Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2007-267417.pdf [ 5.16MB ]
Metadata
JSON: 831-1.0100578.json
JSON-LD: 831-1.0100578-ld.json
RDF/XML (Pretty): 831-1.0100578-rdf.xml
RDF/JSON: 831-1.0100578-rdf.json
Turtle: 831-1.0100578-turtle.txt
N-Triples: 831-1.0100578-rdf-ntriples.txt
Original Record: 831-1.0100578-source.json
Full Text
831-1.0100578-fulltext.txt
Citation
831-1.0100578.ris

Full Text

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, switchedcapacitors, 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  ^  Acknowledgments  xi 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 2.3.1  Impact of wind power at the distribution level  •  14  ••••  15  2.3.2 2.4 2.5  2.6  Impact of wind power at the transmission level  16  Voltage Control at Remote Locations Research Objectives and Approaches  16 ;  19  2.5.1  Problem statement  19  2.5.2  Research objectives  20  2.5.3  Proposed approach  20  Contributions  22  3. Model Description of Grid-Connected Wind Farm System 3.1  Study System  3.2  System Model Components  27  3.2.1  Mechanical components  27  3.2.2  Electrical components  31  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  3.3  3.4  •.  23 •••••  Conventional Voltage Control of Wind Turbine  24  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  Advanced LMI Representation of LQR  79  5.4  5.5  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  Summary of Control Gains  83  6. Simulation Studies 6.1  6.2  86  Small Disturbances  ,•  86  6.1.1  Wind speed variations  86  6.1.2  Load variations  91  6.1.3  Summary  96  Large Disturbances  97  6.2.1  Three-phase fault  97  6.2.2  Summary  7. Conclusion and Future Work 7.1  Conclusion  7.2  Future Work  102  103 103  •••  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  Appendix D  Matlab Script Files  134  Appendix E  Proof of Theorem  140  Appendix F  Effect of Cross-Product Terms  145  ••••  128  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  Table 6.1  Voltage Magnitude Deviations  97  Table 6.2  Comparisons of the Voltage Control Performance (pu)  Chapter 6 102  Appendix A Table A. 1  Wind Power Model Parameters Model Parameters  112  Table A.2  Turbine Controller Parameters  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  Table A. 10  Operating Conditions  .•••  ••••  113  115 116  Vlll  List of Figures  Chapter 1 Figure 1 1  Installed wind power capacity in the Europe, US, and the World  Figure 1.2  Cost of wind energy (Year 2000 US$) and cumulative  • • ••  2 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  Figure 2.2  Diagram  ratio  •••  17  depicting the wind farm interconnection impedance  18  Chapter 3 Figure 3.1  Wind power system considered for dynamic studies  25  Figure 3.2  Block  Figure 3.3  Variable speed wind turbine with DFIG  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  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  diagram showing subsystem input-output variables  25  •••  26  ••  33  4 Q  reference frame Figure 3.12  Block diagram of the voltage source controller modules  Figure 3.13  Turbine  power versus speed tracking characteristic  41 42  Figure 3.14  Block diagram of the rotor-side converter controller  43  Figure 3.15  Block diagram of the grid-side converter controller  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  ••  48  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 t h )  58  Figure 4.4 Figure 4.5  Bode diagrams of the reduced-order model (4 t h ) • Step response of the open-loop and the closed-loop reduced-order system  58 ^  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 with observer  gg  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  Figure 6.1  Wind speed (m/sec)  87  Figure 6.2  Real power set-point for each WT due to wind speed variation  gg  Figure 6.3  Real power output from each WT due to wind speed variation  gg  Chapter 6  Figure 6.4  Real power output from the wind farm due to wind speed variation  ^  Figure 6.5  Reactive power output from each WT due to wind speed variation  g^  Figure 6.6  Real power output from the wind farm due to wind speed variation  pQ  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 controllers  ••••••  93  Figure 6.11  Voltage transient observed at the WT terminals  93  Figure 6.12  Real power output from each WT  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  .•••••  94  due to the fault Figure 6.22  Reactive power output from the wind farm due to the fault  Figure 6.23  Reactive power set-point and maximum at each WT due to the fault  101 ^  Appendix B Figure B. 1  Bode diagrams of the transfer function of the open-loop 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  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  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  ^q ^  123  125 _ 127  Appendix C Figure C.l  Bode diagram of the full-order model (23 t h ) and the reduced-order model (3 r d )  Figure C.2  Nyquist plot of a compensated loop transfer function  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  Figure C.5  Block diagram of the PI controller with distributed anti-windup scheme  130  132  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 A B C, F  system matrix  D,H  feedforward matrix  A  system matrix of the reduced-order model  B  input matrix of the reduced-order model  C  output matrix of the reduced-order model  D  feedforward matrix of the reduced-order model  G  randomly chosen matrix  input matrix output matrix  observer gain matrix  Ke P S  positive-definite Lyapunov matrix  Y  change variable  Z  slack variable  positive-definite matrix  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  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 (m 2 )  C P (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  stator angular speed  cor  rotor angular speed  slip angular speed  <os  mechanical rotor angular speed cob  base angular speed (rad/sec)  tyjiase  base mechanical angular speed (rad/sec)  e  pitch angle (degree)  A  ratio of the rotor blade tip speed and wind speed  P  air density (kg/m 3 )  c  damping ratio  K  phase margin (degree)  Q/7' R W  observer design parameter  ¥  flux  Mathematical Symbols a <0  matrix a is negative definite  a>0  matrix a is positive definite  aeA  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 d B  first  order time derivative 201og 1 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  Superscripts ref  reference  set  set-point  T  transpose  -1  inverse  max min  maximum minimum  Subscripts abc a  phase abc  c  voltage source converter  ca  cable  cl  closed-loop  d  J-axes of reference frame  dc  direct current  f  field winding of synchronous generator  filter gc j  armature of synchronous generator  filter connected to the grid-side converter gain crossover frequency sub systems  k  damper winding of synchronous generator  load  load  max  maximum  mech  mechanical  min  minimum  m  magnetizing  o  steady state  <1  q-axes of reference frame  r s  rotor of generator  TL  transmission line  t  turbine rotor  tr  transformer  stator of generator  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  VSC V pcc  voltage at the grid voltage source converter 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  XVlll  Acknowledgments 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  Rest of World  Europe  o CO Q. (0 O United States 1982  1985  1988  1991  1994  1997  2000  2003  Year  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 online 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 cd) E> 0) c LL1  o TO CL CO  --  to O o  1980  1984  1988  1992  1996  2000  1000  O  2004  Year  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 E112 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 doublyfed 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.  