underlying control optimization problem in terms of  a system of  LMI constraints and matrix equations that are simultaneously solved. The solution of  LMI equations involves a form  of quadratic Lyapunov function  that; if  it exists, not only gives the stability property of  the controlled system but can also be used for  achieving certain performance  specifications. 5.3.1 Formulation of LQR The LQR control design problem looks for  a feedback  controller gain k e 9 \ p X n for  the system x = Ax + Bu, u = -kx (5.7) that minimizes the cost function  [69] J = mm J ^ ( x r Q x + urRu)flfr  (5.8) where Q and R are design parameters, Q e Si"X n is a symmetric non-negative definite matrix and R e 3 l p X p is a symmetric positive definite  matrix. The final  control gain k should satisfy  the following  Lyapunov equation: ( A - B k ) r P + P(A-Bk) + Q + k r R k = 0 (5.9) where P e %nXn > 0 , known as the Lyapunov matrix. The LQR controller minimizes the quadratic function  of  the state xTQx  and that of  the control signal u Ru . These quadratic functions  are often  associated with the energy in the system's state and control signal. Matrices Q and R are the relative weights of  the state dynamics and the control action. For example, choosing Q large and R small will result in a control gain that will attempt to reduce the deviations of  the state at the expense of  a very strong control action. On the other hand, choosing Q small and R large will result in a control gain that will attempt to reduce the control action at the expense of  allowing large state variations. Therefore,  these matrices are chosen to achieve some balance between the desired performance  and the required control action. Since (5.9) is nonlinear and difficult  to solve, the solution to the LQR is found  by the well-known algebraic Riccati equation A r P + PA + Q - P B R _ 1 B r P = 0 l r ^ (5.10) which is linear in variable P and is readily solved numerically using software  [64], The Since in our case all signals are expected to have some noise, the cost function  (5.8) can be rewritten in terms of  expectation as . Moreover, since we have a set of  linear systems representing a range of  operating conditions, the feedback  gain k should now satisfy  a number of  Lyapunov equations, as follows: The Lyapunov equations (5.12) are nonlinear and therefore  difficult  to solve. The problem is even more complicated by the fact,  that these equations for  multiple systems have to be solved simultaneously for  a common Lyapunov matrix P > 0. Instead of  trying to solve (5.11) and (5.12) directly, in the following  Subsections we re-formulate  this problem as a set of  LMIs, which are then solved for  a common matrix P. In doing so, we provide two LMI formulations.  The first  conventional LMI formulation  is based on minimizing the quadratic cost function  [70]. The second method presents an LMI formulation  based on minimizing the upper bound of  the cost function  [71], which is a preferred  approach to dealing with uncertain signals/variables. controller gain is then computed as k = R B P. (5.11) (Aj - B j k f  P + P(A 7 - Byk) + Q + k r R k = 0 (5.12) 5.3.2 Conventional approach Using the H 2 representation of  the LQR problem [61], [67], we would like to find  the state-feedback  gain matrix k that minimizes the following  cost function  in terms of  output y as J  = minjiiTy^yT] (k) 1 L K (5.13) subject to ( A y - B y k ) r P + P ( A / - B / k ) + Q + k r R k < 0 and P > 0 V  J (5.14) We first  formulate  an LMI for  the cost function  (5.13), using the output y as given in [67] y = >1/2 0 R l / 2 (5.15) then substitute (5.15) into (5.13) with u = -kx to obtain .T~ y y = E tr = E • tr x r Qx + u r Ru / r Q V2 ' -RV 2k xx \QV2)T - k ^ R 1 / 2 ) 7 (5.16) where the function  tr stands for  trace, which is the sum of  all its diagonal entries. By utilizing the identity tr{ABC)  = tr(CBA)  and the state covariance matrix Y = E the H 2 representation of  the LQR problem (5.13) can be expressed as xx y y ( ' ' 7 = fr(QY)  + fr  R 1 / 2 k Y k r ( R 1 / 2 ) (5.17) where it is assumed that E of  variables [67] such that X > R 1 / 2 k Y k r ( R 1 / 2 ) 7 W W = R w > 0 and Y = E xx . We then utilize the change (5.18) Let us define  a change of  variables as used in [67] such that k = A Y " 1 a n d k G 3 I P X N , A E P = Y" Y = Y and P e Si nXn and Ye5R nxn (5.19) (5.20) (5.21) Using the well-chosen variable (5.19) and the Schur complement, (5.18) can be put into a matrix inequality as follows: X - j R ^ A j Y - ' j / R 1 / 2 ^ X R 1 / 2 A A V 2 Y >0 (5.22) Finally, the cost function  (5.13) is formulated  as follows: Minimize: J = min {fr(QY)  + /r(X)l ( Y , X ) 1 J (5.23) subject to LMI constraint (5.22). To complete the LMI formulation  of  the H 2 LQR problem, we still need to obtain an LMI for  (5.14). Using the change of  variables (5.19)—(5.21), Equation.(5.14) can be re-formulated as follows: I (Ay-Byk) P + P(Ay-Byk) + Q + krRk -1 = Y Y T  Ay - ArBy + AyY - By A + YQY + ArRA Y < 0 (5.24) = Y7 Ay - ArB y + Ay Y - By A + YQY + ArRA < 0 Then, utilizing the Schur complement, (5.24) can be put into an inequality as follows: Ay Y + YrAy - ByA - A^ By ) AT Yr A - R 1 0 Y 0 -Q" <0 (5.25) with A > 0 and Y > 0 . Finally, LMI formulation  of  the H 2 LQR problem based on the conventional approach with cost function  (5.23) is as follows: Minimize: J = min {;r(QY) + /r(X)| ( Y , X ) 1 1 subject to LMI constraints (5.22) and (5.25). 5.3.3 Cost-guaranteed approach As defined  in [67], the cost-guaranteed approach is to replace the cost function  (5.11) with a certain upper bound when the system is subject to noise. Thus, if  we write the Lyapunov equations (5.12) as a matrix inequality, the solution of  this inequality will be an upper bound. Therefore,  the H j LQR problem based on the cost-guaranteed approach can be defined  as follows: subject to (5.14), which is not an LMI. This optimization problem provides a necessary and sufficient  condition to guarantee the system asymptotic stability. The proof  for  these properties is given in Appendix E. Using the change of  variables (5.20), the bound on the cost function  /r (P) in (5.26) may be changed to Minimize: (5.26) tr (Y 1 ) . We further  introduce a slack matrix variable Z as used in [67] such that Z > Y _ 1 (5.27) which is used to write the following  matrix inequality: -1 Z I Z - I Y >0, and Y > 0 I Y -1 (5.28) where I is an nxn identity matrix, and Z  = Z T  e 3inX n is the symmetric positive definite matrix. Thereafter,  the H 2 LQR problem based on the cost-guaranteed approach is formulated  as follows: Minimize: V = minjfr-(Z))  (5.29) (Z) 1 J subject to (5.25) and (5.28). 5.3.4 Evaluation of controllers The LQR formulations  presented in Subsections 5.3.2 and 5.3.3 have been used for  tuning the controller gains. The numerical solutions were carried out using the LMI Control System Toolbox [68] with the input script files  as given in Appendix D. All tuning parameters and LQR gains are summarized in Table 5.2 in Section 5.5 for  consistency and further comparison. The two controllers resulting from  Subsections 5.3.2 and 5.3.3 are hereafter referred  to as the LQR supervisory (LQRS) and the LQR cost-guaranteed (LQRCG), respectively. The same symmetrical fault  study as described in Section 4.4.3 was used here to compare the system's response with different  controllers. In particular, the voltage transient observed at the PCC for  the system with PID supervisory (Section 4.4.2), LQRS (Section 5.3.2), and LQRCG (Section 5.3.3) controllers is shown in Figure 5.2. As can be seen, the two LQR controllers perform  much better than the PID controller, which has lower order. During the fault,  the PID-supervisory controller resulted in 0.44/sec of  the voltage recovery rate, whereas the LQRS and LQRCG controllers resulted in 0.875/sec of  the voltage recovery rate. While performing  better than the PID controller during the fault,  both the LQRS and LQRCG controllers showed very similar behaviour, with some undesirable oscillations after  the fault. The following  section presents some control modifications  to reduce the oscillatory behaviour and further  improve the transient response of  the system. 