Set-point level  a,  Monitored at Bus A  -M  132kV  BusA  TL  132kV line Load  TR  7  fault  33kV 0  20  40  60  80  100  Distance to the fault in kilometers  TL  Load 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 asflicker. 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 tapchanging 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, hand, when the X/R  which results in a significant increase in V cp. On the other  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) V  gr,d  jXI  higher X/R ratio  V  (b)  I  Zrid  lower X/R ratio  Figure 2.1: Voltage regulation based on the X/R ratio.  jXI  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 and the grid impedance Z GRI D  ZJ  ARM  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 WT  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 LQRLMI-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 modelorder 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-ridethrough 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 socalled 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  3.1  a .  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 inputoutput 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 v  wt3 •  WT2  WT3  0.69/ Q 34.5kVCj  .6  0.69/ 34.5kV  1,..., 9 - bus number WT1,... - wind turbines T1, T2,... - transformers C1, C2, C3-cables TL - transmission line PCC - point of common coupling  WT1  0.69/ 34.5kV C1  9km  9km  100km Local I • -i X Load T T  '  ' T4 "  Utility Grid (Syn. Gen.)  TL  (PCC)  Figure 3.1: Wind power system considered for dynamic studies.  Wind speed input(s) v g,Wtl  14 -< (3.17) t4  TL (3.15) t5  SG (3.18) 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 speed  Mechanical power  v qd,r  Terminal of WT P  Rotor-side converter  DC-link Grid-side converter  VSC Controller  Figure 3.3: Variable-speed wind turbine with DFIG.  g  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 constant. In steady-state for a lossless converter, Pfii depends on the power  3.2  ter  ter  in order to keep the dc-link voltage is equal to Pr and the rotor speed cor  absorbed or generated by the rotor-side converter.  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=^A rvlc p(A,d)  where Pmech  (3.1)  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; C p(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  vw-  ; Rt is the rotor radius in meters; cot is  the turbine rotor speed in rad/sec; and 6 is the blade pitch angle in degrees. The function C p (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 C P (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, are set at minimum pitch to maximize the mechanical power PME CH•  the blades  The variable PGBT is the  set-point value of the output of the wind turbine.  pset  rg  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 C p (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 actual ~1 base ' 1 pu ~ base ' 1 pu  (3.2)  1  The calculation of T mecij and X with the Pmecf, ^mech mech = Kpu V °>t J  1T  and Rt are then rewritten as follows:  Rtcot X =-  and  (3-3)  +A  1 2H t  1F +  1mech  D,'tg  1 s  °h,base  -o  1  CM K,tg + a  Ts — K J  •  2H„  tyfiase  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 R n l  • . qs= s qs +COe¥d  v  S  +  -  1  dX  fqs dt  ~  1 dVds ds = Rslds - <»eVqs + dt  v  v  qr= Rr iqr+®sVdr  (3.4)  1 dWqr + dt  1 dWdr dr = Rr'dr ~ ®s¥qr + cob dt  v  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-caeand yd r  cor is slip electrical angular speed; y/ qs and y/^ are the stator flux linkage; y/ qr 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 =  , vds  . 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 R  t j -HCj -HtJ  l  al  T-W.  -w.—  v  dc  eft anl  f'lter  ia2  ™  '62  '41  v  L  filter  >c2  -•EJ  v  bn\ vcnl  v  cn2 vbn2van2  V  cn2 vbn2 van2  Figure 3.7: Schematic representation of the voltage source converter.  Grid side  DC-link T1  v  T3  £  'a 2 l  C  dc  T6  T4  Js}  4  >  b2 A  duty cycle  d c(t)  triangle signal AAAAM  g c(  0  0N •OFF  A v  v  cnl comparator  >  bn2  g c ( 0 = 4 : (OS  T=0_  Figure 3.8: Representation of the switching function on the grid-side converter.  an2  The output of the comparator is the pulse train g c{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 1 2 3 4 5 . 6  Tl/T4 1 0 0 1 1 0  T2/T5 0 1 0 1 0 1  T3/T6 0 0 1 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  (3.9)  /  3 ,  where 6 C = 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  magnitude terminal voltages van2,  i  t  Vbnh  ar)  d vc„2 of the grid-side converter are described as  v  an2 = da vdc »  (3.10)  v  bn2 = db vdc  v  cn2 = dc vdc  A change of variables to the qd- synchronous reference frame [49] is then applied such that v  qd  where  (3.11)  =T qd(8 e)yabc  (vf  = vqI  T  t  v d and ( «6c) = van2  v  v  bn2  v  cn2  and the qd - transformation  matrix is defined as  cos(0 e )  cos 0o-  e  Tqd( e) = -Z sin(0 c )  In  T  cos  In  sin  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  dc  (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 a n - 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 v  q=vquvdc>  v  d =  (3-14)  v  du vdc  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 ^ - s y n c h r o n o u s 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  1  R  »  T ^  d\ l  dc\  >  0)Li, e  -Idl  qi  <  13  l  di  l  dc2  +  i  v  d\  C-'  Ve C Vq2  •? c  Figure 3.9: Transmission line lumped-parameter qd - model.  A v  q = Vq\ ~ vq2 = RTl}ql  Av  =vd\  d  ~vd2  _C TL d v<l  ,  + — 0 )  (Ofo  = RTL'dl +  r  L j  L r l  (Ob  l . „  e  ut  dt  L  i  d  l  -OeLniql  v  ;7T + ^e^TL vdl (Of, at _ C T L dv d x  dc\ ~  a L  e TL vql  T  COfo at _CTLdvq2  l  T~ + °>e LTL vd2  qc2 ~  *  ;  z  <£2  A v  dt  _ C T L dv d 2 (fy  a> L  e TL vq2  T  at  q = vq\ - vq2 = Rcalql  Av  d  iqcl =  (3.15)  L  +  (Ofr  ,, +  _ C ca Vcl - — T (% dt ;  Av  at  q\ ~ ^qcl  v  d\ =  R  oidcl  at  ^A^rf/  + L c a d l f - coeLcai qi cojj dt  (3.16)  „ ^ „ ^e c ca v ol y  Ltr d lql + 0) L i i e tr d (% dt  +  Avd = vd l - vd 2 = Rtrid l (% dt v  I +  4QaV£/l  =Rtriql  <7 =  di qi  °  CO/,  = vd\ ~ vd 2 = C ca dv q\  C  • CO eLtri qi  (3.17)  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 at  -Ra*a +kl<°eV a)  d¥f = dt  ®b(Efd-  (3.18)  R  f if)  dVk dt where  [» a  h  '/]7'=L1[Vfl  V*  Wfl  with  k1  "0  =  Vk =  - 1 "  1  sq  »  i  «  =  jsd _  Vsq ¥sd V  l  'Wkq }fkd_  and  0  Rkq 0 0 R•kd  , k2 =  »  v  5 =  sq  ySd.  .  kq  h = }kd _  Lamq  0  Lmq  0  0  0  Lamd  0  Lmd  Lmd  0  T-'kqmq  0  0  Lm d  0  Lkdmd  Lmd  0  L = Lmq 0 0  Lmd L  Lmd  fmd  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; Ej d 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.  Exciter  Filter 1  sTp+Ti s  1 + ST r  1 l +  sT  ex  + ^  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 di  koad  gl  CQb dt L  load  - vq2 - Rload lql  di  dl  _v  d2 ~ Rload ldl  t d t v  q2 ~  R l  o qc2'  v  d2 =  ~  ^ekoad^l  (3.19)  + Mekoadiql  R  oidc2  1  where Rq is 10 . The output voltage vector v  b7,pcc  as  2  defines the voltage vector at the PCC  shown in Figure 3.2. The current vector i q ( 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 ,  qdjilter  qa,r set  Q'filter . set \dfilter  8  ^g  Figure 3.12: Block diagram of the voltage source converter controller modules.  In Figure 3.12, PS GET  and QS GET  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, QS GET 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 ou  s H 3  <D  0.6  C  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 qand 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 vd r  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: .  V  = R l  r qr  +  ~1T + ®Wdr (% dt  v  dr = Rr ldr + (Ob  A  at  - Q)syf  (3.20)  '  • set  set  Ol  PI1  V  —*' rO +~H -  a  V  V  qr  +h  qr  +  V,  H p - H PI3 [ — » Q - H pi4 h > Q I  I  g•  dr  +  V  qr  -f  v  dc  V,  dr  +  _  + O — •  VSVDR  qr : set dr  . set  PI2  dr  "dr  >®—>  T  CO IIf s t qr  Figure 3.14: Block diagram of the rotor-side converter controller.  By introducing the new variables  v  qr  Lr digr  R l  r qr  COfr  dt (3.21)  Lty• +  v  'dr ~ Rrldr  (%dt  (3.20) can be simplified to the following: v  *qr=vqr+cos¥dr  (3.22) V  dr  =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 (3.23) (.S) Q ( :.)=Idr( s) 4 K J  V d r(s)  Rr+S^LrlcOb) =  1  Rr+s{L r/03 b)  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  =  (3.24)  v  qr ldr  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  (3.25)  v  qr ldr  V % V  Assuming that slip frequency cos is small and that the controller is being tuned for the nominal operating condition,  =1 on its local base, the term  c l1- 0 ) A s  V here as  f l *-cos  ^  v <°s )  . Hence, (3.25) is rewritten as  f 1  f 1  N V y  I-6>s v ^  ijf is approximated  j  qr  -  V  Lr di qr Rfiqr + dt J  x  (3.26)  j  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 qr  (3.27)  where Rr - {^>Rr  B.2  r  and  T  \  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  (3.28)  v  dr ldr  ~  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  rP'  g =  f 1  (3.29)  v  v °>s J  dr ldr  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'8 -  r1  f 1  > dr  v ^  j  \f  I -a>s  v  r  \  L  Rrid r + r  J  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  v  q, filter ~ Kfilter  1  .  q, filter +  ^filter  dig,filter  . + meLfilter  1  d, filter  (3.32) v  d,filter  D  ~ Kfilter  1  .  L  filter filter  d,  ^d, filter ~J t  T co L  e filter'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 L  q, filter ~ Rfilter  D K  v  d, filter ~ filter  1  q, filter +"  1  .  filter  dig,filter  (Ojj  dt  L  filter  d, filter  (3.33)  did, filter  the modified command voltages of (3.32) can be expressed as  V  hfilter  ~ V'q,filter  + ®eL filter  'd, filter ~ Vd, filter " ®eLfilter  1  1  d, filter  (3.34)  q, filter  The corresponding transfer function can be expressed as  G5(S) = I«/'^S) V  q, filter 0 )  Rf  Ute r  1 + S (l FLHE R  j )  (3.35) „ , &  N  1  d,filter  s  ( )  6(f)=7T Yd, filter 0 )  1 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 setvalues are calculated by the real and reactive power commands P0ter •set q,filter •S€t _ld, filter  v  q,tr  v  ~vd,tr  V  l  Here \ q < j > t r = \ y q > t r transformer; Pfi/  ter  The value for Qflj  ter  d,tr  vdtr J ter  ter  as  -1 tpset  q,tr_  and Qfif  and Qfi[  ,filter  Qfiiter  is the voltage at the low-voltage-side wind turbine are the set-point of the real and reactive power commands.  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  2 cob  J  dt  p  (3  3 7 )  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  (0^  ~Pr=  ^cjllte^dc  ~ [dc,r vdc  (  3  -38)  at  which can be rewritten as dv ^ dc  Qc  where i d c  l  dc, filter  and idc,filter  r  (3.39)  l  dc,r  are  ^ e current in the rotor-side converter and the grid-side  converter, respectively. The transfer function from the current id c, filter  t0  the dc voltage vdc  is therefore as follows: c  W  . - S * « ! L - = - a dc, filter v*^)  (3.40)  sL  dc  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  obtained by Pf{  ter  = vd cis/ c[ fllte  r  .  The real power set-point Ps^l ter  in (3.36) is  • Pfllter  +r\  w  a  b  • set l dcfilter  >-  v  dc  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 G 7 (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  set 2 j,g  where  =  m  m  QJT  max  Qj,c '  X  •>max  Qlc +Q2T+-  max  + Qnl c  AGpcc  (4.1)  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.  +P  * Absorb reactive power  0max  ^(pu)  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 Agp CC 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 4 th -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 4 th -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 (104 th ).  Frequency (rad/sec)  Figure 4.4: Bode diagrams of the reduced-order model (4 th )  The 4 th -order reduced model may be realized in terms of the state variables that are related to the voltage at the PCC as  i =  [Avpcc  A  Av  Vc  (4.3)  Avpcc]7  pcc  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-<  where |Avp CC =  x  (4.4)  (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) where A B  / ;  "o 0  (4.5)  z (t) = Cj\(t) + D,-w(f)  C h Dj are the augmented system matrices -Ci  B  i=  -V  »7  c  i =  I  "1 0 0 0 0" 0 10 0 0  '  D Ii =  "0"  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; Ae D G Si mXP  is the state matrix; B e 3 i n X p is the input matrix; C e 3 i m X n is the output matrix; is the direct feed-through matrix; x e  is the state vector; u e  is the input  signal vector; ze 3l m 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 0 -6.3643  -52.306 13.7505  13.7505 -76.688 > -4.7933 72.8850  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  51.3130  -61.339  -16.300 -71.572  6.52150 0 -14.052  -16.300  -5.6663 -64.272  71.5720  64.2720  -75:609_  1.70940 -4.5492  7.08100  10.4324  -1.8855  6.79060"  "0.04200"  0 -10.294  -55.954  6.46570  -18.208  -7.0810  0  55.9540  -48.792  13.2378  -77.843  Bh - 10.4324  0 -6.4657  13.2378  -4.7963  77.4240  -1.8855  0 -18.208  77.8430  -77.424  -112.34  -6.7906  A,= 0 0  "0 Ah  =  C 0-C t-C h-  1  0  0  0 10  0  0'  0 0  (4.7)  B 0 - 10.2209 -1.7963  -6.4062 >  » /  =  9.55480  =D; =Di =  (4.8)  (4.9)  (4.10)  and the output and the state vector are  =[K"pcc  ~\T  AVpCC  = [ jAvpcc  Avpcc  Avpcc  Avpcc  & 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 5 th -order reduced model.  TABLE 4 . 1 EIGENVALUES, DAMPING RATIO, AND FREQUENCY OF THE 5™-ORDER REDUCED MODEL  Damping ratio  Eigenvalues 0 -22.6±j20.8 -60.9±jl06  -  0.786 0.5  Frequency (rad/sec) 0 30.8 122  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 k t 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]  k  V or  p  +jo)  g c  k  h  d  + ico,  -  G r{jCD  J  gc)  = Xe j6{ Wz c\  (4.12)  eM°>gc) P  s  G r(jco gc)  where Gr(ja> gc)  jk  .  (4.13)  (o gc  is the transfer function of the 4' -order reduced model obtained for the  nominal operating condition, as follows:  Gr{s) =  -0.028s 4 - 2 7 . 7 8 S 3 - 7 3 5 4 s 2 +15618025 + 31501951  (4.14)  s 4 + 167.14s 3 + 21340s 2 + 788654s+ 14079701  By solving (4.12) with the gain crossover frequency (Ogc = 65 rad/sec, the gains of the PID supervisory controller may be expressed as k p =a, k d =bja)gc  , and k t = d c g a j n . To speed  up the settling time, the integral gain k t 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 r  /  filter with an equivalent gain given as  k d-k dt  \ \ 1s +1 . This is usually done to  kd  Vv y  "  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  jAvpcc  Gains  6.7122  Av  pcc  0.4635  Avpcc .  