0.92 1.2 time(sec) Figure 5.2: Comparison of  PID, LQRS, and LQRCG controllers. 5.4 Advanced LMI Representation of LQR As an attempt to further  improve controller performance,  in this section we consider taking into account the cross-product of  the state and control signals [61]. This is accomplished by including these cross-product terms into the LQR cost function  in addition to the quadratic functions  of  the state and control signal, as done in Section 5.3. 5.4.1 Taking into account cross-product terms in the conventional approach Adding the cross-product terms, the cost function  for  the LQR problem (5.16) can be expressed as J  = mini? 00 x^Qx + u r R u + x r N u + u r N r x (5.30) where N e S l n X p satisfies  the condition Q - N r R ] N > 0 . We further  proceed by substituting u = -kx into (5.30) to obtain J  = E m rr< m T" rp rp rrt xl Qx + x k Rkx-x Nkx-x^k'N 7 x = ^ ( Q + k R k r - N k - k r N r \ t r \ E } r{E  xxT  ) (5.31) T  ^ = /r(QY) + fr  R ^ K Y k ^ R 1 / 2 ) -^ (NkY + k r N r Y r ) Hence, the conventional H 2 LQR problem with the cross-product terms becomes J = min (k,Y) tr QY + R! / 2 k Y k r ( R 1 / 2 ) - NkY - k r N r Y T (5.32) subject to (5.25). Note that the second term was already presented in (5.23) as X . However, (5.32) includes kY and therefore  cannot be easily solved. By using the change of  variables shown in (5.19), we obtain ? r ( - N k Y - k r N r Y r ) = / r ( - N k Y - Y r k r N r ) = ^ ( - N A - A r N r ) (5.33) Hence, the advanced H 2 LQR problem based on the conventional approach can be formulated  as follows: Minimize: J  - min (A ,Y ,X) fr(QY  + x)-/r(NA + A r N 3 (5.34) subject to (5.22) and (5.25). 5.4.2 Taking into account cross-product terms in the cost-guaranteed approach Utilizing (5.29) and (5.34), the H 2 LQR problem based on the cost-guaranteed approach can be described as following: J = min\E |"xrQx + u r R u + x r N u + u ^ N 7 * ] ] (k) I L JJ = min ^ r ( Q Y + x ) - / r ( N A + A r N r ) ? r (Z)- / r (NA + A r N r ) (5.35) < min (Z,A) Hence, the advanced H 2 LQR problem based on the cost-guaranteed approach can be formulated  as follows: Minimize'. V = min (Z,A) (5.36) subject  to (5.25) and the slack matrix variable constraint (5.28). 5.4.3 Evaluation of controllers The formulations  presented in Subsections 5.4.1 and 5.4.2 have been used for  tuning the controller gains. The numerical solutions were carried out using the LMI Control System Toolbox [68] with the input script files  as shown in Appendix D. All tuning parameters and gains are summarized in Section 5.5 for  consistency and further  comparison. The two controllers resulting from  Subsections 5.4.1 and 5.4.2 are hereafter  referred  to as the advanced LQR supervisory (ALQRS) and the advanced LQR cost-guaranteed (ALQRCG), respectively. The same symmetrical fault  study as described in Section 4.4.3 was used here to compare the system's response with different  controllers. In particular, the voltage transient observed at the PCC for  the system with different  controllers is shown in Figures 5.3 and 5.4. To get an idea of  what has been gained by taking into account the cross-product terms, Figure 5.3 first compares the system's response with the ALQRS (see Section 5.4.1) versus the LQRCG (see Section 5.3.3). As can be observed in Figure 5.3, the ALQRS does improve performance  and reduces oscillatory behaviour, especially after  the fault  has been cleared. This achievement already well justifies  the extra effort  involved in formulating  and tuning this controller. Performance  of  the ALQRCG (see Section 5.4.2) is depicted in Figure 5.4, wherein this controller is further  compared with the PID supervisory and ALQRS controllers. An interesting observation can be made here. In particular, the ALQRCG controller provides even further  improvement and damping of  the post-fault  oscillations over the ALQRS. The formulation  and design of  the ALQRCG has paid off  with the best transient performance  of the system, which was the goal of  this Chapter. The following  section summarizes controller gains. The computer studies presented in Chapter 6 compare the proposed ALQRCG with traditional control solutions such as PFC, LVC, and PID-supervisory controller. Figure 5.3: Comparison of  LQRCG and ALQRS controllers. Figure 5.4: Comparison of  PID, ALQRS and ALQRCG controllers. 5.5 Summary of Controller Gains The tuning parameters and gains of  the LQR-based controllers are summarized in Table 5.2. In selecting the design parameters, we begin with R = 10 and Q = diag{\,  1, 1, 1, 1). Since these design parameters showed very slow settling time, we increased the entry Q y and Q2 2 speed up the integral action, and decreased R. Table 5.3 summarizes the eigenvalues and damping ratios, along with corresponding frequencies,  of  the closed-loop 5th-order reduced model (A 0 - B 0 k ) . An important observation can be made regarding the data in Table 5.2, namely, that considering the cross-product terms results in noticeably higher derivative and integral gains. This is especially pronounced in the third derivative, where the ALQRCG has the highest gain of  3.3509. The integral gain of  ALQRCG also increases to 2.7747. Thus, better damping and faster  performance  can be expected. TABLE 5.2 DESIGN PARAMETERS AND CONTROL GAINS Conventional Approach Advanced Approach Design parameter LQRS LQRCG ALQRS ALQRCG R 1.2 2.2 . 0.955 4 Q diag{  70, 10, 1, 1, 1) N [0, 0, 0, 0, o f [0.55, 0.55, 0.55, 0.55, 0.55f Gains State LQRS LQRCG ALQRS ALQRCG fAvp CC -5.5182 -5.5099 -5.6543 -5.7190 A v p c c -2.2902 -2.2342 -2.1965 -1.3626 A v p c c 0.8600 0.8917 1.5886 2.7747 A v p c c 0.0660 0.0106 0.1861 0.1217 Avpcc -0.1852 -1.8105 0.1958 3.3509 TABLE 5.3 EIGENVALUES, DAMPING RATIO, AND FREQUENCY LQRS Eigenvalues Damping ratio Frequency (rad/sec) -9.10 1 9.10 -28.3±j20.6 0.808 25.0 -63.4±jl07 0.508 125 LQRCG Eigenvalues Damping ratio Frequency (rad/sec) -9.88 1 9.88 -35.3±jll.6 0.905 37.1 -61.0±j94.9 0.541 113 ALQRS Eigenvalues. Damping ratio Frequency (rad/sec) -10.6 1 10.6 -25.0±jl9.4 0.790 31.7 -68. l±j 110 0.526 130 ALQRCG Eigenvalues Damping ratio Frequency (rad/sec) -11.2 1 11.2 -12.7±j23.6 0.475 26.8 -73.7±jl30 0.493 149 Chapter 6 Simulation Studies We modelled the system depicted in Figure 3.1 and described in Chapter 3 together with the various controllers described in Chapters 4 and 5. We implemented the overall detailed model of  the system using Matlab/Simulink software  [72] was implemented, and carried out computer studies to study the impact of  wind speed variations, load variations at the PCC, and the remote three-phase symmetrical fault.  In the simulation studies presented in this chapter, the proposed advanced LQR-cost-guaranteed controller (ALQRCG) is compared with the PID-supervisory controller, the conventional power factor  control (PFC) and the local voltage control (LVC). 6.1 Small Disturbances 6.1.1 Wind speed variations To study the effect  of  wind speed variations, different  wind speeds were assumed for  the three wind turbines (WTs), and are shown in Figure 6.1. These wind variations are assumed to represent a realistic wind gust that is unavoidable, especially if  the WTs are located apart from  each other, as is in the case of  the system depicted in Figure 3.1. We performed  the computer simulations of  the system with different  controllers and plotted the corresponding results in Figures 6.2 through 6.8. The real power output from  each wind turbine is controlled by the maximum power tracking curve (see Figure 3.13),. which determines the real power set-point in each turbine, as shown in Figure 6.2. As can be observed, the internal controllers in each turbine track the instantaneous power command very well. The actual real power output plotted in Figure 6.3 follows  that of  Figure 6.2, which altogether corresponds to the wind speed trends. The combined real power from  the wind farm  that is injected into the grid is shown in Figure 6.4. As can be seen, the variations are relatively small, which is due to the pitch control action. The reactive power outputs at the terminal of  each wind turbines are shown in Figure 6.5 for the system with different  controllers. When the WTs operate in PFC, they output no additional reactive power to the grid to maintain a unity power factor.  When the LVC is used, the reactive power injected by each WT is somewhat different,  due to the different  wind speeds. However, when any of  the supervisory controllers are used, the commanded reactive power is evenly distributed among the participating WTs, because they all operate somewhat below the limit. In this case, the WTs contribute evenly to the overall reactive power injected by the wind farm,  as depicted in Figure 6.6. The voltage fluctuations  observed at the WT terminals and at the PCC are shown in Figures 6.7 and 6.8, respectively. Overall, these variations are small and therefore  do not represent a concern for  the power quality in the system. Voltage fluctuation  is minimized by the variable-speed wind turbine technology and the action of  the internal controllers. Figure 6.1: Wind speed (m/sec). 3 Q. C o CD o Q. "S <D CC 0.027 0.0265 0.027 0.0265 0.027 -0.0265 -0.027 0.0265 - a y , LVC JP'yC-PID - supervisory ALQRCG I i I I I I i i -0 10 20 30 40 50 60 time(sec) 70 80 90 100 Figure 6.2: Real power set-point for  each WT due to wind speed variation. ( WT1 — 1 WT2 WT3) 3 Q. D Q. 0 <5 1 CL "5 a> CC 0.027 0.0265 0.027 0.0265 0.027 0.0265 0.027 PFC 0.0265 -/••,.. X. LVC ''/py^X/ PID - supervisory fKoC ALQRCG \ , J  Jy\  .A'" / / vV... f\/""'"\. -A^ rv I I I I i i i -10 20 30 40 50 60 time(sec) 70 80 90 100 Figure 6.3: Real power output from  each WT due to wind speed variation. (—WT1 —WT2 WT3) 3 Q. 3 Q. O i_ Q) g O Q. "(5 CD cc 0.078 0.0775 0.077 0.078 0.0775 0.077 0.078 0.0775 0.077 0.078 0.0775 0.077. PFC LVC PID - supervisory ALQRCG 0 10 20 30 40 50 60 70 80 90 100 time(sec) Figure 6.4: Real power output from  the wind farm  due to wind speed variation. Q. 13 Q. "5 o 1— <D o Q. a> > 13 to CD DC 0.00005 -0.00005 0.00005 -0.00005 0.00005 PFC PID - supervisory -0.00005 0.00005 -0.00005. v V A A A A V N / V X ^ ALQRCG V W A a A A A A / V \ 10 20 30 40 50 60 time(sec) 70 80 90 100 Figure 6.5: Reactive power output from  each WT due to wind speed variation. (—WT1 —WT2 WT3) 3 Q. 3 0 1 <D 3 o Q. <1} > O TO CD tr 0.0031 0.003 0.0029 0.0028 0.0031 0.003 0.0029 0.0028 0.0031 . 0.003 0.0029 0.0028 0.0031 0.003 0.0029 0.0028, LVC 40 50 60 time(sec) 100 Figure 6.6: Reactive power output from  the wind farm  due to wind speed variation. 3 Q. CD •5 CO O > 3 CD 0 10 20 30 40 50 60 70 80 90 100 time(sec) Figure 6.7: Voltage fluctuations  due to wind variation, as observed at the WT terminals. 0.905 -0.9049. Figure 6.8: Voltage fluctuations  due to wind variation, as observed at the PCC. 6.1.2 Load variations In the following  simulation study, a sequence of  step-changes in the load impedance is implemented, wherein the load is first  increased by 20%, then decreased by 20%, and finally decreased by further  20%. For test clarity, the wind speed here is assumed constant for  all WTs. The corresponding transient responses of  the system with different  controllers are provided in Figures 6.9 through 6.16. The voltage transients observed at the PCC due to load changes are plotted in Figures 6.9 and 6.10. The performance  of  the four  controllers is compared in Figure 6.9. As can be seen in this figure,  the PFC results in the largest voltage changes. This result is expected for  this control mode, wherein the voltage deviations are somewhat proportional to load changes. The voltages at the WT terminals are plotted in Figure 6.11. When the LVC is used, the changes in the load cause a small transient, but overall the voltages at the WT terminals return to the same set values, which is not the case for  the PFC. However, due to the impedance of  the cables and transformers  connecting the WTs to the PCC, the voltage at the PCC changes with the load variations, as can be seen in Figure 6.9. The real power produced by each WT is depicted in Figure 6.12 and the combined real power injected by the wind farm  is shown in Figure 6.13. As can be observed in these two figures,  there is a very small transient there due to the load changes, but the level of  injected real power returns to the same level which is determined by the wind speed. This behaviour is the same for  all the controllers shown in Figures 6.12 and 6.13, as they all operate using the reactive power only. Reactive power output for  each WT and the farm  are shown in Figures 6.14 and 6.15, respectively. As can be seen from  the figures,  the amount of  injected reactive power depends on the control scheme. The reactive power set values are plotted in Figure 6.16. As can be seen from  these studies, the PID-supervisory and ALQRCG controllers perform  very similarly and better than the PFC or LVC. However, as shown in Figure 6.10, the ALQRCG responds somewhat faster,  which is especially noticeable in the beginning of  each transient. 0.92 3 Q. 0.915 PFC LVC PID-supervisory ALQRCG in 3 0.885 0 2 4 6 8 10 12 . 14 16 time(sec) Figure 6.9: Voltage transient observed at the PCC due to load impedance changes. 0.912 0.91 0.908 S 0.906 O O 0.904 o> to 0.902 CD CT) to o 0.9 > v> m 0.898 0.896 0.894. PID-supervisory ALQRCG 0 8 time(sec) 10 12 14 16 Figure 6.10: Voltage transient observed at the PCC due to load impedance changes: Detailed view of  the PID-supervisory and ALQRCG controllers. PFC 3 Q. 0) •C -t—i to LVC 0.9? - _ v.. 0.9 - F R PID - supervisory 6 8 10 time(sec) 12 14 & 0.92 O) to _ L -§ 0.9 CO 3 f" CQ ALQRCG 0.92 - j ' 0.9 i i i i 1 1 16 Figure 6.11: Voltage transient observed at the WT terminals. D O. H § "o "3 a. ^ o a) 0 a. "ro <D 01 0.027 0.0265 0.027 0.0265 0.027 0.0265 0.027 1 ••• WT1 WT2 WT3 \ J 0.0265. 0 0.08 0.075 0.08 3 CL 3 CL -t—1 o L_ <D o a. ro <D a: 0.075 0.08 0.075 0.08 0.075. 0 LVC PID - supervisory ALQRCG 6 8 10 time(sec) 12' 14 Figure 6.12: Real power output from  each WT. PFC LVC PID - supervisory 8 time(sec) 10 12 14 i i I I p-s/-- I I 16 i y / t lr r r ALQRCG I | I I 16 6 8 10 12 14 16 time(sec) Figure 6.14: Reactive power output from  each WT. PFC 0.02 -0.02 LVC 2 4 6 8 10 12 time(sec) 0.02 E 3 E 'x (0 E 0 Q. 1 <D O) w d) g o Q. d) > o (0 (U OL 8 time(sec) Figure 6.16: Reactive power set-point and maximum at each WT. (- WT1 WT2 WT3, j=1,2,3) 6.1.3 Summary The level of  voltage deviation observed at the PCC depends on the control scheme being used. When the wind turbine operates in the PFC mode, the load changes result in the most noticeable deviations in the voltage level at the PCC. When the LVC mode is used, the voltage fluctuations  are significantly  reduced, because the PCC is relatively close to the wind farm  (9km cable). However, the proposed ALQRCG controller shows best performance.  