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 antiwindup 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  Figure 4.6:  k  i  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 y = CjX + D,u + n  (5.1) ,  (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 = A2 P cc = _ 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  Av  pcc  Av  (5.3)  pcc + •  where A v 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 statefeedback 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  + S T\L  +GR WG T  -S TCIQ~ 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 signal as R w = E wwT and Qn=E  are  J  defined using the expectation E of each noise , 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 = SC 0 Q^ . The observer can be expressed as  x = A 0 x + B 0 w + K e ( z - z) (5.5) A  A  z = C0x 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 G = [1.9515, 2.3081, 2.0722, 1.5768, 0.4277] 7 .  matrix  Applying  these  design  parameters and using the Control System Toolbox [64], the observer gain is obtained as follows:  K  e -SCoQj  61.5334  20.6683  44.7403  48.6022  1.2318  20.6683  26.3330  25.4970  17.7305  5.7823  T  (5.6)  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 t h 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-  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  5.3  Eigenvalues  Damping ratio  Frequency (rad/sec)  -22.0 -31.4 -78.5 -61.5±jl06  1 1 1 0.503  22.0 31.4 78.5 122  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 performancebased 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 rRk = 0 where P e %nXn  (5.9)  > 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 wellknown algebraic Riccati equation ^ rP = 0 A r P + PA + Q - P B R _l 1r B  (5.10)  which is linear in variable P and is readily solved numerically using software [64], The controller gain is then computed as k = R B P. 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 . (5.11)  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: (Aj - B j k f P + P ( A 7 - B y k) + Q + k r R k = 0  (5.12)  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 reformulate 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.  5.3.2 Conventional approach Using the H 2 representation of the LQR problem [61], [67], we would like to find the statefeedback gain matrix k that minimizes the following cost function in terms of output y as (5.13)  J = minjiiTy^yT] (k) 1 L K subject to (Ay-Byk) rP + P(A/-B/k) +Q+krRk<0 V  and  J  (5.14)  P>0  We first formulate an LMI for the cost function (5.13), using the output y as given in [67] >1/2  y=  0  (5.15)  Rl/2  then substitute (5.15) into (5.13) with u = -kx to obtain .T~ = E tr  yy  r r = E x Qx + u Ru  / r  • tr  QV2 2  -RV k  (5.16)  ' xx  2 T  \QV )  1  2  -k^R / )  7  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 xx the H 2 representation of the LQR problem (5.13) can be expressed as  yy  ( ' ' 7 = fr(QY) + fr R 1 / 2 k Y k r ( R 1 / 2 )  (5.17)  where it is assumed that E  = R w > 0 and Y = E xx  WW  . We then utilize the change  of variables [67] such that X>R1/2kYkr(R1/2)7  (5.18)  Let us define a change of variables as used in [67] such that k = AY"  1  k G3IPXN,  and  A E  (5.19)  P = Y"  and  P e Si nXn  (5.20)  Y=Y  and  Ye5R nxn  (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-jR^AjY-'j/R R1/2A  X A V  2  1  /  2  ^ (5.22)  >0  Y  Finally, the cost function (5.13) is formulated as follows: Minimize: J = min {fr(QY) + /r(X)l (Y,X)1  J  subject to LMI constraint (5.22).  (5.23)  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 =Y  -1  Y T Ay - ArBy + AyY - By A + YQY + ArRA Y  <0  (5.24)  = Y 7 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  -R1 0 0 -Q"  A Y  <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:  Minimize:  (5.26)  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 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:  Z-IY  --1 1  Z I  I >0, Y  and  Y>0  where I is an nxn identity matrix, and Z = Z T e 3inX n matrix.  (5.28)  is the symmetric positive definite  Thereafter, the H 2 LQR problem based on the cost-guaranteed approach is formulated as follows: Minimize: V = minjfr-(Z)) 1 J (Z)  (5.29)  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:  5.4  Comparison of PID, LQRS, and LQRCG controllers.  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? x^Qx + u r R u + x r N u + u r N r x  (5.30)  00  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 m  rr< m  rp  T"  rp rrt  J = E xl Qx + x k R k x - x N k x - x ^ k ' N 7 x = ^ ( Q + k R k r - N k - k r N r \}tr{E t r \ E xxT T ^  = /r(QY) + fr R ^ K Y k ^ R 1 / 2 )  (5.31)  )  -^(NkY + k r N r Y r )  Hence, the conventional H 2 LQR problem with the cross-product terms becomes  J = min tr QY + R ! / 2 k Y k r ( R 1 / 2 ) - NkY - k r N r Y  T  (5.32)  (k,Y)  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(-NkY-krNrYr) = /r(-NkY-YrkrNr) =^(-NA-ArNr)  (5.33)  Hence, the advanced H 2 LQR problem based on the conventional approach can be formulated as follows: Minimize: J-  min  fr(QY + x)-/r(NA + A r N 3  (A,Y,X)  subject to (5.22) and (5.25).  (5.34)  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 |"x r Qx + 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 ) < min  (Z,A)  (5.35)  ?r(Z)-/r(NA + A r N r )  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:  5.5  Comparison of PID, ALQRS and ALQRCG controllers.  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 5 th -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 Design parameter  LQRS  R  1.2  LQRCG 2.2 .  ALQRS  ALQRCG  0.955  4  diag{ 70, 10, 1, 1, 1)  Q N  Advanced Approach  [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  Av  pcc  -2.2902  -2.2342  -2.1965  -1.3626  Av  pcc  0.8600  0.8917  1.5886  2.7747  vpcc  0.0660  0.0106  0.1861  0.1217  -0.1852  -1.8105  0.1958  3.3509  A  Avpcc  TABLE 5.3 EIGENVALUES, DAMPING RATIO, AND FREQUENCY  LQRS Eigenvalues -9.10 -28.3±j20.6 -63.4±jl07  Damping ratio 1 0.808 0.508  Frequency (rad/sec) 9.10 25.0 125  LQRCG Eigenvalues -9.88 -35.3±jll.6 -61.0±j94.9  Damping ratio 1 0.905 0.541  Frequency (rad/sec) 9.88 37.1 113  ALQRS Eigenvalues. -10.6 -25.0±jl9.4 -68. l±j 110  Damping ratio 1 0.790 0.526  Frequency (rad/sec) 10.6 31.7 130  ALQRCG Eigenvalues -11.2 -12.7±j23.6 -73.7±jl30  Damping ratio 1 0.475 0.493  Frequency (rad/sec) 11.2 26.8 149  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).  0.027  - a y , 0.0265 LVC  3Q.  0.027  JP'yC-  C o 0.0265 CD  o Q.  PID - supervisory  0.027 -  "S <D CC 0.0265 ALQRCG 0.027  0.0265 0  I  10  20  i  30  I  40  I  I  50 time(sec)  60  I  70  i  i  80  -  90  100  Figure 6.2: Real power set-point for each WT due to wind speed variation. ( WT1 — 1 WT2 WT3) PFC  0.027  /••,.. X.  0.0265  3Q.  0.027  D Q.  0.0265  LVC  ''/py^X/ PID - supervisory  0.027 0 <5 1 CL "5 0.0265 a> CC  fKoC ALQRCG  0.027  \ 0.0265 10  20  -A^rv  , J Jy\ .A"' / / vV... f\/""'"\. I  30  I  40  I  50 time(sec)  I  60  i  70  i  80  i  90  Figure 6.3: Real power output from each WT due to wind speed variation. (—WT1 — W T 2 WT3)  -  100  PFC  0.078 0.0775 0.077  3Q. 3Q.  LVC  0.078 0.0775 0.077  PID - supervisory  0.078  O Q g) 0.0775 O Q. "(5 0.077 i_  CD  cc  ALQRCG  0.078 0.0775 0.077. 0  10  20  30  40  50 time(sec)  60  70  80  90  100  Figure 6.4: Real power output from the wind farm due to wind speed variation. PFC  0.00005  -0.00005 0.00005 Q.  1Q.3 -0.00005 "5 o 0.00005 <D  PID - supervisory  1 —  o  Q. a>  > 13 to -0.00005 CD  DC  v V A A A A V N / V X ^ ALQRCG  0.00005  V W A a A A A A / V \ -0.00005.  