A summary of  steady state values is given in Table 6.1, where it is shown that only the supervisory controllers provide a required voltage tracking at the remote PCC. TABLE 6.1 MAGNITUDE OF VOLTAGE DEVIATIONS Steady State Value PFC LVC PID- Supervisory & ALQRCG High (pu) 0.9154 0.9067 0.90497 Low (pu) 0.8892 0.9021 0.90497 Deviation (%) 2.62 0.46 0 6.2 Large Disturbances From the point of  view of  power system stability, it is desirable to keep the wind farm operational and connected to the grid for  as long as possible, even during large system-wide disturbances. This can be achieved by actively controlling the WTs during the disturbances and attempting to keep the voltage as close as possible to the pre-disturbance level (within the current limits), as well as suppressing the voltage swings that may activate the protection circuitry and prematurely trip the turbine. 6.2.1 Three-phase fault To study system response to a large disturbance, the same symmetrical fault  study as was used in Chapter 5 is presented here, with more detail. In this study, at t -1 s a fault  is assumed upstream in one of  the transmission lines. This fault  is cleared after  0.15s by disconnecting the faulted  line. The transient responses of  the system with different controllers are plotted in Figures 6.17 through 6.23. We analyzed the performance  of  various controllers and their ability to regulate voltage at the PCC in Chapters 4 and 5. For consistency, the voltage transient observed at the PCC is shown again in Figure 6.17. This figure  shows that the final  proposed ALQRCG controller outperforms  the PFC and LVC basic schemes, as well as providing a superior transient response, compared to the standard PID controller employed in the proposed supervisory scheme. For evaluating the performance  of  different  controllers, two time intervals are of  importance, one during the fault  and another after  the fault  has been cleared. During the fault,  the system is stressed by a disturbance, which is also evident from  the transients observed in the real power output of  each individual WT, as shown in Figure 6.18, as well as of  the wind farm,  as shown in Figure 6.19. However, the major differences  among the controllers are found  in terms of  the reactive power provided during and after  the fault,  shown in Figures 6.20 through 6.23. As can be seen in these figures,  the proposed control provides the most reactive power support during the fault,  as well as better damping after  the fault. 0.92 1.2 time(sec) Figure 6.17: Voltage transient observed at the PCC due to the fault. 0.03H 0.025 0.02 ....... W T 1 W T 2 - - - W T 3 0.03 0.025 0.02 LVC 1.2 time(sec) Figure 6.18: Real power output at each WT due to the fault. PFC '°S.9 1 1.1 1.2 1.3 1.4 1.5 time(sec) 0.015r 0 .01 • 0.005-0 • -0.005: 0.015r • WT1 - WT2 WT3 LVC 0.01 0.005 0 -0.005 —vw— "Www 3fSK? smmmtntMAw 1.2 time(sec) Figure 6.20: Reactive power output at each WT due to the fault. ( - - - WT1 — WT2 WT3) 0.95 0.9 0.85 0.95 0.9 : 0.85 0.95 PFC • WT1 WT2 WT3 " I V "'•'f i/.'ivii'.^ '.'i yjtti i v awwuwj t if. LVC 0.9-f : 11» UtUtKfitf£K&'MVi  HWWWU1 MitttVlttl  L'.iVAViV PID - supervisory A ' : vn^WmwMMWU'.U'.s: i:u;mnu tuivt tr-.-'.'I'.'l'.T.U'.'. 1.2 time(sec) Figure 6.21: Voltage transient observed at the terininal of  each WT due to the fault. 3 Q. 3 Q. "3 0 1 0) 5 o Q. a) > o to 0) EC 0.04-0.02-0 = -0 .02 -" V V ^ LVC 1.2 time(sec) Figure 6.22: Reactive power output from  the wind farm  due to the fault. LVC . 1.5 Q. E' E 'x CO E T3 C 03 0 Q. 1 0) U) L_ 0> o Q. <1) > O CD CD CC PID - supervisory 0 0.01 0.005 / \ \ / I „ \ 1 1 1 ALQRCG 8 1.1 1.2 time(sec) 1.3 1.4 1.5 Figure 6.23: Reactive power set-point and maximum at each WT due to the fault. ( - - -WT1 WT2 WT3, j=1,2,3) 6.2.2 Summary Several observations can be made with regard to the performance  of  different  controllers. When the PFC mode was used, the voltage at the PCC depended on the reactive power balance of  the network. When the network impedance was changed, the reactive power level was also changed. The PFC mode enabled zero reactive power consumption by the wind farm  at every event. However, after  the fault,  the voltage at the PCC settled down below the pre-fault  level, due to weaker transmission line impedance. When the LVC mode was in use, the voltage response at the PCC was significantly  improved because of  its partial reactive power contribution. A summary of  the steady state voltages at the PCC after  the fault  is given in Table 6.2, where it is shown that only the supervisory controllers can restore the voltage after  the fault  with zero steady state error. During the transient, the LVC and PID-supervisory controls showed voltage recovery rates of  0.26/sec and 0.44/sec, respectively, while the ALQRCG controller demonstrated a voltage recovery rate of  0.875/sec. An important observation can be made regarding the proposed ALQRCG controller. In particular, although the proposed controller was tuned for  a range of  operating conditions defined  by normal variation of  the load, the overall supervisory ALQRCG controller demonstrated outstanding performance,  even under a severe disturbance such as fault,  with significantly  improved transient performance  and faster  damping of  the voltage swings. TABLE 6.2 COMPARISONS OF VOLTAGE CONTROL PERFORMANCE (PU) Set-point PFC LVC PID- supervisory & ALQRCG 0.90497 0.8918 0.9025 0.90497 Deviation (%) 1.32 0.25 0 (Deviation: deviation between the set-point and the steady-state value in percent) Chapter 7 Conclusion and Future Work 7.1 Conclusion This thesis addresses the operation of  wind power generation systems and their contribution to' voltage control in the network. Voltage control in the system becomes particularly important when wind energy penetration is high. We developed a detailed model of  a candidate industrial site with multiple wind turbines and used it to perform  simulation studies and evaluate alternative control solutions. The goal of  our investigation was to make use of available wind turbine technology, namely the variable-speed doubly-fed  induction generator with power electronic converters, to actively participate in improving voltage control in the system without using additional compensating devices. To ensure reliable operation of  the proposed control scheme, the operating-point-dependent reactive power limit of  each wind turbine was taken into account. The overall supervisory voltage control scheme and the control design methodology developed in this thesis can be applied to larger wind farms  and network configurations. In Chapter 3, we presented the model of  the grid-connected wind farm  to investigate the impact of  wind power on power system dynamics. The overall component modules were represented in the ^^-synchronous reference  frame.  In Chapter 4, based on the models presented in Chapter 3, the transfer  functions  to be used in the design of  the VSC controllers were presented. In Chapter 5, we proposed an innovative supervisory control scheme that allows using a wind farm  for  regulating voltage at a point of  common coupling (PCC) that is remote and/or different  from  the wind farm  grid-connection point. The supervisory scheme acts as a distributed controller and makes use of  reactive power that is available from  all participating wind turbines, while taking into account time-varying operating conditions and the limits of individual turbines to ensure their safe  and reliable operation. Through numerous simulation studies, this supervisory scheme, even with a generic PID controller, outperformed  the traditional control modes of  the wind turbines, such as reactive power support and/or local voltage support, in cases of  both small and large disturbances. As the next step in this research, we investigated several advanced control approaches that would work together with the supervisory control scheme. To enable a linear and robust control framework,  the overall system was represented by a set of  reduced-order linear systems that cover an operating range of  interest determined by variations of  the load. To make the control design applicable to realistic systems, with noise and disturbances in the measured signals, we considered an observer. Several control solutions based on the linear-quadratic-regulator design were investigated. The best controller was designed using the linear-quadratic-regulator and linear-matrix-inequality approach, which takes into account cross-coupling between the state and control inputs, and minimizes an upper bound on the cost-guaranteed objective function. In Chapter 6, we carried out detailed simulation studies that considered the impact of  wind speed variations, load variations, and faults  in the network on the voltage at a point of common coupling. For the case system considered in this thesis, small disturbances such as wind speed and load variations were not likely to represent an objectionable voltage control problem, even when the wind farm  was equipped with traditional controllers. However, the faults  resulted in much larger disturbances that should be mitigated, if  possible. The proposed final  controller was further  compared with traditional control techniques and shown to provide an improved transient response under small disturbances as well as faults.  In particular, the final  controller achieved a faster  response and better-damped behaviour during and after  the fault.  The achieved response is less likely to trigger the protection circuitry and therefore  more likely to ride through the fault  in a favourable  way. 7.2 Future Work The modelling work in this thesis was not validated against the hardware system, which would be of  definite  value. Such validation and tests could be possible if  an industrial partner with an appropriate facility  becomes involved in this research. However, the wind energy facilities  in British Columbia are only in the planning stage at present. Contacts with other provinces may perhaps lead to potential industrial collaborators. Our research goal is to encourage active use of  wind turbines and wind farms  in power system operations. As an extension of  the supervisory controller, more advanced linear controllers, such as a gain scheduling and/or nonlinear controller, could be studied to further improve the system's performance. Beyond the voltage control problem, which was the primary focus  .of  this thesis, the overall theme of  making wind farms  active participants in improving the operation of  power systems can be extended. An area of  application of  a similar supervisory control scheme would be to use the wind farms  for  frequency  control, similar to conventional generation stations, by utilizing real power instead of  (or in addition to) reactive power. This approach would be particularly important for  places with very large wind power penetration or islands with relatively small total inertia of  conventional generators. However, it may require not extracting the maximum possible energy from  the available wind at all times. 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Appendix A System Parameters and Operating Conditions Table A. 1 Wind Power Model Parameters [8] (all quantities are given in per unit on 4MVW base) 1/2 pAr 0.0145 Fixed constant K b 69.5 i j aij 4 4 4.9686e-010 4 3 -7.1535e-008 4 2 1.6167e-006 4 1 -9.4839e-006 4 0 1.4787e-005 3 4 -8.9194e-008 3 3 5.9924e-006 Power coefficient 3 2 -1.0479e-004 3 1 5.705 le-004 Cp{9,X) 3 0 -8.6018e-004 2 4 2.7937e-006 4 4 2 3 -1.4855e-004 = X X i=0j=0 2 2 2.1495e-003 2 1 -1.0996e-002 2 0 1.5727e-002 where 2<A<13 1 4 -2.3895e-005 1 3 1.0638e-003 1 2 -1.3934e-002 1 1 6.0405e-002 1 0 -6.7606e-002 0 4 1.1524e-005 0 3 -1.3365e-004 0 2 -1.2406e-002 0 1 2.1808e-001 0 0 -4.1909e-001 Table A.2 Turbine Controller Parameters [8] (all quantities are given in per unit on 4MVW base) Pitch controller gains KPP 150 Kip 25 Actuator time constant T p (second) 0.01 Time constant T A pc 0.05 Torque controller gains Kpt 3 K it 0.6 Pitch compensator gains Kpc 3 Kic 30 Pitch angle limtation ^max(deg) 27 0mm ( d e g ) 0 Power maximum and minimum pmax rt 1 pmin rt 0.1 Cut-in wind speed (m/s) vw , 3.5 Cut-off  wind speed (m/s) 25 Table A.3 Two-Mass Rotor Model Parameters [8] (all quantities are given in per unit on 100MW base) Rotor inetia constant H t 0.1716 Generator inertia constant H r 0.036 Shaft  stiffness Ktg 11.868 Shaft  damping coefficient Dtg 0.06 Reference  rotor speed oI tef 1.15 Base rotor speed fyfrase 1.335 Table A.4 DFIG and DC Link Parameters [6] (all quantities are given in per unit on 100MW base) Stator resistance Rs 0.1285 Rotor resistance Rr 0.1519 Stator inductance Ls 2.82 Rotor inductance Lr 2.9535 Magnetizing inductance Lm 110.54 DC link capacitance Cdc 0.01 Reference  dc voltage ref Vdc 1 Table A.5 Maximum Operating Limit of  VSC (all quantities are given in per unit on 100MW base) and Qn 0.012 Table A.6 PI Controller Gains of  VSC Proportional gain Integral gain Rotor-side converter PI1 and PI3 0.0426 59.853 PI2 and .PI4 l _ 2.8013 39.376 Grid-side converter PI5 and PI6 0.0018 0.6436 DC link PI7 0.03155 2.3873 Local voltage controller 0.0137 6.7057 Table A.7 Line Parameters [34] (all quantities are given in per unit on 100MW base, 132kV base) Resistance Inductance Capacitance Transformer  (TR1,TR2,TR3) 0.0092 0.233 Transfomer  (TR4) 0.0216 0.539 Transmission line 0.3775 0.6689 0.024 Cable 0.0378 0.0669 0.0016 Grid-side filter 0.000435 0.002696 Table A.8 Thyristor Excitation system [13] Amplifier  gain Tt 0.05 T 0.2 Exciter gain T ±ex 10 Thyristor gain K t 10 Filter gain TR 0 Table A.9 Synchronous Generator Parameters [13] (all quantities are given in per unit on 100MW base, 132kV base) Stator resistance Ra 0.000135 Stator d-axis inductance Ld 0.0815 Stator ^-axis inductance L1 0.0793 J-axis magnetizing inductance Lmd 0.0748 g-axis magnetizing inductance Lmq 0.0725 g-axis mutual inductance Lamq 0.0815 J-axis mutual inductance ^amd 0.0793 q-axis mutual inductance Lkqmq 0.0825 ^-axis mutual inductance Lkdmd 0.1052 g-axis mutual inductance Lfmq 0.0822 Field resistance R f 0.00027 Field inductance L f 0.0074 d-axis damper winding resistance Rkd 0.0013 d-axis  damper winding inductance Lkd 0.0077 g-axis damper winding resistance Rkq 0.000279 ^r-axis damper winding inductance Lkq 0.0327 Electrical angular speed coe 1 Table A. 10 Operating Conditions Case nomina l local load i m p e d a n c e BUS V(PU) P(PU) Q(PU) Grid 9 0.94860 0.07338 0.001595 8 0.93425 0.07303 0.049410 Load 7 (PCC) 0.90497 TR 0.07284 0.044610 WF 0.07738 0.002935 Total 0.15022 0.047555 Load Resistance (• Rload) Reactance ( Xload) 5 1.6 WF BUS V(PU) P(PU) Q (PU) 1 0.90936 0.02671 0 2 0.90912 0.02588 -0.0001888 3 0.90866 0.02671 0 4 0.90843 TR 0.02588 -0.0001888 Cable 0.02586 0.00110100 Total 0.05174 0.00091190 5 0.90728 0.02671 0 6 0.90704 TR 0.02588 -0.0001891 Cable 0.05166 0.00210200 Total 0.07755 0.00191200 Case 50% decrease of the local load impedance BUS V(PU) P(PU) Q (PU) Grid 9 0 . 9 4 8 6 0 0 .18900 0 .072600 8 0 . 9 0 5 8 4 0 .18330 0 . 1 1 0 3 0 0 Load 7 ( P C C ) 0 .84259 TR 0 .18210 0 . 0 8 0 1 9 0 WF 0 .07766 0 .002245 Total 0 .25976 0 . 0 8 2 4 3 5 Load Resistance (• Rload) Reactance . ( Xload  ) 2.5 0 .8 WF BUS V(PU) P (PU) Q (PU) 1 0 .84725 0 .02671 0 2 0 . 8 4 7 0 0 0 .02599 - 0 . 0 0 0 2 1 9 3 3 0 .84651 0 .02671 0 4 0 . 