10  20  30  40  50 time(sec)  60  70  80  90  Figure 6.5: Reactive power output from each WT due to wind speed variation. (—WT1 — W T 2 WT3)  100  0.0031 0.003 0.0029  3Q.  0.0028 0.0031  LVC  0.003 0.0029 0.0028  30 1 <D 3o Q. < } >1 O TO CD  tr  0.0031 . 0.003 0.0029 0.0028 0.0031 0.003 0.0029 0.0028,  40  50 time(sec)  60  100  Figure 6.6: Reactive power output from the wind farm due to wind speed variation.  3Q.  C D •5 CO O > 3  CD  0  10  20  30  40  50 60 time(sec)  70  80  90  100  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 PFC LVC  0.915  3  PID-supervisory ALQRCG  Q.  in 3 0.885  0  2  4  6  8  10  12 .  14  time(sec)  Figure 6.9: Voltage transient observed at the PCC due to load impedance changes.  16  0.912 PID-supervisory  0.91  ALQRCG  0.908 S  0.906  O O 0.904  oto> 0.902 C D CT) to o 0.9 >  v> m  0.898 0.896 0.894.  0  8  12  10  14  time(sec)  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  LVC  3Q. 0.9?  -  0.9  -  0) •C  _  v.. F  -t—i  to & 0.92 _ O) to § 0.9 C O 3 CQ 0.92  - j  R PID - supervisory -  L  f" ALQRCG  '  0.9  i  i  6  i  8  time(sec)  i  10  1  12  1  14  Figure 6.11: Voltage transient observed at the WT terminals.  16  0.027  ••• WT1  1  \  0.0265  D O. H §  0.027  WT2  WT3  14  16  J LVC  "o 0.0265 "3  ^a.  PID - supervisory  0.027  o a) 0a. "ro 0.0265 <D 01  ALQRCG  0.027  i  i  I  I  p-s/-I  0.0265.  0  I 6  8 time(sec)  10  12'  Figure 6.12: Real power output from each WT. PFC  0.08  i  /  y  0.075  LVC  0.08  t  3  CL  0.075  3CL  0.08  o L_ <D o a.  0.075  lr  r  PID - supervisory  -t—1  ro <D a:  0.08  0.075.  0  r  ALQRCG  I  |  8  10  time(sec)  I  I  12  14  16  6  8 time(sec)  10  12  Figure 6.14: Reactive power output from each WT. PFC  0.02  -0.02  LVC  2  4  6  8 time(sec)  10  12  14  16  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 High (pu) Low (pu) Deviation (%)  6.2  PFC  LVC  PID- Supervisory & ALQRCG  0.9154 0.8892 2.62  0.9067 0.9021 0.46  0.90497 0.90497 0  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.  ....... W  0.03H  T 1  WT2---WT3  0.025  0.02 LVC  0.03 0.025  0.02 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 time(sec)  1.3  1.4  1.5  0.015r  • WT1  0.01 •  - WT2  WT3  0.005-  0•  -0.005: 0.015r  LVC  —vw—  "Www 3 S fK ?s m m m t n t M A w  0.01 0.005  0  -0.005  1.2  time(sec)  Figure 6.20: Reactive power output at each WT due to the fault. ( - - - WT1 — WT2 PFC  0.95 0.9 0.85 0.95 0.9 : 0.85 0.95  WT3)  • WT1 "I  V  WT2  WT3 awwuwj t if.  "'•'f i/.'ivii'.^'.'i yjtti i v  LVC : 11» UtUtKfitf£K&'MVi  f  H W W W U 1  MitttVlttl  PID - supervisory  A ' 0.9-  :v n ^ W m w M M W U . 'U . 's : ii: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.  L V i . ' A V V i  0.040.02-  0= 3  " V V ^  -0.02 -  LVC  Q.  3Q.  "3  0 0) 5o Q. a) > oto 0) EC 1  1.2 time(sec)  . 1.5  Figure 6.22: Reactive power output from the wind farm due to the fault. LVC  Q.  E'  E 'x CO  PID - supervisory  /  E  C 03 0Q. L_  \  /  T3  1 0) U) 0>  \  0  o  \  I „  1 1 1  ALQRCG  Q. <1)  > 0.01 O C CD D 0.005 CC  8  1.1  1.2  time(sec)  1.3  1.4  Figure 6.23: Reactive power set-point and maximum at each WT due to the fault. (---WT1  WT2  WT3, j=1,2,3)  1.5  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 0.90497 Deviation (%)  PFC 0.8918 1.32  LVC 0.9025 0.25  PID- supervisory & ALQRCG 0.90497 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 linearquadratic-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. However, the benefit of improving the power system's operation (quality, stability, and reliability) may well justify the cost.  Another application would be to look for control solutions that would enable automatic operation of wind energy resources in stand-alone mode and continue supplying some islands of local loads in cases of wide-area power outages and/or blackouts. This would require a distributed control approach, wherein the voltages and frequencies over a possibly larger area would have to be monitored and integrated into the wind farm control to enable continuous and reliable operation in a given area.  References [1]  J. Goldemberg and T. B. Johansson, World Energy Assessment, 2004. Available: httPV/www.undp.org/energv/docs/WEAOU full.pd f (Sept. 24, 2005).  [2]  L. Flowers, Wind Energy Update, National Renewable Energy Laboratory, Jan. 2005. Available: http://www.eere.energv.gov/windandhydro/windpoweringamerica/pdfs/wpa/wpa upd ate.pdf (Sept. 24, 2005).  [3]  Canadian Wind Energy Association. Available: http://www.canwea.ca/en/WindEnergy.html (July 11 2005).  [4]  G. Hehenberger, "MULTIBRID technology: 5MW wind energy converter for offshore applications," in Proc. World Wind Energy Conference, Germany, 2002.  [5]  ENERCON GmbH, Wind Turbine E-l 12. Available: http://www.enercon.de/en/ home.htm (Jan. 8, 2006).  [6]  T. Ackermann, G. Andersson, and L. Soder, "Distributed generation: A definition," Electrical Power System Research, vol. 57, no. 3, pp. 195-204, Apr. 2001.  [7]  Specifications for Connecting Wind Farms to the Transmission Network, ELT 1999411a, Eltra, 1999.  [8]  N. W. Miller, W. W. Price, J. J. Sanchez-Gasca, Dynamic Modeling of GE 1.5 and 3.6 Wind Turbine Generators, GE Power Systems, Oct. 2003. Available: http://www.easthavenwindfarm.com/filing/high/modeling.pd f CJuly 20, 2004).  [9]  T. Senjyu, N. Sueyoshi, R. Kuninaka, K. Uezato, H. Fujita, and T. Funabashi, "Study on terminal voltage and power factor control of induction generator for wind power generation system," in Proc. International Conference on Power System Technology, vol. 1, Nov. 2004, pp. 753-758.  [10] T. Sun, Z. Chen, and F. Blaabjerg, "Transient analysis of grid-connected wind turbines with DFIG after an external short-circuit fault," in Proc. Nordic Wind Power Conference, Mar. 2004. [11] T. Senjyu, T. Kinjo, H. Fujita, and Aichi, "Analysis of terminal voltage and output power control of wind turbine generator by series and parallel compensation using SMES," in Proc. IEEE 35th Annual Power Electronics Specialists Conference, vol. 6, June 2004, pp. 4278-4284.  [12] Y. Kubota, T. Genji, K. Miyazato, N. Hayashi, H. Tokuda, and Y. Fukuyama, "Verification of cooperative control methods for voltage control equipment on distribution network simulator considering interconnection of wind power generators," in Proc. IEEE Transmission and Distribution Conference and Exhibition, vol. 2, Oct. 2002, pp. 1151-1156. [13] P. Kundur, Power System Stability and Control, Electric Power Research Institute, McGraw-Hill, New York, 1994. [14] M. H. J. Bollen, Understanding Power Quality Problems, John Wiley & Sons Inc., New York, 2000. [15] ABB, Inc., Wind Farm Integration in British Columbia-Stage 1&2: Planning and Interconnection Criteria, Mar. 2005. Available: http://www.bctc.com/generator interconnection/generator interconnection requirem ents (June 11,2005). [16] Wind Energy Delivery and Reliability - Workshop Summary Report, Wind Power Transmission Workshop, Mar. 2005, Toronto, Canada. [17] N. Dizdarevic, M. Majstrovic, and S. Zutobradic, "Power quality in a distribution network after wind power plant connection," in Proc. IEEE Power Systems Conference and Exposition, vol. 2, Oct. 2004, pp. 913-918. [18] T. Ackermann, Wind Power in Power Systems, John Wiley & Sons, Ltd., England, 2005. [19] Z. Lubosny, Wind Turbine Operation in Electric Power Systems, Springer, New York, 2003. [20] K. S. Salman and A. L. J. Teo, "Windmill modeling consideration and factors influencing the stability of a grid-connected wind power-based embedded generator," IEEE Trans, on Power Systems, vol. 18, no. 2, pp. 793-802, May 2003. [21] E. ON Netz, Wind power report, Germany, Sept. 2004. Available: http://www.nowhinashwindfarm.co.uk/EON Netz Windreport e eng.pdf (Apr. 22, 2005). [22] Elsam Engineering, BC Wind Integration System Expansion Study, July 2004. [23] Z. Litipu and K. Nagasaka, "Improve the reliability and environment of power system based on optimal allocation of WPG," in Proc. IEEE Power Systems Conference and Exposition, vol. 1, Oct. 2004, pp. 524-532. [24] C. Eping, J. Stenzel, M. Poller, and H. Muller, "Impact of large scale wind power on power system stability," in Proc. 5th International Workshop on Large-Scale Integration of Wind Power and Transmission Networks for Offshore Wind Farms, Apr. 2005. [25] R. Doherty, E. Denny, and M. O'Malley, "System operation with a significant wind power penetration," in Proc. IEEE Power Engineering Summer Meeting, vol. 1, June 2004, pp. 1002-1007.  [26] I. Erlich and U. Bachmann, "Grid code requirements concerning connection and operation of wind turbines in Germany," in Proc. IEEE Power Engineering Society General Meeting, June 2005, pp. 2230-2234. [27] Z. Chen, "Issues of connecting wind farms into power systems," in Proc. IEEE/PES Transmission and Distribution Conference & Exhibition: Asia and Pacific, Aug. 2005, pp. 1-6. [28] J. J. Grainger and W. J. Stevenson, Power System Analysis, McGraw-Hill, New York, 1994. [29] S. Heier, Grid Integration of Wind Energy Conversion System, Chicester, John Wiley & Sons Inc., UK, 1998. [30] J. F. Manwell, J. G. McGowan, and A. L. Rogers, Wind Energy Explained, John Wiley & Sons Inc., New York, 2002. [31] L. T. Ha and T. K. Saha, "Investigation of power loss and voltage stability limits for large wind farm connections to a subtransmission network," in Proc. IEEE Power Engineering Summer Meeting, vol. 2, June. 2004, pp. 2251-2256. [32] M. P. Palsson, T. Toftevaag, K. Uhlen, and J. O. G. Tande, "Large-scale wind power integration and voltage stability limits in regional networks," in Proc. IEEE Power Engineering Summer Meeting, vol. 2, July 2002, pp. 762-769. [33] T. Gjengedal, "Large scale wind power farms as power plants," in Proc. Nordic Wind Power Conference, Mar. 2004. [34] A Candidate Site of Wind Farm in Vancouver Island, Vancouver, British Columbia, Canada, Powertech Labs, Inc., Vancouver, British Columbia, Canada, Sept. 2004. [35] Y. Xiaofeng and K. Tomsovic, "Application of linear matrix inequalities for load frequency control with communication delays," IEEE Trans, on Power Systems, vol. 19, pp. 1508-1515, Aug. 2004. [36] B. Chaudhuri, R. Majumder, and B. C. Pal, "Wide-area measurement-based stabilizing control of power system considering signal transmission delay," IEEE Trans, on Power Systems, vol. 19, pp. 1971-1979, Nov. 2004. [37] W. Hongxia, K. S. Tsakalis, and G. T. Heydt, "Evaluation of time delay effects to wide-area power system stabilizer design," IEEE Trans, on Power Systems, vol. 19, pp. 1935-1941, Nov. 2004. [38] R. A. Ramos, L. F. C. Alberto, and N. G. Bretas, "Linear matrix inequality based controller design with feedback linearization: Application to power systems," in Proc. IEE Control Theory and Applications, vol. 150, Sept. 2003, pp. 551-556.  [39] B. Chaudhuri, B. C. Pal, A. C. Zolotas, I. M. Jaimoukha, and T. C. Green, "Mixedsensitivity approach to H/sub /spl infin// control of power system oscillations employing multiple FACTS devices," IEEE Trans, on Power Systems, vol. 18, Iss. 3, pp. 1149-1156, Aug. 2003. [40] D. Rerkpreedapong, A. Hasanovic, and A. Feliachi, "Robust load frequency control using genetic algorithms and linear matrix inequalities," IEEE Trans, on Power Systems, vol. 18, pp. 855-861, May 2003. [41] R. Gupta, B. Bandyopadhyay, and A. M. Kulkarni, "Design of power system stabilizer for single-machine system using robust periodic output feedback controller," in Proc. IEE Generation, Transmission and Distribution, vol. 150, Mar. 2003, pp. 211-216. [42] H. Werner, P. Korba, and T. C. Yang, "Robust tuning of power system stabilizers using LMI-techniques," IEEE Trans, on Control Systems Technology, vol. 11, Iss. 1, pp. 147-152, Jan. 2003. [43] D. D. Siljak, D. M. Stipanovic, and A. I. Zecevic, "Robust decentralized turbine/governor control using linear matrix inequalities," IEEE Trans, on Power Systems, vol. 17, pp. 715-722, Aug. 2002. [44] S. Xie, S. L. Xie, Y. Wang, and G. Guo, "Decentralized control of multimachine power systems with guaranteed performance," in Proc. IEE Control Theory and Applications, vol. 147, May 2000, pp. 355-365. [45] B. C. Pal, A. H. Coonick, I. M. Jaimoukha, and H. El-Zobaidi, "A linear matrix inequality approach to robust damping control design in power systems with superconducting magnetic energy storage device," IEEE Trans, on Power Systems, vol. 15, pp. 356-362, Feb. 2000. [46] M. G. Rui, J. M. Castro, and J. Ferreira, "An aggregated wind park model," in Proc. International Conference on Power System Transient, July 1999. [47] C. H. Ong, Dynamic Simulation of Electric Machinery, Prentice Hall, New . Jersey, 1998. [48] J. Morren, S. W. H. de Haan, P. Bauer, and J. T. G. Pierik, "Comparison of complete and reduced models of a wind turbine using doubly-fed induction generator," in Proc. 10th European Conference on Power Electronics and Applications, Germany, Sept. 2003. [49] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems, John Wiley & Sons Inc., New Jersey, 2002. [50] N. Mohan, T. M. Undeland, and W. P. Robbins, Power electronics, John Wiley & Sons Inc., New Jersey, 2003. [51] P. Kundur, M. Klein, G. J. Rogers, and M. S. Zywno, "Application of power system stabilizers for enhancement of overall system stability, IEEE Trans, on Power System, vol. 4, no. 2, pp. 614—626, May 1989.  [52] J. G. Slootweg and W. L. Kling, "Aggregated modeling of wind parks in power system dynamics simulations," in Proc. IEEE Power Tech Conference, vol. 3, June 2003, pp. 23-26! [53] R. Pena, J. C. Clare, and G. M. Asher, "Doubly fed induction generator using backto-back PWM converters and its application to variable-speed wind-energy generation," in Proc. IEEE Electric Power Application, vol. 143, no. 3, 1996, pp. 231-241. [54] S. Muller, M. Deicke, and R. W. Doncker, "Doubly fed induction generator systems for wind turbines," IEEE Industry Applications Magazine, pp. 26-33, May/June 2002. Available: http://www.ieee.org/ias (July 2005). [55] K. J. Astrom and T. Hagglund, Advanced PID Control, Instrumentation, Systems, and Automation Society, Aug. 2005. [56] C. L. Phillips and R. D. Harbor, Feedback Control Systems, Prentice Hall, New Jersey, 1996. [57] A. Tapia, G. Tapia, J. X. Ostolaza, and J. R. Saens, "Modeling and control of a wind turbine driven doubly fed induction generator," IEEE Trans, on Energy Conversion, vol. 18, no. 2, pp. 194-204, June 2003. [58] Z. Saad-Saoud, M. L. Lisboa, J. B. Ekanayake, N. Jenkins, and G. Strbac, "Application of STATCOMs to wind farms," in Proc. IEE Generation, Transmission, and Distribution, vol. 145, Sept. 1998, pp. 511-516. [59] A. Papantoniou and A. Coonick, "Simulation of FACTS for wind farm applications," IEE Colloquium on Power Electronics for Renewable Energy, June 1997, pp. 1-5. [60] P. Cartwright, L. Holdsworth, J. B. Ekanyake, and N. Jenkins, "Coordinated voltage control strategy for a doubly-fed induction generator (DFIG)-based wind farm," in Proc. IEE Generation, Transmission and Distribution, vol. 151, July 2004, pp. 495502. [61] B. D. O. Anderson and J. B. Moore, Optimal Control. Linear Quadratic Methods, Prentice-Hall, New Jersey, 1998. [62] K. Zhou and J. C. Doyle, Essentials of Robust Control, Prentice Hall, New Jersey, 1998. [63] C. T. Chen, Linear System Theory and Design, Oxford University Press, New York, 1999. [64] Control System Toolbox® for use with Matlab®, MathWorks, Jan. 2000. [65] K. Ogata, Modern Control Engineering, Prentice-Hall, New Jersey, 1997. [66] F. L. Lewis and V. L. Syrmos, Optimal Control, John Wiley & Sons, Inc., New York, 1995.  [67] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15, SIAM, Pennsylvania, 1994. [68] LMI Control Toolbox® for use with Matlab®, MathWorks, Jan. 2000. [69]. P. Dorato, C.T. Abdallah, and V. Cerone. Linear Quadratic Control: An Introduction. Prentice-Hall, Englewood Cliffs, New Jersey, 1995. [70] E. Feron, V. Balakrishnan, S. Boyd, and L. El Ghaoui, "Numerical methods for H2 related problems," in Proc. American Control Conference Proceeding, Illinois, June 1992,pp. 2921-2922. [71] R. A. Luke, P. Dorato, and C. T. Abdallah, "Guaranteed gain-phase margins for multi-model control," in Proc. American Control Conference Proceeding, Pennsylvania, June 1998. [72] MATLAB® and Simulink®, MathWorks, Jan! 2000.  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) Fixed constant  Power coefficient C p{9,X) 4  4  X  =X i=0j=0 where 2<A<13  1/2 pAr  0.0145  Kb  69.5  i 4 4 4 4 4 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1 0 0 0 0 0  j 4 3 2 1 0 4 3 2 1 0 4 3 2 1 0 4 3 2 1 0 4 3 2 1 0  a  ij 4.9686e-010 -7.1535e-008 1.6167e-006 -9.4839e-006 1.4787e-005 -8.9194e-008 5.9924e-006 -1.0479e-004 5.705 le-004 -8.6018e-004 2.7937e-006 -1.4855e-004 2.1495e-003 -1.0996e-002 1.5727e-002 -2.3895e-005 1.0638e-003 -1.3934e-002 6.0405e-002 -6.7606e-002 1.1524e-005 -1.3365e-004 -1.2406e-002 2.1808e-001 -4.1909e-001  Table A.2 Turbine Controller Parameters [8] (all quantities are given in per unit on 4MVW base) K  150  K  25  PP  Pitch controller gains Actuator time constant  ip T p (second)  0.01  Time constant  T  0.05  A pc K  Torque controller gains  pt  3  K it  0.6  K  3  pc  Pitch compensator gains  K  30  ic de  27  ^max( g)  Pitch angle limtation  0  0  mm ( d e g ) rpmax t rpmin t  Power maximum and minimum Cut-in wind speed (m/s)  vw  Cut-off wind speed (m/s)  1 0.1  ,  3.5 25  Table A.3 Two-Mass Rotor Model Parameters [8] (all quantities are given in per unit on 100MW base) Rotor inetia constant  Ht Hr  0.1716  Shaft stiffness  K  11.868  Shaft damping coefficient  D  tg  0.06  Reference rotor speed  oI tef  1.15  Base rotor speed  fyfrase  Generator inertia constant  tg  0.036  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  C  0.01  dc ref V dc  Reference dc voltage  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 PI1 and PI3 Rotor-side converter PI2 and .PI4 l _ Grid-side converter PI5 and PI6 DC link PI7 Local voltage controller  Proportional gain 0.0426 2.8013 0.0018 0.03155 0.0137  Integral gain 59.853 39.376 0.6436 2.3873 6.7057  Table A.7 Line Parameters [34] (all quantities are given in per unit on 100MW base, 132kV base) Transformer (TR1,TR2,TR3) Transfomer (TR4) Transmission line Cable Grid-side filter  Resistance 0.0092 0.0216 0.3775 0.0378 0.000435  Inductance 0.233 0.539 0.6689 0.0669 0.002696  Capacitance  0.024 0.0016  Table A.8 Thyristor Excitation system [13] Amplifier gain  Tt  0.05  T  0.2 10  Thyristor gain  ± Tex Kt  Filter gain  TR  0  Exciter gain  10  Table A.9 Synchronous Generator Parameters [13] (all quantities are given in per unit on 100MW base, 132kV base) Stator resistance  R  a  0.000135  Stator d-axis inductance  Ld  0.0815  Stator ^-axis inductance  L  1  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  L  0.0822  Field resistance  R  f  0.00027  Field inductance  L  f  0.0074  d-axis damper winding resistance  R  kd  0.0013  d-axis damper winding inductance  L  kd  0.0077  g-axis damper winding resistance  R  kq  0.000279  ^r-axis damper winding inductance  L  kq  0.0327  Electrical angular speed  coe  fmq  1  Table A. 10 Operating Conditions  Case  Grid  Load  n o m i n a l local l o a d i m p e d a n c e BUS  V(PU)  P(PU)  Q(PU)  9  0.94860  0.07338  0.001595  8  0.93425  0.07303  0.049410  TR  0.07284  0.044610  WF  0.07738  0.002935  Total  0.15022  0.047555  Resistance  Reactance  7 (PCC)  0.90497  Load  WF  R  (• load)  ( Xload)  5  1.6  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  TR  0.02588  -0.0001888  Cable  0.02586  0.00110100  Total  0.05174  0.00091190  0.02671  0  TR  0.02588  -0.0001891  Cable  0.05166  0.00210200  Total  0.07755  0.00191200  4 5 6  0.90843 0.90728 0.90704  Case  Grid  Load  50% decrease of the local load impedance BUS  V(PU)  P(PU)  Q (PU)  9  0.94860  0.18900  0.072600  8  0.90584  0.18330  0.110300  TR  0.18210  0.080190  WF  0.07766  0.002245  Total  0.25976  0.082435  Resistance  Reactance  7 (PCC)  0.84259  Load  WF  (• Rload)  . ( Xload  2.5  0.8  )  BUS  V(PU)  P (PU)  Q (PU)  1  0.84725  0.02671  0  2  0.84700  0.02599  -0.0002193  3  0.84651  0.02671  0  TR  0.02599  -0.0002197  Cable  0.02596  0.0008907  Total  0.05195  0.0006710  0.02671  0  TR  0.02599  -0.0002205  Cable  0.05187  0.00166500  Total  0.07785  0.00144450  4  5  6  0.84626  0.84504  0.84479  Case  Grid  Load  50% increase of the local load impedance BUS  V(PU)  P(PU)  Q(PU)  9  0.94860  0.02793  -0.018460  8  0.94900  0.02782  0.0310500  TR  0.02778  0.0300000  WF  0.07728  0.0031640  Total  0.10506  0.0331640  Resistance  Reactance  7 (PCC)  0.92565  Load  WF  R  (• load)  ( Xload)  7.5  2.4  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  TR  0.02584  -0.0001803  Cable  0.02582  0.00117200  Total  0.05166  0.00099170  0.02671  0  TR  0.02584  -0.0001808  Cable  0.05159  0.00224800  Total  0.07743  0.00206760  4 5 6  0.92904 0.92792 0.92768  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 proportionalplus-integral (PI) controller can be expressed [56] as  1 + GC(5)G/,(5) = 1+  -kp+-L  G p{s)  (B.L)  where G p (5) is the first-order transfer function of a system, and G c (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 Acl(s) = s2 +2gco ns + co2  The gain coefficients of a PI controller can be obtained by comparing (B.l) with (B.2).  (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 40.6339 G2(s) = Ga(S) = 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 k p . 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 k p 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 time (sec)  0.6  0.7  0.8  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 (B.6)  G{ (s) = G3 (S) = 1.1646 + 0.060 Is  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  open-loop closed loop  20  C  n > CO 2  0 -20 90  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 + d  0.060life,  v  (1 + 0.0601*; +\A646k p)s P 1  +^ ^ p nnfinifr  (B.7)  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 ,4 c / O) = s 2 + 3 3 0 ^ y + 165 2  (B.8)  By comparing (B.7) and (B.8), the coefficients of the PI controller are computed as k p =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 139830 G5(s) = G6(s) = 5 + 60.8276  (B.9)  Figure B.5 shows a comparison of the Bode diagrams of the transfer function of the openloop 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 + 300,2:  Ad{s) = s2+1200^5  (B.ll)  By comparing (B.10) and (B.l 1), the proportional and integral gains are computed as 0.0081 and 0.6436, respectively.  