8 4 6 2 6 TR 0 .02599 - 0 . 0 0 0 2 1 9 7 Cable 0 .02596 0 . 0 0 0 8 9 0 7 Total 0 .05195 0 . 0 0 0 6 7 1 0 5 0 . 8 4 5 0 4 0 .02671 0 6 0 .84479 TR 0 .02599 - 0 . 0 0 0 2 2 0 5 Cable 0 .05187 0 . 0 0 1 6 6 5 0 0 Total 0 .07785 0 . 0 0 1 4 4 4 5 0 Case 50% increase of the local load impedance BUS V(PU) P(PU) Q(PU) Grid 9 0.94860 0.02793 -0.018460 8 0.94900 0.02782 0.0310500 Load 7 (PCC) 0.92565 TR 0.02778 0.0300000 WF 0.07728 0.0031640 Total 0.10506 0.0331640 Load Resistance (• Rload) Reactance ( Xload) 7.5 2.4 WF BUS V(PU) P (PU) Q(PU) 1 0.92996 0.02671 0 2 0.92972 0.02584 -0.0001800 3 0.92928 0.02671 0 4 0.92904 TR 0.02584 -0.0001803 Cable 0.02582 0.00117200 Total 0.05166 0.00099170 5 0.92792 0.02671 0 6 0.92768 TR 0.02584 -0.0001808 Cable 0.05159 0.00224800 Total 0.07743 0.00206760 Appendix B Voltage Source Converter Controller Design The idea of  the pole-placement techniques is to make an open-loop system behave as the desired closed-loop system. Suppose that the desired closed-loop characteristic equation for  a system with a proportional-plus-integral (PI) controller can be expressed [56] as where Gp (5) is the first-order  transfer  function  of  a system, and Gc (s)  is a PI controller. The objective with the pole-placement technique is to make the closed-loop characteristic equation (B.l) behave as the desired closed-loop characteristic equation such that 1 + G C ( 5 ) G / , ( 5 ) = 1 + -kp+-L Gp{s) (B.L) Acl(s)  = s2 +2gcons + co2 (B.2) The gain coefficients  of  a PI controller can be obtained by comparing (B.l) with (B.2). B.l PI controller design for  PI2 and PI4 To illustrate the conclusions drawn in this section, the Bode diagrams of  the transfer  function of  the open-loop and closed-loop systems are shown in Figure B.l. Bode diagram 10 10 Frequency (rad/sec) Figure B.l: Bode diagrams of  transfer  function  of  the open-loop and closed-loop system. The transfer  function  of  (3.23) with the numerical values is G2(s)  = Ga(S)  = 40.6339 5 + 6.1723 (B.3) The closed-loop characteristic equation for  (B.3) with a PI controller is described as Ad(s)  = s1 +(6A723  + 40.6339k p)s + 40.6339k i (B.4) The natural frequency  con is chosen as 40 rad/sec, where the phase approaches a constant value and corresponds to a 20 dB per decade change in the magnitude. The damping ratio C, -1 .5 is selected by sweeping from  0.1, which starts to provide a positive proportional gain kp . With this damping ratio, less than 5% overshoot to the step response of  the closed-loop system is obtained. Thus, the desired closed-loop characteristic equation is Ac!(S)  = s2 +2£A>ns + A>2 = s 2 + 1 2 0 s + 4 0 2 (B.5) By comparing (B.4) and (B.5), the gain coefficients  kp and k t are obtained as 2.8013 and 39.3760, respectively. The Bode diagram in Figure B.l shows the improving bandwidth and the improvement of  phase margin for  the closed-loop system. Due to the increased bandwidth, the closed-loop system now features  faster  step response time as seen in Figure B.2. Step response of the open-loop Step response of the closed-loop 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 time (sec) Figure B.2: Comparison of  the step response of  the open-loop and the closed-loop system. B.2 PI controller design for  PI1 and PI3 The transfer  function  of  (3.27) or (3.31) with the numerical values is G{  (s) = G3 (S)  = 1.1646 + 0.060 Is (B.6) Bode diagrams of  the transfer  function  of  the open-loop and closed-loop systems are shown in Figure B.3 to illustrate the conclusions drawn in this section. Bode diagram 60 m" 40 -o CD T3 3 20 C n> CO 2 0 -20 90 open-loop closed loop Frequency (rad/sec) Figure B.3: Bode diagrams of  transfer  function  of  the open-loop and closed-loop system. The closed-loop characteristic equation of  (B.6) with a PI controller can be expressed as Ad{s)  = s2 + (1 + 0.0601*; +\A646k p)s + ^ ^ p -  (B.7) d 0.060life, v P 1 nnfinifr By choosing con as 165 rad/sec where the phase approaches a constant value and corresponds to a 20 dB per decade change in the magnitude, the desired closed-loop characteristic equation is ,4c /O) = s 2 +330^y + 1652 (B.8) By comparing (B.7) and (B.8), the coefficients  of  the PI controller are computed as kp =0.0426 and k t =59.853. The damping ratio £ = 5.5 is selected by sweeping from  5.0, which yields a positive proportional gain. The Bode diagram plotted in Figure B.3 shows the feature  of  pole-zero cancellation, and its closed-loop step response is shown in Figure B.4. Step response of the closed-loop Figure B.4: Step response of  the closed-loop system. B.3 PI controller design for  PI5 and PI6 The transfer  function  of  (3.35) with the numerical values is G5(s)  = G6(s)  = 139830 5 + 60.8276 (B.9) Figure B.5 shows a comparison of  the Bode diagrams of  the transfer  function  of  the open-loop and closed-loop systems. Bode diagram open-loop closed loop 10 10 Frequency (rad/sec) Figure B.5: Bode diagrams of  transfer  function  of  the open-loop and closed-loop system. The closed-loop characteristic equation of  (B.9) with a PI controller can be expressed as Arf(s)  = s 2 + (60.8276 + 139830^)^ + 139830^- (B.10) To have a desired closed-loop system, a>n is chosen as 300 rad/sec, where the phase becomes constant and corresponds to a 20 dB per decade change in the magnitude. The damping ratio is chosen as 2 by sweeping from  1. With this damping ratio, the overshoot in the closed-loop step response is less than 5%, as can be seen in Figure B.6. With these parameters, the desired closed-loop characteristic equation becomes By comparing (B.10) and (B.l 1), the proportional and integral gains are computed as 0.0081 and 0.6436, respectively. Ad{s)  = s2+1200^5 + 300: ,2 (B.ll) 2500 ]| 2000 -g 1500 Step response of the open-loop = 1000 I 500 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 time (sec) 1.15 Step response of the closed-loop <u T3 £ < 0 3 0.5 3 Q. CL 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 time (sec) Figure B.6: Step response of  the open-loop and closed-loop system. B.4 PI controller design for  PI7 The transfer  function  of  (3.40) is GJ(S)  = 376.9911 (B.12) Figure B.7 shows a comparison of  the Bode diagrams of  the transfer  function  of  the open-loop and closed-loop systems. Bode diagram Frequency (rad/sec) Figure B.7: Bode diagrams of  transfer  function  of  the open-loop and closed-loop system. The closed-loop characteristic equation of  the nominal loop (B.12) with a PI controller is as follows: Ad  (5) = s 2 +37699.11k ps + 37699.1 \k t (B. 13) Since the controllers for  the grid-side filter  in the previous section B.3 have been designed up to a frequency  range of  300 rad/sec, the natural frequency  con is chosen as 300 rad/sec. With this parameter, the desired closed-loop characteristic equation becomes 4,/O) = S 2 + 6 0 0 ^ + 3002 (B.14) The damping ratio is chosen as 1.6, where the step response of  the closed-loop system features  less than 5% overshoot, as shown in Figure B.8. With this damping ratio, the proportional and integral gains are 0.03155 and 2.3873, respectively. Step response of the closed-loop 0.8 3" D. J 0.6 Z3 Q. | 0.4 0.2 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 time (sec) Figure B.8: Step response of  the closed-loop system. / Appendix C Local Voltage Controller Design To design a local voltage controller, there is a need to find  to a transfer  function  from  the voltage to the reactive power at the wind turbine terminal. In this thesis, WT3 in Figure 3.1 is selected as a candidate model for  our design of  a local voltage controller since its terminal voltage is close to the terminal voltage of  the wind farm.  To have a representative model, the rest of  system seen from  the WT3 is modelled as an equivalent RL load model. Thus, this representative model has system input, which is reactive power, and system output, which is the magnitude voltage from  this equivalent load. By applying the balanced model-order reduction technique, the 3rd-order reduced transfer  function  is obtained as ^ , N -0.002786s3 + 204.4s2 + 1277000s + 228900000 Gp(s)  = = = (C.l) s +10880s + 2784000s+ 163600000 Bode diagrams of  the full-order  model (23th) and the reduced-order model (3rd) are shown in Figure C.l. The reduced-order transfer  function  approximates the full-order  model in the frequency  range of  interest. Thereafter,  this reduced-order model is considered to represent the plant. Bode Diagram • Reduced (3rd) -Full(23th) 10 10 10 Frequency (rad/sec) Figure C. 1: Bode diagrams of  the full-order  model (23 t h) and the reduced-order model (3rd). In this thesis, we design a PI controller, since it is frequently  used in industry. The compensated loop transfer  function  can be expressed [56] as Gc(s)G p(s)  = kpS  "t* kj Gp(s) (C.2) where Gc(s)  stands for  the controller, Gp(s)  is a nominal plant, and kp and ki are proportional and integral gain, respectively. As in Figure C.2, it is assumed that the compensated Nyquist diagram is to pass through the point 1Z(-180° +(/>m), for  the frequency  a\ , to achieve the phase margin (f>m [56], Or, Gc{jco [)GJj(D [)  = \Z{-m a + 0m) (C.3) Gc(s)Gp(s) Re Figure C.2: Nyquist plot of  a compensated loop transfer  function. If  the angle of  a controller Gc(jo\)  is denoted by 6, then from  (C.3), 9 = Z.G C (M)  = -180° + 4>m ~ ^Gp (M) (C.4) From Gc(ja\)  and (C.3), k-Gc (.ja\  ) = k p - j - ^ = \Gc <Ja\  )| (cos 9+j sin 9) (C.5) where, from  (C.3), \GC(M)\  = Gp(M)  | (C.6) From (C.5), equating real part to real part yields cos 6 k p \GPim)\ (C.7) and equating imaginary part to imaginary part yields o\ sin# (C.8) Gp(M)\ With the chosen design specifications  such that the settling time rs = 0.075 seconds and the phase margin is 85° , the phase-margin frequency,  or the gain-crossover frequency,  is g calculated as o\=— = 9.3321 rad/sec, to yield a specified  settling time r,, . The Ts t a n Qm coefficients  of  kp and k t are then calculated as 0.0137 and 6.7057, respectively. The anti-k-windup gain can be computed as ka=— [55], k p Figure C.3 shows the compensated system Bode diagram, and Figure C.4 shows the step response of  the open-loop and the closed-loop system. In this thesis, the LVC mode is implemented as shown in Figure C.5. Figure C.3: Bode diagrams of  transfer  function  of  the open-loop and closed-loop system. Figure C.4: Step response of  the closed-loop system. set • WT,1 V, WT,1 . set WT,2" V, WT,2 ' V, set WT,3' WT,3 + + + \ i, 1 s ki I ] ) > 1 s ki > W3 ± y ) — K Z ) ^ 1 s ki * ka + ka + k a ' + Figure C.5: Block diagram of  the PI controller with distributed anti-windup scheme. Appendix D Matlab Script Files % Written by Hee-Sang Ko. % UBC Power and Control Lab. in May 2006. % In these codes, the integral state is added in the last row. D.l Matlab Code for  LQRCG and ALQRCG % clear all load systemA load systemB load systemC A=aA; B=aB; C=aC; [n,m]=size(A); [nn,mm]=size(B); % % Need the modification  for  the integral action Ata=zeros(n+1 ,n+1); Atb=zeros(n+1 ,n+1); Atc=zeros(n+1 ,n+1); Ata((l:n),[l:n])=aA; Atb((l:n),[l:n])=bA; Atc((l :n),[l :n])=cA; Ata(n+1,[ 1 :n])=-aC; Atb(n+1,[ 1 :n])=-bC; Atc(n+1,[ 1: n])=-cC; Bta=[aB; -aD]; Btb=[bB; -bD]; Btc=[cB; -cD]; Cta=zeros(2,n+l); Cta(l,l)=l; Cta(2,n+l)=l; Ctb=Cta; Ctc=Cta; Dta=zeros(2,l); % % nstate=size(Ata, 1); ncon=size(Bta,l); % Q=eye(size(Ata,2))* 1; Q(nstate,nstate)=70; Q(1,1)=10; R=eye(size(Bta,2))*2.2; % For ALQRCG 4 Qhalf  = sqrtm(Q); Rhalf=  sqrtm(R); N = ones(nstate,l)*0.55; Posi=Q-N*inv(R)*N'; % check for  the condition of  Q-N'RA-1N>=0 % Start LMI % Define  the problem variables and matrix inequality constraints setlmis([]) % Define  and describe the matrix variables X = lmivar(2, [1 nstate]); Y = lmivar(l, [nstate 1]); Z = lmivar(l, [nstate 1]); % Define  the individual LMIs. See pp.8-11 (Mathworks). LMI_Sys_l = newlmi; lmiterm([LMI_Sys_ 1 1 1 Y], Ata, 1, 's'); %(1,1) block: A1*Y + Y*A1' lmiterm([LMI_Sys_ 1 1 1 X], Bta, -1, 's'); %(1,1) block: -B1*X - X'*A1' lmiterm([LMI_Sys_l 1 2 -Y], 1, Qhalf);  %(1,2) block: Y*QlA(l/2) lmiterm([LMI_Sys_l 1 3 -X],l, Rhalf);  %(1,3) block: X'*RlA(l/2) lmiterm([LMI_Sys_l 2 2 0], -1); %(2,2) block: -eye(3,3) lmiterm([LMI_Sys_l 3 3 0], -1); %(3,3) block: -eye(l,l) LMI_Sys_2 = newlmi; lmiterm([LMI_Sys_2 1 1 Y], Atb, 1, 's'); %(1,1) block: A1*Y + Y*A1' lmiterm([LMI_Sys_2 1 1 X], Btb, -1, 's'); %(1,1) block: -B1*X - X'*A1' lmiterm([LMI_Sys_2 1 2 -Y], 1, Qhalf);  %(1,2) block: Y*QlA(l/2) lmiterm([LMI_Sys_2 1 3 -X],l, Rhalf);  %(1,3) block: X'*RlA(l/2) lmiterm([LMI_Sys_2 2 2 0], -1); %(2,2)block: -eye(3,3) lmiterm([LMI_Sys_2 3 3 0], -1); %(3,3) block: -eye(l.l) LMI_Sys_3 = newlmi; lmiterm([LMI_Sys_3 1 1 Y], Ate, 1, 's'); %(1,1) block: A1*Y + Y*A1' lmiterm([LMI_Sys_3 1 1 X], Btc, -1, 's'); %(1,1) block: -B1*X - X'*A1' lmiterm([LMI_Sys_3 1 2 -Y], 1, Qhalf);  %(1,2) block: Y*QlA(l/2) lmiterm([LMI_Sys_3 1 3 -X],l, Rhalf);  %(1,3) block: X'*RlA(l/2) lmiterm([LMI_Sys_3 2 2 0], -1); %(2,2) block: -eye(3,3) lmiterm([LMI_Sys_3 3 3 0], -1); %(3,3) block: -eye(l,l) % Coupling constraint LMI_Couple = newlmi; lmiterm([-LMI_Couple 1 1 Z],l,l); % (1,1) block: [Z I; I Y] >0 lmiterm([-LMI_Couple 1 2 0],1); % (1,2) block: [Z I; I Y] >0 lmiterm([-LMI_Couple 2 2 Y],l,l); % (2,2) block: [Z I; I Y] >0 % Positive definition  constraint LMI_YPos = newlmi; lmiterm([-LMI_YPos 1 1 Y], 1, 1); % Y > 0 from  [Z I; I Y] >0 % Stroe the internal representation of  the LMI system (pp. 8-6) lmisys = getlmis; % Solve ns = decnbr(lmisys); for  j=l:ns [Xj, Yj, Zj] = defcx(lmisys,  j, X, Y, Z); % c(j) = trace(Zj); c(j) = trace(Zj)-trace(N*Xj)-trace(Xj'*N'); end options = [le-5 0 0 0 0]; [copt,xopt] = mincx(lmisys,c,options); dispC ') disp(") disp('The optimized variable matrix X is ...'); Xstar = dec2mat(lmisys,xopt,X) disp('The optimized variable matrix Y is ...'); Ystar = dec2mat(lmisys,xopt,Y) disp('The optimized variable matrix Z is ...'); Zstar = dec2mat(lmisys,xopt,Z) disp(") disp('The robust-optimal gain is ...'); Kstar = Xstar*inv(Ystar); dispO % No cross-product term (LQRCG) % Cross-product term (ALQRCG) D.2 Matlab Codes for  LQRS and ALQRS % — clear all load systemA load systemB load systemC A=aA; B=aB; C=aC; [n,m]=size(A); [nn,mm]=size(B); % % Need the modification  for  the integral action Ata=zeros(n+1 ,n+1); Atb=zeros(n+l ,n+l); Atc=zeros(n+1 ,n+1); Ata((l :n),[l :n])=aA; Atb((l:n),[l:n]j=bA; Atc((l:n),[l:n])=cA; Ata(n+l,[l:n])=-aC; Atb(n+l,[l :n])=-bC; Atc(n+1, [ 1: n])=-cC; Bta=[aB; -aD]; Btb=[bB; -bD]; Btc=[cB; -cD]; Cta=zeros(2,n+l); Cta(l,l)=l; Cta(2,n+l)=l; Ctb=Cta; Ctc=Cta; Dta=zeros(2,l); % : % .... nstate=size(Ata, 1); ncon=size(Bta, 1); % Q=eye(size(Ata,2))*l; Q(nstate,nstate)=70, Q(l,l)=10; R=eye(size(Bta,2))* 1.2; % For ALQRS -> 0.955 Qhalf  = sqrtm(Q); Rhalf  = sqrtm(R); N = ones(nstate,l)*0.55; Q-N*inv(R)*N' % Start LMI % Define  the problem variables and matrix inequality constraints setlmis([]) % Define  and describe the matrix variables X = lmivar(2, [1 nstate]); Y = lmivar(l, [nstate 1]); M = lmivar(2, [1,1]); % Define  the individual LMIs. See pp.8-11 (Mathworks). LMI_Sys_l = newlmi; lmiterm([LMI_Sys_l 1 1 Y], Ata, 1, 's'); lmiterm([LMI_Sys_l 1 1 X], Bta, -1, 's'); lmiterm([LMI_Sys_l 1 2 -Y], 1, Qhalf); lmiterm([LMI_Sys_l 1 3 -X],l, Rhalf); lmiterm([LMI_Sys_l 2 2 0], -1); lmiterm([LMI_Sys_l 3 3 0], -1); %(1,1) block: A1*Y + Y*A1' %(1,1) block: -B1*X-X'*A1' %(1,2) block: Y*QlA(l/2) %(1,3) block: X'*RlA(l/2) %(2,2) block: -eye(3,3) %(3,3)block: -eye(l,l) LMI_Sys_2 = newlmi; lmiterm([LMI_Sys_2 1 1 Y], Atb, 1, 's'); lmiterm([LMI_Sys_2 1 1 X], Btb, -1, 's'); lmiterm([LMI_Sys_2 1 2 -Y], 1, Qhalf); lmiterm([LMI_Sys_2 1 3 -X],l, Rhalf); lmiterm([LMI_Sys_2 2 2 0], -1); lmiterm([LMI_Sys_2 3 3 0], -1); %(1,1) block: A1*Y + Y*A1' %(1,1) block: -B1*X - X'*A1' %(1,2) block: Y*QlA(l/2) %(1,3) block: X'*RlA(l/2) %(2,2) block: -eye(3,3) %(3,3)block: -eye(l,l) LMI_Sys_3 = newlmi; lmiterm([LMI_Sys_3 1 1 Y], Ate, 1, 's'); lmiterm([LMI_Sys_3 1 1 X], Btc, -1, 's'); lmiterm([LMI_Sys_3 1 2 -Y], 1, Qhalf); lmiterm([LMI_Sys_3 1 3 -X],l, Rhalf); lmiterm([LMI_Sys_3 2 2 0], -1); lmiterm([LMI_Sys_3 3 3 0], -1); % Subject function,  LMI #4: % [ M Sqrt(R)X ] % [ ] > 0 % [X'Sqrt(R) Y ] LMI_Sys_4 = newlmi; lmiterm([-LMI_Sys_4 1 1 M], 1, 1); lmiterm([-LMI_Sys_4 1 2 X], Rhalf,  1); lmiterm([-LMI_Sys_4 2 2 Y], 1, 1); %(1,1) block: A1*Y + Y*A1' %(1,1) block: -B1*X - X'*A1' %(1,2) block: Y*QlA(l/2) %(1,3) block: X'*RlA(l/2) %(2,2) block: -eye(3,3) %(3,3) block: -eye(l,l) % LMI #4: M % LMI #4: X'*sqrt(R) % LMI #4: Y % Stroe the internal representation of  the LMI system (pp.8-6) lmisys = getlmis; % Solve ns = decnbr(lmisys); for  j=l:ns [Xj, Yj, Mj] = defcx(lmisys,  j, X, Y, M); c(j) = trace(Q*Yj) + trace(Mj); %c(j) = trace(Q*Yj) + trace(Mj) - trace(N*Xj) - trace(Xj'*N'); end % No cross-product term (LQRS) % Cross-product term (ALQRS) options = [le-5 0 0 0 0]; [copt,xopt] = mincx(lmisys,c,options); disp(' ') disp(") disp('The optimized variable matrix X is ...'); Xstar = dec2mat(lmisys,xopt,X) disp('The optimized variable matrix Y is ...'); Ystar = dec2mat(lmisys,xopt,Y) disp('The optimized variable matrix M is ...'); Mstar = dec2mat(lmisys,xopt,M) disp(") disp('The robust-optimal gain is ...'); Kstar = Xstar*inv(Ystar) dispC) Appendix E Proof of Theorem Theorem: If  common positive definite  P exists that satisfies  the Lyapunov inequality (5.14) as in (E.l), (Ay- -Byk) 7 , P + P ( A y -Byk) + Q + k r R k <0 (E.l) then the cost function  J of  (5.11) is bounded by the scalar expression £^x(0) rPx(0) = £ ^ r ( x ( 0 ) P x ( 0 ) r ) =<r(X0P) asin(E.2): K = m m j ^ { J ° ( x r Q x + urRu)<fc  J<£[ / r (x (0 )Px (0 ) r ) ] = /r(XoP) (E.2) where j = l,...,N  where N  is the total number of  multiple systems. The variables Q and R are design parameters; Q > 0 is positive semi-definite  matrix, and R is positive definite symmetric matrix. The variable P is the Lyapunov matrix and is positive definite  such that P > 0. The variable X 0 is the expectation of  the covariance of  the stationary random initial vector such that £^x(0)x(0) r ] = X 0 . For proof  of  the theorem, the following  Definition  and Lemma are needed. Definition: Consider the system (5.1) and the cost function  (5.11). If  there exists a control law u and a positive scalar JG such that the closed-loop is stable and the closed-loop value of  the cost function  (5.11) satisfies  J < JQ , then Jq is said to be a guaranteed cost and u is said to be a guaranteed cost controller for  the system (5.1). Lemma [67]: The closed-loop system x = (A-Bk)x + Gw (E.3) for  given system matrices (A,B) is asymptotically stable if  and only if  there exists a positive definite  P > 0 satisfying ( A - B k f  P + P ( A - B k ) < 0 (E.4) where the state noise signal has zero mean £[w] = 0 and symmetric positive definite covariance matrix E T W W = R W > 0 . This lemma can be extended to the multiple systems situation. If  matrix Lyapunov inequality (E.5) is satisfied  by a common positive definite  P for  all the systems, the systems are guaranteed to have asymptotic stability within the linear regions for  which these multiple systems are defined,  i.e., ( A y - - B 7 - k ) r p + p ( A y - B 7 - k ) < 0 (E.5) Proof For convenience, the proof  is carried out based on a single system. The solution of  the minimization of  a cost function V = min (x r Qx + u r R u ) dt (E.6) can be found  from  utilizing the parameter optimization problem solved by the second Lyapunov method [63], [65] such that x r Q x + u r R u = E dt (E.7) where P is the Lyapunov matrix. Substitute u = -kx into (E.7) to obtain E(x TQx + xTk TRkx\  = E - - ( x ' W dt = E ( - X T  P X - X ^ P X ) (E.8) Substitute the system (5.1) with u = -kx into the right-hand side of  (E.8) to obtain xT  ^Q + k^Rkjx 7^ = E -xT  ( ( ( A - B k ) r + G w r ) P - P ( ( A - B k ) + Gw))x (E.9) Since i?[w] = 0, (E.9) can be rewritten as xT  ( (A• -Bk) r P + P ( A - B k ) j x = -E x r E x <0 (E.10) where E is ^Q + k ^ R k j , which is a positive definite  matrix. Thus, there exists a positive definite  P , and by Lemma, there exist stable ( A - B k ) as t —» Equation (E. 10) is then posed in a matrix Lyapunov inequality such that ( A - B k ) P + P ( A - B k ) + E < 0 (E.ll) Multiply each term in the Lyapunov inequality (E.ll) by the system state transition matrix Bk) t f o r ^ g i ef t a n c j e ( A B k V for  the right to obtain e ( A - B k ) r / p ( A _ _ B k ) e ( A - B k ) . + g ( A - B k f / ( A _ B k ) p . ( A - B k ) . + e ( A - B k ) % e ( A - B k ) , < Q (E.12) Equation (E; 12) can be simplified  to dt ' A-Bk)7"/pg(A-Bk)/ <0 (E.13) Take integral to (E.13) to obtain •"o dt dt<0 (E.14) Multiply each term of  (E.14) by the stationary random initial vector x(0) and x(0) and utilize expectation to obtain K x ( 0 f e ( A - B k ^ dt <0 (E.15) Equation (E.l5) can be rewritten as <E x(0) Px(0) (E-l 6) Since £,j^x(0)7, Px(0)J is scalar, (E.l6) can be rewritten as f ^ E x ) * <E x(0) Px(0) = E^tr  (x(O)Px(O)7, = tr (X 0 P) (E.17) with utilizing trace operation property such that tr(ABC)  = tr(CBA)  = tr{CAB).  To avoid the dependency of  the cost function  V on initial conditions, we assume the initial conditions are random variables with zero mean and a covariance equal to the identity such that £ [x(0)x(0) r ] = I and £[x(0)] = 0 (E.l 8) Thus, (E.17) can be reduced E\ J^(x rEx^  =E\ 0xr(Q  + k rRk)x)<# (E.l 9) and hence, (E.2) holds. Appendix F Effect  of the Cross-Product Terms It is well known from  the LQR theory that for  the linear time-invariant system x ( 0 = F x ( 0 + G u ( 0 , x ( 0 ) given (F.l) the determination of  an optimal control with the associated optimal cost function v = J ^ ° ^ x r Q x + u r R u + x r N u + u r N r x d t ( F . 2 ) is reduced as follows: u(t)  = u0(t)  + R~1l$Tx(t)  ( F . 3 ) where u * ( 7 ) = - k x ( 7 ) = R - 1 G ^ P x ( 0 , P is the positive definite  matrix, which is the solution of  the Riccati equation in the LQR problem [61], Now we show how linear quadratic problems with the cross-product terms arise when dealing with linearized systems [61]. Suppose that beginning at x(?0), the optimal control • * u 0 ( 7 ) drives the state along the trajectory x ( t ) . However, for  whatever reason, as shown in $ * Figure F.l the state at time t is not x (t)  but is x ( 7 ) + £ x ( 7 ) with Sx(t)  small. Intuitively, additional optimal control can be expected related to Sx(t). Thus, the expected optimal control with the additional optimal control S\(t)  can be described as follows: * * V (0 = v o ( 0 + *v (0 (F.4) 5x(0 x(0 +8x(0 Figure F. 1: Optimal and neighbouring optimal paths. Hence, comparing (F.4) with (F.3), the control signal £v (0 in (F.4) can be seen to be equivalent to ^ R ^ N ^ x j in (F.3), which is related to the cross-product terms. Therefore,  an important application of  the cross-product terms compounds to the case when an optimal control is in place for  a nonlinear system, but additional closed-loop regulation is required to maintain, as closely as possible, the optimal trajectory in the presence of  disturbances that cause small perturbations from  the trajectory. END 

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