Step response of the open-loop  2500  ]| 2000 -g 1500 =  1000  I  500  0  0  0.01  0.02  0.03  0.04 0.05 time (sec)  0.06  0.07  0.08  0.07  0.08  Step response of the closed-loop  1.15  3Q. <u 3 0.5 T3 CL  <£  0  0  0.01  0.02  0.03  0.04  0.05  0.06  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 openloop 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 p s + 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 ^ + 300 2  The damping ratio  (B.14)  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 time (sec)  0.06  0.07  Figure B.8: Step response of the closed-loop system.  0.08  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 3 rd -order reduced transfer function is obtained as  ^ , N -0.002786s 3 + 204.4s 2 + 1277000s + 228900000 Gp(s) = = = s +10880s + 2784000s+ 163600000  (C.l)  Bode diagrams of the full-order model (23 th ) and the reduced-order model (3 rd ) 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 (23t h ) and the reduced-order model (3 rd ).  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  G c(s)G p(s)  where G c(s)  =  kpS "t* kj  (C.2)  G p(s)  stands for the controller, G p(s)  proportional and integral gain, respectively.  is a nominal plant, and kp  and ki are  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 ~ ^G p  From Gc(ja\)  (M)  (C.4)  and (C.3),  Gc (.ja\ ) = k  p  k- j - ^ = \Gc <Ja\ )| (cos 9+j sin 9)  (C.5)  where, from (C.3),  \GC(M)\  (C.6)  = Gp(M)  |  From (C.5), equating real part to real part yields  cos 6 k p  (C.7)  \GPim)\  and equating imaginary part to imaginary part yields o\ sin#  (C.8)  Gp(M)\  With the chosen design specifications such that the settling time r s = 0.075 seconds and the phase margin is 85° , the phase-margin frequency, or the gain-crossover frequency, is g calculated as o\=— T  s  t a n  = 9.3321 rad/sec, to yield a specified settling time r,, . The Qm  coefficients of k p and k t are then calculated as 0.0137 and 6.7057, respectively. The antikwindup gain can be computed as k a=— [55], kp 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"  1  i,  >  I  ki  s  ]  +  V, WT,2 '  1 >  set V, WT,3'  ki  s  + W3K Z ±) ^ y)—  WT,3  1 *  s  k  a  ki  +  ka  +  ka '  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 Qhalf = sqrtm(Q); Rhalf= sqrtm(R); N = ones(nstate,l)*0.55;  4  % check for the condition of Q-N'RA-1N>=0  Posi=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]); 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'); lmiterm([LMI_Sys_ 1 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*Ql A (l/2) %(1,3) block: X'*Rl A (l/2) %(2,2) block: -eye(3,3) %(3,3) block: -eye(l,l)  LMI_Sys_2 = newlmi; lmiterm([LMI_Sys_2 lmiterm([LMI_Sys_2 lmiterm([LMI_Sys_2 lmiterm([LMI_Sys_2 lmiterm([LMI_Sys_2 lmiterm([LMI_Sys_2  1 1 Y], Atb, 1, 's'); 1 1 X], Btb, -1, 's'); 1 2 -Y], 1, Qhalf); 1 3 -X],l, Rhalf); 2 2 0], -1); 3 3 0], -1);  %(1,1) block: A1*Y + Y*A1' %(1,1) block: -B1*X - X'*A1' %(1,2) block: Y*Ql A (l/2) %(1,3) block: X'*Rl A (l/2) %(2,2)block: -eye(3,3) %(3,3) block: -eye(l.l)  LMI_Sys_3 = newlmi; lmiterm([LMI_Sys_3 lmiterm([LMI_Sys_3 lmiterm([LMI_Sys_3 lmiterm([LMI_Sys_3 lmiterm([LMI_Sys_3 lmiterm([LMI_Sys_3  1 1 Y], Ate, 1, 's'); %(1,1) block: A1*Y + Y*A1' 1 1 X], Btc, -1, 's'); %(1,1) block: -B1*X - X'*A1' 1 2 -Y], 1, Qhalf); %(1,2) block: Y*Ql A (l/2) 1 3 -X],l, Rhalf); %(1,3) block: X'*Rl A (l/2) 2 2 0], -1); %(2,2) block: -eye(3,3) 3 3 0], -1); %(3,3) block: -eye(l,l)  % Coupling constraint LMI_Couple = newlmi; lmiterm([-LMI_Couple 1 1 Z],l,l); lmiterm([-LMI_Couple 1 2 0],1); lmiterm([-LMI_Couple 2 2 Y],l,l);  % (1,1) block: [Z I; I Y] >0 % (1,2) block: [Z I; I Y] >0 % (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 lmiterm([LMI_Sys_l lmiterm([LMI_Sys_l lmiterm([LMI_Sys_l lmiterm([LMI_Sys_l lmiterm([LMI_Sys_l  1 1 Y], Ata, 1, 's'); 1 1 X], Bta, -1, 's'); 1 2 -Y], 1, Qhalf); 1 3 -X],l, Rhalf); 2 2 0], -1); 3 3 0], -1);  %(1,1) block: A1*Y + Y*A1' %(1,1) block: -B1*X-X'*A1' %(1,2) block: Y*Ql A (l/2) %(1,3) block: X'*Rl A (l/2) %(2,2) block: -eye(3,3) %(3,3)block: -eye(l,l)  LMI_Sys_2 = newlmi; lmiterm([LMI_Sys_2 lmiterm([LMI_Sys_2 lmiterm([LMI_Sys_2 lmiterm([LMI_Sys_2 lmiterm([LMI_Sys_2 lmiterm([LMI_Sys_2  1 1 Y], Atb, 1, 's'); 1 1 X], Btb, -1, 's'); 1 2 -Y], 1, Qhalf); 1 3 -X],l, Rhalf); 2 2 0], -1); 3 3 0], -1);  %(1,1) block: %(1,1) block: %(1,2) block: %(1,3) block: %(2,2) block: %(3,3)block:  A1*Y + Y*A1' -B1*X - X'*A1' Y*Ql A (l/2) X'*Rl A (l/2) -eye(3,3) -eye(l,l)  LMI_Sys_3 = newlmi; lmiterm([LMI_Sys_3 lmiterm([LMI_Sys_3 lmiterm([LMI_Sys_3 lmiterm([LMI_Sys_3 lmiterm([LMI_Sys_3 lmiterm([LMI_Sys_3  1 1 Y], Ate, 1, 's'); 1 1 X], Btc, -1, 's'); 1 2 -Y], 1, Qhalf); 1 3 -X],l, Rhalf); 2 2 0], -1); 3 3 0], -1);  %(1,1) block: A1*Y + Y*A1' %(1,1) block: -B1*X - X'*A1' %(1,2) block: Y*Ql A (l/2) %(1,3) block: X'*Rl A (l/2) %(2,2) block: -eye(3,3) %(3,3) block: -eye(l,l)  % 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);  % 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 options = [le-5 0 0 0 0]; [copt,xopt] = mincx(lmisys,c,options);  % No cross-product term (LQRS) % Cross-product term (ALQRS)  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- - B y k ) 7 , P + P ( A y - B y k ) + Q + k r R k < 0 then  the  cost  function  J  of  (5.11)  is  bounded  (E.l)  by  the  scalar  expression  £^x(0) r Px(0) = £ ^ r ( x ( 0 ) P x ( 0 ) r ) =<r(X 0 P) asin(E.2):  K = m m j ^ { J ° ( x r Q x + urRu)<fc J < £ [ / r ( x ( 0 ) P x ( 0 ) r ) ] = /r(XoP)  where j = l,...,N  (E.2)  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 - B k ) 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-Bkf P + P(A-Bk)<0  (E.4)  where the state noise signal has zero mean £ [ w ] = 0 and symmetric positive definite covariance matrix E  WW  T  = RW>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., (Ay--B7-k)rp + p(Ay-B7-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 Q x + 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 xrQx + urRu = E  (E.7)  dt  where P is the Lyapunov matrix. Substitute u = - k x into (E.7) to obtain E(x TQx + xTk TRkx\  =E- - ( x ' W dt  = E ( - X  T  PX-X^PX)  (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 • - B k ) r P + P ( A - B k ) j x = -E x r E x < 0  where E is ^Q + k ^ R k j , which is a positive definite matrix.  (E.10)  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-Bk) P + P(A-Bk) + E < 0  (E.ll)  Multiply each term in the Lyapunov inequality (E.ll) by the system state transition matrix Bk) t f e(A-Bk)  r  or  ^ g ief t a n c j  e  (A  /p(A__Bk)e(A-Bk).  Bk  V for the right to obtain  + g ( A - B k f / ( A _ B k ) p . ( A - B k ) . +e(A-Bk)%e(A-Bk),  (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)  < Q  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^Ex)*  <E x(0) Px(0) = E^tr (x(O)Px(O)7,  = tr ( X 0 P )  with utilizing trace operation property such that tr(ABC) = tr(CBA) = tr{CAB).  (E.17)  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^  and hence, (E.2) holds.  =E\ 0xr(Q  + k rRk)x)<#  (E.l 9)  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)  where  (  F  .  3  )  u* (7) = -kx(7) = R - 1 G^Px(0, P is the positive definite matrix, which is the  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: *  *  (F.4)  V (0 = v o ( 0 + *v(0  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  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0100578/manifest

Comment

Related Items